Numerical prediction of two-dimensionaltransverse injection flows

A. T. Sriram1 and Joseph Mathew2

Department of Aerospace EngineeringIndian Institute of Science,Bangalore 560 012, INDIA

ABSTRACT

Simulations were performed using an up-wind method (Roe scheme) fully coupled withk-ω equations of a set of transverse injection ex-periments which have often been used for val-idation. As had been found before, generally,the simulations agreed with measurements atlow injection pressure ratios but became in-creasingly poor at high pressures. However,through a sequence of steps which increased res-olution and added low Reynolds number correc-tions to the turbulence model, it was found thatimproved solutions could be found which are invery close agreement with experiment even atthe highest injection pressure. Mean flow com-putations with current turbulence models arethus shown to be adequate for the entire rangeof pressure ratios.

Keywords: Turbulence modeling, compress-ible, transverse jets

INTRODUCTION

The flowfield arising from transverse injec-tion of a round sonic jet into a supersonic flow isa basic configuration of interest in the design ofsupersonic combustors or thrust vector controlof supersonic jets. The flow is also of funda-mental interest because it presents separationfrom smooth surfaces, embedded subsonic re-gions, curved shear layers, strong shocks, anunusual development of the injected jet into akidney-shaped streamwise vortex pair, and awake behind the jet. Although the geometryis simple, the flow is complex and is a goodcandidate for assessing the behavior of turbu-lence models for high-speed flow. All such as-sessments of the three-dimensional flow are pre-

ceded by at least some code validation with thecorresponding two-dimensional flow.

Sonic surface

Mach disc

Separation shock

boundary layerTurbulent

Jet induced bow shock

PUVSUVTUV SDV

PDV

Recompressionshock

Figure 1. Slot injection flow features

The two-dimensional flow due to injectionthrough rectangular slots has most of the samefeatures except that the downstream jet devel-opment is simpler. The jet’s wake is also miss-ing because the jet reattaches downstream. Asketch of the flow field is shown in Fig. 1.From the injector, the underexpanded sonic jetexpands rapidly into the supersonic crossflow.Expansion waves reflect from the jet boundary,coalesce and give rise to a Mach disk. While acentral part of the jet traverses the Mach disk,the flow outside this core spills around the disk.The plane jet reattaches downstream and en-closes a recirculation zone. The jet is deflectedby the crossflow, but is also an obstacle to thesupersonic cross flow and hence generates a bowshock ahead. Consistently with the pressurerise across the bow shock, the wall boundarylayer is separated. This upstream separation

1Doctoral student2Senior member, AIAA, Associate Professor

1American Institute of Aeronautics and Astronautics

42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada

AIAA 2004-1099

Copyright © 2004 by Indian Institute of Science. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

has an associated separation shock which in-tersects the bow shock. The separated shearlayer encloses multiple recirculation zones. Arecompression shock arises where the jet at-taches downstream.

This flow is usually considered for valida-tion only and not in devices. In simulationsreported so far, generally, accurate solutionshave been found at low injection pressure ra-tios. At higher levels there is a monotonicloss of accuracy: the point of upstream sep-aration lies closer to the injector in all sim-ulations, and the pressure rise is significantlylarger. In the present work, by a combinationof methods including explicit compressibilitymodeling, a low-Reynolds number correctionand better gridding, significantly better solu-tions have been obtained even at the highestpressure ratio. We may now conclude that,over the full range of pressure ratios for whichdata is available, present turbulence models areadequate.

Experimental data

Three sets of reference experiments of trans-verse injection into supersonic air streams areavailable. The earliest dataset is that of Spaidand Zukoski1. The other two are Aso et al.2

and Aso et al.3. Schlieren photographs from theexperiments are available for qualitative com-parisons, but static pressure along the wall hasbeen the primary measurement used for quan-titative assessments of computations.

