Prediction of turbulent gas–solids flow in curved ducts using the Eulerian–Lagrangian method

The flow of particulate two‐phase flow mixtures occur in several components of solid fuel combustion systems, such as the pressurised fluidised bed combustors (PFBC) and suspension‐fired coal boilers. A detailed understanding of the mixture characteristics in the conveying component can aid in refining and optimising its design. In this study, the flow of an isothermal, dilute two‐phase particulate mixture has been examined in a high curvature duct, which can be representative of that transporting the gas–solid mixture from the hot clean‐up section to the gas turbine combustor in a PFBC plant. The numerical study has been approached by utilising the Eulerian–Lagrangian methodology for describing the characteristics of the fluid and particulate phases. By assuming that the mixture is dilute and the particles are spherical, the governing particle momentum equations have been solved with appropriately prescribed boundary conditions. Turbulence effects on the particle dispersion were represented by a statistical model that accounts for both the turbulent eddy lifetime and the particle transit time scales. For the turbulent flow condition examined it was observed that mixtures with small particle diameters had low interphase slip velocities and low impaction probability with the pipe walls. Increasing the particle diameters (>50 μm) resulted in higher interphase slip velocities and, as expected, their impaction probability with the pipe walls was significantly increased. The particle dispersion is significant for the smaller sizes, whereas the larger particles are relatively insensitive to the gas turbulence. The main particle impaction region, and locations most prone to erosion damage, is estimated to be within an outer duct length of two to six times the duct diameter, when the duct radius of curvature to the duct diameter ratio is equal to unity. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

Numerical computation of turbulent flow through a square-sectioned 90° bend

Finite-volume, semi-elliptic computations are reported of the three-dimensional flow around a 90° square-sectioned bend for which detailed laser-Doppler measurements have been reported by Taylor et al.¹ While the standard k- eddy-viscosity model has been used in the main flow region, in place of the usual “wall functiond”, the mixing-length hypothesis has been employed to resolve the flow in the layer immediately adjacent to the wall. The scheme is successful in predicting the details of the primary and secondary flow fields both within the bend and downstream thereof.     

Low Reynolds number k–ε model for near‐wall flow

A wall‐distance free k–ε turbulence model is developed that accounts for the near‐wall and low Reynolds number effects emanating from the physical requirements. The model coefficients/functions depend non‐linearly on both the strain rate and vorticity invariants. Included diffusion terms and modified C_ε(1,2) coefficients amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Numerical study of the turbulent flow in strongly curved stationary and rotating U‐ducts

A large‐eddy simulation (LES) study is undertaken to explore the complex flow of developing turbulent flow through stationary and rotating U‐ducts with strong curvature. Three flow cases are investigated: stationary (non‐rotating), positive and negative rotational cases. Stationary and positive rotational cases are shown to have similar flow characteristics in terms of the mean velocity variations, although the predicted separation zone is nearly doubled in size for positive rotational case. Unlike the positive rotation for which the laminarization effects are observed, turbulence is significantly enhanced for the negative rotation mainly due to the existence of strong secondary flow. Turbulence is found to be highly anisotropic throughout the duct apart from the far downstream regions of the bend for the negative rotational case. The stress–strain relation seems to be completely invalid in the U‐duct apart from the bend region. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical study of turbulent line puffs via the renormalization group (RNG) k–ϵ model

The time evolution of a line puff, a turbulent non-buoyant element with significant momentum, is studied using the renormalization group (RNG) k–ϵ model. The numerical results show that the puff motion is characterized by a vortex pair flow; the computed flow details and scalar mixing characteristics can be described by self-similar relations beyond a dimensionless time of around 30. The added mass coefficient of the puff motion is found to be approximately unity. The predicted puff flow and mixing rate are substantially similar to those obtained from the standard k–ϵ model and are well supported by experimental data. The computed scalar field reveals significant secondary concentration peaks trailing behind in the wake of the puff. The present results suggest that the overall mixing rate of a puff is primarily determined by the large-scale motion and that streamline curvature probably plays a minor role. © 1998 John Wiley & Sons, Ltd.     

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.     

Evaluation of one‐ and two‐equation low‐Re turbulence models. Part II—Vortex‐generator jet and diffusing S‐duct flows

This second segment of the two‐part paper systematically examines several turbulence models in the context of two flows, namely a vortex flow created by an inclined jet in crossflow, and the flow field in a diffusing S‐shaped duct. The test cases are chosen on the basis of availability of high‐quality and detailed experimental data. The tested turbulence models are integrated to solid surfaces and consist of: Rodi's two‐layer k–ε model, Wilcox's k–ω model, Menter's two‐equation shear–stress‐transport model, and the one‐equation model of Spalart and Allmaras. The objective of the study is to establish the prediction accuracy of these turbulence models with respect to three‐dimensional separated flows with streamline curvature. At the same time, the study establishes the minimum spatial resolution requirements for each of these turbulence closures, and identifies the proper low‐Mach‐number preconditioning and artificial diffusion settings of a Reynolds‐averaged Navier–Stokes algorithm for optimum rate of convergence and minimum adverse impact on prediction accuracy. Copyright © 2003 John Wiley & Sons, Ltd.     

