Two‐level finite element method with a stabilizing subgrid for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method with a stabilizing subgrid of a single node is described, and its application to the Navier–Stokes equation is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems. The results show that the proper choice of the subgrid node is crucial in obtaining stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Copyright © 2008 John Wiley & Sons, Ltd.     

A multigrid method for steady incompressible Navier–Stokes equations based on partial flux splitting

Flux splitting is applied to the convective part of the steady Navier–Stokes equations for incompressible flow. Partial upwind differences are introduced in the split first-order part, while central differences are used in the second-order part. The discrete set of equations obtained is positive, so that it can be solved by collective variants of relaxation methods. The partial upwinding is optimized in the same way as for a scalar convection–diffusion equation, but involving several Peclet numbers. It is shown that with the optimum partial upwinding accurate results can be obtained. A full multigrid method in W-cycle form, using red–black successive under-relaxation, injection and bilinear interpolation, is described. The efficiency of this method is demonstrated.     

An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

A fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher‐order ultra‐parabolic equation. The convergence with respect to the fictitious time (we call the latter ‘internal iterations’) allows one to obtain fully implicit nonlinear scheme in full time steps for the physical‐time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second‐order of approximation. For the solution of the internal iteration, a fractional‐step scheme is proposed based on the splitting of the combination of the Laplace, bi‐harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second‐order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time‐dependent Navier–Stokes equations

This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two‐dimensional non‐stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.     

An implicit edge‐based ALE method for the incompressible Navier–Stokes equations

A new finite volume method for the incompressible Navier–Stokes equations, expressed in arbitrary Lagrangian–Eulerian (ALE) form, is presented. The method uses a staggered storage arrangement for the pressure and velocity variables and adopts an edge‐based data structure and assembly procedure which is valid for arbitrary n‐sided polygonal meshes. Edge formulas are presented for assembling the ALE form of the momentum and pressure equations. An implicit multi‐stage time integrator is constructed that is geometrically conservative to the precision of the arithmetic used in the computation. The method is shown to be second‐order‐accurate in time and space for general time‐dependent polygonal meshes. The method is first evaluated using several well‐known unsteady incompressible Navier–Stokes problems before being applied to a periodically forced aeroelastic problem and a transient free surface problem. Published in 2003 by John Wiley & Sons, Ltd.     

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal shape design for the time‐dependent Navier–Stokes flow

This paper is concerned with the problem of shape optimization of two‐dimensional flows governed by the time‐dependent Navier–Stokes equations. We derive the structures of shape gradients for time‐dependent cost functionals by using the state derivative and its associated adjoint state. Finally, we apply a gradient‐type algorithm to our problem, and numerical examples show that our theory is useful for practical purposes and the proposed algorithm is feasible in low Reynolds number flows. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved projection scheme applied to pseudospectral methods for the incompressible Navier–Stokes equations

An improved projection scheme is proposed and applied to pseudospectral collocation-Chebyshev approximation for the incompressible Navier–Stokes equations. It consists of introducing a correct predictor for the pressure, one which is consistent with a divergence-free velocity field at each time step. The main objective is to allow a time variation of the pressure gradient at boundaries. From different test problems, it is shown that this method, associated with a multistep second-order time scheme, provides a time accuracy of the same order as the temporal scheme used for the pressure, and also improves the prediction of the velocity slip. Moreover, it does not exhibit any numerical boundary layer mentioned as a drawback of fractional steps algorithm, and does not require the use of staggered grids for the velocity and the pressure. Its effectiveness is validated by comparison with a previous time-splitting algorithm proposed by Goda (K. Goda, J. Comput. Phys., 30 , 76–95 (1979)) and implemented by Gresho (P. Gresho, Int. j. numer. methods fluids, 11 , 587–620 (1990)) to finite element approximations. Steady and unsteady solutions for the regularized driven cavity and the rotating cavity submitted to throughflow are also used to assess the efficiency of this algorithm. © 1998 John Wiley & Sons, Ltd.     

