Numerical predictions of flows over backward-facing steps

Predictions are reported for two-dimensional, steady, incompressible flows over rearward-facing steps for both laminar and turbulent conditions. The standard k-? turbulence model was used for the turbulent flow. Attention was focused on obtaining accurate solutions to the differential equations. It is concluded that some of the serious discrepancies that have occurred between prediction and observation, and attributed in earlier studies to the inadequacy of the turbulence model, may have been due to the inaccuracy of the solution.     

Numerical solutions of high-Re recirculating flows in vorticity-velocity form

A numerical method for computing high-Re laminar steady flows is presented. The incompressible Navier-Stokes equations are expressed in terms of vorticity-velocity variables, discretized in space by finite differences on a staggered grid and advanced in time by a scalar alternating direction implicit (ADI) procedure, which allows a fully vectorized computer code. The accuracy and efficiency of the present formulation are discussed in comparison with the standard ω-ψ and u, v, P forms. Numerical results are presented for two test cases: the driven cavity at Re up to 5000 and the backward-facing step at Re up to 800.     

Stream function solution of Navier-Stokes problems with inter-element penalties

In this note, we apply a finite element stream function formulation with inter-element penalties to the Navier-Stokes equations. The approach is an extension of a technique previously introduced for Stokes, flow. The solution is obtained by iterative linearization using successive approximation, and results for a standard numerical test case are given.     

Comparison of several open boundary numerical treatments for laminar recirculating flows

Several open boundary conditions (OBCs) are compared and evaluated in the framework of the SIMPLE algorithm using staggered and non-staggered grid systems. The benchmark laminar flow test cases used for the OBC evaluation are Poiseuille-Benard flow in a channel and stratified backward-facing step flow. The investigated OBCs are linear explicit step space extrapolation, Orlanski's monochromatic wave, and pressure extrapolation. Orlanski's and pressure extrapolation open boundary treatment for unsteady and steady flows, respectively, yield little reflection and has proved to be adequate for engineering calculations.     

Numerical computations of critical flow over a weir

Computing critical flows in hydraulics involves three problems in one: the internal flow problem, the location of the free surface and the determination of the critical flow rate. The subject can involve such difficulties as non-uniqueness, non-existence, ill-conditioning and catastrophes. This paper discusses the difficulties relating to computing critical flows over weirs. A new rapidly convergent method of determining the critical flow rate is presented and various results are shown using it with finite element discretization and with a new streamline shifting method. Numerical results are in good agreement with published data, both numerical and experimental.     

Conservative calculations of non-isentropic transonic flows

The classical potential formulation of inviscid transonic flows is modified to account for non-isentropic effects. The density is determined in terms of the speed as well as the pressure, which in turn is calculated from a second-order mixed-type equation derived via differentiating the momentum equations. The present model differs in general from the exact inviscid Euler equations since the flow is assumed irrotational. On the other hand, since the shocks are not isentropic, they are weaker and are placed further upstream compared to the classical potential solution. Furthermore, the streamline leaving the aerofoil does not necessarily bisect the trailing edge. Results for the present conservative calculations are presented for non-lifting and lifting aerofoils at subsonic and transonic speeds and compared to potential and Euler solutions.     

Calculation of laminar recirculating flows using a local non-staggered grid refinement system

A grid-embedding technique for the solution of two-dimensional incompressible flows governed by the Navier-Stokes equations is presented. A finite volume method with collocated primitive variables is employed to ensure conservation at the interfaces of embedding grids as well as global conservation. The discretized equations are solved simultaneously for the whole domain, providing a strong coupling between regions of different refinement. The formulation presented herein is applicable to uniform or non-uniform Cartesian meshes. The method was applied to the solution of two scalar transport equations, to cavity flows driven by body and shear forces and to a sudden plane contraction flow. The numerical predictions are compared with the exact solutions when available and with experimental data. The results show that neither the convergence rate nor the stability of the method is affected by the presence of embedded grids. Embedded grids provide a better distribution of grid nodes over the computational domain and consequently the solution accuracy was improved. The grid-embedding technique proved also that significant savings in computing time could be achieved.     

