A dynamic subgrid-scale tensorial Eddy viscosity model

In the Navier-Stokes equations the removal of the turbulent fluctuating velocities with a frequency above a certain fixed threshold, employed in the Large Eddy Simulation (LES), causes the appearance of a turbulent stress tensor that requires a number of closure assumptions. In this paper insufficiencies are demonstrated for those closure models which are based on a scalar eddy viscosity coefficient. A new model, based on a tensorial eddy viscosity, is therefore proposed; it employs the Germano identity [1] and allows dynamical evaluation of the single required input coefficient. The tensorial expression for the eddy viscosity is deduced by removing the widely used scalar assumption of the high-frequency viscous dissipation and replacing it by its tensorial counterpart arising in the balance of the Reynolds stress tensor. The numerical simulations performed for a lid driven cavity flow show that the proposed model allows to overcome the drawbacks encountered by the scalar eddy viscosity models. Received November 25, 1997     

A Priori Tests of RANS Models for Turbulent Channel Flows of a Dense Gas

Dense gas effects, encountered in many engineering applications, lead to unconventional variations of the thermodynamic and transport properties in the supersonic flow regime, which in turn are responsible for considerable modifications of turbulent flow behavior with respect to perfect gases. The most striking differences for wall-bounded turbulence are the decoupling of dynamic and thermal effects for gases with high specific heats, the liquid-like behavior of the viscosity and thermal conductivity, which tend to decrease away from the wall, and the increase of density fluctuations in the near wall region. The present work represents a first attempt of quantifying the influence of such dense gas effects on modeling assumptions employed for the closure of the Reynolds-averaged Navier–Stokes equations, with focus on the eddy viscosity and turbulent Prandtl number models. For that purpose, we use recent direct numerical simulation results for supersonic turbulent channel flows of PP11 (a heavy fluorocarbon representative of dense gases) at various bulk Mach and Reynolds numbers to carry out a priori tests of the validity of some currently-used models for the turbulent stresses and heat flux. More specifically, we examine the behavior of the modeled eddy viscosity for some low-Reynolds variants of the \(k-\varepsilon \) model and compare the results with those found for a perfect gas at similar conditions. We also investigate the behavior of the turbulent Prandtl number in dense gas flow and compare the results with the predictions of two well-established turbulent Prandtl number models.     

Turbulent transport in an inclined jet in crossflow

The present study experimentally investigates a turbulent jet in crossflow relevant to film cooling applications. The jet is inclined at 30°, and its mean velocity is the same as the crossflow. Magnetic resonance imaging is used to obtain the full three-dimensional velocity and concentration fields, whereas Reynolds stresses are obtained along selected planes by Particle Image Velocimetry. The critical role of the counter-rotating vortex pair in the mixing process is apparent from both velocity and concentration fields. The jet entrainment is not significantly higher than in an axisymmetric jet without crossflow, because the proximity of the wall inhibits the turbulent transport. Reynolds shear stresses correlate with velocity and concentration gradients, consistent with the fundamental assumptions of simple turbulence models. However the eddy viscosity is strongly anisotropic and non-homogeneous, being especially low along the leeward side of the jet close to injection. Turbulent diffusion acts to decouple mean velocity and concentration fields, as demonstrated by the drop in concentration flux within the streamtube issued from the hole. Volume-averaged turbulent diffusivity is calculated using a mass–flux balance across the streamtube emanating from the jet hole, and it is found to vary slowly in the streamwise direction. The data are compared with Reynolds-Averaged Navier–Stokes simulations with standard k − ε closure and an optimal turbulent Schmidt number. The computations underestimate the strength of the counter-rotating vortex pair, due to an overestimated eddy viscosity. On the other hand the entrainment is increasingly underpredicted downstream of injection. To capture the correct macroscopic trends, eddy viscosity and eddy diffusivity should vary spatially in different ways. Therefore a constant turbulent Schmidt number formulation is inadequate for this flow.     

On harmonic perturbations in a turbulent mixing layer

A theoretical model of harmonic perturbations in a turbulent mixing layer is proposed. The model is based on the triple decomposition method. It is assumed that the instantaneous velocities and pressure consist of three distinctive components: the mean (time average), the coherent (phase average), and the random (turbulent) motion. The interaction between incoherent turbulent fluctuations and large-scale coherent disturbances is incorporated by the Newtonian eddy viscosity model. A slight divergence of the flow is also taken into account, and the results are compared with experimental data. For a high amplitude of the perturbations, the nonlinear feedback to the mean flow is taken into account by means of the coherent Reynolds stresses. The results reveal the possibility of a negative spreading rate of the mixing layer. A simultaneous consideration of the mean flow divergence and nonlinear self-interaction results in Landau-like amplitude equations. It is observed that the nonlinear term in the amplitude equation is not significant at the levels of amplitude considered. The velocity disturbance profiles of the second harmonic are also presented and, at low-level amplitude, they are in good agreement with experiments.     

Performance of RNG Turbulence Modelling for Forced Convective Heat Transfer in Ducts

This investigation concerns numerical calculation of turbulent forced convective heat transfer and fluid flow in straight ducts using the RNG (Re-Normalized Group) turbulence method.

