Deciphering the flow structure of Czochralski melt using Partially Averaged Navier–Stokes (PANS) method

Czochralski melt flow is an outcome of complex interactions of centrifugal, buoyancy, coriolis and surface tension forces, which act at different length and time scales. As a consequence, the characteristic flow structures that develop in the melt are delineated in terms of recirculating flow cells typical of rotating Bénard–Marangoni convection. In the present study, Partially Averaged Navier–Stokes (PANS) method is used for the first time to study an idealized Czochralski crystal growth set-up. It is observed that with a reduction in the PANS filter width, more turbulent scales are resolved and the present PANS model is able to resolve almost all the characteristic flow structures in the Czochralski flow at a comparatively lower computational cost compared with more advanced turbulence modelling tools, such as Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES).     

Evaluation of LES models for flow over bluff body from engineering application perspective

Three SGS stress closure LES models are evaluated for turbulent flow over a square cylinder. Emphasis is placed on solving engineering-application-type problems on affordable computer resources and within reasonable turnaround times. Results are compared with available experimental data and previously published workshop results. Numerical strategies are kept the same for all the cases. Results are also discussed keeping in view limitations of LES methodology of modelling for practical problems and current developments. It is concluded that a one-equation model for subgrid kinetic energy is the best choice.     

Large-Eddy Simulations of Shear Flows

The general framework of large-eddy simulations (LES) is presented first, with Smagorinsky's model. Afterwards, Kraichnan's spectral eddy-viscosity is introduced and it is shown how it can be handled for LES purposes in isotropic turbulence. The spectral eddy viscosity is generalized to a spectral eddy diffusivity. The nonlocal interaction theory is used to discuss the backscatter issue, and a generalization of spectral eddy coefficients is presented. This so-called spectral-dynamic model allows the representation of non-developed turbulence in the subgrid scales. Utilization of these spectral models in physical space is envisaged in terms of, respectively, the structure function and hyperviscosity models. Two applications of these models to shear flows are considered, namely the plane mixing layer and channel flow, with statistical results and information on the topology of coherent vortices and structures presented.     

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.     

Two scale meshfree method for the adaptivity of 3-D stress concentration problems

 In this study, a meshfree method called Reproducing Kernel Particle Method (RKPM) with an inherent characteristic of multi-resolution is modified to develop structural analysis algorithm using two scales. The shape function of RKPM is decomposed into two scales, high and low. The two scale decomposition is incorporated into linear elastic formulation to obtain high and low scale components of von Mises stresses. The advantage of using this algorithm is that the high scale component of von Mises stress indicates the high stress gradient regions without posteriori estimation. This algorithm is applied to the analysis of 2- and 3-dimensional stress concentration problems. It is important to note that the two scale analysis method has been applied to 3-dimensional stress concentration problem for the very first time. Also, the possibility of applying this algorithm to adaptive refinement technique is studied. The proposed method is verified by analyzing typical 2- and 3-dimensional linear elastic stress concentration problems. The results show that the algorithm can effectively locate the high stress concentration regions and can be utilized as an efficient indicator for the adaptive refinement technique. Received 10 January 2000     

Numerical Simulation of Self-Excited Oscillation of a Turbulent Jet Flowing into a Rectangular Cavity

The example of a plane jet flow into a rectangular cavity (“dead end”) is used in comparing the capabilities of different approaches to numerical simulation of self-oscillatory turbulent flows characterized by global quasi-periodic oscillation of all flow parameters. The calculations are performed for two flow modes, of which the first one is statistically steady according to the available experimental data, and the second one is self-oscillatory. In both cases, three approaches are used to describe the turbulence, namely, the method of large eddy simulation (LES) in combination with the subgrid model of Smagorinsky, and steady and unsteady Reynolds averaged Navier-Stokes equations (SRANS and URANS) with two well-known differential models of turbulence. In the case of the first flow mode, all three approaches produce qualitatively similar and quantitatively close results. In the case of the second (self-oscillatory) mode, a steady-state solution of Reynolds equations may only be obtained in half the domain using the symmetry boundary conditions; within the framework of the other two approaches, the solutions turn out to be unsteady-state. In so doing, their characteristics calculated using the LES and URANS methods differ significantly from each other; in the case of URANS, they further depend on the model of turbulence employed. The best results as regards the accuracy of prediction of the amplitude-frequency characteristics of self-oscillation are produced by the use of the LES and three-dimensional URANS methods. A similar inference may be made with respect to the mean flow parameters. From this standpoint, the worst results are those obtained from calculations involving the use of the symmetry boundary conditions on the geometric symmetry plane of the flow.__________Translated from Teplofizika Vysokikh Temperatur, Vol. 43, No. 4, 2005, pp. 568–579.Original Russian Text Copyright © 2005 by D. M. Denisikhina, I. A. Bassina, D. A. Nikulin, and M. Kh. Strelets.     

