Turbulent thermal convection in a finite domain: Part II. Numerical results

A pseudospectral method is used to solve the Boussinesq equations for a fully inhomogeneous turbulent flow. The numerical data are analyzed using the empirical eigenfunction technique. As a result of the underlying inhomogeneity of the flow, the eigenfunctions (structures) are inhomogeneous in all three directions. This is the first instance in which fully three‐dimensional empirical eigenfunctions have been calculated. The generated basis set is extremely efficient at depicting the flow. The first eigenfunction captures almost 60% of the average energy. The eigenfunctions are an optimal basis for capturing the energy of the flow and more than 95% of the energy is captured by the first 100 eigenfunctions. Ten classes of eigenfunctions are present and examples of each are shown. The average Nusselt number for the bounded geometry is found to be lower than that for a correspondong homogeneous case and the physics causing this decrease is analyzed and discussed.

[1]  E. Spiegel,et al.  Time-dependent solutions of multimode convection equations , 1982, Journal of Fluid Mechanics.

[2]  John Whitehead,et al.  Finite bandwidth, finite amplitude convection , 1969, Journal of Fluid Mechanics.

[3]  Steven A. Orszag,et al.  Transition from periodic to chaotic thermal convection , 1982, Journal of Fluid Mechanics.

[4]  R. M. Clever,et al.  Instabilities of convection rolls in a fluid of moderate Prandtl number , 1979, Journal of Fluid Mechanics.

[5]  S. Chandrasekhar.  Hydrodynamic and Hydromagnetic Stability , 1961 .

[6]  Lawrence Sirovich,et al.  Turbulent thermal convection in a finite domain: Part I. Theory , 1990 .

[7]  R. Kessler,et al.  Nonlinear transition in three-dimensional convection , 1987, Journal of Fluid Mechanics.

[8]  J. Massaguer,et al.  Instability of swirl in low-Prandtl-number thermal convection , 1988, Journal of Fluid Mechanics.

[9]  J. Whitehead,et al.  Instabilities of convection rolls in a high Prandtl number fluid , 1971, Journal of Fluid Mechanics.

[10]  R. Ash,et al.  Topics in stochastic processes , 1975 .

[11]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[12]  W. Malkus,et al.  The heat transport and spectrum of thermal turbulence , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[13]  Eric D. Siggia,et al.  Stability of finite‐amplitude convection , 1983 .

[14]  F. Busse,et al.  Non-linear properties of thermal convection , 1978 .

[15]  E. Spiegel,et al.  Numerical solutions of single-mode convection equations , 1977, Journal of Fluid Mechanics.

[16]  J. Lumley.  Stochastic tools in turbulence , 1970 .

[17]  Horst Leipholz,et al.  Instability of Continuous Systems , 1971 .

[18]  David Moore,et al.  Axisymmetric convection in a cylinder , 1976, Journal of Fluid Mechanics.

[19]  Lawrence Sirovich,et al.  An Eigenfunction Analysis of Turbulent Thermal Convection , 1989 .

[20]  S. Balachandar,et al.  Simulations of turbulent thermal convection , 1989 .

[21]  S. Balachandar,et al.  Numerical simulation of high Rayleigh number convection , 1989 .

[22]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[23]  W. Malkus.  Discrete transitions in turbulent convection , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[24]  G. Grötzbach,et al.  Spatial resolution requirements for direct numerical simulation of the Rayleigh-Bénard convection , 1983 .