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AIAA-99-3136 DETERMINATION OF STABILITY AND CONTROL DERIVATIVES USING COMPUTATIONAL FLUID DYNAMICS AND AUTOMATIC DIFFERENTIATION * Michael A. Park George Washington University Joint Institute for the Advancement of Flight Sciences (JIAFS) mikepark@tabdemo.larc.nasa.gov † Lawrence L. Green NASA Langley Research Center, Hampton, Virginia l.l.green@larc.nasa.gov ‡ Raymond C. Montgomery NASA Langley Research Center, Hampton, Virginia r.c.montgomery@larc.nasa.gov § David L. Raney NASA Langley Research Center, Hampton, Virginia d.l.raney@larc.nasa.gov dynamics (CFD) solvers are augmented via automatic Abstract differentiation, to directly calculate the stability and With the recent interest in novel control effectors control derivatives. The CFD forces and moments are there is a need to determine the stability and control differentiated with respect to angle of attack, angle of derivatives of new aircraft configurations early in the sideslip, and aircraft shape parameters to form these design process. These derivatives are central to most derivatives. A subset of static stability and control control law design methods and would allow the derivatives of a tailless aircraft concept have been determination of closed-loop control performance of the computed by two differentiated inviscid CFD codes and vehicle. Early determination of the static and dynamic verified for accuracy with central finite-difference behavior of an aircraft may permit significant approximations and favorable comparisons to a improvement in configuration weight, cost, stealth, and simulation database.
performance through multidisciplinary design. The classical method of determining static stability and control derivatives—constructing and testing wind tunnel Introduction models—is expensive and requires a long lead time for Previous work attempted to determine stability the resultant data. Wind tunnel tests are also limited to derivatives from computational fluid dynamics (CFD) the preselected control effectors of the model. To codes; for example, Finley used an Euler code to overcome these shortcomings, computational fluid compute the forces and moments for a generic configuration from which a subset of the stability * Graduate Student, NASA Langley Research Center, Multidisciplinary derivatives can be inferred by using finite-difference Optimization Branch, MS 159, Hampton, Virginia, Member AIAA methods. Charlton employed a similar method on the Lockheed Martin Tactical Aircraft Systems—Innovative † Research Scientist, Multidisciplinary Optimization Branch, MS 159, ¶ Control Effectors (LMTAS-ICE) configuration. No Senior Member AIAA attempt was made to compute those derivatives ‡ analytically.
Research Scientist, Dynamics and Control Branch, MS 132, Senior The present work proposes to use one or more CFD Member AIAA solvers, augmented via automatic differentiation (AD), to § Research Scientist, Dynamics and Control Branch, MS 132, Member directly calculate static stability and control derivatives AIAA of the LMTAS-ICE configuration. Using exact AD is Copyright 1999 by the American Institute of Aeronautics, Inc. No ¶ The use of trademarks or names of manufacturers in this report is for copyright is asserted in the United States under Title 17, U.S. Code.
accurate reporting and does not constitute an official endorsement, The Government has royalty-free license to exercise all rights under the either expressed or implied, of such products or manufactures by the copyright claimed herein for government purposes. All other rights National Aeronautics and Space Administration reserved by the copyright owner.
American Institute of Aeronautics and Astronautics more robust and removes the requirement of determining University for alpha testing. In general, to apply the optimal step size for the finite-difference calculation. ADIFOR or ADJIFOR to a given FORTRAN 77 code, This AD technique is illustrated with potential and Euler the user is only required to specify those program flow solvers, but can also be extended to Reynolds- variable names that correspond to the independent and averaged Navier-Stokes (RANS) flow solvers to include dependent variables of the target differentiation. Each viscous effects. The potential and Euler flow solver AD tool then determines the variables that require outputs (forces and moments) are differentiated with associated derivative computations, formulates the respect to the angle of attack and the angle of sideslip, appropriate forward or reverse mode derivative yielding a subset of the static stability derivatives. expressions, and generates new FORTRAN 77 code for For control effectiveness, various control effectors the computation of both the original simulation and the can also be tested and optimized by finding the associated derivatives.
sensitivity of aircraft moments to changes in the Currently, some manual processing is required to configuration grid points that define these effectors. formulate the adjoint of iterative and parallel message- Because they are of greater importance to aircraft passing codes due to the prototype nature of the alpha control, only moments were differentiated for control version of ADJIFOR used in this study. These manual effectiveness, although the method is equally applicable manipulations of the code are expected to be greatly to forces. reduced with the formal release of the ADIFOR 3.0 package to the general public. This release is planned for the summer of 1999.
Automatic Differentiation 3–5 Automatic differentiation is a technique for augmenting computer programs with statements for the Computational Fluid Dynamics Solvers computation of derivatives. It relies on the fact that every The PMARC (Panel Method Ames Research Center) function, no matter how complicated, is executed on a potential flow solver is a FORTRAN 77 code that can computer as a (potentially very long) sequence of compute surface pressures, forces, and moments of elementary operations such as additions, multiplications, arbitrary shapes. The code is based on the assumption of and elementary functions such as sine and cosine. By inviscid, irrotational, and incompressible flow, with some repeatedly applying the chain rule of differential calculus boundary layer and compressible corrections available, to the composition of those elementary operations, but not implemented in this study. PMARC also has a derivative information can be computed exactly and in a limited capability to compute solutions of unsteady, completely automated fashion. time-varying flow conditions.
Two approaches for computing derivatives with AD The input file to the program includes the set of grid are the forward mode and the reverse mode. The forward points describing the shape of the geometry as a set of mode applies the chain rule of differentiation to panels. In this study, both the right and left halves of the propagate, equation by equation, derivatives of LMTAS-ICE (Fig.1) configuration are modeled in the intermediate variables with respect to the input variables. PMARC input file, with a total of 2560 panels. PMARC In contrast, the reverse mode (adjoint) propagates, in allows half the aircraft to be described and the solution to reverse through the program, the derivatives of the output be mirrored in the x - z plane. Although describing only variables with respect to the input variables. The forward one half of the configuration would reduce the time and mode is more suited to problems with fewer input memory required for a converged solution, this technique variables than output variables, whereas the reverse mode would not capture the effects of a nonzero angle of is better suited to problems with fewer output variables sideslip. The input file also specifies the flight condition, than input variables. Many hybrids of the forward and certain algorithmic parameters, and the user-defined reverse modes are possible, with complementary position of the reference point about which all moments tradeoffs in required random access memory (RAM), are summed. The forces and moments are also disk space, and execution time. nondimensionalized with a user-specified reference area, The forward method of AD is implemented in the length of the mean aerodynamic chord, and wingspan.
