Convergence Acceleration for Unified-Grid Formulation usingPreconditioned Implicit Relaxation
by
Oshin Peroomian , Sukumar Chakravarthy, Sampath Palaniswamy and Uriel C. GoldbergMetacomp Technologies, Inc.
Westlake Village, CA
Abstract
Improved convergence rates for a unified gridframework are achieved by combining severalconvergence acceleration strategies, which include localimplicit time-stepping, low-speed preconditioning, andrelaxation methods. It is demonstrated that goodconvergence can be achieved on various grid types andtopologies, all speed regimes, and for both inviscid andviscous flows.
Introduction
Over the last three decades Computational FluidDynamics tools have been developed to encompassincreasing levels of complexity in physics, numerics andgeometry. They have been applied to an ever-expandingset of problems. The increasing demands placed on CFDcontinue to force the need for further improvements.Increasing the effectiveness of CFD requires thereduction of the computational cycle time (time fromproblem definition to completion of solution). Part ofthis is being achieved through the use of unified-gridapproaches which greatly reduce the demands placed onthe grid and the grid generation process [1]. Acorresponding improvement in convergence rate to thedesired solution is also required. The combination ofspeed-up and geometrical versatility is not traditionallyaddressed together. In this paper we present some ideas,approaches, implementation details and results of tests todemonstrate our recent progress in this area.
Convergence acceleration has become an important issuein the use of CFD in the design processes. The need forfast turnaround times in a design cycle dictates a fastgrid-to-solution time which, in turn, influences the choiceof grid type, grid generation process, the number of gridpoints, choice of physical models such as turbulencemodels, wall functions, the choice of numericalmethodology, solvers, and the choice of solutionmethodology parameters. Often the ultimately desirablegrid is too fine for rapid computation and is, therefore,coarsened to achieve a faster turnaround time. Manytechniques have been employed to speed up theconvergence rate of numerical methods so that suchmeasures become unnecessary.
Several issues can affect the convergence rate of anumerical method. These can range from the reflectionat boundaries, influence of cell aspect ratio, numericalaccuracy, numerical diffusion, eigenvalue spectrum, etc.A number of techniques have been utilized to remedyadverse effects caused by the above-mentioned factorsand thereby accelerate the convergence rate. Thefollowing is a sample collection of such techniques:
1. Multi-Stage Runge-Kutta (RK) methods [2,3] havebeen developed, possessing stability limits of CFL >1 which not only allow larger time-steps but havebetter residual damping characteristics than thestandard RK methods. A trade-off occurs due tothe fact that 4-5 stages are usually considered andthis increases the computational time per time step.This type of scheme has been adopted by many
unstructured-grid codes since it deals with a singlecell rather than a "line-of-cells".
2. By employing a local time-stepping option, i.e. eachcell's time step based on its own characteristic value(eigenvalue, mesh volume, etc.), one can increasethe rate of convergence. Often the above-mentionedmulti-stage RK methods are coupled with thisapproach.
3. Implicit schemes can be employed to deal withstiffness effects, avoid stability limits associated withexplicit schemes, take larger time steps, andconverge faster to steady state. In structured gridcodes, often a line-implicit approach is adopted. Inunstructured grid codes, however, due to thebookkeeping involved, the most common approachis a point-implicit methodology.
4. Proper and careful choice of inflow/outflow and far-field boundary conditions can help avoid spuriousreflections. These methods can be divided into twocategories. One involves exponential damping ofthe equations [2,8,9] and the other involves solvinga characteristics-based time-dependent equation atthe boundary [4,5,6,7]. For preconditioned forms ofthe equations it is important to use thecorresponding characteristic theory, not thecharacteristics for the original non-preconditionedequations.
5. Relaxation techniques [10,11] can be developed forTVD upwind formulations. These avoidfactorization errors that occur at large values of thetime step and they can be designed to takeadvantage of signal propagation directionsassociated with hyperbolic equations.
6. Multi-grid [12,13] techniques can be interpreted inseveral ways as a convergence accelerationtechnique. One viewpoint is that they helppropagate the solution faster by resorting to partialsolution updates on the cruder levels.
