Numerical modelling of complex turbulent free‐surface flows with the SPH method: an overview

The gridless smoothed particle hydrodynamics (SPH) method is now commonly used in computational fluid dynamics (CFD) and appears to be promising in predicting complex free‐surface flows. However, increasing flow complexity requires appropriate approaches for taking account of turbulent effects, whereas some authors are still working without any turbulence closure in SPH. A review of recently developed turbulence models adapted to the SPH method is presented herein, from the simplistic point of view of a one‐equation model involving mixing length to more sophisticated (and thus realistic) models like explicit algebraic Reynolds stress models (EARSM) or large eddy simulation (LES). Each proposed model is tested and validated on the basis of schematic cases for which laboratory data, theoretical or numerical solutions are available in the general field of turbulent free‐surface incompressible flows (e.g. open‐channel flow and schematic dam break). They give satisfactory results, even though some progress should be made in the future in terms of free‐surface influence and wall conditions. Recommendations are given to SPH users to apply this method to the modelling of complex free‐surface turbulent flows. Copyright © 2006 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.     

A Navier–Stokes solver for complex three-dimensional turbulent flows adopting non-linear modelling of the Reynolds stresses

A non-linear modelling of the Reynolds stresses has been incorporated into a Navier–Stokes solver for complex three-dimensional geometries. A k–ε model, adopting a modelling of the turbulent transport which is not based on the eddy viscosity, has been written in generalised co-ordinates and solved with a finite volume approach, using both a GMRES solver and a direct solver for the solution of the linear systems of equations. An additional term, quadratic in the main strain rate, has been introduced into the modelling of the Reynolds stresses to the basic Boussinesq's form; the corresponding constant has been evaluated through comparison with the experimental data. The computational procedure is implemented for the flow analysis in a 90° square section bend and the obtained results show that with the non-linear modelling a much better agreement with the measured data is obtained, both for the velocity and the pressure. The importance of the convection scheme is also discussed, showing how the effect of the non-linear correction added to the Reynolds stresses is effectively hidden by the additional numerical diffusion introduced by a low-order convection scheme as the first-order upwind scheme, thus making the use of higher order schemes necessary. © 1998 John Wiley & Sons, Ltd.     

An algorithm for rotating axisymmetric flows: model,validation and industrial applications

The study of axisymmetric flows is of interest not only from an academic point of view, due to the existence of exact solutions of Navier–Stokes equations, but also from an industrial point of view, since these kind of flows are frequently found in several applications. In the present work the development and implementation of a finite element algorithm to solve Navier–Stokes equations with axisymmetric geometry and boundary conditions is presented. Such algorithm allows the simulation of flows with tangential velocity, including free surface flows, for both laminar and turbulent conditions. Pseudo‐concentration technique is used to model the free surface (or the interface between two fluids) and the k–ε model is employed to take into account turbulent effects. The finite element model is validated by comparisons with analytical solutions of Navier–Stokes equations and experimental measurements. Two different industrial applications are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

A numerical investigation of 2d,steady free surface flows

Systematic tests have been performed to study the behaviour of a numerical method developed to calculate 2D, steady free surface flows. The Reynolds equations are solved in the physical space by employing a non–orthogonal staggered grid, while the k-ε model is adopted to approximate the Reynolds stresses. The free surface is calculated following an iterative procedure and various parameters that affect convergence and accuracy of the numerical solution have been examined. Calculated results are compared with measured data for two cases, i.e. the wave generation above a bottom topography at various Froude numbers and the free surface formation above a submerged hydrofoil. © 1997 John Wiley & Sons, Ltd.     

A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows

A three‐dimensional, non‐hydrostatic pressure, numerical model with k–ε equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non‐hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non‐hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure–Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element simulation of turbulent Couette–Poiseuille flows using a low Reynolds number k–ϵ model

Developing Couette–Poiseuille flows at Re=5000 are studied using a low Reynolds number k–ϵ two‐equation model and a finite element formulation. Mesh‐independent solutions are obtained using a standard Galerkin formulation and a Galerkin/least‐squares stabilized method. The predictions for the velocity and turbulent kinetic energy are compared with available experimental results and to the DNS data. Second moment closure's solutions are also compared with those of the k–ϵ model. The deficiency of eddy viscosity models to predict dissymmetric low Reynolds number channel flows has been demonstrated. Copyright © 1999 John Wiley & Sons, Ltd.     

