a multigrid method for high speed reactive...

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

A97-32517AIAA-97-2106

A Multigrid Method for High SpeedReactive Flows

Scott G. Sheffer*, Antony Jameson*, and Luigi MartinellrCFD Laboratory for Engineering Analysis and DesignDepartment of Mechanical and Aerospace Engineering

Princeton UniversityPrinceton, New Jersey 08544 U.S.A.

Abstract

This paper presents a multigrid method for com-puting inviscid and viscous high speed steady-statehydrogen/oxygen and hydrogen/air reactive flows.The governing equations for reactive flow are solvedusing an explicit multigrid algorithm while treatingthe chemical source terms in a point implicit man-ner. The CUSP (Convective Upwind and Split Pres-sure) scheme is used to provide necessary artificialdissipation without contaminating the solution. Re-sults indicate good multigrid speedups and adequateresolution of the reaction zone in both inviscid ax-isymmetric and viscous two-dimensional test cases.The method lends itself to efficient parallel computa-tions, thus enabling the calculation of reactive flowswith detailed chemical kinetics.

Nomenclature

C(wjj) convective Euler fluxesD(wij) diffusive and dissipative fluxes

E total energy (internal, chemical and kinetic)f, g Euler flux vectors

fv, gv diffusive flux vectorshi static enthalpy of species iM Mach number

p static pressureq conductive heat flux vectorR mixture gas constant

R(wy) total flux residual for cell i,jT static temperature

u, v Cartesian velocity componentsUdi, Vdi diffusion velocity components for species i

Vij volume of cell i,jw vector of flow variables7 mixture ratio of specific heats

Copyright ©1997 by the Authors. Published by theAIAA Inc. with permission.

* Graduate Student, Student Member AIAAfjames S. McDonnell Distinguished University Professor

of Aerospace Engineering, AIAA Fellow{Assistant Professor of Aerospace Engineering, AIAA

Member

p densitypi density of species iT viscous stress

<j) chemical source term, <9fi cell element and boundary

Introduction

The simulation of high speed chemically reactingflows is a very challenging area for computationalefforts. The presence of shock waves necessitatesgood shock capturing properties, while chemical re-actions require avoiding excessive numerical dissipa-tion so that the solution remains uncontaminated.Viscous effects, heat conduction and species diffu-sion complicate reactive flow calculations, both fromthe increased computational work required and be-cause of possible interactions between the chemistryand these effects. Diffusion of radical species in aboundary layer may significantly alter the result-ing flowfield. Exponential increases and decreasesof radical species in small spatial zones near shockslead to large gradients that must simultaneously becaptured without oscillation and without unneces-sary dissipation. In addition, the stiff nature of thechemical source terms makes the integration of thegoverning equations very difficult and time consum-ing.

The inherent inaccuracy of chemical rate datapresents another challenge to modeling reactiveflows. The turn around time of reactive flow sim-ulations must be made short enough so that vary-ing rate coefficients may be examined to determinewhich set is the most appropriate for modeling aparticular phenomena.

The use of multigrid acceleration for reactive flowcalculations has not been adequately examined.Bussing and Murman [1] explored the use of multi-grid, but only for their one-dimensional calculations.Multigrid techniques may have been thought to betoo dissipative and cause radical species to be moved

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to physically incorrect regions. However, propermultigrid techniques, in which the coarse grids areforced by the fine grid solution, can in fact be usedto compute chemically reacting flows. With the ad-dition of viscous effects and species diffusion, a wayto accelerate convergence is sorely needed. Whileparallelization will decrease the computational timeassociated with a numerical simulation, multigridtechniques represent an untapped potential for con-vergence acceleration of reactive flow calculations.

Because most chemical reactions have characteristictimes much less than those of the convective flowfield, explicit schemes are handicapped in computingsuch flows. The time step for an explicit schemeis proportional to the shortest characteristic time,so that stability restrictions require very short timesteps in reactive flow simulations. This short timestep leads to very long simulation times for steadystate computations.

Several ways to overcome this limitation of explicitschemes have been explored. The first is to use afully implicit scheme for the steady state computa-tion, thus removing stability limitations completely.Another way of reducing the effect of the chemicalterm stiffness is to treat the source term in a pointimplicit manner. In this formulation due to Bussingand Murman [1], the source term at the next timelevel is linearized about the current time level, lead-ing to a fully explicit equation at the cost of a matrixinversion. This action has the effect of precondition-ing the species continuity equations, rescaling thechemical characteristic time so that it is of the sameorder as the convective characteristic time.

