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Copyright ©1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1997A9715871, AIAA Paper 97-0873

Application of low speed preconditioning to the compressible Navier-Stokesequations

Christopher L. ReedLockheed Martin Tactical Aircraft Systems, Fort Worth, TX

Dale A. AndersonTexas Univ., Arlington

AIAA, Aerospace Sciences Meeting & Exhibit, 35th, Reno, NV, Jan. 6-9, 1997

The concept of low-speed time-derivative preconditioning is evaluated for use in production CFD analysis codes.State-of-the-art CFD techniques are implemented in a 2D finite-volume Navier-Stokes code in order to determine theirbehavior with preconditioning. The standard equations and three preconditioners are implemented in the test code using bothcentral-difference and Roe's flux-difference splitting upwind techniques. The test code also includes four solver techniques,two nondimensionalizations, and three flux limiters. Results of numerous tests indicate that properly implemented low-speedpreconditioning produces Mach-number-independent convergence. Test cases indicate that solution accuracy is degradedsignificantly as the Mach number is reduced without preconditioning. It is shown analytically and numerically that thetruncation error of the original equations is significantly higher than that of the preconditioned equations at very low Machnumbers. (Author)

Page 1

APPLICATION OF LOW SPEED PRECONDITIONING TO THECOMPRESSIBLE NAVIER-STOKES EQUATIONS

Christopher L. Reed*Lockheed Martin Tactical Aircraft Systems

Dale A. Andersen^University of Texas at Arlington

ABSTRACTThe concept of low-speed time-derivative precondi-tioning is evaluated for use in production CFD anal-ysis codes. State-of-the-art CFD techniques areimplemented in a two-dimensional finite-volumeNavier-Stokes code in order to determine theirbehavior with preconditioning. The standard equa-tions and three preconditioners are implemented inthe test code using both central-difference andRoe's flux-difference splitting upwind techniques.The test code also includes four solver techniques,two non-dimensionalizations, and three flux limit-ers. Results of numerous tests indicate that prop-erly implemented low-speed preconditioningproduces Mach number independent convergence.Test cases indicate that solution accuracy isdegraded significantly as the Mach number isreduced without preconditioning. It is shown analyt-ically and numerically that the truncation error ofthe original equations is significantly higher thanthat of the preconditioned equations at very lowMach numbers.

INTRODUCTIONComputational Fluid Dynamics (CFD) methodshave matured significantly over the past severalyears. Areas of significant development includetransonic analysis, complex grid generation meth-ods, unstructured grid techniques, upwind methodsand incompressible flow techniques. These tech-niques make a large number of CFD analyses notonly possible, but routine. However, some technol-ogy areas still need significant development. Onesuch area is the analysis of flow fields which arepredominately incompressible but include largeregions of fully compressible flow. Such is the caseof vertical and short take-off and landing (V/STOL)aircraft which make use of propulsive lift.

Techniques for solving fluid dynamic problemshave historically been classified as applicable toeither compressible or incompressible flows1. Thetime-dependent inviscid compressible fluiddynamic equations are classified mathematicallyas hyperbolic. Methods to obtain steady-state solu-tions to these equations assume an initial solutionand then march in time until a steady state isreached. This final solution satisfies the steady-state flow equations. To insure a conservative solu-tion and correct shock wave locations, the conser-vative form of the equations is solved using"conservative variables" (i.e.,p, p«, pv, pw, E) asthe independent variable set. Methods to carry outthese solutions quickly and accurately have beendeveloped to a very high level. These methodsinclude implicit solvers, upwind differencing tech-niques, and a host of domain decompositionschemes.

The incompressible fluid dynamic equations are asubset of the compressible equations. These equa-tions are termed "incompressible" because as theMach number is decreased in the limit to zero, thevariation in density also approaches zero and thetime-derivative of density in the continuity equationvanishes. This causes the equations to become amixed elliptic/parabolic system. For this reason, theincompressible equations are generally solvedusing a segregated approach in which a Poissonequation is solved for pressure and the velocityfield is obtained using a guess-and-correctprocedure2. Since the density is nearly constantand shock wave locations are not a problem, it ismore efficient to use "primitive variables" (i.e.,p, u, v,w,T) as the independent variable set whensolving the incompressible equations3.

* Specialist Senior, Senior Member AIAAt Professor, Department of Aerospace and Mechanical EngineeringCopyright © 1997 by Lockheed Martin Corporation. All rights reserved. Published by the American Institute of Aeronautics and Astronautics,

Inc. with permission.

1American Institute of Aeronautics and Astronautics

A number of authors have advocated a variety oftechniques for solving the fluid dynamic equationsat "all speeds." In reference 4, Volpe discussesusing standard compressible flow codes to solvelow Mach number problems. The results of hisexperiments indicate that as the Mach number wasreduced from 0.1 to 0.001, the convergence wasslowed considerably and solution accuracy wassignificantly affected (errors increased by as muchas a factor of 30). In reference 5, Karki and Patan-kar present a method which uses a segregatedapproach to solve the two-dimensional, steady-state flow equations at all speeds. The resultsobtained are in good agreement with test dataexcept in the vicinity of a smeared shock whichthey attribute to excessive numerical dissipation. Inreference 1, Chen and Pletcher use a coupledModified Strongly Implicit procedure (CMSIP) tosolve the two-dimensional compressible Navier-Stokes equations written in primitive variables.They show excellent agreement with test data for avariety of problems with Mach numbers down to0.01.