7. For low speed flows, when the numerical diffusionand the eigenvalue spectrum of the advection termsbecome large, preconditioning techniques have beenemployed to increase the rate of convergence[14,15,16,17,18,19,20]. By multiplying the timederivative vector in the governing equations by apreconditioning matrix, the eigenvalues of theequations are normalized so that they all have thesame order of magnitude. These concepts havebeen generalized in [19,21] to be suitable for all flow
regimes. For low speed flows, preconditioning alsoleads to much improved accuracy in the solution ona given grid (especially on coarse grids).
Only a few of these approaches have been researched inan unstructured grid framework and none have beenhitherto applied in the unified-grid framework. Theauthors became aware of the very recent work in thisarea by Weiss [35]. The approach shown here issomewhat similar to his; however, the schemes shownwere derived independently and the application arena is aunified-grid framework.
In this paper, several ideas and approaches from theabove-mentioned categories (including some novelimplementations) are explored in various combinations.Illustrative examples are employed to demonstrate theeffectiveness of the method for all flow regimes and gridtopologies.
Numerical Framework
The basic numerical framework in which the proposedscheme is implemented is termed the unified-grid,unified-physics and unified-computing framework.These have been implemented in a software suite calledCFD-H- [22,23] and the user is referred to thesereferences for details of the basic numerical framework.Here we will give only a brief synopsis of this frameworkand methodology.
CFD++ Methodology—general outlineThe numerical framework of CFD++ is based on thefollowing general elements: 1) Unsteady compressibleand incompressible Navier-Stokes equations withturbulence modeling [unified-physics]. 2) Unification ofCartesian, structured curvilinear, and unstructured grids,including hybrids [unified-grid]. 3) Unification oftreatment of various cell shapes including hexahedral,tetrahedral and triangular prism cells (3-d), quadrilateraland triangular cells (2-d) and linear elements (1-d)[unified-grid]. 4) Treatment of multiblock patchedaligned (nodally connected), patched-nonaligned andoverset grids [unified-grid]. 5) Total VariationDiminishing discretization based on a new multi-dimensional interpolation framework. 6) Riemannsolvers to provide proper signal propagation physics,including versions for preconditioned forms of thegoverning equations. 7) Consistent and accuratediscretization of viscous terms using the same multi-dimensional polynomial framework. 8) Pointwiseturbulence models that do not require knowledge ofdistance to walls. 9) Versatile boundary conditionimplementation includes a rich variety of integrated
boundary condition types for the various sets ofequations. 10) Implementation on MPP computersbased on the distributed-memory message-passing modelusing native message-passing libraries or MPI, PVM,etc. [unified-computing].
Preconditioned-Implicit Relaxation
In this paper, several ideas on convergence accelerationhave been brought together to yield a fast methodologyfor all flow regimes. The approach can be labeled as a"preconditioned-implicit-relaxation" scheme. It combinesthree basic ideas: implicit local time-stepping, relaxation,and preconditioning. Preconditioning the equationsideally equalizes the eigenvalues of the inviscid fluxJacobians and removes the stiffness arising from largediscrepancies between the flow and sound velocities atlow speeds. Use of an implicit scheme circumvents thestringent stability limits suffered by their explicitcounterparts, and successive relaxation allows update ofcells as information becomes available and thus aidsconvergence.
