A robust multi‐grid pressure‐based algorithm for multi‐fluid flow at all speeds

This paper reports on the implementation and testing, within a full non‐linear multi‐grid environment, of a new pressure‐based algorithm for the prediction of multi‐fluid flow at all speeds. The algorithm is part of the mass conservation‐based algorithms (MCBA) group in which the pressure correction equation is derived from overall mass conservation. The performance of the new method is assessed by solving a series of two‐dimensional two‐fluid flow test problems varying from turbulent low Mach number to supersonic flows, and from very low to high fluid density ratios. Solutions are generated for several grid sizes using the single grid (SG), the prolongation grid (PG), and the full non‐linear multi‐grid (FMG) methods. The main outcomes of this study are: (i) a clear demonstration of the ability of the FMG method to tackle the added non‐linearity of multi‐fluid flows, which is manifested through the performance jump observed when using the non‐linear multi‐grid approach as compared to the SG and PG methods; (ii) the extension of the FMG method to predict turbulent multi‐fluid flows at all speeds. The convergence history plots and CPU‐times presented indicate that the FMG method is far more efficient than the PG method and accelerates the convergence rate over the SG method, for the problems solved and the grids used, by a factor reaching a value as high as 15. Copyright © 2003 John Wiley & Sons, Ltd.     

A moving mesh BGK scheme for multi‐material flows

In this paper, a moving mesh BGK scheme (MMBGK) for multi‐material flow computations is proposed. The basic idea of constructing the MMBGK is to couple the Lagrangian method, which tracks material interfaces and keeps the interfaces sharp, with a remapping‐free ALE‐type kinetic method within each single material region, where the kinetic method is based on the BGK (Bhatnagar–Gross–Krook) model. Within each single material region, a numerical flux formulation is developed on moving meshes from motion of microscope particles, and the mesh velocity is determined by requiring both mesh adaptation for accuracy and robustness, such that the grids are moving towards to the regions with high flow gradients in a way of diffusive mechanism (velocity) to adjust the distances between neighboring cells, thus increasing the numerical accuracy. To keep the sharpness of material interfaces, the Lagrangian velocity and flux are constructed at the interfaces only. Consequently, a BGK‐scheme‐based ALE‐type method (i.e., the MMBGK scheme) for multi‐material flows is constructed. Numerical examples in one and two dimensions are presented to illustrate the accuracy and robustness of the MMBGK scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

A further work on multi‐phase two‐fluid approach for compressible multi‐phase flows

This paper is to continue our previous work Niu (Int. J. Numer. Meth. Fluids 2001; 36 :351–371) on solving a two‐fluid model for compressible liquid–gas flows using the AUSMDV scheme. We first propose a pressure–velocity‐based diffusion term originally derived from AUSMDV scheme Wada and Liou (SIAM J. Sci. Comput. 1997; 18 (3):633—657) to enhance its robustness. The scheme can be applied to gas and liquid fluids universally. We then employ the stratified flow model Chang and Liou (J. Comput. Physics 2007; 225 :240–873) for spatial discretization. By defining the fluids in different regions and introducing inter‐phasic force on cell boundary, the stratified flow model allows the conservation laws to be applied on each phase, and therefore, it is able to capture fluid discontinuities, such as the fluid interfaces and shock waves, accurately. Several benchmark tests are studied, including the Ransom's Faucet problem, 1D air–water shock tube problems, 2D shock‐water column and 2D shock‐bubble interaction problems. The results indicate that the incorporation of the new dissipation into AUSM⁺‐up scheme and the stratified flow model is simple, accurate and robust enough for the compressible multi‐phase flows. Copyright © 2008 John Wiley & Sons, Ltd.     

