Document
20730?
//j--_ z
Helicopter RotorBlade Computation
in UnsteadyFlows Using
Moving OversetGrids
Jasim Ahmad and Earl P.N. Duque
Reprinted from
Journal of Aircraft
Volume 33, Number 1, Pages 54-60 GIAMA.
A publication of the American Institute of Aeronautics andAstronautics, Inc.
370 L'Enfant Promenade,SW Washington, DC 20024-2518
:z/- _ :2 - 7"//
JOURNAL OF AIRCRAFT Vol. 33, No. 1, January-February 1996
Helicopter Rotor Blade Computation in Unsteady Flows
Using Moving Overset Grids
Jasim Ahmad* Sterling Software, Moffett Field, California 94035-1000 and Earl P. N. Duquet U.S. Army Aeroflightdynamics Directorate, Moffett Field, California 94035-1000 An overset grid thin-layer Navier-Stokes code has been extended to include dynamic motion of helicopter rotor blades through relative grid motion. The unsteady flowfield and airioads on an AH-IG rotor in forward flight were computed to verify the methodology and to demonstrate the method's potential usefulness towards comprehensive helicopter codes. In addition, the method uses the blade's first harmonics measured in the flight test to prescribe the blade motion. The solution was impulsively started and became periodic in less than three rotor revolutions. Detailed unsteady numerical flow visualization techniques were applied to the entire unsteady data set of five rotor revolutions and exhibited flowfield features such as blade vortex interaction and wake roll- up. The unsteady blade loads and surface pressures compare well against those from flight measurements.
Details of the method, a discussion of the resulting predicted flowfield, and requirements for future work are presented. Overall, given the proper blade dynamics, this method can compute the unsteady flowfleld of a general helicopter rotor in forward flight.
Introduction Notable among the comprehensive helicopter analysis codes is CAMRAD/JA by Johnson.6 To account for compressibility, HE accurate computation of a helicopter flowfield is es- three-dimensional effects, and arbitrary blade geometries, sential for proper and efficient airload predictions in for- Strawn and Tung 7 coupled CAMRAD to the full potential ward flight and hover. The constantly changing aerodynamic rotor (FPR) code. Strawn et al. _ demonstrated the capability environment and loads are important features of rotorcrafl to model rotors in forward flight using the coupled CAMRAD aerodynamics. Strong tip vortices in the rotor wakes dominate and FPR by computing the airloads of the Puma rotor.
the fiowfield to produce a highly unsteady and nonuniform Hernandez and Johnson _ used the coupled CAMRAD/JA- induced velocity field at the rotor disk. Recent emphasis on FPR method to compute the AH-1G rotor in forward flight.
numerical simulation procedures for complex flows have made They were able to compare well against flight test data. In it possible to accurately obtain rotor blade aerodynamic char- addition, they found that tip core size specifications have a acteristics by solving the governing differential equations.
large effect upon the predicted rotor loads.
Descriptions of various computational fluid dynamics (CFD) Ramachandran et al. '° developed a method based upon the methods for rotorcraft problems based on the solution of full- vorticity-embedding technique. This method solves the un- potential, Euler, and Navier-Stokes equations can be found steady full potential equation on a Eulerian grid with an over- in the literature. Notable among them is the introduction of set Lagrangian vortical velocity field. He also applied his method embedded or overset grid scheme to the existing single grid to the forward flight of the A H- 1G rotor system with favorable scheme by Duque and Srinivasan.' Also, Srinivasan et al. -'-3 comparisons to flight tests.
ased a thin-layer Navier-Stokes method for forward-flight All of the above methods use single-structured grids. Single simulation of a nonlifting rotor blade as well as rotor in hover.
grids have limited use. For example, it would be very difficult Chen et al.' and Agarwal and Deese -_solved the Euler equa- to use a single-structured grid to investigate the aerodynamic tions.
interference between either a rotor and fuselage or the aero- Unlike fixed wing aircraft, the helicopter airloads depend dynamic interference between a rotor and the rotor hub. In greatly upon the unsteady dynamic motion of the blades. To addition, almost all of the above methods use wake models model the blade motion, comprehensive helicopter analysis to include the influence of the wake. Wake models limit the methods have been developed that use aerodynamic and wake method's generality and require considerable adjustments to models coupled to structural dynamic models. The aerody- their approximations to obtain accurate solutions.