Point-Implicit Time Stepping
When using a time-marching method in Navier-Stokescalculations, one needs to be aware of three types ofstability criteria. The first two are the common CFLcondition and the von-Neumann condition dealing withthe inviscid and viscous parts of the equations,respectively. The other arises due to stiff source termspresent because of turbulence and chemical reactions(not discussed here). When using an explicit scheme(say a multi-stage RK), the first two conditions limit thetime step as follows:
A/ = mini Ax'
where CFL is the Courant-Friedrich-Lewy number(usually around 2-3 for multi-stage RK),VN is the Von-Neumann number (usually around 1.0 for multi-stageRK), Ax is the smallest cell side, v is the kinematicviscosity, and /I™,* is the largest eigenvalue (u+c for theEuler equations). Implicit schemes do not suffer fromthese time-stepping restrictions. In fact, if all terms aswell as boundary conditions are treated implicitly thenfully implicit schemes do not have any linear stabilitylimits. In implicit schemes the RHS is evaluated at leastpartially at the new time level. This gives rise to acoupled set of discrete equations which need to besolved iteratively. Essentially a linear algebra problem(Ax=b) arises where x is the solution vector consisting ofthe dependent variables at all grid points. In structured
grids where a simple bookkeeping exists, a commonpractice is to use approximate factorization coupled witha line-implicit approach. However, this is not so simplein unstructured approaches. Many unstructured-gridcodes use a multi-stage Runge-Kutta time marchingcoupled with a local time-stepping concept in which thetime-step in each cell is computed based on the followingformula and cells are updated using their own maximumallowable time step. These types of approaches arevulnerable to the above stability restrictions. Theapproach taken in CFD++ is a point-implicit one. Thepoint-implicit scheme is a compact scheme which in itssimplest form has the following structure:
In the simplest case, one can carry out a single iterationin which case the Jacobian and RHS calculations arecompletely based on old time level quantities. This typeof scheme is ideal for the unified-grid framework presentin CFD++ and does not suffer from severe stabilityrequirements. This is especially important in problemswhere the viscous stability dominates and causes severerestrictions on explicit schemes (Note that the viscousstability limit goes as Ax1). However one must be awareof several facts when using this implementation of thepoint-implicit scheme:
1 . If the boundary condition treatment is explicit, then,in certain cases, it can create an upper bound on theCFL condition.
2. When using local time stepping, very large CFLsmight not be possible.
3. The scheme can become unstable at large CFLswhen a single iteration is used.
In most of the inviscid flow examples given in the lattersections of this paper, an infinite CFL was possible eventhough the boundary condition treatment was explicit.However, for most of these cases a CFL of 50-100(based on the inviscid criterion) was chosen. For thevery fine grids (several points in the sublayer) used in theturbulent flow calculations, when spatially varying timesteps were used, the CFL was sometimes limited to only5-10.
Preconditioning
The main effect of preconditioning on the compressibleNavier-Stokes equations (at low speeds) is to normalizethe eigenvalues of the flux Jacobian matrix and to
overcome the time-stepping restrictions due to the largediscrepancy between the speed of sound and the fluidvelocity. Preconditioning also modifies the diffusion inthe flux terms so that non-physical solutions on coarsemeshes in low-speed flows are avoided. Preconditioningis implemented by multiplying the time derivative vectorof the dependent variables by a preconditioningmatrix[14,15,16,17,19,21].
Now if this is inverted, one obtains the followingequation:
It is clear that this will change the eigenvalues andeigenvectors of the Jacobian matrices associated withpreconditioned fluxes and thus the diffusion. This alsodestroys the physically correct transient and takes adifferent path to the correct steady state solution whensuch a state exists. If transient response is important,then dual time-stepping techniques[16] can be utilized sothat only the internal (pseudo-time) steps arepreconditioned. The dual time-stepping basically has aphysical time associated with the transient response and asecond "pseudo" time associated with the iterationsbetween each time step
For low speed compressible flows, the preconditioningmodifies the wave speeds of the equations to overcomethe numerical difficulties associated with computation oflow speed flows. In such flows, the velocity is verysmall compared to the speed of sound and thus thestability criterion which is inversely proportional to thespeed of sound (u+c wave with u being very small)becomes influential. In addition, the waves which aretraveling at the velocity of the flow (u waves) travelmuch slower than the u+c and u-c waves, and this fact,combined with the time-step which is dictated by thespeed of sound, implies that a very long time is needed toconverge to a solution. By altering the magnitude of theu+c and u-c waves so that they are comparable with theu waves, the solution is obtained hi much feweriterations or time steps.For example take the 1-D Euler equations in non-conservation form:
' u pc2 0
1 . 0P0 p «,
f \p= 0
7"0 1 00 0 1
/ \Pu
<pj+
t
(u pc2 0I u 0P
(0 p u)
f \pu
l/>>
multiplication of the time derivative term by the inverseof the preconditioning matrix:
= 0
Inverting this matrix and multiplying it by the Jacobianmatrix yields the new equation for the update of theprimitive variables and the corresponding Jacobianmatrix.
