111 the SSME staged combustion cycle, the fuel is partially burned at very high pressure and relatively high temperature in the preburners. The resulting hot gas ...
I .
NASA Technical Memorandum 1 O O d
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Three-Dimensional Incompressible Navier-Stokes Computations of Internal Flows
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I
I
D. Kwak, J. L. C. Chang, S. E. Rogers, and M. Rosenfeld
March 1988 (UA SA-!fl¶-lOtW76) T H R EB-DIB ENSIOU AL INCORPRESSIBLE UAVIER-SII)KES COHPUTATXOYS OF IUTERIAL FLOUS [WASA) 1 4 p CSCL O l C
I88-1 9425
Unclas 63/03
NationalAeronautics and Space Administratii
0131798
NASA Technical Memorandum100076
Three-Dimensional Incompressible Navier-Stokes Computations of Internal Flows D. Kwak, Ames Research Center, Moffett Field, California J. L. C. Chang, Rocketdyne, Rockwell International, Canoga Park, California S. E. Rogers, Sterling Federal Systems, Palo Alto, California M. Rosenfeld, Ames Research Center, Moffett Field, California
March 1988
National Aeronautics and Space Administration Ames Research Center Moffett Field, California 94035
THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES COMPUTATIONS OF INTERNAL FLOWS
D. Kwak,* 3 . L. C. Cliang,t S. E. Rogers,: and M. Roseiifeld$ Allies Research Center
SUMMARY Several incoiiipressible Navier-Stokes solution methods for obtaining steady and unsteady solutions are discussed. Special attention is given to internal flows which iiivolve distinctively different features froiii external flows. The characteristics of tlie flow solvers employing tlie method of pseudocoiiipressil~ilityand a fractional step method are briefly described. The present discussion is limit ed to a primitive variable foriiiulatioii in generalized curvilinear coordinates. ('omputed
results include simple test cases a n d internal flow i n the Space Shuttle main engine hot-gas manifold.
INTRODUCTION Incompressible iiit ernal flows are encountered in many realistic engineering probleiiis such as the flow through impeller passages and duct, flow. Geometric variation for these flows is diverse and iiaturally tlie coiiiputatioiial approach has to take this into account. Internal flows are in general very different froiii the exterial flows. For instance, the boundary layer in external flow is usually very thin compared to tlie characterisitc length of the moving object, while in internal flow the entire flow field is likely to be viscous. The boundary layer is often separated hy abrupt changes in geometry and tlie blockage effect caused by tlie separated zone becomes significant. One of tlie major iiiotivatioiis of the present work was to develop numerical siiiiulat ion capability especially suitable for internal flows in the Space Shuttle iiiairie engine (SSME) power head. To upgrade tlie SSME without increasing the weight arid size of tlie existing engine. i t becanie essential to underst and tlie dyiiaiiiics of the hot -gas flow in t lie engine power head. Bec a m e o f the coiiiplexity of t lie geonietrp, an experillielit a1 approach was extremely difficult as well as time coiisiiniing. ('oniputational siiiiulatioii, therefore. offers an econoniical alternative to coiiipleiiient t lie esperiiiierital work i n analyzing tlie ciirrent configuration, and to suggest new, iiiiproved design possibilities. T h e Reynolds number of tlie primary flow in the power head is of order l o 6 . Because of the high gas teiiiperature, tlie Mach n i i ~ i i h e ris less tliari 0.12. Tlie flow is t urhulent and is practically inco~iipressible.In the past. i ~ i i i i i e r o i i sniiiiierical niethods of siniulatiiig viscous iiicoiiipressible flows have been developed (ref. 1 ). Tlie present paper is a short
* t
Research Scientist. NAS.4 Aiiies Research ('enter. Senior Staff Scientist, Rocketdyne, Rockwell Inter~iatioiial,('anoga Park, C'A. $ Research Scientist. Sterling Federal Sjstenis, Palo -4lto. C-4. 5 XRC' Fellow, N-4S-A .41iies Research ('enter. To be presented at the Twelfth IMAC'S Conference on Scientific Coniputation, Paris, France. July 1 s-22.1988.
suniiiiarg o f our recent effort to develop a1t.ernative met hods for siiiiulating three-diniensional viscous incoiiipressible flows in generalized coordinates. This work is partially sponsored by the N A S A Marshall Space Flight. Center.
