The effect of the choice of the reference pressure location in numerical modelling of incompressible flow

The discontinuity of a finite-element pressure field that is sometimes present in the neighbourhood of the pressure-specification-point is shown to arise either from round-off, or from mistakes in modelling. The implications of this are considered. In particular it restricts grid refinement near the pressure-specification-point. The analysis can be extended to finite-difference calculations, and to other fields governed by equations similar to Poisson's equation.     

Three-dimensional surface water modelling using a mesh adaptive technique

Numerical simulation of open water flow in natural courses seems to be doomed to one- or two-dimensional numerical simulations. Investigations of flow hydrodynamics through the application of three-dimensional models actually have very few appearances in the literature. This paper discusses the development and the initial implementation of a general three-dimensional and time-dependent finite volume approach to simulate the hydrodynamics of surface water flow in rivers and lakes. The slightly modified Navier-Stokes equations, together with the continuity and the water depth equations, form the theoretical basis of the model. A body-fitted time-dependent co-ordinate system has been used in the solution process, in order to accommodate the commonly complex and irregular boundary and bathymetry of natural water courses. The proposed adaptive technique allows the mesh to follow the movement of the water boundaries, including the unsteady free-water surface. The primitive variable equations are written in conservative form in the Cartesian co-ordinate system, and the computational procedure is executed in the moveable curvilinear co-ordinate system. Special stabilizing techniques are introduced in order to eliminate the oscillating behaviour associated with the finite volume formulation. Also, a new and comprehensive approximation for the pressure forces at the faces of a control volume is presented. Finally, results of several tests demonstrate the performance of the finite volume approach coupled with the adaptive technique employed in the three-dimensional time-dependent mesh system.     

Three-dimensional analysis of the flow in a curved hydraulic turbine draft tube

The three-dimensional turbulent flow in a curved hydraulic turbine draft tube is studied numerically. The analysis is based on the steady Reynolds-averaged Navier–Stokes equations closed with the κ-ε model. The governing equations are discretized by a conservative finite volume formulation on a non-orthogonal body-fitted co-ordinate system. Two grid systems, one with 34 × 16 × 12 nodes and another with 50 × 30 × 22 nodes, have been used and the results from them are compared. In terms of computing effort, the number of iterations needed to yield the same degree of convergence is found to be proportional to the square root of the total number of nodes employed, which is consistent with an earlier study made for two-dimensional flows using the same algorithm. Calculations have been performed over a wide range of inlet swirl, using both the hybrid and second-order upwind schemes on coarse and fine grids. The addition of inlet swirl is found to eliminate the stalling characteristics in the downstream region and modify the behaviour of the flow markedly in the elbow region, thereby affecting the overall pressure recovery noticeably. The recovery factor increases up to a swirl ratio of about 0˙75, and then drops off. Although the general trends obtained with both finite difference operators are in agreement, the quantitative values as well as some of the fine flow structures can differ. Many of the detailed features observed on the fine grid system are smeared out on the coarse grid system, pointing out the necessity of both a good finite difference operator and a good grid distribution for an accurate result.     

A spectral method with staggered grid for incompressible Navier-Stokes equations

In order to solve the Navier-Stokes equations by spectral methods, we develop an algorithm using a staggered grid to compute the pressure. On this grid, an iterative process based on an artificial compressibility matrix associates the pressure with the continuity equation. This method is very accurate and avoids naturally most of the effects of parasite modes appearing in classical spectral methods with a velocity—pressure formulation.     