= 0
This new Jacobian matrix has the following eigenvaluestructure:
/ \pu +wt
uf$~^~I
p0
pf>2 0
u 0
P «
Vu
P
1+-
The three eigenvalues of the Jacobian matrix for thissystem are u-c, u, and u+c. Now let us consider the
The convective eigenvalue, u, is untouched, however,the acoustic eigenvalues have now been altered by theintroduction of the p parameter. Note that whenp2 = c2, the old Jacobian matrix and eigenvalues arerecovered. One can see the normalizing effect of thisparameter as u becomes much smaller than c. In thatcase if p is set to u , then all eigenvalues become oforder u and the stiffness due to the large discrepancybetween the sound speed and the flow velocity isremoved. This is the simplest form of preconditioning,but there are many fancier preconditioners in theliterature. As the preconditioner becomes increasinglycomplex, more of the terms in the matrix become non-zero. In the above preconditioning matrix only the 1,1element was altered from an identity matrix, but fancierpreconditioners alter more terms in order to get an eventighter eigenvalue spread. Such preconditioning schemescan be found in [14,15,19]. In fact, the preconditionerdiscussed in [19] is suited for all Mach numbers and
gives rise to the best eigenvalue spread across the Machnumber range.
In a conservative finite-volume scheme the discretizedform of the fluxes in the preconditioned form of theequation must be written as follows:
f _JRiemarm (in-91.)
For example in a Roe's scheme, the diffusion matrix |PA|is based on the right and left eigenvectors of the matrixPA which is obtained by multiplying the preconditioningmatrix, P, with the Jacobian of the inviscid fluxes of thenon-preconditioned system. The P"1 which multiplies thediffusion term is necessary in order to have aconservative scheme. In a very simple (say 1st order RK)method, we can write a time update scheme as follows:
- = -P,(JHS,)A/
where RHS is the right hand side of the non-preconditioned system. One inherent problem with allpreconditioners arises from this eventual multiplicationPipface- The flux fmemann is calculated at each cellface and the P"1 matrix which multiplies the diffusion isbased on an average (say Roe's average) of the cell faceneighboring quantities. The P which multiplies the RHS,however, is computed based on the centroidal value ofthe cell being updated. Thus, in regions of highgradients, />./^e will not necessarily be close toidentity. This can cause problems in convergence as wellas in stability of the scheme. In low speed calculations,these regions of high gradients occur near stagnationpoints. Such points pose yet another problem forpreconditioners: it has been shown that the normalizationparameter in preconditioners must be roughly equal tothe velocity magnitude (\«2 +v2 +w2 ) in order toideally normalize the eigenvalues and to obtain anoptimal condition number. Obviously at stagnationpoints a singularity exists in the method. Darmofal [18]has shown that as one approaches this singularity theeigenvectors of PA become parallel and thus make thescheme unstable. A common fix for this problem is tolimit the method to stay away from stagnation pointsingularities.
The above shows the basic concepts behindpreconditioning and its initial implementation. Therehave been some recent innovations in preconditioningwhich include:
1. Preconditioning based on Mach number so thateigenvalue spread and condition number becomeoptimal at all speeds. [19]
2. Viscous preconditioning which modifies it based oncell Reynolds number[24]
3. Preconditioning based on cell-aspect ratio[25]
In this paper, we have concentrated on inviscidpreconditioners. These are preconditioners which solelytry to modify the inviscid-flux-Jacobian-matrixeigenvalues. We have evaluated three suchpreconditioners:
1. Preconditioner by Turkel [14,15] which narrowlybunches the eigenvalues as the Mach number goestowards zero. The preconditioning matrix for thissystem effectively adds scaled pressure timederivative terms to the momentum equations.
2. Preconditioning by Weiss [16] which makes thelargest eigenvalue only twice as large as theconvective one as the Mach number goes towardszero. As the flow velocity tends towards (andpasses) sonic conditions the scheme tends to thenon-preconditioned scheme.