A computational method has been developed to predict the turbulent Reynolds stresses and turbulent heat fluxes in ducts with different turbulence models. The turbulent Reynolds stresses and other turbulent flow quantities are predicted with the RNG κ?ε model and the RNG non-linear κ-ε model of Speziale. The turbulent heat fluxes are modeled by the simple eddy diffusivity (SED) concept, GGDH and WET methods. Two wall functions are used, one for the velocity field and one for the temperature field. All the models arc implemented for an arbitrary three dimensional duct.

Fully developed condition is achieved by imposing cyclic boundary conditions in the main flow direction. The numerical approach is based on the finite volume technique with a non-staggered grid arrangement. The pressure-velocity coupling is handled by using the SIMPLEC-algorithm. The convective terms are treated by the QUICK, scheme while the diffusive terms are handled by the central-difference scheme. The hybrid scheme is used for solving the κ and ε equations.

The overall comparison between the models is presented in terms of friction factor and Nusselt number. The secondary flow generation is also of major concern.     

Compound wall treatment with low‐Re turbulence model

A generalized treatment for the wall boundary conditions relating to turbulent flows is developed that blends the integration to a solid wall with wall functions. The blending function ensures a smooth transition between the viscous and turbulent regions. An improved low Reynolds number k?ε model is coupled with the proposed compound wall treatment to determine the turbulence field. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. Computations with fine and coarse meshes of a few flow cases yield appreciably good agreement with the direct numerical simulation and experimental data. The method is recommended for computing the complex flows where computational grids cannot satisfy a priori the prerequisites of viscous/turbulence regions. Copyright © 2011 John Wiley & Sons, Ltd.     

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

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

Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier–Stokes Dynamics

We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution. Received 27 April 2001 and accepted 16 December 2001     

New equations for nonlinear acoustics in a low Mach number and weakly heterogeneous atmosphere

Various scalar equations are proposed, modeling the pressure field in the linear and nonlinear acoustical regimes. They are derived by assuming a flow with a small Mach number and a smaller medium heterogeneity. Such assumptions are well satisfied in the atmospheric boundary layer. Further simplifications can be obtained when less intense turbulent fluctuations are superimposed to a sheared mean flow. In the linear regime, a hierarchy of equations with increasing orders of precision is established. A new equation is found where all terms quadratic with respect to the ambient flow are retained, either related to sound convection by the flow, or to the flow inhomogeneity. Numerical solutions indicate that it is more precise than the equations in the literature for small Mach numbers, but less robust for larger negative Mach numbers. Two generalizations of Lilley’s equation incorporate the effects of turbulent fluctuations. Nonlinear terms are of different origins, either thermodynamical, inertial, or related to the flow shear. For a locally plane wave, they simplify into a single term which appears as the classical Westervelt quadratic nonlinearity convected by the flow. Consequently, all linear equations can easily be generalized to nonlinear ones, such as a new Lilley’s equation augmented with acoustical nonlinearities and turbulent flow fluctuations.     

A comparison of primitive variables and streamfunction–vorticity for fem depth-averaged modelling of steady flow

The governing equations for depth-averaged turbulent flow are presented in both the primitive variable and streamfunction–vorticity forms. Finite element formulations are presented, with special emphasis on the handling of bottom stress terms and spatially varying eddy viscosity. The primitive variable formulation is found to be preferable because of its flexibility in handling spatial variation in viscosity, variability in water surface elevations, and inflow and outflow boundaries. The substantial reduction in computational effort afforded by the streamfunction–vorticity formulation is found not to be sufficient to recommend its use for general depth-averaged flows. For those flows in which the surface can be approximated as a fixed level surface, the streamfunction–vorticity form can produce results equivalent to the primitive variable form as long as turbulent viscosity can be estimated as a constant.     

Balance equation of the generalised sub-grid scale (SGS) turbulent kinetic energy in a new tensorial dynamic mixed SGS model

A new dynamic model is proposed in which the eddy viscosity is defined as a symmetric second rank tensor, proportional to the product of a turbulent length scale with an ellipsoid of turbulent velocity scales. The employed definition of the eddy viscosity allows to remove the local balance assumption of the SGS turbulent kinetic energy formulated in all the dynamic Smagorinsky-type SGS models. Furthermore, because of the tensorial structure of the eddy viscosity the alignment assumption between the principal axes of the SGS turbulent stress tensor and the resolved strain-rate tensor is equally removed, an assumption which is employed in the scalar eddy viscosity SGS models. The proposed model is tested for a turbulent channel flow. Comparison with the results obtained with other dynamic SGS models (Dynamic Smagorinsky Model, Dynamic Mixed Model and Dynamic K-equation Model) shows that the tensorial definition of the eddy viscosity and the removal of the local balance assumption of the SGS turbulent kinetic energy considerably improves the agreement between results obtained with Large Eddy simulation (LES) and Direct Numerical Simulations (DNS), respectevely. Received August 26, 1999     

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.     

