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.     

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.     

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.     

The (9,5) HOC formulation for the transient Navier–Stokes equations in primitive variable

We recently proposed an improved (9,5) higher order compact (HOC) scheme for the unsteady two‐dimensional (2‐D) convection–diffusion equations. Because of using only five points at the current time level in the discretization procedure, the scheme was seen to be computationally more efficient than its predecessors. It was also seen to capture very accurately the solution of the unsteady 2‐D Navier–Stokes (N–S) equations for incompressible viscous flows in the stream function–vorticity (ψ – ω) formulation. In this paper, we extend the scope of the scheme for solving the unsteady incompressible N–S equations based on primitive variable formulation on a collocated grid. The parabolic momentum equations are solved for the velocity field by a time‐marching strategy and the pressure is obtained by discretizing the elliptic pressure Poisson equation by the steady‐state form of the (9,5) scheme with the Neumann boundary conditions. In particular, for pressure, we adopt a strategy on the collocated grid in conjunction with ideas borrowed from the staggered grid approach in finite volume. We first apply this extension to a problem having analytical solution and then to the famous lid‐driven square cavity problem. We also apply our formulation to the backward‐facing step problem to see how the method performs for external flow problems. The results are presented and are compared with established numerical results. This new approach is seen to produce excellent comparison in all the cases. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

A fully discrete ALE method over untwisted time–space control volumes

In this paper, a fully discrete high‐resolution arbitrary Lagrangian–Eulerian (ALE) method is developed over untwisted time–space control volumes. In the framework of the finite volume method, 2D Euler equations are discretized over untwisted moving control volumes, and the resulting numerical flux is computed using the generalized Riemann problem solver. Then, the fluid flows between meshes at two successive time steps can be updated without a remapping process in the classic ALE method. This remapping‐free ALE method directly couples the mesh motion into a physical variable update to reflect the temporal evolution in the whole process. An untwisted moving mesh is generated in terms of the vorticity‐free part of the fluid velocity according to the Helmholtz theorem. Some typical numerical tests show the competitive performance of the current method. Copyright © 2016 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.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid vortex method for the simulation of three‐dimensional flows

This paper presents an integral vorticity method for solving three‐dimensional Navier–Stokes equations. A finite volume scheme is implemented to solve the vorticity transport equation, which is discretized on a structured hexahedral mesh. A vortex sheet algorithm is used to enforce the no‐slip boundary condition through a vorticity flux at the boundary. The Biot–Savart integral is evaluated to compute the velocity field, in conjunction with a fast algorithm based on multipole expansion. This method is applied to the simulation of uniform flow past a sphere. Copyright © 2007 John Wiley & Sons, Ltd.     

Multigrid third‐order least‐squares solution of Cauchy–Riemann equations on unstructured triangular grids

In this paper, a multigrid algorithm is developed for the third‐order accurate solution of Cauchy–Riemann equations discretized in the cell‐vertex finite‐volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least‐squares norm. The standard second‐order least‐squares scheme is extended to third‐order by adding a high‐order correction term in the residual. The resulting high‐order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.     

Vorticity—velocity formulation in the computation of flows in multiconnected domains

The thermofluid dynamic fields in two-dimensional multiconnected domains are analysed by solving the Navier–Stokes equations with the Boussinesq approximation in the vorticity–velocity formulation. The need of an integral condition for the pressure to be single-valued on each independent irreducible loop, in analogy with the ω–Ψ formulation, is demonstrated. The field equations are discretized by a finite difference technique and solved at the steady state via an alternating direction implicit method of a scalar type. Two test cases at low Reynolds and Rayleigh numbers are considered: the multiconnected driven cavity and an annulus with isothermal walls and stationary or rotating inner cylinder.     

A VORTICITY–STREAMFUNCTION FORMULATION FOR STEADY INCOMPRESSIBLE TWO-DIMENSIONAL FLOWS

A vorticity–streamfunction formulation for incompressible planar viscous flows is presented. The standard kinematic field equations are discretized using centred finite difference schemes and solved in a coupled way via a Newton-like linearization scheme. The linearized system of partial differential equations is handled through the restarting linear GMRES algorithm, preconditioned by means of an incomplete LU approximate factorization. The proposed solution technique constitutes a fast and robust algorithm for treating laminar flows at high Reynolds numbers. The pressure field is obtained at a subsequent step by solving a convection– diffusion equation in terms of the stagnation pressure, which presents certain advantages compared with the widely used static pressure Poisson equation. Results are shown for a wide variety of applications including internal and external flows.     

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.     

The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. Copyright © 2000 John Wiley & Sons, Ltd.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

A stream function–vorticity formulation‐based immersed boundary method and its applications

A new stream function–vorticity formulation‐based immersed boundary method is presented in this paper. Different from the conventional immersed boundary method, the main feature of the present model is to accurately satisfy both governing equations and boundary conditions through velocity correction and vorticity correction procedures. The velocity correction process is performed implicitly based on the requirement that velocity at the immersed boundary interpolated from the corrected velocity field accurately satisfies the nonslip boundary condition. The vorticity correction is made through the stream function formulation rather than the vorticity transport equation. It is evaluated from the firstorder derivatives of velocity correction. Two simple and efficient ways are presented for approximation of velocity‐correction derivatives. One is based on finite difference approximation, while the other is based on derivative expressions of Dirac delta function and velocity correction. It was found that both ways can work very well. The main advantage of the proposed method lies in its simple concept, easy implementation, and robustness in stability. Numerical experiments for both stationary and moving boundary problems were conducted to validate the capability and efficiency of the present method. Good agreements with available data in the literature were achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Schwarz domain decomposition for the incompressible Navier–Stokes equations in general co‐ordinates

This paper describes a domain decomposition method for the incompressible Navier–Stokes equations in general co‐ordinates. Domain decomposition techniques are needed for solving flow problems in complicated geometries while retaining structured grids on each of the subdomains. This is the so‐called block‐structured approach. It enables the use of fast vectorized iterative methods on the subdomains. The Navier–Stokes equations are discretized on a staggered grid using finite volumes. The pressure‐correction technique is used to solve the momentum equations together with incompressibility conditions. Schwarz domain decomposition is used to solve the momentum and pressure equations on the composite domain. Convergence of domain decomposition is accelerated by a GMRES Krylov subspace method. Computations are presented for a variety of flows. Copyright © 2000 John Wiley & Sons, Ltd.     

A Godunov‐type fractional semi‐implicit method based on staggered grid for dam‐break modeling

A semi‐implicit finite volume model based upon staggered grid is presented for solving shallow water equation. The model employs a time‐splitting scheme that uses a predictor–corrector method for the advection term. The fluxes are calculated based on a Riemann solver in the prediction step and a downwind scheme in the correction step. A simple TVD scheme is employed for shock capturing purposes in which the Minmond limiter is used for flux functions. As a consequence of using staggered grid, an ADI method is adopted for solving the discretized equations for 2‐D problems. Several 1‐D and 2‐D flows have been modeled with satisfactory results when compared with analytical and experimental test cases. The model is also capable of simulating supercritical as well as subcritical flow. Copyright © 2010 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.