References 6-12 discuss a technique called "pre-conditioning" which attempts to scale the eigenval-ues of the coupled fluid dynamic equations in orderto accelerate convergence for low-speed flows.This technique alters the transient part of the equa-tions which are then time-marched to a steady-state solution. When the steady state is reached,the transient part of the equations approaches zero(along with the preconditioning) and a steady-statesolution is obtained. Various results from the abovereferences indicate excellent convergence charac-teristics for the preconditioned equations and goodcomparisons with test data.

The present work evaluates and extends the pre-conditioning concept for use in production CFDcodes. Techniques were developed which allowpreconditioning to be retrofitted into existing CFDanalysis codes. In order to develop techniquesapplicable to a wide range of existing compressibleflow solvers, various state-of-the-art CFD method-ologies had to be considered. This paper dis-cusses the concept of low-speed preconditioningand the implementation of various state-of-the-artCFD techniques.

PRECONDITIONING CONCEPTSTraditional time-dependent solution techniquesusing the compressible Navier-Stokes equationswork poorly at very low Mach numbers. Variousreasons for this are given in the literature6. Thesereasons include round-off error or computer preci-sion problems, factorization errors, and time-stepconstraints due to near infinite acoustic speeds1.The first of these problems can be overcome byselection of appropriate non-dimensionalizationsand increasing computer precision. The secondproblem can be overcome by using alternate solvertechniques which reduce or eliminate factorizationerrors. The last of the above stated problems isactually the most significant and the most difficultto overcome.

For hyperbolic systems of equations, the eigenval-ues of the Jacobian matrix define the speed ofpropagation of characteristic information throughthe solution domain. For an explicit numericalmethod, the CFL condition specifies the maximumallowable time-step based on the maximum eigen-value. This condition ignores the minimum eigen-value which controls the rate of propagation of theslowest characteristic information. When there is alarge disparity in the eigenvalues some informationpropagates near its desired rate while other infor-mation is propagating much slower than desired.Since the equations are coupled, the solution can-not converge to steady-state until the slowest prop-agating waves have stopped moving through thesolution domain. This is what causes the conver-gence of the solution algorithm to becomeextremely slow or "stiff." The disparity between theeigenvalues can be quantified by defining the Con-dition Number (CN) as the ratio of the magnitudeof the maximum to the minimum eigenvalue.

where Cv is the contravarient velocity and c is thespeed of sound. As the Mach number decreases,the condition number increases. It would be desir-able to modify the equations so that the eigenval-ues are "clustered" about some physicallymeaningful propagation speed and do not vary withMach number. Since we are only interested insteady-state solutions, it is reasonable to attemptto modify the time-derivative or transient part of the

American Institute of Aeronautics and Astronautics

equations. This would also allow the modification toapproach zero as the solution reached a steadystate. The modifiedl equations are written as

The eigenvalues are derived from the Jacobianmultiplying the spatial derivative; therefore, werequire [P]-1 to be invertible so that

Now, it is the eigenvalues of the matrix product[P][A] that control the propagation speed. Tomaintain the physical significance of the eigenval-ues, the modified eigenvalues must remain realand maintain the signs of the original eigenvalues.The matrix [P] is selected to provide clustered andsomewhat Mach number independent eigenvalues.

Along with the original equations, three precondi-tioners are considered in this paper. They are theChoi and Merkle inviscid preconditioner6, the Choiand Merkle viscous preconditioner7, and thePletcher and Chen preconditioner12.

The Choi and Merkle inviscid preconditioner isgiven as

1 0 0 00 1 0 00 0 1 0

ooo-i

The eigenvalues of the preconditioned system are

c c2•j(1 + M2) T J-f(M2- I)2 + M2c2

Notice that when M is equal to 1, the third andfourth eigenvalues revert to their unmodified form.Also notice that as M approaches zero the eigen-value dependence on the speed of sound van-ishes. Figure 1 shows a plot of the conditionnumber versus Mach number for both of the Choi

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and Merkle preconditioners. The condition numberof the unmodified equations is also shown for refer-ence. Notice that the condition number becomes aconstant as the Mach number is reduced. This indi-cates that the preconditioned Jacobian is wellbehaved and should produce Mach number inde-pendent convergence at low Mach numbers. Unlikethe original system, the eigenvectors of the precon-ditioned equations are dependent on the M fromthe preconditioning matrix. As M->0 the lefteigenvectors are no longer a set of four linearlyindependent vectors and therefore do not span thesolution space. This is similar to observationsmade in reference 13 for a different preconditioningconcept. For this reason, limits are placed on thevalue of the variable M in the preconditioningmatrix. On the upper end, the value of M is limitedto 1 because the original system condition numberis more favorable at Mach 1 and above. On thelower end, the value is limited to some small num-ber, e, which is generally taken to be an order ofmagnitude smaller than the freestream Mach num-ber. However, this is dependent on the particularflow solution and in many cases may be set to avery small number such as 10~6. Thus, the formulafor M in the preconditioning matrix is


Af = \ Af ,„„,, => e < A/lnca, < 1.01.0=>A/local>1.0

The Choi and Merkle viscous preconditioner isgiven as

0 1 0 00 0 1 0

-c2) o o i + -

The eigenvalues of the preconditioned system areidentical to those of the inviscid preconditioner. Theeigenvectors for this preconditioned system aresomewhat less dependent on M than the inviscidpreconditioner eigenvectors; therefore, this precon-ditioner is somewhat more robust than the inviscidpreconditioner.

The Pletcher and Chen preconditioner is given as

1 00 c2A/2 )0 1 0 00 0 1 0

0 0 0

The eigenvalues of the preconditioned system are

yv(l + yA/2) T + (1 -yA/2)2

The behavior of this set of eigenvalues is similar tothose of the Choi and Merkle preconditioners. Theeigenvectors for this preconditioned system aresimilar to those for the Choi and Merkle inviscidPreconditioner.

