Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.     

A fractional two‐step implicit algorithm using curvilinear collected grids

An efficient fractional two‐step implicit algorithm is reported to simulate incompressible fluid flows in a boundary‐fitted curvilinear collocated grid system. Using the finite volume method, the convection terms are discretized by the high‐accuracy Roe's scheme to minimize numerical diffusion. An implicitness coefficient Π is introduced to accelerate the rate of convergence. It is demonstrated that the proposed algorithm links the fractional step method to the pressure correction procedure, and the SIMPLEC method could be considered as a special case of the fractional two‐step implicit algorithm (when Π=1). The proposed algorithm is applicable to unsteady flows and steady flows. Three benchmark two‐dimensional laminar flows are tested to evaluate the performance of the proposed algorithm. Performance is measured by sensitivity analyses of the efficiency, accuracy, grid density, grid skewness and Reynolds number on the solutions. Results show that the model is efficient and robust. Copyright © 2006 John Wiley & Sons, Ltd.     

Homotopy perturbation method to obtain new solitary solutions with compact support for Boussinesq‐like B(2n, 2n) equations with fully nonlinear dispersion

The aim of this paper is to obtain new solitary solutions with compact support for Boussinesq‐like B(2n, 2n) equations with fully nonlinear dispersion using the homotopy perturbation method (HPM). The special case B(2, 2) is chosen to illustrate the concrete scheme of the HPM in B(2n, 2n) equations. General formulas for the solutions of B(2n, 2n) equations are established. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt

In this paper, we prove in general that the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the homotopy analysis method (HAM) profound in 1992 when ħ = −1. Besides, by using the thin film flows of Sisko and Oldroyd 6-constant fluids on a moving belt as examples, we show that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.: Chaos Solitons and Fractals (2006) in press) are divergent, and thus useless. However, by choosing a proper value of the auxiliary parameter ħ, we give convergent series solution by means of the HAM. These two examples also show that, different from the HPM and other traditional analytic techniques, the HAM indeed provides us with a simple way to ensure the convergence of the solution.     

Steady‐state solution of incompressible Navier–Stokes equations using discrete least‐squares meshless method

A fractional step method for the solution of the steady state incompressible Navier–Stokes equations is proposed in this paper in conjunction with a meshless method, named discrete least‐squares meshless (DLSM). The proposed fractional step method is a first‐order accurate scheme, named semi‐incremental fractional step method, which is a general form of the previous first‐order fractional step methods, i.e. non‐incremental and incremental schemes. One of the most important advantages of the proposed scheme is its capability to use large time step sizes for the solution of incompressible Navier–Stokes equations. DLSM method uses moving least‐squares shape functions for function approximation and discrete least‐squares technique for discretization of the governing differential equations and their boundary conditions. As there is no need for a background mesh, the DLSM method can be called a truly meshless method and enjoys symmetric and positive‐definite properties. Several numerical examples are used to demonstrate the ability and the efficiency of the proposed scheme and the discrete least‐squares meshless method. The results are shown to compare favorably with those of the previously published works. Copyright © 2010 John Wiley & Sons, Ltd.     

A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows

A three‐dimensional, non‐hydrostatic pressure, numerical model with k–ε equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non‐hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non‐hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure–Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

Bubble‐based stabilized finite element methods for time‐dependent convection–diffusion–reaction problems

In this paper, we propose a numerical algorithm for time‐dependent convection–diffusion–reaction problems and compare its performance with the well‐known numerical methods in the literature. Time discretization is performed by using fractional‐step θ‐scheme, while an economical form of the residual‐free bubble method is used for the space discretization. We compare the proposed algorithm with the classical stabilized finite element methods over several benchmark problems for a wide range of problem configurations. The effect of the order in the sequence of discretization (in time and in space) to the quality of the approximation is also investigated. Numerical experiments show the improvement through the proposed algorithm over the classical methods in either cases. Copyright © 2016 John Wiley & Sons, Ltd.     

