Determination of Static and Dynamic Stability and ...

Determination of Static and Dynamic Stability and Control Derivatives with Computational Fluid Dynamics and Automatic Differentiation

by

Michael Andrew Park

B. S. Aerospace Engineering, May 1998, University of Southern California

A Thesis submitted to

The Faculty of

The School of Engineering and Applied Science of The George Washington University in partial satisfaction of the requirements for the degree of Master of Science

August, 2000

Thesis directed by

Robert Sandusky Professor, Professional Engineer

This research was conducted at NASA Langley Research Center

ABSTRACT

With the recent interest in the design of novel aircraft configurations, there is a need to determine the stability and control derivatives of these new aircraft configurations early in the design process. To directly calculate derivatives of forces and moments with respect to angle of attack, angle of sideslip, rotation rate, and control deflections computational fluid dynamics (CFD) solvers were modified and augmented with automatic differentiation. In this report, a subset of the static stability and control derivatives of a highly swept delta configuration have been computed by potential flow, Euler, and turbulent Navier-Stokes calculations to illustrate this current method. Rotation rate derivatives were computed with a turbulent Navier-Stokes code modified in this study to calculate solutions in rotating (noninertial) reference frame. Flow structure was determined with computational flow visualizations that were correlated to angle of attack trends in stability parameters. This new methodology calculated shape-change control effectiveness over the entire configuration, which cannot be directly measured in wind tunnel or flight tests. The computed derivatives compare favorably with central finite-difference derivative approximations of a wind tunnel database and equivalent CFD methods.

ii

ACKNOWLEDGEMENTS I would to thank Lawrence L. Green, my mentor, for all of the help, support, encouragement, and ideas that he has provided over the past two years. Dr. Long Yip and the NASA Langley Research Center ASCOT program provided funding and resources to conduct this research. I also owe a debt of thanks to Dr. Thomas A. Zang for supporting my George Washington University grant, detailed review of my conference papers, and advocating my work to others. Dr. Robert T. Biedron has enabled this work by patiently answering my incessant questions about the inner workings of CFL3D. I would like to thank Dr. James L. Thomas for introducing me to CFD and sparking my interest in the field. Dr. Raymond C. Montgomery and Dr. David L. Raney have shared ideas and given me a great opportunity to collaborate on innovative engineering applications. My instructor and advisor Professor Robert R. Sandusky has provided me with an educational experience by challenging and directing me during the last two years. Dr. Vladislav Klein served on this thesis review committee and his comments and suggestions have improved the quality of this report. And most of all, I must thank my sister and parents for their love, support, and understanding through my years of education.

iii

TABLE OF CONTENTS ABSTRACT.........................................................................................................................ii ACKNOWLEDGEMENTS ................................................................................................iii TABLE OF CONTENTS ....................................................................................................iv LIST OF FIGURES.............................................................................................................vi LIST OF SYMBOLS ........................................................................................................viii 1. INTRODUCTION .......................................................................................................1-1 1.1 Motivation..............................................................................................................1-1 1.2 Previous Work........................................................................................................1-3 1.3 Current Study.........................................................................................................1-4 1.4 Overview and Organization...................................................................................1-5 2. METHODS...................................................................................................................2-1 2.1 The ADIFOR Automatic Differentiation Tool.......................................................2-2 2.2 The PMARC Linear Aerodynamics CFD Code.....................................................2-4 2.2.1 PMARC Modifications for Shape-Change Control Effector Modeling..........2-5 2.3 The CFL3D Reynolds-Averaged Thin-Layer N-S Flow Solver ............................2-6 2.3.1 Noninertial Reference Frame Introduction.....................................................2-8 2.3.2 General Noninertial Frame Motion.................................................................2-9 2.3.3 General Navier-Stokes Source Term Formulation........................................2-11 2.3.4 Noninertial Constant-Rate Source Term Simplification...............................2-13 2.3.5 Constant Rate CFD Boundary Conditions ....................................................2-15 3. EXAMPLE CONFIGURATIONS...............................................................................3-1 3.1 ICE Configuration Description..............................................................................3-1 3.2 Validation Model: 2-D NACA 0012 Airfoil..........................................................3-3 4. RESULTS.....................................................................................................................4-1 4.1 ADIFOR-Generated Derivative Code Validation. .................................................4-2 4.2 ICE Static Computational Flow Visualization.......................................................4-2 4.3 ICE Static Force and Moment Calculation............................................................4-5 4.4 ICE Static Stability Derivatives .............................................................................4-9 iv

4.4.1 Simulator Database, 0–10 deg Angle of Attack............................................4-10 4.4.2 ADIFOR-Generated PMARC, 0–10 deg Angle of Attack............................4-12 4.4.3 ADIFOR-Generated CFL3D – Euler, 0–10 deg Angle of Attack.................4-14 4.4.4 ADIFOR-Generated CFL3D – Turbulent N-S, 0–10 deg Angle of Attack .........................................................................................................4-16 4.4.5 ICE Static Stability Derivative Comparison, 0–10 deg Angle of Attack......4-18 4.4.6 ICE Static Stability Derivative Comparison, 0–30 deg Angle of Attack......4-20 4.5 Constant-Rate Noninertial Dynamic Derivatives.................................................4-23 4.5.1 Euler 2-D NACA 0012 Validation Case.......................................................4-23 4.5.2 ICE ADIFOR-Generated Noninertial CFL3D Body-Axis Rate Derivatives .................................................................................................4-27 4.6 CFD Rotary-Balance Simulation.........................................................................4-30 4.6.1 ICE Velocity Vector Roll, 0 deg Angle of Attack ........................................4-31 4.6.2 ICE Velocity Vector Roll Computational Flow Visualization, 15 deg Angle of Attack ..........................................................................................4-33 4.6.3 ICE Velocity Vector Roll, 15 deg Angle of Attack ......................................4-35 4.6.4 ICE Velocity Vector Roll Computational Flow Visualization, New Moment Reference, 15 deg Angle of Attack..............................................4-37 4.6.5 ICE Velocity Vector Roll, New Moment Reference, 15 deg Angle of Attack .........................................................................................................4-39 4.7 ICE Shape-Change Effectiveness Maps...............................................................4-41 5. COMPUTATIONAL RESOURCES AND PREDICTIVE PERFORMANCE...........5-1 5.1 Computational Resources.......................................................................................5-1 5.2 Relative Predictive Performance............................................................................5-4 6. CONCLUSIONS..........................................................................................................6-1 6.1 Static Flow Visualization, Forces, Moments, and Derivatives ..............................6-2 6.2 Noninertial, Modified CFL3D Rotational Calculations .........................................6-3 6.3 Predictive Performance and Computational Resources .........................................6-5 6.4 Continuous Mold-Line Control Effectiveness .......................................................6-6 7. REFERENCES.............................................................................................................7-1

v

LIST OF FIGURES Fig. 1 ICE configuration...................................................................................................1-3 Fig. 2 Shape-change effector model and PMARC grid modification. .............................2-6 Fig. 3 Inertial and noninertial reference frame.................................................................2-9 Fig. 4 Constant-rate motion of the noninertial frame.....................................................2-14 Fig. 5 ICE configuration three-view. ...............................................................................3-2 Fig. 6 NACA 0012 385 × 97 grid close-up......................................................................3-4 Fig. 7 Low angle of attack ICE static pathlines; 0.6 Mach. .............................................4-3 Fig. 8 Moderate angle of attack ICE static pathlines; 0.6 Mach. .....................................4-4 Fig. 9 High angle of attack ICE static pathlines; 0.6 Mach. ............................................4-4 Fig. 10 ICE configuration and symbol conventions.........................................................4-5 Fig. 11 ICE wind tunnel and CFL3D longitudial force and moment comparison. ..........4-7 Fig. 12 ICE static wind tunnel lateral bias and CFL3D comparison................................4-8 Fig. 13 Sketch of longitudinal ICE derivative estimates from WT data........................4-10 Fig. 14 Wind tunnel derivaites with shaded ±10% range of the y-axis..........................4-11 Fig. 15 Comparison of ADIFOR-generated PMARC static stability derivatives with ±10% wind tunnel shaded range. ...........................................................................4-13 Fig. 16 Comparison of three grid resolutions for ADIFOR-generated Euler CFL3D derivatives with ±10% wind tunnel shaded range................................................4-15 Fig. 17 Comparison of three grid resolutions for ADIFOR-generated turbulent N-S CFL3D derivatives with ±10% wind tunnel shaded range. ...................................4-17 Fig. 18 Static stability derivative comparisons. .............................................................4-19 Fig. 19 Static stability derivatives, 0–30 deg angle of attack.........................................4-22 Fig. 20 2-D NACA 0012 Euler airfoil pitch rate derivtive, 0.1 Mach, α = 0, q = 0......4-24 Fig. 21 2-D NACA 0012 Euler airfoil pitch rate derivtive, 0.5 Mach, α = 0, q = 0......4-24 Fig. 22 CFL3D.NI.AD Euler central differencing verification, NACA 0012 , 0.5 Mach, α = 0, q = 0. ...........................................................................................................4-25 Fig. 23 2-D NACA 0012 Euler airfoil pitch rate derivtive, 0.8 Mach, α = 0, q = 0......4-26 Fig. 24 CFL3D.NI.AD Euler central differencing verification, NACA 0012 , 0.8 Mach, α = 0, q = 0. ...........................................................................................................4-27

vi

Fig. 25 Longitudinal pitch rate derivatives. ...................................................................4-28 Fig. 26 Rolling moment roll and yaw rate derivatives...................................................4-29 Fig. 27 Yawing moment roll and yaw rate derivatives. .................................................4-29 Fig. 28 Side force roll and yaw rate derivatives.............................................................4-29 Fig. 29 ICE CFL3D.NI and rotary-balance comparison, 0 deg angle of attack.............4-32 Fig. 30 ICE CFL3D.NI N-S computational flow visualization, α = 15, Ω = 0. ............4-33 Fig. 31 ICE CFL3D.NI N-S computational flow visualization, α=15, Ω = 0.2. ...........4-34 Fig. 32 ICE CFL3D.NI N-S computational flow visualization, α = 15, Ω = 0.4. .........4-34 Fig. 33 ICE CFL3D.NI and rotary-balance comparison, 15 deg angle of attack...........4-36 Fig. 34 ICE CFL3D.NI N-S computational flow visualization, new moment reference, α = 15, Ω = 0. ........................................................................................................4-37 Fig. 35 ICE CFL3D.NI N-S computational flow visualization, new moment reference, α=15, Ω = 0.2. .......................................................................................................4-38 Fig. 36 ICE CFL3D.NI N-S computational flow visualization, new moment reference, α = 15, Ω = 0.4. .....................................................................................................4-38 Fig. 37 ICE CFL3D.NI and rotary-balance comparison, new moment reference, 15 deg angle of attack. .......................................................................................................4-40 Fig. 38 Reverse mode ADIFOR-generated PMARC pitch effectiveness. .....................4-42 Fig. 39 Reverse mode ADIFOR-generated PMARC roll effectiveness.........................4-42 Fig. 40 Reverse mode ADIFOR-generated PMARC yaw effectiveness. ......................4-42

vii

LIST OF SYMBOLS

B

Position relative to an inertial frame

b

Position relative to a noninertial frame

b

Wingspan of the ICE configuration

C

Origin of a noninertial frame relative to inertial frame

Cf

Coefficient of a force or moment component

Cl, Cm, Cn

Coefficient of rolling, pitching, and yawing moment

CN, CA, CS

Coefficient of normal, axial, and side force

e

Total energy per unit volume

F, G, H

Inviscid flux terms

Fˆ , Gˆ , Hˆ

Inviscid flux terms divided by J

Fv, Gv, Hv

Viscous flux terms

ˆ ,H ˆv Fˆv , G v

Viscous flux terms divided by J

I, J, K

Normal unit vectors of an inertial frame

i, j, k

Normal unit vectors of a noninertial frame

J

Jacobian of the coordinate transformation from Cartesian to generalized coordinate

Q

Vector of the conserved variables

Qˆ

Vector of the conserved variables divided by J

R

Residual

S

Navier-Stokes equation vector source term, general motion

S

Navier-Stokes equation vector source term, constant-rate motion

u

Flow velocity vector

u∞

Freestream velocity

u, v, w

Flow velocity components

Xn

Displacement normal to surface of grid or aircraft skin

viii

Greek α

Angle of attack

β

Angle of sideslip

∆

Change in force or moment coefficient

Θ

General motion, difference in acceleration measured in the inertial and noninertial reference frames

Θx, Θy, Θz

Three components of Θ

Θ

Constant-rate motion, difference in acceleration measured in the inertial and noninertial reference frames

Θ x , Θy , Θz

Three components of Θ

ρ

Density

ω

Noninertial frame rotational rate vector

ωx , ωy , ωz

Noninertial frame rotational rate scalar components

Ω

Nondimensional rotary-balance rotation rate about velocity vector (nondimensionalization

b ) 2u∞

Superscripts

x&

First order derivative with respect to time

&x&

Second order derivative with respect to time

Subscripts xα

Partial derivative with respect to angle of attack

xβ

Partial derivative with respect to angle of sideslip

xp

Partial derivative with respect to nondimensional body axis roll rate (p nondimensionalization

xq

b ) 2u∞

Partial derivative with respect to nondimensional body

ix

axis pitch rate (q nondimensionalization xr

MAC ) 2u ∞

Partial derivative with respect to nondimensional body axis yaw rate (r nondimensionalization

b ) 2u∞

Acronyms 2-D

Two-dimensional

3-D

Three-dimensional

ADIFOR

Automatic Differentiation in FORTRAN

ADJIFOR

Automatic Adjoint Generation in FORTRAN

CFD

Computational fluid dynamics

CFL3D

Computational Fluids Laboratory 3-Dimensional

CFL3D.AD

ADIFOR-generated CFL3D

CFL3D.NI

CFL3D with noninertial, constant-rate modifications

CFL3D.NI.AD

ADIFOR-generated CFL3D with noninertial, constant-rate modifications

F77

FORTRAN 77

HASC

High-Angle-of-Attack Stability and Control

HPCCP

High Performance Computing and Communication Program

ICE

Lockheed Martin Tactical Aircraft Systems—Innovative Control Effectors

MAC

Mean aerodynamic chord

MDOB

Langley Multidisciplinary Optimization Branch

MPI

Message Passing Interface

NAS

National Aerodynamic Simulation

NASA

National Aeronautics and Space Administration

N-S

Navier-Stokes

O2K

Silicon Graphics Origin 2000

PMARC

Panel Method Ames Research Center x

PMARC.AD

ADIFOR-generated PMARC

RAM

Random Access Memory

S-A

Spalart-Allmaras one-equation turbulence model

xi

1. INTRODUCTION

Static and dynamic stability and control derivatives are essential to predicting the open- and closed-loop performance, stability, and controllability of aircraft. These stability and control parameters are also central to most control law design methods. These derivatives are the entire configuration force and moment components differentiated with respect to flow angles, angular rates, and control effector displacements. The static derivatives describe how the aircraft forces and moments vary with steady-state values of the flow angles of attack and sideslip. These static stability derivatives determine the static stability of the configuration. The rotational rate, dynamic derivatives quantify the aerodynamic damping of aircraft motions and are used with static derivatives to predict the open-loop longitudinal short period, lateral pure roll, and lateral Dutch roll behavior of the configuration. The control derivatives are the change in total aircraft forces and moments with respect to changes in configuration shape. 1.1 Motivation The classical method of determining stability and control derivatives, constructing and testing wind tunnel models, is expensive and requires a long lead-time for the resultant data. Wind tunnel tests are also limited to predetermined configurations of the model. It is impractical to construct and test numerous wind tunnel models during the conceptual design studies of aerospace vehicles. Due to cost and time limitations, 1-1

