Author(s): Jun Hong Liang
Linked Author(s):
Keywords: Boltzmann theory; Shallow water equations
Abstract: Boltzmann theory can provide a powerful tool to hydraulic modeling and analysis. Boltzmann theory establishes the mathematical link between the gross properties of matter and the motion of the molecules that make up this matter. The probabilistic moments of the Boltzmann equation are known to provide the Navier-Stokes equations. In addition, the Boltzmann equation has been shown to be applicable in flow problems where the Navier-Stokes equations fail to apply. Moreover, the Boltzmann approach provides the basis for the theoretical deriving of well known phenomenological relations, such as Newton’s law of viscosity. Furthermore, the Boltzmann approach has been successfully exploited by a number of researchers in various fields. Perhaps the two applications that are most relevant to the hydraulic community are: application of Boltzmann theory to turbulence modeling application of Boltzmann theory to numerical modeling. For example, recent research shows that well known turbulence models can be derived from the Boltzmann theory if one exploits the analogy between molecular fluctuations and turbulent fluctuations. This paper focuses on illustrating how the Boltzmann theory can be used to formulate numerical algorithms for shallow water equations in vertical plate. The paper begins by showing the classical shallow water equations in vertical plate are obtainable from the moments of the Boltzmann equation. This connection is then exploited to formulate a numerical model for shallow flows in the vertical plate on the basis of the Boltzmann equation. In particular, since the shallow water equations are moments of the Boltzmann equation, it follows that the discrete form of the shallow water equations can be derived by taking moments of the discrete Boltzmann equation. The Boltzmann-based numerical model is applied to small amplitude waves, large amplitude waves (bores) and to viscous as well as turbulent flows in channels. The advantages of using the Boltzmann-based to formulate numerical algorithms for surface water flows are summarized. The author envisions that much can be learned by exploring the merits of Boltzmann theory in hydraulics.
Year: 2005