The experiments of Spaid and Zukoski1 areof particular value because they were done sys-tematically, cover a large range of injectionpressure ratios, and provide good spatial res-olution of the wall pressure data. Moreover,the precaution of using sidewalls to obtain two-dimensional flow over most of the span hasproved essential when good quality simulationsare to be assessed. They also documented theloss of two-dimensionality at the highest pres-sure ratios in their experiments. The experi-mental configuration of Aso et al.2 was simi-lar to that of Spaid and Zukoski1. Their laterexperiment3 was a repetition with end plates toensure two-dimensionality of the flow over mostof the section.

The simulations presented here are a rel-evant subset (2-d flow, turbulent incomingboundary layer etc.) of the experiments ofSpaid and Zukoski1. The conditions are listed

in Table 1. Also listed is one case from Aso etal.2 which has been used in all previous simula-tions. This case is used for directly comparingpresent results with those of others.

Case pjet/p∞ p0jet/p0∞ mjet Jkg/m-s

1 12.75 1.19 0.10 1.872 24.31 2.27 0.18 3.573 46.90 4.31 0.35 6.774 8.74 0.22 0.03 0.725 17.12 0.43 0.06 1.406 32.44 0.81 0.12 2.657 63.50 1.58 0.23 5.198 17.72 0.31 0.87 1.26

Cases M∞ p∞ T∞ Tjet

Pa K K

1–3 2.61 6566 134.5 2444–7 3.50 3145 86.5 2458 3.75 11090 78.43 249

Table 1. Conditions in slot injection experi-ments. Cases 1–7 are from Spaid and Zukoski1;slot was located at 228.6mm from leadingedge and slot width was 0.2667mm; cross-stream fluid is air and injectant is nitrogen.Case 8 is from Aso et al.2; slot was 330 mmfrom leading edge and slot width was 1mm.J = ρjetu

2jet/ρ∞u2

∞ is the momentum flux ra-tio.

Previous simulations

Rizzetta4 simulated the slot injection exper-iments of Aso et al.2 using an explicit MacCor-mack scheme and k-ε equations. Explicit com-pressibility modeling was provided as recom-mended by Sarkar et al.5. Configurations withslot widths 0.5 mm, 1.0mm and 2.0 mm for thefull range of pressures were simulated. Gener-ally, the wall pressure distributions agreed wellat low injection pressures. Upstream separationlengths and penetration heights agreed well forthe narrowest slot (0.5 mm) and very poorlyfor the widest (2.0 mm). The disparity wasattributed to three-dimensional effects presentin the experiments. Explicit compressibilitymodeling was found to have a significant ef-fect on the upstream wall pressure distribution.

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With modeling, the upstream pressure rise wassharper, its location matched that in the exper-iment, and the elevated pressure region in theseparation bubble was also closer to experimen-tal data. The error in the length of the sepa-ration region was reduced by about one-half.Chenault and Beran6 have pointed out thatRizzetta’s study suffers from two shortcom-ings. First, the experiments were themselvesdeficient in several ways; insufficient numberof pressure taps, and data which are not verysmooth and suffers from three-dimensional ef-fects. Secondly, Rizzetta had not included tur-bulence kinetic energy in the total energy.

Gerlinger et al.7 presented their methodfor simulating injection flows and considered,as examples, the normal injection experimentsof Aso et al.2, helium injection experimentsof Kraemer and Rogers (reported in Weid-ner and Drummond8), and parallel injectionexperiments9. They used central differencingand matrix artificial dissipation with a veloc-ity magnitude scaling to keep dissipation low.Turbulence was modeled using q-ω (q =

√k).

Transport equations for q and ω were decoupledfrom the other transport equations for stability.The q-ω model was chosen because previous k-εsimulations had not been stable. The effects ofcorrections for separated flows and compress-ibility were examined with reference to the ex-periments of Aso et al.2. Three types of cor-rections to the standard turbulence model wereassessed. One is a limit on ω, the second is as-signing a different value of a model constant inthe ω-equation, and the third is the compress-ibility correction due to Sarkar et al.5. Theyfound the effect of the first two modificationsapplied together to be approximately the sameas that due to the third alone. Being slightlybetter, and perhaps marginally more econom-ical, they retained only the first two modifica-tions.

