Second-moment Turbulence Closure Research Articles

A two-scale second-moment turbulence closure based on weighted spectrum integration

A two-scale second-moment turbulence closure has been derived based on the weighted integration of the dynamic equation for the covariance spectrum. The goal is to close the Reynolds stress equations with two additional scalar equations that provide separately the scales of the spectral energy transfer and of the turbulence energy dissipation rate. Such a model should provide better prediction of nonequilibrium turbulent flows. The derivation consists of analytical integration of the wave-number-weighted covariance spectrum using a model of the spectral equations with an assumed simple representation of the shape of the energy spectrum. The resulting closure consists of a set of three tensorial equations, one for the Reynolds stress and two for length scale tensors, the latter representing the energy containing- and dissipative eddies respectively. The trace of the two tensor-scale equations leads to a set of two scalar scale parameters. In the equilibrium limit, the model reduces to the standard second-moment single-scale closure. The approach makes it also possible to derive the scale equations in a more systematic manner as compared with the common single-scale and other multi-scale models. The performance of the model in capturing the scale dynamics is illustrated by predictions of several generic homogeneous and inhomogeneous unsteady flows, demonstrating the expected response of the two scale equations.

Computation of flow and heat transfer in rotating two-pass rectangular channels (AR = 1:1, 1:2, and 1:4) with smooth walls by a Reynolds stress turbulence model

Numerical predictions of three-dimensional flow and heat transfer are presented for rotating two-pass smooth channels with three aspect ratios (AR = 1:1; 1:2; 1:4). Detailed predictions of mean velocity, mean temperature and Nusselt number for two Reynolds numbers ( Re = 10,000 and 100,000) were carried out. A total of fifteen calculations have been performed with various combinations of rotation number, Reynolds number, and coolant-to-wall density ratio. The rotation number and inlet coolant-to-wall density ratio varied from 0.0 to 0.28 and from 0.13 to 0.40, respectively. The focus of this study is to investigate the effect of the channel aspect ratio, the Reynolds number, and the coolant-to-wall density ratio on the nature of the flow and heat transfer. A multi-block Reynolds-averaged Navier–Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method.

First-order conditional moment closure modeling of turbulent, nonpremixed methane flames

Results obtained using a first-order conditional moment closure (CMC) approach to modeling turbulent, nonpremixed flames of methane are presented. Predictions are based on both k– ε and second-moment turbulence closures and use the GRI-Mech 2.11 kinetic scheme involving 279 reactions and 49 species and the GRI-Mech 3.0 scheme applying 325 reactions and 53 species. To provide a comprehensive assessment of the ability of these methods to predict a wide range of flames, comparisons are made with experimental data on three piloted and two unpiloted flames, the majority of which exhibit no local extinction effects. It is concluded that first-order CMC modeling is capable of yielding reliable predictions for flames with little or no extinction effect, with the only exception being NO, which is overpredicted at most locations and stoichiometries. Results derived using the two turbulence closures are in general in close accord, although real space predictions demonstrate the superiority of results derived on the basis of the second-moment turbulence closure. Calculation of flames that exhibit extinction effects demonstrate a gradual deterioration of results with increasing Re, pointing to the requirement for second-order CMC modeling as such effects become more significant. Anomalies are observed in regard to NO, which is significantly overpredicted for the piloted flames, but less so for the unpiloted flames. These results demonstrate a requirement not only to explore second-order CMC modeling approaches for such flames, but also to examine the application of different kinetic schemes, including a detailed investigation of the mechanisms and rates employed for NO chemistry.

A direct-numerical-simulation-based second-moment closure for turbulent magnetohydrodynamic flows

A magnetic field, imposed on turbulent flow of an electrically conductive fluid, is known to cause preferential damping of the velocity and its fluctuations in the direction of Lorentz force, thus leading to an increase in stress anisotropy. Based on direct numerical simulations (DNS), we have developed a model of magnetohydrodynamic (MHD) interactions within the framework of the second-moment turbulence closure. The MHD effects are accounted for in the transport equations for the turbulent stress tensor and energy dissipation rate—both incorporating also viscous and wall-vicinity nonviscous modifications. The validation of the model in plane channel flows with different orientation of the imposed magnetic field against the available DNS (Re=4600,Ha=6), large eddy simulation (Re=2.9×104,Ha=52.5,125) and experimental data (Re=5.05×104 and Re=9×104, 0⩽Ha⩽400), show good agreement for all considered situations.