Incompressible Roe‐like scheme for turbulent flows

In the current study, numerical investigation of incompressible turbulent flow is presented. By the artificial compressibility method, momentum and continuity equations are coupled. Considering Reynolds averaged Navier–Stokes equations, the Spalart–Allmaras turbulence model, which has accurate results in two‐dimensional problems, is used to calculate Reynolds stresses. For convective fluxes a Roe‐like scheme is proposed for the steady Reynolds averaged Navier–Stokes equations. Also, Jameson averaging method was implemented. In comparison, the proposed characteristics‐based upwind incompressible turbulent Roe‐like scheme, demonstrated very accurate results, high stability, and fast convergence. The fifth‐order Runge–Kutta scheme is used for time discretization. The local time stepping and implicit residual smoothing were applied as the convergence acceleration techniques. Suitable boundary conditions have been implemented considering flow behavior. The problem has been studied at high Reynolds numbers for cross flow around the horizontal circular cylinder and NACA0012 hydrofoil. Results were compared with those of others and a good agreement has been observed. Copyright © 2012 John Wiley & Sons, Ltd.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.     

SOME VALIDATION OF STANDARD,MODIFIED AND NON-LINEAR k–ε TURBULENCE MODELS

Standard, modified and non-linear k–ε: turbulence models are validated against three axisymmetric flow problems—flow through a pipe expansion, flow through a pipe constriction and an impinging jet problem—to underpin knowledge about the solution quality obtained from two-equation turbulence models. The extended models improve the prediction of turbulence as a flow approaches a stagnation point and the non-linear model allows for the prediction of anisotropic turbulence. Significantly different values for the non-linear model coefficients are proposed in comparison with values found in the literature. Nevertheless, current turbulence models are still unable to accurately predict the spreading rate of shear layers. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids, 24: 965–986, 1997.     

PREDICTION OF AEROFOIL WAKE SUBJECTED TO THE EFFECTS OF CURVATURE AND PRESSURE GRADIENT

Experimental data on the development of wakes in a straight duct, a curved duct, a curved diffuser and a straight diffuser are compared with computations based on a finite volume scheme incorporating the k– ε model of turbulence. The results show that the computations based on the standard k–ε model are able to satisfactorily capture only the mean velocity profiles. To improve the predictions, several modifications to the model are tried out. Close agreement between experiment and computation as regards the velocity profiles, wake parameters and profiles of the turbulent kinetic energy k and Reynolds shear stress ⌅{uv} is obtained when modification to the model constant C_μ, based on the curvature parameter and the ratio of the production of turbulent kinetic energy to its rate of dissipation, is incorporated. The modified model is also able to capture the asymmetry in the profiles of k and ⌅{uv} caused by the curvature and its enhancement due to the additional presence of an adverse pressure gradient.     

Absorbing boundary technique for open channel flows

An absorbing boundary condition is formulated and applied to the one‐dimensional open channel flow equations in conjunction with an explicit MacCormack scheme. The physical flow domain has been truncated by introducing an artificial pseudo‐boundary. By using an appropriate boundary condition on a truncated domain, it is shown that, for flow containing shocks, the solution can be accelerated to its stationary profile with no loss of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

Development of an artificial compressibility methodology using flux vector splitting

An implicit, upwind arithmetic scheme that is efficient for the solution of laminar, steady, incompressible, two-dimensional flow fields in a generalised co-ordinate system is presented in this paper. The developed algorithm is based on the extended flux-vector-splitting (FVS) method for solving incompressible flow fields. As in the case of compressible flows, the FVS method consists of the decomposition of the convective fluxes into positive and negative parts that transmit information from the upstream and downstream flow field respectively. The extension of this method to the solution of incompressible flows is achieved by the method of artificial compressibility, whereby an artificial time derivative of the pressure is added to the continuity equation. In this way the incompressible equations take on a hyperbolic character with pseudopressure waves propagating with finite speed. In such problems the ‘information’ inside the field is transmitted along its characteristic curves. In this sense, we can use upwind schemes to represent the finite volume scheme of the problem's governing equations. For the representation of the problem variables at the cell faces, upwind schemes up to third order of accuracy are used, while for the development of a time-iterative procedure a first-order-accurate Euler backward-time difference scheme is used and a second-order central differencing for the shear stresses is presented. The discretized Navier–Stokes equations are solved by an implicit unfactored method using Newton iterations and Gauss–Siedel relaxation. To validate the derived arithmetical results against experimental data and other numerical solutions, various laminar flows with known behaviour from the literature are examined. © 1997 John Wiley & Sons, Ltd.     

