Aerodynamic Interaction Effects of a Helicopter Rotor and Fuselage

APPENDIX I

Rotor-Fuselage Interactional Aerodynamics:

An Unsteady Rotor Model

_,,¢ Richard W. Barnweil D. Douglas Boyd, Jr.

Professor Senior Research Associate Virginia Polytechnic Institute and State University Hampton, VA 23681-2199 local load per unit span, dimensionless

Abstract

on 13_2R 3 An unsteady helicopter rotor model suitable for the = polynomial number n,] study of rotor-fuselage interactional aerodynamics has been developed. The model couples a Generalized local pressure, dimensionless on P** Pg Dynamic Wake Theory (GDWT) code with a thin-layer normalized Legendre functions P,Q Navier-Stokes code through an unsteady pressure jump boundary condition. The unsteady boundary condition radial location, dimensionless on R models each rotor blade as a radially varying, time rotor radius, meters R dependent pressure jump between adjacent grid planes.

i-th component of perturbation velocity, A non-rotating cylindrical grid is used for the isolated qi rotor case, and the unsteady boundary condition models dimensionless on _R the blades as a pressure jump traveling around the rotor dimensionless time, dimensionless on disk. To demonstrate the coupling of the two codes, _-I calculations are compared to experimental unsteady and freestream speed, dimensionless on time-averaged inflow data.

V_ g2R Notation normal component of induced velocity, positive downward, dimensionless on g2R ratio of blade area in GDWT to area local mean normal component of A fact g.F w I used in OVERFLOW, dimensionless induced velocity, positive downward, dimensionless on £ZR C local blade chord, dimensionless on R total energy per unit mass per unit vol- coordinate normal to rotor disk plane, Z e 0 ume dimensionless on R additional energy per unit mass per unit coefficients of the normal downwash E add _,f3 v series expansion volume due to Pg, dimensionless effective angle of attack, rad harmonic number m) r eteff mean coning angle, rad apparent mass matrix [M] Mach number M ratio of specific heats Y cosine and sine components of L- local blade pitch, rad O [L_], [L'] matrix mean and first harmonic components of 0 o, 0 c, 0 s local blade pitch, rad component of freestream normal to k rotor disk, dimensionless on fir ,posi- Presented at the American Helicopter Society tive down K.J 54th Annual Forum, Washington, D.C., May 20- 22, I998. Copyright © 1998 by the American Helicopter Society, Inc. All rights reserved.

case. However, the inclusion of the rotor system to the advance ratio (component of freestream computation adds even another level of complexity. If velocity parallel to rotor disk), dimen- only the approximation to time-averaged rotor inflow sionless on f_R effects are desired, several methods have been ellipsoidal coordinates, dimensionless developed [3, 7, 8]. If, however, viscous effects on a dimensionless coordinate along fuselage under the influence of unsteady rotor inflow freestream line, positive pointing effects are needed when separation is expected, only upstream from rotor Navier-Stokes simulations of the entire problem including time-accurate representation of the moving P density, kg / m 3 blade geometry are presently available. For example, pressure function, dimensionless on Meakin [5] used the unsteady, thin-layer Navier-Stokes equations to compute the flow field around a complete p_2 R 2 tiltrotor aircraft, including three-bladed rotors and acceleration and momentum compo- moving overset grid systems for a hypothetical flight nents of pressure function, dimension- condition. Ahmad, et al., [9] also used the thin-layer less on p_2R 2 Navier-Stokes equations to compute the unsteady blade azimuth angle, rad/sec pressures and flowfield on an isolated two-bladed rotor using moving overset embedded grids. In [6, 9] a first rotor rotational speed, rad/sec order time accurate scheme was used which necessitated ()* derivative with respect to the use of very small time steps. Even though solutions in [9] show a converged (periodic) solution in three to five rotor revolutions, 1152 time steps were required per Introduction revolution at a cost of 45 Cray C-90 hours.

Unsteady interactional aerodynamics between a The current work is designed to examine techniques helicopter fuselage and rotor are not well understood.

to reduce the computational times required to provide There exist a number of computational methodologies unsteady calculations that include a rotor model while available to study these interactional effects. These maintaining the capability to calculate viscous flows on v range from superimposing isolated rotor and isolated a fuselage. It is not to replace the full simulations or the fuselage effects to Navier-Stokes simulation of the simpler analyses, but rather to provided an additional entire problem [1-5]. Each of these methods have their tool to the researcher or designer.

respective advantages. For example, linearly superimposing the effects of an isolated rotor and Method isolated fuselage has the advantage of being a quick, simple method and works well when the nonlinear The current method employs a hybrid approach effects, such as flow separation, are negligible. On the which allows a quick assessment of the time-averaged other hand, Navier-Stokes simulations have the and time-accurate rotor-fuselage interactional advantage of including all aspects of the flow field in a aerodynamics. A simple, fast-running Generalized single calculation. Each of these methods also have Dynamic Wake Theory [10, 11] method is used to trim disadvantages. For example, the superposition technique an isolated rotor and generate a time dependent pressure does not work well when the interactional effects are not distribution to represent the rotor blades in the plane of truly linear, (e.g. with flow separation), while full the rotor disk. This set of unsteady, periodic pressures is simulations may take many hours or even days of CPU used in a version of the thin-layer Navier-Stokes code, time to execute. The hybrid method that is presented OVERFLOW [12], which was modified under the here draws from advantages of methods on either end of current effort to include an unsteady pressure jump the spectrum.

boundary condition between adjacent planes in a non- For typical shapes currently used for helicopter rotating cylindrical grid for the isolated rotor.

fuselages, flow separation is a common occurrence. This GDWT is typically the case even if one were to examine the isolated fuselage independent of the rotor system [6].

Details of the GDWT can be found in [10, 11]; For isolated fuselage computations, a steady state however, for completeness, an outline of the theory is solution is typically sought. Therefore, many of the given here. In addition, the current method of time solution acceleration techniques such as multigrid, grid integration of the GDWT equations is presented as is the sequencing, and preconditioning may be employed to current method of trimming the rotor to specified thrust, obtain an economical solution to the isolated fuselage roll, and pitch moments.

%.J the model), ix and 13, of the w velocity component The GDWT [10, 11] begins with the mass conservation equation and the incompressible, expansion: linearized Euler equations: qi, i = 0 (1) qi- V_qi,_ = -t_,i (2) where repeated subscripts imply that the summation convention is being used and commas imply that Closed form expressions for [M], [LC]-1 , [L'] -I , _c, derivatives are being taken. The pressure function, _, and _,s are given in [10, I I]. As described in references can be split into two components, (pA and cP v 10 and 11, the x functions depend on the loading model corresponding respectively to the two terms on the left chosen by the user. The following loading model is hand side of equation (2), each of which satisfies used: Laplace's equation. The boundary conditions are that the pressure function matches the known blade loading (wt- L + 130_tcos_- f_) at the rotor and that it equals zero at infinity.

ixeff = 8 - _ + l.tsinxg (8) When written in ellipsoidal coordinates, the solution to Laplace's equation for the pressure potential 0 = 00 + 0cCOSXl/+0ssin_g (9) on a circular planform, along with the conditions that the pressure perturbation equals zero at infinity and that L = rtc(_ " + I'tsln_ll)2ixeff (I0) the pressure potential is zero at the edge of the planform, ,J_-M 2 can be written in closed form as follows: This loading model includes angle of attack effects due to (1) the freestream component of inflow normal to the 1 --NI --m , t' n (v)a, (,q disk, (2) the induced inflow feedback from the GDWT, ra=0 n=m+l,m+3,...

(3) the mean coning angle of the blades, (4) the pitch rll c -- ms , -- rate, (5) the tangential velocity variations due rotor [X n cos(m_l/) +'t" n sin(my)] (3) rotation, and (6) the increase in lift with Mach number.

where v, rl, and are dimensionless ellipsoidal In equation (10), a lift curve slope of 2_ has been coordinates, P and are normalized Legendre assumed. A very simple stall model is also included in are the unknown coefficients of the method whereby the effective angle of attack is not functions, x_c and z_s allowed to increase above 10 degrees. However, it has the cosine and sine functions and are themselves been found that, for the cases presented here, the results functions of the dimensionless time variable, _. Since from this model are not sensitive to the 10 degree v _a and (pv each satisfy Laplace's equation, equation maximum angle of attack.

Also described in these references is a technique to (3) can be applied to t_ a and (pv separately. Examining generalize equations (6) and (7) to include effects of the only the normal component of the perturbation velocity mean induced inflow on the wake skew angle. It is this and integrating along a streamline from a point on the non-linear version of the GDWT that is used in the rotor disk back to infinity, leads to the following results presented in this paper.

equations for the normal component of perturbation Equations (6) and (7) form a set of first order velocity: ordinary differential equations that can be solved by various techniques. Here, a Jameson-style Runge-Kutta w = -V'_J0 "_"z aq (4) integration [13] is used to solve for ix and 13, and thus for w, at each successive time step. Time integration is " 9_A (5) carded out over a specified number of rotor revolutions.

W = _Z rl=0 That leads to the complete loading and downwash where ( )* denotes a derivative with respect to solution on the rotor disk. This loading information is in dimensionless time and z is the direction normal to the turn used in an outer trim loop for the isolated rotor.

rotor disk. Expanding the w component of velocity in The rotor trim condition is determined using a modified Newton-Raphson technique [14]. In this an expansion similar to that used for cp and technique, the rotor state is first determined based on a superimposing the (_a and _v components results in specified collective, lateral, and longitudinal pitch the following set of first order differential equations for setting. These controls are then each perturbed the unknown, time-dependent coefficients ("states" of independently to form a "derivative matrix" which is where Pg is the dimensionless pressure determined by used to determine the successive collective, lateral, and the local load per unit span (from the GDWT code) longitudinal pitch settings for the rotor based on the divided by the local blade segment chord at the current changes noted in the rotor thrust, roll, and pitch grid point, y is the ratio of specific heats, and A fact is moments due to the control perturbations. These the ratio of the total blade area in the GDWT to the total successive determinations of the pitch settings are area on which the pressure is applied in OVERFLOW in continued until the rotor is trimmed to within a specified order to maintain the same overall thrust between the root-mean-square change in the thrust, roll, and pitch two methods. For the two specified planes where the moments. This trim technique is classified as a modified 2 . boundary condition is applied, one half of the additional V Newton-Raphson technique since the derivative matrix energy term is placed on each plane. The code is then is held fixed throughout the trim procedure rather than executed in the time-accurate mode until a periodic being recalculated at each step of the iteration as it inflow solution is obtained.

would be done in a classic Newton-Raphson method.

Once the rotor has been trimmed, the unsteady Code Coupling loading and unsteady downwash are known for the In the current effort, the GDWT code and entire rotor disk. The unsteady downwash has a OVERFLOW are used together in a "loose" coupling.

frequency content consistent with the specified number That is, the GDWT code and OVERFLOW are both of harmonics and shape functions chosen by the user in stand-alone codes that are executed sequentially. Since the GDWT method. Four harmonics provide good the GDWT code calculates the isolated rotor trim correlation between predicted and measured downwash loading, a trim calculation is not needed in when using a loading model that assumes the blades are OVERFLOW. Thus, OVERFLOW is required only to modeled as a "pressure delta function" or pressure spike execute enough time steps to obtain a periodic solution.

traveling around the rotor azimuth. This is the method The common link between the two codes is that the used in this paper. In addition, since the non-linear same rotor system pressure distribution is used in each; version of the GDWT is used, the calculated loading is the GDWT calculates a pressure distribution that is coupled to the calculated downwash. Thus, the loading, subsequently used in OVERFLOW. However, there is a which initially was two-dimensional, has now been .= difference in how the pressure distributions are used in corrected to account for effects up to a frequency of four each code. The GDWT, as applied here, assumes a that per revolution. It is this discrete, unsteady rotor disk the loading is a delta function at the mid-chord of the pressure distribution which is to be used as a boundary blade, whereas OVERFLOW assumes that the loading is condition in the thin-layer Navier-Stokes code.

applied on grid line which has associated with it an area OVERFLOW "wedge" in the cylindrical grid used for the rotor. In addition, the GDWT uses the linearized Euler equations In this section, the thin-layer Navier Stokes code, and OVERFLOW uses the Navier-Stokes equations.

OVERFLOW, and the method used to couple Therefore, the common pressure distribution when used OVERFLOW to the GDWT code is discussed.

in each code will produce slightly different inflow For purposes of this study, the thin-layer Navier- characteristics due to the slightly different physical Stokes code, OVERFLOW (version 1.7v), was chosen assumptions. Since the GDWT does not include a as a baseline code based on its robustness, its fuselage model, it will be necessary to iterate between convergence times for unsteady cases, and its low Mach the two codes when a fuselage is introduced into the number capabilities. For this study, this version of OVERFLOW solution. An iterative capability has been OVERFLOW was modified to include an unsteady implemented in the codes, but is not necessary in the pressure jump boundary condition. This boundary present work since only the isolated rotor case is shown.

condition is applied between two user specified planes In order to include fuselage effects in the GDWT, which are separated by a user specified "iblanked" an iterative technique has been developed in which the plane• (As is common practice, an "iblanked" region is inflow from the GDWT and OVERFLOW are compared one which is assumed to be outside the computational on a similar frequency basis. Since the GDWT provides domain and in which no computations are performed.)

inflow information only up to user specified frequency, The new boundary condition is implemented as an the inflow calculated by OVERFLOW is filtered to • additional energy term which is added to the quantity match the GDWT frequency for purposes of the 0% in the following form: comparison. A difference is taken between the GDWT inflow and the filtered OVERFLOW inflow. This Eadd = y- 1 difference is then used in the GDWT code as an "inflow GDWT Time-Accurate Results correction" to account for differences in the two codes and assumptions made therein. The GDWT code is re- Figure 2 shows a comparison of the measured and executed, and the entire procedure may be repeated as predicted unsteady inflow ratio with mean inflow ratios needed.

removed for nine experimental measurement locations.

The measurements are located above the rotor plane along 3 radial lines at azimuth angles of 30, 180, and

Experimental Data

300 degrees and at 3 spanwise locations of r/R.=0.60, All of the experimental data used in this paper are 0.78, 0.90. These measurement locations are marked from laser velocimeter inflow measurements obtained at with the letters A-J (excluding the letter I) on figure 1. In v the NASA Langley 14- by 22-Foot Tunnel [15]. Two all of the plots in figure 2, the measured and predicted different blade planforms were used in the test, one data show a general four per revolution oscillation in the tapered and one rectangular. Only the data for a rotor inflow as is expected for a four bladed rotor. The with a 4-bladed, tapered planform is presented in this locations D-F (r/R=0.78) in this figure show that paper. This rotor has constant chord (3.15 inches) over predictions of unsteady inflow peak-to-peak magnitudes the inboard 75 percent of the span and an approximately and phases are predicted well. For all of the plots in this 3-to-1 taper ratio over the outer 25 percent of the span figure, good phase predictions are displayed, but peak- such that the tip chord is approximately one inch. The to-peak predictions are not as good. The measured data blades have a linear twist of-13 degrees, a root cutout at generally show sharp blade passage pulses at the 25 percent span, a solidity of 0.0977 and radius of 32.5 beginning of each oscillation cycle. The predictions inches. The thrust coefficient was 0.0065. Inflow data using the GDWT in this manner are not expected to used here were taken in a plane approximately 2.6 predict these sharp blade passage pulses, since only the inches above the tip path plane of the rotor for all first four harmonics of inflow are being computed (i.e., azimuth stations 30 degrees apart and at a number of only 15 states in the GDWT model are used). However, radial stations. The flight condition chosen for this the discrete loading distribution used in the GDWT to comparison has a 3 degree nose down shaft tilt and an calculate the inflow is not as strongly affected by the advance ratio of 0.23. Data samples were processed at a number of harmonics used. It is these loading values resolution of 128 samples per revolution or about every that are to be used in OVERFLOW. It has been shown 2.8 degrees of blade travel.

that using many more states can improve the correlations of the time-accurate inflow [17]; but the

Results corresponding improvement in the loading distribution

is much smaller and hence much less necessary. The results presented in figures 1-2 give a general confidence GDWT Time-Averaged Results that the GDWT code is working correctly.

In this section, results will be presented from the GDWT code before the coupling with OVERFLOW.

OVERFLOW Time-Averaged Results Figure 1 shows a contour plot comparison between With the loading predictions from the GDWT code, the time-averaged, measured inflow ratio and the OVERFLOW is executed in two stages. First, to GDWT code prediction for the baseline case. For the minimize the number of time steps required to converge GDWT calculations, 4 harmonics and 15 states were to a periodic solution, a mean flow is established by used. It can be seen that many of the major time- using a steady-state computation with a constant averaged flow features are captured, as discussed in pressure jump equivalent to an equal distribution of [16]. An upwash region can be see along the front of the thrust on the rotor. Then, the case is restarted in time- disk in both the measurement and in the prediction. The accurate mode using the GDWT determined unsteady prediction matches the location of the zero induced pressure jump boundary condition. For the cases inflow ratio line well, but the predicted gradients of the presented here, the grid topology consists of a single inflow, especially on the advancing and retreating sides, stretched cylindrical grid which extends 1.5 rotor radii are higher than the gradients of the inflow in the above and below, 2.5 rotor radii ahead and to either side measured data. Other features such as the strong of, and 4.5 rotor radii behind the rotor (see figure 3). A downwash region in the first quadrant are predicted grid size of l15x129x56 was used in the vertical, well. Some of the discrepancies in the inflow azimuthal, and radial directions, respectively. The rotor distribution can be attributed to the fact that there was a portion of the grid is in the vertical center of the grid and fuselage body and hub present when the experiment was occupies a grid of 129x30 (minus the inner 6 radial conducted [16], but the GDWT in this case includes no stations, to account for the blade root cutout). A fuselage or hub effects.

rl,...,, rotor disk where there is no blade loading. Since it is the freestrearrdcharacteristic (Reimann invariant) boundary condition was used at the outer boundaries; a periodic rotor loading and hub/fuselage blockage effects that actually cause the induced inflow flowfield, it can be boundary condition was used at the iF=0 degrees deduced that the near zero inflow region near the center location; an axis boundary condition was used along the of the rotor disk is at least partially caused by the low rotor axis; the new unsteady boundary condition was loading and lack of hub model in the hub region.

used along two planes in the vertical center of the grid (separated by a single, iblanked plane of the same size OVERFLOW Time-Accurate Results as the rotor planes) and extends from the root cutout Figure 5 shows a comparison of the measured location to the rotor edge. For all of the cases examined unsteady inflow and the predicted unsteady inflow. In here, 1408 time steps (a multiple of the number of time these plots, the sharp blade-passage pulses are seen at steps per revolution) were sufficient to obtain a the beginning of each oscillation cycle, and the converged steady state solution. Here, convergence was magnitudes and phases of these pulses are predicted determined by a two order of magnitude drop in the L2- well.

norm of the residual. Though typically a larger residual drop considered crucial to a converged solution, a two orders of magnitude drop in residual is sufficient here

Sensitivities

due to the small values of the initial residual. This To show the sensitivity of the above results" to steady state calculation was then followed by two particular parameters related to the solution procedure, revolutions of unsteady calculations, which was several of the relevant parameters pertaining to the sufficient to obtain periodicity in the solution.

solution procedure are explored. They include the The computations presented use the following number of Newton sub-iterations, the use of viscous methods in OVERFLOW: (1) central difference terms, the time step (azimuthal resolution) used, and the calculations of the right-hand side of the equations, (2) outer boundary condition used. In addition, the amount 3-factor diagonal left-hand side inversion, (3) a matrix of CPU time required for each case is presented.

dissipation scheme, and (4) Newton sub-iterations.

Figure 4 shows a contour plot comparison between Newton Sub-iterations the time-averaged, measured inflow ratio and the In the case presented above, six Newton sub- OVERFLOW prediction which has been filtered to iterations were used at each time step in the unsteady contain the same frequency content as the GDWT portion of the calculations. To demonstrate that six sub- prediction presented in figure 1. For purposes of iterations is sufficient, the baseline case was re-executed comparisons later in the paper, this case is referred to as with ten sub-iterations (instead of six) starting from the the "baseline case". Since the highest frequency same steady state conditions. Figure 6 shows the available in the measured time-averaged contour plot is normalized L2-norm of the conservative variables six per revolution (due to Nyquist cut-off versus Newton sub-iteration number for the last time considerations), this filtering also places the step in the case. It can be seen that there is little OVERFLOW prediction on a frequency basis similar to those measured data. Generally, the major time- advantage to using more than six Newton sub-iterations.

This is further demonstrated by figures 7 and 8, which averaged features seen in figure 1 are also seen in the show that there is virtually no difference in the time- figure 4. It can be seen in figure 4 that the location of the averaged or time-accurate inflows. Thus, the use of six zero induced inflow line is not as well predicted, but the Newton sub-iterations is adequate for the current case.

overall levels are well predicted. Also, in contrast to the results presented in figure 1, the gradients of inflow Viscous Terms (spacings between the contour lines) are predicted well.

As with the GDWT time-averaged results presented Since the case presented here is an isolated rotor earlier, some of the discrepancies in the flow field can be with no surface in the flowfield, viscous terms in the flow solver were not used. To show that these terms are attributed to the fact that the experimental data includes the effects of a fuselage and hub. An example of these not dominant in this particular case, the above baseline discrepancies can be seen on the retreating side of the case was re-executed with all of the thin-layer viscous rotor disk near the center of the contour plot. At that terms activated in the flow solver. Figures 9-10 demonstrate that these terms are not a dominant location in the predicted results, a region of near zero influence in the isolated rotor case. However, these inflow can be seen which is not present in the measured data. Since the predictions do not use a hub or hub wake viscous terms will be required when a fuselage or other solid surface is introduced into the solution.

model, there are no hub blockage effects in the flowfleld and the flow is free to travel through the center of the Time Step Table 1: All of the above cases used 128 time steps per Case CPU (hr:min) revolution. To test the adequacy of this time step (and azimuthal resolution), the entire case were re-executed Time Step 1:38 with a doubled time step, i.e., with 64 azimuth steps per revolution. The resulting time averaged and time Outer Boundary 4:06 accurate calculations are shown in figures 11-12. Both of these figures show that 64 time steps per revolution is adequate to capture the gross features of the time Concluding Remarks averaged and time accurate inflow. Due to the (1) An unsteady helicopter rotor model using an similarities between the 64 time step per revolution case unsteady pressure jump boundary condition has been and the original 128 time step per revolution case, it developed and included in the thin-layer Navier-Stokes, appears that the 128 time step per revolution case is overset grid code, OVERFLOW.

quite adequate for these predictions.

(2) Even though isolated rotor calculations are Outer Boundary presented here for the purposes of demonstrating the In all of the above cases, the outer boundary is model, excellent agreement is shown between time relatively close in proximity to the rotor disk. This raises averaged and unsteady measured and predicted inflow a question as to the effect of the outer boundary on the ratios at a plane above the rotor disk, and CPU solution. To demonstrate the effect of the outer requirements for the isolated rotor method are found to boundary on the solution, the boundary condition there be reasonable.

was changed from a Reimann invariant (freestream/ characteristic) condition to simply a freestream (3) It has been demonstrated that suitable condition and the baseline case was re-executed. Figures parameters were chosen with regard to the number of 13-14 show the results of the computations with the Newton sub-iterations used, the viscous terms used, the freestream outer boundary condition. Figure 13 shows a time step used, and the outer boundary conditions used.

good match with the baseline time averaged plot. In figure 14, it can be seen that there is a slight blade-to- blade difference at the beginning of the revolution which disappears by the end of the revolution. This is evidence Acknowledgments that, formally, one more revolution should be calculated, This work is supported by the NASA Langley however, for the remaining blade passages, it can be Research Center Subsonic Aerodynamics Branch under seen that the gross effects of the outer boundary V Grant Number NCC- 1245.

condition on the unsteady inflow are small. This implies that the solution is relatively insensitive to what is References happening at the outer boundary.

CPU times [1] Berry, J.D., "A Method of Computing the Aerodynamic Interactions of a Rotor-Fuselage All of the steady and unsteady calculations using V Configuration in Forward Flight," Doctor of Philosophy OVERFLOW were run on a Cray C-90. Table 1 shows a Thesis, Georgia Institute of Technology, Atlanta, GA, comparison of the CPU times for each of the cases May 1990.

presented above. As can be seen in the table, the CPU time required is relatively inexpensive.

[2] Berry, I.D., Letnikov, V.B., Bavykina, I., Chaffin, M.S., "A Comparison of Interactional Table 1: v Aerodynamics Methods for a Helicopter in Low Speed Forward Flight," Proceedings of the 23rd European Case CPU (hr:min) Rotorcraft Forum, Volume 1, pp. 33.1-33.9, September 16-18, 1997, Dresden, Germany.

Baseline 4:16 Newton Sub-iteration 5:39 [3] Chaffin, M.S., Berry, J.D., "Navier-Stokes Simulation of a Rotor Using a Distributed Pressure Disk Viscous Terms 5:01 Method," 51st Annual Forum Proceedings, Volume 1, American Helicopter Society, pp 112-136, May 1995.

V [14] Johnson, W., "A Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics Part [4] Boyd, Jr., D.D., Brooks, T.F., Burley, C.L., Jolly, Jr., J.R., "Aeroacoustic Codes for Rotor Harmonic and I: Analytical Development;' NASA TM 81182, June 1980.

BVI Noise - Camrad.Modl/H S: Methodology and Users' Manual" NASA TM 110297.

[15] Althoff, S.L., Elliott, J.W., Sailey, R.H., "Inflow Measurement Made with a Laser Velocimeter [5] Meakin, R., "Moving Body Overset Grid on a Helicopter in Forward Flight," Volumes V, NASA Methods for Complete Aircraft Tiltrotor Simulations," TM 100545, April 1988.

Paper AIAA-93-3350, Presented at the l lth AIAA Computational Fluid Dynamics Conference, June 6-9, 1993, Orlando, FL. [16] Peters, D.A., He, C.J., "Correlation of Measured Induced Velocities with a Finite-State Wake Model," Presented at the 45th Annual National Forum [6] Chaffin, M.S., Berry, J.D., "Navier-Stokes and Potential Theory Solutions for a Helicopter Fuselage of the American Helicopter Society, Boston, MA., May 1989.

and Comparison With Experiment," NASA TM 4566, June 1994.

[17] He, C.J., "Development and Application of a Generalized Dynamic Wake Theory For Lifting Rotors," [7] Rajagopalan, R.G., Mathur, S.J., "Three Ph.D. Thesis, Aerospace Engineering Department, Dimensional Analysis of a Rotor in Forward Flight," Journal of the American Helicopter Society, July 1993. Georgia Institute of Technology, Atlanta, GA, July 1989.

r _ [8] Zori, L.A.J., Rajagopalan, R.G., "Navier-Stokes Calculations of Rotor-Airframe Interaction in Forward Flight," Presented at the 48th Annual Forum of the American Helicopter Society, Washington, D.C., June 1992.

[9] Ahmad, J., Duque, Earl, P.N., "Helicopter Rotor Blade Computation in Unsteady Flows Using Moving Embedded Grids," AIAA Paper 94-1922, Presented at the 12th AIAA Applied Aerodynamics Conference, Colorado Springs, CO, June 20-22, 1994.

[10] Peters, D.A., Cheng, J.H., "Finite State Induced Inflow Models Part II: Three Dimensional Rotor Disk," Journal of Aircraft, Volume 32, Number 2, March-April 1995.

[11] Peters, D.A., Boyd, Jr., D.D., Cheng, J.H., "Finite State Induced Inflow Model for Rotors in Hover and Forward Flight," Presented at the 43rd Annual Forum of the American Helicopter Society, St. Louis, MO, May 18-20, 1987.

[12] Buning, P.G., lespersen, D.C., Pullium, T.H., o , Chan, W.M., Slotnick, J.P., Krist, S.E., Renze, K.J., "OVERFLOW User's Manual: Version 1.7v" June 11, 1997.

[13] Waiters, R.W., "Class Notes for AOE 6145 - Computational Fluid Dynamics I," Department of Aerospace and Ocean Engineering, Virginia Polytechnic and State University, Fall 1994.

v

v

Measured Predicted / 0 ,.\ _0"__.202__" / 0.012 0 012 \\\,\\, Figure 1: Measured and GDWT Predicted Time Averaged Inflow Ratio V L , V ............. Measured dR = 0.60 -- Predicted Location C Location A Location B 0.03 0.03[ 0.02 0.02 _" , _t i t 0.01 , _, _ _Jt r " _| _ I "'_ l l j ,, 0 ' _ ' -0,01 ' " -0.01 ,,, , r , _ , tf -0.02 -0.02 I - . . , oo3_ _6 - _0 ½5o -o% .... 9'o _6 2_d '3_o dR = 0.78 Location F Location D Location E 0.03 0.03 0.03 C I G) 0.02 0.02 0.02 ,It , I 1 i I I l l b I It I t I 0.01 0.01 0.01 _ _' h .o 0 0 o I' I -0.01 -0.01 -0.01 "_ d u -0.02 -0.02 -0.02 -0.03 -o.o_ .... 90 .... 180'" 270'' 360 -oo% .... _ .... 1_0 2"_c; 3_o .... 90 .... 180'' 270 r ' 360 dR = 0.90 Location J Location G Location H 0.03 0.03 _,, 0.03 0.02 0.02 0.01 O.Ol 0.01 I , I ,l,=l ¢_ o C 0 ,', _ ,' .

