AIAA 91-0173 Sensitivity Analysis Applied to the Euler Equations: A Feasibility Study with Emphasis on Variation of Geometric Shape Arthur C. Taylor, 111, Vamshi Mohan Korivi and Gene W. Hou Old Dominion University Norfolk, VA

29th Aerospace Sciences Meeting January 7-10, 1991IRen0, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S. W., Washington, D.C. 20024

Sensitivity Analysis Applied to the Euler Equations: A Feasibility Study with Emphasis on Variation of Geometric Shape

Arthur C. Taylor, III* Vamshi ~ o h a i ~ k r i v i t

Gene W. Hou*

ABSTRACT

Department of Mechanical Engineering and Mechanics Old Dominion University

Norfolk, Virginia

An approximation method is presented for efficiently pre- dicting the changes which occur in a steady-state numerical so- lution of the Euler equations as a consequence of a small change in the geometric shape of the domain. Using a Taylor's series expansion about a known steady-state solution for one geometric shape, the neighboring steady-state solutions for similar geomet- ric shapes are estimated. The technique is implemented using an upwind cell-centered finite volume formulation. The method is successfully applied to two test problems, including a sub- sonic nozzle (M, = 0.85), and a supersonic inlet (M, = 2.0). In each of these two test cases, conventional numerical solu- tions are first obtained for flow through three slightly different geometric shapes. After defining one of these three solutions to be the "known" solution, the approximation method is used to generate predicted solutions for the remaining two shapes. In both test cases, and for all geometric shape variations which are tested, the predicted solutions compare very well with the conventional numerical solutions, and are computationally less expensive to obtain.

1.0 Introduction

In recent years, great advances have been made in the ability of researchers to generate detailed accurate numerical solutions to the governing equations of fluid flow, as the discipline known as CFD (Computational Fluid Dynamics) has evolved into be- ing. These advances in CFD have roughly paralleled the devel- opment of powerful modem supercomputers. The truly practical benefits of the development of CFD software will only be fully realized, however, when these codes are made to be efficient tools for engineers who must use them in a design environ- ment. The present research is motivated by a belief that the task of making CFD to be a more effective and efficient design tool can be accomplished (at least in part) through the use of a technique known as sensitivity analysis (a subject which is well understood in the field of computational structural mechanics).

*Assistant Professor, Member AIAA ?Graduate Research Assistant $Associate Professor, Member AIAA

Copyright O 1991 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

At the most fundamental level, sensitivity analysis is founded on the principle (from basic numerical analysis) that information about the behavior of an unknown function in the neighborhood of a known point can be efficiently and accurately approximated if the slopes in the neighborhood of the known point are defined, and can be determined at the known point. Roughly speaking, a sensitivity analysis involves the calculation of slopes, known as sensitivity coefficients. More specifically, derivatives of the response(s) of a particular system of interest are taken with respect to the design variable(s), or any indepen- dent variable@), and are calculated. For the designer, accurate calculation of sensitivity coefficients can be used in many ways, including 1) the generation of better initial guesses for analysis, 2) function approximation to predict trends in the response of a system as a consequence of changes in the design variable(s), and 3) for use in design optimization. In the application of the principles of sensitivity analysis in fluid flow calculations, researchers have been and continue to be active, and refs. [I] through [6] are given to provide additional background material on the subject.

A method is proposed herein for predicting the changes which occur in a steady-state numerical solution to the full equations of fluid flow as a consequence of a small change in the geometric shape of the computational domain. Actual implementation and testing of the method is accomplished for the Euler equations in 2D (i.e., testing in 3D and/or for viscous flow has not been attempted to date). In short, the method em- ploys a Taylor's series expansion about a known steady-state numerical solution, where each physical (x,y) coordinate of the computational mesh is defined to be an independent variable. The technique then requires the construction and evaluation of the Jacobian matrix of the complete discrete steady-state resid- ual vector using derivatives taken with respect to the complete vector of physical (x,y) coordinates of the computational mesh. In addition, the technique requires the construction and evalua- tion of the Jacobian matrix of the discrete residual vector with derivatives taken with respect to the complete vector of field variables, where this Jacobian matrix is a well-known part of any implicit formulation for integration of the governing equa- tions in time. Once the complete steady-state numerical solution for a single geometric shape is determined, a single linear sys- tem of algebraic equations involving these Jacobian matrices is then solved for each slightly different geometric shape varia- tion for which an estimated value of the complete steady-state numerical solution might be desired.

There are two immediate primary goals of the present re- search. The first goal is to test the proposed approximation method, so that the potential benefits of the technique (if any) can be assessed, particularly with respect to the accuracy and computational efficiency which can be achieved when using the procedure. The second primary goal of this research is that it might serve as a feasibility study by which the potential merits of applying the principles of sensitivity analysis to modem CFD methods might be judged a priori, especially where variation of geometric shape is a design variable (as is often the case). That is, since both sensitivity analysis and the approximation method of the present research depend entirely on the use of slopes to predict changes in a system's response(s), the success of the present research implies that future applications of sensitivity analysis to the numerical solution of the full equations of fluid flow are feasible.

Of particular interest and concern in this research is the use of slopes to predict solution changes in the presence of discontinuities in the flow field, where these slopes are not defined (e.g., shock waves in supersonic flow). Although in the discretization of the governing equations these discontinuities are typically approximated by a continuously differentiable set of algebraic equations, nevertheless, the slopes which are found in a numerical solution in the neighborhood of shocks can fluctuate wildly. Thus a potential difficulty with the technique is identified at the outset of the research, the effect of which should vary depending on the particular flow field, and the strength of the discontinuities.