wind tunnel model construction and wind tunnel tests are minimized in the prototyping phase. Therefore, the measurement of aircraft stability and control parameters on preliminary configuration aerodynamic forces and moments is limited. A high fidelity stability and control parameter prediction tool is therefore sought. Early determination of the static and dynamic behavior of an aircraft configuration may permit significant improvement in configuration weight, cost, stealth, and performance through multidisciplinary design. Analytical and empirical methods are often employed to predict these stability and control derivative values. These estimation techniques are not suited to unconventional configurations or high-speed, compressible flows dominated by viscous effects. Evaluating unconventional configurations and control effectors is of growing interest due to the design and analysis of next generation attack, transport, and reusable launch vehicles. Examples of these new unconventional designs are the Lockheed Martin Tactical Aircraft Systems—Innovative Control Effectors (ICE) * configuration, the blended wing body, and the X-33. The use of computational fluid dynamics (CFD) methods is of growing practicality due to increases in computational power of computers and parallel algorithm development. CFD methods have the promise of allowing rapid prototyping and design cycles. High fidelity CFD calculations can be employed to compliment wind tunnel studies, by providing wind tunnel test engineers with verification of correct test measurements during wind tunnel tests. Precomputing wind tunnel test results would allow a real-time decision making ability to structure wind tunnel investigations. These CFD calculations can also separate the effects of contributing factors that are difficult to measure in wind tunnel or free flight tests. For example, the effects of configuration mold-line perturbations on total aerodynamic force and moments can be computed. Also, CFD methods can model rotating aircraft aerodynamics in high Mach and Reynolds number flows where corresponding measurements in wind tunnels equipped with moving force measurement rigs are limited.

*

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an official endorsement, either expressed or implied, of such products or manufactures by the National Aeronautics and Space Administration

1-2

A methodology of using high fidelity, Euler and turbulent Navier-Stokes (N-S) calculations gives improved capability in predicting these dynamic stability and other derivative values in compressible flow on conventional or unconventional designs. This application of Euler and N-S calculations has additional potential for providing engineers with insight, gained from higher fidelity codes, into aircraft dynamics at the preliminary design stage, when control surface size and preliminary control laws are being evaluated. CFD methods may eventually satisfy the ultimate goal of wind tunnel measurements, which is the correct prediction of full-scale behavior. 1.2 Previous Work Previous work has been attempted to determine stability derivatives from high-fidelity CFD codes; for example, Finley1 used an Euler code to compute the forces and moments for a generic configuration from which a subset of the stability derivatives can be inferred by using finite-difference methods. Charlton2 employed a similar method on the ICE configuration (Fig. 1), which included N-S solutions. Godfrey and Cliff 3 used the sensitivity equation method to compute a number of derivatives, for a flying wing configuration, including derivatives of aircraft forces and moments with respect to angle of attack and sideslip.

Fig. 1 ICE configuration. The effects of rotary motion on CFD computed aerodynamics has also been studied

with

moving-grid

time-dependent

cases

and

noninertial

calculations.

Considerable previous work performed on turbomachinery has demonstrated noninertial, rotating reference frame fluid mechanics as a means to greatly reduce computational time of time-dependent CFD (for an example see Ref. 4). Kandil and Chuang5,6 and Menzies

1-3

and Kandil 7 have demonstrated noninertial reference frame calculations for general motions on rolling aircraft stability problems. Limache and Cliff 8 devised an efficient scheme for the special case of constant-rate motion and applied this technique to stability and control work with a 2-D, unstructured grid code and the sensitivity equation method. 1.3 Current Study The purpose of this study is to apply improved differentiation techniques to CFD methods to improve the prediction of wind tunnel measurements. These CFD methods include potential flow, Euler, and N-S with turbulence modeling. The relative predictive preformance of these methods is also correlated to the required computaional resources (memory, number of processors, and execution time). The present work employs two CFD solvers, augmented via automatic differentiation, 9 to directly calculate static stability and control derivatives of the ICE configuration (Fig. 1). The application of automatic differentiation eliminates the errors introduced by approximate finite difference methods and requires less time than generating exact derivative calculation code manually. Using exact automatic differentiation is more robust than finite difference methods and removes the requirement of determining the optimal step size for the finite difference calculation. In this study, an automatic differentiation technique is illustrated with a potential flow solver and a code that solves either Euler flow equations or thin-layer N-S equations coupled with field equation turbulence modeling to include viscous effects. The flow solver outputs (forces and moments) of either code are differentiated with respect to angle of attack and the angle of sideslip, yielding a subset of the static stability derivatives. The potential flow solver was also differentiated with respect to deflections of continuous, mold-line control effectors. These mold-line control effectors will be described in more detail later in this work. The investigations into static behavior (e. g. static stability derivatives) are extended to rotational rate effects by the modification of a three-dimensional (3-D) N-S code to perform steady-state calculations in a rotating, noninertial reference frame. These CFD solutions of dynamic behavior has the potential to reduce the reliance on forced motion wind tunnel and spin tunnel tests for calculating these derivatives.

1-4

This work adopts the Limache and Cliff approach for the special case of constant-rate motion. However, the noninertial modifications are applied to a 3-D code in this study, so the effects of roll, pitch, and yaw rate can be modeled on an entire configuration. This noninertial modification of a 3-D code also allows rotary-balance tests to be simulated. To expand this study to body-axis rate derivatives an automatic differentiation technique is employed to generate derivative calculation code. The automatic differentiation technique, employed in this study is used in place of the sensitivity method of Limache and Cliff.3,8 A two-dimensional (2-D) NACA 0012 test case is performed to validate the current method with the existing Limache and Cliff approach. Computational determination of these static and dynamic derivatives is cheaper and faster than performing wind tunnel measurements and will aid rapid prototyping and multidisciplinary design. This study is an initial investigation of this computational stability and control prediction technique. Comparisons are presented among the current work, experimental measurements, and similar computational methods. 1.4 Overview and Organization First, the automatic differentiation technique used in this study is described. Then, the CFD methods are detailed with the code modifications performed in this study. The properties of the ICE example configuration are then shown. A summary of the validation of the automatic differentiation technique is presented. ICE flow structure is investigated with computational flow visualizations that were correlated to angle of attack trends in stability parameters. Computed static force and moment coefficients are compared to wind tunnel measurements. Next, automatically differentiated codes are employed to calculate static stability derivatives (force and moment coefficients differentiated with respect to angle of attack and sideslip). These computed static stability derivatives are then compared to wind tunnel central difference derivative estimates. The code modifications to calculate the aerodynamics of bodies experiencing constant-rate, dynamic motions are validated with a NACA 0012 airfoil example. At the completion of this validation, rate derivatives are presented for the ICE configuration. ICE rotary-balance tests are simulated with noninertial CFD. These rotary-balance

1-5

simulations include computational flow visualizations, forces and moments. The effect of a different rotation and moment reference center location on rotary-balance simulations is also illustrated with computed forces, moments, and computational flow visualizations. Effectiveness maps of continuous mold-line control effectors are presented. The relative predictive performance of these codes is compared to their execution requirements. Finally, conclusions are presented.

1-6

2. METHODS

Two codes were used in this study. The first, PMARC 10 (Panel Method Ames Research Center), is a fast executing potential flow code that is suitable for initial stability and control estimates. The second, CFL3D11 (Computational Fluids Laboratory 3-Dimensional) can run in ether Euler or turbulent N-S mode. The ICE configuration employed in this study is a challenging aerodynamic problem for a potential flow solver such as PMARC because of highly swept leading edges. This highly swept plan form can give rise to vortical flow at moderate to high angles of attack (>5 deg). Vortical flow is not modeled in the PMARC potential flow solver, but is modeled in the CFL3D Euler and N-S calculation. PMARC analysis is most suited to the low angle of attack region (0 to 6 deg) of the flight envelope, where the airflow is attached to the aircraft and where most cruise portions of a flight take place. This investigation was extended to a steady-state CFL3D Euler and turbulent N-S solution when PMARC results showed nonideal correlation with wind tunnel measurements at larger (>6 deg) angles of attack. Steady-state Euler and N-S calculations were more suited to these higher angles of attack and angles of sideslip (6 to 15 deg) due to expected vortical flow structures. Highly separated and time varying flow conditions (>15 deg) may require the investment of time-dependent N-S calculations to correctly predict derivatives. In spite of anticipated difficulties at higher angles of attack, PMARC was chosen for the initial work in this study because of its potential for rapid 2-1

code execution. Also, only steady-state solutions of the CFL3D Euler and turbulent N-S mode were performed in the interest of reducing the computational resources required of this study. This study required derivative information from these CFD codes (e. g. stability and control derivatives). There are a number of ways to generate derivative information from a FORTRAN 77 (F77) code and its results. The finite difference method does not require code modification, but the finite difference results are plagued with dependency on the finite difference step size. These finite difference methods can also be extremely inefficient for a large number of independent variables. For example, a central difference method requires two code analysis executions per independent variable, which is unacceptable for tens, hundreds, or thousands of independent variables. Complex number theory12,13 is an eloquent method for derivatives than can be implemented quickly in FORTRAN, because it only requires a limited amount of code modifications. Unfortunately, Complex number theory is not applicable to derivatives with respect to more than one independent variable at a time. The hand-coded sensitivity equation method is exact and robust, but requires extensive hand programming, debugging, and validation time.3,8 Automatic differentiation9 is a technique for augmenting computer programs with statements for the computation of derivatives. It relies on the fact that every function, no matter how complicated, is executed on a computer as a (potentially very long) sequence of elementary operations such as additions, multiplications, and elementary functions such as sine and cosine. By repeatedly applying the chain rule of differential calculus to the composition of those elementary operations, derivative information can be computed exactly and in a completely automated fashion. This study employs the differentiation method of the ADIFOR14,15 (Automatic Differentiation of FORTRAN) tool. 2.1 The ADIFOR Automatic Differentiation Tool ADIFOR14,15 it the automatic differentiation process applied to the PMARC and CFL3D codes in this study. This tool can apply two approaches for computing derivatives with automatic differentiation, which are the forward (direct) mode and the reverse (adjoint) mode. The forward mode applies the chain rule of differentiation to propagate,

2-2

equation by equation, derivatives of intermediate variables with respect to the input variables. In contrast, the reverse mode reorganizes these derivative statements to propagate, backward through the program, the derivatives of the output variables with respect to the input variables. A reverse mode derivative calculation first requires a forward pass through the analysis of the code so that intermediate values of the function calculation can be recorded. These recorded intermediate values are employed during a subsequent reverse pass through reverse mode ADIFOR-generated code to finally compute the reverse mode derivatives. The forward mode of automatic differentiation is more suited to problems with fewer input variables than output variables, whereas the reverse mode is better suited to problems with fewer output variables than input variables. Many hybrids of the forward and reverse modes are possible, with complementary tradeoffs in required random access memory (RAM), disk space, and execution time. The forward and the reverse method of automatic differentiation are implemented in the current version 3.0 ADIFOR tool. The reverse mode of ADIFOR 3.0 is sometimes referred to as ADJIFOR (Automatic Adjoint Generation in FORTRAN), which is demonstrated in Ref. 16. The ADIFOR tool has been developed jointly by the Center for Research on Parallel Computation at Rice University and the Mathematics and Computer Sciences Division at Argonne National Laboratory. Both the forward and the reverse techniques are available in the current ADIFOR 3.0 package. In general, to apply ADIFOR to a given F77 code, the user is only required to specify those program variable names that correspond to the independent and dependent variables of the target differentiation. Each automatic differentiation tool then determines the variables that require associated derivative computations, formulates the appropriate forward or reverse mode derivative expressions, and generates new F77 code for the computation of both the original analysis and the associated derivatives. Vectors of independent and dependant variables can be differentiated once or twice to yield Jacobian and Hessian matrices. Hessian matrices are only currently available with the forward mode, whereas Jacobian matrices are available with the forward and reverse mode. ADIFOR is intended to only handle standard F77 code, but some F77 extensions are allowed. In previous releases of ADIFOR, some manual processing is required to

2-3

formulate the forward and reverse of parallel message-passing codes, which employ F77 extensions. Manual manipulation was also required to efficiently handle the iterative nature of certain codes for increased differentiated-code performance over a standard application of ADIFOR (CFD codes are often iterative). These manual manipulations of the code have been greatly reduced with the current formal release of the ADIFOR 3.0 package. The CFL3D code differentiated in this study employs Cray style pointer statements for dynamic memory allocation and the MPI libraries to facilitate ease of use and efficient, scalable parallelization. These implementations are not standard F77 features, and therefore previous releases of ADIFOR cannot handle the code without manual preprocessing and postprocessing. The latest release of ADIFOR has reduced or eliminated much of the manual processing associated with the MPI libraries; techniques for handling the dynamic memory allocation libraries are being developed. 2.2 The PMARC Linear Aerodynamics CFD Code The PMARC potential flow solver is a F77 code that can compute surface pressures, forces, and moments of arbitrary shapes. The code is based on the assumptions of inviscid, attached, irrational, and incompressible flow. Some boundary layer and compressible corrections available, but not implemented in this study. PMARC also has a limited capability to compute solutions of unsteady, time-varying flow conditions. The input file to the program includes the set of grid points describing the shape of the geometry as a set of panels. The input file also specifies the flight condition, certain algorithmic parameters (e. g. convergence criteria), and the user-defined position of the reference point about which all moments are summed. The forces and moments are also nondimensionalized with a user-specified reference area, length of the mean aerodynamic chord (MAC), and wingspan. Nondimensionalization with respect to dynamic pressure is handled implicitly in the formulation of the code. All runs were performed with the assumption of incompressible flow or low-speed flight conditions. The original PMARC code uses swap or scratch files to record intermediate values during operation. This disk usage allows PMARC to solve problems with a large number of panels on machines with limited RAM capacity. Previous releases of ADIFOR

2-4

ignore F77 read and write statements and are therefore unable to follow the dependency of variables through scratch file read and write operations. To allow the application of the older versions of ADIFOR, the PMARC code was modified to eliminate the need to use scratch files during execution. See Ref. 17 for more details of this PMARC modification. The ADIFOR forward mode of differentiation was used to compute the stability derivatives because there are two inputs—angle of attack and angle of sideslip— compared to the six output forces and moments. The ADIFOR reverse mode is employed to calculate the derivatives used for guidance in the optimization of control effector size and placement, because the hundreds or thousands of independent variables representing normal displacements of CFD grid points greatly outnumber the three dependent variables (aircraft moment components). The derivatives of the three moment component coefficients are each evaluated separately in three code executions. 2.2.1 PMARC Modifications for Shape-Change Control Effector Modeling A novel shape-change effector is a device that generates mild control moments by slightly deforming the shape of a flying body. This slight change in the body shape can be due to a physical deflection of the skin or a change in local flow streamlines due to a flow control device. These slight shape changes are generated without opening gaps in the skin of the aircraft. To begin a multidisciplinary design of this style of shape-change control effectors, sensitivity information is sought to provide design guidance for the placement and sizing of discrete shape-change control effectiveness. This study's objective is to develop a technique for direct and efficient calculation of the sensitivity of total aircraft moments to small, continuous aircraft mold-line displacements. A normal perturbation of a grid point is a model for a small, local, deployable shape change control effector. Outward surface perturbations in the local normal direction were chosen to mimic an inflatable device. To model these shape-change devices, PMARC was modified to allow each of the grid points describing the configuration surface to be perturbed a distance, Xn , in a direction normal to the surface. The PMARC ICE configuration grid is shown on the left of Fig. 2.

