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     

LES prediction of space-time correlations in turbulent shear flows

We compare the space-time correlations calculated from direct numerical simulation(DNS) and large-eddy simulation(LES) of turbulent channel flows.It is found from the comparisons that the LES with an eddy-viscosity subgrid scale(SGS) model over-predicts the space-time correlations than the DNS.The overpredictions are further quantified by the integral scales of directional correlations and convection velocities.A physical argument for the overprediction is provided that the eddy-viscosity SGS model alone does not includes the backscatter effects although it correctly represents the energy dissipations of SGS motions.This argument is confirmed by the recently developed elliptic model for space-time correlations in turbulent shear flows.It suggests that enstrophy is crucial to the LES prediction of spacetime correlations.The random forcing models and stochastic SGS models are proposed to overcome the overpredictions on space-time correlations.     

Shroud heat transfer measurements inside a heated multiple rotating cavity with axial throughflow

Experimental measurements of heat transfer are made from the inner peripheral surface of a rotating test rig designed to be similar to a gas turbine high pressure compressor internal air system. The test rig comprises a number of annular discs sealed at their periphery by a shroud. An axial throughflow of cooling air enters the test rig and flows through the annular section between the disc bores and a central shaft. Tests were carried out for the following range of rotational speeds and axial throughflow rates: 540 < N_R < 10,800 rev/min and (corresponding to the range of rotational and axial Reynolds numbers 4 × 10⁵ < Re < 7.7 × 10⁶ and 3.3 × 10⁴ < Re_z < 2.2 × 10⁵).

The shroud Nusselt numbers are found to depend on the shroud Grashof number. They are relatively insensitive to changes in axial Reynolds number and two geometrically similar cavities give similar values of Nusselt number. The heat transfer from the shroud is governed by the mechanism of free convection. It is recommended that a modified form of a correlation for Rayleigh–Bénard convection in a gravitational force field be used, with appropriate modification, to predict shroud heat transfer.     

A Non-Linear SGS Model Based On The Spatial Velocity Increment

A new subgrid scale (SGS) modelling concept for large-eddy simulation (LES) of incompressible flow is proposed based on the three-dimensional spatial velocity increment δ u _i . The new model is inspired by the structure function formulation developed by Métais and Lesieur [39] and applied in the context of the scale similarity type formulation. First, the similarity between the SGS stress tensor τ _ij and the velocity increment tensor Q _ij = δ u _i δ u _j is analyzed analytically and numerically using a priori tests of fully developed pipe flow at Re _τ = 180. Both forward and backward energy transfers between resolved and unresolved scales of the flow are well predicted with a SGS model based on Q _ij . Secondly, a posteriori tests are performed for two families of turbulent shear flows. LES of fully developed pipe flow up to Re _τ = 520 and LES of round turbulent jet at Re _D = 25000 carried out with a dynamic version of the model provide promising results that confirm the power of this approach for wall-bounded and free shear flows.     

A dynamic two-equation Sub Grid Scale model

In this paper we demonstrate that the transport equation of the generalised subgrid scale (SGS) turbulent stress tensor is form-invariant but not frame-indifferent under Euclidean transformations of the frame. A new closure equation between the generalized SGS turbulent stress tensor and the resolved kinematic quantities is proposed. The closure equation at the basis of the proposed model (Two-Equation Model, TEM): a) respects the principle of the turbulence frame indifference [1]; b) takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales; c) removes any balance assumption between the production and dissipation of SGS turbulent kinetic energy; d) assumes scale similarity in the definition of the second-order tensor representing the turbulent velocity scales. In the proposed model: a) the closure coefficient C which appears in the constitutive equation is uniquely determined without using Germanos dynamic procedure [2]; b) the generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy (which is calculated by means of its balance equation) and the modified Leonard tensor; c) the viscous dissipation of the generalized SGS turbulent kinetic energy is calculated by solving the balance equation. The proposed model is tested for a turbulent channel flow at Reynolds numbers (based on friction velocity and channel half-width) ranging from 180 to 2340.Received: 11 February 2004, Accepted: 20 August 2004, Published online: 22 February 2005PACS: 02.60.Cb, 47.27.Eq, 47.11. + j Correspondence to: F. Gallerano     

