Approximate Analysis and Sensitivity Analysis Methods for Viscous Flow Involving Variation of Geometric Shape

Arthur C. Taylor 111' Vamshi Mohan ~ o r i v i ~

Gene W. HOU+

Department of Mechanical Engineering and Mechanics Old Dominion University

Norfolk, Virginia 235294247

ABSTRACT

Changes in steady-state numerical solutions of the thin- layer Navier-Stokes equations in response to small variations in geometric shape are estimated using an approximation method which is developed herein. In addition, this approximation method is extended to become a general procedure for calcu- lating aerodynamic shape sensitivity derivatives by direct dif- ferentiation of the algebraic equations which approximate the governing equations. The methods are successfully applied in 2D using an upwind cell-centered finite volume procedure for a low Reynolds number (REL = 100) laminar flow through a double-throat nozzle, where the flow is accelerated from Mach 0.10 at the inflow to about Mach 2.80 at the outflow. Calcu- lations reported herein using the approximation method show good agreement between the predicted numerical solutions, and the conventional numerical solutions. In addition, the aerodynamic sensitivity derivatives which are calculated us- ing the method of direct differentiation match those calculated using the method of finite differences (i.e., "brute force"), but are computationally less expensive to obtain.

1.0 Introduction

One theme of the present study will be the development and testing of a general procedure for efficiently and accurately estimating the changes which occur in a known steady-state numerical solution of the governing equations of viscous flow in response to small changes in the geometric shape of the computational domain. The purpose of this part of the study is that a technique be developed where significant computational resources can be saved in the event that solutions are needed of the flow over or through a large number of closely related geometric shapes. The approximation method employs the use of slopes which are calculated by direct differentiation of the algebraic equations which model the governing equations.

Assslam Professor, Member AlAA Graduate Research Assistant

Assoc~air Professor, Mcmbcr AlAA Copyr~ght 8 1991 by the Arner~can Inst~tule of Aeronaudcs and Aslronauucs. All rtghls reserved

A second theme of the present study focuses on a closely related discipline known as sensitivity analysis, a subject which also requires the calculation of slopes, known as sensitivity derivatives, where the derivatives of the response of a system are taken with respect to the design variables of interest (or with respect to any independent variables of interest). It is demonstrated that the approximation method mentioned pre- viously is easily extended to become a general procedure for use in the direct calculation of aerodynamic sensitivity deriva- tives taken with respect to arbitrary variations of geometric shape. In a design environment, an accurate knowledge of the sensitivity derivatives of a given system can subsequently be applied in several useful ways. These include 1) function ap- proximation to predict trends in the behavior of a system in response to changes in the design variables, 2) generation of better initial guesses for further analysis, and 3) design opti- mization. In the application of the principles of approximate analysis and sensitivity analysis to Computational Fluid Dy- namics (CFD), researchers have been and continue to be active, and Refs. [11 through [8] are given as a partial list in order to provide additional background material.

In the present work, the recently completed research of Refs. [7] and [8] is successfully extended from inviscid flows (i.e., the Euler equations) to the treatment of viscous flows. The present work is demonstrated for the severe test case of a low Reynolds number, laminar, transonic internal flow through a double-throat nozzle, where the flow is accelerated from a freestream value of the Mach number of 0.10, to a Mach number at the exit of the nozzle which exceeds 2.80.

The remainder of the work is organized as follows: After this introduction, the next section is a presentation of theory, a section which is further subdivided into a review of the governing equations, a review of the implicit upwind cell- centered finite volume approach, and the basic approximation and sensitivity equations (i.e., the central foci of the present work) are presented. In the next section, the test problem is described, and the computational results are presented from the application of the approximation method and the sensitivity equations to the test problem. The final major section of the paper is a summary of the work where conclusions are given. In addition, there are three appendices.

2.0 Presentation of Theory

2.1 Governing Equations

The governing equations in this research are the 2-D thin- layer Navier-Stokes equations, given as:

where:

R(Q) is called the residual, and is clearly equal to zero for a steady-state solution. Q is a vector of conserved variables, p is density, u and v are velocity components in Cartesian coordinates, and eo is total energy (i.e., eo = e + w, where e is the specific internal energy of the fluid). The inviscid flux vectors, l ( Q ) and 6(Q) are given by:

A transformation to generalized ( E , q ) coordinates from Cartesian (x,y) coordinates has been made in Eq. (I), where (x. J,.. 77x, 7iy are metric terms, and J is the determinant of the Jacobian matrix of this transformation. The vectors F(Q) and G(Q) are given by:

P, the pressure, is evaluated using the ideal gas law:

and -/ is the specific heat ratio, taken to be 1.4. The thin-layer viscous terms in generalized coordinates are given by:

where:

The molecular viscosity is given by p , Stokes' hypothesis for the bulk viscosity (X=2p/3) has been used, a is the speed of sound, Pr is the Prandtl number (taken to be 0.72), and ReL is the Reynolds number. Nondimensionalization of Eq. (1) is with respect to p, and U,, the freestream density and velocity, respectively. The physical coordinates (x,y) are nondimensionalized by a reference length, L, and the viscosity is nondimensionalized by p,, the molecular viscosity of the freestream. The nondimensional molecular viscosity can be computed using Sutherland's law and a reference temperature, I ,, the static temperature of the freestream. For additional kimplicity in this work, however, the molecular viscosity is taken to be constant, equal to that of the freestream.