2-5

Grid surface “Tent” displaced surface Normal displacement (Xn)

Fig. 2 Shape-change effector model and PMARC grid modification. The local normal direction (denoted as the dotted arrow) is determined by taking the cross vector product of vectors connecting pairs of neighboring grid points (solid lines), see Fig. 2. As this grid point is displaced along the normal vector, the surface CFD grid which models the aircraft skin is stretched to a tetrahedral, “tent” shape (outlined with dashed line). This “tent” shape is used to model a shape-change control effector. The reverse mode of the ADIFOR is employed to augment the modified PMARC code with exact, total aircraft moment derivatives with respect to the normal displacement of each grid point (ICE has over a thousand grid points). These derivatives of total aircraft moments computed at each grid point by the new ADIFOR-generated PMARC code are then interpolated over the surface of the configuration to yield control effectiveness maps. These control effector derivatives are taken at the limit as the “tent” or “bump” effector displacement goes to zero. 2.3 The CFL3D Reynolds-Averaged Thin-Layer N-S Flow Solver The Euler and turbulent N-S calculations were performed with CFL3D. 11 The CFL3D code is a F77 Reynolds-averaged thin-layer N-S flow solver for structured-volume grids. CFL3D was written primarily at NASA Langley Research Center and is undergoing continuous development and improvement. The code has the ability to compute inviscid Euler, laminar N-S, and turbulent N-S calculations on structured-volume grids. The code employs parallelization by decomposing the

2-6

computational domain into many separate blocks. These individual blocks are analyzed in separate processes that communicate with each other by means of the Message Passing Interface (MPI) standard. The code can compute solutions for steady-state or time-dependent cases, with time-dependent solutions requiring much more execution time than steady-state cases. The time-dependent solutions can be computed with either a static or moving grid. To reduce execution time requirements, CFL3D is modified in this study to compute steady-state forces and moments of bodies in simultaneous constant-speed translation and constant-rate rotation. These CFL3D modifications include adding a source term to the residual calculation and modifying the boundary and initial conditions. The noninertial CFL3D modifications are detailed in the next section. This version of CFL3D modified for noninertial, constant-rate rotation calculations is referred to CFL3D.NI (CFL3D Noninertial) in this study. The automatic differentiation of this code was initially performed with the version 2.0 of ADIFOR, so the MPI functionality of the code was handled manually to allow differentiation. The analysis portion of the code was separated from the variable array sizing routines that use C language FORTRAN extensions to dynamically allocate memory for work arrays. Only the analysis portion of the code was differentiated. The current ADIFOR 3.0 release now handles the majority of the CFL3D MPI implementation, so the manual processing of the code due to MPI has been virtually eliminated. The ADIFOR tool is anticipated to be improved to handle the entire 1.1 version of MPI, which will eliminate all manual code modification of the CFL3D MPI implementation for ADIFOR processing. CFL3D is a higher fidelity code than PMARC, used to investigate a dramatic change in the wind tunnel database derivatives between 5 and 7.5 deg angle of attack that was not predicted by the potential flow code. The Euler and viscous mode calculations were performed on different computational grids. The Euler grids have near unity grid cell aspect ratio to work most effectively with the multigrid algorithm. The viscous grids have cells clustered in the neighborhood of the boundary layer to resolve the large velocity gradients in the viscous boundary layer. The Euler grid and the viscous grid were split into many zones or blocks to utilize the parallel nature of CFL3D. Grids are

2-7

coarsened internally in CFL3D by recursively removing every other grid point in the three grid index directions. This yields separate grids or multigrid levels that are used by the code. Converged results are presented for each of the various grid fineness levels. The force and moment derivative calculations for both PMARC and CFL3D were performed with Silicon Graphics Origin 2000 (O2K) computers located at NASA Ames and Langley Research centers. The computer resources were provided by National Aerodynamic Simulation (NAS), High Performance Computing and Communication Program (HPCCP), and the NASA Langley Research Center Multidisciplinary Optimization Branch (MDOB). 2.3.1 Noninertial Reference Frame Introduction Performing calculations in a noninertial reference frame is a way of reducing the computational requirements of obtaining a solution on a rigid, accelerating CFD grid. These solutions can be time-dependent or steady-state calculations, although only steady-state solutions were performed in this study. Implementing noninertial calculations in CFL3D requires a source term to be added to the existing steady-state N-S formulation and an increment to the standard CFL3D freestream boundary conditions. The next section details the derivation of the source term and freestream boundary condition increment. In the first step of these noninertial modifications, the difference in acceleration described by vectors referenced to inertial and noninertial reference frames will be discussed. This difference will be used to form a source term to enable noninertial calculations of general motions. A noninertial general motion formulation is also presented in Ref. 5–7. Then these equations and source terms will be simplified for the special case of constant-rate motion. 8 Also the noninertial, freestream boundary conditions will be derived. The addition of this noninertial source term and freestream boundary conditions to CFL3D is all that is required to perform the noninertial calculations. The initialization of the flow field in CFL3D is also improved to speed convergence by adding a rotary component to the standard freestream seed conditions.

2-8

2.3.2 General Noninertial Frame Motion There are two reference frames depicted in Fig. 3, the inertial reference frame (denoted with uppercase symbols) and the noninertial frame (denoted with lowercase symbols). y, ωy, j

B(X, Y, Z), b(x, y, z) Y, J x, ωx, i C(X, Y, Z) X, I

z, ωz , k

Z, K

Fig. 3 Inertial and noninertial reference frame. The CFD grid (depicted as a cube) is embedded in the noninertial reference frame. Positions relative to each of these two reference frames are quantified by three scalar quantities (X, Y, Z and x, y, z) that describe location along three orthonormal unit vectors (I, J, K and i, j, k). Note that bold face type indicates vector symbols. The inertial frame is fixed in space and the noninertial frame can translate and rotate with the rotation described with three scalar components (ωx, ωy, ωz) of the rotation vector ω . In this noninertial formulation, the CFD grid is fixed to noninertial reference frame. Therefore, the stationary grid formulation of the CFL3D N-S equations is already coded in this new formulation with local (lower-case) variables. However, the existing stationary grid N-S equations in CFL3D are only formulated for a nonmoving (inertial) frames of reference. Therefore, the existing, stationary grid N-S equations are not correctly formulated for noninertial frames because the CFD grid and its associated reference frame are rotating (e. g., accelerating) in this study to simulate aircraft rotational motion. To correctly modify stationary grid N-S equations to calculate valid solutions with a translating and rotating CFD grid, the relation between the descriptions of the

2-9

same point in both reference frames (b and B) must be sought. Note that this derivation is performed with all three of the unit normal vectors of both systems parallel (e. g. I and i), which removes the necessity of a rotational coordinate transform between the inertial and noninertial reference frames. There are two points of interest in Fig. 3, The origin of the noninertial frame, C, expressed as a function of the inertial coordinates (X, Y, Z) and a fluid particle, B and b, expressed in the inertial and noninertial coordinates (X, Y, Z and x, y, z, respectively). It is very easy to express the relation of the position of a point in both coordinate systems by addition of vectors.

B = C +b

(1)

Now to find the relationship between the instantaneous velocity of a point expressed in both coordinate systems. The velocity is found by differentiating the expression for the vector relation of position Equ. (1) with respect to time. Note that there will be an added complexity to computing the derivative of any vector quantity expressed in the noninertial frame, e. g., b, because the unit normal vectors (i, j, k) are changing as a function of time due to rotation. To find the derivative of b the product rule is used on the multiplication of the scalar components (x , y, z) and unit vectors (i , j, k). The time derivative of the unit vectors (e. g.

di ) is the velocity of the end point of the vectors (e. dt

g. ω × i ).18 dB dC db = + dt dt dt

(2)

b = xi + yj + zk

(3)

dB d C dx dy dz di dj dk = + i + j + j +x +y +z dt dt dt dt dt dt dt dt

(4)

B& = C& + b& + ω× b

(5)

The relation between acceleration expressed in both frames of reference is determined in a fashion similar to the velocity relationship. The acceleration is computed by differentiating the velocity relation. The time derivative of the b and b& terms are determined in the same fashion as the derivative of the b term was derived in the velocity

2-10

relation, Equ. (5). This relation is valid for any vector quantity expressed in a noninertial frame. dB& dC& db& d ( ω × b ) = + + dt dt dt dt

(6)

db & = b + ω× b dt

(7)

db& && = b + ω × b& dt

(8)

&& = C&&+ b&&+ ω× b& + d ω × b + ω × db B dt dt

(9)

&& = C&&+ b&&+ ω× b& + ω & × b + ω× b& + ω × ( ω × b ) B

(10)

&& = C&& + b&& + 2ω × b& + ω × ( ω× b )+ ω &× b B

(11)

The difference in acceleration measured in each frame (Θ) is computed by subtracting the acceleration of a fluid particle in the noninertial frame, b&&, from the && . acceleration of the same particle expressed in the inertial frame, B

&& − b&& = C&& + 2ω × b& + ω × ( ω× b )+ ω &× b Θ=B

(12)

By accounting for this difference in acceleration (Θ), equations expressed with vectors expressed in a noninertial frame of reference can correctly model the total acceleration of a fluid particle in the inertial frame. 2.3.3 General Navier-Stokes Source Term Formulation The difference in acceleration between an inertial and noninertial frame of reference is employed to form a source term correction to the N-S equation in CFL3D. To illustrate the formulation of the source term, the existing implementation of the N-S equation in CFL3D must be examined. In the CFL3D code, the N-S equation is expressed in a regular-spaced, Cartesian coordinate system. The generalized grid coordinate system that defines the problem is internally mapped by CFL3D to this regular-spaced, Cartesian grid with a coordinate transform. The N-S equation written in the regular-spaced, Cartesian grid, coordinate system as:11 ∂Qˆ ∂ ( Fˆ − Fˆv ) ∂ (Gˆ − Gˆ v ) ∂ ( Hˆ − Hˆ v ) + + + =0 ∂t ∂ξ ∂η ∂ζ 2-11

(13)

The Jacobian (J) of the coordinate transformation from the Cartesian to the generalized coordinate system is shown here. J=

∂ ( ξ ,η ,ζ , t ) ∂ ( x, y, z , t )

(14)

Where Q is a vector of the conserved variables. The conserved variables are a combination of density, ρ, velocity components (u, v, w) and total energy per unit volume, e. The vector Qˆ is the conserved variables divided by J. Q 1 T Qˆ = = [ ρ ρ u ρ v ρ w e ] J J

(15)

The inviscid flux terms are F, G, and H and the viscous flux terms are Fv, Gv, and Hv.

ˆ, H ˆ, F ˆv , Gˆ v ,and Hˆ v flux terms are created by dividing by J in the same manner The Fˆ , G as Q. For a nondeforming mesh (J is constant with respect to time) the solution is advanced in time with the residual, R. 1 ∂Q = R(Q ) J ∂t

(16)

 ∂ ( Fˆ − Fˆv ) ∂(Gˆ − Gˆ v ) ∂( Hˆ − Hˆ v )  R(Q ) = −  + +  ∂ξ ∂η ∂ζ  

(17)

The residual is computed as:

To enable the noninertial calculations, a source term is added to the standard CFL3D residual calculation. 1 ∂Q = R(Q ) + S J ∂t

(18)

The source term is a vector with 4 nonzero components (the continuity equation is not affected). S=

T ρ  0 Θ x Θ y Θ z ( u gΘ )  J

(19)

The momentum equation source terms (Θx, Θy, and Θz) are the three components of the && − b&& ). The “work” done by this pseudo-acceleration must be pseudo-acceleration (Θ, B

included in the energy equation. The energy equation source term (u gΘ ) is the scalar

2-12

product of the local velocity, b& or u = [u

v

w] , with the pseudo-acceleration. The T

N-S equations in CFL3D are written in conservative form, so the vector source term must include the volume of each computational (J−1 ) cell and the density, ρ, of the fluid. The conserved variables (e. g. ρ, ρu, ρv, ρw, e) in this noninertial formulation are expressed in relative variables (referenced to the noninertial frame) and not absolute variables (referenced to the inertial frame).

0   && &  C + 2ω × b + ω × ( ω × b ) + ω& × b x ρ  && & &× b S =  C + 2ω × b + ω × ( ω× b )+ ω y J  C&& + 2ω × b& + ω × ( ω × b ) + ω& × b z   b& g C&& + 2ω × b& + ω × ( ω × b ) + ω& × b 

( ( (

) ) )

(

)

         

(20)

2.3.4 Noninertial Constant-Rate Source Term Simplification The goal of this study is to model constant-rate CFD grid rotation and translation with a steady-state CFD calculation. Therefore the general, noninertial source term, Equ.

& (20), is simplified for the special case of constant-rate motion. 8 In this formulation, the ω term is zero because the rotation rate is assumed to be constant by this constant-rate, steady-state formulation. The acceleration of the origin of the noninertial, grid fixed reference frame, C&& , can be formulated, because the path of the origin through space is specified by the constant-rate motion. 8 The noninertial reference frame is following a curved path (Fig. 4, the dashed arrow) through inertial space as it simultaneously translates and rotates. y, ωy, j

ω × -u ∞ -u∞

C(X, Y, Z) z, ωz, k

2-13

x, ωx, i

Fig. 4 Constant-rate motion of the noninertial frame. The origin of the noninertial reference frame must accelerate to follow this curved path. The reference frame origin acceleration (dotted arrow) is found in the same fashion as the unit normal vectors are differentiated. The freestream velocity (u∞) is expressed as a noninertial vector (it changes inertial direction with grid as they rotate together) and it is therefore differentiated in the same fashion as any vector quantity expressed in noninertial coordinates. The expression for the acceleration of a grid that is moving in curved path with constant rotation rate is: C&& = ω × C&

(21)

C&& = ω × −u∞

(22)

Note that this reference frame origin acceleration is zero when u∞ is parallel to ω. Also,

& , is zero because the rotation rate, the rotational acceleration of the noninertial frame, ω ω, is constant in a constant-rate formulation.

