Large eddy simulation of turbulent flows in external flow field using three-step FEM–BEM model

An innovative computational model is presented for the large eddy simulation (LES) modeling of multi-dimensional unsteady turbulent flow problems in external flow field. Based on the LES principles, the model uses a pressure projection method to solve the Navier–Stokes equations in transient condition. The turbulent motion is simulated by Smagorinsky sub-grid scale (SGS) eddy viscosity model. The momentum equation of the flow motion is solved using a three-step finite element method (FEM). The external flow field is simulated using a boundary element method (BEM) by solving a pressure Poisson equation that assumes the pressure as zero at the infinity. Through extracting the boundary effects on a specified finite computational domain, the model is able to solve the infinite boundary value problems. The present model is used to simulate the flows past a two-dimensional square rib and a three-dimensional cube at high Reynolds number. The simulation results are found to be reasonable and comparable with other models available in the literature even for coarse meshes.     

亚格子尺度湍流特性研究

采用大涡模拟方法模拟了雷诺数ReH=18,400的后台阶湍流流动,研究了亚格子尺度湍流动能和湍流耗散的特性。给出了后台阶湍流流动的流场结构以及亚格子湍动能和亚格子湍耗散的空间分布结果,比较了大涡模拟预报湍流粘性以及等效计算粘性。研究表明,亚格子尺度湍动能和亚格子湍耗散随着流动在空间的发展而呈现减弱趋势,回流区内亚格子湍动能和耗散较弱;在台阶截面(y/H=1处)亚格子湍动能和耗散最大。亚格子湍动能小于脉动动能统计量,亚格子粘性小于等效湍流模型粘性预报结果。     

Large‐eddy simulation of separated leading‐edge flow in general co‐ordinates

An incompressible separated transitional boundary‐layer flow on a flat plate with a semi‐circular leading edge has been simulated and a very good agreement with the experimental data has been obtained, demonstrating how this technique may be applied even when finite difference formulae are used in the periodic direction. The entire transition process has been elucidated and vortical structures have been identified at different stages during the transition process. Efficient numerical methods for the large‐eddy simulation (LES) of turbulent flows in complex geometry are developed. The methods used are described in detail: body‐fitted co‐ordinates with the contravariant velocity components of the general Navier–Stokes equations discretized on a staggered mesh with a dynamic subgrid‐scale model in general co‐ordinates. The main source of computational expense in simulations for incompressible flows is due to the solution of a Poisson equation for pressure. This is especially true for flows in complex geometry. Fourier techniques can be employed to speed up the pressure solution significantly for a flow which is periodic in one dimension. With simple conditions fulfilled, it is possible to Fourier transform a discrete elliptic equation such as the Poisson equation for the pressure field, decomposing the problem into a set of two‐dimensional problems of similar type (Poisson‐like). Even when a complex geometry and body‐fitted curvilinear co‐ordinates are used in the other two dimensions, as in the present case, the resulting Fourier‐transformed 2D problems are much more efficiently solved than the 3D problem by iterative means. Copyright © 2000 John Wiley & Sons, Ltd.     

A priori evaluation of dynamic subgrid models of turbulence in square duct flow

A priori tests of two dynamic subgrid-scale (SGS) turbulence models have been performed using a highly resolved direct numerical simulation database for the case of turbulent incompressible flow in a straight duct of square cross-section. The model testing is applied only to the homogeneous flow direction where grid filtering can be applied without the introduction of commutation errors. The first SGS model is the dynamic (Smagorinsky/eddy viscosity) SGS model (DSM) developed by Germano et al. [1] while the second is the dynamic two parameter (mixed) model (DTM) developed by Salvetti and Banerjee [2]. For the Smagorinsky model we have used both the Fourier cut-off filter and a modified Gaussian filter which has the property that it removes aliasing errors in consistent a priori model-testing for spectral-based datasets. Results largely consistent with those found for plane channel flow are observed but with some slight differences in the corner regions. As found in prior studies of this sort, there is a very poor correlation of the modelled and exact subgrid-scale dissipation in the case of the DSM. The DSM over-predicts subgrid-scale dissipation on average. Instantaneously, the model provides an inaccurate representation of subgrid-scale dissipation, in general underestimating the magnitude by approximately one order of magnitude. On the other hand, the DTM shows excellent agreement with the exact SGS dissipation over most of the duct cross-section with a correlation coefficient of approximately 09.     

