Periodicity of localized traveling waves in Poiseuille-Rayleigh- Benard flow

Abstract

In the cold winter, there is a singular phenomenon on the surface of the lake, and people can see that the bubble will continuously bubble out from the surface of the lake. Scientists have studied this phenomenon, and found that the convective motion is caused by the temperature difference between the surface and the bottom of the lake. The convective motion is very common and also of importance to our lives. As the results of the British scholar Rayleigh and French scholar Benard have a significant impact on the study of convective movement, the researchers use the Rayleigh-Benard to name the convective motion. Scholars have conducted a lot of research on the Rayleigh-Benard convection, with which the mechanism of the occurrence and development of the Rayleigh-Benard convection not only can be understood, but also can be provided to help our life. The experimental theory of the Rayleigh-Benard convection is very simple. Firstly, a liquid is added between two thin metal plates; secondly, the temperature of the upper plate is kept constant with the lower plate heated, thereby the temperature difference formed between two plates used to heat the liquid; finally, the liquid is heated till it exceeds a critical value, which can be formed on the Rayleigh-Benard convection. With the development of computer technology, the researchers use the computer software to carry out the numerical simulation of the fully two-dimensional dynamics equations to the Rayleigh-Benard convection, with rich research results obtained. By changing the shape of the cavity, exerting different boundary conditions and heating methods, and applying different experimental materials, the researchers have studied the convection motion and obtained the growth characteristics, the change process and the physical characteristics of the pattern. The two-dimensional numerical simulation of the fully hydrodynamic equations is used to study periodicity of localized traveling wave in Poiseuille-Rayleigh-Benard flow, and it is found that the stable period of localized traveling wave is influenced by the horizontal flow Reynolds number and the reduced Rayleigh number. For Prandtl number Pr =6.99, the reduced Rayleigh number r =5, and the horizontal flow Reynolds number Re =6, with the periodicity of localized traveling wave found. By studying the stable period of localized traveling wave at Pr =6.99, the result indicates that the stable period of localized traveling wave reduces with increasing Re and increasing r , and that its increase rate for a small reduced Rayleigh number is greater than that for a lager reduced Rayleigh number. For the reduced Rayleigh number r =8, the stable period of localized traveling wave is analyzed at Pr =0.72, 0.0272, which is different with the variation of the stable period of localized traveling wave with the horizontal flow Reynolds number when the Prandtl number Pr =6.99. The stable period increases with increasing Re for Pr =0.72, 0.0272, with the increasing rate of Pr =0.72 being larger than Pr =0.0272.