In this work, the point-implicit formulation of Buss-ing and Murman is combined with an explicit multi-grid solver [11] using CUSP dissipation to computehigh speed reactive flows. The fully explicit algo-rithm is parallelized using the MPI standard on anIBM SP-2 to achieve high parallel speedups.

Governing Equations

The two-dimensional equations for chemically react-ing flow can be written in a Cartesian coordinatesystem (x,y) as:

dw | 8(f - fv) | 0(g - gv) = ^dt dx dy

where w is the vector of flow variables, f and gare the convective flux vectors, fv and gv are thediffusive flux vectors and u> is the vector of sourceterms. Consider a control volume fi with boundaryd£l. The equations of motion of the fluid can then

be written in integral form as

— / / w dxdy + j> (fdy - gdx) =

i (fvdy-gvdx) + fjudxdy, (2)

where w is the vector of flow variables

w =

f, g are the Euler flux vectors

f =piU

pu2 +ppuv

pEu + pu

_ puvg ~~ » pv2 + p

pEv + pv

fv, gv are the diffusive flux vectors including speciesdiffusion effects

fv =

gv =

rxxrxy

TXXU + TxV

TyxU + TyyV

and <jj is the chemical source vector

In these equations, i = 1, . . . , N and N is the num-ber of species. For a thermally perfect gas, pressuremay be determined from

p = PRT

where R is the mixture gas constant. The density isfound from

N

Temperature may be determined from the followingrelation,

where hi are the individual species enthalpies whichdepend solely on temperature for a thermally perfectgas.

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Chemical Model

One strength of the current work is that the re-sulting code can use an arbitrary chemical model,thus allowing for quick comparison of reaction sets.Therefore, several different chemistry models forhydrogen-air combustion are used in this paper.The first is a reduced equation model for hydrogen-air combustion due to Evans and Schexnayder [3]involving seven species and eight reactions. Nitro-gen in this set is treated as an inert diluent. Thesecond group of rate equations is the nine species,nineteen reaction modified model of Jachimowski[5] and Wilson and MacCormack [19]. Again, ni-trogen is treated as an inert diluent because reac-tions involving nitrogen have been determined tobe negligible below Mach 5 [13]. Another chem-istry model is that of Yetter et al. [22, 9, 21]. Thismechanism, which is part of a broader mechanismfor the CO/H20/02 system, contains eight reactingspecies and 21 reactions. The last model was oneproposed by Westbrook [18] which includes eight re-acting species and seventeen reactions for hydrogen-oxygen combustion.

Numerical Model

source terms leads, in general, to a time step restric-tion due to the stability limitation of the explicitscheme. This time step can be orders of magni-tude less than the time step of the convective terms.Treating the source terms implicitly removes the sta-bility criterion at the expense of a matrix inversion.The point implicit treatment has the result of reduc-ing the stiffness of the problem by effectively rescal-ing the characteristic time of the reactions so thattheir magnitudes are commensurate with the con-vective characteristic time.

This treatment necessitates the inversion of thesource term Jacobian matrix with dimension TV x Nwhere N is the number of species present in the flow.An inversion for the momentum and energy equa-tions is not necessary due to the absence of chemicalsource terms in those equations. This N x N inver-sion is done only during the first stage of each timestep of the solver and is retained and used for thesucceeding four stages. This time-saving step has noeffect on the results of the computation.

Because the chemical source terms have been treatedimplicitly, the time step limitation of the explicittime integration scheme depends solely on the spec-tral radius of the flux Jacobian.

Flow Equations

The governing equations are solved using a conserva-tive second-order accurate finite volume formulationin which the chemical source terms are treated pointimplicitly.

When the integral governing equations 2 are inde-pendently applied to each cell i, j in the domain, weobtain a set of coupled ordinary differential equa-tions of the form

(3)

where JJ) are the convective Euler fluxes,D(wy) includes the diffusive fluxes and the artificialdissipation fluxes added for numerical stability rea-sons and uVij are the chemical source terms. Thisequation (3) can be rewritten as follows (drop thei,j subscripts for clarity):

dtl (4)

where R is the sum of the two flux contributions.The governing ordinary differential equations aresolved using a standard five-stage time steppingscheme [12].

