Reynolds stress closure applied to axisymmetric,impinging turbulent jets

A second-order, single-point closure model for calculating the transport of momentum in turbulent flows is extended to cover flows that are close to solid surfaces. In such flows the proximity of a solid boundary directly influences the fluctuating pressure field within the main body of the flow and leads to a dampening of velocity fluctuations normal to the wall. These effects are accommodated through the incorporation of an additional contribution in the modelled form of the redistributive fluctuating pressure term used in the Reynolds stress transport equation. Predictions of the extended closure model are compared with available data in configurations where an air jet impinges orthogonally onto a plane surface. The inclusion of the wall reflection model is shown to result in superior predictions of mean velocities, and normal and shear stresses. In particular, normal-to-wall velocity fluctuations and shear stresses are successfully damped resulting in agreement with observations.     

Application of a second-moment closure model to simulate the turbulent dispersion from an elevated line source

In this study, a second-moment closure model and an algebraic stress and flux model are used to simulate the turbulent dispersion from an elevated line source located in a turbulent boundary layer. In the modeling of the Reynolds stress equation, several models for the triple velocity correlation are examined by comparing their predicting results with the direct numerical simulation data. Predicted mean and fluctuating correlation quantities of velocity and temperature by the second-moment closure model and the algebraic model are compared with the experimental data. From the comparisons, it is found that the second-moment closure is capable of reproducing the experimental results satisfactorily. However, predictions of the algebraic stress and flux model are quite poor. Received on 12 June 1998     

Modeling fluid-particle interaction in dilute-phase turbulent liquid-particle flow simulation

The present work examines the predictive capability of a two-fluid CFD model that is based on the kinetic theory of granular flow in simulating dilute-phase turbulent liquid-particle pipe flows in which the inter-stitial fluid effect on the particle fluctuating motion is significant.The impacts of employing different drag correlations and turbulence closure models to describe the fluid-particle interactions(i.e.drag force and long-range interaction)are examined at both the mean and fluctuating velocity levels.The model pre-dictions are validated using experimental data of turbulent liquid-particle flows in a vertical pipe at different particle Reynolds numbers(ReP > 400 and ReP < 400),which characterize the importance of the vortex shedding phenomenon in the fluid-phase turbulence modulation.The results indicate that(1)the fluctuating velocity level predictions at different ReP are highly sensitive to the drag correlation selec-tion and(2)different turbulence closure models must be employed to accurately describe the long-range fluid-particle interaction in each phase.In general,good agreement is found between the model predic-tions and the experimental data at both the mean and fluctuating velocity levels provided that appropriate combinations of the drag correlation and the turbulence closure model are selected depending on Rep.     

Current capabilities for modelling turbulence in industrial flows

This paper summarizes some of the more widely applied or promising schemes for computing heat and momentum transport in industrially relevant flows. Such flows typically involve complex flow domains, severe pressure gradients and regions of flow separation and reattachment. Models tuned by reference to equilibrium, simple shear flows cannot in general be relied upon to predict accurately the effects of these complexities on the transport processes. The main conclusions drawn are that second-moment closure offers a far more reliable basis for computing non-equilibrium turbulent flows than eddy-viscosity schemes, especially in flows with very complex strain fields or those substantially affected by external force fields. Moreover, where significant variations in shear stress occur across the near-wall viscosity-affected sublayer, the usual practice of employing wall functions needs to be replaced by at least a two-equation model in order to capture, even qualitatively, the consequent effects on wall heat transfer or skin friction coefficient.     

Coupled RANS–LPT modelling of dilute,particle-laden flow in a duct with a 90° bend

A dilute, particle-laden flow in a square duct with a 90° bend is modelled using a RANS approach, coupled to a second-moment turbulence closure, together with a Lagrangian particle tracking technique, with particle dispersion modelled using a stochastic approach that ensures turbulence anisotropy. Detailed predictions of mean and fluctuating fluid and particle velocities are validated through comparisons of predictions with experimental measurements made for gas–solid flows in a vertical-to-horizontal flow configuration. Reasonable agreement between predicted first and second moments and data is found for both phases, with the consistent application of anisotropic and three-dimensional modelling approaches resulting in predictions that compare favourably with those of other authors, and which provide fluctuating particle velocities in acceptable agreement with data.     

