Differential Reynolds Stress Model Research Articles

An assessment of turbulence models applied to the simulation of a two-dimensional submerged jet

This paper is concerned with the investigation of the performance of different turbulence models in the numerical prediction of transient flow caused by a confined submerged jet. Several widely used models, i.e., the standard k– ε, RNG k– ε, low Reynolds number k– ε models and the differential Reynolds stress model, as included in CFD codes, were compared with each other for a two-dimensional, incompressible, turbulent jet flow and with reported experimental data. A flapping oscillation was predicted regardless of the model used. A chosen Strouhal (St) number definition brought the fundamental frequencies from both the experiments and computations into close proximity. However, different turbulence models have exhibited quite different behaviours in terms of the frequency and regularity of the oscillation and in terms of the scale and duration of the vortices generated. All versions of the k– ε model yielded regular oscillations, which agree with experimental observations. On the other hand, the Reynolds stress (RS) model produced a complex pattern but a slower dissipation of vortices. In addition, some aspects of gridding and inflow representation are also discussed.

A Numerical Study of Vortex Breakdown in Turbulent Swirling Flows

Solutions to the incompressible Reynolds-averaged Navier–Stokes equations have been obtained for turbulent vortex breakdown within a slightly diverging tube. Inlet boundary conditions were derived from available experimental data for the mean flow and turbulence kinetic energy. The performance of both two-equation and full differential Reynolds stress models was evaluated. Axisymmetric results revealed that the initiation of vortex breakdown was reasonably well predicted by the differential Reynolds stress model. However, the standard K-ε model failed to predict the occurrence of breakdown. The differential Reynolds stress model also predicted satisfactorily the mean azimuthal and axial velocity profiles downstream of the breakdown, whereas results using the K-ε model were unsatisfactory. [S0098-2202(00)01601-1]

Computational fluid dynamics modelling of an entrained flow biomass gasifier

A mathematical model, based on the Computational Fluid Dynamics package CFX4, has been developed to study the flow within an entrained flow biomass gasifier. The gasifier is designed to convert sawdust and chopped cotton gin trash into a low calorific value gas which can be burned in a modified engine to run a generator. Calculations of the flowfield are performed using the standard k– ϵ model and a Differential Reynolds Stress Model (DSM). In line with current thinking, it is shown that the k– ϵ model gives unphysical results for complex swirling flows, whereas the DSM model performs well. Particle tracking was performed to determine typical trajectories for the biomass and char and the results used to determine means of avoiding slagging in the gasifier base. The simulations have proved to be very useful to the designers who are now using the model to optimise the design.

Scaling of the turbulent boundary layer along a flat plate according to different turbulence models

Abstract At sufficiently large Reynolds number the turbulent boundary layer along a flat plate under zero pressure gradient can be split up in an inner and outer layer. The classical theory says that a law-of-the-wall holds in the inner layer, and a defect law in the outer layer. It is shown that four different types of commonly used turbulence models (an algebraic, k – ϵ , k – ω and a differential Reynolds-stress model) all reproduce the classical similarity scalings for Re θ above about 10 4 . This was verified by numerically solving the turbulent boundary-layer equations for Reynolds numbers (based on the momentum-loss thickness) in between 300 and 5×10 7 . The boundary-layer solution in the outer layer is shown to converge to the similarity solution of a defect-layer equation. All turbulence models considered give a wall function and defect law that is close to Direct Numerical Simulations of Spalart (1988) and new high-Reynolds-number experiments by Fernholz et al. (1995) . An exception is the algebraic model that gives a too thin boundary layer.

Numerical and experimental study of swirling flow in a model combustor

The present paper describes a numerical and experimental investigation of strongly swirling flow in a water model combustion chamber equipped with a swirler of special design. The turbulence models used for the numerical calculations are the standard κ—ε model, the RNG κ—ε model and a differential Reynolds stress model (DRSM). In the water model, local mean velocity components and normal stresses are measured using a laser Doppler anemometer. Comparison of numerical predictions against experimental data reveals the superiority of the DRSM over the standard and RNG κ—ε models. The DRSM captures all the major features of the swirling flow, while the other two models do not. For instance, both the experimental data and the DRSM predictions reveal complex, interesting flow behaviour: a corner recirculation zone, and a central toroidal recirculation zone connected to a central reverse zone, which persists all the way to the outlet of the chamber. However, the other two turbulence models predict that the swirling flow evolves into a solid-body-rotation-type flow downstream. The RNG κ—ε model gives very little improvement over the standard κ—ε model for the swirling flow case considered.

Present Status of Second-Order Closure Turbulence Models. I: Overview

An overview of the second-order closure turbulence models is presented in this paper. Models studied include the k-ε-eddy viscosity model, k-ε-nonlinear Reynolds stress model, differential Reynolds stress model, k-ε algebraic stress model, near-wall second-order closure model, low-Reynolds number model, two-layer model, and multiscale model, which cover the efforts of scientists and engineers over the past 50 years. However, at the present time, there exists no unified turbulence model. Each model applies successfully to some turbulent flows, while it predicts unsatisfactory results for other flows, especially for flows that are very different from those for which the models were calibrated. To improve the prediction accuracy and the applicability of the existing turbulence models, modifying, or even remodeling, the ε equation, the dissipation rate of turbulent kinetic energy, and pressure-strain terms of the Reynolds stress, uiuj¯, equations is necessary.

