Second-moment Closure Research Articles

Computational Modelling of Turbulent Mixing in Confined Swirling Environment Under Constant and Variable Density Conditions

A high-intensity swirling flow in a model combustor subjected to large density variations has been examined computationally. The focus is on the Favre-averaged Navier–Stokes computations of the momentum and scalar transport employing turbulence models based on the differential second-moment closure (SMC) strategy. An updated version of the basic, high-Reynolds number SMC model accounting for a quadratic expansion of both the pressure–strain and dissipation tensors and a near-wall SMC model were used for predicting the mean velocity and turbulence fields. The accompanied mixing between the annular swirling air flow and the central non-swirling helium jet was studied by applying three scalar flux models differing mainly in the model formulation for the pressure-scalar gradient correlation. The computed axial and circumferential velocities agree fairly well with the reference experiment [So et al., NASA Contractor Report 3832, 1984; Ahmed and So, Exp. Fluids 4 (1986) 107], reproducing important features of such a weakly supercritical flow configuration (tendency of the flow core to separate). Although the length at which the mixing was completed was reproduced in reasonable agreement with the experimental results, the mixing activity in terms of the spreading rate of the shear/mixing layer, that is its thickness, was somewhat more intensive. Prior to these investigations, the model applied was validated by computing the transport of the passive scalar in the non-swirling (Johnson and Bennet, Report NASA CR-165574, UTRC Report R81-915540-9, 1981) and swirling (Roback and Johnson, NASA Contractor Report 168252, 1983) flow in a model combustor.

Modelling turbulent separated flow in the context of aerodynamic applications

In contrast to rapid advances in computing, numerical methods and visualisation, the predictive capabilities of statistical models of turbulence are limited and improve only slowly, despite much intensive research in the recent past. The intuitive nature of turbulence modelling, its strong reliance on calibration and validation, the extreme sensitivity of model performance to seemingly minor variations in modelling details and flow conditions, and the fact that the non-local dynamics of turbulence are not well captured by single-point closure, all conspire to make turbulence modelling an especially demanding component of CFD, but one that is crucially important for the correct prediction of complex flows. This applies in particular to separation from streamlined bodies, which is, from a computational point of view, the most challenging flow feature in aeronautical CFD. This paper reviews some aspects of the foundation and application of turbulence models to flows that relate to aeronautical practice, with particular emphasis being placed on turbulence-transport models at a closure level higher than that based on the Boussinesq-viscosity hypothesis. Following a review of basic modelling issues, including aspects of linear-eddy-viscosity two-equation modelling, some recent experience and current work on predicting separation from continuous surfaces with non-linear eddy-viscosity models and second-moment closure are reported. The predictive performance of several anisotropy-resolving models is illustrated by reference to computational solutions for a number of flows, both two- and three-dimensional, some compressible and others incompressible.

Prediction of Cascade Flows With Innovative Second-Moment Closures

We report on the performances of two second-moment turbulence closures in predicting turbulence and laminar-to-turbulent transition in turbomachinery flows. The first model considered is the one by Hanjalic and Jakirlic (HJ) [Comput. Fluids, 27(2), pp. 137–156 (1998)], which follows the conventional approach with damping functions to account for the wall viscous and nonviscous effect. The second is an innovative topology-free elliptic blending model, EBM [R. Manceau and K. Hanjalic, Phys. Fluids, 14(3), pp. 1–11 (2002)], here presented in a revised formulation. An in-house finite element code based on a parallel technique is used for solving the equation set [Borello et al., Comput. Fluids, 32, pp. 1017–1047 (2003)]. The test cases under scrutiny are the transitional flow on a flat plate with circular leading edge (T3L ERCOFTAC-TSIG), and the flow around a double circular arc (DCA) compressor cascade in quasi-off-design condition (i=−1.5°) [Zierke and Deutsch, NASA Contract Report 185118 (1989)]. The comparison between computations and experiments shows a satisfactory performance of the HJ model in predicting complex turbomachinery flows. The EBM also exhibits a fair level of accuracy, though it is less satisfactory in transition prediction. Nevertheless, in view of the robustness of the numerical formulation, the relative insensitivity to grid refinement, and the absence of topology-dependent parameters, the EBM is identified as an attractive second-moment closure option for computation of complex 3D turbulent flows in realistic turbomachinery configurations.

