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MASTERTITLE. CO,%tPL'TATION OF HIGH REYNOLDS NU.EER INTERNAL/ EXTERNAL FLOWS

AUTHOR(S): .. C. Cline and R. G. Wilmnth

SSUBMITTED TO

AIAA -14th Fluid and Plasma DvnaTrics Conference,

Palo Alto, CA, 'rune 22-23, 1981

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COMPUTATION OF HIGH REYNOLDS NUMBER INTERNAL/EXTERNAL FLOWS

by

Michae! CllueLos Alamos Nati-nal Laboratory

Los Alamos, NM 87545

and

Richard G. WilmothNASA :.anglev Research Center

Hampton, VA 23665

Abstract

A general, user oriented computer program,

called VNAP2, has been developed to calculate highRemolds number, internal/external flows. VNAP2solves the two-dimensional, time-dependent Navier-Stokes equations. The turbulence is modeled ritheither a mixing-length, a one transport equation, ora two transport equation. model. Interior gridonints are computed using the explicit MacCormackscheme with special procedures to speed up the cal-culation Li, the fine grid. All boundary conditionsare calculated using a reference tlane chara,teris-ttc scheme with the viscous terns treated as sourceterms. Several Internal, external, and internal/ex-ternai flow calculations are presented.

Introduction

The computation of high ,eyrolds number flowshas become a major tool In the analvsts and designof aerosvace vehicles. While 9avier-Stokes solu-tions for complete vehicle configurations are stillbeyond the limits of present-day computers, corputs-tlonal techni q ues are used routi pely in the analysisand design of vsrious individual components, e.gairfoils, wing-body combinarious, inlets and noz-zles. Most of these analvses use either purely in-viscid or the so-called patched viscous-invlscldtechniques due to theit greater comput r.tional efft-ciencv and ease of use uver the more exact Navier-Stokes solution methods. These approximate tech-niques often vteld results of sur ,.risingly high ac-

curacv. I However, their use is generally limited to

problems involving weak viscous-in^iscid tnterzc-tions or to strong interacrtons that are suffletent-ly well understood to be modeled empirically.

Navier-Stokes (N-S) solution methods, on theother band, are not subject to these fur..:amentalliml:ationa and are applicable to more generalclasses of flow prc:,lems. However, the N-S methodshave .sot found widespread use for design purposesdue primarily to their expense and difficulty ofuse Computer run tiers of several hocrs a:z notuncommon in so^/ing a high-Remolds numoer proble.in which the viscous laPer must be well resolved.Furtherm. re, application of N-S methods is oftenviewed by the engineer sa an arr requiring extenriveknowledge of the numerical algorithm and considera-ble trial and error t:) obtain a correct, convergedsolution. The latter is often the result of at-tempting to use a N-S computer code which has beenwritten to solve a very specific class of problems

Croup T-i, Member AIAA

*A Propulsion Aerod ynamics RranchThis paper is declared a work of the U.S. Governmentand therefore '.s in the public domain.

end ma y not be sufficientl y general, yell-document-ed, and user-oriented. Clearly, as more efficientN-S algorithms and larger, faster computers becomeavailable, mora attention mist be given to the de-velopment of user-oriented N-S codes if the y are to

receive practical application.

The purpose of this paver is to describe a N-S

convuter program, VNAP2, 2 which has evolved over a

period of several sears for soling a relativelyvldo class of steadv and unstead y , internal and ex-

ternal flow problems. VNAP2 is a acdlfied version

of V4AP 1 and solves the two-dimensional (axJsvmm^c

ric), time-dependent, compressible Navier-Stokesequations. Both sit.gle and dual flowing strears may

be solved. The flow bounda , ies may be arbitrary

curved solid walls, inflow/outflow boundar.es, orfree-)et enveiopes. Turbulent as well as laminarand inviscid flows may be tread. Some typical in-ternal end external flow geometries that ma y besolveu are shown in Pig. 1. Although the ^NAP2 codehas been applied mainly to nozzle and inlet fl^vs,the relatively general treatment of geometries andflow houndaries allows a variety of 3ther p^oblemsto be solved, a.g , airfoils, flow-though nacelles,free-shear flows and free- jet expansions.