2.2 S~atial Discretization and Im~licit Formulation

The governing equations are solved in their alternative integral conservation law form using an upwind cell-centered finite volume formulation (only an overview of this method is presented here, with additional details found in Refs. [9] through [15]), where the residual at each cell becomes a balance of inviscid and viscous fluxes across cell interfaces. As an example, this flux balance for the jk" cell in a typical computational grid is given by Eq. (9), for a steady-state solution, and for A( = Aq = 1:

where subscripts j,k refer to the E.7 directions, respectively, and subscripts j + refer to the < = constant cell interfaces of the jkth cell, subscripts k + refer to the q = constant cell interfaces of the jkth cell. (All references to quantities which are evaluated at the cell interfaces will therefore require only a single subscript, either j 5 or k i 1.) Rjk is the discrete rep-

% resentation of the residual at the jkt cell. Upwind evaluation of the inviscid fluxes is accomplished by upwind interpolation of the field variables, Q, from the approximate cell centers to the cell interfaces, where the flux vector splitting proce- dure of van Leer [16] is employed. A third-order accurate upwind biased inviscid flux balance is used in the streamwise (I) and in the normal (17) directions. The finite volume equiv- alent of second-order accurate central differences is used for the viscous terms. The resulting discrete higher-order accurate algebraic approximate representation of the residual at each cell depends locally on cell-centered values of the vector Q at nine cells. That is, for the jkth cell:

When written for each cell (including boundary condition relationships) and assembled globally, this can be expressed as:

{WQ*)) = (0) (11)

where {Q"} is called the "root" (i.e., the steady-state value of the field variables). Therefore, Eq. (11) represents a large coupled system of nonlinear algebraic equations, and thus finding a steady-state solution to the governing equations has been replaced (approximately) by the problem of finding the root, {Q*}, of this set of algebraic equations.

The governing equations are discretized in time using the Euler implicit method, followed by a Taylor's series lineariza- tion of the discrete equations about the known time level. This results in a large system of linear equations at each time step, given as:

Equations (12) and (13) represent the fundamental implicit formulation for integrating the governing equations in time to steady-state. In these equations, n is the time iteration index, and ("AQ] is the incremental change in the field variables between the known (nth) and the next (n+lth) time levels. The matrix, r&] is diagonal, and contains the time term. As a consequence of Eq. (lo), the large Jacobian matrix, [v] is sparse and has a banded structure, with nine non- zero diagonals, the individual elements of which are 4x4 block coefficient matrices. In addition to its use in Eq. (12) above, this important Jacobian matrix plays another central role in this study, which will be shown later.

In principle, Eq. (12) can be repeatedly solved directly (using Eq. (13) to update the field variables), as the solution is advanced to steady-state, and for very large time steps, the direct method represents Newton's root finding procedure for nonlinear equations. The direct method however is not necessarily the most efficient procedure with respect to overall CPU tlme [17j,[18], and the large storage requirements of the method make its use not feasible in 3D. Therefore, more commonly, an iterative algorithm is selected for use in the repeated solution of Eq. (12). Popular choices of these iterative algorithms include approximate factorization (AF) [19], conventional relaxation algorithms [13],[14],[20], the strongly implicit procedure (SIP) [21], and the preconditioned conjugate gradient method 1221, to name a few.

2.3 The Basic Approximation Technique

In this section, the fundamental approximation method is presented which is used in the present work for estimating the changes which occur in a steady-state numerical solution of the governing equations in response to small variations in the geometric shape of the computational domain. Much of the material in this section is taken from Ref. [7 ] , where the method is developed and applied to inviscid flow.

When obtaining a conventional numerical solution for a fluid flow problem, the geometric shape of the computational domain is defined entirely by the computational mesh which is used in the calculations. A computational mesh is of course defined entirely by the (x,y) coordinates of the intersection points of the grid lines of the mesh. Henceforth, any vector representing the complete set of (x,y) coordinates of a compu- tational mesh will be represented symbolically as (XI .

In the discussion which follows, consider a particular fluid flow problem for which the computational mesh, {Xi), has been defined, and for which a conventional steady-state solution, {Q;}, is known. Recall that {Q;) is the "root" of a set of coupled nonlinear algebraic equations, given as:

Note in Eq. (14) that the functional dependence of the discrete algebraic residual on the physical (x,y) coordinates of the computational mesh (which defines the geometric shape) is now emphasized explicitly. Consider next a second fluid flow problem which is similar to the first in all respects, except that the geometric shape is slightly different, and therefore a

second computational mesh, (Xz), is defined. The steady-state solution for the second problem, (Qa), of course satisfies Eq. (15), such that:

Using a Taylor's series expansion about {R(Q;, X I ) ) , where the complete vector of physical (x,y) coordinates, {X) , is taken to be a vector of independent variables, and neglecting higher order terms, the result is:

where: {AQ*} = { Q a l - {Q;}

{ A X ) = {X,} - { X I )

Next, considering steady-state solutions only (i.e., using Eqs. (14) and (15)), Eq. (16) becomes:

Equation (18) represents a central relationship in this study, and thorough understanding of this simple equation and its potential usefulness is a primary mission of this pa- per. Note that there are two principal ways that the equation might be exploited, depending on which of the two vectors, either { A Q * ) or { A X ) , is taken to be "unknown". In all of the work presented herein, {AQ"} (i.e., the predicted incre- mental change in the steady-state solution) is taken to be the basic unknown, and the geometric shape change, { A X ) , (and thus the entire right-hand side vector of Eq. (18)) is known. The second alternative is potentially useful in the inverse de- sign problem, where the unknown change of geometric shape, { A x } , which is required to produced a specified (i.e., known) change in the field variables, {AQ*), is estimated (although this proposed technique has not been tested to date). Hence- forth, the remainder of the discussion will assume that {AQ*} is defined to be the unknown vector.

The use of Eq. (18) involves two large sparse banded Ja- cobian matrices. On the left-hand side of Eq. (18), the terms

of the Jacobian matrix, [ d R ( ~ ~ x l ) ] , are well understood and

documented, as they are also used in implicit time integration / relaxation algorithms for the numerical solution of the gov- erning equations. Specifically, a comparison of Eq. (12) with Eq. (18) will reveal that the large left-hand side Jacobian ma- trices of these two equations are identical (for very large time steps). Therefore, in principle, the approximation method of Eq. (18) is ideally suited for implementation together with an implicit code which uses Newton iteration (i.e., the direct solver algorithm with very large time steps) to determine the known steady-state solution, {Q;). With this ideal scenario, solutions to Eq. (18) could be very rapidly obtained using sim- ply forward and backward substitution operations, where thz

previously LU factored left-hand side Jacobian mamx could already be stored in memory following the final Newton iter- ation, during the determination of {Q;).

The use of the direct solver approach is not in widespread use, however, in typical CFD codes, for reasons which have been previously discussed. Therefore, in implementing the present approximation method, typically it will be necessary to actually construct the terms of the Jacobian matrix on the left-

( [-]) Then, a complete hand side of Eq. (18) i.e.,

algorithm for the solution of q. (18) must be applied, at least for the first geometric shape variation, { A X ) , for which a predicted change in the solution, {AQ*) , is desired.