Geometrical interpretation of the multi‐point flux approximation L‐method

In this paper, we first investigate the influence of different Dirichlet boundary discretizations on the convergence rate of the multi‐point flux approximation (MPFA) L‐method by the numerical comparisons between the MPFA O‐ and L‐method, and show how important it is for this new method to handle Dirichlet boundary conditions in a suitable way. A new Dirichlet boundary strategy is proposed, which in some sense can well recover the superconvergence rate of the normal velocity. In the second part of the work, the MPFA L‐method with homogeneous media is studied. A systematic concept and geometrical interpretations of the L‐method are given and illustrated, which yield more insight into the L‐method. Finally, we apply the MPFA L‐method for two‐phase flow in porous media on different quadrilateral grids and compare its numerical results for the pressure and saturation with the results of the two‐point flux approximation method. Copyright © 2008 John Wiley & Sons, Ltd.     

The capturing of free surfaces in incompressible multi‐fluid flows

By treating it as a contact discontinuity in the density field, a free surface between two immiscible fluids can be automatically ‘captured’ by the enforcement of conservation laws. A surface‐capturing method of this kind requires no special tracking or fitting treatment for the free surface, thereby offering the advantage of algorithm simplicity over the surface‐tracking or the surface‐fitting method. A surface‐capturing method based on a new multi‐fluid incompressible Navier–Stokes formulation is developed. It is applied to a variety of free‐surface flows, including the Rayleigh–Taylor instability problem, the ship waves around a Wigley hull and a model bubble‐rising problem to demonstrate the validity and versatility of the present method. Copyright © 2000 John Wiley & Sons, Ltd.     

On the composite Riemann problem for multi‐material fluid flows

We develop an Eulerian fixed grid numerical method for calculating multi‐material fluid flows. This approach relates to the class of interface capturing methods. The fluid is treated as a heterogeneous mixture of constituent materials, and the material interface is implicitly captured by a region of mixed cells that have arisen owing to numerical diffusion. To suppress this numerical diffusion, we propose a composite Riemann problem (CRP), which describes the decay of an initial discontinuity in the presence of a contact point between two different fluids, which is located off the initial discontinuity point. The solution to the CRP serves to calculate multi‐material no mixed numerical flux without introducing any material diffusion. We discuss the CRP solution and its implementation in the multi‐material fluid Godunov method. Numerical results show that a simple framework of the CRP greatly improves capturing material interfaces in the Godunov method and reproduces many of the advantages of more complicated interface tracking multi‐material treatments. Copyright © 2014 John Wiley & Sons, Ltd.     

Accurate and robust adaptive mesh refinement for aerodynamic simulation with multi‐block structured curvilinear mesh

A multi‐block curvilinear mesh‐based adaptive mesh refinement (AMR) method is developed to satisfy the competing objectives of improving accuracy and reducing cost. Body‐fitted curvilinear mesh‐based AMR is used to capture flow details of various length scales. A series of efforts are made to guarantee the accuracy and robustness of the AMR system. A physics‐based refinement function is proposed, which is proved to be able to detect both shock wave and vortical flow. The curvilinear mesh is refined with cubic interpolation, which guarantees the aspect ratio and smoothness. Furthermore, to enable its application in complex configurations, a sub‐block‐based refinement strategy is developed to avoid generating invalid mesh, which is the consequence of non‐smooth mesh lines or singular geometry features. A newfound problem of smaller wall distance, which negatively affects the stability and is never reported in the literature, is also discussed in detail, and an improved strategy is proposed. Together with the high‐accuracy numerical scheme, a multi‐block curvilinear mesh‐based AMR system is developed. With a series of test cases, the current method is verified to be accurate and robust and be able to automatically capture the flow details at great cost saving compared with the global refinement. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

A finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite element assembly strategies on multi‐core and many‐core architectures

We demonstrate that radically differing implementations of finite element methods (FEMs) are needed on multi‐core (CPU) and many‐core (GPU) architectures, if their respective performance potential is to be realised. Our numerical investigations using a finite element advection–diffusion solver show that increased performance on each architecture can only be achieved by committing to specific and diverse algorithmic choices that cut across the high‐level structure of the implementation. Making these commitments to achieve high performance for a single architecture leads to a loss of performance portability. Data structures that include redundant data but enable coalesced memory accesses are faster on many‐core architectures, whereas redundancy‐free data structures that are accessed indirectly are faster on multi‐core architectures. The Addto algorithm for global assembly is optimal on multi‐core architectures, whereas the Local Matrix Approach is optimal on many‐core architectures despite requiring more computation than the Addto algorithm. These results demonstrate the value in making the correct choice of algorithm and data structure when implementing FEMs, spectral element methods and low‐order discontinuous Galerkin methods on modern high‐performance architectures. Copyright © 2012 John Wiley & Sons, Ltd.     