The effect of the stability of mixed finite element approximations on the accuracy and rate of convergence of solution when solving incompressible flow problems

The stability of two different mixed finite element methods for incompressible flow problems are theoretically analysed. The effect of the stability of the mixed approximation on the accuracy and the rate of convergence of solution is assessed for two non-trivial problems. The numerical results presented indicate that if the stability of the mixed approximation is not guaranteed then both pressure and velocity solutions are markedly less accurate. In one of the cases considered the ultimate convergence of both the pressure and the velocity solutions is seriously in doubt.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

A finite element for incompressible plane flows of fluids with memory

Flows of fluids with single-integral memory functionals are considered. Evaluation of the stress at a material point involves the deformation history of that point, and a dominant computational cost in finite element approximation is the construction of streamlines. It is shown that the simple crossed-triangle macro-element is in many ways an ideal finite element for the difficult non-linear, non-self-adjoint problem. The question as to whether this element produces convergent velocity and pressure solutions is addressed in the light of its failure to satisfy the discrete LBB condition. The effect of the element's ill-disposed (‘spurious’) pressure modes is discussed, and a pressure smoothing scheme is given which gives good results in Newtonian and non-Newtonian flows at various Reynolds and Deborah numbers. As an example of the element's success in modelling such flows, the problem of pressure differences in flows over transverse slots is studied numerically. The results are compared with experimental observations of such flows. The effect of fluid memory on the relation between first normal-stress differences and pressure differences is investigated.     

Finite element implementation of boundary conditions for the pressure Poisson equation of incompressible flow

In this paper we address the problem of the implementation of boundary conditions for the derived pressure Poisson equation of incompressible flow. It is shown that the direct Galerkin finite element formulation of the pressure Poisson equation automatically satisfies the inhomogeneous Neumann boundary conditions, thus avoiding the difficulty in specifying boundary conditions for pressure. This ensures that only physically meaningful pressure boundary conditions consistent with the Navier-Stokes equations are imposed. Since second derivatives appear in this formulation, the conforming finite element method requires C¹ continuity. However, for many problems of practical interest (i.e. high Reynolds numbers) the second derivatives need not be included, thus allowing the use of more conventional C⁰ elements. Numerical results using this approach for a wall-driven contained flow within a square cavity verify the validity of the approach. Although the results were obtained for a two-dimensional problem using the p-version of the finite element method, the approach presented here is general and remains valid for the conventional h-version as well as three-dimensional problems.     

Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re = 800 stable?

A detailed case study is made of one particular solution of the 2D incompressible Navier–Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear-stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re = 800 is steady—and it is stable, to both small and large perturbations.     

A modified finite element method for solving the time-dependent,incompressible Navier-Stokes equations. Part 1: Theory

Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via 1-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection-dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid-driven cavity at Re ≤ 10,000, flow past a circular cylinder at Re ≤ 400, and the simulation of a heavy gas release over complex topography.     

On the implementation of symmetric and antisymmetric periodic boundary conditions for incompressible flow

In this paper we consider symmetric and antisymmetric periodic boundary conditions for flows governed by the incompressible Navier-Stokes equations. Classical periodic boundary conditions are studied as well as symmetric and antisymmetric periodic boundary conditions in which there is a pressure difference between inlet and outlet. The implementation of this type of boundary conditions in a finite element code using the penalty function formulation is treated and also the implementation in a finite volume code based on pressure correction. The methods are demonstrated by computation of a flow through a staggered tube bundle.     

Development of a computer program for the prediction of flow in a rotating cavity

A computer program has been developed to predict laminar source-sink flow in a rotating cylindrical cavity. Although the program is based on a standard finite difference technique for recirculating flow, it incorporates two novel features. Step changes in grid size are employed to obtain sufficient resolution in the boundary layers and special treatment is given to the solution of the pressure correction equations, in the ‘SIMPLE’ algorithm, in order to improve the convergence properties of the method. Results are presented both for the flow in an infinite rotating cylindrical annulus and a finite rotating cylindrical cavity, with the inner cylindrical surface acting as a uniform source and the outer cylinder as a sink. These show good agreement with existing analytical solutions and illustrate some of the problems associated with the computation of rapidly rotating flows.     

