Calculation of complex near-wall turbulent flows with a low-Reynolds-number k-ϵ model

An improved low-Reynolds-number k-? model has been formulated and tested against a range of DNS (direct numerical simulation) and experimental data for channel and complex shear layer flows. The model utilizes a new form of damping function adopted to account for both wall proximity effects and viscosity influences and a more flexible damping argument based on the gradient of the turbulent kinetic energy on the wall. Additionally, the extra production of the inhomogeneous part of the viscous dissipation near a wall has been added to the dissipation equation with significantly improved results. The proposed model was successfully applied to the calculation of a range of wall shear layers in zero, adverse and favourable pressure gradients as well as backward-facing-step separated flows.     

Numerical investigation of fully developed turbulent fluid flow and heat transfer in a square duct

Three-dimensional fully developed turbulent fluid flow and heat transfer in a square duct are numerically investigated with the author's anisotropic low-Reynolds-number k-ε turbulence model. Special attenton has been given to the regions close to the wall and the corner, which are known to influence the characteristics of secondary flow a great deal. Hence, instead of the common wall function approach, the no-slip boundary condition at the wall is directly used. Velocity and temperature profiles are predicted for fully developed turbulent flows with constant wall temperature. The predicted variations of both local wall shear stress and local wall heat flux are shown to be in close agreement with available experimental data. The present paper also presents the budget of turbulent kinetic energy equation and the systematic evaluation for existing wall function forms. The commonly adopted wall function forms that are valid for two-dimensional flows are found to be inadequate for three-dimensional turbulent flows in a square duct.     

Prediction of turbulent wall shear flows directly from wall

The fully elliptic Reynolds-averaged Navier–Stokes equations have been used together with Lam and Bremhorst's low-Reynolds-number model, Chen and Patel's two-layer model and a two-point wall function method incorporated into the standard k-? model to predict channel flows and a backward-facig step flow. These flows enable the evaluation of the performance of different near-wall treatments in flows involving streamwise and normal pressure gradients, flows with separation and flows with non-equilibrium turbulence characteristics. Direct numerical simulation (DNS) of a channel flow with Re =3200 further provides the detailed budgets of each modelling term of the k and ?-transport equations. Comparison of model results with DNS data to evaluate the performance of each modelling term is also made in the present study. It is concluded that the low-Reynolds-number model has wider applicability and performs better than the two-layer model and wall function approaches. Comparison with DNS data further shows that large discrepancies exist between the DNS budgets and the modelled production and destruction terms of the ? equation. However, for simple channel flow the discrepancies are similar in magnitude but opposite in sign, so they are cancelled by each other. This may explain why, even when employing such an inaccurately modelled ?-equation, one can still predict satisfactorily some simple turbulent flows.     

A parametric study of LES on laminar-turbulent transitional flows past an airfoil

Low-Reynolds-number aerodynamic performance of small-sized air vehicles is an area of increasing interest. In this study, low-Reynolds-number flows past an SD7003 airfoil are investigated to understand important viscous features of laminar separation and transitional flow followed by the complicated behavior of the flow reattachment process. In order to satisfy the three-dimensional (3D) requirement of the code, a simple “3D wing” is constructed from a two-dimensional (2D) airfoil. A parametric study of large eddy simulation (LES) on the airfoil flows at Re = 60,000 is performed. Effects of grid resolution and sub-grid scale (SGS) models are investigated. Although 3D effects cannot be accurately captured owing to the limitation of the grid resolution in the spanwise direction, the preliminary LES calculations do reveal some important flow characteristics such as leading-edge laminar separation and vortex shedding from the primary laminar separation bubble on the low-Reynolds-number airfoil.     

