Finite element computation of a turbulent flow over a two-dimensional backward-facing step

The present paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a (k-ε) turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ε is proposed as a numerical treatment for physically meaningless values of k and ε. Results obtained for a recirculating flow in a two-dimensional channel with a sudden expansion in width are presented and compared with experimental values.     

Control volume finite element solution of confined turbulent swirling flows

A control volume finite element method that uses a triangular grid has been applied for solving confined turbulent swirling flows. To treat the velocity-pressure coupling, the vorticity-streamfunction formulation has been used. For turbulence effects the k-? model has been adopted. Consistent with the use of wall functions in the near-wall regions, a boundary condition for the calculation of the vorticity at computational boundaries is proposed and used effectively. The discretized equations are obtained by making use of an exponential interpolation function. Its use has been beneficial in reducing numerical diffusion. Comparisons of the current predictions with available experimental and numerical data from the literature showed generally fair agreement.     

Finite element computation of a turbulent flow over a two-dimensional backward-facing step

This paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a k-? turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ? is proposed as a numerical treatment for physically meaningless values of k and ?. Results obtained for a recirculating flow in a two-dimensional channel with a sudden expansion in width are presented and compared with experimental values.     

On the finite-element calculation of turbulent flow using the k-ϵ model

Although the finite-element (FE) method has been successful in analysing complex laminar flows, a number of difficulties can arise when two-equation turbulence models (e.g. the k-? model) are incorporated. This work describes a particular FE discretization of the k-? model and reports its performance in recirculating flow. Severe problems encountered in attempts to obtain convergence of the numerical scheme are isolated and analysed, and methods by which the problems can be overcome are suggested. Insight gained in this work has enabled a practical turbulent flow FE code to be constructed which is robust and efficient. This code is the subject of a further paper.     

Computation of turbulent asymmetric wake

The development of asymmetric wake behind an aerofoil in turbulent incompressible flow has been computed using finite volume scheme for solving two-dimensional Navier-Stokes equations along with the k-ε model of turbulence. The results are compared with available experimental data. It is observed that the computed shift of the point of minimum velocity with distance is sensitive to the prescribed value of the normal component of velocity at the trailing edge of the aerofoil. Making the model constant C_u as a function of streamline curvature and changing the production term in the equation for ε, has only marginal influence on the results.     

On a finite element CFD algorithm for compressible,viscous and turbulent aerodynamic flows

This paper develops and analyses individual construction aspects of an efficient and accurate finite element algorithm for prediction of viscous and turbulent flow fields of impact in aerodynamics. The theoretical construction employs a Taylor weak statement (TWS) for coincident embedding of stability mechanisms within a classic Galerkin finite element formulation of semi-discrete approximation error orthogonalization. A wide variety of the stabilizing mechanisms of independently derived CFD algorithms are contained within the TWS theory. An implicit construction that meets the requirement of efficient convergence to steady state is developed. The theoretical asymptotic error estimates of the TWS finite element algorithm for supersonic and viscous boundary layer flows are verified. Application to a three-dimensional turbulent flow is cited.     

Finite volume methods for laminar and turbulent flows using a penalty function approach

A penalty function, finite volume method is described for two-dimensional laminar and turbulent flows. Turbulence is modelled using the k-? model. The governing equations are discretized and the resulting algebraic equations are solved using both sequential and coupled methods. The performance of these methods is gauged with reference to a tuned SIMPLE-C algorithm. Flows considered are a square cavity with a sliding top, a plane channel flow, a plane jet impingement and a plane channel with a sudden expansion. A sequential method is employed, which uses a variety of dicretization practices, but is found to be extremely slow to converge; a coupled method, evaluated using a variety of matrix solvers, converges rapidly but, relative to the sequential approach, requires larger memory.     

Numerical aspects of calculation of confined swirling flows with internal recirculation

Predictions were performed for two different confined swirling flows with internal recirculation zones. The convection terms in the elliptic governing equations were discretized using three different finite differencing schemes: hybrid, quadratic upwind interpolation and skew upwind differencing. For each flow case, calculations were carried out with these schemes and successively refined grids were employed. For the turbulent flow case the k-ε turbulence model was used. The predicted cases were a laminar swirling flow investigated by Bornstein and Escudier, and a turbulent low-swirl case studied by Roback and Johnson. In both cases an internal recirculation zone was present. The laminar case is well predicted when account is taken of the estimated radial velocity component at the chosen inlet plane. The quadratic upwind interpolation and skew upwind schemes predict the main features of the internal recirculation zone also with a coarse grid. The turbulent case is well predicted with the coarse as well as the finer grids, the skew upwind and quadratic upwind interpolation schemes yielding results very close to the measurements. It is concluded that the skew upwind scheme reaches grid independence slightly before the quadratic upwind scheme, both considerably earlier than the hybrid scheme.     