Combined three‐dimensional lattice Boltzmann method and discrete element method for modelling fluid–particle interactions with experimental assessment

A general algorithmic framework is established in this paper for numerical simulations of three‐dimensional fluid–particle interaction problems with a large number of moving particles in turbulent flows using a combined lattice Boltzmann method (LBM) and discrete element method (DEM). In this approach, the fluid field is solved by the extended three‐dimensional LBM with the incorporation of the Smagorinsky turbulence model, while particle interactions are modelled by the DEM. The hydrodynamic interactions between fluid and particles are realized through the extension of an existing two‐dimensional fluid–particle hydrodynamic interaction scheme. The main computational aspects comprise the lattice Boltzmann formulation for the solution of fluid flows, the incorporation of a large eddy simulation‐based turbulence model within the framework of the three‐dimensional LBM for turbulent flows, the moving boundary condition for hydrodynamic interactions between fluid and moving particles, and the discrete element modelling of particle‐particle interactions. To assess the solution accuracy of the proposed approach, a much simplified laboratory model of vacuum dredging systems for mineral recovery is employed. The numerical results are compared with the experimental data available. It shows that the overall correspondence between numerical results and experimental measurements is good and thus indicates, to a certain extent, the solution accuracy of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

Some predictions of the attached eddy model for a high Reynolds number boundary layer

Many flows of practical interest occur at high Reynolds number, at which the flow in most of the boundary layer is turbulent, showing apparently random fluctuations in velocity across a wide range of scales. The range of scales over which these fluctuations occur increases with the Reynolds number and hence high Reynolds number flows are difficult to compute or predict. In this paper, we discuss the structure of these flows and describe a physical model, based on the attached eddy hypothesis, which makes predictions for the statistical properties of these flows and their variation with Reynolds number. The predictions are shown to compare well with the results from recent experiments in a new purpose-built high Reynolds number facility. The model is also shown to provide a clear physical explanation for the trends in the data. The limits of applicability of the model are also discussed.     

Steady and unsteady incompressible flow in a double driven cavity using the artificial compressibility (AC)‐based characteristic‐based split (CBS) scheme

In this paper, the explicit characteristic‐based split (CBS) scheme has been employed to solve both steady and unsteady flows inside a non‐rectangular double driven cavity. This problem is recently suggested as a benchmark problem for incompressible flows. Both unstructured and structured meshes have been employed in the present study to make sure that the predicted results are as close to reality as possible. The results obtained show the existence of steady state at lower Reynolds numbers (?1000) and transient states at higher Reynolds numbers. The flow approaches a turbulent state as the Reynolds number is increased to 10 000. Copyright © 2005 John Wiley & Sons, Ltd.     

Mathematical models for the numerical study of turbulent flows

Abstract

This paper presents (1) a brief overview of the mathematical models used in the numerical study of turbulent flows; (2) a K‐? model of turbulence; and (3) extensions of the K‐? model to account for some of the effects of compressibility, low Reynolds number, streamline curvature, and preferential stress dissipation.     

An investigation of the Prandtl number effect on turbulent heat transfer in channel flows by large eddy simulation

Summary Fully developed turbulent channel flow with passive heat transfer has been calculated to investigate the turbulent heat transfer by use of the large eddy simulation (LES) approach coupled with dynamic subgrid-scale (SGS) models. The objectives of this study are to examine the effectiveness of the LES technique for predicting the turbulent heat transfer at high Prandtl numbers and the effects of the Prandtl number on the turbulent heat transfer in a fully developed turbulent channel flow. In the present study, the Prandtl number is chosen as 0.1 to 200, and the Reynolds number, based on the central mean velocity and the half-width of the channel, is 10⁴. Some typical cases are computed and compared with available data obtained by direct numerical simulation (DNS), theoretical analysis and experimental measurement, respectively, which confirm that the present approach can be used to predict the heat transfer satisfactorily, even at high Prandtl numbers. To depict the effect of the Prandtl number on turbulent heat transfer, the distributions of mean value and fluctuation of resolved flow temperatures, the heat transfer coefficient, turbulent heat fluxes, and some instantaneous iso-thermal sketches are analyzed.     