namics are typically based upon lifting-line theory and depend The work summarized herein uses an alternative gilding upon either linear methods or two-dimensional experimental method known as overset gilds or the Chimera method Z_ to airfoil data with corrections for unsteady and three-dimen- allow for blade motions and to accurately compute the vortical sional effects.
wake from first principles. The overset-gild scheme greatly simplifies arbitrary blade motions, which is important in Presented as Paper 94-1922 at the AIAA 12th Applied Aerody- achieving trimmed flight conditions, as well as in predicting namics Conference, Colorado Springs, CO, June 20-22, 1994; re- airloads accurately in an effort to develop a comprehensive ceived Sept. 23, 1994; revision received May 10, 1995; accepted for helicopter analysis method. Also, with overset grids one can publication June 24, 1995. This paper is declared a work of the U.S.
more easily discretize the domain with simple well-defined Government and is not subject to copyright protection in the United grids that accurately compute the rotor wake.
States.
Until recently, few methods were able to solve the transient *Research Scientist, NASA Ames Research Center, MS 258-1.
flow about multiple bodies moving relative to one another.
Member AIAA.
Meakin'-" computed the unsteady flowfield of the V-22 tiltro- tResearch Scientist, NASA Ames Research Center, Aviation and tor aircraft, including the rotor rotation, by solving the un- Troop Command. Member AIAA.
AHMAD AND DUOUE Table 2 Computed and flight test rotor force Acknowledgments and moment coefficients The authors would like to thank K. Ramachandran for providing the blade surface geometry and all of his subsequent CT, Ce, c_,,, c_,, thrust torque longitudinal lateral help in this work. We would also like to thank Jeffry L. Cross for providing valuable information regarding the flight test.
Computed 0.00455 0.000227 - 0.00022 0.000008 Finally, the authors wish to express their appreciation to W.
Flight 0.00464 0.0002196 0.0 0.0 J. McCroskey for his help and encouragement.
p. = 0.19. Mr = 0.65, Re = 9.73 x l0 n, ando" = 0.0651.
References 0.00"1.
_Duque, E. P. N., and Srinivasan, G. R., "'Numerical Simulation 0.006.
of a Hovering Rotor Using Embedded Grids." Proceedings of the o.o0_ 48th AHS Annual Forum and Technology Display (Washington. DC).
American Helicopter Society. 1992.
0.004 2Srinivasan, G. R., Baeder, J. D., Obayashi, S., and McCroskey, l W. J., "Flowfield of a Lifting Rotor in Hover: Navier-Stokes Sim- oAm3_ ulation," AIAA Journal, Vol. 30, No. 10, 1992, pp. 2371-2378.
O.O_, _Srinivasan, G. R., and Baeder, J. D., "'TURNS: A Free-Wake go la0 270 m Azimuth Euler/Navier-Stokes Numerical Method for Helicopter Rotors," AIAA Journal, Vol. 31, No. 5, 1993, pp. 959-962.
Fig. 11 Time history of rotor thrust.
_Chen, C. L., and McCroskey, W. J., "'Numerical Solutions of Forward Flight Rotor Flow Using an Upwind Method," AIAA Paper revolution at various radial locations. The figures show strong 89-1846, June 1989.
blade vortex interaction for the advancing blade around azi- SAgarwal, R. K., and Deese, J. E., -A Euler Solver for Calculating the Flowfield of a Helicopter Rotor in Hover and Forward Flight," muthal locations of 70-90 deg and on the retreating side AIAA Paper 87-1427, June 1987.
around 270 deg. The loads are underpredicted for the ad- "Johnson, W., CAMRAD/JA ; A Comprehensive Analytical Model vancing side. At the retreating side, loads are slightly under- of Rotorcraft Aerodynamics and Dynamics, Vol. I. Johnson Acro- predicted for the cases presented here. Blade-vortex inter- nautics. Palo A_to, CA, 1988.