In this paper, the methodology used in develop-ing VRAr2 as a user-orienced, production-type compu-ter program is presented and score results obtainedin solving a variety of flow problems are snows.Although the results, in some case:, point out theneed for further improvements in both numerical andphysical modeling, the y do illustrate, tc a greatextent, that practical Wavier-Stokes applicationsare possible.

Gcverni Jg Equations

The VNAP2 code solves the two-dimensional tax-isvmmetric), time-dependent , Navier-Stokes equA-tions. The turbulence 1s modeled using either tnixing-length, a one transport equation or a rwotransport equrtion model The mixing-length model

employs the Lrunder et al" morel for free shear lay-

ere _^a the Cebecl-Smith model for boundary laypro.

The one equation model is tha, of Daly 6 while the

two equation model is the Jones-Launder 7-10

model.

For details of there turbulence models, including

boundary conditions, see Ref.2. VNAP2 employs anexplicit artificial viscosity to stabilize the cal-culations for anock waves. This artifit*ial viscosi-ty is used in place of the fourth-order smoothing

usually employed by MacCormack. 11 For details of the

governing equations see Re.'. 2.

Physical ana :omoutatiunal Flow Spaces.

The physical flow space geometr y is shown InFig. 2. The rlw is from left to right. The upoerboundar y , called the wall, can be either a aolidboundary, a free jet boundary , or an arbitrary sub-sonic (notsal to the bounda:v) inflow/ourflcv brntnd-jrv. The lover boundary , called the centerbody, canbe either a solid boundary or a plare (line) of svm-metrv. The geometry can be either a single f2 wingstream or, if the dual 'low s pace --ells are present,a dual °lowing stream. lne dual flow space walls,shown in Fig. 2, may begin in the interior and con-tLnue to the exit (inlet geometry), mav begin at theinlet and terminate in the interior (afterbody geom-etry), as shown iii Fig. 2, or mav begin and end inthe interior (airfoil gaometry). All of the aboveboundaries may be Arbitrary curved boundaries pro-vided -he v coordinate is a sing_t value function ofX. Th_s single value function of x requires dualflow space walls, that begin or terminate in the in-tericr, do so with pointed ends. The points can bevery blunt, but cannot be vertical walls The leftboundary is a subso n ic, suverson'c, or mixes iuilowboundary while the right boundar y is a subsonic, su-perscnic, or alxed outflow boundary or a subsonicinflow boundary.

The phy sical space grid has the following orop-erties: one set of grid lines are straight and inthe y direction with arbitrary spacing in the x di-rection; the second set of grid lines approximatelyfollow the +all and centerbody contours; the Ayspacing of these grid lines is arbitrary at one xlocation and Ls proportional to those values at anyother x lo=ation.

The x, v physical space is mapped into a rec-tangular ;, n computational space ar shown in Fig.2. The mapping is carried out in two parts - thefirst part aml_ the physical space to a rectangularcomoutational space while the second maps the varia-ble grid _omputational space to a uuiform grid cosyputatlonal space. Both the upper a,id lower dualflow space walls colla pse to the the same grid linein the computational space, as shown in Fig. 2. Theflaw variables at C,e g.-id points on the upper dualflow space wall mre stored :n the regular solutionarra y while the v-riables st the lower dual flowspace wsl: are stored in a dummy array. These flowvariables are coatlnually switched between these towarrays during the calculation. P-r details of thetransformations, see Ref. 2.

Numerical Method

The computational plane grid points are dividedup Into interior and boundary points. The boundarygrid points are further divided up into left bounds-.v, right boundary, wall, centerbody, tad dual flowsome wall points (see Fig. 2).

