A comparative study of central and upwind difference schemes using the primitive variables

The use of the velocity-pressure formulation of the Navier-Stokes equations for the numerical solution of fluid flow problems is favoured for free-surface problems, more involved flow configurations, and three-dimensional flows. Many engineering problems involve such features in addition to strong inertial effects. The computational instabilities arising from central-difference schemes for the convective terms of the governing equations impose serious limitations on the range of Reynolds numbers that can be investigated by the numerical method. Solutions for higher Reynolds numbers Re > 1000 could be reached using upwind-difference schemes. A comparative study of both schemes using a method based on the primitive variables is presented. The comparison is made for the model problem of the driven flow in a square cavity. Using a central scheme stable solutions of the pressure and velocity fields were obtained for Reynolds numbers up to 5000. The streamfunction and vorticity fields were calculated from the resulting velocity field and compared with previous solutions. It is concluded that total upwind differencing results in a considerable change in the flow pattern due to the false diffusion. For practical calculations, by a proper choice of a small amount of partial upwind differencing the vorticity diffusion near the walls and the global features of the solutions are not sigificantly altered.     

A compact finite difference method on staggered grid for Navier–Stokes flows

Compact finite difference methods feature high‐order accuracy with smaller stencils and easier application of boundary conditions, and have been employed as an alternative to spectral methods in direct numerical simulation and large eddy simulation of turbulence. The underpinning idea of the method is to cancel lower‐order errors by treating spatial Taylor expansions implicitly. Recently, some attention has been paid to conservative compact finite volume methods on staggered grid, but there is a concern about the order of accuracy after replacing cell surface integrals by average values calculated at centres of cell surfaces. Here we introduce a high‐order compact finite difference method on staggered grid, without taking integration by parts. The method is implemented and assessed for an incompressible shear‐driven cavity flow at Re = 10³, a temporally periodic flow at Re = 10⁴, and a spatially periodic flow at Re = 10⁴. The results demonstrate the success of the method. Copyright © 2006 John Wiley & Sons, Ltd.     

Benchmark solutions for the incompressible Navier–Stokes equations in general co-ordinates on staggered grids

Benchmark problems are solved with the steady incompressible Navier–Stokes equations discretized with a finite volume method in general curvilinear co-ordinates on a staggered grid. The problems solved are skewed driven cavity problems, recently proposed as non-orthogonal grid benchmark problems. The system of discretized equations is solved efficiently with a non-linear multigrid algorithm, in which a robust line smoother is implemented. Furthermore, another benchmark problem is introduced and solved in which a 90° change in grid line direction occurs.     

Simulation of flow around a thin,flexible obstruction by means of a deforming grid overlapping a fixed grid

This paper presents a numerical method for simulation of coupled flows, in which the fluid interacts with a thin deformable solid, such as flows in cardiovascular valves. The proposed method employs an arbitrary Lagrangian–Eulerian (ALE) method for flow near the solid, embodied in the outflow domain in which the mesh is fixed. The method was tested by modelling a two‐dimensional channel flow with a neo‐Hookean obstacle, an idealization of the coupled flow near a cardiovascular valve. The effects of the Reynolds number and the dimensionless elastic modulus of the material on the wall shear stress, the size of the downstream reverse flows, and the velocity and pressure profiles were investigated. The deformation of the obstacle, the pressure drop across the obstacle, and the size of the top reverse flow increased as the Reynolds number increased. Conversely, increasing the elastic modulus of the obstacle decreased the deformation of the obstacle and the size of the top reverse flows, but did not affect the pressure drop across the obstacle over the range studied. Copyright © 2007 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.     

Orthogonal grid generation for Navier–Stokes computations

Body conforming orthogonal grids were generated using a fast hyperbolic method for aerofoils, and were used to solve the Navier–Stokes equation in the generalized orthogonal system for the first time for time accurate simulation of incompressible flow. For grid generation, the Beltrami equation and the definition equation for the orthogonality are solved using a finite difference method. The grids generated around aerofoils by this method have better orthogonality than the results published by earlier investigators. The Navier–Stokes equation at Reynolds numbers of 3000 and 35 000 for NACA 0012 and NACA 0015 respectively, have been solved as an application. The obtained results match quite well with the corresponding experimental results. © 1998 John Wiley & Sons, Ltd.     