SOLUTION METHODS A ilia jor probleiii of solving tlie iiicoiiipressible Navier-Stokes equations conies from tlie lack of a pressure teriii in the contiiiuity equation. In realistic t liree-diniensional applications, satisfying cont iniiitg i n a reasoiiable aiiiount of coiiiputi~igtime becomes a priiiiary issue i n addition to accuracy and rohustness. For ccniveriience and flexibility, a priiiiitive variable foriiiulatioii in generalized curvilinear coordinates is chosen. and our discussion is limited to this formulation. I n this section, three computer codes recently developed at N A S A Allies Research Center and related solution iiiethods used in the codes are summarized. Derivation of equations and algorit hniic (let ails can be found in full-length versions of the authors’ publications (refs. 2-6). Pseudocompressibility Met hod Recent advances in the state of the art in coiiiputatioiial fluid dyiiaiiiics ((‘FD) have been iiiade i n conjunction wit 11 conipressible flow computations. Therefore, it is of significant interest to lie able to use some of these coiiipressible flow algorithms. To do this, the artificial coiiipressibility ~iiethotl(ref. 7’) can he used. In this forniulation, tlie continuity equation is modified by adding a t iiiie-derivative o f t lie pressure teriii resulting i n a hyperbolic-paraholic type of time-dependent system of equations. This method was originally intended for tlie steady-state computation of incompressible flows. However, hy introducing a pseudotime iteration, this can be made time accurate (refs. 6 and 8). In an incompressible flow, a disturbance in the pressure causes waves which travel with infinite speed. When pseudocompressibilita- is introduced, waves of finite speed result in which t lie iiiagiii t ude of t lie speed depends up011 tlie iiiagnii ude of tlie pseudoconipressibility. In a t rile inconipressible flow. the pressure field is affected instantaiieously by a disturbance in the flow, h i i t wit 11 pseudoconipressibility, there will be a time lag between the disturbance and its effect 0 1 1 the pressure field (ref. 3 ) . The interaction of tlie pressure wave propagation and the viscous field is especialll- proiioiiiiced i n iiiterrial flows (ref. 3). Various applicatioiis evolved froiii this concept hale been reported for obtaining steady-state solutioris (refs. 2-4, 9 and 10).
INS3D Cod-e- By conht ructing a pseudocompressil>le form of governing equations, fast , iniplici t scheiiies developed for compressible flows, such as the approxiniate-factorization sclieiiie (ref. 11 ) can be impleiiieiited. The present code is writ ten for obtaining steady-state solutions. The 5paiial diwretization uqeq second-order central differencing with additional nuiiierical diqsipat ion iernis. This code has lxen validated and ~ i u ~ i i e r o ut liree-dimensional s problenis have been \oI\c(l u h i i i g this code (refs. 2-4, 12 and 1 3 ) .
INSUP Code-
To obtain time-accurate solutions using the pseudocoiiipressibility formu-
lation. i t is necessary to satisfy continuity at each time step by subiteration in pseudotime. To use a large time step in the pseudotime iteration, an upwind differencing scheme based on flux-
difference splitting is used coiiibined with an implicit line relaxation scheme. This renioves the factorization error and the need for nu~iiericaldissipation terms. The code has been validated and excellent results have been obtained (ref. 6).
