Review of viscous internal flows models

B. V. Rogov ; I. A. Sokolova

Matematičeskoe modelirovanie, Tome 14 (2002) no. 1, pp. 41-72.

Résumé

Review of wide spread simplified models for numerical investigation of viscous flow through nozzles and tunnels is presented. The most attention of the review is turned to details of stationary flow calculations in wide ranges of Mach number at moderate and large Rynolds numbers, in particular viscous flow through Laval nozzle. The commonly used parabolized models and common numerical methods for their calculations are presented, and new models of smooth channel approximation with effective numerical non-iterating and marching methods for them as well. The hierarchy of simplified models for internal flows is established and comparative analysis of internal and similar external models for viscous flows is presented. The analogy in numerical methods of the approaches is found.

Bibliographie

[1] Alshina E. A., Kalitkin N. N., Rogov B. V., Sokolova I. A., “O tochnosti kvaziodnomernoi modeli gladkogo kanala”, Matem. modelirovanie, 13:10 (2001), 120–124

[2] Aslanov T. D., Byrkin A. P., Schennikov V. V., “Chislennyi raschet vnutrennikh techenii vyazkogo gaza s ispolzovaniem uravnenii Nave–Stoksa”, Uch. zap. TsAGI, 12:3 (1981), 44–54

[3] Astrov V., Levin L., Pavlov E., Khristianovich S. A., “O raschete sopla Lavalya”, PMM, 7:1 (1943), 3–24

[4] Arkhangelskaya L. A., Skurin L. I., “Ispolzovanie metoda globalnykh iteratsii po davleniyu dlya resheniya uravnenii Nave–Stoksa”, Vestn. S.-Peterburg. Un-ta. Ser. 1, 3:15 (1994), 70–74 | Zbl

[5] Bezmenov V. Ya., Byrkin A. L., Gorenbukh P. I. i dr., “Issledovanie techeniya gaza v giperzvukovykh soplakh pri bolshikh chislakh Reinoldsa na osnove uproschennykh uravnenii Nave–Stoksa”, Uch. zap. TsAGI, 20:4 (1989), 53–61

[6] Botkin V. A., Egorov I. V., Ivanov D. V., “Primenenie metoda Nyutona k raschetu vnutrennikh sverkhzvukovykh otryvnykh techenii”, PMTF, 1997, no. 1, 30–42

[7] Belotserkovskii O. M., Byrkin A. P., Mazurov A. P., Tolstykh A. I., “Raznostnyi metod povyshennoi tochnosti dlya rascheta techenii vyazkogo gaza”, ZhVM i MF, 22:6 (1982), 1480–1490 | MR

[8] Borisov A. V., Kovenya V. M., “Primenenie neyavnoi raznostnoi skhemy dlya rascheta vnutrennikh techenii vyazkogo gaza”, Chisl. metody mekh. splosh. sredy, 7:4 (1976), 36–47 | MR

[9] Byrkin A. P., Mezhirov I. I., “O raschete techeniya gaza v giperzvukovom sople s uchetom vliyaniya vyazkosti (pryamaya zadacha)”, Uch. zap. TsAGI, 2:1 (1971), 33–41

[10] Byrkin A. P., Timofeeva T. A., Tolstykh A. I., “Primenenie kompaktnykh skhem tretego-chetvertogo poryadka dlya rascheta techeniya gaza v soplakh s bolshimi sverkhzvukovymi chislami $M$ na osnove uproschennykh uravnenii Nave–Stoksa”, Uch. zap. TsAGI, 19:6 (1988), 28–39, (a) | MR

[11] Byrkin A. P., Tolstykh A. I., “Kompaktnye skhemy tretego i chetvertogo poryadkov v zadachakh o vnutrennikh techeniyakh vyazkogo i nevyazkogo gazov”, Zh. vychisl. matem. i matem. fiz., 28:8 (1988), 1234–1251, (b) | MR | Zbl

[12] Byrkin A. P., Klyuev V. N., Sosunov A. Yu., Tolstykh A. I., “Chislennoe modelirovanie techeniya gaza v soplakh Lavalya i techenie tormozheniya v kanalakh na osnove polnykh uravnenii Nave–Stoksa”, Uch. zap. TsAGI, 24:1 (1993), 87–97, (a)