ADIFOR (Automatic Differentiation of FORTRAN) All runs were performed with the assumption of tool. The reverse mode is used in another tool, incompressible flow or low-speed flight conditions.
ADJIFOR (Automatic Adjoint Generation in The original PMARC code uses swap or scratch files FORTRAN). The ADIFOR and ADJIFOR tools have to record intermediate values during operation. This disk been developed jointly by the Center for Research on usage allows PMARC to solve problems with a large Parallel Computation at Rice University and the number of panels on machines with limited RAM Mathematics and Computer Sciences Division at capacity. ADIFOR and ADJIFOR ignore FORTRAN Argonne National Laboratory. Both techniques are read and write statements and are therefore unable to available in the prototype ADIFOR 3.0 package recently follow the dependency of variables through scratch file provided to the NASA Langley Research Center by Rice read and write operations. To allow the application of American Institute of Aeronautics and Astronautics AD, the PMARC code was modified to eliminate the averaged thin-layer Navier-Stokes flow solver for need to use scratch files during execution. These structured-volume grids. The analysis for this study was modifications entailed replacing scratch file read and performed in an inviscid, Euler mode. The single- write operations with equivalent operations to common processor sequential version was differentiated into a blocks during AD code generation. If sufficient RAM is version with angle of sideslip as the only independent in not available for executing the code with the common the derivative calculation. The code was only blocks in the AD-generated code, the scratch file differentiated with respect to angle of sideslip to operations can be reactivated following code generation. investigate a change in the simulation database To further minimize the use of RAM, scratch file derivatives between 5 and 7.5 deg angle of attack. Work operations for the new AD-created variables containing is ongoing to automatically differentiate the latest derivative information can also be inserted into the code prerelease version of CFL3D. This work includes a in a fashion complementary to the original variables. parallel execution scheme that would allow faster The forward mode of differentiation (ADIFOR) was derivative determination in both angle of attack and angle used to compute the stability derivatives because there of sideslip. The volume grid used to model the airflow are two inputs—angle of attack and angle of sideslip— about both halves of the ICE configuration contains compared to the six output forces and moments. The 3.1 million grid points.
reverse mode (ADJIFOR) is employed to calculate the The force and moment calculations were performed derivatives used for guidance in the optimization of with a single processor of the HPCCP Silicon Graphics control effector size and placement, because the Origin 2000 with mesh sequencing and multigrid thousands of independent variables greatly outnumber convergence acceleration on three levels. Timings are the six dependent variables. The independent variables of approximately 0.98 second per coarse-grid iteration, 18 these derivative calculations are the normal seconds per medium-grid iteration, and 230 seconds per displacements of each of the surface-defining grid points. fine-grid iteration. The total time for a function Normal displacements were chosen to mimic an evaluation was 18 hours. The sequential version required inflatable device. The outputs are the three moment 825 Mbytes RAM. The inclusion of angle of sideslip coefficients, each evaluated in three separate code derivatives caused a 500 percent increase in the executions. sequential execution time and required 1600 Mbytes of Typical execution times for the original PMARC RAM. The timing difference is possibly due to poor code are about four minutes on one R10000 processor of cache memory usage and AD overhead. This difference a Silicon Graphics Octane. The forward ADIFOR- in execution time is further magnified by the use of generated PMARC aerodynamic analysis, including aggressive compiler options, which are more effective on angle of attack and angle of sideslip derivatives, required the original code. A parallel execution scheme employing approximately 12 minutes for the same single-processor 8 processors has demonstrated a 600 percent speedup execution scheme. The calculations were performed in from 18 to 3 hours for determining forces and moments.
64-bit arithmetic with versions of codes that had all The differentiation of this parallel code may allow these scratch file operations replaced with common blocks. stability derivatives to be calculated quickly enough to The RAM requirements were 160 Mbytes for the integrate this derivative determination into a design PMARC function and 480 Mbytes for the function plus process.
its derivatives with respect to angle of attack and angle of sideslip. The control derivatives of each separate moment LMTAS-ICE with respect to the normal displacement of all the grid The LMTAS-ICE is a proposed configuration that is points is determined with the ADJIFOR reverse mode.
used by NASA to test the synergy of low-observable This calculation was performed on the NASA Langley technologies with new types of control effectors. The High Performance Computing and Communication goal is a lightweight, low-observable, and highly Program (HPCCP) sixteen-processor Silicon Graphics maneuverable aircraft. The basic layout of the aircraft is Origin 2000 in 20 minutes with one processor. The a tailless, highly swept delta shape with integrated wing, memory requirements are 171 Kbytes RAM and 6 body, and propulsion systems (see Fig.1). One of the Gbytes of disk storage. Information from each of the over goals of the LMTAS-ICE program is to investigate a fifty iterations required to solve the aerodynamic wide range of conventional and novel control effectors interference coefficient matrix is stored to disk. This for their impact on controllability, weight, and stealth.
information is used to formulate the adjoint solution.
The LMTAS-ICE configuration is a challenging The code can be modified to iterate the adjoint solution aerodynamic problem for a potential flow solver such as to save disk space with an increased cost in code PMARC because of a sharp leading edge and predicted execution time.
high Mach numbers near the wing tips at cruise The CFL3D (Computational Fluids Laboratory conditions. The combination of sharp leading edges and a 3-Dimensional) code is a FORTRAN 77 Reynolds- highly swept planform can give rise to vortical flow at American Institute of Aeronautics and Astronautics moderate to high angles of attack. Vortical flow is not The deltas ( ∆ ), changes in angle of attack and modeled in the PMARC potential flow solver, but is sideslip, are chosen between 0.0001 and 5 deg (see Table modeled in the CFL3D Euler calculation. To further 1). The value of the computed derivative is plotted as a resolve these flow effects and to model some novel function of the logarithm of ∆ (Fig. 2). The dashed control effectors such as the synthetic jets, a fine-grid horizontal line is the ADIFOR-computed derivative. The RANS solution may be required. PMARC analysis is symbols CN , CA , and CS represent the nondimensional most suited to the low angle of attack region (0 to 6 deg) coefficients of aerodynamic forces referenced to the body of this aircraft’s flight envelope, where the airflow is axes in the directions upward, aft, and toward the right attached to the aircraft and where most cruise portions of wing tip as shown in Fig.1. Figure 2a, 2b, and 2d show a a flight take place. This investigation was extended to an comparison of the force derivatives. The symbols Cl , Euler code when the PMARC-calculated lateral Cm , and Cn represent the nondimensional coefficients of derivatives showed nonideal (possibly unacceptable) moments in the roll, pitch, and yaw axes as shown in correlation with the simulation database at larger angles Fig.1. Figure 2c, 2e, and 2f show the moment derivative of attack. Euler calculations may be more suited to these comparisons.
