Numerical simulation of liquid sloshing in a partially filled container with inclusion of compressibility effects

A numerical scheme of study is developed to model compressible two-fluid flows simulating liquid sloshing in a partially filled tank. For a two-fluid system separated by an interface as in the case of sloshing, not only a Mach-uniform scheme is required, but also an effective way to eliminate unphysical numerical oscillations near the interface. By introducing a preconditioner, the governing equations expressed in terms of primitive variables are solved for both fluids (i.e. water, air, gas etc.) in a unified manner. In order to keep the interface sharp and to eliminate unphysical numerical oscillations in unsteady fluid flows, the non-conservative implicit Split Coefficient Matrix Method (SCMM) is modified to construct a flux difference splitting scheme in the dual time formulation. The proposed numerical model is evaluated by comparisons between numerical results and measured data for sloshing in an 80% filled rectangular tank excited at resonance frequency. Through similar comparisons, the investigation is further extended by examining sloshing flows excited by forced sway motions in two different rectangular tanks with 20% and 83% filling ratios. These examples demonstrate that the proposed method is suitable to capture induced free surface waves and to evaluate sloshing pressure loads acting on the tank walls and ceiling

Discontinuous Galerkin Methods for inviscid low Mach number flows

In this work we present two preconditioning techniques for inviscid low Mach number flows. The space discretization used is a high-order Discontinuous Galerkin finite element method. The time discretizations analyzed are explicit and implicit schemes. The convective physical flux is replaced by a flux difference splitting scheme. Computations were performed on triangular and quadrangular grids to analyze the influence of the spatial discretization. For the preconditioning of the explicit Euler equations we propose to apply the fully preconditioning approach: a formulation that modifies both the instationary term of the governing equations and the dissipative term of the numerical flux function. For the preconditioning of the implicit Euler equations we propose to apply the flux preconditioning approach: a formulation that modifies only the dissipative term of the numerical flux function. Both these formulations permit to overcome the stiffness of the governing equations and the loss of accuracy of the solution that arise when the Mach number tends to zero. Finally, we present a splitting technique, a proper manipulation of the flow variables that permits to minimize the cancellation error that occurs as an accumulation effect of round-off errors as the Mach number tends to zero

Preconditioned conjugate-gradient methods for low-speed flow calculations

An investigation is conducted into the viability of using a generalized Conjugate Gradient-like method as an iterative solver to obtain steady-state solutions of very low-speed fluid flow problems. Low-speed flow at Mach 0.1 over a backward-facing step is chosen as a representative test problem. The unsteady form of the two dimensional, compressible Navier-Stokes equations is integrated in time using discrete time-steps. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. The new iterative solver is used to solve a linear system of equations at each step of the time-integration. Preconditioning techniques are used with the new solver to enhance the stability and convergence rate of the solver and are found to be critical to the overall success of the solver. A study of various preconditioners reveals that a preconditioner based on the Lower-Upper Successive Symmetric Over-Relaxation iterative scheme is more efficient than a preconditioner based on Incomplete L-U factorizations of the iteration matrix. The performance of the new preconditioned solver is compared with a conventional Line Gauss-Seidel Relaxation (LGSR) solver. Overall speed-up factors of 28 (in terms of global time-steps required to converge to a steady-state solution) and 20 (in terms of total CPU time on one processor of a CRAY-YMP) are found in favor of the new preconditioned solver, when compared with the LGSR solver

Flux Splitting for stiff equations: A notion on stability

For low Mach number flows, there is a strong recent interest in the development and analysis of IMEX (implicit/explicit) schemes, which rely on a splitting of the convective flux into stiff and nonstiff parts. A key ingredient of the analysis is the so-called Asymptotic Preserving (AP) property, which guarantees uniform consistency and stability as the Mach number goes to zero. While many authors have focussed on asymptotic consistency, we study asymptotic stability in this paper: does an IMEX scheme allow for a CFL number which is independent of the Mach number? We derive a stability criterion for a general linear hyperbolic system. In the decisive eigenvalue analysis, the advective term, the upwind diffusion and a quadratic term stemming from the truncation in time all interact in a subtle way. As an application, we show that a new class of splittings based on characteristic decomposition, for which the commutator vanishes, avoids the deterioration of the time step which has sometimes been observed in the literature

A Global Preconditioning Method for the Euler Equations

This study seeks to validate a recently introduced global preconditioning technique for the Euler equations. Energy and enthalpy equations are nondimensionalized by means of a reference enthalpy, resulting in increased numerical accuracy for low-speed flows. A cellbased, finite volume formulation is used, with Roe flux difference splitting and both explicit and implicit time integration schemes. A Newton-linearized iterative implicit algorithm is implemented, with Symmetric Gauss-Seidel (LU/SGS) nested sub-iterations. This choice allows one to retain time accuracy, and eliminates approximate factorization errors, which become dominant at low speed flows. The linearized flux Jacobians are evaluated by numerical differentiation. Higher-order discretization is constructed by means of the MUSCL approach. Locally one-dimensional characteristic variable boundary conditions are implemented at the farfield boundary. The preconditioned scheme is successfully applied to the following traditional test cases used as benchmarks for local preconditioning techniques: point disturbance, flow angle disturbance, and stagnation point arising from the impingement of two identical jets. The flow over a symmetric airfoil and a convergentdivergent nozzle are then simulated for arbitrary Mach numbers. The preconditioned scheme greatly enhances accuracy and convergence rate for low-speed flows (all the way down to M ≈ 10E − 4). Some preliminary tests of fully unsteady flows are also conducted

An Investigation of Flux-Splitting Algorithms for Chemically Reacting Flows

This paper presents an investigation of seven different flux splitting algorithms for the discretization of inviscid fluxes, which are the primary source for the non-linear behavior (eg. shocks, contact discontinuities). The aim of the present work is to enhance the accuracy and robustness of CHEM, a three-dimensional flow solver, which is capable of simulating a wide range of flow conditions, including chemical non-equilibrium. Five different test cases cases are considered and thoroughly analyzed. The overall goal is to find a numerical scheme that can meet some stringent specifications of efficiency, accuracy and robustness on the widest possible spectrum of flow conditions