Chenault and Beran6 examined whether im-provements can be realized by using Reynoldsstress transport models (RSTM). Two-equationmodels, like the k-ε, are eddy viscosity modelsand do not account for anisotropy in Reynoldsstresses. Since the injector region has severalflow structures which cause Reynolds stresses todevelop in different ways (across shocks, shock-shear layer interactions, in separated flows), itis natural to expect RSTM to provide betterquality predictions. Also, in spite of the fiveadditional transport equations, these compu-

tations are becoming tractable. For compari-son, a k-ε model was also used; the k-equationwas obtained as a contraction of the Reynoldsstress transport equations for consistency. Ex-plicit compressibility modeling was not used.The basic numerical scheme was similar to theone used here in that it consisted of a Roescheme with MUSCL, but the limiter was thatof Venkatakrishnan. They simulated the ex-periments of Aso et al.2, even though severaldeficiencies were noted, and that of Spaid andZukoski1. It would have been preferable to usedata from Aso et al.3 which had been preparedmore carefully to overcome the earlier deficien-cies.

While predictions generally degrade with in-creasing injection pressure ratio, other charac-teristics also play significant roles. For exam-ple, in Chenault and Beran6 (Figures 4(b) and6(c) in their paper) wall pressure distributionsat approximately the same injection pressureratio (pjet/p∞) but from the two different ex-periments exhibit noticeably different solutionswith RSTM and k-ε. The upstream separa-tion and pressure rise occurs earlier for theRSTM simulation of the Aso et al.2 experiment,whereas the simulations agree for the Spaid andZukoski1 experiment. Chenault and Beran6

have also provided distributions of Reynoldsstresses in the interior and pointed out that theRSTM model showed more rapid changes andis therefore more responsive to the changes inflow structure, but interior flow details were notavailable in the experiments to establish the ac-curacy of these distributions. It is discouragingthat clear improvements have not attended thecloser approximation (physical and mathemat-ical) expected from RSTM. They speculatedthat the poor performance at high pressuresmay be due to external factors such as the in-adequacy of turbulence models, and differencesbetween the boundary conditions which may bereadily specified at the slot (uniform flow) andactual conditions in the experiment.

PRESENT SIMULATIONS

Simulations have been performed over theentire range of conditions in Table 1 and alsothose in Aso et al.3. In every case, solutionswere obtained using the Roe and MacCormackschemes, and with Baldwin-Lomax and k-ωmodels. Explicit compressibility modeling wasused in all k-ω computations. Generally, com-putations using the Roe scheme with the k-ω

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model were robust. So, most of the results pre-sented here are from these computations. Dis-cussions of the performance of the other meth-ods are also available10.

The Favre-averaged Navier-Stokes equa-tions for compressible turbulent flow have beenintegrated to steady states with a six-stage, lowstorage Runge-Kutta method. The basic for-mulation uses standard procedures which areavailable in textbooks11. The equations werewritten in divergence form for the conservationvariables ρ, ρu, ρv, ρE, ρk, and ρω, where ρ isthe density, u and v are velocity componentsin the streamwise (x) and transverse (y) di-rections, E is the sum of internal and kineticenergy (mean and turbulence), k is the turbu-lence kinetic energy and ω is the specific dis-sipation. For the finite volume equations, in-viscid fluxes were obtained using Roe’s approx-imate Riemann solver. Variables were extrap-olated with the MUSCL procedure for higher-order accuracy. Initially, second-order solutionswere computed. Later, it was found that athigh injection pressure ratios the additional res-olution provided by a third-order formulationis beneficial. The min-mod limiter was used forthe second-order computations and van Albadalimiter for the third-order ones. Viscous fluxeswere obtained using the spatial integration pro-cedures given in Peyret and Taylor12. Explicitcompressibility modeling has been done as sug-gested by Wilcox13 In preliminary calculationsit was found that, as in Rizzetta4, the correc-tion reduced the difference in upstream sep-aration length by about half. Especially athigher pressures, including this correction alsoprovides some stabilization of the computationsalso.