[13] Byrkin A. P., Verkhovskii V. P., “K raschetu profilirovannykh giperzvukovykh sopl aerodinamicheskikh trub s uchetom vliyaniya vyazkosti”, Uch. zap. TsAGI, 24:1 (19936), 172–176, (b)

[14] Vasilevskii S. A., Tirskii G. A., Utyuzhnikov S. V., “Chislennyi metod resheniya uravnenii vyazkogo udarnogo sloya”, DAN SSSR, 290:5 (1986), 1058–1061

[15] Vasilevskii S. A., Tirskii G. A., Utyuzhnikov S. V., “Chislennyi metod resheniya uravnenii vyazkogo udarnogo sloya”, Zh. vychisl. matem. i matem. fiz., 27:5 (1987), 741–750

[16] Vetluttskii V. N., Muchnaya M. I., “Raschet vyazkogo techeniya v giperzvukovom sople”, Izv. AN SSSR. Mekh. zhidkosti i gaza, 1977, no. 4, 29–35

[17] Voinovich P. A., “Marshevyi metod rascheta techenii vyazkogo gaza v kanalakh”, Zh. vychisl. matem. i matem. fiz., 24:6 (1984), 944–947 | MR | Zbl

[18] Voinovich P. A., Golovachev Yu. P., Fursenko A. A., “Marshevyi metod rascheta smeshannykh techenii vyazkogo gaza”, Chisl. metody dinam., vyaz. zhidkosti, Novosibirsk, 1983, 90–95

[19] Voinovich P. A., Fursenko A. A., “Metod globalnykh iteratsii dlya rascheta smeshannykh techenii vyazkogo gaza”, Differents. uravneniya, 20:7 (1984), 1151–1156 | MR | Zbl

[20] Volkov V. A., Gidaspov V. Yu., Kozelko A. N., Pirumov U. G., “Marshevyi polisetochnyi algoritm rascheta sverkhzvukovykh statsionarnykh techenii gaza na estestvenno adaptirovannykh setkakh”, Matem. modelirovanie, 8:6 (1996), 121–127 | MR | Zbl

[21] Gershbein E. A., Peigin S. V., Tirskii G. A., “Sverkhzvukovoe obtekanie tel pri malykh i umerennykh chislakh Reinoldsa”, Itogi nauki i tekhniki. Mekhanika zhidkosti i gaza, 19, VINITI, M., 1985, 3–85

[22] Golovachev Yu. P., Chislennoe modelirovanie techenii vyazkogo gaza v udarnom sloe, Nauka. Fizmatlit, M., 1996 | Zbl

[23] Golovachev Yu. P., Fursenko A. A., “Marshevyi metod rascheta techenii vyazkogo gaza”, ZhVM i MF, 21:6 (1981), 1592–1596 | Zbl

[24] Denisenko O. V., “Metod rascheta sverkhzvukovykh sopl pri silnom vliyanii vyazkosti”, Uch. zap. TsAGI, 13:4 (1982), 71–80 | MR

[25] Denisenko O. V., Provotorov V. P., “Issledovanie techenii vyazkogo gaza pri umerennykh chislakh Reinoldsa”, Trudy TsAGI, 2269, 1985, 111–127

[26] Egorov I. V., Ivanov D. V., “Primenenie polnostyu neyavnykh monotonnykh skhem dlya modelirovaniya ploskikh vnutrennikh techenii”, ZhVM i MF, 36:12 (1996), 91–107 | MR | Zbl

[27] Egorov I. V., Ivanov D. V., “Modelirovanie khimicheski neravnovesnogo techeniya gaza v kanale peremennogo secheniya”, Matem. modelirovanie, 9:11 (1997), 85–100 | Zbl

[28] Egorov Yu. E., Strelets M. Kh., Shur M. L., “Primenenie metoda masshtabirovaniya szhimaemosti dlya rascheta statsionarnykh techenii vyazkikh gazov i gazovykh smesei v soplakh Lavalya”, Matem. modelirovanie, 2:10 (1990), 3–12 | Zbl

