Simulating hydrostatic and non‐hydrostatic oceanic flows

The thin aspect ratio of oceanic basins is simultaneously a complication to contend with when developing ocean models and an opportunity to simplify the equations of motion. Here we discuss these two aspects of this geometric feature in the context of hydrostatic and non‐hydrostatic ocean models. A simple analysis shows that the horizontal viscous operator in the hydrostatic primitive equations plays a central role in the specification of boundary conditions on the lateral vertical surfaces bounding the domain. The asymptotic analysis shows that for very thin aspect ratios the leading‐order flow cannot be closed unless additional terms in the equations are considered, namely either the horizontal viscous forces or the non‐hydrostatic pressure forces. In either case, narrow boundary layers must be resolved in order to close the circulation properly. The computational cost increases substantially when non‐hydrostatic effects are taken into account. Copyright © 2008 John Wiley & Sons, Ltd.     

Comment on ‘generalized potential flow theory and direct calculation of velocities applied to the numerical solution of the Navier–Stokes and the Boussinesq equations’

In a recent paper a generalized potential flow theory and its application to the solution of the Navier–Stokes equation are developed.¹ The purpose of this comment is to show that the analysis presented in that paper is in general not correct. We note that the theoretical development of Reference 1 is in fact an extension—although not cited—of some work first done by Hawthorne for steady inviscid flow.² Hawthorne's solution is correct, and his analysis, which we briefly describe, provides a useful introduction to this note.     

Modelling and simulation of fires in vehicle tunnels

Applying a low‐Mach asymptotic for the compressible Navier–Stokes equations, we derive a new fluid dynamics model,which should be capable to model large temperature differences in combination with the low‐Mach number limit. The model is used to simulate fires in vehicle tunnels, where the standard Boussinesq‐approximation for the incompressible Navier–Stokes seems to be inappropriate due to the high temperatures developing in the tunnel. The model is implemented using a modified finite‐difference approach for the incompressible Navier–Stokes equations and tested in some realistic fire events. Copyright © 2004 John Wiley & Sons, Ltd.     

Compressible viscous flow calculations using compatible finite element approximations

We discuss in this paper the numerical simulation of compressible viscous flows by a combination of finite element methods for the space approximation, an implicit second-order multistep scheme for the time discretization and GMRES iterative methods for solving the non-linear problems encountered at each time step. Numerical results corresponding to flows around aerofoils and aerospace vehicles illustrate the possibilities of these methods.     

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.     

A semi‐implicit finite difference method for non‐hydrostatic,free‐surface flows

In this paper a semi‐implicit finite difference model for non‐hydrostatic, free‐surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi‐hydrostatic flows. The governing equations are the free‐surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free‐surface are integrated by a semi‐implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright © 1999 John Wiley & Sons, Ltd.     

Mixed‐order interpolation for the Galerkin coarse‐grid approximations in algebraic multigrid solvers

An empirical investigation is made of AMG solver performance for the fully coupled set of Navier–Stokes equations. The investigation focuses on two different FV discretizations for the standard driven cavity test problem. One is a collocated vertex‐based discretization; the other is a cell‐centred staggered‐grid discretization. Both employ otherwise identical orthogonal Cartesian meshes. It is found that if mixed‐order interpolation is used in the construction of the Galerkin coarse‐grid approximation (CGA), a close‐to‐optimum mesh‐independent scaling of the AMG convergence is observed with similar convergence rates for both discretizations. If, on the other hand, an equal‐order interpolation is used, convergence rates are mesh‐dependent but the scaling differs in each case. For the collocated‐grid case, it depends both on the mesh size, h (or bandwidth Q～h^?1) and on the total number of grids, G, whereas for the staggered‐grid case it depends only on Q. Comparing the two characteristics reveals that the Q‐dependent parts are very similar; it is only in the G‐dependent convergence for the collocated‐grid case that they differ. This takes the form of stepped reductions in the AMG convergence rate (implying step reductions in the quality of the Galerkin CGA that correlate exactly with step increases in G). These findings reinforce previous evidence that, for optimum mesh‐independent performance, mixed‐order interpolations should be used in forming Galerkin CGAs for coupled Navier–Stokes problems. Copyright © 2010 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.     