The computational domain and a represen-tative grid is shown in Fig. 2. For clarity, onlyone-half the grid lines in x and y have beenshown. Clustering is provided in the boundarylayer and for the region where the jet undergoesthe initial expansion and the Mach disk forms.It is important to provide enough resolutionwhere the jet expands and then compresses toform the Mach disk because these expansionand compression regions are connected to theupstream shocks and separations. The Carte-sian mesh has been generated with differentstretching functions14 in the two directions.

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−0.2

0

0.2

0.4

x/xc

y/x c

M∞

p∞

T∞

Mjet

, pjet

, Tjet

Figure 2. Computational region and grid. Forclarity, only one-half the gridlines in each di-rection have been shown.

At the inflow boundary, flow is supersonicand uniform at the given Mach number M∞.All quantities were specified. The turbulentkinetic energy follows from assuming the tur-bulence level Tu = (2/3k∞)1/2/u∞ to be 0.005as in Rizzetta4. Specific dissipation ω wasobtained by equating molecular and eddy vis-cosities. At the upper and right boundariesoutflow conditions (extrapolation) were used.The wall (y = 0) is a no-slip, adiabatic bound-ary except at the slot where sonic conditions areprescribed. Pressure was extrapolated from theinterior. Turbulent kinetic energy was set tozero at the wall, and the hydraulically-smooth-surface condition was used to calculate ω. Atthe slot, velocity components u and v, pressureand temperature were specified at the appropri-ate sonic condition. The values k = 100 m2/s2

and ω = 5 × 105 s−1 were also prescribed.(Gerlinger et al.7 have recommended that avalue for ω be prescribed rather than taking itsslot-normal derivative to be zero.) The inflowboundary was located slightly upstream of theplate leading edge and the boundary layer wasallowed to grow naturally. Symmetry condi-tions were used on y = 0 between the inflowplane and the plate. From a uniform state atinflow conditions, the flow is allowed to developuntil a steady solution is obtained.

General features of the computations

Since the solution is of a steady flow, thesize of the time derivative term is an indicatorof whether the required solution has been ob-tained or not. Here, closeness to steady statehas been measured by calculating changes be-tween successive steps divided by the change atthe initial step. For each conserved variable f ,the error

ε(f) =‖fn+1 − fn‖‖f1 − f0‖

was monitored. Both the L∞ and L1 norms

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over all computational cells have been used. Atypical evolution of the error for a calculationusing the Roe scheme and k-ω model is shownin Fig. 3. The sudden increase after about70,000 steps corresponds to the step at whichlimiters on k and ω were removed. These sim-ple limiters were needed to preserve positivityof k and ω during initial steps.

2 4 6 8 10 12

x 104

10−8

10−6

10−4

10−2

100

Number of iterations

ε ∞

Figure 3. Scaled residue history in L∞.——: ε(ρ); – – –: ε(ρE).

Grid resolution was assessed by comparingsolutions at three levels of refinements in boththe directions. The flow is from Aso et al.2 (case8 of Table 1.). At successive refinements therewere 180, 240 and 360 cells in the streamwisedirection, and 60, 120 and 180 cells in the wall-normal direction. The resulting wall pressuredistributions are shown in Fig. 4. It can be in-ferred that a grid of about 240×120 points is ad-equate, since the improvements with additionalcells are not significant. Grid refinement in thestreamwise direction affected the upstream so-lution only, whereas wall-normal refinement af-fected the solution everywhere. The grid cho-sen for most of simulations had 240× 120 cells.There were 23 points in the slot, and the dis-tance in wall units y+ to the first cell centerabove the wall was about 0.5.