[29] Egorov Yu. E., Strelets M. Kh., Shur M. L., Granitsy primenimosti parabolicheskikh modelei dlya chislennogo issledovaniya techenii v soplakh Lavalya. 1. Priblizhenie uzkogo kanala, Preprint Gos. in-ta prikladnoi khimii, No 4, L., 1991

[30] Zhmakin A. I., Makarov Yu. N., Chislennoe issledovanie medlennykh neizotermicheskikh techenii vyazkogo gaza v kanalakh, Fiz.-tekhn. in-t AN SSSR. Prepr. No 821, 1983 | Zbl

[31] Ivanov M. Ya., Kraiko A. N., “Chislennoe reshenie pryamoi zadachi o smeschannom techenii v soplakh”, Izv. AN SSSR. Mekh. zhidk. i gaza, 1969, no. 5

[32] Kalitkin N. N., Rogov B. V., Sokolova I. A., “Vysokotochnyi metod rascheta vyazkikh techenii v sople Lavalya”, Matem. modelirovanie, 9:7 (1997), 81–92 | MR | Zbl

[33] Kalitkin N. N., Rogov B. V., Sokolova I. A., “Dvukhstadiinyi marshevyi metod rascheta vyazkikh techenii cherez soplo Lavalya”, Matem. modelirovanie, 11:7 (1999), 95–117

[34] Kalitkin N. N., Rogov B. V., Sokolova I. A., “Reshenie pryamoi zadachi sopla Lavalya metodom globalnykh iteratsii po napravleniyam linii toka”, DAN, 370:1 (2000), 46–49

[35] Kalitkin N. N., Rogov B. V., Sokolova I. A., “Effektivnyi metod rascheta vyazkikh techenii so znachitelnym iskrivleniem linii toka”, DAN, 374:2 (2000), 190–193

[36] Karataev S. T., Koterov V. N., “Chislennyi metod rascheta sverkhzvukovykh techenii vyazkogo gaza”, Zh. Vychisl. Matem. i Matem. Fiz., 30:4 (1990), 586–600 | MR | Zbl

[37] Kiselev A. S., Sternin L. E., “Kompaktnaya raznostnaya skhema so skalyarnymi progonkami dlya integrirovaniya uravnenii gazovoi dinamiki”, ZhVM i MF, 39:1 (1999), 154–162 | MR | Zbl

[38] Kibel I. A., Teoreticheskaya gidromekhanika, Chast 2, eds. Kochin N. E., Kibel I. A., Roze N. V., 1963 | Zbl

[39] Kovalev V. L., Krupnoe A. A., Tirskii G. A., “Reshenie uravnenii vyazkogo udarnogo sloya metodom prostykh globalnykh iteratsii po gradientu davleniya i forme udarnoi volny”, Dokl. RAN, 338:3 (1994), 333–336 | Zbl

[40] Kovalev V. I,, Luschik V. T., Sizova V. I., Yakubenko A. E., “Trekhparametricheskaya model turbulentnosti: chislennoe issledovanie pogranichnogo sloya v sople s zavesnym okhlazhdeniem”, Mekhanika zhidkosti i gaza, 1992, no. 1, 48–57

[41] Kovenya V. M., Chernyi S. T., “Metod resheniya statsionarnykh uproschennykh uravnenii vyazkogo gaza”, Chisl. metody mekh. splosh. sredy, 10:1 (1979), 71–87

[42] Kovenya V. M.,Chernyi S. T., Metod resheniya statsionarnykh uproschennykh uravnenii vyazkogo gaza, Institut Prikladnoi i teoreticheskoi mekhaniki. Preprint No 42, 1981 | MR | Zbl

[43] Kovenya V. M., Yanenko N. N., Metod rasschepleniya v zadachakh gazovoi dinamiki, Nauka, Novosibirsk, 1981 | MR | Zbl

[44] Kovenya V. M., Tarnavskii G. A., Chernyi S. T., Primenenie metoda rasschepleniya v zadachakh aerodinamiki, Nauka, Novosibirsk, 1990