Chemical Source Terms

The chemical source vector LJ was treated in a pointimplicit manner [1]. An explicit treatment of the

Numerical Dissipation

The Convective Upwind and Split Pressure (CUSP)scheme provides excellent resolution of shocks athigh Mach numbers at a reasonable computationalcost [6, 7]. The CUSP splitting is combined with aLED or ELED limiter to achieve second order ac-curacy in smooth regions with oscillation-free cap-ture of shocks and large gradient regions. CUSP hasbeen shown to be an accurate and effective dissi-pation scheme for viscous flows [17] and high speedreactive flows [15, 14].

Boundary Conditions

The surface boundary is modeled as an adiabatic,non-catalytic inert surface. For Euler calculations,flow tangency is enforced at the the surface while ano-slip boundary condition is used for viscous flows.Due to the supersonic nature of the flow, outflowboundary quantities are extrapolated from the inte-rior and inflow quantities are taken to be free streamvalues.

Free stream values of radical species are set to amass fraction of 1 x 10~12. Varying this value didnot affect the results.

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Parallelization

The MPI standard is used to parallelize the codeon an IBM SP-2. A static domain decompositionwith two-level halos for flow quantities and one-levelhalos for grid information is used. The current pointimplicit scheme will achieve a higher level of speedupas compared to a convective flow code alone due tothe larger number of operations that take place percell [14].

Multigrid Convergence Acceleration

Multigrid acceleration has been applied quite suc-cessfully to the solution of both the non-reactive Eu-ler and Navier-Stokes equations [11, 12, 17]. How-ever, the application of multigrid methods to reac-tive flow calculations has been limited. Bussing andMurman [1] reported success in using Ni's multigridmethod for one-dimensional reactive flow calcula-tions. Additional examples of multigrid accelerationfor reactive flows are lacking.

The approach taken in this paper is to use a pre-viously validated multigrid solver [11, 12, 17] andinclude chemical source terms on all levels [14]. Thecoarse grid corrections to the species densities arelimited to ensure that no mass fraction becomes neg-ative. Varying degrees of underrelaxation can beused to enhance the convergence rate while captur-ing the sharp gradients and large radical growth re-gions that characterize reactive flow problems.

Results and Discussion

The formulation described in this paper was ap-plied to both inviscid axisymmetric flows over spher-ical blunt bodies and two-dimensional viscous rampflows.

Stoichiometric hydrogen/oxygen flow over an ax-isymmetric spherical tip projectile at M = 3.55 wassimulated using the reduced chemistry model (sixspecies, eight reactions) of Evans and Schexnayder[3]. This corresponds to an experiment conductedby Lehr [10] which is shown in Figure 1. The di-ameter of this projectile is 15 mm. The free streamtemperature is 292 K and the free stream pressureis 24800 Pa. The grid in this case is 64 x 64 cells.The bow shock in front of the body raises the tem-perature of the flow so that, after an induction zone,the flow reacts. As shown in Figure 2, the lengthof the induction zone varies depending on the postshock temperature which corresponds to the rela-tive strength of the shock. Temperature is approx-imately constant in the induction zone as radicals

Figure 1: Experimental shockwave and reactionfront: M = 3.55 hydrogen/oxygen, from Lehr [10].

Figure 2: Temperature contours: M = 3.55 hydro-gen/oxygen. Contour levels in K: min: 250, max:3130, inc: 93. Grid size: 64 x 64. Circles indicateexperimental shock and heat release locations.

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build up to the necessary levels to react and pro-duce water vapor. The production of water vapor isaccompanied by heat release which raises the tem-perature and, because pressure is nearly constantacross the reaction zone, lowers the density. Fig-ure 3 shows the temperature along the stagnationstreamline computed using the Evans and Schexnay-der reaction model and also the Jachimowski eightspecies, nineteen reaction modified model. The eight

3500

3000-

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00x/R

Figure 3: Temperature along stagnation streamlinefor two different chemical models: M = 3.55 hydro-gen/oxygen; projectile surface at x/R = 0.00. Gridsize: 64 x 64.