In principle, Eq. (18) can be solved by either 1) direct LU factorization followed by forward and backward substitution, or 2) it can be solved by a choice of any of a number of well- known iterative methods, many of which have been mentioned previously. The former choice has the very important advan- tage of requiring only a single LU factorization of the left-hand side coefficient matrix, which of course can then be reused for an unlimited number of geometric shape variations,{AX). If direct solution of Eq. (18) is not feasible because of stor- age restrictions (which is generally the case for practical 3-D problems, even on large supercomputers), then an iterative al- gorithm will be the only recourse, and consequently the ability to reuse the LU factorization of the full coefficient matrix (for multiple geometric shape variations) will be lost.

In deriving the terms of the global Jacobian matrix,

laR(;kx1)] , it is noted that the discrete residual for each L

cell in the homain is functionally dependent locally on only eight (x,y) coordinate points of the computational grid. For the jkLh cell, this is expressed as:

where in the above the 2 element vectors, xl through !is, represent the (x,y) coordinates of these eight local grid points (e.g., x l = ( x , , y ,) T , etc.). For the jkth cell, these eight grid coordinates are illustrated in Fig. 1. As a consequence of Eq.

aa(g: ,x l ) . (191, the matrix, [ ax 1 , is very sparse and has a banded

structure consisting of eight non-zero diagonals, the individual elements of which are block 4x2 coefficient matrices. To illustrate this further, the single linear four component vector equation which is associated with the jkth cell of the domain is isolated from the global system of equations given by Eq. (18), and is given below as Eq. (20):

The nine 4x4 coefficient matrices, [A] through [I], on the left-hand side of Eq. (20) are each associated with a particular one of the nine non-zero diagonals (previously mentioned) of

the matrix. [*I. In addition, these matrices are con-

structed of linear combinations of the 4x4 inviscid and viscous flux Jacobian matrices, with derivatives taken with respect to the field variables. On the right-hand side of Eq. (20), the eight 4x2 coefficient matrices, [Wl] through [W8] are each associ- ated with a particular one of the eight previously mentioned

non-zero diagonals of the matrix, ~ R ( Q ; , X , ) [ ax /,. Furthermore,

these matrices are constructed of inear corn inations of the 4x2 inviscid and viscous flux Jacobian matrices, with deriva- tives taken with respect to the local grid coordinates. Addi- tional details concerning the construction of these matrices, [Wl] through [W8], are given in Appendix A.

All boundary conditions in this research are consistently linearized, and are included in the global linear systems which are solved. This applies to the fundamental implicit formula- tion for integrating the equations in time (Eqs. (12) and (13)), as well as to the basic approximation method of the repre- sent research (Eq. (18)). In particular, consistent boundary condition linearization is included as an integral part of the global Jacobian matrices on both sides of Eq. (18). That is, boundary condition relationships are linearized with respect to their dependence on the field variables as well as with respect to their dependence, if any, on the physical (x,y) coordinates. To insure the best possible results, it is emphasized that fully consistent boundary condition linearization should not be con- sidered optional in the application of Eq. (18) (as it typically is in the integration of the equations in time).

2.4 Fundamental Geometric Shape Sensitivity Equations

In this section, it is shown that the basic approximation equation discussed in the previous section (Eq. (18)) can easily be extended for use in the computation of aerodynamic shape sensitivity derivatives taken with respect to variation of geometric shape. The resulting procedure is very general in nature, and represents the same approach which was developed and successfully tested in Ref. [8] for inviscid flow (i.e., the Euler Equations).

The geometric shape of a computational domain is de- fined by the (x,y) coordinates on the boundary of the com- putational mesh upon which calculations are made. In the present research, it is assumed that these (x,y) coordinates on the boundaries of the computational domain are represented parametrically, as expressed by Eq. (21):

where Xb represents the vector of (x,y) coordinates on the boundary (or boundaries) of interest, and Xb is of course a relatively small subset of the larger vector, X (Recall that X represents the complete vector of (x,y) coordinates of the entire computational mesh.) The vector p represents a vector of independent geomemc shape (design) variables which control the shape of the domain over /through which the fluid flows.

The computational mesh (i.e., the vector X) for a compu- tational domain is typically obtained as the principal output of a mesh generation program, where the principal input to the program is the vector of (x,y) coordinates on the boundaries of the mesh, (i.e., xb). Additional input to the mesh generation program typically includes a vector of parameters, 7, which exerts some control over the distribution of points throughout the domain (e.g., stretching of the grid near solid walls to re- solve viscous gradients); however, it will be assumed herein that 3 is independent of the vector p (i.e., ;i # 7 (p)). There- fore, the operation of a typical mesh generation program is expressed functionally as:

At the present, one important point to be understood from Eq. (22) is that the entire vector of (x,y) points of the computational mesh is functionally dependent on the vector P. In short, if one or more of the geometric shape design variables change, then in general, glJ the points in the computational mesh will move.

The fundamental equations which are applied in the present research for calculation of aerodynamic shape sensi- tivity derivatives are derived in one step from the previously described approximation method (Eq. (18)) by application of the Calculus. Dividing both sides of Eq. (18) by a small but finite change in the k'" element of the vector, p, and taking the limit as this finite change becomes infinitesimally small, the result is:

where in the above, Pk represents the kth element of the vector of geometric shape design parameters, p. The global Jacobian matrices, [g] and 1%) are of course understood from previous discussions. The so ution of Eq. (23) yields the vector, ($1, which represents the sensitivity derivatives of the complete vector of field variables taken with respect to the k'" geometric shape design parameter, Dk. The vector, {el, represents the sensitivity of the entire computational mesh with derivatives taken with respect to Pk.

Note that Eq. (23) can and must be solved one time for each different geometric shape design variable (i.e., for each element of p) of interest. However, in the event that direct solution of Eq. (23) is employed, the LU factorization of the left-hand side coefficient matrix needs to be performed only once, and can be reused for as many solutions of the system as are required. This reuse of the LU factorization represents a potentially large savings in computational work, particularly if the number of geometric shape design variables of interest is large.