Fractional-Step Method T h e fractional-step net hod caii he used for time-dependent coiiiput at ions of the inco~iipressible Navier-Stokes equations (refs. 5 and 14-16). Here, the discretized equations are advanced in time by decoupling the solution of tlie nioiiientuiii equation from that of tlie continuity equation. T h e c o ~ i i ~ i i oapplication ~i of this method is done by two steps. The first step is to solve for an auxiliary velocity field usiiig the nio~iient11111 equation in which the pressure-gradient t ern1 caii lie coiiiput ed from t lie pressure i l l the previous time step (ref. Is), or can be excludecl eiit irelj (ref. 16). I n the second step, the pressure which maps the auxiliary velocity onto a divergencefree velocity field is computed. Various other operator splittiiigs can be adopted by treating the nionient iim equation as a conibination of convection, pressure, and viscous ternis.
LNSFS Code-
A generalized flow solver based on this approach using a staggered grid has heen developed (ref. 5 ) . The goveriii~igeqiiaitons are discretized co~iservativelyusiiig finite volunies. Ratlier t lian choosing t lie ('artesian velocity conil~oneiitsas dependent variables, the voluiiie fluxes over tlie faces of the conipiit ational cells are used. They are equivalent to the cont ravariaiit velocity coniponenf s described in a staggered grid. This procedure, combined wit 11 accurate and consisteiit a~iproxiniatioiisof the geometric quantities, satisfies the discretized mass conservation equatioir exact IF. A nc)veI four-color ZEBRA scheme is devised for solving t lie Poisson equation for presslire correction. Several coniliutational results have l ~ e coiiipared n witli ex~ieriiiients and other iiiiiilerical soliitions in reference 5 .
COMPUTED RESULTS
Validation of Flow Solvers T h e three flow solvers described above have 11een validated liy coniput iiig various basic fluid dynaiiiics ~~roldeiiis. A few examples are listed i n this section. figure I , vortex shedding from a circular cylinder is .suiiiiiiarized by a plot of St milia1 ~ i u ~ i i b eas r a function of the Reynolds nuiiiber lmsetl on rliaiiieter of tlie cyinder. Strouhal nuniber is tlie diiii~~isioiiless frequency of the vortex shedding a11d a good agree~iientis see11 betiwen the coiiil)iited r - u l t ~ obtaiiied by INSITP and t h e esperiiiielit* 11). K o \ ~ ~ s z n a(ref. y 1 7 ) and Hoshko (ref. 1s). 111
Tn figiire 2. tlie i i i i i e euoliitioii of separation leirgih ior f l o ~ vover a circiilar cylinder a t Reynold'; number of 40 is coinpared w i t h one other coniliut at ion (ref. 1 9 ) and wit 11 sollie experiniental ~.e>ult s (ref. 20). T h e coniputed results i n figure 2 are ohtaiiietl ilsing INSFS.
SSME Flow Analysis the SSME staged combustion cycle, the fuel is partially burned at very high pressure and relatively high temperature in the preburners. T h e resulting hot gas is used to run the turbine and is then discharged from the gas turbine to the annular turnaround duct ( T A D ) , and experiences a 180" turn before it diffuses into the fuel bowl. This assembly is called the hot-gas manifold ( I1 (; 11). Tlie SShIE flow siiiiulat ion was performed using INS3D code. 111
To simulate this flow undergoing a 180" U-turn, it is necessary to incorporate strong curvature effect in the t urbuleiice niodeling. Several levels of turbulence models have been inipleiiierited i n INS3D code. However. for the present problem, a length scale deternii~iedby the point of mininiuni vorticity is used. This length scale is incorporated into an extended Prandtl-Karniann iiiising-lengt 11 theory. The coiiibiiiation of these automatically account for curvature effect. Full details of this model are given i n references 13 and 21. I n figure 3 ( a ) ,the grid in the 180" turn region is shown. In figure 3(b). computed pressure coefficients for the inner and the outer walls are compared w i t h experiment (ref. 22). I n figure 4, t.lie coiiiput.er model of the current bhree-duct, SSME power head is shown. This niodel is chosen t.o analyze critical areas where the dynamics of t.he hot-gas flow is expect.ed t.o have a significant effect. 