Turbulent boundary-layer analysis using finite elements

The use of finite element methods for turbulent boundary-layer flow is relatively recent and of limited extent.¹ In the present study, we extend the group variable approach of Fletcher and Fleer^2,3 to treat turbulent boundary layer flows with heat transfer using a two-equation turbulence model. The main concepts in the formulations include a Dorodnitsyn-type transformation which uses a velocity component as the transverse variable, a ‘variational’ formulation for the transformed equations using special test functions and development of a two-equation turbulence model in terms of the turbulent kinetic energy and turbulence dissipation rate as additional field variables. Several numerical test cases have been examined comparing the results with finite difference calculations and comparing the two-equation turbulence model with an algebraic turbulence model.     

Examination of wall damping for the k-ε turbulence model using direct simulations of turbulent channel flow

Handler, Hendricks and Leighton have recently reported results for the direct numerical simulation (DNS) of a turbulent channel flow at moderate Reynolds number. These data are used to evaluate the terms in the exact and modelled transport equations for the turbulence kinetic energy k and the isotropic dissipation function ε. Both modelled transport equations show significant imbalances in the high-shear region near the channel walls. The model for the eddy viscosity is found to yield distributions for the production terms which do not agree well with the distributions calculated from the DNS data. The source of the imbalance is attributed to the wall-damping function required in eddy viscosity models for turbulent flows near walls. Several models for the damping function are examined, and it is found that the models do not vary across the channel as does the damping when evaluated from the DNS data. The Lam-Bremhorst model and the standard van Driest model are found to give reasonable agreement with the DNS data. Modification of the van Driest model to include an effective origin yields very good agreement between the modelled production and the production calculated from the DNS data, and the imbalance in the modelled transport equations is significantly reduced.     

Application of the Dorodnitsyn finite element method to swirling boundary layer flow

The Dorodnitsyn finite element method for turbulent boundary layer flow with surface mass transfer is extended to include axisymmetric swirling internal boundary layer flow. Turbulence effects are represented by the two-layer eddy viscosity model of Cebeci and Smith¹ with extensions to allow for the effect of swirl. The method is applied to duct entry flow and a 10 degree included-angle conical diffuser, and produces results in close agreement with experimental measurements with only 11 grid points across the boundary layer. The introduction of swirl (w_e/u_e = 0.4) is found to have little effect on the axial skin friction in either a slightly favourable or adverse pressure gradient, but does cause an increase in the displacement area for an adverse pressure gradient. Surface mass transfer (blowing or suction) causes a substantial reduction (blowing) in axial skin friction and an increase in the displacement area. Both suction and the adverse pressure gradient have little influence on the circumferential velocity and shear stress components. Consequently in an adverse pressure gradient the flow direction adjacent to the wall is expected to approach the circumferential direction at some downstream location.     

Introduction of turbulent model in a mixed finite volume/finite element method

The aim of this work is to present a new numerical method to compute turbulent flows in complex configurations. With this in view, a k-? model with wall functions has been introduced in a mixed finite volume/finite element method. The numerical method has been developed to deal with compressible flows but is also able to compute nearly incompressible flows. The physical model and the numerical method are first described, then validation results for an incompressible flow over a backward-facing step and for a supersonic flow over a compression ramp are presented. Comparisons are performed with experimental data and with other numerical results. These simulations show the ability of the present method to predict turbulent flows, and this method will be applied to simulate complex industrial flows (flow inside the combustion chamber of gas turbine engines). The main goal of this paper is not to test turbulence models, but to show that this numerical method is a solid base to introduce more sophisticated turbulence model.     

On steady state computation of turbulent flows using k–ε models approximated by the time splitting method

The time splitting method is frequently used in numerical integration of flow equations with source terms since it allows almost independent programming for the source part. In this paper we will consider the question of convergence to steady state of the time splitting method applied to k–ε turbulence models. This analysis is derived from a properly defined scalar study and is carried out with success for the coupled k–ε equations. It is found that the time splitting method does not allow convergence to steady state for any choice of finite values of the time step. Numerical experiments for some typical turbulent compressible flow problems support the fact that the time splitting method is always nonconvergent, while its nonsplitting counterpart is convergent. Copyright © 2005 John Wiley & Sons, Ltd.     