TWO-DIMENSIONAL DEVELOPMENTIn order to develop the techniques and evaluate thesuitability of incorporating preconditioning intoexisting Navier-Stokes codes, a two-dimensionaltest code was developed. This code was used toevaluate preconditioning concepts, as well as totest preconditioning with a variety of numericalsolvers, dissipation methods and boundary condi-tions. We begin by describing the finite-volume dis-cretization applied to the test code.

Finite-Volume DiscretizationThe non-dimensional, conservative, differentialform of the two-dimensional Navier-Stokes equa-tions, excluding body forces and external heataddition, can be written in vector form using Carte-sian coordinates as

where Q is the conservative variables, P is theflux vector and $ is the spatial vector given below.

= u + vj

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In the above equations, p is the fluid density, uand v are the Cartesian components of velocity,and E is the total energy per unit volume. The vari-


able m indicates a summation over i and j . 8im isthe Kronecker delta. The pressure, p , is related tothe conserved variables by the ideal gas law.

p = y-

The viscous stress terms and heat flux terms are

= JL/ (2)

= JLr

The ratio of specific heats is denoted by y. r is thestatic temperature, u, is the viscosity coefficientand K is the thermal conductivity. These equationshave been non-dimensionalized using theapproach discussed in reference 2.

x - xL' '-I- t = L/V

P = ——Y

The second method was used for the majority ofthe test cases shown. A comparison of computedresults using each of these methods is shown inthe results section.

The integral form of the Navier-Stokes equationscan be obtained by integrating Eq. (1) over an ele-mental control volume. Rearranging the terms andapplying Green's Theorem (or the Divergence The-orem in a plane) yields the finite-volume form.

faces

(3)

Here, the first term represents the rate of change ofQ over some fixed control volume Q, and the sec-ond term represents the flux balance through thesides of the same control volume.

Since the conservative fluxes are easily con-structed from primitive variables and the viscousterms and upwinding require primitive variables, itis more convenient to express the Navier-Stokesequations in terms of primitive variables. Thoughwe have used both {p, u, v, p} and {p, u, v, T}primitives, the former will be used here. Using thechain rule, Eq. (3) can be transformed.

faces

= £_ T=T-TJ

R =

where °° denotes freestream conditions, the tilderepresents dimensional quantities, and L is thecharacteristic length used to compute the Reynoldsnumber,

Re =

Pressure in the test code has been non-dimension-alized in two ways in order to determine the mostsuitable technique. The standard method is

where

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These equations are no longer conservative intime, but the steady-state fluxes remain conserva-tive. Now applying the preconditioning to the time-derivative yields

The alternate method suggested by severalauthors

faces

Rearranging yields

faces

It is very important for proper shock wave positionthat the right-hand-side of the equation remainconservative, so we cannot move the transforma-tion inside the difference expression there. Thisalso makes it much easier to retrofit precondition-ing into an existing code since the majority of theflux calculations do not have to be altered. The onlyflux requiring modification for preconditioning is theartificial dissipation term as will be shown later.

The implicit formulation of the equations is given as

[7 + ̂ (8^+DjP)]A0 = -^ (4)

faces _> _^[(F,. + Fv) • Ao] +

For either upwind or central-difference, the fluxterms can be broken into three separate parts: theinviscid flux, the viscous flux, and the artificial dissi-pation flux. The following paragraphs describeeach of the terms in this equation.

The left-hand-side of the equation contains theJacobians for the inviscid and dissipation fluxterms. The viscous Jacobians are not included inthe present work. Note that the inviscid Jacobianhas been multiplied by the preconditioning matrix.The term D\f is the implicit dissipation term basedon the preconditioned system and depends onwhether we are using upwind or central difference.These matrices are shown in reference 1 4 for eachpreconditioner implemented in the two-dimensionaltest code.

The first summation on the right-hand-side of theequation contains the inviscid and viscous fluxterms. In either a central-difference algorithm or aRoe's flux difference split upwind algorithm theinviscid flux vector, F; , at a cell face is computedas

where F, and Fr are the flux vectors from the leftand right sides of the face. In central-difference orfirst-order upwind schemes, Ft and Fr are con-structed from the primitive variables located at thecell centers on the left and right sides of the spe-cific cell face.

For an upwind calculation, Roe's flux differencesplitting technique is used16. Higher-order accuracyis obtained by extrapolating the flow variables tothe face from each side16. It is possible that inregions of strong gradients in the flow that extrapo-lations can cause non-physical results. For thisreason, the extrapolations are limited, whichreduces the spatial accuracy to first-order inregions of strong gradients. Two different limitershave been implemented in the two-dimensionaltest code. They are the superbee/minmod limiter16and Thompson's smooth symmetric limiter17.

The viscous flux term, Fv, is composed of viscousstress and heat flux terms which are shown in Eq.(2). This term is computed using second-ordercentral differences.

