Applications of Computational Methods for Dynamic Stability and Control Derivatives

AIAA 2004-0377

Annlications of

- -1- r - - - -- -- - - - - --

Computational Methods for

Dynamic Stability and

Control Derivatives

Lawrence L. Green

NASA Langley Research Center

Hampton, VA

Angela M. Spence

Mississippi State University

Mississippi State, MS

42nd AIAA Applied Aerospace Sciences

Conference and Exhibit

January 5-8, 2004

Reno, NV

AIAA 2004-0377

APPLICATIONS OF COMPUTATIONAL METHODS FOR

DYNAMIC STABILITY AND CONTROL DERIVATIVES

Lawrence L. Green * NASA Langley Research Center, Hampton, V A Angela M. Spence-- Mississippi State University, Mississippi State, MS ABSTRACT Because of their dynamic capabilities, these S&C Initial steps in the application of a low-order facilities may be expensive to build, operate. andlor ............. ,.,.1 ....... , ................ r1 ,........,.TY\.."l1t .... t;",., .... l fll11rl rl"n':UTI1 .. c (r~n\ pU.I"". 111,",l'I\.J~ ............ 11.1-''-&'""'".\. ..... ". II\"&.~ -,J .I"IIII~U \ _ .. A..-I maintain. The range of motions that can be performed code to the calculation of aircraft dynamic stability and in such facilities may be limited by the wind tunnel control (S&C) derivatives are documented. Several walls or by the test rig's kinematic and vibrational capabilities, unique to CFD but not unique to this restrictions. In addition, many conventional dynamic particular demonstration, are identified and test facilities do not provide the capability to determine demonstrated in this paper. These unique capabilities damping and cross derivatives individually; instead, complement conventional S&C techniques and they these facilities can onll provide measurements of include the ability to: 1) perform maneuvers without combined derivatives. The measurement of these the flowlkinematic restrictions and support interference quantities in pairs is not the preferred situation; it is commonly associated with experimental S&C simply a kinematic reality of the facilities available. A lO facilities, 2) easily simulate advanced S&C testing companion paper details several computational techniques. 3) compute exact S&C derivatives with techniques to separate such paired dynamic derivatives.

uncertainty propagation bounds, and 4) alter the flow Some facilities are not equipped with a "slip ring" physics associated with a particular testing technique capability and thus, cannot perform continuous rotary from those observed in a wind or water tunnel test in motions; instead they rely on oscillatory motions to order to isolate effects. Also presented are discussions provide brief periods of steady rotational motion.

about some computational issues associated with the The difficulty of many experimental S&C simulation of S&C tests and selected results from facilities to adequately model the flight characteristics numerous surface grid resolution studies performed of today's aircraft over the entire flight envelope, the during the course of the study. cost of developing better S&C testing facilities, and the recent advent of novel morphing aircraft ll l3 configurations - all contribute to the difficulty of INTRODUCTION l2 29 Aircraft stability and control (S&C) predicting S&C derivatives. Prior research efforts - derivatives quantify the aerodynamic forces and using CFD to predict S&C derivatives have made some moments used to model aircraft dynamics. S&C progress toward bridging the capability gap, but these l5 derivatives are used to predict, for example, the efforts are very time consuming and have failed to longitudinal short period, lateral pure roll, lateral Dutch make a significant impact in the day to day business of roll, spin behaviors, and handling qualities sensed by S&C prediction. Because both CFD and experimental · ~o ~l h h' . d' d pilots for a given configuration. Historically, aircraft stu les' .. ave t elr umque lsa vantages, aerospace d static and dynamic S&C derivatives have been companies spend millions of dollars fixing S&C determined through wind tunnel tests, water tunnel problems discovered during certification flight tests or l s tests, or with empirical formulations - based on prior production use. The current situation for S&C tests. However, most of the wind tunnels and all of the prediction is similar to the situation seen before CFD water tunnels used for S&C tests operate at low free methods were widely used to accurately predict stream velocities and Reynolds numbers. These aerodynamic performance.

limitations restrict the range of flight conditions that CFD offers capabilities and techniques that complement experimental S&C testing techniques for can be adequately simulated.

Many S&C testing facilities provide unique obtaining S&C derivatives. With CFD, support and dynamic capabilities that are not typically available in the wind tunnels used for aircraft performance tests. - Senior Research Scientist, Multidisciplinary Optimization Branch, MIS 159, Senior Member AIAA This material is declared a work of the U.S. .- Engineering Co-op Student, Multidisciplinary Government and is not subject to copyright protection Optimization Branch, MIS 159, Student Member in the United States. AIAA AIAA 2004-0:177 wall interference effects can be either modeled or develop computational methods to predict the S&C eliminated in order to assess their impact. Because the derivatives of interest; and create a long-term plan for vehicle orientations and movements within the flow synergistic usc of experimental facilities, testing can be arbitrarily specified to obtain time-dependent techniques, and CFD methods to close the gap between solutions, CFD can easily simulate advanced S&C CFD and experimental S&C activities. This paper is testing techniques (possibly including those for which among the first to be published under this new effort.

no testing facility currently exists). By applying Results presented in this paper are relevant to I2 16 various differentiation techniques - . 32. 33 to CFD furthering S&C applications of CFD. The results arc codes, researchers can obtain exact derivatives of the computed by using a low-order panel method code on a simplified F-16XL lighter configuration. This work forces and moments throughout the time history of a maneuver. CFD can independently eliminate or refine represents the first of many steps needed in the various physics features within a flow to isolate application of CFD to S&C problems. The particular effects. Finally, CFD can help explain the configuration was chosen because it has a wide variety causes and types of separated flows affecting S&C of measured data available. The panel code was chosen, among other reasons, for its quick run time, prediction.

ease of use, extensive motion capabilities and its ability In theory, applying CFD to S&C prediction is not only feasible, but straight forwardl; one would to represent three-dimensional bodies within time- simply grid the vehicle of interest and perform the dependent flows.

required calculations. In practice, many issues must be In some ways, the use of a low-order panel addressed before CFD can accurately predict S&C method CFD code to compute the forces and moments derivatives, including: of a highly swept configuration like the F-16XL at high angles of attack is a worst-case scenario. The vortical • the cultural differences that exist between the and viscous effects, which dominate the F-16XL CFD and S&C communities, evidenced by differences in language, notation, and flowfield, at high angles of attack, are not modeled in the code. The authors of this paper certainly do not expectations of the two groups advocate the uninformed use of a low-order panel • the significant increase in computer resources method code for the computation of viscous, vortical, (execution time, memory, and disk space) bursting-vortical, or separated flows. However, the required to address the aircraft S&C derivative 1S code was used anyway because of the great potential to needs . 16 beyond the substantial resources preliminary design that could result from a successful already required for standard aircraft CFD application. The focus of this paper is to demonstrate performance calculations several computational techniques that advance the • the need for uncertainty quantification of S&C ability of CFD to compute S&C derivati ves; this paper derivatives used in flight simulations and also raises a number of significant issues associated multidisciplinary design efforts and the further 34 38 with S&C computations. The authors do advocate increase in computational resources - using the appropriate level of physics fidelity with associated with computing S&C predictions respect to the problem posed, which can only be with uncertainty included assessed by examining all available levels of fidelity.