A numerical study of turbulent flow over a two‐dimensional hill

Calculations of mean velocities and Reynolds stresses are reported for the recirculating flow established in the wake of two‐dimensional polynomial‐shaped obstacles that are symmetrical about a vertical axis and mounted in the water channel downstream of a fully developed channel flow for Re=6×10⁴. The study involves calculations of mean and fluctuating flow properties in the streamwise and spanwise directions and include comparisons with experimental data [Almeida GP, Durão DFG, Heitor MV. Wake flows behind two‐dimensional model hills. Experimental Thermal and Fluid Science 1993; 7: 87–101] for flow around a single obstacle with data resulting from the interaction of consecutive obstacles, using two versions of the low‐Reynolds number differential second‐moment (DSM) closure model. The results include analysis of the turbulent stresses in local flow co‐ordinates and reveal flow structure qualitatively similar to that found in other turbulent flows with a reattachment zone. It is found that the standard isotropization of production model (IPM), based on that proposed by Gibson and Launder [Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics 1978; 86(3): 191–511], with the incorporation of the wall reflection model of Craft and Launder [New wall‐reflection model applied to the turbulent impinging jet. AIAA Journal 1992; 32(12): 2970–2972] predicts the mean velocities quite well, but underestimates the size of the recirculation region and turbulent quantities in the shear layer. These inadequacies are circumvented by adopting a new cubic Reynolds stress closure scheme based on that more recently developed by Craft and Launder [A Reynolds stress closure designed for complex geometries. International Journal of Heat and Fluid Flow 1996; 17: 245–254] which satisfies the two component limit (TCL) of turbulence. In this model the geometry‐specific quantities, such as the wall‐normal vector or wall distance, are replaced by invariant dimensionless gradient indicators. Also, the model captures the diverse behaviour of the different components of the stress dissipation, ε_ij, near the wall and uses a novel decomposition for the fluctuating pressure terms. Copyright © 2001 John Wiley & Sons, Ltd.     

Eulerian prediction of the fluid/particle correlated motion in turbulent two-phase flows

An approximate equation governing the turbulent fluid velocity encountered along discrete particle path is used to derive the fluid/particle turbulent moments required for dispersed two-phase flows modelling. Then, closure model predictions are compared with results obtained from large-eddy simulation of particle fluctuating motion in forced isotropic fluid turbulence.     

COMPUTATIONAL STUDIES OF IMPINGING JETS USING K-ϵ TURBULENCE MODELS

This paper reports numerical modelling of impinging jet flows using Rodi and Malin corrections to the k–ϵ turbulence model, carried out using the PHOENICS finite volume code. Axisymmetric calculations were performed on single round free jets and impinging jets and the effects of pressure ratio, height and nozzle exit velocity profile were investigated numerically. It was found that both the Rodi and Malin corrections tend to improve the prediction of the hydrodynamic field of free and impinging jets but still leave significant errors in the predicted wall jet growth. These numerical experiments suggest that conditions before impingement significantly affect radial wall jet development, primarily by changing the wall jet's initial thickness.     

The structure of a three-dimensional boundary layer subjected to streamwise-varying spanwise-homogeneous pressure gradient

A spatially-evolving three-dimensional boundary layer, subjected to a streamwise-varying spanwise-homogeneous pressure gradient, equivalent to a body force, is investigated by way of direct numerical simulation. The pressure gradient, prescribed to change its sign half-way along the boundary layer, provokes strong skewing of the velocity vector, with a layer of nearly collateral flow forming close to the wall up to the position of maximum spanwise velocity. A wide range of flow-physical properties have been studied, with particular emphasis on the near-wall layer, including second-moments, major budget contributions and wall-normal two-point correlations of velocity fluctuations and their angles, relative to wall-shear fluctuations. The results illustrate the complexity caused by skewing, including a damping in turbulent mixing and a significant lag between strains and stresses. The study has been undertaken in the context of efforts to develop and test novel hybrid LES–RANS schemes for non-equilibrium near-wall flows, with an emphasis on three-dimensional near-wall straining. Fundamental flow-physical issues aside, the data derived should be of particular relevance to a priori studies of second-moment RANS closure and the development and validation of RANS-type near-wall approximations implemented in LES schemes for high-Reynolds-number complex flows.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

