Explicit formulations for evaluation of velocity gradients using boundary‐domain integral equations in 2D and 3D viscous flows

In this paper, explicit boundary‐domain integral equations for evaluating velocity gradients are derived from the basic velocity integral equations. A free term is produced in the new strongly singular integral equation, which is not included in recent formulations using the complex variable differentiation method (CVDM) to compute velocity gradients (Int. J. Numer. Meth. Fluids 2004; 45 :463–484; Int. J. Numer. Meth. Fluids 2005; 47 :19–43). The strongly singular domain integrals involved in the new integral equations are accurately evaluated using the radial integration method (RIM). Considerable computational time for evaluating integrals of velocity gradients can be saved by using present formulation than using CVDM. The formulation derived in this paper together with those presented in reference (Int. J. Numer. Meth. Fluids 2004; 45 :463–484) for 2D and in (Int. J. Numer. Meth. Fluids 2005; 47 :19–43) for 3D problems constitutes a complete boundary‐domain integral equation system for solving full Navier–Stokes equations using primitive variables. Three numerical examples for steady incompressible viscous flow are given to validate the derived formulations. Copyright © 2007 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.     

Remarks on the links between low‐order DG methods and some finite‐difference schemes for the Stokes problem

In this paper we demonstrate that some well‐known finite‐difference schemes can be interpreted within the framework of the local discontinuous Galerkin (LDG) methods using the low‐order piecewise solenoidal discrete spaces introduced in (SIAM J. Numer. Anal. 1990; 27 (6): 1466–1485). In particular, it appears that it is possible to derive the well‐known MAC scheme using a first‐order Nédélec approximation on rectangular cells. It has been recently interpreted within the framework of the Raviart–Thomas approximation by Kanschat (Int. J. Numer. Meth. Fluids 2007; published online). The two approximations are algebraically equivalent to the MAC scheme, however, they have to be applied on grids that are staggered on a distance h/2 in each direction. This paper also demonstrates that both discretizations allow for the construction of a divergence‐free basis, which yields a linear system with a ‘biharmonic’ conditioning. Both this paper and Kanschat (Int. J. Numer. Meth. Fluids 2007; published online) demonstrate that the LDG framework can be used to generalize some popular finite‐difference schemes to grids that are not parallel to the coordinate axes or that are unstructured. Copyright © 2008 John Wiley & Sons, Ltd.     

A transformation‐free HOC scheme for incompressible viscous flows on nonuniform polar grids

We recently proposed a transformation‐free higher‐order compact (HOC) scheme for two‐dimensional (2‐D) steady convection–diffusion equations on nonuniform Cartesian grids (Int. J. Numer. Meth. Fluids 2004; 44 :33–53). As the scheme was equipped to handle only constant coefficients for the second‐order derivatives, it could not be extended directly to curvilinear coordinates, where they invariably occur as variables. In this paper, we extend the scheme to cylindrical polar coordinates for the 2‐D convection–diffusion equations and more specifically to the 2‐D incompressible viscous flows governed by the Navier–Stokes (N–S) equations. We first apply the formulation to a problem having analytical solution and demonstrate its fourth‐order spatial accuracy. We then apply it to the flow past an impulsively started circular cylinder problem and finally to the driven polar cavity problem. We present our numerical results and compare them with established numerical and analytical and experimental results whenever available. This new approach is seen to produce excellent comparison in all the cases. Copyright © 2009 John Wiley & Sons, Ltd.     

Spectral element analysis of herringbone‐grooved journal bearings with groove–ridge discontinuity

Many researchers have investigated the performance of herringbone‐grooved journal bearings (HGJBs). However, few have yet mentioned the issue of film thickness discontinuities in HGJBs with a finite number of grooves. Most studies have involved the application of a finite difference method for discretization. The present work utilizes the spectral element method to calculate the pressure distribution and dynamic coefficients of HGJBs, in which the thickness of the fluid film changes abruptly in the groove–ridge region. Conservation of mass is adopted to solve the problem. Additionally, the present method can be adopted for grooves with curvy geometry. The numerical results were compared with the analytical solution for a one‐dimensional slider bearing and an HGJB. It also shows that for the case of HGJB, the numerical result by the present method is more accurate than the numerical results found in the literature (Trans. ASME J. Tribol. 2000; 122 : 103–109, Int. J. Numer. Meth. Heat Fluid Flow 2002; 12 : 518–540). Furthermore, employing the present method with the Elrod algorithm can improve the accuracy of deriving loads of HGJBs when cavitation occurs. In addition, the result displays the efficiency of the present method by observing the CPU time. Therefore, the approach can be employed to compute the critical mass of a HGJB. The influence of changing groove angle, groove depth, groove width, and the eccentricity on the critical mass are discussed. Observing the variations in critical mass shows that when the eccentricity is small, a larger groove angle, a lower groove depth, and smaller groove width correspond to a higher critical mass of the HGJB. Copyright © 2010 John Wiley & Sons, Ltd.     

