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.     

A diffuse-interface immersed boundary method for simulation of compressible viscous flows with stationary and moving boundaries

In this paper, a diffuse-interface immersed boundary method (IBM) is proposed for simulation of compressible viscous flows with stationary and moving boundaries. In the method, the solution of flow field and the implementation of boundary conditions are decoupled into two steps by applying the fractional step technique, ie, the predictor step and the corrector step. Firstly, in the predictor step, the intermediate flow field is resolved by a recently developed gas kinetic flux solver (GKFS) without consideration of the solid boundary. The GKFS is a finite volume approach that solves the Navier-Stokes equations for the flow variables at cell centers. In GKFS, the inviscid and viscous fluxes are evaluated as a single entity by reconstructing the local solution of continuous Boltzmann equation. Secondly, in the corrector step, the intermediate flow field is corrected by the present diffuse-interface IBM. During this process, the velocity field is firstly corrected by the implicit boundary condition–enforced IBM so that the no-slip boundary condition can be accurately satisfied. After that, the density correction is made by an iterative approach with the help of the continuity equation. Finally, the correction of the temperature field is made in the same way as that of the velocity field. Good agreements between the present simulations and the reference data in literature demonstrate the reliability of the proposed method.     

High-order ghost-point immersed boundary method for viscous compressible flows based on summation-by-parts operators

A high-order immersed boundary method is devised for the compressible Navier-Stokes equations by employing high-order summation-by-parts difference operators. The immersed boundaries are treated as sharp interfaces by enforcing the solid wall boundary conditions via flow variables at ghost points. Two different interpolation schemes are tested to compute values at the ghost points and a hybrid treatment is used. The first method provides the bilinearly interpolated flow variables at the image points of the corresponding ghost points and the second method applies the boundary condition at the immersed boundary by using the weighted least squares method with high-order polynomials. The approach is verified and validated for compressible flow past a circular cylinder at moderate Reynolds numbers. The tonal sound generated by vortex shedding from a circular cylinder is also investigated. In order to demonstrate the capability of the solver to handle complex geometries in practical cases, flow in a cross-section of a human upper airway is simulated.     

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 sharp‐interface immersed boundary framework for simulations of high‐speed inviscid compressible flows

A new finite‐volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high‐speed inviscid compressible flows. The IB method adopts a sharp‐interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid‐fluid interface. We show that proposed reconstruction leads to second‐order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift‐off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape‐changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.     

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.     

Numerical simulation of compressible multifluid flows using an adaptive positivity-preserving RKDG-GFM approach

The Runge-Kutta discontinuous Galerkin method together with a refined real-ghost fluid method is incorporated into an adaptive mesh refinement environment for solving compressible multifluid flows, where the level set method is used to capture the moving material interface. To ensure that the Riemann problem is exactly along the normal direction of the material interface, a simple and efficient modification is introduced into the original real-ghost fluid method for constructing the interfacial Riemann problem, and the initial conditions of the Riemann problem are obtained directly from the solution polynomials of the discontinuous Galerkin finite element space. In addition, a positivity-preserving limiter is introduced into the Runge-Kutta discontinuous Galerkin method to suppress the failure of preserving positivity of density or pressure for the problems involving strong shock wave or shock interaction with material interface. For interfacial cells in adaptive mesh refinement, the data transfer between different grid levels is achieved by using a L² projection approach along with the least squares fitting. Various numerical cases, including multifluid shock tubes, underwater explosions, and shock-induced collapse of a underwater air bubble, are computed to assess the capability of the present adaptive positivity-preserving RKDG-GFM approach, and the simulated results show that the present approach is quite robust and can provide relatively reasonable results across a wide variety of flow regimes, even for problems involving strong shock wave or shock wave impacting high acoustic impedance mismatch material interface.     

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.     