Large eddy simulation of turbulent open channel flow with heat transfer at high Prandtl numbers

Summary. Large eddy simulation of a fully developed turbulent open channel flow with heat transfer is performed. The three-dimensional filtered Navier-Stokes and energy equations are numerically solved using a fractional-step method. Dynamic subgrid-scale (SGS) models for the turbulent SGS stress and heat flux are employed to close the governing equations. The objective of this study is to analyze the behavior of turbulent flow and heat transfer in turbulent open channel flow, in particular for high Prandtl number, and to examine the reliability of the LES technique for predicting turbulent heat transfer near the free surface. The turbulent open channel flow with constant difference of temperature imposed on the free surface and bottom wall is calculated for the Prandtl number (Pr) from 1 up to 100, the Reynolds number (Re) 180 based on the wall friction velocity and the channel depth. To illustrate the turbulent flow and heat transfer behaviors, some typical quantities, including the mean velocity, temperature and their fluctuations, heat transfer coefficients, turbulent heat fluxes, and flow structures of velocity and temperature fluctuations, are exhibited and analyzed.     

A particle–particle Reynolds stress transportation model of swirling particle-laden-mixtures turbulent flows

On the basis of the gas–particle Euler–Euler two-fluid approach, a new particle–particle Reynolds stress transportation model is proposed for closing the constitution equations of particle-laden-mixtures turbulent flows. In this model, binary particle-particle interaction originating from large-scale particle turbulent diffusions are fully considered in view of an extension closure idea of second-order-moment disperse gas–particle turbulent flows. The binary-particles turbulent flows with different density and same diameter are numerically simulated. The number density, the time-averaged velocity, the fluctuation velocity, the multiphase fluctuation velocity correlations, the normal and the shear Reynolds stress are obtained. Simulated results are in good agreement with experimental data. Binary mixture system has a unique transportation behavior with a stronger anisotropy due to particle inertia and multiphase turbulence diffusions. Fluctuation velocity correlation of axial–axial gas–particle is about twice larger than those of axial–axial particle–particle interaction. Moreover, both normal and shear Reynolds stress are redistributed.     

A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations

We present a scheme for solving two‐dimensional semilinear reaction–diffusion equations using an expanded mixed finite element method. To linearize the mixed‐method equations, we use a two‐grid algorithm based on the Newton iteration method. The solution of a non‐linear system on the fine space is reduced to the solution of two small (one linear and one non‐linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows: Computational issues

This paper presents essential numerical procedures in the context of the coupled lattice Boltzmann (LB) and discrete element (DE) solution strategy for the simulation of particle transport in turbulent fluid flows. Key computational issues involved are (1) the standard LB formulation for the solution of incompressible fluid flows, (2) the incorporation of large eddy simulation (LES)‐based turbulence models in the LB equations for turbulent flows, (3) the computation of hydrodynamic interaction forces of the fluid and moving particles; and (4) the DE modelling of the interaction between solid particles. A complete list is provided for the conversion of relevant physical variables to lattice units to facilitate the understanding and implementation of the coupled methodology. Additional contributions made in this work include the application of the Smagorinsky turbulence model to moving particles and the proposal of a subcycling time integration scheme for the DE modelling to ensure an overall stable solution. A particle transport problem comprising 70 large particles and high Reynolds number (around 56 000) is provided to demonstrate the capability of the presented coupling strategy. Copyright © 2007 John Wiley & Sons, Ltd.     

Large eddy simulation of particle deposition and resuspension in turbulent duct flows