A virtual boundary method with improved computational efficiency using a multi‐grid method

The flow around spherical, solid objects is considered. The boundary conditions on the solid boundaries have been applied by replacing the boundary with a surface force distribution on the surface, such that the required boundary conditions are satisfied. The velocity on the boundary is determined by extrapolation from the flow field. The source terms are determined iteratively, as part of the solution. They are then averaged and are smoothed out to nearby computational grid points. A multi‐grid scheme has been used to enhance the computational efficiency of the solution of the force equations. The method has been evaluated for flow around both moving and stationary spherical objects at very low and intermediate Reynolds numbers. The results shows a second order accuracy of the method both at creeping flow and at Re=100. The multi‐grid scheme is shown to enhance the convergence rate up to a factor 10 as compared to single grid approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Performance of algebraic multi‐grid solvers based on unsmoothed and smoothed aggregation schemes

A comparison is made of the performance of two algebraic multi‐grid (AMG0 and AMG1) solvers for the solution of discrete, coupled, elliptic field problems. In AMG0, the basis functions for each coarse grid/level approximation (CGA) are obtained directly by unsmoothed aggregation, an appropriate scaling being applied to each CGA to improve consistency. In AMG1 they are assembled using a smoothed aggregation with a constrained energy optimization method providing the smoothing. Although more costly, smoothed basis functions provide a better (more consistent) CGA. Thus, AMG1 might be viewed as a benchmark for the assessment of the simpler AMG0. Selected test problems for D'Arcy flow in pipe networks, Fick diffusion, plane strain elasticity and Navier–Stokes flow (in a Stokes approximation) are used in making the comparison. They are discretized on the basis of both structured and unstructured finite element meshes. The range of discrete equation sets covers both symmetric positive definite systems and systems that may be non‐symmetric and/or indefinite. Both global and local mesh refinements to at least one order of resolving power are examined. Some of these include anisotropic refinements involving elements of large aspect ratio; in some hydrodynamics cases, the anisotropy is extreme, with aspect ratios exceeding two orders. As expected, AMG1 delivers typical multi‐grid convergence rates, which for all practical purposes are independent of mesh bandwidth. AMG0 rates are slower. They may also be more discernibly mesh‐dependent. However, for the range of mesh bandwidths examined, the overall cost effectiveness of the two solvers is remarkably similar when a full convergence to machine accuracy is demanded. Thus, the shorter solution times for AMG1 do not necessarily compensate for the extra time required for its costly grid generation. This depends on the severity of the problem and the demanded level of convergence. For problems requiring few iterations, where grid generation costs represent a significant penalty, AMG0 has the advantage. For problems requiring a large investment in iterations, AMG1 has the edge. However, for the toughest problems addressed (vector and coupled vector–scalar fields discretized exclusively using finite elements of extreme aspect ratio) AMG1 is more robust: AMG0 has failed on some of these tests. However, but for this deficiency AMG0 would be the preferred linear approximation solver for Navier–Stokes solution algorithms in view of its much lower grid generation costs. Copyright © 2001 John Wiley & Sons, Ltd.     