The remainder of this manuscript is organized as follows: After this introduction, the next section involves the presentation of theory, which includes a review of the governing equations, a review of the implicit upwind cell-centered finite volume procedure, and the approximation method (i.e., the central focus of the present work) is presented. After that, the results from the application of the approximation method to two test problems are given, including a subsonic and also a supersonic test case. The final major section of the paper is a summary of the work, and includes the conclusions.

2.0 Presentation of Theory

2.1 Governing Equations

The governing equations in this research are the 2-D Euler 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., eg = e + w, where

e is the specific internal energy of the fluid). The flux vectors, F (Q) and G (Q) are given by:

A transformation to generalized (E,r]) coordinates from Cartesian (x,y) coordinates has been made in eq. 1, where €,, t y , r],, qy are mebic 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 7 is the specific heat ratio, taken to be 1.4.

2.2 S ~ a t i a l Discretization

Computationally, the governing equations are solved in this research in their alternate, integral conservation law form using a cell-centered finite volume formulation. Only an overview of the method will be given here, with details on the implementa- tion of this procedure given in refs. [7] through [13], with [7] in particular being a comprehensive treatment of the subject. It should be noted that the approximation method of the present re- search is not restricted exclusively to use with the cell-centered finite volume approach. but is applicable with other numerical techniques as well.

The finite volume method has three important features to be noted. First, the integral form of the equations is valid in the presence of discontinuities in the flow field, and thus accurate shock capturing capability of the method is assured. Addition- ally, the metric terms are quickly and accurately evaluated as the direction cosines of the cell interfaces, and 1/J of eq. 1 is evaluated as the area (volume in 3-D) of the cells. Finally, it is straightforward to show that the method can exactly maintain a freestream flow on arbitrary meshes.

In the finite volume method, flux derivatives are evaluated as a balance of fluxes across cell faces. As an example, this balance of fluxes for the jkh cell in a typical computational grid is given by eq. 3, for a steady solution, and for A( = Ar] = 1:

where subscripts j,k refer to the (, r] directions, respectively, and subscripts j f 4 refer to the € = constant cell interfaces of the jkh cell, subscripts k f refer to the r] = constant cell interfaces of the jkh cell. (All references to quantities which are evaluated at the cell interfaces will therefore require only a single subscript, either j f 4 or k f I.) Rjk is the discrete

3, representation of the residual at the jk cell. From eq. 3, clearly a method is needed for the evaluation

of all flux terms at the cell interfaces, in order to complete

the discretization (in space) of the residual, R(Q), given by eq. 2. In the present work, the flux terms are evaluated using the upwind method of Van Leer [14], although other continuously differentiable upwind methods could also have been chosen. Briefly, with Van Leer's method, the flux vectors are split into two parts according to the signs (either + or - ) of the eigenvalues of the Jacobian matrices of the respective split- fluxes. For example, the flux ej++ (Q,++) of eq. 3 is divided as:

?i++ ( ~j+i) = @A+ (Q.&) +?,;+ (QAJ such that the 4 x 4 split-flux Jacobian matrix [$&Ij+! has

only non-negative real eigenvalues. and , + has only non- I,+ positive real eigenvalues. Of course, the flux vectors at the remaining three cell faces are treated similarly.

Evaluation of the split-fluxes at the cell interfaces is accom- plished through upwind interpolation of the conserved variables from the approximate cell centers, using the interpolating poly- nomials shown below, for the j + & cell interface:

In eqs. 4 and 5 , the missing second subscript on all cell-centered values of the vector Q is simply k, and has been dropped for notational convenience. The symbols AC and VI are the forward and backward difference operators, respectively, in the ( direction (i.e., AcQl = Q,+I - Q, . VEQl = QI - Q1-1). In addition, q5( and K( are parameters which control the accuracy of the spatial discretization in the ( direction, where:

dC = 0 : First order upwind interpolation

q5t = 1 : Higher order interpolation, controlled by K~ where K( may take on values in the range : -1.0 5 K~ 5 +I .0

Special cases are:

K~ = -1.0, fully upwind differencing

K~ = 113, upwind biased 3rd order accurate K < = + 1.0, standard central difference scheme

Although only the split-flux evaluation on the j + & cell inter- face has been explicitly illustrated, flux evaluation on the re- maining three cell interfaces is of course handled similarly. For all test results to be presented herein, a 2nd order accurate fully upwind flux balance (K< = -1 0 ) is uscd in the streamwise direction, and a third order accurate Flux balance (K,, = 1/3) is used in the normal direction.

Using the balance of fluxes given by eq. 3, together with the previously described procedures for evaluation of these fluxes, the discretization of the residual. R(Q), of eq. 2 is complete. Therefore, a discrete representation of the residual, RJk(Q), can be written for each cell in the domain. Furthermore, it is easy to show that RJk(Q) will in general be functionally dependent on

the cell-centered values of the conserved vector, Q, at nine cells in the domain, as expressed by eq. 6, immediately below:

This nine-point "molecule" representation of eq. 6 above is illustrated 'in fig. 1, for the jkhcell in a typical computational mesh. (The meaning of the letters A through I in the figure will become apparent later. Furthermore, it is obvious that adjustments must be made to this nine-point molecule for cells which are adjacent to the boundaries.)

When eq. 6 is written for each cell in the domain, the result is a system of simultaneous nonlinear algebraic equations, given by eq. 7, below, for a steady-state solution:

where (Q*) is called the "root" (i.e., the steady-state solution). Therefore, finding a steady-state solution to the governing equa- tions has been replaced (approximately) by finding the root of a set of simultaneous nonlinear, coupled, algebraic equations.