0 1 1 t.he overall performance of t,he SSME. Mult.iple zone coniput,at.ions were perfornied using t.lie grid shown. From t,his comput,at,ional flow analysis and also from experiments, t.he cent.er duct, of the current, t,hree-duct,HGM is found t.o t.ransfer a. liniit,ed aniount. of mass flow (about, 10% of the tot,al flow). .Also t.he t.ransverse pressure gradient, reinairis large toget.her with a large bubble of separat.ion aft.er t,he 180" t.urn. To reduce this large separat,ion bubble^ a paraniet.ric st,udy is performed to find t,he best. possible configuration. To improve the qua1it.y of the flow? a large-area, two-duct. design concept has heen developed. In addition, the c1uct.s are chosen to have an ellipt,ical shape t.o distribute t.Iie mass flow evenly t.o t.he main inject.or region. A grid for a t,wo-duct, niodel is showil i n figiire -5. Coiiiput,a.t.ionalsiniulation of this new two-duct. configurat.ion shows t.hat. the separation I,ubble existing in blie present. design is pract,ically renioved i n the new configurat,ion. This is confiriiied by experiiiient,~.
Tlie niost significant aspect of tlie present study is to pinpoi~it the locations where flow experiences the niost energy loss. ,411 important iiieasure of the energy losses is the mass-weighted average t o t a1 presslire alc)ng the flow. Figure G illustrates the decreasing coefficient of the massn-eighted total pressiire aloiig the centerline of tlie TAD, the fuel bowl, and the transfer duct. The discontinuities s1iow.n i n the figure correspond to the entrance of t h e duct where energy fluxes are computed over cliflvrent planes. 111 the figure. three different HGM configurations are conipared. The initial two-duci design shows 28% less total pressure drop compared to tlie current threeduct version. After fine-t uiiing tlie two-duct configuratiori coniputationally, the pressure drop decreased even further t o 36% less than the original configuration. This final confi,guration is tlieii tested using cold air flow, wliicli shows 40% reduction in pressure loss. Further details of t h i q work can be foillid i n references 12, 13, and 23.
CONCLUDING REMARKS This paper presents a summary of incoinpressible Navier-Stokes flow-solver development work. Coiiiputed results on several validation probleiris are conipared with experilrielits and other coniputations. Computational results of the SSME power head are favorably compared with test dai a, and offer iiiforiiiatiori not readily available from experiments. The results show that C'FD can reduce the developnient time and cost by suggesting the best possible configurations for filial verification by experinients. The SSME application provides an example of how the present CFD capabilities can be integrated into the aerospace design process. Further study of the SSME is i n progress, and ilie total performance i~riprovenrent~ will be coiiipared i n the future.
REFERENCES 1. Ferziger, J . H.: Iricoriipressible turbule~it.flows. J . Comp. Phys., vol. 69, pp. 1-48, 1987
2. Iiwak,D.; Chang, J . L. C.; Shanks, S. P.; and Chakravartliy, S.: A three-dimensional incoiiipressible Navier-Stokes flow solver using primitive variables. AIAA J., vol. 24, no. 3, pp. 390-396, 1986. 3 . C:haiig, J . L. ('.;and Kwak, D.: 011the method of pseudo coiiipressibilit,y for numerically solving iiiconipressible flows. AIAA Paper 84-0252, Reno, Nevada, 1984.
4. Rogers, S. E.; C h n g , J . L. C.; and Kwak, D.: A diagonal algorithm for the met,hod of pseudocoiiipressihi1it.y. J. Comp. Pliys., vol. 73, no. 2, pp. 364-379, Deceiiiber 1987. 5. Rosenfeld, AI.; Kwak, D.; aiid Vinokur, M.: A Solution niethod for unst,eady, iiicoiiipressible Nayier-Stokes equatioiis i n generalized coordinate systems. AIAA Paper 88-0718, AIAA 26th Aerospace Sciences Meeting, Reno, Nevada, January 11-14, 1988.