A γ‐DGBGK scheme for compressible multi‐fluids

The paper presents a Discontinuous Galerkin γ‐BGK (γ‐DGBGK) method for compressible multicomponent flow simulations by coupling the discontinuous Galerkin method with a γ‐BGK scheme based on WENO limiters. In this γ‐DGBGK method, the construction of the flux in the DG method is based on the kinetic scheme which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation at cell interfaces. WENO limiters are used to obtain uniform high‐order accuracy and sharp non‐oscillatory shock transition, and time accuracy obtained by integration for the flux function at the cell interface. Numerical examples in one and two space dimensions are presented to illustrate the robust and accuracy of the present scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

A novel non‐upwind,interconnected, multi‐grid,overlapping numerical procedure for problems involving fluid flow

A novel numerical procedure for heat, mass and momentum transfer in fluid flow is presented. The new scheme is passed on a non‐upwind, interconnected, multi‐grid, overlapping (NIMO) finite‐difference algorithm. In 2D flows, the NIMO algorithm solves finite‐difference equations for each dependent variable on four overlapping grids. The finite‐difference equations are formulated using the control‐volume approach, such that no interpolations are needed for computing the convective fluxes. For a particular dependent variable, four fields of values are produced. The NIMO numerical procedure is tested against the exact solution of two test problems. The first test problem is an oblique laminar 2D flow with a double step abrupt change in a passive scalar variable for infinite Peclet number. The second test problem is a rotating radial flow in an annular sector with a single step abrupt change in a passive scalar variable for infinite Peclet number. The NIMO scheme produced essentially the exact solution using different uniform and non‐uniform square and rectangular grids for 45 and 30° angle of inclination. All other schemes were unable to capture the exact solution, especially for the rectangular and non‐uniform grids. The NIMO scheme was also successful in predicting the exact solution for the rotating radial flow, using a uniform cylindrical‐polar coordinate grid. Copyright © 2008 John Wiley & Sons, Ltd.     

Advances in the simulation of multi‐fluid flows with the particle finite element method. Application to bubble dynamics

In this work, we extend the Particle Finite Element Method (PFEM) to multi‐fluid flow problems with the aim of exploiting the fact that Lagrangian methods are specially well suited for tracking interfaces. We develop a numerical scheme able to deal with large jumps in the physical properties, included surface tension, and able to accurately represent all types of discontinuities in the flow variables. The scheme is based on decoupling the velocity and pressure variables through a pressure segregation method that takes into account the interface conditions. The interface is defined to be aligned with the moving mesh, so that it remains sharp along time, and pressure degrees of freedom are duplicated at the interface nodes to represent the discontinuity of this variable due to surface tension and variable viscosity. Furthermore, the mesh is refined in the vicinity of the interface to improve the accuracy and the efficiency of the computations. We apply the resulting scheme to the benchmark problem of a two‐dimensional bubble rising in a liquid column presented in Hysing et al. (International Journal for Numerical Methods in Fluids 2009; 60 : 1259–1288), and propose two breakup and coalescence problems to assess the ability of a multi‐fluid code to model topology changes. Copyright © 2010 John Wiley & Sons, Ltd.     

An SPH multi‐fluid model based on quasi buoyancy for interface stabilization up to high density ratios and realistic wave speed ratios

We introduce a smoothed particle hydrodynamics (SPH) concept for the stabilization of the interface between 2 fluids. It is demonstrated that the change in the pressure gradient across the interface leads to a force imbalance. This force imbalance is attributed to the particle approximation implicit to SPH. To stabilize the interface, a pressure gradient correction is proposed. In this approach, the multi‐fluid pressure gradients are related to the (gravitational and fluid) accelerations. This leads to a quasi‐buoyancy correction for hydrostatic (stratified) flows, which is extended to nonhydrostatic flows. The result is a simple density correction that involves no parameters or coefficients. This correction is included as an extra term in the SPH momentum equation. The new concept for the stabilization of the interface is explored in 5 case studies and compared with other multi‐fluid models. The first case is the stagnant flow in a tank: The interface remains stable up to density ratios of 1:1000 (typical for water and air), in combination with artificial wave speed ratios up to 1:4. The second and third cases are the Rayleigh‐Taylor instability and the rising bubble, where a reasonable agreement between SPH and level‐set models is achieved. The fourth case is an air flow across a water surface up to density ratios of 1:100, artificial wave speed ratios of 1:4, and high air velocities. The fifth case is about the propagation of internal gravity waves up to density ratios of 1:100 and artificial wave speed ratios of 1:4. It is demonstrated that the quasi‐buoyancy model may be used to stabilize the interface between 2 fluids up to high density ratios, with real (low) viscosities and more realistic wave speed ratios than achieved by other weakly compressible SPH multi‐fluid models. Real wave speed ratios can be achieved as long as the fluid velocities are not very high. Although the wave speeds may be artificial in many cases, correct and realistic wave speed ratios are essential in the modelling of heat transfer between 2 fluids (eg, in engineering applications such as gas turbines).     