On the eddy viscosity model of periodic turbulent shear flows

Physical argument shows that eddy viscosity is essentially different from molecular viscosity. By direct numerical simulation, it was shown that for periodic turbulent flows, there is phase difference between Reynolds stress and rate of strain. This finding posed great challenge to turbulence modeling, because most turbulence modeling, which use the idea of eddy viscosity, do not take this effect into account. The project supported by the National Natural Science Foundation of China (19732005) and Liu Hui Center for Applied Mathematics of Nankai & Tianjin University     

A von Neumann–Smagorinsky turbulent transport model for stratified shear flows

A simple subgrid turbulent diffusion model based on an analogy to the von Neumann–Richtmyer artificial viscosity is explored for use in modelling mixing in turbulent stratified shear flow. The model may be more generally applicable to multicomponent turbulent hydrodynamics and to subgrid turbulent transport of momentum, composition and energy. As in the case of the von Neumann artificial viscosity and many subgrid-scale models for large-eddy simulation, the turbulent diffusivity explicitly depends on the grid size and is not based on a quantitative model of the unresolved turbulence. In order to address the issue that it is often not known a priori when and where a flow will become turbulent, the turbulent diffusivity is set to zero when the flow is expected to be stable on the basis of a Richardson/Rayleigh–Taylor stability criterion, in analogy to setting the von Neumann artificial viscosity to zero in expanding flows. One-dimensional predictions of this model applied to a simple shear flow configuration are compared to those obtained using a K–ε model. The density and velocity profiles predicted by both models are shown to be very similar.     

A non‐linear k–ε model with realizability for prediction of flows around bluff bodies

The incompressible flow around bluff bodies (a square cylinder and a cube) is investigated numerically using turbulence models. A non‐linear k–ε model, which can take into account the anisotropy of turbulence with less CPU time and computer memory then RSM or LES, is adopted as a turbulence model. In tuning of the model coefficients of the non‐linear terms are adjusted through the examination of previous experimental studies in simple shear flows. For the tuning of the coefficient in the eddy viscosity (=C_μ), the realizability constraints are derived in three types of basic 2D flow patterns, namely, a simple shear flow, flow around a saddle and a focal point. C_μ is then determined as a function of the strain and rotation parameters to satisfy the realizability. The turbulence model is first applied to a 2D flow around a square cylinder and the model performance for unsteady flows is examined focussing on the period and the amplitude of the flow oscillation induced by Karman vortex shedding. The applicability of the model to 3D flows is examined through the computation of the flow around a surface‐mounted cubic obstacle. The numerical results show that the present model performs satisfactorily to reproduce complex turbulent flows around bluff bodies. Copyright © 2003 John Wiley & Sons, Ltd.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of a differential equation for turbulent viscosity to the analysis of plane non-self-similar flows

A differential equation of the kinetic-energy balance of turbulence is used in a number of papers to close the equations describing average motion in turbulent flows. On the basis of this relation, a differential equation for turbulent viscosity is obtained herein. Numerical computations are carried out for incompressible non-self-similar turbulent and transition flows in awake, a jet, and a boundary layer; universal constants in the equation for the viscosity are refined. The flow in a wake and boundary layer with high longitudinal pressure gradients is investigated by analytical and numerical methods. Dimensionless criteria determining the nature of the effect of the pressure gradient on the average flow and turbulent viscosity are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–127, September–October, 1971.The author is greateful to I. P. Smirnov, S. Yu. Krasheninnikov, and V. B. Kuz'mich for aid in compiling the program for the numerical computations and to L. L. Bychkov for processing the computational results and plotting the graphs.     

Application of the method of near-wall boundary conditions to an investigation of turbulent flows with longitudinal pressure gradients

The dynamic and thermal characteristics of steady near-wall boundary layers in flow deceleration regions are studied on the basis of differential turbulencemodels. The method of transferring the boundary conditions from the wall into the flow is tested for flows with variable longitudinal pressure gradients. Using differential turbulence models in the transition and low-Reynolds-number regions near surfaces the effect of the parameters of highly turbulent free stream on the development of dynamic processes in the developed turbulent boundary layer in the flow deceleration region is studied. The calculated profiles of the velocity, the kinetic energy of turbulence, the friction and thermal conductivity coefficients, and the temperature factor are compared with the experimental data in the cases in which the boundary conditions are preassigned both on the wall and in the flow. The effect of an intermediate boundary condition on the results of the calculations is analyzed.     