-0.01 -0.01 0.02 -0.01 / _o N-- -0,02 -0.02 I_ -0.02 -0.03 -o.% ..._ .... 1'=0 2-_6 _o 0.036.... _b .... 1_ ' _,-_6 ' "_o .... 90 .... 18(_ ' _ 2"70'' 360 Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 2: Measured and GDWT Predicted Time Histories of Induced Inflow for Azimuth Angles of 30, 180, and 300 degrees at Three Radial Stations of r/R = 0.60, 0.78, 0.90 %,.,/ v Figure 3: Schematic of Isolated Rotor Grid Topology as Viewed from Behind and Above Rotor on the Retreating Side V Measured Predicted (Baseline) Figure 4: Measured and OVERFLOW-GDWT Predicted (Baseline) Time Aver- aged Inflow Ratio ,_..# ............. Measured Predicted (Baseline) dR = 0.60 Location B Location C Location A 0.03 0.03 0.03 V , 0.02 0.02 0.02 I 1/ I II q ' '_ _' 0.01 = 0.01 r ,I I , _ 0.01 !, I Ii J _, i /I

°

-0.01 - • t iI _ -0.01 m -0.02 -0.02 -0.02 C .... 90''' '180'' 2"/0'' 360 -0.03(_ .... go .... 180'' 2-/(_'' 3;0 dR = 0.78 Location F Location D Location E 0.03 0,03[ 0"03 I (U 0.02 0.02_ I t II L k I I I ! 0.01 0.01 _ _ _, _ i I i [ f i _ O I t L I I 0 f I i I o I : : ' ,, -0.01 -0.01 'l d i, -0.02 li= -0.02 e,, -o,o2[ ,, °0.03 - - 0 q 0 _ 0 . . . . . . . . . . . . . . 9 0 1 8 0 2 7 0 ' ' _ 0 -0% .... g'o .... i_6 r_6 3;o dR = 0.90 Location J Location H Location G 0.03 0.03 0.03 ¢- 0.02 0.02 _) 0.02 =E 0.01 = 0.01 I r_ I I .o 4d

0 o o o

m 0 -0,01 -0.01 t O -0.02 -0.02 -0.02 .... z .... I .....

-0.03 -0% .... go _0 2-_d 3_o 9(3 180 27'(_ :360 -0.o3_ .... gb .... 18(_'' 2"I(_'' :360 Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 5: Measured and OVERFLOW-GDWT Predicted (Baseline) Time Histories of Induced Inflow for Azimuth Angles of 30, 180, and 300 degrees at Three Radial Stations of r/R = 0.60, 0.78, 0.90 1 2 3 4 5 6 7 8 9 10

S ub-iteration N umber

Figure 6: Normalized L-2 norm of Conservative Variables vs. Newton Sub-iteration Number v %J v Baseline with 10 Baseline Newton Sub-iterations M.e

.o24 / ,\

M,F

\

Z_?2J

Figure 7: Time Averaged Inflow Ratio for Baseline vs. Baseline with 10 Newton Sub-iterations v v ............. Baseline Baseline with 10 Newton Subiterations dR = 0.60 Location B Location C Location A 0.03 0.03 0.03 w 0.02 0.02 ._ 0.02 = 0.01 0.010 0.01 -0.01 -0.01 _ -0.01 %,,, m -0.02 ',_ -0.02 -0.02 C -0.03 -0.03_ .... 9'0 .... 180'' 2-_d'' _360 .... 9_'" %o " 2_0" 3_o dR = 0.78 Location F Location D Location E O.OC 003 0.03 C t: 0.02 _) 0.0-" = 0.01 0.01 ._o %.,/ -0.01 -0.01 • -0.01 0.010 -0.02 _= -0.02 -0.02 C -003 _. .... _,, , , = .... , .... , -0.03_ .... 90 .... 180'' 270'' 360 -°°°6 .... _o l_d 2_d" 3_o 0 90 180 270 360 dR = 0.90 Location H Location J Location G 0.03 0.03 0.03 0.02 0.02 It II _t 0.02 = 0.01 0.01 0.010 -0.01 -0.01 _ -0.01 O -0.02 -0.02 _---0.02 C -0.03 .... 9'o 1_o '2->o 3_o -0% .... 9'o 1_o 2->d" 3_o -0.o3_ .... gb'" i_0" 2_6" 3_o Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 8: Time Histories of Induced Inflow for Baseline vs. Baseline with 10 Newton Sub-iteration for Azimuth Angles of 30, 180, and 300 degrees at Three Radial Stations of r/R = 0.60, 0.78, 0.90 Baseline with Viscous Baseline Terms Included

/V_D -w,F ,x

.o24 / '\

0 s

%\_\ ? J Y

Figure 9: Time Averaged Inflow Ratio for Baseline vs. Baseline with Viscous Terms Included KJ ............. Baseline Baseline with Viscous Terms Included dR = 0.60 Location A Location B Location C v

o o3r o o3 r o o_ F

= 0.0 0.01 0.0

_ o

o o

.o.o, F ..... O.Ol .O.Ol F _ ,_ ,_

-v-

_.o,o_ r _ _o.o__ .o.o__

. . ,_L• _ .... ' ' ' ' ' ' .... ' 003_ ..... 0%--9'0. h_O___6.½_o "°030 go 180 2",,0 300 ' '"9'0" "I_6 _,_d 3_0 r/R = 0.78 Location D Location E Location F 0.0 0.0 _ 10_ 10_/_ _ _j_ _ 0,0: °°'I °°'F /I /I tt /I oo,

° oo,_ oo_ F , , , , oo_ r

-°% 'H 9'o- _;d ½_6 - 3;o -o.% .... ,:' '.0 '_;6 _,-_0' 3;o .o.o3_ _2_6" 3;o dR = 0.90 Location G Location J Location H 0.03 0.03 t- 0.03[ 0.02 0.02 i 0+01 0.01 0.01 .2 0 0 0.021- n- -0.01 -0.01 -0.01 _o -0.02 -0.02 '_ -0.02 -0.03 -0.03 .... 9'o l&d 2-_d ' 3&o -o.% .... 9'o' _$6' 2"_0' 3_o r,,,,j Ref. Blade Location :tef. Blade Location Ref. Blade Location Figure 10: Time Histories of Induced Inflow for Baseline vs. Baseline with Viscous Terms Included for Azimuth Angles of 30, 180, and 300 degrees at Three Radial Stations of r/R = 0.60, 0.78, 0.90 ==: ,%# Baseline with Doubled Time Step Baseline (Halved Azimuth Resolution) l 0.012 Figure 11: Time Averaged Inflow Ratio for Baseline vs. Baseline with Doubled Time Step (Halved Azimuth Resolution) ............. Baseline Baseline with Doubled Time Step (Halved dR = 0.60 Azimuthal Resolution) Location B Location C Location A 0.03 0.03 0"03 I k.J 0.02 o.o21: 0.02 = 0.01 0.01 _, _, _ 0.01 xx _x _x _' -0.01 -0.01 _ -0.01 -0.02 -0.02 -0.02 r" -0.03 .... gO .... 180'' 270'' 360 -0.03_ -o.% .... 9'0 '_' '1_" 2_d" s;o dR = 0.78 %,J Location F Location D Location E 0.03 0.03 C 0"03 I (U 0.02 G) 0.02

:I

0.01 i 0.01 / • /- / l I 1 _ I ._o -O.Ol -0.01 _ .O.Ol

° I

r-- -0.02 -0.02 e- "0-02 I -0.03 -0.03_ -o% .... 9_ .... 15d" 2_0" s_o .... 9'0 .... 1_6 2_6" 's_o 90 180 270 360 dR = 0.90 Location J Location H Location G 0.03 0.03 0.03 0.02 0.02 ._ 0.02 = 0.01 0.01

g: o

-0.01 i -0.01 -O.Ol O -0.02 -0.02 -0.02 ¢,,,,, -0.03 .... 9'o '_ ;_6 2->0 3_o -o% ,, '9'o''' '_6'' 2-_d" 3;o -o.% .... 9'o _0 _6 _o Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 12: Time Histories of Induced Inflow for Baseline vs. Baseline with Doubled Time Step (Halved Azi- muthal Resolution) for Azimuth Angles of 30, 180, and 300 degrees at Three Radial Stations of r/R = 0.60, 0.78, 0.90 Baseline with Freestream Baseline Outer Boundary Condition Figure 13: Time Averaged Inflow Ratio for Baseline vs. Baseline with Freestream %,," Outer Boundary Condition ............. Baseline Baselinewith Free- Stream Outer dR = 0.60 Boundary Condition Location B Location C Location A 0.03 0,03 e" 0"03 I V 0.02 0.02 t- 0.01 J 0.01 c

o

-0.o 1 -0.oi _ -O.Ol -0.02 _- -o.o2 -0.02 C °°36 .... ;o.... I;6 2_6" 3_o -o.o3; .... 9_ .... 1;6 2"_6" _o dR = 0.78 Location D Location E Location F , O.010_ 0.01f,, _ _ _ _ 0.010

°0:F, ::k :r

-0.01 f " -0.01 -0.01 , " 0 go I00 270 360 -0.03_'I;d- Z_0- 3;0 -0.%.... 90 ........... iao 27o 3' " ;o dR = 0.90 Location H Location J Location G 0.03 0.03 0.03[ t_ 0.02 4} 0.02 o.01 = 0.01 0.01 O t_ o -0.01 _ -0.01 -0.01 -0.02 -0,02 _ -0.02 -0.03_ .... 9'0 .... i_6" 2+6" 3;o °°36 .... 9'o 1;6 2_d" _;o Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 14: Time Histories of Induced Inflow for Baseline vs. Baseline with Freestream Outer Boundary Condition for Azimuth Angles of 30, 180, and 300 degrees at Three Radial Stations of r/R = 0.60, 0.78, 0.90

J k.2 k/

APPENDIX II

ROTOR/FUSELAGE UNSTEADY

INTERACTIONAL AERODYNAMICS:

A NEW COMPUTATIONAL MODEL

By David Douglas Boyd, Jr.

DISSERTATION SUBMITTED TO THE FACULTY OF THE VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN AEROSPACE ENGINEERING COMMITTEE: Richard W. Barnwell, Chair Henry E. Jones William Devenport Russell H. Thomas William H. Mason July 27, 1999 v,_.* Blacksburg, Virginia Keywords: Rotorcraft, Aerodynamics, Computational Fluid Dynamics, Unsteady Flow v Copyright (_)1999, David Douglas Boyd, Jr.

ROTOR/FUSELAGE UNSTEADY

v INTERACTIONAL AERODYNAMICS:

A NEW COMPUTATIONAL MODEL

V David Douglas Boyd, Jr.

v ABSTRACT A new unsteady rotor/fuselage interactional aerodynamics model has been developed. This model loosely couples a Generalized Dynamic Wake Theory (GDWT) to a Navier-Stokes solution procedure. This coupling is achieved using a newly developed unsteady pressure jump bound- ary condition in the Navier-Stokes model. The new unsteady pressure jump boundary condition models each rotor blade as a moving pressure jump which travels around the rotor azimuth and is applied between two adjacent planes in a cylindrical, non-rotating grid. Comparisons are made between predictions using this new model and experiments for an isolated rotor and for a coupled rotor/fuselage configuration.

v V ,ll,,..j M.'

Acknowledgments

V I would like to thank the personnel in the Subsonic Aerodynamics Branch and the U.S. Army Joint Research Program Office, Aeroflightdynamics Directorate, of the NASA Langley Research Center. In particular, I would like to thank Ms. Susan Gorton, and Mr. Edgar Waggoner for their interest in and support of this research effort, which was funded under Grant Number NCC-1245.

I also would like to thank Dr. Richard Barnwell, Mr. Mark Chaffin, and Dr. Henry Jones, for the many useful discussions on computational fluid dynamics and its application to rotorcraft. A special thank you goes to my wife, Kim, and son, David, who have seen me through this entire experience.

v

Nomenclature

English symbols aoo freestream speed of sound [m/sec] rn m an, bn series expansion coefficients [see Appendix A] local area ratio between blade and computational cell A(?)

i the imaginary number, v/-Z--f c local blade chord normalized by R g coefficient function [see Appendix A] M.,- force and moment coefficients

Cl,Cm

pressure coefficient

Cp

modified pressure coefficient unsteady component of modified pressure coefficient e; thrust coefficient

Cr

mean thrust coefficient CT roll moment coefficient

G

mean roll moment coefficient

CL

cM pitch moment coefficient

CM

mean pitch moment coefficient stagnation energy per unit mass [m2/sec 2] eo

P,_,ft

inviscid fluxes (see equations (4.3) to (4.5))

L, dv,_v viscous fluxes (see equations (4.3) to (4.5))

hinge offset location normalized by R V iv V

/-/2 coefficient function [see Appendix A]

[Lq quasi-steady inflow matrix, cosine component

quasi-steady inflow matrix, sine component

[Lq

L sectional blade lift [N/m] M local Mach number v freestream Mach number Moo Nr number of azimuthal time steps per rotor revolution P,p pressure [N/m 2] Poo freestream pressure [N/m 2] normalized, associated Legendre functions of the 1st kind normalized, associated Legendre functions of the 2nd kind ith component of perturbation velocity [m/sec] qi cartesian components of heat flux [J/(m 2 sec)] qx,qy,qz vector of conservative variables = [9, 9u, Ov, 9w, 9co] r ith conservative variable Qi f radial location on disk, normalized by R F radial location on disk [m] R rotor radius [m] R gas constant [J/(kg K)] t time [sec] T temperature [K] nondimensional time, normalized by £2 V_ freestream velocity, normalized by f2R v,. induced inflow velocity [m/sec] elements of mass flow matrix

v2

v cartesian velocities [m/sec] W local normal component of velocity at rotor disk, normalized by D..R local normal component of w Wl v V Greek symbols effective angle of attack [rad] _e// rotor shaft tilt angle [rad] blade mean coning angle [rad] first harmonics of rotor flapping [rad]

f lc,f ls

blade mean coning angle [rad] -?" -r induced inflow coefficients _j, [3j ratio of specific heats Y Dirac delta function [see Appendix A] 8q AP pressure jump [N/m 2] E user specified tolerance X freestream component normal to disk, normalized by _2R, positive down Xi induced inflow ratio = Vi/(f2R) time averaged induced inflow ratio _i,mean mean component of Xi generic time averaged function [see Appendix A] pressure function, normalized by 9_2R 2 radial expansion shape function v, rl,gt dimensionless ellipsoidal coordinates _t _t on rotor disk phase of first harmonic of blade pitch [rad] local air density [kg/m 3] P freestream air density [kg/m 3] poo rotor advance ratio = V=/_R /./ induced inflow in/.t direction, normalized by _R, positive downstream ,ui mean component of Pi 13i,mean local blade pitch [rad] built-in blade twist function [rad] Oo blade collective pitch [rad] Oc, Os cosine and sine components of blade pitch [rad] Ol magnitude of first blade pitch harmonic [rad] vi -mc cosine part of pressure expansion of m th harmonic and n th polynomial "C n sine part of pressure expansion of mth harmonic and n th polynomial stress tensor [m/sec] "r, ij COn rigid flap natural frequency [cycles/revolution] rotor rotational speed [rad/sec] Subscripts add additional component i ith component derivative with respect to ith direction ,i n,j,q polynomial number coordinate along freestream line, positive pointing upstream oo freestream quantities Superscripts harmonic number m,r,p u unsteady component ,...7 vii Operators & Acronyms

A(.) operator used to denote an incremental value of (.)

S(.) filter operator

(.)_ double factorial [see Appendix A]

derivative with respect to ?

(.)

C)

reference quantities GDWT Generalized Dynamic Wake Theory RLM rotor loading model RFFM rotor/fuselage flowfield model V V ,,j Vlll

Contents

Abstract ii "_ 111 Acknowledgments ,wo Nomenclature vm 1 Introduction 1 2 Rotor Loading Model: Generalized Dynamic Wake Theory 12 ix 3 GDWT Validation 19 4 Rotor/Fuselage Flowfield Model: OVERFLOW 61 J v 5 Coupling Model 69 6 Results: Isolated Fuselage 74 i X xd 7 Results: Isolated Rotor 85 '%.-* 8 Results: Rotor/Fuselage 94 xi

9 Summary 145

Bibliography 152 A Filtering Operation Vita 157 V xii

List of Figures

1.1 Analysis Types for Coupled Solutions ......................... i0 1.2 Current Hybrid Method ................................ i 1 3.1 ROBIN Fuselage in the NASA Langley Research Center 14- by 22-Foot Subsonic 3.2 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangu- 3.3 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.4 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangu- 3.5 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.6 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangu- 3.7 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.8 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangu- XII1 3.9 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.10 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered 3.11 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.12 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered 3.13 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.14 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered 3.15 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.16 Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered 3.17 Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced 3.19 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values E_=.

3.20 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.21 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values v 3.22 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values xiv 3.23 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.24 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values

v

3.25 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.26 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.27 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.28 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.29 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.30 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.31 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.32 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.33 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values 3.34 Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values v 5.1 Current Hybrid Method ................................

XV llL I 6.1 ROBIN Fuselage in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel .........................................

v 6.4 Lift, Drag, and Sideward Direction Force Coefficients. Steady State Computation.. 82 6.6 Measured and Predicted Pressure Coefficients vs Vertical Location for c_ = 0 ° and M_ = 0.1265 ......................................

7.1 IRTS in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel... 90 7.3 Measured and Predicted Induced Inflow in Two Directions at Location J for the 7.4 Measured and Predicted Time Averaged Induced Inflow Ratio for the IRTS (Mea- 8.1 IRTS/Fuselage Configuration in the NASA Langley Research Center 14- by 22- Foot Subsonic Tunnel .................................

8.2 Schematic of a Periodic Grid and Replacement for Periodic Grid in LU-SGS Scheme. 110 8.3 Lift, Drag, and Sideward Direction Force Coefficients, Time Averaged Computation. 111 8.4 Measured and Predicted Unsteady Modified Pressure Coefficient on the Top Cen- 8.5 Measured and Predicted Unsteady Modified Pressure Coefficient on the Retreating 8.6 Measured and Predicted Induced Inflow in Two Directions for an Isolate Rotor and 8.7 Measured and Predicted Time Averaged Induced Inflow Ratio from Time Accurate xvi

8.8 MeasuredandPredicted Time AveragedParallelInducedInflow Ratio from Time

8.9 MeasuredandPredicted Lateral andLongitudinal Time Averaged InducedInflow

8.10 MeasuredandPredicted UnsteadyModified Pressure Coefficienton theTop Cen-

8.11 MeasuredandPredicted Unsteady Modified Pressure CoefficientontheRetreating

8.12 MeasuredandPredicted InducedInflow in Two Directionsfor anIsolateRotor and

8.13 Measured andPredicted Time Averaged Induced Inflow Ratio from Time Accurate

8.14 Measured and Predicted Time Averaged Parallel Induced Inflow Ratio from Time 8.15 Measured and Predicted Lateral and Longitudinal Time Averaged Induced Inflow 8.16 Measured and Predicted Unsteady Modified Pressure Coefficient on the Top Cen- 8.17 Measured and Predicted Unsteady Modified Pressure Coefficient on the Retreating 8.18 Measured and Predicted Induced Inflow in Two Directions for an Isolate Rotor and 8.19 Measured and Predicted Time Averaged Induced Inflow Ratio from Time Accurate 8.20 Measured and Predicted Time Averaged Parallel Induced Inflow Ratio from Time 8.21 Measured and Predicted Lateral and Longitudinal Time Averaged Induced Inflow xvii 8.22 Isolated Fuselage and Rotor/Fuselage, Time Averaged Surface Pressure Coefficients 130 8.24 Time Averaged Surface Pressure Coefficient on Complete Rotor/Fuselage Config- 8.26 Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Config- 8.28 Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Config- 8.29 Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Config- L_ 8.31 Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Config- 8.32 Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Config- 8.34 Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Config- 8.35 Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Config- XVlll

List of Tables

6.1 Computational Grid System .............................

yz_ V V xix

K,J

Chapter I

Introduction

1.1 Motivation

It is well known that rotary wing aircraft aerodynamics are complicated. Unlike fixed wing aircraft, on which a steady-state flight condition typically implies steady-state aerodynamics, a rotary wing aircraft experiences a significant unsteady aerodynamic environment in all flight conditions, even in level, unaccelerated flight, due to the presence of the rotating wings (rotors). This aerodynamic environment includes the aerodynamic interactions, which are inherently unsteady and complex, between the rotor(s) and the fuselage. One example of the complexity associated with these inter- actions is the problem of flow separation phenomena. Whereas fixed wing aircraft typically have little significant flow separation in steady-state flight due to their streamlined fuselage shapes, ro- tary wing aircraft typically have blunt aft regions that are conducive to flow separation. Even in hover, the flow induced by the rotor(s) impinging on the fuselage tends to separate on the underside of the fuselage which is often a blunt surface. Sheridan and Smith [i] discuss many other examples and categories of rotorcraft interactional aerodynamics. They also state in their conclusions that: "... it will be necessary to develop tractable theories and analytical methods to account for all these phenomena. Interactional aerodynamics of the airframe is not as neatly packaged as rotor aerodynamics. Many of the interactions involve viscous processes, and in some aspect semi-empirical techniques may always be needed. But a start must be made in developing the required mathematical models so that we can cope with these problems adequately in the vehicle design phase."

't,..-' David D. Boyd, Jr. Chapter 1. Introduction 2 In short, the above passage calls for the development of methods to address the coupled ro- tor/fuselage interactional aerodynamic effects, whereas previously, the rotor effects and fuselage effects were treated in isolation.

In the years since Sheridan and Smith's paper [I], many types of analyses have been developed and used in the prediction of the unsteady interactional aerodynamic characteristics of rotorcraft.

Figure 1.1 is a graphic that depicts and categorizes several of the major methodologies.

In the area of relatively low computational expense and complexity (see figure 1.1), singularity methods have been used. These methods typically use singularities, such as a lifting line to repre- sent the rotor blade, a system of vortices to represent the wake, and source, doublet, and/or vortex panels to model the fuselage. In a relatively inexpensive and computationally efficient manner, these methods are able to capture low order effects on each component due to the other compo- nent, such as the mean downwash on fuselage due to the rotor or the mean inflow at the rotor disk due to the fuselage. But, since the fuselage is typically modeled using a panel method, calcula- tion of some interactional aerodynamic effects, such as flow separation due to rotor downwash, is difficult. In cases where viscous effects are predominant, the viscous flow effects must be either ignored, specified a priori, or determined by coupling the method to a boundary layer type model.

On the other end of the computational expense and complexity scale, at relatively high computa- tional expense and complexity, are the methods involving computational fluid dynamics (CFD), in particular, Navier-Stokes methods. These types of methods have been used to calculate the entire flow field of the complete rotorcraft configuration, all in one computation. Even though these com- putational methods are theoretically able to capture all of the interactional aerodynamic couplings between the rotor(s) and fuselage, their computational expense is prohibitive for routine use.

There is a lack of methods available in the literature which fall between the singularity methods and the Navier-Stokes CFD methods for studying unsteady interactional aerodynamics of rotor- craft. The current research is motivated by the lack of available hybrid methods which are compu- tationally efficient, yet are able to capture primary interactional aerodynamic effects between the rotor and fuselage. Figure 1.1 shows, with a dotted ellipse labeled "Hybrid Methods", where the current work falls on the computational expense and complexity scale.

David D. Boyd, Jr. Chapter 1. Introduction 3

1.2 Literature Review

As discussed previously, unsteady rotor/fuselage interactionaI aerodynamics generally fall into three categories: (1) singularity methods, (2) CFD methods, and (3) hybrid methods. In the fol- lowing sections, a brief review of each is given.

V 1.2.1 Singularity Methods .%¢, Singularity methods are typically characterized by the use of a source, doublet, and/or vortex panel representation of the fuselage, a lifting line or lifting surface representation of the rotor, and a vor- tex lattice model representation of the rotor wake. For rotorcraft analyses, these methods are used to compute the flowfield of the complete vehicle. Johnson [2] provides an extensive discussion of singularity methods used for rotorcraft analyses up through the year 1986. Since that time, other singularity methods have been developed as well. Egolf and Lorber [3] used a source panel de- scription of the fuselage, a lifting line blade model, and a prescribed vortex wake description of the rotor/fuselage system to model the unsteady rotor/fuselage flowfield. The prescribed vortex wake was prevented from cutting through the fuselage by displacing, in an a priori manner, the segments of the vortices that would otherwise have been inside the fuselage. No attempts were made to model the wake of the fuselage or the flow separation from the fuselage. Only limited comparisons were made to experimental data. Mavris, et al. 14] used a doublet representation of r , the fuselage, a lifting line blade model, and a free vortex wake description of the rotor/fuselage system. No modeling was used for the fuselage wake or fuselage flow separation. Also, vortex wake filaments that are inside the fuselage were excluded from the computations. Comparisons with experimental pressures show good agreement along the top of the fuselage, but agreement degrades on the sides of the fuselage. Mavris, et al. [4] attribute these discrepancies to flow sepa- ration on the fuselage and to inadequate vortex-surface interaction predictions. Berry [5] combined a fuselage source panel representation, a source-dipole representation for the rotor, and a distorting vortex-lattice representation of the rotor wake to model the rotor/fuselage system. Comparisons are made to measured time averaged and unsteady inflow velocities; no fuselage pressure comparisons are made which include the rotor influence. Quackenbush, et al. [6] used a source/doublet de- scription of the fuselage, a vortex-lattice model for the rotor blades and a novel "Constant Vorticity Contour (CVC)" free wake model; fuselage flow separation was not modeled. Close surface/vortex interactions were modeled using selective remeshing of the curved vortex elements and using an - j David D. Boyd, Jr. Chapter 1. Introduction 4 "Analytical/Numerical Matching (ANM)" scheme. Computational efficiency was improved by us- ing "fast vortex methods" for wake-on-wake and wake-on-body computations. Generally good agreement with measured results are demonstrated for time averaged induced velocities above the rotor disk and for time averaged and unsteady pressures on the top centerline of the fuselage.

Crouse [7] used a source panel description of the fuselage, a lifting line blade model, and a free tip vortex wake model without an inboard wake model to represent the rotor/fuselage system. Vortex wake elements that cross the fuselage surface are handled by splitting them into smaller segments, and shifting the collocation points of these smaller segments such that they are at a specified min- imum distance from the fuselage surface. This method is similar in concept to that used by Egolf and Lorber [3] as discussed above. Good comparisons of unsteady pressures were shown on the top centerline of the tail boom of the fuselage. Boyd, et al. [8] included the open-loop effects of a fuselage, represented by a non-lifting fuselage source panel method, on the rotorcraft trim in a comprehensive rotorcraft analysis. This effect was implemented in the comprehensive analysis as an additional rotor inflow distribution plus an additional rotor wake distortion due to the presence of the fuselage. Effects of the rotor on the fuselage were not modeled. Though some computations have proved successful and can be computationally efficient, all of these singularity methods suffer from the inability to predict some of the rotor/fuselage interactional effects. For example, methods that use a source panel description of the fuselage do not have the capability to determine the lift or the lift change on a fuselage due to the rotor. Also, quantities such as flow separation and drag must either be ignored, must be specified a priori, or must be determined by coupling the method to a boundary layer model.

v 1.2.2 CFD Methods In recent years, unsteady calculations on complete rotorcraft configurations using CFD methods have become possible. In addition, there are several degrees of complexity that can be modeled with CFD. For rotorcraft applications, methods have been developed to solve the full potential equation, the Euler equations, and the Navier-Stokes equations.

Chen and Bridgeman [9] coupled the three dimensional boundary layer equations to the full potential equations in a blade-fixed coordinate system for an isolated rotor. The three dimensional boundary layer equations assumed that the surface curvature effects were negligible and included additional terms in the x- and z-momentum equations to account for centrifugal and Coriolis forces in the boundary layer due to blade rotation. These equations were coupled by using a modified David D. Boyd, Jr. Chapter 1. Introduction 5 tangency boundary condition. This modified boundary condition enforces a velocity component ("transpiration velocity") normal to the surface which "deflects the inviscid flow from the body surface thus simulating the displacement of the inviscid flow due to the momentum defect in the boundary-layer" [9]. Good comparisons were shown for integrated drag quantities (torque) on a non-lifting isolated rotor in hover for a range of hover tip Mach numbers. Also, good chordwise pressure coefficient comparisons were shown for two radial stations in a non-lifting, forward flight condition. Bridgeman, et al. [10] solved the unsteady, full potential equation coupled to a three dimensional boundary layer model for isolated rotors in hover and forward flight. This method is similar to that presented in [9], with a number modeling improvements. Though it is possible conceptually to include a fuselage in these full potential computations, this would be difficult in practice due to the blade-fixed coordinate systems typically used in such analyses. Thus, interac- tional aerodynamic computations would be difficult to compute using these existing tools.

Recent solutions to the Euler equations for isolated rotor applications have used unstructured grid techniques [11, 12, 13] to refine the grid system efficiently to better capture wake structures such as tip vortices. These techniques may be extended easily to include a fuselage body. How- ever, since the Euler equations do not include viscous terms, computation of any viscous effects (e.g., viscous drag on a fuselage) would, like the full potential equation examples above, still re- quire coupling the method to a boundary layer analysis.