species, nineteen reaction model produced a slightlydifferent shock location, while the point of heat re-lease is close to the Evans and Schexnayder model.The only difference between these two calculationswas the chemistry model used: both models used thesame algorithm and grid and were converged to thesame level of accuracy. Thus, even though both sim-ulations are converged, different solutions result be-cause the mathematical models were slightly differ-ent in the modeling of the chemistry. The differencein shock location could be explained by a differencein relaxation characteristics behind the shock due tothe chemical models. In addition, the heat releaseprofiles lead to slightly different static pressures be-hind the heat release region, which could result indifferent shock positions. Figure 3 indicates that thenumerical dissipation scheme is providing the neces-sary dissipation in regions of steep gradients, suchas the shock, to prevent oscillations and preservemonotonicity while still allowing sharp resolution.

However, upon examining Figure 2, one may no-tice that the numerical solution differs slightly fromthe experimental shock location and heat releasefront. Thus, another simulation was performed ona 64 x 128 cell grid which had twice the number ofgrid cells in the normal direction as the 64 x 64 gridhad. The Evans and Schexnayder model was used

for the chemistry, since it produced a shock loca-tion that was closer to the body on the 64 x 64 grid.Temperature contours for this finer grid simulationare presented in Figure 4 along with experimentalshock and heat release front locations. As can beseen, the agreement between the experimental re-sults and the numerical simulation is much better onthis finer grid. The increased resolution has movedthe shock closer to the body, increased the length ofthe induction zone and moved the heat release frontcloser to the body. Density and pressure contoursare presented in Figures 5 and 6.

Figure 4: Temperature contours: M = 3.55 hydro-gen/oxygen. Contour levels in K: min: 250, max:3130, inc: 93. Grid size: 64 x 128. Circles indicateexperimental shock and heat release locations.

The temperature along the stagnation streamlinefor the 64 x 128 grid is shown in Figure 7. As inthe case of the 64 x 64 cell grid, the flow passesthrough a shock which is followed by an inductionzone where radicals are created due to the highertemperature. After sufficient amounts of radicalshave been formed, water vapor is created, increasingthe temperature and further hastening the reaction.

The mass fractions of the reactants and products forthe Evans model simulation on the 64 x 128 grid areplotted in Figure 8. As expected, the reactants havealmost constant mass fraction through the shock andonly begin to be consumed in the heat release region.Water vapor is not formed in any appreciable quan-tity until the heat release region. It should also benoted that the mass fraction profiles are monotonic;no overshoots in the mass fraction of these primarycomponents is observed, again indicating good per-formance of the CUSP dissipation scheme.

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2£3

Tem

pera

!

ttW

3000

2500-

2000

1500-

lUvU

500-

o-

-- 4 --4--.-- -i-f/............. .4.. .......... L...... ._| _.j_. ._;...._. / ...

. . . . , ....... . . . . . . : ..... ..... . /..

•-^•-1--j^7jtj-—. . . . . 1 . . . . . . : . / _ j ........ ........ ...., ........ ....... . .......

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00x/R

Figure 5: Density contours: M = 3.55 hydro-gen/oxygen. Contour levels in kg/m3: min: 0.05,max: 0.6, inc: 0.016. Grid size: 64 x 128.

Figure 7: Temperature along stagnation streamline:M = 3.55 hydrogen/oxygen; projectile surface atx/R = 0.00. Grid size: 64 x 128.

i.U

0.8

0.6

0.4-

0.2-

o.o-"

! i | > priS]..... ......... ........ ......... ...... j ..^p

........................... . . . . . . . :.......... . . . . . . . . . . . . . . . . . | ! ^ -

.. |._ _ ._ . . ^ . . . . . |_ . f |_ ._

....... .... ...... ; .......... ...J

! 1 1 A.|

Figure 6: Normalized pressure contours: M = 3.55hydrogen/oxygen. Contour levels (P/POO)- min: 1.0,max: 16.5, inc: 0.5. Grid size: 64 x 128.

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00X/R

Figure 8: Mass fractions along stagnation stream-line: M = 3.55 hydrogen/oxygen. Shock location isat x/R = 0.22; heat release region begins at x/R =0.065; projectile surface at x/R = 0.00. Grid size:64 x 128.