Applicability of the boundary-layer approximation to the calculation of a plane turbulent mixing layer

Calculation of a mixing layer-a classical problem in the theory of jet flows-is usually performed with a boundary-layer approximation. If the velocity of one of the flows is small, then even in the case of an incompressible fluid the angle of expansion of the mixing layer reaches 20?, and in the case of flows with different densities this angle is even larger. Therefore, there is some doubt about the smallness of the neglected terms which refer to differentiation in the flow direction. In the present paper the problem of the mixing of two semiinfinite flows is solved both with a boundary-layer approximation and by using a complete system of Reynolds equations (the molecular viscosity is assumed small compared to the turbulent viscosity; Re→∞).     

A variant of the low-Reynolds-number two-equation turbulence model applied to variable property mixed convection flows

Previous work has demonstrated that the low-Reynolds-number model of Launder and Sharma (1974) offers significant advantages over other two-equation turbulence models in the computation of highly non-universal buoyancy-influenced (or “mixed convection”) pipe flows. It is known, however, that the Launder and Sharma model does not possess high quantitative accuracy in regard to simpler forced convection flows. A variant of the low-Reynolds-number scheme is developed here by reference to data for constant property forced convection flows. The re-optimized model and the Launder and Sharma formulation are then examined against experimental measurements for mixed convection flows, including cases in which variable property effects are significant.     

Nonlinear turbulence models for predicting strong curvature effects

Prediction of the characteristics of turbulent flows with strong streamline curvature, such as flows in turbomachines, curved channel flows, flows around airfoils and buildings, is of great importance in engineering applications and poses a very practical challenge for turbulence modeling. In this paper, we analyze qualitatively the curvature effects on the structure of turbulence and conduct numerical simulations of a turbulent Uduct flow with a number of turbulence models in order to assess their overall performance. The models evaluated in this work are some typical linear eddy viscosity turbulence models, nonlinear eddy viscosity turbulence models （NLEVM） （quadratic and cubic）, a quadratic explicit algebraic stress model （EASM） and a Reynolds stress model （RSM） developed based on the second-moment closure. Our numerical results show that a cubic NLEVM that performs considerably well in other benchmark turbulent flows, such as the Craft, Launder and Suga model and the Huang and Ma model, is able to capture the major features of the highly curved turbulent U-duct flow, including the damping of turbulence near the convex wall, the enhancement of turbulence near the concave wall, and the subsequent turbulent flow separation. The predictions of the cubic models are quite close to that of the RSM, in relatively good agreement with the experimental data, which suggests that these models may be employed to simulate the turbulent curved flows in engineering applications.     

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.     

Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes

The results of direct numerical simulation of turbulent flows of non-Newtonian pseudoplastic fluids in a straight pipe are presented. The data on the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances are obtained for steady turbulent flows at the Reynolds numbers of 10⁴ and 2×10⁴. As distinct from Newtonian fluid flows, the viscous shear stresses turn out to be significant even far from the wall. In power-law fluid flows the mechanism of the energy transport from axial to transverse component fluctuations is suppressed. It is shown that with decrease in the fluid index the turbulent transfer of the momentum and the velocity fluctuations between the wall layer and the flow core reduces, while the turbulent energy flux toward the wall increases. The earlier-proposed models for the average viscosity and the non-Newtonian one-point correlations are in good agreement with the data of direct numerical simulation.     

Inner-outer decomposition of wall shear stress fluctuations in turbulent channels

Fluctuating wall shear stress in turbulent channel flows is decomposed into small-scale and large-scale components. The large-scale fluctuating wall shear stress is computed as the footprints of the outer turbulent motions, and the small-scale one is obtained by subtracting the large-scale one from the total, which fully remove the outer influences. We show that the statistics of the small-scale fluctuating wall shear stress is Reynolds number independent at the friction Reynolds number larger than 1000, while which is Reynolds number dependent or the low-Reynolds-number effect exists at the friction Reynolds number smaller than 1000. Therefore, a critical Reynolds number that defines the emergence of universal small-scale fluctuating wall shear stress is proposed to be 1000. The total and large-scale fluctuating wall shear stress intensities approximately follow logarithmic-linear relationships with Reynolds number, and empirical fitting expressions are given in this work.     

A Dissipation Rate Equation for Low-Reynolds-Number and Near-Wall Turbulence

Two-equation turbulence models are usually formulated for specific flow types and are seldom validated against a variety of flows to account for near-wall and low-Reynolds-number effects simultaneously. In addition to low-Reynolds-number effects, near-wall flows also experience wall blocking, which is absent in free flows. Consequently, near-wall modifications to two-equation models could be quite different from low-Reynolds-number corrections. Besides, it is known that existing two-equation models perform poorly when used to calculate plane wall jets and two-dimensional backstep flows. These problems could be traced to the modeling of the dissipation rate equation. In this paper an attempt is made to improve the modeling of the dissipation rate equation so that it could successfully predict both free and wall-bounded shear flows including plane wall jets and backstep flows. The predictions are compared with experimental and direct numerical simulation data whenever available. Most of the data used are obtained at low Reynolds numbers. Good correlation with data is obtained. Therefore, for the first time, a model capable of correctly predicting free and wall-bounded shear flows, backstep flows, and plane wall jets is available. Received: 12 December 1995 and accepted 12 November 1996     

Dynamical eigenfunction decomposition of turbulent channel flow

The results of an analysis of low-Reynolds-number turbulent channel flow based on the Karhunen-Loéve(K-L) expansion are presented. The turbulent flow field is generated by a direct numerical simulation of the Navier-Stokes equations at a Reynolds number Re,= 80 (based on the wall shear velocity and channel half-width). The K-L procedure is then applied to determine the eigenvalues and eigenfunctions for this flow. The random coefficients of the K-L expansion are subsequently found by projecting the numerical flow field onto these eigenfunctions. The resulting expansion captures 90% of the turbulent energy with significantly fewer modes than the original trigonometric expansion. The eigenfunctions, which appear either as rolls or shearing motions, posses viscous boundary layers at the walls and are much richer in harmonics than the original basis functions. Chaotic temporal behaviour is observed in all modes and increases for higher-order eigenfunctions. The structure and dynamical behaviour of the eigenmodes are discussed as well as their use in the representation of the turbulent flow.     

Energy-based decomposition of friction drag in turbulent square-duct flows

The generation of friction drag in turbulent duct flows has direct connection with statistical quantities and corresponding turbulence dynamics in the duct cross-section. In this study, we generalize the RD identity (Renard and Deck, 2016) to a ‘two-dimensional’ form which we exploit to decompose the mean friction drag in turbulent square-duct flows into contributions associated with viscosity, turbulence and cross-stream convection. The friction Reynolds number of the duct flows ranges from 220 to 2000. The scaling, spatial distribution and local normalization of the contributions to friction are investigated and compared with those in pipe and channel flows. As in other canonical flows, we find logarithmic growth of the turbulent contribution in contrast to the viscous one, the former thus becoming dominant at high enough Reynolds numbers. Whereas cross-stream convection has no net effect on friction, its contribution may be locally comparable to the other two, hence may be responsible for redistribution of friction along the duct perimeter.     

A FENE-P k–ε turbulence model for low and intermediate regimes of polymer-induced drag reduction

A new low-Reynolds-number k–ε turbulence model is developed for flows of viscoelastic fluids described by the finitely extensible nonlinear elastic rheological constitutive equation with Peterlin approximation (FENE-P model). The model is validated against direct numerical simulations in the low and intermediate drag reduction (DR) regimes (DR up to 50%). The results obtained represent an improvement over the low DR model of Pinho et al. (2008) [A low Reynolds number k–ε turbulence model for FENE-P viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics, 154, 89–108]. In extending the range of application to higher values of drag reduction, three main improvements were incorporated: a modified eddy viscosity closure, the inclusion of direct viscoelastic contributions into the transport equations for turbulent kinetic energy (k) and its dissipation rate, and a new closure for the cross-correlations between the fluctuating components of the polymer conformation and rate of strain tensors (NLT_ij). The NLT_ij appears in the Reynolds-averaged evolution equation for the conformation tensor (RACE), which is required to calculate the average polymer stress, and in the viscoelastic stress work in the transport equation of k. It is shown that the predictions of mean velocity, turbulent kinetic energy, its rate of dissipation by the Newtonian solvent, conformation tensor and polymer and Reynolds shear stresses are improved compared to those obtained from the earlier model.