The second summation on the right-hand-side ofEq. (4) is the explicit artificial dissipation. The artifi-cial dissipation term has been separated from theother flux terms because this term requires specialtreatment due to the preconditioning and therequirement for conservation. This term must bebased on the preconditioned Jacobian and not sim-ply multiplied by the preconditioning matrix. This isbecause we have modified the eigenvalues andeigenvectors of the system and both upwind andcentral-difference dissipations are based on thecharacteristic speeds and direction of propagationof information through the solution. According toreference 9, central-difference dissipations shouldbe applied to the variables which are beingupdated, which in this case are the primitive vari-ables. Experiments performed during the currentresearch have proved this to be true for best con-vergence. At the same time, we have a require-ment to maintain a conservative scheme so thatshock waves will be positioned properly. These twoconflicting requirements were overcome in thepresent work by applying two transformations.First, the artificial dissipation flux at a cell-face is


computed using the techniques described in thefollowing paragraphs. These fluxes are then trans-formed to conservative form and the flux balancefor the cell computed. The resulting flux differenceis transformed back to non-conservative form. Thissame technique was used for both the central-dif-ference and the upwind dissipations. Numericalexperiments verify that using this technique pro-vides accurate shock capturing. Meanwhile, basingthe dissipation directly on the conservative vari-ables inhibited convergence, and ignoring the con-servative requirement by dropping the twotransformations, resulted in incorrect shock loca-tions. Note that the transformation inside the sum-mation is based on the properties at a cell facewhile the transformation outside of the summationis based on cell-centered data.

The form of the central-difference dissipationmodel used in the current work is a blending ofsecond-difference and fourth-difference dissipa-tion terms as discussed in reference 1 8. The sec-ond-difference terms are used to preventoscillations at shock waves, while the fourth-differ-ence terms are important for stability and conver-gence to a steady state. Reference 1 4 provides thedetails of the central-difference artificial dissipationflux calculation.

For upwind we use Roe's flux-difference splitting.In this case, the Padp term in Eq. (4) can be writtenas

where AQ = QT-Q\. The preconditioned Jaco-bian of the inviscid equations can be expressed asthe matrix product

PA = [R][Kp][L]

where [R] and [L] are any valid right and lefteigenvector matrices and [~kp] is a diagonal matrixof the eigenvalues. The Roe averaged dissipationJacobian, \PA\ , is constructed by using the abso-lute values of the eigenvalues in the [X^] matrixand reconstructing PA .

\PA\ =

The dissipation term can then be expressed as

= AFj + AF2 + AF3

Carrying out the calculations yields the followingdissipation fluxes for the Choi and Merkle inviscidpreconditioner.

= I

Ap + ACv(M2-l)pCv}

C2M2

-T} -T][2c2M2

and,

+ Y] [2c2M2-r

Y = 2pACvc2Af2

The dissipation fluxes for the other implementedpreconditioning concepts were obtained in thesame way. The resulting dissipation fluxes aregiven in reference 14.

Time Integration SchemesTo determine the behavior of preconditioners withvarious standard time-integration schemes, fourdifferent techniques were programmed in the two-dimensional test code. These techniques weretwo-stage Runge-Kutta, approximate factorization,approximate LU factorization, and point-implicit.The implementation of these time-integrationschemes is described in reference 14.

Boundary ConditionsA number of different boundary conditions havebeen implemented in the two-dimensional testcode. These include characteristic inflow/outflow,slip wall, and no-slip wall. Each of these boundaryconditions is based on the preconditioned systemof equations and are described in detail in refer-ence 14.

Preconditioner Mach Number FunctionEach of the preconditioners require a limiting of theMach number in the preconditioning matrix. Werefer to the limited Mach number as the precondi-tioner Mach number function. For the two Choi andMerkle preconditioners the Mach number functionis given as

M = M local ~1.0'

For the Pletcher and Chen preconditioner, theMach number function is given as

M = \ Miocal => e < Mlocal < Vl/Y

•/T/Y => Miocai * TiTyIt was also noted in the previous section, that theeigenvectors of the preconditioned Jacobian matrixwere functions of this Mach number function. Thereason this function is limited on the lower end is toprevent the eigenvectors from vanishing andbecoming non-independent sets. Another problem

encountered during this research was an erraticbehavior of the local Mach number during the initialhigh transient phase of some solutions. During thisphase, in certain regions of the flow field, such asnear stagnation points, the Mach number maychange as much as two orders of magnitude fromone grid cell to the next. This can cause significantconvergence problems. Again, this can be attrib-uted to the dependence of the eigenvectors on thelocal Mach number. Since the solution isexpressed in terms of the eigenvectors, the erraticbehavior of the local Mach number also causes thesolution to change erratically leading to conver-gence problems. The technique used to alleviatethis problem was to smooth the preconditionerMach number function. At each iteration, the pre-conditioner Mach number is calculated and limitedas shown above. After that, a smoothing operatoris applied which takes a weighted average of theMach number function at each cell with its neigh-bors. This technique provided improved robustnessparticularly for problems such as the MAC A 0012airfoil test case to be discussed later.

For the Choi and Merkle viscous preconditioner,the Mach number function was also limited toimprove the stability in low Reynolds numberregions. The concept of this limiting was obtainedfrom references 7 and 19 and is described in refer-ence 14.

ANALYSIS TEST CASESA number of two-dimensional test cases were ana-lyzed to prove the preconditioning concept, quan-tify convergence improvements and validate theaccuracy of the solutions. This section describesthese test cases and the results obtained.

Inviscid Bump in a ChannelThe simplest test case analyzed was an inviscidbump in a channel. This test case is similar to testcases reported in references 5 and 6. The geome-try consisted of a channel that was 1 unit tall and 4units long. The bump was placed on the lower sur-face of the channel beginning 1.5 units down-stream of the inflow plane and was 1 unit long. Thebump had a maximum height of 0.1 units. Thegeometry and grid are shown in figure 2. Theboundary conditions for the upper and lower sur-faces of the channel were specified as slip wall


Figure 2: Inviscid bump in a channel geometry andgrid.

boundaries. The inflow and outflow planes wereboth specified to be characteristic inflow/outflowboundaries.