A meshless method for numerical simulation of depth‐averaged turbulence flows using a k‐ϵ model

A three‐dimensional numerical model is developed to analyze free surface flows and water impact problems. The flow of an incompressible viscous fluid is solved using the unsteady Navier–Stokes equations. Pseudo‐time derivatives are introduced into the equations to improve computational efficiency. The interface between the two phases is tracked using a volume‐of‐fluid interface tracking algorithm developed in a generalized curvilinear coordinate system. The accuracy of the volume‐of‐fluid method is first evaluated by the multiple numerical benchmark tests, including two‐dimensional and three‐dimensional deformation cases on curvilinear grids. The performance and capability of the numerical model for water impact problems are demonstrated by simulations of water entries of the free‐falling hemisphere and cone, based on comparisons of water impact loadings, velocities, and penetrations of the body with experimental data. For further validation, computations of the dam‐break flows are presented, based on an analysis of the wave front propagation, water level, and the dynamic pressure impact of the waves on the downstream walls, on a specific container, and on a tall structure. Extensive comparisons between the obtained solutions, the experimental data, and the results of other numerical simulations in the literature are presented and show a good agreement. Copyright © 2015 John Wiley & Sons, Ltd.     

Solid‐particle erosion in the tube end of the tube sheet of a shell‐and‐tube heat exchanger

Erosion is one of the major problems in many industrial processes, and in particular, in heat exchangers. The effects of flow velocity and sand particle size on the rate of erosion in a typical shell‐and‐tube heat exchanger were investigated numerically using the Lagrangian particle‐tracking method. Erosion and penetration rates were obtained for sand particles of diameters ranging from 10 to 500 µm and for inlet flow velocities ranging from 0.197 to 2.95 m/s. A flow visualization experiment was conducted with the objective of verifying the accuracy of the continuous phase calculation procedure. Comparison with available experimental data of penetration rates was also conducted. These comparisons resulted in a good agreement. The results show that the location and number of eroded tubes depend mainly on the particle size and velocity magnitude at the header inlet. The rate of erosion depends exponentially on the velocity. The particle size shows negligible effect on the erosion rate at high velocity values and the large‐size particles show less erosion rates compared to the small‐size particles at low values of inlet flow velocities. The results indicated that the erosion and penetration rates are insignificant at the lower end of the velocity range. However, these rates were found to increase continuously with the increase of the inlet flow velocity for all particle sizes. The particle size creating the highest erosion rate was found to depend on the flow velocity range. Copyright © 2005 John Wiley & Sons, Ltd.     

Use of two‐step split‐operator approach for 2D shallow water flow computation

The objective of this paper is to present a methodology of using a two‐step split‐operator approach for solving the shallow water flow equations in terms of an orthogonal curvilinear co‐ordinate system. This approach is in fact one kind of the so‐called fractional step method that has been popularly used for computations of dynamic flow. By following that the momentum equations are decomposed into two portions, the computation procedure involves two steps. The first step (dispersion step) is to compute the provisional velocity in the momentum equation without the pressure gradient. The second step (propagation step) is to correct the provisional velocity by considering a divergence‐free velocity field, including the effect of the pressure gradient. This newly proposed method, other than the conventional split‐operator methods, such as the projection method, considers the effects of pressure gradient and bed friction in the second step. The advantage of this treatment is that it increases flexibility, efficiency and applicability of numerical simulation for various hydraulic problems. Four cases, including back‐water flow, reverse flow, circular basin flow and unsteady flow, have been demonstrated to show the accuracy and practical application of the method. Copyright © 1999 John Wiley & Sons, Ltd.     

Compact high‐resolution algorithms for time‐dependent advection on unstructured grids

A technique for constructing monotone, high resolution, multi‐dimensional upwind fluctuation distribution schemes for the scalar advection equation is presented. The method combines the second‐order Lax–Wendroff scheme with the upwind positive streamwise invariant (PSI) scheme via a fluctuation redistribution step, which ensures monotonicity (and which is a generalization of the flux‐corrected transport approach for fluctuation distribution schemes). Furthermore, the concept of a distribution point is introduced, which, when related to the equivalent equation for the scheme, leads to a ‘preferred direction’ for the limiting procedure, and hence to a new distribution of the fluctuation, which retains second‐order accuracy from the Lax–Wendroff scheme, even when the solution contains turning points. Experimental comparisons show that the new method compares favourably in terms of speed, accuracy and robustness with other, similar, techniques. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient curvilinear non‐hydrostatic model for simulating surface water waves