Interior Grid Points

The interior grid points are computed using the

unsplit MacCormack sclheme. 12 The governing equa-tions are left in nonconservation form. In order toimprove the computational efficiency for highRe ynolds number flows, the grid points in fine partof the grid may be subcvcled. This is uccomp_ishedby first computing the grid points in the coarsepart of the grid for one time step at. "text, thegrid points in the fine grid are calculated k times,where k is an integer, with a time step At/k. The

grid points at the edge of the fine Crid r-quire aspecial procedure, because one of their neighboringpoints is calculated as part of the coarse grid.Except far the first subcvcled time step, this pointis unknown. However, the values at t and t+At areknown from the coarse grid solution and, so, thevalues between t and t+at are determined by linearinterpolation.

In order to further improve the computations.efficiency, a special procedure is employed to in-crease the allowable time step in the subcycled par,of the grid. This procedure allows the removal ofthe sound speed from th_ time step C-F-L condition.Procedures that accomplished this hcve been proposed

by Harlow and Aw den 13 and MacCormack. 14 The proce-dure of 9arlw, and Amsden is an implicit scheme thatremoves the sound speed, in both the x and y direc-tions, by an implicit trea-ment of 'he mass equationand the pressure gradient terms in the momentumequations. MacCormack'u procedure is explicit andremoves the sound speed in only one direction.(MacCormack's procedure also includes an implicitprocedure to remove the viscous dlffusi,,n restric-tion from the time step C-F-L condition.) Becauseexplicit schemes art easier to program for efficientcomputation on v-ctor computers and becaus- highRe ynolds number flows usually required fine gridspacing in only one direction, it was decided to usea procedure aimilar to the' of MacCormack.

MacCormack's procedure is based on the assump-t'-on that the velocity component, in the coordinatedi.-ction with the fine grid spacing, is negligiblecompared to the sound speed. This allows the gov-e_'ning equations to be simplified. MacCormack thenapplies the Method of Characteristics to these sim-plified equations. However, for flwc over bodieswith large amounts of curvcture as well as many freesh_ r flows this assumption is quests — able. Be-cause VNAP2 is intended to be a gen-7-il code forsolving a vide variety of problems, this assumptionwas felt to be too restrictive. Therefore, the maindifference between MacCormack's scheme and the onepresented below is the removal of ,.his restriction.This procedure, however, does asaume that the tiow,in the y direction is subsonic.

The procedure here is to separate the governingequations into tvo parts. The first part consistsof the Mach Line characteristic compatibility equa-tions while the streamline compatibility equationsand all viscous terms make up the secor.i part.Because the sound speed limitation is due to thefirst part, the se,:ond part can be computed by thestandard MacCormack scheme. The first part of thegoverning equations is -olved by a special proce-dure. This special procedure consists of selectingan increased time step based on ay/lvl instead ofAy/i vj+a) where v is the velocity component in they d,:ectior. and a is the sound speed. The Mach linecharactertstics are then extended back from the so-lution print at the increased time step. Thesecharacteilstics will intersect the previous solutionplane outside the computation stet of the MacCormackscheme. By calculating these characteristic inter-secting points, special differenceE using this in-creased domain of dependence can be determined andused in the first part of the governing equations.These special differences use a larger Ay than theMacCormack scheme and, therefore, ire lesb accurate.However, this procedure is only used in boundary orfree shear layers where the viscous terms dominatethe solution. This special procedure decreases the

as inviscid flows than VNAP? can solve more accu-

rately and with significantly less amounts of compu-

ter time All zases were rur rith the unmolifiedVNAP2 code utilizing o,ly s +mall data file. Atthis time, very few pLrametric studies to determinethe optimum turbulence model parameters, ini-ial-data st,rface quantities and grid point distribu-tions, have been carried out. ks a resuJt, the ac-curacv and efficiency of these results do not neces-sarily rerresent the optimal use of the VNAP2 code.