A general optimal formulation for the dynamic Smagorinsky subgrid‐scale stress model

In this paper, a general optimal formulation for the dynamic Smagorinsky subgrid‐scale (SGS) stress model is reported. The Smagorinsky constitutive relation has been revisited from the perspective of functional variation and optimization. The local error density of the dynamic Smagorinsky SGS model has been minimized directly to determine the model coefficient C_S. A sufficient and necessary condition for optimizing the SGS model is obtained and an orthogonal condition (OC), which governs the instantaneous spatial distribution of the optimal dynamic model coefficient, is formulated. The OC is a useful general optimization condition, which unifies several classical dynamic SGS modelling formulations reported in the literature. In addition, the OC also results in a new dynamic model in the form of a Picard's integral equation. The approximation tensorial space for the projected Leonard stress is identified and the physical meaning for several basic grid and test‐grid level tensors is systematically discussed. Numerical simulations of turbulent Couette flow are used to validate the new model formulation as represented by the Picard's integral equation for Reynolds numbers ranging from 1500 to 7050 (based on one half of the velocity difference of the two plates and the channel height). The relative magnitudes of the Smagorinsky constitutive parameters have been investigated, including the model coefficient, SGS viscosity and filtered strain rate tensor. In general, this paper focuses on investigation of fundamental mathematical and physical properties of the popular Smagorinsky constitutive relation and its related dynamic modelling optimization procedure. Copyright © 2005 John Wiley & Sons Ltd.     

Mathematical and Physical Constraints on Large-Eddy Simulations

We examine the mathematical constraints on the filtering operation in large-eddy simulation (LES), and the physical constraints on the formulation of subgrid scale (SGS) models. The mathematical constraints on filtering arise from the assumption of commutativity between the filtering operation and the differential operators: this breaks down for finite domains and for non-uniform filter kernels. The physical constraints on the modelling are twofold; invariance principles require any model to be invariant under a group of transformations: the most stringent of these being the principle of frame indifference. In addition, realizability requires that the SGS model should have the same mathematical properties as the term it replaces, which leads to further constraints on the modelling. Finally, we investigate the effects of commutation and realizability for two simulated cases: forced homogeneous isotropic turbulence in a box and fully developed channel flow. In particular, we look for correlations between the commutation errors and the non-realizable regions, and the large-scale resolved flow structures in the simulations. Received: 17 June 1996 and accepted 3 April 1997     

Usability of explicit filtering in large eddy simulation with a low‐order numerical scheme and different subgrid‐scale models

Based on a priori tests, in large eddy simulation (LES) of turbulent fluid flow, the numerical error related to low‐order finite‐difference‐type methods can be large in comparison with the effect of subgrid‐scale (SGS) model. Explicit filtering has been suggested to reduce the error, and it has shown promising results in a priori studies and in some simulations with fourth‐order method. In this paper, the effect of explicit filtering on the total simulation error is studied together with a second‐order scheme, where the numerical error should be even larger. The fully developed turbulent channel flow between two parallel walls is used as a test case. Rather simple SGS models are applied, because these models are most likely used in practical applications of LES. Explicit filtering is here applied to the non‐linear convection term of the Navier–Stokes equations, four three‐dimensional filter functions are applied, and the effect of filtering is separated from the effect of SGS modelling. It is shown that the effect of filtering is rather large and smooth filters introduce an additional error component that increases the total simulation error. Finally, filtering via subfilter‐scale modelling is applied, and it is shown that this approach performs better. However, the large‐frequency components of the resolved flow field are not as effectively damped as when the non‐linear convection term is filtered. Copyright © 2007 John Wiley & Sons, Ltd.     