An evaluation of a 3D free‐energy‐based lattice Boltzmann model for multiphase flows with large density ratio

In this paper, the 3D Navier–Stokes (N–S) equation and Cahn–Hilliard (C–H) equations were solved using a free‐energy‐based lattice Boltzmann (LB) model. In this model, a LB equation with a D3Q19 velocity model is used to recover continuity and N–S equations while another LB equation with D3Q7 velocity model for solving C–H equation (Int. J. Numer. Meth. Fluids, 2008; 56 :1653–1671) is applied to solve the 3D C–H equation. To avoid the excessive use of computational resources, a moving reference frame is adopted to allow long‐time simulation of a bubble rising. How to handle the inlet/outlet and moving‐wall boundary conditions are suggested. These boundary conditions are simple and easy for implementation. This model's performance on two‐phase flows was investigated and the mass conservation of this model was evaluated. The model is validated by its application to simulate the 3D air bubble rising in viscous liquid (density ratio is 1000). Good agreement was obtained between the present numerical results and experimental results when Re is small. However, for high‐Re cases, the mass conservation seems not so good as the low‐Re case. Copyright © 2009 John Wiley & Sons, Ltd.     

Comments on ‘Simplex space–time meshes in finite element simulations’

Some comments are provided on the citations offered in a recent paper (M. Behr, Int. J. Numer. Meth. Fluids 2008; 57 :1421–1434) that describes space–time finite element computations of advection of ‘Gaussian hills’, including computations with mesh refinement in the time direction. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical modelling of 3D stratified free surface flows: a case study of sediment dumping

A three‐dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds‐averaged Navier–Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ‐coordinate system with the use of operator‐splitting method (Int. J. Numer. Meth. Fluids 2002; 38 :1045–1068). The numerical model is validated against the one‐dimensional diffusion problem and the two‐dimensional density‐gradient flow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratified flow. For the two‐dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium‐size sediments. The model is also employed to study the dumping of sediments in a three‐dimensional environment with the presence of free surface. It is found that during the descending process an annulus‐like cloud is formed for fine sediments whereas a plate‐like cloud for medium‐size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Instabilities during batch sedimentation in geometries containing obstacles: A numerical and experimental study

Batch sedimentation of non‐colloidal particle suspensions is studied with nuclear magnetic resonance flow visualization and continuum‐level numerical modelling of particle migration. The experimental method gives particle volume fraction as a function of time and position, which then provides validation data for the numerical model. A finite element method is used to discretize the equations of motion, including an evolution equation for the particle volume fraction and a generalized Newtonian viscosity dependent on local particle concentration. The diffusive‐flux equation is based on the Phillips model (Phys. Fluids A 1992; 4 :30–40) and includes sedimentation terms described by Zhang and Acrivos (Int. J. Multiphase Flow 1994; 20 :579–591). The model and experiments are utilized in three distinct geometries with particles that are heavier and lighter than the suspending fluid, depending on the experiment: (1) sedimentation in a cylinder with a contraction; (2) particle flotation in a horizontal cylinder with a horizontal rod; and (3) flotation around a rectangular inclusion. Secondary flows appear in both the experiments and the simulations when a region of higher density fluid is above a lower density fluid. The secondary flows result in particle inhomogeneities, Rayleigh–Taylor‐like instabilities, and remixing, though the effect in the simulations is more pronounced than in the experiments. Published in 2007 by John Wiley & Sons, Ltd.     