Treatment of numerical diffusion in strong convective flows

A three-dimensional extension of the QUICK scheme adapted for the finite volume method and non-uniform grids is presented to handle convection-diffusion problems for high Peclet numbers and steep gradients. The algorithm is based on three-dimensional quadratic interpolation functions in which the transverse curvature terms are maintained and the diagonal dominance of the coefficient matrix is preserved. All formulae are explicitly given in an appendix. Results obtained with the classical upwind (UDS), the simplified QUICK (transverse terms neglected) and the present full QUICK schemes are given for two benchmark problems, one two-dimensional, steady state and the other three-dimensional, unsteady state. Both QUICK schemes are shown to give superior solutions compared with the UDS in terms of accuracy and efficiency. The full QUICK scheme performs better than the simplified QUICK, giving even for coarse grids acceptable results closer to the analytical solutions, while the computational time is not affected much.     

Use of pseudo-concentrations to follow creeping viscous flows during transient analysis

Creeping viscous flows are followed through finite element meshes by use of pseudo-concentrations which define material position. The concentrations, assumed to be transported only by convection, serve as material markers. Illustrations are presented related to industrial forming processes and the slow deformation of geological structures.     

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.     

A new,reliable family of split-operator methods for computing reacting flows

A new, reliable family of split-operator methods is presented for the computation of reacting flows. The methods may be inconsistent or consistent depending upon their applicability. Analyses are carried out for flow combusting according to general multicomponent, multireaction linear or non-linear chemical kinetic models, and conditions for guaranteed stability/convergence are established. A special chemical splitting parameter can also be embedded in the methods enabling accelerated convergence of an iterative finite-difference solution. Computed results highlight the capabilities and properties of the methods.     

Adaptive finite element methods for high-speed compressible flows

We describe an adaptive finite element algorithm for solving the unsteady Euler equations. The finite element algorithm is based on a Taylor/Galerkin formulation and uses a very fast and efficient data structure to refine and unrefine the grid in order to optimize the approximation. We give a general version of the method which can be applied to moving grids with sliding interfaces and we present the results for a transient supersonic calculation of rotor-stator interaction.     

A numerical method to simulate turbulent cavitating flows

The objective of this paper is to develop a numerical method for simulating multiphase cavitating flows on unstructured grids. The multiphase medium is represented using a homogeneous mixture model that assumes thermal equilibrium between the liquid and vapor phases. We develop a predictor–corrector approach to solve the governing Navier–Stokes equations for the liquid/vapor mixture, together with the transport equation for the vapor mass fraction. While a non-dissipative and symmetric scheme is used in the predictor step, a novel characteristic-based filtering scheme with a second order TVD filter is developed for the corrector step to handle shocks and material discontinuities in non-ideal gases and mixtures. Additionally, a sensor based on vapor volume fraction is proposed to localize dissipation to the vicinity of discontinuities. The scheme is first validated for simple one dimensional canonical problems to verify its accuracy in predicting jump conditions across material discontinuities and shocks. It is then applied to two turbulent cavitating flow problems – over a hydrofoil using RANS and over a wedge using LES. Our results show that the simulations are in good agreement with experimental data for the above tested cases, and that the scheme can be successfully applied to both RANS and LES methodologies.     

Development of new triangular elements for free surface flows

New finite elements have been developed to simulate steady and unsteady two-dimensional free surface flows. The depth-averaged velocity components with the free surface elevation have been used as independent variables in the model. The differences between the various elements presented lie in the choice of velocity approximation. The Newton–Raphson method has been used to solve the non-linear system of equations. Emphasis is put on bench-mark examples to assess the accuracy and efficiency of the elements. A simple stable new element tested herein shows promising advantages for industrial finite element codes.     

Prediction of incompressible flow separation with the approximate factorization technique

This paper describes one application of the approximate factorization technique to the solution of incompressible steady viscous flow problems in two dimensions. The velocity-pressure formulation of the Navier-Stokes equations written in curvilinear non-orthogonal co-ordinates is adopted. The continuity equation is replaced with one equation for the pressure by means of the artificial compressibility concept to obtain a system parabolic in time. The resulting equations are discretized in space with centred finite differences, and the steady state solution obtained by a time-marching ADI method requiring to solve 3 x 3 block tridiagonal linear systems. An optimized fourth-order artificial dissipation is introduced to damp the numerical instabilities of the artificial compressibility equation and ensure convergence. The resulting solver is applied to the prediction of a wide variety of internal flows, including both streamlined boundaries and sharp corners, and fast convergence and good results obtained for all the configurations investigated.     