actions at outboard stations seem stronger as indicated by the rStrawn, R., and Tung, C., "'The Prediction of Transonic Loading negative lift.
on Advancing Helicopter Rotors," NASA TM-88238, United States Table 2 lists the computed rotor force and moment coef- Army Aviation Systems Command, USAVSCOM TM 86-A-1, April 1986.
ficients. As shown, the rotor is not trimmed in pitch, but essentially trimmed in roll. The power is overpredicted by _Strawn, R., Desopper, A., Miller. J., and Jones, A.. "'Correlation of PUMA Airloads-Evaluation of CFD Prediction Methods," Fif- approximately 15%.
teenth European Rotorcraft Forum. Amsterdam, Sept. 1989.
Figure 11 illustrates the time variation of the integrated "Hernandez, F., and Johnson, W., "'Correlation of Airloads on a rotor thrust for the fifth rotor revolution. The solid line pre- Two-Bladed Helicopter Rotor," International Specialist Meeting on sents the predicted azimuthal variation while the dashed line Rotorcraft Acoustics and Rolor Fluid Dynamics, PA. Oct. 1991.
represents the time averaged or gross rotor thrust. The gross "'Ramachandran. K., Schlechtriem. S.. Caradonna. F. X., and thrust agrees within 1.8% of the gross weight of the aircraft Steinhoff, J. S., "Free-Wake Computation of Helicopter Rotor Flow- in flight.
field in Forward Flight," AIAA Paper 93-3079, July 1993.
"Steger. J. L., Doughcrty, F. C.. and Benek. J. A., "'A Chimera Conclusions and Future Work Grid Scheme." Advances in Grid Generation, edited by K. N. Ghia and U. Chin. American Society of Mechanical Engineers FED-5, The unsteady forward-flight flowfield of the AH-1G heli- 1983, pp. 59-69.
copter's two-bladed rotor system was computed by solving _-'Meakin. R. "'Moving Body Overset Grid Methods for Complete the thin-layer Navier-Stokes equations on moving overset AircraIt Tiltrotor Simulations," AIAA Paper 93-3350, July 1993.
grids, The resulting flowfield visualizations and airload com- t_Srinivasan, G. R.. and Ahmad. J. U., "'Navier-Stokes Simulation parisons verify the method's versatility for rotoreraft prob- of Rotor-Body Flowfield in Hover Using Overset Grids," Nineteenth lems. The method efficiently obtained periodic solutions within European Forum, Cernobbio, (Como) Italy, 1993.
three rotor revolutions. Unsteady streak lines showed the _Yoon, S., and Jameson, A., "'An LU-SSOR Scheme for the Euler significant blade vortex interactions and wake roll-up. In ad- and Navier-Stokes Equations," AIAA Paper 87-0600, Jan. 1987.
_SVan Leer, B.. Thomas, J. L.. Roe, P. L., and Newsome. R. W., dition, the streak-line patterns showed that grid resolution "'A Comparison of Numerical Flux Formulas for the Euler and Na- needs improvement in the rotor wake to reduce diffusion in vier-Stokes Equations," AIAA Paper 87-1104. June 1987.
the wake. Blade surface pressures compared well with the _"Baldwin, B. S., and Lomax, H., "'Thin-Layer Approximation and flight test data, especially at the retreating side. This com- Algebraic Model for Separated Turbulent Flow,'" AIAA Paper 78- parison was somehow surprisingly better in the inboard sec- 0257, Jan. 1978.
tion. Leading-edge surface pressure irregularities point to er- _Johnson, W., Helicopter Theory. Princeton Univ. Press, Prince- rors in the original airfoil coordinate definitions. The spanwise ton, N J, 1980 and integrated load data also correlates fairly well with the eSAmirouche, F. M. L., Computational Methods in Multibody Dy- flight airload data and previous computations. namics, Prentice-Hall, Englewood Cliffs, NJ, 1992.