In a typical application, the solution vector, {F} of Eq. (23) will represent far more sensitivity informatlon than is actually required, and it is only a relatively small subset

of (8) which is extracted for further use in additional sensitivity equations which support Eq. (23), and which are specific to the particular problem of interest. For example, ~f the sensitivity derivatives of aerodynamic loads (forces)

are of interest, then only the subset of information which is needed to describe the sensitivity of the pressure and skin friction coefficients along the solid wall boundaries of the computational domain will be taken from the vector. {g}. and used. The specific ancillary sensitivity relationships which support Eq. (23) for the calculation of the sensitivity of aerodynamic forces on solid walls are given in Ref. [8] where they are successfully applied for inviscid flow, and are also repeated here for convenience as Appendix B.

In principle, the elements of the vector [ g) can be corn- puted analytically by direct differentiation o Eq. (22) (for each element of p), and for demonstration purposes, this is done in the example problems of Ref. [8], where additional discus- sion on the treatment of these terms is found. However, in the present work, the terms of {@! are approximated using finite differences (i.e., using the mes generation program and central differences, centered across the computational mesh of the baseline geometry, with Lak = 3~0.00001). It is pos- tulated that approximating the elements of {g) with finite differences will in general be a reliable and accurate approach. Furthermore, it is believed that this approximate treatment of these grid sensitivity terms does not severely diminish the over- all analytical approach represented by Eq. (23), and should not introduce some of the problems which might be associated with a finite difference (i.e., "brute force") approach on the fluids equations. This is because typically the equations of mesh gen- eration are by design "smoother" than are the equations which model the fluid flow, and are also move efficient to solve.

3.0 Computational Results

3.1 Description of the Test Problem

The test problem chosen for the present work is that of laminar flow through a double-throat nozzle. The geometry and computational grid which is used are illustrated in Fig. 2. Additional discussion concerning this test problem is found in Refs. [23], [24], and [25]. The freestream Mach number (M ) is taken to be 0.10, and the Reynolds number (REL =

4) for all calculations to be presented is selected to be 1&: The reference length (L) is selected to be one-half of the height separating the upper and lower walls at the smaller of the two throats, where the smaller throat is located at x=O in the figures. Note from Fig. 2 that the nozzle is a total of 26 reference lengths long.

The geometric shape of the test problem is defined us- ing expressions of the form given by Eq. (21). A detailed description of the resulting parametric representation of this shape is given in Ref. [25], and is repeated here for con- venience as Appendix C. As a consequence of this particular choice of representations of the shape, there is defined a total of ten geometric shape (design) parameters which might be considered, and hence the vector p (of Eqs. (21) and (22)) is composed of ten elements, P1 through Plo, each being de- fined in Appendix C. Only the first element, PI, is considered herein in order to demonstrate the applicability of the methods to viscous flow, which is the central objective of this work.

Treatment of the remaining nine elements of p is of course a straightforward extension of the methodology.

The centerline of the nozzle is a plane of symmetry, and for this reason it is only necessary to compute the flow through either the lower or the upper half of the nozzle (the lower half is chosen in the present work). A computational grid is used with 171 points (uniformly spaced) in the x direction, and 38 points are used in the y direction (with grid stretching to resolve gradients in the vicinity of solid walls). Boundary conditions in all calculations are applied as follows:

1) On the lower solid wall boundary, no-slip is applied to the velocity, the density is extrapolated, and the wall tem- perature is specified to be that of the stagnation temperature of the freestream.

2) On the upper boundary of the computational domain (i,e., along the centerline of the nozzle) flow symmetry bound- ary conditions are applied.

3) On the left (inflow) boundary, the entropy and stagna- tion enthalpy are held fixed to be that of the freestream. The v component of velocity is set to zero, and the u component of velocity is extrapolated.

4) On the right (outflow) boundary, all variables are extrapolated.

For the geometry, computational grid, reference and boundary conditions described above, a conventional steady- state numerical solution is obtained using the approximate fac- torization (AF) time integration algorithm, where freestream conditions throughout the domain are used as the initial guess. Convergence is defined to be when the L2 norm of the discrete residual is reduced to machine zero (a reduction of approxi- mately twelve orders-of-magnitude). The complete pressure contours for the solution are shown in Fig. 3a, and the com- plete Mach number contours are shown in Fig. 3b. It is noted that the solution shows the flow to be accelerated from M x 0.10 (subsonic) on the inflow boundary to M 2 2.8 (super- sonic) at some points on the outflow boundary. Henceforth, the steady-state solution shown in Figs. 3a and 3b is referred to as the "baseline" solution.

3.2 Application of the Approximation Method

With the baseline steady-state numerical solution discussed in the previous section, the approximation method of Eq. (18) is used to estimate neighboring solutions for variations of the baseline geometric shape shown in Fig. 2. Recall that Eq. (18) is general in nature, and is applicable to arbitrary variations of geometric shape which are "small". Small variations in the first element, PI , of the vector p, are selected for study. As defined in Appendix C, the parameter also represents the half-height, H, of the nozzle measured at the smaller of the two throats. For the test results to be shown, changes in this parameter H are made holding all other geometric shape parameters fixed, although clearly this restriction is not required in the application of Eq. (18). It is noted that changes in the value of this particular parameter (holding all others fixed) results in a uniform change along the entire length of the nozzle in the height of the separation between the upper

and lower walls (i.e., a change in H results in a uniform change in the distances from solid wall boundaries to the centerline of the nozzle). In the results to follow, changes in H (henceforth referred to as AH) have been tested for three geometric shape variations involving 2%, 5%, and also 10% increases in the baseline value of this parameter.

In addition to the application of the approximation method, for comparison purposes, conventional steady-state numerical solutions are also obtained for all of the geometric shape vari- ations which are tested. As is the case when obtaining the baseline solution, this is done using the approximate factoriza- tion time integration algorithm, and convergence is defined to be a reduction of the discrete residual to machine zero. In ob- taining these conventional (as well as the predicted) numerical solutions for the geometric shape variations of interest, care must be taken not to inadvertently change the value of the Reynolds number. Since the Reynolds number has been de- fined using the half-height of the baseline nozzle at the smaller throat as the reference length, it is important that the reference length of this baseline geometry also be used to nondimen- sionalize the physical x,y grids coordinates of the geometric shape variations. For example, for the AH = 2% geometric shape variation, the nondimensional half-height of the smaller throat will be H = 1.02.