The analysis of the channel flow sensitivity to the parameters of the k–ε method

This paper deals with the problem of using sensitivity analysis for fluid mechanics solutions to the constants of the standard k–ε method for 2D, incompressible and steady flows. The problem is described and analysed on the basis of a channel flow. Sensitivity coefficients of the following properties were determined: a pressure, two components of a velocity, a turbulence kinetic energy, a dissipation rate of turbulence kinetic energy and a turbulence dynamic viscosity. The calculated property values depend on five model constants that are parameters of the sensitivity analysis in this paper. Sensitivity coefficients are derivatives of the above properties, for individual parameters. In this paper these coefficients are determined using a finite difference approximation to the sensitivities coefficients. The author of this paper compares three models of the boundary layer with regard to the sensitivity of properties to the parameters. Irrespective of the boundary layer model used here, the analysis of sensitivity coefficients for the channel flow properties shows that the most sensitive property is the turbulence dissipation rate. Next properties of consequence, although of significantly smaller values of sensitivity coefficients, are the turbulence viscosity and the turbulence kinetic energy. All flow properties are mostly sensitive to the C_µ parameter. One of the final conclusions in this paper is that the analysis of sensitivity coefficient fields allows the reliable checking of results and indicates those areas most prone to calculation difficulties. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical and experimental investigation of turbulent flow in a rectangular duct

Details of the turbulent flow in a 1:8 aspect ratio rectangular duct at a Reynolds number of approximately 5800 were investigated both numerically and experimentally. The three-dimensional mean velocity field and the normal stresses were measured at a position 50 hydraulic diameters downstream from the inlet using laser doppler velocimetry (LDV). Numerical simulations were carried out for the same flow case assuming fully developed conditions by imposing cyclic boundary conditions in the main flow direction. The numerical approach was based on the finite volume technique with a non-staggered grid arrangement and the SIMPLEC algorithm. Results have been obtained with a linear and a non-linear (Speziale) k–ε model, combined with the Lam–Bremhorst damping functions for low Reynolds numbers. The secondary flow patterns, as well as the magnitude of the main flow and overall parameters predicted by the non-linear k–ε model, show good agreement with the experimental results. However, the simulations provide less anisotropy in the normal stresses than the measurements. Also, the magnitudes of the secondary velocities close to the duct corners are underestimated. © 1998 John Wiley & Sons, Ltd.     

OPERATOR–SPLITTING COMPUTATION OF TURBULENT FLOW IN AN AXISYMMETRIC 180° NARROWING BEND USING SEVERAL k—ϵ MODELS AND WALL FUNCTIONS

This paper describes a finite element implementation of an operator-splitting algorithm for solving transient/steady turbulent flows and presents solutions for the turbulent flow in an axisymmetric 180° narrowing bend, a benchmark problem dealt with at the 1994 WUA-CFD annual meeting. Three k–ϵ based models are used: the standard linear k–ϵ model, a non-linear k–ϵ model and an RNG k–ϵ model. Flow separation after the bend, as observed in the experiment, is predicted by the RNG model and by both the linear and non-linear k–ϵε models with van Driest mixing length wall functions. Good agreement with experimental data of pressure distribution on bending walls is obtained by the present numerical simulation. Results show that there is very little difference between the linear and non-linear k–ϵε models in terms of predicted velocity fields and that the non-linearities mainly affect the distribution of turbulent normal stress and pressure, in analogy to the effect of second-order viscoelastic fluid models on laminar flow. Both the linear and non-linear k–ϵε models fail to predict any flow separation if logarithmic wall functions are used.     

Assessment of variants of the k–ϵ turbulence model for engine flow applications

This paper is concerned with simulation of the mean flow and turbulence evolution in a model engine and comparison of the behaviour of certain important turbulence parameters, namely the intensity, length scale and dissipation time scale, as predicted by three variants of the k–? model developed for application to strongly compressible flows. The predictions pertain to the axisymmetric, disc-chamber, four-stroke, Imperial College model engine operating at 200 rpm and compression ratios of 3·5 and 6·7. The paper analyses the predicted variations of these parameters during the induction, compression and expansion strokes and identifies the versions that produce the most consistent and physically plausible variations. The significance, to the turbulence evolution, of the ratio of the turbulence dissipation time scale to the time scale of compression/expansion is also discussed. It is concluded that on these grounds the Morel–Mansour and El Tahry versions are, and the Watkins version is not, suitable for engine applications.     

On steady state computation of turbulent flows using k–ε models approximated by the time splitting method

The time splitting method is frequently used in numerical integration of flow equations with source terms since it allows almost independent programming for the source part. In this paper we will consider the question of convergence to steady state of the time splitting method applied to k–ε turbulence models. This analysis is derived from a properly defined scalar study and is carried out with success for the coupled k–ε equations. It is found that the time splitting method does not allow convergence to steady state for any choice of finite values of the time step. Numerical experiments for some typical turbulent compressible flow problems support the fact that the time splitting method is always nonconvergent, while its nonsplitting counterpart is convergent. Copyright © 2005 John Wiley & Sons, Ltd.     

