Predictive Capabilities of an Improved Cubic k–ε Model for Inert Steady Flows

Through an improved ε transport equation, a major quality enhancement of the cubic k–ε model, earlier developed in[13], is obtained. The ε-equation of [13],yielding good results for wall-bounded and rotating flows, is combined with the one derived by Shih et al. [20], which produces good results for free shear flows (e.g. the plane jet–round jet anomaly is resolved).Results are presented for the following flows: fully developed stationary and rotating channel and pipe, backward-facing step, sudden pipe expansion, smooth channel expansion and contraction, plane and round jet. Heat transfer predictions in turbulent impinging jets are also discussed. Accurate results are obtained for the mean flow quantities for all test cases, without case dependent model tuning. This revised version was published online in July 2006 with corrections to the Cover Date.     

Prediction of Flow and Heat Transfer through Stationary and Rotating Ribbed Ducts Using a Non-linear <Emphasis Type="Italic">k</Emphasis>−<Emphasis Type="Italic">ε</Emphasis> Model

The present paper deals with the prediction of three-dimensional fluid flow and heat transfer in rib-roughened ducts of square cross-section, which are either stationary, or rotate in orthogonal mode. The main objective is to assess how a recently developed variant of a cubic non-linear k−ε model (proposed by Craft et al. Flow Turbul Combust 63:59–80, 1999) can predict three-dimensional flow and heat transfer characteristics through stationary and rotating ribbed ducts. The present paper discusses turbulent air flow and heat transfer through two different configurations, namely: (I) a stationary square duct with “in-line” normal and (II) a square duct with normal ribs in a “staggered” arrangement under stationary and rotating conditions, with the axis of rotation normal to the flow direction and parallel to the ribs. In this paper the flow and thermal predictions of the linear k−ε model (EVM) are also included, as a set of baseline predictions. The mean flow predictions show that both linear and non-linear k−ε models can successfully reproduce most of the measured data for stream-wise and cross-stream velocity components. Moreover, the non-linear model is able to produce better results for the turbulent stresses. The heat transfer predictions show that both EVM and NLEVM2, the more recent variant of the non-linear k−ε, with the algebraic length-scale correction term, overestimate the measured Nusselt numbers for both geometries examined. While the EVM with the differential length-scale correction term underestimates heat transfer levels, the Nusselt number predictions with the NLEVM2 and the ‘NYP’ term are in close agreements with the measured data. Comparisons with our earlier work, Iacovides and Raisee (Int J Heat Fluid Flow, 20:320–328, 1999), show that the NLEVM2 thermal predictions are of similar quality to those of a second-moment closure.     

Nonlocal turbulent transport in a buoyant vortex ring during the ascent of a thermal in a stratified atmosphere

The flow initiated by a hot gas cloud (thermal) in a stratified atmosphere is calculated on the basis of theκ-ε turbulence model and the transport model for the Reynolds stresses and turbulent fluxes and the results obtained are compared The nonlocal nature of the turbulent transport in a vortex ring and its effect on certain flow characteristics are explained In particular, the calculations carried out using the Reynolds stress model show much slower cooling of the temperature-vortex torus than those based calculated on theκ-ε-model Modification of theκ-ε-model to take the effect of curvature of the streamlines approximately into account makes it only partially possible to reproduce the results obtained on the basis of the Reynolds stress model Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 12–20, January–February, 1999. The research was carried out with support from the Russian Foundation for Basic Research (project No. 95-01-00544a).     

Simulation of Wind Flow Around a Building with a k–ε Model

The three-dimensional numerical simulation of airflow around a building using a k–ε two-equation turbulence model is presented in this paper. Several cases of numerical simulation of airflow around a building are carried out to estimate the influence of mesh spacing on simulated results. The accuracy of simulations is examined by comparing the predicted results with wind-tunnel experiments. It is confirmed that numerical simulations by means of the k–ε model reproduce the velocity fields well when using fine mesh resolution. In the latter part of the paper, the simulation method is applied to predict the flow field around a building with different width-to-height ratios, under light wind conditions. Received 16 June 1999 and accepted 20 July 2000     

Investigation of flow and heat transfer in corrugated-undulated plate heat exchangers