Internal Flow

The internal flow case is nozzle B-3 of Ref. 19and is the planar, converging-diverging nozzle shownin Fig. 4. The flow is from left to right with the

phy sical space grid enclosed by the dashed line.The Revnolds number based on the throat height is

7.7 w 10 5 . At the left boundary, p is set equal .:o

200.6 kPa (29.1 psla), T T is set equa. to 300 R and

9 is set equal to 0. At the right boundary - extra-poiation is used when the flow is supersonic. Wherthe flow +i subsonic outflow, p is set equal to101.4 kFa (14.7 psis). For subsonic inflow, in ad-dition to specifying p, v is set equal to 0 and p isset equal to the average of the Fall and midplanevalues. The wall is a no-slip boundary and the re-

sults presenced i.ere emp loyed the tw, equation tur-bulence T^Sel. The initial conditions consisted of1-D, inviscid flow that wns sonic at the throat end

subsonic downstream.

The physical space grid, Mach number and turbu-lence energy con^ours .re shown in Fig. 5. The et-per'mental data are from Ref. l y . From Fig. 5, wo

see that at this pressure ratio, the flow separatedcreating a reverse flow region. The wall and mid-plane pressures are shown it Fig. 6. This calcula-tion used a 45 by 21 grid and required 3000 timesteps (30,000 subcycled time steps) and 2.1 hours ofcpu time (CDC-7600) to reach steadv state.

^_xternal Flow

The external flow cases are conflgs. 1 and 3 ofRef. 20 and are the ax:syaimetric, boattail afterbodyflows, with solid bodies simulating the exhaust jet,shown in Fig. 7. TFe flow is from left to rightwith the physical space grid enclosed by the dashedline. The Reynolds number based on x at the left

boundary is 1.05 x 107 . The first case (L - 27.0

cm) is config. 3 while the second (L - 12.2 cm) isconfig. 1. Both cases consisted of the same flowconditions. The left boundary inflow profiles of pT

and TT for a free stream Mach nuaoer of 0.8 were de-

termined using the same inviscid/boundary layer pro-cedure ew-loyed by Ref. 21. The flow angle d wLsset equal to 0. At the right boundary p was setequal to the free stream value. The wall is an ar-bitrary inflow/outflow boundary . For outflow, p isset equal to the free stream value. When inflow oc-

cu r s, in addition to specifying p, u and p are setequal to their free stream values. The centerbody

is a :o-slip boundary . Both - ;.culstions employed

the extended interval time smoothing. The initialconditions consisted of extending the inflow pro-files downstream to the right boundary.

The physical space grid, pressure and Mach num-ber contours for config. 3 (L - 27.0 cm), employingthe mixing-length turbulence model, are shown in

Fig. d while t?a surface pressure is shown in Fig.

9. T'.e phveical space grid, Mach number and turbu-lence energy contours for config. 1 (L - 12.2 cm),employing the two equation turbulence model, areshorn in Fig. 10 while the ,urface pressure is shownin Fi; ,.. 11. The experimental data, for both cases,are ft-)m Ref. 20. From these figures, we see thatthe flow for config. 3 remained attached while s:!pa-ration uccured for config. 1. The mixing-lengthmodel produced slightly better results for confi,,t.3. However, :he two equation model more accuratelypredictee she )ressure plateau 'n config. 1, but un-derpredicted the amount of upstream expansion.

Swanson` found that a relaxation or lag model im-

proved the pressure plateau prediction of the mix-ing-length model, but at the erpense of also under-predicting the amount of upstream expansion. Tneconfig. 3 calculation, employing the sdxing-lengthmodel, used a 40 by 25 grid and required 75 1) timesteps (15,000 subcycled time steps) and 1.0 hours ofCPC time (CDC-7600) to reach steady state. Theconfig. 1 calculation, employing the mixing-lengthmodel, used z. 47 tT 29 grid and reauired 750 ti-sesteps (17,000 subcycled time steps) and 2.1 hours ofCPU time (CDC-760(j) to reach stead; state. The twoequation model computational times were 1.4 hoursfor config. 3 and 3.7 hours for config. 1.