[45] Kopchenoe V. I., Laskin I. N., “Ob odnoi konechno-raznostnoi skheme dlya chislennogo resheniya parabolizovannykh uravnenii Nave–Stoksa”, Zh. vychisl. matem. i matem. fiz., 36:2 (1996), 126–137 | MR

[46] Kuznetsov A. E., Strelets M. Kh., Shur M. L., “Raschet statsionarnykh trekhmernykh techenii vyazkogo gaza v kanalakh i soplakh”, Teplofiz. vysokikh temperatur, 31:3 (1993), 395–405

[47] Kuznetsova L. V., Pavlov B. M., “Primenenie uravnenii Nave–Stoksa k issledovaniyu techeniya vyazkogo gaza v sople Lavalya”, Vychisl. metody i programmir., no. 30, M., 1979, 120–130

[48] Lapin Yu. V., Nekhamkina O. A., Pospelov V. A. i dr., “Chislennoe modelirovanie vnutrennikh techenii vyazkikh khimicheski reagiruyuschikh gazovykh smesei”, Itogi nauki i tekhn. Ser. mekhan. zhidkosti i gaza, 19, VINITI, M., 1985, 86–185

[49] Lapin Yu. V., Strelets M. Kh., Vnutrennie techeniya gazovykh smesei, Nauka, M., 1989

[50] Levin V. Ya., Nigodyuk V. E., Pirumov U. G. i dr., “Issledovanie techenii v soplakh Lavalya pri nizkikh chislakh Reinoldsa”, Izv. AN SSSR. Mekh. zhidkosti i gaza, 1980, no. 3, 90–97

[51] Luschik V. T., Sizov V. P., Yakubenko A. E., “K ispolzovaniyu priblizheniya uzkogo kanala dlya rascheta turbulentnogo techeniya v soplakh zhidkostnykh raketnykh dvigatelei”, TVT, 31:5 (1993), 752–758

[52] Mazurov A. P., “Chislennyi raschet techeniya vyazkogo gaza v ploskom sople GPVRD”, Uch. zap. TsAGI, 24:3 (1993), 63–75 | MR

[53] Mamina M. P., Pospelov V. A., Khodzhiev S., “Reshenie pryamoi zadachi sopla Lavalya v priblizhenii “uzkogo kanala” metodom ustanovleniya”, Gidroaerodinamika, L., 1983, 26–30

[54] Markachev Yu. E., “Iteratsionnye algoritmy rascheta statsionarnogo techeniya vyazkogo gaza v sople Lavalya v priblizhenii “uzkogo kanala””, Trudy TsAGI, 2024, 1979, 3–16

[55] Medvedev A. K., Fomin V. M., “Analiz dvizheniya vyazkogo reagiruyuschego gaza v uzkom kanale”, Teplofizika i aeromekhanika, 2:4 (1995), 323–332

[56] Mikhailov V. V., “Metod rascheta sverkhzvukovykh sopel s uchetom vliyaniya vyazkosti”, Izv. AN SSSR. Mekh. zhidkosti i gaza, 1969, no. 1, 69–72

[57] Muchnaya M. I., Ispolzovanie uproschennykh uravnenii Nave–Stoksa dlya rascheta vyazkogo techeniya v giperzvukovom sople, Preprint in-ta teor. i prikl. mekh. SO AN SSSR, No 17, Novosibirsk, 1981

[58] Muchnaya M. I., “Raschet techeniya v profilirovannykh giperzvukovykh soplakh s pomoschyu uproschennykh uravnenii Nave–Stoksa”, Chisl. metody mekh. splosh. sredy, 13:5 (1982), 145–148

[59] Muchnaya M. I., “Issledovanie techenii v giperzvukovykh soplakh v ramkakh uproschennykh uravnenii Nave–Stoksa”, Izv. AN SSSR. Mekhan. zhidkosti i gaza, 1986, no. 6, 20–26

[60] Muchnaya M. I., “Chislennoe issledovanie techeniya gaza v giperzvukovykh soplakh pri vysokikh chislakh Reinoldsa”, Izv. RAN. Mekhan. zhidkosti i gaza, 1999, no. 1, 161–164 | Zbl

[61] Pirumov U. G., “Raschet techeniya v sople Lavalya”, Izv. AN SSSR. Mekh. zhidkosti i gaza, 1967, no. 5