基于人工神经网络的亚格子应力建模

亚格子(SGS)应力建模在湍流大涡模拟(LES)中有着极为重要的作用. 传统亚格子应力模型存在相对误差较大、耗散过强等问题. 近年来, 计算机技术的发展使得人工神经网络(ANN)等机器学习方法逐渐成为亚格子应力建模型的新研究范式. 本文着重考虑滤波宽度及雷诺数影响, 在不可压缩槽道湍流中建立了亚格子应力的ANN模型. 该模型以滤波后的直接数值模拟(fDNS)流场物理量及滤波尺度为输入信息, 相应滤波尺度下的亚格子应力为输出量. 通过对不同滤波尺度及不同雷诺数数据的训练, ANN模型能够给出与直接数值模拟(DNS)高度吻合的亚格子应力. 此外, 模型在亚格子耗散等非ANN建模量上也有着优异的预测性能, 与基于DNS获得的对应物理量的相关系数大都在0.9以上, 较梯度模型及Smagorinsky模型有明显提升. 在后验测试中, ANN模型对流向平均速度剖面的预测同样优于梯度模型、Smagorinsky模型及隐式大涡模拟(ILES)等传统LES模型. 在脉动速度均方根预测方面, 除了某些法向位置外ANN模型的性能整体上相对其他3个模型有所提升. 然而, 随着网格尺度的增大ANN模型预测的结果与fDNS结果的偏差逐渐增大. 总之, ANN方法在发展高精度亚格子应力模型上具有很大的潜力.      

Assessment of the modulated gradient model in decaying isotropic turbulence

A recently introduced nonlinear model undergoes evaluations based on two isotropic turbulent cases: a University of Wiscosion-Madison case at a moderate Reynolds number and a Johns Hopkins University case at a high Reynolds number. The model uses an estimation of the subgrid-scale (SGS) kinetic energy to model the magnitude of the SGS stress tensor, and uses the normalized velocity gradient tensor to model the structure of the SGS stress tensor. Testing is performed for the first case through a comparison between direct numerical simulation (DNS) results and large eddy simulation (LES) results regarding resolved kinetic energy and energy spectrum. In the second case, we examine the resolved kinetic energy, the energy spectrum, as well as other key statistics including the probability density functions of velocities and velocity gradients, the skewness factors, and the flatness factors. Simulations using the model are numerically stable, and results are satisfactorily compared with DNS results and consistent with statistical theories of turbulence.     

Numerical simulation of the noise generated by a low Mach number, low Reynolds number jet

In this paper we study the sound field produced by a turbulent round jet with a Mach number of 0.6 based on the centerline velocity and the ambient speed of sound c_∞. The turbulent flow field is found by solving the fully compressible Navier–Stokes equations with help of high-order compact finite difference schemes. It is shown that the simulated flow field is in good agreement with experiments. The corresponding sound field has been obtained with help of the Lighthill equation using two different formulations for the Lighthill stress tensor T_ij. In the first formulation of T_ij the fluctuating density is taken into account. In the second formulation the density is assumed to be constant. As an additional check we have also performed an acoustic calculation using a formulation in which a homogeneous wave equation is solved. The boundary conditions for this homogeneous wave equation are obtained from the numerical simulation of the Navier–Stokes equation. The results obtained with both formulations of the Lighthill stress tensor are nearly identical. This implies that an incompressible formulation of the conservations laws could be used to predict jet noise at low Mach numbers.     

Implicit large-eddy simulation applied to turbulent channel flow with periodic constrictions

The subgrid-scale (SGS) model in a large-eddy simulation (LES) operates on a range of scales which is marginally resolved by discretization schemes. Accordingly, the discretization scheme and the subgrid-scale model are linked. One can exploit this link by developing discretization methods from subgrid-scale models, or the converse. Approaches where SGS models and numerical discretizations are fully merged are called implicit LES (ILES). Recently, we have proposed a systematic framework for the design, analysis, and optimization of nonlinear discretization schemes for implicit LES. In this framework parameters inherent to the discretization scheme are determined in such a way that the numerical truncation error acts as a physically motivated SGS model. The resulting so-called adaptive local deconvolution method (ALDM) for implicit LES allows for reliable predictions of isotropic forced and decaying turbulence and of unbounded transitional flows for a wide range of Reynolds numbers. In the present paper, ALDM is evaluated for the separated flow through a channel with streamwise-periodic constrictions at two Reynolds numbers Re = 2,808 and Re = 10,595. We demonstrate that, although model parameters of ALDM have been determined for isotropic turbulence at infinite Reynolds number, it successfully predicts mean flow and turbulence statistics in the considered physically complex, anisotropic, and inhomogeneous flow regime. It is shown that the implicit model performs at least as well as an established explicit model.      