A momentum coupling method for the unsteady incompressible Navier–Stokes equations on the staggered grid

A new numerical method is developed to efficiently solve the unsteady incompressible Navier–Stokes equations with second-order accuracy in time and space. In contrast to the SIMPLE algorithms, the present formulation directly solves the discrete x- and y-momentum equations in a coupled form. It is found that the present implicit formulation retrieves some cross convection terms overlooked by the conventional iterative methods, which contribute to accuracy and fast convergence. The finite volume method is applied on the fully staggered grid to solve the vector-form momentum equations. The preconditioned conjugate gradient squared method (PCGS) has proved very efficient in solving the associate linearized large, sparse block-matrix system. Comparison with the SIMPLE algorithm has indicated that the present momentum coupling method is fast and robust in solving unsteady as well as steady viscous flow problems. © 1998 John Wiley & Sons, Ltd.     

Numerical simulation on an interaction of a vortex street with an elliptical leading edge using an unstructured grid

An algorithm for a time accurate incompressible Navier–Stokes solver on an unstructured grid is presented. The algorithm uses a second order, three‐point, backward difference formula for the physical time marching. For each time step, a divergence free flow field is obtained based on an artificial compressibility method. An implicit method with a local time step is used to accelerate the convergence for the pseudotime iteration. To validate the code, an unsteady laminar flow over a circular cylinder at a Reynolds number of 200 is calculated. The results are compared with available experimental and numerical data and good agreements are achieved. Using the developed unsteady code, an interaction of a Karman vortex street with an elliptical leading edge is simulated. The incident Karman vortex street is generated by a circular cylinder located upstream. A clustering to the path of the vortices is achieved easily due to flexibility of an unstructured grid. Details of the interaction mechanism are analysed by investigating evolutions of vortices. Characteristics of the interactions are compared for large‐ and small‐scale vortex streets. Different patterns of the interaction are observed for those two vortex streets and the observation is in agreement with experiment. Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of vertical grid variability on a free surface flow model

A previously developed numerical model that solves the incompressible, non‐hydrostatic, Navier–Stokes equations for free surface flow is analysed on a non‐uniform vertical grid. The equations are vertically transformed to the σ‐coordinate system and solved in a fractional step manner in which the pressure is computed implicitly by correcting the hydrostatic flow field to be divergence free. Numerical consistency, accuracy and efficiency are assessed with analytical methods and numerical experiments for a varying vertical grid discretization. Specific discretizations are proposed that attain greater accuracy and minimize computational effort when compared to a uniform vertical discretization. Published in 2007 by John Wiley & Sons, Ltd.     

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.     

A numerical model for aerated‐water wave breaking

This work presents a numerical model designed for the simulation of water‐wave impacts on a structure when aeration of the liquid phase is considered. The model is based on a multifluid Navier–Stokes approach in which all fluids are assumed compressible. The numerical method is based on a finite volume algorithm in space and a second order Runge–Kutta method in time. A validation of this model is performed. It shows a good accuracy for acoustic and shock wave propagation in a bubbly liquid and for wave breaking. Copyright © 2011 John Wiley & Sons, Ltd.     

Discretization of incompressible vorticity–velocity equations on triangular meshes

This paper describes a new approach to discretizing first- and second-order partial differential equations. It combines the advantages of finite elements and finite differences in having both unstructured (triangular/tetrahedral) meshes and low-order physically intuitive schemes. In this ‘co-volume’ framework, the discretized gradient, divergence, curl, (scalar) Laplacian, and vector Laplacian operators satisfy relationships found in standard vector field theory, such as a Helmholtz decomposition. This article focuses on the vorticity–velocity formulation for planar incompressible flows. The algorithm is described and some supporting numerical evidence is provided.     