_"Chan, W. M., Chiu, 1. T., and Buning, P. G., "'User's Manual Although the rotor was only partially trimmed, these results for the Hyperbolic Grid Generator and the HGUI Graphical User demonstrate the method's capabilities. To trim the rotor prop- Interface," NASA TM 108791, Oct. 1993.
erly, one would need to include blade dynamics by tightly -"Cross, J. F., and Watts. M. E., "'Tip Aerodynamics and Acoustics coupling the unsteady load prediction to a suitable dynamics Test," NASA Ref. Pub. 1179. Dec. 1988.
model, which is a straightforward extension to the present :'Cross, J. F., and Tu. W.. "Tabulation of Data from the Tip formulation. This coupling would also allow for aeroelastic Aerodynamics and Acoustics Test," NASA TM 102280. Nov. 199(I.
effects such as torsion and bending. The inclusion of a fuselage :-'Lane. D. A., "'UFAT--A Particle Tracer for Time-Dependent through additional overset grids would further improve the Flow Fields," Proceedings of IEEE Visualization "94 (Washington, solutions.
DC), 1994, pp. 257-264.
AHMAD AND DUOUE 59 1.5 Figure 9 gives the surface pressure distributions at a radial 1.0" station of r/R = 0.97. Again, the solid line is the computed solution while the symbols represent flight data. The lower 0.5" surface pressures are well predicted for both advancing and 0,0" retreating sides, except at 300 deg. The upper surface ad- e vancing and retreating sides have a consistent trend. For the -0.S' advancing blade, the upper surface consistently underpredicts -1.0 the leading-edge pressure towards the leading edge. On the retreating side, the upper surface pressures compare well with -12; the experiment with some small overshoots. Hernandez and Johnson 9 found similar behavior in their predicted pressure 1.5 distributions.
1.0 At all radial stations, the calculated pressure distributions depict irregular behavior towards the leading-edge region; 0.5 especially at 90 deg in Figs. 8 and 9. A preliminary analysis 0,0 of the airfoil coordinates of the associated surface slopes and e inviscid two-dimensional airfoil calculations suggests that er- -0.5 rors in the input blade geometry's leading-edge region causes -1,0 these irregularities. The same effects appear in the results of Hernandez and Johnson, 9 who used the same input data. This -1.5 b) discrepancy needs to be corrected in future work.
Figure 10 compares the unsteady measured (dashed line) 3.O and calculated (solid line) airloads of the fifth computed rotor 2.0 1.4 r_ 1.0 1.2 ' 0.0 1.0 U c -1.0 t9 0.$ C) -2.n O.4 O.2 3.0 0.0 3eo N 180 27O a) Azimuth 1.o ,,, 1.0 • , u 0,0
l
0.0_ -1.0 O O.4 -20 0,0 0.2 0.4 0.5 0.8 1,0 O,2 d) Chord 0.O Fig. 9 Computed and measured blade surface pressure coefficient 90 1110 270 360 distributions, r/R = 0.97; K, flight; --, computation. Ik = a) 90, b) Azimuth b) 105, c) 270, and d) 300 deg.
0.0.
blade roots descend further than those from the tips. This behavior illustrates the wake roll-up process. It also gives more insight into the wake diffusion as the particles continue to spread in the far-field background grid. This region has no
intergrid boundary points; therefore, the wake diffusion is 5
inherent to the solution and can only be controlled by ade- quate grid resolution.
"0"20 90 180 270 360 The fuselage, hub, and other control surfaces have a sig- c) Azimuth nificant effect on the blade loads. Although this simulation neglects those important components, the overall load and O.6 surface pressure comparisons have fair agreement with the flight test. The following load comparisons give considerable O-4 confidence in the computations.
Calculated and measured pressure coefficients are pre-
r_ o.2
sented in Figs. 8 and 9 for different azimuthal and radial locations. Figure 8 shows the pressure distribution at the in- 0.0 board radial station 0.6. The solid line is the computed so- lution while the symbols represent the experimental value.
-0.2 Note that the scales are different for each pressure plot. To- gO 180 270 360 wards the upper surface leading edge, the inboard stations' d) Azimuth pressure distributions either overshoot or underpredict the lrtg. 10 Computed and measured blade normal force coefficients pressure, while the _, = 180-deg position compares fairly well showing the Made-vortex interaction; ---, flight; --, computation.
r/R = a) 0.60, b) 0.75, c) 0.97, and d) 0.99.
with a difference in the trailing-edge pressures.