An efficient curvilinear non‐hydrostatic free surface model is developed to simulate surface water waves in horizontally curved boundaries. The generalized curvilinear governing equations are solved by a fractional step method on a rectangular transformed domain. Of importance is to employ a higher order (either quadratic or cubic spline function) integral method for the top‐layer non‐hydrostatic pressure under a staggered grid framework. Model accuracy and efficiency, in terms of required vertical layers, are critically examined on a linear progressive wave case. The model is then applied to simulate waves propagating in a canal with variable widths, cnoidal wave runup around a circular cylinder, and three‐dimensional wave transformation in a circular channel. Overall the results show that the curvilinear non‐hydrostatic model using a few, e.g. 2–4, vertical layers is capable of simulating wave dispersion, diffraction, and reflection due to curved sidewalls. Copyright © 2010 John Wiley & Sons, Ltd.     

A dual‐time central‐difference interface‐capturing finite volume scheme applied to cavitation modelling

This paper describes a central‐difference interface‐capturing scheme applied to the prediction of flows with cavitation. Compressible cavitation schemes based on standard central‐difference solvers have been previously described, but the current scheme uses an incompressible formulation only previously implemented with an upwind solver. The central‐difference solver offers significant advantages in computational time compared with upwind schemes. Regions of cavitation are captured rather than tracked. This means that there is no need for complex tracking and reconstruction procedures for the interface of the cavitation region. The use of such schemes on an arbitrarily unstructured mesh is no more complicated than on its structured counterpart. Results for a number of test cases are presented, with comparisons made with both experimental data and other numerical solutions. Copyright © 2010 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.     

A high‐order mass‐lumping procedure for B‐spline collocation method with application to incompressible flow simulations

This paper presents new developments of the staggered spline collocation method for cost‐effective solution to the incompressible Navier–Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high‐accuracy of the B‐spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high‐order generalization of the mass‐lumping technique of the finite‐element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi‐consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier–Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd.     

Kernel‐based vector field reconstruction in computational fluid dynamic models

Kernel‐based reconstruction methods are applied to obtain highly accurate approximations of local vector fields from normal components assigned at the edges of a computational mesh. The theoretical background of kernel‐based reconstructions for vector‐valued functions is first reviewed, before the reconstruction method is adapted to the specific requirements of relevant applications in computational fluid dynamics. To this end, important computational aspects concerning the design of the reconstruction scheme like the selection of suitable stencils are explained in detail. Extensive numerical examples and comparisons concerning hydrodynamic models show that the proposed kernel‐based reconstruction improves the accuracy of standard finite element discretizations, including Raviart–Thomas (RT) elements, quite significantly, while retaining discrete conservation properties of important physical quantities, such as mass, vorticity, or potential enstrophy. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical analysis of higher-dimensional dispersive long-wave equations

In this paper, numerical analysis of the (2+1)-dimensional dispersive long-wave equation (DLWE) is studied by using the homotopy perturbation method (HPM). For this purpose, the available analytical solutions obtained by multiple traveling-wave solution will be compared to show the validity and accuracy of the presented numerical algorithm. The obtained results prove the convergence and accuracy of the HPM for the numerically analyzed (2+1)-dimensional DLWE system.     

A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

A new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.     

A high‐order triangular discontinuous Galerkin oceanic shallow water model

A high‐order triangular discontinuous Galerkin (DG) method is applied to the two‐dimensional oceanic shallow water equations. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high‐order Lagrange polynomials on the triangle using nodal sets up to 15th order. Both the area and boundary integrals are evaluated using order 2N Gauss cubature rules. The use of exact integration for the area integrals leads naturally to a full mass matrix; however, by using straight‐edged triangles we eliminate the mass matrix completely from the discrete equations. Besides obviating the need for a mass matrix, triangular elements offer other obvious advantages in the construction of oceanic shallow water models, specifically the ability to use unstructured grids in order to better represent the continental coastlines for use in tsunami modeling. In this paper, we focus primarily on testing the discrete spatial operators by using six test cases—three of which have analytic solutions. The three tests having analytic solutions show that the high‐order triangular DG method exhibits exponential convergence. Furthermore, comparisons with a spectral element model show that the DG model is superior for all polynomial orders and test cases considered. Copyright © 2007 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.