Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Some uses of Green's theorem in solving the Navier–Stokes equations

This paper gives a review of methods where Green's theorem may be employed in solving numerically the Navier–Stokes equations for incompressible fluid motion. They are based on the concept of using the theorem to transform local boundary conditions given on the boundary of a closed region in the solution domain into global, or integral, conditions taken over it. Two formulations of the Navier–Stokes equations are considered: that in terms of the streamfunction and vorticity for two-dimensional motion and that in terms of the primitive variables of the velocity components and the pressure. In the first formulation overspecification of conditions for the streamfunction is utilized to obtain conditions of integral type for the vorticity and in the second formulation integral conditions for the pressure are found. Some illustrations of the principle of the method are given in one space dimension, including some derived from two-dimensional flows using the series truncation method. In particular, an illustration is given of the calculation of surface vorticity for two-dimensional flow normal to a flat plate. An account is also given of the implementation of these methods for general two-dimensional flows in both of the mentioned formulations and a numerical illustration is given.     

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.     

Creeping flow analyses of free surface cavity flows

Two industrially important free surface flows arising in polymer processing and thin film coating applications are modelled as lid-driven cavity problems to which a creeping flow analysis is applied. Each is formulated as a biharmonic boundary-value problem and solved both analytically and numerically. The analytical solutions take the form of a truncated biharmonic series of eigenfunctions for the streamfunction, while numerical results are obtained using a linear, finite-element formulation of the governing equations written in terms of both the streamfunction and vorticity. A key feature of the latter is that problems associated with singularities are alleviated by expanding the solution there in a series of separated eigenfunctions. Both sets of results are found to be in extremely good agreement and reveal distinctive flow transformations that occur as the operating parameters are varied. They also compare well with other published work and experimental observation.     

THE NO-SLIP BOUNDARY CONDITION IN FINITE DIFFERENCE APPROXIMATIONS

A finite difference method for the Navier–Stokes equations in vorticity –streamfunction formulation is proposed to resolve the difficulty of the lack of a vorticity boundary condition at a no-slip boundary. It is particularly suitable for flows in regions with complicated geometries. Convergence with second-order accuracy in vorticity and velocity is established. In numerical experiments the convergence rates agree with theoretical predictions. Test results for the two-dimensional driven cavity problem and for the flow in expansion and contraction channels are given.     

Transonic viscous-inviscid interaction by a finite element method

A method is outlined for solving two-dimensional transonic viscous flow problems, in which the velocity vector is split into the gradient of a potential and a rotational component. The approach takes advantage of the fact that for high-Reynolds-number flows the viscous terms of the Navier-Stokes equations are important only in a thin shear layer and therefore solution of the full equations may not be needed everywhere. Most of the flow can be considered inviscid and, neglecting the entropy and vorticity effects, a potential model is a good approximation in the flow core. The rotational part of the flow can then be calculated by solution of the potential, streamfunction and vorticity transport equations. Implementation of the no-slip and no-penetration boundary conditions at the walls provides a simple mechanism for the interaction between the viscous and inviscid solutions and no extra coupling procedures are needed. Results are presented for turbulent transonic internal choked flows.     

Finite difference and finite element methods for mhd channel flows

A Galerkin finite element method and two finite difference techniques of the control volume variety have been used to study magnetohydrodynamic channel flows as a function of the Reynolds number, interaction parameter, electrode length and wall conductivity. The finite element and finite difference formulations use unequally spaced grids to accurately resolve the flow field near the channel wall and electrode edges where steep flow gradients are expected. It is shown that the axial velocity profiles are distorted into M-shapes by the applied electromagnetic field and that the distortion increases as the Reynolds number, interaction parameter and electrode length are increased. It is also shown that the finite element method predicts larger electromagnetic pinch effects at the electrode entrance and exit and larger pressure rises along the electrodes than the primitive-variable and streamfunction–vorticity finite difference formulations. However, the primitive-variable formulation predicts steeper axial velocity gradients at the channel walls and lower axial velocities at the channel centreline than the streamfunction–vorticity finite difference and the finite element methods. The differences between the results of the finite difference and finite element methods are attributed to the different grids used in the calculations and to the methods used to evaluate the pressure field. In particular, the computation of the velocity field from the streamfunction–vorticity formulation introduces computational noise, which is somewhat smoothed out when the pressure field is calculated by integrating the Navier–Stokes equations. It is also shown that the wall electric potential increases as the wall conductivity increases and that, at sufficiently high interaction parameters, recirculation zones may be created at the channel centreline, whereas the flow near the wall may show jet-like characteristics.     

NUMERICAL STUDY OF HIGH-REYNOLDS-NUMBER FLOW PAST A BLUFF OBJECT

A semi-explicit finite difference scheme is proposed to study unsteady two-dimensional, incompressible flow past a bluff object at high Reynolds number. The bluff object comes from a class of elliptical cylinders in which the aspect ratio and the angle of attack are two controlled parameters. Associated with the streamfunction–vorticity formulation, the interior vorticity, streamfunction and wall vorticity are updated in turn for each time step. The streamfunction and wall vorticity are solved by means of a multigrid method and a projection method respectively. In regard to the vorticity transport equation, implicitness is merely associated with the diffusion operator, which can be made semi-explicit via approximate factorization. Low-diffusive upwinding is devised to handle the convection part. Numerical results are reported for Reynolds numbers up to 40,000. Comparisons with other numerical or physical experiments are included.     

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.     

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.     

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.     

