Numerical simulation of a single rising bubble by VOF with surface compression

The capability of the direct volume of fluid method for describing the surface dynamics of a free two‐dimensional rising bubble is evaluated using quantities of a recently published benchmark. The model equations are implemented in the open source computational fluid dynamics library OpenFOAM®. Here, a main ingredient of the numerical method is the so‐called surface compression that corrects the fluxes near the interface between two phases. The application of this method with respect to two test cases of a benchmark is considered in the main part. The test cases differ in physical properties, thus in different surface tension effects. The quantities centre of mass position, circularity and rise velocity are tracked over time and compared with the ones given in the benchmark. For test case one, where surface tension effects are more pre‐eminent, deviations from the benchmark results become more obvious. However, the flow features are still within reasonable range. Nevertheless, for test case two, which has higher density and viscosity ratios and above all a lower influence of the surface tension force, good agreement compared with the benchmark reference results is achieved. This paper demonstrates the good capabilities of the direct volume of fluid method with surface compression with regard to the preservation of sharp interfaces, boundedness, mass conservation and low computational time. Some limitation regarding the occurrence of parasitic currents, bad pressure jump prediction and bad grid convergence have been observed. With these restrictions in mind, the method is suitable for the simulation of similar two‐phase flow configurations. Copyright © 2012 John Wiley & Sons, Ltd.     

A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations

A fourth‐order compact finite difference scheme on the nine‐point 2D stencil is formulated for solving the steady‐state Navier–Stokes/Boussinesq equations for two‐dimensional, incompressible fluid flow and heat transfer using the stream function–vorticity formulation. The main feature of the new fourth‐order compact scheme is that it allows point‐successive overrelaxation (SOR) or point‐successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature. Copyright © 2003 John Wiley & Sons, Ltd.     

Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier–Stokes equations

This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier–Stokes equations within the DFG high‐priority research program flow simulation with high‐performance computers by Schafer and Turek (Vol. 52, Vieweg: Braunschweig, 1996). The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid method proves to be the best solver. Copyright © 2002 John Wiley & Sons, Ltd.     

A spectral/hp least‐squares finite element analysis of the Carreau–Yasuda fluids

A least‐squares finite element model with spectral/hp approximations was developed for steady, two‐dimensional flows of non‐Newtonian fluids obeying the Carreau–Yasuda constitutive model. The finite element model consists of velocity, pressure, and stress fields as independent variables (hence, called a mixed model). Least‐squares models offer an alternative variational setting to the conventional weak‐form Galerkin models for the Navier–Stokes equations, and no compatibility conditions on the approximation spaces used for the velocity, pressure, and stress fields are necessary when the polynomial order (p) used is sufficiently high (say, p > 3, as determined numerically). Also, the use of the spectral/hp elements in conjunction with the least‐squares formulation with high p alleviates various forms of locking, which often appear in low‐order least‐squares finite element models for incompressible viscous fluids, and accurate results can be obtained with exponential convergence. To verify and validate, benchmark problems of Kovasznay flow, backward‐facing step flow, and lid‐driven square cavity flow are used. Then the effect of different parameters of the Carreau–Yasuda constitutive model on the flow characteristics is studied parametrically. Copyright © 2016 John Wiley & Sons, Ltd.     

Free (open) boundary condition: some experiences with viscous flow simulations

The free (or open) boundary condition (FBC, OBC) was proposed by Papanastasiou et al. (A new outflow boundary condition, International Journal for Numerical Methods in Fluids, 1992; 14:587–608) to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. In the present work, implementation of the FBC has been tested in several benchmark problems of viscous flow in fluid mechanics. The FEM is used to provide numerical results for both cases of planar and axisymmetric domains under laminar, isothermal or non‐isothermal, steady‐state conditions, for Newtonian fluids. The effects of inertia, gravity, compressibility, pressure dependence of the viscosity, slip at the wall, and surface tension are all considered individually in the extrudate‐swell benchmark problem for a wide range of the relevant parameters. The present results extend previous ones regarding the applicability of the FBC and show cases where the FBC is inappropriate, namely in the extrudate‐swell problem with gravity or surface‐tension effects. Particular emphasis has been given to the pressure at the outflow, which is the most sensitive quantity of the computations. In all cases where FBC is appropriate, excellent agreement has been found in comparisons with results from very long domains. The formulation for Picard‐type iterations is given in some detail, and the differences with the Newton–Raphson formulation are highlighted regarding some computational aspects. Copyright © 2011 John Wiley & Sons, Ltd.     