3. Preconditioning by Bram van Leer [19,21] whichnarrowly bunches the eigenvalues at all Machnumbers. In fact this yields the most optimal spreadin eigenvalues and the most optimal conditionnumber obtainable. This is achieved by writing theequations in a stream aligned coordinate system andfinding the optimal preconditioning matrix in that setof variables.
Detailed analysis has been conducted on thesepreconditioners. The most robust was found to be thatproposed by Weiss. This preconditioner suffers onlyfrom the stagnation point singularity which can becircumvented easily by limiting the preconditionedparameter away from stagnation points. Thispreconditioner does not suffer from flow anglesensitivity, sonic point singularity, or outer boundarysensitivity. The result of our study is presented in thetable below. Following the tabular format presented in[25], we present the results of our analysis.
The relaxation scheme in CFD++ is implemented basedon the point-implicit method. The simple point implicitscheme described earlier goes through a single iterationand updates the cell average value. In implementing therelaxation scheme, several iterations are done using a lefthand side (LHS) update between iterations. The basicscheme (single iteration) is as follows:
Now we rewrite this scheme as flows:
where n represents the old time level and * and **represent iteration steps. One can see that if oneiteration is carried out with the * level being equal to then-th level the basic scheme is recovered. The relaxationscheme implemented in CFD++ is obtained by expandingthe Jacobian (^RIfsy^) into the central and neighbor
cell effects:
where the subscript c denotes the central cell and fndenotes the face neighbors. After expanding the LHS inthe above manner the scheme is then written as follows:
d(RHS)
\d(RHS) -RHS"f»
Note that the neighbor effects have been taken to theright hand side of the equation. This scheme isimplemented in a Gauss-Seidel fashion, i.e. using newiteration level solutions as they become available in theneighbor-effect term. In using the relaxation scheme oneshould be aware of the following properties:
1. Number of sweeps and sweep direction. In theabove implementation of this point-implicitrelaxation scheme, the number of sweeps can playan important role in its convergence properties.Obviously if only one sweep is performed some ofthe cells will not feel the effect of the relaxation.Too many iterations are futile because the RHS isnot updated between sweeps and the solution isadvanced using the cell and neighbor LHScontributions. The sweep direction can be forward,backward or both. Usually a two-sweep forward-backward implementation yields adequate results.
2. Cell ordering in which the relaxation sweeps aredone. The cell ordering in the relaxation scheme canalso play a significant role in its convergenceproperties. Several options have been considered:
a) Sweep order according to cell numbersb) "Boundaries-on-in" ordering of cellsc) Ordering starting from a specific boundaryd) Checkerboard (or colored) scheme
numberinge) Cell ordering based on eigenvalue
Depending on the problem under consideration, aspecific cell-ordering will aid in accelerating theconvergence properties of the scheme. Forexample, in an inviscid supersonic flow problem, it isadvantageous to sweep in the "free-stream"direction.
3. The "goodness" of the Jacobian. One must keep inmind that the iterations are advanced using anupdate of the LHS only. Thus, if the Jacobian is avery rough estimate, then too many iterations mightmake the scheme unstable.
ResultsMore than ten problems were chosen in order todemonstrate the convergence characteristics of thisimplicit-relaxation scheme. Since this scheme wasimplemented in the unified-physics, unified-grid andunified-computing framework, the problems tend tocover a vast array of physical and topological ranges.They include compressible and incompressible flowregimes; inviscid, laminar and turbulent physics,structured, unstructured, and hybrid meshes. SinceWeiss' preconditioning was finally chosen andimplemented, only the flows which are incompressible orat a low-Mach number have been simulated with thepreconditioning option.
Inviscid Flow over Cylinder—A Mach 0.4 inviscidflow over a cylinder is calculated using a variety of meshtopologies. The goal here is to demonstrate the easewith which the relaxation scheme operates within theunified-grid framework. Four grids were chosen for thisanalysis: the first was a 26x26 quadrilateral gridclustered near the surface of the cylinder, the second wasa finer version of the same grid (51x51), the third atriangular mesh grid and the fourth a hybrid grid whichhas a hierarchically subdivided section near the top of thecylinder. Figures la, b, and c show these meshtopologies. In all cases, the relaxation scheme was setup so that it operated at a local CFL of 50 with 2forward-backward boundaries-on-in LHS sweeps.Figure 2 shows the convergence characteristics on thefour mesh topologies compared to a 2nd order Runge-Kutta and 1st order point-implicit schemes. Nearly fourorders of magnitude drop in the residual is realizedwithin 150 steps. One should note that the leveling offseen in the residual of the hierarchical mesh is due to thetruncation error mismatch on the boundaries of the twogrids.