6. Rogers, S. E.; aiid Kwak, D.: A n upwind differencing scheme for the time-accurat,e incoiiipressible Navier-Stcokes equations. AIAA Paper 88-2583, AIAA Gtli Applied Aerodynaiiiics Conference, Williamsberg, VA, Julie 6-8, 1988. 7. (%orin, A. J.: A iiuiiierical met hod for solving incompressible viscous flow problems. J. ('oiiip. Phys., vol. 2, pp. 12-26, 1967. S. hlerkle, C'. L.; and Athavale, M.: Tiiiie-accurate uiisteady incoiiipressible flow algoritlinis based 0 1 1 artificial compressibility. AIAA Paper 87-1 1.37, AIAA X t h (!oiiiput,ational Fluid Dyiiaiiiics C'oiiference, Honoliilu. Hawaii, June 9-11, 1987.
9. ('hai, D.; and hlerkle, c'. L.: Applicat.ion of time-it.erative scheiiies to iiicoiiipressible flow iZIA-4 J., vol. 23. no. 10. pp. 1.518-1524, 198.5. 10. Soli, W. Y.: De\-eloping fluid flow in a curved duct of square cross-section aiid its fully developed dual solutions. J . Fluid Mech., to appear in Septeniber 1988. 11. Beam, R. hl.; and N'ariiiing, R. F.: A n iinplicit finit,e-difterence algoritliiii for hyperbolic s;ysteiiis in conservation-law f o r m J . of Coinp. Pliys., vol. 22, pp. 8;-110, 1976.
12. C'liang, J . L. C.;Kwak, D.; and Dao, S. C.: A three diniensio~ialii~coiiipressil~le flow siiiiulation iiietliod and its application to the Space Shuttle main engine. Part I. Laminar flow. A I A A Paper 8.5-017.5, Reno, Nev.. Jan. 1985. 13. ('hang. J . L. ('.;Iiwak. D.; Dao, S. (1.; and Rosen. R.: A three-dimensional inconipressible flow siiiiulation method and its application to the Space Shuttle main engine. Part 11. Turlnlent flow. A I X A Paper 85-1670, AIAA 18th Fluid Dynaiiiics and Plasiiiadyiiaiiiics aiid Laser Co~iference.('incinnati. Ohio, .July 16-1H. 198.5. 1-1. ('lioriii, A. J .: Nuiiierical solution of Navier-St.okes equat,ions. Mat,lieiiiat,icsof (loiiiput.ation, uol. 22, 1 1 0 . 104, pp. 745-762, 1968.
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15. Dw~yer.H. S.; Soliman, M.; and Hafez, M.: Time accurate solutions of tlie Navier-Stokes equations for reacting flows. Proceedings of t,he 10th IIit,ernational Cloriference on Nunierical
hlethods in Fluid Dynamics, Beijing, China, June 1986, pp. 247-251, Springer-Verlag.
16. Kim, J.; arid Moiii, P.: Application of a fractional-step method t o incompressible NavierStokes equations. J. Coinp. Phys., vol. 59, no. 2, pp. 308-323, 1985.
17. Kovasznay, L. S. G.: Hot wire iiivest,igation of the wake behind cylinders at low Reynolds ~iulnhers.Proc. Roy. Soc., A , vol. 198, pp. 174-190, 1949. 1s. Rosliko. A.: On the development of turbulent wakes from vortes streets. NACA Rep. 1191, 1953.
19. C'outanceau, M.; and Bouard. R.: Experiiiieiital deteriiiinatioii of the main features of the viscous flow in tlie wake of a circular cylinder in uiiiforiii t r a n ~ l a t ~ i o nPart . 2. unsteady flow. J . Fluid hlech.. vol. i 9 , pp. 2.57-2i2, 19'ii.