[62] Pirumov U. G., Roslyakov G. S., Techeniya gaza v soplakh, MGU, M., 1978

[63] Polyanskii A. F., Skurin L. I., “Iteratsionno-marshevyi metod integrirovaniya sistem uravnenii Nave-Stoksa dlya gaza”, Vestn. S.-Peterburg. Un-ta. Ser. 1, 1998, no. 1, 87–92

[64] Rogov B. V., Sokolova I. A., “Kvaziodnomernaya model techeniya khimicheski reagiruyuschikh smesei gazov v gladkikh iskrivlennykh kanalakh peremennogo secheniyayu”, Matem. modelirovanie, 6:12 (1994), 38–56 | Zbl

[65] Rogov B. V., Sokolova I. A., “Uravneniya dlya techenii vyazkikh gazov v izognutykh ploskikh kanalakh peremennogo secheniya”, Matem. modelirovanie, 7:11 (1995), 39–54 | MR | Zbl

[66] V. Rogov, I. A. Sokolova, “Asymptotic Accuracy of Smooth -Channel Approximation for Viscous Flows”, Physics Doklady, 42:11 (1997), 619–623 | MR

[67] Rogov B. V., Sokolova I. A., “Giperbolicheskaya model vyazkikh smeshannykh techenii”, Dokl. RAN, 378:5 (2001), 628–632 | MR

[68] Ryzhov O. S., “O gazovykh techeniyakh v soplakh Lavalya”, PMM, 22:3 (1958)

[69] Savelev A. D., Tolstykh A. I., “Algoritmy rascheta techenii vyazkogo gaza, osnovannye na kompaktnykh approksimatsiyakh tretego poryadka”, ZhVM i MF, 27:11 (1987), 1709–1724 | MR

[70] Skovorodko P. A., Yakovenko V. A., “Chislennoe issledovanie techeniya vyazkogo gaza v sople Lavalya v priblizhenii uzkogo kanala”, Izv. SO RAN Sibirskii fiziko-tekhnicheskii zhurnal, 1992, no. 1, 17–23

[71] Strelets M. Kh., Shur M. L., “Metod masshtabirovaniya szhimaemosti dlya rascheta statsionarnykh techenii vyazkogo gaza pri proizvolnykh chislakh Makha”, ZhVM i MF, 28:2 (1988), 254–266 | MR | Zbl

[72] Sych V. M., “Raschet iskrivlennoi pristenochnoi turbulentnoi strui”, Uch. zap. TsAGI, 16:3 (1985), 58–68

[73] Surzhikov S. T., “Radiatsionno-gazodinamicheskaya model sopla s lokalnym nagrevom”, Matem. modelirovanie, 9:9 (1997), 54–74 | Zbl

[74] Tirskii G. A., Utyuzhnikov S. V., “Sovremennye gazodinamicheskie modeli vneshnikh i vnutrennikh zadach sverkh- i giperzvukovoi aerodinamiki”, Modelirovanie v mekhanike, 7:2 (1993), 5–28 | MR

[75] Tolstykh A. I., “O chislennom raschete sverkhzvukovogo obtekaniya zatuplennykh tel potokom vyazkogo gaza”, Zhurnal Vych. Mat. i Mat. Fiz., 6:1 (1966), 113–120

[76] Falkovich S. V., “Ob odnom klasse sopel Lavalya”, PMM, KhI:2 (1947), 223–230 | MR | Zbl

[77] Frankl F. I., “K teorii sopla Lavalya”, Izvestiya AN SSSR. Ser. Matematika, 9:5 (1945), 387–422 | MR | Zbl

[78] Khirshel E., Kordulla V., Sdvigovoe techenie szhimaemoi zhidkosti. Chislennyi raschet pogranichnogo sloya, Mir, M., 1987

[79] Khristianovich S. A., Mekhanika sploshnoi sredy, Nauka, M., 1981

[80] Khristianovich S. A., “Priblizhennoe integrirovanie uravnenii sverkhzvukovogo techeniya gaza”, PMM, KhI:2 (1947), 215–222