Measurement and simulation of viscous dissipation in the wave affected surface layer

In this study we compare turbulence parameters from field observations and model simulations specifically under the influence of weak to moderate wind forcing and breaking short waves. The experiment was performed during 12 days under very weak stratification at a fetch-limited lake in Switzerland. The near surface observations were obtained by using a quasi-free rising profiler which measured small scale shear and temperature fluctuations. We used a two-equation k-epsilon. turbulence model with an algebraic second-moment Closure scheme. The one-dimensional numerical model was extended to consider breaking waves by a shear-dependent parameterisation. The agreement of observed and simulated turbulence quantities is very promising. Especially well simulated is the enhanced turbulence level in the wave-affected-surface-layer (WASL) of a few din thickness. The logarithmic slope of the turbulent dissipation rate in this WASL was found to vary between -2.1 and -1.7. Below the WASL the classic law-of-the-wall was well reproduced by the data and the model.

A Lagrangian stochastic model for dispersion in stratified turbulence

In this paper we discuss the development of a Lagrangian stochastic model (LSM) for turbulent dispersion of a scalar (species). Given any tensorally linear second-moment closure (SMC) turbulence model we show how to derive a mathematically equivalent set of stochastic differential equations (SDEs), i.e., the second-moment equations constructed from these SDEs are exactly the same (within a realizability constraint) as the given SMC. This set of equations forms the LSM. Both turbulence anisotropy and buoyancy effects are incorporated by this method. In order to achieve the correct critical Richardson number and to obtain the simplest Lagrangian formulation, a revised set of constants for the isotropization of production model is proposed. They improve agreement with experiments. The LSM is applied to homogeneous shear flow with varying degrees of stratification. Our model is shown to capture important physics associated with buoyant flows and we also parametrize our results. Finally the form of the current model is compared with a few of the well-known Lagrangian stochastic models.

Predictions of flow and heat transfer in multiple impinging jets with an elliptic-blending second-moment closure

We present numerical computations of flow and heat transfer in multiple jets impinging normally on a flat heated surface, obtained with a new second-moment turbulence closure combined with an elliptic blending model of non-viscous wall blocking effect. This model provides the mean velocity and turbulent stress fields in very good agreement with PIV measurements. The exploration of several simpler closures for the passive thermal field, conducted in parallel, confirmed that the major prerequisite for the accurate prediction of the temperature field and heat transfer is to compute accurately the velocity and stress fields. If this is achieved, the conventional anisotropic eddy-diffusivity model can suffice even in complex flows. We demonstrate this in multiple-impinging jets where such a model combination provided the distribution of Nusselt number over the solid plate in good agreement with experiments. Extension of the elliptic blending concept to full second-moment treatment of the heat flux and its truncation to a quasi-linear algebraic model is also briefly discussed.

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.

On the equilibrium states predicted by second moment models in rotating, stably stratified homogeneous shear flow

The structural equilibrium behavior of the general linear second-moment closure model in a stably stratified, spanwise rotating homogeneous shear flow is considered with the aid of bifurcation analysis. A closed form equilibrium solution for the anisotropy tensor aij, dispersion tensor Kij, dimensionless scalar variance θ2̄/k(S/Sθ)2, and the ratio of mean to turbulent time scale ε/Sk is found. The variable of particular interest to bifurcation analysis, ε/Sk is shown as a function of the parameters characterizing the body forces: Ω/S (the ratio of the rotation rate to the mean shear rate) for rotation and Rig (the gradient Richardson number) for buoyancy; it determines the bifurcation surface in the ε/Sk−Ω/S−Rig space. It is shown, with the use of the closed form solution, that the Isotropization of Production model does not have a real and stable equilibrium solution when rotational and buoyant forces of certain magnitudes are simultaneously imposed on the flow. When this occurs, time integration of the turbulence model results in a diverging solution. A new set of scalar model coefficients that is consistent with experimental data, predicts turbulence decay past the critical gradient Richardson number Rigcr=0.25, and ensures the existence of stable, real solutions for all combinations of rotation and buoyancy is proposed.