Calculation of quasi-three-dimensional incompressible viscous flows by the finite element method

This paper discusses the calculation of quasi-three-dimensional incompressible viscous flow by FEM. The Reynolds-averaged Navier-Stokes equations are solved in curvilinear co-ordinates by the reduced integration and penalty method (RIP). Streamline upwind artificial viscosity (SUAV) and the Baldwin-Lomax algebraic model of turbulence are used. Time discretization is by the general implicit θ-method.     

On penalty function methods in the finite-element analysis of flow problems

In this paper the penalty function method is reviewed in the general context of solving constrained minimization problems. Mathematical properties, such as the existence of a solution to the penalty problem and convergence of the solution of a penalty problem to the solution of the original problem, are studied for the general case. Then the results are extended to a penalty function formulation of the Stokes and Navier-Stokes equations. Conditions for the equivalence of two penalty-finite element models of fluid flow are established, and the theoretical error estimates are verified in the case of Stokes's problem.     

Development of projection and artificial compressibility methodologies using the approximate factorization technique

Predictions for two-dimensional, steady, incompressible flows under both laminar and turbulent conditions are presented. The standard k-? turbulence model is used for the turbulent flows. The computational method is based on the approximate factorization technique. The coupled approach is used to link the equations of motion and the turbulence model equations. Mass conservation is enforced by either the pseudocompressibility method or the pressure correction method. Comparison of the two methods shows a superiority of the pressure correction method. Second- and fourth-order artifical dissipation terms are used in order to achieve good convergence and to handle the turbulence model equations efficiently. Several internal and external test cases are investigated, including attached and separated flows.     

Segregated finite element algorithms for the numerical solution of large-scale incompressible flow problems

This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solution algorithms for the simulation of large-scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregated solution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solution algorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non-symmetric advection–diffusion type equations resulting from the segregated algorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solution algorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.     

Variable penalty method for finite element analysis of incompressible flow

A new scheme is applied for increasing the accuracy of the penalty finite element method for incompressible flow by systematically varying from element to element the sign and magnitude of the penalty parameter λ, which enters through ?.v + p/λ = 0, an approximation to the incompressibility constraint. Not only is the error in this approximation reduced beyond that achievable with a constant λ, but also digital truncation error is lowered when it is aggravated by large variations in element size, a critical problem when the discretization must resolve thin boundary layers. The magnitude of the penalty parameter can be chosen smaller than when λ is constant, which also reduces digital truncation error; hence a shorter word-length computer is more likely to succeed. Error estimates of the method are reviewed. Boundary conditions which circumvent the hazards of aphysical pressure modes are catalogued for the finite element basis set chosen here. In order to compare performance, the variable penalty method is pitted against the conventional penalty method with constant λ in several Stokes flow case studies.     

Fully developed flow in a curved pipe of arbitrary curvature ratio

It is generally assumed in curved pipe flow analyses that the curvature ratio, δ, of the pipe is very small, in which case the flow depends on a single parameter, the Dean number. This is not the case if δ is not very small. To determine the importance of this effect we have numerically solved the full Navier-Stokes equations, in primitive variable form, for arbitrary values of δ. A factored ADI finite-difference scheme has been used, employing Chorin's artificial compressibility technique. The results show that the central-difference calculation on a staggered grid is stable, without adding artificial damping terms, due to coupling between pressure and velocity. A spatially variable time step is used with a fixed Courant number.     

Solution of the incompressible mass and momentum equations by application of a coupled equation line solver

This paper describes an iterative technique for solving the coupled algebraic equations for mass and momentum conservation for an incompressible fluid flow. The technique is based on the simultaneous solution for pressure and velocity along lines. In a manner similar to ADI methods for a single variable, the solution domain is entirely swept line-by-line in each co-ordinate direction successively until a converged solution is obtained. The tight coupling between the equations that is guaranteed by the method results in an economical solution of the equation set.     

Finite element solution of the streamfunction-vorticity equations for incompressible two-dimensional flows

The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework of the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution of two well-known benchmark problems: flow in a lid-driven cavity at Re ? 10,000 and flow over a backward-facing step at Re = 800.