Finite element analysis of air flow around permeable sand fences

A numerical study of the turbulent air flow in a trench trap and the turbulent flow around a permeable sand fence is reported in this paper. The two-dimensional modified k–ε turbulence model proposed by Kato and Launder is used to predict the turbulent characteristics of the air flow. The discretization method for the governing equations is the three-step Taylor/Galerkin finite element method proposed by the authors. For the flow in a trench trap the numerical results are compared with experimental data obtained under realistic conditions using a large wind tunnel. For the air flow around a permeable sand fence a pressure loss model is used to represent the effect of the porosity of the fence on the flow field. © 1997 John Wiley & Sons, Ltd.     

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.     

Implementation of physical boundary conditions into computational domain in modelling of oscillatory bottom boundary layers

This paper discusses the importance of realistic implementation of the physical boundary conditions into computational domain for the simulation of the oscillatory turbulent boundary layer flow over smooth and rough flat beds. A mathematical model composed of the Reynolds averaged Navier–Stokes equation, turbulent kinetic energy (k) and dissipation rate of the turbulent kinetic energy (ε) has been developed. Control‐volume approach is used to discretize the governing equations to facilitate the numerical solution. Non‐slip condition is imposed on the bottom surface, and irrotational main flow properties are applied to the upper boundary. The turbulent kinetic energy is zero at the bottom, whereas the dissipation rate is approaching to a constant value, which is proportional to the kinematic viscosity times the second derivative of the turbulent kinetic energy. The output of the model is compared with the available experimental studies conducted in oscillatory tunnels and wave flume. It is observed that the irrotational flow assumption at the upper boundary is not realistic in case of water tunnels. Therefore, new upper boundary conditions are proposed for oscillatory tunnels. The data of wave flume show good agreement with the proposed numerical model. Additionally, several factors such as grid aspect ratio, staggered grid arrangement, time‐marching scheme and convergence criteria that are important to obtain a robust, realistic and stable code are discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Calculation of turbulent fluid flow and heat transfer in ducts by a full Reynolds stress model

A computational method has been developed to predict the turbulent Reynolds stresses and turbulent heat fluxes in ducts by different turbulence models. The turbulent Reynolds stresses and other turbulent flow quantities are predicted with a full Reynolds stress model (RSM). The turbulent heat fluxes are modelled by a SED concept, the GGDH and the WET methods. Two wall functions are used, one for the velocity field and one for the temperature field. All the models are implemented for an arbitrary three‐dimensional channel. Fully developed condition is achieved by imposing cyclic boundary conditions in the main flow direction. The numerical approach is based on the finite volume technique with a non‐staggered grid arrangement. The pressure–velocity coupling is handled by using the SIMPLEC‐algorithm. The convective terms are treated by the van Leer scheme while the diffusive terms are handled by the central‐difference scheme. The hybrid scheme is used for solving the ε equation. The secondary flow generation using the RSM model is compared with a non‐linear k–ε model (non‐linear eddy viscosity model). The overall comparison between the models is presented in terms of the friction factor and Nusselt number. Copyright © 2003 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.     

A two‐scale low‐Reynolds number turbulence model

In this study, a two‐scale low‐Reynolds number turbulence model is proposed. The Kolmogorov turbulence time scale, based on fluid kinematic viscosity and the dissipation rate of turbulent kinetic energy (ν, ε), is adopted to address the viscous effects and the rapid increasing of dissipation rate in the near‐wall region. As a wall is approached, the turbulence time scale transits smoothly from a turbulent kinetic energy based (k, ε) scale to a (ν, ε) scale. The damping functions of the low‐Reynolds number models can thus be simplified and the near‐wall turbulence characteristics, such as the ε distribution, are correctly reproduced. The proposed two‐scale low‐Reynolds number turbulence model is first examined in detail by predicting a two‐dimensional channel flow, and then it is applied to predict a backward‐facing step flow. Numerical results are compared with the direct numerical simulation (DNS) budgets, experimental data and the model results of Chien, and Lam and Bremhorst respectively. It is proved that the proposed two‐scale model indeed improves the predictions of the turbulent flows considered. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Direct Numerical Simulation of Turbulent Flow over a Rectangular Trailing Edge