Navier-Stokes computations have also been developed for use in rotorcraft analysis. While time averaged Navier-Stokes methods have been developed and are quite practical to use [14, 15, 16, 17, 18], routine computations for unsteady flows on full configurations are not yet practical. Even so, several of these computations [19, 20] are found in the literature. Meakin [19] used a thin-layer Navier-Stokes method to calculate the flowfield around the Bell/Boeing V-22 Osprey tiltrotor air- craft, including the fuselage and the rotor, for a fictitious flight condition. Though this was a full aircraft simulation, the purpose of the computation was to demonstrate the feasibility of using a new domain connectivity algorithm for the moving, dynamic, overset grids for such a computa- tion. Since the computations were performed to demonstrate a technology, no comparisons are made with experimental quantities. Srinivasan and Ahmad [20] used a Navier-Stokes scheme to calculate the quasi-steady flowfield for a hovering rotor mounted on a whirl tower. Due to the quasi-steady nature of the hovering condition, the equations were solved in the blade-fixed coor- dinate frame, using a momentum source term in the equations to account for the centrifugal force of the blade rotation. This simulation was also a feasibility study, so only a comparison of the predicted and measured mean thrust values were presented. For this simulation, which utilized David D. Boyd, It. Chapter I. Introduction 6 approximately 1.3 million grid points, a computational time of 14 Cray-YMP hours was quoted.

Ahmad and Duque [21] used a thin-layer Navier-Stokes method with embedded, moving, overset grids to demonstrate the ability to calculate the unsteady flowfield of an isolated, two-bladed rotor in forward flight. Blade surface pressures at several radial and azimuthal and local normal load co- efficients are compared to flight test data. These comparisons match reasonably well. Even though this computation did not include a fuselage body, the chimera grid scheme would render the task of including a body feasible, though at an additional computational cost. This isolated rotor compu- tation required substantial computational resources; it required approximately 45 Cray C-90 hours and generated 40 Gigabytes of data.

Though most of these Navier-Stokes methods are suitable for computations over complete air- craft, including the helicopter rotor, all of the moving grid computations suffer from the require- ment to re-compute the grid domain connectivity information at every time step; this technique is known as a "dynamic chimera scheme" and has two distinct disadvantages. First, regenerating the grid domain connectivity at every time step can be as computationally expensive as, or even more computationally expensive than, the actual flow solution. In addition, in some instances, the time step is restricted not by the flow or flow solver, but by the moving grid domain connectivity requirement that a "hole point" not become a "field point" at any time step [19]. This requirement can potentially limit the time step not to physical phenomena, but to grid cell size.

In addition to time step issues discussed above, other factors place limits on the current Navier- Stokes computations for rotorcraft. One issue is the numerical dissipation of concentrated vortices.

It is well known that, in certain flight conditions, blade tip vortices can have a large effect on the rotor aerodynamics and that these vortices need to be computed in the flowfield over several rotor revolutions. Numerical studies discussed in the literature suggest that a 5th order scheme using 14 points across the vortex core produces satisfactory results for a vortex that is well-aligned with the grid [22]. However, in a typical rotorcraft simulation, the vortex location is not known apriori and thus the vortex in general will not be aligned well with the grid. Therefore a more strict resolution requirement is imposed on the numerical scheme [22]. For current methods either a prohibitively dense grid must be used to assure that the vortex is resolved well spatially, or grid adaption must be used to refine the grid in the regions that contain the vortices. Both methods are computationally expensive.

Another such issue is turbulence modeling. Many of the turbulence models in current use were developed for wall bounded flows. They are not well suited to the three dimensional, non-isotropic turbulence associated with rotor blade tip vortices.

David D. Boyd, Jr. Chapter I. Introduction 7 1.2.3 Hybrid Methods Considering the computational expense of current CFD methods, one possible approach to examin- ing unsteady rotor/fuselage interactional aerodynamics is to use hybrid methods. For unsteady ro- tor/fuselage aerodynamics, several hybrid methods have been developed. One such hybrid method, developed by Steinhoff, et al. [23], modified the Navier-Stokes equations by adding a "vorticity confinement" term to the momentum equations. This new term is used to prevent, or counter- act, numerical diffusion of concentrated vortical regions by "convecting" vorticity back toward the centroids of concentrated vorticity regions in the flowfield. This particular method is well suited for inclusion of a fuselage body. Boyd and Bamwell [24] first introduced a hybrid unsteady rotor model which weakly couples a Generalized Dynamic Wake Theory (GDWT) [25, 26, 27, 28] to a thin-layer Navier-Stokes model, OVERFLOW [29]. Extensive induced inflow comparisons were made between Laser Doppler Velocimeter (LDV) measurements and predictions. Even though the computations were for an isolated rotor, excellent agreement was found with measured quanti- ties. Also presented was an outline of a method to couple a fuselage into the calculations using the overset grid capabilities in OVERFLOW. This new model uses the GDWT to obtain unsteady loading and unsteady induced inflow on the rotor, and then applies the unsteady loading inside OVERFLOW as a new unsteady pressure-jump, actuator disk-type, boundary condition. Another hybrid method, building on the previous literature [24], is developed in this research.

1.3 Present Approach

The objective of the current research is to develop an efficient, hybrid, unsteady computational model appropriate to the study of unsteady rotor/fuselage interactional aerodynamics.

In examining fully CFD, unsteady, moving grid methods for complete rotorcraft, it can be ob- served that small time steps are needed for method stability, for capturing aerodynamic effects that are on the order of the rotor blade chord size, and for proper usage of the dynamic chimera grid scheme. However, to capture the primary effects of rotor/fuselage interactional aerodynamics, chordwise aerodynamics on the rotor blade are of less importance than the gross loading on the rotor blade itself. This can be seen by the successes of some of the singularity methods which use lifting line rotor blade models (i.e., no chordwise loading distribution on the rotor blade) discussed previously in the "Literature Review" section above. In addition, fully CFD methods compute the rotor loading internally and require a number of rotor revolutions to obtain a periodic solution. The David D. Boyd, Jr. Chapter I. Introduction 8 combined requirements of needing very small time steps and of needing several rotor revolutions to obtain a periodic solution are a large contributor to the computational expense of these methods.

A hybrid method is developed here which reduces the computational expense by separating the rotor loading calculation from the CFD component of the computation. This hybrid method is k.l depicted schematically in figure 1.2. From the figure, it can be seen that there are three components to this hybrid method: 1. Rotor Loading Model, 2. Rotor/Fuselage Flowfield Model, 3. Coupling Model.

1.3.1 Rotor Loading Model The present approach separates the rotor loading model from the rotor/fuselage flowfield model.

Splitting the procedure into these two separate models allows an otherwise computationally expen- sive element, the rotor loading computation, to be accomplished using a efficient, simplified model apart from the CFD computation. To compute the rotor loading, the GDWT [25, 26, 27, 28], which will be discussed in detail in a subsequent chapter, is used here. Previous implementations of the GDWT have focused on calculation of the unsteady inflow for an isolated rotor. As a significant advance over previous models, the unsteady rotor portion of the model uses the GDWT to calculate unsteady infow and unsteady loading. The unsteady loading on the rotor is determined in the form of a AP, or "pressure jump", across the rotor disk. This zSaois then used as a boundary condition in the rotor/fuselage flowfield model.

v 1.3.2 Rotor/Fuselage Flowfield Model The unsteady loading, as determined by the GDWT, is then used in conjunction with a Navier- Stokes model, in this case, OVERFLOW, to compute the time dependent flow over the fuselage including effects of a helicopter rotor. The loading is used in the Navier-Stokes method as an unsteady boundary condition in the flowfield. This boundary condition is effectively an unsteady actuator disk model. Though there will be concentrated regions of vorticity near the edges of an unsteady actuator disk model used as a boundary condition in this manner, these are not true David D. Boyd, Jr. Chapter 1. Introduction 9 "tip vortices" and thus internal structure of these flow features is of secondary importance. This alleviates the need to develop prohibitively dense grids, use grid adaption, or use higher order schemes to resolve these concentrated vorticity regions. As such, turbulence modeling of the inside of these vortex structures becomes less important as well. So, by modeling the rotor as an unsteady actuator disk in OVERFLOW, several computationally expensive requirements typically needed for full CFD rotorcraft modeling are diminished. Using the above model, the Navier- Stokes method is then used to compute the periodic flowfield of the rotor/fuselage combination.

Further details of this component will be discussed in a subsequent chapter.

1.3.3 Coupling Model With the completion of the Navier-Stokes method, there are two solutions which were obtained with the same AP distribution: the GDWT solution, which is for an isolated rotor, and the OVER- FLOW solution, which contains both the rotor (as a boundary condition) and the fuselage body.

The primary difference is that one solution contains a fuselage and the other does not. Since the loading in the GDWT depends on the rotor inflow, and since these inflow values are influenced by the presence of the fuselage, a method of coupling the two codes has been developed to account for the fuselage effects in the GDWT (and thus the rotor loading model). Since the fuselage effects on the rotor are assumed to consist of low frequency effects (as compared to the higher frequency blade-passage effects), the method employed here differences the time averaged inflow generated in the two successive solutions, and uses this difference as an "inflow correction" to the GDWT.

This coupled process continues until only small differences are seen between successive iterations.

v David D. Boyd, Jr. Chapter 1. Introduction 10 / J O L) r Computational Expense Figure 1.1" Analysis Types for Coupled Solutions.

David D. Boyd, Jr. Chapter 1. Introduction 11 Rotor/Fuselage _P(r,_,t) Rotor Loading Flowfield lab.-- laB""- Model Model (GDWT) (OVERFLOW) %=J Coupling Model (Inflow Corrections) Figure 1.2: Current Hybrid Method.

Chapter 2

V

Rotor Loading Model: Generalized

Dynamic Wake Theory

M.;

2.1 Introduction

The Generalized Dynamic Wake Theory (GDWT) of Peters, Boyd, and He [26], Peters and He [25], and He [27] is used in the present approach to obtain unsteady loading that is to be used in conjunction with OVERFLOW as discussed previously. In a later chapter, the OVERFLOW and the coupling technique between OVERFLOW and the GDWT will be discussed. Even though the GDWT is spelled out in detail in the literature, the present chapter will describe the GDWT for background purposes and describe the particular manner in which the theory is implemented for the current research.

2.2 General Description

The GDWT is a theory that was originally designed to pose the issue of unsteady aerodynamics of a helicopter rotor in a state-space form. This type of state-space form is desirable for inclusion in a rotor stability analysis since stability analyses for the rotor dynamics are usually presented in a state-space form as well. With the aerodynamics and dynamics of the rotor stated in similar forms, K.J the solution of the system of equations is simplified.

David D. Boyd, Jr. Chapter 2. Generalized Dynamic Wake Theory 13

2.3 GDWT Outline

There are several aerodynamic concepts which are combined in the development of the GDWT.

First, the acceleration potential derived by Kinner [30] for circular wing planforms is used with slight modification. The original acceleration potential derived by Kinner is a general form for the solution to Laplace's equation (inviscid, linear, potential flow) in ellipsoidal coordinates. These modifications to Kinner's acceleration potential function (or pressure function), applied by Peters, Boyd, and He [26], Peters and He [25], and He [27], are made to eliminate terms in the potential that are not compatible with boundary conditions associated with a rotor. This modified accel- eration potential, _, is then expressed using Legendre functions and transcendental functions in ellipsoidal coordinates as follows: _(v,_,_,t") =_1 _ _ /_n (V)_n(irl)[_nc(t-)cos(m_)+ -_ns(t-)sin(m_/)] (2.1) 2 m=O n=m+l,m+3,...

mc and ms terms are general coefficients of the pressure function and are, in general, where the "_n % functions of time and are determined from the loading on the rotor. The ellipsoidal coordinate system used here can be seen in figure 2.1. This figure shows a view of the xz plane with represen- tative 11 and v values labeled. The _ coordinate (not shown in figure 2.1) is an angular, azimuthal coordinate, measured around the z-axis. The "rotor disk" is defined by the following conditions: n = 0 (2.2) v = v/'i -72 (2.3) %.., = V (2.4) where ? is the radial coordinate on the rotor disk, measured from the axis of rotation, and gt is the angular, azimuthal coordinate measured about the axis of rotation. Equation (2.1) is effectively an expression for all admissible functions of loading on a rotor. To establish a link between loading on the rotor and induced inflow, the continuity equation and the linearized, incompressible Euler equations are used as follows: 'K,J qi,i "- 0 (2.5) David D. Boyd, Jr. Chapter 2. Generalized Dynamic Wake Theory 14 qi-Vo.qi,_ = -d_,i (2.6) ql = u (2.7) q2 = v (2.8) q3 = w (2.9) V

(2.1o)

where the summation convention is assumed over the index i, and _ is the coordinate pointing upstream along a streamline• Using equations (2.5) and (2.6), it can be shown that the pressure function, _, satisfies Laplace's equation. Also, since these equations are linear, the pressure func- tion can be split into a component associated with each of the two terms on the left hand side of equation (2.6). Each of these components, in tum, also satisfies Laplace's equation• As such, solu- tions can be derived for each component, then combined into a complete solution. The quantities used here are assumed to be total quantities, not perturbations. Also, it is assumed that only the velocity normal to the rotor disk is of interest and that it is of the following form: (2.11) w(?,lg, t-') = _ _ _(?) [_(t-)cos(rV)+ _(t-) sin(rV)] r=O j=r+l,r+3,...

With these assumptions, a closed form set of first order, non-linear differential equations, in state- -r -r space form, can be established for the induced inflow coefficients aj, 13j. These equations are as follows: - m _ -mc (2.12) -m _1._[Lc] -1 _n T'n o_ n

} {}

{

{ }"

(2.13)

=

+[L'] -l

'I I:}

°'• /

{}

With these equations, the unsteady aerodynamics and induced inflow are cast in the time domain and in a state-space form. The problem has now been reduced to the computation of the states of -m -m the model, an, [3 n , given a loading on the rotor. This is a non-linear model since the loading and the inflow are coupled through mass flow parameter, Vm, in equations (2.12) and (2.13). As described David D. Boyd, Jr. Chapter 2. Generalized Dynamic Wake Theory 15 in reference [26], the mass flow parameter Vm, which accounts for the energy added to the flow by the rotor, takes the place of the Vo, term that results from equation (2.6) in order to extent the theory and to cover the case of hover, where Vo. approaches zero.

2.4 Solution Procedure

Though the details of the GDWT are discussed in the literature, few details are provided for the solution of this set of first order, non-linear differential equations. The procedure used in the current research is described here.

First, for a given set of rotor collective, lateral, and longitudinal pitch controls settings, the blade loading may be determined by any theory that can generate a loading given velocity (inflow) infor- mation. In the current research, as was done in previous literature [26, 25, 27], a two dimensional strip theory is employed. At first, this may seem to be an unnecessary restriction to a two dimen- sional theory. It has been shown [26, 25, 27] that this is not a restriction since the inflow and the loading are coupled. So, three dimensional effects, such as load reduction at the blade tip, are K.2 included in this theory. The equations of the strip theory used in this research are outlined below: (2.14) = Otw(F ) +00+0ccos_g+0ssinlg

o(7,v)

= O-(w+psin_s+_ol_COS_g-O.5cO+Z_i) (2.15) (Xef f ?+psin_ L r_c(? + p sin_g)2aeff (2.16) p_22R 3 F Equation (2.16) gives the lift at a particular point on the rotor disk. Through the effective angle of attack, C_eff, this loading theory includes local effects of the blade pitch (0), of the total induced inflow velocity (w), of the inflow due to the shaft tilt (psincXs), of the inflow due to the mean blade coning angle ([3o/acos_), of the velocity at the 3/4 chord point due to blade pitch rate (0.5c0), and of the inflow correction determined by the coupling scheme (z_i), described in a later chapter. In addition, the lift curve slope is assumed to be 2n per radian, the Prandtl-Glauert compressibility correction is applied to the lift, and a simple stall model is used which limits the maximum angle of attack to 10 degrees. Now, given the blade pitch settings of the rotor and the rotor operat- V ing conditions, the local sectional loading can be determined from equations (2.14), (2.15), and David D. Boyd, Jr. Chapter 2. Generalized Dynamic Wake Theory 16 (2.16). With the lift distribution determined, equations (2.12) and (2.13) are solved using a 4-stage Jameson-style Runge-Kutta technique [31] until periodic induced inflow is obtained; this usually occurs within two rotor revolutions. No blade dynamics model is used in this study and the blade hinge offset is assumed to be at the center of rotation (i.e., the flap hinge is at the center of rotation).

With the above calculation complete, the mean thrust coefficient and mean hub moment coeffi- cients can be determined from the following: 4 f2_'c°Cd , (2.17) 4 f2_ lc

CM - 3v%Jo z2 dv (2.18)

4 f2rc ls eL -- 3V/_ J0 z2 dv (2.19) Since typically the initial pitch settings do not produce the desired thrust and moment coef- ficients on the rotor, they must be adjusted in subsequent iterations until the desired thrust and moment coefficients are obtained. This is known as "trimming" the isolated rotor; this accounts only for the desired loading on the isolated rotor and does not include any "feedback" forces from any other source (such as a fuselage). The trim procedure used in the current research for the isolated rotor is a modified Newton-Raphson technique adapted from the literature [32]. The fol- lowing equation is used to determine the pitch setting "corrections" which are used to iteratively trim the isolated rotor: -1

I OCt) (ac )

Ngo

OCM)

acM

(2.20)

ACM

An c

NTo

{ }

A0o }

ACL

A0s

ac<] (ac<

kNgo )

where the matrix of partial derivatives is called the "derivative matrix" and each partial derivative is determined by using a one-sided forward difference formula and by using an independent per- turbation to each of the initial pitch settings. Each row in the derivative matrix is computed by an independent perturbation of each corresponding pitch setting and solving equations (2.12) and (2.13). Once computed, the derivative matrix is held unchanged throughout the subsequent rotor David D. Boyd, Jr. Chapter 2. Generalized Dynamic Wake Theory 17 trimming process. The ACt, ACM, and ACL are the changes in the current thrust and moment coef- ficients required to match the desired values. The trimming process is considered complete when value of the following is true: %,d v/(ACT) 2 Jr- (ACM) 2 -at- (ACL) 2 < E (2.21) where e is a specified tolerance. Now, with the trim task complete, the unsteady loading and unsteady induced inflow are known.

At the end of the trim process, the unsteady loading is in the form of a sectional loading (i.e., force per unit span). However, for use in OVERFLOW, the loading needs to be in the form of a pressure that will be applied to a grid point in the flowfield. The sectional loading is converted to a pressure (force per unit area) using the assumption that the force is evenly distributed over the chordwise and spanwise extent of the local blade section of interest. These pressures are now ready for use in OVERFLOW, which will be discussed in a later chapter.

Since the current research employs the GDWT in a manner in which it is not normally used, new computer coding has been developed here to compute required quantities from this theory, in combination with solution methods that have not previously been used with the GDWT. To validate that the new theory and models have been implemented correctly, a validation study is presented in the next chapter.

David D. Boyd, Jr. Chapter 2. Generalized Dynamic Wake Theory 18 v=1.0 v=0.4

/ :oo

v=-0.4_ v _--1.___-.0. 8 V Figure 2. i: Ellipsoidal Coordinates.

Chapter 3

GDWT Validation

Introduction

3.1

Though there are numerous examples of the GDWT published in the literature, the present combi- nation of the GDWT with the present solution procedures has not been explored in the past. Also, computer coding to do the actual computations with these combinations of theory and solution pro- cedures was not available for the purposes of the current research. As such, validation is required to determine that the current model matches previously published literature on the subject. This validation effort is described in this chapter.

3.2 Experimental Setup

All of the experimental data used for the validation effort in this chapter was obtained from laser velocimeter inflow measurements made in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel. The experimental setup and data used for this chapter is described in detail in references [33, 34, 35, 36]; for completeness, the experiments are described here in brief.

Two different planforms were used in this test, one rectangular and one tapered. A summary of the two geometries is given in table 3.1 and a picture of the configuration, installed in the tunnel, is shown in figure 3.1. In this figure, the rotor/fuselage configuration consists of a generic helicopter fuselage body, known as the ROtor Body INteraction (ROBIN) fuselage [37], and one of the two David D. Boyd, Jr. Chapter 3. GDWT Validation 20 rotor configurations, described above. Inflow data was taken on these two rotor configurations at several advance ratios (i.e., forward speed divided by the rotor tip speed) for 12 azimuthal stations located 30 degrees apart (starting from an azimuth location of 0 ° , directly downstream of the rotor hub) and at a number of radial stations at each of these azimuth stations. Data samples were processed at a resolution of 128 samples per rotor revolution at each measurement location. For this chapter, the data is presented in two formats. The first format is the time averaged data, which is obtained by temporally averaging the unsteady data at each measurement location. The second format is time accurate data, which is the unsteady data before it is time averaged.

There are two advance ratios used in this chapter. The first,/1 = 0.15, can be though of as boundary between (1) very low speed flight, where the fuselage flowfield is completely dominated by the rotor wake, and (2) moderate forward flight, where the fuselage flowfield is no longer dominated by rotor wake interactions. The second, p = 0.23 can be thought of as a moderate forward flight case. At both speeds, pressure pulse effects on the fuselage from the passing rotor blades is felt.

The time averaged data will be presented in the form of contour plots of induced inflow ratio (i.e., magnitude of induced inflow velocity divided by rotor tip speed) mapped over the entire rotor disk. The time accurate induced inflow ratio will be presented as time histories which depict the induced inflow ratio at a particular fixed point over the rotor disk. For the contour plots of the measured data, it is worth noting that, even though the temporal resolution in the time accurate measurements corresponds to approximately 2.8 ° of blade travel, the spatial resolution in the mea- sured data is limited. This spatial resolution limit is imposed by the limited number of azimuthal locations at which measurements were taken over the rotor disk. For the contour plots, the impli- cation is that, since there were only twelve azimuthal measurement locations, the spatial frequency content of the induced inflow data in the azimuthal direction is limited by the Nyquist criterion to six harmonics per rotor revolution. In addition, there are some measurement locations at which no data is available.

For the sole purpose of making consistent contour plots of measured data, which are presented subsequently, the time averaged data for induced inflow ratio over the rotor disk has been linearly interpolated onto a grid consisting of seventeen evenly spaced radial stations from the 20% ra- dius location to the tip location and onto twelve evenly spaced azimuth locations around the rotor disk. Interpolation onto a regular grid in this manner serves to "full-in" measured data that is not available at particular locations over the rotor disk.

David D. Boyd, Jr. Chapter 3. GDWT Validation 21 Table 3.2 lists the matrix of cases, measured and predicted, that a will be used for the validation effort in this chapter.

3.3 Time Averaged Induced Inflow

The following two sections present the time averaged, measured and predicted induced inflow ratio for several configurations. The quantities presented here are ones that have been derived from the measurements and predictions by time averaging the time accurate induced inflow ratio. Also, it should be noted here that the measured quantities include the effects of the ROBIN fuselage geometry and the predicted quantities do not include these effects (i.e., the predictions are for an isolated rotor).

3.3.1 Rectangular Planform Figure 3.2 shows a spatial contour plot comparison between the time averaged, measured and predicted induced inflow ratio for the rectangular planform at an advance ratio of 0.15, for a range of harmonics in the GDWT. Figure 3.2a is the measured data, including effects of the fuselage body.

Figure 3.2b shows the GDWT predicted data using only two harmonics. Here, only the nominally linear streamwise gradient of induced inflow is predicted. Figure 3.2c shows the GDWT predicted data using four harmonics. In this figure, it can be seen that the major features of the measured data are predicted well. For example, the upwash on the forward section of the disk is predicted, though it does not cover as much of the forward region of the disk as in the measured data. The aft downwash regions, concentrated in the first and fourth rotor quadrants (see figure 3.18 for quadrant definitions), are also predicted. Figure 3.2d shows the GDWT predicted data using eight harmonics. In this figure, all of the same features are present as in figure 3.2c, but the details of the induced inflow are slightly different. Figure 3.3 shows the same data as in figure 3.2, but these are lateral and longitudinal subsets of the data and show the radial variation of measured and predicted induced inflow. Figure 3.3a shows measured and predicted lateral variation of the induced inflow ratio for 2, 4, and 8 harmonics; the advancing and retreating sides of the rotor disk are labeled. Figure 3.3b shows measured and predicted longitudinal variation of the inflow ratio for 2, 4, and 8 harmonics; the forward and aft portions of the rotor disk are labeled. Figures 3.4 and 3.5 show the same case as the previous two figures, except these predictions use 128 azimuth David D. Boyd, Jr. Chapter 3. GDWT Validation 22 steps per revolution instead of 64. No significant differences are apparent between the two different azimuthal resolutions; this shows that, for this particular case, 128 azimuth steps per revolution is a sufficient azimuthal resolution.

Figures 3.6 to 3.9 show the same contour plots and lateral/longitudinal plots as in figures 3.2 to 3.5, but for the higher advance ratio of 0.23. For this advance ratio, as with the lower advance ratio, little difference is seen between the use of 4 and 8 harmonics and little difference is seen between the use of 64 azimuthal steps and 128 azimuthal steps.

3.3.2 Tapered Planform Figures 3.10 to 3.17 represent the same cases as in figures 3.2 to 3.9, but instead, using the tapered planform rotor. In comparison to the computations on the rectangular planform, the extent of the upwash region on the forward portion of the disk is better predicted based on the contour plots. Also, from the lateral and longitudinal plots, it can be seen that fewer harmonics are needed to predict the induced inflow behavior at the tip for the tapered planform. These findings for the contour plots and for the lateral and longitudinal plots are in agreement with findings in the M-/ published literature [25].

3.4 Time Accurate Induced Inflow

The previous sections presented the time averaged quantities derived from the experimental and predicted data. The following two sections present the unsteady induced inflow ratio components associated with the cases presented above except for the cases corresponding to two harmonics.

These two harmonic cases are excluded at this point because using two harmonics does not ade- quately represent the time averaged induced inflow ratio distribution over the rotor disk. In order to show the time dependent quantities more clearly, the local time averaged quantities have been removed from each of the following plots. Figure 3.18 shows the measurement locations where the comparisons of the unsteady induced inflow will be made. The radial and azimuthal positions are shown for locations A through J. Locations A through I are used in this chapter. Location J will not be used in this chapter. It is, however, used in chapters 7 and 8 and is shown here for later reference.

v_ David D. Boyd, Jr. Chapter 3. GDWT Validation 23 3.4.1 Rectangular Planform Figure 3.19 shows the unsteady component of induced inflow ratio at particular points on the rotor disk. The black dotted curves represent the measured data and the solid red lines represent the pre- dicted data. In many of the measured data, a distinct set of four pulses per revolution can be seen.

These pulses represent blade passages past the measurement location. The predictions at these measurement locations also show a periodic signature that generally matches the measurements in phase of the signals. Comparing figure 3.19 to figure 3.20 shows that for this configuration and this azimuthal resolution, the eight harmonic prediction matches the phase and amplitude better than the four harmonic prediction. Figures 3.21 and 3.22 show that the same holds for the 128 azimuth steps per revolution case. Figures 3.23 through 3.26 show that these results also hold for the higher advance ratio of 0.23. In these cases, it can be seen that, even using eight harmonics, the sharp, high frequency waveform of the measured inflow pulses is not matched well.

3.4.2 Tapered Planform Figures 3.27 through 3.34 show the same features and results for the tapered blades as was pre- sented for the rectangular blades above. As with the rectangular blade results, there is a typical, high frequency four per revolution pulse (waveform) indicative of blade passages seen at the mea- surement locations; the phase of these pulses generally matches the phase of the measured data.

Also, as with the rectangular blade predictions, the eight harmonic results predict the amplitude better than the four per revolution results.

3.5 Observations

From the evidence presented in this chapter, the following observations can be made.

Even though completely different methodology and computer coding from previous litera- ture was used in the implementation and solution of the equations of the GDWT, the current results match well with the previously published literature [25, 27].

• The best overall choice for number of harmonics, considering both the advance ratio and configuration variations, is eight. This conclusion is influenced more by the time averaged David D. Boyd, Jr. Chapter 3. GDWT Validation 24 predictions than the time accurate predictions. This "weighting" toward the time averaged computations is influenced by the fact that the coupling model (discussed later in chapter 5) uses the time averaged quantities, not the time accurate quantities.

• Even with eight harmonics, it is not possible to simulate the sharpness of the blade passages.

• The induced inflow ratio predicted by the GDWT is relatively insensitive to the number of azimuthal steps used.

Based on the study presented in this chapter, and the observations presented in this section, it has been established that the new GDWT implementation is able to reproduce measured data in a manner comparable to previous literature. Thus, referring back to figure 1.2, the rotor loading model has been established. As such, the values of the AP pressure jump are now ready to be used in the rotor/fuselage model, which will be discussed in the next chapter.

k.,.J David D. Boyd, Yr. Chapter 3. GDWT Validation 25 Table 3.1: Rotor Geometries Property Rectangular Tapered 0.8255 meters 0.8606 meters radius 0.0800 meters 0.0660 meters root chord 0.0660 meters 0.0254 meters tip chord 3:1 past 0.75R (no taper) taper ratio number of blades 0.24R 0.24R root cutout location 0.06R 0.06R flap/lag hinge location (none) (none) sweep of quarter-chord v NACA 0012 NACA 0012 airfoil section -13" _8 ° twist (none) (none) precone 0.0065 0.0065 nominal thrust coefficient 0.0977 0.0977 solidity 0.55 0.55 nominal tip Mach number 1" 1o approx, mean coning angle 3 ° nose down 3 ° nose down shaft tilt David D. Boyd, It. Chapter 3. GDWT Validation 26 Table 3.2: Test and Prediction Matrix Advance Ratio No. Azimuths No. Harmonics Configuration 0.15 64 2,4,8 rectangular 0.15 128 2,4,8 rectangular 0.23 64 2,4,8 rectangular 0.23 128 2,4,8 rectangular 0.15 64 2,4,8 tapered 0.15 128 2,4,8 tapered 0.23 64 2,4,8 tapered 0.23 128 2,4,8 tapered L .

v David D. Boyd, ]r. Chapter 3. GDWT Validation 27 Figure 3.1: ROBIN Fuselage in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel(1986).

v David D. Boyd, Jr. Chapter 3. GDWT Validation 28 downwash ................. upwash predicted measured 2 harmonics b) predicted predicted 4 harmonics 8 harmonics ^_O_?t ) __,/o. (d) Figure 3.2: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangular Planform, p -- 0.15, 64 Azimuths.