This test case was run using the unmodified equa-tions as well as with each implemented precondi-tioner at a variety of inflow Mach numbers rangingfrom 0.2 to 0.0000001. This test case was also runwith upwind and central difference and with each ofthe different solver techniques. Figure 3 shows

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Figure 3: Preconditioning produces Mach numberindependent convergence.

results produced when the Choi and Merkle invis-cid preconditioner is used and illustrates that thispreconditioning produces Mach number indepen-dent convergence. Figure 4 shows the unmodifiedequations and their Mach dependent convergence.Figure 5 compares the convergence of the varioussolver techniques using the Choi and Merkle invis-cid preconditioner. The point implicit scheme con-verges fastest, followed by the approximatefactorization, approximate LU, and then the explicittwo-stage Runge-Kutta. Figure 6 compares upwind

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Figure 5: Solver comparisons with Choi and Merkleinviscid preconditioner (upwind).

and central difference convergence again for theChoi and Merkle inviscid preconditioner. Forapproximate factorization, the central-differencescheme converges the fastest. For point implicit,the upwind scheme converges faster. Figure 7compares the convergence of the implementedpreconditioners using the upwind approximate fac-torization solver technique. Notice that the viscous


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Cho! and Merkle InvscidChoi and Merkle Viscous

—•— PletcherandChen

0 250 500 750 1000 1250 1500Iteration

Figure 7: Convergence comparison for theimplemented preconditioners.

and inviscid Choi and Merkle preconditioners con-verge at about the same rate while the Pletcherand Chen preconditioner converges at a slightlyslower rate.

As discussed previously, the two-dimensional testcode was non-dimensionalized in two slightly differ-ent ways in order to determine which technique

was most suitable to low speed analyses. The dif-ference between the non-dimensionalizations wasthe manner in which the pressure was non-dimen-sionalized. The first technique uses the standardnon-dimensionalization shown in reference 2, whilethe second technique uses a gage pressure orpressure coefficient as the non-dimensional pres-sure. Several analyses were conducted for theinviscid bump in a channel test case to comparethe differences between using the standard pres-sure non-dimensionalization and the gage pres-sure non-dimensionalization. All of these analyseswere executed in 64 bit precision on a SiliconGraphics Power Indigo 2 workstation with anR8000 CPU. Figures 8 and 9 compare the conver-

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gence of the solution with the standard and gage-pressure non-dimensionalizations using theupwind-difference technique and the point-implicitsolver at seven different Mach numbers rangingfrom 0.1 to 0.0000001. Notice that with the stan-dard non-dimensionalization that as the inflowMach number is reduced the solution converges toa progressively higher residual level. But, with thegage-pressure non-dimensionalization, each of theMach numbers produce identical convergence his-tories. Other combinations of differencing andsolver produced similar results. The inviscid bumpin a channel test case is a relatively simple casewith excellent convergence characteristics. Consid-ering this and the results of these numerical exper-


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iments, we would limit analysis with the standardnon-dimensionalization to Mach numbers no lessthan 0.0001.

NAG A 0012 Airfoil AnalysisThe geometry and pressure distribution for incom-pressible flow around a NACA 0012 airfoil wasobtained from reference 20. The angle-of-attack ofall test cases was zero so that a symmetry planecould be taken through the airfoil and a smaller gridused. The grid topology and geometry for this testcase is shown in figure 10. The topology wasselected for simplicity in setting boundary condi-tions and to place as many grid points as possibleon the airfoil surface. A number of solutions werecomputed for this configuration in order to deter-mine the behavior and accuracy of the solutions forthe preconditioned equations. Figure 11 shows theconvergence of the test code using an upwindapproximate factorization solver for the originalsystem and each of the implemented precondition-ers at an inflow Mach number of 0.01. The Pletcherand Chen preconditioner oscillates in the earlystages of the solution but all of the preconditionersout-perform the standard equations. Figure 12compares the convergence histories of the varioussolvers using the Choi and Merkle inviscid precon-ditioner at Mach 0.01. In this case, the approximatefactorization and the point implicit schemes per-form nearly the same and the approximate LUsolver is slightly slower. Figure 13 compares the

Figure 10: Grid for the NACA 0012 airfoil analysis.

1000 2000 3000Iteration

4000 5000

Figure 11: Convergence comparison of theimplemented preconditioners for the NACA 0012.

convergence of the upwind versus central differ-ence dissipation techniques using the Choi andMerkle inviscid preconditioner. All four casesbehave essentially the same. Figure 14 comparesthe computed pressure distributions at Mach 0.01to the analytical results from reference 20. Notethat at this low Mach number the solution of theunmodified equations has lost significant accuracy


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"-s-Wv^xV-VS^— '

——— — Runge-Kutti

_._._. Approximat_._. Point Implic

^

^L*

1sLUj Factorizationit

t i i i

2500 5000 7500 10000Iteration

Figure 12: Convergence comparison of theimplemented solvers for the NACA 0012.

10'

-10-3

I510*cc

10-'

10-'

AF Central Diff.AF Upwind

_._._. Port Imp. CentralDrtt,—. —. Point Imp. Upwind

1000 2000 3000Iteration

4000 5000

Figure 13: Comparison of upwind versus centraldifference for the NACA 0012.

while the solution of the preconditioned equationsapparently retains accuracy. This phenomenon willbe discussed in detail in a later section.

In order to verify that preconditioning does notdegrade the accuracy of flow solutions computedfor transonic Mach numbers, solutions for the flowaround a NACA 0012 airfoil were computed for a

-1.0

-0.5

Abbott and DoenhoffStandard EquationsChoi and MerHelnviscidChoi and MerMe ViscousPtetcherandChen

1.00

Figure 14: Computed Cp comparisons of theimplemented preconditioners for the NACA 0012 atMach 0.01.

freestream Mach number of 0.8. This producestransonic flow over the airfoil with a maximumMach number of 1.2 and a shock wave approxi-mately seventy percent along the chord. Figure 15compares the pressure coefficient distributions forsolutions computed with and without the precondi-tioner. Notice that the shock wave locations are vir-tually identical in both cases.