Internal/External Flow

The first two cases are the two external flowcases presented above, lut with the solid simulatnrareplaced by the exhcust jets. The geometry is ehownin Fig. 12 with the physical space grid enclosed bythe dashed 'lines. The left external boundary, wailand r1!ht boundary are the same as the externalcases. At the left internal boundary, PT is set

equal to :32.4 kPa (19.2 psia), T T is set equal tr

300 R and 9 is set equal to 0. The free streampressure I s 65.2 kPa (9.45 psi&). The centerbody isthe flow centerline, while the dual flow space wallsare no-slip boundaries. Again, both calculatio-.aemployed the extended interval time smoothing. Onlythe two equation turbulence model was employed forthese tao cases. The initial conditions for thesecases consisted of the external flow solutions pre-sented above along with the 1-D, inviscid flow solu-tion for the nozzle.

The physical space grid and Mach number con-tours for config. 3 (L ^ 27.0 cm) are shown in Fig.13. The external surface pressure is shown in Fig.14 while the total pressure profiles for the shearlayer, produced by the interaction between the ex-haust jet unto the external flow, are shown in Fig.15. The physical space grad and Mach number con-tours for config. 1 (L - 12.2 cm) are shown in Fig.16. The external surface pressure is shmm in Fig.17 while the total pressure profiles for the shearlayer are shown in Fig. 18. The experimental data,for both cases, are trom Refs. 20 and 22. FromFigs. 14 and 17, we see that the computed solutions,for both cases, underpredicted the shear layerspreading rate. The same trends were fo• -nd in re-

sults generated by i patched method. 23 The config.3

calculation used a 40 by 38 grid and required 300time steps (40,000 subcycled time steps) and 5.6hours of CPU time to reach steady state. The con-fig. 1 calculation used a 47 by 42 grid and required300 time steps (40,000 subcycled time steps) and9.0 hours of CPU time (CDC-7600) to reach steadystate.

4

The ldst rase is the NACA 1-69-100 inlet ofRef. 24 and is =shown in Fig. 19. ine flow is fromLeft to right with the physical space grid enclosedby the dashed line. The Revnolda number based on

more efficient solution algorithms for ver y nonuni-form grid point distributions.

Acknowledgment

the maximum external diameter is 6.1 n 10 f, . At theleft boundary , PT was set equal to 101.4 kPa (14.7

psia), TT was set equal to 294.4 K And 9 was set

equal to 0. The free stream pressure is 66.5 Us(9.64 psia) which produces a free stream Mach numherof 0.8. The wall is the arbitrar y inflow/outflowboundary and uses the same boundary conditions asthe crevious two cases At the right externalboundary, the pressure was set equal to the measuredvalues. In the experiments of Ref. 24, the internalflat rate war controll1 d by a throttling mechanismwell downstream of the right boundary In order notto compute this rather extensive flow region, avdlue of pressure was specified at thr- righr inter-nal houndary suc'. that the 'nviscid, 1-D mass flowequalled the experimental value. The centerbody isthe flow centerline, while the dual flow space wallsare no-slip b- idarles. The extended interval timesmoothing was employed. The initial conditions cor,-sisted of 1-1), inviscid floc.

