Approximate Wall Boundary Conditions in the Large-Eddy Simulation of High Reynolds Number Flow

The near-wall regions of high Reynolds numbers turbulent flows must be modelled to treat many practical engineering and aeronautical applications. In this review we examine results from simulations of both attached and separated flows on coarse grids in which the near-wall regions are not resolved and are instead represented by approximate wall boundary conditions. The simulations use the dynamic Smagorinsky subgrid-scale model and a second-order finite-difference method. Typical results are found to be mixed, with acceptable results found in many cases in the core of the flow far from the walls, provided there is adequate numerical resolution, but with poorer results generally found near the wall. Deficiencies in this approach are caused in part by both inaccuracies in subgrid-scale modelling and numerical errors in the low-order finite-difference method on coarse near-wall grids, which should be taken into account when constructing models and performing large-eddy simulation on coarse grids. A promising new method for developing wall models from optimal control theory is also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Prediction of turbulent wall shear flows directly from wall

The fully elliptic Reynolds-averaged Navier–Stokes equations have been used together with Lam and Bremhorst's low-Reynolds-number model, Chen and Patel's two-layer model and a two-point wall function method incorporated into the standard k-? model to predict channel flows and a backward-facig step flow. These flows enable the evaluation of the performance of different near-wall treatments in flows involving streamwise and normal pressure gradients, flows with separation and flows with non-equilibrium turbulence characteristics. Direct numerical simulation (DNS) of a channel flow with Re =3200 further provides the detailed budgets of each modelling term of the k and ?-transport equations. Comparison of model results with DNS data to evaluate the performance of each modelling term is also made in the present study. It is concluded that the low-Reynolds-number model has wider applicability and performs better than the two-layer model and wall function approaches. Comparison with DNS data further shows that large discrepancies exist between the DNS budgets and the modelled production and destruction terms of the ? equation. However, for simple channel flow the discrepancies are similar in magnitude but opposite in sign, so they are cancelled by each other. This may explain why, even when employing such an inaccurately modelled ?-equation, one can still predict satisfactorily some simple turbulent flows.     

A Dissipation Rate Equation for Low-Reynolds-Number and Near-Wall Turbulence

Two-equation turbulence models are usually formulated for specific flow types and are seldom validated against a variety of flows to account for near-wall and low-Reynolds-number effects simultaneously. In addition to low-Reynolds-number effects, near-wall flows also experience wall blocking, which is absent in free flows. Consequently, near-wall modifications to two-equation models could be quite different from low-Reynolds-number corrections. Besides, it is known that existing two-equation models perform poorly when used to calculate plane wall jets and two-dimensional backstep flows. These problems could be traced to the modeling of the dissipation rate equation. In this paper an attempt is made to improve the modeling of the dissipation rate equation so that it could successfully predict both free and wall-bounded shear flows including plane wall jets and backstep flows. The predictions are compared with experimental and direct numerical simulation data whenever available. Most of the data used are obtained at low Reynolds numbers. Good correlation with data is obtained. Therefore, for the first time, a model capable of correctly predicting free and wall-bounded shear flows, backstep flows, and plane wall jets is available. Received: 12 December 1995 and accepted 12 November 1996     

A numerical method for steady and nonisothermal viscoelastic fluid flow for high Deborah and Péclet numbers

A combined finite element/streamline integration method is presented for nonisothermal flows of viscoelastic fluids. The attention is focused on some characteristic problems that arise for numerical simulation of flows with high Deborah and Péclet numbers. The two most important problems to handle are the choice of an outflow boundary condition for not completely developed flow and the treatment of the dissipative term in the temperature equation. The ability of the numerical method to handle high Deborah and Péclet numbers will be demonstrated on a contraction flow of an LDPE melt with isotropic and anisotropic heat conductivity. The influence of anisotropic heat conduction and the difference between the stress work and mechanical dissipation will be discussed for contraction flows. Received:  4 February 1997 Accepted: 23 October 1997     

Progress in subgrid-scale transport modelling for continuous hybrid non-zonal RANS/LES simulations

