Numerical model of a turbulent flow behind a heated grid in a wind tunnel

M. K. Вaev ; G. G. Chernykh

Matematičeskoe modelirovanie, Tome 23 (2011) no. 10, pp. 44-64.

Résumé

The closure of Corrsin equation is implemented using the gradient hypothesis linking a mixed two-point correlation moment of the third-order with two-point correlation function of the second-order of passive scalar field. Based on the closed system of Kolmogorov and Yaglom equations the numerical model of locally isotropic turbulence was constructed. Under the assumption of constancy of Loitsiansky and Corrsin invariants the self-similar solution of the Corrsin equation, corresponding to an infinitely large Reynolds and Peclet numbers, was constructed. A numerical model of the dynamics of turbulence and temperature fluctuations behind a heated grid located in a wind tunnel is constructed on the basis of closed Karman–Howarth and Corrsin equations

Détail

Keywords: locally isotropic and isotropic turbulence, Karman-Howarth and Corrsin equations, numerical modeling.
Mots-clés : Kolmogorov and Yaglom equations

Bibliographie

[1] Monin A. S., Yaglom A. M., Statisticheskaya gidromekhanika, v. 2, Izd. vtoroe, pererabotannoe i rasshirennoe, Gidrometeoizdat, Sankt-Peterburg, 1996

[2] Hasselmann K., “Zur Deutung der dreifachen Geschwindigkeits korrelationen der isotropen Turbulenz”, Dtsh. hydrogr. Zs., 1958, no. 11, 207–217 | DOI

[3] Kostomakha V. A., “Eksperimentalnoe modelirovanie izotropnoi turbulentnosti”, Nestatsionarnye zadachi mekhaniki sploshnykh sred (Dinamika sploshnoi sredy), 70, SO AN SSSR, In-t gidrodinamiki, 1985, 92–104

[4] Kostomakha V. A., Struktura izotropnoi i lokalno izotropnoi turbulentnosti, Diss. na soisk. uch. st. kand. fiz.-mat. nauk, Novosibirsk, 1986

[5] Millionschikov M. D., “Izotropnaya turbulentnost v pole turbulentnoi vyazkosti”, Pisma v ZhETF, 10:8 (1969), 406–411

[6] Millionschikov M. D., “O strukture koeffitsienta turbulentnoi vyazkosti dlya izotropnoi turbulentnosti”, Pisma v ZhETF, 11:5 (1970), 203–206

[7] Lytkin Yu. M., Chernykh G. G., “Ob odnom sposobe zamykaniya uravneniya Karmana-Khovarta”, Dinamika sploshnoi sredy, 27, SO AN SSSR, In-t gidrodinamiki, 1976, 124–130 | MR

[8] Lytkin Yu. M., Chernykh G. G., “Raschety korrelyatsionnykh funktsii v izotropnoi turbulentnosti”, Dinamika sploshnoi sredy, 35, In-t gidrodinamiki SO AN SSSR, 1978, 74–88 | MR

[9] Batchelor G. K., Townsend A. A., “Decay of isotropic turbulence in initial period”, Proc. Roy. Soc., A193 (1948), 539–558 | Zbl

[10] Townsend A. A., The structure of turbulent shear flow, Second edition, Cambridge University Press, 1976 | MR | Zbl

[11] Domaradzki J. A., Mellor G. L., “A simple turbulence closure hypothesis for the triple-velocity correlation functions in homogeneous isotropic turbulence revisited”, J. Fluid Mech., 140 (1984), 45–61 | DOI | Zbl

[12] Akatnov N. I., “O zamykanii uravneniya vtorykh dvukhtochechnykh momentov, opisyvayuschego vyrozhdenie odnorodnoi izotropnoi turbulentnosti”, Gidrodinamika, sb. nauch. trudov, LGTU, L., 1990, 31–39

[13] Akatnov N. I., Bystrova E. N., “Raschety nekotorykh kharakteristik odnorodnoi turbulentnosti na osnove resheniya uravneniya Karmana-Khovarta, zamknutogo posredstvom poluempiricheskoi modeli”, Teplofizika vysokikh temperatur, 37:4 (1999), 895–903

[14] Oberlack M., Peters N., “Closure of the Two-Point Correlation Equation as a Basis for Reynolds Stress Models”, Applied Scientific Research, 51 (1993), 533–538 | DOI | Zbl

[15] Onufriev A. T., Opisanie turbulentnogo perenosa. Neravnovesnye modeli, ucheb. posobie, Mosk. fiz.-tekhn. in-t, M., 1995