,he nhysicaJ space grid, Mach number and turbu-lence energy contours are shown in Pig. 20, whilethe surface pressure is shown in Fig. 21. The ex-perimental data are from Ref. 24. Prom the turbu-lence energy contours in ?ig. 20, we see that theflow over the inlet is initially laminar, but quick-lv becomes turbulent. The flow transitions first onthe interior surface. deference 24 did not give thetransition locations. The surface pressures in Rig.21, are for the first FZ of the inlet. This smallarea of interest, combined with the thin laminarbounddry layer, required a very fine grid spacing inboth coordinate directions. Because of the finegrid spacing in the a dlrecLion, the subcyling op-tion was not used The differences in surface pres-sure, between theory and experiment, at the inlettip are probably the result of too coarse a grid.The difference, between theory and experiment, nearthe internal, right boundary is most likely due tothe approximate treatment of the internal flow atthis boundary. This calculation was also made usingthe mixing-length turbulence model, however, thismodel significantly overpredicts the level of turbu-lence upstream of the inlet. This is because thedownstream blockage due to the presence of 'he inletcreates a weak shear layer profile upstream of theinlet that extends a large distance in the crossstream direction. Therefore, the mixing length mod-el predict. very large mixing lengths. This pro-duces turbulent viscosities, upstream of the inlet.,that are several orders of magnitude larger than themolecular value.As a result, the mixing-length modelsolution 1.s not presented here. This calculat_onused a 53 by 44 grid and required 3000 time stepsand 1.0 hours of CPU time (CDC-7600) to reach steadystag.

Conclusions

A general, user oriented computer program forcomputing high Revnolds number flows has been pre-sented. Six high Reynolds number flow calculationswere described. These computed results show thatpractical Xavier-Stokes applications are possible.However, these results also indicate the need forbetter turbulence modeling for separated flows and

This work was supported by the Propulsion Aero-dynamics Branch u.° the NASA Langley Research Center(LRC) and the U.S. Department of Energy. The ai-

thor g wood like to thank R. C. Swanson a..d L. E.Putnam of the LRC and B. J. Daly of the Lo g AlamosNational Laboratory for many ' • elpful discussions.

References

1. "Computation of Viscous-In-iscid Interaction,AGARD Symposium, Colorado .prings, CO, Sept.29-Oct 1, 1980, AGARD-CP-291

2. Cline, M. C., "VNAP2: A Computer Program forComputation of Two-Dimension&:., Time-Dependent,Compressible, Turbulent Flow," Los Alamos Na-tional Laboratory report LA-R871, Aug. L981.

3. Cline, M. C., "VNAP: A Computer Program fnrComputation of Two-Dimensional, Time-DepenatuL,Compressible, Viscous, Internal Flows,' LosAlamos National Laboratory report LA-7326,Nov. 1978.

4. Launder, B. E., Morse, A., Rodi, W., snd Spald-ing, D. B., "The Prediction of Free Shear Flows- A Comparison of the Performance of Six Tu.bu-lence Models," NASA SP-321, Vol. I, pp. 361-426, 1973.

5. Cebecl, T , Smith, A. M. 0., and Moslnskis. G.,"Calculation of Compressible Ad!abatic Turbu-lent Boundary Layers," AIAA J. R, pp. 1974-1982, Nov. 1970.

6. Daly, B. J., unpubl'shed notes, Los Alamos Na-tional Laboratory, 1972.

7. Jones, W. P. and Launder, B. E., "The Predic-tion of Lamrinarization With a Two-Equation Mod-el of Turbulence," Int. J. Heat Mass Transfer15, pp. 301-314, 1972.

8. Jones, W. P. and Launder, B. E., "The Calcula-tion of Low-Revnolds-Number Phenomena with aTwo-Equation Model of Turbulence," Inc. J. HeacMass Transfer 16, pp. 1119-1130, 1973.

9. Launder, B. E. and Sharma, B. I., "Applicationof the Energy-Dissipation Model of Turbulenceto the Calculation of Plow Near a SpinningDisc," Letters Heat lass Transfer 1, pp. 131-137, 1974.

10. Han)alic, K. and Launder, B. E., "Contributionrowards a Re ynolds-Stress Closure for Low-Reynolds-Number Turbulence," J. Fluid Mech. 74,pp. 593-610, 1876.