• the absence of a set of acceptable test cases Producing quantifiable and justifiable error bounds on that the CFD community must solve and the the accuracy of results is a critical area that must be degree of accuracy required for CFD to addressed when using CFD to predict S&C derivatives.

become a credible alternative to wind and This paper demonstrates a grid resolution study and water tunnel testing uncertainty propagation techniques that can be used as To address these concerns, a team was a model for providing error bounds in the future. Error recently formed at the National Aeronautics and Space bounds help identify the limitations of the current Administration (NASA) Langley Research Center method; eventually, researchers will be able to (LaRC). The Computation Methods for Stability and compare the error bounds from all levels of fidelity and Control (COM SAC) team includes key researchers in determine the required fidelity for any S&C the CFD and S&C communities devoted to developing application.

and applying CFD methods for S&C needs. A recent workshop hosted by the COMSAC team devoted to TECHNICAL APPROACH this topic was well attended, and the topic gained broad This study involves the application of the support from the aerodynamic community. The automatic differentiation of FORTRAN (ADIFOR) COMSAC team is currently examining many research tool, version 2.0D,40. 41 to the Panel Method Ames endeavors in this arena. 12-29 COMSACs mission is to: Research Center (PMARC) code,42 version 14.10.

understand the causes and types of separated flows ADIFOR 2.0D was applied to PMARC by using a wide affecting S&C prediction; usc this understanding to AIAA 2004-0377 variety of input variahles as potential independent Computational Fluid Dynamics Code variahles of differentiation. The PMARC body and The demonstrations presented here employ the wind axis forces and moments were used as the PM ARC code,4:! a low-order panel method CFD code dependent variables of differentiation. Since that was modified by applying AD to enable efficient uncertainty estimates of all S&C parameters (forces computation of exact first and second force and and moments and their derivatives) are normally moment derivatives with respect to a wide variety of requested by most control law designers, the code inputs. This panel method CFD code was chosen computation of both first and second derivatives with for its fast execution and moderate computational respect to selected independent variables of resource requirements, ease of use, ability to simulate a differentiation are demonstrated; the latter are used to wide variety of steady and time-dependent rotary and enable the propagation of known or assumed input oscillatory aircraft motions, and its amenahility to processing by AD to compute derivatives that are exact variable uncertainties through the code to determine the uncertainty effects on the computed forces and to the formulation accuracy of the code solution.

Second derivatives have been included to allow for moments. The relationship for uncertainty .16 propagation . .18 as a function of normally distributed. input variable uncertainty propagation through the code random variables is with respect to the output forces and moments and their first derivatives. From the combination of the above mathematical techniques and the panel method choice, j a demonstration of the computation of aircraft dynamic

(Yj = i(dF (Y;j,j=J,n

S&C derivatives, with propagated uncertainty hounds, ;=1 dx; could be rapidly performed. Sample results are

m = number of random inputs

presented for the F-16XL fighter configuration modeled with a coarse surface grid, shown in three-

n = number of output functions (1)

view in Figure I.

(Yi = standard deviation of

The PMARC,42 version 14.10 potential flow solver is a FORTRAN 77 code that can compute the random inputs surface pressures, forces, and moments of arbitrary shapes. The code assumes inviscid, irrotational, and In this case, the output function F may be any of the incompressible flow. Boundary layer and compressible computed body or wind axis forces and moments, or corrections are available but are not implemented in their first derivatives with respect to parameters such as this study. PMARC can also compute solutions of the angle of attack. angle of sideslip, and the steady unsteady, time-varying flow conditions for a wide rotational rates. Original and ADIFOR-generated variety of aircraft motions. The original PMARC code PM ARC code versions were executed for an array of has been modified to allow for the input of a free oscillatory and rotary aircraft motions to compare with stream Mach number, angle of attack, and angle of data generated for a 10% scale F-16XL model tested sideslip in degrees, rather than simply the input of free in the NASA LaRC 14- by 22-Foot Subsonic Tunnel.

stream velocity components. The code was also Other data from wind tunnel (18% scale model)43 and modified to allow ADIFOR processing by replacing the water tunnel (2.5% scale model)44. 45 tests is also use of scratch 110 files with common block access, and available, but was largely ignored for this study. The by inserting a few lines of code into the program to computational F-16XL geometry, as shown in Figure I, identify independent and dependent variables of was modified from that of the actual aircraft. The differentiation.

engine inlet was faired over with a smooth, bulbous The input file to the program uses a set of grid surface and some detail near the aft end of the aircraft points describing the shape of the geometry as a set of was simplified; this same geometry was used in the panels. In this study, both the right and left halves of a water tunnel tests. Limited comparisons with modified F-16XL configuration are modeled in the 9 43 experimental data . are presented in this paper. Other PM ARC input geometry file. The gray-scale in Figure configurations, including a generic transport wing- I reflects different surface patches in the PMARC body. a Boeing 747 wing-body-pylon-nacelle geometry input file, however these patches cannot be configuration, and an F/A-18 AlB fighter were also readily identified from the figure. The surface executed during the course of the present studies; the resolution, with 984 surface points and 566 surface results of these studies may be the subject of a follow- panels, is moderately coarse by CFD standards, yet the on publication. Only F-16XL data is shown in this

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vehicle is still clearly identifiable. Higher surface paper.

resolution grid files were available and used. Several such studies were performed and are discussed in the sections "Grid Resolution Studies" and "Results".

AIAA 2004-0377 PMARC allows the user to descrihe only half the functions evaluations (i.e .. sine or cosine). By aircraft and mirror the solution in the x-;: plane. repeatedly applying the chain rule of differential However. this technique does not capture the effects of calculus to the composition of those elementary a nonzero angle of sideslip. which are of interest in the operations. derivative information can he computed S&C arena. The input file also specifies the flight exactly and automatically.

condition. certain algorithmic parameters. and the user- The two approaches for computing derivatives defined position of the reference point about which all with AD are the forward mode and the reverse mode.