Multi-symplectic method for generalized Boussinesq equation

The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.     

A 3D AGGLOMERATION MULTIGRID SOLVER FOR THE REYNOLDS–AVERAGED NAVIER–STOKES EQUATIONS ON UNSTRUCTURED MESHES

An agglomeration multigrid strategy is developed and implemented for the solution of three-dimensional steady viscous flows. The method enables convergence acceleration with minimal additional memory overhead and is completely automated in that it can deal with grids of arbitrary construction. The multigrid technique is validated by comparing the delivered convergence rates with those obtained by a previously developed overset-mesh multigrid approach and by demonstrating grid-independent convergence rates for aerodynamic problems on very large grids. Prospects for further increases in multigrid efficiency for high-Reynolds-number viscous flows on highly stretched meshes are discussed.     

Discontinuous Galerkin methods for the Navier–Stokes equations using solenoidal approximations

An interior penalty method and a compact discontinuous Galerkin method are proposed and compared for the solution of the steady incompressible Navier–Stokes equations. Both compact formulations can be easily applied using high‐order piecewise divergence‐free approximations, leading to two uncoupled problems: one associated with velocity and hybrid pressure, and the other one only concerned with the computation of pressures in the elements interior. Numerical examples compare the efficiency and the accuracy of both proposed methods. Copyright © 2009 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.     

Iterative solution techniques for the stokes and Navier-Stokes equations

The linear system arising from a Lagrange-Galerkin mixed finite element approximation of the Navier–Stokes and continuity equations is symmetric indefinite and has the same block structure as a system arising from a mixed finite element discretization of a Stokes problem. This paper considers the iterative solution of such a system, comparing the performance of the one-level preconditioned conjugate residual method for indefinite matrices with that of a more traditional two-level pressure correction approach. Asymptotic estimates for the amount of work involved in each method are given together with the results of related numerical experiments.     

An analysis and comparison of the time accuracy of fractional‐step methods for the Navier–Stokes equations on staggered grids

Fractional‐step methods solve the unsteady Navier–Stokes equations in a segregated manner, and can be implemented with only a single solution of the momentum/pressure equations being obtained at each time step, or with the momentum/pressure system being iterated until a convergence criterion is attained.The time accuracy of such methods can be determined by the accuracy of the momentum/pressure coupling, irrespective of the accuracy to which the momentum equations are solved. It is shown that the time accuracy of the basic projection method is first‐order as a result of the momentum/pressure coupling, but that by modifying the coupling directly, or by modifying the intermediate velocity boundary conditions, it is possible to recover second‐order behaviour. It is also shown that pressure correction methods, implemented in non‐iterative or iterative form and without special boundary conditions, are second‐order in time, and that a form of the non‐iterative pressure correction method is the most efficient for the problems considered. Copyright © 2002 John Wiley & Sons, Ltd.     

A SEGREGATED SOLUTION ALGORITHM FOR INCOMPRESSIBLE FLOWS IN GENERAL CO-ORDINATES

To analyse an incompressible Navier–Stokes flow problem in a boundary- fitted curvilinear co-ordinate system is definitely not a trivial task. In the primitive variable formulation, choices between working variables and their storage points have to be made judiciously. The present work engages contravariant velocity components and scalar pressure which stagger each other in the mesh to prevent even–odd pressure oscillations from emerging. Now that smoothness of the pressure field is attainable, the remaining task is to ensure a discrete divergence-free velocity field for an incompressible flow simulation. Aside from the flux discretizations, the indispensable metric tensors, Jacobian and Christoffel symbols in the transformed equations should be approximated with care. The guiding idea is to get the property of geometric identity pertaining to these grid-sensitive discretizations. In addition, how to maintain the revertible one-to-one equivalence at the discrete level between primitive and contravariant velocities is another theme in the present staggered formulation. A semi-implicit segregated solution algorithm felicitous for a large-scale flow simulation was utilized to solve the entire set of basic equations iteratively. Also of note is that the present segregated solution algorithm has the virtue of requiring no user-specified relaxation parameters for speeding up the satisfaction of incompressibility in an optimal sense. Three benchmark problems, including an analytic problem, were investigated to justify the capability of the present formulation in handling problems with complex geometry. The test cases considered and the results obtained herein make a useful contribution in solving problems subsuming cells with arbitrary shapes in a boundary-fitted grid system.     