The general boundary element method and its further generalizations

In this paper, the basic ideas of the general boundary element method (BEM) proposed by Liao [in Boundary Elements XVII, Computational Mechanics Publications, Southampton, MA, 1995, pp. 67–74; Int. J. Numer. Methods Fluids, 23 , 739–751 (1996), 24 , 863–873 (1997); Comput. Mech., 20 , 397–406 (1997)] and Liao and Chwang [Int. J. Numer. Methods Fluids, 23 , 467–483 (1996)] are further generalized by introducing a non‐zero parameter . Some related mathematical theorems are proposed. This general BEM contains the traditional BEM in logic, but is valid for non‐linear problems, including those whose governing equations and boundary conditions have no linear terms. Furthermore, the general BEM can solve non‐linear differential equations by means of no iterations. This disturbs the absolutely governing place of iterative methodology of the BEM for non‐linear problems. The general BEM can greatly enlarge application areas of the BEM as a kind of numerical technique. Two non‐linear problems are used to illustrate the validity and potential of the further generalized BEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Large eddy simulation of turbulent flows via domain decomposition techniques. Part 2: applications

The present paper discusses the application of large eddy simulation to incompressible turbulent flows in complex geometries. Algorithmic developments concerning the flow solver were provided in the companion paper (Int. J. Numer. Meth. Fluids, 2003; submitted), which addressed the development and validation of a multi‐domain kernel suitable for the integration of the elliptic partial differential equations arising from the fractional step procedure applied to the incompressible Navier–Stokes equations. Numerical results for several test problems are compared to reference experimental and numerical data to demonstrate the potential of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

A novel fully‐implicit finite volume method applied to the lid‐driven cavity problem—Part II: Linear stability analysis

A novel finite volume method, described in Part I of this paper (Sahin and Owens, Int. J. Numer. Meth. Fluids 2003; 42 :57–77), is applied in the linear stability analysis of a lid‐driven cavity flow in a square enclosure. A combination of Arnoldi's method and extrapolation to zero mesh size allows us to determine the first critical Reynolds number at which Hopf bifurcation takes place. The extreme sensitivity of the predicted critical Reynolds number to the accuracy of the method and to the treatment of the singularity points is noted. Results are compared with those in the literature and are in very good agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

A method to compute ship exhaust plumes with waves and wind

Thermal and concentration transport models are implemented in CFDShip‐Iowa version 4.5, a semi‐coupled solver for air/water free surface flow (Int. J. Numer. Meth. Fluids 2008; 58 (6):591–624), to investigate the exhaust plume around ship superstructures. An incompressible, variable density approximation is implemented where the density can change in all governing equations due to temperature variations only. The thermal and concentration models are tested for the cases of steady and unsteady flow with thermal and solution transport in a 2D square cavity, and for a 3D thermal plume in an open environment, showing good agreement between computational results and experimental data. To test the method in an extreme motions condition, the exhaust plume of the ONR Tumblehome model DTMB 5613 is studied, showing complicated vortical structures in air including a pair of counter‐rotating vortices downstream of the stack for cross‐flow, and bended bird‐plume shape in the symmetry plane and varying arc‐shape in axial sections both for temperature and NO_x concentration fields. Effects of smoke exhaust speed and wind speed on the temperature and concentration distributions are studied. Finally, a smoke downwash computation is performed for a ship free to move in 6 degrees of freedom in a sea state 8 condition. Copyright © 2010 John Wiley & Sons, Ltd.     

QALE‐FEM for modelling 3D overturning waves

A further development of the QALE‐FEM (quasi‐arbitrary Lagrangian–Eulerian finite element method) based on a fully nonlinear potential theory is presented in this paper. This development enables the QALE‐FEM to deal with three‐dimensional (3D) overturning waves over complex seabeds, which have not been considered since the method was devised by the authors of this paper in their previous works (J. Comput. Phys. 2006; 212 :52–72; J. Numer. Meth. Engng 2009; 78 :713–756). In order to tackle challenges associated with 3D overturning waves, two new numerical techniques are suggested. They are the techniques for moving the mesh and for calculating the fluid velocity near overturning jets, respectively. The developed method is validated by comparing its numerical results with experimental data and results from other numerical methods available in the literature. Good agreement is achieved. The computational efficiency of this method is also investigated for this kind of wave, which shows that the QALE‐FEM can be many times faster than other methods based on the same theory. Furthermore, 3D overturning waves propagating over a non‐symmetrical seabed or multiple reefs are simulated using the method. Some of these results have not been found elsewhere to the best of our knowledge. Copyright © 2009 John Wiley & Sons, Ltd.     

Comments on ‘Adiabatic shock capturing in perfect gas hypersonic flows’

Some comments are provided on the shock‐capturing techniques and stabilization parameters used in a recent paper (B.S. Kirk, Int. J. Numer. Meth. Fluids 2009; DOI: 10.1002/fld.2195 ) in conjunction with the SUPG formulation of compressible flows. Copyright © 2010 John Wiley & Sons, Ltd.     