 An experimental and numerical investigation of heat transfer and fluid flow was conducted for corrugated-undulated plate heat exchanger configurations under transitional and weakly turbulent conditions. For a given geometry of the corrugated plates the geometrical characteristics of the undulated plates, the angle formed by the latter with the main flow direction, and the Reynolds number were made to vary. Distributions of the local heat transfer coefficient were obtained by using liquid-crystal thermography, and surface-averaged values were computed; friction coefficients were measured by wall pressure tappings. Overall heat transfer and pressure drop correlations were derived. Three-dimensional numerical simulations were conducted by a finite-volume method using a low-Reynolds number k–ɛ model under the assumption of fully developed flow. Computed flow fields provided otherwise inaccessible information on the flow patterns and the mechanisms of heat transfer enhancement. Received on 5 February 1999     

Turbulent flow simulation of liquid jet emanating from pressure-swirl atomizer

The performances of three linear eddy viscosity models (LEVM) and one algebraic Reynolds stress model (ARSM) for the simulation of turbulent flow inside and outside pressure-swirl atomizer are evaluated by comparing the interface position with available experimental data and by comparing the turbulence intensity profiles at the atomizer exit. It is found that the turbulence models investigated exhibit zonal behaviors, i.e. none of the models investigated performs well throughout the entire flow field. The turbulence intensity has a significant influence on the global characteristics of the flow field. The turbulence models with better predictions of the turbulence intensity, such as Gatski-Speziale’s ARSM model, can yield better predictions of the global characteristics of the flow field, e.g. the reattachment lengths for the backward-facing step flow and the sudden expansion pipe flow, or the discharge coefficient, film thickness and the liquid sheet outer surface position for the atomizer flows. The standard k–ε model predicts stronger turbulence intensity as compared to the other models and therefore yields smaller film thickness and larger liquid sheet outer surface position. In average, the ARSM model gives both quantitatively and qualitatively better results as compared to the standard k–ε model and the low Reynolds number models.     

An Attempt to Combine Large Eddy Simulation with the k-ε Model in a Channel-Flow Calculation

Large eddy simulation (LES) is combined with the Reynolds-averaged Navier–Stokes (RANS) equation in a turbulent channel-flow calculation. A one-equation subgrid-scale model is solved in a three-dimensional grid in the near-wall region whereas the standard k–ε model is solved in a one-dimensional grid in the outer region away from the wall. The two grid systems are overlapped to connect the two models smoothly. A turbulent channel flow is calculated at Reynolds numbers higher than typical LES and several statistical quantities are examined. The mean velocity profile is in good agreement with the logarithmic law. The profile of the turbulent kinetic energy in the near-wall region is smoothly connected with that of the turbulent energy for the k–ε model in the outer region. Turbulence statistics show that the solution in the near-wall region is as accurate as a usual LES. The present approach is different from wall modeling in LES that uses a RANS model near the wall. The former is not as efficient as the latter for calculating high-Reynolds-number flows. Nevertheless, the present method of combining the two models is expected to pave the way for constructing a unified turbulence model that is useful for many purposes including wall modeling. Received 11 June 1999 and accepted 15 December 2000     

Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows

Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re _λ between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)³. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other. The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does. In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138, 2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which with increasing Reynolds number approach the empirical values.     

A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow

This paper presents a manufactured solution (MS), resembling a two-dimensional, steady, wall-bounded, incompressible, turbulent flow for RANS codes verification. The specified flow field satisfies mass conservation, but requires additional source terms in the momentum equations. To also allow verification of the correct implementation of the turbulence models transport equations, the proposed MS exhibits most features of a true near-wall turbulent flow. The model is suited for testing six eddy-viscosity turbulence models: the one-equation models of Spalart and Allmaras and Menter; the standard two-equation k–ε model and the low-Reynolds version proposed by Chien; the TNT and BSL versions of the k–ω model.     

Turbulence Modelling of Unsteady Turbulent Flows Using the Stress Strain Lag Model

This paper reports the application of a recently developed turbulence modelling scheme known as the C _as model. This model was specifically developed to capture the effects of stress-strain misalignment observed in turbulent flows with mean unsteadiness. Earlier work has reported the approach applied within a linear k-ε modelling framework, and some initial testing of it within the k-ω SST model of Menter (AIAA J 32:1598–1605, 1994). The resulting k-ε-C _as and SST-C _as models have been shown to result in some of the advantages of a full Reynolds Stress transport Model (RSM), whilst retaining the computational efficiency and stability benefits of a eddy viscosity model (EVM). Here, the development of the the high-Reynolds-number version of the C _as model is outlined, with some example applications to steady and unsteady homogeneous shear flows. The SST-C _as form of the model is then applied to further, more challenging cases of 2-D flow around a NACA0012 aerofoil beyond stall and the 3-D flow around a circular cylinder in a square duct, both being flows which exhibit large, unsteady, separated flow regions. The predictions returned by a range of other common turbulence modelling schemes are included for comparison and the SST-C _as scheme is shown to return generally good results, comparable in some respects to those obtainable from far more complex schemes, for only moderate computing resource requirements.     