& terms are known for constant-rate motion, the expression, Now that the C&& and ω Equ. (12), for the difference between the acceleration computed in the inertial frame and the noninertial frame (CFD grid) can be simplified ( Θ ).

&& − b&& = ω × − u + 2ω × b& + ω × ( ω × b ) Θ=B ∞

(23)

The constant-rate source term becomes: S=

T ρ 0 Θ x Θ y Θ z ( u gΘ )    J

(24)

0   &  ω × −u∞ + 2ω × b + ω × ( ω × b ) x ρ & S =  ω × − u∞ + 2ω × b + ω × ( ω × b ) y J  ω × −u∞ + 2ω × b& + ω × ( ω × b ) z   b& g ω × − u∞ + 2ω × b& + ω × ( ω × b ) 

( ( (

where b = [ x

y

) ) )

(

z ] and b& = [ u v w] or u.

2-14

)

         

(25)

2.3.5 Constant Rate CFD Boundary Conditions For freestream boundary conditions, the fluid particles are at rest in the inertial frame, therefore their velocity, B& ∞ , is zero. The expression for the fluid velocity at the CFD grid freestream boundary conditions is b& ∞ . The aircraft, the noninertial reference frame, and the CFD grid are all translating together at a velocity, C& , which is negative the freestream velocity, u∞ . Starting with Equ. (5):

B& ∞ = 0

(26)

0 = C& + b& ∞ + ω × b ∞

(27)

b& ∞ = − C& − ω × b ∞

(28)

C& = −u∞

(29)

b& ∞ = u∞ − ω × b ∞

(30)

Therefore, the freestream boundary conditions can be described as the combination of a uniform flow component, u∞ , and a rigid body rotation component,

ω × b ∞ . In the modified version of CFL3D, CFL3D.NI, these freestream boundary conditions are also applied throughout the flow field when the computations are started to initialize the flow field. The near field or aircraft surface boundary conditions of CFL3D remain unchanged in this noninertial formulation.

2-15

3. EXAMPLE CONFIGURATIONS

Results of this study are presented for two example configurations. The first is the ICE configuration, which is a complete 3-D example. The vast majority of results are presented for the ICE configuration. The second example is the 2-D NACA 0012 airfoil, which is employed for validation of the current methods. 3.1 ICE Configuration Description The Lockheed Martin Tactical Aircraft Systems ICE configuration is a proposed configuration that is used by NASA to test the synergy of low-observable technologies with new types of control effectors. 19,20 The goal of the ICE program is a lightweight, low-observable, and highly maneuverable aircraft. The basic layout of the aircraft is a tailless, highly swept delta shape with integrated wing, body, and propulsion systems (Fig. 5).

3-1

Fig. 5 ICE configuration three-view.

This unconventional shape with no vertical tails does not lend itself to classical stability and control derivative estimation techniques that apply empirical formulas to aircraft shape parameterizations. This configuration was chosen for this study because a particular emphasis of the ICE program is to investigate a wide range of conventional and novel control effectors for their impact on controllability, weight, and stealth. Also, it has been used for a shape-change effector 6-degree of freedom simulator by Raney, et. al. 21 The full-scale version of the ICE configuration has a reference area of 808.6 ft2 . The root chord of the configuration in 517.5 in. or 43.125 ft. The mean aerodynamic chord (MAC) has a length of 345 in. or 28.75 ft. The wing span (b) of the configuration is 450 in. or 37.5 ft. The leading edge sweep of the ICE configuration is 65 deg and the aspect ratio is 1.74.19 The moment reference center of the ICE is located longitudinally at 39% of the MAC and a distance of 16% of the MAC below the body. Three different grids were employed for the panel method, Euler, and N-S portions of the study. In the PMARC portion of the study, both the right and left halves of the ICE configuration are modeled in the PMARC input file, with a total of 2560 surface panels (PMARC grid shown on the left of Fig. 2). PMARC allows half the aircraft to be 3-2

described and the solution to be mirrored in the x-z plane. Although describing only one half of the configuration would reduce the time and memory required for a converged solution, this technique would not capture the effects of a nonzero angle of sideslip. The Euler and viscous mode calculations were performed on different computational grids. Both full-span volume grids contain 2.9 million cells. The Euler grid has near unity grid cell aspect ratio to work most effectively with the multigrid algorithm. The Euler grid was stretched to form a viscous grid, which clustered cells in the neighborhood of the boundary layer to resolve the large velocity gradients in the viscous boundary layer. The Euler grid and the viscous grid were split into 18 zones or blocks to utilize the parallel nature of CFL3D. Both grids are coarsened internally by CFL3D to two additional levels (365,000 and 46,000 cells) by recursively removing every other grid point in the three grid index directions. This yields a total of three grids or multigrid levels that are used by the code. Converged results are presented for each of the three grid fineness levels. 3.2 Validation Model: 2-D NACA 0012 Airfoil The 2-D NACA 0012 airfoil example is used in this study to validate the current ADIFOR-generated noninertial CFL3D (CFL3D.NI.AD) with previously published results by Limache and Cliff.8 The 2-D grid dimensions are 385 × 97 (36,672 cells); the grid is coarsened internally by CFL3D to 193 × 49, 97 × 25, and 49 × 13. The 385 × 97 2-D NACA 0012 grid is shown in Fig. 6.

3-3

Fig. 6 NACA 0012 385 × 97 grid close-up. The NACA 0012 pitching moment center is located at the leading edge of the airfoil. Note that the grid cells have an equal streamwise distribution near the upper and lower surface of airfoil, Fig. 6. The grid is therefore best suited to subsonic flow and not the resolution of shock gradients for transonic calculations.

3-4

4. RESULTS

The results of this study focus on the ICE configuration. These results are separated into subsections. The first subsection (Sec. 4.1) explains the validation of the ADIFOR-generated codes, then Sec. 4.2 provides computational flow visualizations of the ICE configuration at various angle of attack. These flow visualizations illustrate the effects of angle of attack on the symmetric flow structure of the ICE configuration. These computational flow visualizations are employed to explain angle of attack treads in the calculated forces, moments, and their derivatives. The longitudinal forces and moment (Sec. 4.3) are presented next, followed by static stability derivatives (Sec. 4.4). Then constant-rate dynamic derivatives are presented in Sec. 4.5 for a 2-D NACA 0012 airfoil, which is mentioned for validation of the ADIFOR-generated noninertial CFL3D (CFL3D.NI.AD) with previously published results. Then constant-rate dynamic stability derivatives are presented for the ICE configuration. Rotary-balance tests (including computational flow visualizations) are simulated by means of noninertial, modified CFL3D (Sec. 4.6). Also, two different rotation and moment reference centers are examined for their effect on calculated computational flow visualizations, forces, and moments. Finally, the continuous mold-line control derivatives, computed by reverse mode ADIFOR-generated PMARC (Sec. 4.7) are presented. Before these results were generated, the ADIFOR-generated codes were verified for correct ADIFOR processing.

4-1

4.1 ADIFOR-Generated Derivative Code Validation. The first step in verifying the accuracy of the ADIFOR-generated versions of the CFD codes was to compare the forces and moments of the ADIFOR-generated CFD codes to the forces and moments of the original version of the codes. This ensured that the original analysis of the code was not corrupted in some way by ADIFOR during the new derivative code generation. This comparison showed no significant discrepancies. The static stability derivatives of the forward mode ADIFOR-generated PMARC code were then compared with second-order central-difference derivative approximations of PMARC runs at perturbed angle of attack and angle of sideslip. The comparison showed that these derivatives matched to within numerical accuracy as the step size was significantly reduced. A more complete set of results and analysis (including reverse mode ADIFOR-generated PMARC) is provided in Park, et. al. 17 A finite-difference study of ADIFOR-generated CFL3D with noninertial modifications

was

performed

for

the

NACA

0012

(Sec.

4.5.1).

Also,

a

ADIFOR-generated CFL3D with noninertial modifications NACA 0012 test case was performed to validate the noninertial modification and ADIFOR processing with an existing method (Sec. 4.5.1). An exhaustive finite-difference study of ADIFOR-generated CFL3D

was

not

deemed

necessary

when

favorable

comparison

between

ADIFOR-generated CFL3D, ADIFOR-generated PMARC, and the wind tunnel data derivatives were found.

4.2 ICE Static Computational Flow Visualization The ICE configuration (Fig. 1 and Fig. 5) flow structure was examined with pathlines. These pathlines are shown to illustrate the changes in airflow structure with increasing angles of attack. The pathlines in this study were created by the Fieldview visualization package. Fieldview calculated the pathlines by integrating the velocities determined from conserved variables stored in CFL3D output files. Pathlines for the starboard half-span of the ICE configuration are shown in Fig. 7–Fig. 9. The pathlines were seeded slightly ahead of the sharp leading edge (just outside the boundary layer).

4-2

This view is forward of the configuration looking aft at the upper surface of the ICE configuration. The view point for these visualizations is held fixed with respect to the configuration as angle of attack varies and is located on the center plane of the configuration and elevated approximately seven deg above horizontal. The pathlines were computed from a full-span N-S CFL3D solution on a grid with approximately 3 million cells. These symmetric solutions in Fig. 7–Fig. 9 were calculated at 0.6 Mach, zero deg angle of sideslip, zero rotational rate, and various angles of attack. These ICE CFL3D solutions were computed with the S-A turbulence model at a Reynolds number of 2,490,000 per foot or 71,760,000 per mean aerodynamic chord. Three levels of multigrid were used for the fine grid solution of the 0–15 deg angle of attack cases and multigrid was disabled for the fine grid solution of the 20–30 deg angle of attack cases. The structure of the symmetric flow depicted in Fig. 7–Fig. 9 aids interpretation of the subsequent figures, which depict force, moment, and stability derivative information at these flight conditions. Note the attached flow at 5 deg angle of attack (Fig. 7a). Weak leading edge vortical flow was present at 10 deg angle of attack (Fig. 7b).

Initial vortex structure

a) Angle of attack = 5 deg

b) Angle of attack = 10 deg

Fig. 7 Low angle of attack ICE static pathlines; 0.6 Mach.

4-3

Vortex burst structure

a) Angle of attack = 15 deg

b) Angle of attack = 20 deg

Fig. 8 Moderate angle of attack ICE static pathlines; 0.6 Mach. This initial leading edge vortex structure gained strength at 15 deg angle of attack (Fig. 8a). A vortex burst developed near the trailing edge at 20 deg angle of attack (Fig. 8b). This vortex burst structure is identified by an abrupt streamwise increase in vortex diameter. The initial vortex burst structure intensified and moved forward at 25 and 30 deg angles of attack (Fig. 9a and Fig. 9b).

Vortex burst structure

Vortex burst structure

a) Angle of attack = 25 deg

b) Angle of attack = 30 deg

Fig. 9 High angle of attack ICE static pathlines; 0.6 Mach.

4-4

4.3 ICE Static Force and Moment Calculation The orientation of the six forces and moments, two flow angles, and three body-axis rotational rates is shown in Fig. 10. A comparison of longitudinal forces and moment at 0.6 Mach, zero deg angle of sideslip, and zero rotational rate is shown in Fig. 11. A comparison of lateral force and moments at 0.6 Mach, zero deg angle of sideslip, and zero rotational rate is shown in Fig. 12. The comparison is the wind tunnel data (solid line, WT) present in the ICE simulator database19 and turbulent N-S CFL3D (dashed line). These ICE CFL3D solutions were computed with the S-A turbulence model at a Reynolds number of 2,490,000 per ft or 71,760,000 per MAC. The wind tunnel model test at 0.6 Mach number is estimated to have a Reynolds number of 2,500,000 per ft or 4,000,000 per MAC. The ICE wind tunnel model has a MAC of approximately 1.6 ft. Three levels of multigrid were used for the fine grid solution of the 0–15 deg angle of attack cases and multigrid was disabled for the fine grid solution of the 20–30 deg angle of attack cases. Coefficients of normal force, axial force, and pitching moment are shown in Fig. 11a–c, respectively. Coefficients of side force, rolling moment, and yawing moment are shown in Fig. 12a–c, respectively.

Fig. 10 ICE configuration and symbol conventions. 4-5

Fig. 11 ICE wind tunnel and CFL3D longitudial force and moment comparison. shows good agreement between WT and CFL3D. The WT data has more detail because it was measured at approximately 1 deg increments, which were smaller than the 5 deg increments of the CFL3D calculations. There is no flow visualization information available for the WT data, but the CFL3D pathlines will be used to infer the effects of flow structure on the CFL3D calculations, which may also indicate the flow structure effects on WT measurements. Note that the initiation and strengthening of vortical flow between 5 and 15 deg angles of attack (Fig. 7 and Fig. 8a) increased the normal force (Fig. 11a) and decreased pitching moment (Fig. 11c). The increasing strength of the vortex flow and the forward movement of the burst location over the wing between 20 and 30 deg angles of attack (Fig. 8b and Fig. 9), increased the pitching moment (Fig. 11c) which resulted in static longitudinal instability above 15 deg angle of attack. Note that CFL3D captures the radical change in Cm measured by WT (Fig. 11c). Additional sources for CFD computed static ICE longitudinal forces and moment are provided for Splitflow in Ref. 2, PMARC in Ref. 17, and HASC in Ref. 22. Fig. 12 is presented to illustrate the wind tunnel measured value of the lateral static force and moment coefficients at zero deg angle of sideslip for the ICE configuration (solid line). The CFL3D CFD code predicts a zero value for the lateral force and moments as expected for the symmetric ICE configuration (dashed line). There are local peaks in these nonzero wind tunnel measurements at approximately 11 and 23 deg angle of attack. As will be seen in the next section, the magnitudes of the WT nonzero measurements are similar to the magnitudes of the lateral angle of sideslip derivatives. These nonzero measurements in the wind tunnel data may be indicative of model misalignments or asymmetries.

4-6

Fig. 11 ICE wind tunnel and CFL3D longitudial force and moment comparison.

4-7

Fig. 12 ICE static wind tunnel lateral bias and CFL3D comparison.

