An eddy viscosity model with near‐wall modifications

An extended version of the isotropic k–ε model is proposed that accounts for the distinct effects of low‐Reynolds number (LRN) and wall proximity. It incorporates a near‐wall correction term to amplify the level of dissipation in nonequilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and the Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Comparisons indicate that the present model is a significant improvement over the standard eddy viscosity formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Low Reynolds number k–ε model for near‐wall flow

A wall‐distance free k–ε turbulence model is developed that accounts for the near‐wall and low Reynolds number effects emanating from the physical requirements. The model coefficients/functions depend non‐linearly on both the strain rate and vorticity invariants. Included diffusion terms and modified C_ε(1,2) coefficients amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

Large eddy simulation using a dynamic mixing length subgrid‐scale model

A novel dynamic mixing length (DML) subgrid‐scale model for large eddy simulations is proposed in this work to improve the cutoff length of the Smagorinsky model. The characteristic mixing length (or the characteristic wave number) is dynamically estimated for the subgrid‐scale fluctuation of turbulence by the cutoff wave‐number, k_c, and the dissipation wave‐number, k_d. The dissipation wave number is derived from the kinetic energy spectrum equation and the dissipation spectrum equation. To prove the promise of the DML model, this model is used to simulate the lid‐driven cubical cavity with max‐velocity‐based Reynolds numbers 8850 and 12,000, the channel flows with friction‐velocity‐based Reynolds numbers 180, 395, 590, and 950, and the turbulent flow past a square cylinder at the higher Reynolds number 21,400, respectively, compared with the Smagorinsky model and Germano et al.'s dynamic Smagorinsky model. Different numerical experiments with different Reynolds numbers show that the DML model can be used in simulations of flows with a wide range of Reynolds numbers without the occurrence of singular values. The DML model can alleviate the dissipation of the Smagorinsky model without the loss of its robustness. The DML model shows some advantages over Germano et al.'s dynamic Smagorinsky model in its high stability and simplicity of calculation because the coefficient of the DML model always stays positive. The characteristic mixing length in the DML model reflects the subgrid‐scale fluctuation of turbulence in nature and thus the characteristic mixing length has a spatial and temporal distribution in turbulent flow. Copyright © 2011 John Wiley & Sons, Ltd.     

Near‐wall turbulence modelling with enhanced dissipation

A low‐Reynolds number k–ε turbulence model is proposed that incorporates diffusion terms and modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional k–ε model, it requires no wall function/distance parameter that bridges the near‐wall integration. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2003 John Wiley & Sons, Ltd.     

A new second‐moment closure approach for turbulent swirling confined flows

An improved anisotropic model for the dissipation rate—ε—of the turbulent kinetic energy (k), to be used together with a non‐linear pressure‐strain correlations model, is proposed. Experimental data from the open literature for two confined turbulent swirling flows are used to assess the performance of the proposed model in comparison to the standard ε transport equation and to a linear approach to model the pressure‐strain term that appears in the exact equations for the Reynolds‐stress tensor. For the less strongly swirling flow the predictions show much more sensitivity to the εtransport equation than to the pressure‐strain model. In opposition, for the more strongly swirling flow, the results show that the predictions are much sensitive to the pressure‐strain model. Nevertheless, the improved εtransport equation together with the non‐linear pressure strain model yield predictions in good agreement with experiments in both studied cases. Copyright © 2003 John Wiley & Sons, Ltd.     

On the performance of sudden‐expansion pipe without/with cross‐flow injection: experimental and numerical studies