2.3 Basic Implicit Formulation and Linearization

Having discretized R(Q) of eq. 1 in space at each cell, the Euler implicit discretization of the equations in time is given by: (2) = {R"+~(Q))

where {"AQ) is the incremental change in the cell-centered values of the vector Q between the next (n +lh) time level and the current known (n") time level. That is:

Linearizing in time about the known nh time level using a Taylor series expansion, eq. 8 becomes:

where p&] is a diagonal matrix, and a is a large Jaco- I 4 I bian matrix. Clearly eq. 9 represents a system of simultaneous algebraic equations which is now linear in {" AQ}.

It has been previously established from eq. 6 and fig. 1 that RJk(Q) depends on cell-centered values of the vector Q at only nine cells in the domain (instead of all the cells), and for

this reason [q] is not a full matrix, but instead is very sparse, and has a banded structure. To illustrate this further, the linear vector equation which is associated with the jk& cell in the domain is now isolated from the global system of equations given by eq. 9, and is written below as eq. 10:

It is straightforward to show that the 4 x 4 coefficient ma- trices [A]" through [I]" are constructed of linear combinations of the 4 x 4 split-flux Jacobian matrices (which are evaluated at the cell interfaces). and matrix [BIn includes the time term. Re- turning momentarily to fig. 1, it is seen that eq. 10 can also be represented as a nine-point molecule. The left-hand side of eq. 10 reflects the nine-point molecule in a linear sense, while the right-hand side of the equation reflects the same molecule in a nonlinear sense. It is therefore this nine-point "molecule" struc- ture of the left-hand side of eq. 10 which gives the coefficient matrix of the global linear system (i.e., eq. 9) a sparse, banded structure. Finally, it is noted that in this research all boundary conditions are consistently linearized and are included in the linear system of eq. 9.

In order to write eq. 9 more compactly, the following relationship is defined, given by eq. 11, below:

The fundamental implicit formulation for integrating the gov- erning equations in time may now be written as the following two step iterative procedure, given by eqs. 12 and 13. below:

Step 2 {Q"+'} = {Qn) + {"AQ) n = 1 , 2 , 3 ... (13)

where n is the time step index. In addition, an initial condition, (QO), is needed to begin the iterative procedure.

In principle, eq. 12 can be repeatedly inverted directly using a banded solver which takes advantage of the fact (in terms of both computational method and computer storage) that outside of the bandwidth, all of the elements are zero (as the solution is advanced in time to a steady-state, using eq. 13 to update the solution at each iteration). For very large time steps, it is clear from eq. 11 that the iterative procedure of eqs. 12 and 13 reduces exactly to the well known Newton's root finding method for nonlinear algebraic equations. Furthermore, it is shown in [15],[16] that Newton's method will converge uadratically to

the solution of (R(Q*)) = (0) provided I ) thatq-] is the true Jacobian matrix of the discrete residual, (Rn(Q)), including consistently linearized boundary conditions, and 2) the initial guess, ( Q O ] is sufficiently close to the root, (Q* 1.

For fluid calculations, however, direct inversion of eq. 12 can often be impractical, even when using modem supercom- puters, because of excessively large storage requirements in per- forming the reduction of the matrix [V (Q)]", particularly in 3- D. Furthermore, when the storage restriction is not a limiting factor for a given problem, solution by repeated direct reduction of eq. 12 is not necessarily the most efficient solution procedure with respect to overall CPU time [15],[17]. As an alternative to the direct method, there are many possible choices of algorithms available for use in the repeated solution of the large system of

linear equations given by eq. 12, including conventional relax- ation algorithms [ l 11 ,[I21 ,[I 81, approximate factorization meth- ods [19], SIP (the Strongly Implicit Procedure) [20], and also the preconditioned conjugate gradient method [21], to mention some of the better known. Most importantly, these algorithms circumvent the excessive storage problem of the direct solver approach.

Summarizing briefly, it has been shown that finding a steady-state solution to the governing equations can be replaced (approximately) with the problem of finding the root of a set of coupled, simultaneous, nonlinear, algebraic equations, given by eq. 7. In turn, finding the root of this nonlinear set of equations is replaced by the problem of finding the solutions to an itera- tive sequence of purely linear simultaneous algebraic equations, given by eqs. 12 and 13.

2.4 The Basic Approximation Technique

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 de- fined entirely by the (x,y) coordinates of the intersection points of the grid lines of the mesh. Henceforth, any vector repre- senting the complete set of (x,y) coordinates of a computational mesh will be represented symbolically as (X).

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 {Qi} 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 identical to the first in all respects, except that the geometric shape is different (yet similar), and therefore a second computational mesh, {X2) , is defined. The steady- state solution for the second problem, {Q;} , of course satisfies eq. 15, such that:

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

where:

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

Equation 17 represents the central hypothesis of this re- search, and thorough understanding of this simple relationship and its potential usefulness is a primary mission of this paper. Note that there are two principal ways that the equation might be exploited, depending on which of the two vectors, either {AQ*) or { A X ) , is taken to be "unknown". In all of the work presented herein, {AQ* ) (i.e., the predicted incremental change in the steady-state solution) is taken to be the basic un- known, and the geometric shape change, {AX) , (and thus the entire right-hand side vector of eq. 17) is known. The second alternative is potentially useful in the inverse design problem, where the unknown change of geometric shape. {AX) . 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). Henceforth, the remain- der of the discussion will assume that {AQ*) is defined to be the unknown vector.