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moments are summed. The forces and moments are The forward mode applies the chain rule of also nondimensionalized with a user-specified differentiation to propagate, equation hy equation.

reference area. mean aerodynamic chord length. and derivatives of intermediate variahles with respect to the input variables. In contrast, the reverse (adjoint) mode wingspan.

The PMARC code includes a wide variety of propagates, in reverse through the program, the derivatives of the output variables with respect to the possihle aircraft motions: input variahles. The forward mode is hetter suited to • straight translation with imposed angles prohlems with fewer input variables ~han outpu~

of attack ( a) and sideslip ( f3 )

variahles, whereas the reverse mode IS hetter sUited to • pure rotary motions about the three axes prohlems with fewer output variahles than input described as constant rotational rates (p, variables. Many hybrids of the forward and reverse q, and r) modes are possible, with complementary tradeoffs in • planar oscillatory motions in hoth required random access memory (RAM), disk space, translation and rotation about each of and execution time.

three axes. each described hy an The forward method of AD is implemented in amplitude and frequency the ADIFOR, version 2.0D . 41 tool. The reverse mode • coning motions (illustrated suhsequently) and some second derivative capability are also used within advanced experimental S&C implemented in the ADIFOR, version 3.0. Both tools facilities were developed jointly by the Center for Research on In contrast to the limitations of some experimental Parallel Computation at Rice University and the facilities, coning motions can be simulated within Mathematics and Computer Sciences Division at PM ARC with either continuous (i.e., a "slip ring") or Aroonne National Laboratory. Both tools have been limited roll motion capability (oscillatory test motion).

us:d with the PM ARC code, but only results using the In addition. the authors also use two other ADIFOR 2.0D tool are shown in this paper. In general, capabilities of the CFO code that are not general~y ..

to apply ADIFOR to a given FORTRAN 77 code, the available to S&C experimentalists. These capabilities user is only required to specify the independent and are: the choice of either rigid or flexible wakes, and the dependent variable names of the target differentiation ability to somewhat arbitrarily prescribe the time and a differentiation starting point in the program. The stepping characteristics of the solution. Although AD tool then determines the variahles that require flexible wakes are preferred because they more derivative computations. It formulates the appropriate accurately model the flow physics involved, the rigid forward or reverse mode derivative expressions for wakes were more robust for some of the motions of these variahles and generates new FORTRAN 77 code interest and also lend themselves better to flow for the computation of both the original simulation and visualization. Wakes are currently attached to the wing the associated derivatives. The ADIFOR-generated trailing edges and the upper vertical tail trailing edge.

code is compiled (with special libraries) and executed The time stepping characteristics of a given solution like the original code.

can be defined to match the data sampling rate of a The forward mode of differentiation was used particular S&C facility, if desired. A caveat of this to compute the S&C derivatives because generally, technique is that the size of the shed wake panels in the only a few input, or independent, variables are PM ARC solution is proportional to the chosen time identified for differentiation within a given code step, which alters the flow physics to some extent if generation (compared with 12 output, or dependent, sufficiently small time steps are not used.

variables). The reverse mode of AD may be employed to calculate the S&C derivatives for morphing vehicles, Automatic Differentiation as in References 12 and13, where the potentially Automatic differentiation . 41. 46 (AD) is a thousands of independent variables (perhaps, each technique for derivative computation. AD relies on the surface grid point, along with the 30 or so flight fact that every function, no matter how complicated, is condition and orientation variahles) would greatly executed on a computer as a (potentially very long) outnumber the 12 dependent variahles.

sequence of elementary arithmetic operations and AIAA 2004-0377 Grid Resolution Studies 3. The Y -Z coordinate directions of the Three assumptions were made at the start of FINEST_ VOL2 grid were swapped to this work: I) a panel method CFD code with a coarse conform with PMARC requirements.

surface grid could be used in conceptual or preliminary 4. Alternate grid points in each of the I-J-K design whereas Euler I Navier-Stokes computations are coordinate directions were removed to likely to require so much execution time as to be produce another volume grid incompatible with conceptual or preliminary design, (MEDIUM_VOL, details presented in 2) the surface grid shown in Figure I was probably too Table I).

coarse to be accurate but represented a good starting 5. Again, alternate grid points in each of the point, and 3) grid resolution could be increased at any I-J-K coordinate directions were removed time, once the computational processes were to produce yet another volume grid established, to smoothly increase the accuracy of the (COARSE_ VOL).

results. It remains to be seen if the tirst assumption is Surface grids consisting of 56 patches were extracted valid. However, as illustrated in Figure 2 of the as subsets from each of the volume grids as follows: Results section of the paper, the second assumption is 1. A surface grid containing 113.856 grid valid; this surface grid inaccurately approximates even points and 108,736 grid panels the most basic of S&C parameters, such as the lift (FINEST_SRF) was obtained from the variation with angle of attack, even at low angles of FINEST_ VOLl grid.

attack where a panel method formulation should be 2. A reduced surface grid containing valid and vertical was not expected to affect the MEDIUM_SRF was obtained from the computation. This led the authors to, among other MEDIUM_VOL grid.

things, systematicalIy examine the surface grid 3. A further reduced surface grid resolution requirements through a variety of studies, (COARSE_SRF) was obtained from the essentialIy testing the validity of the third assumption. COARSE_VOL grid.

A considerable amount of processing and 4. Individual I-and J-grid lines from the analysis was involved in obtaining the surface grids to COARSE_SRF grid were selected to be be used for the current surface grid resolution studies, retained through two further sessions of which were obtained as subsets of volume grids. The coarsening (COARSE_SRF2 and F-16XL volume grid history is as folIows: COARSE_SRF3, respectively).

I. The first grid was a 30-block, structured, All the above surface grids required some I-J half-configuration volume grid containing reordering of grid lines to conform with the PMARC 1,502,138 grid points (FINEST_ VOLl); geometry convention for "correct side out" patches.

it was taken from previous Euler studies The majority of runs performed during this study used of References 39 and 43. Preliminary the COARSE_SRF3, I-J surface grid, or variations of analysis using an earlier version of this grid (described subsequently). Details about alI the PMARC and an EulerlNavier-Stokes volume and surface grids are presented in Table I.

code was performed with this grid, but it Six types of surface grid resolution studies provided unsuitable convergence for were conducted with the PMARC code. The studies Navier-Stokes solutions in studies related used: to References 15 and 16 and was I. ExternalIy generated grids for a therefore abandoned. symmetric, unswept, untapered, parabolic 2. A new 36-block, structured, fulI arc wing with an aspect ratio of two; configuration volume grid containing alpha = 10 degrees (37 cases executed).