The negatively buoyant turbulent wall jet: performance of alternative options in RANS modelling

The paper describes the application of different levels of turbulence closure and near-wall treatment to the computation of a 2D downward-directed wall jet that encounters a slow, upward-moving flow. The working fluid is water and the two streams may be at the same temperature or the wall-jet fluid may be hotter, leading to significant buoyant effects. The distance of penetration of the wall jet is found to be highly dependent on the turbulence model employed. It is established first that the new analytical wall function (AWF) developed by the authors [Int. J. Heat Fluid Flow 23 (2002) 148] leads to flow predictions in close agreement with a so-called `low-Reynolds-number' treatment where computations extend all the way to the wall. However, for some test cases, both sets of calculations (employing an eddy-viscosity model) indicate too great a penetration of the wall-jet into the opposing stream. The use of the AWF in conjunction with a second-moment closure, particularly one which satisfies the two-component-limit, gives generally closer agreement.

A new perspective on realizability of turbulence models

In the second-moment-closure (SMC) method of turbulence modelling, measures to ensure a realizable turbulence model are currently limited to constraining the Reynolds stress to physically plausible values. These constraints address neither the realizability of the other statistical moments (e.g. pressure–strain correlation) nor the underlying causes of unrealizable Reynolds stress. For achieving increased consistency with flow physics in SMC, we propose the additional requirement that the closure model for each of the unclosed statistical moments in the Reynolds stress equation be individually realizable. We then proceed to derive two realizability constraints on the rapid-pressure statistics: (i) the rapid pressure-gradient variance must be positive which leads to the requirement that the tensor must be positive semi-definite, and (ii) the rapid pressure–strain correlation closure must satisfy the Schwarz inequality. Calculations with currently popular models show that unrealizable rapid-pressure–strain correlation precedes unrealizable Reynolds stress. It is also demonstrated that when the Launder, Reece and Rodi (LRR) rapid-pressure–strain correlation model is modified (truncated) to satisfy the new constraints, Reynolds stress realizability is always preserved. These findings clearly indicate that an unrealizable closure model is the cause of Reynolds stress realizability violation and highlight the importance of the new constraints.

Airflow pattern and temperature distribution in a typical refrigerated truck configuration loaded with pallets

This work is part of a research activity aiming to improve and to optimise air-distribution systems in refrigerated vehicles in order to decrease the temperature differences throughout palletised cargos. This condition is essential in order to preserve the quality, safety and shelf life of perishable products. The present study reports on the numerical and experimental characterization of airflow within a semi-trailer enclosure loaded with pallets. The experiments were carried out on a reduced-scale (1:3.3) model of a refrigerated-vehicle trailer. The performance of ventilation and temperature homogeneity were characterized with and without supply air duct systems. Both configurations are extensively used in refrigerated transport. The numerical modelling of airflow was performed using the Computational Fluid Dynamics (CFD) code Fluent and a second-moment closure, the Reynolds stress model (RSM). The results obtained using the RSM model showed good agreement with the experimental data. Numerical and experimental results clearly show the importance of air ducts in decreasing temperature differences throughout the cargo.

타원방정식에 의한 벽면 부근의 난류열유속 모형화

A new second-moment closure model for turbulent heat fluxes is proposed on the basis of the elliptic equation. The new model satisfies the near-wall balance between viscous diffusion, viscous dissipation and temperature-pressure gradient correlation, and also has the characteristics of approaching its respective conventional high Reynolds number model far away from the wall. The predictions of turbulent heat transfer in a channel flow have been carried out with constant wall heat flux and constant wall temperature difference boundary conditions respectively. The velocity field variables are supplied from the DNS data and the differential equations only for the mean temperature and the scalar flux are solved by the present calculations. The present model is tested by direct comparisons with the DNS to validate the performance of the model predictions. The prediction results show that the behavior of the turbulent heat fluxes in the whole region is well captured by the present model.<br/>

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 &amp; Sons, Ltd.

