LOW DIMENSIONAL MODELING OF TURBULENT THERMAL CONVECTION

Park HM

Korean Journal of Chemical Engineering, Vol.13, No.2, 136-143, March, 1996

The Karhunen-Loeve decomposition is used to obtain a low dimensional model describing the dynamics of turbulent thermal convection in a finite box. The Karhunen-Loeve decomposition is a procedure for decomposing a stochastic field in an optimal way such that the stochastic field can be represented with a minimum number of degree of freedom. Numerical data for the turbulent thermal convection, generated by a pseudo-spectral method for the case of Pr=0.72 and aspect ratio=2, are processed by means of Karhunen-Loeve decomposition to yield a set of empirical eigenfunctions. A Galerkin procedure employing this set of empirical eigenfunctions reduces the Boussinesq equation to a small number of ordinary differential equations. this low dimensional model obtained from numerical data at the reference Rayleigh number of 70 times the critical Rayleigh number is found to predict turbulent convection reasonably well over a range of Rayleigh numbers around the reference value.

Keywords:Turbulence;Thermal Convection;Karhunen-Loeve Decomposition;Direct Spectral Simulation;Low Dimensional Dynamics

[References]

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McLaughlin JB, Orszag, S.A., J. Fluid Mech., 122, 123 (1982)
Park H, Sirovich L, Phys. Fluids A, 2, 1659 (1990)
Silveston PL, Forsch. Ing. Wes., 24, 29 (1958)
Sirovich L, Quar. Appl. Mach., XLV(3), 561 (1987)
Sirovich L, Park H, Phys. Fluids A, 2, 1649 (1990)
Sirovich L, Rodriguez JD, Phys. Lett. A, 190(5), 211 (1987)
Sirovich L, Quar. Appl. Math., XLV, 583 (1987)

[1] Aubry N, Holmes P, Lumley JL, Stone E, J. Fluid Mech., 192, 115 (1988)

[2] Brown GL, Roshko A, J. Fluid Mech., 64, 775 (1974)

[3] Canuto C, Hussaini MY, Quateroni A, Zang T, "Spectral Methods in Fluid Dynamics," Springer-Verlag (1988)

[4] Castaing B, Gunaratne G, Heslot F, Kadanoff L, Libchaber A, Thomas S, Wu XZ, Zaleske S, Zanetti G, J. Fluid Mech., 204, 1 (1989)

[5] Chandrasekhar S, "Hydrodynamic and Hydromagnetic Stability," Oxford, Clarendon Press (1961)

[6] Constantine P, Foias C, Manley OP, Temam R, J. Fluid Mech., 150, 427 (1985)

[7] Deane AE, Sirovich L, J. Fluid Mech., 222, 231 (1991)

[8] Garon AM, Goldstein RJ, Phys. Fluids, 16, 1818 (1973)

[9] Grotzbach G, J. Comput. Phys., 491, 241 (1983)

[10] Herring JH, Wyngaard J, "Direct Numerical Simulation of Turbulent Rayleigh-Benard Convection," in Fifth Symposium on Turbulent Shear Flows, 39, Springer, Berlin (1986)

[11] Kessler R, J. Fluid Mech., 174, 357 (1987)

[12] Lumley JL, "The Structure of Inhomogeneous Turbulent Flows, in Atmospheric Turbulence and Radio Wave Propagation," ed. Yaglom, A.M. and Tatarski, V.I., pp. 166-176, Nauka, Moscow (1967)

[13] McLaughlin JB, Orszag, S.A., J. Fluid Mech., 122, 123 (1982)

[14] Park H, Sirovich L, Phys. Fluids A, 2, 1659 (1990)

[15] Silveston PL, Forsch. Ing. Wes., 24, 29 (1958)

[16] Sirovich L, Quar. Appl. Mach., XLV(3), 561 (1987)

[17] Sirovich L, Park H, Phys. Fluids A, 2, 1649 (1990)

[18] Sirovich L, Rodriguez JD, Phys. Lett. A, 190(5), 211 (1987)

[19] Sirovich L, Quar. Appl. Math., XLV, 583 (1987)