Low-Speed Flow over Cylinder—A Mach 0.01 flowover the same topology as above was calculated usingthe preconditioned form of the relaxation scheme. A16x16 grid was used for all calculations. Figure 3 showsthe convergence histories using a 2nd order Runge-Kuttascheme, a 1st order point-implicit scheme and therelaxation scheme. The best results were obtained with alocal CFL number of 50 and the relaxation scheme using2 forward-backward boundaries-on-in LHS sweeps. Inthat case, seven orders of magnitude residual drop wasachieved in less than 100 steps.
Oblique Shock Simulation—Two geometries wereused to simulate an oblique shock at Mach 2.0. The first
case involved a supersonic flow over a 10 degree rampusing a 51x51 grid. Figure 4 shows the convergencehistory using a 2nd order RK method, a first order point-implicit (CFL 50) method and the relaxation schemeusing a CFL of 50 and two forward-backward LHSsweeps in the streamwise direction. Again, therelaxation scheme improved the convergence by leapsand bounds. The second case involved a simplerectangular domain where a Mach 2.0 flow impingesonto the lower surface at an angle causing an obliqueshock to develop. Three mesh topologies were used tosimulate the flow. The first was a simple Cartesianmesh, the second a triangular mesh and the third a meshwith a mixture of triangular and quadrilateral elements.Figures 5a, b, c show the mesh topologies. Figure 6shows the convergence histories on the three grids usingthe relaxation scheme at CFL 50 and 2 forward-backward boundary-on-in LHS sweeps. Theconvergence histories are roughly the same with all threegrids showing a drop of 4 orders of magnitude in theresidual in roughly 200 steps.
Supersonic Laminar Flow over Flat Plate—A Mach2.0 laminar flow over a flat plate at Re = 1.85xl06/mwas calculated using a 51x101 stretched quadrilateral-mesh grid (Ay^ = 0.0001, >>+«100). Thecomputational grid was 0.31m in the x-direction andO.lm in the y-direction. Figure 7 shows the convergencehistory using a point-implicit (CFL 50), the relaxationscheme using CFL 50 and 2 forward LHS sweeps in thestreamwise direction, the other relaxation scheme usingeigenvalue-based cell-ordering and 2 forward-backwardLHS sweeps. Roughly six orders of magnitude drop inthe residual was achieved in 400 steps by the eigenvalue-based cell ordering even in this highly stretched grid.
Incompressible Flow over a Flat Plate—Anincompressible laminar flow over a flat plate wassimulated using the preconditioned version of therelaxation scheme with primitive variable updating [16].A grid similar to that of the above case was used. Figure8 shows the convergence history of the relaxationscheme using CFL 25 and 2 forward-backwardboundaries-on-in LHS sweeps. Roughly five orders ofmagnitude drop in the residual was achieved in 400steps. This example was provided to show how easilythe scheme can be adapted to a methodology where theupdate-variables are different from the conservationvariables.