[16] Chernykh G. G., Korobitsina Z. L., Kostomakha V. A., “Numerical simulation of Isotropic Turbulence Dynamics”, International Journal of Computational Fluid Dynamics, 10:2 (1998), 173–182 | DOI | MR | Zbl

[17] Ling S. C., Huang T. T., “Decay of weak turbulence”, Phys. Fluids, 1970, no. 13, 2912–2920 | DOI

[18] Millionschikov M. D., “Vyrozhdenie odnorodnoi izotropnoi turbulentnosti v vyazkoi neszhimaemoi zhidkosti”, DAN SSSR, 22:5 (1939), 236–240

[19] Loitsyanskii L. G., “Nekotorye osnovnye zakonomernosti izotropnogo turbulentnogo potoka”, Trudy TsAGI, 440, 1939, 3–23

[20] Onufriev A. T., Pyrkova O. A., Zadacha o zatukhanii slaboi turbulentnosti v odnorodnom izotropnom potoke, preprint No 2005-1, MFTI, Dolgoprudnyi, 2005

[21] Onufriev A. A., Onufriev A. T., Pyrkova O. A., Priblizhennoe reshenie zadachi o zatukhanii odnorodnoi izotropnoi turbulentnosti s uchetom yavleniya peremezhaemosti v potoke, preprint No 2008-1, MFTI, Dolgoprudnyi, 2008

[22] Frost V. A., “Raschety vyrozhdeniya izotropnoi turbulentnosti s ispolzovaniem approksimatsii Khasselmana”, Trudy Vserossiiskoi shkoly-seminara “Aerofizika i fizicheskaya mekhanika klassicheskikh i kvantovykh sistem”, IPMekh RAN, M., 2008, 171–175

[23] Grebenev V. N., Oberlack M. A., “Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence”, Mathematical Physics, Analysis and Geometry, 12:1 (2009), 1–18 | DOI | MR | Zbl

[24] Barenblatt G. I., Gavrilov A. A., “K teorii avtomodelnogo vyrozhdeniya odnorodnoi izotropnoi turbulentnosti”, ZhETF, 65:2(8) (1973), 806–813

[25] Korneyev A. I., Sedov L. I., “Theory of isotropic turbulence and its comparison with experimental data”, Fluid Mechanics-Soviet Research, 1976, no. 5, 37–48

[26] Shumann U., Patterson J., “Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence”, J. Fluid Mech., 88 (1978), 685–709 | DOI

[27] George W., “The decay of homogeneous isotropic turbulence”, Phys. Fluids, A4 (1992), 1492–1509 | MR | Zbl

[28] Speziale C., Bernard P., “The energy decay in self-preserving isotropic turbulence revisited”, J. Fluid Mech., 241 (1992), 645–667 | DOI | MR | Zbl

[29] Chasnov J., “Computation of the Loitsiansky integral in decaying isotropic turbulence”, Phys. Fluids, 5:11 (1993), 2579–2581 | DOI | Zbl

[30] Mansour N., Wray A., “Decay of isotropic turbulence at low Reynolds number”, Phys. Fluids, 6:2 (1994), 808–814 | DOI | Zbl

[31] Metais O., Lesieur M., “Spectral large-eddy simulation of isotropic and stably stratified turbulence”, J. Fluid Mech., 239 (1992), 157–194 | DOI | MR | Zbl

[32] Corrsin S., “The Decay of Isotropic Temperature Fluctuations in an Isotropic Turbulence”, J. Aeronautical Sciences, 18:12 (1951), 417–423 | MR | Zbl

[33] Mills R. R., Kistler A. L., O'Brien V., Corrsin S., Turbulence and temperature fluctuations behind a heated grid, Tech. note/NACA, No 4288, Washington, 1958

[34] Warhaft Z., “Passive scalars in turbulent flows”, Annu. Rev. Fluid Mech., 32 (2000), 203–240 | DOI | MR | Zbl

[35] Antonia R. A., Smalley R. J., Zhou T., Anselment F., Danaila L., “Similarity solution of temperature structure functions in decaying homogeneous isotropic turbulence”, Phys. Rev. E, 69 (2004), 016305-1–016305-11 | DOI

[36] Akatnov N. I., Bystrova E. N., “Ispolzovanie modeli osesimmetrichnoi turbulentnosti dlya rascheta statisticheskikh kharakteristik pulsatsionnogo dvizheniya v potoke s odnorodnoi neizmennoi skorostyu sdviga osrednennogo dvizheniya”, Teplofizika vysokikh temperatur, 38:4 (2000), 600–608

[37] Frost V. A., Ivenskikh N. N., Krasitskii V. P., The problem of stochastic description of turbulent micromixing and combustion on the base of two-point probability distribution functions, preprint No 699, Institute for Problems in Mechanics RAS, Moscow, 2002, 28 pp.