Laminar Flat Plate AnalysisBoth of the previous test cases were inviscid. Pre-conditioning must also be applicability to viscouscases. A geometrically simple yet significant testcase is laminar flow over a flat plate. Analyticalvelocity profiles can be obtained from a Blasiussolution21. These profiles can be compared to pro-files obtained from the two-dimensional test codeto determine the accuracy of the solutions pro-duced for the viscous cases.

The grid was 85 X 75 with 25 points on the plateand 31 points in front and 31 points aft of the plate.The normal spacing was such that it providedapproximately 20 points in the boundary layer for aReynolds number of 10,000. The grid is shown infigure 16.

This test case was run only with the Choi andMerkle viscous preconditioner and the Pletcherand Chen preconditioner. Figure 17 shows the con-


Standard EquationsPreconditioned Equations

1.00

Figure 15: Computed Cp comparisons of thepreconditioned and non-preconditioned equationsat Mach 0.8.

Standard Equations. Choi and Merkfe Viscous

_._._, pietcherandChcn

5000 10000 15000Iteration

20000

Figure 17: Convergence comparisons fora Mach 0.1laminar flat plate.

Figure 16: Laminar flat plate grid.

vergence histories for the original equations andthe preconditioned equations at Mach 0.1. Theunmodified equations required 20,000 iterations toreduce the residual to the same level attainedusing the preconditioned equations after 2,500 iter-ations. Comparisons with a Blasius profile, shownin figure 18 , indicate relatively good agreement forall cases. Figure 19 shows the convergence histo-ries of the solutions of the original equations andthe preconditioned equations at Mach 0.01. In thiscase, the results using the preconditioned equa-tions were essentially the same as the Mach 0.1case, but the solution of the unmodified equations

0.08

0.07

0.06

0.0

BlasiusStandard EquationsChoi and MeiMe ViscousPletcherandChen

0.25 0.50 0.75 1.00Velocity

Figure 18: Velocity profiles for the laminar flat plateat Mach 0.1.

required 400,000 iterations to reach the same con-vergence level. Again, comparisons with a Blasiusprofile are shown in figure 20. Notice the degrada-tion in the solution of the unmodified equations atMach 0.01 even though it has converged fiveorders of magnitude in residual norm.


Standard EquationsChoi and MerMe Viscous

-.-.-. PletcherandChen

0 100000 200000 300000 400000Iteration

Figure 19: Convergence comparison for a Mach 0.01laminar flat plate.

• Blasius©— Standard EquationsS—— Choi and Merkte ViscousA—— PletcherandChen

>0.04

OJ 1.000.50 0.75Velocity

Figure 20: Velocity profiles for the laminar flat plateat Mach 0.01.

Strongly Converging NozzleThe goal of this project is to produce a code thatcan handle incompressible and compressibleregions in the same solution. For this reason, astrongly converging nozzle test case is includedhere. This test case is similar to that reported inreference 7. The geometry and grid for the nozzleare shown in figure 21. The geometry shown has

Figure 21: Geometry and grid for the stronglyconverging nozzle analysis.

an area ratio (AR) of 10 between the inflow and thethroat. The grid is 45 X 31. When the solution isconverged, the inflow Mach number is approxi-mately 0.03 while the exit is supersonic. Figure 22shows the convergence history for the strongly

1000 3000 40002000Iteration

Figure 22: Convergence comparison of theimplemented preconditioners for the stronglyconverging nozzle analysis.

converging nozzle with the unmodified equationsand the implemented preconditioners using a cen-tral-difference approximate factorization solver.Using the Choi and Merkle preconditionerimproved convergence over the original equationsby a factor of almost four.

SOLUTION ACCURACYIt was pointed out that low Mach number solutionsof the pressure distribution over a NACA 0012 air-foil became increasingly inaccurate as the Machnumber was reduced when the original equations


were used. However, the solutions computed usingthe preconditioned equations remained accurateeven as the Mach number was reduced. Variousauthors have suggested that preconditioning pro-duces more accurate solutions than the originalequations'9'22'23). However, none of the cited refer-ences provide a detailed description this accuracyphenomenon.

One possible explanation to the low Mach numberaccuracy differences relates to the scaling of thespatial truncation error. To illustrate this, considerthe one-dimensional preconditioned non-conserva-tive form of the Euler equations.

where

dp̂dt

dt dx

3%^dt dxdu=-dx

(5)

dp ~^- = 0.dx

Notice, that only the last equation has changed dueto the preconditioning. Suppose we take thesteady-state part of the original pressure equationand determine the truncation error.

PC ^r— T M-, — UK dx dx

Using second-order central-differences for thederivative terms and retaining only the first fourterms of the series expansion, we obtain

[A] =

« p 0

0 u l-P

0 pc2 «

and

If the preconditioner is the identity matrix, then theunmodified equations are recovered and the indi-vidual equations are

du

dp tdu— 4- or-^—dt PC dx

-\ "»pd*

dp n+ u~ = 0.

Now, consider the Choi and Merkle inviscid pre-conditioner which is given as

[P] =1 0 00 1 00 0 A/2

The preconditioned individual equations are

28« dj> _pCdx+Udx ~

This equation is called the modified equation2. Theright-hand-side of this equation is the truncationerror (T.E. ) since it represents the differencebetween the original partial differential equationand the finite-difference approximation. So, thetruncation error of the non-preconditioned equationis

(T.E.)np = -Ax21

3d p"

Now taking the steady-state part of the precondi-tioned pressure equation (from Eq. (5)) and per-forming the same manipulations, the truncationerror is

-Ax23

d p

By inspecting the two equations above we candetermine the difference.