higher angles of attack and angles of sideslip (6 to 15 For small ∆ , the central-difference calculations are deg). Highly separated and time-varying flow conditions equal to the ADIFOR values of the static stability (>15 deg) may require the investment of time-dependent derivatives for more than six significant figures. The RANS calculations to correctly predict derivatives.
difference between the central-difference approximation In spite of these concerns, PMARC was chosen for and ADIFOR derivative value grows larger with ∆ . A log the initial work in this study because of its potential for scale for ∆ was used in the figures. In this ADIFOR rapid code execution and the option to solve either steady study, the central-differences are relatively insensitive to or unsteady flows. The LMTAS-ICE was chosen because the size of ∆ . Previous ADIFOR finite-difference of the existence of a wind-tunnel-based aerodynamic 3–5 validation studies done with viscous Navier-Stokes database and a desire to optimize control effectors for the calculations showed a much greater sensitivity to the ∆ configuration. There is also a desire to develop a size. This sensitivity is possibly due to both the nonlinear "seamless" aircraft control scheme, where aircraft natures of the transonic Navier-Stokes codes for the attitude is controlled with slight mold-line distortions problems investigated and the uses of first order one- rather than with hinged moving surfaces. These novel sided finite-difference approximations in the previous control effectors are designed to keep the flow attached studies. One-sided finite-difference or central-difference and therefore lend themselves to the linear aerodynamic approximation comparisons were not made to the theory of PMARC. The ADJIFOR modification of CFD ADIFOR-generated CFL3D Euler derivatives because of code is well suited to the task of finding guidance for the the computer time required for the many lengthy optimal definition of these continuous shape-morphing executions. Instead, the validity of the derivatives was controls.
inferred by direct comparisons to PMARC and wind-tunnel-based derivatives. Each PMARC Validation of AD central-difference approximation required four by Central Finite Differences executions (two angle of attack and two angle of sideslip The first step in verifying the accuracy of the perturbations) of the PMARC code. Therefore, the time ADIFOR-generated version of PMARC was to compare required for the code execution of a central-difference the forces and moments of the ADIFOR-generated approximation was approximately 16 minutes. In PMARC code to the forces and moments of the original contrast, all the derivatives can be computed in a single version of PMARC. This comparison showed no run (about 12 minutes) of the ADIFOR-generated significant discrepancies. The derivatives were then PMARC code. This difference in processing time checked for accuracy using second order required for these two approaches will central-difference approximations. The cruise condition become more significant in problems with more of 4.39 deg angle of attack ( α ), 0 deg angle of sideslip independent variables or more time-consuming, higher fidelity flow calculations.
( β ), and 0.60 Mach was chosen as the point about which to perform the finite differencing. By, treating the forces and moments PMARC outputs as a function ( P ) of angle Force and Moment Comparison of attack or sideslip ( A ), the central-difference value of to Simulation Database the force or moment derivative is calculated as The ADIFOR application of PMARC has been shown to produce the correct derivatives of its calculated forces
( ) ( ) ∆ − − ∆ + ∂ A P A P P
and moments. The ADIFOR-generated PMARC and ) ( ∆ + = O CFL3D codes are now further verified by comparing 2 ∆ ∂ A these results to the simulation database derived from American Institute of Aeronautics and Astronautics wind tunnel data (see Fig. 3 and Tables 2–4). Note that deg angle of attack. At higher angles of attack the the data at angle of sideslip not equal to zero (Tables 3 presumed predominance of vortical flow patterns starts to and 4) is shown as the difference between the displaced affect the lateral-directional coefficients dramatically.
(Lateral displacements to nonzero angles of sideslip were angle of sideslip ( β ) data and data taken at zero angle of not performed with CFL3D, but its accuracy can be sideslip for the same angle of attack. All calculations were performed at a Mach number of 0.60. inferred from the angle of sideslip derivatives computed with ADIFOR.) In contrast, CFL3D correctly predicted The wind tunnel data that is presented for comparison to the PMARC results was taken from an aerodynamic the change in lift slope, including changes due to leading edge vortical flow. Note that a ∆ CD for CFL3D could database used for a flight simulator. The way that the wind tunnel data was gathered and reduced to form the also be estimated independent of the calculation done for database needs to be researched. Further understanding PMARC, but this estimation was not performed due to of the methods used to convert the wind tunnel data into the similarities between the results of these two codes.
the simulation aerodynamic database may help to explain some of the differences between the aerodynamic Derivative Comparison database and PMARC in the lateral coefficients ( CS , Cl , to Simulation Database and Cn Fig. 3d–3f). These differences could also be The longitudinal forces and moments shown in Fig. 3 highlighting flow properties not modeled in PMARC.
are differentiated with respect to angle of attack, and the In an effort to improve the correlation between wind lateral forces and moments are differentiated with respect tunnel data and the values from PMARC and CFL3D, an to angle of sideslip. These derivative values are shown in estimate was sought of the drag component missing from Fig. 5. The derivatives plotted for PMARC are the AD- the inviscid code calculations. This delta coefficient of generated values. The CFL3D derivatives are drag ( ∆ CD ) was estimated by subtracting the axial force central-difference approximations for the longitudinal coefficient ( CA ) of the wind tunnel data from the cases, and AD-generated values for the lateral cases.
corresponding value in PMARC when the normal force (Work is under way for AD-generated CFL3D coefficients of both are near zero (see Fig. 3b). The ∆ CD longitudinal derivatives, but these derivatives were not calculation is made when the normal force is near zero so available in time for this publication.) Second order that any lift-induced drag component of ∆ CD is central differencing is performed on the simulation minimized. In the absence of wind tunnel data, an database between 2 deg positive and 2 deg negative angle estimate of ∆ CD can be based on theoretical skin 0 of sideslip for the lateral cases, and a weighted central friction calculations. The ∆ CD oriented along the wind 0 method is used for angle of attack derivatives because of vector is resolved into the CA , CN , and CS directions for the nonuniform spacing between points in the all angles of attack and angles of sideslip. The force and longitudinal cases. CFL3D provides a better prediction of moment reference center for the configuration is located the normal force slope (Fig. 5a), but both CFL3D and a distance ( ∆ Z ) below the centroid of lateral area, where PMARC have difficulty with the pitching moment derivative (Fig. 5c), possibly indicating the need for a the ∆ CD is assumed to act. Therefore, the ∆ CD 0 0 viscous Navier-Stokes calculation. ADIFOR-generated increment to the forces also causes additional moments.