This paper describes a direct numerical simulation (DNS) study of turbulent flow over a rectangular trailing edge at a Reynolds number of 1000, based on the freestream quantities and the trailing edge thickness h; the incoming boundary layer displacement thickness δ^* is approximately equal to h. The time-dependent inflow boundary condition is provided by a separate turbulent boundary layer simulation which is in good agreement with existing computational and experimental data. The turbulent trailing edge flow simulation is carried out using a parallel multi-block code based on finite difference methods and using a multi-grid Poisson solver. The turbulent flow in the near-wake region of the trailing edge has been studied first for the effects of domain size and grid resolution. Then two simulations with a total of 256 × 512 × 64 (∼ 8.4×10⁶) and 512 × 1024 × 128 (∼ 6.7×10⁷) grid points in the computational domain are carried out to investigate the key flow features. Visualization of the instantaneous flow field is used to investigate the complex fluid dynamics taking place in the near-wake region; of particular importance is the interaction between the large-scale spanwise, or Kármán, vortices and the small-scale quasi-streamwise vortices contained within the inflow boundary layer. Comparisons of turbulence statistics including the mean flow quantities are presented, as well as the pressure distributions over the trailing edge. A spectral analysis applied to the force coefficient in the wall normal direction shows that the main shedding frequency is characterized by a Strouhal number based on h of approximately 0.118. Finally, the turbulence kinetic energy budget is analysed. Received 4 March 1999 and accepted 27 October 2000     

AN ADAPTIVE FINITE ELEMENT METHOD FOR A TWO-EQUATION TURBULENCE MODEL IN WALL-BOUNDED FLOWS

This paper presents an adaptive finite element method for solving incompressible turbulent flows using a k–ϵ model of turbulence. Solutions are obtained in primitive variables using a highly accurate quadratic finite element on unstructured grids. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, pressure and turbulence variables. The efficiency and convergence rate of the methodology are evaluated by solving problems with known analytical solutions. The method is then applied to turbulent flow over a backward-facing step and predictions are compared with experimental measurements. © 1997 John Wiley & Sons, Ltd.     

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.     

Finite volume computation of the turbulent flow over a hill employing 2D or 3D non-orthogonal collocated grid systems

A general numerical method for the solution of the complete Reynolds-averaged Navier-Stokes equations for 2D or 3D flows is described. The method uses non-orthogonal co-ordinates, Cartesian velocity components and a pressure-velocity-coupling algorithm adequate for non-staggered grid systems. The capability of the method and the overall performance of the κ–? eddy viscosity model are demonstrated by calculations of 2D and 3D flow over a hill. Solution error estimations based on fine grids, e.g. 320 × 192 control volumes, together with comparisons with standard turbulence model modifications, low-Reynoldsnumber or streamline curvature effects, have allowed the investigation of model drawbacks in predicting turbulent flows over surface-mounted hills.     

Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes

The results of direct numerical simulation of turbulent flows of non-Newtonian pseudoplastic fluids in a straight pipe are presented. The data on the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances are obtained for steady turbulent flows at the Reynolds numbers of 10⁴ and 2×10⁴. As distinct from Newtonian fluid flows, the viscous shear stresses turn out to be significant even far from the wall. In power-law fluid flows the mechanism of the energy transport from axial to transverse component fluctuations is suppressed. It is shown that with decrease in the fluid index the turbulent transfer of the momentum and the velocity fluctuations between the wall layer and the flow core reduces, while the turbulent energy flux toward the wall increases. The earlier-proposed models for the average viscosity and the non-Newtonian one-point correlations are in good agreement with the data of direct numerical simulation.     

AN IMPROVED NAVIER–STOKES PROCEDURE FOR ANALYSIS OF TWO-DIMENSIONAL AEROFOILS IN LAMINAR AND TURBULENT FLOWS

An accurate and robust Navier–Stokes procedure to predict the complex flow about an aerofoil has been developed. Much improvement over existing methods is achieved in various aspects of the solution procedure. The computational grid generated by conformal mapping, which is not only orthogonal but aligned with the inviscid streamlines, keeps the equations simple and minimizes the error due to false diffusion. Formal second-order accuracy is ensured by employing the QUICK scheme for the convective derivatives in the full Navier–Stokes and turbulence transport equations. To treat the separated region properly and to better resolve the flow field in the wake, the two-layer k–ε turbulence model is incorporated. The onset of transition is triggered in a unique fashion to warrant the smooth transition to turbulent flow. Sample calculations for various aerofoil sections show that the prediction is improved substantially over those by existing methods. The details of the flow extending to the wake, such as the surface pressure distribution, C_Lmax, the velocity fields and the Reynolds stress profiles, are found to be in excellent agreement with the data. © 1997 John Wiley & Sons, Ltd.     