Employment of the second‐moment turbulence closure on arbitrary unstructured grids

AbstractThe paper presents a finite‐volume calculation procedure using a second‐moment turbulence closure. The proposed method is based on a collocated variable arrangement and especially adopted for unstructured grids consisting of ‘polyhedral’ calculation volumes. An inclusion of 23k in the pressure is analysed and the impact of such an approach on the employment of the constant static pressure boundary is addressed. It is shown that this approach allows a removal of a standard but cumbersome velocity–pressure –Reynolds stress coupling procedure known as an extension of Rhie‐Chow method (AIAA J. 1983; 21: 1525–1532) for the Reynolds stresses. A novel wall treatment for the Reynolds‐stress equations and ‘polyhedral’ calculation volumes is presented. Important issues related to treatments of diffusion terms in momentum and Reynolds‐stress equations are also discussed and a new approach is proposed. Special interpolation practices implemented in a deferred‐correction fashion and related to all equations, are explained in detail. Computational results are compared with available experimental data for four very different applications: the flow in a two‐dimensional 180o turned U‐bend, the vortex shedding flow around a square cylinder, the flow around Ahmed Body and in‐cylinder engine flow. Additionally, the performance of the methodology is assessed by applying it to different computational grids. For all test cases, predictions with the second‐moment closure are compared to those of the k–εmodel. The second‐moment turbulence closure always achieves closer agreement with the measurements. A moderate increase in computing time is required for the calculations with the second‐moment closure. Copyright © 2004 John Wiley & Sons, Ltd.

A Numerical Study of Flow and Heat Transfer in Rotating Rectangular Channels AR=4 With 45 deg Rib Turbulators by Reynolds Stress Turbulence Model

Computations were performed to study three-dimensional turbulent flow and heat transfer in stationary and rotating 45 deg ribbed rectangular channels for which experimental heat transfer data were available. The channel aspect ratio (AR) is 4:1, the rib height-to-hydraulic diameter ratio e/Dh is 0.078 and the rib-pitch-to-height ratio P/e is 10. The rotation number and inlet coolant-to-wall density ratios, Δρ/ρ, were varied from 0.0 to 0.28 and from 0.122 to 0.40, respectively, while the Reynolds number was fixed at 10,000. Also, two channel orientations (β=90deg and 135 deg from the rotation direction) were investigated with focus on the high rotation and high density ratios effects on the heat transfer characteristics of the 135 deg orientation. These results show that, for high rotation and high density ratio, the rotation induced secondary flow overpowered the rib induced secondary flow and thus change significantly the heat transfer characteristics compared to the low rotation low density ratio case. A multi-block Reynolds-Averaged Navier-Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method.

Prediction of Flow and Heat Transfer in Rotating Two-Pass Rectangular Channels With 45-deg Rib Turbulators

Abstract Numerical predictions of three-dimensional flow and heat transfer are presented for a rotating two-pass rectangular channel with 45-deg rib turbulators and channel aspect ratio of 2:1. The rib height-to-hydraulic diameter ratio e/Dh is 0.094 and the rib-pitch-to-height ratio P/e is 10. Two channel orientations are studied: β=90deg and 135 deg, corresponding to the mid-portion and the trailing edge regions of a turbine blade, respectively. The focus of this study is twofold; namely, to investigate the effect of the channel aspect ratio and the channel orientation on the nature of the flow and heat transfer enhancement. A multi-block Reynolds-averaged Navier-Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method. The numerical results compare reasonably well with experimental data for both stationary and rotating rectangular channels with rib turbulators at Reynolds number (Re) of 10,000, rotation number (Ro) of 0.11 and inlet coolant-to-wall density ratio (Δρ/ρ) of 0.115.