CFL3D lateral derivatives are similar to the ADIFOR- The equations used to increment the various PMARC and CFL3D forces and moments shown in Fig. 4 and generated PMARC values at 2.5 and 5 deg angle of attack. The lateral derivatives of the ADIFOR-generated Tables 5a–5c are as follows: CFD codes agree reasonably well with wind tunnel data below 6 deg angle of attack; then at higher angles of CD = ∆ 00868 . 0 attack the wind tunnel data changes radically while Z = ∆ = = ft 0 . 9 ft 37.5 Span ft 28.75 MAC PMARC remains linear.
= = Sideslip of Angle Attack of Angle β α The ADIFOR-generated CFL3D code converged to values near those of the wind tunnel at 7.5 deg angle of CD CN CN ∆ × + = sin α attack for the coarse and medium grids. The 7.5 deg CD CA CA ∆ × + = cos α angle of attack fine-grid derivatives converged to CD CS CS ∆ × × − = sin cos β α significantly different values (Fig 5e–5f), possibly indicating the simulation of different flow physics on this CD Z Cl Cl ∆ × × × ∆ − = sin cos Span / β α finer mesh than the two coarser meshes. The prediction CD Z Cm Cm ∆ × × ∆ + = cos MAC / α that different flow phenomenon are being modeled is Cn Cn = based on truncation error arguments. The truncation error of a CFD code is based on the difference between the Following the addition of these increments, the discretized partial differential equation representing the PMARC values show excellent trend agreement up to 6 flow field in the code and the original continuous partial American Institute of Aeronautics and Astronautics differential equation. If the CFD solution is converging process. The use of these interpolation methods, forced to to the same exact solution as the mesh is refined, the satisfy the values of the function and its derivatives, may truncation error of a CFD solution should decrease in a allow the calculations to be performed at fewer flight predictable manner. Because CFL3D is based on a conditions and still have the same resolution of aircraft second order method, the truncation error should vary forces and moments, as more numerous, function-only linearly with a square of a characteristic length of the calculations.
mesh. The forces, moments, and their derivatives are computed in the ADIFOR-generated CFL3D code as a PMARC Control Effectiveness Derivatives function of the discretized flow equations. Therefore, In this study, ADJIFOR-generated control effective- The forces, moments, and their derivatives should vary in ness derivatives plotted as a function of control a predictable second order manner as the mesh is refined.
placement are used to provide guidance for the optimal The CFL3D-calculated forces, moments, and placement of shape change control effectors. The shape derivatives at lower angles of attack converged in a of the aircraft is described in the PMARC input file as a predictable manner as the mesh size was sequenced. The set of panels, each designated by four corner grid points.
forces and moments of the fine grid at 7.5 deg angle of The shape is modified by moving each of these grid attack converged to values indicative of a second order points in and out a small amount along a vector normal to reduction in truncation error, whereas the derivatives did the surface. The derivative of the forces and moments of not. Therefore, this flight condition of 7.5 deg angle of the configuration is calculated with respect to the normal attack is still under investigation.
displacement of these grid points. Optimal placement of The heightened sensitivity of lateral force and these controls is located in areas of large gradients of the moment coefficients to angle of sideslip (particularly at forces and moments. The placement of these effectors angle of sideslip equal to zero) at this higher angle of must allow for coordinated control of all three moments attack can also be seen in the simulation database. (This simultaneously because they may be highly coupled. A data is not shown here due to paper length constraints.)
deployment scheme that allows the control force to be The wind tunnel measurements, from which this applied in a linear fashion with input command would be simulation database was derived, were performed every 2 desirable from a pilot or control designers perspective, deg, between positive and negative 10 deg angle of because the control force of these effectors can be very sideslip. Therefore, the angle of sideslip derivatives at nonlinear with deflection height and simultaneous angle of sideslip equal to zero were computed between application of nearby effectors. Identifying the optimal positive and negative 2 deg angle of sideslip or a locations would permit a certain deflection of the skin to finite-difference step of 4 deg. The forces and moments have the largest effect on aircraft control. Control exhibit strongly nonlinear behavior as angle of sideslip is effectors that may be used in this manner are flexible, varied at 7.5 deg angle of attack.
inflatable surfaces, shape memory alloys, and Finite-difference approximations of the lateral piezoelectrics.
derivatives in the simulation database can be interpreted This study focuses on the three moment coefficients to be very sensitive to step size and to the angle of because of their greater importance to closed-loop sideslip about which the derivative approximation is controller design. Figure 6 shows the ADJIFOR- centered, due to the nonlinear nature of the wind tunnel generated values of the control effectiveness contours data. Using a finite-difference method with a large step interpolated over the aircraft surface. Figure 6a–6c shows size (4 deg) masks the actual behavior of these nonlinear the control effectiveness of an upper surface deflection functions of angle of sideslip. From previous experience, on the three moments in the roll, pitch, and yaw axes, it is expected that the AD-generated derivatives for CFD respectively. Figure 6e–6f shows the control codes that simulate the required flow physics present in effectiveness of deflecting the lower surface of the the full scale or wind tunnel models will prove to be configuration on the same three moments. Areas shaded more accurate than derivatives of wind tunnel in black and white offer the greatest amount of control measurements obtained by coarse finite differences.
