Poiseuille-Rayleigh-Bénard流动中对流斑图的分区和成长

采用SIMPLE算法对二维流体力学基本方程组进行了数值模拟,研究了Poiseuille-Rayleigh-Bénard流动中对流斑图的分区、成长及水平流动对不同斑图特征物理量的影响.结果表明,上下临界雷诺数Re_u,Re_l将流动分成三个区域,即行波区、局部行波区、水平流区.Re_u和Re_l随着相对瑞利数r的增大而增大.在对流斑图的成长阶段,三种斑图随时间的成长过程是不同的,但对流圈都是从下游区开始成长;特征物理量随着时间的变化也是不同的,行波对流和局部行波对流的最大垂直流速wmax和努塞尔数Nu经过指数增长阶段后进入周期变化的稳定阶段;水平流斑图的w_(max)和Nu经过缓慢增长后又缓慢降到稳定值.三种斑图的w_(max)和Nu随雷诺数Re增大而减小,不同斑图区域有不同的变化规律.本文给出了Re_u和Re_l随r的变化关系式及不同斑图的w_(max)和Nu随着Re的变化关系式.

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

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-Bénard 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-Bénard flow model possess certain representative and theoretical significance and practical value. So far, the investigation on the Poiseuille-Rayleigh-Bénard 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-Bénard 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-Bénard 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 Re_u and Re_l, i.e., traveling wave zone, localized traveling wave zone, and horizontal flow zone, where each of the Re_l and Re_u 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 w_max 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 w_max and Nu of horizontal flow pattern decrease down to a stable constant after slow increase. The values of w_max 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 Re_l and Re_u varying with r and formulas for computing the w_max and Nu varying with Re in different convection patterns are suggested.