Showing papers on "MUSCL scheme published in 2008"

A second-order semi-implicit hybrid scheme for one-dimensional Boussinesq-type waves in rectangular channels

Abstract: A numerical procedure is proposed for the solution of the Boussinesq equations in rectangular channels over a horizontal bed. The Boussinesq equations account for non-hydrostatic effects in free-surface flows. The proposed approach is a predictor-corrector procedure where the hyperbolic part of the equations is treated using a higher-order estimate of the variable slope of the Godunov-type, MUSCL scheme. The proposed slope estimate is third-order accurate on irregular grids and fourth-order accurate on regular grids. The dispersive, non-hydrostatic terms are treated using a semi-implicit discretization. A stability analysis of the proposed predictor-corrector procedure shows that optimal accuracy is achieved when the implicitation parameter is set equal to 0.5. This analysis is confirmed by numerical experiments, whereby the propagation of a solitary wave and the development of an undular bore are reproduced and compared successfully with the available analytical solutions. The proposed method is shown to be of a good level of accuracy without being too sensitive to the numerical parameters. Its applicability in practical problems is illustrated by a comparison with laboratory experiments. Copyright (C) 2008 John Wiley & Sons, Ltd.

A simple high‐resolution advection scheme

Abstract: A simple, robust, mass-conserving numerical scheme for solving the linear advection equation is described. The scheme can estimate peak solution values accurately even in regions where spatial gradients are high. Such situations present a severe challenge to classical numerical algorithms. Attention is restricted to the case of pure advection in one and two dimensions since this is where past numerical problems have arisen. The authors' scheme is of the Godunov type and is second-order in space and time. The required cell interface fluxes are obtained by MUSCL interpolation and the exact solution of a degenerate Riemann problem. Second-order accuracy in time is achieved via a Runge-Kutta predictor-corrector sequence. The scheme is explicit and expressed in finite volume form for ease of implementation on a boundary-conforming grid. Benchmark test problems in one and two dimensions are used to illustrate the high-spatial accuracy of the method and its applicability to non-uniform grids.

Volume fraction flux approximation in a two-fluid flow

Abstract: The broken dam problem flow is tested to check accuracy of different procedures for gas-liquid interface resolution based on solution of the additional equation for the volume fraction of liquid phase. The study is focused on the numerical schemes used to approximate advection fluxes of this equation. In particular, the MUSCL scheme with QUICK interpolants and compressive minmod TVD limiters with the slope modification technique for the volume fraction fluxes is applied, as well as the upwind-downwind donor acceptor procedure designed in the VOF method. As the first stage, the quite simple and explicit procedure adopting the artificial compressibility method is used to solve the velocity and pressure equations. Computations are initially performed with a careful grid and time step independence studies. Importance of the wall boundary condition is also discussed. To present free surface motion, results of numerical investigation are shown in terms of contour plots for the volume fraction at successive times, as well as surge front and column height positions versus time.

Time-linearized time-harmonic 3-D Navier–Stokes shock-capturing schemes

Abstract: In the present paper, a numerical method for the computation of time-harmonic flows, using the time-linearized compressible Reynolds-averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux-vector-splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen-turbulence-Reynolds-number hypothesis, to approximate eddy-viscosity perturbations. The resulting linear system is solved using a pseudo-time-marching implicit ADI-AF (alternating-directions-implicit approximate-factorization) procedure with local pseudo-time-steps, corresponding to a matrix-successive-underrelaxation procedure. The stability issues associated with the pseudo-time-marching solution of the time-linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time-nonlinear computations for 3-D shock-wave oscillation in a square duct, for various back-pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock-capturing capability of the time-linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.

Generalized Riemann Problems: From the Scalar Equation to Multidimensional Fluid Dynamics

Abstract: The Generalized Riemann Problem (GRP) is the initial-value problem for nonlinear hyperbolic systems of (quasi) conservation laws, in one space dimension. The initial data in this case are piecewise linear, with possible jump discontinuities (of the unknowns and their slopes). The classical Riemann Problem (RP) serves as a primary “building block” in the construction of many numerical schemes (most notably the Godunov scheme). Likewise, the GRP plays a key role in the design of second-order high-resolution schemes (e.g., the MUSCL scheme). The analytic study of the GRP, both for scalar conservation laws and for systems, leads to an array of “GRP schemes” which generalize the Godunov method and at the same time are explicit, robust numerical algorithms, capable of resolving complex multidimensional fluid dynamical problems. (M. Ben-Artzi and J. Falcovitz, “Generalized Riemann Problems in Computational Fluid Dynamics”, Cambridge University Press, 2003). The paper reviews the basic theory in the scalar case, with special attention to the surprising complexity of scalar 2-D “Riemann-type” problems (the Guckenheimer equation). The GRP analysis is then extended to the case of quasi 1-D compressible, inviscid, non-isentropic flow. The basic analytic facts and the resulting numerical algorithms are outlined. Special attention is devoted to the “Acoustic Approximation”, which is a very simple (yet second-order) modification of the Godunov scheme. Some simulations of rather complex two-dimensional flows are presented and compared with experimental data.

Assessment of Large-Eddy Simulation for Separated Internal Flow

Abstract: This paper presents a systematic numerical investigation of dierent Implicit Large-Eddy Simulations (ILES) for massively separated flows. Three numerical schemes, a third-order MUSCL scheme, a fifth-order MUSCL scheme and a ninth-order accurate WENO method, are tested in the context of separation from a gently curved surface. The case considered here is a simple wall-bounded flow that consists of a channel with hill-type curvature on the lower wall. The separation and reattachment locations, velocity and stress profiles are presented and compared against solutions from classical LES simulations.

Volume fraction flux approximation in a two-fluid flow *

Abstract: The broken dam problem flow is tested to check accuracy of different procedures for gas-liquid interface resolution based on solution of the additional equation for the volume fraction of liquid phase. The study is focused on the numerical schemes used to approximate advection fluxes of this equation. In particular, the MUSCL scheme with QUICK interpolants and compressive minmod TVD limiters with the slope modification technique for the volume fraction fluxes is applied, as well as the upwind- downwind donor acceptor procedure designed in the VOF method. As the first stage, the quite simple and explicit procedure adopting the artificial compressibility method is used to solve the velocity and pressure equations. Computations are initially performed with a careful grid and time step independence studies. Importance of the wall boundary condition is also discussed. To present free surface motion, results of numerical investigation are shown in terms of contour plots for the volume fraction at successive times, as well as surge front and column height positions versus time.