Non-linear Eddy-viscosity Models Research Articles

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.

102 1方程式非線形渦動粘性モデルの開発(流体 I,2.学術講演)

102 1方程式非線形渦動粘性モデルの開発(流体 I,2.学術講演)

A hybrid LES/RANS approach using an anisotropy-resolving algebraic turbulence model

In order to derive a possible path for developing a large eddy simulation (LES) applicable to high Reynolds-number turbulent flows, a hybrid approach connecting LES with the Reynolds-averaged Navier–Stokes (RANS) modeling in the near-wall region is studied. In contrast to most of the previous studies that have employed linear eddy-viscosity models, in this study, an advanced non-linear eddy-viscosity model is introduced to resolve the near-wall stress anisotropy more correctly. To investigate the model performance in detail, the proposed model is applied to fully-developed plane channel flows with various grid resolutions and at various Reynolds numbers. The grid resolution in the streamwise ( x) and spanwise ( z) directions ranges from 10 1 to 10 3 in wall unit (Δ x + or Δ z +), while y wall + ∼ 1 in the wall-normal ( y) direction. The present model provides encouraging results for further development of this kind of hybrid LES/RANS model.

Modelling two- and three-dimensional separation from curved surfaces with anisotropy-resolving turbulence closures

The ability of non-linear eddy-viscosity and second-moment models to predict separation from two- and three-dimensional curved surfaces is examined on the basis of computations for two flows that are geometrically akin: one separating from periodically-spaced, two-dimensional `hills' in a plane channel, and the other from a three-dimensional hill in a duct. One major objective is to examine whether the predictive performance in 3-d (three-dimensional) conditions relates to that in 2-d (two-dimensional) flow. In the former, the separation pattern is far more complicated, being characterised by multiple vortical structures associated with `open' separation. The predicted separation behaviour in the 2-d flow differs significantly from model to model, with only one non-linear model among those examined performing well, this variant formulated to adhere to the two-component wall limit. In 3-d separation, none of the models gives a credible representation of the complex multi-vortical separation pattern.

A turbulence model study of separated 3D jet/afterbody flow

Three-dimensional RANS calculations and comparisons with experimental data are presented for subsonic and transonic flow past a non-axisymmetric (rectangular) nozzle/afterbody typical of those found in fast-jet aircraft. The full details of the geometry have been modelled, and the flow domain includes the internal nozzle flow and the jet exhaust plume. The calculations relate to two free-stream Mach numbers of 0-6 and 0-94 and have been performed during the course of a collaborative research programme involving a number of UK universities and industrial organisations. The close interaction between partners contributed greatly to the elimination of computational inconsistencies and to rational decisions on common grids and boundary conditions, based on a range of preliminary computations. The turbulence models used in the study include linear and non-linear eddy-viscosity models. For the lower Mach number case, the flow remains attached and all of the turbulence models yield satisfactory pressure predictions. However, for the higher Mach number, the flow over the afterbody is massively separated, and the effect of turbulence model performance is pronounced. It is observed that non-linear eddy-viscosity modelling provides improved shock capturing and demonstrates significant turbulence anisotropy. Among the linear eddy-viscosity models, the SST model predicts the best surface pressure distributions. The standard k -ε model gives reasonable results, but returns a shock location which is too far downstream and displays a delayed recovery. The flow field inside the jet nozzle is not influenced by turbulence modelling, highlighting the essentially inviscid nature of the flow in this region. However, the resolution of internal shock cells for identical grids is found to be dependent on the solution algorithm -specifically, whether it solves for pressure or density as a main dependent variable. Density-based time-marching schemes are found to return a better resolution of shock reflection. The paper also highlights the urgent need for more detailed experimental data in this type of flow.

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.

Simulation of turbulent flows around a circular cylinder using nonlinear eddy-viscosity modelling: steady and oscillatory ambient flows

Steady incident flow past a circular cylinder for sub- to supercritical Reynolds number has been simulated as an unsteady Reynolds-averaged Navier–Stokes (RANS) equation problem using nonlinear eddy-viscosity modelling assuming two-dimensional flow. The model of Craft et al. (Int. J. Heat Fluid Flow 17 (1996) 108), with adjustment of the coefficients of the ‘cubic’ terms, predicts the drag crisis at a Reynolds number of about 2×10 5 due to the onset of turbulence upstream of separation and associated changes in Strouhal number and separation positions. Slightly above this value, at critical Reynolds numbers, drag is overestimated because attached separation bubbles are not simulated. These do not occur at supercritical Reynolds numbers and drag coefficient, Strouhal number and separation positions are in approximate agreement with experimental measurements (which show considerable scatter). Fluctuating lift predictions are similar to sectional values measured experimentally for subcritical Reynolds numbers but corresponding measurements have not been made at supercritical Reynolds numbers. For oscillatory ambient flow, in-line forces, as defined by drag and inertia coefficients, have been compared with the experimental values of Sarpkaya (J. Fluid Mech. 165 (1986) 61) for values of the frequency parameter, β= D 2/ νT, equal to 1035 and 11240 and Keulegan–Carpenter numbers, KC= U 0 T/ D, between 0.2 and 15 (D is cylinder diameter, ν is kinematic viscosity, T is oscillation period, and U 0 is the amplitude of oscillating velocity). Variations with KC are qualitatively reproduced and magnitudes show best agreement when there is separation with a large-scale wake, for which the turbulence model is intended. Lift coefficients, frequency and transverse vortex shedding patterns for β=1035 are consistent with available experimental information for β≈250−500. For β=11240, it is predicted that separation is delayed due to more prominent turbulence effects, reducing drag and lift coefficients and causing the wake to be more in line with the flow direction than transverse to it. While these oscillatory flows are highly complex, attached separation bubbles are unlikely and the flows probably two dimensional.

Numerical simulation of buoyant plume dispersion in a stratifice atmosphere using a lagrangian stochastic model

In the present paper, numerical simulations of buoyant plume dispersion in a neutral and stable atmospheric boundary layer have been carried out. A Lagrangian Stochastic Model (LSM) with a Non-Linear Eddy Viscosity Model (NLEVM) for turbulence is used to generate a Reynolds stress field as an input condition of dispersion simulation. A modified plume-rise equation is included in dispersion simulation in order to consider momentum effect in an initial stage of plume rise resulting in an improved prediction by comparing with the experimental data. The LSM is validated by comparing with the prediction of an Eulerian Dispersion Model (EDM) and by the measured results of vertical profiles of mean concentration in the downstream of an elevated source in an atmospheric boundary layer. The LSM predicts accurate results especially in the vicinity of the source where the EDM underestimates the peak concentration by 40% due to inherent limitations of gradient diffusion theory. As a verification study, the LSM simulation of buoyant plume dispersions under a neutral and stable atmospheric condition is compared with a wind-tunnel experiment, which shows good qualitative agreements.

Application of a Higher Order GGDH Heat Flux Model to Three-Dimensional Turbulent U-Bend Duct Heat Transfer

A higher order version of the generalized gradient diffusion hypothesis (HOGGDH) for turbulent heat flux is applied to predict heat transfer in a square-sectioned U-bend duct. The flow field turbulence models coupled with are a cubic nonlinear eddy viscosity model and a full second moment closure. Both of them are low Reynolds number turbulence models. The benefits of using the HOGGDH heat flux model are presented through the comparison with the standard GGDH.

Investigation of a Two-Equation Turbulent Heat Transfer Model Applied to Ducts

This investigation concerns numerical calculation of fully developed turbulent forced convective heat transfer and fluid flow in ducts over a wide range of Reynolds numbers. The low Reynolds number version of a non-linear eddy viscosity model is combined with a two-equation heat flux model with the eddy diffusivity concept. The model can theoretically be used for a range of Prandtl numbers or a range of different fluids. The computed results compare satisfactory with the available experiment. Based on existing DNS data and calculations in this work the ratio between the time-scales (temperature to velocity) is found to be approximately 0.7. In light of this assumption an algebraic scalar flux model with variable diffusivity is presented.

An investigation of wall-anisotropy expressions and length-scale equations for non-linear eddy-viscosity models

New closure approximations are proposed, within the framework of non-linear eddy-viscosity modeling, which aim specifically at an improved representation of near-wall anisotropy in both shear and stagnation flows. The main novel element is the introduction of tensorial terms, alongside strain and vorticity, which depend on wall-direction indicators and which procure the correct asymptotic near-wall behavior of the Reynolds stresses. The newly formulated non-linear constitutive equation for the Reynolds stresses is combined with low-Reynolds-number forms of equations for the rate of dissipation ε or the specific dissipation ω, the latter incorporating a number of new features into the established form of the equation. The predictive performance of three model variants is investigated by reference to three test flows: a plane channel flow, a separated flow in a channel with periodic hill-shaped obstacles on one wall and a plane impinging jet. It is shown that the new model elements result in a substantially improved representation of the Reynolds-stress field at the wall, especially in the wall-normal Reynolds stress. One of the variants includes the use of the modified ω-equation, and it is shown that this model performs especially well in the presence of separation.

Development of Curvature Sensitive Nonlinear Eddy-Viscosity Model

Current widely used turbulence closures, except the full second-moment closure, cannot properly account for the suppression of turbulence by convex curvature and the enhancing effect by concave curvature. Particularly, the failure of the explicit algebraic stress model (EASM) to capture the curvature strain effects in curved flows can be attributed to the equilibrium assumption embodied in EASM. Thus, the algebraic stress model (ASM) approach is reexamined in the framework of a generalized curvilinear coordinate system. In the approach, the equilibrium assumption applies only to the gradient part of convective transport while the curvature-related algebraic terms in the Reynolds stress convection process are retained. In this way, a curvature sensitive ASM is derived and is further transformed into an explicit nonlinear eddy-viscosity model in an orthogonal coordinate system. With this newly developed explicit ASM to predict the curved flow in a two-dimensional U duct, it is observed that the turbulence suppression at the convex wall is successfully captured

On Application of Nonlinear k-ε Models for Internal Combustion Engine Flows

The linear k-ε model, in its different formulations, still remains the most widely used turbulence model for the solutions of internal combustion engine (ICE) flows thanks to the use of only two scale-determining transport variables and the simple constitutive relation. This paper discusses the application of nonlinear k-ε turbulence models for internal combustion engine flows. Motivations to nonlinear eddy viscosity models use arise from the consideration that such models combine the simplicity of linear eddy-viscosity models with the predictive properties of second moment closure. In this research the nonlinear k-ε models developed by Speziale in quadratic expansion, and Craft et al. in cubic expansion, have been applied to a practical tumble flow. Comparisons between calculated and measured mean velocity components and turbulence intensity were performed for simple flow structure case. The effects of quadratic and cubic formulations on numerical predictions were investigated too, with particular emphasis on anisotropy and influence of streamline curvature on Reynolds stresses.

Heat transfer in turbulent impinging jets with a non-linear k-epsilon model

This paper contains a numerical study of the convective heat transfer of a turbulent jet, impinging onto a flat plate. The low-Reynolds standard k - epsilon model (Yang [1]) (further 'YS'), dramatically overpredicts the heat transfer at the stagnation point (e.g. Behnia [2, 3]). A non-linear eddy-viscosity model, based on (Merci [4]), is presented, in which there is no overprediction. The model differs from the YS model in two aspects: the constitutive law, relating the turbulent stresses to the local mean velocity gradients, is non-linear, and the transport equation for e is altered (and physically more correct). Both the epsilon transport equation and the constitutive law are important to improve the results. For the turbulent heat flux, the linear gradient hypothesis is used. Results are presented for a constant turbulent Prandtl number and for a Prandtl number which is a function of the local eddy viscosity. Differences between these results are small. Results are discussed for turbulent jets with variable distance between the jet nozzle exit and the flat plate. The local Nusselt number profiles on the flat plate are compared to experimental data (Baughn [5, 6], Cooper [7], Craft [8]). The flow field is studied in terms of mean velocities and turbulent stresses. Globally, the presented non-linear eddy-viscosity model yields accurate flow field predictions. The heat transfer at the stagnation point is well predicted, and the shape of the profiles of the local Nusselt number is improved, though still not in perfect agreement with the experimental data.

An Improved Turbulence Model for Rotating Shear Flows.