[81] Shevelev Yu. D., Trekhmernye zadachi teorii laminarnogo pogranichnogo sloya, Nauka, M., 1977

[82] Chernyi S. G., “O vybore sistemy koordinat dlya chislennogo resheniya uproschennykh uravnenii Nave–Stoksa”, Chisl. metody mekh. splosh. sredy, 13, no. 1, Novosibirsk, 1982, 132–146

[83] Anderson D. A., Tannehill J. C., Pletcher R. H., Computational Fluids Mechanics and Heat Transfer, Hemisphere, New York, 1984 ; Anderson D., Tannekhil Dzh., Pletcher R., Vychislitelnaya gidromekhanika i teploobmen, Mir, M., 1990 | MR | Zbl | Zbl

[84] Anderson O. L., “Calculation of internal viscous flows in axisymmetric ducts at moderate to high Reynolds numbers”, Computers and Fluids, 8 (1980), 391–441 | DOI | MR

[85] Blottner F. G., “Numerical solution of slender channel laminar flows”, Comp. Meth. Appl. Mech. Eng., 11 (1977), 319–339 | DOI | Zbl

[86] Briley W. R., “Numerical method for predicting three-dimensional steady viscous flow in ducts”, J. Comp. Phys., 14 (1974), 8–28 | DOI | Zbl

[87] Briley W.R., McDonald H., “Three dimensional viscous flows with large secondary velocity”, J. Fluid Mech., 144 (1984), 47–77 | DOI | Zbl

[88] Casulli V., Greenspan D., “Pressure method for the numerical solution of transient, compressible fluid flows”, Int. J. Numer. Meth. Fluids, 4 (1984), 1001 | DOI | Zbl

[89] Cebeci T. (ed.), Numerical and Physical Aspects of Aerodynamic Flows, Springer-Verlag, Berlin, 1982 | MR | Zbl

[90] Chen K. H., Pletcher R. H., “Primitive variable, strongly implicit calculation procedure for viscous flows at all speeds”, AIAA Journal, 29:8 (1991), 1241–1249 | DOI

[91] Cheng H. K., “The blunt-body problem in hypersonic flow at low Reynolds number”, IAS, 1963, 63–92

[92] Choi Y. H., Merkle C. L., “The application of preconditioning in viscous flows”, J. Comput. Phys., 105:2 (1993), 207–223 | DOI | MR | Zbl

[93] Cline M. C., “Computation of two-dimensional, viscous nozzle flow”, AIAA Journal, 14:3 (1976), 295–296 | DOI

[94] Davis R. T., “Numerical solution of the hypersonic viscous shock layer equation”, AIAA J., 8:5 (1970), 843–851 | DOI | Zbl

[95] Davis R. T., Flugge-Lozt I., “Second-order boundary-layer effects in hypersonic flow past axysymmetric blunt bodies”, J. Fluid Mech., 20:4 (1964), 593–623 | DOI | MR | Zbl

[96] Davis R. T., Werle M. J., Wornom S. F., “A Consistent Formulation of Compressible Boudary-Layer Theory with Second-Order Curvature and Displacment Effects”, AIAA Journal, 8:9 (1970), 1701–1703 | DOI | Zbl

[97] Davis R. T., Rubin S. G., “Non-Navier Stokes viscous flow computations”, Computers and Fluids, 8:1 (1980), 101–131 | DOI | MR | Zbl

[98] Ebrahimi H. B., Gilbertson M., “Two and three dimensional parabolized Navier–Stokes code for scramjet combustor, nozzle, and film cooling analysis”, AIAA, 1992, 92–0391

[99] Fletcher C. A. J., Computational Techniques for Fluid Dynamics, V. 2, Springer-Verlag, Berlin, 1988; Fletcher K., Vychislitelnye metody v dinamike zhidkostei, v 2-kh t., Mir, M., 1991

[100] Gao Zhi, “Simplified Navier-Stokes Equations (SNSE)”, Scienta Sinica. Ser. A, 31:3 (1988), 322–339 | MR

[101] Ghia K. N., Sokhey J. S., “Laminar incompressible viscous flow in curved ducts of rectangular cross sections”, Journal of Fluids Engineering, 99 (1977), 640–648