A refined r‐factor algorithm for TVD schemes on arbitrary unstructured meshes

A refined r‐factor algorithm for implementing total variation diminishing (TVD) schemes on arbitrary unstructured meshes, referred to henceforth as a face‐perpendicular far‐upwind interpolation scheme for arbitrary meshes (FFISAM), is proposed based on an extensive review of the existing r‐factor algorithms available in the literature. The design principles, as well as the respective advantages and disadvantages, of the existing algorithms are first systematically analyzed before presenting the FFISAM. The FFISAM is designed to combine the merits of various existing r‐factor algorithms. The performance of the FFISAM, implemented in 10 classical TVD schemes, is evaluated against four two‐dimensional pure‐advection benchmark test cases where analytical solutions are available. The numerical results clearly show that the FFISAM leads to a better overall performance than the existing algorithms in terms of accuracy and convergence on arbitrary unstructured meshes for the 10 classical TVD schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

A stress‐based least‐squares finite‐element model for incompressible Navier–Stokes equations

In this paper we present a stress‐based least‐squares finite‐element formulation for the solution of the Navier–Stokes equations governing flows of viscous incompressible fluids. Stress components are introduced as independent variables to make the system first order. Continuity equation becomes an algebraic equation and is eliminated from the system with suitable modifications. The h and p convergence are verified using the exact solution of Kovasznay flow. Steady flow past a large circular cylinder in a channel is solved to test mass conservation. Transient flow over a backward‐facing step problem is solved on several meshes. Results are compared with that obtained using vorticity‐based first‐order formulation for both benchmark problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Buoyancy modelling with incompressible SPH for laminar and turbulent flows

This work aims to model buoyant, laminar or turbulent flows, using a two‐dimensional incompressible smoothed particle hydrodynamics model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work [1], we extend the unified semi‐analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds‐averaged Navier–Stokes approach to treat turbulent flows. The k ? ? turbulence model is used, where buoyancy is modelled through an additional term in the k ? ? equations like in mesh‐based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or to impose heat flux (Neumann) wall boundary conditions in incompressible smoothed particle hydrodynamics. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock‐exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a finite volume approach using an open‐source industrial code. Copyright © 2015 John Wiley & Sons, Ltd.     

A short note on the reattachment length for BFS problem

The standard backward‐facing step flow problem is solved for steady state laminar case using stream function‐vorticity method. The steady state results are obtained as the asymptotic solution of the transient formulation. The primary reattachment length is studied and the discrepancy in the v velocity is reported. A method for determining appropriate locations for comparison is proposed. The energy equation is solved and found to be in good agreement with benchmark results. Copyright © 2005 John Wiley & Sons, Ltd.     

A new r‐ratio formulation for TVD schemes for vertex‐centered FVM on an unstructured mesh

The r‐ratio is a parameter that measures the local monotonicity, by which a number of high‐resolution and TVD schemes can be formed. A number of r‐ratio formulations for TVD schemes have been presented over the last few decades to solve the transport equation in shallow waters based on the finite volume method (FVM). However, unlike structured meshes, the coordinate directions are not clearly defined on an unstructured mesh; therefore, some r‐ratio formulations have been established by approximating the solute concentration at virtual nodes, which may be estimated from different assumptions. However, some formulations may introduce either oscillation or diffusion behavior within the vertex‐centered (VC) framework. In this paper, a new r‐ratio formulation, applied to an unstructured grid in the VC framework, is proposed and compared with the traditional r‐ratio formulations. Through seven commonly used benchmark tests, it is shown that the newly proposed r‐ratio formulation obtains better results than the traditional ones with less numerical diffusion and spurious oscillation. Moreover, three commonly used TVD schemes—SUPERBEE, MINMOD, and MUSCL—and two high‐order schemes—SOU and QUICK—are implemented and compared using the new r‐ratio formulation. The new r‐ratio formulation is shown to be sufficiently comprehensive to permit the general implementation of a high‐resolution scheme within the VC framework. Finally, the sensitivity test for different grid types demonstrates the good adaptability of this new r‐ratio formulation. Copyright © 2015 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.     