A PERFORMANCE COMPARISON OF THE STANDARD k-ϵ MODEL AND A DIFFERENTIAL REYNOLDS STRESS MODEL FOR A BACKWARD-FACING STEP

A differential Reynolds stress model (RSM) based on Launder, Reece, and Rodi (LRR) is formulated with appropriate wall Junctions and applied to predict the backward-facing step problem of Driver and Seegmiller. Numerical prediction obtained with the LRR, with and without “wall reflection” terms in the pressure strain model, are compared with the results of standard k-e model of Launder and Spalding for the step problem. The results demonstrate that both LRR models, i.e., with and without wall reflection terms, are capable of capturing the secondary bubble near the step, as observed in the experiment, whereas the standard k-ϵ model fails to predict the secondary bubble. In addition, the mean velocity profiles obtained with the LRR models agree better with the experimental data than those by the k-ϵ model, particularly inside the recirculating flow region. It also emerges from the present study that, with proper wall functions, LRR model is capable of predicting recirculating flows at least as well as the original LRR model does without the “wall reflection” terms.

Scaling of Equilibrium Boundary Layers Under Adverse Pressure Gradient Using Turbulence Models

The large Reynolds number behavior of four commonly used turbulence models was investigated by numerically solving the boundary-layer equations up toReo = 108 for turbulent boundary layers under an adverse pressure gradient with an equilibrium parameter (3 = (^*/rw)(dp/djt:) in the range between 0 and 200. All models reproduce the same classical scalings in the inner and outer layer. The solution in the outer layer asymptotes to the defect law described by the defect-layer equation as derived by Tennekes and Lumley (Tennekes, H., and Lumley, J. L., A First Course in Turbulence, MIT Press, Cambridge, MA, 1972, pp. 186-188) and not to the one derived by Wilcox (Wilcox, D. C., Turbulence Modeling, DCW Industries, Inc., La Canada, CA, 1993, pp. 110-121). The asymptotic state is obtained at a higher Reynolds number for increasing f3 value. The turbulence models show that the same outer-edge velocity can generate two different equilibrium boundary layers. Comparison with existing experiments shows that the differential Reynolds stress model is very accurate; the algebraic model and the k-ui model are also reasonably good. The k-e model is not accurate for these adverse-pressure gradient boundary layers, and it gives a far too large wall-shear stress.

Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients

Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.

Improved prediction of turbulent separated flow with a low-Reynolds-number Nonlinear κ − ϵ model

A low-Reynolds-number k - e turbulence model is proposed and applied to the turbulent backstep flows of different expansion ratios in this paper. The comparisons with the measurements show that the predictions of the present nonlinear k - e model are greatly improved over that of the standard model for all turbulent quantities and are better than the predictions of some partial differential Reynolds stress models for Reynolds stress components.

Comparison of turbulence models for attached boundary layers relevant to aeronautics

With a single numerical method the performance of three classes of turbulence models is compared for different types of attached boundary layers, for which direct numerical simulations or experiments are available in the literature. The boundary-layer equations are solved with the following turbulence models: an algebraic model, two-equation models (k-e andk-ω), and a differential Reynolds-stress model. The test cases are the channel flow, and boundary layers with zero, favourable and adverse streamwise pressure gradient. The differential Reynolds-stress model gives the best overall performance, whereas the performance of the algebraic model and thek-ω model is reasonably good. The performance of thek-e model is less good for boundary layers with a non-zero streamwise pressure gradient, but it can easily be improved by an additional source term in the e equation, which is also applied in the considered differential Reynolds-stress model.

Large-eddy simulation of turbulent anisochoric flows

The objective of this study is to develop a large eddy simulation model for nonreacting as well as chemically reacting flows at high Reynolds numbers. This paper focuses on the large eddy simulation model applied to the nonreacting case and the evaluation of its predictive capabilities. Further, results from the large eddy simulations are used to investigate the large-scale structures present in unsteady wake flows. The large eddy simulation model contains subgrid-scale models for both cross and Reynolds quantities to account for both outscatter (i.e., transfer from small scales to larger scales) transfer between large scales and dissipation. To evaluate the large eddy simulation model, results from two- and three-dimensional large eddy simulations of nonreacting flow in a rectilinear channel with a triangular shaped bluff body are compared with experimental measurements and with results from two-dimensional simulations made with the k-e model and a differential Reynolds stress model. The investigations indicate that the results obtained with the large eddy simulation model is in good agreement with experimental data. Comparison of results obtained with the large eddy simulation model and results obtained with the k-e model and the differential Reynolds stress model clearly reveals the known weaknesses of the latter methods.