Modified ghost fluid method for three-dimensional compressible multimaterial flows with interfaces exhibiting large curvature and topological change

In order to capture the material interface dynamics, especially under the impact of strong shocks, the key feature of the modified ghost fluid method (MGFM) is to construct a multimaterial Riemann problem normal to the interface and use its solution to define interface conditions. However, such process sometimes may not be easily or accurately implemented when the multidimensional interfaces come into contact or undergo significant deformations. In this article, a three-dimensional interface treating procedure is developed for a wide range of compressible multimaterial flows. It utilizes the MGFM with an explicit approximate Riemann problem solver to define interface conditions. More importantly, a weighted average technique is designed to optimize the treatment for interfaces exhibiting large curvature and topological change. This remedies two defects of the traditional approach in these extreme cases. One is that the normal directions of interfacial ghost nodes may not be easily calculated. The other is that the interface conditions may not be accurately defined. The numerical methodology is validated through several typical problems, including gas-liquid Riemann problem and shock-bubble/droplet interaction. These results indicate that the developed method is capable of dealing with interfacial evolutions in three dimensions, especially when interfaces undergo merger, fragmentation, and other complex changes.     

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).     

水/气多介质问题的界面处理方法

针对不可压缩可压缩水/气多介质问题, 提出一种新的界面处理方法。在可压缩水/气界面处构造Riemann问题, 在水中设音速趋于无穷大, 求解Riemann问题得到不可压缩可压缩水/气界面处流体的准确流动状态; 然后以此状态结合GFM(ghost fluid method)方法分别为2种流体定义界面边界条件, 将两相流问题转化为单相流问题计算, 通过求解level set方程来跟踪界面的位置。对各种不同的界面边界条件定义方法进行了比较, 数值模拟结果表明算法能准确地捕捉各类间断的位置, 证明了算法的有效性和稳健性。     

真实虚拟流方法在多介质可压缩流动模拟中的应用

为了克服原始虚拟流方法(ghost fluid method,GFM)在处理激波与大密度比流体-流体（气-水）界面相互作用时遇到的困难,采用真实虚拟流法(real ghost fluid method,RGFM)处理流体界面附近的虚拟点,结合HLLC(Harten-Lax-Van Leer with contact discontinuities)格式求解Euler方程,采用五阶WENO(weighted essentially nonoscillatory)格式求解level set输运方程。通过一维和二维算例的物质界面捕捉研究,证明RGFM在处理小密度比界面问题时优于GFM,同时RGFM还可用于求解激波与大密度比物质界面相互作用问题。计算表明,将RGFM引入到本文算法中,可精确捕捉到激波与界面（气-气、气-水界面）相互作用的变化细节,包括大密度比界面的剧烈变形和破碎,并具有较高的计算分辨率。     

Moving least squares reconstruction for sharp interface immersed boundary methods

We propose a new approach for reconstructing velocity boundary conditions in sharp-inerface immersed boundary (IB) methods based on the moving least squares (MLS) interpolation method. The MLS is employed to not only reconstruct velocity boundary conditions but also to calculate the pressure and velocity gradients in the vicinity of the immersed body, which are required in fluid structure interaction problems to obtain the force exerted by the fluid on the structure. To extend the method to arbitrarily complex geometries with nonconvex shaped boundaries, the visibility method is combined with the MLS method. The performance of the proposed curvilinear IB MLS (CURVIB-MLS) is demonstrated by systematic grid-refinement studies for two- and three-dimensional tests and compared with the standard CURVIB method employing standard wall-normal interpolation for reconstructing boundary conditions. The test problems are flow in a lid-driven cavity with a sphere, uniform flow over a sphere, flow on a NACA0018 airfoil at incidence, and vortex-induced vibration of an elastically-mounted cylinder. We show that the CURVIB-MLS formulation yields a method that is easier to implement in complex geometries and exhibits higher accuracy and rate of convergence relative to the standard CURVIB method. The MLS approach is also shown to dramatically improve the accuracy of calculating the pressure and viscous forces imparted by the flow on the body and improve the overall accuracy of FSI simulations. Finally, the CURVIB-MLS approach is able to qualitatively capture on relatively coarse grids important features of complex separated flows that the standard CURVIB method is able to capture only on finer grids.     

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).