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.     

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.     

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.     

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.     

Simulation of vortex shedding past a square cylinder with different turbulence models

This paper presents the results of numerical simulations of vortex shedding past a free-standing square cylinder at Re_D=22 000, obtained with different turbulence models. Using wall functions, the standard k–ε model is compared with a modification suggested by Kato and Launder (Proc. 9th Symp. Turbulent Shear Flows, Kyoto, 10-4-1 (1993)). In addition, both versions are used in a two-layer approach, in which the flow close to the cylinder is computed with a locally more suitable one-equation turbulence model and only outside the viscous near-wall layer with the two mentioned high-Re model versions. To allow a comparison, the simulations are performed first using the same computational domain and boundary conditions as in previous investigations. Then results are presented that were obtained on a computational domain and with boundary conditions more suitable for a comparison with the experiments. © 1998 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.     

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.     

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.     

Evaluation of erosion in a two‐way coupled fluid–particle system

In this study, the effects of flow turbulence intensity, temperature, particle sizes and impinging velocity on erosion by particle impact are demonstrated numerically. Underlying turbulent flow on an Eulerian frame is described by the compressible Reynolds averaged Navier–Stokes equations with a RNG k–ε turbulence model. The particle trajectories and particle–wall interactions are evaluated by a Eulerian–Lagrangian approach in a two‐way coupling system. An erosion model considering material weight removal from surfaces is used to predict erosive wear. Computational validation against measured data is demonstrated satisfactorily. The analysis of erosion shows that the prevention of erosion is enhanced by increasing the effects of flow temperature and turbulence intensity and reducing particle inertial momentum. Copyright © 2001 John Wiley & Sons, Ltd.     

A multigrid method with higher‐order discretization schemes

The implementation of the multigrid method into the SIMPLE algorithm presents interesting aspects concerning the mass fluxes conservation on coarser grids, the k–ε turbulence model and the higher‐order discretization schemes. Higher‐order discretization schemes for the convection terms are increasingly used in order to guarantee accuracy in demanding engineering applications. However, when used in single‐grid algorithms, their convergence is considerably slower compared with the first‐order schemes. Unbounded higher‐order schemes offer maximum accuracy, but quite often they do not converge due to their oscillatory behaviour. This paper demonstrates the dual function of the multigrid method: reduction of CPU time and stabilization of the iterating procedure, making it possible to perform computations with the third‐order accurate QUICK scheme in all cases. The method is applied to the calculation of two‐ and three‐dimensional flows with or without turbulence modelling. The results show that the convergence rate of the present algorithm does not deteriorate when QUICK is used and that, if applied on complex engineering cases, large gains in computational time can be achieved. Copyright © 2001 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.     

Numerical and experimental investigation of turbulent flow in a rectangular duct

Details of the turbulent flow in a 1:8 aspect ratio rectangular duct at a Reynolds number of approximately 5800 were investigated both numerically and experimentally. The three-dimensional mean velocity field and the normal stresses were measured at a position 50 hydraulic diameters downstream from the inlet using laser doppler velocimetry (LDV). Numerical simulations were carried out for the same flow case assuming fully developed conditions by imposing cyclic boundary conditions in the main flow direction. The numerical approach was based on the finite volume technique with a non-staggered grid arrangement and the SIMPLEC algorithm. Results have been obtained with a linear and a non-linear (Speziale) k–ε model, combined with the Lam–Bremhorst damping functions for low Reynolds numbers. The secondary flow patterns, as well as the magnitude of the main flow and overall parameters predicted by the non-linear k–ε model, show good agreement with the experimental results. However, the simulations provide less anisotropy in the normal stresses than the measurements. Also, the magnitudes of the secondary velocities close to the duct corners are underestimated. © 1998 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.     

Boussinesq cut‐cell model for non‐linear wave interaction with coastal structures

Boussinesq models describe the phase‐resolved hydrodynamics of unbroken waves and wave‐induced currents in shallow coastal waters. Many enhanced versions of the Boussinesq equations are available in the literature, aiming to improve the representation of linear dispersion and non‐linearity. This paper describes the numerical solution of the extended Boussinesq equations derived by Madsen and Sørensen (Coastal Eng. 1992; 15 :371–388) on Cartesian cut‐cell grids, the aim being to model non‐linear wave interaction with coastal structures. An explicit second‐order MUSCL‐Hancock Godunov‐type finite volume scheme is used to solve the non‐linear and weakly dispersive Boussinesq‐type equations. Interface fluxes are evaluated using an HLLC approximate Riemann solver. A ghost‐cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The model is validated for solitary wave reflection from a vertical wall, diffraction of a solitary wave by a truncated barrier, and solitary wave scattering and diffraction from a vertical circular cylinder. In all cases, the model gives satisfactory predictions in comparison with the published analytical solutions and experimental measurements. Copyright © 2007 John Wiley & Sons, Ltd.     