[102] Graves R. A., Jr., Hamilton H. H., “Simple numerically generated orthogonal coordinate system for viscous flow”, AIAA Journal, 19:2 (1981), 133–134 | DOI

[103] Greenberg J. B., Presser C., “A fully automatic method for predicting internal reacting flow”, J. Comput. Phys., 40:2 (1981), 361–375 | DOI | Zbl

[104] Helliwell W. S., Dickinson R. P., Lubard S. C., “Viscous flow over arbitrary geometries at high angle of attack”, AIAA, 1980, 80–0064

[105] Himansu A., Rubin S. G., “Multigrid acceleration of a relaxation procedure for the reduced Navier-Stokes equations”, AIAA Journal, 26:9 (1988), 1044–1051 | DOI

[106] Hirsch C., Numerical computation of internal and external flows, V. 1, 2, Wiley and Sons, New York, 1988

[107] Kaushik S., Rubin S. G., “Pressure based flux-split solutions for incompressible and compressible internal flows”, Computers and Fluids, 27:1 (1998), 71–94 | DOI | MR | Zbl

[108] Kim M. D., Thareja R. R., Lewis C. H., “Three-dimensional viscous flow-field computations in a streamline coordinate system”, J. Spacecraft and Rockets, 19:1 (1982), 41–46 | DOI

[109] Kim M. D., Thareja R. R., Lewis C. H., “Three-dimensional viscous flow-field computations in a streamline coordinate system”, AIAA, 401 (1981)

[110] Kobayashi M. H., Pereira I. C. F., “Characteristic-base pressure correction at all speeds”, AIAA Journal, 34:2 (1996), 272–280 | DOI | Zbl

[111] Kreiss H. O., Oligger J., “Comparison of accurate methods for the integration of hyperbolic equations”, Tellus, 24 (1972), 199–215 | MR

[112] Kreskovsky J. P., Shamroth S. J., “An implicit marching method for the two-dimensional reduced Navier–Stokes equations at arbitrary Mach number”, Computer Methods in Applied Mechanics and Engineering, 13 (1978), 307–334 | DOI | Zbl

[113] Miller J. H., Tannehill J. C., Lawrence S. L., Edwards T. A., “Parabolized Navier–Stokes code for hypersonic flows in thermo-chemical equilibrium or nonequiliobrium”, Computers and Fluids, 27:2 (1998), 199–215 | DOI | MR | Zbl

[114] Patankar S. V., Spalding D. B., “A calculation procedure for heat, mass, and momentum transfer in threedimensional parabolic flow”, Intern. Journal of Heat and Mass Transfer, 15 (1972), 1787–1806 | DOI | Zbl

[115] Patnaik G., Guirguis R. H., Boris J. P., Oran E. S., “A barely implicit correction for flux-corrected transport”, J. Comput. Phys., 71 (1987), 1 | DOI | Zbl

[116] Prabhu D. K., Tannehill J. C., Marvin J. G., “A new PNS code for three-dimensional chemically reacting flows”, AIAA, 1987, 1472

[117] Rae W. J., “Some numerical results on viscous low-density nozzle flows in the slender-channel approximation”, AIAA, 9:5 (1971), 811–820 | DOI

[118] Roberts D. W., Forester C. K., “Parabolic procedure for flows in ducts with arbitrary cross sections”, AIAA, 17:1 (1979), 33–40 | DOI | Zbl

[119] Rogov B. V., Sokolova L. A., “Smooth Channel Approximation of Viscouse Flows through Laval Nozzle”, VIII International Conference on Methods of Aerophysical Research. ICMAR 96, Proceedings. Part 1 (Sept. 3–7, 1996, Novosibirsk), 1996, 180–185

[120] Rogov B. V., Sokolova L. A., “Efficient Simplified Model For Internal Viscous Flow”, AIAA, 1998, 98–2493

[121] Rogov I. A., Sokolova L. A., “Turbulent Chemical-Reacting Gases Flows Through Curved Smooth Wall Channels”, Proceedings of International Conference on the Methods of Aerophysical Research (29 June–3 July, 1998, Novosibirsk), 1998, 179–184