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The growth and destruction of radical species is de-picted in Figure 9 for the Evans model. This figure

problem without compromising the accuracy of thesolution.

o.o

o1£1o

-5.0-

-10.0-

-12.50.35 0.30 0.25 0.20 0.15

X/R0.10 0.05 0.00

I»

Figure 9: Logarithm of radical mass fractions alongstagnation streamline: M = 3.55 hydrogen/oxygen.Shock location is at x/R = 0.22; heat release regionbegins at x/R = 0.065; projectile surface at x/R =0.00. Grid size: 64 x 128.

shows the logarithm of the mass fractions of the rad-ical species. Exponential growth can be observed inthe induction zone, which is to be expected. Aftera small region of slow growth, the radical mass frac-tions grow approximately nine orders of magnitudein a very short distance, which is a testament to theability of the CUSP splitting and the flux limiterto capture large changes in the conservative vari-ables without oscillation and without undue damp-ing. The mass fraction of the radicals does not de-crease greatly after the heat release zone, even withthe formation of a great deal of water vapor. Thisis due to the conversion of the reactants into radicalspecies which then combine directly to form water.

Upon examination of Figure 4, it can be seen thatthe agreement between the computation and exper-iment is quite good. This computation also agreesfavorably with the simulation of Yungster et al. [23]who limited the cell Damkohler number so that theheat release was spread out among two to three cells.No such limitation is necessary using the current for-mulation.

Figures 10 and 11 show the convergence historiesfor the six species, eight reaction 64 x 64 cell calcu-lation without and with multigrid acceleration. Theconvergence histories present the root mean squaredresidual of the density of water and the number ofsupersonic cells in the domain versus the number ofcycles. The initial increase in the density residual isdue to the production of water as the flow initiallyreacts and the bow shock and reaction zone moveoutward from the body. The use of multigrid pro-vides a significant convergence acceleration in this

.00 50000 1000.00 1500.00 2000.00 2500.00 3000.00

Work

Figure 10: Single grid convergence history: M3.55 hydrogen/oxygen. Grid size: 64 x 64.

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Figure 11: Multigrid convergence history: M = 3.55hydrogen/oxygen. Grid size: 6.4 x 64.

The single grid calculation converged about six or-ders of magnitude in 2000 iterations, consuming 995seconds (wall clock) using four processors of an IBM


SP-2. The multigrid calculation, using only two lev-els of multigrid for this 64 x 64 mesh, converged four-teen orders of magnitude in 2000 iterations while re-quiring 1250 seconds on four processors. The samelevel of convergence (which results in a solution thatdoes not change) as the single grid case using multi-grid takes about 800 iterations at a cost of 500 sec-onds, almost halving the computational time. Theconverged solutions with and without multigrid ac-celeration are virtually identical due to the correctforcing of the coarse grid by the fine grid solution.No unphysical diffusion of radical species is seen.The convergence history for the multigrid 64 x 128cell simulation is shown in Figure 12. Multigridagain provides a significant convergence accelera-tion.

I- °

.00 500.00 100000 1SOO.OO 200000 Z50OOO 3000.00

Work

Figure 12: Multigrid convergence history: M = 3.55hydrogen/oxygen. Grid size: 64 x 128.

A two dimensional viscous reactive test case wastaken from an experiment performed by Fielding [4].In this experiment (Figure 13), a wedge of half an-gle 6.34° was placed in a free stream of partiallyreacted hydrogen and air. The free stream Machnumber was 2.10, the free stream temperature wasapproximately 569 K and the pressure was 0.060 at-mospheres. The object of the experiment was tosee if radical-seeded hydrogen-air mixtures wouldreact at low pressures over a wedge after passingthrough an oblique shock. The mass fractions ofthe inlet flow were determined by Fielding using aone dimensional reacting gas code and are given inTable 1. Hydrogen had been injected into the airstream upstream of the ramp, partially reacted andthen expanded so that the flow constituents became

0.5625" 0.5"

Figure 13: Experimental setup for M = 2.10 viscouswedge.

Species Mass FractionH202H2ON2H0OHHO2H2O2

6.86 x 10--5

1.75 x ID"1

2.86 x 10~2

7.84 x 10"1

2.33 x 10~4

9.66 x 10~4

1.65 x 10~4

3.97 x 10-3

4.76 x 10~10

Table 1: Free stream mass fractions for M — 2.10hydrogen-air viscous wedge flow.

frozen. It was assumed that nitrogen was an inertdiluent. Due to the low pressure, the Reynolds num-ber at the end of the ramp is approximately 33000so that fully laminar flow may be assumed. Speciesdiffusion and heat conduction effects are included inthe simulation. The nine species, 21 reaction modelof Yetter et al. was used because it had been themodel utilized to generate the composition of theincoming flow.