In addition to the baseline numerical solution, Fig. 4a shows a comparison between the results of the conventional numerical solution and the results of the approximation method (Eq. (18)), for AH = 2%, where pressure (PIP,) along the lower wall is plotted vs x for 1.0 5 x < 9.0. The pressure is plotted along only a portion of the length of the wall because the pressure over the entire length of the nozzle changes by an order-of-magnitude, which would present a scaling problem in the plots in clearly comparing the relatively small changes in the solutions which occur on account of the shape variations. Figure 4b represents a similar comparison as is shown in Fig. 4a, except that the centerline pressure is plotted (again, for 1.0 5 x < 9.0). Figure 4c is a comparison of the skin friction coefficient, Cf (defined in Appendix B) vs x along the lower wall, for 2.91 < x < 5.5, also for AH = 2%.

Figures 5a, 5b, and 5c show similar results to those pre- sented in Figs. 4a, 4b, and 4c, respectively, except a larger variation, AH = 5%, is considered. Figures 6a, 6b, and 6c are also similar, except the results for AH = 10% are shown. In particular, note in Fig. 6c the success of the method in accu- rately predicting flow separation. As can be seen from these figures, the comparisons between the conventional numerical solutions and the predicted numerical solutions obtained us- ing the approximation method are generally good. Of course, some of the results for the AH = 10% test case (which is not a small variation) have clearly been pushed beyond the capabilities of the linear approximation method.

The results which are shown here for the wall pressure comparisons were selected to represent the "worst case" com- parisions. The explanation for the difficulties encountered in these regions is the presence of shock wave interactions on the wall and centerline, as seen in Figs. 3a and 3b. Other results not shown (i.e., results for the remaining x locations)

were seen to be at least as good and were usually significantly better than those results which have been shown here.

3.3 Calculation of Aerodynamic Shape Sensitivity Derivatives

Aerodynamic sensitivity derivatives taken with respect to the parameters which define the geometric shape of the test problem (Fig. 2) are computed for the baseline geometry us- ing Eq. (23). In addition, for comparison purposes, all sen- sitivity derivatives are also computed using finite difference approximations (i.e., the "brute force" method). Specifically, central differences are used across the baseline numerical so- lution, with an incremental change in the independent variable (i.e., LIPk) of kO.00001, and the conventional numerical solu- tions to the governing equations which must be computed on either side of the baseline solution (in order to form the central difference approximations) are converged to machine zero.

In the results yet to be shown, sensitivity derivatives with respect to the single geometric shape parameter, H, are con- sidered. Recall that this parameter represents the half-height of the nozzle at the smaller of the two throats, and is also PI , the first element of the vector of geometric shape variables, p, defined in detail in Appendix C. Figure 7a shows the com- puted sensitivity derivatives of the pressure coefficient, Cp, with derivatives taken with respect to this parameter, H, along the lower wall of the nozzle (i.e., 3 along the lower wall is plotted vs x). In this figure, a comparison is made of these sensitivity derivatives computed using the direct differentia- tlon method of Eq. (23), to the same derivatives computed us- ing the method of brute force finite difference approximations. Figure 7b shows similar results to those shown in Fig. 7a, ex- cept % vs x is plotted along the centerline of the nozzle. As expected, the agreement between the results obtained using the direct differentiation method of Eq. (23) and those obtained using finite difference approximations is excellent. Note in the figures the expected severe fluctuations in the sensitivity derivatives, and in particular the sudden reversals of the alge- braic sign of these slopes in regions of shock interations. This is consistent with the difficulties which were encountered in these regions in the results of the approximation method given in the previous section.

Of particular interest and concern in the design of a nozzle is typically the production of thrust. The sensitivity derivatives of the total thrust, T, which is generated by this nozzle, taken with respect to the elements of the vector (although not accomplished to date for the present geometry, but demon- strated in Ref. [8] for inviscid flow) can be computed using the simple ancillary sensitivity relationships given in Appendix B. In designing a nozzle of this type under the current flow conditions, it is hypothesized that an optimum design might be found (within the constraints of the given ten geometric shape parameters) using standardized engineering optimiza- tion schemes which employ as input an accurate knowledge of these sensitivity derivatives.

3.4 CPU Time

In the course of the present study, the following points were noted concerning the computational resources which were required for the calculations, most of which were performed on a Cray-2 computer:

1) The approximation method (Eq. (18)) generates pre- dicted numerical solutions significantly more efficiently com- pared to the cost in CPU time of generating conventional nu- merical solutions. The exact amount of savings of course de- pends on how tightly the conventional numerical solutions are converged. This savings will be particularly large when mul- tiple geometric shape variations are of interest (since the LU factorization of the left-hand coefficient matrix of Eq. (18) can be reused for an unlimited number of geometric shape variations, resulting in an enormous savings). The potential for CPU savings through the use of Eq. (18) to generate bet- ter initial guesses for further analysis was not investigated in the present study, but was considered in a limited study for an inviscid flow problem in Ref. [7], where a 20% net CPU savings was reported.

2) In evaluating the CPU requirements for the brute force finite difference approach and the direct differentiation method of Eq. (23) for computing sensitivity derivatives, it is difficult to make a precise quantitative comparison. This is because the CPU time required for the finite difference approach can vary greatly, depending on numerous factors (e.g., how tightly and how rapidly the conventional numerical solutions are con- verged for subsequent use in the finite difference approach). However, it was noted that the sensitivity derivatives were ob- tained using Eq. (23) with a significant savings in CPU time over the finite difference approach. As with the approximation method, this savings will increase substantially as the number of design variables increases.

3) The computer memory requirement for solving Eq. (18) and / or Eq. (23) is very large even for 2D problems, and pushes the limits of modem supercomputers, making a direct solution not feasible for 3D applications.

4.0 Summary and Conclusions

A method has been developed herein based on a Taylor's series expansion for efficiently estimating the changes which occur in a steady-state numerical solution of the thin-layer Navier-Stokes equations in response to small (but very gen- eral) variations of geometric shape. With one simple addi- tional step, the approximation method is extended to become a useful and practical procedure for efficiently and accurately computing aerodynamic shape sensitivity derivatives by di- rect differentiation of the algebraic equations which model the fluid flow. The work represents the successful completion of an initial feasibility study, where ideas previously developed and tested for inviscid flow are directly extended and applied to viscous flow.