Several features of the flow field can be iden-tified clearly by examining Mach number con-tours (Fig. 5.). The expanding jet, Mach diskwith a downstream subsonic pocket, bow, sepa-ration and reattachment shocks are visible. Thejet boundary up to the Mach disk, and the sep-aration zones (PUV, SUV, PDV, SDV sketchedin Fig. 1) can be seen. Between the sonic sur-face and the wall lie subsonic regions. So itis necessary to predict the growth of the initialexpanding flow of the jet, and separation zones,

accurately to get a good prediction of wall pres-sures which is often the only quantity measuredin experiments.

(a)

0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

(b)

0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure. 4. Effect of grid refinement, case 8.(a) Along the wall, – – –: 180 × 120;– · – · –: 240 × 120; ——: 360 × 120.(b) Normal to the wall, – – –: 240 × 60;– · – · –: 240 × 120; ——: 240 × 180.

Figure 5. Mach number distribution, case 7

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Comparisons with previous results

Wall pressure data from several other in-vestigations are compared with the presentsimulation of case 8 in Fig.6. They include thek-ε solutions of Rizzetta4 and Chenault andBeran6, and the RSTM solution of Chenaultand Beran6. The q-ω data of Gerlinger et al.7

was close to Rizzetta4 and is not shown. Thesolution of Rizzetta4 is quite different from thatof Chenault and Beran6 who have pointed outsome reasons for the differences. The presentsolution is quite close to the RSTM solutioneverywhere, both upstream and downstream.The RSTM solution does not pick up the pres-sure spike (seen in other simulations and in theexperiment). However, all simulations showa sharper pressure rise than the experiment,and differ from the experiment by about thesame amount in the pressure level. We mayconclude that the present simulation predictsthe same flow as other simulations. The flow inthe experiment may have been transitional—aqualitatively different flow.

0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure 6. Comparison with simulations of otherinvestigators. ——: present simulation; +++:RSTM; – – –: k − ε Chenault and Beran6; – ·– · –: k−ε Rizzetta4; •••: Experiment, case 20.

Wall pressure distributions

A comparison of wall pressure distributionsover the full pressure range is presnted in theFigs. 7 and 8. Generally, the predictions at thelower injection pressures are very good. ForM∞ = 2.61 and pressure ratio of 46, thereis about a 10% difference between predictionand experiment in the length of the upstream

separation zone and a similar difference in theplateau pressure level.

(a)

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

(b)

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

(c)

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure 7. Wall pressures for Spaid and Zukoski1

experiments at M∞ = 2.61 and pjet/p∞ =12.75, 24.31, 46.09 in (a), (b) and (c), respec-tively. •••: Experiment; ——–: Computation.

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The performance is quite similar at thehigher Mach number but one may say that thedifferences begin to appear at slightly higherpressure ratios. There is a monotonic fall in ac-curacy with increase in pressure ratio at bothfreestream Mach numbers. The deficiency is inthe prediction of the upstream part. Generally,

the downstream distribution is quite accurate.

All these solutions are summarized in Fig.9which shows the distance to the upstream sep-aration point in all the simulations and in theexperiment. For the experiment, this is the dis-tance at the slot midspan.

(a) (b)

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

(c) (d)

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure 8. Wall pressures for Spaid and Zukoski1 experiments at M∞ = 3.50 and pjet/p∞ = 8.74,17.12, 32.44, 63.50 in (a), (b), (c) and (d), respectively. • • •: Experiment; ——: Computation.

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(a) (b)

0 10 20 30 40 500

50

100

150

200

pjet

/p∞

x sep/w

0 10 20 30 40 50 60 700

50

100

150

200

250

pjet

/p∞

x sep/w

Figure 9. Distance to upstream separation point (a): M∞ = 2.61; (b): M∞ = 3.50.• • •: Experiment; —�—: Computation.