effectiveness, but should be avoided because they often These Euler derivative calculations are expensive located near each other indicating an unacceptable level executions in terms of CPU time, but these processes of sensitivity to effector location. Areas with colors in the may be more applicable to the design process if speed red to pink range have larger contiguous areas, allowing improvement can be achieved by increased use of RAM a number of grid points to work together to generate a caching or parallel code execution. This derivative positive change in moment. Areas with colors ranging information may also be combined with function from blue to violet offer the same benefits for generating information to form response surfaces. Splines or other negative moments. The red and violet areas of the aircraft interpolation methods can be used to structure these leading edge and slightly aft of the middle of the wing response surfaces from the high-fidelity flow simulation are currently being targeted for effector placement.
data. These response surfaces can be used in the design Differentiating a new grid with more numerous, evenly American Institute of Aeronautics and Astronautics distribution panels many produce a smoother data set. It should be noted that theses comparisons are not The control derivative discontinuities near the trailing strictly valid because the proposed effectors represent edge of the model may be due to a greater sensitivity of relatively large, discrete changes to the surface, while the the code near locations that it is enforcing the Kutta ADJIFOR derivatives are valid for individual, condition. incrementally small, continuous changes in the surface shape. Nevertheless, the ADJIFOR derivatives of the A MATLAB application was developed to systematically investigate the design and effectiveness of three aircraft moments provide information useful to the these mold-line changing control effectors. The controls designer.
application reads in the coordinates, the ADJIFOR- Figure 8 shows the calculated control effectiveness generated PMARC control effectiveness, and directions values in terms of the change in the moment coefficients.
normal to the grid points for the ICE configuration. In The resulting change in moment coefficient of the this MATLAB tool, the aircraft configuration is shown aircraft ( ∆ Cl , ∆ Cm , and ∆ Cn ) due to a deflection of a with the control effectiveness derivatives interpolated single point is depicted in Fig. 8. These plots help to over the body, similar to Fig. 6. portray how the effectiveness of deflecting a single point The MATLAB tool allows the user to select a series to a height of 0.2 ft varies over the upper surface of the of grid points and corresponding deflection heights for a PMARC grid. Chord location is varied while holding the proposed effector. An estimate for the effector is span location fixed at column 17 in Fig. 8a–8c. Figure calculated by multiplying the ADJIFOR-generated 8d–8f, conversely, varies span location while holding the PMARC effectiveness of each selected grid point by its chord location fixed at chord row 2.
specified normal displacement and then summing the The finite-difference and ADJIFOR-derived products. The application also has the ability to construct effectiveness follow similar trends, with the an input file for PMARC with the deflected geometry.
finite-difference generally predicting greater Undeflected PMARC results are subtracted from the effectiveness. Due to the similar trend information deflected PMARC calculation to determine the proposed between the two estimates, the ADJIFOR-derived effectiveness of a unit displacement of the effector on prediction is able to give high fidelity guidance to forces and moments. The effectiveness of a finite optimally place these effectors but not predict the displacement of these effectors is then compared to the effectiveness of large discrete deployments.
ADJIFOR-generated PMARC data and its prediction of In a previous paper, the ICE configuration was effectiveness.
shown to have satisfactory longitudinal control and The subsequent figures show comparisons of damping characteristics. A preliminary stability PMARC finite-difference calculations with the linear augmentor was designed to compensate for buildup of ADJIFOR-generated PMARC control unsatisfactory lateral-directional characteristics by using effectiveness for the upper aircraft skin. Figure 7a shows shape-morphing control effectors. In a follow-on paper, a half span of the grid used as input to PMARC and the the surface description of the control effector and its MATLAB tool. Some grid points are annotated to maximum normal displacement may be investigated and describe where on the 41-span location by 17-chord quantified to show the feasibility of generating these location grid the comparisons are being made. The panels moments. The original study used a shape-change device affected by the deflected grid points are highlighted in that was chosen arbitrarily with only the aid of gray. Figure 7b–7d shows the deflected surface engineering judgment. The process used in the previous finite-difference value divided by the study will be repeated to test the control effectors ADJIFOR-generated PMARC estimate. Figure 7b shows designed with information derived from an adjoint bump height effectiveness prediction (for location 17, 2) formulation of the PMARC CFD code.
is not very linear with bump height, but the ratio does approach 1.0, as expected, for small displacements. The Conclusions prediction of the effectiveness of spanwise extensions of The ADIFOR-generated CFD codes agree with the deflected grid points from point 17, 2 outboard is shown original code forces and moments. The differentiated in Fig. 7c. Similar chordwise extensions of deflected grid PMARC code outputs compare well to points from 17, 2 aft are shown in Fig. 7d. Both these central-finite-difference approximations with step sizes extensions are under predicted by ADJIFOR-generated smaller than 0.5 deg. The PMARC longitudinal PMARC data because of the nonlinearity of the coefficients CN , CA , and Cm , incremented with ∆ CD , effectiveness with deflection height and the relatively compare quantitatively with the wind tunnel data up to 6 large deflection used. The spanwise extensions shown in deg angle of attack. PMARC lateral coefficients CS , Cl , Fig. 7c were performed along row 2, very near the and Cn compare qualitatively with wind tunnel data with leading edge with a bump height of 0.2 ft. The chordwise a constant difference between zero and 6 deg. Wind extensions shown in Fig. 7d had a deflection of 0.2 ft and tunnel data trends change drastically above 6 deg angle were performed at a constant span location of column 17.
of attack, whereas the PMARC results remain linear.
American Institute of Aeronautics and Astronautics This linear region below 6 deg angle of attack is the References primary area of interest for the initial optimization of Finley, D., “Euler Technology Assessment Program shape morphing control effectors. CFL3D predicts a for Preliminary Aircraft Design Employing SPLITFLOW change in forces, moments, and derivatives above 6 deg Code With Cartesian Unstructured Grid Method,” NASA angle of attack. The ADIFOR-generated CFL3D code CR-4649, Mar. 1995.
requires lengthy execution time, but may be placed into a Charlton, E., “Numerical Stability and Control design loop if speed improvements can be obtained from Analysis Towards Falling-Leaf Prediction Capabilities of increased use of RAM caching or an implementation of Splitflow for two Generic High-Performance Aircraft parallel code.
Models,” NASA CR-1998-208730, Sept. 1998.
