A complete boundary integral formulation for compressible Navier–Stokes equations

A complete boundary integral formulation for compressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for wall pressure and wall skin friction of two‐dimensional compressible laminar viscous flow around airfoils are in good agreement with field numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Finite volume solution of the Navier–Stokes equations in velocity–vorticity formulation

For the incompressible Navier–Stokes equations, vorticity‐based formulations have many attractive features over primitive‐variable velocity–pressure formulations. However, some features interfere with the use of the numerical methods based on the vorticity formulations, one of them being the lack of a boundary conditions on vorticity. In this paper, a novel approach is presented to solve the velocity–vorticity integro‐differential formulations. The general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so‐called generalized Biot–Savart formula combined with a fast summation algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well‐known fractional step approaches are used to solve the vorticity transport equation. The paper describes in detail how we accurately impose no normal‐flow and no tangential‐flow boundary conditions. We impose a no‐flux boundary condition on solid objects by the introduction of a proper amount of vorticity at wall. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential‐flow boundary condition. As application examples, the impulsively started flows through a flat plate and a circular cylinder are computed using the method. The present results are compared with the analytical solution and other numerical results and show good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

Three-dimensional incompressible Navier–Stokes equations on non-orthogonal staggered grids using the velocity–vorticity formulation

This paper is concerned with the numerical resolution of the incompressible Navier–Stokes equations in the velocity–vorticity form on non-orthogonal structured grids. The discretization is performed in such a way, that the discrete operators mimic the properties of the continuous ones. This allows the discrete equivalence between the primitive and velocity–vorticity formulations to be proved. This last formulation can thus be seen as a particular technique for solving the primitive equations. The difficulty associated with non-simply connected computational domains and with the implementation of the boundary conditions are discussed. One of the main drawback of the velocity–vorticity formulation, relative to the additional computational work required for solving the additional unknowns, is alleviated. Two- and three-dimensional numerical test cases validate the proposed method. © 1998 John Wiley & Sons, Ltd.     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

A velocity–vorticity formulation of the Navier–Stokes equations is presented as an alternative to the primitive variables approach. The velocity components and the vorticity are solved for in a fully coupled manner using a Newton method. No artificial viscosity is required in this formulation. The pressure is updated by a method allowing natural imposition of boundary conditions. Incompressible and subsonic results are presented for two-dimensional laminar internal flows up to high Reynolds numbers.     

Fundamental solutions of the streamfunction–vorticity formulation of the Navier–Stokes equations

A new boundary element procedure is developed for the solution of the streamfunction–vorticity formulation of the Navier–Stokes equations in two dimensions. The differential equations are stated in their transient version and then discretized via finite differences with respect to time. In this discretization, the non-linear inertial terms are evaluated in a previous time step, thus making the scheme explicit with respect to them. In the resulting discretized equations, fundamental solutions that take into account the coupling between the equations are developed by treating the non-linear terms as in homogeneities. The resulting boundary integral equations are solved by the regular boundary element method, in which the singular points are placed outside the solution domain.     

Solution of the incompressible Navier–Stokes equations on domains with one or several open boundaries1

The aim of this paper is to develop a methodology for solving the incompressible Navier–Stokes equations in the presence of one or several open boundaries. A new set of open boundary conditions is first proposed. This has been developed in the context of the velocity–vorticity formulation, but it is also emphasized how it can be formally extended to the equations in primitive variables. The case of a domain involving several independent open boundaries is considered next. An influence matrix technique is applied such that the inlet mass flux is split onto the several outlets in order to enforce the prescribed mean pressure at each outlet. Both approaches are validated by numerical test cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Pseudostress–velocity formulation for incompressible Navier–Stokes equations

This paper presents a numerical algorithm using the pseudostress–velocity formulation to solve incompressible Newtonian flows. The pseudostress–velocity formulation is a variation of the stress–velocity formulation, which does not require symmetric tensor spaces in the finite element discretization. Hence its discretization is greatly simplified. The discrete system is further decoupled into an H ( div ) problem for the pseudostress and a post‐process resolving the velocity. This can be done conveniently by using the penalty method for steady‐state flows or by using the time discretization for nonsteady‐state flows. We apply this formulation to the 2D lid‐driven cavity problem and study its grid convergence rate. Also, computational results of the time‐dependent‐driven cavity problem and the flow past rectangular problem are reported. Copyright © 2009 John Wiley & Sons, Ltd.     