Navier-Stokes computations of horseshoe vortex flows

The computation of the incompressible three-dimensional turbulent viscous flow about an aerofoil/flat plate junction is investigated. An iterative, fully decoupled technique is applied to the Reynolds-averaged Navier-Stokes equations (RANSEs) written in a non-orthogonal curvilinear body-fitted co-ordinate system. The results of the computations are compared with well-documented experiments.     

A continuous shape sensitivity equation method for unsteady laminar flows

This paper presents the application of a general shape sensitivity equation method (SEM) to unsteady laminar flows. The formulation accounts for complex parameter dependence and is suitable for a wide range of problems. The flow and sensitivity equations are solved on 3D meshes using a Streamline-Upwind Petrov Galerkin (SUPG) finite element method. In the case of shape parameters, boundary conditions for sensitivities depend on the flow gradient at the boundary. Therefore, an accurate recovery of solution gradients is crucial to the success of shape sensitivity computations. In this work, solution gradients at boundary points are extracted using the Finite Node Displacement (FiND) method on which the finite element discretization is enriched locally via the insertion of nodes close to the boundary points. The normal derivative of the solution is then determined using finite differences. This approach to evaluate shape sensitivity boundary conditions is embedded in the continuous SEM. The methodology is applied to the flow past a cylinder in ground proximity. First, the proposed method is verified on a steady state problem. The computed sensitivity is compared to the actual change in the solution when a small perturbation is imposed to the shape parameter. Then, the study investigates the ability of the SEM to anticipate the unsteady flow response to changes in the ground to cylinder gap. A reduction of the gap causes damping of the vortex shedding while an increase amplifies the unsteadiness.     

Turbulent characteristics of shear-thinning fluids in recirculating flows

 A miniaturised fibre optic Laser-Doppler anemometer was used to carry out a detailed hydrodynamic investigation of the flow downstream of a sudden expansion with 0.1–0.2% by weight shear-thinning aqueous solutions of xanthan gum. Upstream of the sudden expansion the pipe flow was fully-developed and the xanthan gum solutions exhibited drag reduction with corresponding lower radial and tangential normal Reynolds stresses, but higher axial Reynolds stress near the wall and a flatter axial mean velocity profile in comparison with Newtonian flow. The recirculation bubble length was reduced by more than 20% relative to the high Reynolds number Newtonian flow, and this was attributed to the occurrence further upstream of high turbulence for the non-Newtonian solutions, because of advection of turbulence and earlier high turbulence production in the shear layer. Comparisons with the measurements of Escudier and Smith (1999) with similar fluids emphasized the dominating role of inlet turbulence. The present downstream turbulence field was less anisotropic, and had lower maximum axial Reynolds stresses (by 16%) but higher radial turbulence (20%) than theirs. They reported considerably longer recirculating bubble lengths than we do for similar non-Newtonian fluids and Reynolds numbers. Received: 23 February 1999/Accepted: 28 April 1999     

Multiscale computations of thin films in multiphase flows

In multiphase flows thin films are often encountered when fluid masses collide. These films can become very thin and in direct numerical simulations (DNS) it is often impractical to resolve their thickness fully, even with adaptive grid refining. Here we examine the collision of a fluid drop with a wall and develop a multiscale approach to compute the flow in the film between the drop and the wall. By using a semi-analytical model for the flow in the film we capture the evolution of films thinner than the grid spacing reasonably well.     

Periodic solutions of rigid body–viscous flow interaction

This paper describes the work on extending the finite element method to cover interactions between a viscous flow and a moving body. The problem configuration of interest is that of an arbitrarily shaped body undergoing a simple harmonic motion in an otherwise undisturbed incompressible fluid. The finite element modelling is based on a primitive variables representation of the Navier-Stokes equations using curved isoparametric elements. The non-linear boundary conditions on the moving body are obtained using Taylor series expansion to approximate the velocities at the fixed finite element grid points. The method of averaging is used to analyse the resulting periodic motion of the fluid. The stability of the periodic solutions is studied by introducing small perturbations and applying Floquet theory. Numerical results are obtained for several example body shapes and compared with published experimental results. Good agreement is obtained for the basic non-linear phenomenon of steady streaming.