Evaluation of one‐ and two‐equation low‐Re turbulence models. Part I—Axisymmetric separating and swirling flows

This first segment of the two‐part paper systematically examines several turbulence models in the context of three flows, namely a simple flat‐plate turbulent boundary layer, an axisymmetric separating flow, and a swirling flow. The test cases are chosen on the basis of availability of high‐quality and detailed experimental data. The tested turbulence models are integrated to solid surfaces and consist of: Rodi's two‐layer k–ε model, Chien's low‐Reynolds number k–ε model, Wilcox's k–ω model, Menter's two‐equation shear‐stress‐transport model, and the one‐equation model of Spalart and Allmaras. The objective of the study is to establish the prediction accuracy of these turbulence models with respect to axisymmetric separating flows, and flows of high streamline curvature. At the same time, the study establishes the minimum spatial resolution requirements for each of these turbulence closures, and identifies the proper low‐Mach‐number preconditioning and artificial diffusion settings of a Reynolds‐averaged Navier–Stokes algorithm for optimum rate of convergence and minimum adverse impact on prediction accuracy. Copyright © 2003 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.     

Finite element implementation of two‐equation and algebraic stress turbulence models for steady incompressible flows

The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε turbulence model, or any other two‐equation model. The finite element discretization is based on the SUPG method together with a discontinuity capturing technique to deal with sharp internal and boundary layers. The iterative strategy consists of several nested loops, the outermost being the linearization of the Navier–Stokes equations. The basic k–ε model is used for the implementation of an algebraic stress model that is able to account for the effects of rotation. Some numerical examples are presented in order to show the performance of the proposed scheme for simulating directly steady flows, without the need of reaching the steady state through a transient evolution. Copyright © 1999 John Wiley & Sons, Ltd.     

SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITH k–ε TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow- water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard k–ε turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and ε. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substi- tuting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver. Evaluation of horizontal gradients in the σ-co-ordinate system requires high-order derivatives which can cause spurious flows and this is avoided by obtaining these gradients in real space. In this paper the method is applied to parallel oscillatory (tidal) flow in deep and shallow water and compared with field measurements. It is then applied to current flow about a conical island of small side slope where vortex shedding occurs and velocities are compared with data from the laboratory. Computed concentration distributions are also compared with dye visualization and an example of the influence of temperature on plume dispersion is presented. © 1997 John Wiley & Sons, Ltd.     

Reynolds‐stress modelling of M = 2.25 shock‐wave/turbulent boundary‐layer interaction

M = 2.25 shock‐wave/turbulent‐boundary‐layer interactions over a compression ramp for several angles (8, 13 and 18°) at Reynolds‐number Re=7 × 10³ were simulated with three low‐Reynolds second‐moment closures and a linear low‐Reynolds standard k–ε model. A detailed assessment of the turbulence closures by comparison with both mean‐flow and turbulent experimental quantities is presented. The Reynolds‐stress model which is wall‐topology free and which uses an optimized redistribution closure, is in good agreement with experimental data both for wall‐pressure and mean‐velocity profiles. Detailed analysis of three components of the Reynolds‐stress tensor (comparison with measurements and transport‐equation budgets) provides a critical evaluation of full Reynolds‐stress models for the separated supersonic compression ramp. The discrepancy observed in the shock‐wave foot region, between computations and measurements for the Reynolds‐stresses profiles, could be explained by considering the experimental shock‐wave oscillation and directions for future modelling work are indicated. Copyright © 2007 John Wiley & Sons, Ltd.     

Computational study on turbulent flows around modern tanker hull forms

To enhance the applicability of CFD techniques to the hull form design of modern commercial ships, an efficient and robust numerical method for turbulent flow calculation is developed. The preprocessor is composed of hull form presentation, surface mesh generation, and field grid generation. The finite‐volume method is utilised to solve the Reynolds‐averaged Navier–Stokes equations. Three k–? turbulence models, i.e., the standard k–? model (SKE), the RNG‐based k–? model (RNG), and the realisable k–? model (RKE), are evaluated to investigate the difference caused by the turbulence model. The developed numerical method is applied to two practical VLCC hull forms with the same forebody and the slightly different afterbodies in order to find out whether CFD can capture the difference of stern flow due to hull form variation. It is found that RKE successfully predict the strength and location of bilge vortex, while SKE and RNG fail. However, all three models can provide the right information on the nominal wake difference between two hull forms. Copyright © 2002 John Wiley & Sons, Ltd.