The methods are applied to the test problem of a low Reynolds number (REL = 100) laminar flow through a double- throat nozzle, where the flow is accelerated from about Mach

0.10 on the inflow to about Mach 2.80 on the outflow. Agree- ment was good between the conventional numerical solutions and the predicted numerical solutions obtained using the ap- proximation method, and were obtained with a significant sav- ings in CPU time. The sensitivity derivatives of the pressure along the lower wall and also the centerline of the nozzle (i.e., derivatives taken with respect to the first geometric shape de- sign variable) were computed using the methods developed herein. These sensitivity derivatives agreed very well in com- parison with the same derivatives computed using the method of brute force finite differences, but were obtained with a sig- nificant savings in computational cost.

5.0 Acknowledgments

This research was supported in part by grant number DMC-865-7917 from the National Science Foundation. Ap- preciation is expressed to Dr. Perry A. Newman, Dr. Henry E. Jones, and Dr. E. Carson Yates, Jr., of NASA Langley Research Center, for many lengthy and helpful discussions which contributed to this work.

6.0 References

Yates, E.C., Jr., and Desmarais, R., "Boundary Inte- gral Method for Calculating Aerodynamic Sensitivities with Illustration for Lifting Surface Theory," in Pro- ceedings of the International Symposium of Boundary Element Methods (IBEM 89), published by Springer- Verlag, Oct. 2-4, 1989, East Hartford, Conn.

Elbanna, H.M., and Carlson, L.A., "Determination of Aerodynamic Sensitivity Coefficients in the Transonic and Supersonic Regimes," AIAA Journal, Vol. 27, No. 6, June, 1990, pp. 507-518, also AIAA Paper 89-0532.

Yates, E.C., Jr., "Aerodynamic Sensitivities from Sub- sonic, Sonic, and Supersonic Unsteady, Nonplanar Lifting-Surface Theory," NASA TM-100502, Septem- ber, 1987.

Sobieszczanski-Sobieski, J., "The Case For Aerody- namic Sensitivity Analysis". In "Sensitivity Analysis in Engineering," NASA CP-2457, 1987.

Bristow, D.R., and Hawk, J.D., "Subsonic Panel Method For The Efficient Analysis of Multiple Ge- ometry Perturbations," NASA CP-3528, 1982.

Jameson, A., "Aerodynamic Design Via Control The- ory," NASA CR-181749, also ICASE Report No. 88-64, November, 1988.

Taylor, A.C. 111, Korivi, V.M., and Hou, G.W., "Sen- sitivity Analysis Applied to The Euler Equations: A Feasibility Study with Emphasis on Variation of Geo- metric Shape," AIAA Paper 91-0173.

Taylor, A.C. 111, Hou, G.W., and Korivi, V.M., "A Methodology for Determining Aerodynamic Sensitivity Derivatives With Respect to Varia- tion of Geometric Shape", Proceedings of the

AIAA/ASME/ASCE/AHS/ASC 32nd Structures, Struc- tural Dynamics, and Materials Conference, April 8-10, 1991, Baltimore, MD, also AIAA Paper 91-1 101.

Walters, R.W., and Thomas, J.L., "Advances in Up- wind Relaxation Methods," in State of the Art Surveys of Computational Mechanics, ed. A.K. Noor, ASME Publication, 1988.

10. Thomas, J.L., Van Leer, B., and Walters, R.W., "Im- plicit Flux-Split Schemes for the Euler Equations," AIAA Journal, Vol. 28, No. 6, June, 1990, pp. 973-974, also AIAA Paper 85-1680.

11. Walters, R.W., and Dwoyer, D.L., "An Efficient Iter- ation Strategy for the Euler Equations," AIAA Paper 85-1529.

12. Newsome, R.W., Walters, R.W., and Thomas, J.L., "An Efficient Iteration Strategy for UpwindRelaxation solutions to the Thin-Layer Navier-Stokes Equations," AIAA Journal, Vol. 27, No. 9, September, 1989, pp. 1165-1166, also AIAA Paper 87-1 113.

13. Thomas, J.L., and Walters, R.W., "Upwind Relaxation Algorithms for the Navier-Stokes Equations," Journal, Vol. 25, No. 4, April 1987, pp. 527-534.

14. Napolitano, M., and Walters, R.W., "An Incremen- tal Block-Line-Gauss-Seidel Method for the Navier- Stokes Equations, AIAA Journal, Vol. 24, No. 5, May 1986, pp. 77&776.

15. Walters, R.W., and Dwoyer, D.I., "Efficient Solutions to the Euler Equations for Supersonic Flow with Em- bedded Subsonic Regions," NASA Technical Paper 2523, January 1987.

16. Van Leer, B., "Flux-Vector Splitting for the Euler Equations," ICASE Report 82-30, September 1982 (also Lecture Notes in Physics, Vol. 170, 1982, pp. 507-5 12).

17. Riggins, D.W., and Walters, R.W., and Pelletier, D., "The Use of Direct Solvers for Compressible Flow Computations," AIAA Paper 88-0229.

18. Hafez, M., Palaniswamy, S., and Mariani, P., "Calcu- lations of Transonic Flows with Shocks Using New- ton's Method and Direct Solver, Part 11," AIAA Paper 88-0226.

19. Beam, R.M. and Warming, R.F., "An Implicit Factored Scheme for the Compressible Navier-Stokes Equa- tions," AIAA Journal, Vol. 16, April 1978, pp. 393402.

20. Taylor, A.C., 111, Ng, W.F., and Walters, R.W., "Upwind Relaxation Algorithms for the Navier- Stokes Equations Using Inner Iterations," AIAA Paper 89-1954.

21. Walters, R.W., Dwoyer, D.L., and Hassan, H.A., "A Strongly Implicit Procedure For The Compressible Navier-Stokes Equations", AIAA Journal, Vol. 24, No. 1, January, 1986, pp. 6-12.

22. Venkatakrishnan, V., "Preconditioned Conjugate Gra- dient Method For The Compressible Navier-Stokes Equations", AIAA Paper 90-0586.

23. Viviand, H., "Comparison of Numerical Solutions to Internal Flow in a Double Throat Nozzle," GAMM Committee for Numerical Methods in Fluid Mechan- ics, France, December, 1985.