Flow and heat transfer in rotating two-pass rectangular channels (AR=2) by Reynolds stress turbulence model

Numerical predictions of three-dimensional turbulent flow and heat transfer are presented for a rotating two-pass smooth rectangular channel with channel aspect ratio of 2:1. The focus of this study is to investigate the effect of rotation, channel orientation and the sharp 180° turn on the flow and heat transfer distributions. Two channel orientations are studied: β=90° and β=135°. A multi-block Reynolds-averaged Navier–Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure. The Reynolds number ( Re) is fixed at 10,000 while the rotation number ( Ro) is varied from 0 to 0.22. Two inlet coolant-to-wall density ratios Δρ/ ρ are studied (0.115 and 0.22). The numerical results are compared with the experimental data for both stationary and rotating two-pass rectangular channels.

Elliptic blending model: A new near-wall Reynolds-stress turbulence closure

A new approach to modeling the effects of a solid wall in one-point second-moment (Reynolds-stress) turbulence closures is presented. The model is based on the relaxation of an inhomogeneous (near-wall) formulation of the pressure–strain tensor towards the chosen conventional homogeneous (far-from-a-wall) form using the blending function α, for which an elliptic equation is solved. The approach preserves the main features of Durbin’s Reynolds-stress model, but instead of six elliptic equations (for each stress component), it involves only one, scalar elliptic equation. The model, called “the elliptic blending model,” offers significant simplification, while still complying with the basic physical rationale for the elliptic relaxation concept. In addition to model validation against direct numerical simulation in a plane channel for Reτ=590, the model was applied in the computation of the channel flow at a “real-life” Reynolds number of 106, showing a good prediction of the logarithmic profile of the mean velocity.

Comparative Analysis of Four Second-Moment Turbulence Closure Models for the Oceanic Mixed Layer

Abstract In this comparative study, four different algebraic second-moment turbulence closure models are investigated in detail. These closure schemes differ in the number of terms considered for the closure of the pressure–strain correlations. These four turbulence closures result in the eddy-diffusivity principle such that the closure assumptions are contained in dimensionless so-called stability functions. Their performance in terms of Prandtl number, Monin–Obukhov similarity theory, and length scale ratios are first tested against data for simple flows. The turbulence closure is then completed by means of a k–ϵ two-equation model, but other models such as the two-equation model by Mellor and Yamada could also be used. The concept of the steady-state Richardson number for homogeneous shear layers is exploited for calibrating the sensitivity of the four models to shear and stable stratification. Idealized simulations of mixed layer entrainment into stably stratified flow due to surface stress and due to...

Flow and Heat Transfer in a Rotating Square Channel With 45 deg Angled Ribs by Reynolds Stress Turbulence Model

Numerical predictions of three-dimensional flow and heat transfer are presented for a rotating square channel with 45 deg angled ribs as tested by Johnson et al. (1994). The rib height-to-hydraulic diameter ratio e/Dh is 0.1 and the rib pitch-to-height ratio (P/e) is 10. The cross section of the ribs has rounded edges and corners. The computation results are compared with the experimental data of Johnson et al. (1994) at a Reynolds number (Re) of 25,000, inlet coolant-to-wall density ratio Δρ/ρ of 0.13, and three rotation numbers (Ro) of 0.0, 0.12, and 0.24. A multiblock numerical method has been employed with a near-wall second-moment turbulence closure model. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method. Pressure is computed using a hybrid SIMPLER/PISO approach, which satisfies the continuity of mass and momentum simultaneously at every time step. The second-moment solutions show that the secondary flows induced by the angled ribs, rotating buoyancy, and Coriolis forces produced strong nonisotropic turbulent stresses and heat fluxes that significantly affected flow fields and surface heat transfer coefficients. The present near-wall second-moment closure model provided an improved flow and heat transfer prediction.