4-8

4.4 ICE Static Stability Derivatives Fig. 13–Fig. 19 show flow angle static stability derivatives for the longitudinal and lateral forces and moments of the ICE configuration at 0.6 Mach, zero deg angle of sideslip, and zero rational rate. For the low angle of attack range (0–10), four derivative sources are shown. Each derivative source is depicted in a separate figure and then compared in Fig. 18. The high angle of attack range (0–30 deg) is shown in Fig. 19. The derivatives are presented in units of deg-1. Fig. 14 shows wind tunnel19 based simulation database central-difference derivatives (CD-WT). Fig. 15

is ADIFOR-generated forward mode PMARC

(PMARC.AD) values. Fig. 16 and Fig. 17 are the Euler (CFL3D.AD Eul) and turbulent N-S (CFL3D.AD N-S) ADIFOR-generated CFL3D results, respectively. Fig. 18 shows a comparison of the 2 deg CD-WT, PMARC.AD derivatives, and fine grid CFL3D.AD N-S derivatives. Fig. 19 shows a comparison of CFL3D.AD N-S and CD-WT for 0–30 deg angles of attack. The longitudinal forces and moments (CN, CA, and Cm) are differentiated with respect to angle of attack (α) and the lateral forces and moments (CS, Cl, and Cn) are differentiated with respect to angle of sideslip (β). The orientation of these forces, moments, and flow angles is shown in Fig. 10. The symbols CN, CA, and CS represent the nondimensional coefficients of aerodynamic forces referenced to the body axes in the directions upward, aft, and toward the right wing tip which are depicted in Fig. 14–Fig. 19 subplots a, b, and d, respectively. The symbols Cl, Cm, and Cn represent the nondimensional coefficients of moments in the roll, pitch, and yaw axes shown in subplots c, e, and f, respectively. The shaded areas shown in Fig. 14–Fig. 18 are centered about the centrally differenced wind tunnel data and extend ±10% of the y-axis (data range) of the plots. These shaded areas are provided for reference to wind tunnel derivatives. The y-axes and shaded areas are identical in Fig. 14–Fig. 18. The width of the shaded band was chosen arbitrarily, but was indicative of desired wind tunnel prediction accuracy. Charlton reports, the prediction of force and moment values desired by control engineers is roughly within ±10% of wind tunnel measured value data range.2

4-9

4.4.1 Simulator Database, 0–10 deg Angle of Attack The static stability derivatives calculated from central-finite-difference estimates of the ICE wind tunnel database are presented in Fig. 14. The wind tunnel database longitudinal data was measured at nonuniform break points. To yield derivatives, the longitudinal data was center differenced for each pair of break points and plotted at the average angle of attack of the pair of the break points, which is consistent with the central difference approximation. Fig. 13 graphically sketches this process with the squares being the original longitudinal data and the circles denoting the angle of attack of the central-difference calculation. The wind tunnel database lateral data is central differenced between ±2, ±4, and ±6 deg angle of sideslip to yield derivative values at angle of

force or moment (C f)

Coefficient of

sideslip equal to zero.

dCf / dα Cf(α) dCf dα dα / 2

dα / 2

Angle of attack (α)

Fig. 13 Sketch of longitudinal ICE derivative estimates from WT data. The ±2 deg centrally differenced lateral derivatives is assumed to be the most accurate for small perturbations. Note that the lateral derivative values in Fig. 14 are dependent on the step size of the finite difference calculation. This step size dependence is possibly due to (possibly flow) nonlinearities in the measured force and moment coefficients. The spread in lateral, angle of sideslip derivatives (due to varying step size) at constant angle of attack can indicate the nonlinearities or uncertainties in the centrally differenced data set. The shaded areas shown in Fig. 14, and subsequent figures, is centered about the longitudinal and lateral ±2 deg centrally differenced wind tunnel data and extends ±10% of the range of the y-axis.

4-10

Fig. 14 Wind tunnel derivaites with shaded ±10% range of the y-axis.

4-11

4.4.2 ADIFOR-Generated PMARC, 0–10 deg Angle of Attack The derivatives were plotted for PMARC.AD in Fig. 15. These are relatively inexpensive computations as compared to CFL3D and CFL3D.AD. Much of the data trends below 6 deg angle of attack are captured with these calculations. Note that PMARC results are essentially a linear function of angle of attack. The nonlinear effects of vortical flow is not modeled. The lateral derivatives of the PMARC.AD codes agree reasonably well with wind tunnel data below 6 deg angle of attack; at higher angles of attack the wind tunnel data changes radically while PMARC.AD remains linear. To more precisely predict derivative values and the change in wind tunnel data derivative trends above 6 deg angle of attack a higher fidelity aerodynamic modeling is required.

4-12

Fig. 15 Comparison of ADIFOR-generated PMARC static stability derivatives with ±10% wind tunnel shaded range.

4-13

4.4.3 ADIFOR-Generated CFL3D – Euler, 0–10 deg Angle of Attack The derivatives were plotted for both the Euler and turbulent N-S modes of CFL3D.AD. The CFL3D.AD Eul results are shown in Fig. 16. The three grid sizes are depicted: coarse (C), medium (M), and fine (F). CFL3D.AD Eul over predicts the effect of vortical flow on normal force slope (Fig. 16a). Both CFL3D.AD Eul and PMARC.AD have difficulty with the pitching moment derivative (Fig. 16c and Fig. 16c), indicating the need for a viscous Navier-Stokes calculation. CFL3D.AD Eul lateral derivatives are similar to the PMARC.AD values at 2.5 and 5 deg angle of attack. CFL3D.AD Eul converged to values near those of the wind tunnel at 7.5 deg angle of attack for the coarse and medium grids. The 7.5 deg angle of attack fine-grid derivatives converged to significantly different values (Fig. 16e–Fig. 16f), indicating the modeling of a different flow structure on this finer mesh than the two coarser meshes. The computation flow visualizations using the N-S mode of CFL3D in Fig. 7 predict a transition from attached flow to vortical flow between 5 and 10 deg angles of attack. The 7.5 angle case for CFL3D.AD Eul may be transitioning from attached to vortical flow structure between the medium (M) and fine (F) grid resolutions. Attached and vortical flow structures have different sensitivity to angle of attack or sideslip perturbations, which result in different static stability derivatives. Three levels of multigrid are used for the CFL3D.AD Eul fine-grid calculations, which may conflict with the change in flow structure between the medium and fine grid resolutions. Instead of focusing excess attention on these Euler calculations which have known difficulties with incipient separation conditions, resources where focused on performing turbulent N-S calculations.

4-14

Fig. 16 Comparison of three grid resolutions for ADIFOR-generated Euler CFL3D derivatives with ±10% wind tunnel shaded range.

4-15

4.4.4 ADIFOR-Generated CFL3D – Turbulent N-S, 0–10 deg Angle of Attack CFL3D.AD N-S (shown in Fig. 17) had much better differentiated flow solver convergence properties than the CFL3D.AD Eul. These ICE CFL3D.AD N-S solutions were computed with the S-A turbulence model at a Reynolds number of 2,490,000 per foot or 71,760,000 per mean aerodynamic chord. Three levels of multigrid were used for the fine grid solution. The CFL3D.AD N-S also results show a very good comparison with the wind tunnel database. The normal force and axial force derivatives are well predicted by CFL3D.AD N-S at 5 deg angle of attack and lower. The CFL3D.AD N-S pitching moment derivative value is offset from the CD-WT but similar inflections in the data set are shown. The drastic change in the lateral derivatives at 7.5 deg angle of attack is correctly modeled by CFL3D.AD N-S, but the comparison at 10 deg is degraded for CS and Cn. The 10 deg angle of attack case is near a significant peak in lateral force and moment coefficient measurement of the wind tunnel at zero deg angle of sideslip and 11 deg angle of attack (Fig. 12). This significant peak in lateral wind tunnel measurement may affect the centrally differenced wind tunnel derivative estimate.

4-16

Fig. 17 Comparison of three grid resolutions for ADIFOR-generated turbulent N-S CFL3D derivatives with ±10% wind tunnel shaded range.

4-17

4.4.5 ICE Static Stability Derivative Comparison, 0–10 deg Angle of Attack To facilitate comparison, the 2 deg CD-SIM, PMARC.AD, and fine-grid CFL3D N-S are plotted together on Fig. 18. The Euler results were omitted due to convergence difficulties at 7.5 deg angle of attack. The ADIFOR-generated PMARC results give good qualitative trend information below 6 deg angle of attack. The ADIFOR-generated turbulent N-S CFL3D results have very good agreement to the wind tunnel database derivatives up to 5 deg longitudinally and 7.5 deg laterally. Note the N-S results are able to model the drastic change in the wind tunnel lateral derivatives at 5 deg angle of attack. Finite-difference approximations of the lateral derivatives in the wind tunnel database can be interpreted to be very sensitive to step size and to the angle of sideslip about which the derivative approximation is centered, due to the nonlinear nature of the wind tunnel data. Using a finite-difference method with a large step size (2, 4, or 6 deg) may mask the actual small-disturbance behavior of these nonlinear functions of angle of sideslip.

From

previous

experience,

it

is

expected

that

the

automatic

differentiation-generated derivatives for CFD codes that simulate the required flow physics present in the full scale or wind tunnel models will prove to be more accurate than derivatives of wind tunnel measurements obtained by coarse finite differences.

4-18

Fig. 18 Static stability derivative comparisons.

4-19

4.4.6 ICE Static Stability Derivative Comparison, 0–30 deg Angle of Attack The results presented in Fig. 19 are an extension to higher angles of attack of the Fig. 14 and Fig. 17 values. Fig. 19a–c depict the longitudinal angle of attack derivatives. The CD-WT derivatives have more variation than the fine-grid CFL3D.AD N-S derivatives, but the general trends of CD-WT are shown by CFL3D. AD. The CD-WT derivatives had large peaks at 12–15 and 21–24 deg angles of attack. These angle of attack ranges are where flow structure is changing as predicted by the change computational flow visualizations between Fig. 7b and Fig. 8a and between Fig. 8b and Fig. 9a. These peaks in Fig. 19 correspond to the angle of attack ranges for large peaks in WT measured lateral forces and moments at zero angle of sideslip in Fig. 12. Fig. 19d–f shows the comparison among the lateral angle of sideslip derivatives for three central-finite-difference estimates from wind tunnel data19 (CD-WT) and an ADIFOR-generated CFL3D solution (CFL3D.AD). The CFL3D.AD derivatives are dashed lines and the lateral CD-WT derivatives are the symbols with a central-finite-difference step of ±2, ±4, and ±6 deg angle of sideslip for the circle, square, and diamond, respectively. The ±2 deg CD-WT data is connected with solid lines because the smallest central-finite-difference step (±2) is presumed to be the most accurate of the three finite difference step sizes for small sideslip disturbances. All three finite difference step sizes are shown to give an indication of the nonlinearities or measurement noise in the wind tunnel data. The derivatives in Fig. 19 are presented in the units of deg−1 . The effects of the vortical flow structure (Fig. 7–Fig. 9) can be seen clearly in the lateral force and moments angle-of-sideslip derivatives (Fig. 19d–f). The initiation and strengthening of vortical flow between 5 and 10 deg angles of attack (Fig. 7) can be interpreted to have sharply influenced the angle-of-attack trends of CSβ and Cnβ (Fig. 19a and Fig. 19c) computed by CD-WT and CFL3D.AD. Then, the derivatives CSβ and Cnβ (Fig. 19a and Fig. 19c) dramatically reversed angle-of-attack trends above 10 deg angle of attack. The CFL3D.AD Clβ (Fig. 19b) derivative showed excellent agreement with CD-WT for 0 to 15 deg angles of attack. The Clβ comparison deteriorated at higher (20– 30 deg) angles of attack. 4-20

As angle of sideslip varies, each wing experiences different effective leading-edge sweep angles. Due to the highly swept (65 deg) leading edge of the ICE configuration the vortical flow field over the wing may be sensitive to changes in effective leading-edge sweep angle. Therefore, the calculation of a vortex burst structure that formed symmetrically at 20 deg angle of attack (Fig. 8b) may be produced asymmetrically at lower (10–15 deg) angles of attack. An asymmetric, bursting vortex structure may have been responsible for the dramatically reversed angle-of-attack trends in the lateral derivatives (Fig. 19d–f). The ICE configuration does not have any vertical surfaces, so the magnitude of CSβ and Cnβ (Fig. 19d and Fig. 19f) was reduced as compared to a configuration with vertical surfaces. The small magnitude of CSβ and Cnβ may have hindered measurement accuracy and exacerbated comparison of CD-WT with CFL3D.AD. The wind tunnel lateral force and moment coefficient measurements at zero deg angle of sideslip had significant peaks at 11 and 23 deg angles of attack (Fig. 12). CFL3D N-S calculated zero lateral coefficients at zero angle of sideslip for this symmetric configuration. These significant peaks in lateral wind tunnel measurement may have affected CD-WT and degraded the comparison of CD-WT and CFL3D.AD N-S (especially CS and Cn).

4-21

Fig. 19 Static stability derivatives, 0–30 deg angle of attack.

4-22

4.5 Constant-Rate Noninertial Dynamic Derivatives The two examples in the constant-rate dynamic stability derivative study were a 2-D Euler study of NACA 0012 airfoil and a 3-D turbulent N-S calculation on the ICE configuration. The NACA 0012 study will be detailed first, because it was used for initial validation by comparisons to existing methods. 4.5.1 Euler 2-D NACA 0012 Validation Case The NACA 0012 study focused on the effect of pitch rate on the coefficients of normal force and pitching moment (Fig. 20–Fig. 24) at zero deg angle of attack and zero pitch rate. The derivatives, CNq (a subplots) and Cmq (b subplots), of these force and moment coefficients with respect to nondimensional pitch rate were computed by the ADIFOR-generated, noninertial CFL3D code (CFL3D.NI.AD) and finite difference of noninertial CFL3D (CFL3D.NI). The pitch rate derivatives are nondimensionalized by dividing by the airfoil chord and multiplying by two times the freestream velocity. The NACA 0012 pitching moment center is located at the leading edge of the airfoil. The convergence history of the derivative values is shown in Fig. 20 through Fig. 24. The discontinuities in the derivative convergence history are due to mesh sequencing from a coarser to a finer mesh every 500 iterations. A maximum of three levels of multigrid was employed on the finer meshes. The 2-D grid dimensions (49 × 13, 97 × 25, 193 × 49, and 385 × 97) are denoted for each mesh sequencing level. Up to three levels of multigrid was used for the NACA 0012 case. The derivative values are compared to results computed by a similar method published by Limache and Cliff (SFLOW),8 a panel method (QUADPAN), 23 and a vortex lattice method (VORLAX). 24 These 2-D NACA 0012 cases shown in Fig. 20 through Fig. 24 were chosen for initial validation of CFL3D.NI.AD. To improve convergence, a blend of half standard CFL3D and half CFL3D low-Mach-number preconditioning was applied for the 0.1 Mach (Fig. 20) case. This preconditioning option was not applied to the 0.5 Mach (Fig. 21 and Fig. 22) or 0.8 Mach (Fig. 23 and Fig. 24) cases. Note that the CFL3D.NI.AD

4-23

derivative values are in excellent agreement with the SFLOW values. For the 0.1 and 0.5 Mach cases, the differences between CFL3D.NI.AD and SFLOW, although small, are most likely due to the formulation differences between the flow solvers in CFL3D.NI.AD and SFLOW. The SFLOW code employs the hand-coded sensitivity equation technique and an unstructured grid discretization, whereas CFL3D.NI.AD is an automatically differentiated structured grid formulation.