Supersonic Turbulent Flow over Ramp—This casecorresponds to the Mach 2.84 flow past a 24 degreeramp [26]. The impinging shock on the boundary layerupstream of the ramp corner causes flow separation to
occur ahead of the corner and a subsequent reattachmentoccurs downstream of it. In problems of this type theseparation bubble shape and size affect many flowquantities such as pressure, velocity and skin friction. A99x99 grid was used with clustering in the x-direction(Axnn^O.lSo) at the ramp corner, and clustering in the ydirection (AymnrO.OOOlSo) at the wall, where 50 =0.023m is the measured boundary layer thicknessupstream of the shock. This grid has at least eight cellsin the viscous sublayer and yf < I. The inflow staticpressure was 24 Kpa and the total temperature 276 K.Although a direct comparison with the experimental datarequires inflow profiles, we have chosen here to initializewith free stream quantities since that is a biggerchallenge to overcome in terms of convergence. Athree-equation pointwise k-e-R, turbulence model wasused [27] to model the turbulent flow features. Since theemphasis of this paper is not on the accuracy of results,but on convergence aspects, the reader is referred to[27,28] for results obtained with this turbulence modelon this specific problem. The cell-ordering employed forthis case was based on the eigenvalue magnitude. Asdescribed above, in this type of ordering more iterationsare done in regions with higher eigenvalues than in thosewith smaller ones. This enables faster convergence sincemore sweeps are done in the "fine-grid" region. Figure 9shows the convergence history for this problem. Thesteeper line represents more iterations in the boundarylayer region, i.e. roughly 10 sweeps in that region. Inboth cases the CFL was ramped from 1.0 to 10.0 inabout 300 steps and the maximum number of LHSsweeps per step was no more that 5 times the number ofcells. Even on this fine grid (50 times finer than a gridtypically used with wall-functions), roughly six orders ofmagnitude drop in residuals was achieved in about 1200steps.
Transonic Flow over a 2-D Bump— This casecorresponds to the experiments conducted in [29]. A121x121 grid, shown in Figure lOa, was used for thecalculations. The grid is highly stretched at the lower andupper walls with a minimum spacing of Ay = 4xlO"6m.This grid is also extremely fine with several grid pointswithin y+ - 1. A Mach 0.615 flow was used to initializethe flowfield and Goldberg's three equation model [27]was used to model the turbulent flow features. The flowdownstream of the bump separates, causing a lambda-shock to develop near the end of the bump geometry,seen in the Mach number contours given in Figure lOb.The eigenvalue-based cell ordering was used for thiscalculation along with 2 forward-backward LHS sweepsat CFL 15. Figure 11 shows the convergence history
with nearly four and a half orders of magnitude residualdrop in about 1200 steps.
Turbulent Reattaching Shear Layer- This case istaken from the experiments conducted by Samimy et. al.[30]. The geometry involves a backward-facing stepfollowed by a ramp and a constant area section. Figure12a shows the grid used for this calculation, consisting of19500 quadrilateral elements. A free shear layer isformed off the step, which eventually reattaches to theslanted wall (ramp). This problem is more complicatedthan the previous one. Here a large recirculatingsubsonic flow pocket (cavity flow) interacts with a freeshear layer and a compression structure. The inflow wasat Mach 2.46 and was initialized with boundary layerprofiles (3.12mm thickness). The inflow stagnationpressure and temperature were 528.1 Kpa and 297 K,respectively. The interior of the domain was initializedwith Mach 2.46 flow conditions everywhere except forthe small triangular region downstream of the step,where the velocity was set to zero and pressure wasmatched to that of the inflow. Goldberg's three equationmodel [27] was used for this case too. Pressure andvelocity contours are given in Figures 12 b and c for thesake of completeness; however, once again the user isreferred to [28] for a detailed discussion. Figure 13shows the convergence history for a relaxation schemewith one LHS sweep and no particular cell ordering, anda relaxation scheme with eigenvalue-based cell orderingand 2 forward-backward LHS sweeps at CFL 2.0. Inroughly 2800 steps, the eigenvalue based scheme enabledfive and a half orders of magnitude drop in residual. Thiscase is quite challenging since in its transient stages verystrong shocks and expansions develop in the flowfield.This limits the CFL to a low value, however, even withthat restriction, the relaxation scheme has shown markedimprovement over tests run using AF and point-implicitmethods (not shown).
Low Subsonic Flow over Slanted Backward FacingStep—This case is taken from the experimentsconducted in [31]. The basic geometry is comprised of a45 degree backward facing step followed by a very longchannel section. Figure 14a shows the 151x101 gridused for the calculation. The grid is clustered near thelower and upper walls. Goldberg's three equation model[27] was used also in this case. The Mach 0.1 flow wasused to initialize the flowfield and no inflow profileswere used. A large separation bubble is present behindthe slanted backward facing step, as shown in Figure14b. The preconditioned version of the relaxationscheme with 2 backward-forward LHS sweeps at CFL50 was used for this case. Figure 15 shows theconvergence history. We see that four orders drop in the
magnitude of the residual is achieved in about 1000steps.