The fundamental solution method for incompressible Navier–Stokes equations

A complete boundary integral formulation for incompressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch show good agreement with available experimental data. © 1998 John Wiley & Sons, Ltd.     

Towards a transparent boundary condition for compressible Navier–Stokes equations

A new artificial boundary condition for two‐dimensional subsonic flows governed by the compressible Navier–Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow, as well as a convection velocity, is used to properly discretize the terms corresponding to the entering waves. Numerical tests on various classical model problems, whose solutions are known, and comparisons with other boundary conditions (BCs), show the efficiency of the BC. Direct numerical simulations of more complex flows over a dihedral plate are simulated, without creation of acoustic waves going back in the flow. Copyright © 2001 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.     

Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method

We prove convergence of the finite element method for the Navier–Stokes equations in which the no‐slip condition and no‐penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier–Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks

A numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a ‘universal approximator’ based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of ‘finite element‐type’ discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of , Re = 400 for the driven cavity flow with a density of and Rayleigh number Ra = 1 000 000 for the natural convection flow with a density of . Copyright © 2001 John Wiley & Sons, Ltd.     

Stretched Cartesian grids for solution of the incompressible Navier–Stokes equations

Two Cartesian grid stretching functions are investigated for solving the unsteady incompressible Navier–Stokes equations using the pressure–velocity formulation. The first function is developed for the Fourier method and is a generalization of earlier work. This function concentrates more points at the centre of the computational box while allowing the box to remain finite. The second stretching function is for the second‐order central finite difference scheme, which uses a staggered grid in the computational domain. This function is derived to allow a direct discretization of the Laplacian operator in the pressure equation while preserving the consistent behaviour exhibited by the uniform grid scheme. Both functions are analysed for their effects on the matrix of the discretized pressure equation. It is shown that while the second function does not spoil the matrix diagonal dominance, the first one can. Limits to stretching of the first method are derived for the cases of mappings in one and two directions. A limit is also derived for the second function in order to prevent a strong distortion of a sine wave. The performances of the two types of stretching are examined in simulations of periodic co‐flowing jets and a time developing boundary layer. Copyright © 2000 John Wiley & Sons, Ltd.     

A high‐order scheme for the incompressible Navier–Stokes equations with open boundary condition

We present a numerical scheme to solve the incompressible Navier–Stokes equations with open boundary condition. After replacing the incompressibility constraint by the pressure Poisson equation, the key is how to give an appropriate boundary condition for the pressure Poisson equation. We propose a new boundary condition for the pressure on the open boundary. Some numerical experiments are presented to verify the accuracy and stability of scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Fourth‐order finite difference scheme for the incompressible Navier–Stokes equations in a disk

We develop an efficient fourth‐order finite difference method for solving the incompressible Navier–Stokes equations in the vorticity‐stream function formulation on a disk. We use the fourth‐order Runge–Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth‐order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

A multi‐energy‐level lattice Boltzmann model for the compressible Navier–Stokes equations

In this paper, we propose a new lattice Boltzmann model for the compressible Navier–Stokes equations. The new model is based on a three‐energy‐level and three‐speed lattice Boltzmann equation by using a method of higher moments of the equilibrium distribution functions. As the 25‐bit model, we obtained the equilibrium distribution functions and the compressible Navier–Stokes equations with the second accuracy of the truncation errors. The numerical examples show that the model can be used to simulate the shock waves, contact discontinuities and supersonic flows around circular cylinder. The numerical results are compared with those obtained by traditional method. Copyright © 2007 John Wiley & Sons, Ltd.     

Upwind discretization of the steady Navier–Stokes equations

A discretization method is presented for the full, steady, compressible Navier–Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non-linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme), results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of the convection terms is derived. On the basis of this higher-order scheme, to preserve monotonicity, a new limiter is constructed. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave–boundary layer interaction. The results obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.