Developing turbulent flow in smooth pipes

A numerical and experimental investigation of steady incompressible developing turbulent flow in smooth pipes is presented. Finite difference techniques are used to solve simultaneously the vorticity transport and stream function equations utilising a modified form of the Van Driest effective viscosity model. The numerical solutions are verified experimentally using air as a working fluid at pipe Reynolds 1 × 10⁵, 2 × 10⁵ and 3 × 10⁵.     

Turbulent free surface flow simulation using a multilayer model

Based on the steady hydrodynamic equations, a multilayer (ML) model has been formulated for simulating turbulent flow in open channels. The model is imposed on a general curvilinear co-ordinate system with non-staggered finite volume discretization. The turbulent quantities in the model are described by the layer-averaged K-ε turbulence model with standard coefficients. Assuming a vertical hydrostatic pressure distribution, a depth correction scheme, originating in the Rhie and Chow approach for confined flows, is incorporated into the SIMPLE procedure to compute the water surface. Using the multilayer model, flows in a 180° channel bend, near a groin, and in straight open channels are computed. The results are compared with experimental data and with calculations of a depth-averaged model (DAV) having three-dimensional effect corrections. The comparisons show that the predictions of the ML model on mean flow values are in good agreement with the available data and are better than those of the DAV model. The vertical distribution of the turbulent energy dissipation rate is also shown to agree well with the open-channel measurements.     

Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems

A numerical solution for shallow-water flow is developed based on the unsteady Reynolds-averaged Navier–Stokes equations without the conventional assumption of hydrostatic pressure. Instead, the non-hydrostatic pressure component may be added in regions where its influence is significant, notably where bed slope is not small and separation in a vertical plane may occur or where the free-surface slope is not small. The equations are solved in the σ-co-ordinate system with semi-implicit time stepping and the eddy viscosity is calculated using the standard k–ϵ turbulence model. Conventionally, boundary conditions at the bed for shallow-water models only include vertical diffusion terms using wall functions, but here they are extended to include horizontal diffusion terms which can be significant when bed slope is not small. This is consistent with the inclusion of non-hydrostatic pressure. The model is applied to the 2D vertical plane flow of a current over a trench for which experimental data and other numerical results are available for comparison. Computations with and without non-hydrostatic pressure are compared for the same trench and for trenches with smaller side slopes, to test the range of validity of the conventional hydrostatic pressure assumption. The model is then applied to flow over a 2D mound and again the slope of the mound is reduced to assess the validity of the hydrostatic pressure assumption. © 1998 John Wiley & Sons, Ltd.     

An extended κ–ϵ finite element model

An extended κ–? model (to include low-Reynolds-number regions) employing weighting functions is presented. Wall functions for the near-wall zones are developed giving correct boundary values for the Shear stress and κ–?. A finite element model using a penalty formulation for incompressible turbulent flow is applied to Solve a flow between two plates. Results with mesh boundaries situated in the near-wall region and a: the wall are compared with measured values.     

Transition from vortex to wall driven turbulence production in the Taylor–Couette system with a rotating inner cylinder

Axisymmetrically stable turbulent Taylor vortices between two concentric cylinders are studied with respect to the transition from vortex to wall driven turbulent production. The outer cylinder is stationary and the inner cylinder rotates. A low Reynolds number turbulence model using the k‐ω formulation, facilitates an analysis of the velocity gradients in the Taylor–Couette flow. For a fixed inner radius, three radius ratios 0.734, 0.941 and 0.985 are employed to identify the Reynolds number range at which this transition occurs. At relatively low Reynolds numbers, turbulent production is shown to be dominated by the outflowing boundary of the Taylor vortex. As the Reynolds number increases, shear driven turbulence (due to the rotating cylinder) becomes the dominating factor. For relatively small gaps turbulent flow is shown to occur at Taylor numbers lower than previously reported. Copyright © 2002 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.