A sharp interface method for high-speed multi-material flows: strong shocks and arbitrary materialpairs

A general framework is developed for solving high-speed and high-intensity multi-material interaction problems on adaptively refined Cartesian meshes. The framework is applicable for interfaces separating materials with very different properties and in the presence of strong shocks. A sharp interface treatment is maintained through a modified Ghost Fluid Method. The embedded boundaries are tracked and represented with level sets. A tree-based Local Mesh Refinement scheme is employed to efficiently resolve the desired physics. Results are shown for situations that cover varied combination of materials (fluids, rigid solids and deformable solids) with careful benchmarking to establish the validity and the versatility of the approach.     

High-resolution algorithms of sharp interface treatment for compressible two-phase flows

In this paper, a kind of arbitrary high order derivatives (ADER) scheme based on the generalised Riemann problem is proposed to simulate multi-material flows by a coupling ghost fluid method. The states at cell interfaces are reconstructed by interpolating polynomials which are piece-wise smooth functions. The states are treated as the equivalent of the left and right states of the Riemann problem. The contact solvers are extrapolated in the vicinity of contact points to facilitate ghost fluids. The numerical method is applied to compressible flows with sharp discontinuities, such as the collision of two fluids of different physical states and gas–liquid two-phase flows. The numerical results demonstrate that unexpected physical oscillations through the contact discontinuities can be prevented effectively and the sharp interface can be captured efficiently.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of interface and free surface flows in a viscous fluid using adapting quadtree grids

A new adaptive quadtree method for simulating laminar viscous fluid problems with free surfaces and interfaces is presented in this paper. The Navier–Stokes equations are solved with a SIMPLE‐type scheme coupled with the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, 1997) volume of fluid (VoF) method and PLIC reconstruction of the volume fraction field during refinement and derefinement processes. The method is demonstrated for interface advection cases in translating and shearing flow fields and found to provide high interface resolution at low computational cost. The new method is also applied to simulation of the collapse of a water column and the results are in excellent agreement with other published data. The quadtree grids adapt to follow the movement of the free surface, whilst maintaining a band of the smallest cells surrounding the surface. The calculation is made on uniform and adapting quadtree grids and the accuracy of the quadtree calculation is shown to be the same as that made on the equivalent uniform grid. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulations of viscous flows using a meshless method

This paper uses the element‐free Galerkin (EFG) method to simulate 2D, viscous, incompressible flows. The control equations are discretized with the standard Galerkin method in space and a fractional step finite element scheme in time. Regular background cells are used for the quadrature. Several classical fluid mechanics problems were analyzed including flow in a pipe, flow past a step and flow in a driven cavity. The flow field computed with the EFG method compared well with those calculated using the finite element method (FEM) and finite difference method. The simulations show that although EFG is more expensive computationally than FEM, it is capable of dealing with cases where the nodes are poorly distributed or even overlap with each other; hence, it may be used to resolve remeshing problems in direct numerical simulations. Flows around a cylinder for different Reynolds numbers are also simulated to study the flow patterns for various conditions and the drag and lift forces exerted by the fluid on the cylinder. These forces are calculated by integrating the pressure and shear forces over the cylinder surface. The results show how the drag and lift forces oscillate for high Reynolds numbers. The calculated Strouhal number agrees well with previous results. Copyright © 2008 John Wiley & Sons, Ltd.     

可压流数值模拟中当地DFD方法的改进和应用

阐述了求解守恒型Euler 方程的当地DFD (Domain-Free Discretization) 方法的改进和应用。DFD 离散策略的核心,是解域内点上控制方程的离散形式可与解域外的一些点相关。通过边界附近解的近似形式,外部相关点上的流动变量值得到确定并强加相应的边界条件。与最初的当地DFD方法不同,在解的近似形式构建中,采用了CCST技术 (Curvature-Corrected Symmetry Technique),因此外部相关点上的密度和切向速度分别由等熵和等总焓关系确定。空间离散采用Galerkin 有限体积格式。最后,给出了固定和运动物体可压缩绕流的数值模拟结果,以验证改进的当地DFD方法的可靠性和数值解精度的提高。     

The development of a Cartesian cut cell method for incompressible viscous flows

This paper describes the extension of the Cartesian cut cell method to applications involving unsteady incompressible viscous fluid flow. The underlying scheme is based on the solution of the full Navier–Stokes equations for a variable density fluid system using the artificial compressibility technique together with a Jameson‐type dual time iteration. The computational domain encompasses two fluid regions and the interface between them is treated as a contact discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures. The Cartesian cut cell technique is used for fitting the complex geometry of solid boundaries across a stationary background Cartesian grid which is located inside the computational domain. A time accurate solution is achieved by using an implicit dual‐time iteration technique based on a slope‐limited, high‐order, Godunov‐type scheme for the inviscid fluxes, while the viscous fluxes are estimated using central differencing. Validation of the new technique is by modelling the unsteady Couette flow and the Rayleigh–Taylor instability problems. Finally, a test case for wave run‐up and overtopping over an impermeable sea dike is performed. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical implementation of Aristov–Pukhnachev's formulation for axisymmetric viscous incompressible flows