The paper explores the possibilities that different turbulence closures offer, for in‐depth analysis of a complex flow. The case under investigation is steady, turbulent flow in a pipe with sudden expansion without/with normal‐to‐wall injection through jets. This is a typical geometry where generation of turbulence energy takes place, due to sudden change in boundary conditions. This study is aimed at investigating the capability of a developed computational program by the present authors with three different turbulence models to calculate the mean flow variables. Three two‐equation models are implemented, namely the standard linear k ? ε model, the low Reynolds number k ? ε model and the cubic nonlinear eddy viscosity (NLEV) k ? ε model. The performance of the chosen turbulence models is investigated with regard to the available data in the literature including velocity profiles, turbulent kinetic energy and reattachment position in a pipe expansion. In order to further assess the reliability of the turbulence models, an experimental program was conducted by the present authors also at the fluid mechanics laboratory of Menoufiya University. Preliminary measurements, including the surface pressure along the two walls of the expansion pipe and the pressure drop without and with the presence of different arrangements of wall jets produced by symmetrical or asymmetrical fluid cross‐flow injection, are introduced. The results of the present studies demonstrate the superiority of the cubic NLEV k ? ε model in predicting the flow characteristics over the entire domain. The simple low Reynolds number k ? ε model also gives good prediction, especially when the reattachment point is concerned. The evaluation of the reattachment point and the pressure‐loss coefficient is numerically addressed in the paper using the cubic NLEV k ? ε model. The results show that the injection location can control the performance of the pipe‐expansion system. It is concluded that the introduction of flow injection can increase the energy loss in the pipe expansion. The near‐field turbulence structure is also considered in the present study and it is noticed that the turbulence level is strongly affected by the cross‐flow injection and the jet location. Copyright © 2011 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.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

A numerical study of turbulent flow over a two‐dimensional hill

Calculations of mean velocities and Reynolds stresses are reported for the recirculating flow established in the wake of two‐dimensional polynomial‐shaped obstacles that are symmetrical about a vertical axis and mounted in the water channel downstream of a fully developed channel flow for Re=6×10⁴. The study involves calculations of mean and fluctuating flow properties in the streamwise and spanwise directions and include comparisons with experimental data [Almeida GP, Durão DFG, Heitor MV. Wake flows behind two‐dimensional model hills. Experimental Thermal and Fluid Science 1993; 7: 87–101] for flow around a single obstacle with data resulting from the interaction of consecutive obstacles, using two versions of the low‐Reynolds number differential second‐moment (DSM) closure model. The results include analysis of the turbulent stresses in local flow co‐ordinates and reveal flow structure qualitatively similar to that found in other turbulent flows with a reattachment zone. It is found that the standard isotropization of production model (IPM), based on that proposed by Gibson and Launder [Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics 1978; 86(3): 191–511], with the incorporation of the wall reflection model of Craft and Launder [New wall‐reflection model applied to the turbulent impinging jet. AIAA Journal 1992; 32(12): 2970–2972] predicts the mean velocities quite well, but underestimates the size of the recirculation region and turbulent quantities in the shear layer. These inadequacies are circumvented by adopting a new cubic Reynolds stress closure scheme based on that more recently developed by Craft and Launder [A Reynolds stress closure designed for complex geometries. International Journal of Heat and Fluid Flow 1996; 17: 245–254] which satisfies the two component limit (TCL) of turbulence. In this model the geometry‐specific quantities, such as the wall‐normal vector or wall distance, are replaced by invariant dimensionless gradient indicators. Also, the model captures the diverse behaviour of the different components of the stress dissipation, ε_ij, near the wall and uses a novel decomposition for the fluctuating pressure terms. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

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.     

A new hybrid turbulence modelling strategy for industrial CFD

This paper presents a new strategy for turbulence model employment with emphasis on the model's applicability for industrial computational fluid dynamics (CFD). In the hybrid modelling strategy proposed here, the Reynolds stress and mean rate of strain tensors are coupled via Boussinesq's formula as in the standard k–εmodel. However, the turbulent kinetic energy is calculated as the sum of the normal Reynolds‐stress components, representing the solutions of the appropriate transport equations. The equations governing the Reynolds‐stress tensor and dissipation rate have been solved in the framework of a ‘background’ second‐moment closure model. Furthermore, the structure parameter C‐µ has been re‐calculated from a newly proposed functional dependency rather than kept constant. This new definition of C‐µ has been assessed by using direct numerical simulation (DNS) results of several generic flow configurations featuring different phenomena such as separation, reattachment and rotation. Comparisons show a large departure of C‐µ from the commonly used value of 0.09. The model proposed is computationally validated in a number of well‐proven fluid flow benchmarks, e.g. backward‐facing step, 180° turn‐around duct, rotating pipe, impinging jet and three‐dimensional (3D) Ahmed body. The obtained results confirm that the present hybrid model delivers a robust solution procedure while preserving most of the physical advantages of the Reynolds‐stress model over simple k–εmodels. A low Reynolds number version of the hybrid model is also proposed and discussed. Copyright © 2003 John Wiley & Sons, Ltd.     