Geometrical Properties of the Resolved-Scale Velocity and Temperature Fields Predicted using Large-Eddy Simulation

In this paper, local geometrical properties of the velocity and temperature fields of combined forced and natural convection in a vertical slot are studied using large-eddy simulation based on both numerical and analytical approaches. Previous studies on turbulence geometrical statistics appearing in the literature have primarily focused on either isothermal or isotropic turbulent flows; whereas in this work, we extend the scope of research to investigation of a wall-bounded thermal flow. In particular, we focus on studying the resolved helicity, enstrophy generation, local vortex stretching, and a variety of characteristic geometrical alignment patterns between the resolved velocity, vorticity, temperature gradient, subgrid-scale heat flux and the eigenvectors of the resolved strain rate tensor. In order to quantify the effect of buoyancy on the geometrical properties of the thermal flow field, a systematic comparative analysis has been conducted based on three different flow regimes (i.e., viscous sublayer, buffer layer and logarithmic layer) in both the hot and cold wall regions. The near-wall restriction on the geometrical property of the thermal flow field has been analyzed and some interesting wall-limiting geometrical alignment patterns in the form of Dirac delta functions are also reported.     

A secant-viscosity composite model for the strain-rate sensitivity of nanocrystalline materials

In order to address the strain-rate sensitivity of nanocrystalline solids, a secant-viscosity composite model is developed in this article. The microgeometry of the composite is taken to consist of the grain-interior phase and the grain-boundary affected zone (GBAZ) as suggested by Schwaiger et al. [Schwaiger, R., Moser, B., Dao, M., Chollacoop, N., Suresh, S., 2003. Some critical experiments on the strain-rate sensitivity of nanocrystalline nickel. Acta Mater. 51, 5159–5172], while the constituent properties are modeled by a unified viscoplastic constitutive law. The drag stress of the grain interior is assumed to follow the Hall–Petch relation, but that of the GBAZ is independent of grain size, d. Then in terms of the secant viscosity of the constituent phases, the strain-rate sensitivity of the nanocrystalline solid is determined with the help of a linear viscous comparison composite and a field-fluctuation approach. To test the applicability of the developed model, it is applied to predict the strain-rate effect of a nanocrystalline Ni, and the grain-size dependence of its stress–strain relations. Our theoretical calculations indicate that the tensile strength of a nanocrystalline Ni with d = 40 nm is about five times that of a microcrystalline one with d = 10 μm under the same strain rate of , and that the nanocrystalline Ni exhibits a much stronger strain-rate effect. These predictions are found to be consistent with the experimental data of Schwaiger et al. Possible grain-size softening with further grain-size reduction such as reported in molecular dynamic simulations is also demonstrated.     

A new development of the dynamic procedure in large-eddy simulation based on a Finite Volume integral approach. Application to stratified turbulence

A Finite Volume-based large-eddy simulation method is proposed along with a suitable extension of the dynamic modelling procedure that takes into account for the integral formulation of the governing filtered equations. Discussion about the misleading interpretation of FV in some literature is addressed. Then, the classical Germano identity is congruently rewritten in such a way that the determination of the modelling parameters does not require any arbitrary averaging procedure and thus retains a fully local character. The numerical modelling of stratified turbulence is the specific problem considered in this study, as an archetypal of simple geophysical flows. The original scaling formulation of the dynamic sub-grid scale model proposed by Wong and Lilly (Phys. Fluids 6(6), 1994) is suitably extended to the present integral formulation. This approach is preferred with respect to traditional ones since the eddy coefficients can be independently computed by avoiding the addition of unjustified buoyancy production terms in the constitutive equations. Simple scaling arguments allow us not to use the equilibrium hypothesis according to which the dissipation rate should equal the sub-grid scale energy production. A careful a priori analysis of the relevance of the test filter shape as well as the filter-to-grid ratio is reported. Large-eddy simulation results are a posteriori compared with a reference pseudo-spectral direct numerical solution that is suitably post-filtered in order to have a meaningful comparison. In particular, the spectral distribution of kinetic and thermal energy as well as the viscosity and diffusivity sub-grid scale profiles are illustrated. The good performances of the proposed method, in terms of both evolutions of global quantities and statistics, are very promising for the future development and application of the method.     