In the present work a finite‐difference technique is developed for the implementation of a new method proposed by Aristov and Pukhnachev (Doklady Phys. 2004; 49 (2):112–115) for modeling of the axisymmetric viscous incompressible fluid flows. A new function is introduced that is related to the pressure and a system similar to the vorticity/stream function formulation is derived for the cross‐flow. This system is coupled to an equation for the azimuthal velocity component. The scheme and the algorithm treat the equations for the cross‐flow as an inextricably coupled system, which allows one to satisfy two conditions for the stream function with no condition on the auxiliary function. The issue of singularity of the matrix is tackled by adding a small parameter in the boundary conditions. The scheme is thoroughly validated on grids with different resolutions. The new numerical tool is applied to the Taylor flow between concentric rotating cylinders when the upper and lower lids are allowed to rotate independently from the inner cylinder, while the outer cylinder is held at rest. The phenomenology of this flow is adequately represented by the numerical model, including the hysteresis that takes place near certain specific values of the Reynolds number. Thus, the present results can be construed to demonstrate the viability of the new model. The success can be attributed to the adequate physical nature of the auxiliary function. The proposed technique can be used in the future for in‐depth investigations of the bifurcation phenomena in rotating flows. Copyright © 2009 John Wiley & Sons, Ltd.     

A ghost-cell immersed boundary method for large-eddy simulations of compressible turbulent flows

A methodology to perform a ghost-cell-based immersed boundary method (GCIBM) is presented for simulating compressible turbulent flows around complex geometries. In this method, the boundary condition on the immersed boundary is enforced through the use of ‘ghost cells’ that are located inside the solid body. The computations of variables on these ghost cells are achieved using linear interpolation schemes. The validity and applicability of the proposed method is verified using a three-dimensional (3D) flow over a circular cylinder, and a large-eddy simulation of fully developed 3D turbulent flow in a channel with a wavy surface. The results agree well with the previous numerical and experimental results, given that the grid resolution is reasonably fine. To demonstrate the capability of the method for higher Mach numbers, supersonic turbulent flow over a circular cylinder is presented. While more work still needs to be done to demonstrate higher robustness and accuracy, the present work provides interesting insights using the GCIBM for the compressible flows.     

A scaling analysis of added-mass and history forces and their coupling in dispersed multiphase flows

Accurate momentum coupling model is vital to simulation of dispersed multiphase flows. The overall force exerted on a particle is divided into four physically meaningful contributions, i.e., quasi-steady, stress-gradient, added-mass, and viscous-unsteady (history) forces. Time scale analysis on the turbulent multiphase flow and the viscous-unsteady kernel shows that the integral representation of the viscous-unsteady force is required except for a narrow range of particle size around the Kolmogorov length scale when particle-to-fluid density ratio is large. Conventionally, the particle-to-fluid density ratio is used to evaluate the relative importance of the unsteady forces (stress-gradient, added-mass, and history forces) in the momentum coupling. However, it is shown from our analysis that when particle-to-fluid density ratio is large, the importance of the unsteady forces depends on the particle-to-fluid length scale ratio and not on the density ratio. Provided the particle size is comparable to the smallest fluid length scale (i.e., Kolmogorov length scale for turbulence or shock thickness for shock-particle interaction) or larger, unsteady forces are important in evaluating the particle motion. Furthermore, the particle mass loading is often used to estimate the importance of the back effect of particles on the fluid. An improved estimate of backward coupling for each force contribution is established through a scaling argument. The back effects of stress-gradient and added-mass forces depend on particle volume fraction. For large particle-to-fluid density ratio, the importance of the quasi-steady force in backward coupling depends on the particle mass fraction; while that of the viscous-unsteady force is related to both particle mass and volume fractions.     

A multigrid pseudo-spectral method for incompressible Navier–Stokes flows

A pseudo-spectral solver with multigrid acceleration for the numerical prediction of incompressible non-isothermal flows is presented. The spatial discretization is based on a Chebyshev collocation method on Gauss–Lobatto points and for the discretization in time the second-order backward differencing scheme (BDF2) is employed. The multigrid method is invoked at the level of algebraic system solving within a pressure-correction method. The approach combines the high accuracy of spectral methods with efficient solver properties of multigrid methods. The capabilities of the proposed scheme are illustrated by a buoyancy driven cavity flow as a standard benchmark case. To cite this article: K. Krastev, M. Schäfer, C. R. Mecanique 333 (2005).     