The partially integrated transport modelling (PITM) method can be viewed as a continuous approach for hybrid RANS/LES modelling allowing seamless coupling between the RANS and the LES regions. The subgrid turbulence quantities are thus calculated from spectral equations depending on the varying spectral cutoff location [Schiestel, R., Dejoan, A., 2005. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theoretical and Computational Fluid Dynamics 18, 443–468; Chaouat, B., Schiestel, R., 2005. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Physics of Fluids, 17 (6)] The PITM method can be applied to almost all statistical models to derive its hybrid LES counterpart. In the present work, the PITM version based on the transport equations for the turbulent Reynolds stresses together with the dissipation transport rate equation is now developed in a general formulation based on a new accurate energy spectrum function E(κ) valid in both large and small eddy ranges that allows to calibrate more precisely the c_sgs2 function involved in the subgrid dissipation rate _sgs transport equation. The model is also proposed here in an extended form which remains valid in low Reynolds number turbulent flows. This is achieved by considering a characteristic turbulence length-scale based on the total turbulent energy and the total dissipation rate taking into account the subgrid and resolved parts of the dissipation rate. These improvements allow to consider a large range of flows including various free flows as well as bounded flows. The present model is first tested on the decay of homogeneous isotropic turbulence by referring to the well known experiment of Comte-Bellot and Corrsin. Then, initial perturbed spectra E(κ) with a peak or a defect of energy are considered for analysing the model capabilities in strong non-equilibrium flow situations. The second test case is the classical fully turbulent channel flow that allows to assess the performance of the model in non-homogeneous flows characterised by important anisotropy effects. Different simulations are performed on coarse and refined meshes for checking the grid independence of solutions as well as the consistency of the subgrid-scale model when the filter width is changed. A special attention is devoted to the sharing out of the energy between the subgrid-scales and the resolved scales. Both the mean velocity and the turbulent stress computations are compared with data from direct numerical simulations.     

Non-Equilibrium Mixing of Turbulence Scales Using a Continuous Hybrid RANS/LES Approach: Application to the Shearless Mixing Layer

A new subgrid-scale (SGS) model based on partially integrated transport method (PITM) is applied to the case of a turbulent spectral non-equilibrium flow created by the mixing of two turbulence fields of differing scales: the shearless mixing layer. The method can be viewed as a continuous hybrid RANS/LES approach. In this model the SGS length scale is no longer given by the size of the discretization step, but is dynamically estimated using an additional transport equation for the dissipation rate. The results are compared to those corresponding to the classical model of Smagorinsky and to the experimental data of Veeravalli and Warhaft. A method for creating an anisotropic analytical pseudo-random field for inflow conditions is also proposed. This approach based on subgrid-scale transport modelling combined with anisotropic inlet conditions gives better results for the prediction of the shearless mixing layer.     

Predicting Buoyant Shear Flows Using Anisotropic Dissipation Rate Models

This paper examines the modeling of two-dimensional homogeneous stratified turbulent shear flows using the Reynolds-stress and Reynolds-heat-flux equations. Several closure models have been investigated; the emphasis is placed on assessing the effect of modeling the dissipation rate tensor in the Reynolds-stress equation. Three different approaches are considered; one is an isotropic approach while the other two are anisotropic approaches. The isotropic approach is based on Kolmogorov's hypothesis and a dissipation rate equation modified to account for vortex stretching. One of the anisotropic approaches is based on an algebraic representation of the dissipation rate tensor, while another relies on solving a modeled transport equation for this tensor. In addition, within the former anisotropic approach, two different algebraic representations are examined; one is a function of the Reynolds-stress anisotropy tensor, and the other is a function of the mean velocity gradients. The performance of these closure models is evaluated against experimental and direct numerical simulation data of pure shear flows, pure buoyant flows and buoyant shear flows. Calculations have been carried out over a range of Richardson numbers (Ri) and two different Prandtl numbers (Pr); thus the effect of Pr on the development of counter-gradient heat flux in a stratified shear flow can be assessed. At low Ri, the isotropic model performs well in the predictions of stratified shear flows; however, its performance deteriorates as Ri increases. At high Ri, the transport equation model for the dissipation rate tensor gives the best result. Furthermore, the results also lend credence to the algebraic dissipation rate model based on the Reynolds stress anisotropy tensor. Finally, it is found that Pr has an effect on the development of counter-gradient heat flux. The calculations show that, under the action of shear, counter-gradient heat flux does not occur even at Ri = 1 in an air flow. This revised version was published online in July 2006 with corrections to the Cover Date.     