NUMERICAL SIMULATION OF TURBULENT FLOW OF AGITATED LIQUID WITH PITCHED BLADE IMPELLER

The turbulent flow field in an agitated system with baffles was solved numerically using the standard k-e model, an algebraic Reynolds stress model (ASM) and a differential Reynolds stress model (RSM). The commercial software FLOW3D (CFDS, Harwell Laboratories, 1991) was used for this purpose. The aim of the study was to investigate the influence of the impeller boundary conditions and turbulence models to the agreement with experimentally obtained laser-Doppler anemometry data. The boundary conditions for the impeller discharge used in the numerical calculations were obtained as whole-cycle-ensemble averages from experimental LDA-measurements (Fort et al., 1992). Since measurements of the rate of dissipation of turbulent kinetic energy ( ϵ) was not available the dissipation rate per unit mass in the impeller discharge was estimated from the expression: where k is the turbulent kinetic energy per unit mass and L the macroscale of turbulence in the pitched blade impeller discharge. The macroscale of turbulence (L) in the impeller boundary condition for e was varied in order to optimize the fit of theoretically obtained profiles of turbulent kinetic energy with experimental data. The constant A was fixed to 0.85 according to Wu and Patterson (1989). The optimal values of L for the different turbulence models were compared with the projected height of the impeller blade (h). All three components of the mean velocity were compared with experimental data for the optimal ratio of L/h for six radial cross-sections in the tank. The mean velocity field obtained from simulations showed good agreement with experimental data for all models, with somewhat better agreement for the k — e model. An optimal value of the ratio L/h was found to be equal to 2.0 for the k — ϵ model and 1.3 for the ASM. However, no such optimal value for the RSM could be determined in this study.

Numerical analysis of swirling non-reacting and reacting flows by the Reynolds stress differential method

So et al.'s isothermal He-air mixing experiment and Wilhelmi's propane-air diffusion-controlled combustion experiment were analyzed with a differential Reynolds stress model. For both experiments, the IPC (isotropization of production and convection) model of Fu et al. for the rapid term improved the normal stress distribution in the large swirl velocity region and the Hanjalic and Launder model for the diffusion term improved the normal stress distribution near the centreline, in comparison with the IP (isotropization of production) model for the rapid term and the Daly and Harlow model for the diffusion term. For Wilhelmi's experiment, the IPC model and the Hanjalic and Launder model yielded improved mixture fraction distributions near the centreline. The intensity of axial and swirl velocities near the centreline was still underestimated, however, and the model requires further improvement.

The Reynolds-stress model of turbulence applied to the natural-convection boundary layer along a heated vertical plate

The turbulent natural-convection boundary layer for air along a heated vertical plate is investigated numerically with an algebraic (ASM) and fully differential Reynolds-stress model (RSM). From the literature a set of model constants is selected, in such a way that the wall-heat transfer and mean-flow structure are predicted in close agreement with the experimental data. Sensitivity tests on RSM constants show which constants dominate the mean-flow prediction, and which constants only affect turbulence quantities. Wall modifications are employed to improve predictions of the near-wall turbulence. RSM calculations of the turbulence quantities agree well with available experimental data. ASM results are poorer, but still in qualitative agreement with experiments. Hence, in natural-convection boundary layers, the local-equilibrium assumption has only limited applicability. Furthermore, the eddy-viscosity concept used in the k− ε model (KEM) is tested. The KEM gives good mean-flow results, but for a good prediction of the detailed turbulence structure the RSM is needed.

Possibility of enhancing the erosive activity of a cavitation void under the simultaneous action of continuous and pulsed sound : Bashkirov, V.I., Sukhanov, L.I., 18 (April–June 1973) 503–504

So et al.'s isothermal He-air mixing experiment and Wilhelmi's propane-air diffusion-controlled combustion experiment were analyzed with a differential Reynolds stress model. For both experiments, the IPC (isotropization of production and convection) model of Fu et al. for the rapid term improved the normal stress distribution in the large swirl velocity region and the Hanjalic and Launder model for the diffusion term improved the normal stress distribution near the centreline, in comparison with the IP (isotropization of production) model for the rapid term and the Daly and Harlow model for the diffusion term. For Wilhelmi's experiment, the IPC model and the Hanjalic and Launder model yielded improved mixture fraction distributions near the centreline. The intensity of axial and swirl velocities near the centreline was still underestimated, however, and the model requires further improvement.

Directionality of high-intensity acoustic radiation : Ostrovskii, L.A., Fridman, V.E., 18 (April–June 1973) 478–481

So et al.'s isothermal He-air mixing experiment and Wilhelmi's propane-air diffusion-controlled combustion experiment were analyzed with a differential Reynolds stress model. For both experiments, the IPC (isotropization of production and convection) model of Fu et al. for the rapid term improved the normal stress distribution in the large swirl velocity region and the Hanjalic and Launder model for the diffusion term improved the normal stress distribution near the centreline, in comparison with the IP (isotropization of production) model for the rapid term and the Daly and Harlow model for the diffusion term. For Wilhelmi's experiment, the IPC model and the Hanjalic and Launder model yielded improved mixture fraction distributions near the centreline. The intensity of axial and swirl velocities near the centreline was still underestimated, however, and the model requires further improvement.