Two‐dimensional prediction of time dependent,turbulent flow around a square cylinder confined in a channel

This paper presents two‐dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder–Sharma low‐Reynolds number k–ε model and a non‐linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×10⁴. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid‐point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third‐order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k–ε does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second‐order Crank–Nicolson scheme. The unsteady computations using the linear and non‐linear k–ε models significantly improve the velocity field predictions. However, the linear k–ε shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non‐linear k–ε model brings the two‐dimensional unsteady predictions of the time‐averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 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.     

Experimental validation of two depth‐averaged turbulence models

A finite volume turbulence model for the resolution of the two‐dimensional shallow water equations with turbulent term is presented. After making a finite volume discretization of the depth‐averaged k–ε equations in conservative form, the q–r equations, that give stability to the process, are obtained. Wall and inlet boundary conditions for the turbulent equations and wall conditions for the hydrodynamic equations are discussed. A comparison between the k–ε and q–r models and some experimental results is made. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 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.     

Numerical prediction of a turbulent curved wake and comparison with experimental data

Numerical studies of the curved wake of a NACA 0012 airfoil of chord length 0.150 m are presented. The airfoil is placed in air at 10 m/s in the straight section of a duct of 0.457 m × 0.457 m cross‐section followed by a 90° bend with a mean radius‐to‐height ratio of 1.17. The trailing edge is located at one chord length upstream of the bend entry plane. The authors' own measurements are used to define the boundary conditions and for comparison with the predicted results. The numerical models are based on the time‐averaged, three‐dimensional conservation equations of fluid flow, incorporating the k–ε, RNG k–ε, realizable k–ε and the Reynolds stress turbulence models. The results show that the models are capable of predicting the effects of curvature on the wake development. However, quantitative differences between prediction and experiment exist. The results obtained using the Reynolds stress model show better agreement with the experimental data, compared with the k–ε based models, but not consistently for all parameters. There are also better predictions by the RNG k–ε and realizable k–ε models compared with the standard k–ε model. The predicted results using the RNG k–ε are closer to experimental data than the realizable k–ε. Copyright © 2005 John Wiley & Sons, Ltd.     

An algorithm for rotating axisymmetric flows: model,validation and industrial applications

The study of axisymmetric flows is of interest not only from an academic point of view, due to the existence of exact solutions of Navier–Stokes equations, but also from an industrial point of view, since these kind of flows are frequently found in several applications. In the present work the development and implementation of a finite element algorithm to solve Navier–Stokes equations with axisymmetric geometry and boundary conditions is presented. Such algorithm allows the simulation of flows with tangential velocity, including free surface flows, for both laminar and turbulent conditions. Pseudo‐concentration technique is used to model the free surface (or the interface between two fluids) and the k–ε model is employed to take into account turbulent effects. The finite element model is validated by comparisons with analytical solutions of Navier–Stokes equations and experimental measurements. Two different industrial applications are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow

This paper presents a manufactured solution (MS), resembling a two-dimensional, steady, wall-bounded, incompressible, turbulent flow for RANS codes verification. The specified flow field satisfies mass conservation, but requires additional source terms in the momentum equations. To also allow verification of the correct implementation of the turbulence models transport equations, the proposed MS exhibits most features of a true near-wall turbulent flow. The model is suited for testing six eddy-viscosity turbulence models: the one-equation models of Spalart and Allmaras and Menter; the standard two-equation k–ε model and the low-Reynolds version proposed by Chien; the TNT and BSL versions of the k–ω model.     