An Approximate Roe-Type Riemann Solver for a Class of Realizable Second Order Closures

A realizable, objective second-moment turbulence closure, allowing for an entropy characterisation, is analyzed with respect to its convective subset. The distinct characteristic wave system of these equations in non-conservation form is exposed. An approximate solution to Ihe associated one-dimensional Riemann problem is constructed making use of approximate jump conditions obtained by assuming a linear path across shock waves. A numerical integration method based on a new approximate Riemann solver (flux-difference-splitting) is proposed for use in conjunction with either unstructured or structured grids. Test calculations of quasi one-dimensional flow cases demonstrate the feasibility of the current technique even where Euler-based approaches fail.

A Modified Model for Diffusion in Second-Moment Turbulence Closures

A study has been carried out to determine the relative roles of the three diffusion sub-processes contained in the Lumley (1978) diffusion model. The three sub-processes are described as being the production of turbulent transport, the third-order pressure-velocity process, which regulates the relative magnitudes of the turbulent transport components, and the pressure-diffusion. The present work describes a unique method for calibrating the model based on an analysis of zero-mean-shear turbulence. On the basis of the analysis, and using recent direct numerical simulation and experimental data, the coefficients in the Lumley (1978) model are modified such that the model gives the correct behavior in the diffusive limit. The modified model was then validated by carrying out CFD predictions for three benchmark flows of engineering interest. The modified Lumley (1978) diffusion model has two clear advantages over the more commonly used Daly and Harlow (1970) model. First, unlike the Daly and Harlow (1970) model, the modified Lumley (1978) model is mathematically correct and second, it was shown in the present validation that the modified Lumley (1978) model gives the most consistently reasonable predictions.

Second-Moment Turbulence Closures for CFD: Needs and Prospects*

Differential second-moment (Reynolds-stress) turbulence closure models (DSM) have long been expected to replace the currently popular two-equation k — e and similar eddy viscosity models (EVM) as the industrial standard for Computational Fluid Dynamics (CFD). Yet, despite almost three decades of development and indisputable progress, only a few commercial CFD vendors offer DSM as a modelling option. Even fewer industrial users recognize the natural superiority of the DSM. These models, used and researched mainly within academic community, are still viewed as a development target rather than as a proven and mature technique for solving complex flow phenomena. This paper gives an overview of the rationale for employing more advanced models for the computation of complex flows and transport processes. It also discusses reasons for their slow adoption by the CFD community. Physical arguments are briefly given; these illustrate a higher degree of exactness inherent in the second-moment closure approach. The superiority of these models is demonstrated by a series of computational examples, provided by author's co-workers who used either the same or very similar computational methods and model(s). Examples include several nonequilibrium flows, attached and with separation and reattachment, flow impingement and stagnation, longitudinal vortices, secondary motion, swirl, system rotation. The modelling of molecular effects, both near and away from a solid wall and associated laminar-lo-turbulent and reverse transition are also discussed in view of the need for an advanced closure approach particularly when wall phenomena are in focus. Numerical aspects associated with the application of second-moment closure are then discussed, together with current practice used to overcome numerical problems and to reconcile the need for advanced models with unavoidably increasing computational challenge. Several examples related to the automotive industry illustrate the applicability of DSM to real complex flows which have industrial relevance.

Low-Reynolds-number second-moment closure without wall-reflection redistribution terms

Abstract A full second-moment turbulence closure applicable right up to a wall is proposed. The closure adopts a quasi-linear rapid model, but does not contain wall-reflection redistribution terms. We test the turbulence model in fully-developed channel flow, boundary layers in zero, adverse and favorable pressure gradients, and plane and round jets. The predictions are generally in good agreement with experiments and DNS, though discrepancies are present in normal stress profiles. The result encourages further testing of the model in various complex flows to establish a simple low-Reynolds-number second-moment closure for both wall and free flows without wall-reflection redistribution terms.

