A numerical study of three-dimensional Benard convection is carried out to determine the amplitude of convection as a function of Rayleigh number, Prandtl number and horizontal plan forms of convection cell. We treat three types of plan forms, namely, square, rectangle and roll, all of which have the same horizontal wavenumber. The calculations are carried out for air and water in the range of R≤8Rc, where R is prescribed Rayleigh number and Rc is the critical Rayleigh number.
The upper and lower boundaries are assumed to be free-slip and conducting. The variables such as velocity, temperature and pressure, are expanded in a series consisting of the eigenfunctions of the linear stability problem and the system is truncated to take into account only a limited number of terms. The amplitudes of the eigen-functions are evaluated by numerical integration of resulting non-linear equations.
The main results are summerized as follows.
(1) In all cases considered, the system achievs a steady state, a single mode being dominant.
(2) The non-dimensional amplitudes of convection increase except perturbation temperature as R increases.
(3) The velocity pattern depends on Prandtl number more markedly than the temperature pattern does in the three-dimensional convection. On the other hand, in the two-dimensional convection, the dependences of velocity and temperature patterns on Prandtl number are very small.
(4) The dependence of vertical heat flux on horizontal plan forms appears more markedly for air than for water. In the case of air Nusselt number for the two-dimensional convection is larger than that for three-dimensional convection.
(5) The reversed gradient of horizontally averaged temperature is found in the middle layer at high Rayleigh number. The gradient is larger for the two-dimensional convection than for the three-dimensional convection.
(6) The energy budget of convection indicates that the most characteristic difference between threedimensional and two-dimensional convections is found in the conversion rate of kinetic energy through inertial term, that is, the conversion rate for two-dimensional convection is much smaller than that for three-dimensional convection.
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