The flow was simulated on a grid with 128 cells inthe streamwise direction and 96 in the direction nor-mal to the plate. Symmetry boundary conditionswere implemented along the centerline of the flowupstream of the wedge so that the governing equa-tions were solved on only half of the experimentaldomain. The grid was clustered near the leadingedge of the wedge to properly resolve the growthof the boundary layer and the shock attachment.Cells were also clustered near the ramp to resolvethe boundary layer structure adequately; approxi-mately 32 cells were within the boundary layer.

Figure 14 presents contours of density for this flow.The relatively thick boundary layer is immediatelynoticeable. Because of this thick boundary layer,the shock near the leading edge of the plate is at amuch higher angle than the region away from the

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ramp. The growing displacement thickness from theboundary layer causes the main flow to see an effec-tively curved wall and thus, weak expansion wavesare formed to adjust the curvature of the shock. Itis also important to point out the absence of largeheat release regions even though the domain near theleading edge experiences a relatively large compres-sion. The shock angle obtained from the numericalsimulation is 35.0° while the experimental shock an-gle is 34.5°.

Figure 14: Density contours:gen/air 6.34° viscous ramp.

M = 2.10 hydro-Contour levels in

kg/m3. min: 0.033, max: 0.056, inc: 0.00129.

The pressure field (Figure 15) yields trends similarto the density field. There is curvature of the shocknear the leading edge and expansion waves may beseen in the region behind the oblique shock. Very lit-tle change in the pressure is seen through the bound-ary layer.

The lack of progress of the reaction may be under-stood by viewing Figure 16, which shows contoursof the logarithm of OH mass fraction. It is imme-diately obvious that the inlet mass fractions are notin an equilibrium state, as rapid reaction takes placein the free stream flow, producing more OH. Behindthe shock, the increased pressure and temperatureand slightly increased residence time cause faster for-mation of OH, but the post shock temperature andpressures are not sufficient to allow large quantitiesof OH to strip hydrogen atoms from H2 moleculesand form water vapor. Near the ramp surface, itcan be observed that some OH is being consumedand transformed to water. This is due to the highertemperature in the boundary layer near the wall, andbecause the molecules in that region have a longerresidence time at those higher temperatures due tothe lower velocity near the wall. The maximum wa-ter mass fraction is approximately 3%, which is notmuch greater than the free stream water mass frac-

Figure 15: Normalized pressure contours: M = 2.10hydrogen/air 6.34° viscous ramp. Contour levels

): min: 1.0, max: 3.06, inc: 0.0857.

tion from Table 1. This small amount of water pro-duction and heat release is not sufficient to causethermal runaway, greater reaction and greater heatrelease. These results agree qualitatively with the

Figure 16: Logarithm of OH mass fraction contours:M = 2.10 hydrogen/air 6.34° viscous ramp. Con-tour levels: min: -3.33, max: -2.65, inc: 0.057.

experimental findings. Excited OH fluorescence wasobserved in the region behind the shock and near theleading edge, but large regions of heat release wereabsent.

Figures 17 and 18 show convergence histories for thiscalculation without and with multigrid acceleration.The single grid calculation converged approximatelythree orders of magnitude in 2000 iterations. Thecomputational cost of this calculation was 4578 sec-onds using six processors of an IBM SP-2. Themultigrid calculation with two levels of grids con-verged six orders of magnitude after 2000 iterations

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in 7992 seconds of wall clock time. The multigridcalculation achieved the same convergence level asthe single grid calculation in approximately 2550 sec-onds, which is 1.80 times faster than the single gridcalculation. The results of both simulations are vir-tually identical. As in the inviscid case presentedpreviously, no unphysical diffusion of species is seendue to the correct forcing of the coarse grid.

d"siJ

s i

Figure 17: Single grid convergence history: M =2.10 hydrogen/air 6.34° viscous ramp. Grid size:128 x 96.