J y
58 AHMAD AND DUOUE
..
f
,) "it
Fig. 7 Computed flow visualization using unsteady streak-line particles. Particles are released at inboard and outboard blade tip every 5 deg of azimuthal rotation. Traces shown for total five rotor revolutions. Flow periodicity is approximately established after second revolution: a) top view and b) perspective view.
1.5 moment and out of balance in pitching (longitudinal) hub moment. The important point is that the overall method is 1.0 stable, it predicts airloads that agree well with the flight test, O.5 and it lays the foundation for more complete trim computa-
o.o
tions by including an appropriate dynamic model.
Figure 6 shows one of the blade's normal force coefficient -O.5 time history at r/R = 0.97. The solid line is the computation, -1.0 whereas the dashed line is the experiment. The figure shows ¥=9_' that the normal force has an initial transient that exists for -1.5 the first rotor revolutions and then becomes periodic.
2.5 Figures 7a and 7b illustrate unsteady particle streak-line 2.0 patterns after five rotor revolutions. The unsteady flowfield 1.5 analysis tool (UFAT) by Lane 2: at NASA Ames Research 1.0 Center computed these unsteady streak lines. Rakes of par-
u
ticles were released every, 5 deg of rotation along a plane of 0.0.
points just behind each blade's tip and root trailing edge. Each -0.5' rake has a different shading to distinguish them from each -1.0- other.
Figure 7a illustrates a top view of the highly complex flow- -1.5 0.0 0.2 0.4 0.6 0.8 1.0 field caused by the wakes interacting with the blades. The Chord particles released from a leading-blade cross over the follow- Fig. 8 Surface pressure coefficient r/R = 0.60; A, flight; _, com- ing blade, showing evidence of blade vortex interaction. The putation.
particles also show evidence of excessive wake diffusion. Ide- ally, the tip vortices should maintain their core radius and intensity. The computed streakline particles remain close to process tends to degrade the solution because of the required interpolation at the boundaries and the disparate grid reso- each other initially, but when they encounter the following lutions. The number of intergrid boundary interpolations can blade they tend to diffuse. This behavior indicates that the be reduced by either changing the intermediate grid positions overset grid interpolations may be affecting the solution.
With the present grids, once the wake leaves the blade it or consolidating the two separate intermediate grids into one with more effectively distributed points.
must pass through five intergrid boundaries (blade grid to The streak-line patterns in Fig. 7 also demonstrate the rotor intermediate grid to background grid back to intermediate wake descent and roll-up. The particles released from the and then another intermediate to blade). The interpolation
AHMAD AND DUOUE 57
ing their pitch and flap motions the blade grids create holes within the intermediate grid as shown in Figs. 3 and 4.
Flight Test Conditions The flight conditions chosen as a validation case comes from the flight tests documented by Cross and Watts 2° and Cross and Tu. 2' This test was chosen because of its extensive load survey, acoustic measurements, and detailed blade harmonics data. In addition, select flight conditions have been computed by two other numerical methods. 9.'° Both methods showed good correlations with the flight test data.
The flight tests were performed with the AH-1G helicopter at NASA Ames Research Center. The rotor is a two-bladed rectangular-planform teetering rotor with the operational loads survey (OLS) symmetrical airfoil section, and a linear twist of - 10 deg from root to tip. Each blade has an AR of 9.8.
The forward-flight case chosen is test point no. 2157 re- ported in Cross and Watts.-" This flight condition has an ad- vance ratio of 0.19, hover tip Mach number of 0.65, flight Reynolds number of 9.73 x 108, and a rotor thrust coefficient Fig. 4 Blade grid with intergrid hole boundary.
equal to 0.00464. These conditions correspond to a forward speed of 82 kn at a rotor rotation rate of 315.9 rpm.
The first blade harmonics reported by the flight test were used to prescribe the rotor blade motions. Table 1 lists the blade harmonics.