Assessment of ghost-cell based cut-cell method for large-eddy simulations of compressible flows at high Reynolds number

Large-eddy simulations (LES) of high Reynolds number flows are performed using a non-body conformal method in conjunction with a wall model. We use a simple wall function to model the wall-shear stress and the truncation error of the numerical discretization to model the sub-grid scale turbulence (implicit LES), although these can be easily replaced if necessary. The validation cases are: turbulent flow through an inclined channel, turbulent flow over a wavy surface, and supersonic flow over a circular cylinder. Since the near-wall grids are naturally coarse, the key is to use a method that is capable of capturing the flow dynamics accurately in the vicinity of the interface. Towards the purpose, we develop a Cartesian cut-cell method, referred to as the ghost-cell based cut-cell method (GC-CCM), in the context of fully compressible solutions of Navier–Stokes equations. This method employs ghost-cells inside the solid interface such that the local spatial reconstruction remains consistent everywhere including in the vicinity of the boundary. In order to capture the near-wall flow behavior more accurately with coarse grids, this method decomposes cell faces of merged cells and computes fluxes through each decomposed segment separately. The objective of this work is to qualify whether the proposed method can accurately represent the high Reynolds number flows in the vicinity of immersed interfaces. To analyze the performance of the proposed method, we compare the results to the corresponding numerical results from the two other non-body conformal methods, namely the ghost-cell based immersed boundary method (GCIBM) and standard cut-cell method (S-CCM), that are implemented in the same numerical solver. The comparison demonstrates that the proposed method is capable of capturing near-wall flows relatively accurately with coarse grids.     

Effective eddy viscosities in implicit modeling of decaying high Reynolds number turbulence with and without rotation

We assess the applicability of the numerical dissipation as an implicit turbulence model. The nonoscillatory finite volume numerical scheme MPDATA developed for simulations of geophysical flows is employed as an example of a scheme with an implicit turbulence model. A series of low resolution simulations of decaying homogeneous turbulence with and without Coriolis forces in the limit of zero molecular viscosity are performed. To assess the implicit model the long-time evolution of turbulence in the simulations is investigated and the numerical velocity fields are analyzed to determine the effective spectral eddy viscosity that is attributed to the numerical discretization. The detailed qualitative and quantitative comparisons are made between the numerical eddy viscosity and the theoretical results as well as the intrinsic eddy viscosity computed exactly from the velocity fields by introducing an artificial wave number cutoff. We find that the numerical dissipation depends on the time step and exhibits contradictory dependence on rotation: it is overestimated for rapid rotation cases and is underestimated for nonrotating cases. These results indicate that the numerical dissipation may fail to represent the effects of the physical subgrid scale processes unless the parameters of a numerical scheme are carefully chosen.     

Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

With the two-scale expansion technique proposed by Yoshizawa,the turbulent fluctuating field is expanded around the isotropic field.At a low-order two-scale expansion,applying the mode coupling approximation in the Yakhot-Orszag renormalization group method to analyze the fluctuating field,the Reynolds-average terms in the Reynolds stress transport equation,such as the convective term,the pressure-gradient-velocity correlation term and the dissipation term,are modeled.Two numerical examples:turbulent flow past a backward-facing step and the fully developed flow in a rotating channel,are presented for testing the efficiency of the proposed second-order model.For these two numerical examples,the proposed model performs as well as the Gibson-Launder (GL) model,giving better prediction than the standard k-ε model,especially in the abilities to calculate the secondary flow in the backward-facing step flow and to capture the asymmetric turbulent structure caused by frame rotation.     

KOLMOGOROV BEHAVIOR OF NEAR-WALL TURBULENCE AND ITS APPLICATION IN TURBULENCE MODELING

The near-wall behavior of turbulence is re-examined in a way different from that proposed by Hanjalic and Launder¹ and followers^2,3,4,5. It is shown that at a certain distance from the wall, all energetic large eddies will reduce to Kolmogorov eddies (the smallest eddies in turbulence). All the important wall parameters, such as friction velocity, viscous length scale, and mean strain rate at the wall, are characterised by Kolmogorov microscales. According t o this Kolmogorov behavior of near-wall turbulence, the turbulence quantities, such as turbulent kinetic energy, dissipation rate, etc. at the location where the large eddies become “Kolmogorov” eddies, can be estimated by using both direct numerical simulation (DNS) data and asymptotic analysis of near-wall turbulence. This information will provide useful boundary conditions for the turbulent transport equations. As a n example, the concept is incorporated in the standard κ - εmodel which is then applied t o channel and boundary layer flows. Using appropriate boundary conditions (based on Kolmogorov behaviour of near-wall turbulence), there is no need for any wall-modification to the κ - ε equations (including model constants). Results compare very well with the DNS and experimental data.     