A mass‐conserving Level‐Set method for modelling of multi‐phase flows

A mass‐conserving Level‐Set method to model bubbly flows is presented. The method can handle high density‐ratio flows with complex interface topologies, such as flows with simultaneous occurrence of bubbles and droplets. Aspects taken into account are: a sharp front (density changes abruptly), arbitrarily shaped interfaces, surface tension, buoyancy and coalescence of droplets/bubbles. Attention is paid to mass‐conservation and integrity of the interface. The proposed computational method is a Level‐Set method, where a Volume‐of‐Fluid function is used to conserve mass when the interface is advected. The aim of the method is to combine the advantages of the Level‐Set and Volume‐of‐Fluid methods without the disadvantages. The flow is computed with a pressure correction method with the Marker‐and‐Cell scheme. Interface conditions are satisfied by means of the continuous surface force methodology and the jump in the density field is maintained similar to the ghost fluid method for incompressible flows. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast nested multi‐grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi‐grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body‐fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub‐division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell‐cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell‐centered finite volume discretization, Roe's flux splitting, a least‐squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell‐cutting. Several cycling strategies, such as the saw‐tooth, W‐ and V‐cycles, have been studies. The V‐cycle has been found to be the most efficient. In general, the multi‐grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.     

CFD‐based multi‐objective optimization method for ship design

This paper concerns development and demonstration of a computational fluid dynamics (CFD)‐based multi‐objective optimization method for ship design. Three main components of the method, i.e. computer‐aided design (CAD), CFD, and optimizer modules are functionally independent and replaceable. The CAD used in the present study is NAPA system, which is one of the leading CAD systems in ship design. The CFD method is FLOWPACK version 2004d, a Reynolds‐averaged Navier–Stokes (RaNS) solver developed by the present authors. The CFD method is implemented into a self‐propulsion simulator, where the RaNS solver is coupled with a propeller‐performance program. In addition, a maneuvering simulation model is developed and applied to predict ship maneuverability performance. Two nonlinear optimization algorithms are used in the present study, i.e. the successive quadratic programming and the multi‐objective genetic algorithm, while the former is mainly used to verify the results from the latter. For demonstration of the present method, a multi‐objective optimization problem is formulated where ship propulsion and maneuverability performances are considered. That is, the aim is to simultaneously minimize opposite hydrodynamic performances in design tradeoff. In the following, an overview of the present method is given, and results are presented and discussed for tanker stern optimization problem including detailed verification work on the present numerical schemes. Copyright © 2006 John Wiley & Sons, Ltd.     

A mass‐fraction‐based interface‐capturing method for multi‐component flow

An interface‐capturing method based on mass fraction is developed to solve the Riemann problem in multi‐component compressible flow. Equations of mass fraction with modified form, which is derived from conservative equations of mass, are employed here to capture the interface. By introducing mass fraction into Euler equations system, as well as other conservative coefficients, a quasi‐conservative numerical model is created. Numerical examples show that the mass fraction model performs well not only in multi‐component fluids modeled by simple stiffened gas equation of state (EOS) but also in that modeled by complex Mie–Grüneisen EOS. Moreover, the mass fraction model is applied to Riemann problem with piecewise EOS; the expression of which depends on density. It is found that the mass fraction model can well adapt to the analytic change in piecewise EOS and produce accuracy solutions with fewer unknown quantities, and the model can be easily extended to m‐component fluid mixture by using only m + 4 equations with no additional conditions. Copyright © 2013 John Wiley & Sons, Ltd.