Effects of the Similarity Model in Finite-Difference LES of Isotropic Turbulence Using a Lagrangian Dynamic Mixed Model

A Lagrangian dynamic formulation of the mixed similarity subgrid (SGS) model for large-eddy simulation (LES) of turbulence is proposed. In this model, averaging is performed over fluid trajectories, which makes the model applicable to complex flows without directions of statistical homogeneity. An alternative version based on a Taylor series expansion (nonlinear mixed model) is also examined. The Lagrangian models are implemented in a finite difference code and tested in forced and decaying isotropic turbulence. As comparison, the dynamic Smagorinsky model and volume-averaged formulations of the mixed models are also tested. Good results are obtained, except in the case of low-resolution LES (32³) of decaying turbulence, where the similarity coefficient becomes negative due to the fact that the test-filter scale exceeds the integral scale of turbulence. At a higher resolution (64³), the dynamic similarity coefficient is positive and good agreement is found between predicted and measured kinetic energy evolution. Compared to the eddy viscosity term, the similarity or the nonlinear terms contribute significantly to both SGS dissipation of kinetic energy and SGS force. In order to dynamically test the accuracy of the modeling, the error incurred in satisfying the Germano identity is evaluated. It is found that the dynamic Smagorinksy model generates a very large error, only 3% lower than the worst-case scenario without model. Addition of the similarity or nonlinear term decreases the error by up to about 50%, confirming that it represents a more realistic parameterization than the Smagorinsky model alone.     

Regularization modeling for LES of separated boundary layer flow

Regularization models for the turbulent stress tensor are applied to mixing and separated boundary layers. The Leray and the NS-α models in large-eddy simulation (LES) are compared to direct numerical simulation (DNS) and (dynamic) eddy-viscosity models. These regularization models are at least as accurate as the dynamic eddy-viscosity model, and can be derived from an underlying dynamic principle. This allows one to maintain central transport properties of the Navier-Stokes equations in the model and to extend systematically toward complex applications. The NS-α model accurately represents the small-scale variability, albeit at considerable resolution. The Leray model was found to be much more robust, allowing simulations at high Reynolds number. Leray simulations of a separated boundary layer are shown for the first time. The strongly localized transition to turbulence that arises under a blowing and suction region over a flat plate was captured accurately, quite comparable to the dynamic model. In contrast, results obtained with the Smagorinsky model, either with or without Van Driest damping, yield considerable errors, due to its excessive dissipation.     

A new dynamic subgrid-scale model using artificial neural network for compressible flow

The subgrid-scale (SGS) kinetic energy has been used to predict the SGS stress in compressible flow and it was resolved through the SGS kinetic energy transport equation in past studies. In this paper, a new SGS eddy-viscosity model is proposed using artificial neural network to obtain the SGS kinetic energy precisely, instead of using the SGS kinetic energy equation. Using the infinite series expansion and reserving the first term of the expanded term, we obtain an approximated SGS kinetic energy, which has a high correlation with the real SGS kinetic energy. Then, the coefficient of the modelled SGS kinetic energy is resolved by the artificial neural network and the modelled SGS kinetic energy is more accurate through this method compared to the SGS kinetic energy obtained from the SGS kinetic energy equation. The coefficients of the SGS stress and SGS heat flux terms are determined by the dynamic procedure. The new model is tested in the compressible turbulent channel flow. From the a posterior tests, we know that the new model can precisely predict the mean velocity, the Reynolds stress, the mean temperature and turbulence intensities, etc.     

Variational formulation of ideal fluid flows according to gauge principle

On the basis of the gauge principle of field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian Λ_A including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate a. Without Λ_A, the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian Λ_A yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the Λ_A characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian a space and the Eulerian x space. In contrast, with translation symmetry alone, there is an arbitrariness in the transformation between these spaces.     

A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD

An eddy-viscosity model based on Durbin’s elliptic relaxation concept is proposed, which solves a transport equation for the velocity scales ratio instead of , thus making the model more robust and less sensitive to grid nonuniformities. Computations of flows and heat transfer in a plane channel, behind a step and in a round impinging jet show all satisfactory results.