Riblet modelling using a second-moment closure

A low Reynolds number second-moment closure has been used to calculate a turbulent boundary layer which develops over a riblet surface with zero pressure gradient. The calculated mean velocity distributions compare favourably with measurements. Calculated Reynolds stresses away from the riblet surface region are also in agreement with measurements. In the vicinity of the riblets, the model reflects the increased anisotropy of the Reynolds stress tensor inadequately. Possible reasons for this shortcoming are discussed and suggestions for improving the model are made.     

A Computational Algorithm for Second-Moment Closure Turbulence Modelling

A computational formulation is proposed for second-moment closure turbulence models, especially suited to models intended to ensure physical realizability. It enables to cast the quite complicated model equations in a compact form. It is specifically applied here to a two-dimensional parabolized flow, though it lends itself to extension to more complex flows. An effective computational algorithm is proposed, based on a staggered grid and a block tridiagonal solver. The algorithm is applied to a turbulent mixing layer, and the comparison between the predictions obtained by standard modelling tools and a realizable second-moment closure clearly points out the superiority of the latter.     

The development of the large-scale structures in round impinging jets exiting long pipes at two Reynolds numbers

Measurements of the two-point and two-time correlation of the fluctuating wall pressure were performed in radial wall jets formed by impinging jets exiting a long pipe with Reynolds numbers of 23,300 and 50,000, and nozzle-to-plate distances of 2.0 diameters. The results from the two flows were compared to examine how the three-dimensionality and the development of the large-scale structures in the near field of the radial wall jet changed as the Reynolds number of the impinging jet varied. The measurements indicated that the large-scale structures were more two-dimensional, more periodic, and more prominent in the higher Reynolds number flow.This revised version was published online in November 2004 with corrections to the acknowledgement.     

BOUNDARY TREATMENT AND AN EFFICIENT PRESSURE ALGORITHM FOR INTERNAL TURBULENT FLOWS USING THE PDF METHOD

The generalized Langevin model, which is used to model the motion of stochastic particles in the velocity–composition joint probability density function (PDF) method for reacting turbulent flows, has been extended to incorporate solid wall effects. Anisotropy of Reynolds stresses in the near-wall region has been addressed. Numerical experiments have been performed to demonstrate that the forces in the near-wall region of a turbulent flow cause the stochastic particles approachi ng a solid wall to reverse their direction of motion normal to the wall and thereby, leave the near-wall layer. This new boundary treatment has subsequently been implemented in a full-scale problem to prove its validity. The test problem considered here is that of an isothermal, non-reacting turbulent flow in a two-dimensional channel with plug inflow and a fixed back-pressure. An efficient pressure correction method, developed in the spirit of the PISO algorithm, has been implemented. The pressure correction strategy is easy to implement and is completely consistent with the time- marching scheme used for the solution of the Lagrangian momentum equations. The results show remarkable agreement with both k–ϵ and algebraic Reynolds stress model calculations for the primary velocity. The secondary flow velocity and the turbulent moments are in better agreement with the algebraic Reynolds stress model predictions than the k– ϵ predictions. © 1997 by John Wiley & Sons, Ltd.     

Numerical computation of an impinging jet with uniform wall suction

The paper presents numerical predictions of a turbulent axisymmetric jet impinging onto a porous plate, based on a finite volume method of solving the Navier-Stokes equations for an incompressible air jet with the K–ε turbulence model. The velocity and pressure terms of the momentum equations are solved by the SIMPLE (semi-implicit method for pressure-linked equation) method. In this study, non-uniform staggered grids are used. The parameters of interest include the nozzle-to-wall distance and the suction velocity. The results of the present calculations are compared with available data reported in the literature. It is found that suction effects reduce the boundary layer thickness and increase the velocity gradient near the wall.     