The third test case consisted of a stoichiometrichydrogen/air flow over a 10° viscous ramp. Thefreestream Mach number was 4.0, the freestreamtemperature was 1200 K and the freestream pres-sure was one atmosphere. The 2 cm ramp was pre-ceded by a 1 cm solid wall section. The effects of vis-cosity, heat conduction and species diffusion are allincluded in this computation. This is a common vis-cous/reactive test case, with several published com-putations available [2, 20, 16, 8] for turbulent reac-tive flows. However, because of the lack of concreteknowledge regarding the effect of turbulence on com-bustion, in this work the flow was computed assum-ing fully laminar flow. This will produce different re-sults than those obtained using a turbulence model,but a grid convergence study may still be pursuedto determine the robustness of the numerical model.In addition, most of the researchers who attemptedthis computation used grids leading to solutions thatwere severely underresolved, both in the boundarylayer region and in the reaction zone. Thus, anyresults that were obtained must be viewed in thislight.

500.00 1000.00 1500.00

Work

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Figure 18: Multigrid convergence history: M — 2.10hydrogen/air 6-34° viscous ramp. Grid size: 128 x96.

This flow was computed using Westbrook's ninespecies, seventeen reaction hydrogen-air mechanismbecause other researchers have used this model forthis test case. This is a very challenging flow becauseof the myriad of physical phenomena that must beresolved accurately. The formation of the bound-ary layer must be captured well, without unneces-sary dissipation, so that the displacement thicknessis correct. The conduction of heat must also be accu-rate so that the effects of viscous dissipation deep inthe boundary layer are felt correctly in the sectionsof the boundary layer far from the wall. Proper res-olution of the interaction of the shock and boundarylayer and the oscillation free capture of the shockare essential, as is the correct modeling of speciesdiffusion. The free stream temperature is not highenough to initiate a reaction, but the combination ofhigh temperature in the boundary layer, diffusion ofradicals in the direction normal to the wall and theoblique shock cause a reaction front to form past theshock in the ramp region. Any inaccuracy will causethe location of the reaction front to be incorrect.

An assiduous grid convergence study was under-taken for this flow. Three grids were used to simu-late this flow and to determine the robustness of thealgorithm. All three grids had cells clustered nearthe wall to properly capture the laminar boundarylayer, with approximately 32 cells within the bound-ary layer. In addition, cells were clustered near thestart of the ramp, where shock-boundary layer in-teraction may lead to interesting phenomena. We


would expect any artificial diffusion of mass, mo-mentum and energy to decrease as the grid reso-lution increases. In addition, the modeling of con-vective and diffusive transport is second order accu-rate and thus, the accuracy of the physical trans-port should increase as the grid spacing becomessmaller. Figures 19, 20 and 21 show the temper-ature in the flow field for three grids of size 64 x 96,64 x 108 and 128 x 156, with the first number of cellsalong the wall and the second number of cells nor-mal to the wall. As was expected, the formation

Figure 19: Temperature: M — 4.0 hydrogen/air10° viscous ramp, 64 x 96 grid. Temperature range:1200-3500 K.

Figure 20: Temperature: M = 4.0 hydrogen/air 10°viscous ramp, 64 x 108 grid. Temperature range:1200-3500 K.

of the boundary layer and the attendant viscous dis-sipation raises the temperature near the wall whichcauses various radical species to form. This leadsto the formation of some water vapor very near thewall before the shock, but does not greatly affectthe flowfield. One may also notice the formation ofa weak oblique shock at the left edge of the domaindue to the displacement thickness of the boundarylayer. The interaction of the stronger oblique shockand the boundary layer at the ramp corner sepa-rates the boundary layer for a short distance and

Figure 21: Temperature: M = 4.0 hydrogen/air 10°viscous ramp, 128 x 156 grid. Temperature range:1200-3500 K.

causes a small recirculation zone in that region. Be-yond the oblique shock, the increased temperatureand pressure lead to faster radical production andincreased diffusion of those radicals and heat to theunreacted bulk flow behind the shock. The resultis a reaction front that forms and gradually movesaway from the wall. The reaction front may be seenin a different way by viewing the mass fraction of wa-ter in Figure 22 for the 128 x 156 grid. Comparing

Figure 22: Water mass fraction: M = 4.0 hydro-gen/air 10° viscous ramp, 128 x 156 grid. Mass frac-tion range: 0.00-0.18.