(a) (b)

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure 10. Wall pressures from the lowest and highest injection pressure cases of Spaid and Zukoski1

at M∞ =3.50. (a) pjet/p∞ = 8.74; (b) pjet/p∞ = 63.50. • • •: Experiment −4 − 4 − 4: Chenaultand Beran6 with k-ε model; ——: Present simulation.

Improving predictions

Several conclusions were drawn from anexamination of the comprehensive set of simu-lations of slot injection. First, the Roe schemewas found to be robust and generally accu-rate over the full range. Residue fell continu-ously through many orders. Simulations wereaccurate at the lower injection pressures butdeteriorated at higher pressures. The differ-ences appeared to be quantitative only and notqualitative: the shapes of the computed andmeasured pressure distributions remain similareven as the differences increase. So it seemedreasonable to expect that improved predictions

may be possible. Since simulations of case7 (M∞ = 3.5, pjet/p∞ = 63.5) showed thelargest errors, it is the best case to measurethe possible improvements. Before consideringstrategies for improvement, it is noted that thepresent simulation (k-ω) agrees closely withthat of Chenault and Beran6 (k-ε) as shown inFig. 10.

Grid refinement

From an examination of the computed flowfield, it was clear that at the lower injectionpressures the Mach disk forms at roughly theheight of the incoming boundary layer. The

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poor performance at higher injection pressureis partly due to the lack of resolution in theregion where the Mach disk forms. Althoughinitial grid refinement studies at a fixed pres-sure ratio suggested that the 240 × 120 gridshould be adequate, a finer grid seems neces-sary at the higher injection pressure. Figure 11shows that the 360 × 180 grid results in a smallimprovement.

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure 11. Effect of grid refinement at high in-jection pressure. Case 7 of Table 1.; Roe schemewith standard k–ω model. • • •: ExperimentSpaid and Zukoski1; ——: 240 × 120 grid; – ·– · –: 360 × 180.

Low-Reynolds-number correction

Low-Reynolds number versions of the k-ε have been used in literature to account forwall damping. Different wall damping func-tions were used in Rizzetta4, Gerlinger et al.7

and Chenault and Beran6 for slot injection sim-ulations. The presence of recirculation zonessuggests that a low-Reynolds number versionmight be helpful. The version given in Wilcox13

has been used here. The eddy viscosity is writ-ten as µt = α∗ρk/ω, where α∗ = 1 for thestandard k-ω model. The model coefficient βis unchanged, but β∗ and α get modified:

α∗ =α∗o + Ret/Rk

1 + Ret/Rkα =

59α∗

αo + Ret/Rω

1 + Ret/Rω

β∗ =9

1005/18 + (Ret/Rβ)4

1 + (Ret/Rβ)4

with α∗o = β/3, αo = 1/10, Rβ = 8, Rk = 6,Rω = 27/10. The turbulence Reynolds numberRet = ρk/(ωµ). Figure 12 shows that some

improvements are realized by using this low-Reynolds number version on the 360 × 180grid.

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure 12. Effect of turbulence model and or-der of scheme at high injection pressure. Case7 of Table 1.; Roe scheme with k–ω models on360 × 180 grid. • • •: Experiment Spaid andZukoski1; - – –: Second order scheme with stan-dard k–ω model; ——: Second order schemewith low-Reynolds-number k–ω model; – · – · –:Third order scheme with low-Reynolds-numberk–ω model.

Higher order scheme

The simulations presented above have usedthe MUSCL procedure to obtain fluxes ofsecond-order accuracy. We now consider theeffect of increasing the accuracy to third or-der. The limiter was also changed from min-mod to van Albada, which allows variation inthe slopes, because it provides for a slightly lessdissipation consistent with the use of a higher-resolution scheme. With these further changes,the computed solution (Fig. 12) has come muchcloser to the experiment in some ways. The up-stream separation point has moved much closerto that in the experiment. The pressure rise inthe reference solution (Fig. 10(b)) was moregradual and the peak level was higher. Theinitial pressure rise is sharper and as in experi-ment, but there is a further weak increase whichis not found in the experiment. The down-stream distribution degrades slightly, but thedifferences between experiment and simulationremain quite small in all cases anyway.