(T.E.)np =

This says that the truncation error for the non-pre-conditioned system is I/A/2 times the truncationerror of the preconditioned system. So, even


though the partial differential equations of the origi-nal and preconditioned systems should producethe same steady-state solution, the discretizedresults differ. In fact, the truncation error of the non-preconditioned system will be significantly higherthan that of the preconditioned system at very lowMach numbers.

To verify this result, a numerical experiment hasbeen conducted. The symmetric NACA 0012 airfoilsolution discussed previously was computed againin order to conduct a grid spacing error study.Using the same geometry shown previously, a 201X 121 grid was developed. This grid was set-up sothat it could be halved three times to produce 101X 61, 51 X 31, and 26 X 16 grids. The matrix ofanalyses are shown in table 1. This matrix includes

Table 1: Run log for the grid spacing errorstudy

Very Fine201X121

Mach 0.1S.E.*

Mach 0.05S.E.

Mach 0.01S.E.

Mach 0.1RE.*

Mach 0.05RE.

Mach 0.01RE.

Fine101 X61

Mach 0.1S.E.

Mach 0.05S.E.

Mach 0.01S.E.

Mach 0.1RE.

Mach 0.05RE.

Mach 0.01RE.

Medium51X31

Mach 0.1S.E.

Mach 0.05S.E.

Mach 0.01S.E.

Mach 0.1RE.

Mach 0.05RE.

Mach 0.01RE.

Coarse26X16

Mach 0.1S.E.

Mach 0.05S.E.

Mach 0.01S.E.

Mach 0.1RE.

Mach 0.05RE.

Mach 0.01RE.

* Standard Equationst Preconditioned Equations

variations in grid spacing and Mach number. Eachsolution was also computed with and without pre-conditioning. The residual was reduced by at leastfive orders of magnitude in each case. To obtainthe solution error, the difference in each of theprimitive variables was summed on the locationsrepresenting the minimum grid size (26 X 16). Thevery fine grid solution was taken as zero error.

Figure 23 is a plot of solution error versus the aver-age spacing of each analysis. Both preconditionedand non-preconditioned, and at all Mach numbers,the solution error is reduced as the average grid

0.0150

0.0125

0.0100

§0.0075UJ

0.0050

0.0025

I i

_ .fl-_..*_._..£_.

M=0.01 Not PreconditionedM=0.01 Preconditioned

< M=0.05 Not Preconditioned• M=0.05 Preconditioned< M=0.10 Not Preconditioned< M=0.10 Preconditioned

°'000t).2 0.3 0.4 0.5 0.6 0.7 0.8 0.9delta Xavg

Figure 23: Solution error versus average gridspacing.

spacing is reduced. The non-preconditioned solu-tions show a greater reduction in solution error withgrid spacing, especially at Mach 0.01. It appearsthat the slope of the error curves increase as theMach number is reduced for the non-precondi-tioned cases. However, for the preconditionedcases the solution error does not vary with Machnumber. The preconditioned cases also show sig-nificantly lower error than the non-preconditionedsolutions.

Figure 24 shows the same results plotted as solu-tion error versus Mach number. The individualcurves represent the various grid densities. Noticethat for the non-preconditioned cases the solutionerror increases significantly as the Mach number isreduced. This effect is much more pronounced asthe grid becomes coarser. Notice that the solutionerror remains constant with Mach number for eachgrid when the solutions are preconditioned.

These experiments appear to validate the analyti-cal result discussed previously. Without precondi-tioning the truncation error increases as the Machnumber is decreased. This also explains the resultsdiscussed by Volpe in reference 4, where heobserved that low Mach number solutionsimproved as the grid was refined. However, thiscurrent work shows that with a fixed grid, precondi-


0.0150

0.0125

0.0100

k.

§0.0075UJ

0.0050

0.0025

0.0000,

\\\

: ••*

— 0 — Fine Grid Not Preconditkmed0 Fine Grid Preconditioned

- ••- • Medium Grid Not Preconditioned- •€]- • Medium Grid Preconditioned_..*_., Coarse Grid Not Preconditioned-..A-., coarse Grid Preconditioned

\\\\\\

*^

MB___

\\*-..

^»^

-4=

•-»...

— -

-*.,.,

" - - l

— ——— f

\

\

0.1 JMach Number

Figure 24: Solution error versus Mach number.

tioning keeps the truncation error constant to verylow Mach numbers and removes the requirementto overly refine the grid.

An interesting observation may be made from theresults shown in figure 24. At Mach 0.05 the samesolution accuracy is obtained for a preconditionedsolution using a grid of 26 X 16 and a non-precon-ditioned solution using a grid of 101 X 61. Thismeans that preconditioning allows the number ofgrid points to be reduced by a factor of 16 for thatcase in order to achieve the same accuracy.

CONCLUSIONSThree preconditioning concepts were imple-mented in a two-dimensional test code. This codewas set-up to develop implementation techniquesand to test various aspects of the preconditioners.Both central-differencing with an added artificialviscosity and upwind-differencing using Roe's flux-difference split technique were implemented in thetest code. Three different flux limiters, minmod,superbee, and smooth symmetric, were used withthe higher-order upwind method. Four differentnumerical solvers were implemented including two-stage Runge-Kutta explicit, approximate LU,approximate factorization, and point-implicit withsub-iterations. Two different non-dimensionaliza-tions were tested where the difference between thenon-dimensionalizations was in the use of a gagepressure.