Weak Solutions and Attractors for Three-Dimensional Navier–Stokes Equations with Nonregular Force

A three-dimensional Navier–Stokes equation is considered. The forcing term is the derivative of a continuous function; the case of white noise is also considered. The aim is to prove the existence of weak solutions and to construct an attractor for the corresponding shift dynamical system in path space, following an idea of Sell.     

A simple and efficient error analysis for multi‐step solution of the Navier–Stokes equations

A simple error analysis is used within the context of segregated finite element solution scheme to solve incompressible fluid flow. An error indicator is defined based on the difference between a numerical solution on an original mesh and an approximated solution on a related mesh. This error indicator is based on satisfying the steady‐state momentum equations. The advantages of this error indicator are, simplicity of implementation (post‐processing step), ability to show regions of high and/or low error, and as the indicator approaches zero the solution approaches convergence. Two examples are chosen for solution; first, the lid‐driven cavity problem, followed by the solution of flow over a backward facing step. The solutions are compared to previously published data for validation purposes. It is shown that this rather simple error estimate, when used as a re‐meshing guide, can be very effective in obtaining accurate numerical solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

The Lagrangian approach of advective term treatment and its application to the solution of Navier—Stokes equations

A one-dimensional transport test applied to some conventional advective Eulerian schemes shows that linear stability analyses do not guarantee the actual performances of these schemes. When adopting the Lagrangian approach, the main problem raised in the numerical treatment of advective terms is a problem of interpolation or restitution of the transported function shape from discrete data. Several interpolation methods are tested. Some of them give excellent results and these methods are then extended to multi-dimensional cases. The Lagrangian formulation of the advection term permits an easy solution to the Navier-Stokes equations in primitive variables V, p, by a finite difference scheme, explicit in advection and implicit in diffusion. As an illustration steady state laminar flow behind a sudden enlargement is analysed using an upwind differencing scheme and a Lagrangian scheme. The importance of the choice of the advective scheme in computer programs for industrial application is clearly apparent in this example.     

The steady Navier–Stokes/energy system with temperature‐dependent viscosity—Part 1: Analysis of the continuous problem

In this first part we propose and analyse a model for the study of two‐dimensional incompressible Navier–Stokes equations with a temperature‐dependent viscosity. The flow is supposed in a mixed convection regime and considers an outflow region, leading to a strongly coupled problem between the Navier–Stokes and energy equations, which will be justified theoretically. The coupling in the continuous problem is treated by an outer temperature fixed point strategy. Existence results for a particular variational formulation follows from this study. Further, a particular uniqueness result for small data is also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Integrals of Motion of an Incompressible Fluid Occupying the Entire Space

This paper studies integral relations to which the solutions of the Navier–Stokes equations or Euler equations satisfy in the case of fluids filling the entire threedimensional space. The existence of these relations is due to a rapid decrease of the velocity field at infinity (but not too rapid in order that the required asymptotic forms are reproduced with time). Of special interest are the integrals of motion whose density depends quadratically on the velocities or their derivative with respect to the coordinates. Such integrals (conservation laws) for the Navier–Stokes equations were recently found by Dobrokhotov and Shafarevich. In the present paper, new conservation laws are obtained, which are quadratic in the derivatives of the velocity and lead to identities that link the averaged and pulsation characteristics of ree turbulent flows.