11. Hung, C. M. and MacCorstack, R. W., "NumericalSolutions of Supersonic and H9Personic LaminarCompression Corner Flows," AIAA J. 14, pp. 475-481, April 1976.

12. MacCormack, R. W., "The Effect of Viscosity inhypervelocity Impact Cratering," AIAA Paper 69-354, Cincinnati, Ohio, April 1969.

19. Mason, M. L., Putnam, L. E. and Re, R. J., "ne

Effect of Throat Contouring on Two-Dimensional,Converging-Diverging Nozzles at Static Condi-tions,' NASA TP-1704, Aug. 1980.

20. Reubush, D. E., "Experimental Study of the Ef-fectiveness of Cylindrical Plume Simulrtors forPredicting Jet-On Boattail Drag at Mach Nt-mbersup to 1.30," NASA TN D-7795, 1974.

21. Swanson, R. C., "Numerical Solutions of theNavier-Stokes Equations for Transonic AfterbodyFlows," NASA-7P-1784, Dec. 1980.

22. Mason, M. L. and Putnam, L. E., "Pitot PressureMeasurements In Flow Fields Behind Circular-ArcNoz-ie9 With Exhaust Jets at Subsonic Free-Stream Mach Numbers," NASA TM-80169, Dec. 1979.

23. Vilmoth, ct. G., "Viscous-Inviscid Calculationsof Jet Entrainment Effects on the Subsonic FlowOver Nozzle Afterbodies," NASA-TP-1626, April1980.

24. Re, R. J., "An Investigation of Several NACA 1-Series Inlets at Mach Numbers From 0.4 to 1.29for Mass-Flow Ratios Near 1.0," NASA TM R-3324,Dec. 1975.

13

Harlot, F. H. and Amsden, A. .., "A Numerical

Fluid Dvuamics Cr:culation Method for Ail FlowSpeeds," J. of (,omp. Phys. 8, o)• :47 - 213,Oct. 1971.

14 MacCormack, F. u., "A Rapid Solver fcr Hyper-bolic Systems of Equations," Lecture Notes inPhysics 59, pp. 307-317, 1976.

15

Serra, R. A., "Determination. of Internal GasFlows by a Transient Numerical Technique," AIMJ. 10, 603-611, May 1972.

16

1liger, J. and Sundstrom, A. "Theoretical and?ractical Aspects of Some Initial BoundaryValue Problems in Fluid Dynamics,' SIA11 J. ofApplied Mathematics 35, pp. 419-445, Nov. 19-8.

17

Moretti, G. and Abbett, M., "A Time-DependentComputational Method for Blunt Body Flows,"AIM J. 4, pp. 2136-2141, Dec. 1966.

18

Rudy, D. H. and S=ri'cwerda, J. C., "A Nonre-flecting Outflow Boundary Condition for Subson-ic Navier-Stokes Calculatione, - J. of Comp.Phys. 36, pp. 55-70, June 1980.

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INTERNAL

SINGLE FLOWS

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NOZZLE

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Fig. 1, Typical internal, external and internal/external geometries.

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Upper Dual Flow Sdoca Wali

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Fig. 4. Internal flow geometry.

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Fig. 2. Physical and computational spac,^s.

MACH NUMBER

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iFig. 5. Internal flow physical space grad,

Mach number and turbulence Pnergv

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0.98

0 250 500 750 1000 1250

TIME STEPS

Fig. 3. Pressure ratio vs time step forsubsonic flow in a converging duct.

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0.8

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Fig. 7. Exr.ernal flow geometry.

06

0.4 -

Y/L0.2

^MIOPLANE

0 --

- TWO EQUATION 141100^

_XPERIMENT

0.e I

0.6-P/rTJ

0.4-

0.2 r

?0.4 .0.2 0 0.2 0.4 0.6 0.8 1.0X/L

Fib. 6. internal flow wall (top) and midplane Fig. 8. Externsl flow (config. 3) physical

pressure ratio ( L-5.96 cm). space grid, pressure and Ma ^ h number

contours.