Results and Discussion The results were computed on the Numerical Aerodynamic Simulation Facility's Cray C-90 computer located at NASA Ames Research Center. The time-accurate calculation im- pulsively starts from freestream conditions with the viscous no-slip boundary condition applied at the blade surfaces. The time steps for this rotor blade system correspond to approx- imately 0.3125 deg of rotation or a nondimensional time step of 0.082. Each rotor revolution requires about 1152 time steps.
The complete unsteady computation required a total of 45 h of single processor CPU time for five complete rotor revo- lutions producing approximately 40 Gbytes of flowfield data.
As noted in the previous section, the computed results used the first blade harmonics as measured by the flight test. How- ever, the blade collective was corrected to the value recom- mended by Ramachandran et al., _° which allowed them to Fig. 5 Global background grid with blade system.
match the measured value of thrust. No further trim of the rotor was attempted in the present investigation with the result that the computed thrust was about 1.8% too low. In addition, To improve interpolation, each blade grid lies within an the rotor was approximately balanced in rolling (lateral) hub intermediate grid. The intermediate grids are Cartesian with points concentrated at the blade's vicinity as shown in Fig. 4.
The grids extend from the hub rotation axis to approximately Table I Blade first barnmnics, five blade chords from the upper surface, seven chords from test point 2157 the lower, four chords in front, seven chords behind, and five chords from the tip. Each grid contains 75 points spanwise, 0,,. 0,,. O,,. _,_. B,c.
75 points chord, and 41 points from bottom to top. Figure 4 deg deg deg deg deg illustrates boundary planes within an intermediate grid with 6.0 -5.5 1.7 -0.15 2.13 the blade grid cutting through it.
= 11.19. Mr = 0.65. Re = 9.73 x 10_.and The global background grid shown in Fig. 5 completes the Cr = 0.(_64.
overset grid system. The global grid extends to four blade radii from the hub center upstream, downstream, and to the sides. The grid also extends two blade radii above the blade 0.6 ................................................................ = ............... .
and two and one-half radii below. The grid consists of a total r/R .= 0.97 i of 95 x 95 x 51 points with points clustered vertically in the 0.4 rotor disk vicinity.
.......... "! ............ !........... i........... i............. i
The entire moving overset system totals 1.62 million grid points. During the grid motions, the background grid remains stationary as the blade and intermediate grids rotate together 0.0 through it. Figure 5 highlights the relative positions of the intermediate and blade grids with respect to the background
I
grid. The intermediate grids move only in rotation about the -0.2 spin axis and subsequently create hole regions within the back- Azimuth (_r_) ground grid. The blade grids also rotate about the spin axis, but then also pitch and flap about those respective axes. Dur- Fig. 6 Time history of normal force coefficient.
56 AHMAD AND DUOUE Blade Motion The method assumes rigid blade motions in flap and pitch.
The periodic blade motion for pitch and flap as a function of blade azimuth can be described by a Fourier series '°'7 as shown in Eqs, (7) and (8): Pitch " i) 0 = 0o + 0,_ cos q, + 0,_ sin _b + 0.. cos 2_h (7) + 0_ sin 2s + --- Flap
To
/3 = t3,, + /3,_ cos _0 + /3,_ sin q, + 132.. cos 2_b
(8)
+ /3z_ sin 2s + -..
Using only the mean and first blade harmonics, Eulerian angles prescribe the blade motion to the flow solver. Euler parameters or Eulerian angles '8 are useful and convenient ways to express motion of rotating bodies in terms of the fixed inertial frame.