Particle deposition and resuspension in a horizontal, fully developed turbulent square duct flow at four flow bulk Reynolds numbers (10,320, 83k, 215k and 250k) is simulated by applying large eddy simulation coupled with Lagrangian particle tracking technique. Forces acting on particles includes drag, lift, buoyancy and gravity. Four particle sizes are considered with the diameters of 5?μm, 50?μm, 100?μm and 500?μm. Results obtained for the fluid phase are in good agreement with the available experimental and numerical data. Predictions for particles show that particle size, flow Reynolds number and the duct (celling, floor and vertical) walls play important roles in near-wall particle deposition and resuspension. For the smallest particle (5?μm), the particle deposition rates in duct ceiling, floor and vertical walls are found to be similar with each other and all increase with the flow Reynolds number, while the particle resuspension tends to occur in the middle wall regions and corners of the duct with less influenced by the flow Reynolds number. The ceiling deposition rate gradually decreases with particle size while the floor and vertical wall deposition rates both increase with particle size. The ceiling particle deposition rate increases with Reynolds number while the floor deposition rate decreases with it. The vertical deposition rate for the small particles (5–50?μm) increases with the flow Reynolds number obviously, while for the large particles (100–500?μm) that becomes insensitive. In addition, the flow Reynolds number is found to have an obvious effect on particle resuspension while the effect of particle size on particle resuspension decreases with Reynolds number. Eventually, a dynamic analysis was conducted for particles deposition and resuspension in turbulent duct flows.     

Improved version of the synthetic eddy method for setting nonstationary inflow boundary conditions in calculating turbulent flows

One method for generating synthetic turbulence, i.e., the synthetic eddy method, is tested by means of the example of canonical turbulent shear flows (plane-channel and boundary-layer flows). A modification of the method, differing from the original version by the determination of the linear scale of synthetic eddy structures, is proposed. The synthetic field of turbulent fluctuations evolves more quickly to the physically realistic one when the modified method is used instead of the original one. The friction coefficient and profiles of the average velocity and Reynolds stresses also deviate less and recover faster if the modified method is used.     

Effects of higher wall roughness on dense particle–laden dispersion behaviors

To analyze the effects of higher wall roughness on dense particle–laden dispersion behaviors under reduced gravity environments, a dense gas–particle two-phase second-order-moment turbulent model are developed. In this model, the wall roughness function and the kinetic theory of granular flows are coupled and closed. Anisotropy of gas–solid two-phase stresses and the interaction between gas–particle are fully considered using two-phase Reynolds stress model and the two-phase correlation transport equation. Numerical simulation test is validated by Sommerfeld and Kussin (2003) experiments data with higher wall roughness 8.32 μm. Predicted results showed that the particle concentration distribution, particle fluctuation velocity, particle temperature and particle collision frequency are greatly affected by higher wall roughness, as well as particle Reynolds stress and interactions between gas and particle turbulent flows are redistributed. Under microgravity conditions, particle temperature and collision frequency are greatly less than those of earth and lunar gravity.     

A stream function implicit finite difference scheme for 2D incompressible flows of Newtonian fluids

The development of a new algorithm to solve the Navier–Stokes equations by an implicit formulation for the finite difference method is presented, that can be used to solve two‐dimensional incompressible flows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady fluid mechanics problems: cavity‐driven flow with Re=100, 400 and 1000 and flow in a sudden expansion with expansion ratio H/h=2 and Re=50, 100 and 200. The results obtained for the stream function are compared with values obtained by different available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

Discrete direct and adjoint sensitivity analysis for arbitrary Mach number flows

Parallel discrete direct and adjoint sensitivity analysis capabilities are developed for arbitrary Mach flows on mixed‐element unstructured grids. The discrete direct and adjoint methods need a consistent and complete linearization of the flow‐solver to obtain accurate derivatives. The discontinuous nature of the commonly used unstructured flux‐limiters, like Barth–Jespersen and Venkatakrishnan, make them unsuitable for sensitivity analysis. A modification is proposed to make these limiters piecewise continuous and numerically differentiable, without compromising the monotonicity conditions. An improved version of Symmetric Gauss–Seidel that significantly reduces the computational cost is implemented. A distributed‐memory message passing model is employed for the parallelization of sensitivity analysis solver. These algorithms are implemented within a three‐dimensional unstructured grid framework and results are presented for inviscid, laminar and turbulent flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Level set X‐FEM non‐matching meshes: application to dynamic crack propagation in elastic–plastic media

This paper develops two aspects improving crack propagation modelling with the X‐FEM method. On the one hand, it explains how one can use at the same time a regular structured mesh for a precise and efficient level set update and an unstructured irregular one for the mechanical model. On the other hand, a new numerical scheme based on the X‐FEM method is proposed for dynamic elastic–plastic situations. The simulation results are compared with two experiments on PMMA for which crack speed and crack path are provided. Copyright © 2006 John Wiley & Sons, Ltd.