Fig. 20 2-D NACA 0012 Euler airfoil pitch rate derivtive, 0.1 Mach, α = 0, q = 0.

Fig. 21 2-D NACA 0012 Euler airfoil pitch rate derivtive, 0.5 Mach, α = 0, q = 0.

To verify that CFL3D.NI.AD was producing correct derivatives it was compared to center finite difference estimates of CFL3D.NI cases at 0.5 Mach (Fig. 22). The center difference step is ±0.0001 nondimensional pitch rate for the circles and ±0.0010

4-24

nondimensional pitch rate for the squares. Note the excellent agreement between the center difference estimates and the ADIFOR-generated code values at 0.5 Mach.

Fig. 22 CFL3D.NI.AD Euler central differencing verification, NACA 0012 , 0.5 Mach, α = 0, q = 0. The 0.8 Mach case (Fig. 23) shows poor convergence properties. The poor convergence of CFL3D.NI.AD at 0.8 Mach may be due to the interaction of a shock, the flux limiter implemented in CFL3D, and the automatic differentiation technique. The CFL3D smooth flux limiter11 tuned to κ = 1/3 was employed for the NACA 0012 study. This poor convergence may be due to the automatic differentiation technique attempting to formulate the continuous derivative of a shock and flux limiter, which does not have a continuous derivative. The 0.8 Mach case is also the worst comparison to SFLOW. Only the final value for SFLOW is quoted in Ref. 3; therefore the SFLOW 0.8 Mach case may or may not be fully converged. The convergence was not improved by disabling multigrid calculations or performing additional iteration cycles. At 0.8 Mach, the final value of CFL3D.NI.AD and SFLOW differ in normal force pitch rate derivative by 4.4% (Fig. 23a) and in pitching moment pitch rate derivative by 8.9% (Fig. 23b).

4-25

Fig. 23 2-D NACA 0012 Euler airfoil pitch rate derivtive, 0.8 Mach, α = 0, q = 0. To further verify that CFL3D.NI.AD was producing correct derivatives and to examine differentiated flow solver convergence difficulties the differentiated code was compared to center finite difference estimates of CFL3D.NI cases at 0.8 Mach (Fig. 24). The center difference step is ±0.0001 nondimensional pitch rate for the circles, ±0.0005 for the squares, ±0.0010 for the diamonds, and ±0.0020 for the triangles. Note the inconclusive trends in the derivative value with variations in finite difference step size. This is a difficult derivative problem to calculate with finite difference methods due to the discrete nature of the flux limiter in the presence of the shock. Note that the smallest finite difference size (±0.0001) is the closest, but this may be a fortuitous match. Also note that the majority of the values presented for 0.8 Mach are within 10% of each other, which may be an acceptable prediction capability for the difficult problem of transonic flow sensitivities.

4-26

Fig. 24 CFL3D.NI.AD Euler central differencing verification, NACA 0012 , 0.8 Mach, α = 0, q = 0. 4.5.2 ICE ADIFOR-Generated Noninertial CFL3D Body-Axis Rate Derivatives Figures Fig. 25 through Fig. 28 show the ICE dynamic derivatives computed by the DYNAMIC 25 code and fine-grid CFL3D.NI.AD in N-S mode. The DYNAMIC code utilized strip theory and the results of the high-angle-of-attack stability and control (HASC)22,26 prediction code to calculate the dynamic derivatives. The HASC code employs VORLAX24 and empirical corrections to predict configuration forces and moments at various flow angles and rotational rates. The derivatives were computed at zero rotational rate, zero angle of sideslip, and various angles of attack. The CFL3D.NI.AD dynamic derivatives were computed assuming rotations about the moment center of the configuration, which according to the test data is located slightly below the body. The moment reference center of the ICE is located longitudinally at 39% of the MAC and a distance of 16% of the MAC below the body. These ICE CFL3D.AD N-S solutions were computed with the S-A turbulence model at a Reynolds number of

4-27

2,490,000 per foot or 71,760,000 per mean aerodynamic chord length. Three levels of multigrid were used on the 0–15 deg angle of attack cases. Multigrid was disabled for the fine-grid 20–30 deg angle of attack cases. The longitudinal pitch rate derivatives CNq and Cmq are shown in Fig. 25a and Fig. 25b, respectively. The longitudinal dynamic derivatives were nondimensionalized by dividing by the MAC (345 in.) and multiplying by two times the freestream velocity.

Fig. 25 Longitudinal pitch rate derivatives. DYNAMIC results are not provided for the normal force pitch rate (q) derivative (Fig. 25a), so only CFL3D.NI.AD are shown. Note that the CFL3D.NI.AD calculation of Cmq (Fig. 25d) was consistently more negative than the combined analytical and vortex lattice method of DYNAMIC and VORLAX. This trend agrees with those of both SFLOW and CFL3D.NI.AD when compared to VORLAX for the 2-D NACA 0012 case (Fig. 20b and Fig. 21b). CNq increased and Cmq decreased with increasing angle of attack for attached flow (0–6 deg angles of attack). These angle of attack trends reversed in vortical flow (6–15 deg angles of attack) and another inflection point is seen at the inception of bursting vortex flow (15–20 deg angles of attack). The rolling moment dynamic derivatives Clp and Clr are shown in Fig. 26a and Fig. 26b, respectively. The yawing moment dynamic derivatives Cnp and Cnr are shown in Fig. 27a and Fig. 27b, respectively. The side force dynamic derivatives CSp and CSr are shown in Fig. 28a and Fig. 28b, respectively. The lateral dynamic derivatives were nondimensionalized by dividing by the wingspan (450 in.) and multiplying by two times

4-28

the freestream velocity. There was no forced, oscillatory motion wind tunnel data for comparison.

Fig. 26 Rolling moment roll and yaw rate derivatives.

Fig. 27 Yawing moment roll and yaw rate derivatives.

Fig. 28 Side force roll and yaw rate derivatives. 4-29

The roll rate (p) derivatives (Fig. 26a, Fig. 27a, and Fig. 28a) showed a reversal of angle of attack trends at 5 deg angle of attack. The reversals of the q and p derivative trends at 5 deg angle of attack corresponded to the indication of vortical flow at 10 deg angle of attack in Fig. 7b. These p derivatives also a change in angle of attack trends at 15–20 deg angle of attack, which was slightly below the indication of vortex bursting in the static pathlines (Fig. 8b). A roll rate creates differential angles of attack on each wing, which may induce asymmetric vortical burst structures at lower angles of attack than a zero-roll-rate, symmetric case. The yaw rate (r) derivatives (Fig. 26b, Fig. 27b, and Fig. 28b) had consistent trends in angle of attack at 15 deg angle of attack and lower. These trends became less consistent at 20, 25, and 30 deg angles of attack, which corresponded with the initial indication of a symmetric vortex burst structure in Fig. 8b and Fig. 9. The CFL3D.NI.AD differentiated flow solver had convergence difficulties at 20, 25, and 30 deg angles of attack. The 30 deg angle of attack case never reached a steady-state value, so an average of the last two thousand iterations is presented. A similar technique for extracting ADIFOR-generated derivatives from a code that exhibits nonsteady, oscillatory behavior is provided in Ref. 27. These convergence difficulties may have been due to the presence of bursting vortex structures, with their inherent unsteadiness and increased sensitivity to disturbances. These high angle-of-attack conditions may be more suitable to a time-accurate solution, but in the interest of minimizing computational resource requirements, that approach was not attempted in this study. 4.6 CFD Rotary-Balance Simulation The noninertial, modified CFL3D (CFL3D.NI) code was used to perform velocity vector rolls on ICE configuration (Fig. 29–Fig. 33). In these velocity vector rolls, the rotation vector was parallel to the freestream velocity vector; this condition simulated a wind tunnel rotary-balance (ROT-BAL) test. The ICE configuration was rotated about the moment reference center of the configuration in this study, which according to test data is located slightly below the body. The moment reference center of the ICE is located longitudinally at 39% of the MAC and a distance of 16% of the MAC below the body.

4-30

Mach 0.3 was chosen to simulate the incompressible conditions of the low-speed, rotary-balance tests. These ICE CFL3D.NI N-S solutions were computed with the S-A turbulence model at a Reynolds number of 2,490,000 per foot or 71,760,000 per MAC length. Assuming the rotary-balance test velo city was 30 ft per sec. (1 lb. per square ft. dynamic pressure), the Reynolds number per foot of the rotary-balance test at sea level would be 190,000. The Reynolds number per MAC length of the rotary-balance test is estimated to be 300,000. Three levels of multigrid were used on the 0 and 15 deg angle of attack fine-grid solutions for all rotational rates. The change (∆) in force or moment coefficient between cases nonrotating and rotating about the velocity vector is shown in Fig. 29 and Fig. 33. Computational flow visualizations are shown in Fig. 30–Fig. 32. The rotation rate (Ω ) about the velocity vector was nondimensionalized by multiplying by the wingspan b (450 in.) and dividing by two times the freestream velocity (u∞), with a positive rotational rate indicating the starboard wing was descending. 4.6.1 ICE Velocity Vector Roll, 0 deg Angle of Attack Fig. 29 shows a comparison of wind tunnel rotary-balance data19 (ROT-BAL, solid line) and CFL3D.NI (dashed line) at 0.3 Mach, zero deg angle of attack, and zero deg angle of sideslip. Note that nonlinear effects with rotational rate were modeled in CFL3D.NI. The change in normal force (∆CN, Fig. 29a) due to rotation seen in ROT-BAL was not well predicted by CFL3D.NI. The change in axial force (∆CA, Fig. 29b) due to rotation was very similar between ROT-BAL and CFL3D.NI, especially for rotational rates less than 0.2. The shape of the change in pitching moment (∆Cm, Fig. 29c) due to rotation measured by ROT-BAL was predicted by CFL3D.NI, but there was a constant offset of approximately 0.002. The change in side force (∆CS, Fig. 29d) due to rotation was assumed to be an odd function as calculated by CFL3D.NI, but ROT-BAL showed inconclusive trends. The change in rolling moment (∆Cl, Fig. 29e) due to rotation was the best lateral comparison of CFL3D.NI with ROT-BAL. The nonlinear increase in the damping or restoring rolling moment with the larger rotational roll rates was also modeled. The change in yawing moment (∆Cn, Fig. 29f) due to rotation calculated by CFL3D.NI was much greater in magnitude than the ROT-BAL measured effects. 4-31

Fig. 29 ICE CFL3D.NI and rotary-balance comparison, 0 deg angle of attack.

4-32

4.6.2 ICE Velocity Vector Roll Computational Flow Visualization, 15 deg Angle of Attack Fig. 30 through Fig. 32 show the ICE configuration at 0.3 Mach, 15 deg angle of attack and zero deg angle of sideslip, performing velocity vector rolls at various rotational rates. The ICE configuration is rotated about the moment reference center to perform the CFL3D.NI solutions. These pathlines are shown to illustrate the changes in airflow structure with increasing rotational rate. Pathlines for both halves of the ICE configuration are shown in Fig. 30–Fig. 32. The pathlines were seeded slightly ahead of the sharp leading edge (just outside the boundary layer). This view is forward of the configuration looking aft at the upper surface of the ICE configuration. The view point is held fixed with respect to the configuration and is located on the center plane of the configuration and elevated approximately seven deg above horizontal.

Fig. 30 ICE CFL3D.NI N-S computational flow visualization, α = 15, Ω = 0. The 0.2 and 0.4 rotational rate cases (Fig. 31 and Fig. 32) showed a much tighter vortex core on the ascending, port wing than the descending, starboard wing. The block arrows in Fig. 31 and Fig. 32 depicted the direction of the rotation of the aircraft about the velocity vector. The 0.4 rotational rate case (Fig. 32) depicted a vortex burst on the descending, starboard wing.

4-33

Fig. 31 ICE CFL3D.NI N-S computational flow visualization, α=15, Ω = 0.2.

Fig. 32 ICE CFL3D.NI N-S computational flow visualization, α = 15, Ω = 0.4. From this point of view, the vortex wakes in Fig. 31 and Fig. 32 appear to be converging, but actually were spiraling around the rotation vector.

4-34

4.6.3 ICE Velocity Vector Roll, 15 deg Angle of Attack Fig. 33 shows a comparison of wind tunnel rotary-balance data19 (ROT-BAL, solid line) and CFL3D.NI (dashed line) at 0.3 Mach, 15 deg angle of attack, and zero deg angle of sideslip. Note that nonlinear effects with rotational rate were modeled in CFL3D.NI. The increase in normal force (∆CN, Fig. 33a) due to rotation was very similar between ROT-BAL and CFL3D.NI. The change in axial force (∆CA, Fig. 33b) due to rotation was very similar in magnitude between ROT-BAL and CFL3D.NI, but opposite in sign. The change in pitching moment (∆Cm, Fig. 33c) due to rotation was assumed an even function as calculated by CFL3D.NI, but ROT-BAL showed inconclusive trends. The change in side force (∆CS, Fig. 33d) due to rotation was assumed to be an odd function as calculated by CFL3D.NI, but ROT-BAL showed inconclusive trends. The change in rolling moment (∆Cl, Fig. 33e) due to rotation was the best lateral comparison of CFL3D.NI with ROT-BAL. The ROT-BAL measured effect of rotational rate on rolling moment decreased from 0 to 15 deg angle of attack where the CFL3D.NI predicted an increase in rotational effect from 0 to 15 deg angle of attack. The change in yawing moment (∆Cn, Fig. 33f) due to rotation calculated by CFL3D.NI was much greater in magnitude and opposite in sign of the ROT-BAL trend. The nonlinear effects with rotational rate as calculated by CFL3D.NI in Fig. 33 can be correlated to the calculation of a vortex burst structure over the descending wing as illustrated in Fig. 31 and Fig. 32. The difference between ROT-BAL and CFL3D.NI (Fig. 29 and Fig. 33) may be due a number of factors. A possible explanation of ROT-BAL asymmetries may be model asymmetries or installation misalignments. The poor comparisons of ROT-BAL and CFL3D.NI may be due to rotation about different locations for the experimental and computational cases. The CFL3D.NI code simulated rotation about the reported moment center of the configuration, which is outside the model. The ROT-BAL tests may or may not have rotated the model about that moment reference center location. The Reynolds number of the ROT-BAL tests are estimated to be two hundred times smaller than the CFL3D.NI simulations. This may have a significant effect on the 15 deg angle of attack CFD predicted separated, busting vortex flow and the measured versus calculated force and moment coefficients. 4-35

Fig. 33 ICE CFL3D.NI and rotary-balance comparison, 15 deg angle of attack.