[38] Babenko V. A., Frost V. A., Zavisimost skalyarnoi dissipatsii ot khimicheskikh prevraschenii v turbulentnykh potokakh, preprint No 840, IPMekh RAN, 2007, 33 pp.

[39] Krasitskii V. P., Frost V. A., “Molekulyarnyi perenos v turbulentnykh potokakh”, Izv. RAN. MZhG, 2007, no. 2, 46–58 | MR

[40] Babenko V. A., Frost V. A., “Modeling of Turbulent Reacting Flows on the Base of Equation for Scalar Field Correlational Function”, International Journal of Heat and Mass Transfer, 52:13–14 (2009), 3314–3319 | DOI | Zbl

[41] Baev M. K., Chernykh G. G., “Chislennoe modelirovanie turbulentnogo techeniya za nagretoi reshetkoi”, Prikladnaya mekh. i tekhn. fiz., 50:3 (2009), 118–126

[42] Baev M. K., Chernykh G. G., “On Corrsin Equation Closure”, J. of Engineering Thermophysics, 19:3 (2010), 154–169 | DOI | MR

[43] Kolmogorov A. N., “Rasseyanie energii pri lokalno izotropnoi turbulentnosti”, DAN SSSR, 32:1 (1941), 19–21

[44] Yaglom A. M., “O lokalnoi strukture polya temperatury v turbulentnom potoke”, DAN SSSR, 69:6 (1949), 743–746 | MR | Zbl

[45] Zhu Y., Antonia R. A., Hosokava I., “Refined similarity hypotheses for turbulent velocity and temperature fields”, Phys. Fluids, 7:7 (1995), 1637–1648 | DOI

[46] Antonia R. A., Zhu Y., Anselment F., Ould-Rous M., “Comparison between the sum of second-order velocity structure functions and the second order temperature structure function”, Phys. Fluids, 8:11 (1996), 3105–3111 | DOI

[47] Saddoughi S. G., Veervalli S. V., “Local isotropy in turbulent boundary layers at high Reynolds number”, J. Fluid Mech., 268 (1994), 333–372 | DOI

[48] Gibson C. H., Schwarz W. H., “The universal equilibrium spectra of turbulent velocity and scalar fields”, J. Fluid Mech., 16:3 (1963), 365–384 | DOI | MR | Zbl

[49] Demidovich B. P., Maron I. A., Shuvalova E. Z., Chislennye metody analiza, GIFML, M., 1963

[50] Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L., Metody splain-funktsii, Nauka, M., 1980 | MR

[51] Demin V. F., Dobrolyubov L. V., Stepanov V. A., Sistemy programmirovaniya na algole, Nauka, M., 1977 | Zbl

[52] Golitsyn G. S., “O strukture turbulentnosti v oblasti malykh masshtabov”, Prikladnaya matematika i mekhanika, 24:6 (1960), 1124–1129 | Zbl

[53] Sreenivasan K. R., “On local isotropy of passive scalars in turbulent shear flows”, Proc. R. Soc. Lond. A, 434 (1991), 165–182 | DOI | Zbl

[54] Loitsyanskii L. G., Mekhanika zhidkosti i gaza, Nauka, M., 1970 | MR

[55] Sedov L. I., Metody podobiya i razmernosti v mekhanike, Nauka, M., 1977 | MR

[56] Landau L. D., Lifshits E. M., Gidrodinamika, Izd. trete, pererabotannoe, Nauka, M., 1986 | MR

[57] Klimentenok U. A., Korobitsyna Zh. L., Chernykh G. G., “O chislennoi realizatsii asimptoticheskogo resheniya Loitsyanskogo–Millionschikova”, Matematicheskoe modelirovanie, 7:1 (1995), 69–80 | MR | Zbl

[58] Davidson P. A., “Was Loitsyansky correct? A review of arguments”, J. of Turbulence, 1 (2000), 2–14 http://www.jot.iop.org | DOI | MR

[59] Grebenev V. N., Filimonov M. Yu., “O sokhranenii invarianta Loitsyanskogo v modeli dinamiki odnorodnoi izotropnoi turbulentnosti”, Vestnik NGU. Ser.: Matematika, mekhanika, informatika, 14:4 (2009), 28–45 | MR

[60] Samarskii A. A., Vvedenie v teoriyu raznostnykh skhem, Nauka, M., 1971 | MR | Zbl