Further refinement to a very fine grid of580 × 270 points did not show significant im-provement (Fig. 13). The Mach number distri-

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bution in Fig. 5 is this solution on the 580×270grid. All flow features—Mach disk, bow shock,separation shock, recompression shock and sep-arated regions—have been well resolved. Avery clean solution has been obtained.

0.7 0.8 0.9 1 1.1 1.20

1

2

3

4

x/xc

p/p ∞

Figure 13. Effect of further grid refinement.Case 7 of Table 1.; Third-order Roe schemewith low-Reynolds number k–ω models on 360× 180 grid. • • •: Experiment Spaid andZukoski1; – – –: 360× 180 grid; —-: 580× 270grid.

Although a seemingly elementary concept,this study has highlighted the need for ade-quate resolution to obtain good quality solu-tions. The resolution and the locations where itis most needed changes with injection pressureratio. Clearly the divergence in solution accu-racy was not due to external factors such asthe approximation of the non-uniform injectorvelocity profile, or a large scale unsteadinesswhich would require a Large Eddy Simulationto compute accurately.

CONCLUSIONS

A flow which is considered for validation of2-dimensional high speed turbulent flow com-putations has been simulated over the full rangeof injection pressure ratios for which experi-mental data is available. As had been foundby other investigators, simulations were accu-rate at low pressures and monotonically di-verged with increasing injection pressure. Forthe Spaid & Zukoski1 experiments, noting thatthe differences between computed and mea-sured pressures seem to only quantitative andnot qualitative, a sequence of steps were taken

and improved solutions were obtained. It turnsout that the most important requirement is ad-equate resolution of the flow structure in regionbetween the wall and the Mach disk which lieswell above the incoming boundary layer at highinjection pressures. The differences betweencomputation and experiment, even at the high-est pressures, are now small enough to concludethat, over this entire range, present turbulencemodels are quite sufficient.

References

[1] Spaid, F. W., and Zukoski, E. E. Astudy of the interaction of gases jets fromtransverse slots with supersonic externalflows. AIAA J. Vol. 6, No. 2, 1968, 205–212.

[2] Aso, S., Okuyama, S., Kawai, M., andAndo, Y. Experimental study on the mix-ing phenomena in supersonic flows withslot injection. AIAA Paper 91–0016, 1991.

[3] Aso, S., Okuyama, S., Ando, Y., andFujimori, T. Two-dimensional and three-dimensional mixing flow fields in super-sonic flow induced by injected secondaryflows through traverse slot and circularnozzle. AIAA Paper 93–0489, 1993.

[4] Rizzetta, D. P. Numerical simulation ofslot injection into turbulent supersonic su-personic stream. AIAA J. Vol. 30, No. 10,1992, 2434–2439.

[5] Sarkar, S., Erlebacher, G., Hus-saini, M. Y., and Kreiss, H. O. Theanalysis and modeling of dilatational termsin compressible turbulence. NASA CR-181959, 1989.

[6] Chenault, C., and Beran, P. S. k-ε and Reynolds stress turbulence modelcomparisons for two-dimensional flows.AIAA J. Vol. 36, No. 8, 1998, 1401–1412.

[7] Gerlinger, P., Algermissen, J., andBruggemann, D. Numerical simula-tion of mixing for turbulent slot injection.AIAA J. Vol. 34, No. 1, 1996, 73–8.

[8] Weidner, E. H., and Drummond, J. P.Numerical study of staged fuel injection forsupersonic combustion. AIAA J. Vol. 20,No. 10, 1982, 1426–1431.

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[9] Burrows, M., and Kurkov, A. P. An-alytical and experimental study of super-sonic combustion of hydrogen in a vitiatedairstream. NASA TM-X–2828, 1973.

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