8

MACH RIMER

OA -- ---^ (1.4 -- - ----

02O 2 0----0 0_

Co 0 — _ 0 — — _- -- —.

O cI

0.2 — M11N0-LENOTH MID L -0.2- TWO EOUA"ION MOM-'

O ExPEWENT I — M1:IiG-LENGTH MODELTWO EOTION MODEL

O.A .0.^W

Q O EXPE11YENTI

I_ I I _ I 1 I I I-

I I I0 0.4 at 1.2 1.6 2.0 2.t2.0 32 3.6

1 /0

Fig. 9. Ex t ernal flow (config. 3) surface

pressure coefficient (D-15.2' cm).

L, I I L-1 ' I ___ I -. 1 _J

0 02 nA OF 08 1.0 12 1.1 16 1.8

1io

Fig. 11. External flow (config.. l; Purface

pressure coefficient (D-15 :4 cm)

I ^

I '

121.9 cm ^I 762tm ^

16.^"QT811cm------_-

Fig 12. Internal /externat flow afterbodvgeometry.

TURBULENCE ENERGY

MACH NUMBIA

Fig. 10. External flow (config. 1) physicalspace grid Mach number and turbulence Fig. 13. Internal/external flow (config. 3)

energy contours. physical space grid and Mach number.

9

MACH "IUFIDER

0.4 - - --T _-_^ - T-4- -r- - -r-;-T -T

X /r,.0.13 X/r,+2.13 X/ri•4.13

0.2 O i 2,0 -

Cp 0 . C

02 0I 1.6I

O U iI

• 0.4 r, .,j

0.8 0^- 0

0 0.. CA 1.2 1.6 2.0 0.4

X/D

0.4 0.8 1.2 1.6 u8 1.2 1.6 O .d 11 1.6

<ig. 14. Interne.!! .vernal floe(config. 3) external

Pr/Pt•

surfa,e pressure

coefficient ( D-15.24 cm). Fig. 15. Internal / ex: ernal flow ( config. i) to ,-. t,s;eure -atioprofiles in the shear layer ( x is meas,.re, from theexit and r,.3.81 ca).

°ig. 16. "eternal/external flow (config. 1) physical space grid and Mach number

0.4

0.2 -2.0

1.6

1.2

r/ri

0.8

G.4

00.4

X/rj•0.1 X/r, - 2.0 X/rJ.4.0

OO

O D0 0

o° ° °/0 a r00

lkc^O °8

R 1 1) 1 t

O0

I t

0000co 0

0-D2-

0 Co

•0.4 - 0'A171AA,

-0.6 -

^............

'D 02 . 14 0.6 0.8 1.0X/D

Fig. 11. Internal/external flow :ig. 18(config. 1) external96rface pressurecoefficient (D-15.24 cm).

0.8 1.2 1.6 0.8 1.2 1.6 0.8 2 1.6PT /P T -

Internal/external flow (cunfig. 1) total pressure-atio profiles in the shear layer (x is measuredfrom the exit and r,.3.81 ca).

10

1.0

I III ^

I ^I ab7cm

I ^ I

45.72 cm ---^I

— 48.72 cm --+1

I I I! - t-----^-- --- - - ---40 .9 co

inACH YUMI?R

Fig. 19. Interral/external flow inset geometry,

0.8 \^^Ps/PT^

"XPTm

-0.4 — TWO EQUATION MODEL0 O O EXTERNAL COWLI F XPER

O INTERNAL COWL `

TURBULENCE: ENERGY

0.2 1

0 ^— —L0 0.01 0.02 0.03 0.04 0.05 0.06

X/L

F. 21. Internal/external flow (inlet) surfacepres • ., r o 1 L-45 72 cm) .

Fig. 20. Internal/external flow (inlet) physicalspace grid, Mach number and turbulenceenei gy .