Calculation of Reynolds stresses in turbulent supersonic flows

The baseline numerical procedure of interest in this study combines flux vector splitting, flux difference splitting and an explicit treatment of the diffusion terms of the flow equations. The viscous terms are treated explicitly to preserve the wave propagation properties of the Euler fluxes and permit splitting. The experience with this scheme has been limited to laminar or, at best, ‘eddy viscosity’ flows. In this paper the applicability of the scheme is extended to include the calculation of turbulent Reynolds stresses in supersonic flows. The schemes and our implementation are discussed. Both laminar and turbulence subsets of the Reynolds/Favre-averaged equations are tested, with a discussion of relative performance. The test problem for turbulence consists of a zero-pressure-gradient supersonic boundary layer as well as a supersonic boundary layer experiencing the combined effects of adverse pressure gradient, bulk compression and a concave streamline curvature. Excellent agreement with experimental measurements is observed for most of the quantities compared, which suggests that the numerical procedures presented in this paper are potentially very useful.     

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.     

Two-Equation Turbulence Modelling of a Transitional Separation Bubble

Calculations of the Reynolds averaged equations using two different turbulence models have been compared with direct numerical simulation of a transitional separation bubble. Three methods of transition modelling were investigated. The first had no transition adjustment, the second involved fixing the transition point at the location observed in the simulation and the third was a direct transformation of a method proposed by Wilcox [1] which involved sensitising the eddy viscosity and transport equations to the local turbulent Reynolds number. The models captured the general features of the flow but were unable to show the recovery behaviour of the flow behind the bubble. Reasons for the failure are discussed using apriori analysis of terms in the model equations. This revised version was published online in July 2006 with corrections to the Cover Date.     

湍流边界层外区相干结构的三维波模型

根据流动稳定性理论,提出了一种三维波模型来描述湍流边界层外区大尺度相干结构。计算所得流线图和等涡量线图较罗纪生,周恒（1993）的二维波模型更符合实验结果。说明该三维模型能够较好地反映湍流边界层外区大尺度相干结构的物理特征。     

A Gas-Kinetic Scheme for Turbulent Flow

RANS simulations may not provide accurate results for all flow conditions. The interaction between a shock wave and a turbulent boundary layer is an example which may still be difficult to simulate accurately. Beside the inability to reproduce physical phenomena such as shock unsteadiness, the argument is put forward that the conventional numerical schemes, based on the Navier-Stokes equations, may be unable to generate a physically consistent turbulent stress tensor in the presence of large unresolved scales of motion. A large ratio between unresolved and resolved scales of motion, a sort of Knudsen number based on turbulent fluctuations, might introduce inaccuracies for which the turbulence model is not accountable. In order to improve the accuracy of RANS simulations, researchers have suggested various ad-hoc modifications to standard turbulence models which limit eddy viscosity or the turbulent stress tensor in the presence of strong gradients. Gas-kinetic schemes might be able to improve RANS predictions in shocklayers by removing or limiting the errors caused by the large scales ratio. These schemes are a class of their own; in the framework of a finite-volume or finite-elements discretizations, they model the numerical fluxes on the basis of the Boltzmann equation instead of the Navier-Stokes equations as is conventionally done. In practical terms, these schemes provide a higher accuracy and, more importantly, an in-built “multiscalar” mechanism, i.e. the ability to adjust to the size of unresolved scales of motion. This property makes them suitable for shock-capturing and rarefied flow. Gas-kinetic scheme may be coupled to a conventional RANS turbulence model; it is shown that the turbulent stress tensor is naturally adjusted as a function of the unresolved-to-resolved scales ratios and achieves a higher physical consistency than conventional schemes. The simulations shown - well-known benchmark cases with strong shock-boundary layer interactions - have been obtained with a standard two-equation turbulence model (k- ω). It is shown that the gas-kinetic scheme provides good quality predictions, where conventional schemes with the same turbulence model are known to fail.     

Modelling turbulence around and inside porous media based on the second moment closure

To predict turbulence in porous media, a new approach is discussed. By double (both volume and Reynolds) averaging Navier–Stokes equations, there appear three unknown covariant terms in the momentum equation. They are namely the dispersive covariance, the macro-scale and the micro-scale Reynolds stresses, in the present study. For the macro-scale Reynolds stress, the TCL (two-component-limit) second moment closure is applied whereas the eddy viscosity models are applied to the other covariant terms: the Smagorinsky model and the one-equation eddy viscosity model, respectively for the dispersive covariance and the micro-scale Reynolds stress. The presently proposed model is evaluated in square rib array flows and porous wall channel flows with reasonable accuracy though further development is required.     

湍流边界层复涡黏模式的实验研究

在开口式循环水槽底部湍流边界层外区引入周期性扰动。利用X型热膜探针在扰动下游进行测量。用实验的方法研究了周期性大尺度结构下壁湍流涡黏模式中涡黏系数的形式，结果发现和周期性扰动对应的变形率及与之对应的雷诺应力间存在着相位差。这是目前许多最终导致涡黏系数的湍流模式理论都没有考虑到的一个重要因素。