[122] Rogov B. V., Sokolova I. A., “Fast Numerical Method for Turbulent Flows Through A Laval Nozzle”, Proceedings of 'Finite-Difference Methods: Theory and Application. CFDM 987, V. 3, eds. A. A. Samarskii, Minsk, Belarus, 1998, 47–52 | MR

[123] Rogov B. V., Sokolova I. A., Turbulent Combustion Flow Through Cross Section Channel, Proceedings of 5-th ASME/JSME Joint Thermal Engineering Conference March 15–19, 1999. Can Diego, California. 1999, AJTE 99-6132

[124] Rogov B. V., Sokolova I. A., “Numerical model for subsonic and supersonic viscous flow with strong curvature streamlines”, Proceedings of International Conference on the Methods of Aerophysical Research, Part 3 (10–15 July, 2000, Novosibirsk), 2000, 112–117

[125] Rothmayer A. P., Davis R. T., “Massive Separation and Dynamic Stall on a Cusped Trailing Edge Airflot”, Numerical and Physical Aspects in Aerodynamic Flows, ed. T. Sebeci, Springar-Verlag, NY, 1985

[126] Rubin S. G., Reddy D. R., “Analysis of global pressure relaxation for flows with strong interaction and separation”, Computer and Fluids, 11:4 (1983), 281–306 | DOI | Zbl

[127] Rubin S. G., “Reduced Navier–Stokes Euler pressure relaxation anf flux-vector splitting”, Computer and Fluids, 16:4 (1988), 485–490 | DOI | Zbl

[128] Rubin S. G., Tannehill J. C., “Parabolized/reduced Navier–Stockes computational techniguies”, Annual Review of Fluid Mechanics, 24 (1992), 117–144 | MR | Zbl

[129] Rubin S. G., Khosla P. K., “Polynomial interpolation method for viscous flow calculation”, Journal of Comput. Physics, 24 (1977), 217–224 | DOI | MR

[130] Rubin S. G., Khosla P. K., “Reduced Navier–Stokes computations for compressible viscous flows”, Comput. Syst. Engng., 1 (1990), 549–562 | DOI

[131] Sesterhenn J., Muller B., Thomann H., “Flux-vector splitting for compressible low Mach number flow”, Computers and Fluids, 22:4–5 (1993), 441–451 | DOI | MR | Zbl

[132] Shuen J. S., Chen K. H., Choi Y., “A time-accurate algorithm for chemical non-equilibrium viscous flows at all speeds”, AIAA, 1992, 92–3639

[133] Srinivasan K., Rubin S. G., “Segmented multigrid domain decomposition procedure for incompressible viscous flow”, International Journal of Numerical Methods in Fluids, 15 (1992), 1333–1335 | DOI

[134] TenPas P. W., Pletcher R.H., “Coupled space-marching method for the Navier–Stokes equations for subsonic flows”, AIAA, 29:2 (1991), 219–226 | DOI

[135] Turkel E., “Review of preconditioning methods for fluid dynamics”, Applied Numerical Mathematics, 12 (1993), 257–284 | DOI | MR | Zbl

[136] Volpe G., “On the use and accuracy of compressible flow codes at low Mach numbers”, AIAA, 1991, 91–1662

[137] Vigneron Y. C., Rakich J. V., Tannehill J. C., “Calculation of supersonic viscous flow over delta wings with sharp subsonic leading edges”, AIAA, 1978, 78–1137

[138] Wadawadigi G., Tannehill J. C., Buelow Ph. E., Lawrence S. L., “A three-dimensional upwind PNS code for chemically scramjet flowfields”, AIAA, 1992, 92–2898

[139] Weiland C., Schroder W., Menne S., “An external insight into hypersonic flow phenomena using numerical methods”, Computer Fluids, 22:4–5 (1993), 407–426 | DOI | Zbl

[140] Williams J. C., “Viscous compressible and incompressible flow in slender channels”, AIAA, 1:1 (1963), 186–195 | DOI | MR | Zbl

[141] Yamasaki N., Veda Y., Namba M., “Numerical simulation of supersonic flows with chemical reactions”, Mem. Fac. Eng. Kyushu Univ., 52:4 (1992), 433–462 | MR