David D. Boyd, Yr. Chapter 3. GDWT Validation 29 ........... predicted, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonics o o o o omeasured 0.1 0.1 Longitudinal Lateral 0.08 0.08 f 0.06 0.06 r !

0.04 0.04-- ..o ..... _-_ 0.02 0.02 o o ...E.._"-

.... o

)h o 0 -0.02 -0,02 -0.04 -0.04 -0.06 •0.06 retreating advancing -0.08 -0.08 forward aft side side 1 I I I 1 I I I I I I I I I I I I I I I -0.1 -0.1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -1 -0.8 -0,6 -0,4 -0.2 0 0.2 0.4 0,6 0.8 I (a) dR (b) dR Figure 3.3: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Rectangular Planform,/1 = 0.15, 64 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 30 downwash ................. upwash predicted measured 2 harmonics

V

o. 0.04 b) (a) predicted predicted 4 harmonics 8 harmonics _-_ o__ Figure 3.4: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangular Planform, p = 0.15,128 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 31 ........... predicted, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonlcs o o o o omeasured 0.1; Longitudinal Lateral 0.08 0.08

o,[

0,06 0"06 r 0.04 0.04 o o ....... Z-.f 0.02 0.02 0 -0.02 -0.02 -0.04 -0.04 -0,06 -0,06 retreating advancing -0.08 -o.o8 forward aft side side I I I I I l [ I I ] -0.1 i t ! I I I I I I I -0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -08 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 dR (a) dR (b) %.J Figure 3.5: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Rectangular Planform, p = 0.15, 128 Azimuths.

m¸ t David D. Boyd, Jr. Chapter 3. GDWT Validation 32 downwash ................. upwash predicted measured 2 harmonics %.j (b) predicted predicted 4 harmonics 8 harmonics "'! '_' _'_¢ "'::::::" _o o,_ C) d) Figure 3.6: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangular Planform, p = 0.23, 64 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 33 ........... predicted, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonics o o o o omeasured 0.1 Longitudinal Lateral 0.08 0.08

ol f

0.06 0.06 _- 0.04 0.04 F" 0.02 0.02 !

-0.02 -0.04 -0.04 -0,06 -0.06 retreating advancing -0.08 -o.o8 forward aft side side -0.1 I I I I I I I I I I I I I I I l ' i l I 43.1 -1 -0.8 -0.6 -0,4 43.2 0 0.2 0.4 0.6 0.8 1 -I -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 z dR (a) dR (b) Figure 3.7: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Rectangular Planform, p = 0.23, 64 Azimuths.

David D. Boyd, It. Chapter 3. GDWT Validation 34 downwash ................. upwash predicted measured 2 harmonics = ; V b)

a)

predicted predicted 4 harmonics 8 harmonics Figure 3.8: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Rectangular Planform, p = 0.23, 128 Azimuths.

_t.j David D. Boyd, Jr. Chapter 3. GDWT Validation 35 ........... predicted, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonics o o o o omeasured 0,1 Longitudinal 0.08 o if ,atera, 0,06 0.06 _- s,,,,.

0.04 0.04 0.02 0.02 _ _ .......

-0,02 -0,02 -0,04 -0,04 -0.06 -0.06 retreating advancing -0,08 aft -o_08 - forward side side i I I i I l -0.1 ' I I I I I I I I I I I I I I I -0.1 0.2 0,4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 -1 -0.8 -0.6 -0,4 -02 0 0.2 0.4 0,6 0.8 1 dR (a) dR (b) ,%j Figure 3.9: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Rectangular Planform, p = 0.23, 128 Azimuths.

kJ David D. Boyd, Jr. Chapter 3. GDWT Validation 36 downwash ................. upwash predicted measured 2 harmonics b) (a) predicted predicted 4 harmonics 8 harmonics V

\ tT _v

v

_ i/_ (c) (d)

Figure 3.10: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered Plan- form,/_ = 0.15, 64 Azimuths.

David D. Boyd, It. Chapter 3. GDWT Validation 37 ........... predicted, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonics o o o o omeasured 0.1 Longitudinal 0.08 o::f Latera, 0.06 0.06 F 0.04 0.04 I- ..........

0.02 -0.02 -0,04 -0.04 -0.06 .0.06 retreating advancing -0.08 -0.08 forward aft side side I I _ I I I I I I I I ] 1 I I I 1 I I I -0.1 -0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0,6 0.8 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 08 1 -1 dR (b) dR (a) Figure 3.11: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Tapered Planform,/.t = 0.15, 64 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 38 downwash ................. upwash predicted measured | 2 harmonics

V

_0._,o_ (a) (b) predicted predicted 4 harmonics 8 harmonics %.J Figure 3.12: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered Plan- form, p = 0.15, 128 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 39 ........... predicted_, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonics o o o o omeasured 0.1 Longitudinal 0 08 !

f ,atera, 0.06 0.06 _- 0.04 0.02 -0.02 -0.02 - Q_ -0,04 -0.04 - -0.06 -0.06 retreating advancing -0.08 -oo8 forward aft side side l l ! I I I I I I 1 -o.1 _ = L = _ _ = _ _ -0,1 -1 -0,8 -0.6 -0.4 -0.2 0 0,2 0.4 0.6 0,8 1 -1 -0.8 -0,6 -0.4 -0.2 0 0.2 0.4 0,6 0.8 1 dR (a) dR (b) -,,.J Figure 3.13: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Tapered Planform, p = 0.15, 128 Azimuths.

== , David D. Boyd, Jr. Chapter 3. GDWT Validation 40 downwash ................. upwash predicted measured 2 harmonics

b)

predicted predicted 4 harmonics 8 harmonics V

_-_>-" °','17' "J:_

/Y.'_" o \;°i",i!_i_

Figure 3.14: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered Plan- form, p = 0.23, 64 Azimuths.

David D. Boyd, 7r. Chapter 3. GDWT Validation 41 ........... predicted, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonics o o o o omeasured 0.1 m Longitudinal 0.08 - o if Late., 0.06 ..,.0 0.06 F 0.04 0.02 V ............

-0.02

-o.o_ r _

v -0.04 -0.04 -0.06 -0.06 retreating advancing -0.08 -0.06 side side forward aft l I I I I I I I I I I I I 1 I [ I I I 1 -0.1 -0.1 -1 -0.8 -0.6 -0.4 -02 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 dR (b) dR (a) Figure 3.15: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Tapered Planform, ,u -- 0.23, 64 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 42

downwash upwash predicted measured 2 harmonics

V

(b) predicted predicted 4 harmonics 8 harmonics Figure 3.16: Measured and GDWT Predicted Time Averaged Induced Inflow Ratio; Tapered Plan- form, p = 0.23, 128 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 43 ........... predicted, 2 harmonics .............................. predicted, 4 harmonics predicted, 8 harmonics o o o o omeasured 0.1 0.1 Lateral Longitudinal 0.08 0.08 0,06 0.O6 .y... 0 0.04 0.04 0.02 0.02 t 11 ° I -0.02 .iilr.

-0.04 -0.06 -0.06 - retreating advancing -0.08 side side -0,08 - forward aft I I I i i l i i i i I I I 1 I I -0,1 I I I I -0.1 -1 -0.8 -0.6 -04 -0.2 0 02 0.4 06 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0,6 0.8 1 dR (a) dR (b) Figure 3.17: Measured and GDWT Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio; Tapered Planform, p = 0.23, 128 Azimuths.

David D. Boyd, Jr. Chapter 3. GDWT Validation 44 V _=180 ° Quadrant 2 Quadrant 3 \ \ \ \ / \ \ \ \ / / \ \ / / \ / I r/R=0.80 I\\ I /J _=84 ° I I I \H _ \ / / I \ iliA I I _=30_ \ \ .=B Quadrant I Quadrant 4 u/=30 ° r/R=0.90 r/R=0.78 dR=0.60 Figure 3.18: Locations Used in Comparisons.

David D. Boyd, Jr. Chapter 3. GDWT Validation 45 ................... measured predicted = 30 ° Location B oo, Location C °_ I Location A l

!

C m . . .i .... i . , . , i i | i i I -0.05 .0.06 ' 90 180 270 380 .... _ .... _ .... _.,_,I -0.05 .... _ .... t .... _ ....

gO 180 270 360 180 270 _= 180 ° 0.05 Location E 005 Location F I ° I Location D

I

m ..... | . . , ........ , .- .qll , ,t .0.05 -0.05 "g()' "180' '270 '3_0 -0"05 0 "_)" "180" 270 360 .... _,.... ,;,;_o .... 3;o _, = 300 ° 0.05 I" 0.05 Location G oo5 Location H Location I n- e- .0,05 ,i i_0 -0'06 0 .... _) .... 180"'' 270' '3_1 -0"0504 "g0" "t II0" "270" .... _,.... ,;o .... :_o .... _o Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.19: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform, ,u = 0.15, 64 Azimuths, 4 Harmonics.

David D. Boyd, Yr. Chapter 3. GDWT Validation 46 ................... measured predicted v = 30 ° o.os Location C o.o5 Location B o.o, Location A I

!

_ -0,05 .... J .... I .... I. ,.=1 _ . _ I ' ' I . . . . I . . . . l . . . . I -0,05 .... ' .... ' .... ',,,,I 0 gO 1110 270 ilO 180 270 3(10 0 90 I _0 270 360 _= 1800 v o_ Location E o- [ Location F o.. Location D +% .... ,_ .... i;o.... ,;o .... 3;o °"o .... _ .... ,_'+" '_o +,_o +.% .... ,; ...............

180 270 3t_ = 300 ° ¢. 0.05 Location I

=-

Location G o.o,[ Location H o.o_

=

n- i .... i .... i .... i .... i

+

m . = . . i . • . • . .

-0.05 Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.20: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform,/a = 0.15, 64 Azimuths, 8 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 47 ................... measured predicted _s = 30 ° 0o5 Location C !°°_ I Location A 005 Location B • . . | . . , t . . . | , . | -¢1.06 0 "90" 180 '270' "3_0 _s= 180 ° o.o5 Location F = o.= Location D oo, Location E =.

13: _= .... • .... , .... | .... | -0,05 -0.05 .... ' .... i .... | .... i _+051 .... ,;, .... ,;o .... _o....

gO 180 270 360 0 90 180 270 3B0 _s = 300 ° ¢. 0.05 Location G 0,5 Location H oo_ Location I v =.

=r.

==

m .... • .... | .... i .... i *0.05 .... " .... | .... | .... " ,.0.0_ _o% .... ,_ .... ,;o .... ,;o"" "_0 0 gO 1110 270 Be0 Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.21: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform, p = 0.15, 128 Azimuths, 4 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 48 . . . . . . . . . . . . . . . . . . . measured predicted = 30 ° Location B 0.05 Location C __o.05 Location A o.o5 I

o

m . . i ...i... i -i

4.05

"0'050 " gO _ 180 _270" "0"050 .... go=''" 'I 801 .... 270'= ,,,13go 27O 380 = 180 ° = J C 0.05 Location D 0,05 o05 Location F Location E =,

J

C m -0.05 -0.05 "gO' 'leO" _'2_0" ' "_0 "0"050' 'gO' 1gO '" "3_0 • • . , . . | .... | . • . , . . . 12_ O" .... _ .... 180 .... 270 .... 3eO = 300 ° = 0.05 Location G o.05 Location H o.o_ Location I m -0,05 .... ' .... _ .... ' .... ' -0.05 '''' .... ' .... =.,.,I -0"050 .... _ .... ' .... ' .... ' 0 gO 1110 270 360 iN) t00 270 360 Ref. Blade Location Ref. Blade L(_,at_n Ref. Blade Location Figure 3.22: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform, p = 0.15, 128 Azimuths, 8 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 49 ................... measured predicted _P = 30 ° = oo, Location A oo5 Location B 005 Location C

=

m . • . i . . i . . . i . . i .... n .... | .... z.,ll!

.... | .... n " ' ' ' I ' ' ' ' I ,,0.0_ -0.05 0 " 90" 't 80 "270" '_0 -0.05 O0 teO 270 _0 1 _ _0 360 = 180 ° C 0.05 Location D 005 Location F Location E oo5 CO @ |

J

.... , .... | .... i .... n -0.05 -0.05 .... • .... | .... u,lull gO 180 270 _0 .o% .... _ .... ,;o .... 2;0 .... 3;0 90 180 270 3e0 = 300 ° 0.05 oo5 Location H Location I j o.= Location G z g:

!

m .... . .... | .... | .... i -0.05 .... • .... "" • "J I .... = .0.05 0 90 180 270 360 -0.05 0 " "g()" ' '180 270" _ 1_ 27O _ Ref. Blade L(_.,at_n Ref. Blade Location Ref. Blade Location Figure 3.23: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform, g --- 0.23, 64 Azimuths, 4 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 50 ................... measured predicted = 30 ° oo, Location C oos Location B _o.o_ Location A l v _ .... i . ..| .... | .... i "°°So oo' t_o 27o 3co . ,, i , . ,i .... I .... _ -0.05 0 "90' 180 270 3 -0.o, ...; .... ,_.. _,o . .i t , J_ = 180 ° o0_ Location F 0.05 Location E = o.o_ Location D |

o%

-0. . _ , .1 .... I j , ii I .... , -0.05 .... ' .... ' .... ''_1 0 90 1110 270 360 "0'050 .... 90 .... 180""' "2_/0 ''" '360 = 300 ° 0.05 Location G oo, Location H o.o_ Location I

=

O t_

J

t- m-0.0 5 .... = .... _i , h , I .... ' -0.05 -0'050 .... 90 .... 1_0 .... 2"_0" ' ' _1_0 90 180 270 380 .... ,_ .... ,;o"' "_,o'"'_o Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.24: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform, p = 0.23, 64 Azimuths, 8 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 51 ................... measured predicted = 30 ° C 0.05 0o5 Location C Location A o.o, [ Location B

o

_ -0.05 .... ' .... _ .... '.,,,I ..0.1_ 0 90 180 270 360 .. i ., | .... i ' ' '360 .o.o6 • '9o' ,oo _o 0 9O 180 270 3(I0 _g = 180 ° Location D t" 0.05 o.o_ Location F o,05 Location E m o =E I c .... I .... i .... | .... , -0.05 gO 180 270 380 .oo, ° .... ,_ .... ,;,...._o...

_' = 300 ° C 0.05 Location G oo_ Location H o.o, F Location I I-

!

!

.... • .... i .... i .... | -0.05 ..... , -..i,,,,i .0.05 .... _ .... ,;o .... 2;0 .... _0 ..0.05 ' _" "1_" 270 _ Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.25: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform,/a = 0.23, 128 Azimuths, 4 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 52 ................... measured predicted _I' = 30 ° 0._ C 0,05 Location A Location B o.05 Location C

o

g0

-0.05 •0.05 '' "_ .... ' .... ' .... ' 9O 180 270 360 _= 180 ° 0,05 005 Location F Location E

_J°i

e- "°°So .... _) .... 1_o .... 2;0' " ' "_0 _,.0, .... _, .... ,;o'"'_o "'' "_;o = 300 ° C 0.05 Location G 0.05 Location H 005 Location I

T

=.o

rr

!

m-0.06 .... ' .... ' • - • ,.I ....

_0 180 270 _0 • °'o .... _,.... ,;o .... 2;o _o _.o_.... ._ .... ,;o .... _o .... _o Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.26: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Rectangular Planform,/a = 0.23, 128 Azimuths, 8 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 53 ................... measured predicted ',I.' = 30 ° Location C oo,[ Location A 005 Location B

_° °

. . . i .. | , , ,I , , , ,!

-0.05 • gO' "I 80' 270 360 0o_ Location F

Jl

.... , , . | .... | .... | -0.05 90' "t80 270 380

o__.._ .... ,_ .... _o,,,,_ o._, ;_;oo: ° -

0.05 Location I !°_ I Location G 0o5 Location H

=.o

_ L

-o-!- ;o ,;o _o _ _-o ,; ,;o ,;o _o .... .... .... .... .... .... ....

-O.05 0 " ' _)" ' '180" "270' Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.27: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform, p = 0.15, 64 Azimuths, 4 Harmonics.

David D. Boyd, It. Chapter 3. GDWT Validation 54 ................... measured predicted = 30 ° 0.05 _, 0.05 005 Location B Location C Location A

o

o

P- C .... | .... : .... : .... | m -0,05 -0,05 ''*' .... E''*ll'':'l -0.05 0 90 180 270 380 0 90 180 270 360 = 1 80 ° Location D e" 0.05 Location F 005 Location E 005 I= c -0.05 -0.05 -0.05 .... 90 .... 1801 .... 2701... '3aOI .... 9_) .... I_0 .... 270 .... 3_3 = 300 ° ¢,,,, 0.05 Location G oo, Location H 005 Location I rr !

..0.05 -0.05 .... 90 .... 180 .... 270 .... 3_0 Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.28: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform, p = 0.15, 64 Azimuths, 8 Harmonics.

David D. Boyd, It. Chapter 3. GDWT Validation 55 ................... measured predicted = 30 ° oos Location C ,.. oo5 oos Location B Location A n- O

==

• | . . . | . . . i , . | _ -0.05 .... ' .... z .... I''_=! ..0.05 ''' ' ..., .... , .... t 0 90 180 270 360 'gO" 180 270 3(10 "0"050 ' " " 90 '180 '270' '360 _= 180 ° oo5 Location F j °°5[ Location D °°5 LocationE

o

-o.o_ .... _,.... ,;,...._.--_ .o.o_ .... _,.... ,;o .... _o .... _, _,, . il ,-- • | ,i13_ 0 -0.05 0 "90' '180 '270' = 300 ° o= Location I °°51 Location G o.o_ Location H

_o o

,,0.05 .00% .... _,.... ,;o .... _o.... _ "% .... _ .... ,;o .... ,;o_o 0 gO I00 27O 38O Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.29: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform, p = 0.15, 128 Azimuths, 4 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 56 ................... measured predicted _P=30 ° Location C _e°°_[ Location A 0.0`[ Location B 005

o

.... | .... I .... i,j,ii "0"650 90 180 270 360 "0'050 90 180 270 360 "0"0` I _ _0 3e0 _Y = 180 ° e" 0.05 Location F Location D 00` Location E 00` rr =_.o, ,o ,o .... _,o'" - . ........... "00, .... _ .... ,;o .... _o.... -,;o "0"0`0 " "_>" ' '180" '270' .. , = ., i 1113_ 0 = 300 ° Location I _ o.0`f Location G oo, Location H oos C"0o%_. _. - ',;o .... =;o "'=;o _'0`o.... _,.... ,;o .... _;0'" _ "0"0` "90' 't80" 270 360 Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.30: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform, p = 0.15, 128 Azimuths, 8 Harmonics.

rCj David D. Boyd, Jr. Chapter 3. GDWT Validation 57 ................... measured predicted = 30 ° Location B 0.05 Location C __o.o_ Location A o.o5 m -0.06 .... ' .... ' .... i .... _ -0.05 0 I)0 1110 270 3_ = 180 ° Location F Location E 0.o_ I °i Location D o.o_

g: °

I

.'0,05 0 90 180 27O _ gO 180 270 360 : 300 ° o.o5 Location I o.o, [ Location G o.o, [ Location H _c_o.o_ .... _,.... &_;o_o _.o%.... _,.... ,_o .... _,o_o Ref, Blade Location Ref. Blade Location Ref. Blade Location Figure 3.31: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform, _u = 0.23, 64 Azimuths, 4 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 58 ................... measured predicted = 30 ° oo5 Location C c 0.05 Location A 0.05 Location B -0.05 ' ' 90' IIR} 270 340 .0'%.... _,.... ,;o .... _;o'"'_o .0.o_ .... ;, .... ,;o .... _o....

= 180 ° 0.05 Location F 005 Location E j o I Location D

o

v e- .... , .... | .... i .... | .... i .... . .... , .... i -0.05 °'_o .... _ .... ,_o .... 2_0 .... _0 -O.O5 90 180 270 360 90 t 00 270 3_ : 300 ° Kw. 0.05 Location G o.o_ Location H oo5 Location I K¢ m •0.05 .............. ' .... ' -0.05 .... _,.... ,;,,.... _o'_ -0% .... _,.... ,;o .... 2;o .... _, 0 gO 100 27O 3eo Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.32: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform,/_ = 0.23, 64 Azimuths, 8 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 59 ................... measured predicted = 30 ° Location B 0,05 Location C Location A 0.o5 | °i

!

... | ...i .... i .... | m -0.05 -0.05 • 90" 180 27O 380 -0050 go _80 27o' _o t80 270 360 = 180 ° Location D oo_ ¢,, 0.05 o05 Location F Location E rr r- _ -0.05 .... ' .... i .... 'll::! .... | .... , .... i .... | -0.05 -0050 .... _.... ,;0 .... 2;o .... _o 90 180 270 80 180 270 380 = 300 ° Location H 0o5 Location I v o _ _.05' .... ' .... ' • • , , I _ _ , , ! 41.05 | , . , | | .... _ .... ,80' '_,0 ';o 0 90 t80 270 360 Ref. Blade L_t_n Ref. Blade Location Ref. Blade Location Figure 3.33: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform, p = 0.23, 128 Azimuths, 4 Harmonics.

David D. Boyd, Jr. Chapter 3. GDWT Validation 60 ................... measured predicted = 300 LocaUon C Location B oo_ Location A 0.o5 _-0._ .... ' .... = .... _.._,1 -0.05 .,._ .... i .... | .... i .0.050 .... _) .... = .... = ....

180 270 3_ tSO 270 360 '_ = 180 ° Location D 0.o5 oo_ Location F Location E

go

e" m .0.05 -0.05 .... _ .... 180= .... 270_ ' ' ' _3BO' `0.05( _g = 3000 C 0.05 Location G o.o_ Location H oos Location I n- O c -0.05 .... 90 ' .... 180 ' .... 270 _,,_,1 360 `0.06 .... 90 .... 180 = .... 270 =''" '360= -0.05 .... 90 ' .... 180' .... 270' .... 300 ' Ref. Blade Location Ref. Blade Location Ref. Blade Location Figure 3.34: Measured and GDWT Predicted Unsteady Induced Inflow Ratio With Mean Values Removed; Tapered Planform, p = 0.23, 128 Azimuths, 8 Harmonics.

,,..,, k.)

Chapter 4

Rotor/Fuselage Flowfield Model:

OVERFLOW

4.1 Introduction

A rotor loading model which computes a distributed AP pressure jump value on the isolated rotor disk has been established and discussed extensively in chapters 2 and 3. From figure 1.2, it can be seen that these AP values are used in the rotor/fuselage flowfield model to compute the entire flow- field of the combined rotor/fuselage system. The rotor/fuselage flowfield is computed by solving the Navier-Stokes equations in conjunction with conservation of mass and energy. The Reynolds averaged Navier-Stokes equations in Cartesian coordinates are as follows: _)___.__Q + O(/_-Fv) + _)(G- Gv) + _)(_r-_v) 0 (4.1) bt Ox 0y _z where, V P pu (4.2) pv , pw peo , V David D. Boyd, Jr. Chapter 4. OVERFLOW 62 9u 2 + P _xx L (4.3) P = puv "Cxz

IPul

puw pu(eo + P) (ux_ + V_y + wT_z) - qx "Cyx pvu (4.4) = pv2+p =

I

"Cyz pvw pv(eo + p) ( U'_yx + v%, + WT, yz) -- qy pw T, zx pwu (4.5) 9wv Hv = 'r,zy 9w 2 + P "Czz 9w(eo + P) (u'Czx + v'%, + wT,=) -- qz In the above set of equations, [3 is the density, u, v, w are the velocity components, p is the pressure, e0 is the total energy per unit volume, 'gij are components of the stress tensor, and qx, qy, qz are components of the heat flux.

The quantities above are all considered to be averaged, or mean-flow, quantities. Also, Newto- nian flow is assumed. Thus, the stress tensor is proportional to the velocity gradients in the flow.

The constant of proportionality is composed of the sum of two components. The first component is the molecular viscosity, Pl, which accounts for laminar or molecular viscosity, and is a property of the fluid. The second component is the turbulent eddy viscosity, Pt, which accounts for the turbulence in the flow and is computed using a turbulence model.

The above system of equations is then closed by introducing the perfect gas law as follows: (4.6) p = pRT where 9_ is the gas constant and T is the temperature.

= V David D. Boyd, Jr. Chapter 4. OVERFLOW 63 For the current rotor/flowfield model, a time accurate, Reynolds averaged Navier-Stokes (RANS) tool known as OVERFLOW [29] is used to solve the above set of equations. OVERFLOW is a computer code that uses a finite difference technique to solve the compressible, RANS equations in generalized coordinates. Even though OVERFLOW is technically a RANS solver, it is typically used in a thin-layer mode by ignoring the viscous terms that are not associated with the direc- tion normal to surfaces in the flowfield; that approach is used here as well. The code employs a chimera (overset) grid scheme [38], which is helpful for complex geometries. Several solution procedure options are available in this code. A second or fourth order central difference scheme with a second and fourth order artificial dissipation scheme is used for the convective and viscous terms. A Pulliam-Chaussee scalar diagonal inversion [39] is typically used for the left-hand of the equations, though other options are possible, namely a Beam-Warming [40] scheme and a Lower Upper Symmetric Gauss SeideI (LU-SGS) scheme [41]. For steady state computations, a local time stepping scheme and a multigrid scheme [42] are implemented to accelerate convergence.

Also, for steady state computations at low Mach numbers, a low Mach number preconditioning scheme [42] is used. For unsteady computations, a "Newton sub-iteration" scheme [43] is im- plemented for each time step to reduce linearization and factorization errors and to increase the time accuracy of the scheme to approximately second order. Also available are several turbulence %./ models and an extensive set of boundary condition options. Available turbulence models include a Baldwin-Lomax model, a Baldwin-Barth model, a Spalart-Allmaras model, and a k - co model; the Spalart-Allmaras turbulence model is used for all computations presented here. Boundary condi- tion options include conditions for inviscid walls, viscous walls, periodic grids, symmetry planes, singular axes, inflow, outflow, characteristic conditions, etc. For both steady and unsteady com- putations, all boundary conditions are applied explicitly; some of these boundary conditions may also be applied to any region inside the volume of the computational grid, not just at the outer grid boundary faces.

v This chapter describes the manner in which OVERFLOW is used for the rotor/fuselage flowfield model depicted in figure 1.2.

4.2 New Boundary Conditions

For the purposes of the current research, OVERFLOW has been extended to include two new, novel, explicit boundary conditions which use the pressure jump previously computed by the David D. Boyd, Jr. Chapter 4. OVERFLOW 64 GDWT to model a helicopter rotor. One of the boundary conditions is used for time averaged computations and the other is used for unsteady, time accurate computations. Both are applied to two planes of a non-rotating, cylindrical grid with an "iblanked plane" in between. Figure 4.1a shows a top view schematic of the rotor disk used in the new boundary conditions. This schematic represents the non-rotating, cylindrical portion of the grid that is used to represent the rotor disk.

This "rotor portion" of the grid is a subset of the much larger overset volume grid set that is used to represent the entire rotor/fuselage flowfield. This grid subset is where the new boundary conditions are applied.

The shaded area in this figure represents a small region of the cylindrical grid; the rectangle outlined by the dark lines represents a rotor blade. Both of these regions are enlarged in the figure for clarity of presentation. Figure 4. lb shows an edge view of this same schematic. In this view, five planes can be seen. These planes consist of an upper rotor plane, an iblanked plane, a lower rotor plane, and two planes, labeled A and B, which are used in the formulation of the boundary conditions.

Althotlgh the focus of the current research is on unsteady interactions, it is necessary to have both the time averaged and time accurate boundary conditions discussed above. These boundary conditions are used in a complementary manner as follows. First, the steady state flowfield around the isolated fuselage is determined. Then, using this isolated fuselage computation as a starting point, the steady state flowfield of the fuselage, including the time averaged rotor model, can be determined using the time averaged rotor boundary condition. The previous two stages are used to set up the steady state flow features so that the unsteady computations can be used as a final stage, which is executed until a periodic solution is obtained. This "building block" approach of using a steady state computation as an initial condition to a time averaged computation, and in turn, using the time averaged computation as an initial condition to the time accurate computation, is used to reduce the computational resources required relative to using a completely time accurate computation from inception.

Next, the details of each of the new boundary conditions will be described.

v David D. Boyd, Jr. Chapter 4. OVERFLOW 65 4.2.1 Time Averaged Boundary Condition As stated above, the boundary conditions are applied on two planes, separated by an "iblank plane 1''. Use of the "iblank plane" between the upper and lower rotor planes prevents the artif- ical dissipation in OVERFLOW from being activated by the new pressure jump imposed across the two planes.

The steps in the application of this boundary condition are as follows: ° Identify the two planes (labeled A and B in figure 4.1b) surrounding the upper and lower rotor planes, 2. Average the conservative flow quantities in planes A and B as follows: (Qi,A + Qi,B) (4.7) Qi,avg : where Qi are given in equation (4.2), , Replace the existing conservative flow quantities in the upper and in the lower rotor planes with these average values, 4. Add an "additional energy term" to the fifth conservative flow quantity (Qs).

Steps 1, 2, and 3 above are relatively straightforward and are accomplished with application of existing boundary conditions. Step 4 is the new step in this process introduced here and is described below.