The use of eq. 17 involves two large sparse banded Jacobian matrices. On the left-hand side of eq. 17, the terms of the Jaco-

bian matrix. [-I, are well understood and documented.

as they are also used in mplicit time integration / relaxation al- gorithms for the numerical solution of the governing equations. Specifically, a comparison of eq. 12 with eq. 17 will reveal that the large left-hand side Jacobian matrices of these two equations are identical (for very large time steps, from eq. 11). Therefore, in principle, the approximation method of eq. 17 is ideally suited for implementation in conjunction 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. 17 could be very rapidly obtained using simply forward and backward substitu- tion operations, where the previously LU factored left-hand side Jacobian matrix could already be stored in memory following the final Newton iteration during the determination of {Qi).

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-hand side of eq. 17 ( i.e.. [-I. men, a complete

algorithm for the solution of eq. 17 must applied, at least for the first geometric shape variation, { A x ) , for which a predicted change in the solution, {A&*), is desired.

In principle, eq. 17 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 advantage of requiring only a single LU factorization of the left-hand side

coefficient matrix, which of course can then be reused for an un- limited number of geometric shape variations, {AX). If direct solution of eq. 17 is not feasible because of storage restrictions (which is generally the case for practical 3-D problems, even on large supercomputers), then an iterative algorithm will be the only recourse, and consequently the ability to reuse the LU fac- torization of the full coefficient matrix (for multiple geometric shape variations) will be lost.

In this research, it is necessary to derive and construct the

terms of the large Jacobian matrix, iw], of the right-

hand side of eq. 17. The subsequent iscussion will outline this procedure, and the resulting terms are given for the specific nu- merical method which is chosen for application in the present work (i.e., the upwind cell-centered finite volume method using Van Leer's flux-vector splitting. in 2D). It is straightforward to show that the discrete residual for each cell in the computational domain depends on the physical (x,y) coordinates of the four vertices of the cell. This dependence can be written as follows for the jkh cell of the computational domain, where the coor- dinates of the four vertices of this cell are labelled in fig. la:

- + +Gk+{ (33. 32) + dk;+ (33, 32) -G+ k-f (34- 31) - G;-{ (34- 3,)

where the 2 element vectors 31, 32, 33, 34 are given by (see Fig. la):

Because of this dependence of the discrete residual for each cell on only four grid points of the computational mesh, the matrix, [w] is not a full matrix. but is instead very sparse.

and has a banded structure. To illustrate this further, the single linear four component vector equation which is associated with the jka cell in the domain is isolated from the global system of equations given by eq. 17, and is written below as eq. 18:

Note that the 4x4 coefficient matrices [A] through [a on the left-hand side of eq. 18 are identical to those found on the left- hand side of eq. 10. The 4x2 coefficient matrices on the right- hand side of eq. 18 (given by [Wl], [W2]. [W3]. [W4]) are constructed of linear combinations of 4x2 split-flux Jacobian matrices, with derivatives taken with respect to the physical (x,y) coordinates. A detailed presentation of these terms, while straightforward in principle, is algebraically complex. and is

therefore omitted here. but given as an appendix. The global

Jacobian matrix. [v] of eq. 17, is therefore constructed

of four diagonals, where the elements of these four diagonals are 4x2 block coefficient matrices, and there is one such diag- onal which is associated with each of the four 4x2 coefficient matrices on the right-hand side of eq. 18.

As mentioned previously, 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 formulation for integrating the equations in time (eqs. 11. 12. and 13) as well as to the basic approximation method of the present research (eq. 17). In particular, consistent boundary condition linearization is included as an integral part of the global Jacobian matrices on both sides of eq. 17. That is. boundary condition relationships are linearized with respect to their dependence on the field variables as well as with respect to their dependence on the physical (x,y) coordinates.

For many boundary condition types, there will be no de- pendence of the boundary conditions on the physical (x,y) co- ordinates. In these cases, derivatives of boundary condition relationships taken with respect to the coordinate points of the computational mesh will be zero, and thus only zero contribu-

aa(y;,%) tions to the Jacobian matrix, [ 1. will k made. How-

ever, one important example of boundary conditions which & depend (locally) on the coordinates of the grid points is the flow tangency boundary condition, which is used on the test problems to be presented. Consistent linearization of the flow tangency boundary condition relationships will therefore result in non- zero contributions to the right-hand side coefficient matrix of eq. 17 (as well as to the left-hand side, of course).

It is emphasized at this point that fully consistent boundary condition linearization should not be considered optional in the application of eq. 17 (as it typically is in the integration of the equations in time, eqs. 11, 12, and 13). Failure to properly account for the boundary conditions can significantly degrade the quality of the predicted results which are obtained when using the approximation method. (This fact was confirmed by numerical experimentation during the preliminary testing of eq. 17, where as expected, significantly poorer results were obtained when boundary condition linearization was simply neglected.)

3.0 Results 3.1 Test Problem 1 - Subsonic Nozzle (M, = 0.85)

The first test case geometry is a family of similarly shaped nozzles, where the entire flow field is found to be subsonic, with a freestream Mach number (M,) of 0.85. The basic nozzle shape is illustrated in fig. 2. where the ramp angle. 4, is a single parameter which can be varied to generate different (yet closely related) geometries. Ramp angles of 5.0'. 5.5'. and 7.0° were selected for the test calculations of this research, where of these three similar shapes, the 5' geometry was arbitrarily chosen to serve as what henceforth may be referred to as the "baseline" geometry.