6,293,908 grid points (FINEST_ VOL2) 2. Externally generated grids for was developed for a modified F-16XL. configurations similar to the F-16XL An engine inlet fairing was added and tip wing plus fuselage carry-through missiles and launchers were removed. (COARSE3_SRF) with parabolic arc

This configuration had improved grid wing sections; alpha = 10 degrees (45

resolution above the wing upper surface cases executed).

3. Internally generated grids based on the to enable users to capture votical flow over the upper wing surface with original F-J6XL configuration (COARSE3_SRF) input to PMARC, EulerlNavier-Stokes computations.

Studies using this grid have not yet been modified through input choices to

PMARC; alpha = 10 degrees (34 cases

performed.

executed).

AIAA 2004-0377 4. Externally generated grids starting from flow feature. However, several attempts were made to higher resolution surfaee geometries simulate the effects of vortical flow hy attaching (FINESLSRF, MEDIUM_SRF, and additional wakes to the upper wing surface ncar the COARSE_SRF): alpha = 10 degrees (3 leading edge and/or over the mid-chord of the wing.

cases executed or attempted). Two such attempts are shown in Figure 3; the figure 5. Externally generated grids for includes the same lift force coefficient data as in Figure configurations similar to the F-16XL 2, hut with two additional curves representing the wing plus part of the fuselage with attempts to add additional wake to the wing upper surface, indicated with dashed lines. Indeed, the data parabolic arc wing sections: alpha = 30 degrees (22 cases executed). comparisons could he improved at high angles of attack (30 to 50 degrees) hy adding wakes separating 6. Externally generated grids for configurations similar to the F-16XL from the mid-chord of the wing. However, such results required knowing either where the separation wing plus part of the fuselage with parabolic arc wing sections: small might occur or what value of the lift or normal force amplitude forced pitch oscillation about coefficient was to he ohtained. The same technique alpha = 0 degrees (30 cases exeeuted). significantly overpredicted the lift and normal force coefficient at 10 to 20 degrees angle of attack, The cases in group 4 failed to produce any useful results: cases in groups 5 and 6 produced results similar indicated hy the dashed curves extending off the top of to the cases in group 2. Only the results from groups I, the figure. This technique would need to he tailored for 2, and 3 are reported in this paper. The surface and specific flow conditions. The cases should he repeated volume grid history/usage is summarized in Table I. with both Euler and Navier-Stokes codes to truly assess the impact of vortex formation and bursting.

RESULTS The effect of uncertainty propagation through As mentioned previously, Figure I shows a the CFD code was also examined; the same cases three-view of the computational geometry of the shown in Figure 2 were executed with an ADIFOR- F-16XL used for most of these calculations: the gray- generated Hessian (second derivative) code. For this scale indicate different surface patches in the PMARC case, derivatives of the forces and moments and their geometry COARSE3_SRF input file. The surface first derivatives with respect to angle of attack were resolution, consisting of 984 surface points and 566 differentiated again with respect to angle of attack.

surface panels, is moderately coarse by CFD standards. Uncertainties with respect to the angle of attack were Results are first presented for the static forces and computed and propagated through the code, moments, then for dynamic situations. Figure 2 shows proportional to variable gradients with respect to the a comparison of computed lift, pitching moment, and uncertain variable, via Equation 1. The results from normal force coefficient data with measured data . 43 this study are shown as the error bars in Figures 3 and using this geometry. Circles in the figure are the lift 4 and in Tahle 2. Figure 3 shows the computed lift force coefficient, triangles are the pitching moment coefficient data of Figure 2, with added error bars coefficient, and squares are normal force coefficient; showing the uncertainty in C due to a one-sigma L open symbols represent the computation and solid uncertainty (for illustration purposes) in the angle of symbols are the data from Reference 9, as well as the attack (AOA) for an assumed uncertainty in the solid diamonds representing the lift force coefficient measured angle of attack given by CT = 0.1 degrees, AOA from Reference 43. The computed data reflects the determined from the measured data. Figure 4 shows general character of the measured data, but the the lift curve slope, C , for the data in Figure 2, with La computed data differs significantly with both sets of added error bars showing uncertainty due to a one- measured data. It was assumed that most of the sigma uncertainty in the angle of attack. Figures 3 and differences in the lift and normal force coefficient 4 indicate that the propagated uncertainties due to computed/measured data comparisons could be uncertainties in angle of attack account for some, but attributed to: I) at high angles of attack, the inability of only a small part, of the differences in the computed the PMARC code to model the vortical and viscous and measured lift coefficient. The data shown in Table flows, 2) uncertainty propagation due to uncertainties 2 illustrates a two-variahle (angle of attack and pitch in the angle of attack. and 3) the surface resolution rate, Q) uncertainty assessment. The table shows the used in the PMARC computations for all angles of variable name, the uncertain variable name, the attack. Minor differences between the tested geometry computed output variable sigma (from Eq. 1), and one-, and the computational geometry may also contribute to two-, and three-sigma (representing 84.1, 97.7, and the differences in the forces and moments.

99.9% , respectively) low/high uncertainty hounds for Vortex-generated lift is known to occur over each variable, as well as the mean value for each the wing,39 and the PMARC code cannot model this AIAA 2004-0377 variahle. Table 2 illustrates a typical preliminary of magnitude greater than either the lift force or uncertainty analysis for the F-16XL configuration. The pitching moment coefficient. Figure 7 shows the same lift force coefficient data as in Figure 5, hut plotted . . . C dC L d dCI ( . h .

uncertaInlies In L' --, an --' WIt a In with curves of constant chordwise resolution (varying da dq span wise resolution) around the airfoil section.

degrees and q in degrees per second) are presented as Comparing Figures 5 and 7, the minimum lift force calculated from first and second derivatives of C with boundary in Figure 5 is simply a composite of all the L solutions with only four panels chordwise. In contrast, respect to a and q, assuming (Tin!'u/ = 0.1 for hoth a the maximum boundary in Figure 5 is a composite of and q. The same capahility is already availahle for all the leftmost points on the curves in Figure 7 (those of the forces and moments in both the wind and body point with minimum span wise resolution for a given axis systems; results can he easily computed for any of chordwise resolution). A similar hut inverse behavior the independent variabies of interest.