Numerical investigation of corner angle and wing number effects on fluid flow characteristics of a turbulent stellar jet

In this research the fluid dynamics characteristics of a stellar turbulent jet flow is studied numerically and the results of three dimensional jet issued from a stellar nozzle are presented. A numerical method based on control volume approach with collocated grid arrangement is employed. The turbulent stresses are approximated using k–ε and k–ω models with four different inlet conditions. The velocity field is presented and the rate of decay at jet centerline is noted. Special attention is drawn on the influence of corner angle and number of wings on mixing in stellar cross section jets. Stellar jets with three; four and five wings and 15–65° corner angles are studied. Also the effect of Reynolds number (based on hydraulic diameter) as well as the inflow conditions on the evolution of the stellar jet is studied. The Numerical results show that the jet entrains more with corner angle 65° and five wings number. The jet is close to a converged state for high Reynolds numbers. Also the influence of the inflow conditions on the jet characteristics is so strong.     

Effect of couple stresses on the flow in a constricted annulus

The flow of an incompressible couple stress fluid in an annulus with local constriction at the outer wall is considered. This configuration is intended as a simple model for studying blood flow in a stenosed artery when a catheter is inserted into it. The effects couple stress fluid parameters α and σ, height of the constriction (ε), and ratio of radii (k) on the impedance and wall shear stresses are studied graphically. Graphical results show that the resistance to the flow as well as the wall shear stress increases as the ratio of the radii increases and decreases as the couple stress fluid parameters increases.     

'T-RANS' Simulation of Deterministic Eddy Structure in Flows Driven by Thermal Buoyancy and Lorentz Force

The paper reports on the application of the Time-dependent Reynolds-Averaged Navier–Stokes (T-RANS) approach to analysing the effects of magnetic force and bottom-wall configuration on the reorganisation of a large coherent structure and its role in the transport processes in Rayleigh–Bénard convection. The large-scale deterministic motion is fully resolved in time and space, whereas the unresolved stochastic motion is modelled by a `subscale' model for which the conventional algebraic stress/flux expressions were used, closed with the low-Re number (k)-(ε)-(θ²) three-equation model. The applied method reproduces long-term averaged mean flow properties, turbulence second moments, and all major features of the coherent roll/cell structure in classic Rayleigh–Bénard convection in excellent agreement with the available DNS and experimental results. Application of the T-RANS approach to Rayleigh–Bénard convection with wavy bottom walls and a superimposed magnetic field yielded the expected effects on there organisation of the eddy structure and consequent modifications of the mean and turbulence parameters and wall heat transfer. This revised version was published online in August 2006 with corrections to the Cover Date.     

Simulation of the thermal method for reducing turbulent friction

The results of calculating a supersonic turbulent boundary layer on a heated surface on the basis of the algebraic two-parameter (k-ε) and four-parameter (k-ε-θ ²-ε ₆) models of turbulence are compared with experimental data. Emphasis is placed on the ability of the models to predict the behavior of the friction and heat-transfer coefficients on a heated surface. The optimal model of turbulence is chosen. The possibility of improving the efficiency of viscous drag reduction by localizing the regions of heat addition to the boundary layer is demonstrated on the basis of numerical calculations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–68, January–February, 1998. This research was carried out with financial support from the International Scientific and Technological Center (project No. 199).     

Revisiting URANS Computations of the Backward-facing Step Flow Using Second Moment Closures. Influence of the Numerics

Previous studies have shown that Unsteady Reynolds-Averaged Navier–Stokes (URANS) computations are able to reproduce the vortex shedding behind a backward-facing step. The aim of the present work is to investigate not only the quantitative predictions of the URANS methodology concerning the characteristic frequencies, but also the amplitude of the energy of the resolved eddies, by using the Elliptic Blending Reynolds Stress Model. This innovative low-Reynolds number second moment closure reproduces the non-viscous, non-local blocking effect of the wall on the Reynolds stresses, and it is compared to the standard k − ε and LRR models using wall-functions. Consistent with previous studies, in the 2D computations shown in the present article, the vortex shedding is captured with the correct Strouhal number, when second moment closures are used. To complete these previous analyses, we particularly focus here on the energy contained in the unsteady, resolved part and its dependency on the numerical method. This energy is less than 5% of the total energy and is strongly dependent on the mesh. Using a refined mesh, surprisingly, a steady solution is obtained. It is shown that this behaviour can be linked to the very small spatial oscillations at the step corner, produced by numerical dispersion, which act as perturbations that are sufficient to excite the natural mode of the shear layer, when the local Peclet number, comparing convection and diffusion effects, is high enough. This result suggests that URANS is not appropriate to quantitatively predict the amplitude of the large-scale structures developing in separated shear-layers, and that URANS results must be interpreted with care in terms of temporal variations of forces, temperatures, etc., in industrial applications using marginally fine meshes.     