A convergence study of two ocean models applied to a density driven flow

A three‐dimensional, baroclinic and rotational benchmark for hydrostatic coastal ocean models is suggested. The computational domain is a quadratic basin 200×200 km horizontally and 200 m deep. The lateral boundaries are closed. Solutions to the problems are estimated with two different sigma co‐ordinate models, both for the diagnostic and the prognostic case. Grid‐ and time‐converged results are presented. For the transports, convergence within two significant digits is reported. Copyright © 2001 John Wiley & Sons, Ltd.     

Monolithic Newton‐multigrid solution techniques for incompressible nonlinear flow models

We present special Newton‐multigrid techniques for stationary incompressible nonlinear flow models discretized by the high order LBB‐stable Q₂P₁ element pair. We treat the resulting nonlinear and the corresponding linear discrete systems by a fully coupled monolithic approach to maintain high accuracy and robustness, particularly with respect to different rheological behaviors and also regarding different problem sizes and types of nonlinearity. Here, local pressure Schur complement techniques are presented as a generalization of the classical Vanka smoother. The discussed methodology is implemented for the well‐known flow around cylinder benchmark configuration for generalized Newtonian as well as non‐Newtonian flows including non‐isothermal, shear/pressure dependent and viscoelastic effects.Copyright © 2012 John Wiley & Sons, Ltd.     

Temporal accuracy analysis of phase change convection simulations using the JFNK‐SIMPLE algorithm

The incompressible Navier–Stokes and energy conservation equations with phase change effects are applied to two benchmark problems: (1) non‐dimensional freezing with convection; and (2) pure gallium melting. Using a Jacobian‐free Newton–Krylov (JFNK) fully implicit solution method preconditioned with the SIMPLE (Numerical Heat Transfer and Fluid Flow. Hemisphere: New York, 1980) algorithm using centred discretization in space and three‐level discretization in time converges with second‐order accuracy for these problems. In the case of non‐dimensional freezing, the temporal accuracy is sensitive to the choice of velocity attenuation parameter. By comparing to solutions with first‐order backward Euler discretization in time, it is shown that the second‐order accuracy in time is required to resolve the fine‐scale convection structure during early gallium melting. Qualitative discrepancies develop over time for both the first‐order temporal discretized simulation using the JFNK‐SIMPLE algorithm that converges the nonlinearities and a SIMPLE‐based algorithm that converges to a more common mass balance condition. The discrepancies in the JFNK‐SIMPLE simulations using only first‐order rather than second‐order accurate temporal discretization for a given time step size appear to be offset in time. Copyright © 2007 John Wiley & Sons, Ltd.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.     

Mobile and immobile fluid transport: Coupling framework

We introduce a solver method for mobile and immobile transport regions. The motivation is driven by transport processes in porous media (e.g. waste disposal, chemical deposition processes). We analyze the coupled transport‐reaction equation with mobile and immobile areas. We apply analytical methods, such as Laplace‐transformation, and for the numerical methods we apply Godunov's scheme, see (Mat. Sb. 1959; 47 :271–306; Finite Volume Methods for Hyperbolic Problems. Cambridge University Press: Cambridge, 2002). The method is based numerically on flux‐based characteristic methods and is an attractive alternative to the classical higher‐order TVD methods, see (J. Comput. Phys. 1993; 49 :357–393). In this paper, we will focus on the derivation of analytical solutions for general and special solutions of the characteristic methods that are embedded in a finite‐volume method. At the end of the paper, we illustrate the higher‐order method for different benchmark problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Implicit LES of free and wall‐bounded turbulent flows based on the discontinuous Galerkin/symmetric interior penalty method