was observed for the pitching moment. These results suggest. as expected, that four panels chord wise The final clement contrihuting to the (around the airfoil) are too few; the resulting wing discrepancy between measured and computed data in sections are modeled only as a composite of forward Figure 2 was the effect of surface resolution. A series and rearward facing wedges, resulting in large errors in of grid resolution studies, described previously. was the predicted forces and moments. More panels performed. The results of these studies are shown in chord wise appear to smoothly and significantly Figures 5 through 10. Figure 5 shows an envelope of improve the fidelity of the PMARC solution. Put computed lift coefficient data of various fidelities for a another way, the results only with only four panels symmetric. unswept, untapered. parabolic arc wing chordwise never asymptotically approach the expected

with an aspect ratio of two, where alpha = 10 degrees

value with increased surface resolution for the cases (group I grid resolutions studies) at various surface studied. A discontinuous approach to the analytic resolutions, plotted as a function of the Log 1 0 (log solution is observed; the surface grid resolution must base 10) of the number of surface panels. The be "good enough" to begin an asymptotic approach to envelope of solutions includes data from 37 cases analytic value. A similar conclusion was reported in (group I grid resolution studies) with the total surface Reference 37. The F-16XL grid used for most of the (full span) resolution ranging from 8 to 4096 panels studies has 10 panels chordwise around the airfoil increasing in even multiples of 2, with distributions section and 8 panels span wise for each wing semispan; ranging from 2 to 256 panels chordwise, and 2 to 64 Based upon this result alone, one might expect the grid panels span wise, although not all possible shown in Figure 1 to be sufficient to accurately combinations within that range were executed due to approximate the measured results, which not the case.

time and memory limitations on the available Figure 8 shows an envelope of lift coefficient computers. Also shown in Figure 5 is an analytic data for wing shapes similar to the F-16XL plan form computation of the lift coefficient taken from (group 2 grid resolution studies, 45 cases). Behavior Reference 2. The envelope of computed lift coefficient similar to that seen in Figure 5 is observed, but grid data demonstrates grid independence at dense surface independence is not achieved to the same degree. The resolutions (thousands of surface panels); the data drag force and pitching moment data envelopes for become both independent of the total number of these cases are similar to those of Figures 6. More surface panels and their distributions span wise and panels, above a certain minimum threshold chordwise, chordwise, as well as agreeing well with the analytic improve the accuracy of the computed data. However, solution. These results provide great evidence that the an important limiting effect in accuracy due to the PMARC code can produce useful results for some more complex F-16XL wing geometry, compared to geometries and flight conditions at sufficiently dense the swept tapered wing of Figures 5-7, is observed in surface resolutions.

Figure 8.

Figure 6 illustrates the envelopes of data for Figure 9 shows a scatter plot (rather than an the drag and the pitching moment coefficients. Similar envelope of data) for the 34 cases of group 3 grid to Figure 5, grid independence is observed in drag resolution studies. In this study, the total number of coefficient. The apparent lack of grid independence at surface panels ranged from 504 to 2486, with the dense surface resolutions in the pitching moment is nominal grid containing 566 panels. The distribution simply an artifact of the magnified vertical scale of the panels varied from run to run, with surface

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compared with Figure 5. Nearly the same level of grid resolution increasing either over the whole vehicle, independence is achieved for both the lift force and only the wings, or select regions of the wings. This was pitching moment coefficient data. The convergence a systematic study performed with the goal of matching achieved in the drag coefficient is actually two orders the lift coefficient at 10 degrees angle of attack from AIAA 2004-0377 Refen:nce 9. Many executions beyond those 34 the body rotates about the velocity vector. which is reported here were performed that resulted in true aligned with oncoming flow stream. Figure 14 "outlier solutions" with unreasonably large lift illustrates an oscillatory coning (or, inclined axis roll coeflicients (greater than 2.0); the same can also be OAR» motion where the body rotates around an axis

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said of the previous two grid resolution studies, not aligned with the velocity vector. The motion is although more "outlier solutions" occurred is the named for the oscillatory behavior of the angles of studies associated with Figure 9. Also shown in Figure attack and sideslip observed during the motion. These 9 are the values of the lift coeflicient obtained from types of motions can only be performed in a few wind References 9 and 54. Most of the computed results in and water tunnels globally. Figures 15 and 16 illustrate the CFD simulation of these two coning motions. In Figure 9 fall outside of the range of the two reference solutions, and no grid independence, nor any approach Figure IS, the original geometry is rotated to an initial pitch orientation angle, here 30.8 degrees, before the to the expected value. is discernable. Taken with previous results. this suggests that surface grid CFD solution is performed. A body axis roll maneuver is then performed with the rotated geometry and with smoothness, in addition to the surface grid resolution and distribution, plays a big role in the accuracy of the the velocity vector aligned with the rotational vector, as above in Figure I I. The motion could also be PMARC results for lift coeflicient. In contrast, as shown in Figure 10, nearly any grid distribution or performed with an unrotated body and imposed angle of attack. but the body-axis forces differ between the resolution may predict the expected pitching moment for this flow condition to acceptable accuracy. two scenarios. In Figure 16, an angle of attack of 50 degrees has been imposed on the body axis roll of the To summarize, several important effects have been identified in Figures 5-10 which strongly affect rotated geometry illustrated in Figure IS. The angle of attack in Figure 16, roll rates. and the forward speeds the accuracy of the PMARC results: I) the total grid density, 2) the grid distribution spanwise and of Figures II, 12, IS, and 16 were chosen simply to illustrate the maneuvers and do not reflect actual tunnel chordwise, 3) minimum thresholds of surface resolution may exist for obtaining accurate results, 4) test conditions, although the body rotation angle of 30.8 was chosen to match experimental data available configuration geometric complexity, and 5) grid smoothness. It is important to realize that once a grid for the F-16XL tested in the NASA LaRC 14- by 22- has been obtained for a given configuration within a Foot Subsonic Tunnel. Figures IS and 16 are included given study, the user may have little, if any, means to simply to illustrate the ease with which advanced improve upon any of these aspects of the grid, which testing techniques can be simulated through CFD.

significantly affect the quality of the results. At this Obviously, the maneuvers should be performed with a grid of sufficient density, using guidance obtained from point, it is assumed that a grid of sufficient quality can be obtained and validated for use. The remainder of the above grid resolutions studies, and with a code that the figures illustrate in a qualitative sense what could is able to capture all the relevant flow physics.

be done with such a grid for S&C applications. Typical execution times for the F-16XL geometry in the original PMARC code are about 3.25 Figure I I shows the original vehicle geometry, with wake trailing from the top of the minutes each on a R 12000 dual processor of a Silicon Graphics® Octane®. The forward mode ADIFOR- vertical tail, during a body axis roll maneuver. For all computations, other wakes (not shown in Figure II) generated PMARC aerodynamic analysis, including the computation of angle of attack and sideslip derivatives, were also attached to the trailing edges of both wings.