Calculation of complex near-wall turbulent flows with a low-Reynolds-number k-ϵ model

An improved low-Reynolds-number k-? model has been formulated and tested against a range of DNS (direct numerical simulation) and experimental data for channel and complex shear layer flows. The model utilizes a new form of damping function adopted to account for both wall proximity effects and viscosity influences and a more flexible damping argument based on the gradient of the turbulent kinetic energy on the wall. Additionally, the extra production of the inhomogeneous part of the viscous dissipation near a wall has been added to the dissipation equation with significantly improved results. The proposed model was successfully applied to the calculation of a range of wall shear layers in zero, adverse and favourable pressure gradients as well as backward-facing-step separated flows.     

Computation of normal impinging jets in cross-flow and comparison with experiment

The PHOENICS code has been used to model the flow field surrounding subsonic and underexpanded jets impinging on a ground plane in the presence of a cross-flow, for cases with both a fixed ground plane and a ‘rolling road’. The standard k-ε turbulence model is used, without correction factors. It is confirmed that this overpredicts the free jet entrainment rate; the wall jet spreading rate is slightly underpredicted but the initial thickness is too high. Agreement with experiment is, nevertheless, much better than for previous calculations, showing the importance of the extent of the grid used. The ground vortex formed in cross-flow is shown to move with varying effective velocity ratio and with rolling road operation in the same manner as experimentally observed. Ground vortex self-similarity is also accurately predicted with the numerical modelling.     

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.     

Numerical simulation of axisymmetric turbulent flow in combustors and diffusers

Numerical studies of turbulent flow in an axisymmetric 45° expansion combustor and bifurcated diffuser are presented. The Navier-Stokes equations incorporating a k–? model were solved in a non-orthogonal curvillinear co-ordinate system. A zonal grid method, wherein the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. However, treatment of the zonal boundaries required special handling. Boundary overlap and interpolating techniques were used and an adjustment of the flow variables was required to assure conservation of mass flux. Three finite differencing methods—hybrid, quadratic upwind and skew upwind—were used to represent the convection terms. Results were compared with existing experimental data. In general, good agreement between predicted and measured values was obtained.     

Numerical modelling of complex turbulent free‐surface flows with the SPH method: an overview

The gridless smoothed particle hydrodynamics (SPH) method is now commonly used in computational fluid dynamics (CFD) and appears to be promising in predicting complex free‐surface flows. However, increasing flow complexity requires appropriate approaches for taking account of turbulent effects, whereas some authors are still working without any turbulence closure in SPH. A review of recently developed turbulence models adapted to the SPH method is presented herein, from the simplistic point of view of a one‐equation model involving mixing length to more sophisticated (and thus realistic) models like explicit algebraic Reynolds stress models (EARSM) or large eddy simulation (LES). Each proposed model is tested and validated on the basis of schematic cases for which laboratory data, theoretical or numerical solutions are available in the general field of turbulent free‐surface incompressible flows (e.g. open‐channel flow and schematic dam break). They give satisfactory results, even though some progress should be made in the future in terms of free‐surface influence and wall conditions. Recommendations are given to SPH users to apply this method to the modelling of complex free‐surface turbulent flows. Copyright © 2006 John Wiley & Sons, Ltd.     

Applications of the artificial compressibility method for turbulent open channel flows

A three‐dimensional (3‐D) numerical method for solving the Navier–Stokes equations with a standard k–ε turbulence model is presented. In order to couple pressure with velocity directly, the pressure is divided into hydrostatic and hydrodynamic parts and the artificial compressibility method (ACM) is employed for the hydrodynamic pressure. By introducing a pseudo‐time derivative of the hydrodynamic pressure into the continuity equation, the incompressible Navier–Stokes equations are changed from elliptic‐parabolic to hyperbolic‐parabolic equations. In this paper, a third‐order monotone upstream‐centred scheme for conservation laws (MUSCL) method is used for the hyperbolic equations. A system of discrete equations is solved implicitly using the lower–upper symmetric Gauss–Seidel (LU‐SGS) method. This newly developed numerical method is validated against experimental data with good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulating two-dimensional turbulent flow by using the κ–ϵ model and the vorticity–streamfunction formulation

A numerical procedure to solve turbulent flow which makes use of the κ–? model has been developed. The method is based on a control volume finite element method and an unstructured triangular domain discretization. The velocity-pressure coupling is addressed via the vorticity-streamfunction and special attention is given to the boundary conditions for the vorticity. Wall effects are taken into account via wail functions or a low-Reynolds-number model. The latter was found to perform better in recirculation regions. Source terms of the κ and ε transport equations have been linearized in a particular way to avoid non-realistic solutions. The vorticity and streamfunction discretized equations are solved in a coupled way to produce a faster and more stable computational procedure. Comparison between the numerical predictions and experimental data shows that the physics of the flow is correctly simulated.