Verification testing in computational fluid dynamics: an example using Reynolds‐averaged Navier–Stokes methods for two‐dimensional flow in the near wake of a circular cylinder

Verification testing was performed for various Reynolds‐averaged Navier–Stokes methods for uniform flow past a circular cylinder at Re= 5232. The standard and renormalized group (RNG) versions of the k–ε method were examined, along with the Boussinesq, Speziale and Launder constitutive relationships. Wind tunnel experiments for flow past a circular cylinder were also performed to obtain a comparative data set. Preliminary studies demonstrate poor convergence for the Speziale relationship. Verification testing with the standard and RNG k–ε models suggests that the simulations exhibit global monotonic convergence for the Boussinesq models. However, the global order of accuracy of the methods was much lower than the expected order of accuracy of 2. For this reason, pointwise convergence ratios and orders of accuracy were computed to show that not all sampling locations had converged (standard k–ε model: 19% failed to converge; RNG k–ε model: 14% failed to converge). When the non‐convergent points were removed from consideration, the average orders of accuracy are closer to the expected value (standard k–ε model: 1.41; RNG k–ε model: 1.27). Poor iterative and global grid convergence was found for the RNG k–ε/Launder model. The standard and RNG k–ε models with the Boussinesq relationship were compared with experimental data and yielded results significantly different from the experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Cubic eddy‐viscosity turbulence models for strongly swirling confined flows with variable density

An investigation on the predictive performance of cubic eddy‐viscosity turbulence models for strongly swirling confined flows with variable density is presented. Comparisons of the prediction with the experiments show some improvements of cubic models over the linear k–ε model. The linear k–ε model does not contain any mechanism to represent the interaction of swirl and density variation and as a consequence it performs poorly. With appropriate modelling, two‐equation cubic turbulence models can capture the subcritical nature of the flow, represent the azimuthal velocity profiles of combined forced‐free vortex motion, and predict the combined effects of swirl and density variation fairly well. However, the calibration of model coefficients is still a topic of investigation. Further amendments are also needed for the equations of k and ε to take into account the effects of swirl and density gradients correctly. Copyright © 2004 John Wiley & Sons, Ltd.     

AN ALGEBRAIC STRESS FINITE ELEMENT MODEL OF TURBULENT FLOW

This paper describes a finite element numerical model for the simulation of both steady and truly transient turbulent flow in two dimensions. All elements of the model and computational approach were chosen, however, for ease of applicability in the future to fully three-dimensional flows. The turbulent mean flow is described by the Reynolds- averaged Navier–Stokes equations. The well-known two-equation K–ε model is the base for the representation of turbulence quantities. From three candidate algebraic stress models, Rodi's model was chosen for implementation after preliminary tests on turbulent channel flow. The scheme was then tested at length on flow past a backward-facing step and flow past a box. Comparisons were made with the computed and experimental results of other investigators. For the backward-facing step problem the model appears to equal or improve upon the accuracy of prediction s of earlier finite element codes. The frequency of vortex shedding from the corners of the box in terms of the Strouhal number is predicted well. © 1997 by 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.     

A fully coupled Navier–Stokes solver for calculation of turbulent incompressible free surface flow past a ship hull

This paper deals with the calculation of free surface flow of viscous incompressible fluid around the hull of a boat moving with rectilinear motion. An original method used to avoid a large part of the theoretical problems connected with free surface boundary conditions in three‐dimensional Navier–Stokes–Reynolds equations is proposed here. The linearised system of convective equations for velocities, pressure and free surface elevation unknowns is discretised by finite differences and two methods to solve the fully coupled resulting matrix are presented here. The non‐linear convergence of fully coupled algorithm is compared with the velocity–pressure weakly coupled algorithm SIMPLER. Turbulence is taken into account through Reynolds decomposition and k–ε or k–ω model to close the equations. These two models are implemented without wall function and numerical calculations are performed up to the viscous sub‐layer. Numerical results and comparisons with experiments are presented on the Series 60 CB=0.60 ship model for a Reynolds number Rn=4.5×10⁶ and a Froude number Fn=0.316. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical modeling of cohesive sediments dynamics in estuaries: Part I—Description of the model and simulations in the Po River Estuary