Figure 12 shows the vehicle, again with wake trailing required approximately 6.29 minutes for the same single-processor execution scheme. The second from the top of the vertical tail, during a forced oscillation in roll maneuver. The reader should note derivative calculation, required for the uncertainty the two reversals in rotary direction illustrated by the propagation analysis, required for the same two wake. In this case, the only useful "pure rotary" independent variables required about 14.10 minutes.

portion of the motion Occurs between the two reversals All the calculations were performed in 64-bit after start-up transients have diminished; this is typical arithmetic. The RAM requirements were about 130 of how the maneuver might be performed in a wind MB for the PMARC function and 213 MB for the tunnel test where the faCility does not have "slip ring" function plus its derivatives with respect to angle of attack and angle of sideslip. The second derivative capability. CFD offers the ability to perform the roll maneuver, without any support or wall interference code required about 481 MB of storage. Late in the effects, in either the pure rotary or in forced oscillation studies, the PMARC code was found to execute three modes. to five times faster per case on a PC cluster, composed Figures 13 and 14 illustrate two advanced of a host and 32 worker nodes. Each worker node uses experimental techniques used to determine dynamic a 2193 Mhz central processing unit (CPU), the Linux S&C derivatives. Figure 13 shows a coning motion; AIAA 2004-0377 2.4.18 operating system. and a Lahey/Fujitsu Optimizing Fortran 95 compiler. REFERENCES I. Nelson, Robert c., Flight Stability and CONCLUSIONS Automatic Control (Second Edition), McGraw The Panel Method Ames Research Center Hill International Editions Aerospace Science & (PMARC) code, version 14.10, has been differentiated Technology Series, New York, New York, 1998.

in several versions to provide exact first and second 2. Nicolai, Leland M., Fundamentals of Aircraft derivatives of the forces and moments with respect to Design, METS, Inc. Xenia, Ohio, 1975.

wide variety of inputs. The original and differentiated 3. Roskam, Jan, Airplane Flight Dynamics and codes have been executed to simulate a variety of Automatic Flight Controls. Parts 1 and II, motions typically used in experimental facilities to DARcorporation, Lawrence, Kansas, 2001.

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using reasonable amounts of computer time and 5. Roskam, Jan, Advanced Aircraft Analysis memory. The results generally reflect the trends of the (Software Package and Documentation), measured F-16XL data, but the lift and normal force DARcorporation, Lawrence. Kansas, 200 I.

coefficient data were substantially different from the 6. McDonnell Douglas Astronautics Company, measured data. St. Louis Division, The USAF Stability and Additional wakes separating from the mid Control DATCOM, Volume I, Users Manual, chord upper surface region of the wing could be used McDonnell Douglas Astronautics Company, St.

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this technique requires some knowledge about the 7. Lan, Edward c., "VORSTAB - A Computer expected flowfield characteristics. Propagated Program for Calculating Lateral-Directional uncertainty bounds were also shown to account for a Stability Derivatives with Vortex Flow Effect," small part of the difference between the computed and NASA Contractor Report 172501, January 1987.

measured data at low angles of attack. In addition, the 8. Tavoularis, S., "Equivalence Between results of numerous surface grid resolution studies Sideslip and Roll In Wind-Tunnel Model were reported. The surface grid resolution studies Testing," AIAA J. Aircraft. Vol. 36, No.5, pages 895-896,May, 1999.

show that a minimum of eight panels chord wise around the airfoil sections of a wing are necessary to obtain 9. Klein, V., Murphy. P. c., Curry, T. J., and meaningful approximations to the expected forces and Brandon, J. M., "Analysis of Wind Tunnel moments. The results are strongly influenced by the Longitudinal Static and Oscillatory Data of the F- 16XL Aircraft," NASAlTM-97-206276.

surface resolution in both the chordwise and spanwise directions, as well as the geometric complexity of the 10. Green, Lawrence L., Spence, A. M., and body and the smoothness of the grid distribution. For Murphy, P. c.: Computational Methods for smooth distributions of surface panels, grid Dynamic Stability and Control Derivatives.

AIAA 2004-0015, January 5-8. 2004.

independence of results and agreement with known analytic solutions can be obtained with sufficiently II. Dorsett. K. M., and Mehl, D. R., "Innovative Control Effectors (ICE)," Wright Laboratory dense surface resolutions. In the absence of smooth surface resolution, grid independence of the results Report, WL-TR-96-3043, Jan. 1996.

cannot be achieved; in fact, the results in such cases do 12. Raney, D. L., Montgomery, R. c., and Green, not appear to even converge to common values. L. L., and Park, M. A., "Flight Control Using Finally, the work reported here demonstrates Distributed Shape-Change Effector Arrays," AIAA 2000-1560, April 2000.

some potential for using very coarse computational models to predict S&C characteristics of complex 13. Park, M., Green, L., Montgomery, R., and Raney, D., "Determination of Stability and vehicles, especially when measured data is available to calibrate the computations. Such coarse computational Control Derivatives Using Computational Fluid Dynamics and Automatic Differentiation," AIAA models may be useful in the conceptual or preliminary design arena, but they represent only the first phase of paper 99-3136, June 1999.

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14. Limache. A. c., and Cliff, E. M., applying CFD codes to aircraft S&C problems. The work also demonstrated several computational "Aerodynamic Sensitivity Theory for Rotary Stability Derivatives," AIAA Paper 98-4313, techniques that can be used to further the application of CFD to stability and control applications. Aug. 1998.

AIAA 2004-0377 15. Park, Michael A. and Green, Lawrence L., 29. Alhright. A. E., Dixon, C J., Hegedus, M. C, "Steady-State Computation of Constant "Modification and Validation of Conceptual Rotational Rate Dynamic Stahility Derivatives", Design Aerodynamic Prediction Method AIAA 200-4321, August 2000.

HASC95 With VTXCHN."NASA CR-47 12, 16. Park. Michael A., Determination of Static Mar. 1996.

and Dynamic Stahility and Control Derivatives 30. Brandon, J.M., and J.V. Foster: Recent With Computational Fluid Dynamics and Dynamic Measurements and Considerations for Automatic Differentiation, Master Thesis. The Aerodynamic Modeling of Fighter Airplane George Washington University, August 2000. Configurations, AIAA 98-4447, August 1998.

17. Gainer, Thomas G., "A Discrete-Vortex 31. Klein, V., and Noderer. K.D.: Modeling of Aircraft Unsteady Aerodynamic Characteristics", Method for Studying the Wing Rock of Delta Wings," NASAfTP-2002-211965, Decemher Part I - Postulated Models, NASA TM 109120, 2002. May, 1994; Part 2 - Parameters Estimated From 18. Melville, Reid. "Aeroelastic Instahility of Wind Tunnel Data, NASA TM 110161, April Tactical Aircraft in Nonlinear Flow Regeims," 1995; Part 3 - Parameters Estimated From Flight AIAA 2002-2970, June, 2002. Data, NASA TM I 10259, May 1996.