NUMERICAL INVESTIGATION OF TURBULENT SHALLOW RECIRCULATING FLOWS BY A QUASI-THREE-DIMENSIONAL k–ϵMODEL

A quasi-three-dimensional multilayer k– ϵ model has been developed to simulate turbulent recirculating flows behind a sudden expansion in shallow waters. The model accounts for the vertical variation in the flow quantities and eliminates the problem of closure for the effective stresses resulting from the depth integration of the non-linear convective accelerations found in the widely used depth- integrated models. The governing equations are split into three parts in the finite difference solution: advection, dispersion and propagation. The advection part is solved using the four-node minimax–characteristics method. The dispersion and propagation parts are treated by the central difference method, the former being solved explicitly and the latter implicitly using the Gauss–Seidel iteration method. The relative effect of bed-generated turbulence and transverse shear-generated turbulence on the recirculating flow has been studied in detail. In comparison with the results computed by the depth-integrated k–ϵ model, the results computed by the present model are found to be closer to the reported data.     

Numerical analysis of developing turbulent flow in a 180° bend tube by an algebraic Reynolds stress model

A numerical analysis has been performed for three‐dimensional developing turbulent flow in a 180° bend tube with straight inlet and outlet section used by an algebraic Reynolds stress model. To our knowledge, numerical investigations, which show the detailed comparison between calculated results and experimental data including distributions of Reynolds stresses, are few and far between. From this point of view, an algebraic Reynolds stress model in conjunction with boundary‐fitted co‐ordinate system is applied to a 180° bend tube in order to predict the anisotropic turbulent structure precisely. Calculated results are compared with the experimental data including distributions of Reynolds stresses. As a result of this analysis, it has been found that the calculated results show a comparatively good agreement with the experimental data of the time‐averaged velocity and the secondary vectors in both the bent tube and straight outlet sections. For example, the location of the maximum streamwise velocity, which appears near the top or bottom wall in the bent tube, is predicted correctly by the present method. As for the comparison of Reynolds stresses, the present method has been found to simulate many characteristic features of streamwise normal stress and shear stresses in the bent tube qualitatively and has a tendency to under‐predict its value quantitatively. Judging from the comparison between the calculated and the experimental results, the algebraic Reynolds stress model is applicable to the developing turbulent flow in a bent tube that is known as a flow with a strong convective effect. Copyright © 2005 John Wiley & Sons, Ltd.     

A novel implicit numerical treatment for non‐linear turbulence models using high and low Reynolds number formulations

Non‐linear turbulence models can be seen as an improvement of the classical eddy‐viscosity concept due to their better capacity to simulate characteristics of important flows. However, application of non‐linear models demand robustness of the numerical method applied, requiring a stable discretization scheme for convergence of all variables involved. Usually, non‐linear terms are handled in an explicit manner leading to possible numerical instabilities. Thus, the present work shows the steps taken to adapt a general non‐linear constitutive equation using a new semi‐implicit numerical treatment for the non‐linear diffusion terms. The objective is to increase the degree of implicitness of the solution algorithm to enhance convergence characteristics. Flow over a backward‐facing step was computed using the control volume method applied to a boundary‐fitted coordinate system. The SIMPLE algorithm was used to relax the algebraic equations. Classical wall function and a low Reynolds number model were employed to describe the flow near the wall. The results showed that for certain combination of relaxation parameters, the semi‐implicit treatment proposed here was the sole successful treatment in order to achieve solution convergence. Also, application of the implicit method described here shows that the stability of the solution either increases (high Reynolds with non‐orthogonal mesh) or preserves the same (low Reynolds number applications). Additional advantages of the procedure proposed here lie in the possibility of testing different non‐linear expressions if one considers the enhanced robustness and stability obtained for the entire numerical algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

Time‐dependent simulation of cavitating flow with k − ℓ turbulence models

A compressible, multiphase, one‐fluid Reynolds‐averaged Navier–Stokes solver has been developed to study turbulent cavitating flows. The interplay between turbulence and cavitation regarding the unsteadiness and structure of the flow is complex and not well understood. This constitutes a critical point to accurately simulate the dynamic behavior of sheet cavities. In the present study, different formulations based on a k ? ? transport‐equation model are investigated and a scale‐adaptive formulation is proposed. Numerical results are given for a Venturi geometry and comparisons are made with experimental data. The scale‐adaptive model shows several improvements compared with standard turbulence models. Copyright © 2011 John Wiley & Sons, Ltd.     

SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITH k–ε TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow- water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard k–ε turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and ε. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substi- tuting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver. Evaluation of horizontal gradients in the σ-co-ordinate system requires high-order derivatives which can cause spurious flows and this is avoided by obtaining these gradients in real space. In this paper the method is applied to parallel oscillatory (tidal) flow in deep and shallow water and compared with field measurements. It is then applied to current flow about a conical island of small side slope where vortex shedding occurs and velocities are compared with data from the laboratory. Computed concentration distributions are also compared with dye visualization and an example of the influence of temperature on plume dispersion is presented. © 1997 John Wiley & Sons, Ltd.