24. Thomas, J.L., Walters, R.W., Van Leer, B., and Rum- sey, C.L., "An Implicit Flux-Split Algorithm for the Navier-Stokes Equations," GAMM Committee for Nu- merical Methods in Fluid Mechanics, France, Decem- ber, 1985.

25. Bristeau, M.O., Glowinski, R., Periaux, J., and Vi- viand, Henri (Eds.), "Numerical Simulation of Com- pressible Navier-Stokes Flows", Notes On Numerical Fluid Mechanics, (Proceedings of a GAMM Work- shop, Numerical Methods in Fluid Mechanics), Vol. 18, pp. 5-10, published by Friedr. Vieweg, & Sohn, Braunschweig, Germany, 1987.

7.0 Appendix A - Viscous Derivatives, Additional Details

The purpose of this appendix is to provide additional in- formation on the construction of the eight coefficient matrices, [Wl] through [W8] of Eq. (20). Only the viscous contribu- tion to the terms is considered here; the inviscid contribution (for the van Leer's flux-vector splitting in 2D) is presented in detail in Ref. [7]. As stated previously, these eight matrices are constructed of linear combinations of the 4x2 inviscid and viscous flux Jacobian matrices, with derivatives taken with re- spect to the x,y grid coordinates. This linear combination is easily obtained by differentiation of Eq. (19) with respect to each of the eight local coordinates, xl through 98, of Fig. 1 which, neglecting the inviscid contribution, gives

Inviscid + Terms

Inviscid

Inviscid

3 ~ : ~ ~ G V Inviscid W 4 1 = - Is] + + Terms

Note from Eq. (19) that the inviscid fluxes do not depend on the four coordinates, xs, 26, X7,Xg, and thus the four matrices, [W5], [W6], [W7], [W8], are zero for inviscid flow.

In order to evaluate analytically the derivatives in Eq. (24) above, it is convenient to first express the discrete thin-layer viscous flux vector, G;' in the following form:

kt,'

In the above, the terms through ?+b6 are func- k+ 3 k t 4

tions of the field variables, Q, at the k+l and k cell centers, where the missing subscript j has been dropped for notational convenience. These terms are:

The terms dl.++ through 44k++ are functions of the local grid coordinates, xl ,XZ, 23, %4,%5, x6, and can be evaluated as follows:

In addition:

The term, Jk+; is in fact the determinate of the Jacobian matrix

of the coordinate transformation, evaluated at the k + i cell face of the jkh cell, and is evaluated as the inverse of the average area of the two cells across the cell face. Therefore, Vk+l and Vk are the areas of the two cells on either side of the 1; + cell face. These areas each are computed as one- half the cross-product of two vectors which represent the two diagonals joining the four vertices of a cell, such that:

Similar relationships for the thin-layer viscous flux vector on the k-a cell face (is. , G:' ) can be easily obtained

k - 4

from the above expressions for the k + i cell face (Eqs. (25) through (31)) through the following simple adjustments to these expressions:

1) The coordinates, %l ,%4, %3, % 4 , % 5 , %s, are everywhere replaced by the coordinates, G,xl, x4, x7, j12, x 3 , re- spectively. (Note: It is critical that these coordinates be replaced one for one using the exact ordering listed here.)

and are assumed to be nondimensionalized by L, the reference length. The convention is established that as one moves along the surface in the direction of increasing the index, i, then the solid surface is on the right, and the fluid is on the left.

Nondimensional pressure (Cp) and skin friction (Cf) co- efficients which are associated with this ith element on the boundary are defined as follows:

where in the above, Pbi and q,i are the pressure and shear stress, respectively, which are associated with i" element on the boundary, and ;p,UL is the dynamic pressure of the freestream. Nondimensional force coefficients in the x and y directions, given as Cxi and Cyi, respectively, for the i" surface element, are given as:

The total force coefficients in the x and y directions are of course given by summing the above expressions over the total number of elements of interest, NE, and the result is:

2) The indices involving k are everywhere decremented Sensitivity derivatives of these force coefficients taken by one. That is, k+l becomes k, k++ becomes k-$, with respect to ,& are given as: and k becomes k-1 . NE

= { ( b i t - ybi) + CP. At this point, all of the terms of the viscous flux vec- tors for the jkth cell are expressed as explicit functions of the

apk eight local grid coordinates, x l , ? ~ , % 3 , % 4 , %5, 2 6 , x 7 , xs. There- N E

fore evaluating analytically the terms of the Jacobian matrices on the right-hand side of Eq. (24) can be accomplished in a

NE straightforward manner using the fundamental rules of differ-

= 5 {% ( apk entiation, and these details are omitted here.

8.0 Appendix B - Ancillary Sensitivity Relationships

The purpose of this appendix is to include some impor- tant extra terms and simple relationship which might often be needed in computing sensitivity derivatives, in order to make the presentation of the methods more complete. Specifically, expressions are given for generalized aerodynamic force coef- ficients in 2D, and their sensitivity derivatives. Although not actually employed in the present work, one application of these expressions would be to compute the sensitivity derivatives of the thrust of the double-throat nozzle which is used herein as the example problem.

Figure 8 illustrates the ith element (oriented at an arbitrary angle in space) which is located on the boundary of the geomet- ric shape of interest, over / through which the fluid is passing. In the figure, the coordinates (xb,, ybi ) and (xbi+, , ybitl ) are the physical (x,y) coordinates at either end of this i" element,

. (35)

Note in the above expressions that terms such as %,$, etc., are evaluated as being elements of the vector { g ) of the previously discussed right-hand side of Eq. (23), and

ac and also $ are obtained from the vector {g) by apk solution of Eq. (23). (Note: Since Ct involves radients of

+c,. the velocity at the i" element of the boundary, then ;7y: involves terms from both .) Note that if {%) and Eq. (35) is applied over the entire upper and lower walls of the geometry of the test problem, then -% represents the sensitivity derivative of the thrust of the nozzle (taken with respect to Pk), where the algebraic sign reversal accounts for the convention in the figures that a positive thrust acts in the direction opposite to the positive x axis.