Anderson, W. K., Newman, J. C, Whitfield, 19. Jouannet, C and Krus, P., "Lift Coefficient 32.

Predictions for Delta Wing Under Pitching D. L., and Nielsen, E. J.: Sensitivity Analysis for Motions," AIAA 2002-2969, June 2002.

the Navier-Stokes Equations on Unstructured 20. Kay, Jacoh, "Acquiring and Modeling Meshes Using Complex Variables". AIAA paper Unsteady Aerodynaimc Characteristics," AIAA 99-3294, 1999.

2000-3907, August 2000. 33. Vatsa, V. N.: Computation of Sensitivity 21. Katz, Joseph, and Schiff. Lewis B., Derviatives of Navier-Stokes Equations Using "Modeling Aerodynamic Responses to Aircraft Complex Variahles. Fifth Symposium on the Maneuvers-A numerical Validation," AIAA J. Large-Scale Analysis and Design and ISE, Aircraft, Vol. 23, No.1, January 1986. Williamsburg, Virginia. Oct. 12-15, 1999.

van Dam, C P., DalBello, T.. and Chao, D. 34. Hemsch, M., "Statistical Analysis of CFD 22.

D., "Prediction of Flows About Forebodies at Solutions from the Drag Prediction Workshop," High-Angle-of-Attack Dynamic Conditions," AIAA 2002-0842, January 2002.

Final Report of NASA Langley Research Center 35. Hosder, S., Grossman, B., Haftka. R. T., Grant NAG-I-2006, May 2000. Mason, W. H., and Watson, L. T., Observations 23. Youngren, H. H., Bouchard, E. E., on CFD Simulation Uncertainties, AIAA 2002- Coopersmith, R. M., and Miranda, L. R., 5531, September 2002.

"Comparison of Panel Method Formulations and 36. Green, L., Lin, H.-Z., and Khalessi, M., its Influence on the Development of QUADPAN, "Probabilistic Methods for Uncertainty an Advanced Low Order Method," AIAA Paper Propagation Applied to Aircraft Design,", AIAA 83-1827, July 1983. Paper 2002-3140, June 2002.

24. Miranda. L. R., Elliott, R. D., and Baker, W. 37. Eca, L., Vaz, G. B., and Falco de Campos, A.

M., "A Generalized Vortex Lattice Method of C, "Verfication Study of Low and Higher-Order Subsonic and Supersonic Flow Applications," Potential Based Panel Methods for 20 Foils," NASA CR-2865, Dec. 1977. AIAA 2002-3112, June 2002.

25. Simon, 1. M., "Dynamic Derivative Data for 38. Putko, M. M., Taylor. III, A. C, Newman, P.

High-Angle-of-Attack Simulation," AIAA paper A., and Green, L. L., "Approach for Input 92-4355, Aug. 1992. Uncertainty Propagation and Robust Design in 26.

Blake, W. B., Dixon, C J., and Adler, CO., CFD Using Sensitivity Derivatives," Journal of "Development of a High-Angle-of-Attack Fluids Engineering, published by the Society of Stability and Control Prediction Code," AIAA Mechanical Engineers, Vol. 124, No.1. March Paper 92-4354, Aug. 1992. 2002, pp. 60-69 (see also.AIAA Paper 2001- 27.

Ahn, T .. Kim, C, Rho, O. H., and Kim, H.-J., 2528, June 2001).

"Dynamic Stall Control Using Aerodynamic Lessard, Wendy B .. "Subsonic Analysis of 39.

Sensitivity Analysis." AIAA 2001-0255, January 0.04-Scale F-16XL Models Using an 2001. Unstructured Euler Code," NASA Technical

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28.

1988 Report to the Aerospace Profession Paper 3597, October 1996.

XXXIII, Thirty-Third Symposium Proceedings, September 1989.

AIAA 2004-0377 40. Bischof. C, Carle, A .. Corliss, G., Griewank. 43. Hahne. David E, "Low-Speed Aerodynamic A., and Hovland. P .. "ADIFOR-Generating Data for an O.IS-Scale Model of an F-16XL With Derivative Codes from Fortran Programs," Various Leading-Edge Modifications," Scie/ltijic Programming, No. I, 1992, pp. 1-29. NASAffM-1999-209703, December 1999.

41. Bischof, C, Carle, A, Khademi, P., and 44. Kramer, Brian R.: Experimental Evaluation Mauer, A, "Automatic Differentiation of of Superposition Techniques Applied to Dynamic FORTRAN," IEEE Computational Science & Aerodynamics. AIAA 2002-0700, January 14-17,

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Engineering, Fall 1996. 2002.

42. Ashby, Dale L., Dudley, Michael Roo Iguchi, 45. Eidetics Corporation: Aircraft Dynamics: Steve K., Browne, Lindsey, and Katz, Joseph. Nonlinear Math Modeling. Annual Report June "Potential Flow Theory and Operation Guide for 2002. Contract N AS 1-0 1091.

the Panel Code PMARC," NASA TM 102851. 46. Carle. A and Fagan, M., "Overview of january i 991. (Unofficiaiiy revised in i 999 for Adifor .~.O." Department of Computationai and PMARC version 14). Applied Mathematics, Rice University, CAAM- TR 00-02. Jan. 2000 .

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AIAA 2004-0377 Table I Summary of F-16XL geometries. grids, and grid usages.

Grid Description Grid Size Surface Resolution Grid Usage Half configuration 30 blocks Not used for the FINEST_VaLl volume grid w/tip 1,502.138 grid Not applicable current studies missiles & launchers points Full configuration 36 blocks Future Euler and 6.293.908 grid Not applicable FINEST_ VOL2 volume grid w/inlet Navier-Stokes fairing points code use planned Future Euler and Subset of 36 blocks MEDIUM_VOL Not applicable Navier-Stokes 837,924 grid points FINEST_ VOL2 grid code use planned Future Euler and Subset of 36 blocks COARSE_VOL MEDIUM_ VOL Not applicable Navier-Stokes 118,216 grid points grid code use planned Surface grid Execution extracted from 56 patches attempted, FINEST_SRF 108,736 surface panels FINEST_ VOL2 113,856 grid points exceeded volume grid available RAM Execution Surface grid attempted, extracted from 56 patches MEDIUM_SRF 27,184 surface panels exceeded MEDIUM VOL 29,772 grid points - available disk volume grid space Execution Surface grid completed, extracted from 56 patches unreasonably COARSE_SRFI 6,796 surface panels COARSE VOL 8,1 18 grid points large forces and - volume grid moments computed Surface grid extracted from 56 patches Not used for the COARSE_SRF2 1,744 surface panels COARSE_VOL 2,434 grid points current studies volume grid Surface grid Used extensively extracted from 56 patches COARSE_SRF3 566 surface panels for the current COARSE_VOL 984 grid points studies volume grid Table 2 Sample uncertainty analysis for F-16XL all values of (Jinpur = 0.1 .