An advanced switching parameter for a hybrid LES/RANS model considering the characteristics of near-wall turbulent length scales

In this study, we proposed an idea for an advanced switching parameter used in a hybrid approach connecting large eddy simulation (LES) with Reynolds-averaged Navier–Stokes modeling [the hybrid LES/RANS (HLR) model]. Although the HLR model is promising way to predict engineering turbulent flows, an important problem is that RANS is always adopted in the near-wall region, even if the grid resolution is fine enough for LES. To overcome this difficulty, the switching parameter proposed here introduced knowledge of the Kolmogorov microscale that is thought to be reasonable for representing the near-wall turbulence. This parameter enabled the present HLR model to be smoothly replaced by a full LES if a grid resolution was fine enough in the near-wall region. To confirm model performance, the present HLR model was applied to numerical simulations of a periodic hill flow as well as fundamental plane channel flows. The model generally provided reasonable predictions for these test cases that include complex turbulence with massive flow separation.     

Recent progress in the development of the Elliptic Blending Reynolds-stress model

The Elliptic Blending Reynolds Stress Model (EB-RSM), originally proposed by Manceau and Hanjalić (2002) to extend standard, weakly inhomogeneous Reynolds stress models to the near-wall region, has been subject to various modifications by several authors during the last decade, mainly for numerical robustness reasons. The present work revisits all these modifications from the theoretical standpoint and investigates in detail their influence on the reproduction of the physical mechanisms at the origin of the influence of the wall on turbulence. The analysis exploits recent DNS databases for high-Reynolds number channel flows, spanwise rotating channel flows with strong rotation rates, up to complete laminarization, and the separated flow after a sudden expansion without and with system rotation. Theoretical arguments and comparison with DNS results lead to the selection of a recommended formulation for the EB-RSM model. This formulation shows satisfactory predictions for the configurations described above, in particular as regards the modification of the mean flow and turbulent anisotropy on the anticyclonic or pressure side.     

A New Low-Reynolds-Number One-Equation Model of Turbulence

In this study, we propose a new Low-Reynolds-Number (LRN)one-equation model, which is derived from an LRN two-equation(k-ε) model. The derivation of the transport equation, in principle, is based on the assumption that the turbulent structure parameter remains constant. However, the relation for the turbulent structure parameter a ₁(=|− |/k) is modified to account for near-wall turbulence. As a result, the present one-equation model contains a term which takes the near-wall limiting behavior explicitly into account. Thus, the present model provides the correct wall-limiting behavior of turbulence in the vicinity of the wall and can be applied to the analysis of heat transfer. The validity of the present model is tested in channel flows, boundary layer flows with and without pressure gradient, plane wall jet, and flow with separation and reattachment. The calculated results showed good agreement with the direct numerical simulation (DNS) and experimental data. This revised version was published online in July 2006 with corrections to the Cover Date.     

Modifications for an explicit algebraic stress model

An extension of the explicit algebraic stress model, developed by Gatski and Speziale [Gatski TB, Speziale CG. On the explicit algebraic stress models for complex turbulent flows. Journal of Fluid Mechanics 1993; 254: 59–78] is proposed. The extension implicates some essential characteristics of second‐order closure models. The strain‐dependent coefficients are modified, resulting in an alleviation of the numerical instabilities involved in the model. A new near‐wall damping function f_μ in the eddy viscosity relation is introduced. To enhance dissipation in near‐wall regions, the model constant C_ϵ1 is modified and an extra positive source term is included in the dissipation equation. In addition, a realizable time scale is incorporated to remove the wall singularity. Computed results show that the modified Gatski–Speziale (MGS) model predictions are in better agreement with the direct numerical simulation (DNS) and experimental data than those of the original Gatski–Speziale (OGS) model. Copyright © 2001 John Wiley & Sons, Ltd.     