Two-Hole Injection - This case corresponds to theUVA two-hole transverse injector topology. Table 1,taken from McDaniel et. al [32], shows the geometrycharacteristics.
Test section heightTest section widthLength of measurement domainEnd of nozzle contourStep locationStep height1st injector location2nd injector locationDiameter D
Figure 16 shows a sketch of the UVA two-holetransverse injector. The topology consists of an inflowregion followed by a backward-facing step after whichtwo transverse injectors are located. The computationalgrid used to simulate the flowfield was derived from afine grid (400K cells, Sekar[33]). This mesh wascoarsened everywhere except near the hole regions and agrid with only 70K cells was obtained. 2-D slices of thecoarse grids on the lower wall (downstream of thebackward-facing step) are shown in Figures 17a and b.The Mach number of the primary flow was 2.089 andthat of the jets was 1.183. The flow conditions weresimilar to those found in McDaniel et al. [32]. Theturbulence model invoked for this calculation is a one-equation pointwise eddy viscosity model given inGoldberg [34]. Figures 18a and b show the pressure inthe cross-stream direction over the two holes ascompared to the data from [32]. Figure 19 shows theconvergence history for this case using a point-implicitscheme (CFL 1.0) and the relaxation scheme using 2LHS forward-backward sweeps at CFL 3.0 (rampedfrom 1.0). Nearly five orders of magnitude drop inresiduals was achieved by the relaxation scheme in about1800 steps.
Low Speed Parachute—A Mach 0.01 3-D flow past aparachute was computed using the preconditionedversion of the relaxation scheme. The grid used for thecalculation is comprised of 10500 cells. Figure 20 showsthe solution in one of the symmetry planes and Figure 21shows the convergence history with the relaxationscheme using 2 LHS forward-backward sweeps at CFL5.0. A four order of magnitude drop in residuals wasachieved in nearly 500 steps.
Inviscid Flow over Space Shuttle Orbiter—A Mach2.1 inviscid flow at a 10 degree angle-of-attack wassimulated over the space shuttle orbiter geometry. Themesh consisted of roughly 270,000 tetrahedral elements.The relaxation scheme was used with 2 forward-backward boundaries-on-in LHS sweeps. The local CFLwas ramped from 1.0 to 5.0 between steps 1 and 300.Figures 22a and b show the pressure contours and grid inthe symmetry plane, respectively. Figure 23 shows theconvergence history for the scheme. Five orders ofmagnitude drop in the residual was achieved in roughly900 steps. The relaxation scheme performed rather welleven with a large number of grid points in the mesh (asalso seen in the two-hole case).
Conclusions
This paper has presented a preconditioned-implicitrelaxation scheme that shows very nice convergenceproperties for a variety of problems encompassing flowspeeds from low subrsonic to supersonic and gridtopologies from single block structured to multiblockunstructured.
Currently, a multigrid option is being added to this point-implicit relaxation scheme which should improve theconvergence characteristics even further.
Acknowledgments
This work was partially funded by Air Force SBIRF33615-97-C-2732. The authors wish to thank Drs.Mark Hagenmaier and Balu Sekar of WrightLaboratories for their invaluable guidance, and the AirForce for its support through the SBIR program.
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Figure 12 a) Mesh used in reattaching shear layerFigure 9. Convergence histories for Mach 2.84 calculation b) Pressure contours c) Streamwiseturbulent flow past a 24 degree ramp. velocity contours and streamlines.
\7
Figure 10 a) 121x121 Grid for transonic flow overa 2-D bump, b) Mach number contours.
Figure 13 Convergence histories for turbulentreattaching shear layer.
** 10°° JZl**."" S8<* *** Figure 14 a) 151x101 grid used for Mach 0.16 flownumber oftflrnorv t& s **/
over a slanted backwards facing step, b) streamwiseFigure 11 Convergence history for transonic flow velocity contours and streamlines,over a 2-D bump.
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