Examination of Diffusion Modeling Using Zero-Mean-Shear Turbulence

An examination of diffusion modeling in second-moment turbulence closures is presented. The main objectives areto gain a better understanding oftheprocesses that are grouped into thenetdiffusivetransportoftheReynolds stresses, Dij, and to assess existing models for Dij. The benchmark case of zero-mean-shear (ZMS) turbulence is used as a guide. The analysis of ZMS turbulence shows that the pressure ‐velocity processes in Dij play a central role in establishing the anisotropy level in diffusive turbulence. Existing models for Dij based solely on the triple velocity correlation are shown to be inadequate in the diffusive limit. The present evaluation of diffusion models using ZMS turbulence and subsequent analysis of ZMS turbulence show that the Lumley (Lumley, J. L., “ Computational Modelling of Turbulent Flows,” Advances in Applied Mechanics , Vol. 18, Academic, New York, 1978,pp.123 ‐176)modelisthemostviableexisting diffusionmodel.Amodie cationto Lumley’ smodelthatenables exact predictions in the diffusive limit is suggested.

Modelling isolated wave-turbulence interactions in the stable atmospheric boundary layer

The interaction between a large-scale isolated wave and turbulence is considered in the context of the stable atmospheric boundary layer. A two-dimensional model with a second-moment turbulence closure is used to simulate the interaction numerically. Results obtained using other closures are discussed. the initial form of the wave-like disturbance is suggested by observations, which are described in a companion paper. the turbulence is shown to exert only a small influence on the wave, while the turbulence itself remains close to production-dissipation equilibrium. Wave-like components of the turbulence fluxes vary roughly linearly with wave amplitude, but overall the wave is strongly nonlinear and depends heavily on the background flow. Only one term in the equations for the quadratic turbulence quantities varies in a simple way with wave amplitude; this term represents a net transfer of energy from the wave to the turbulence.

Expanding the limits of “equilibrium” second-moment turbulence closures

The paper discusses possibilities for refinements of conventional “equilibrium” second-moment turbulence closure models, aimed at improving model performances in predicting turbulent flows of greater complexity. In focus are the invariant modelling of the low-Re-number and wall proximity effects, as well as extra strain-rates and control of the turbulence length-scale. In addition to satisfying most of the basic physical constraints, the main criterion for model validation was the quality of reproduction of flow and turbulence details, particularly, in the vicinity of a solid wall, in a broad variety of non-equilibrium flows featured by different phenomena. It is demonstrated that the new model, which includes several new modifications, but also some proposed in the past, can satisfactorily reproduce a range of attached and separating flows with strong time- or space-variations or abrupt changes of boundary conditions. Cases considered cover a wide range of Re-numbers involving in some cases also the laminar-to-turbulent or reverse transition.

Computational method of second-moment turbulence closures in complex geometries

This paper presents a finite volume numerical method to calculate turbulent flows using second-moment turbulence closures. The numerical method is formulated for a collocated grid arrangement and on body-fitted coordinate grids. A special interpolation scheme is derived to resolve the potential decoupling between momentum and Reynolds stress transport equations. This interpolation scheme naturally leads to stability-promoting terms in the momentum equations in the form of anisotropic eddy viscosities. Issues such as the implementation of boundary conditions with a body-fitted coordinate grid are also discussed. Applications are given to demonstrate the robustness of the numerical method and the performance of a second-moment turbulence closure. Computational results are compared with available experimental data and to those of the standard two-equation turbulence model. Good results have been obtained with the second-moment turbulence closure.

Realizability of second-moment closure via stochastic analysis

It is shown that realizability of second-moment turbulence closure models can be established by finding a Langevin equation for which they are exact. All closure models currently in use can be derived formally from the type of Langevin equation described herein. Under certain circumstances a coefficient in that formalism becomes imaginary. The regime in which models are realizable is, at least, that for which the coefficient is real. The present method does not imply unrealizable solutions when the coefficient is imaginary, but it does guarantee realizability when the coefficient is real; hence, this method provides sufficient, but not necessary, conditions for realizability. Illustrative computations of homogeneous shear flow are presented. It is explained how models can be modified to guarantee realizability in extreme non-equilibrium situations without altering their behaviour in the near-equilibrium regime for which they were formulated.