This paper presents the second validation step of a compressible discontinuous Galerkin solver with symmetric interior penalty (DGM/SIP) for the direct numerical simulation (DNS) and the large eddy simulation (LES) of complex flows. The method has already been successfully validated for DNS of an academic flow and has been applied to flows around complex geometries (e.g. airfoils and turbomachinery blades). During these studies, the advantages of the dissipation properties of the method have been highlighted, showing a natural tendency to dissipate only the under‐resolved scales (i.e the smallest scales present on the mesh), leaving the larger scales unaffected. This phenomenon is further enhanced as the polynomial order is increased. Indeed, the order increases the dissipation at the largest wave numbers, while its range of impact is reduced. These properties are spectrally compatible with a subgrid‐scale model, and hence DGM may be well suited to be used for an implicit LES (ILES) approach. A validation of this DGM/ILES approach is here investigated on canonical flows, allowing to study the impact of the discretisation on the turbulence for under‐resolved computations. The first test case is the LES of decaying homogeneous isotropic turbulence (HIT) at very high Reynolds number. This benchmark allows to assess the spectral behaviour of the method for implicit LES. The results are in agreement with theory and are even slightly more accurate than other numerical results from literature, obtained using a pseudo‐spectral (PS) method with a state‐of‐the‐art subgrid‐scale model. The second benchmark is the LES of the channel flow. Three Reynolds numbers are considered: Re_τ=395, 590 and 950. The results are compared with DNS of Moser et al. and Hoyas et al., also using PS methods. Both averaged velocity and fluctuations are globally in good agreement with the reference, showing the ability of the method to predict equilibrium wall‐bounded flow turbulence. To show that the method is able to perform accurate DNS, a DNS of HIT at Re_λ=64 and a DNS of the channel flow at Re_τ=180 are also performed. The effects of the grid refinement are investigated on the channel flow at Re_τ=395, highlighting the improvement of the results when refining the mesh in the spanwise direction. Finally, the modification of the ILES parameters, that is the Riemann solver and of the SIP coefficient, is studied on both cases, showing a significant influence on the choice of the Riemann solver. Copyright © 2015 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.     

A novel Cartesian cut cell solver for incompressible viscous flow in irregular domains

A Cartesian cut cell solver with solution‐based adaptive mesh refinement is developed for simulating viscous, incompressible flows with arbitrary complex geometries. The cut cells are automatically generated using Volume CAD (VCAD), a framework for storing geometric and material attribute data. Unlike earlier cut cell methods, this solver organizes the cutting patterns into only six categories and further subdivides the resulting pentagon into two quadrilaterals, such that mesh data can be stored by uniform data structure and the post‐processing of flow data can be handled conveniently. A novel method is proposed to treat minuscule cut cells without the process of cell merging. A collocated finite volume method, which can be used even when multiple cell shapes and orthogonal and non‐orthogonal grids exist in the decomposition, is employed to discretize the Navier–Stokes equations. A modified SIMPLE‐based smoothing pressure correction scheme is applied in this cut cell method to suppress checkerboard pressure oscillations caused by collocated arrangement. The solver is first used to simulate a channel flow to demonstrate its calculation accuracy expressed with L₁ and L_∞ norm errors and then the method is utilized to solve three benchmark problems of flow and heat transfer within irregular domains to verify its feasibility, efficiency, accuracy and potential in engineering applications. Copyright © 2010 John Wiley & Sons, Ltd.     

Integrated radial basis functions‐based differential quadrature method and its performance

In this paper, indirect radial basis function networks (IRBFN) proposed by Nam and Tranh (Neural Networks 2001; 14 (2):185–199; Appl. Math. Modelling 2003; 27 :197–220) are incorporated into the differential quadrature (DQ) approximation of derivatives. For simplicity, this new variant of RBF‐DQ approach is named as iRBF‐DQ method. The proposed approach is validated by its application to solve the one‐dimensional Burger's equation, and simulate natural convection in a concentric annulus by solving Navier–Stokes equations. It was found that as compared to the benchmark data, the iRBF‐DQ approach can provide more accurate results than the original RBF‐DQ method. Copyright © 2006 John Wiley & Sons, Ltd.