Uncertainty Computed 99.9% 97.7% 84.1% Mean 84.1% 97.7% 99.9% Variable Name Variable sigma Low Low Low Value High High High dCUdALPDEG ALPDEG 4.72E-05 2.02E-02 2.03E-02 2.03E-02 2.04E-02 2.04E-02 2.05E-02 2.05E-02 dCUdALPDEG Q 4.97E-05 2.02E-02 2.03E-02 2.03E-02 2.04E-02 2.04E-02 2.05E-02 2.05E-02 dCUdALPDEG ALPDEG+Q 6.B5E-05 2.04E-02 2.05E-02 2.06E-02 2.02E-02 2.02E-02 2.03E-02 2.04E-02 CL ALPDEG 2.04E-03 3.97E-01 3.99E-01 4.01 E-01 4.03E-01 4.05E-01 4.07E-01 3.95E-01 dCUdQ ALPDEG 2.41E-02 2.42E-02 4.97E-05 2.39E-02 2.40E-02 2.40E-02 2.41 E-02 2.42E-02 Q dCUdQ 1.23E-03 2.04E-02 2.16E-02 2.2BE-02 2.41E-02 2.53E-02 2.65E-02 2.7BE-02 dCUdQ ALPDEG+Q 1.23E-03 2.04E-02 2.16E-02 2.2BE-02 2.41E-02 2.53E-02 2.65E-02 2.7BE-02 CL ALPDEG 2.41 E-03 4.03E-01 4.06E-01 4.0BE-01 3.94E-01 3.96E-01 3.9BE-01 4.01 E-01 CL TOTAL 3.15E-03 4.01 E-01 4.04E-01 4.07E-01 4.10E-01 3.91 E-01 3.95E-01 3.9BE-01 AIAA 2004-0377

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. .. >,~~\.~

~ Figure I. Three-view of F-16XL computational model.

, I3 AIAA 2004-0377 2.0 0.6 1.5 0.5 z 01.0 "C 0.4 c ~0.5 Envelope ~0.3 Envelope == Nk:olai JO.O b.- - -0 - CL PMARC COMPUTATION 0.2 0 - -lS,.- - eM PMARC COMPUTATION 0' - - 0- - eN PM"RC COMPUTATION --+-- CL NASA I TM-91-20621e DATA -0.5 ~ eM NASA/TM·97·20627fi DATA 0.1 ~ eN NASA/TM·'7·2t1e276DATA -.....- CL HAHNE DATA. RUN 180 Q 0.0 2 3 4 -1. 20 0 20 40 60 80 100 Alpha (degrees) Log10(Res) Figure 2. Computed static lift. moment. and Figure 5. Envelope of computed lift coefficient data at normal force coefficient data and measured data. various surface resolutions for AR = 2, symmetric, parabolic wing with analytic data.

1.5 0.10 0.05 1.0 0.00 :::ii: ~0.5 ~-0.05 c o --- PMARC DATA. TE WAKES ONLY -0.10 - - -b.- - - PMARC DATA. +LlNE 3 WAKES 0.0 - - -v- - - PMARC DATA. +LlNE 4 WAKES --+-- KLEIN·MURPHY DATA -0.15 --+-- HAHNE DATA. RUN '90 -0.20 2 3 4 -0'~20 0 20 40 60 80 100 Alpha (degrees) Log10(Res) Figure 3. Computed static lift coefficient data, Figure 6. Envelopes of computed drag and pitching including uncertainty propagation error bars, and moment coefficient data at various surface resolutions, measured data. for AR = 2, symmetric, parabolic wing.

0.05 PMARC DATA, TE WAKES ONLY KLEIN-MURPHY DATA 0.6 0.04 <> 0.03 0.5 ~ ________ ~~~~<>~~

.-

0.02 !!

..

-!!

0.4 0.0' !.

-- 4 ;- 0.00 ---1-- 8 .c .l!- --0-- , • ~0.3 "f. -0.01 --0-- 32 U -b--- 64 -0.02 0.2 ----v-- '28 --<>-- 25.

-0.03 --- Nicolai 0.1 -0.04 -0.05 -20 0 20 40 60 80 '00 0.0 2 3 4 Alpha (degnt •• ) Log10(Res) Figure 4. Computed static lift curve slope, C La ' Figure 7. Variation in computed moment coefficient with surface resolution for AR = 2, symmetric, coefficient data, including uncertainty propagation parabolic wing and fixed chord wise resolutions.

error bars, and measured data.

AIAA 2004-0377 0.6 0.5 0.4 ~0.3

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0.2 --0-- elMin Envelope 0.1 ____ ClMex Envelope 0.0

0 234

Logl0(Res) Figure 8. Envelope of computed lift coefficient data at Figure II. Sample body axis pure rotary roll various surface resolutions maneuver, with trailing vertical tail wake.

for configurations similar to F-J6XL.

2.0 o F·18XL V.riations -- NASA/TM·97·206276 DATA 1.5 - - - - - - - HAHNE DATA, RUN 190 1.0 o ~O.S~-------------------..-~------ 0.0 o o -0.5 Reversal -1.0 2 3 4 Logl0(Res) Points Figure 9. Variation in computed lift coefficient with Figure 12. Sample body axis oscillatory roll maneuver, surface resolution with trailing vertical tail wake.

for the F-J6XL configuration.

2.0 6. F-18XL V.rilltions -- NASA I TM·.7·206276 DATA 1.5 1.0 :E 0 . 5 0.0 -0.5 -1.0 1 2 3 4 I Logl0(Res) Figure J O. Variation in computed pitching moment coefficient with surface resolution

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for the F-16XL configurations.

AIAA 2004-0377 ..

a=8 a=8

Figure 13 . Conventional coning motion. Figure 15. Computational illustration of pure coning motion.

a=8-A

Figure 16. Computational illustration of oscillatory Figure 14 . Oscillatory coning , or inclined axis roll coning, or inclined axis roll OAR), motion.

(JAR), motion.