Despite the variation of the ramp angle, it is noted that the three similar test geometries are otherwise defined (in this research) to be identical in three respects:

The height of the entrance (i.e.. the inflow bound- ary) in each case is the same, where one-half of this height is defined to be a reference length. (See

fig. 2)

In each case, the upper half of the nozzle is taken to be the mirror image of the lower, and the centerline of the nozzle is thus a plane of symmetry. (See fig. 2)

The total length of each nozzle is four reference lengths. In addition, the inclined ramp starts and ends two and three reference lengths, respectively, from the inflow boundary. (See fig. 2)

Three computational grids were generated, one for each of the three chosen test geometries. Each grid used 41 points in the streamwise direction, and 31 points in the normal direction. Grid stretching was not employed. Only the lower half of the domain was actually gridded, where because of symmetry, it is only necessary to actually compute the solution for either the upper or lower half of the nozzle. Boundary condition specifications were identical in all three test cases, and are outlined as follows:

On the lower wall boundary, flow tangency was enforced.

On the upper boundary of the computational do- main (i.e., on the centerline of the nozzle), flow symmetry was enforced. (Note: In reality, the boundary conditions which were enforced on the centerline were identical to those of the lower wall.)

On the inflow boundary, both entropy and stagna- tion enthalpy were held constant to be that of the freestream, and the v component of velocity was fixed to be zero. The static pressure was extrap- olated.

On the outflow boundary, density as well as both components of velocity were extrapolated. The static pressure was held constant, and specified to be that of the freestream (i.e., P/P, was specified to be 1.0).

Conventional steady-state numerical solutions were first ob- tained for each of the three proposed test geometries using the well-known spatially-split approximate factorization (AF) time integration algorithm [19]. Convergence to the steady-state so- lution was in all cases taken to be when the Lz norm of the discrete residual was reduced six orders of magnitude. The pressure contours from the conventional numerical solution for the baseline (I$=5.0°) geometry are shown in fig. 3.

Using the known numerical solution for the 5.0° nozzle, the approximation method of the present research was applied to the 5.5' and to the 7.0' geometries, and thus a predicted numerical solution was generated for each of these two shapes. Solution

of the linear system of eq. 17 was accomplished by direct LU factorization followed by forward and backward substitution using a vectorized banded matrix solver [15]. Figures 4a and 4b are plots of the wall and centerline static pressure for the 5.5' nozzle. A comparison is made in each of these two plots between the conventional steady-state numerical solution, and the predicted solution using the methods of the present work. (In addition, these figures also include the numerical solution for the baseline 5.0° nozzle). Figures 5a and 5b illustrate the same results for the 7.0' nozzle as do figs. 4a and 4b for the 5.5' nozzle. Figures 6a and 6b are complete pressure contours plots from the conventional numerical solution and the predicted solution, respectively, for the 5.5' test case. Figures 7a and 7b are the same as for figs. 6a and 6b, but are for the 7.0' test case.

For this subsonic test case, and for both of the two geo- metric shape variations (i.e.. the 5.5' and 7.0' nozzles) which were tested, the approximation method of the present research was clearly successful in generating accurate estimates of the resulting changes in a steady-state solution which occur as a consequence of changes in geometric shape. From the wall and centerline pressure plots in particular, it is noted that somewhat greater accuracy of prediction was achieved in the 5.5' case than in the 7.0'. This of course is an expected result, as the 7.0' case represents a greater shaper variation from the baseline geometry (i.e., the 7.0' case is a more severe test case).

3.2 Test Problem 2 - Su~ersonic Inlet (M,, = 2.0)

The second test case geometry is a family of similarly shaped inlets (diffusers), where the entire flow field is found to be supersonic, with a freestream Mach number (M,) of 2.0. The basic inlet shape is again illustrated by fig. 2, where ramp angles (4) of 5.0°, 5S0, and 7' were selected for the test calculations, and the 5' geometry was again arbitrarily chosen to serve as the baseline geometry. The identical three grids which were used in the previous subsonic nozzle test case were reused in the present supersonic inlet test case. Therefore, the previously studied subsonic nozzle problems were converted to be supersonic inlet problems simply by increasing the freestream Mach number, and by changing the specification of inflow and outflow boundary conditions as follows:

(1) On the inflow boundary, all variables were held fixed, and specified to be that of the freestream.

(2) On the outflow boundary, all variables were ex- trapolated.

(3) Boundary conditions on the upper and lower sur- face of the computational domain were the same as for the subsonic test case.

In the supersonic test case, conventional steady-state nu- merical solutions were obtained for the three proposed test ge- ometries using the vertical line Gauss-Seidel (VLGS) relaxation algorithm [l l] , with repeated line relaxation sweeps in the pos- itive streamwise direction only. Under certain specific restric- tions which were applicable to this supersonic test case and which are explained fully in ref. [13], this VLGS algorithm is exactly equivalent to Newton's root finding method. Conver- gence to the steady-state solution was in all cases taken to be when the Lz norm of the discrete residual was reduced to six orders of magnitude. The pressure contours from the conven- tional numerical solution for the baseline (4=5.0°) geometry are shown in fig. 8.

Using the known numerical solution for the 5.0' nozzle, the approximation method of the present research was applied to the 5.5' and the 7.0' geometries, and thus a predicted numerical solution was generated for each of these two shapes. Direct solution of the linear system of eq. 17 for the supersonic test case was accomplished using a single sweep of the VLGS algorithm in the positive streamwise direction (the explanation of this follows from the previous paragraph ). Figures 9a and 9b are plots of the wall and centerline static pressure for the 5.5' nozzle. A comparison is made in each of these two plots between the conventional steady-state numerical solution, and the predicted solution using the methods of the present work. (In addition, these figures also include the numerical solution for the baseline 5.0' nozzle). Figures 10a and lob illustrate the same results for the 7.0' nozzle as do figs. 9a and 9b for the 5.5' nozzle. Figures 1 l a and 1 l b are complete pressure contours plots from the conventional numerical solution and the predicted solution, respectively, for the 5.5' test case. Figures 12a and 12b are the same as for figs. 1 l a and l lb , but are for the 7.0' test case.