A symmetry-preserving Cartesian grid method for computing a viscous flow past a circular cylinder

This article deals with a numerical method for solving the unsteady, incompressible Navier–Stokes equations in domains with arbitrarily-shaped boundaries, where the boundary is represented using the Cartesian grid approach. We introduce a novel cut-cell discretization which preserves the spectral properties of convection and diffusion. Here, convection is discretized by a skew-symmetric operator and diffusion is approximated by a symmetric, positive-definite coefficient matrix. Such a symmetry-preserving discretization conserves the kinetic energy (if the dissipation is turned off) and is stable on any grid. The method is successfully tested for an incompressible, unsteady flow around a circular cylinder at $\mathit{Re}=100$. To cite this article: R. Verstappen, M. Dröge, C. R. Mecanique 333 (2005).     

A multi-dimensional time-marching method of characteristics for viscous and heat-conducting flows

Summary A numerical scheme is presented which employs the characteristic surfaces in space-time for solving Navier-Stokes equations for compressible fluid flow. We consider the general case of a three-dimensional flow, a simplification of which yields the equations of the two-dimensional case. Emphasis is put on the method itself. We apply it to simulate a laminar hypersonic flow around a circular cylinder of a five-components gas mixture of nitrogen and oxygen with thermally perfect constituents and at chemical nonequilibrium. First, the partial differential equations are transformed into a standard form with directional derivatives, enabling to attain the compatibility conditions, including the viscosity terms. These conditions are discretized by approximating their integrals along the corresponding characteristic surfaces. The result is an explicit time-marching numerical scheme. Using a grid fitted between the shock and the cylinder, and starting from roughly estimated initial conditions, a steady solution is searched. A comparison is made with the solution obtained under the assumption of a perfect gas. Received 6 April 1999; accepted for publication 13 May 1999     

A finite element method for compressible viscous fluid flows

This work presents a mixed three‐dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady‐state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid building‐block and gridless method for compressible flows

A hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

A modified adaptive grid method for recirculating flows

In this study a method of equidistribution of a weight function for grid adaption is modified to produce a smoother grid which yields a more accurate solution. In the original scheme the weight function was estimated on each grid independently and a large variation in the values of the, weight function could generate a highly skewed and non-uniform grid which produced large errors. In this study the weight function is smoothed by coupling neighbouring weight functions. Abrupt changes in the weight function are alleviated and a smoother grid distribution is obtained. With relatively minor modifications of the original weight function it is demonstrated in this study that the solution can be improved. The test cases presented are the one-dimensional convection-diffusion equation, a laminar polar cavity flow, a laminar backwardfacing step flow and a turbulent reacting sudden expansion pipe flow. Numerical efficiencies ranging from factors of five to 10 are achieved over uniform grid methods.     

A second‐order time accurate finite element method for quasi‐incompressible viscous flows

A finite element method for quasi‐incompressible viscous flows is presented. An equation for pressure is derived from a second‐order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second‐order time accurate scheme and a least‐squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second‐order accurate method and our previous first‐order accurate formulation is shown. The method is also demonstrated in the computation of the leaky‐lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

An implicit Lagrangian lattice Boltzmann method for the compressible flows

In this paper, we propose a new Lagrangian lattice Boltzmann method (LBM) for simulating the compressible flows. The new scheme simulates fluid flows based on the displacement distribution functions. The compressible flows, such as shock waves and contact discontinuities are modelled by using Lagrangian LBM. In this model, we select the element in the Lagrangian coordinate to satisfy the basic fluid laws. This model is a simpler version than the corresponding Eulerian coordinates, because the convection term of the Euler equations disappears. The numerical simulations conform to classical results. Copyright © 2006 John Wiley & Sons, Ltd.     

A CWENO ghost fluid method for compressible multimaterial flow

In this work a new ghost fluid method (GFM) is introduced for multimaterial compressible flow with arbitrary equation of states. In previous researches, it has been shown that accurate wave decomposition at the interface by solving a Riemann problem alleviates the shortcomings of the standard GFM in dealing with the impingement of strong waves onto the interface but these Riemann‐based GFM are not consistent with the framework of the central WENO scheme in which the emphasis is to avoid solving Riemann problems at control volume faces and enjoy the black box property (being independent of equation of state). The aim of this work is to develop a new GFM that is completely consistent with the methodology behind central schemes; that is, it enjoys a black box property. The capabilities of the proposed GFM method is shown by solving various types of multimaterial compressible flows including gas–gas, gas–water and fluid–solid interfaces interacting with strong shock waves in one and two space dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

A coupled WC-TL SPH method for simulation of hydroelastic problems

A coupled weakly compressible (WC) and total Lagrangian (TL) smoothed particle hydrodynamics (SPH) method is developed for simulating hydroelastic problems. The fluid phase is simulated using WCSPH method, while the structural dynamics are solved using TLSPH method. Fluid and solid components of the method are validated separately. A sloshing water tank problem is solved to test the WCSPH method while oscillation of a thin plate and large deformation of a cantilever beam are simulated to test the TLSPH method. After validating each component, the coupled WC-TL SPH scheme is used to simulate two benchmark hydroelastic problems. The first test case shows the evolution of water column with an elastic boundary gate, and the second one investigates the breaking water column impact on elastic structures. The agreement between WC-TL SPH results and literature data shows the ability of the proposed method in simulating hydroelastic phenomena.