The ADJIFOR-generated PMARC code gave Bischof, C., Corliss, G., Green, L., Griewank, A., superior insight into the placement of relatively large Haigler, K., and Newman, P., “Automatic Differentiation displacement, discrete, and multi-effector shape-change of Advanced CFD Codes for Multidisciplinary Design,” control devices. ADJIFOR generally underpredicted Journal on Computing Systems in Engineering , Vol.3, effectiveness for these larger displacements because the No. 6, 1993, pp. 625–637.
code is only strictly valid over incrementally small, Carle, A., Green, L., Bischof, C., and Newman, P., individual displacements, but was almost always of the “Applications of Automatic Differentiation in CFD,” correct sign.
AIAA Paper 94-2197, June 1994.
The application of automatic differentiation to CFD Carle, A., Fagan, M., and Green, L., “Preliminary codes has great potential for predicting stability and Results From the Application of Automated Adjiont control derivatives. The calculation of these derivatives Code Generation to CFL3D” AIAA Paper 98-4807, Sept.
can now occur in the early design phase to influence and 1998.
improve the configuration.
Ashby, D., Dudley, M., Iguchi, S., Browne, L., and Katz J., “Potential Flow Theory and Operation Guide for Acknowledgements the Panel Code PMARC_12,” NASA Ames Research The authors would like to thank Thomas Zang and Center, Moffett Field, CA, Dec. 1992.
Jim Batterson for their support and diligent review; Scott, M., Montgomery, R., and Weston, R., HPCCP for the use of their Origin 2000; Bob Weston for “Subsonic Maneuvering Effectiveness of High aid in running PMARC; Bob Biedron for his invaluable Performance Aircraft Which Employ Quasi-Static Shape help with the CFL3D source code; Alan Carle, a Change Devices,” SPIE, 1998 International Symposium developer of ADIFOR and ADJIFOR, for his guidance in on Smart Structures and Materials , paper 3326-24, pp.
implementing these tools. 223–233.
Michael Park is supported by a NASA grant to Gainer and Hoffman, “Summary of Transformation George Washington University and would like to thank Equations and Equations of Motion Used in Free-Flight his advisor, Professor Robert Sandusky, for his guidance. and Wind Tunnel Data Reduction and Analysis,” NASA- SP-3070, 1972.
American Institute of Aeronautics and Astronautics Table 1 ADIFOR and central-difference derivative comparison 1a Angle of attack derivatives Delta CN CA Cm α α α ADIFOR 0.035454709062101 -0.003267609090179 -0.000829079154041 0.00010 0.035454709131141 -0.003267609093372 -0.000829079179268 0.00100 0.035454709061831 -0.003267609089643 -0.000829079157029 0.01000 0.035454708332922 -0.003267609023218 -0.000829079133067 0.10000 0.035454255577332 -0.003267559873352 -0.000828895343906 0.50000 0.035452260331233 -0.003267367165219 -0.000828709841046 1.00000 0.035448144567092 -0.003266980803803 -0.000829166439763 2.00000 0.035426249591663 -0.003265017618858 -0.000828574228044 3.00000 0.035390453908999 -0.003261680510116 -0.000827783833800 4.00000 0.035340142420115 -0.003257029240917 -0.000826603207321 5.00000 0.035275928055248 -0.003251137916411 -0.000825303188473 1b Angle of sideslip derivatives Delta CS Cl Cn β β β ADIFOR -0.000008538661361 -0.000834076090398 -0.000416633127904 0.00010 -0.000008538660777 -0.000834076088413 -0.000416633128362 0.00100 -0.000008538661417 -0.000834076089846 -0.000416633127798 0.01000 -0.000008541627900 -0.000834068593914 -0.000416632548211 0.10000 -0.000008540791200 -0.000834067092253 -0.000416631995080 0.50000 -0.000008539216973 -0.000834039640050 -0.000416612530736 1.00000 -0.000008539749531 -0.000833899327802 -0.000416547971397 2.00000 -0.000008532680933 -0.000833404628431 -0.000416295502544 3.00000 -0.000008525603110 -0.000832545220898 -0.000415871827408 4.00000 -0.000008510866398 -0.000831363321756 -0.000415281716613 5.00000 -0.000008493765165 -0.000829844366410 -0.000414523104020 Table 2 Force and moment coefficients at 0 degrees of sideslip.
Angle of Attack CN CA Cm CS Cl Cn 2a Simulation database 1.16 0.00649 0.01010 0.00454 0.00100 -0.00008 0.00004 1.62 0.02551 0.00917 0.00405 0.00096 -0.00003 0.00005 2.70 0.06726 0.00671 0.00297 0.00084 0.00010 0.00009 3.71 0.10607 0.00385 0.00205 0.00085 0.00026 0.00008 4.68 0.14412 0.00051 0.00126 0.00085 0.00042 0.00007 5.66 0.18022 -0.00316 0.00063 0.00074 0.00053 0.00005 6.66 0.21742 -0.00750 -0.00002 0.00074 0.00064 0.00006 7.70 0.25738 -0.01238 -0.00114 0.00039 0.00073 0.00007 8.74 0.30058 -0.01725 -0.00216 0.00015 0.00074 0.00011 9.67 0.34156 -0.02053 -0.00289 -0.00062 0.00062 0.00042 2b PMARC values 1.16 0.01178 0.00142 0.00132 0.00000 0.00000 0.00000 1.62 0.02783 0.00054 0.00120 0.00000 0.00000 0.00000 2.70 0.06566 -0.00189 0.00077 0.00000 0.00000 0.00000 3.71 0.10128 -0.00465 0.00017 0.00000 0.00000 0.00000 4.68 0.13564 -0.00773 -0.00060 