A new BML-based RANS modelling for the description of gas turbine typical combustion processes

The work is concentrated on the formulation and validation of integral models within RANS framework for the numerical prediction of the premixed and partially premixed flames occurring in gas turbine combustors. The premixed combustion modelling is based on the BML approach coupled to the mixing transport providing variable equivalence ratio. Chemistry is described by means of ILDM model solving transport equations for reaction progress variables conditioned on the flame front. Multivariate presumed PDF model is used for the turbulence-chemistry interaction treatment. Turbulence is modelled using the second moment closure (SMC) and the standard κ-ε model as well. The influence of non-gradient turbulent transport is investigated comparing the gradient diffusion closure and the solution of the scalar flux transport equations. Different model combinations are assessed simulating several premixed and partially premixed flame configurations and comparing results to the experimental data. The proposed model provides good predictions particularly in combination with SMC.

Joint effects of geometry confinement and swirling inflow on turbulent mixing in model combustors: a second-moment closure study

An isothermal, incompressible, swirling flow in three generic combustor configurations featuring different inflow structure with respect to the circumferential velocity type (both configurations with annular and central swirling jet were considered), combustor confinement (in terms of expansion ratio ER) and swirl intensity (S) was studied computationally by means of Reynolds Averaged Navier-Stokes method (RANS), using second-Moment Closure (SMC) models. The work focuses on the investigation of the combined effects of the above-mentioned flow parameters on the mixing between a swirling annular jet (representing air stream) and the non-swirling inner jet (representing fuel) within a combustor. Both the basic high-Reynolds number SMC model, modified to account for the non-linearities in the pressure scrambling and dissipation processes, and a low-Reynolds number SMC model, accounting separately for the viscous and non-viscous wall blockage, were applied. Inflow conditions are computationally generated, rather than prescribed. In the course of the inflow data generation, a number of the equilibrium swirling flows in the concentric annulus and pipe geometries, from which coaxial and central swirling jets expand into a combustion chamber, are computed. The SMC results reproduced all important mean flow and turbulent features in good agreement with available experimental and LES data.

Recent Studies in Turbine Blade Cooling

Gas turbines are used extensively for aircraft propulsion, land‐based power generation, and industrial applications. Developments in turbine cooling technology play a critical role in increasing the thermal efficiency and power output of advanced gas turbines. Gas turbine blades are cooled internally by passing the coolant through several rib‐enhanced serpentine passages to remove heat conducted from the outside surface. External cooling of turbine blades by film cooling is achieved by injecting relatively cooler air from the internal coolant passages out of the blade surface in order to form a protective layer between the blade surface and hot gas‐path flow. For internal cooling, this presentation focuses on the effect of rotation on rotor blade coolant passage heat transfer with rib turbulators and impinging jets. The computational flow and heat transfer results are also presented and compared to experimental data using the RANS method with various turbulence models such as k‐ε, and second‐moment closure models. This presentation includes unsteady high free‐stream turbulence effects on film cooling performance with a discussion of detailed heat transfer coef‐ ficient and film‐cooling effectiveness distributions for standard and shaped film‐hole geometry using the newly developed transient liquid crystal image method.

Recent progress in nonlinear eddy-viscosity turbulence modeling

This article presents recent progresses in turbulence modeling in the Unit for Turbulence Simulation in the Department of Engineering Mechanics at Tsinghua University. The main contents include: compact Non-Linear Eddy-Viscosity Model (NLEVM) based on the second-moment closure, near-wall low-Re non-linear eddy-viscosity model and curvature sensitive turbulence model. The models have been validated in a wide range of complex flow test cases and the calculated results show that the present models exhibited overall good performance.

On an Improved Model for the Turbulent PBL

Abstract Cheng, Canuto, and Howard have presented a second-moment closure PBL model that appears to overcome some of the shortcomings of the existing Mellor and Yamada–type closure models. This note demonstrates that with a slight readjustment in the closure constants of the latter, more specifically the Kantha and Clayson closure model, they can be made to yield results very much similar to that of the Cheng et al. model.