DIFFRACTION OF A SOLITARY WAVE BY A THIN WEDGE PART 2. EFFECTS OF VISCOUS BOUNDARY LAYERS

Based on the study on the Mach reflection of a solitary wave in [3] , we continue to investi- gate effects of the boundary layers on the bottom and the vertical side wall. By using matched asymptotic methods, the two-dimensional KdV equation is modified to account for effects of viscosity. Numerical simulation of the problem shows that the effects of side wall are important while the effects of the bottom can be neglected. The results including the side wall's effects agree satisfactorily with those of Melville's experiments. Finally, we establish the simplified concept of the side wall effect and conclude that it repre- sents the physical reason for the discrepancy between the experiments and the previous calculations based on the inviscid fluid flow theory.

Oscillatory Dynamics of a Spherical Solid in a Liquid in an Axisymmetric Variable Cross Section Channel

The dynamics of a solid spherical body in an oscillating liquid flow in a vertical axisymmetric channel of variable cross section is experimentally studied.It is shown that the oscillating liquid leads to the generation of intense averaged flows in each of the channel segments.The intensity and direction of these flows depend on the dimensionless oscillating frequency.In the region of studied frequencies,the dynamics of the considered body is examined when the primary vortices emerging in the flow occupy the whole region in each segment.For a fixed frequency,an increase in the oscillation amplitude leads to a phase-inclusion holding effect,i.e.,the body occupies a quasi-stationary position in one of the cells of the vertical channel,while oscillating around its average position.It is also shown that the oscillating motion of a liquid column generates an averaged force acting on the body,the magnitude of which depends on the properties of the body and its position in the channel.The quasi-stationary position is determined by the relative density and size of the body,as well as the dimensionless frequency.The behavior of the body as a function of the amplitude and frequency of fluid oscillation and relative size is discussed in detail.Such findings may be used in the future to control the position of a phase inclusion and/or to strengthen mass transfer effects in a channel of variable cross section by means of fluid oscillations.

Modeling wave propagation across rock masses using an enriched 3D numerical manifold method

The three-dimensional numerical manifold method(3DNMM) method is further enriched to simulate wave propagation across homogeneous/jointed rock masses. For the purpose of minimizing negative effects from artificial boundaries, a viscous nonreflecting boundary, which can effectively absorb the energy of a wave, is firstly adopted to enrich 3DNMM. Then, to simulate the elastic recovery property of an infinite problem domain, a viscoelastic boundary, which is developed from the viscous nonreflecting boundary, is further adopted to enrich 3DNMM. Finally, to eliminate the noise caused by scattered waves, a force input method which can input the incident wave correctly is incorporated into 3DNMM. Five typical numerical tests on P/S-wave propagation across jointed/homogeneous rock masses are conducted to validate the enriched 3DNMM. Numerical results indicate that wave propagation problems within homogeneous and jointed rock masses can be correctly and reliably modeled with the enriched 3DNMM.

The Hilbert expansion of the Boltzmann equation in the incompressible Euler level in a channel

The study of the hydrodynamic limit of the Boltzmann equation with physical boundary is a challenging problem due to the appearance of the viscous and Knudsen boundary layers.In this paper,the hydrodynamic limit from the Boltzmann equation with the specular reflection boundary condition to the incompressible Euler equations in a channel is investigated.Based on the multi-scaled Hilbert expansion,the equations with boundary conditions and compatibility conditions for interior solutions,and viscous and Knudsen boundary layers are derived under different scaling,respectively.Then,some uniform estimates for the interior solutions,and viscous and Knudsen boundary layers are established.With the help of the L2-L∞ framework and the uniform estimates obtained above,the solutions to the Boltzmann equation are constructed by the truncated Hilbert expansion with multiscales,and hence the hydrodynamic limit in the incompressible Euler level is justified.