In this method, the blade rotates about its spin axis at some given rotational rate. At each time step, the blade rotates through by an increment of _bthat results in a change in pitch and flap. The incremental change in the blade position is then imposed by transforming the position vector through succes- sive matrix multiplications as shown in Eq. (9): T n > Fig. 2 Moving overset grid schematic. r = [A][B][C] x.¢_. = T_o,,_ (9) The transformation matrix T consists of the rotation ma- In overset grids, the quality of the solution interpolation at trices A, B, and C. The matrices A, B, and C represent the the boundaries depends on the relative grid cell ARs, skew- various coordinate rotations. See Amirouche TM for details of ness, and clustering, etc. It also depends on the bounda_'s the transformation matrices.
proximity to high flow gradient regions. While forming the grids, one must ensure that the boundaries have adequate Grid System overlap, nearly equal cell ARs, and low skewness. Boundaries The grid system consists of five overset grids; one for each should also avoid high flow gradients.
rotor blade and one intermediate transition grid for each blade With moving overset grids, individual grids move with their to help connect them to the single global background grid.
appropriate grid motion. As the grids move, the holes and The rotor grid is of C-H topology with clustering near the tip, hole boundaries change with time. Figure 2 provides a sche- root, and leading and trailing edges. Beyond the tip and in- matic for a helicopter blade in rotation, with one blade grid board of the root chord the surface grid collapses to a slit and omitted for clarity. As the blades rotate from some time state forms a singular plane that extends approximately one chord T,, to another time T,, the grid attached to the blades rotate length beyond either side. Each rotor blade grid has 119 total along with them. Subsequently, the holes change with the chordwise points and 45 points in the spanwise direction that blade rotation as shown.
lie on the body. In the normal direction, the first grid point To determine the grid's changing connectivity and hole points, is 0.00003 chords off the body and extends approximately 0.75 the code known as domain connectivity functions in three chord from the surface in all directions. The hyperbolic grid dimensions (DCF3D) by Meakin '2 was employed. DCF3D generator by Chan et al. '9 generated the resulting volume grid uses inverse mapping of the computational space to limit the illustrated in Fig. 3.
search time and to compute hole and outer-boundary inter- polation stencils. The major expense in DCF3D is the creation of the inverse maps. However, the maps are independent of the relative orientation of the grids and so it repeatedly uses the maps during the grid movement.
During the flowfield solution process, intergrid boundaries are constantly created due to the grid movement. After each flow solution time step, grid connectivity data has to be de- termined. Currently, the flow solver (Chi-TURNS) and the connectivity algorithm (DCF3D) are separate Fortran codes.
A UNIX C-Shell script is used to join the two methods.
With each time step, DCF3D obtains the latest connectivity data and hole points. Relevant intergrid boundary data are written to disk. The hole-fringe points are then checked to Hole Boundary ensure that they are introduced to the solution gradually. The flow solver then begins the solution process by reading in the DCF3D files from disk, solving the equations of motion, and then moving the grids to the next time step. Once complete, the solution and grids are written to disk and the whole process continues until the desired number of time steps have been Fig. 3 Blade grid cutting through intermediate grid.
completed.
AHMAD AND DUOUE 55 Back round Grid steady thin-layer Navier-Stokes equations on moving overset grids. To accomplish this calculation, he developed a method
iiiiiiai , iii iiiiiiii
that determines the interpolation coefficients between the overset grids. His domain connectivity algorithm (DCF) was verified in detail in his work. But the set of flight conditions he considered were hypothetical and did not include the dy- namic pitch, flap, or lag motions of the rotor blades. The method presented here accounts for the blade motion by ap- propriately moving the overset grid system according to the -- Hole Fringe Points blade harmonics measured in a flight test.
In summary, a numerical method that computes the un- Airfoil Outer Boundary Poims steady flowfield of a helicopter rotor in forward-flight or hover Fig. I Schematic of overset grid system.
is presented. The method solves the thin-layer Navier-Stokes equations on a system of moving overset grids and uses the blade harmonics measured in flight to prescribe the blade The matrices L and U are formed by performing either motions. Comparisons between computed and flight-test un- backward or forward differences on the appropriate flux Ja- steady blade loads and surface pressures of the AH-1G heli- cobians A, B, or C as shown in Eqs. (3). In Eqs. (3), At is copter are presented. Numerical flow visualizations exhibit the time step, A_ and V_ represent forward and backward flowfield features such as blade-vortex interaction and wake differences, respectively, 1 is the identity matrix, and _ is the roll-up. This article provides details of the method, a discus- spectral radius. The matrix D is a diagonal matrix that com- sion of the resulting predicted flowfield, and demonstrates pletes the back-solve process. This decomposition scheme re- the capability to model the unsteady flowfield of an arbitrary duces the required floating point operations in comparison to helicopter rotor in forward flight.