4-36

4.6.4 ICE Velocity Vector Roll Computational Flow Visualization, New Moment Reference, 15 deg Angle of Attack Fig. 34 through Fig. 36 show the ICE configuration at 0.3 Mach, 15 deg angle of attack and zero deg angle of sideslip, performing velocity vector rolls at various rotational rates. A new moment reference center is employed in this section to investigate the effect of this location on the flow visualization. The ICE configuration is rotated about the moment reference center to perform the CFL3D.NI solutions. This moment reference center is located longitudinally at 39% of the MAC and the center of the body in the z-direction. This new location is 28% of the MAC higher than the original moment reference center of Fig. 30–Fig. 32.

Fig. 34 ICE CFL3D.NI N-S computational flow visualization, new moment reference, α = 15, Ω = 0.

4-37

Fig. 35 ICE CFL3D.NI N-S computational flow visualization, new moment reference, α=15, Ω = 0.2.

Fig. 36 ICE CFL3D.NI N-S computational flow visualization, new moment reference, α = 15, Ω = 0.4. The 0.0, 0.2, and 0.4 rotational rate cases (Fig. 35 and Fig. 36) showed a similar vortex core on the descending, starboard wing. The ascending, port wing expirenced a

4-38

reattachment of the vortex on the upper surface of the wing at 0.2 and 0.4 rotational rate. The reattachment was a result of the reduction in local angle of attack of this wing due to the effective angle of attack generated by the rotational motion of the configuration. The block arrows in Fig. 35 and Fig. 36 depicted the direction of the rotation of the aircraft about the velocity vector. The 0.4 rotational rate case (Fig. 32) depicted a small vortex on the lower surface of the ascending, port wing. Also, a stagnation area was noted on the upper surface of the ascending, port wing leading edge near the wing tip. From this point of view, the vortex wakes in Fig. 35 and Fig. 36 appear to be converging, but actually were spiraling around the rotation vector. 4.6.5 ICE Velocity Vector Roll, New Moment Reference, 15 deg Angle of Attack Fig. 37 shows a comparison of wind tunnel rotary-balance data19 (ROT-BAL, solid line) and CFL3D.NI (dashed line) at 0.3 Mach, 15 deg angle of attack, and zero deg angle of sideslip. The ICE configuration is rotated about the moment reference center to perform the CFL3D.NI solutions. This moment reference center of the CFL3D.NI solutions are located longitudinally at 39% of the MAC and the center of the body in the z-direction. This new location is 28% of the MAC higher than the original moment reference center of Fig. 33. Note that this change in moment reference center as compared to Fig. 33 had a small effect on the calculated changes in forces and moments due to rotation. A small improvement in rolling moment comparison in Fig. 37 was calculated over the original moment reference center rolling moment comparison in Fig. 33. A different effect on Cm (Fig. 33c and Fig. 37c) was noted for the two reference location as well as an increased nonlinearity in Cn (Fig. 37f).

4-39

Fig. 37 ICE CFL3D.NI and rotary-balance comparison, new moment reference, 15 deg angle of attack.

4-40

4.7 ICE Shape-Change Effectiveness Maps The ICE shape-change control effectiveness maps are the one ADIFOR reverse mode example in this study. The reverse mode ADIFOR-generated PMARC control effectiveness derivatives plotted as a function of control placement and are used to provide guidance for the optimal placement of shape-change control effectors. Optimal placement of these controls is located in areas of large gradients of the forces and moments. The placement of these effectors must allow for coordinated control of all three moments simultaneously because they may be highly coupled. A deployment scheme that allows the control force to be applied in a linear fashion with input command would be desirable from a pilot or control designers perspective, because the control force of these effectors can be very nonlinear with deflection height and simultaneous application of nearby effectors.17 Second derivatives with respect to grid point displacement would be desired to better estimate nonlinear control effectiveness with respect to grid point displacement. Identifying the optimal locations would permit a certain deflection of the skin to have the largest effect on aircraft control. Control effectors that may be used in this manner are flexible, inflatable surfaces, shape memory alloys, and piezoelectrics. This study focuses on the three moment coefficients because of their greater importance to closed-loop controller design than force coefficients. Fig. 38 though Fig. 40 show the reverse mode ADIFOR-generated PMARC values of the control effectiveness contours interpolated over the aircraft surface. The “a” subplots of these figures show the control effectiveness of an upper surface deflection on the three moments in the roll, pitch, and yaw axes, respectively. The “b” subplots of these figures show the control effectiveness of deflecting the lower surface of the configuration on the same three moments.

4-41

Fig. 38 Reverse mode ADIFOR-generated PMARC pitch effectiveness.

Fig. 39 Reverse mode ADIFOR-generated PMARC roll effectiveness.

Fig. 40 Reverse mode ADIFOR-generated PMARC yaw effectiveness. 4-42

The control effectiveness maps contained isolated bands or points, often near the edges of the configuration, which had extremely large sensitivity values. The color range of these maps has been reduced to show more detail in the center of the configuration and saturate the areas shaded in black and white. These black and white areas offer the greatest amount of control effectiveness, but should be avoided because they often located near each other indicating an unacceptable level of sensitivity to effector location. Areas with a yellow color (light gray in black and white) have larger contiguous areas, allowing a number of grid points to work together to generate a positive change in moment for outward displacements. Areas with a red color (dark gray in black and white) offer the same benefits for generating negative moments outward displacements. The red and yellow areas of the aircraft leading edge and slightly aft of the middle of the wing are currently being targeted for effector placement.21 Differentiating a new grid with more numerous, evenly distribution panels many produce a smoother data set. The control derivative discontinuities near the trailing edge of the model may be due to a greater sensitivity of the code near locations that it is enforcing the Kutta condition. The effect of effector placement is the focus of this study and the nonlinear effects of finite effector heights are not considered. Partial derivatives are computed, so the secondary effect of interaction or interference between adjacent displaced “bumps” or grid points is neglected. The effects of bump height and bump interaction, especially in the streamwise direction, have been found considerable in PMARC finite difference studies (Park, et al. 17 ). In the presence of these assumptions, These PMARC finite difference studies also showed that ADIFOR-generated PMARC control effectiveness maps gave superb placement guidance. This guidance information was of the correct sign except near zero derivative values, which have low effectiveness and are therefore not of interest to a controls engineer. The ADIFOR-generated PMARC control effectiveness estimates were also found to correctly predict relative effectiveness of spatially separated, finite effectors, if not their absolute effectiveness.

4-43

5. COMPUTATIONAL RESOURCES AND PREDICTIVE PERFORMANCE

To successfully incorporate the methodologies of this study in to a multidisciplinary design optimization framework, the execution time of the processes used in this study must be detailed. Also, these execution times should be correlated to the relative predictive performance of these codes. 5.1 Computational Resources. The computational resources required for the solutions of the ICE configuration in this study are shown in Table 1. These timing runs were performed on three different O2K computers. The O2K machines were a four-processor NASA Langley Multidisciplinary Optimization Branch computer with 4 Gb RAM (Cases 1–6), A 16-processor NASA Langley HPCCP computer with 12 Gb RAM (Cases 7–10), and a 64-processor NASA Ames NAS computer with 16 Gb RAM (Cases 11–14). The column labeled “Independents” indicates whether function only (zero independents) or original analysis plus derivatives. Two “Independents” is original analysis plus derivatives with respect to angle of attack and angle of sideslip. Five “Independents” is original analysis plus derivatives with respect to angle of attack and angle of sideslip, roll rate, pitch rate, and yaw rate.

5-1

Table 1 Execution Time and RAM for ICE Configuration. Case

Description

Independents

Processors

Wall Time

Total RAM

1a PMARC linear aerodynamics: 4-processor O2K 1

PMARC

0

1 (4)

4 min

160 Mb

2

PMARC.AD

2 (α, β)

1 (4)

12 min

480 Mb

3

Center-finite-difference PMARC

2 (α, β)

1 (4)

20 min

160Mb

1b CFL3D nonlinear aerodynamics: 4-processor O2K 4

CFL3D N-S

0

4 (4)

13 hours

1468 Mb

5

CFL3D.NI N-S

0

4 (4)

13.5 hours

1468 Mb

6

Time-accurate CFL3D estimate

0

4 (4)

175 hours

†

†

1468 Mb

1c CFL3D nonlinear aerodynamics: 16-processor O2K 7

CFL3D N-S

0

14 (16)

4 hours

1800 Mb

8

CFL3D.NI.AD N-S, 0–15 α

5 (α, β, p, q, r)

14 (16)

31 hours

9828 Mb

9

CFL3D.NI.AD N-S, 20–30 α

5 (α, β, p, q, r)

14 (16)

90–120 hours

9828 Mb

10

Center-finite-difference CFL3D.NI

5 (α, β, p, q, r)

14 (16)

44 hours

1800 Mb

1d CFL3D nonlinear aerodynamics: 64-processor O2K 11

CFL3D N-S

0

19 (64)

5.75 hours

1700 Mb

12

CFL3D.AD Eul, 0–10 α

2 (α, β)

19 (64)

30.5 hours

4600 Mb

13

CFL3D.AD N-S, 0–10 α

2 (α, β)

19 (64)

41.1 hours

5300 Mb

14

Center-finite-difference CFL3D N-S

2 (α, β)

19 (64)

28.75 hours

1700 Mb

The column labeled “Processors” indicates the number of O2K processors employed for the calculations. The parenthesized number in “Processors” column is the maximum number of processors on the specific machine the execution was performed. The PMARC code uses a sequential execution scheme and therefore only one processor was required. The CFL3D executions employed a parallel execution scheme with one master processor and the remaining processors as workers, so the number of actual worker processors is one less than the number quoted in the “Processors” column. The “Wall Time” column provides the total time for each analysis. This “Wall Time” is not the node-hours. By means of a batch queuing system, the 16-processor and

†

Estimate

5-2

64-processor O2K total wall times were achieved through multiple 45 min and 105 min runs, respectively. The 16-processor and 64-processor O2K computers had significant shutdown and restart overhead (approximately 10%), which adversely affects total wall time for these examples. The “RAM” column is the sum of the random access memory required by all the processors. The cases 1–3 are for PMARC and ADIFOR-generated PMARC linear aerodynamics codes. Case 1 is the computational requirements of the original PMARC code. Case 2 is the computational requirements of the ADIFOR-generated PMARC to produce the original analysis of the code and derivatives with respect to angle of attack and sideslip. Case 3 is the requirements of five Case 1 executions to determine the analysis and derivatives with respect to angle of attack and sideslip by means of a central-finite-difference method. Cases 4–6 are intended to highlight the reduction in wall time due to noninertial CFL3D calculations. Note that CFL3D.NI (Case 5) required 0.5 hour (3.8%) more execution time than the original CFL3D (Case 4) steady-state execution wall time for the ICE configuration with the S-A turbulence model. The corresponding wall time increase for 2-D and 3-D Euler calculations due to noninertial modifications was approximately 15%. The noninertial modifications had a larger penalty for Euler than turbulent N-S solutions because N-S and S-A solutions required more calculations per iteration than Euler solutions. The increased calculations per iteration of the turbulent N-S solution masked the same number of noninertial modification calculations per iteration of the turbulent N-S and Euler solutions. A time-accurate CFL3D solution (Case 6) that would emulate a CFL3D.NI solution was estimated to require approximately 175 hours, or more than an order of magnitude increase in wall time over a CFL3D.NI calculation. Cases 7–10 were performed on a 16-processor O2K. The central-finite-difference wall time (Case 10) was calculated by multiplying the CFL3D time by 11 or one analysis and ten perturbed solutions for the five independents. The slight overhead for CFL3D.NI solutions over CFL3D solutions is ignored. The central-difference estimate required 42% more wall time than CFL3D.NI.AD (Case 8) between 0 and 15 deg angles of attack. Compared to the 0–15 deg angle of attack solutions (Case 8), CFL3D.NI.AD required three to four times the wall time at 20, 25, and 30 deg angle of attack (Case 9), due to

5-3

differentiated flow solver convergence difficulties. The vortex burst structures at the higher (20–30 deg) angles of attack (Fig. 8b and Fig. 9) may have been responsible for the convergence difficulties. Cases 11–14 were performed on a 64-processor O2K. The 19-processor Case 11 had a longer execution time than the 14-processor Case 7. Cases 1–10 were all preformed on a dedicated machine, whereas Cases 11–14 were performed on a machine that was also running other parallel codes, which were all competing for processing, message passing, and disk services. The load-balancing of the processors on each of the three O2K computers was different, which may affect a wall time comparison between machines. The N-S Case 13 required 35% more execution time than the Euler case 12. The center-finite-difference CFL3D (Case 14) requires 26% less wall time than the CFL3D.AD (Case 13). The ADIFOR-generated CFL3D executed in less wall time than central differencing for five independents (Cases 8 and 10), but The ADIFOR-generated CFL3D required more wall time than central differencing than two independents (Cases 12 and 14). ADIFOR-generated codes generally perform better, with respect to wall time requirements, than central-difference methods for larger numbers of independents. 5.2 Relative Predictive Performance. To provide a qualitative evaluation of the various ADIFOR-generated codes used in this study, their wall time requirements and prediction capability is highlighted in Table 2. “Wall Times” are approximate times for a 16-processor O2K with 12 Gb RAM. The prediction capability of the various codes as compared to wind tunnel data is classified as Excellent, Good, or poor. Excellent prediction generally is within 10% of the plotted data range and/or displaying similar trends and inflections. Good prediction is often within 10%–20% of the plotted data range and under or over predicting data trends or inflections. Poor prediction is differences over 20% of plotted range and opposite trends or unpredicted inflections. Some leniency in assigning the classification (Excellent, Good, Poor) is granted for cases at angles of attack with transitions between flow structures or highly separated conditions.

