Explicit formulations and performance of LSFD method on Cartesian mesh

Performance of the LSFD method is compared with conventional FD schemes. Generally, 9-point stencils for 2D cases and 27-point stencils for 3D cases are used for the approximation of the first and second order derivatives obtained with conventional central difference schemes. When the same stencils are used, explicit LSFD formulations for approximation of the first and second order derivatives are presented. The LSFD formulations are actually a combination of conventional central difference schemes along relevant mesh lines. It has been found that LSFD formulations need much less iteration steps than the conventional FD schemes to converge, and the ratio of mesh spacing in the x and y directions is an important parameter in the LSFD application, with a great impact on stability of LSFD computation.

A high order boundary scheme to simulate complex moving rigid body under impingement of shock wave

In the paper, we study a high order numerical boundary scheme for solving the complex moving boundary problem on a fixed Cartesian mesh, and numerically investigate the moving rigid body with the complex boundary under the impingement of an inviscid shock wave. Based on the high order inverse Lax-Wendroff(ILW) procedure developed in the previous work(TAN, S. and SHU, C. W. A high order moving boundary treatment for compressible inviscid flows. Journal of Computational Physics, 230(15),6023–6036(2011)), in which the authors only considered the translation of the rigid body,we consider both translation and rotation of the body in this paper. In particular, we reformulate the material derivative on the moving boundary with no-penetration condition, and the newly obtained formula plays a key role in the proposed algorithm. Several numerical examples, including cylinder, elliptic cylinder, and NACA0012 airfoil, are given to indicate the effectiveness and robustness of the present method.

AN L^(∞) SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS

A second order accurate method in the infinity norm is proposed for general three dimensional anisotropic elliptic interface problems in which the solution and its derivatives,the coefficients,and source terms all can have finite jumps across one or several arbitrary smooth interfaces.The method is based on the 2D finite element-finite difference(FEFD)method but with substantial differences in method derivation,implementation,and convergence analysis.One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differential equations and the jump conditions.A finite element discretization whose coefficient matrix is a symmetric semi-positive definite is used away from the interface;and the maximum preserving finite difference discretization whose coefficient matrix part is an M-matrix is constructed at irregular elements where the interface cuts through.We aim to get a sharp interface method that can have second order accuracy in the point-wise norm.We show the convergence analysis by splitting errors into several parts.Nontrivial numerical examples are presented to confirm the convergence analysis.

Large eddy simulation of tip leakage cavitating flow focusing on cavitation-vortex interaction with Cartesian cut-cell mesh method

In the present paper, flow configurations of cavitating flow around a straight NACA0009 foil with a gap between the foil tip and sidewall are investigated numerically by large eddy simulation（LES） coupled with Zwart-Gerber-Belamri（ZGB） cavitation model. A Cartesian cut-cell method is used for mesh generation, which is of good orthogonality and high quality. A good agreement is obtained between simulation and experiment. Two influencing factors on vorticity distributions, the interaction between different vortices and the occurrence of cavitation, are discussed in detail based on the numerical results. A series of ？-shaped loops are observed during the development of the induced vortex, which is a result of the instability of vortex pair. This finding may provide a new viewpoint to control the evolution of tip-leakage vortex（TLV） cavitation. Moreover, it is found that the dilatation term plays a much more important role in the evolution of TLV cavitation compared with that in sheet cavitation.

Two-Grid Immersed Finite Volume Element Methods for Semi-Linear Elliptic Interface Problems with Non-Homogeneous Jump Conditions

In this paper,we propose an immersed finite volume element method for solving the semi-linear elliptic interface problems with non-homogeneous jump conditions.Furthermore,two-grid techniques are used to improve the computational efficiency.In this way,we only need to solve a non-linear system on the coarse grid,and a linear system on the fine grid.Numerical results illustrate that the proposed method can solve the semi-linear elliptic interface problems efficiently.Approximate secondorder accuracy for the solution in the L¥norm can be obtained for the considered examples.

A Full Eulerian Fluid-Membrane Coupling Method with a Smoothed Volume-of-Fluid Approach

A novel full Eulerian fluid-elastic membrane coupling method on the fixed Cartesian coordinate mesh is proposed within the framework of the volume-of-fluid approach.The present method is based on a full Eulerian fluid-(bulk)structure coupling solver(Sugiyama et al.,J.Comput.Phys.,230(2011)596–627),with the bulk structure replaced by elastic membranes.In this study,a closed membrane is considered,and it is described by a volume-of-fluid or volume-fraction information generally called VOF function.A smoothed indicator(or characteristic)function is introduced as a phase indicator which results in a smoothed VOF function.This smoothed VOF function uses a smoothed delta function,and it enables a membrane singular force to be incorporated into a mixture momentum equation.In order to deal with a membrane deformation on the Eulerian mesh,a deformation tensor is introduced and updated within a compactly supported region near the interface.Both the neo-Hookean and the Skalak models are employed in the numerical simulations.A smoothed(and less dissipative)interface capturing method is employed for the advection of the VOF function and the quantities defined on the membrane.The stability restriction due to membrane stiffness is relaxed by using a quasi-implicit approach.The present method is validated by using the spherical membrane deformation problems,and is applied to a pressure-driven flow with the biconcave membrane capsules(red blood cells).

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time.The method presented here uses immersed finite element(IFE)functions for the discretization in spatial variables that can be carried out over a fixedmesh(such as a Cartesianmesh if desired),and this featuremakes it possible to reduce the parabolic equation to a system of ordinary differential equations(ODE)through the usual semi-discretization procedure.Therefore,with a suitable choice of the ODE solver,this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured(Cartesian)mesh.Numerical examples are presented to demonstrate features of this new method.