Continuation of travelling‐wave solutions of the Navier–Stokes equations

An efficient way of obtaining travelling waves in a periodic fluid system is described and tested. We search for steady states in a reference frame travelling at the wave phase velocity using a first‐order pseudospectral semi‐implicit time scheme adapted to carry out the Newton's iterations. The method is compared to a standard Newton–Raphson solver and is shown to be highly efficient in performing this task, even when high‐resolution grids are used. This method is well suited to three‐dimensional calculations in cylindrical or spherical geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

Algebraic multigrid methods for the solution of the Navier–Stokes equations in complicated geometries

The application of standard multigrid methods for the solution of the Navier–Stokes equations in complicated domains causes problems in two ways. First, coarsening is not possible to full extent since the geometry must be resolved by the coarsest grid used. Second, for semi-implicit time-stepping schemes, robustness of the convergence rates is usually not obtained for convection–diffusion problems, especially for higher Reynolds numbers. We show that both problems can be overcome by the use of algebraic multigrid (AMG), which we apply for the solution of the pressure and momentum equations in explicit and semi-implicit time-stepping schemes. We consider the convergence rates of AMG for several model problems and demonstrate the robustiness of the proposed scheme. © 1998 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.     

SIMPLE‐type preconditioners for cell‐centered,colocated finite volume discretization of incompressible Reynolds‐averaged Navier–Stokes equations

This paper contains a comparison of four SIMPLE‐type methods used as solver and as preconditioner for the iterative solution of the (Reynolds‐averaged) Navier–Stokes equations, discretized with a finite volume method for cell‐centered, colocated variables on unstructured grids. A matrix‐free implementation is presented, and special attention is given to the treatment of the stabilization matrix to maintain a compact stencil suitable for unstructured grids. We find SIMPLER preconditioning to be robust and efficient for academic test cases and industrial test cases. Compared with the classical SIMPLE solver, SIMPLER preconditioning reduces the number of nonlinear iterations by a factor 5–20 and the CPU time by a factor 2–5 depending on the case. The flow around a ship hull at Reynolds number 2E9, for example, on a grid with cell aspect ratio up to 1:1E6, can be computed in 3 instead of 15 h.Copyright © 2012 John Wiley & Sons, Ltd.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust semidirect finite difference methods for solving the Navier–Stokes and energy equations

Semidirect solution techniques can be an effective alternative to the more conventional iterative approaches used in many finite difference methods. This paper summarizes several semidirect techniques which generally have not been applied to the Navier–Stokes and energy equations in finite difference form. The methods presented use both successive substitution and Jacobian-based updates as well as two variations of Broyden's full matrix update. A hybrid method is also presented, as is a norm-reducing search technique that can be used to enhance the convergence characteristics of any semidirect approach. These methods have been compared with the well known iterative methods SIMPLE and SIMPLER. The comparison was performed on the natural convection and driven cavity problems. The semidirect methods proved to be reliably convergent without the need for a priori specification of variable under-relaxation factors, which was necessary with the iterative methods. Natural convection and driven cavity solutions have been readily obtained with the proposed methods for Rayleigh and Reynolds numbers up to 10⁹ and 10⁶ respectively. Of the semidirect techniques, the hybrid approach was the most robust. From an arbitrary zero initial guess this method was able to obtain a solution to the natural convection problem for Rayleigh numbers three orders of magnitude larger than was possible with the Newton-Raphson update. The computational effort required by the semidirect methods is comparable to that required by the iterative methods; however, the memory requirements can be significantly greater.     

Dealing with pressure: FEM solution strategies for the pressure in the time‐dependent Navier–Stokes equations

This paper presents a survey of solution methods for calculating the pressure in the time‐dependent Navier–Stokes equations. The primary focus is on the treatment of the pressure‐Poisson equation deriving from index‐1 DAE formulations of the Navier–Stokes equations. Based on extensive operational experience with a variety of solution strategies, the combination of a stabilized pressure‐Poisson operator with an A‐conjugate projection and SSOR preconditioned conjugate gradient method has been found to yield the overall best performance relative to the resolve cost of a high performance direct solver. Copyright © 2002 John Wiley & Sons, Ltd.     

An overlapping control volume method for the Navier–Stokes equations on non‐staggered grids

An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd.     

A Taylor series‐based finite volume method for the Navier–Stokes equations

A Taylor series‐based finite volume formulation has been developed to solve the Navier–Stokes equations. Within each cell, velocity and pressure are obtained from the Taylor expansion at its centre. The derivatives in the expansion are found by applying the Gauss theorem over the cell. The resultant integration over the faces of the cell is calculated from the value at the middle point of the face and its derivatives, which are further obtained from a higher order interpolation based on the values at the centres of two cells sharing this face. The terms up to second order in the velocity and the terms up to first order in pressure in the Taylor expansion are retained throughout the derivation. The test cases for channel flow, flow past a circular cylinder and flow in a collapsible channel have shown that the method is quite accurate and flexible. Copyright © 2008 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.