SPH modelling of depth‐limited turbulent open channel flows over rough boundaries

A numerical model based on the smoothed particle hydrodynamics method is developed to simulate depth‐limited turbulent open channel flows over hydraulically rough beds. The 2D Lagrangian form of the Navier–Stokes equations is solved, in which a drag‐based formulation is used based on an effective roughness zone near the bed to account for the roughness effect of bed spheres and an improved sub‐particle‐scale model is applied to account for the effect of turbulence. The sub‐particle‐scale model is constructed based on the mixing‐length assumption rather than the standard Smagorinsky approach to compute the eddy‐viscosity. A robust in/out‐flow boundary technique is also proposed to achieve stable uniform flow conditions at the inlet and outlet boundaries where the flow characteristics are unknown. The model is applied to simulate uniform open channel flows over a rough bed composed of regular spheres and validated by experimental velocity data. To investigate the influence of the bed roughness on different flow conditions, data from 12 experimental tests with different bed slopes and uniform water depths are simulated, and a good agreement has been observed between the model and experimental results of the streamwise velocity and turbulent shear stress. This shows that both the roughness effect and flow turbulence should be addressed in order to simulate the correct mechanisms of turbulent flow over a rough bed boundary and that the presented smoothed particle hydrodynamics model accomplishes this successfully. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd     

Karhunen–Loeve representations of turbulent channel flows using the method of snapshots

For three‐dimensional flows with one inhomogeneous spatial coordinate and two periodic directions, the Karhunen–Loeve procedure is typically formulated as a spatial eigenvalue problem. This is normally referred to as the direct method (DM). Here we derive an equivalent formulation in which the eigenvalue problem is formulated in the temporal coordinate. It is shown that this so‐called method of snapshots (MOS) has some numerical advantages when compared to the DM. In particular, the MOS can be formulated purely as a matrix composed of scalars, thus avoiding the need to construct a matrix of matrices as in the DM. In addition, the MOS avoids the need for so‐called weight functions, which emerge in the DM as a result of the non‐uniform grid typically employed in the inhomogeneous direction. The avoidance of such weight functions, which may exhibit singular behaviour, guarantees satisfaction of the boundary conditions. The MOS is applied to data sets recently obtained from the direct simulation of turbulence in a channel in which viscoelasticity is imparted to the fluid using a Giesekus model. The analysis reveals a steep drop in the dimensionality of the turbulence as viscoelasticity is increased. This is consistent with the results that have been obtained with other viscoelastic models, thus revealing an essential generic feature of polymer‐induced drag reduced turbulent flows. Published in 2006 by 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 PREDICTION OF SEMI-CONFINED JET IMPINGEMENT AND COMPARISON WITH EXPERIMENTAL DATA

The standard k–ε eddy viscosity model of turbulence in conjunction with the logarithmic law of the wall has been applied to the prediction of a fully developed turbulent axisymmetric jet impinging within a semi-confined space. A single geometry with a Reynolds number of 20,000 and a nozzle-to-plate spacing of two diameters has been considered with inlet boundary conditions based on measured profiles of velocity and turbulence. Velocity, turbulence and heat transfer data have been obtained using laser–Doppler anemometry and liquid crystal thermography respectively. In the developing wall jet, numerical results of heat transfer compare to within 20% of experiment where isotropy prevails and the trends in turbulent kinetic energy are predicted. However, stagnation point heat transfer is overpredicted by about 300%, which is attributed directly to the turbulence model and inapplicability of the wall function.     

Applications of URANS on predicting unsteady turbulent separated flows

Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practi cal challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS （URANS）. Some typical linear and nonlinear eddy viscosity turbulence models （LEVM and NLEVM） and a quadratic explicit algebraic stress model （EASM） are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding.