M.A. Leschziner

Assessment of turbulence-transport models including non-linear rng eddy-viscosity formulation and second-moment closure for flow over a backward-facing step

Abstract A computational study is performed in which the predictive capabilities of a range of eddy-viscosity and second-moment-closure models are examined by reference to a separated flow behind a backward-facing step in an expanding channel. The models include three second-moment-closure variants, all being of the ‘Launder-Reece-Rodi’ type, two RNG k—ϵ forms, one combining the RNG approach with a non-linear eddy-viscosity formulation, and a low-Re k—ϵ model. The study demonstrates that to achieve a solution similar to that returned by second-moment closure, the RNG formulation needs to be implanted into a non-linear eddy-viscosity framework; neither returns, on its own, the correct behaviour, not even for mean-flow features. Moreover, relatively minor variations within second-moment closure—specifically, such relating to wall-induced effects on turbulence isotropisation and to stress diffusion—can significantly alter the overall performance of the closure. All models specifically designed to return realistic solutions for normal stresses seriously over-estimate anisotropy.

Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions

High-resolution large-eddy simulation is used to investigate the mean and turbulence properties of a separated flow in a channel constricted by periodically distributed hill-shaped protrusions on one wall that obstruct the channel by 33% of its height and are arranged 9 hill heights apart. The geometry is a modification of an experimental configuration, the adaptation providing an extended region of post-reattachment recovery and allowing high-quality simulations to be performed at acceptable computing costs. The Reynolds number, based on the hill height and the bulk velocity above the crest is 10595. The simulated domain is streamwise as well as spanwise periodic, extending from one hill crest to the next in the streamwise direction and over 4.5 hill heights in the spanwise direction. This arrangement minimizes uncertainties associated with boundary conditions and makes the flow an especially attractive generic test case for validating turbulence closures for statistically two-dimensional separation. The emphasis of the study is on elucidating the turbulence mechanisms associated with separation, recirculation reattachment, acceleration and wall proximity. Hence, careful attention has been paid to resolution, and a body-fitted, low-aspect-ratio, nearly orthogonal numerical grid of close to 5 million nodes has been used. Unusually, the results of two entirely independent simulations with different codes for identical flow and numerical conditions are compared and shown to agree closely. Results are included for mean velocity, Reynolds stresses, anisotropy measures, spectra and budgets for the Reynolds stresses. Moreover, an analysis of structural characteristics is undertaken on the basis of instantaneous realizations, and links to features observed in the statistical results are identified and interpreted. Among a number of interesting features, a distinct ‘splatting’ of eddies on the windward hill side following reattachment is observed, which generates strong spanwise fluctuations that are reflected, statistically, by the spanwise normal stress near the wall exceeding that of the streamwise stress by a substantial margin, despite the absence of spanwise strain.

Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions

Large eddy simulations are presented for the flow in a periodic channel segment, which is laterally constricted by hill-shaped obstructions on one wall, having a height of 33% of the unconstricted channel. The Reynolds number, based on channel height, is 21,560. Massive separation thus arises on the hills’ leeward sides, the length of which is about 50% of that of the periodic segment. After reattachment, the flow is allowed to recover over about 30% of the segment length before being strongly accelerated over the windward side of the next hill. The principal challenge of this flow arises from the separation on the curved hill surface and the fact that the reattachment point, and hence the whole flow, are highly sensitive to the separation process. Simulations were performed with three grids, six subgrid-scale models and eight practices of approximating the near-wall region in simulations on the two coarser grids. These were supported by wall-resolved and wall-function simulations for fully-developed channel flow. The principal objective is to identify the sensitivity of the predictive accuracy to resolution and modelling issues. Coarse-grid simulations are judged by reference to data derived from two independent highly-resolved simulations obtained over identical meshes of close to 5 million nodes. Similarly, coarser-grid simulations were also performed with the two codes to enhance confidence in the results. The principal message emerging from the simulations is that grid resolution, especially in the streamwise direction around the mean separation position, has a very strong influence on the reattachment behaviour and hence the whole flow. This has serious implications for even more challenging high-Reynolds-number cases in which separation occurs from gently curved surfaces. The near-wall treatment, including the details of the numerical implementation of the wall laws, is also shown to be influential, most prominently on the coarsest grid. The application of the no-slip conditions at the wall at which separation occurs is found to cause substantial errors, especially in conjunction with poor streamwise resolution, even if the wall-nearest grid nodes are within the semi-viscous sublayer, in the range 5≲y+≲15. The sensitivity to subgrid-scale modelling is shown to be more modest, with those models returning relatively low subgrid-scale viscosity giving the closest accord with the highly-resolved simulation.

A general non-orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure, Part 1: Computational implementation

Abstract A computational procedure has been developed for predicting separated turbulent flows in complex two-dimensional and three-dimensional geometries. The procedure is based on the fully conservative, structured finite volume framework within which the volumes are non-orthogonal and collocated such that all flow variables are stored at one and the same set of nodes. To ease the task of discretization and to enhance the conservative property of the scheme, a Cartesian or datum-line-adapted decomposition of the velocity field has been used. The solution algorithm is iterative in nature, approaching the steady solution with the aid of a pressure-correction scheme. Convection is approximated with a range of schemes, among them higher-order upstream-weighted approximations and a TVD-type MUSCL form, the last applied principally to the transport equations governing turbulence properties. Effects of turbulence are represented either by two-equation eddy-viscosity models or by a full Reynolds-stress-transport closure. The former category includes both high- and low-Reynolds-number variants in two- as well as three-dimensional conditions. To achieve a stable implementation of the Reynolds-stress equations, a special interpolation practice, analogous to that of Rhie and Chow for momentum [1], has been introduced within the general framework. The procedure has been formulated so as to apply to both incompressible and compressible flows. The latter may contain shocks and highly supersonic regions. To achieve this range of applicability, the retarded-density concept has been combined with the basic pressure-based algorithm to capture shock waves. A ‘Full Approximation Multigrid’ scheme for convergence acceleration has been incorporated and applied in conjunction with all turbulence models including second-moment closure. The present first part of a twin paper focuses on numerical and turbulence-modelling issues. In Part 2, computational results are presented for six representative applications out of fifteen recently predicted with the algorithm within an extensive validation exercise.

Practical evaluation of three finite difference schemes for the computation of steady-state recirculating flows

Abstract The paper examines the performance of three steady-state finite difference formulations, namely: 1. (i) the hybrid central/upwind differencing scheme, 2. (ii) the hybrid central/skew-upwind differencing scheme and 3. (iii) the quadratic, upstream-weighted differencing scheme. Schemes (ii) and (iii) were recently proposed as superior alternatives to the first for discretizing the convection terms in the numerical simulation of recirculating flows. The discretization methods used in schemes (ii) and (iii) specifically aim at a substantial reduction or elimination of the artificial diffusion from which the first scheme suffers in the simultaneous presence of convective dominance and streamline-to-grid skewness. Computations are presented for a number of test configurations, both linear and nonlinear. It is shown that in all cases schemes (ii) and (iii) yield similar results which are superior to those obtained with scheme (i), although the former schemes are found to suffer to a limited extent from boundedness problems. The schemes are finally applied to two confined, laminar, recirculating flows for which experimental data are available. It is found that in these cases artificial diffusion is essentially uninfluential because of a number of features particular to laminar flows of the kind examined.