As stated above, both boundary conditions use the pressures from the GDWT with a few slight modifications. The first new boundary condition time averages the unsteady pressures at each radial and azimuthal location, multiplies by the ratio of the local blade area to the local computational cell area, A(f), to maintain the same level of thrust between the two methods, and divides by (T- 1) to convert the pressures into an energy-like term that is compatible with the Q5 variable above. This conversion to a Q5 compatible quantity can be combined into one expression as follows: A(_) Nr [(pe0)add](P,V) -- (NT)(T-- 1) _ AP(r'V't) (4.8) t=l IThe term "iblank" refers to a technique employed in many methods that use the chimera scheme. This technique involves intentionally excluding certain points (here, a certain plane) from the computation of the flowfield.

David D. Boyd, Jr. Chapter 4. OVERFLOW 66 where Nr is the number of time steps used in one revolution. The value of the left side of equa- tion (4.8) is then split in half. One half of the term is added to the pe0 equation for the lower plane of the rotor grid at the current azimuth and radial location, the other half is subtracted from the pe0 equation for the upper plane in the rotor grid at the current azimuth and radial location; this effectively adds energy to the flowfield, to mimic an actuator disk. The splitting of the addi- tional pe0 term and placement on either side of an "iblank plane" prevents the artificial dissipation in OVERFLOW from acting on the effective additional pressure jump; the artifical dissipation in OVERFLOW is designed so that it will not operate across an iblank region. Without this iblank plane, the artificial dissipation would operate on the pressure jump, smoothing the pressure jump unnecessarily.

4.2.2 Time Accurate Boundary Condition The new time accurate boundary condition is applied in a manner similar to that used for the time averaged boundary condition above. The differences are that (1) the pressures are not time averaged before they are converted to energy-like terms, and (2) the pressures are evenly distributed v : over all azimuthal grid lines that cross the blade chord at the local blade radial station. Similar to the time averaged boundary condition (with exception (1) above), the following additional energy- like term is determined: (4.9)

(y-- 1)

As in the time averaged boundary condition above, the value of the left side of equation (4.9) is then split in half. One half of the term is added to the pe0 equation for the lower plane of the rotor grid, the other half is subtracted from the pe0 equation for the upper plane in the rotor grid.

As discussed above, splitting the additional term and applying it on either side of an iblank plane prevents the artifical dissipation term from acting on the pressure jump. Also, at each radial station on the blade, the number of grid lines that lie on the chord are computed based on the chord length K.J and the arclength between successive azimuthal grid lines. The additional energy term above then is distributed evenly over these grid lines. This distribution allows the shape of the blade to be better modeled in the cylindrical grid.

David D. Boyd, Jr. Chapter 4. OVERFLOW 67

4.3 Summary

Referring to figure 1.2, a rotor/fuselage flowfield model has been introduced in this chapter. This new model consists of a Navier-Stokes method in which several new boundary conditions have been developed. These boundary conditions use information from the GDWT to model an addition of energy to the flowfield as in an actuator disk method, but in an time accurate setting. In the next chapter, the third box in figure 1.2, "coupling model", will be discussed.

%,J N.; David D. Boyd, Jr. Chapter 4. OVERFLOW 68 V® ) A B lower rotor plane '_blanked" plane upper rotor plane (b) Figure 4.1: Rotor Schematic for New Boundary Conditions.

Chapter 5

Coupling Model

5.1 Introduction

Since the pressure jump determined by the isolated rotor loading model depends on the inflow ratio distribution over the rotor disk, and since the presence of a fuselage alters this inflow ratio distribution, some type of coupling model is needed to adjust the inflow ratio and pressure jump distributions to reflect the effects of the fuselage. The current coupling method adopts an "inflow correction" model. In this type of model, the loading on the rotor is determined based on the inflow at the rotor disk, including additional inflow, or inflow corrections, generated by the presence of the fuselage. Presently, the forces and moments on the fuselage are not accounted for directly in the computation of the rotor forces; the rotor forces are, however, computed using inflow corrections to account for the presence of the fuselage. Since typically, for a model in a wind tunnel, the rotor is trimmed to a specified thrust, independent of the forces acting on the fuselage, this modeling assumption is valid here.

In previous chapters, discussions of the rotor loading model and the rotor/fuselage flowfield model were presented. In this chapter, the lower box of figure 1.2, the "coupling model" will be discussed.

David D. Boyd, Jr. Chapter 5. Coupling, Model 70

5.2 Isolated Rotor Configurations

Previous literature [24] used a hybrid method similar to that presented here. The current research builds on the idea presented in reference [24], which is a subset of the current work. This subset is contained inside the dashed box region in figure 5.1. For an isolated rotor computation, no coupling model, or feedback loop, is needed to provide fuselage correction information to the rotor loading model. Therefore, an isolated rotor computation is determined by a single pass through the method, stopping at the end of the rotor/fuselage flowfield computation. This was the method employed in reference [24].

5.3 Rotor/Fuselage Configurations

When there is a fuselage present in the rotor/fuselage flowfield model, a correction scheme is needed to feed information back into the isolated rotor loading model to account for the fuselage presence. The direction of the arrows in figure 5.1 indicates that the correction scheme should reflect the effects of the fuselage on the rotor system, as opposed to the rotor effects on the fuselage.

V In observing effects of a fuselage on the rotor system, it can be noted that time scales associated with the flow around the fuselage are much larger than those associated with the rotation of the rotor system. The effect of the fuselage on the surrounding fluid is one of slow displacement.

However, due to its rotation, the rotor blade will pass this fluid particle at a much higher speed.

Thus, for a given time increment, the rotor blade will traverse far more distance than a fluid particle traveling along the fuselage.

Based on these scale differences, it is expected that the presence of the fuselage produces effects on the rotor disk that are not spatially or temporally concentrated when viewed in the fuselage fixed, spatial frame of reference. However, when this effect is viewed in the rotor blade rotating frame of reference, the rotor blade "sees" the spatial distribution of inflow generated by the presence of the fuselage as a slowly varying (temporal) inflow quantity. This leads to the conclusion that the %..,, influence of the fuselage on the rotor system can be viewed as a time averaged perturbation: to the isolated rotor configuration in the fuselage fixed reference frame.

With the view that the fuselage effect on the rotor system can be considered to be a time averaged 1Note that here, a perturbation is not necessarily small.

David D. Boyd, Jr. Chapter 5. Coupling Model 71 infow ratio correction distributed over the rotor disk, an efficient coupling method which accounts for the primary fuselage effects on the rotor inflow ratio, can be developed. The coupling method developed here is of that type.

To determine the inflow corrections to be fed back into the rotor loading model, a difference is taken between the time averaged, filtered inflow ratio from the rotor/fuselage flowfield model and the time averaged inflow ratio from the GDWT:

A_i "-- ( i ) (5.1)

where F(') is a filtering operation performed on _,RPFM. To show the reason for the filtering oper- ation on the _RFFM term, it is necessary to examine the frequency limitations of each component in the method. For the rotor loading model component, the spatial frequency content of the induced inflow ratio determined by the GDWT, _,/RLM, is limited to the number of harmonics chosen in the method. For example, it was shown in chapter 3 that eight harmonics of rotor inflow are sufficient for that model in this context. Therefore inflow ratio information, time averaged or time accurate, is limited to eight harmonics (in this example). Note that this limitation does not imply that the loading distribution computed by the GDWT is limited to the given number of harmonics. This is because the GDWT computes the inflow ratio (up to a specified number of harmonics) given an._._y loading distribution. This computed inflow ratio influences the loading distribution through the w term in equation (2.15). Thus, the otherwise two dimensional loading distribution reflects inflow corrections up to the number of harmonics in the model.

While the GDWT limits the number of inflow harmonics computed to a specified value, the inflow computed by the rotor/fuselage flowfield model, _,RiFFM, is only limited by the spatial reso- lution found in the rotor grid. For example, a typical rotor grid used in this method [24] uses 128 azimuthal grid lines per rotor revolution. Using the Nyquist cutoff concept, this means that the limit of the frequency content in the azimuthal direction is 64 harmonics. If A_,i were computed and used in the rotor loading model without the filtering operation, inflow corrections (and there- fore loading distribution corrections) would be made inside the GDWT that are inconsistent with K.J the inflow corrections made internal to the model using the w term in equation (2.15). To eliminate this inconsistency, the _RFFM term is filtered to match the frequency content of the _./RLMterm.

A consistent filtering operation, F('), is derived from the GDWT method and is given in Ap- pendix A. Applying this filtering, F(_,_'FM), then computing A_, i from equation (5.1) above, this term may be included in equation (2.15) to complete the coupling loop. With this coupling method, David D. Boyd, Jr. Chapter 5. Coupling Model 72 all of the boxes of figure 1.2 have been described. The next few chapters will discuss applications of entire method.

David D. Boyd, Jr. Chapter 5. Coupling Model 73 -- l

I

Rotor/Fuselage AP(r,_,t) Rotor Loading Flowfield

I

Model Model (GDWT)

(OVERFLOW) I

I

Coupling Model (Inflow Corrections) Figure 5.1" Current Hybrid Method.

r_j

Chapter 6

i

Results: Isolated Fuselage

6.1 Introduction

Chapters 1, 2, 4, and 5, have described the components of the new computational model for ro- tor/fuselage unsteady interactional aerodynamics and how they are used. In the next three chapters, a full configuration will be constructed from basic components: (1) an isolated fuselage, (2) an iso- lated rotor, and (3) a rotor/fuselage combination. The subject of this chapter is the isolated fuselage component. Even though, strictly speaking, an isolated fuselage model is not a new computational model, it is necessary to establish that the computations can be compared favorably to measured v data.

6.2 Experimental Setup

The experimental setup for the comparisons in this isolated fuselage chapter are described in detail in Freeman and Mineck [37]. However, for completeness, the setup will be discussed here in brief.

A test was conducted in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel, in which steady state surface pressures were obtained at one hundred and sixty-two locations on a generic helicopter fuselage shape for a number of flight conditions with and without a rotor in- stalled. This fuselage shape, which is derived mathematically using "super-ellipse" equations [37], is known as the ROtor Body INteraction (ROBIN) fuselage (see figure 6.1). In the photograph David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 75 shown in figure 6.1, the ROBIN is sting mounted, with the rotor installed. For the isolated fuselage portion of the test, the blades were removed, but the hub was left in place. Though test condi- tions included configurations with and without the rotor installed, only steady state pressures were measured on the fuselage. As such, only the isolated fuselage pressures will be used here.

6.3 Computational Grid System

A system of thirteen grids, combined using the chimera grid capabilities in OVERFLOW, is used to represent the ROBIN fuselage geometry and flowfield. The sting mount, rotor hub, and wind tunnel walls are not modeled. A listing of all of grid names and sizes are displayed in table 6.1. From table 6.1 it can be seen that five of these grids involve surface grids associated with the fuselage.

These surface grids are shown and labeled in figure 6.2. Note that, for clarity of presentation, only every third grid line is plotted in figure 6.2. All remaining grids are field volume grids. Figure 6.3 shows a view of the grids immediately surrounding the fuselage. For clarity of presentation, every second grid line in this figure has been removed. Also seen is the positioning of the rotor grid with respect to the fuselage. For the isolated fuselage computations presented in this chapter, the rotor grid is simply another volume grid (i.e., there is no rotor boundary condition applied on the rotor grid). The far field grid extents approximately 2.5 fuselage lengths upstream of the fuselage nose, downstream of the fuselage tail, to the left and right of the fuselage, and above and below the fuselage.

These grids were developed such that, with minimal effort and minimal grid modification, all configurations used in the current research (including an isolated fuselage, an isolated rotor, and a rotor/fuselage combination) could be modeled simply by "grid replacement". As an example, when converting from a rotor/fuselage configuration to an isolated rotor configuration, only the grids containing the fuselage need to be replaced; these are replaced by regular volume grids that maintain the outer boundary shape of the fuselage grids. All other grids remain the same.

These grids were also designed to maintain a "double fringe" overlap between all overlapping grids (except for the outer field grid) to maintain the second and fourth order artificial dissipation scheme used in OVERFLOW. In addition, grid spacings were refined to maintain viscous spacings at the fuselage surface such that there is at least one grid point in the laminar sub-layer region of the boundary layer and to maintain approximately 1.5 million total grid points in order to keep the computational expense reasonable.

David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 76 One novel feature of this grid system is the manner in which the rotor grid is included. In previous isolated fuselage studies of the ROBIN fuselage [44, 45] the grid system did not include a rotor grid, thus a hyperbolic grid generator was used to "grow" the volume grids out from the surface grids without regard to the location of the outer boundaries of these grids. However, due to the geometry of the outer boundaries of grids produced by a hyperbolic grid generator, inclusion of a cylindrical rotor grid by the technique of "hole cutting" [46], while maintaining a double fringe chimera scheme, is difficult. This problem arises due to the lack of control of the outer boundary shape of grids generated by hyperbolic grid generation methods. To alleviate this problem, an elliptic grid generation method is used to control the outer boundary shape of some grids. With this technique, the overlap of the grids can be better controlled to produce a double fringe overlap and the amount of "hole cutting" can be substantially reduced.

For the current research, a hybrid set of grids generated by hyperbolic and elliptic grid generators [47, 48, 49, 50] is used. For volume grids in which no control was needed over the outer boundary shape, a hyperbolic grid generator was used. If control over the outer boundary geometry was desired, an elliptic grid generator was used.

6.4

Steady State Pressure Prediction

As discussed previously, the first step in the OVERFLOW computations is to execute the isolated v fuselage configuration in a steady state mode. For the current computations, the default settings recommended for OVERFLOW [29] are used, except where noted. These defaults include central differencing of the right hand side of the equations, scalar diagonal inversion of the left hand side of the equations, a matrix dissipation scheme, low Mach number preconditioning, multigrid, full multigrid (mesh sequencing), local time stepping with a minimum CFL number constraint, and the Spalart-Allmaras turbulence model. The case presented here is for an angle of attack of 0* and a freestream Mach number of 0.1265.

Figure 6.4 shows the force coefficients in the lift, drag, and sideward directions as a function of iteration number in OVERFLOW. These force and moment coefficients are defined as follows [51]: (6.1) David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 77 (6.2) Q,_ = = (6.3) where a list and derivation of reference quantities for these is given in reference [51]. Though these do not directly correspond to the standard lift, drag, and side force coefficients, they do provide a useful insight into the convergence behavior of the OVERFLOW iteration from an engineering standpoint.

Figure 6.5 shows the pressure tap locations on the ROBIN fuselage. The locations are shown with respect to the downstream coordinate (x-direction). Along the top of the figure, the lowercase letters indicate the locations where predictions are compared to measurements in the subsequent figure.

Figure 6.6 compares measured and predicted pressure coefficients vs the vertical location on the fuselage at various places stations along the length of the fuselage. The vertical axis in figure 6.6 is plotted in the standard negative Cp format and the horizontal axis is the vertical location (z coordinate) of the current section; the solid lines are the predictions and the symbols are the measured values.

The pressure coefficient used here is defined in equation (6.4). From a practical standpoint, computation of Cp inside OVERFLOW is accomplished using equation (6.5) and equation (4.2), v taking into account the nondimensionalizations used in OVERFLOW.

P-P_ Cp- ½PV_ (6.4) 2(q,-1) [ Q2+Q2+Q_ 1 ] Cp = M2 _ Q5 - 2Q1 T("/- 1) (6.5) In figure 6.6 it can be seen that predicted pressure coefficients match the experimental data well in most areas. At locations (f) and (g), discrepancies can be seen on the upper and lower portions of the fuselage. These differences could be due to the presence of the hub and sting mount in the experimental setup. In this figure, predicted Cp values are plotted from both sides of the fuselage for this symmetric flight condition. In this figure, predicted pressure coefficients from both sides of the fuselage are plotted since the computation included the full fuselage instead of a half body David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 78 fuselage plus a symmetry condition. As, expected, no difference is seen between the two sides.

In addition to matching well the experimental data, these predictions are consistent with similar predictions in the literature [45], which used other Navier-Stokes methods (codes), for the same configuration and flight condition.

6.5 Observations

OVERFLOW has the capability to predict the steady state pressure coefficients on an isolated fuselage configuration in an incompressible flight condition, and these predictions are consistent with similar computations in the literature.

V David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 79 Table 6.1" Computational Grid System Grid Name/Description Dimensions Fuselage Grid 93 x 117 x 25 30x 117x33 Nose Grid 28x 117x33 Tail Grid 29x 111 x25 Collar Grid 41 x41 x25 Top Grid 55 x 50 x 12 Lower Field Grid 55 x 50 x 22 Upper Field Grid 42 x 70 x 43 Nose Field Grid 30 x 39 x 29 Tail Field Grid 55 x 32 x 30 Left Field Grid 55 x 32 x 30 Right Field Grid 93 x 82 x 26 Outer Field Grid 37 x 129 x 43 Rotor Grid David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 80 Figure 6.1" ROBIN Fuselage in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel.

V H Tail Collar Nose _ Fuselage Figure 6.2: ROBIN Fuselage Surface Grids.

David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 81

z-::I3--- _ ..... ±

_ _1 i I_

\

Figure 6.3: Grid System Used in Computations.

V David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 82 Lift Direction Force Drag Direction Force 0.02 Side DlrecUon Force (_ 0.01 .0.01 -J-O.02 0 - , , , i I I J l i I _ i i i I 500 1000 1500

Iteration Number

Figure 6.4: Lift, Drag, and Sideward Direction Force Coefficients. Steady State Computation.

David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 83 h !

c d e f g a b _"_ .............. . _ ............ i:,i,

Tll

P P .o .o .o .o .o .o .o .o 0 Ob ,_ M O 0 0 M _ _ 0 0 M,_ 0 M _ ,IbbM X location Figure 6.5: Pressure Tap Locations.

David D. Boyd, Jr. Chapter 6. Results: Isolated Fuselage 84 • • • • measured predicted 0.5 05 0.5 0.25 0.25 0.25 -Cp o -0.25 .025 -025 (b) (c) °-'_._5 `0.1 -oo5_o' ._oli"o'_5 -o.._ 5._i:o'_ o 0_o3 0',5 0.5 0.5 0.5 025 025 0.25 0 0 -Cp o -025 -025 .025 (0 (e) (d) -'_15 `01.0.050 005 0., '0_5 0.5 0.5 0.25 0.25 025 BOO • o -Cp o -0.25 -025 -0.25 (i) (g) (h) %'15 ._ i :0'_ o0 o5 oli ci'15 %.'i5:o'.i :0'o5_"oo5 01io',5 -o..,_,;_,.o.i.:0.05 ooos oli 0'_5 z(m) z(m) z(m) Figure 6.6: Measured and Predicted Pressure Coefficients vs Vertical Location for o_ = 0 ° and M_ = 0.1265.

Chapter 7

Results: Isolated Rotor

7.1 Introduction

Chapter 6 showed that the pressure distribution could be predicted on the isolated ROBIN fuselage configuration for a representative forward flight condition of a rotorcraft. This chapter will employ the isolated rotor method as described in chapter 5. Previous predictions using the current model [24] compared well with time averaged and time accurate laser velocimeter (LV) data. In that work, the predictions were for an isolated rotor, whereas the experimental data contained the effects of a fuselage. Subsequent to those predictions, new experimental data has been acquired on an isolated rotor configuration (described below). Comparisons between these data and the current method are the subject of this chapter.

7.2

Experimental Setup

For comparisons in this section, results from an isolated rotor configuration in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel are shown. The isolated rotor test system (IRTS) [52, 53] is shown in figure 7.1. Here, it can be seen that the rotor is suspended from the ceiling of the tunnel from a tapered, cylindrical shaped sting. A three component laser velocimeter (LV) system was used to measure the three components of velocity at a point at an azimuth of 84 ° , a radial station of r/R = 0.81, and one blade chord above the tip path plane of the rotor. The rotor was trimmed to a nominal 0 ° flapping angle (e.g., the first flapping harmonics are [31c = _Is = 0 ° ), David D. Boyd, Yr. Chapter 7. Results: Isolated Rotor 86 a nominal thrust coefficient of 0.0064, a nominal shaft tilt, cq, of 3° nose down, and a freestream Mach number of 0.1265. The LV data was processed at 128 samples per rotor revolution.

7.3 Computational Grid System

As discussed in chapter 6 for the isolated fuselage configuration, a grid system was developed that, among other things, had the ability to be used for an isolated rotor configuration with minimal changes to the grid system. For this isolated rotor configuration, several minor changes were made to the grids. These changes are as follows: 1. The nose grid has been eliminated, 2. The tail grid has been eliminated, 3. The collar grid has been eliminated, 4. The top grid has been eliminated, 5. The fuselage grid has been replaced by a volume field grid that maintains the outer boundary shape of the original fuselage grid.

Since the nose, tail, collar, and top grids from items 1, 2, 3, and 4 above were "overset" into other background grids, it was only required to delete these grids and leave the remaining background grids unchanged. The only new grid in the system is the volume grid that replaces the fuselage grid. All other grids in the system are unchanged. In essence, what remains, is a set of volume grids, identical to the full configuration, minus the fuselage surface. However, now, the new rotor boundary condition in OVERFLOW will be applied in the rotor grid as discussed in chapter 4.

As with the isolated fuselage configuration, the sting mount and the wind tunnel walls are not modeled. This new grid system is shown in figure 7.2; every second grid line has been removed from this figure for clarity. The farfield extent of the grid is the same as that in figure 6.3.

7.4 Time Averaged Computation

The isolated rotor computations will be started with a prediction of the time averaged flowfietd due to the rotor. For this, the time averaged rotor boundary condition discussed in chapter 4 is used.

David D. Boyd, 7r. Chapter 7. Results: Isolated Rotor 87 This is done to minimize the number of time accurate computations that must be executed. Since this time averaged flow field is only an intermediate stage of the computation, and since there is no time averaged induced inflow data for the isolated rotor test stand available, no comparisons are made to experimental data.

7.5 Time Accurate Computation

Once the time averaged computation discussed above has been obtained, OVERFLOW is restarted in a unsteady mode and the method is continued until a periodic solution is achieved. The state of "periodicity" in this context refers to the lack of differences in the induced inflow ratio predictions for successive blade passages. Boyd and Bamwell [24] explored the sensitivities of several param- eters in OVERFLOW for the time accurate computations of this type. Following the conclusions of that paper, the computations are executed with six Newton sub-iterations and no viscous terms.

Since fourth order central flux differencing in space is now available in OVERFLOW, that is used as well.

Figure 7.3 compares the measured and predicted unsteady induced inflow in directions parallel and perpendicular to the rotor tip path plane at location "J" as shown in figure 3.18. This position is located radially at r/R = 0.80 and azimuthally at g = 84 ° . Figure 7.3a is the induced inflow comparison in the direction parallel to the tip path plane (the u-velocity, or "inplane" velocity) and figure 7.3b is the induced inflow comparison in the direction perpendicular to the tip path plane (w-velocity, or "out-of-plane" velocity). For clarity, both of the comparisons show only the unsteady components of induced inflow. It can be seen that the unsteady u-velocity comparison is excellent in magnitude, phase, and waveform shape. The w-velocity predictions match the phase and waveform shape well, but slightly under-predicts the magnitude of the pulse. The slight discrepancies in the phase for both plots may be explained by realizing that the location of the entire experimental signal has an error band that is equal to +½ of the azimuthal resolution at which the data was acquired. This error band is equivalent to 4-1.4 ° in azimuth. Here, the data has been plotted at the center of the error band. Also, the position of the measurement location in space and the position of the point in space used for the prediction comparisons do not exactly coincide since there is not a computational grid point that falls exactly on the measurement location; the closest point in the grid to the measurement location is shown. Comparing the coordinates for location J in the measurements and in the predictions, the error in radial location is less than 1% David D. Boyd, Jr. Chapter 7. Results: Isolated Rotor 88 of the rotor radius; the error in azimuthal location is less than 0.4 ° , and the vertical location error is negligible (<< 1% of the rotor radius).

Even though the distributed time averaged data was not measured in this experiment, it is worth- while to examine this quantity from the predictions in the same manner as done by Boyd and Bamwell [24]. Figure 7.4 shows a contour plot comparison of the time averaged induced in- flow (w-velocity). Figure 7.4a shows the out-of-plane component of the measured, time averaged induced inflow ratio for the rectangular planform rotor shown in chapter 3. This is the same exper- imental data from figure 3.8. As discussed in chapter 3, this experimental data contains a fuselage, whereas, the current prediction does not. Figure 7.4b shows the current, predicted, time aver- aged induced inflow ratio which as been filtered to roughly match the experimental data frequency content as discussed in Appendix A.

Comparing these two figures, it can be seen that the prediction contains the same general features of the measured data. It can be seen that there is an upwash on the front of the disk, that there is a downwash at the rear of the disk with more concentrated downwash in the first and fourth quadrants, and that the magnitudes of the downwash are similar. These features and the level of prediction shown are similar to those shown in reference [24] for a different rotor system. In addition, figure 7.4b shows a curious feature that has been seen before in the literature for both the current method and for the purely GDWT [25]. This feature, seen on the retreating side of the rotor in the fourth quadrant near the blade root, appears to be a region where the induced inflow is quite small compared to the induced inflow surrounding that region; this feature is not seen in the measured data. That feature can be explained by examining the rotor loading in that region. Since, physically, the rotor induced inflow is generated as a fluid reaction to the rotor loading distribution, and since the rotor loading is very small in that region due to the very low dynamic pressure there, the induced inflow is small in that region. Chapter 8 will show that this feature is affected by the isolated rotor assumption made in previous investigations.

7.6 Observations

The inplane and out-of-plane components of unsteady induced inflow at points above, but still close to, the rotor plane are well predicted and are consistent with previous literature. In addi- tion, though time averaged results are not available for the isolated rotor configuration, the time averaged predicted results are consistent with previously published time averaged results at similar David D. Boyd, Jr. Chapter 7. Results: Isolated Rotor 89 conditions for similar rotor systems.

v David D. Boyd, Jr. Chapter 7. ResuIts: Isolated Rotor 90 Figure 7.1" IRTS in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel.

David D. Boyd, Jr.

Chapter 7. Results: Isolated Rotor

\

Figure 7.2: Grid System Used in Computations.

David D. Boyd, Jr. Chapter 7. Results: Isolated Rotor 92 measured predicted Out-of-plane Induced Inflow 0.03 Inplane Induced Inflow 0.03 0.02 0.02 0.01

|

¢: 0.01

y0

_-0.01 "4".o01 (b) -0.02 (a) -0.02 I I I ] l I I z J l t , , , I _ , , J I .... I .... I , , , , I .... I -0.030 -°°30 9o lao 270 aso 90 180 270 360 Ref. Blade Location (o) Ref. Blade L_tion(°) v Figure 7.3: Measured and Predicted Induced Inflow in Two Directions at Location J for the IRTS.

David D. Boyd, Jr. Chapter 7. Results: Isolated Rotor 93 -- downwash ................. upwash measured predicted f 0 "'_ ,S'_''"

I

_(a) (b) t_j Figure 7.4: Measured and Predicted Time Averaged Induced Inflow Ratio for the IRTS (Measured Data Includes Fuselage).

_s

Chapter 8

Results: Rotor/Fuselage

t

8.1 Introduction

Chapters 6 and 7 showed various predictions made with the current computational model on the two separate components of a full configuration: an isolated fuselage and an isolated rotor. In _rr this chapter, these two components are combined into a single unit using the entire computational model. For comparisons between measured and predicted quantities, several experimental data sets are used since there are no available data sets that contain measurements of all quantities of interest.

8.2 Experimental Setup

V The experimental setup used is a combination of the IRTS discussed in chapter 7 and the 2-meter ROBIN fuselage [33, 34, 35, 36]. This configuration is shown in figure 8.1. In this particular test, no LV data were taken as was done in the IRTS test discussed previously. Instead, unsteady fuselage pressures were measured at several locations on the fuselage. The locations included a row of pressure taps near the top centerline of the fuselage. This row of taps nominally followed the top centerline, but were offset 0.25 inches to the advancing side of the fuselage. This offset was needed since the construction of the fuselage shell consisted of two halves which overlapped on the centerline of the fuselage. In addition, there were six unsteady pressure taps at fuselage station "e" (see figure 6.5) at several vertical locations on the left and right sides of the fuselage. Three of David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 95 these taps were on the left side and three on the right side of the fuselage. The unsteady pressure data was processed at 256 points per rotor revolution.

8.3 Computational Grid System

The grid system for the combined rotor/fuselage configuration is similar to that used in chapter 6, figure ??. Since the rotor grid was already included in the isolated fuselage configuration, no grid changes are needed to proceed directly from the isolated fuselage steady state results in chapter 6 to the time averaged rotor/fuselage configuration. However, for the time accurate computations, slight modifications were made to the grid which do not affect the grid shape or distribution of points. These modifications were made to accommodate a solution scheme in OVERFLOW that is very efficient in the unsteady mode.

For the unsteady rotor/fuselage configuration computations in OVERFLOW, the LU-SGS scheme is used. However, current implementation of this scheme does not allow for the use of spatially periodic boundary conditions. In OVERFLOW, spatially periodic grids and spatially periodic boundary conditions, have identical first and last planes. Figure 8.2(a) represents a schematic of a spatially periodic grid with k being the index in the periodic direction. It can be seen that k --- 1 and k ---- kmax are actually the same point; for a three dimensional grid, these would be planes.

To account for the fact that the LU-SGS scheme cannot be used with spatially periodic boundary conditions, the periodic grids in the rotor/fuselage grid system (excluding the rotor grid), are con- verted to grids that can still pass information to themselves through an extended overlap region.

These grids then use a new boundary condition combination, created by using several existing OVERFLOW boundary conditions in succession, to pass information spatially in a manner similar to the traditional spatially periodic boundary condition. In this new boundary condition combina- tion, the grids are slightly modified so that an overlap, or one-to-one match, of four grid planes is made in each of the computational grids' spatially periodic direction. This overlap of four grid planes is used to replace the single overlap plane used in the spatially periodic boundary condition.

