Second-moment closure for modelling the near-wall turbulence in 3D mean flows

A second-moment closure for the near-wall turbulence is proposed. The limiting behaviour of this closure near a wall is consistent with that of the exact Reynolds-stress transport equations, and it converts asymptotically into a high-Reynolds-number closure remote from the wall. The closure is applied to a pressure-driven 3D transient channel flow. The predicted results are in fair agreement with the DNS data. The project supported by the National Natural Science Foundation of China     

Assessment of Turbulence Models for Predicting the Interaction Region in a Wall Jet by Reference to LES Solution and Budgets

Highly-resolved LES and experimental data for a plane wall jet are used to study the characteristics of turbulence-closure proposals, mainly within the framework of second-moment-transport modelling. The study is motivated by the observed importance of diffusive Reynolds-stress transport in the interaction region between the outer shear layer and the near-wall layer of the wall jet, which gives the near-wall flow characteristics that are very different from those of a conventional boundary layer. Comparisons are presented for mean-flow quantities, second moments and budgets. Also included are a priori studies of approximations for the pressure-velocity interaction, pressure-fluctuation-driven transport and turbulent transport of the Reynolds stresses by triple correlations, the last observed to contribute significantly to the stress budgets. The study reveals major defects in the closure approximations for the pressure-velocity interaction terms, especially in the near-wall region. These defects result in a poor representation by the particular second-moment closures investigated of even the integral and mean-flow characteristics of the wall jet.     

Tensorial characterisation of directional data in micromechanics

Information being dealt with in micro-mechanics is massive. Most of them are directional data. Macro-scale physical laws embedded with micro-scale fundamentals need to be developed in terms of the statistics of the micro-scale variable in a frame-indifferent form. Mathematical techniques and theories for characterizing the statistics of directional data with tensors are hence demanded. This is the main concern of the current paper. Starting with the general theory established in Kanatani (1984) of describing the directional distributions of orientations, mathematical formulations have been extended to address the directional distributions of vector-valued directional data, which is the most common data type being dealt with in micromechanical investigations. For vector-valued directional data, statistical analyses are required in regarding to both their directional probability density distribution and their representative values along each direction. The technique used here is to approximate these directional distributions by polynomials in unit directional vector n. The coefficients are in tensorial form and determined from observed directional data by applying the least square error criterion. These coefficient tensors serve as macro-scale variables representing the statistics of the micro-scale directional data, and are referred to as direction tensor. Orthogonal decompositions are addressed so that the coefficient tensor of different orders can be determined independently from each other. The coefficient tensors in the orthogonal decompositions are referred to as deviatoric direction tensor. The choice of sufficient approximation order is suggested. As an example, a general form of the stress–force–fabric relationship is derived for demonstrating the application of the proposed mathematical theory in the micro-mechanical investigation of the behaviour of granular materials.     

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ─ DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

In this two-part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically-based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N-dimensional (N-D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2- and 3-D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point-group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point-group symmetries are derived.     

Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

This paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of , where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two‐dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76× the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8× slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. Copyright © 2014 John Wiley & Sons, Ltd.     

Assessment of the SSG pressure-strain model in two-dimensional turbulent separated flows

Data from a number of benchmark two-dimensional turbulent separated flows are used to assess the performance of the Speziale, Sarkar and Gatski model [28] for the fluctuating pressure-strain correlations and, in particular, to determine whether the model can reproduce the effects of rigid boundaries without recourse to ad-hoc corrections. Comparative predictions are also obtained with a standard Reynolds-stress closure in which such correlations are necessary and these serve to distinguish the models' true performance from spurious computational artifacts. The models were implemented in a finite-volume method which solves the Navier-Stokes equations on non-orthogonal grids with colocated storage arrangement. Details of the numerical treatments adopted to obtain stable and fast-converging solutions are described and the outcome of rigorous grid-independence checks are reported for each test case. It is found that both models yield essentially similar results for the mean flow and turbulence fields. The implications of this result to the choice of Reynolds-stress closure for complex engineering simulations are discussed.     

Assessing the Role of Curvature Effects in the Representation of Anisotropy Transport in Algebraic Reynolds-Stress Modelling Applied to Separated Flow

A “curvature correction”, in the sense used herein, is model fragment that is intended to account for some proportion of the error that is introduced into explicit algebraic Reynolds-stress models as a consequence of ignoring the convective transport of stress anisotropy within a framework in which the velocity vector and stress tensor are represented in terms of Cartesian components in highly-curved flow conditions. The present paper examines the ability of such corrections to represent this error, based on a-priori investigations of representative 2-D and 3-D massively separated flows for which the ‘exact’ level of the stress transport is known. In essence, the corrections reflect the assumption that the convection of the Reynolds-stress components, or the associated strain-tensor components, expressed in terms of curvature-oriented coordinates, is negligible. The analysis shows, first, that in general recirculating flows, the contribution of anisotropy transport to the stress balance is generally small, so that any form of related correction is of little consequence. Second, the variants of curvature correction examined correlate poorly with the real anisotropy convection. Thus, while these curvature corrections are useful in very particular conditions, such as flow in highly-curved ducts, they are not generality effective – indeed, possibly counterproductive – and cannot be recommended for inclusion in general numerical schemes.     

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     

From single-scale turbulence models to multiple-scale and subgrid-scale models by Fourier transform

A theoretical method based on mathematical physics formalism that allows transposition of turbulence modeling methods from URANS (unsteady Reynolds averaged Navier–Stokes) models, to multiple-scale models and large eddy simulations (LES) is presented. The method is based on the spectral Fourier transform of the dynamic equation of the two-point fluctuating velocity correlations with an extension to the case of non-homogenous turbulence. The resulting equation describes the evolution of the spectral velocity correlation tensor in wave vector space. Then, we show that the full wave number integration of the spectral equation allows one to recover usual one-point statistical closure whereas the partial integration based on spectrum splitting gives rise to partial integrated transport models (PITM). This latter approach, depending on the type of spectral partitioning used, can yield either a statistical multiple-scale model or subfilter transport models used in LES or hybrid methods, providing some appropriate approximations are made. Closure hypotheses underlying these models are then discussed by reference to physical considerations with emphasis on identification of tensorial fluxes that represent turbulent energy transfer or dissipation. Some experiments such as the homogeneous axisymmetric contraction, the decay of isotropic turbulence, the pulsed turbulent channel flow and a wall injection induced flow are then considered as typical possible applications for illustrating the potentials of these models.      

Some developments in computational modeling of turbulent flows

In this paper, some recent developments of two turbulence closure schemes at ICOMP, NASA Lewis will be discussed. One is the Reynolds-stress algebraic equation model and the other is the Reynolds-stress transport equation model. Various model constraints required by the rapid distortion theory, the invariant theory and the realizability principle, etc. will be described in the model development. The models discussed are for high-turbulent Reynolds number flows, so that the near-wall turbulence and the low-Reynolds-number turbulence are not discussed here.