Application of fourth‒order difference method to prediction of turbulence control in two‒dimensional channel with surface roughness

The fourth‒order finite difference method is combined with the vorticity–streamfunction formulation in generalized co‒ordinates. Direct numerical simulations are performed for channel flows with and without surface roughness at a Reynolds number of 10⁴. The present results are in good agreement with those of the pseudospectral method with respect to the flow in a smooth channel. It is shown that the present method predicts well the precise change in the flow with the channel length and roughness height. The turbulence is generally weakened by the roughness. Laminarization is also accomplished under the appropriate condition. © 1997 John Wiley & Sons, Ltd.     

Incompressible flows with combustion simulated by a preconditioning method using multigrid acceleration and MUSCL reconstruction

A numerical algorithm for the steady state solution of three‐dimensional incompressible flows is presented. A preconditioned time marching scheme is applied to the conservative form of the governing equations. The preconditioning matrix multiplies the time derivatives of the system and circumvents the eigenvalue‐caused stiffness at low speed. The formulation is suitable for constant density flows and for flows where the density depends on non‐passive scalars, such as in low‐speed combustion applications. The k–ε model accounts for turbulent transport effects. A cell‐centred finite volume formulation with a Runge–Kutta time stepping scheme for the primitive variables is used. Second‐order spatial accuracy is achieved by developing for the preconditioned system an approximate Riemann solver with MUSCL reconstruction. A multi‐grid technique coupled with local time stepping and implicit residual smoothing is used to accelerate the convergence to the steady state solution. The convergence behaviour and the validation of the predicted solutions are examined for laminar and turbulent constant density flows and for a turbulent non‐premixed flame simulated by a presumed probability density function (PDF) model. Copyright © 2001 John Wiley & Sons, Ltd.     

An unsteady incompressible Navier–Stokes solver for large eddy simulation of turbulent flows

An unsteady incompressible Navier–Stokes solver that uses a dual time stepping method combined with spatially high‐order‐accurate finite differences, is developed for large eddy simulation (LES) of turbulent flows. The present solver uses a primitive variable formulation that is based on the artificial compressibility method and various convergence–acceleration techniques are incorporated to efficiently simulate unsteady flows. A localized dynamic subgrid model, which is formulated using the subgrid kinetic energy, is employed for subgrid turbulence modeling. To evaluate the accuracy and the efficiency of the new solver, a posteriori tests for various turbulent flows are carried out and the resulting turbulence statistics are compared with existing experimental and direct numerical simulation (DNS) data. Copyright © 1999 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.     

Shallow flow modelling using curvilinear depth-averaged stream function and vorticity transport equations

Most receiving water, such as lakes and open reservoirs, have large plan dimensions with respect to their depth. In such cases, the flow may be nearly two-dimensional and the depth-averaged Reynolds equations are appropriate. This paper presents a new version of the governing equations in curvilinear depth-averaged stream function and vorticity transport (ψ, ω) form appropriate for non-orthogonal computational meshes. The equations are discretized using finite differences and solved using successive over-relaxation for the depth-averaged stream function equation and an alternating direction implicit scheme for the vorticity transport equation. Results from the numerical model are validated against data from flow past a backward facing step and jet-forced flow in a circular reservoir. The results indicate that the (ψ, ω) form of the shallow water equations may be useful for applications where the free surface can either be assumed horizontal, or is know a priori.     

A boundary integral equation method for Navier-Stokes equations—application to flow in annulus of eccentric cylinders

A novel Navier-Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non-linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.     

AN ARTIFICIAL BOUNDARY CONDITION FOR TWO-DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOWS USING THE METHOD OF LINES

We design an artificial boundary condition for the steady incompressible Navier–Stokes equations in streamfunction–vorticity formulation in a flat channel with slip boundary conditions on the wall. The new boundary condition is derived from the Oseen equations and the method of lines. A numerical experiment for the non-linear Navier–Stokes equations is presented. The artificial boundary condition is compared with Dirichlet and Neumann boundary conditions for the flow past a rectangular cylinder in a flat channel. The numerical results show that our boundary condition is more accurate.     

DNS of flows with helical symmetry

Some flows such as the wakes of rotating devices often display helical symmetry. We present an original DNS code for the dynamics of such helically symmetric systems. We show that, by enforcing helical symmetry, the three-dimensional Navier–Stokes equations can be reduced to a two-dimensional unsteady problem. The numerical method is a generalisation of the vorticity/streamfunction formulation in a circular domain, with finite differences in the radial direction and spectral decomposition along the azimuth. When compared to a standard three-dimensional code, this allows us to reach larger Reynolds numbers and to compute quasi-steady patterns. We illustrate the importance of helical pitch by some physical cases: the dynamics of several helical vortices and a quasi-steady vortex flow. We also study the linear dynamics and nonlinear saturation in the Batchelor vortex basic flow, a paradigmatic example of trailing vortex instability. We retrieve the behaviour of inviscid modes and present new results concerning the saturation of viscous centre modes.     

Effects of a transverse flows with variable viscosity in micropolar fluids

A regular perturbation analysis is presented for three laminar natural convection flows in micropolar fluids in liquids with temperature dependent viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface. While these flows have well-known power-low similarity solutions when the fluid viscosity is taken to be constant, they are non-similar when the viscosity is considered to a function of temperature. A single similar flow, that adjacent to a vertical isothermal surface, is also analysed for comparison in order to estimate the extent of validity of perturbation analysis. The formulation used here provides a unified treatment of variable viscosity effects on those four flows. Computed first-order perturbation quantities are presented for all four flows. Numerical results for velocity, angular velocity and thermal functions has been shown graphically or tabulated for different values of micropolar parameters. Received on 20 October 1997