A schematic of this is shown in figure 8.2(b). With these new overlapped planes, the "copy-to" and "copy-from" boundary conditions in OVERFLOW are used to pass information from one end of the computational grid to the other to simulate a spatially periodic condition. In table 8.1, the "copy-to" and "copy-from" columns show which information is passed between computational grid planes in the computational k-direction (the spatially periodic direction used in these grids).

Chapter 8. Results: Rotor Fuselage David D. Boyd, Jr.

Table 8.1: LU-SGS "Periodic" Boundary Condition Copy-from Copy-to (k value) (k value) kmax- 3 kmax - 2 3 kmax- I 4 kmax There were some configuration anomalies in the experimental setup for the rotor/fuselage con- figuration that were also incorporated into the grid system. First, due to offsets in the wind tunnel floor and ceiling, the rotor center of rotation was offset approximately two inches to the advanc- ing side of the centerline of the fuselage. This placed the rotor rotation center at approximately the advancing side edge of the pylon atop the fuselage. Second, the fuselage shell was yawed at approximately 0.76 ° nose-left. Both of these anomalies have been accounted for in the grid system.

8.4 Time Averaged Computation

The isolated fuselage configuration results shown in chapter 6 are used as a starting point for the time averaged rotor computations with the combined configuration. These calculations involve using the time averaged rotor boundary condition discussed in chapter 4. As before, this portion of the calculation is an intermediate step between the steady state isolated fuselage computation and the unsteady rotor/fuselage configuration. It uses the same grid system defined above for the steady state computation. Since this is an intermediate stage of the computation, only the force convergence history will be examined to show that this portion of the computation is well behaved.

Figure 8.3 is a continuation of figure 6.4 with the new portion beginning at iteration number 601 and ending at iteration number 1100.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 97

8.5 Time Accurate Computation

For the time accurate computations, OVERFLOW is executed in an unsteady mode using the LU- SGS scheme to invert the left hand side of the equations using the new overlapping grid scheme to eliminate the spatially periodic boundary conditions. The unsteady rotor model boundary condition is applied as discussed in chapters 4 and 7. These computations are executed until a periodic result is obtained in the unsteady pressures on the fuselage.

8.6 Coupled Model Predictions

In subsequent sections, the new, coupled, computational model predictions will be presented for several iterations. The first subsection will show the unsteady fuselage pressure predictions for the first iteration I vs the corresponding measured quantities from the current rotor/fuselage configura- tion experiment. The second subsection will compare measured and predicted, unsteady induced inflow ratios from the first iteration. The third subsection will compare predicted, time averaged induced inflow ratio contours from the first iteration, which have been derived from the unsteady model, to the corresponding measured induced inflow ratio contours, taken from the experimental data presented in chapter 3. The fourth subsection will compare measured and predicted lateral and longitudinal induced inflow ratios. Subsequent sections and subsections will show the comparisons for the second and third iterations, including the coupling. Finally, an examination of the pressure contours on the fuselage surface will be made, as well as an examination of the convergence of the method in terms of the trim pitch settings as a function of iteration.

8.6.1 Iteration I Unsteady Fuselage Pressures Figure 8.4 shows a comparison between the measured and predicted unsteady modified pressure coefficient on the top centerline of the ROBIN fuselage for various downstream locations (x- direction). Plotted on the vertical axis is the negative of the modified pressure coefficient, C'p.

lHere, an iteration refers to one pass through the rotor loading model followed by one pass through the ro- tor/fuselage flowfield model. Typically, only two rotor revolutions are required in each rotor/fuselage flowfield model iteration.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 98 Along the horizontal axis is the reference blade location in the rotor azimuthal coordinates. Since the measurement location is fixed in the non-rotating frame, this horizontal axis can also be viewed as a temporal axis spanning one rotor revolution. The modified pressure coefficient used in these comparisons is defined by the following: C,p 100(P- P_) (8.1)

= -p(nR)2

where the typical velocity in the denominator of the Cp definition had been changed to D.R. This modified Cp is designated C'p. Using a superscript u simply denotes that this is the unsteady compo- nent. This change in definition is needed since, for a rotorcraft, a freestream velocity approaching zero is possible as the hover condition is approached. If the normal definition of Cp were used, the values of Cp would approach infinity as the hover condition is approached. The factor of 1O0 on the C'p simply serves as a convenient factor to scale the modified pressure coefficient function.

Comparing the standard Cp definition and the C'p definition above, the following relation holds between Cp and 64p: !

Cp= lOOp_Cp (8.2) where/z_ is the freestream advance ratio. To show the relation between these quantities more clearly, consider a perfect fluid. For a perfect fluid, stagnation occurs at -Cp = -1.0. Using the current flight condition for the rotor/fuselage configuration, and using equation (8.2), this equates to -C'p = -5.29. Referring this to figure 8.4, a value of -64p = - 1.0 corresponds to -Cp ,_ -0.19.

Examining features in figure 8.4, it can be seen in that there is a dominant blade passage event at a frequency of four pulses per rotor revolution in the measured pressure signature. Here, the phase of the pressure signal is well matched, and the magnitude is slightly over-predicted.

Figure 8.5 shows the measured and predicted -C'p _ as a function of the reference blade location.

Whereas figure 8.4 showed the predictions on the top centerline of the fuselage, figure 8.5 shows the predictions on the retreating and advancing sides of the fuselage (i.e., the left and right sides) for various vertical locations at the same downstream location. It can be seen that the magnitude and phase are predicted well for the advancing side locations; however, the retreating side amplitudes are slightly over-predicted.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 99 Time Accurate Induced Inflow Ratios Figure 8.6 shows the in-plane and out-of-plane components of unsteady induced inflow for the same location as that in figure 7.3. Figure 8.6, however, is a comparison between the unsteady induced inflow velocities for an isolated rotor (obtained from figure 7.3) and that for the first iteration of the method, including the fuselage. It can be seen that the presence of the fuselage has little effect on the unsteady induced inflow components at these particular locations.

Time Averaged Induced Inflow Ratios At this point, it is worthwhile to check the validity of the time averaged results, obtained by time averaging the time accurate calculations, by comparing them to measured data.

Figure 8.7 shows a comparison between the measured and predicted induced inflow ratio per- pendicular to the rotor tip path plane. For each of the predictions, the same pressure distribution is used in the rotor boundary condition. Figures 8.7b and 8.7c show the predicted induced inflow ra- tio for the isolated rotor and the predicted induced inflow ratio for the rotor/fuselage combination, using the same unsteady rotor boundary condition and pressure information. Several improve- ments can be seen in the induced inflow ratio prediction between the isolated rotor prediction and the rotor/fuselage combination. First, the inflow anomaly, seen in the isolated rotor prediction in the fourth quadrant near the blade root, is not present in the rotor/fuselage combination predic- tion. The induced inflow in this region now matches the measured data in that region. Second, the induced inflow near the forward portion of the rotor disk matches the experimental data well.

For example, examining the contour line of "zero" induced inflow in the measured data and in the rotor/fuselage combination prediction, it can be seen that the upwash on the forward section of the disk now extends toward the center of the rotor. This upwash region is generated by the presence of the fuselage and is therefore not predicted in the isolated rotor prediction.

Figure 8.7d shows the difference between the induced inflow ratio for the rotor/fuselage combi- nation and that of the isolated rotor. This difference shows the effects of the fuselage on the time averaged flowfield. As expected, there is an increased upwash near the nose of the fuselage and forward section of the pylon and an increased downwash behind the pylon. The regions of addi- tional induced inflow in this figure show the source of the improved induced inflow predictions shown in figure 8.7c.

Figure 8.8 shows a comparison between the measured and predicted induced inflow ratio parallel David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage l O0 to the rotor tip path plane for the same conditions shown above. Figures 8.8b and 8.8c show the predicted induced inflow ratio parallel to the tip path plane for the isolated rotor and the predicted induced inflow ratio parallel to the tip path plane for the rotor/fuselage combination. Comparing the figures, it can be seen that the rotor/fuselage combination better predicts the induced inflow ratio. For example, examining the induced inflow contour line with a value of 0.015, it can be seen that the isolated rotor prediction is almost symmetric left-to-right, whereas the measured and predicted induced inflow ratios are both skewed at about 45 ° counterclockwise to the oncoming flow. Here, again, it can be seen that inclusion of the fuselage is necessary to correctly predict the time averaged induced inflow ratio.

Figure 8.8d shows the difference between the induced inflow ratio parallel to the tip path plane for the rotor/fuselage combination and that of the isolated rotor. This difference, again, shows the effect of the fuselage on the time averaged flowfield and the sources of the improved predictions with the fuselage included. As expected, there is a decrease in u-velocity near the nose of the fuselage and an acceleration near the rear of the fuselage pylon. It can be seen that, to capture the u-velocity distribution accurately, the fuselage must be included in the computation.

Lateral/Longitudinal Induced Inflow Ratios Figure 8.9 compares the lateral and longitudinal subsets of the measured and predicted induced inflow ratio. The measured data comes from figure 8.7a, while the predicted data is taken from the combined rotor/fuselage case in 8.7c. Comparing the prediction here to the prediction using 8 harmonics in figure 3.9, it can be seen that the longitudinal induced inflow ratio is well predicted using the combined rotor/fuselage model. Whereas the peak-to-peak amplitude of the lateral in- duced inflow ratio was over-predicted by the GDWT isolated rotor model, it is well predicted for the combined rotor/fuselage model. However, there is a slight over-prediction of the inflow ratio near the advancing and retreating blade tip regions.

8.6.2 Iteration 2 Unsteady Fuselage Pressures With the first iteration complete, induced inflow corrections are computed as described in chapter 5. With these induced inflow corrections, the GDWT is used to recompute the rotor loading.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 101 With this new rotor loading, OVERFLOW is used to recompute the rotor/fuselage flowfield. This computation, or "iteration 2", is presented in the current and subsequent sections.

Figure 8.10 shows the top centerline modified pressure coefficient in a manner similar to figure 8.4. Comparison of these two figures shows only very small differences.

Figure 8.11 shows the the modified pressure coefficient on the retreating and advancing sides of the fuselage in a manner similar to figure 8.5. Comparison of these two figures, again, shows only very small differences.

Time Accurate Induced Inflow Ratios Figure 8.12 shows the in-plane and out-of-plane components of unsteady induced inflow for the same location as that in figure 7.3. As was seen in the first iteration, the fuselage has little influence on the unsteady components of induced inflow at this point in the ftowfield.

Time Averaged Induced Inflow Ratios Figure 8.13 shows the same information as figure 8.7, but this is the second iteration. Figures 8.13a and 8.13b show the identical information as that in figures 8.7a and 8.7b. Plots 8.13c and 8.13d are the new parts of this figure. Comparing figures 8.13c and 8.7c, it can be observed that there are only small changes between these figures except on the advancing side of the rotor disk in the first quadrant. In the first iteration, the time averaged induced inflow in that region was over-predicted.

In the second iteration, it is seen that the time averaged induced inflow quantities are now closer to the measured values. However, overall, the shape of the contours did not change significantly.

Comparing figures 8.13d and 8.7d, it can be observed that there are only small changes between these difference plots. Thus the effect of the fuselage on the time averaged induced inflow is quite similar between the first and second iterations.

Figure 8.14 shows the same information as figure 8.8, except it is for the second iteration. As before, figures 8.14a and 8.14b show the same information as figures 8.8a and 8.8b. Comparing figures 8.14c and 8.8c, it can be seen, again that there are only small differences between the first and second iterations. This time, however, the small differences that do occur are on the advancing side in the first part of the second quadrant of the rotor disk. A similar conclusion holds for figures figures 8.14d and 8.8d.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 102 Lateral/Longitudinal Induced Inflow Ratios Figure 8.15 compares the lateral and longitudinal subsets of the measured and predicted induced inflow ratio in a manner similar to figure 8.9. The measured data comes from figure 8.13a (and is the same as in figure 8.7a), while the predicted data is taken from the combined rotor/fuselage case in 8.13c. The predictions shown in this figure are similar to those shown in figure 8.9, and similar trends are present.

Even though differences are small between the first and second iterations, a third iteration is also presented to show that the iteration process has converged.

8.6.3 Iteration 3 Unsteady Fuselage Pressures After iterating once again through the coupling technique, the rotor loading model, and the ro- tofffuselage flowfield model, the third iteration is complete. Figures 8.16 and 8.17 once again show the unsteady modified pressure coefficient on both the top centerline and the sides of the fuselage. As with the comparison between the first and second iterations, there is no significant difference between this iteration and the previous iteration.

Time Accurate Induced Inflow Ratios Again comparing two components of unsteady induced inflow ratio for the current iteration to the isolated rotor results in figure 8.18 shows no significant difference between the two results. In V_ addition, there are no significant differences between this iteration and the previous iteration.

Time Averaged Induced Inflow Ratios Examining the in-plane and out-of-plane time averaged induced inflow ratios in figures 8.19 and 8.20, as was done for the second iteration, it can be seen that there are no significant differences between the current iteration and the previous iteration.

Since there are no significant differences between the second and third iteration in the unsteady David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 103 modified pressure coefficient, in the time accurate inflow ratio, or in time averaged induced inflow ratio, it can be concluded that the iteration process is complete.

Lateral/Longitudinal Induced Inflow Ratios Figure 8.21 compares the lateral and longitudinal subsets of the measured and predicted induced inflow ratio in a manner similar to figures 8.9 and 8.15. The measured data comes from figure 8.19a (and is the same as in figures 8.7a and 8.13a), while the predicted data is taken from the combined rotor/fuselage case in 8.19c. Again, the predictions shown in this figure are similar to those shown in figures 8.9 and 8.15, and similar trends are present.

8.7 Examination of Pressure Contours

In the preceding sections, a detailed examination of the unsteady component of the modified pres- sure coefficient, of the time averaged inflow ratio, and of the unsteady inflow ratio was presented for several iterations of the current method. For the final result (at the end of the third iteration), it is constructive to examine pressure distribution on the fuselage with and without the effects of the rotor. Figure 8.22(a) is a contour plot of surface pressure coefficient on the isolated fuselage configuration presented in chapter 6. It can be seen there is a typical stagnation region on the nose of the fuselage.

Figure 8.22(b) is a contour plot of the time averaged surface pressure coefficient for the ro- tor/fuselage configuration. This plots contains data from the "iteration Y' above. As with figure 8.22(a), there is a stagnation region on the fuselage nose region and behind the pylon. However, comparing figures (a) and (b), it can be seen that the gross time averaged effect of the rotor is to increase the pressure coefficient on the sides of the fuselage and downstream of the pylon. Figures 8.23 and 8.24 are top views of figures 8.22(a) and (b), respectively.

Figure 8.25 shows a set of surface pressure coefficient contour plots similar to those presented in figure 8.22. For comparison purposes, figure 8.25(a) shows the same data as presented in figure 8.22(b). Figure 8.25(b) shows the surface pressure coefficient for the instant at which the reference blade is at the _g = 0° location. All four blades are plotted as rectangles extending from the blade root at r/R = 0.24 to the blade tip at r/R = 1.0 and are drawn "to-scale" in their actual position and orientation with respect to the fuselage; The blade chord is drawn "to-scale" as well. In this contour David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 104 plot from the unsteady computation, a large pressure pulse can be seen on the upper surface of the tail boom. This pressure pulse is a direct result of the rotor blade at the V = 0° location passing over the tail boom. Figure 8.26 shows this pressure pulse in a top view of figure 8.25(b).

Figures 8.27 to 8.35 present surface pressure coefficient contour plots at azimuth locations of V = 15" ,30 ° ,45* ,60* ,75 ° , and 90 ° . These plots again show a pressure pulse traveling on and around the fuselage. It can be seen that this pulse motion is correlated with the motion of the blade. This point is made especially clear in the "Top View" plots presented here. Examination of this set of plots reveals that, locally, the unsteady surface pressure coefficient can be substantially higher than the time averaged surface pressure coefficient and that the pulses are typically of short temporal duration. In addition, the asymmetry of the unsteady loading can be seen clearly in the "Top View" figures.

Iteration Effects on Rotor Trim

8.8

The effect of the coupled computation on the rotor trim can be assessed by examining the pitch control settings required at each stage of the iteration at the end of the trim procedure in the rotor loading computation. These blade pitch settings are a function of rotor azimuth, are referenced to the 0.75R radial location on the blade, and are defined as follows: v (8.3) 0 (V) = 00 + 0c cos V + 0s sin V where 00 is the collective pitch, 0c is the longitudinal pitch, and 0s is the lateral pitch. The first row in table 8.2 shows the measured collective, longitudinal, and lateral pitch settings, along with the magnitude and phase of the longitudinal/lateral combination. All quantities in table 8.2 are in degrees. The magnitude of the first harmonic of pitch, 01, and phase of the first harmonic of pitch, V1 can be defined as follows: e(V) = 00+01cos(V-Vl) (8.4) 01 = _/0_+0_ (8.5) V1 = arctan(0oc ) (8.6) David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 105 where _gl is placed in the range 0 ° < gq < 360 ° .

Table 8.2: Pitch Control Settings [deg] as a Function of Iteration Number 00 0c 0s 01 _1 Type Iteration 8.16 1.52 -4.13 4.40 290.2 Measured N/A 6.84 1.45 -3.42 3.71 293.0 Isolated Rotor 8.06 2.73 -3.69 4.59 306.5 Rotor+Fuselage 8.02 2.67 -3.71 4.57 305.7 Rotor+Fuselage From this table, it can be seen that the collective pitch setting matches the measured collective pitch for the rotor/fuselage combination cases, but is under-predicted for the isolated rotor case as expected. It can also be seen that the magnitude of the first harmonic of pitch is well matched for all of the iterations, especially for the iterations that include the fuselage in the computation. A plot of these quantities vs iteration number are shown in figure 8.36. From this figure, it can be seen that these settings have converged in just two iterations. For the computations that include the complete configuration, a phase difference of approximately 16* can be seen. The explanation of this phase difference can be attributed to the assumption that the blade flap hinge offset is at the center of rotation. To show this, it is only necessary to examine the effect of the blade hinge offset on the rotor flap response. Gessow and Myers [54] showed that the rigid flapping natural frequency of a rotor is a function of blade hinge offset and is given by the following formula: (8.7) where 03,, is the flapping natural frequency in cycles per revolution and h is the hinge offset location as a fraction of rotor radius. Examination of equation (8.7) for h = 0 shows that the flap natural frequency con ---- 1.0 per revolution and for h = 0.06, the flap natural frequency is COn= 1.044 per revolution. The difference in these two natural frequencies is A03,, = 0.044 per revolution. Since there are 360 ° in one revolution, the natural frequency difference equates to approximately 16" of rotor azimuth. This means that in the experiment, which included the blade hinge offset, the phasing of the first harmonic of pitch should preceed the predicted phasing by approximately 16 ° of rotor azimuth. This is precisely the amount seen in table 8.2.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 106

8.9 Resource Usage Summary

Table 8.3 lists the resource usage for each component of the current rotor/fuselage combination computations using the GDWT with 8 harmonics and 128 azimuth steps per revolution. The items listed are the CPU times in hours and minutes [hr:min] required for each iteration stage, main memory required in mega-words [mw], and which machine was used for each phase.

Table 8.3: Resource Usage as a Function of Iteration Number Type CPU [hr:min] [ Memory [mw] Machine <0:07 Cray C-90 GDWT (with 8 harmonics, 128 azimuths) 3:43 Cray C-90 Isolated Fuselage 2:16 Cray C-90 Time Averaged, Isolated Rotor L , 1:52 Cray C-90 Time Accurate, Isolated Rotor (per rev.)

3:38 Cray C-90 Time Averaged, Rotor/Fuselage 2:42 Cray C-90 Time Accurate, Rotor/Fuselage (per rev.)

%., Each of the rows in table 8.3 is for a particular component of the method for this particular case.

Since execution times will differ for each particular case, these should only be used as reference quantities. It should be noted that OVERFLOW does not have a convergence criterion that halts execution at a particular convergence level. It is executed for a specified number of iterations or time steps. Each of the components involving time accurate computations are given as the CPU time required to execute each rotor revolution. For the case presented here, each time accurate stage of the computation was executed for two complete rotor revolutions to assure periodicity, even though periodicity may be achieved in fewer actual blade passage events. Thus, the CPU times above are conservative values for this case. Also, note that the GDWT is executed in a conservative manner at the same resolution required by OVERFLOW. Thus, CPU time for the GDWT computations is conservative as well. To compute the total execution time for the isolated rotor results and the first iteration of the rotor/fuselage configuration, the following formulae can be used: David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 107 Isolated Rotor: CPU Time = GDWT + (Time Averaged, Isolated Rotor) + 2(Time Accurate, Isolated Rotor) = 6:07 (Cray C-90) Rotor/Fuselage, Iteration 1: CPU Time = (Isolated Fuselage) + GDWT + (Time Averaged, Rotor/Fuselage) + 2(Time Accurate, Rotor/Fuselage) = 12:52 (Cray C-90) where the factor of two on the time accurate components indicates that two complete revolutions were executed for this particular case.

8.10 Observations

This chapter has presented results from the full hybrid method for a rotor/fuselage configuration.

From the evidence presented above, several conclusions can be drawn as follow: The unsteady modified pressure coefficient on the top centerline of the fuselage and on the sides of the fuselage shown above, are insensitive to the small trim changes made by the time averaged induced inflow corrections which account for the presence of the fuselage.

The in-plane and out-of-plane unsteady induced inflow velocity components are also insen- sitive to the small trim changes made by the time averaged induced inflow corrections which account for the presence of the fuselage.

The time averaged induced inflow velocities are improved through the iteration/coupling process presented above. For the case presented here, it appears that one iteration is a good approximation to the correct solution and that two iterations is sufficient to capture the time averaged and time accurate induced inflow effects.

• The primary effect of the fuselage on the trimmed blade pitch settings is on the collective David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 108 pitch (for this particular case). The presence of the fuselage improves the pitch setting pre- dictions over those from the isolated rotor case.

• For this flight condition, the primary effects of the rotor on the fuselage are a higher time averaged surface pressure coefficient below and downstream of the rotor and short duration surface pressure pulses imposed by the individual blade passages over the fuselage surfaces.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 109 Figure 8.1: IRTS/Fuselage Configuration in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 110 \ (a) kmax-1 kmax kmax-5 _ kmax-1 (b) kmax-4 kmax-2 kmax-3 Figure 8.2: Schematic of a Periodic Grid and Replacement for Periodic Grid in LU-SGS Scheme.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 111 Lift Direction Force Drag Direction Force 0.02 Side Direction Force

J

im 0.01 -0.01 -

E

m ' ' | i I I I I I I , , , , I ,,J .0.02 0 .

500 1000 1500 Iteration Number Figure 8.3: Lift, Drag, and Sideward Direction Force Coefficients, Time Averaged Computation.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 112 measured predi_ed 3 x=0.0941 x=0.2007 l x=0.0517 -1 -2 "30 90 180 270 360 -3 0 ....................

90 180 270 360 270 360 x=1.1620 x=1.3450 ,u il x=0.2563 1 1 "Cp 0 -1 -1 -1 -2 -2 -2 .3 / .... I .... I .... l.,.ll _30 .... , .... ,_,,,i .... , -3; .... I .... _ .... I .... I 90 180 270 360 o 90 180 270 360 90 180 270 360 Ref. Blade Loc. (°) Ref. Blade Loc. (°) Ref. Blade Loc. (o) Figure 8.4: Measured and Predicted Unsteady Modified Pressure Coefficient on the Top Centerline of the ROBIN Fuselage, Iteration 1.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 113 measured predicted Retreating Side Retreating Side Retreating Side 3 z=-0.07 3 z=0.06 3 z=0.12 2 2 1 1 'g -Cp 0 -I -1 -1 -2 -2 -2 . , ! .... | . , . • ...... i .....

-3o .... _'o'"i_62_o"3_o

-3; .... _o 18o 2_o 3;o

"30 '9C)' 180 2"/(_ 360 s Advancing Side 3 Advancing Side 3 Advancing Side z=-0.08 z=0.06 z=0.12 2 2 2 1 1 -C_ o v -I -1 -1 -2 -2 -2 -3o............. 2_6....

-3; ................... ' -3_ .................... 90 180 360 g0 180 270 360 90 180 270 360 Ref. Blade L_. (o) Ref. Blade Loc. (o) Ref. Blade Loc. (o) Figure 8.5: Measured and Predicted Unsteady Modified Pressure Coefficient on the Retreating and Advancing Sides of the ROBIN Fuselage, Iteration 1.

v David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 114 measured -- predicted, Isolated rotor -- predicted, rotor+fuselage Inplane Induced Inflow Out-of-plane Induced Inflow 0.03 0.03 I 0.02 0.02 f 0.01 ¢0.01 el

t

=L o 0

| I M- -0.01 -0.01 -0.02

(a) -0.02

(b) , , , , I t , = J I , [ , , I , _ , , I `0.03 -0.03(_ ' ' ' ' I , _ j _ I , , , , I , , ! r | 90 180 270 360 90 180 270 360 Ref. Blade Location (°) Ref. Blade Location(°) Figure 8.6: Measured and Predicted Induced Inflow in Two Directions for an Isolate Rotor and a Rotor/Fuselage Combination, Iteration 1.

= , David D. Boyd, Ir. Chapter 8. Results: Rotor Fuselage 115 downwash upwash Predicted Inflow Ratio Measured Inflow Ratio Isolated Rotor (a) (b) .. o.o4 Predicted Inflow Ratio Predicted Inflow Ratio Difference Rotor + Fuselage (d) Figure 8.7: Measured and Predicted Time Averaged Induced Inflow Ratio from Time Accurate Computations, Iteration 1.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 116 downwash upwash Predicted Parallel Inflow Ratio Isolated Rotor Measured Parallel Inflow Ratio _" (a) (b) Predicted Parallel Inflow Ratio Predicted Parallel Inflow Ratio Difference Rotor + Fuselage (d) __ (c) Figure 8.8: Measured and Predicted Time Averaged Parallel Induced Inflow Ratio from Time Accurate Computations, Iteration 1.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 117 predicted o o o o omeasured 0.1 Lateral F" Longitudinal 0.08 0.08

o' f

0.06 0.06 r 0.04 0.04 r 0.02 0.02 _.j o 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 retreating advancing -0.08 side side -0.o8 forward aft i i I I I I I I I I -0,1 -0.1 I I I I I I, I I 1 I -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0,6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 dR (a) dR (b) Figure 8.9: Measured and Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio from Time Accurate Computations, Iteration 1 David D. Boyd, It. Chapter 8. Results: Rotor Fuselage 118 measured predicted 3 x=0.0517 x=0.0941 x=0.2007 2 2 -1 -1 -2 -2 - , , , , i .... I .... I .... i -3_ .... 9'o ...............

0 90 180 270 360 180 270 36O v 3 x=0.2563 x=1.1620 3 x=1.3450

-c: °°

-1 -1 -2 -2 -3 0 ....................

-3_ .............. 90 18o 27o""3_o 90 180 270 360 -30 90 180 270 360 Ref. Blade Loe. (o) Ref. Blade Loc. (o) Ref. Blade Loc. (°) Figure 8.10: Measured and Predicted Unsteady Modified Pressure Coefficient on the Top Center- line of the ROBIN Fuselage, Iteration 2.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 119 measured predicted Retreating Side Retreating Side Retreating Side a z=-0.07 a z= 0.06 3 z=0.12 'u -Cp o -1 -1 -1 -2 -2 -2 ..... I .... | , , ,

3o _ 18o 270 3_o

-35 .... 9'o' i_o......... -3(

270 360 .... 9_ .... 1801 .... 2701 .... 360| a Advancing Side a Advancing Side

3 Advancing Side

z--0.08 z=0.06

z=0.12 -C_ o -1 -1 -1 -2 -2 -2 • ...... | .... I -3" .... i .... , .... i .... , -30 ......... , ...... ,,,= 90 180 270 360 o

90 180 270 360 % 9'o 18o '2-;0 360

Ref. Blade Loc. (o) Ref. Blade Loc. (o) Ref. Blade Loc. (°) Figure 8.11" Measured and Predicted Unsteady Modified Pressure Coefficient on the Retreating and Advancing Sides of the ROBIN Fuselage, Iteration 2.

r_ David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 120 measured predicted, Isolated rotor -- predicted, rotor+fuselage Inplane Induced Inflow Out-of-plane Induced Inflow 0.03 0.03 - 0.02 0.0; ¢O.Ol ¢ 0.01 el

!

t

_ o

\

\

I M" -0.01 V

(a) -0.02

(b) -0.030 , , , i I J , , , I .... I .... I 9O 180 270 360 Ref. Blade Location (°) Ref. Blade Location(°) Figure 8.12: Measured and Predicted Induced Inflow in Two Directions for an Isolate Rotor and a Rotor/Fuselage Combination, Iteration 2.

= %7# David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 121 downwash " Predicted Inflow Ratio Measured Inflow Ratio Isolated Rotor IV ................. upwash

oo

oo & "_ • _ o 0 0 (a) _o2__ (b) Predicted Inflow Ratio Predicted Inflow Ratio Difference Rotor + Fuselage

7" " , 1

(d) Figure 8.13: Measured and Predicted Time Averaged Induced Inflow Ratio from Time Accurate Computations, Iteration 2.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 122 downwash - Predicted Parallel Inflow Ratio upwash Measured Parallel Inflow Ratio Isolated Rotor _V .................