Computation of highly swirling confined flow with a Reynolds stress turbulence model

The ability of a turbulence model to capture the interaction between swirl and the turbulent stress field is, therefore, crucial to the predictive performance of the computatinal scheme as a whole. A finite-volume procedure is used here to contrast the performance of the k-epsilon eddy-viscosity model with that of a Reynolds-stress transport closure. It is shown that the former returns a seriously excessive level of turbulent diffusion and misrepresents the experimentally observed flow characteristics. In contrast, the Reynolds-stress model successfully captures the subcritical nature of the flow by returning significantly lower levels of the shear stress components and predicts velocity and turbulence fields that are in good agreement with corresponding measurements. 22 references.

A new low-Reynolds-number nonlinear two-equation turbulence model for complex flows

A new nonlinear, low-Reynolds-number k–e turbulence model is proposed. The stress–strain relationship is formed by successive iterative approximations to an algebraic Reynolds-stress model. Truncation of the process at the third iteration yields an explicit expression for the Reynolds stresses that is cubic in the mean velocity gradients and circumvents the singular behaviour that afflicts the exact solution at large strains. Free coefficients are calibrated – as functions of y∗ – by reference to direct numerical simulation (DNS) data for a channel flow. By using the nonlinear stress–strain relationship, the sublayer behaviour of all turbulent stresses is reproduced. The extension to nonequilibrium conditions is achieved by sensitising the model coefficients to strain and vorticity invariants on the basis of formal relations derived from the algebraic Reynolds-stress model. The new model has been applied to a number of complex two dimensional (2-D) flows, and its performance is compared to that of other linear and nonlinear eddy-viscosity closures.

Reynolds-stress-transport modeling for compressible aerodynamics applications

Progress is reported in the development of a nonlinear Reynolds-stress-transport model for compressible, turbulent flow. The focus is on a variation of a particular cublc model that does not require the usual topography-related parameters, such as normal-to-wall vectors. However, certain wall-proximity corrections that have been used in the model to replace conventional wall-reflection terms display the wrong response to shocks, which are falsoly interpreted as localized regions of strong inhomogeneity. A modified cubic variant is proposed that allows integration across the semiviscous sublayer and incorporates additional constraints to guard against unphysical response of the pressure-strain model in the vicinity of shock waves. The modified model is applied to both two- and three-dimensional compressible flows, involving shock-wave/boundary-layer interaction, and is shown to yield generally favorable results

A MULTIBLOCK IMPLEMENTATION OF A NON‐ORTHOGONAL, COLLOCATED FINITE VOLUME ALGORITHM FOR COMPLEX TURBULENT FLOWS

A multiblock algorithm for general 2D and 3D turbulent flows is introduced and applied to three cases : a compressor cascade passage, a two-element high-lift aerofoil and a round-to-square transition duct. The method is a generalization of a single-block scheme which is based on a non-orthogonal, fully collocated finite volume framework, applicable to incompressible and compressible flows and incorporating a range of turbulence transport models, including second-moment closure. The multiblock implementation is essentially block-unstructured, each block having its own local co-ordinate system unrelated to those of its neighbours. Any one block may interface with more than one neighbour along any one block face. Interblock communication is handled by connectivity matrices and effected via a two-cell overlap region along block boundaries in which halo data reside. The algorithm and the associated data communication are explained in detail, and its effectiveness is verified, with particular reference to improved numerical resolution and parallel computing.

Advanced Turbulence Modelling of Separated Flow in a Diffuser

The paper describes an investigation into the predictive performance of linear and non-linear eddy-viscosity models and differential stress-transport closures for separated flow in a nominally two-dimensional, asymmetric diffuser. The test case forms part of a broader collaborative exercise between academic and industrial partners. It is demonstrated that advanced turbulence models using strain-dependent coefficients and anisotropy-resolving closure offer tangible advantages in predictive capability, although the quality of their performance can vary significantly, depending on the details of closure approximations adopted. Certain features of the flow defy resolution by any of the closures investigated. In particular, no model resolves correctly the flow near the diffusers inclined wall immediately downstream of the inlet corner, which may reflect the presence of a “flapping” motion associated with a highly-localised process of unsteady separation and reattachment.