Results of the numerous tests indicate that low-speed preconditioning as implemented in the testcode can produce Mach number independent con-vergence down to extremely low Mach numbers.For an inviscid bump in a channel test case, identi-cal convergence rates were obtained for a series ofMach numbers ranging from 0.2 to 0.0000001. Theuse of a gage pressure non-dimensionalizationwas required for very low Mach numbers (belowMach 0.0001).

The preconditioning was equally effective usingeither central-difference or Roe's flux split upwind-differencing. Even at very low Mach numbers bothtechniques provided accurate inviscid results andgood convergence. Of the three flux limiters tested,the minmod limiter provided the most consistentconvergence.

Of the four solvers implemented, the implicitschemes were, as expected, the best. The point-implicit scheme with sub-iterations was consis-tently as good as or better in convergence than theother implicit schemes. Also, the sub-iterations inthe point-implicit scheme remove any problemswith factorization errors that may be encounteredusing the approximate LU or approximate factoriza-tion.

The test cases indicated that solution accuracywas degraded significantly as the Mach numberwas reduced without preconditioning. Solutionerrors became very pronounced below Mach 0.05.It was shown analytically that the increase in thetruncation error of the original equations isinversely proportional to the square of the Machnumber. A numerical study was performed whichsupported the analytical result and established thelink between the truncation error and the low Machnumber solution accuracy problem.

REFERENCES

1. Chen, K. and Pletcher, R., "A PrimitiveVariable, Strongly Implicit CalculationProcedure for Viscous Flows at All Speeds,"AIAA Paper 90-1521, June, 1990.

2. Anderson, D. A., Tannehill, J. C. and Pletcher,R. H., Computational Fluid Mechanics andHeat Transfer, Hemisphere PublishingCorporation, New York, 1984.


3. Merkle, C. L, Venkateswaran, S. and Buelow,P. E. O., "The Relationship Between Pressure-Based and Density-Based Algorithms," AIAAPaper 92-0425, Jan. 1992.

4. Volpe, G., "On the Use and Accuracy ofCompressible Flow Codes at Low MachNumbers," AIAA Paper 91-1662, June, 1991.

5. Karki, K. C. and Patankar, S. V., "PressureBased Calculation Procedure for ViscousFlows at all Speeds in ArbitraryConfigurations," AIAA Journal, Vol. 27, No. 9,Sept. 1989, pp. 1167-1174.

6. Choi, D. and Merkle, C. L., "Application ofTime-Iterative Schemes to IncompressibleFlow," AIAA Journal, Vol. 23, No. 10, Oct. 1985,pp.1518-1524.

7. Choi, Y. H. and Merkle, C. L, "The Applicationof Preconditioning in Viscous Flows," Journalof Computational Physics, 105, 1993, pp. 203-223.

8. Turkel, E., Fiterman, A. and van Leer, B.,"Preconditioning and the Limit to theIncompressible Flow Equations," NASAContractor Report 191500, July 1993.

9. Turkel, E., "Review of Preconditioning Methodsfor Fluid Dynamics," NASA Contractor Report189712, Sept. 1992.

10. Turkel, E., "Preconditioned Methods for Solvingthe Incompressible and Low SpeedCompressible Equations," Journal ofComputational Physics, Vol. 72, No. 2, 1987,pp. 277-298.

11. Weiss, J. M. and Smith, W. A., "PreconditioningApplied to Variable and Constant DensityTime-Accurate Flows on UnstructuredMeshes," AIAA Paper 94-2209, June 1994.

12. Pletcher, R. H. and Chen, K.-H., "On Solvingthe Compressible Navier-Stokes Equations forUnsteady Flows at Very Low Mach Numbers,"AIAA-93-3368-CP, June 1993, pp. 765-775.

13. Darmofal, D. L. and Schmid, P. J., 'Theimportance of eigenvectors for localpreconditioners of the Euler equations," AIAA-95-1655-CP, June 1995, pp. 102-117.

14. Reed, C. L., Low Speed PreconditioningApplied to the Compressible Navier-StokesEquations, Ph.D. dissertation, The Universityof Texas at Arlington, 1995.

15. Roe, P. L., "Characteristic-Based Schemes forthe Euler Equations," Annual Review of FluidMechanics, Vol. 18, 1986, pp. 337-365.

16. Karman, S. L. Jr., "SPLITFLOW: A 3DUnstructured Cartesian/Prismatic Grid CFDCode for Complex Geometries," AIAA Paper95-0343, Jan. 1995.

17. Thompson, D. S., "A Zonal Method forExtending the Applicability of the PNSEquations - Year 5," General DynamicsContractor Report, TEES #1263915, UTA CFDReport 89-05, 1990.

18. Swanson, R. C. and Turkel, E., "On Central-Difference and Upwind Schemes," Journal ofComputational Physics, 101, 1992, pp. 292-306.

19. Venkateswaran, S., Deshpande, M. andMerkle, C. L., 'The Application ofPreconditioning to Reacting FlowComputations," AIAA-95-1673-CP, June 1995,pp. 306-316.

20. Abbott, I. H. and von Doenhoff, A. E., Theory ofWing Sections, Dover Publications, Inc., NewYork, 1959.

21. White, F. M., Viscous Fluid Flow, McGraw-Hill,New York, 1974, p. 265.

22. Fiterman, A., Turkel, E. and Vatsa, V,"Pressure Updating Methods for the Steady-State Fluid Equations," AIAA-95-1652-CP, June1995, pp. 68-76.

23. Van Leer, B., Mesaros, L., Tai, C.-H. andTurkel, E., "Local Preconditioning in aStagnation Point," AIAA-95-1654-CP, June1995, pp. 88-101.


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