9.0 Appendix C - Parameterization of the Nozzle Shape

The purpose of this appendix is to describe the parametric representation of the shape of the nozzle which is tested herein, where this material is also given in Ref. [25]. Because of symmetry, it is only neccessary to parameterize either the lower or the upper solid wall surface. The upper wall is selected here. The wall is described using five polynomial arcs, as illustrated in Fig. 9. In all cases, continuity of slope is enforced at the transition point from one arc to the next. Continuity of curvature is also preserved at the transition point from arc I1 to arc 111 and from arc I11 to arc IV, but is discontinuous as the transition from arc I to arc I1 and from arc IV to arc V. Using Fig. 9, these five arcs are defined as

follows:

Arc I11 (X3 < X 5 X4)

from Eq. (36), the following is deduced:

where in the above and henceforth the subscript x indicates differentiation with respect to X (i.e., Yx = $$).

Arc I1 (X2 5 X 5 X3)

(38) From Eq. (38), the following is noted:

Arc I (XI < X < X2)

Arc IV (X4 < X 5 X5)

where:

Zl = (X - X4)/(X5 - X4) B = 4D - 3Yx(X4)

C = -3D + 2Yx(X4) (42)

D = [Y&) - Y(X4)]/(X5 - X4)

Note that X5 is the location of the second throat, and Y(X5) represents the half-height of this second throat.

Arc V (X5 < X < Xs)

where: z2 = (X - X5)/(X6 - X5) (44)

As a consequence of the above relationships, there are ten geometric shape parameters which can be defined from these expressions, which in the present work will also define the elements of the vector p, of Eqs. (21) and (22). These parameters are given below, including their numerical values for the baseline test geometry.

where: - p1 = H -

p2 = X3 = -

P3 = X4 -

P4 = A =

.is = X2 =

p s = x l =

p 7 = x 5 =

Pa = Y(X5) =

p9 = Xs =

P l 0 = Y(X6) =

I., Fig. 1 - Illustration of the jkth Cell of a Computational Mesh

-12 .00 - 8 . 7 5 -5. 5 0 -2. 2 5 1 . 0 0 4'. 25 7'. 50 1 0 . 7 5 1 4 . 0 0 X A X I S

Fig. 2 - Double-Throat Nozzle, Computational Mesh

Fig. 3a - Pressure Contours, Baseline Numerical Solution, REL = 100

Fig. 3b - Mach Contours, Baseline Numerical Solution, REL = 100

Baseline Numerical Solution

A Conventional Numerical Solution

+ Predicted Numerical Solution

30 2 00 2 00 4 00 5 00 6 00 7 00 8 00 9 OG X W4LL D I S T A N C E

Fig. 4a - Lower Wall Pressure, Predicted Vs. Conventional Numerical Solution, AH = 2%

O Baseline Numerical Solution

A Conventional Numerical Solution

+ Predicted Numerical Solution

I

I0 2 00 3 00 4 00 5 00 6 00 7 00 , ,

8 00 9 00 X WALL D I S T A N C E

Fig. 4b - Centerline Pressure, Predicted Vs. Conventional Numerical Solution, AH = 2%

469

Q Baseline Numerical Solution

A Conventional Kumerical Solution

+ Predicted Kumerical Solution

0 I: 2 . 50 3 00 3 50 4 00 4 . 50 5 . 0 0 5 . 50 6 00

X WALL D I S T A N C E

Fig. 4c - Lower Wall Skin Friction, Predicted Vs. Conventional Numerical Solution, AH = 2%

Baseline Kumerical Solution

A Conventional Numerical Solution

+ Predicted Kumerical Solution

0

m C

0

0 G

0 1 I 1 I I , 1 . 0 0 2 00 3 0 0 4 . 0 0 5 . 0 0 6 00 7 . 0 0 8 00 9 00

X WALL D I S T A N C E

Fig. 5a - Lower Wall Pressure, Predicted Vs. Conventional Numerical Solution, AH = 5%

0 Baseline Numerical Solution

A Conventional Numerical Solution

+ Predicted Numerical Solution

00 2 00 3 0 0 4 00 5 00 6 00 7 00 8 00 9 00 X WALL D I S T A N C E

Fig. 5b - Centerline Pressure, Predicted Vs. Conventional Numerical Solution, AH = 5%

Baseline Numerical Solution

Conventional Xumerical Solution

Predicted Numerical Solution

5 0 3 ' 00 3'. 50 4: 00 4'. 50 5: 0 0 5' . 50 6' 00 X WALL D I S T A N C E

Fig. 5c - Lower Wall Skin Friction, Predicted Vs. Conventional Numerical Solution, AH = 5%

471

6 Baseline Numencal Solution

A Conventional Numerical Solution

+ Predicted Kumerical Solution

I , I I I

00 7

2 . 0 0 3 0 0 4 . 0 0 5 00 6 0 0 7 . 0 0 8 0 0 9 . 0 0 X WALL D I S T A N C E

Fig. 6a - Lower Wall Pressure, Predicted Vs. Conventional Numerical Solution, AH = 10%

I I , 1-

3 0 2 00 3 00 4 00 5 00 6 00 7 0 0 8 00 9 0 0 X WALL D I S T A N C E

Fig. 6b - Centerline Pressure, Predicted Vs. Conventional Numerical Solution, AH = 10%

472

0 Baseline Kumerical Solution

A Conventional Numerical Solution

I I I

5 0 3 . 0 0 3 50 4 . 0 0 4 . 50 5 . 0 0 5 . 50 6 . 00 X WALL D I S T A N C E

Fig. 6c - Lower Wall Skin Friction, Predicted Vs. Conventional Numerical Solution, AH = 10%

2 00 -8 75 -5 50 -2 25 1 00 I

4 2 5 7 50 10 75 14 00 X WALL D I S T A N C E

Fig. 7a - Sensitivity Derivatives, g, Lower Wall

(3 Analytical Appsoarll, Eq. (23)

t Finite Differcncr Approach

3 I 2 00 -8. 75 -5. 50 -2 2 5 1 . 0 0 4 . 2 5 7 50 1 0 7 5 1 4 0 0

X WALL D I S T A N C E

Fig. 7b - Sensitivity Derivatives, $$f-, Centerline

Arc I

Solid Wall

b b i , ybi)

Fig. 8 - Illustration of the ith Element on a Boundary

Fig. 9 - Double-Throat Nozzle, Description of the Boundary Shape