Shear‐improved Smagorinsky modeling of turbulent channel flow using generalized Lattice Boltzmann equation

Generalized Lattice Boltzmann equation (GLBE) was used for computation of turbulent channel flow for which large eddy simulation (LES) was employed as a turbulence model. The subgrid‐scale turbulence effects were simulated through a shear‐improved Smagorinsky model (SISM), which is capable of predicting turbulent near wall region accurately without any wall function. Computations were done for a relatively coarse grid with shear Reynolds number of 180 in a parallelized code. Good numerical stability was observed for this computational framework. The results of mean velocity distribution across the channel showed good correspondence with direct numerical simulation (DNS) data. Negligible discrepancies were observed between the present computations and those reported from DNS for the computed turbulent statistics. Three‐dimensional instantaneous vorticity contours showed complex vortical structures that appeared in such flow geometries. It was concluded that such a framework is capable of predicting accurate results for turbulent channel flow without adding significant complications and the computational cost to the standard Smagorinsky model. As this modeling was entirely local in space it was therefore adapted for parallelization. Copyright © 2010 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.     

Large eddy simulation of homogeneous shear flows with several subgrid‐scale models

In this article, large eddy simulation is used to simulate homogeneous shear flows. The spatial discretization is accomplished by the spectral collocation method and a third‐order Runge–Kutta method is used to integrate the time‐dependent terms. For the estimation of the subgrid‐scale stress tensor, the Smagorinsky model, the dynamic model, the scale‐similarity model and the mixed model are used. Their predicting performance for homogeneous shear flow is compared accordingly. The initial Reynolds number varies from 33 to 99 and the initial shear number is 2. Evolution of the turbulent kinetic energy, the growth rate, the anisotropy component and the subgrid‐scale dissipation rate is presented. In addition, the performance of several filters is examined. Copyright © 2005 John Wiley & Sons, Ltd.     

Turbulence in a three‐dimensional wall‐bounded shear flow

A new turbulent flow with distinct three‐dimensional characteristics has been designed in order to study the impact of mean‐flow skewing on the turbulent coherent vortices and Reynolds‐averaged statistics. The skewing of a unidirectional plane Couette flow was achieved by means of a spanwise pressure gradient. Direct numerical simulations of the statistically steady Couette–Poiseuille flow enabled in‐depth explorations of the turbulence field in the skewed flow. The imposition of a modest spanwise gradient turned the mean flow about 8° away from the original Couette flow direction and this turning angle remained nearly the same over the entire cross section. Nevertheless, a substantial non‐alignment between the turbulent shear stress angle and the mean velocity gradient angle was observed. The structure parameter turned out to slightly exceed that in the pure Couette flow, contrary to the observations made in some other three‐dimensional shear flows. Coherent flow structures, which are known to be associated with the Reynolds shear stress in near‐wall regions, were identified by the λ₂‐criterion. Instantaneous and ensemble‐averaged vortices resembled those found in the unidirectional Couette flow. In the skewed flow, however, the vortex structures were turned to align with the local mean‐flow direction. The conventional symmetry between Case 1 and Case 2 vortices was broken due to the mean‐flow three‐dimensionality. The turning of the coherent vortices and the accompanying symmetry‐breaking gave rise to secondary and tertiary turbulent shear stress components. By averaging the already ensemble‐averaged shear stresses associated with Case 1 and Case 2 vortices in the homogeneous directions, a direct link between the educed near‐wall structures and the Reynolds‐averaged turbulent stresses was established. These observations provide evidence in support of the hypothesis that the structural model proposed for two‐dimensional turbulent boundary layers remains valid also in flows with moderate mean three‐dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessment of two‐equation turbulence modelling for high Reynolds number hydrofoil flows

This paper presents an evaluation of the capability of turbulence models available in the commercial CFD code FLUENT 6.0 for their application to hydrofoil turbulent boundary layer separation flow at high Reynolds numbers. Four widely applied two‐equation RANS turbulence models were assessed through comparison with experimental data at Reynolds numbers of 8.284×106 and 1.657×107. They were the standard k–εmodel, the realizable k–εmodel, the standard k–ωmodel and the shear‐stress‐transport (SST) k–ωmodel. It has found that the realizable k–εturbulence model used with enhanced wall functions and near‐wall modelling techniques, consistently provides superior performance in predicting the flow characteristics around the hydrofoil. Copyright © 2004 John Wiley & Sons, Ltd.     

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.