Three‐dimensional numerical simulation of Marangoni instabilities in liquid bridges: influence of geometrical aspect ratio

Oscillatory Marangoni convection in silicone oil–liquid bridges with different geometrical aspect ratios is investigated by three‐dimensional and time‐dependent numerical simulations, based on control volume methods in staggered cylindrical non‐uniform grids. The three‐dimensional oscillatory flow regimes are studied and compared with previous experimental and theoretical results. The results show that the critical wavenumber (m), related to the azimuthal spatio‐temporal flow structure, is a monotonically decreasing function of the geometrical aspect ratio of the liquid bridge (defined as the ratio of length to diameter). For this function, a general correlation formula is found, which is in agreement with the previous experimental findings. The critical Marangoni number and the oscillation frequency are decreasing functions of the aspect ratio; however, the critical Marangoni number, based on the axial length of the bridge, does not change much with the aspect ratio. For each aspect ratio investigated, the onset of the instability from the axisymmetric steady state to the three‐dimensional oscillatory one is characterized by the appearance of a standing wave regime that exhibits, after a certain time, a second transition to a travelling wave regime. The standing wave regime is more stable for lower aspect ratios since it lasts for a long time. This behaviour is explained on the basis of the propagation velocity of the disturbances in the liquid phase. For this velocity, a general correlation law is found as a function of the aspect ratio and of the Marangoni number. Copyright © 2001 John Wiley & Sons, Ltd.     

Colocated schemes for the incompressible Navier–Stokes equations on non‐smooth grids for two‐dimensional problems

The accuracy of colocated finite volume schemes for the incompressible Navier–Stokes equations on non‐smooth curvilinear grids is investigated. A frequently used scheme is found to be quite inaccurate on non‐smooth grids. In an attempt to improve the accuracy on such grids, three other schemes are described and tested. Two of these are found to give satisfactory results. Copyright © 2000 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.     

On numerical errors and turbulence modeling in tip vortex flow prediction

The accuracy of tip vortex flow prediction in the near‐field region is investigated numerically by attempting to quantify the shortcomings of the turbulence models and the flow solver. In particular, some turbulence models can produce a ‘numerical diffusion’ that artificially smears the vortex core. Low‐order finite differencing techniques of the convective and pressure terms of the Navier–Stokes equations and inadequate grid density and distribution can also produce the same adverse effect. The flow over a wing and the near‐wake with the wind tunnel walls included was simulated using 2.5 million grid points. Two subset problems, one using a steady, three‐dimensional analytical vortex, and the other, a vortex obtained from experiment and propagated downstream, were also devised in order to make the study of vortex preservation more tractable. The method of artificial compressibility is used to solve the steady, three‐dimensional, incompressible Navier–Stokes equations. Two one‐equation turbulence models (Baldwin–Barth and Spalart–Allmaras turbulence models), have been used with the production term modified to account for the stabilizing effect of the nearly solid body rotation in the vortex core. Finally, a comparison between the computed results and experiment is presented. Published in 1999 by John Wiley & Sons, Ltd.     

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.     

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.     

An efficient algorithm for solving the incompressible fluid flow equations

The present paper reports on a modified pressure implicit predictor corrector type scheme for solving the flow governing equations, in which a consistent formulation is combined with a multi-grid solver for the pressure correction. In addition a parabolic sublayer (PSL) approach for the treatment of the flow in the vicinity of solid walls is critically evaluated in terms of accuracy and computational efficiency. The lid-driven cavity flow is chosen as the test case and results are presented for Reynolds numbers ranging from 100 to 1000. Predictions with the proposed scheme indicate substantial computational savings and fairly good agreement when compared with previous work. The PSL approach reduces the computing time, but with increasing Reynolds numbers the accuracy of the solutions tends to deteriorate.     

Implicit weighted essentially non‐oscillatory schemes for the incompressible Navier–Stokes equations

A class of lower–upper/approximate factorization (LUAF) implicit weighted essentially non‐oscillatory (ENO; WENO) schemes for solving the two‐dimensional incompressible Navier–Stokes equations in a generalized co‐ordinate system is presented. The algorithm is based on the artificial compressibility formulation, and symmetric Gauss–Seidel relaxation is used for computing steady state solutions while symmetric successive overrelaxation is used for treating time‐dependent flows. WENO spatial operators are employed for inviscid fluxes and central differencing for viscous fluxes. Internal and external viscous flow test problems are presented to verify the numerical schemes. The use of a WENO spatial operator not only enhances the accuracy of solutions but also improves the convergence rate for the steady state computation as compared with using the ENO counterpart. It is found that the present solutions compare well with exact solutions, experimental data and other numerical results. Copyright © 1999 John Wiley & Sons, Ltd.