The present work contributes to the numerical modeling of complex turbulent multiphasic fluid flows occurring in estuarine channels. This research finds its motivation in the increasing need for efficient management of estuaries by taking into account the complex turbulent stratified flows encountered in estuaries and costal zones. A time‐dependent, 3D finite element model of suspended sediment transport taking into account the effects of cohesiveness between sediments is presented. The model estuary is the forced time‐dependent winds, time elevation at open boundaries and river discharge. To cope with the stiffness problems a decoupling method is employed to solve the shallow‐water equations of mass conservation, momentum and suspended sediment transport with the conventional hydrostatic pressure. The decoupling method partitions a time step into three subcycles according to the physical phenomena. In the first sub‐cycle the pure hydrodynamics including the k–ε turbulence model is solved, followed by the advection–diffusion equations for pollutants (salinity, temperature, suspended sediment concentration, (SSC)), and finally the bed evolution is solved. The model uses a mass‐preserving method based on the so‐called Raviart–Thomas finite element on the unstructured mesh in the horizontal plane, while the multi‐layers system is adopted in vertical with the conventional conforming finite element method, with the advantage that the lowermost and uppermost layers of variable height allow a faithful representation of the time‐varying bed and free surface, respectively. The model has been applied to investigate the SSC and seabed evolution in Po River Estuary (PRE) in Italy. The computed results mimic the field data well. Copyright © 2007 John Wiley & Sons, Ltd.     

A PRESSURE-BASED ALGORITHM FOR HIGH-SPEED TURBOMACHINERY FLOWS

The steady state Navier–Stokes equations are solved in transonic flows using an elliptic formulation. A segregated solution algorithm is established in which the pressure correction equation is utilized to enforce the divergence-free mass flux constraint. The momentum equations are solved in terms of the primitive variables, while the pressure correction field is used to update both the convecting mass flux components and the pressure itself. The velocity components are deduced from the corrected mass fluxes on the basis of an upwind-biased density, which is a mechanism capable of overcoming the ellipticity of the system of equations, in the transonic flow regime. An incomplete LU decomposition is used for the solution of the transport-type equations and a globally minimized residual method resolves the pressure correction equation. Turbulence is resolved through the k–ε model. Dealing with turbomachinery applications, results are presented in two-dimensional compressor and turbine cascades under design and off-design conditions. © 1997 John Wiley & Sons, Ltd.     

Prediction of developing turbulent flow in 90°‐curved ducts using linear and non‐linear low‐Re k–ε models

This paper reports the outcome of applying two different low‐Reynolds‐number eddy‐viscosity models to resolve the complex three‐dimensional motion that arises in turbulent flows in ducts with 90^° bends. For the modelling of turbulence, the Launder and Sharma low‐Re k–ε model and a recently produced variant of the cubic non‐linear low‐Re k–ε model have been employed. In this paper, developing turbulent flow through two different 90^° bends is examined: a square bend, and a rectangular bend with an aspect ratio of 6. The numerical results indicate that for the bend of square cross‐section the curvature induces a strong secondary flow, while for the rectangular cross‐section the secondary motion is confined to the corner regions. For both curved ducts, the secondary motion persists downstream of the bend and eventually slowly disappears. For the bend of square cross‐section, comparisons indicate that both turbulence models can produce reasonable predictions. For the bend of rectangular cross‐section, for which a wider range of data is available, while both turbulence models produce satisfactory predictions of the mean flow field, the non‐linear k–ε model returns superior predictions of the turbulence field and also of the pressure and friction coefficients. Copyright © 2006 John Wiley & Sons, Ltd.     

PREDICTION OF AEROFOIL WAKE SUBJECTED TO THE EFFECTS OF CURVATURE AND PRESSURE GRADIENT

Experimental data on the development of wakes in a straight duct, a curved duct, a curved diffuser and a straight diffuser are compared with computations based on a finite volume scheme incorporating the k– ε model of turbulence. The results show that the computations based on the standard k–ε model are able to satisfactorily capture only the mean velocity profiles. To improve the predictions, several modifications to the model are tried out. Close agreement between experiment and computation as regards the velocity profiles, wake parameters and profiles of the turbulent kinetic energy k and Reynolds shear stress ⌅{uv} is obtained when modification to the model constant C_μ, based on the curvature parameter and the ratio of the production of turbulent kinetic energy to its rate of dissipation, is incorporated. The modified model is also able to capture the asymmetry in the profiles of k and ⌅{uv} caused by the curvature and its enhancement due to the additional presence of an adverse pressure gradient.