Unsaturated Transport with Linear Kinetic Sorption Under Unsteady Vertical Flow

We consider transport of a solute obeying linear kinetic sorption under unsteady flow conditions. The study relies on the vertical unsaturated flow model developed by Indelman et al. [J. Contam. Hydrol. 32 (1998), 77–97] to account for a cycle of infiltration and redistribution. One of the main features of this type of transport, as compared with the case of a continuous water infiltration, is the finite depth of solute penetration. In the infiltration stage an analytical solution that generalizes the previous results of Lassey [Water Resour. Res. 24 (1988), 343–350] and Severino and Indelman [J. Contam. Hydrol. 70 (2004), 89–115] is derived. This solution accounts for quite general initial solute distributions in both the mobile and immobile concentration. When the redistribution is also considered, two timescales become relevant, namely: (i) the desorption rate k⁻¹, and (ii) the water application time t_ap. In particular, we have assumed that the quantity ε =(k t_ap)⁻¹ can be regarded as a small parameter so that a perturbation analytical solution is obtained. At field-scale the concentration is calculated by means of the column model of Dagan and Bresler [Soil Sci. Soc. Am. J. 43 (1979), 461–467], i.e. as ensemble average over an infinite series of randomly distributed and uncorrelated soil columns. It is shown that the heterogeneity of hydraulic properties produces an additional spreading of the plume. An unusual phenomenon of plume contraction is observed at long times of solute propagation during the drying period. The mean solute penetration depth is studied with special emphasis on the impact of the variability of the saturated conductivity upon attaining the maximum solute penetration depth.     

Experimental study and mathematical simulation of the characteristics of a turbulent flow in a straight circular pipe rotating about its longitudinal axis

The vorticity formed in the cross section of a turbulent flow in a straight circular pipe rotating about its longitudinal axis decreases the values of the turbulent stresses, turbulence energy, and dissipation rate along the pipe. The results of laboratory experiments and calculations by the second-order closure model of turbulent transfer are presented. On the whole, the model using a system of transport equations yields better agreement with experimental data than the models with algebraic relations for second-order moments. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 103–116, March–April, 1998.     

Numerical prediction of turbulent flow in a conical diffuser usingk-ε model

Fully developed incompressible turbulent flow in a conical diffuser having a total divergence angle of 8° and an area ratio of 4∶1 has been simulated by ak-ε turbulence model with high Reynolds number and adverse pressure gradient. The research has been done for pipe entry Reynolds numbers of 1.16×10⁵ and 2.93×10⁵. The mean flow velocity and turbulence energy are predicted successfully and the advantage of Boundary Fit Coordinates approach is discussed. Furthermore, thek-ε turbulence model is applied to a flow in a conical diffuser having a total divergence angle of 30° with a perforated screen. A simplified mathematical model, where only the pressure drop is considered, has been used for describing the effect of the perforated screen. The optimum combination of the resistance coefficient and the location of the perforated screen is predicted for high diffuser efficiency or the uniform velocity distribution.     

AN EXTENDED k-ε MODEL FOR NUMERICAL SIMULATION OF WIND FLOW AROUND BUILDINGS

It is assumed in this paper that for a high Reynolds number nearly homogeneouswind flow, the Reynolds stresses are uniquely related to the mean velocity gradientsand the two independent turbulent scaling parameters k and E. By applying dimensionalanalysis and owing to the Cayley-Hamilton theorem for tensors, a new turbulenceenclosure model so-called the axtended k-ε model has been developed. The coefficientsof the model expression were detemined by the wind tunnel experimental data ofhomogeneous shear turbulent flow. The model was compared with the standard k-εmodel in in composition and the prediction of the Reynold’s normal Stresses. Using thenew model the numerical simulation of wind flow around a square cross-section tallbuilding was performed. The results show that the extended k-ε model improves theprediction of wind velocities around the building the building and wind pressures on the buildingenvelope.     

On a Generalized Nonlinear K–ε Model and the Use of Extended Thermodynamics in Turbulence

A resent extension of the nonlinear K–ε model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–ε model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–ε type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–ε model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed. Received 19 February 1998 and accepted 23 October 1998