Showing papers on "MUSCL scheme published in 2013"

Two-Dimensional Slope Limiters for Finite Volume Schemes on Non-Coordinate-Aligned Meshes

Abstract: In this paper we develop a new limiter for linear reconstruction on non-coordinate-aligned meshes in two space dimensions, with focus on Cartesian embedded boundary grids. Our limiter is inherently two dimensional and linearity preserving. It separately limits the $x$ and $y$ components of the gradient, as opposed to a scalar limiter which limits all components simultaneously with one scalar. The limiter is based on solving a tiny linear program (LP) on each cell, using a very efficient version of the simplex method. A variety of computational results on triangular and embedded boundary meshes are presented. They demonstrate that the LP limiter successfully removes oscillations and significantly increases solution accuracy compared to a scalar limiter.

On the optimization of flux limiter schemes for hyperbolic conservation laws

Abstract: A classic strategy to obtain high-quality discretizations of hyperbolic partial differential equations is to use flux limiter (FL) functions for blending two types of approximations: a monotone first-order scheme that deals with discontinuous solution features and a higher order method for approximating smooth solution parts. In this article, we study a new approach to FL methods. Relying on a classification of input data with respect to smoothness, we associate specific basis functions with the individual smoothness notions. Then, we construct a limiter as a linear combination of the members of parameter-dependent families of basis functions, and we explore the possibility to optimize the parameters in interesting model situations to find a corresponding optimal limiter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

A general approach to enhance slope limiters on non-uniform rectilinear grids

Abstract: Most slope limiter functions in high-resolution finite volume methods to solve hyperbolic conservation laws are designed assuming one-dimensional uniform grids, and they are also used to compute slope limiters in computations on non-uniform rectilinear grids. However, this strategy may lead to either loss of total variation diminishing (TVD) stability for 1D linear problems or the loss of formal second-order accuracy if the grid is highly non-uniform. This is especially true when the limiter function is not piecewise linear. Numerical evidences are provided to support this argument for two popular finite volume strategies: MUSCL in space and method of lines in time (MUSCL-MOL), and capacity-form differencing. In order to deal with this issue, this paper presents a general approach to study and enhance the slope limiter functions for highly non-uniform grids in the MUSCL-MOL framework. This approach extends the classical reconstruct-evolve-project procedure to general grids, and it gives sufficient conditions for a slope limiter function leading to a TVD stable, formal second-order accuracy in space, and symmetry preserving numerical scheme on arbitrary grids. Several widely used limiter functions, including the smooth ones by van Leer and van Albada, are enhanced to satisfy these conditions. These properties are confirmed by solving various one-dimensional and two-dimensional benchmark problems using the enhanced limiters on highly non-uniform rectilinear grids.

Force analysis of underwater object with supercavitation evolution

Abstract: Supercavitation generally occurs as a result of flow acceleration along underwater body surface and is numerically investigated in this study using a compressible Navier-Stokes equations solver. Here, the supercavitating flow is assumed to be the homogeneous mixture of pure liquid water and vapour which are in kinematic and thermodynamic equilibrium. Liquid phase and cavitation are modeled by Tait equation of state (EOS) and isentropic one-fluid formulation, respectively. Convective terms of the governing equations are numerically integrated using Godunov-type, cell-centered finite volume MUSCL scheme on unstructured triangular mesh, whereas time integration is handled with the second-order accurate Runge-Kutta approach. Our interest is focused on the force analysis of traveling object with the formation, growth, evolution and even collapse of supercavity enveloping the object. It is found that skin friction drag exerted on the object can be reduced significantly by the formation of supercavity where viscosity of vapour is much smaller than that of liquid water. It is also observed that form drag acting on the object is influenced by the supercavitation. Collapse of supercavity over the body due to external perturbation not only damages underwater object but also alters form drag on it.

Simulation of Supercavitating Flow Accelerated by Shock

Abstract: In this study, supercavitating flow impacted by shock is numerically investigated. The physical model based on inviscid compressible Euler equations is adopted by assuming that two-phase regime is a homogeneous mixture of liquid and vapor which remains in kinematic and thermodynamic equilibrium and behaves as one single fluid. The thermodynamic properties of pure liquid and two-phase mixture are modeled by Tait equation of state (EOS) and isentropic formulation, respectively. The physical model provides a macroscopic description of phase transition and is hyperbolic in time. The inviscid fluxes of governing equations are discretized using cell-centered finite volume MUSCL scheme on triangular mesh, while time integration is performed with two-stage Runge–Kutta scheme. The numerical method is second-order accurate both in space and time. The assembled computational fluid dynamics (CFD) code is validated against analytical solution. Numerical simulation is carried out to study the interaction between supercavitation and pressure waves including shock wave as well as the wave propagation in single- and two-phase media. The subsequent time evolution of supercavity impacted by a pressure wave is strongly influenced by freestream flow speed and pressure wave strength. For a relatively low upstream flow velocity, the supercavity over underwater object may be deformed by a weak explosion wave. However, the supercavitation bubble formed at a high freestream speed appears much more stable. Under strong pressure wave, the supercavity experiences deformation along its boundary which may lead to local collapse. A well-developed supercavity profile is recovered soon after the wave exits the computational domain.

A Surface Flow Routing Algorithm Based on Shallow Water Equation with Kinematic Wave Approximation

Abstract: A two-dimensional numerical model is developed to simulate turbulent shallow-water flow. The model is based on two-dimensional depth-averaged Navier-Stokes equations. A second-order Godunov-type upwind finite volume scheme with augmented HLLC Riemann solver is implemented. The conservative variables near the edges of cells are linearly reconstructed by the MUSCL scheme. The reconstructions are based on the primitive variables. The time marching scheme is second-order TVD Runge-Kutta scheme which can prevent the occurrence of oscillation in every intermediate stage. The model uses first-order approximations for the wet-dry fronts and boundaries which make the solution as robust as possible. An additional flux is calculated to keep the scheme well-balanced. To provide body-fitted mesh, the Cartesian cut-cell method is adopted. The ߢ−ߝ turbulence model is implemented as the turbulence model closure. The model is tested against several laboratory experiments and field measurements. In all test cases, the simulated results agree well with the observations.