Validation of turbulence models in a simulated air-conditioning unit

Details are given of a study to obtain experimental data in an idealized environment for the purpose of evaluating the corresponding computational predictions and which supplement parallel measurements made in actual packaged air-conditioning units. The system consisted of a purpose-built low-speed wind tunnel with a working section constructed to reproduce particular features of the real units. In the experiment, both the mean velocity profiles and turbulence properties of the flow are obtained from triple-hot-wire anemometry measurements. A numerical model, based on finite volume methodology, was used to obtain the solution of the Reynolds-averaged Navier–Stokes equations for incompressible isothermal flow. The Reynolds stress terms in the equations are calculated using the standard k–ϵ model and second-moment closure (Reynolds stress) models. The accuracy of the two models was evaluated against the experimental measurements made 10 mm downstream of a baffle. The results show that the standard k–ϵ model gave the better agreement except in regions of strong recirculation. © 1997 John Wiley & Sons, Ltd.     

Vortical structures and pressure pulsations in draft tube of a Francis-99 turbine at part load: RANS and hybrid RANS/LES analysis

Recognizing the limitations of the conventional linear-eddy-viscosity (LEVM) Reynolds-averaged Navier–Stokes (RANS) models to reproduce complex three-dimensional unsteady flows in hydraulic machinery, we performed a comparative assessment of a second-moment (Re-stress model, RSM) RANS closure and a hybrid RANS/LES method in capturing the flow and vortical structures in the draft tube of a Francis hydroturbine at off-design conditions. Considered is a case of part load (PL) at a flow rate of only 35% of the best efficiency point (BEP) characterised by multiple unsteady vortex systems. Despite some remaining uncertainties in generating the inflow conditions, both approaches reproduced reasonably well the measured mean velocity and the rms of its fluctuations, as well as the pressure spectrum with peaks detecting the precessing vortex core. In contrast to the common LEVMs, the Re-stress closure showed sufficient receptivity to intrinsic unsteadiness and reproduced well the overall flow and vortical patterns as well as the associated pressure pulsations in accord with the experiments. The hybrid RANS/LES method gave similar predictions as the RSM, but resolving a wider range of scales, which however, showed no significant effect on the dynamics of the dominant processing vortex core and the pressure pulsations.     

One Equation Model for Turbulent Channel Flow with Second Order Viscoelastic Corrections

A modified second order viscoelastic constitutive equation is used to derive a k–l type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by P _w , for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori P _w could be either positive or negative. When P _w is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, P _w  < 0 leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the k–l formulation rather than from the closure for P _w . An alternative closure for P _w , mimicking the viscoelastic stress work predicted by DNS using the Finitely Extensible Nonlinear Elastic-Peterlin fluid model, which is mostly characterized by P _w  > 0 but has also a small region of negative P _w in the buffer layer, was also successfully tested. This second model for P _w leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.     

Application of a reflection-free DSM to turbulent flow and heat transfer in a square-sectioned U-bend

The paper reports the outcome of applying two differential second-moment (DSM) closures to resolve the complex three-dimensional motion that arises in turbulent flow in a square-sectioned duct passing around a 180°C bend. The initial results showed (in accord with a number of recent studies) that, with the same underlying closure hypotheses, a DSM scheme produces better agreement with experiment than does the corresponding algebraic second-moment (ASM) treatment, although the differences were acceptably small. Thereafter, applications are reported for a new type of DSM that employs no wall-reflection terms. This leads to markedly better predictions of the turbulence field and thus of the wall heat transfer than the conventionally adopted version.     

Experience Gained Using Second-Moment Closure Modeling for Transitional Flows Due to Boundary Layer Separation

This paper presents detailed information on the experience gained during the attempts to model a set of transitional flows due to boundary layer separation. These flows are developed on a flat plate with a semi-circular leading edge and they have been coded by the ERCOFTAC Special Interest Group on Transition, as T3L flows. Different freestream velocities and turbulence intensities configure these transitional flows and, by consequence, govern the transition mechanism, resulting in larger or smaller transitional regions. The modeling of the T3L flows has been performed by adopting a low-Reynolds number second-moment closure turbulence model. The results showed satisfactory agreement with the experimental measurements, although some difficulties regarding successful convergence have been faced. The final conclusion is that turbulence modeling can present quite accurate results for transitional flows without any additional use of ad-hoc modifications or additional equations based on various transition models and intermittency transport modeling.