/o.o_---./.. _ \ 7, \

,., j

Predicted Parallel Inflow Ratio Predicted Parallel Inflow Ratio Difference Rotor + Fuselage o. 00,3

vo_._:_o.oo_. //

(C) "_J (d) Figure 8.14: Measured and Predicted Time Averaged Parallel Induced Inflow Ratio from Time Accurate Computations, Iteration 2.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 123 predicted o o o o omeasured Longitudinal 0.08 o, f Lateral 0.06 F 0.04 F )h o 0.02 "_ -0.02 - -0.04 -0.06 retreating advancing -0.08 side side -o,o8 forward aft I 1 1 I 1 1 I 1 1 1 -0.1 -0.1 I I t 1 ! 1 I I I | -1 -0.8 -0,6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 dR (a) dR (b) Figure 8.15: Measured and Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio from Time Accurate Computations, Iteration 2 David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 124 measured predicted il x=0.0517 3 x=0.0941 i l x=0.2007 'u -Cp o_ o 30 90 18o 27o 360 30 .... 9'oi;o"2_63;o 30 90 180 270 360 z x=1.1620 x=1.3450 il x=0.2563 C -1 -1

u -2

-2 .... 1 .... Illll|,ll,I 30 90 180 270 360 -3o.... 9'o"'i;62_o"_o % 90 180 270 360 Ref, Blade Loc. (o) Ref. Blade L_. (°) Ref. Blade Loc. (o) Figure 8.16: Measured and Predicted Unsteady Modified Pressure Coefficient on the Top Center- line of the ROBIN Fuselage, Iteration 3.

David D. Boyd, It. Chapter 8. Results: Rotor/F-bselage 125 measured predicted

Retreating Side Retreating Side Retreating Side

3 z=-0.07 a z=0.06 a z=0.12

2 2 -C_ o

o_

-1 -1 -1 -2 -2 ....... i .... | • ........ s .... | r ...... | .... ill |

3o _ i$o 270 3_

-3o _ i_ 270 380

% _o 18o 270 35o

Advancing Side Advancing Side 3rAdvancing Side

3 z=-0.l_8 3 z=0.06 IE z=0.12

'U -Cp o -1 -1 -1 -2 -2 -2 .... | ...... | -3" .... ' .... l .... i .... ,

-3( .... I 0 .... i .... i .... ! % 9o i_6 2_o 39o

180 270 360 0 9O 180 270 360

Ref. Blade Loc. (o)

Ref. Blade Loc. (o)

Ref. Blade Loc, (o)

Figure 8.17: Measured and Predicted Unsteady Modified Pressure Coefficient on the Retreating and Advancing Sides of the ROBIN Fuselage, Iteration 3.

David D. Boyd, Jr. Chapter 8. ResuIts: Rotor Fuselage 126 -- measured -- predicted, Isolated rotor predicted, rotor+fuselage Inplane Induced Inflow Out-of-plane Induced Inflow 0.03 -- 0.0_ 0.02 0.0; 0.01 =0.01 U

E

I | =_ -0.01 -0.01 -0.02

(a) -0,02

(b) , , , , I j , _ , I , , , _ I _ , , I I -0.03(_ ' ' ' ' I , _ , , I , , , , I = = ] L I 90 180 270 360 -o.o3_ 90 180 270 360 Ref. Blade Location (o) Ref. Blade Location(°) Figure 8.18: Measured and Predicted Induced Inflow in Two Directions for an Isolate Rotor and a Rotor/Fuselage Combination, Iteration 3.

David D. Boyd, Yr. Chapter 8. Results: Rotor Fuselage downwash Measured Inflow Ratio Isolated Rotor V Predicted Inflow Raiio ........... upwash _,'" __o "",, _-O.o,. / J )_ \a'_l ( _> 0 0 (a) "_02__._ (b) Predicted Inflow Ratio Predicted Inflow Ratio Difference Rotor + Fuselage "0.01 0

s "J

(d) o- (c) Figure 8.19: Measured and Predicted Time Averaged Induced Inflow Ratio from Time Accurate Computations, Iteration 3.

David D. Boyd, Jr.

Chapter 8. Results: Rotor Fuselage downwash ................. upwash Measured Parallel Inflow Ratio Isolated Rotor _V Predicted Parallel Inflow Ratio Predicted Parallel Inflow Ratio Predicted Parallel Inflow Ratio Difference Rotor + Fuselage

\ °

(c, \°_--S_°-_ _._ (d, Figure 8.20: Measured and Predicted Time Averaged Parallel Induced Inflow Ratio from Time Accurate Computations, Iteration 3.

David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 129 predicted o o o o omeasured 0.1 -- Longitudinal 0.08 0.08 o l f Lateral 0.06 0.06 F 0.04 0.04 F 0.02 )h 0 0.02"_ -0.02 - -0.02 -0.04 - -0.04 -0.06 - -0.06 retreating advancing -0.08 - side side -o.o8 forward aft 1 I l [ l I i i f t -0,1 I I I I I I t t i i -0.1 ' -1 -0.8 -0.6 -0.4 .02 0 0.2 0.4 0.6 0.8 1 -1 .08 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 dR (a) dR (b) Figure 8.21" Measured and Predicted Lateral and Longitudinal Time Averaged Induced Inflow Ratio from Time Accurate Computations, Iteration 3 David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 130 Isolated Fuselage Time Averaged from Unsteady Computation (a) "(b) accelerating 8tagrmting Cp -0.5-0.4-0,3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.22: Isolated Fuselage and Rotor/Fuselage, Time Averaged Surface Pressure Coefficients David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 131 Top View Isolated Fuselage accelerating stagnating -0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 P Figure 8.23: Surface Pressure Coefficient on Isolated Fuselage Configuration, Top View David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 132 Top View Time Averaged from U nsteady Computation accelerating stagnating

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.24: Time Averaged Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, Top View -¢J i | David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 133 Time Averaged from Unsteady Computation Unsteady Computation _=0 ° (a) (b) accelerating stagnating

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.25: Time Averaged and Time Accurate Surface Pressure Coefficients, _ = 0 ° David D. Boyd, Jr. Chapter 8. Results: Rotor/Fuselage 134 Top View _= 0 ° accelerating stagnating V

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.26: Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, _ = 0° , Top View David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage .135 Unsteady /,_ Computation //

(a)

U nsteady Computation _g = 30 ° (b) accelerating stagnating

Cp

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.27: Time Accurate Surface Pressure Coefficients at _g = 15 ° and _g= 30 ° David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 136 Top View _= 15 ° accelerating stagnating

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.28: Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, _ = 15 ° , Top View David D. Boyd, It. Chapter 8. Results: Rotor Fuselage 137 Top View _g = 30 ° accelerating stagnating

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.29: Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, _ = 30 ° , Top View David D. Boyd, Jr. Chapter 8. Results: Rotor/Fuselage 138 Unsteady Computation ¥- 4S 0 L.

Jnsteady Computation ¥. 60 °

(b)

Figure 8.30: Time Accurate Surface Pressure Coefficients at ¥ = 45" and ¥ _ 60" David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 139 Top View _g = 45 ° accelerating stagnating ._

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.31" Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, _ = 45 ° , Top View David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 140 Top View = 60 ° accelerating stagnating

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.32: Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, _ = 60 ° , Top View David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 141 Unsteady Computation gt = 75 ° (a) Unsteady Computation = 90 ° (b) accelerating stagnating

C

-0.5-0.4-0.3-0.2-0.1 O.O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 P Figure 8.33: Time Accurate Surface Pressure Coefficients at _ = 75 ° and _g = 90 ° David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage 142 Top View ¥ = 75 ° accelerating _agnaling

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 8.34: Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, _ = 75 ° , Top View

David D. Boyd, Jr.

Chapter 8. Results: Rotor Fuselage 143 m v Top View = 90 ° v accelerating stagnating

Cp

-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0,9 1.0 Figure 8.35: Time Accurate Surface Pressure Coefficient on Complete Rotor/Fuselage Configura- tion, _ = 90 ° , Top View David D. Boyd, Jr. Chapter 8. Results: Rotor Fuselage v {Z 0o m A 0c [] 0 O, J J [ = • v A v A J .c J J ¢- o 0 I I I -6 2 3 4 iteration # v Figure 8.36: Pitch Settings as a Function of Iteration Number

Chapter 9

v

Summary

v_j In this research, an efficient and accurate, hybrid method, combining a rotor loading model, a rotor/fuselage flowfield model, and a coupling technique, has been developed. This method uses the GDWT for the rotor loading model, OVERFLOW for the rotor/fuselage flowfield model, and a new coupling technique developed in this research. The motivations for development of such a I , model have been discussed and the relationships between the current model and other models in use today have been discussed.

A description of the Rotor Loading Model used for the first component of the model has been presented. In addition, the solution procedures employed in this model have been discussed, along with a validation study of the Rotor Loading Model. In this study, it was shown that predicted time averaged inflow distributions over a rotor disk matched measured time averaged inflow distri- butions well for a several flight conditions and for several rotor configurations. The sensitivity of the results to parameters (such as the number of azimuth steps and the number of harmonics used) was shown. In addition, predicted time accurate inflow quantities were compared to measured quantities. These comparisons showed that the predictions of the unsteady quantities do not match the experimental values well in waveform, but are comparable in magnitude and phase. It is also noted here that these predictions match well with previously published predictions in the literature for these rotor configurations.

A discussion of the method used for the second component of the current model, the Ro- tor/Fuselage Flowfield Model was also provided. The method used to solve the Navier-Stokes equa- tions was discussed; theory and implementation of the new time averaged and time accurate bound- ary conditions is discussed as well. These new rotor boundary conditions are a unique feature of the %J

David D. Boyd, Jr.

Chapter 9. Summary current model, and this is the first time this type of condition has been used for such a computation.

M,I A coupling model was developed, which links the Rotor Loading Model and the Rotor/Fuselage Flowfield Models through the time averaged induced inflow of the rotor. This coupling model is a unique feature of the current model. The derivation of a novel inflow filtering technique has been L • V presented also.

Computations which use the model presented in the first several chapters are compared with experimental data. These comparisons include results for an isolated fuselage, an isolated rotor, and a rotor/fuselage configuration. Predictions of the pressure coefficient on the surface of the isolated fuselage were shown to match experimental data well. Predictions of the time average and time accurate inflow above the rotor tip path plane for the isolated rotor configuration were shown also to match experimental data well.

Predictions of time averaged and time accurate inflow velocities above the rotor disk and predic- tions of unsteady pressure coefficients on the top centerline of the fuselage were shown to match experimental data well. A unique feature of this model is that predictions of the pressure coefficient on the sides of the fuselage match well with measured data. This is a significant accomplishment; kd previous methods have been unable to match unsteady pressures on the sides of the fuselage.

V

v

Bibliography

v Sheridan RF. and R.R Smith. Interactional Aerodynamics - A New Challenge to Helicopter

[1]

Technology. Washington, D.C., May 1979. Proceedings of the 35th Annual American Heli- copter Society Forum, Paper Number A79-49109; Preprint Number 79-59.

[2] W. Johnson. Recent Developments in Rotary-Wing Aerodynamic Theory. AIAA Journal, 24(8): 1219-1244, August 1986.

[3] T.A. Egolf and RF. Lorber. An Unsteady Rotor/Fuselage Interactional Method. Arlington, TX, February 25-27, 1987. Presented at the American Helicopter Society Specialists' Meet- ing on Aerodynamics and Aeroacoustics.

D.N. Mavris, S.G. Liou, N.M. Komerath, and H.M. McMahon. Measurement and Computa-

[4]

tion of the Velocity Field of a Cylinder in the Wake of a Rotor in Forward Flight. Buffalo, NY, June 12-14, 1989. Presented at the 20th AIAA Fluid Dynamics, Plasma Dynamics, and Lasers Conference, Paper AIAA-89-1844.

[5] J.D. Berry. A Method of Computing the Aerodynamic Interactions of a Rotor-Fuselage Con- figuration in Forward Flight. PhD thesis, Georgia Institute of Technology, Atlanta, GA, May 1990.

T.R. Quackenbush, C-M.G. Lain, D.B. Bliss, and A. Katz. Computational Methods for the

[6]

Analysis of Rotor Wake/Airframe Interactions. C.D.I. Report No. 91-02, 1991.

G.L. Crouse, Jr. An Analytical Study of Unsteady Rotor Fuselage Interaction in Hover and [7] Forward Flight. PhD thesis, University of Maryland, College Park, MD, November 1992.

[8] D.D. Boyd, Jr., T.F. Brooks, C.L. Burley, and J.R. Jolly, Jr. Aeroacoustic Codes for Rotor Harmonic and BVI Noise - Camrad.Mod 1/HIRES: Methodology and Users' Manual. NASA TM 207640, 1998.

Bibliography 148 David D. Boyd, Jr.

[9] C.S. Chen and J.O. Bridgeman. Three-Dimensional Viscous Rotor How Calculations Using v Boundary Layer Equations. Amsterdam, Netherlands, September 12-15, 1989. Presented at the 15th European Rotorcraft Forum, Paper Nr.: 13.

[10] J.O. Bridgeman, D.S. Pilchard, and EX. Caradonna. The Development of a CFD Potential Method for the Analysis of Tilt-Rotors. In Presented at the AHS Technical Specialists Meet- ing on Rotorcrafi Acoustics and Fluid Dynamics, Philadelphia, PA, October 15-17, 1991.

[I 1] R.C. Strawn. Wing-Tip Vortex Calculations with an Unstructured Adaptive-Grid Euler Solver. In Proceedings of the 47th Annual American Helicopter Society Forum, pages 65-76, v Pheonix, AZ, May 1991.

[12] T.J. Barth. A 3-D Upwind Euler Solver for Unstructured Meshes. Honolulu, HI, June 1991.

Paper AIAA-91-1548.

[ 13] R.C. Strawn and T.J. Barth. A Finite-Volume Euler Solver for Computing Rotary-Wing Aero- dynamics on Unstructured Meshes. In Proceedings of the 48th Annual American Helicopter Society Forum, pages 419-428, Washington, D.C., June 1992.

[14] LAd. Zori and R.G. Rajagopolan. Navier-Stokes Calculations of Rotor-Airframe Interaction in Forward Hight. pages 489-512, Washington, D.C., June 1992. Proceedings of the 48th Annual American Helicopter Society Forum.

[ 15] R.G. Rajagopolan and Z. Zhaoxing. Performance and Flow Field of a Ducted Propellor. Mon- terey, CA, July 10-12, 1989. Presented at the AIAA/ASME/SAE/ASEE 25th Joint Propulsion Conference, Paper AIAA-89-2673.

[16] R.G. Rajagopolan and S.J. Mathur. Three Dimensional Analysis of a Rotor in Forward Flight.

Journal of the American Helicopter Society, pages 14-25, July 1993.

[17] M.S. Chaffin and J.D. Berry. Navier-Stokes Simulation of a Rotor Using a Distributed Pres- sure Disk Method. In Proceedings of the 51st Annual American Helicopter Society Forum, volume I, pages 112-136, Ft. Worth, TX, May 1995.

[18] J.D. Berry, V.B. Letnikov, I. Bavykina, and M.S. Chaffin. A Comparison of Interactional Aerodynamics Methods for a Helicopter in Low Speed Forward Flight. In Proceedings of the 23rd European Rotorcrafi Forum, volume I, pages 33.1-33.9, Dresden, Germany, September 16-18, 1997.

David D. Boyd, Jr. Bibliography 149 [ 19] R. Meakin. Moving Body Overset Grid Methods for Complete Aircraft Tiltrotor Simulations.

Orlando, FL, June 6-9, 1993. Presented at the llth AIAA Computational Fluid Dynamics Conference, Paper AIAA-93-3350.

[20] G.R. Srinivisan and J.U. Ahmad. Navier-Stokes Simulation of Rotor-Body Flowfield in Hover Using Overset Grids. Cemobbio, France, September 14-16, 1993. Presented at the 19th European Rotorcraft Forum, Paper No. C15.

[21] J. Ahmad and Earl EN. Duque. Helicopter Rotor Blade Computation in Unsteady Hows Using Moving Embedded Grids. In 12th AIAA Applied Aerodynamics Conference, Colorado Springs, CO, June 20-22, 1994. Paper AIAA-94-1922.

[22] L. Tang and J.D. Baeder. Time-Accurate Euler Simulations of Vortex Convection. In Proceed- ings of the 52th Annual American Helicopter Society Forum, pages 1489-1499, Washington, . T D.C., June 4-6 1996.

[23] J. Steinhoff, W. Yonghu, T. Mersch, and H. Senge. Computational Vorticity Capturing: Ap- plication to Helicopter Rotor Flow. In 30th Aerospace Sciences Meeting and Exibit, Reno, NV, January 6-9, 1992. Paper AIAA-92-0056.

[24] D.D. Boyd, Jr. and R.W. Bamwell. Rotor-Fuselage Interactional Aerodynamics: An Un- steady Rotor Model. In Proceedings of the 54th Annual American Helicopter Society Forum, volume I, pages 23-44, Washington, D.C., May 20-22, 1998.

[25] D.A. Peters and C.J. He. Correlation of Measured Induced Velocities with a Finite-State Wake Model. pages 533-550, Boston, MA, May 1989. Proceedings of the 45th Annual American Helicopter Society Forum.

[26] D.A. Peters, D.D. Boyd, Jr., and C.J. He. Finite State Induced Inflow Model for Rotors in Hover and Forward Flight. pages 839-866, St. Louis, MO, May 18-20, 1987. Proceedings of the 43rd Annual American Helicopter Society Forum.

[27] C.J. He. Development and Application of a Generalized Dynamic Wake Theory For Lifting Rotors. PhD thesis, Georgia Institute of Technology, Atlanta, GA, July 1989.

[28] D.A. Peters and C.J. He. Finite State Induced Inflow Models Part II: Three Dimensional Rotor Disk. Journal of Aircraft, Volume 32((2)):323-333, March-April 1995.

V Bibliography 150 David D. Boyd, Yr.

[29] RG. Buning, D.C. Jespersen, T.H. Pulliam, W.M. Chan, J.R Slotnick, S.E. Krist, and K.J.

= , Renze. OVERFLOW User's Manual: Version 1.8, February 23 1998.

[30] W. Kinner. The Potential Theory of Airfoils of Circular Planform. Ingenieur-Archiv, 8(1):47- 80, February 1937.

[31] C. Hirsch. Numerical Computation of Internal and External Flows. Volume 2: Computational Methods for Inviscid and Viscous Flows. John Wiley and Sons, 1990.

[32] W. Johnson. A Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics.

v Part I: Analytical Development. NASA TM 81182, June 1980.

[33] J.W. Elliot, S.L. Althoff, and R.H. Sailey. Inflow Measurement Made With a Laser Velocime- ter on a Helicopter Model in Forward Flight - Volume I: Rectangular Planform Blades at an Advance Ratio of 0.15. NASA TM 100541, April 1988.

[34] J.W. Elliot, S.L. Althoff, and R.H. Sailey. Inflow Measurement Made With a Laser Velocime- ter on a Helicopter Model in Forward Flight - Volume II: Rectangular Planform Blades at an Advance Ratio of 0.23. NASA TM 100542, April 1988.

[35] S.L. Althoff, J.W. Elliot, and R.H. Sailey. Inflow Measurement Made With a Laser Ve- locimeter on a Helicopter Model in Forward Flight - Volume IV: Tapered Planform Blades at an Advance Ratio of 0.15. NASA TM 100544, April 1988.

[36] S.L. Althoff, J.W. Elliot, and R.H. Sailey. Inflow Measurement Made With a Laser Ve- locimeter on a Helicopter Model in Forward Flight - Volume V: Tapered Planform Blades at an Advance Ratio of 0.23. NASA TM 100545, April 1988.

[37] C.E. Freeman and R.E. Mineck. Fuselage Surface Pressure Measurements of a Helicopter Wind-Tunnel Model With a 3.15-Meter Diameter Single Rotor. NASA TM 80051, March 1979.

[38] J.A. Benek, RG. Buning, and J.L. Steger. A 3-D Chimera Grid Embedding Technique.

Cincinnati, OH, July 15-17, 1985. Paper AIAA-85-1523-CE [39] T.H. Pulliam and D.S. Chaussee. A Diagonal Form of an Implicit Approximate Factorization Algorithm. Journal of Computational Physics, 39, 1981.

V Bibliography 151 David D. Boyd, Jr.

[40] R.M. Beam and R.E Warming. An Implicit Finite-Difference Algorithm for Hyperbolic System in Conservation Law Form. Journal of Computational Physics, 22:87-109, 1976.

[41] M. Kandula and EG. Buning. Implementation of LU-SGS Algorithm and Roe Upwinding Scheme in OVERFLOW Thin-Layer Navier-Stokes Code. In 25th AIAA Fluid Dynamics v Conference, Colorado Springs, CO, June 20-23, 1994. Paper AIAA-94-2357.

[42] D. Jespersen, T.H. Pulliam, and EG. Buning. Recent Enhancements to OVERFLOW. 1997.

Paper AIAA-97-0644.

v [43] T.H. Pulliam. Time Accuracy and the use of Implicit Methods. 1993. Paper AIAA-93-3360- CE M.S. Chaffin and J.D. Berry. Navier-Stokes and Potential Theory Solutions for a Helicopter [44] Fuselage and Comparison With Experiment. NASA TM 4566, June 1994.

J.D. Berry, M.S. Chaffin, and E.EN. Duque. Helicopter Fuselage Aerodynamic Predictions: [45] Navier-Stokes and Panel Method Solutions and Comparison With Experiment. San Fran- cisco, CA, January 19-21, 1994. Presented at the 1994 Annual American Helicopter Society Aeromechanics Specialist Conference.

N.E. Suhs and R.W. Tramel. PEGSUS 4.0 User's Manual, 1991.

[46] W.M. Chan. Innovative Software Streamlines Overset Grid Generation. volume 3, NAS [47] Systems Division, Ames Research Center, Mail Stop 258-6, Moffett Field, California, 94035- 1000, May-June 1998. NAS News.

J.E Steinbrenner, J.R. Chawner, and C.L. Fouts. The GRIDGEN 3D Multiple Block Grid [48] Generation System. WRDC-TR-90-3022, October 1989. Wright Research and Development Center Report.

S.J. Alter and K.J. Weilmuenster. The Three-Dimensional Multi-Block Advanced Grid Gen- [49] eration System (3DMAGGS). NASA TM 108985, May 1993.

S.J. Alter. The Volume Grid Manipulator (VGM): A Grid Reusability Tool. NASA CR 4772, [50] April 1997.

W.M. Chan and EG. Buning. User's Manual for FOMOCO Utilities - Force and Moment [51] Computation Tools for Overset Grids. NASA TM 110408, July 1996.

Bibliography 152 David D. Boyd, It.

[52] J.F. Meyers, S.A. Gorton, and J.D. Berry. Instantaneous Doppler Global Velocimetry Mea- surements of a Rotor Wake: Lessons Learned. Lisbon, Portugal, July 13-16, 1998. Presented at the 9th International Symposium on Applications of Laser Techniques to Fluid Mechanics.

[53] G.A. Fleming and S.A. Gorton. Measurement of Rotorcraft Blade Deformation using Projec- V tion Moir6 Interferometry. Ancona, Italy, June 6-19, 1998. Presented at the 3rd International Conference on Vibration Measurements by Laser Techniques.

[54] A. Gessow and G.C. Myers, Jr. Aerodynamics of the Helicopter. Fredrick Ungar Publishing, Co., New York, NY, 1952.

v

Appendix A

v

Filtering Operation

A consistent filtering operation can be derived from the GDWT. This appendix gives a derivation of the filtering operation, derived using various functions introduced in the GDWT development of reference [28]. This operation takes a quantity, A(L _g), which is a function of the radial and azimuthal coordinate, and derives the expressions for coefficients of an infinite series given below.

The A(L _) quantity is then filtered by truncating the infinite series to a finite number of harmonics and shape functions.

Starting with a given quantity that is a function of the radial and azimuthal coordinates, one can express that quantity as a double summation over the harmonics and shape functions in the GDW"I' as follows: A(?,lg) = ZZ _nm(?) [anmC°S(m_l/) +bm sin(m_t)l (A.I) m rI where A is the given function, m is the harmonic number (m = 0, 1,2...), n is the shape function number (n = m + 1, m + 3,...), _ and _ are the radial and azimuthal coordinates, _ is a function m and bn m are the unknown coefficients. These coefficients could be, of ? from the GDWT, and a n in general, a function of time. In that case, the process outlined below would be followed at each discrete time step of interest.

Multiply equation (A. 1) by cos(p_) and integrate over _t from 0 to 2n to get the following: David D. Boyd, Jr. Appendix A. Filtering Operation 154

fo2' A V)cos(m) dV

"0 "* -m - m 2n = _,_n (r)[an f c°s(mgt)c°s(Plll) dv mn 0 2re (A.2) +b m f sin(mgt)cos(pv)dv] Using the orthogonality relations for sine and cosine functions, the right hand side of equation v (A.2) can be rewritten. The resulting form of equation (A.2) can be written as follows: f02rtA(?,lp')cos(pV) d/l/ = EZ_m(?)am[e(m)_)mp] m n v (A.3) = _$_(?)aPng(p) where m = 0, 1,2,..., 0% for both equations above, and n = m + 1, m + 3,..., _o, for the first equation of (A.3) and n --= p + 1, p + 3,..., _o, for the second equation of (A.3). Also, _mp is the Dirac delta function. The function O(p) is given as follows: 2== V (A.4) rc for p#O c(P)= 2_ for p=O Multiplying equation (A.3) by [_(?). ?. x/]--:-_] and integrating from ? = 0 to ? = 1 gives the following: Integral A fo'_qP(?)? 1V/'i-----_-?2 [f0ZnA(?,/g)cos(pll/)dll/ d?: _n [J0 _nP(?)_qP(?)?V/1-rZd anPc(P)

(A.5)

Into "Integral A", substitute the definition of $ in terms of Legendre functions, and perform a change of variables from ? to v using the fact that v = v/[ - ?2. The resulting "Integral A" compo- nent of the above equations becomes: David D. Boyd, Jr. Appendix A. Filtering Operation 155

f0 _ ;_(_)_(_) _vq- _2 d?

1 -p -p

= a-

o TV_V_"

i

= _ fPPn(v)P_(v)dv (A.6) = 4V/_q_mp where H_ is defined in the GDWT as follows: (A.7) Hnm = (n+m- 1)!!(n-m- 1)!!

(n+m)!!(n-m)!!

where the double factorial is defined in Peters, et al. [26], as follows: (n)(n-2)(n-4)...(2) for n=even (n)(n-2)(n-4)...(1) for n=odd 1 for n=0 (n)!!= 1 for n=-i - l for n =-3 Substituting equation (A.6) back into equation (A.5) gives the following: fol_)pq(F)r l_/_-_2[fo2_A(F, lg)cos(p_l)d1411 dr = _n[__mp __ ___a p (A.8) - _pqq q Solving equation (A.8) for the unknown coefficient aq p one gets: (A.9) aP-_7_c(p----_ f0 _P(')r lk/7-_--r2 [f02rCA("lJ'/)c°s(P'k[/)d/[/l dr David D. Boyd, Jr. Appendix A. Filtering Operation 156 noting that p = 0, I, 2,..., oo and q = p + 1,p + 3,... ,oo. Equation (A.9) gives the unknown a_ coefficients of equation (A. 1) given a function A(?, _). A similar derivation can be performed for the bp coefficients, which results in the following: f0 1 (A a0, noting that p = 1,2,..., oo and q = p + 1, p + 3,..., oo. For this application, the integrals in equations (A.9) and (A.10) are computed using the trapezoidal rule.

In order to filter the function to a contain a particular frequency content, it is necessary to truncate the summation to limit the number of aq p and bq p coefficients to a specified number of harmonics such that p = 0, 1,2,... ,Pmax for the a_ coefficient and p = 1,2,... ,Pmax for the bq p coefficient. In this implementation, the above limits set the number of shape functions to q = p + 1,p + 3,...,Pmax • Renaming the indices, and applying truncation to the summations, the filtered value of A(?,_), shown below as A(_, _), is as follows: mmax mmax+ l (A.11) _nm(?) [anmCOS(mV) + bnmsin(mV)] m ?1 where, m is the harmonic number index, n is the shape function index, and mmax is the number of harmonics used. It should be noted that the number of shape functions used is equal the number of harmonics used, plus one (nmax = mmax q- 1). Thus, compacting the above notation and referring to the terminology of chapter 5, the following filtering operation is defined: .q"(A) = .g, (A. 12)

Vita

David Douglas Boyd, Jr. (Doug) was born in on He gradu- ated from Tucker High School, Tucker, Georgia, in 1983. He then entered the Georgia Institute of Technology where he received his Bachelor of Science and Master of Science degrees in Aerospace Engineering in 1987 and 1988, respectively. Thereafter, from 1990 to 1996, he was employed by Lockheed Engineering and Sciences Company where he provided support to the Fluid Mechanics and Acoustics Division at the NASA Langley Research Center in the area of rotorcraft comprehen- sive analysis and rotorcraft acoustic predictions. Currently, he is employed by Virginia Polytech- nic Institute and State University, working on a cooperative research agreement with the Subsonic Aerodynamics Branch of NASA Langley Research Center exploring the unsteady, aerodynamic interactional effects between a helicopter rotor and fuselage. Doug is a member of the American Helicopter Society and the American Institute of Aeronautics and Astronautics. To date, he has been author and/or coauthor of eleven technical publications.