0.00000 0.00000 0.00000 5.66 0.17047 -0.01128 -0.00156 0.00000 0.00000 0.00000 6.66 0.20610 -0.01534 -0.00273 0.00000 0.00000 0.00000 7.70 0.24319 -0.02002 -0.00414 0.00000 0.00000 0.00000 8.74 0.28026 -0.02518 -0.00576 0.00000 0.00000 0.00000 9.67 0.31337 -0.03019 -0.00737 0.00000 0.00000 0.00000 American Institute of Aeronautics and Astronautics Table 3 Force and moment coefficients at 2 degrees of sideslip Angle of Attack CN CA Cm CS Cl Cn 3a Simulation database 0.00 -0.00058 -0.00056 -0.00003 -0.00190 0.00071 -0.00029 2.50 -0.00047 0.00033 -0.00003 -0.00159 -0.00050 -0.00048 5.00 -0.00037 0.00121 -0.00004 -0.00127 -0.00172 -0.00067 7.50 0.00024 0.00045 -0.00011 -0.00008 -0.00231 -0.00130 3b PMARC Values 0.00 0.00004 0.00001 0.00000 -0.00087 0.00033 -0.00060 2.50 -0.00006 0.00001 0.00000 -0.00038 -0.00081 -0.00073 5.00 -0.00017 0.00002 0.00000 0.00010 -0.00194 -0.00086 7.50 -0.00028 0.00003 0.00000 0.00058 -0.00308 -0.00099 Table 4 Force and moment coefficients -2 degrees of sideslip Angle of Attack CN CA Cm CS Cl Cn 4a Simulation database 0.00 -0.00010 -0.00002 0.00005 0.00201 -0.00062 0.00031 2.50 0.00076 -0.00019 -0.00006 0.00160 0.00055 0.00049 5.00 0.00161 -0.00035 -0.00017 0.00121 0.00172 0.00067 7.50 0.00516 0.00021 -0.00070 -0.00059 0.00202 0.00135 4b PMARC Values 0.00 0.00004 0.00001 0.00000 0.00087 -0.00033 0.00060 2.50 -0.00006 0.00001 0.00000 0.00038 0.00081 0.00073 5.00 -0.00017 0.00002 0.00000 -0.00010 0.00194 0.00086 7.50 -0.00028 0.00003 0.00000 -0.00058 0.00308 0.00099 Table 5 PMARC force and moment coefficients with drag addition Angle of Attack CN CA Cm CS Cl Cn 5a 0 degrees of sideslip 1.16 0.01196 0.01010 0.00404 0.00000 0.00000 0.00000 1.62 0.02807 0.00922 0.00392 0.00000 0.00000 0.00000 2.70 0.06607 0.00678 0.00349 0.00000 0.00000 0.00000 3.71 0.10184 0.00402 0.00288 0.00000 0.00000 0.00000 4.68 0.13635 0.00092 0.00211 0.00000 0.00000 0.00000 5.66 0.17133 -0.00264 0.00114 0.00000 0.00000 0.00000 6.66 0.20711 -0.00671 -0.00003 0.00000 0.00000 0.00000 7.70 0.24435 -0.01142 -0.00145 0.00000 0.00000 0.00000 8.74 0.28158 -0.01660 -0.00307 0.00000 0.00000 0.00000 9.67 0.31483 -0.02163 -0.00469 0.00000 0.00000 0.00000 5b 2 degrees of sideslip 0.00 0.00004 0.00001 0.00000 -0.00117 0.00026 -0.00060 2.50 -0.00006 0.00001 0.00000 -0.00069 -0.00088 -0.00073 5.00 -0.00017 0.00002 0.00000 -0.00020 -0.00202 -0.00086 7.50 -0.00028 0.00003 0.00000 0.00028 -0.00315 -0.00099 5c -2 degrees of sideslip 0.00 0.00004 0.00001 0.00000 0.00117 -0.00026 0.00060 2.50 -0.00006 0.00001 0.00000 0.00069 0.00088 0.00073 5.00 -0.00017 0.00002 0.00000 0.00020 0.00202 0.00086 7.50 -0.00028 0.00003 0.00000 -0.00028 0.00315 0.00099 American Institute of Aeronautics and Astronautics Wing Characteristics 2 2 Area ... 75.12 m (808.6 ft ) Span ... 11.43 m (37.5 ft) Aspect Ratio ... 1.74 Leading Edge Sweep ... 65 deg CN CA CS α Cm CI β Relative Wind V Cn Fig. 1 LMTAS-ICE.
American Institute of Aeronautics and Astronautics −6 x 10 0.036 ADIFOR −8.5 0.0358 Central difference −8.52 0.0356 α β −8.54 CN CS 0.0354 −8.56 0.0352 ADIFOR Central difference −8.58 0.035 −4 −2 0 −4 −2 0 10 10 10 10 10 10 ∆ angle of attack, deg ∆ angle of sideslip, deg a) Normal force derivative d) Side force derivative −3 −4 x 10 x 10 −8.28 −3.25 −8.3 −3.26 −8.32 α β −8.34 Cl CA −3.27 −8.36 −3.28 ADIFOR −8.38 ADIFOR Central difference Central difference −8.4 −4 −2 0 −4 −2 0 10 10 10 10 10 10 ∆ angle of attack, deg ∆ angle of sideslip, deg b) Axial force derivative e) Rolling moment derivative −4 −4 x 10 x 10 −8.24 −4.14 −8.26 −4.15 −8.28 −4.16 α β Cn Cm −4.17 −8.3 −4.18 ADIFOR −8.32 ADIFOR Central difference −4.19 Central difference −8.34 −4 −2 0 −4 −2 0 10 10 10 10 10 10 ∆ angle of attack, deg ∆ angle of sideslip, deg c) Pitching moment derivative f) Yawing moment derivative Fig. 2 Comparison of central differences to ADIFOR−generated PMARC derivatives.
American Institute of Aeronautics and Astronautics −3 x 10 0.35 3 Sim. data, β = 2 Sim. data, β = 0 PMARC, β = 2 0.3 Wind tunnel, β = −2 PMARC, β = 0 PMARC, β = −2 CFL3D, β = 0 0.25 0.2 CN CS 0.15 −1 0.1 −2 0.05 0 −3 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg a) Normal force coefficients d) Side force coefficients −3 x 10 0.02 4 0.01 ∆ CD estimate Sim. data, β = 2 PMARC, β = 2 −0.01 0 Cl CA Sim. data, β = −2 Sim. data, β = 0 PMARC, β = −2 PMARC, β = 0 −0.02 CFL3D, β = 0 −2 −0.03 −0.04 −4 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg b) Axial force coefficients e) Rolling moment coefficients −3 −3 x 10 x 10 6 2 Sim. data, β = 2 PMARC, β = 2 Cn Cm Sim. data, β = −2 −2 PMARC, β = −2 Sim. data, β = 0 PMARC, β = 0 −4 −1 CFL3D, β = 0 −6 −8 −2 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg c) Pitching moment coefficients f) Yawing moment coefficients Fig. 3 Comparison of CFD forces and moments to simulation database.