the results of the three grid simulations yields in-teresting insights into the resolution needed for thisflow. The third grid (128 x 156) has twice the reso-lution of the first grid (64 x 96) in both the normaland streamwise directions outside of the boundarylayer. The result is that the reaction front is unphys-ically diffused on the coarser grid (due to inadequateresolution of the reaction zone, artificial diffusion ofradical species and energy and inadequate resolutionfor diffusive transport) and is captured quite well onthe finest grid. The curvature of the reaction frontat the right edge of the domain may be an actualphysical phenomenon or it may be a numerical ar-

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tifact. If it is a spurious numerical phenomenon, itis probably due to one of two causes. First, it maybe that the supersonic extrapolation boundary con-ditions at the outflow plane are contaminating thesolution. Second, the numerical dissipation may becausing unphysical diffusion of radical species andheat. However, it is unlikely, given the resolution ofthis grid in that particular area, that the artificialdissipation is the cause. Thus, in order to test theaccuracy of the boundary condition, an additionalsimulation was performed in which the outflow planeof the simulation domain was moved approximately0.9 cm downstream. The computational size of thisadditional simulation was also 128 x 156 cells, with32 cells again within the boundary layer.

The results of this extended domain simulation arepresented in Figures 23 through 25. The flow fieldtemperature is depicted in Figure 23, where the solidline within the simulation domain indicates the for-mer outflow plane location. Comparing this figureto Figure 21, one can see that the temperature inthe smaller domain and the extended domain matchexceptionally well in the original computational re-gion. Water mass fraction is shown in Figure 24,while pressure may be viewed in Figure 25. The ex-tension of the domain makes it evidently clear thatthe curvature of the reaction front observed in Fig-ures 21 and 22 is indeed a true physical phenomenon.In addition, the simulation on the extended domainindicates that the use of the supersonic extrapola-tion boundary conditions at the outflow plane doesnot compromise the accuracy of the solution in theregion of the boundary. In this flow, radicals andheat in the boundary layer diffuse outward normalto the wall and eventually, along with the increasedtemperature and pressure behind the oblique shock,cause the reaction to proceed in the inviscid regionbehind the oblique shock. This reaction front cou-ples with the shock toward the outflow plane andchanges the angle of the discontinuity in the flowdue to the pressure and heat release behind theshock/reaction front.

Other researchers' previously published results forthis flow were calculated on grids with half as manycells in the normal direction as the finest grid pre-sented here. These simulations show a coupled shockreaction front located much closer to the ramp cor-ner in which the shock curves away from the wall.While these other simulations were computed withturbulence models of varying complexity, it is prob-ably the underresolution of the reaction area whichleads to the greatest disparity between those resultsand those presented here. If the reaction area is un-derresolved and the numerical scheme is too dissipa-tive, then a probable result would be for the reactionzone to move upstream to the shock position. Thiswould be caused by unphysical diffusion of radical

Figure 23: Temperature for extended domain: M =4.0 hydrogen/air 10° viscous ramp, 128 x 156 grid.Temperature range: 1200-3500 K. Solid line inthe computational domain indicates original outflowboundary.

Figure 24: Water mass fraction for extended do-main: M = 4.0 hydrogen/air 10° viscous ramp,128 x 156 grid. Mass fraction range: 0.00-0.18.

Figure 25: Normalized pressure for extended do-main: M = 4.0 hydrogen/air 10° viscous ramp,128 x 156 grid. Normalized pressure range: 0.00-6.00.

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species and thermal energy upstream, allowing thereaction to commence earlier and thus move towardthe shock. In fact, early simulations of this flowfield with the current method using a very coarsegrid were characterized by the reaction front mov-ing upstream and away from the wall toward theshock location.

Conclusions

An accurate solver for the steady-state Euler andNavier-Stokes equations with chemical reactions hasbeen developed, and preliminary results have beenpresented. The CUSP dissipation scheme yields ac-curate capture of shocks, reaction zones and reactionfronts for both inviscid axisymmetric and viscoustwo-dimensional test cases while the use of multi-grid acceleration techniques leads to a significantdecrease in computational time without sacrificingaccuracy.

Acknowledgements

This work was funded by AFOSR-URI F49620-93-1-0427. The first author was supported in part bya Fannie and John Hertz Foundation/Princeton Re-search Center Fellowship.

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