In the present study, we construct a turbulence model based on a low-Reynolds-number nonlinear k-ε model for turbulent flows in a rotating channel. Two-equation models, in particular the non-linear k-ε model, are very effective for solving various flow problems encountered in engineering applications. In channel flows with rotation, however, the explicit effects of rotation only appear in the Reynolds stress components. The exact equations for k and ε do not have any explicit terms concerned with the rotating effects. Moreover, a Coriolis force vanishes in the momentum equation for a fully developed channel flow with spanwise rotation. Consequently, in order to predict rotating channel flows, after proper revision the Reynolds stress equation model (RSM) or the non-linear eddy viscosity model (NLEVM) should be used. In this study, we improve the non-linea k-ε model so as to predict rotating channel flows. In the modelling, the wall-limiting behaviour of turbulence is also considered. First, we evaluated the non-linear k-ε model using the direct numerical simulation (DNS) database for a fully developed rotating turbulent channel flow. Next, we assessed the non-linear k-ε model at various rotation numbers. Finally, based on these assessments, we reconstruct the non-linear k-ε model to calculate rotating shear flows.

Investigation of Anisotropy-Resolving Turbulence Models by Reference to Highly-Resolved LES Data for Separated Flow

The predictive properties of several non-linear eddy-viscosity models are investigated by reference to highly-resolved LES data obtained by the authors for an internal flow featuring massive separation from a curved surface. The test geometry is a periodic segment of a channel constricted by two-dimensional (2D) `hills' on the lower wall. The mean-flow Reynolds number is 21560. Periodic boundary conditions are applied in the streamwise and spanwise directions. This makes the statistical properties of the simulated flow genuinely 2D and independent from boundary conditions, except at the walls. The simulation was performed on a high-quality, 5M-node grid. The focus of the study is on the exploitation of the LES data for the mean-flow, Reynolds stresses and macro-length-scale. Model solutions are first compared with the LES data, and selected models are then subjected to a-priori studies designed to elucidate the role of specific model fragments in the non-linear stress-strain/vorticity relation and their contribution to observed defects in the mean-flow and turbulence fields. The role of the equation governing the length-scale, via different surrogate variables, is also investigated. It is shown that, while most non-linear models overestimate the separation region, due mainly to model defects that result in insufficient shear stress in the separated shear layer, model forms can be derived which provide a satisfactory representation of the flow. One such model is identified. This combines a particular quadratic constitutive relation with a wall-anisotropy term, a high-normal-strain correction and a new form of the equation for the specific dissipation ω = ∈/k.

Validation of Cubic Nonlinear Eddy Viscosity Turbulence Models for Engine Steady Intake Flows.

This paper presents discussions on predicting turbulent flow in a simplified internal combustion engine geometry consisting of an intake port, valve and cylinder. The low-Reynolds-number linear k-e, cubic nonlinear k-e and k-e-A2 turbulence models are applied to the flow field computation and evaluated against experimental data. The results suggest that the cubic nonlinear eddy viscosity modelling of turbulence is very successful for predicting mass flow rates and in-cylinder velocity distributions.

Development of curvature sensitive nonlinear eddy-viscosity model

Development of curvature sensitive nonlinear eddy-viscosity model

Realizability for the Reynolds Stress in Nonlinear Eddy-Viscosity Model of Turbulence

The non-negativeness of the normal stress is one of the realizability conditions for the Reynolds stress. This condition is investigated to improve turbulence models. A statistical theory called th...

Computation of Particle and Scalar Transport for Complex Geometry Turbulent Flows

The prediction of particle and scalar transport in a complex geometry with turbulent flow driven by fans is considered. The effects of using different turbulence models, anisotropy, flow unsteadiness, fan swirl, and electrostatic forces on particle trajectories are shown. The turbulence models explored include k−l, zonal k−ε/k−l, and nonlinear eddy viscosity models. Particle transport is predicted using a stochastic technique. A simple algorithm to compute electrostatic image forces acting on particles, in complex geometries, is presented. Validation cases for the particle transport and fluid flow model are shown. Comparison is made with new smoke flow visualization data and particle deposition data. Turbulence anisotropy, fan swirl, and flow unsteadiness are shown to significantly affect particle paths as does the choice of isotropic turbulence model. For lighter particles, electrostatic forces are found to have less effect. Results suggest, centrifugal forces, arising from regions of strong streamline curvature, play a key particle deposition role. They also indicate that weaknesses in conventional eddy viscosity based turbulence models make the accurate prediction of complex geometry particle deposition a difficult task. Axial fans are found in many fluid systems. The sensitivity of results to their modeling suggests caution should be used when making predictions involving fans and that more numerical characterization studies for them could be carried out (especially when considering particle deposition). Overall, the work suggests that, for many complex-engineering systems, at best (without excessive model calibration time), only qualitative particle deposition information can be gained from numerical predictions.