American Institute of Aeronautics and Astronautics −3 x 10 0.35 3 Sim. data, β = 2 Sim. data, β = 0 PMARC, β = 2 0.3 Wind tunnel, β = −2 PMARC, β = 0 PMARC, β = −2 CFL3D, β = 0 0.25 0.2 CN CS 0.15 −1 0.1 −2 0.05 0 −3 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg a) Normal force coefficients d) Side force coefficients −3 x 10 0.02 4 0.01 Sim. data, β = 2 PMARC, β = 2 Cl CA Sim. data, β = −2 −0.01 PMARC, β = −2 Sim. data, β = 0 −2 PMARC, β = 0 −0.02 CFL3D, β = 0 −0.03 −4 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg b) Axial force coefficients e) Rolling moment coefficients −3 −3 x 10 x 10 6 2 Sim. data, β = 2 PMARC, β = 2 0 0 Cn Cm Sim. data, β = −2 PMARC, β = −2 −2 Sim. data, β = 0 −1 PMARC, β = 0 −4 CFL3D, β = 0 −6 −2 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg c) Pitching moment coefficients f) Yawing moment coefficients Fig. 4 Comparison of CFD forces and moments with drag addition to simulation database.
American Institute of Aeronautics and Astronautics −3 x 10 0.046 1.5 fine−grid CD−sim 0.044 ADIFOR−PMARC CD−sim ADIFOR−PMARC CD−CFL3D ADIFOR−CFL3D 0.042 0.5 med−grid α β 0.04 CN CS 0.038 −0.5 0.036 0.034 −1 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg a) Derivative of normal force coefficients d) Derivative of side force coefficients −3 −4 x 10 x 10 −2 5 fine−grid −2.5 −3 −5 −3.5 α β Cl CA −4 −10 med−grid −4.5 CD−sim CD−sim −15 ADIFOR−PMARC ADIFOR−PMARC med−grid −5 ADIFOR−CFL3D CD−CFL3D −5.5 −20 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg b) Derivative of axial force coefficients e) Derivative of rolling moment coefficients −3 −3 x 10 x 10 0 0 −0.2 −0.5 −0.4 −1 −0.6 α β CD−sim Cn Cm ADIFOR−PMARC −0.8 −1.5 CD−sim CD−CFL3D ADIFOR−PMARC −1 ADIFOR−CFL3D −2 −1.2 fine−grid −2.5 −1.4 0 2 4 6 8 10 0 2 4 6 8 10 Angle of attack, deg Angle of attack, deg c) Derivative of pitching moment coefficients f) Derivative of yawing moment coefficients Fig. 5 Comparison of central differences to ADIFOR−generated PMARC derivatives.
American Institute of Aeronautics and Astronautics −5 −5 x 10 x 10 6 6 4 4 10 10 2 2 ∂ Cl/ ∂ x ∂ Cl/ ∂ x n n 0 0 0 0 −2 −2 −10 −10 Spanwise axis, ft Spanwise axis, ft −4 −4 0 10 20 30 40 50 0 10 20 30 40 50 Chordwise axis, ft Chordwise axis, ft −6 −6 a) Roll sensitivity of upper surface d) Roll sensitivity of lower surface −4 −4 x 10 x 10 2 2 1 1 10 10 ∂ Cm/ ∂ x ∂ Cm/ ∂ x n n 0 0 0 0 −10 −10 Spanwise axis, ft Spanwise axis, ft −1 −1 0 10 20 30 40 50 0 10 20 30 40 50 Chordwise axis, ft Chordwise axis, ft −2 −2 b) Pitch sensitivity of upper surface e) Pitch sensitivity of lower surface −5 −5 x 10 x 10 4 4 2 2 10 10 ∂ Cn/ ∂ x ∂ Cn/ ∂ x n n 0 0 0 0 −10 −10 Spanwise axis, ft Spanwise axis, ft −2 −2 0 10 20 30 40 50 0 10 20 30 40 50 Chordwise axis, ft Chordwise axis, ft −4 −4 c) Yaw sensitivity of upper surface f) Yaw sensitivity of lower surface Fig. 6 ADJIFOR−generated control effectiveness contours.
American Institute of Aeronautics and Astronautics 0 8 (1,1) Cl Cm (17,2) 10 Cn (20,2) (17,5) (34,2) (17,13) Chordwise axis, ft (41,1) Finite difference / adjoint estimate (41,17) (1,17) 0 5 10 15 20 25 17 17,18 17,18,19 17,18,19,20 Spanwise axis, ft Spanwise locations a) ICE half−span grid c) Deployment of spanwise adjacent bumps 3 9 Cl 2.5 Cm Cn 1.5 Cl Cm Cn Finite difference / adjoint estimate Finite difference / adjoint estimate 0.5 1 0 0.1 0.2 0.3 0.4 0.5 2 2,3 2,3,4 2,3,4,5 Normal displacement, ft Chordwise locations b) Variation in bump height: pt. 17,2 d) Deployment of chordwise adjacent bumps Fig. 7 Comparison of ADJIFOR−generated PMARC control estimates with PMARC finite differences.
American Institute of Aeronautics and Astronautics −5 −6 x 10 x 10 6 10 FD ∆ Cl 2 Est ∆ Cl 4 FD ∆ Cl Est ∆ Cl Change in rolling moment coefficient Change in rolling moment coefficient −2 −2 2 3 5 7 9 11 13 10 17 18 24 26 34 Chordwise location Spanwise location a) Chordwise single bumps d) Spanwise single bumps −5 −5 x 10 x 10 4 6 −2 FD ∆ Cm −4 Est ∆ Cm −6 FD ∆ Cm −8 Est ∆ Cm Change in pitching moment coefficient Change in pitching moment coefficient −10 0 2 3 5 7 9 11 13 10 17 18 24 26 34 Chordwise location Spanwise location b) Chordwise single bumps e) Spanwise single bumps −5 −5 x 10 x 10 2 1 FD ∆ Cn 1.5 0.5 Est ∆ Cn 0.5 FD ∆ Cn −0.5 Est ∆ Cn −1 −0.5 −1.5 −1 Change in yawing moment coefficient Change in yawing moment coefficient −1.5 −2 2 3 5 7 9 11 13 10 17 18 24 26 34 Chordwise location Spanwise location c) Chordwise single bumps f) Spanwise single bumps Fig. 8 Comparison of ADJIFOR−generated PMARC estimates with PMARC finite differences.
American Institute of Aeronautics and Astronautics