For this supersonic test case, and for both of the two geometric shape variations (i.e.. the 5.5' and 7.0' inlets) which were tested, the approximation method of the present research was clearly successful in generating accurate estimates of the resulting changes in a steady-state solution which occur as a consequence of changes in geometric shape. Again, the expected result is noted that somewhat geater accuracy of prediction was achieved in the 5.5' case than in the more severe 7' case. In addition, when comparing the overall accuracy of the predicted results in the subsonic test case to that of the supersonic case, it is noted that somewhat greater success was achieved in the former test problem. This is attributed to the belief that the supersonic test problem represents a more severe test problem due to the presence of shock waves in the flow field.

3.3 CPU Time

A short study was undertaken to evaluate (at least in part) the CPU time which was required for some of the calculations which are reported herein. However, because of the vast differ- ences in computational resources which are recquired for dif- ferent types of flow problems, it should not be inferred that the comparisons which are to to be presented in this limited study can be accurately extrapolated to every new flow problem. All

CPU times which are reported were obtained in single precision on an eight processor Cray-YMP computer, and convergence to steady-state in all cases was defined to be a six orders of magnitude reduction in the norm of the residual.

The subsonic test case with the q5 = 7.0" ramp geometry was chosen for CPU time comparisons. Using the approximate factorization (AF) time integration algorithm with a constant Courant number of 10, the conventional numerical solution for the "baseline", t$ = 5.0" ramp geometry required 533 iterations and 84.02 CPU seconds to achieve steady-state convergence. where freestream conditions were used as an initial start-up guess. Then, using the steady-state solution for the 4 = 5.0" geometry as an initial guess, the conventional numerical solution for the t$ = 7.0" geometry required an additional 454 iterations and 71.89 CPU seconds to achieve convergence (again using the AF algorithm with a Courant number of 10).

In using the approximation method of the present research to produce an estimated solution for the subsonic, q5 = 7.0" ge- ometry (using the conventional numerical solution for the t$ = 5.0" geometry for the prediction), then 5.06 CPU seconds were recquired for the complete set-up and solution of eq. 17. Of this total CPU time for the set-up and solution of eq. 17, the CPU time was measured which was required just for the for- ward and backward substitution operations (following complete LU factorization) and also including the CPU time for the large matrix-vector multiplication operation on the right-hand side of the equation. This time was measured to be only 0.037 CPU sec- onds, which of course represents the computational work which would be required to produce a complete estimated solution for each additional geometric shape variation which might follow the prediction for the initial shape variation. Therefore, the im- proved computational efficiency of the approximation method in generating predicted numerical solutions when compared to the cost of generating conventional numerical solutions is clearly demonstrated.

The predicted solution (using eq. 17) for the q5 = 7.0" ge- ometry was used next as an initial start-up guess for generating the conventional numerical solution for this geometry (again using the AF algorithm with a Courant number of 10 ). There were 331 additional iterations and 52.01 CPU seconds needed to achieve convergence (to the required tolerance, discussed pre- viously). This represents a net CPU savings of 14.82 seconds when compared to the far simpler procedure of merely using the conventional numerical solution for the q5 = 5.0" geome- try as an initial guess. Therefore, as a technique for generat- ing improved initial conditions for subsequent use in obtaining converged conventional numerical solutions, the approximation method produced a 20.61% net savings in computational cost in this test problem.

4.0 Summary and Conclusions

An approximation method is presented for the efficient pre- diction of changes which occur in the steady-state numerical solution of the governing equations of fluid flow as a conse- quence of small changes in the geometric shape of the com- putational domain. The technique is implemented in 2D for inviscid flow (i.e., for the Euler equations). The method is

based on a Taylor's series expansion (for multiple equations in multiple variables) about a known steady-state numerical solu- tion. where each physical (x,y) coordinate in the computational mesh is taken to be an independent variable.

The technique is successfully applied to two test problems including 1) a subsonic nozzle (M, = 0.85) and 2) a supersonic inlet (M, = 2.0) . In all test cases, the ability of the method to accurately predict changes in the steady-state solutions (as a consequence of geometric shape variations) was verified. Although the full potential for gains in computational efficiency through use of the method has not yet been fully assessed, it was found in a brief comparative study of CPU times that the procedure resulted in about a 20% net CPU savings when used as a technique for producing a better initial guess for obtaining a conventional numerical solution. However, the predicted numerical solutions were obtained significantly more efficiently than were the corresponding conventional numerical solutions.

The results of this research confirm the possibility of using slopes to accurately predict changes which occur the field vari- ables of a given flow problem, in response to small changes in the constraints on the problem, and at least for some problems, this is possible even in the presence of discontinuities in the flow field. On the basis of this study, the feasibility of applying the methods of sensitivity analysis to calculations involving the full equations of fluid flow is (at least in part) confirmed.

5.0 Acknowledgements

This research was supported in part by grant number DMC- 865-7917 from the National Science Foundation.

6.0 References

Yates, E. C., Jr., and Desmarais, R., "Boundary Integral Method for Calculating Aerodynamic Sensitivities with 11- lustration for Lifting Surface Theory", in Proceedings of the International Symposium of Boundary Element Meth- ods (IBEM 89). published by Springer-Verlag. Oct. 2 4 , 1989, East Hartford, Corn.

