High Accuracy Computations of the Hybrid Burnett Equations
A non-dimensional quantity that is used to describe fluid flows is the Knudsen numbers Kn. Kn = l/L where l is the mean free path of the particles and L a typical length scale. For Knudsen number much smaller than 1 the particles can be described as a continuum. For these applications, Navier-Stokes equations are sufficient to describe the flow. For larger Knudsen values the second order Burnett equations can be used. Doing a linear stability analysis on the Burnett equations it can be shown that the equations are not stable for all wave numbers. A method of stabilizing the equations has been suggested by Lars Söderholm, see (Söderholm, 2006). This results in the Hybrid Burnett equations. For the purpose of investigating the ability for the Hybrid Burnett equations to describe physical flows, the phenomenon of shock waves will be numerically treated for different cases.
A high amplitude sinusoidal wave will be integrated forward in time using Runge-Kutta fourth order scheme as it travels along the x-axis. Fourier’s method will be used for spatial discretization. This wave will with time form a shock wave. The high gradient in the wave profile will result in a reflected wave that is traveling in the opposite direction of the original wave. As the amplitude of the original wave and the self reflected wave decay we will have two small amplitude waves where non-linear effects can be neglected. From this old age wave the amplitude will be compared with the analytical solutions of the Burgers’ equation. The dependence of the amplitude of the self reflected wave of parameters such as the Reynolds number and Knudsen numberwill be investigated.
Source Type:Master's Thesis
Date of Publication:01/01/2009