block tridiagonal methods: Methodology L = I + IB_.k._At(-A[,j + V_A2 - Bj_k., + V.,,Bg Algorithm
- CA., + v,C:)
The single-grid flow solver developed by Srinivasan et al. 2 with overset-grid modifications implemented by Duque and D = I + IB_.,.tAt(A2 + By + (72 + A2 Srinivasan t was employed in the study. Calculations by Sri-
+ B. + + C;),._., (3)
nivasan and Ahmad _3 have demonstrated the method's ver- satility by computing the hovering flowfield of a rotor and U = I + IBj.k.tAt(AT._. , + A_A2 + BT.k., whirl stand. The method solves the thin-layer Navier-Stokes + A.B2 + CT.kj + A_C2) equations shown in Eq. (1): whereA-" = A -- oz.
d Q + 0 op+ o G = Re_, 0 With overset grids, the additional array IB marks the hole
0t _ _ + 0,7 0_ _ _¢ (a)
regions within the grid interior by taking on the value of 0.
Outside of the hole and in the valid regions of the flowfield where Q = J-z(p, pp., pv, pro, e) is the vector of the conserved the IB array equals 1. The IB array removes the hole regions variables, density, momentum: and energy, scaled by the from the solution set as shown in Eqs. (2).
transformation Jacobian, and E, /_, G, and S are the scaled The flux terms use a Roe upwind-biased scheme for all inviscid and viscous flux vectors.
three coordinate directions with higher-order MUSCL-type With overset grids a sequence of grids are placed such that limiting to model the shocks accurately? 5 The resulting flux they lie arbitrarily within a primary grid. For example, Fig.
differences are shown in Eq. (4): 1 provides an example where an airfoil curvilinear grid lies within a background Cartesian grid. The airfoil grid captures = V_E2 + A eE2 + V.F 2 + A.F2 + V_G 2 + A_G2 features such as the boundary layers, tip vortices, and shocks,
(4)
etc. The background grid surrounds the airfoil grid and carries the solution to the far field. The background grid was gen- At the hole fringe points the flux evaluations reduce to erated with some knowledge of the airfoil's surface and outer second order at the resulting interior boundaries by modifying boundary locations. Consequently, some of the background the primitive variable evaluations with the IB array. Equation grid points lie within the airfoil's solid body regions and must (5) shows the _'-coordinate flux evaluations: be removed from the solution. Once removed, hole regions remain within the interior of the larger background grid and
-#(QL, QR) = '}-[£'(QR) + E(QL) - IA(Q_, QR)[(Q, - Q,.)]
create a set of boundary points known as hole-fringe points.
(5) The airfoil grid interpolates data to the background grid at the background's hole-fringe points. Conversely, the back- ground grid interpolates data to the airfoil grid at the airfoil The primitive variable quantities Q are obtained by first- outer boundary points. order extrapolation functions at boundaries or third order away from boundaries. As shown in Eqs. (6), these quantities Within each separate overset grid, different flow solver drop to first order through the use of the IB array: methods may be used to solve the governing equations of motion. The current method uses an implicit upwind method for all of the individual grids. The method advances the so- QL = Q_ + ['(Qj - Qj-,) + I(Qj+, - Q,)llBj.,IBj_, lution in time using the lower-upper symmetric Gauss-Siedel QR = Qj÷, - [_(aj÷, - Qj) + l(aj÷, _ Q,.,)]IB,_,IBj+, (LU-SGS) implicit scheme by Yoon and Jameson 14 shown
in Eqs. (2). The scheme is third-order accurate in space and (6)
first-order accurate in time: Finally, the flow solver assumes fully turbulent flow for the Q* = - IBAtL- _t blades and inviscid flow in background grids. The simple al- LD-tUAQ = -IBAt_ s°'v_"-------Lg Q** = DQ* (2) gebraic turbulence model of Baldwin and Lomax _6 is used to AQ = U-'Q** estimate the eddy viscosity.