Elbanna, H. M., and Carlson, L.A.,"Determination of Aero- dynamic Sensitivity Coefficients in the Trarisonic and Su- personic Regimes". AIAA paper 89-0532.

Yates, E. C., Jr., "Aerodynamic Sensitivities from Subsonic, Sonic, and Supersonic Unsteady, Nonplanar Li fting-Surface Theory", NASA TM-100502, September, 1987.

Sobieszczanski-Sobieski, J.. "The Case For Aerodynamic Sensitivity Analysis". In "Sensitivity Analysis in Engineer- ing", NASA CP-2457, 1987.

Bristow, D. R., and Hawk, J. D., "Subsonic Panel Method For The Efficient Analysis of Multiple Geometry Perturba- tions." NASA CP-3528. 1982.

Jameson. A., "Aerodynamic Design Via Control Theory", NASA CR-181749, also ICASE Report No. 8 8 6 4 , Novem- ber, 1988.

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

Thomas, J. L., Van Leer, B., and Walters, R. W.. "Implicit Flux-Split Schemes for the Euler Equations." AIAA Paper 85-1680.

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

Newsome, R. W.. Walters, R. W., and Thomas, J. L., "An Efficient Iteration Strategy for Upwind/Relaxation Solutions to the Thin-Layer Navier-Stokes Equations," AIAA Paper 87-1 13.

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

Napolitano, M., and Walters, R. W., "An Incremental Block-Line-Gauss-Seidel Method for the Navier-Stokes Equations, AIAA Journal , Vol. 24, No. 5, May 1986, pp. 770-776.

Walters, R. W., and Dwoyer, D. l., "Efficient Solutions to the Euler Equations for Supersonic Flow with Embedded Subsonic Regions," NASA Technical paper 2523, January 1987.

Van Leer. B., "Flux-Vector Splitting for the Euler Equa- tions," ICASE Report 82-30, September 1982 (aslo Lec- ture Notes in Physics, Vol. 170, 1982, pp. 507-512.).

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

Ortega, J. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970, pp. 312-313.

Hafez, M., Palaniswamy, S., and Mariani, P., "Calcula- tions of Transonic Flows with Shocks Using Newton's Method and Direct Solver, Part 11," AIAA Paper 88-0226.

Taylor, A. C., 111 and Ng, W. F., and Walters, R. W., "Up- wind Relaxation Algorithms for the Navier-Stokes Equa- tions Using Inner Iterations," AIAA Paper 89-1954.

B e h , R. M. and Warming, R. F., "An Implicit Factored Scheme for the Compressible Navier-Stokes Equations," AIAA Journal, Vol. 16, April 1978, pp. 393402.

Schneider, G. E., and Zedan, M.. "A Modified Strongly Implicit Procedure for the Numerical Solution of Field Problems," Numerical Heat Transfer, Vol. 4, 1981, pp. 1-19.

Venkatakrishnan, V., "Preconditioned Conjugate Gradient Method For The Compressible Navier-Stokes Equations". AIAA paper 904586.

7.0 Appendix

The purpose of this appendix is to complete the presentation of the details of the terms which comprise the 4 x2 coefficient matrices [W1]. [Wz], [W3]. [W4] from the left-hand side of eq. 18. As stated earlier, these matrices are constructed of linear combinations of 4 x 2 split-flux Jacobian matrices. with derivatives taken with respect to the physical (x,y) coordinates, and it is easy to show that these linear combinations are:

a ~ + (i4# X I ) a e - (i4, i l ) - [ a 4 ] k-4 - [ a i l ] k-f

8 ~ - (%q, XI)

k-f

Before taking derivatives, the terms of the split-flux vectors in 2D (according to Van Leer [14]) are given below as:

where in the above, v is the gradient operator, and i, j' are unit vectors in the x,y directions respectively.

The evaluation of F* proceeds as follows:

where in the above, fi is the full flux vector in the < direction, which is defined earlier in the text. For IME I < 1, (where the component of flow in the F direction is subsonic), the following for F* is used:

The terms of the split-flux vectors, G* are identical to that given above for F*. except that the coordinate q everywhere replaces the coordinate E in these terms.

Using fig. l a for the jkh cell, it is staightforward to verify the following geometric relationships for use in evaluation of the metric terms in a finite volume sense, for v E , ~ 7 7 = 1:

(3) = y+-y1 J k-f

Using the above relationships, all split-flux vectors of the jkh cell can be written as explicit functions of the x,y coordinates of the jkh cell. This in turn permits derivation of analytical expressions for the individual terms of the split-flux Jacobian matrices, with derivatives taken with respect to these x,y co- ordinates, which is the desired result. The individual terms of the 4 x 2 split-flux Jacobian matrices,

be presented below. The results for these

be generalized to include the split-flux Jacobian matrices which result from the remaining three cell faces. Therefore, these terms are:

For ME > 1.0 :

where in the above, F and G are the full flux vectors in Cartesian coordinates (defined earlier in the text). For MC = -1.0, the resulting terms are identical to these above, except that k+ replaces F-, and F- replaces @+.

For IMCI < 1.0' the resulting terms are algebraically more complicated, and are given as:

All of the preceding results for the j+1/2 cell face can be written for the j-112 cell face simply by everywhere replacing the coordinates, XI , y 1, xz, yz, with the coordinates, q, y4, xg, y3. respectively.

The preceding results for the j+1/2 cell face can be written for the k+1/2 cell face by:

1. Everywhere replacing F with G.

2. Everywhere replacing the coordinate ( with the coordinate

rl.