Fast solvers for models of ICEO microfluidic flows

We demonstrate the performance of a fast computational algorithm for modeling the design of a microfluidic mixing device. The device uses an electrokinetic process, induced charge electroosmosis (J. Fluid Mech. 2004; 509 ), by which a flow through the device is driven by a set of polarizable obstacles in it. Its design is realized by manipulating the shape and orientation of the obstacles in order to maximize the amount of fluid mixing within the device. The computation entails the solution of a constrained optimization problem in which function evaluations require the numerical solution of a set of partial differential equations: a potential equation, the incompressible Navier–Stokes equations, and a mass‐transport equation. The most expensive component of the function evaluation (which must be performed at every step of an iteration for the optimization) is the solution of the Navier–Stokes equations. We show that by using some new robust algorithms for this task (SIAM J. Sci. Comput. 2002; 24 :237–256; J. Comput. Appl. Math. 2001; 128 :261–279), based on certain preconditioners that take advantage of the structure of the linearized problem, this computation can be done efficiently. Using this computational strategy, in conjunction with a derivative‐free pattern search algorithm for the optimization, applied to a finite element discretization of the problem, we are able to determine optimal configurations of microfluidic devices. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical study on heavy rigid particle motion in a plane wake flow by spectral element method

Experimental particle dispersion patterns in a plane wake flow at a high Reynolds number have been predicted numerically by discrete vortex method (Phys. Fluids A 1992; 4 :2244–2251; Int. J. Multiphase Flow 2000; 26 :1583–1607). To address the particle motion at a moderate Reynolds number, spectral element method is employed to provide an instantaneous wake flow field for particle dynamics equations, which are solved to make a detail classification of the patterns in relation to the Stokes and Froude numbers. It is found that particle motion features only depend on the Stokes number at a high Froude number and depend on both numbers at a low Froude number. A ratio of the Stokes number to squared Froude number is introduced and threshold values of this parameter are evaluated that delineate the different regions of particle behavior. The parameter describes approximately the gravitational settling velocity divided by the characteristic velocity of wake flow. In order to present effects of particle density but preserve rigid sphere, hollow sphere particle dynamics in the plane wake flow is investigated. The evolution of hollow particle motion patterns for the increase of equivalent particle density corresponds to that of solid particle motion patterns for the decrease of particle size. Although the thresholds change a little, the parameter can still make a good qualitative classification of particle motion patterns as the inner diameter changes. Copyright © 2008 John Wiley & Sons, Ltd.     

Resolution sensitivity of momentum‐exchange and immersed boundary methods for solid–fluid interaction in the lattice Boltzmann method

In the lattice Boltzmann method (LBM), the mechanism of fluid–solid interaction can be effectively captured by appropriately enforcing the no‐slip conditions in shear direction, and bounce‐back of the non‐equilibrium distribution portion in the normal direction at fluid–solid interfaces. Among various solid–fluid interaction schemes being proposed for LBM in recent decades, two simple fluid–solid interaction methods—the momentum exchange algorithm (MEA) and the immersed boundary scheme (IBS)—were developed based on the above concept. In this paper, MEA and IBS are implemented in a D2Q9 LBGK system and applied to measure the wall correction factors of drag force upon a stationary circular particle midway in the Poiseuille channel flow at very low Reynolds number and drag coefficients at low to moderate Reynolds numbers. MEA and IBS are also employed to compare the fluid‐induced torque over the cylinder in the Taylor–Couette flow, and the steady velocity of a particle settling under the influence of gravity inside a tube. The above experiments show that IBS seems to be more accurate and less demanding on lattice resolution. Copyright © 2010 John Wiley & Sons, Ltd.     

A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part II: application to the cavity filling and sloshing problems

Finite element analysis of fluid flow with moving free surface has been performed in 2‐D and 3‐D. The new VOF‐based numerical algorithm that has been proposed by the present authors (Int. J. Numer. Meth. Fluids, submitted) was applied to several 2‐D and 3‐D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools; the orientation vector to represent the free surface orientation in each cell and the baby‐cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2‐D and 3‐D cavity filling and sloshing problems in order to demonstrate the versatility and effectiveness of the scheme. The proposed numerical algorithm resolved successfully the free surfaces interacting with each other. The simulated results demonstrated applicability of the proposed numerical algorithm to the practical problems of large free surface motion. It has been also demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non‐uniform grid systems and can be extended to 3‐D free surface flow problems without additional efforts. Copyright © 2003 John Wiley & Sons, Ltd.