Wall-Distance-Free Turbulence Models Applied to Incompressible Flows

Abstract

A comparative study of low-Re wall-distance-free (WDF) turbulence models in incompressible flows is presented. The study includes the WDF k-? and the three-equation WDF k-?-γ as well as two k-? models which invoke the distance from the wall. The models are implemented in conjunction with a characteristics-based method and an implicit unfactored scheme. Comparison with direct-numerical-simulation data reveals that the WDF models provide much more accurate results for the dissipation rate, especially in the near wall region. A grid refinement study further reveals that the models which explicitly involve the distance from the wall cannot capture the correct turbulence dissipation rate behaviour in the near-wall region even on the finest grid, where grid-independent solution is achieved. However, the low-Re WDF models require slightly more iterations than the other k - ? models to converge. Results are presented for channel, flat plate and backward-facing step flows.     

Explicit algebraic stress modelling of homogeneous and inhomogeneous flows

Capability of the explicit algebraic stress models to predict homogeneous and inhomogeneous shear flows are examined. The importance of the explicit solution of the production to dissipation ratio is first highlighted by examining the algebraic stress models performance at purely irrotational strain conditions. Turbulent recirculating flows within sudden expanding pipes are further simulated with explicit algebraic stress model and anisotropic eddy viscosity model. Both models predict better stress–strain interactions, showing reasonable shear layer developments. The anisotropic stress field are also accurately predicted by the models, though the anisotropic eddy viscosity model of Craft et al. returns marginally better results. Copyright © 2005 John Wiley & Sons, Ltd.     

A new low-Reynolds-number k--fμ model for predictions involving multiple surfaces

A new k--f_μ turbulence model is proposed, in which the near-wall effect without reference to distance and the non-equilibrium effect are incorporated. In this model, the non-local near-wall effect in a general coordinate system is taken into account by the f_μ equation, and the local anisotropy in strongly strained turbulent flows in introduced in the equation. The near-wall characteristics of the present model are ascertained from the DNS data. Also, the validation is explored to separate and reattaching flows. The backward-facing step flow is selected for benchmark test of the present model and the response of the model to flows involving complex surfaces is assessed. The predictions of the present model are checked with the existing measurements. The model performance is shown to be generally satisfactory.     

An Investigation of SGS Stress Anisotropy Modeling in Complex Turbulent Flow Fields

An anisotropy-resolving subgrid-scale (SGS) model for large eddy simulation was investigated. Primary attention was given to the predictive performance of the SGS model in the case of complex turbulence with flow impingement and/or flow separation. The SGS model was constructed by combining an isotropic linear eddy-viscosity model with an extra anisotropic term. Since the extra anisotropic term was modeled to prevent undesirable energy transfer between the grid-scale and SGS parts, the model is expected not to seriously affect computational stability. To validate the model performance for complex turbulent flow fields, the SGS model was applied to numerical simulations of a plane impinging jet and 3-D diffuser flow as well as fundamental plane channel flows. The SGS model provided reasonable predictions for these test cases. Furthermore, the predicted SGS stress components were decomposed into linear and anisotropic parts and their roles were investigated in detail. The usefulness of the present anisotropy-resolving SGS model in practical engineering applications was thus described.     

A low Reynolds number variant of partially-averaged Navier-Stokes model for turbulence

A low Reynolds number (LRN) formulation based on the Partially Averaged Navier-Stokes (PANS) modelling method is presented, which incorporates improved asymptotic representation in near-wall turbulence modelling. The effect of near-wall viscous damping can thus be better accounted for in simulations of wall-bounded turbulent flows. The proposed LRN PANS model uses an LRN k-ε model as the base model and introduces directly its model functions into the PANS formulation. As a result, the inappropriate wall-limiting behavior inherent in the original PANS model is corrected. An interesting feature of the PANS model is that the turbulent Prandtl numbers in the k and ε equations are modified compared to the base model. It is found that this modification has a significant effect on the modelled turbulence. The proposed LRN PANS model is scrutinized in computations of decaying grid turbulence, turbulent channel flow and periodic hill flow, of which the latter has been computed at two different Reynolds numbers of Re = 10,600 and 37,000. In comparison with available DNS, LES or experimental data, the LRN PANS model produces improved predictions over the standard PANS model, particularly in the near-wall region and for resolved turbulence statistics. Furthermore, the LRN PANS model gives similar or better results - at a reduced CPU time - as compared to the Dynamic Smagorinsky model.