F x * u)] 3. Everywhere replacing the coordinates, XI , y 1. XZ, yz. with the coordinates, x3, y3, x2, yz, respectively.

+ v] - f:... * [(A * ") * 7 l 0 F l ~ v F I 4. The right-hand side of each resulting expressions is

multiplied by - 1 .O.

Finally, the required terms from the k-112 cell face can be obtained from the expressions for the k+1/2 cell face simply by everywhere replacing the coordinates, x3, y3, xz, yz, with the coordinates, xq, y4, xl , yl , respectively.

- S trearnwise Direction

j - 2 j - l j j+ l j+2

Fig. 1 -Nine-Point "Molecule" Representation of Eqs. 6 and 10

.I 1 J - 7 .j.k '- j+J-

d+ I l k

/ X 4 1 y 4 - k j d i I / A > o n s t 7 --L I C = onstant

t - constant

L. Fig. l a -Illustration of jkfi Cell of a Typical Computational Mesh

12

Inflow

Fig. 2 -Subsonic Nozzle / Supersonic Inlet Test Geometry

Fig. 3 -Pressure Contours, &5.0°, Subsonic Nozzle, M, = 0.85, Numerical Solution

0 m -

O 5.0' Numerical Solution A 5.5' Numerical Solution

0 N

+ 5.5' Predicted Solution -

01 0 8 I I - .- -- 7 - 1 0 . 0 0 0 . 50 1 . 0 0 1 . 5 0 2 . 0 0 2 . 50 3 . 0 0 3 . 50 4 . 0 0

X WALL DISTANCE

Fig. 4a -Lower Wall Pressure, Subsonic Nozzle, M, = 0.85, 4 = 5.5'

I O 5.0" Numerical Solution 0 hl

A 5.5' Numerical Solution

- + 5.5' Predicted Solution

W LT 3 v," c n . W - [L R

W 2: H . 1- [L

&-IF--- I-- I , 1 I 0 . 0 0 0 . 50 1 . 0 0 1 . 5 0 2 . 0 0 2 . 50 3 . 0 0 3 50 4 . 0 0

X WALL DISTANCE

Fig. 4b -Centerline Pressure, Subsonic Nozzle, M, = 0.85, 4 = 5.5'

0-t --- I I 11 I I I 0.00 0.50 1.00 1.50 2.00 2. 50 3.00 3.50 4.00

X WALL D I S T A N C E

Fig. 5a -Lower Wall Prcssure, Subsonic Nozzle, M, = 0.85, 4 = 7.0'

o O 6.0' Numerical Solution N

... A 7.0' Numerical Solution + 7.0' Predicted Solution

W LT 3 WE: c n . W - LT a.

w 2 z H . 1 - lx W I- Z

0 P..

o 0.00 L l 0.50 1.00 I 1 . I 50 2 00 2. I 50 3.00 I 3.50 I 4.00 I

X WALL D I S T A N C E

Fig. 5b -Centerline Prcssurc, Subsonic Nozzle, M, = 0.85, 4 = 7.0'

Fig. 6a -Pressure Contours, 4 = 5.5', Subsonic Nozzle, M, = 0.85, Numerical Solution

Fig. 6b -Pressure Contours, 4 = 5.5'. Subsonic Nozzle, M, = 0.85, Predicted Solution

Fig. 7a -Pressure Contours, q5 = 7.0'. Subsonic Nozzle, M, = 0.85, Numerical Solution

Fig. 7b --Pressure Contours, 4 = 7.0'. Subsonic Nozzle, M, = 0.85, Predicted Solution

Fig. 8 -Pressure Contours, 4 = 5.0°, Supersonic Inlet, M, = 2.0, Numerical Solution

O 5.0' Numericul Solution A 5.5O Numerical Solution + 5.5' Predicted Solution

?? o 0.00 0. 50 1.00 1.50 2.00 2. 50 3.00 3. 50 4.00

X WALL D I S T A N C E

Fig. 9a -Lower Wall Pressure, Supersonic Inlet, M, = 2.0, 4 = 5.5'

O 5.0' Numerical Solution A 5.5' Numerical Solution + 5.5' Predicted Solution

07- I I I I I - - - --

1-1 0.00 0. 50 1.00 1 . 50 2 00 2. 50 3.00 3. 50 4.00

X WALL D I S T A N C E

Fig. 9b -Centerline Pressure, Supersonic Inlet, M, = 2.0, 4 = 5.5'

O 5.0' Nurncricul Solution A 7.0' Numerical Solution + 7.0' Prcdictcd Solution

LO ' - ,C -- -- 0 . 0 0 0 . 50 1 .00 1.50 2 . 0 0 2 . 5 0 3 . 0 0 3 . 50 4 .00

X WALL D I S T A N C E

Fig. 10a -Lower Wall Pressure, Supersonic Inlet, M, = 2.0, 4 = 7.0°

"--,- o 0 . 0 0 0 .50 1.00 1. 50 2 . 0 0 2 . I 50 3 . 0 0 I 3 . I 50 1 4 . 0 0

X WALL D I S T A N C E

Fig. lob --Centerline Pressure, Supersonic Inlet, M, = 2.0, 4 = 7.0'

Fig. l l b -Pressure Contours, 4 = 5 S 0 , Supersonic Inlet, M, = 2.0, Predicted Solution

Fig. 12a -Pressure Contours, 4 = 7.0°, Supersonic Inlct, M, = 2.0, Numerical Solution

Fig. 12b -Pressure Contours, 4 = 7.0°, Supersonic Inlet, M, = 2.0, Predicted Solution