Partition and growth of convection patterns in Poiseuille-Rayleigh-Bnard flow

Abstract

The natural phenomena which we are familiar with, such as the convections in reservoir, ocean, atmosphere, etc., all occur in nonequilibrium open systems away from heat equilibria. The Poiseuille-Rayleigh-Bnard flow in a horizontal fluid layer heated from below has always been a typical experimental system for studying the nonlinear problem and the pattern formation. The experimental system can be accurately described by the full hydrodynamic equations. Therefore, the researches of the convection spatiotemporal structure, stability and the nonlinear dynamics by using the Poiseuille-Rayleigh-Bnard flow model possess certain representative and theoretical significance and practical value. So far, the investigation on the Poiseuille-Rayleigh-Bnard flow in a horizontal layer heated from below has concentrated mainly on the stability and made remarkable progress. However, a partition of convection pattern and growths of different patterns in the Poiseuille-Rayleigh-Bnard flow have been seldom studied in theory. By using a two-dimensional numerical simulation of the fully hydrodynamic equations in this paper, the research is conducted on the partition of convection pattern, growth and the effects of horizontal flow on the characteristic parameters of different patterns in the Poiseuille-Rayleigh-Bnard flow in a rectangular at an aspect ratio of 10. The SIMPLE algorithm is used to numerically simulate the two-dimensional fully hydrodynamic equations. The basic equations are solved in primitive variables in two-dimensional staggered grids with a uniform spatial resolution based on the control volume method. The power law scheme is used to treat the convective-diffusive terms in the discrete formulation. Results show that a flow zone is divided into three zones by the upper and lower critical Reynolds numbers Reu and Rel, i.e., traveling wave zone, localized traveling wave zone, and horizontal flow zone, where each of the Rel and Reu is a function of reduced Rayleigh number r and increases with increasing r. In the growth stage of the convection pattern, the growth processes of three kinds of patterns with time are different, but the convection rolls all start to grow from the downstream. The variations of characteristic parameters with time are also different, with maximum vertical velocity wmax and Nusselt number Nu of traveling wave and localized traveling wave entering into the stable stage of the cycle variation after the exponential growth stage, and the wmax and Nu of horizontal flow pattern decrease down to a stable constant after slow increase. The values of wmax and Nu of three types of patterns decrease with increasing Reynold number Re, with different laws being in the different pattern areas. In this paper, formulas for computing the Rel and Reu varying with r and formulas for computing the wmax and Nu varying with Re in different convection patterns are suggested.