APPENDIX m

IT I II

m

AIAA 2000-0256

A Computational Model for Rotor-Fuselage

Interactional Aerodynamics

D. Douglas Boyd, Jr. and Richard W. Barnwell

Virginia Polytechnic Institute and State University

Virginia Consortium of Engineering and Sciences Universities

Hampton, Virginia

Susan Althoff Gorton

Aeroflightdynamics Directorate (AvRDEC)

U.S. Army Aviation and Missile Command

NASA Langley Research Center

Hampton, Virginia

38th Aerospace Sciences

Meeting & Exhibit

January 10-13, 2000/Reno, NV

FI [ lill IIIII I For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344 AIAA-2000-0256 A COMPUTATIONAL MODEL FOR ROTOR-FUSELAGE INTERACTIONAL AERODYNAMICS D. Douglas Boyd, Jr.* and Richard W. Barnwell t Virginia Polytechnic Institute and State University Virginia Consortium of Engineering and Sciences Universities Hampton, Virginia Susan Althoff Gorton _ Aeroflightdynamics Directorate (AvRDEC) U.S. Army Aviation and Missile Command NASA Langley Research Center Hampton, Virginia Abstract When designing a new rotorcraft, as with any flight vehicle, an understanding of the aerodynamic environ- A novel unsteady rotor-fuselage interactional aerody- ment, including aerodynamic interaction of the different namics model has been developed. This model loosely vehicle components, is essential. These interactional ef- couples a Generalized Dynamic Wake Theory (GDWT) fects have been known and categorized for many years.

to a thin-layer Navier-Stokes solution procedure. This This paper focuses on the "rotor-fuselage" and "fuselage- coupling is achieved using an unsteady pressure jump rotor" subsets of the categories offered by Sheridan and boundary condition in the Navier-Stokes model. The Smith.t In practice, information on specific interactional new unsteady pressure jump boundary condition mod- effects may be obtained using any combination of wind els each rotor blade as a moving pressure jump which tunnel testing and/or computational modeling.

travels around the rotor azimuth and is applied between two adjacent planes in a cylindrical, non-rotating grid.

Wind tunnel testing has been relied upon heavily in Comparisons are made between measured and predicted designing new rotorcraft and diagnosing and correcting time-averaged and time-accurate rotor inflow ratios. Ad- aerodynamic anomalies discovered on actual flight vehi- ditional comparisons are made between measured and cles because computational modeling of rotorcraft aero- predicted unsteady surface pressures on the top center- dynamics is still in its infancy and lags well behind the line and sides of the fuselage.

computational capabilities used for fixed wing vehicle modeling. Several factors have led to this situation. One Introduction of these is the fact that, as mentioned above, even in It is well known that rotorcraft aerodynamics is a com- level, unaccelerated flight, a rotorcraft is operating in an plicated topic. Due to the combination of various sys- unsteady aerodynamic environment due to the rotation tems associated with rotorcraft, these aerodynamic phe- of the rotor system. A fixed wing aircraft in the same nomena are unsteady, even in level, unaccelerated flight.

situation would be in a steady state environment. The Complicating these issues are the facts that typical rotor- computational implication of this is that a complete ro- craft in service today have bluff aft regions, which can torcraft simulation would necessarily be a time-accurate lead to large regions of flow separation, and that there can computation, whereas the fixed wing simulation could be significant aerodynamic interaction or interference be- be a steady-state computation. Another factor is asso- tween the rotating and non-rotating components of the ciated with the vastly different time and length scales system.

associated with rotorcraft. Some unsteady aerodynamic events, such as blade-vortex interaction, occur at length • Senior Research Associate, AIAA Member scales that are a small fraction of a blade chord and at t Professor, AIAA Fellow _; Aerospace Engineer time scales that are equivalent to a tiny fraction of a rotor Copyright _)2000 by the American Institute of Aeronautics and revolution. To capture these effects, very small time steps Astronautics, Inc. No copyright is asserted in the United States under would be required. However, determining the trim state Title 17,U.S. Code. The U.S. Government has a royalty-free license to of a rotorcraft requires balancing the gross forces on the exercise all fights under the copyright claimed herein for government purposes. All other rights are reserved by the copyright owner. rotorcraft that have a length scale on the order of the ro- American Institute of Aeronautics and Astronautics AIAA-2000-0256 nant. However, a disadvantage is that, for computing rotor-fuselage interactional effects that include viscous effects, a boundary layer coupling model must be em- ployed with these methods. To fully integrate the vis- cous computation, Navier-Stokes methods should be em- ployed. Only a few examples of Navier-Stokes com- t_ putations are present in the literature. In one of these, E o Meakin 14 used the Navier-Stokes equations to compute o?., the time-accurate flowfield around a V-22 tiltrotor vehi- cle, including the rotor. This computation was primar- ily geared toward demonstrating moving, chimera grid technology and is not currently a practical capability due Computational Expense to the large CPU times required. In general, solutions to the Navier-Stokes equations for interactional aerody- namics problems, where everything is modeled in one Figure I. Analysis types for coupled solutions.

large computation, are not currently practical for routine use.

tor radius (i.e., many chord lengths) over a relatively long Hybrid Methods time scale, equivalent to a number of rotor revolutions.

With the expense of Navier-Stokes methods for com- The computational implication of these vastly different pete rotorcraft out of reach for routine computations, time scales is that a time-accurate simulation would need a practical, engineering solution is to use a hybrid ap- to be executed for many time steps.

proach. In hybrid approaches, several different methods There are a number of methods available for compu- complement each other. For example, Steinhoff, et al. 15 tation of the interactional aerodynamic effects associated combined a vorticity capturing method with a Navier- with rotorcraft. Figure 1 categorizes these methods into Stokes method to reduce artificial dissipation effects on three areas: Singularity Methods, Hybrid Methods, and rotor wake vortices, which in turn relaxes the grid reso- Computational Fluid Dynamic (CFD) Methods. Each of lution requirement to resolve and maintain a rotor wake these methods has been used in the past for computa- vortex in the solution procedure. Boyd and Barnwel116 tion of rotorcraft interactional aerodynamics, and each first introduced a hybrid method that loosely couples a method has advantages and disadvantages.

Generalized Dynamic Wake Theory 17-2° (GDWT) with Singularity Methods a Navier-Stokes method. Boyd 3 extended that method Singularity methods typically use linear superposi- to include both a fuselage and a rotor and computed un- tions of solutions of Laplace's equation (i.e., source, steady fuselage surface pressures and unsteady inflow for sink, doublet, vortex elements) to model systems that a complete configuration.

may include the fuselage, the fuselage wake, the rotor The current work uses the method of Boyd 3 and blades, and the rotor wake. Johnson 2 provides an ex- presents results using that method. Below, a brief de- tensive discussion of singularity methods used for ro- scription of the method is provided for completeness.

torcraft analyses up through the year 1986. Boyd 3 dis- cusses other examples of analyses along these lines that Com.putational Method have been published since that time. These analyses have The current computation method is a hybrid shown varying degrees of success. It is apparent from method that loosely couples the GDWT to a Navier: these references that (1) one of the primary advantages Stokes method, OVERFLOW. The details of this of these methods is that they are typically computation- coupling can be found in Boyd, 3 but a brief outline is ally efficient and (2) one of the primary disadvantages is presented here.

the inability to adequately account for viscous effects.

As discussed earlier, determination of the gross load- CFI) Methods ing and rotor trim requires many revolutions of the ro- In recent years, CFD methods, including methods to tor. As such, this computationally expensive portion of solve the full potential equation, the Euler equations, and the method is separated from the CFD portion of the the Navier-Stokes equations, have become available. _t4 computation. This separation greatly reduces the time In general, the full potential and Euler methods, like the spent on time-accurate computations in the CFD portion singularity methods, have the advantage that they are rel- of method. Based on the above assumption, the cur- atively efficient computationally and are quite useful in rent method splits the interactional aerodynamics prob- some applications where viscous effects are not domi- lem into three distinct pieces: (1) the Rotor Loading American Institute of Aeronautics and Astronautics AIAA-2000-0256 used to represent the rotor. The predetermined pressure distribution is applied as an additional term in the energy equation as follows: A(_)AP a(pe0) - (l) y-1 Rotor Loading _D (r,v/,t) Rotor/Fuselage Flowfleld Model --- Model where equation (1) is in terms of the non-dimensional (GDWT) (OVERFLOW) quantities used in OVERFLOW and A(?) is the ratio be- tween the local actual blade area and the local compu- tational cell area at a given radial station on the blade.

This ratio is used to maintain the correct overall thrust.

The additional conservative energy term in equation (I) is then split into two parts. One half of the term is applied to the "upper rotor plane" (see figure 3b) and the negative of the other half of the term is applied to the "lower ro- tor plane". This procedure effectively creates a pressure jump between two planes in the rotor grid, separated by Figure 2. Current hybrid method.

an "iblanked plane" which ensures that the artificial dis- sipation terms, which operate on a pressure discontinu- Model, (2) the Rotor/Fuselage Flowfield Model, and (3) ity, do not modify the input pressure distribution at the the Coupling Model. The arrangement of these pieces is rotor plane. All remaining flow quantities on the upper and lower rotor planes are determined by averaging the shown in figure 2.

quantities at planes "A" and "B" in figure 3b. Figure 3a Rotor Loading Model shows a top view of the rotor grid used in figure 3b. In To reduce the computational expense of the entire pro- this top view, a rectangular section is used to represent cess, a model is used to determine the loading distribu- the actual blade area, and a shaded wedge represents the tion on and the trim state of the helicopter rotor. The computational area (these areas are not to scale). Only model used here is based on the GDWT as discussed one blade is represented in this figure.

above. This model uses a solution of the Laplace equa- For a multibladed rotor, one of these computational tion for a isolated, circular wing developed by Kinner. 2t wedges exists for each blade. A radially varying, addi- Essentially, Kinner's solution provides admissible accel- tional conservative energy term is applied along each of eration potential functions on the circular wing. To deter- these computational wedges for each blade. At each time mine the unknown coefficients in Kinner's solution, Pe- step in the time-accurate solution procedure, the pres- ters, Boyd, He, 18 He, 17 and Peters and He, t9 used the lin- sure jump "travels" around the rotor azimuth direction, earized Euler equations, the continuity equation, and spe- one grid line per time step. This unsteady boundary con- cial rotor boundary conditions, to relate the Kinner accel- dition effectively represents the rotor blades as a pres- eration potential to the induced inflow at the rotor disk.

sure jump traveling around the rotor azimuth on a non- For the current research, the resulting closed form ma- rotating, cylindrical grid.

trix equations are iteratively solved in conjunction with a Using the chimera grid techniques available in OVER- modified Newton-Raphson trim technique to determine FLOW, the above rotor grid is combined with other grids the unsteady induced inflow, the trim state, and the un- which represent the fuselage and the remaining flowfield.

steady loading distribution of the isolated rotor.

OVERFLOW then solves the time-accurate, thin-layer With the solution of the GDWT for the isolated rotor, Navier-Stokes equations on this set of grids, along with the "Rotor Loading Model" portion of figure 2 is com- the unsteady, pressure jump boundary condition. The so- plete. In figure 2 it can be seen that the pressure (loading) lution procedure is executed until the initial transients are distribution from the Rotor Loading Model is used in the removed and a periodic flowfield is obtained.

"Rotor/Fuselage Flow field Model".

Since the specified pressure jump was originally de- termined by an isolated rotor model, the pressure jump Rotor/Fusela_,e Flowfield Model boundary condition does not represent the combined Now, with a known pressure distribution on the rotor rotor-fuselage system. Therefore, once a periodic solu- disk, a Rotor/Fuselage Flowfield Model is used to solve tion has been obtained with the original pressure jump the Navier-Stokes equations. For this model, a thin-layer, boundary condition, an "Inflow Correction" method is Navier-Stokes code (OVERFLOW 22) has been modified used to account for the presence of the fuselage in the to include an unsteady boundary condition. For this new Rotor Loading Model. Discussion of this method is be- boundary condition, a cylindrical, non-rotating grid is American Institute of Aeronautics and Astronautics AIAA-2000-0256 v Figure 4. Laser velocimeter experiment, NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel.

The LV measurements were processed at an azimuthal resolution of approximately 2.8*. Comparisons to both ) the time-averaged and the time dependent measured data will be made subsequently.

(b) lower rotor plane 'Iblanked" plane upper rotor plane Figure 3. Schematic of new boundary condition.

yond the scope of this paper, but is discussed in detail in Boyd. 3 Figure 2 shows the location of the "Coupling Model (Inflow Corrections)" portion of the model.

With these inflow corrections, the GDWT model is re- executed to obtain a new unsteady pressure jump bound- Figure 5. Unsteady surface pressure experiment, NASA ary condition that has been corrected to account for the Langley Research Center 14- by 22-Foot Subsonic Tun- nel.

presence of the fuselage. This cycle is repeated until there is no significant solution change between iterations.

The second experiment ("Experiment 2") used here was carried out by the third author and her colleagues, Results again using the ROBIN fuselage with the same rectan- Experimental Setup gular rotor system. The primary difference in the con- Results from the computational method discussed figuration between the first and second experiments is above will be compared to experimental data. The ex- that, in the first experiment, the rotor drive system was periments used here are discussed in other references, 3'29 contained inside the fuselage shell, whereas, in the sec- but are discussed briefly here for completeness. There ond experiment, the rotor and fuselage were mounted on are two experiments that are used here. The first ex- separate systems. That is, in the second experiment, the periment ("Experiment 1"), reported by Elliott, Ahhoff, rotor drive system was mounted to the tunnel ceiling and and Sailey, 23 used a Laser Velocimetry (LV) system to the fuselage was sting mounted on a post attached to the measure the induced inflow in a plane that was one ro- tunnel floor (see figure 5). This experiment was con- ducted in two phases: (1) an isolated rotor configuration tor blade chord above the rotor tip path plane. These measurements were carried out for the combination of a (with the fuselage lowered to the tunnel floor) and (2) a rotor/fuselage configuration (with the fuselage in place).

generic helicopter fuselage (known as the ROtor Body INteraction (ROBIN) fuselage) and a four-bladed, rect- In the first phase of this test, unsteady inflow measure- ments were made at a limited number of locations on the angular rotor system in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel (see figure 4). advancing side of the rotor, one chord above the tip path American Institute of Aeronautics and Astronautics AIAA-2000-0256 line with a value of 0.015 shows that the induced inflow

plane. Inthesecond phase of theexperiment, unsteady

surface pressures were measured along thetop centerline is asymmetric about the fore-aft plane of the rotor. Figure ofthefuselage and atseveral locations onthe sides ofthe 6b shows the predicted, time-averaged induced inflow ra- fuselage. Thedata presented here area small subset of tio parallel to the rotor tip path plane for the isolated ro- the total data taken inthesecond experiment. Subsequent tor configuration. Although the magnitudes are similar to the measured values, the inflow distribution does not

comparisons will bemade tothese unsteady inflow and

match the measured distribution well. Here, unlike the

unsteady surface pressure data.Table 1listsseveral of

theoperating conditions and rotor parameters associated measured data, the predicted induced inflow is somewhat

withboth Experiments 1and 2. symmetric between the advancing and retreating sides of

the rotor. Figure 6c shows the predicted, time-averaged induced inflow ratio parallel to the rotor tip path plane for Table 1. Operating conditions and rotor parameters.

the full rotor-fuselage configuration. It is seen that the fuselage has a large impact on the inflow distribution. As Property Value with the isolated rotor configuration, the magnitude of Blade planform Rectangular the inflow matches the measured data well. In addition, radius 0.8606 meters the distribution of inflow now matches the experimental root chord 0.0660 meters data well, including the asymmetric pattern seen in the 0.0660 meters tip chord measured data. Figure 6d shows the difference between number of blades 4 the full rotor-fuselage configuration and the isolated ro- root cutout location 0.24R tor configuration. This difference plot shows the effect 0.06R flap/lag hinge location of the fuselage on the in-plane induced inflow. As would airfoil section NACA 0012 be expected for a fuselage, there is a deceleration of the twist -8* flow over the forward portion of the rotor disk due to the nominal thrust coefficient 0.0065 upward slope of the nose of the fuselage and a subse- 0.0977 solidity quent re-direction of the flow. Over the rear portion of 0.55 nominal hover Mtip the rotor disk, there is an acceleration of the flow due to I* approx, mean coning angle the downward slope of the rear portion of the pylon.

shaft tilt 3* nose down Figure 7a shows the measured, time-averaged induced inflow ratio perpendicular to the rotor tip path plane from Experiment 1. These measured data show several typi- Induced Inflow Comparisons cal features of time-averaged induced inflow. First, there Time-Averaged Induced Inflow is an upwash on the forward portion of the rotor disk.

Once the iteration procedure has concluded, as dis- Second, there is an increased downward inflow toward cussed in Boyd, 3 comparisons between a number of the rear portion of the disk with concentrations in the quantities are possible. For these comparisons, the cur- first and fourth rotor quadrants. Figure 7b shows the pre- rent model was executed with and without a fuselage in dicted, time-averaged induced inflow ratio perpendicular the solution procedure. As shown in Boyd and Barn- to the rotor tip path plane for the isolated rotor config- well 16 and Boyd, 3 the current model is also applicable to uration. This configuration exhibits many of the same an isolated rotor configuration (i.e., no fuselage).

features as the measured data. For example, there is an First, a comparison is presented between the measured upwash on the forward portion of the rotor disk, but that and predicted, time-averaged induced inflow. Inflow ra- upwash is not as prominent as in the measured data. Fig- tio is defined as the local velocity divided by the rotor tip ure 7c shows the predicted, time-averaged induced in- speed. The measurement data are from Experiment 1 at flow ratio perpendicular to the rotor tip path plane for the a plane that is one blade chord above the tip path plane full rotor-fuselage configuration. The magnitude as well of the rotor at a rotor advance ratio of/z = 0.23. The as the inflow distribution is well matched when the fuse- predicted results are from the same location above the lage is included in the computation. Figure 7d shows the rotor tip path plane and are at the same operating condi- difference between the full rotor-fuselage configuration tion used in Experiment 1. The rotor tip speed is used to and the isolated rotor configuration. Again, this figure make the data and predicted results nondimensional. displays features that are expected due to the presence of Figure 6a shows the measured, time-averaged induced a fuselage. For example, there is an increased upwash inflow ratio parallel to the rotor tip path plane from Ex- over the forward portion of the disk as the flow is de- periment 1. These experimental data show an induced in- flected upward over the nose of the fuselage, and there is flow pattern that is not symmetric between the advancing an increased downwash at the rear of the rotor disk, just and retreating sides of the rotor. For example, the contour aft of the pylon, as the flow accelerates downward just American Institute of Aeronautics and Astronautics AIAA-2000-0256 behind the fuselage pylon.

Figure 9 shows a comparison of the unsteady compo- nent of the measured and predicted modified pressure co- Time-Accurate Induced Inflow efficient on the top centerline of the fuselage at various The previous section showed that the time-averaged stations along the length of the 2 meter long fuselage.

induced inflow in the parallel and perpendicular direc- The location of the reference blade is plotted along the tions (relative to the rotor tip path plane) are well pre- horizontal axis, and the negative of the modified pres- dicted by the current unsteady method. This section will sure coefficient is plotted along the vertical axis. Since present comparisons of the measured and predicted un- this is a four-bladed rotor, a dominant pressure pulse can steady inflow data corresponding to the same flight con- be seen at a frequency of four pulses per rotor revolution.

ditions used in Experiment 1. The measured data pre- This is indicative of the four blades individually passing sented here is from the first phase of Experiment 2 (iso- over each measurement location. It can be seen that the lated rotor configuration).

phase of each of the predictions matches the measured Figure 8 shows the measured and predicted unsteady phase well; however, the amplitudes are slightly over- induced inflow ratios. These inflow ratios are at an az- predicted.

imuthal location of_ = 84" and a blade radial location of Figure 10 shows a comparison of the unsteady com- r/R = 0.80. Both the isolated rotor and combined rotor- ponent of the measured and predicted modified pressure fuselage configuration are shown. Both components are coefficient on the left and right sides (retreating and ad- well predicted, especially the inplane component. For vancing sides, respectively) of the fuselage at a constant this particular location, the presence of the fuselage has downstream location ofx = 0.8809 meters (x/L _ 0.44) only a minor impact on the predicted unsteady induced for several vertical locations. Again, the reference blade inflow. Previous literature has shown 3,16 that these in- location is on the horizontal axis, and the negative of the duced inflow ratios are typically well predicted over the modified pressure coefficient is on the vertical axis. The entire rotor disk.

retreating side comparisons show that the unsteady pres- Unsteady Surface Pressure sures are slightly overpredicted, while the advancing side In Experiment 2, unsteady surface pressure measure- unsteady pressures are well matched in magnitude and ments were made for the same flight configuration and phase.

the same flight conditions as in Experiment 1. These measurements were made along the top centerline of the Conclusions fuselage and at several locations on the sides of the fuse- A novel computational model for unsteady rotorcraft lage. Comparisons are made here between the measured interactional aerodynamics has been presented. This and predicted unsteady surface pressures along the top new hybrid model couples a rotor loading model and a centerline and at several locations on the advancing and rotor/fuselage flowfield model in a manner that is effi- retreating sides of the fuselage. These pressure taps on cient and capable of predicting time-averaged and time- the sides of the fuselage were located at several vertical accurate rotor inflow ratios and unsteady surface pres- locations and at a constant 44% of the fuselage length.

sures on the fuselage due to blade passages.

For these comparisons, a modified pressure coefficient is used. This modified pressure coefficient is defined in References equation (2) and is used to avoid numerical problems as- Sheridan P.F. and R.P. Smith. Interational Aerody- sociated with the definition of the standard pressure coef- namics - A New Challenge to Helicopter Technol- ficient when the freestream velocity approaches zero (as would be the case in hover). ogy. Washington, D.C., May 1979. Presented at the 35th Annual American Helicopter Society Forum.

C'p = IO0(P- P.)

.

113 D.R 2 (2) W. Johnson. Recent Developments in Rotary-Wing Aerodynamic Theory. AIAA Journal, 24(8):1219- In equation (2), P is the local pressure, P_ is the 1244, August 1986.

freestream pressure, p is the ffeestream density, _ is the rotor tip speed, and the factor of 100 is included for nu- .

D.D. Boyd, Jr. Rotor-Fuselage Interaction Aerody- merical convenience. For reference, equation (3) shows namics: A New Computation Model. PhD thesis, the relation between the standard pressure coefficient and Virginia Polytechnic Institute and State University, the modified pressure coefficient used here.

July, 1999.

e,

= lOO Cp (3)

.

C.S. Chen and J.O. Bridgeman. Three-Dimensional In equation (3),/.t** is the standard rotor advance ratio and Viscous Rotor Flow Calculations Using Boundary Cp is defined in the usual way.

Layer Equations. Amsterdam, Netherlands, Septem- American Institute of Aeronautics and Astronautics AIAA-2000-0256 ber 12-15, 1989. Presented at the 15th European 15. J. Steinhoff, W. Yonghu, T. Mersch, and H. Senge.

Rotorcraft Forum.

Computational Vorticity Capturing: Application to Helicopter Rotor Flow. In 30th Aerospace Sciences 5. J.O. Bridgeman, D.S. Prichard, and EX. Caradonna.

Meeting and Exibit, Reno, NV, January 6-9, 1992.

The Development of a CFD Potential Method for the Paper AIAA-92-0056.

Analysis of Tilt-Rotors. In Preceedings of the AHS Technical Specialists Meeting on Rotorcraft Acous- 16. D.D. Boyd, Jr. and R.W. BarnweI1. Rotor-Fuselage tics and Fluid Dynamics, Philadelphia, PA, October Interactional Aerodynamics: An Unsteady Rotor 15-17, 1991.

Model. In Preceedings of the 54th AnnuaI American Helicopter Society Forum, volume I, pages 23--44, 6. R.C. Strawn. Wing-Tip Vortex Calculations with an Washington, D.C., May 20-22, 1998.

Unstructured Adaptive-Grid Euler Solver. In Pre- ceedings of the 47th Annual American Helicopter 17. C.J. He. Development and Application of a General- Society Forum, Pheonix, AZ, May 1991.

ized Dynamic Wake Theory For Lifting Rotors. PhD thesis, Georgia Institute of Technology, Atlanta, GA, 7. T.J. Barth. A 3-D Upwind Euler Solver for Un- July 1989.

structured Meshes. Honolulu, HI, June 1991. Paper AIAA-91-1548.

18. D.A. Peters, D.D. Boyd, Jr., and C.J. He. Finite State Induced Inflow Model for Rotors in Hover and For- 8. R.C. Strawn and T.J. Barth. A Finite-Volume Euler ward Flight. St. Louis, MO, May 18-20, 1987. Pre- Solver for Computing Rotary-Wing Aerodynamics sented at the 43rd Annual American Helicopter So- on Unstructured Meshes. In Preceedings of the 48th ciety Forum.

Annual American Helicopter Society Forum, Wash- ington, D.C., June 1992.

19. D.A. Peters and C.J. He. Correlation of Measured Induced Velocities with a Finite-State Wake Model.

9. L.A.J. Zori and R.G. Rajagopolan. Navier-Stokes Boston, MA, May 1989. Presented at the 45th An- Calculations of Rotor-Airframe Interaction in For- nual American Helicopter Society Forum.

ward Flight. Washington, D.C., June 1992. Pre- sented at the 48th Annual American Helicopter So- 20. D.A. Peters and C.J. He. Finite State Induced In- ciety Forum.

flow Models Part II: Three Dimensional Rotor Disk.

Journal of Aircraft, Volume 32(Number 2), March- 10. R.G. Rajagopolan and Z. Zhaoxing. Performance April 1995.

and Flow Field of a Ducted Propellor. Mon- terey, CA, July 10-12, 1989. Presented at the 21. W. Kinner. The Potential Theory of Airfoils of AIAA/ASME/SAE/ASEE 25th Joint Propulsion Con- Circular Planform. Ingenieur-Archiv, 8(1):47-80, ference.

February 1937.

11. R.G. Rajagopolan and S.J. Mathur. Three Dimen- 22. EG. Buning, D.C. Jespersen, T.H. Pulliam, W.M.

sional Analysis of a Rotor in Forward Flight. Jour- Chan, J.E Slotnick, S.E. Krist, and K.J. Renze.

nal of the American Helicopter Society, July 1993.

OVERFLOW User's Manual: Version 1.8, February 23 1998.

12. M.S. Chaffin and J.D. Berry. Navier-Stokes Simu- lation of a Rotor Using a Distributed Pressure Disk 23. J.W. Elliott, Susan L. Althoff, and R.H. Sailey. In- Method. In Preceedings of the 51st Annual Ameri- flow Measurement Made With a Laser Velocimeter can Helicopter Society Forum, volume I, pages 112- on a Helicopter Model in Forward Flight - Volume 136, Ft. Worth, TX, May 1995.

17: Rectangular Planform Blades at an Advance Ra- tio of 0.23. NASA TM 100542, April 1988.

13. J.D. Berry, V.B. Letnikov, I. Bavykina, and M.S.

Chaffin. A Comparison of Interactional Aerodynam- ics Methods for a Helicopter in Low Speed Forward Flight. In Proceedings of the 23rd European Rotor- craft Forum, volume I, pages 33.1-33.9, Dresden, Germany, September 16-18, 1997.

14. R. Meakin. Moving Body Overset Grid Methods for Complete Aircraft Tiltrotor Simulations. Orlando, FL, June 6-9, 1993. Presented at the llth AIAA Computational Fluid Dynamics Conference.

American Institute of Aeronautics and Astronautics AIAA-2000-0256 (a) Data (b) Prediction for Isolated Rotor V.

.% "o_ (c) Prediction for Rotor+Fuselage (d) Difference of (¢) and (b)

// °t _\

\o; ;/:J

Figure 6. Measured and predicted time averaged parallel induced inflow ratio from time accurate computations.

(a) Data (b) Prediction for Isolated Rotor

,v_

_,, _ _,,_ ____

-- u_;W;,W_" __C--

(d) Difference of (c) and (b) (c) Prediction for Rotor+Fuselage Figure 7. Measured and predicted time averaged perpendicular induced inflow ratio from time accurate computations.

American Institute of Aeronautics and Astronautics AIAA-2000-0256 ........................ measured ......... predTcted, isolated rotor -- predicted, rotor+fuselage Inplane Induced Inflow 0.o3 Out-of-plane Induced Inflow 0.03 0,02 0.02 I = oo, [- . j'_ fi , 0,01

: o0, L

(a)

-0.02 [- (b) -0.02!

r T , ! i | i i I i i 1 J I i i i z I -0.03 '

.oo3;.... :o .... ,_o .... _o.... 3_o

90 180 270 360 Ref. Blade Location (o) Ref. Blade Location(°) Figure 8. Measured and predicted induced inflow in two directions for an isolated rotor and a rotor/fuselage combina- tion.

American Institute of Aeronautics and Astronautics AIAA-2000-0256 .................... measured predicted x=0.0517 1 _I x=0.2007 I x=0.0941 'U -Cp -1 0 90 180 270 360 x=0.2563 _I x=1.1620 1 I x=1.3450 'tl 0 ._ " " 0 " ' • -Cp 0 -1 -20 .... ,..._= .... , .... , 90 80 270 360 " 0 90 180 270 360 - 0 90 180 270 360 Ref. Blade Loc. (°) R ef. Blade Loc. (o) Ref. Blade Loc. (°) Figure 9. Measured and predicted unsteady modified pressure coefficient on the top centerline of the ROBIN fuselage.

"x" denotes the distance in meters from the nose of the 2 meter long fuselage.

Retreating Side 2_ z=0.12 _ z=0.06 " 0 90 180 270 360 " 0 90 180 270 360 Advancing Side I z=0.12 I z=0.06 , " 0 90 180 270 360 Ref. Blade Loc. (o) Ref. Blade Loc. (°) Ref. Blade Loc. (°) Figure 10. Measured and predicted unsteady modified pressure coefficient on the retreating and advancing sides of the ROBIN fuselage. "z" denotes the distance measured in meters from the horizontal reference line of the fuselage.

American Institute of Aeronautics and Astronautics