20. Cbllins, W. M.; and Dennis? S. C. R.: Flow past. an iiiipulsively st,art,edcircular cylinder. J . Fluid Mecli. vol. GO, pp. 10.5-127, 1973. 21. C'liang, J. L. C.; arid Kwak, D.: A iiuiiierical study of turbulent indernal sliear layer flow in axisymnietric ['-duct. AIAA Paper 88-0.596, Reno, Nev., January 11-14, 1988. 22. Sliarnia, L.; Osteriiiier, B.; Nguyen, L.; Daiig, P.; arid O'C'o111ior,G.: Turbulence nieasureiiieiit s in an axisyiiiiiieteric turnaround duct air flow niodel. Rocketdyne Division, Rockwell Iriteriiatiorial, Report RSS-876.3. ATIJ-87-5237, October 1987.
23. Ya.ng, R-J.; C'hang, J . L. C.; arid h w a k , D.: A Navier-Stokes siniulation of the Space Shuttle iiiaiii engine hot gas nianifold. A I A A Paper 87-0368, AIAA 25t,li Aerospace Sciences Meeting, Reno, Nevada, January 12-15, 19x7.
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-k. Computation (INSUP)
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Experiment Kovasanay (17) Roshko (18)
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400
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Figure 1. Vortex shedding from a circular cylinder. 2.5
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Computation ----- INSFS Collins & Dennis
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Experiment Coutanceau & Bouard
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Figure 2. Time evolution of separation length for flow over a circular cylinder at R=40.
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Experiment (22) 0 0
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Figure 3. Flow in an axysymmetric U-duct; ) grid in U-turn region, b) static pressure distribution at ~ ~ = 1 0 ~ .
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G:;jc!ifINAL PAGE IS OF POOR QUALITY
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Figure 4. Grid for the current 3-duct SSME power head; a ) horizontal view (cross-section B-B), b) vertical view (cross-section A-A). A
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Figure 5. Grid for the new 2-duct SSME power head; a) horizontal view (cross-section B-B), b) vertical view (cross-section A-A).
10
0
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PROVED DESIGN RANSFER DUCTS)
COMPUTATION COLD AIR TEST
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36% IMPROVEMENT 40% IMPROVEMENT
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Figure 6. Comparison of pressure loss in 3-duct and 2-duct design.
11
Report Documentation Page 3. Recipient's Catalog No.
2. Government Accession No.
1. Report No.
NASA TM-100076 5. Report Date
March 1988 6. Performing Organization Code
8. Performing Organization Report No.
7. Author(s)
.
A-881 11
D. Kwak, J.L.C. Chang, S.E.Rogers, and M. Rosenfeld
10. Work Unit No.
505-60-01
9. Performing Organization Name and Address
11. Contract or Grant No.
NASA Ames Research Center Moffett Field, CA 94035 13. Type of Report and Period Covered ~~~~
~~
12. Sponsoring Agency Name and Address
Technical Memorandum National Aeronautics and Space Administration
14. Sponsoring Agency Code
Washington, D.C. 20546 15. Supplementary Notes
Point of Contact: Dochan Kwak, Ames Research Center, MS 258-1 Moffett Field, CA 94035 (41 5 ) 694-6743 or FTS 464-6743
16. Abstract
Several incompressible Navier-Stokes solution methods for obtaining steady and unsteady solutions are discussed. Special attention is given to internal flows which involve distinctively different features from external flows. The characteristics of the flow solvers employing the method of pseudocompressibility and a fractional step method are briefly described. The present discussion is limited to a primitive variable formulation in generalized curvilinear coordinates. Computed results include simple test cases and internal flow in the Space Shuttle main engine hot-gas manifold.
18. Distribution Statement
17. Key Words (Suggested by Author(s))
Unclassified-Unlimi ted
Incompressible Navier-Stokes equations Internal flow
Subject Category - 03 19. Security Classif. (of this report)
Unclassified
20. Security Classif. (of this page)
Unclassified
21. No. of pages
12
22. Price
A02