5-4

Table 2 Qualitative Performance of ADIFOR-Generated Codes. Description

Wall Time

Performance in Different Flow Structures

Attached flow

Vortical Flow

Vortex Breakdown

0–5 deg α

6–15 deg α

>15 deg α

PMARC

4 min

Good

Poor

CFL3D N-S

4 hours

Excellent

Excellent

PMARC.AD

12 min

Good

Poor

CFL3D.AD Euler

30 hours

Excellent

Poor

CFL3D.AD N-S

40 hours

Excellent

Excellent

CFL3D.NI.AD N-S 0–15α

30 hours

Excellent

Excellent

CFL3D.NI.AD N-S >15α

90 hours

Good

Good

Good

5-5

6. CONCLUSIONS

Computational fluid dynamics (CFD) codes were modified in this study to model the effects of rotational rate and continuous mold-line control effectors. An automatic differentiation process (ADIFOR) was applied to these modified CFD codes to produce static and dynamic stability and control derivatives. These ADIFOR-generated, modified CFD codes provide a new capability to a designer. Results were presented primarily for the Lockheed Martin Tactical Aircraft Systems—Innovative Control Effectors (ICE) configuration to illustrate this current method. The CFD code modifications included the derivation of a source term to include the CFL3D Navier-Stokes code to perform steady-state solutions on CFD grids translating and rotating (e. g. accelerating) in a general fashion. The simplification of this term for constant-rate motion was detailed along with the corresponding boundary and initial flow conditions for this specific CFD problem. The steady-state mode of the modified version of the CFL3D code (denoted CFL3D.NI) efficiently modeled constant-rate motion. A linear-aerodynamics panel flow code (PMARC) was modified to model continuous mold-line control effectors with a normal displacement of each of the surface grid points of the model surface description. The application of automatic differentiation to CFD codes has great potential for predicting stability and control derivatives. The calculation of these derivatives can now

6-1

occur in the early aircraft design phase to influence and improve the configuration through multidisciplinary design. 6.1 Static Flow Visualization, Forces, Moments, and Derivatives Symmetric vortical flow structures for the ICE configuration at 0.6 Mach, zero angle of sideslip, and zero rotational rate were identified by means of computational flow visualizations of turbulent CFL3D Navier-Stokes (N-S) calculations at 5–30 deg angles of attack. The nature of these vortical flow structures was correlated to the behavior of calculated and measured forces, moments, angle-of-sideslip derivatives, and rotational rate derivatives at 0–30 deg angles of attack. The flow structure on the upper surface of the ice configuration was seen to be attached at 5 deg angle of attack. Vortical flow was predicted by means of computational flow visualization for 6–15 deg angles of attack. At higher angles of attack (15–30 deg) the upper surface vortex was predicted to develop a burst structure. Static forces and moments of the ICE configuration were calculated by the turbulent N-S mode of CFL3D and compared to wind tunne l (WT) measurements for 0.6 Mach and zero deg angle of sideslip. Good longitudinal comparison was noted and significant nonzero values in the WT measured lateral side force, rolling moment, and yawing moment was illustrated. These nonzero lateral WT measurements at zero angle of sideslip had a similar magnitude as subsequent calculated angle of sideslip derivatives. The CFL3D code predicted zero static lateral force and moment coefficients, as expected, for the symmetric configuration. ADIFOR-generated versions of the CFL3D (CFL3D.AD) and PMARC (PMARC.AD) codes were employed to calculate exact angle of attack and angle of sideslip derivatives at 0.6 Mach. These exact CFD derivatives were compared to centrally differenced WT derivatives. ADIFOR-generated PMARC derivative values compare qualitatively with wind tunnel data with a constant difference between zero and 6 deg angle of attack. Wind tunnel data trends change drastically above 6 deg angle of attack (due to the initiation of vortical flow), whereas the PMARC results predicted a linear relation to angle of attack. This linear region below 6 deg angle of attack is the primary

6-2

area of interest for the cruise and mild maneuvering portions of flight, possibly making these relatively inexpensive computations helpful to the designer. The ADIFOR-generated CFL3D code in both Euler and turbulent Navier-Stokes modes predicted a change in derivative values above 6 deg angle of attack, although the Euler mode derivatives had convergence difficulties at 7.5 deg angle of attack. General longitudinal angle of attack derivative trends are predicted by CFL3D.AD N-S for 0–30 deg angle of attack. The differentiated turbulent Navier-Stokes mode showed an excellent comparison with lateral derivatives up to 7.5 deg angle of attack for CSβ and Cnβ and 15 deg angle of attack for Clβ . The comparison degraded for bursting vortex structures above 15 deg angle of attack. The lateral bias in the zero angle of sideslip wind tunnel measurements may affect the centrally differenced wind tunnel derivative estimates and their comparison to CFD methods. 6.2 Noninertial, Modified CFL3D Rotational Calculations An initial application of ADIFOR to CFL3D with constant-rate noninertial modifications to compute dynamic stability derivatives was completed. This application was initially validated for a 2-D NACA 0012 Euler case by comparison to the SFLOW code, a similar formulation. ADIFOR-generated noninertial CFL3D (CFL3D.NI.AD) derivatives of a 2-D NACA 0012 airfoil showed excellent comparison with existing CFD methods (e. g. SFLOW) at 0.1 and 0.5 Mach. The convergence and comparison of CFL3D.NI.AD with SFLOW at 0.8 Mach was hindered by the presence of a shock structure, but was demonstrated to be at least as good as (if not better than) existing central-differencing derivative methods. The validated CFL3D.NI.AD was then applied to the ICE configuration. Body-axis rate derivatives were calculated at 0.6 Mach and zero deg angle of sideslip for 0–30 deg angles of attack. Derivatives of force and moment coefficients with respect to all three body-axis rotational rates were computed for the full 3-D ICE configuration. The CFL3D.NI.AD results were compared to a strip theory method (DYNAMIC). The angle of attack trends in these computed derivatives and their comparisons to DYNAMIC were discussed.

6-3

Wind tunnel rotary-balance tests were compared to CFL3D.NI velocity vector roll calculations at 0.3 Mach and 0 deg angle of attack. Nonlinear effects with respect to nondimensional rotational rate Ω were modeled. The change in rolling moment due to rotation ∆Cl exhibited the best comparison between the rotary-balance measurements and the CFL3D.NI calculation. This was the parameter of greatest interest for rotary-balance tests. Flow visualization techniques were applied to computational solutions for ICE velocity vector rolls at 0.3 Mach and 15 deg angle of attack; these visualizations depicted an asymmetric vortex burst structure at a nondimensional roll rate of 0.4. The effect of these asymmetric vortical flow structures was observed in the nonlinear effects of rotation rate on forces and moments. The prediction of the measured ∆Cl with computed ∆Cl was degraded at 15 deg angle of attack as compared to 0 deg angle of attack. The computational solution showed an increase in effect of rotational rate on ∆Cl between 0 and 15 deg angle of attack, whereas the measured values showed and decrease in the effect of rotational rate on ∆Cl. The effect of rotation center and moment reference center location was studied with computational flow visualization and the computed forces and moments. The computational flow visualization of rotation about a new rotation center location showed a dramatic change in flow structure. This dramatic change in flow structure had a slight, and favorable, change in computed forces and moment. A slightly improved comparison of calculated rolling moment to measured rolling moment was noted for the new rotation and moment reference center. The Reynolds number of the rotary-balance tests are estimated to be two hundred times smaller than the CFL3D.NI simulations. This may have a significant effect on the 15 deg angle of attack comparison. At 15 deg angle of attack, the computational flow visualization predicted separated, busting vortex flow for the original rotation center location. Computational flow visualization predicted an even more complicated incipient separation and reattachment flow conditions for the new rotation center location. Incipient separation conditions can be extremely sensitive to Reynolds number effects. Performing a CFL3D.NI rotary-balance simulation at rotary-balance wind tunnel test

6-4

Reynolds number would be an excellent topic for future investigations in to this methodology. The application of ADIFOR to noninertial, constant-rate calculations was demonstrated for compressible and viscous flows on an unconventional configuration. This new CFL3D capability proved to be an accurate method to complement or reduce dependency on forced-motion rotary or oscillatory wind tunnel measurements. This noninertial reference frame modification to CFL3D also has direct application to turbomachinery studies. The noninertial reference frame theory utilized to formulate the source terms in CFL3D.NI can easily be extended to include angular or translational acceleration terms to model more generalized aircraft or grid motions. Implementing a general noninertial source term and boundary conditions allows oscillatory motion or time-dependent cases to be performed without computing grid metrics at each time step. The application of ADIFOR to the modified version of CFL3D has great promise as a rotary derivative prediction tool for stability and control work in design studies and multidisciplinary design frameworks. 6.3 Predictive Performance and Computational Resources The predictive performance of the various codes to calculate reference values was evaluated. These predictions were categorized into excellent, good, and poor. These categories correspond to differences between the calculated and reference values less than 10%, 10%–20%, and greater than 20%, respectively, of the data range or y-axis of the plots. The execution wall time of the calculations for this study was quoted for three different SGI Origin 2000 (O2K) computers. These calculations would be a computationally expensive segment of a multidisciplinary loop. The generation of a 0 through 30 angle of attack database including static forces, moments, and their derivatives with respect to angle of attack, angle of sideslip, and three body axis rotation rate would require approximately 16 days on a 16-processor O2K computer. Assuming linear speed-up the same calculations could be performed on a dedicated 512-processor in approximately half a day.

6-5

6.4 Continuous Mold-Line Control Effectiveness The reverse mode ADIFOR-generated PMARC code gave superior insight into the placement of shape-change control devices. This guidance information was of the correct sign except near zero derivative values, which have low effectiveness and are therefore not of interest to a controls engineer. The ADIFOR-generated PMARC control effectiveness estimates were also found to correctly predict relative effectiveness of spatially separated finite grid point displacements and an estimate of absolute effectiveness of small displacements. Second derivatives with respect to grid point displacement would be desired to better estimate nonlinear control effectiveness with respect to grid point displacement. This new capability to model continuous mold-line shape change effectors can be extended to Euler or turbulent N-S codes to include physics that model compressibility and viscous boundary layer effects (separation).

6-6

7. REFERENCES

1

Finley, D., “Euler Technology Assessment Program for Preliminary Aircraft Design Employing SPLITFLOW Code With Cartesian Unstructured Grid Method,” NASA CR-4649, Mar. 1995. 2

Charlton, E., “Numerical Stability and Control Analysis Towards Falling-Leaf Prediction Capabilities of Splitflow for two Generic High-Performance Aircraft Models,” NASA CR-1998-208730, Sept. 1998. 3

Godfrey, A. G. and Cliff, E. M., "Direct Calculation of Aerodynamic Force Derivatives: A Sensitivity-Equation Approach," AIAA Paper 98-0393, Aerospace Sciences Meeting, January 1998. 4

Chen, J. P., Ghosh, A. R., Sreenivas, K., and Whitfield, D. L., "Comparison of Computations Using Navier-Stokes Equations in Rotating and Fixed Coordinates for Flow Through Turbomachinery," AIAA Paper 97-0878, Jan. 1997. 5

Kandil, O. A., and Chuang, H. A., “Unsteady Vortex-Dominated Flows around Maneuvering Wings over a wide Range of Mach Numbers,” AIAA Paper 88-0317, Jan. 1988. 6

Kandil, O. A., and Chuang, H. A., “Unsteady Navier-Stokes Computations Past Oscillating Delta Wing at High Incidence,” AIAA Journal, Vol. 28, No. 9, Sept. 1990, pp. 1565–1572. 7

Menzies, M., and Kandil, O., “Natural Rolling Responses of a Delta Wing in Transonic and Subsonic Flows,” AIAA Paper 98-3391, July 1996.

7-1

8

Limache, A., C. and Cliff, E. M., "Aerodynamic Sensitivity Theory for Rotary Stability Derivatives," AIAA Paper 98-4313, Aug. 1998. 9

Bischof, C., Carle, A., Khademi, P., and Mauer, A., “Automatic Differentiation of FORTRAN,” IEEE Computational Science & Engineering, Fall 1996. 10

Ashby, D., Dudley, M., Iguchi, S., Browne, L., and Katz J., “Potential Flow Theory and Operation Guide for the Panel Code PMARC_12,” NASA Ames Research Center, Moffett Field, CA, Dec. 1992. 11

Krist, E., Biedron, R., and Rumsey, C., "CFL3D User's Manual (Version 5.0,)" NASA TM-1998-20844, June 1998. 12

Vatsa, Veer N., “Computation of Sensitivity Derivatives of Navier-Stokes Equations Using Complex Variables,” Fifth Symposium on the Large-Scale Analysis and Design and ISE, Williamsburg, Virginia, Oct. 12-15, 1999. 13

Anderson, W. K., Newman, J. C., Whitfield, D. L., Nielson, E. J., “Sensitivity analysis for the Navier-Stokes equations on unstructured meshes using complex variables.” AIAA Paper 99-3294, 1999. 14

Bischof, C. Carle, A., Corliss, G., Greewank, A., and Hovland, P., "ADIFOR— Generating Derivative Codes from Fortran Programs," Scientific Programming, No. 1, 1992, pp. 1–29. 15

Carle, A., Green, L., Bischof, C., and Newman, P., “Applications of Automatic Differentiation in CFD,” AIAA Paper 94-2197, June 1994. 16

Carle, A., Fagan, M., and Green, L., “Preliminary Results from the Application of Automated Adjiont Code Generation to CFL3D,” AIAA Paper 98-4807, Sept. 1998. 17

Park, M., Green, L., Montgomery, R., Raney, D., "Determination of Stability and Control Derivatives Using Computational Fluid Dynamics and Automatic Differentiation," AIAA paper 99-3136, 17th Applied Aerodynamics Conference, June 1999. 18

Meirovitch, L., "Methods of Analytical Dynamics," McGraw-Hill, New York, NY, 1970. 19

Dorsett, K. M., and Mehl, D. R., “Innovative Control Effectors (ICE),” Wright Laboratory Report, WL-TR-96-3043, Jan. 1996. 20

Scott, M., Montgomery, R., and Weston, R., “Subsonic Maneuvering Effectiveness of High Performance Aircraft Which Employ Quasi-Static Shape Change Devices,” SPIE,

7-2

1998 International Symposium on Smart Structures and Materials, paper 3326-24, pp. 223–233. 21

Raney, D., Montgomery, R., Green, L., Park, M., "Flight Control using Shape-Change Effector Arrays," AIAA paper 2000-1560, April 2000. 22

Albright, A. E., Dixon, C. J., Hegedus, M. C., “Modification and Validation of Conceptual Design Aerodynamic Prediction Method HASC95 With VTXCHN,” NASA CR-4712, Mar. 1996. 23

Youngren, H. H., Bouchard, E. E., Coopersmith, R. M., and Miranda, L. R., “Comparison of Panel Method Formulations and its Influence on the Development of QUADPAN, an Advanced Low Order Method,” AIAA Paper 83-1827, July 1983. 24

Miranda, L. R., Elliott, R. D., and Baker, W. M., “A Generalized Vortex Lattice Method of Subsonic and Supersonic Flow Applications,” NASA CR-2865, Dec. 1977. 25

Simon, J. M., “Dynamic Derivative Data for High-Angle-of-Attack Simulation,” AIAA Paper 92-4355, Aug. 1992. 26

Blake, W. B., Dixon, C. J., and Adler, C. O., “Development of a High-Angle-of-Attack Stability and Control Prediction Code,” AIAA Paper 92-4354, Aug. 1992. 27

Green, L. L., Newman, P. A., and Haigler, K. J., “Sensitivity Derivatives for Advanced CFD Algorithm and Viscous Modeling Parameters Via Automatic Differentiation,” AIAA Paper 93-3321, July, 1993.

7-3