For a fixed geometric configuration, hydrodynamic instabilities and bifurcation processes of laminar isothermal planar opposed jet flows with symmetric and slightly asymmetric inlet boundary conditions are investigated numerically using a high-resolution approach based on spectral element method. In current configuration, when inlet boundary conditions are symmetric, in the range of the Reynolds number considered (Re <= 200), multiple new symmetry-breaking bifurcations are observed and four new flow patterns are identified. Their hydrodynamic characteristics are analyzed, in particular their symmetries. In addition, the case that inlet boundary conditions are slightly asymmetric is investigated. It is found that bifurcation processes are extremely sensitive to this small symmetry-breaking imperfection and much different from those in the symmetric case. Furthermore, model equations are constructed by symmetry consideration to explain the numerical results based on hydrodynamic equations.

A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz, and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

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We introduce a unified particle framework which integrates the phase-field method with multi-material simulation to allow modeling of both liquids and solids, as well as phase transitions between them. A simple elastoplastic model is used to capture the behavior of various kinds of solids, including deformable bodies, granular materials, and cohesive soils. States of matter or phases, particularly liquids and solids, are modeled using the non-conservative Allen-Cahn equation. In contrast, materials—made of different substances—are advected by the conservative Cahn-Hilliard equation. The distributions of phases and materials are represented by a phase variable and a concentration variable, respectively, allowing us to represent commonly observed fluid-solid interactions. Our multi-phase, multi-material system is governed by a unified Helmholtz free energy density. This framework provides the first method in computer graphics capable of modeling a continuous interface between phases. It is versatile and can be readily used in many scenarios that are challenging to simulate. Examples are provided to demonstrate the capabilities and effectiveness of this approach.

The paper aims to estimate the thickness of the boundary layer for the planar MHD system with vanishing shear viscosity . Under some conditions on the initial and boundary data, we show that the thickness is of the order | In |. Note that this estimate holds also for the Navier-Stokes system so that it extends the previous works even without the magnetic effect.