Large-Eddy Simulation (LES) of Large Hydrocyclones

Fluid dynamics theory applied to the hydrocyclone flow field has progressed in past experimental and theoretical investigations from a two-dimensional Prandtl mixing-length turbulence modeling to the current level of three-dimensional large-eddy simulation (LES) and differential-stress approaches. The LES approach eliminates the explicit empiricism that is imposed in the κ-ε model. Turbulence in hydrocyclones is anisotropic. Since only the subgrid scales are modeled in LES, the anisotropy is largely taken care of in this approach. This article presents validation of the LES model with laser-anemometry data collected on 75 mm and 250 mm hydrocyclones. Verification with experimental data on mass split, axial velocity, tangential velocity, root-mean-squared velocity, air-core profile, and size classification clearly stands as a proof that LES can be applied to even larger hydrocyclones.     

Appraisal of energy recovering sub-grid scale models for large-eddy simulation of turbulent dispersed flows

Current capabilities of Large-Eddy Simulation (LES) in Eulerian–Lagrangian studies of dispersed flows are limited by the modeling of the Sub-Grid Scale (SGS) turbulence effects on particle dynamics. These effects should be taken into account in order to reproduce accurately the physics of particle dispersion since the LES cut-off filter removes both energy and flow structures from the turbulent flow field. In this paper, we examine the possibility of including explicitly SGS effects by incorporating ad hoc closure models in the Lagrangian equations of particle motion. Specifically, we consider candidate models based on fractal interpolation and approximate deconvolution techniques. Results show that, even when closure models are able to recover the fraction of SGS turbulent kinetic energy for the fluid velocity field (not resolved in LES), prediction of local segregation and, in turn, of near-wall accumulation may still be inaccurate. This failure indicates that reconstructing the correct amount of fluid and particle velocity fluctuations is not enough to reproduce the effect of SGS turbulence on particle near-wall accumulation.     

Data-based stochastic subgrid-scale parametrization: an approach using cluster-weighted modelling

A new approach for data-based stochastic parametrization of unresolved scales and processes in numerical weather and climate prediction models is introduced. The subgrid-scale model is conditional on the state of the resolved scales, consisting of a collection of local models. A clustering algorithm in the space of the resolved variables is combined with statistical modelling of the impact of the unresolved variables. The clusters and the parameters of the associated subgrid models are estimated simultaneously from data. The method is implemented and explored in the framework of the Lorenz '96 model using discrete Markov processes as local statistical models. Performance of the cluster-weighted Markov chain scheme is investigated for long-term simulations as well as ensemble prediction. It clearly outperforms simple parametrization schemes and compares favourably with another recently proposed subgrid modelling scheme also based on conditional Markov chains.     

An algebraic variational multiscale-multigrid method for large-eddy simulation: generalized-α time integration,Fourier analysis and application to turbulent flow past a square-section cylinder

This article studies three aspects of the recently proposed algebraic variational multiscale-multigrid method for large-eddy simulation of turbulent flow. First, the method is integrated into a second-order-accurate generalized-α time-stepping scheme. Second, a Fourier analysis of a simplified model problem is performed to assess the impact of scale separation on the overall performance of the method. The analysis reveals that scale separation implemented by projective operators provides modeling effects very close to an ideal small-scale subgrid viscosity, that is, it preserves low frequencies, in contrast to non-projective scale separations. Third, the algebraic variational multiscale-multigrid method is applied to turbulent flow past a square-section cylinder. The computational results obtained with the method reveal, on the one hand, the good accuracy achievable for this challenging test case already at a rather coarse discretization and, on the other hand, the superior computing efficiency, e.g., compared to a traditional dynamic Smagorinsky modeling approach.     

Orientational effects in geophysical fluid dynamics

The approach, accounting the interaction between essentially different scales of turbulent fluid motions on a rotating sphere, is developed. Representation of the large-scale oceanic circulation as a turbulent flow with orientational effects due to synoptic (mesoscale) eddies is the basis of this technique. The scale of these eddies is much less than that of the considered problem, their energetic significance however is important. Such representation leads to employment of nontrivial angular momentum equation, i.e. to micropolar or asymmetrical hydrodynamics. It allows to define a function which would characterize the averaged vorticity of mesoscale (subgrid) motions and does not relate to the mean flow vorticity. Using this parameterization, the calculations with barotropic and 2-layer global ocean models have been carried out. The model results do show an increase of basic ocean gyre transports.     

Calculation of the statistical characteristics of a turbulent flow under the action of external forces

A study is made of the statistical characteristics of isotropic velocity and scalar fields in supply of the kinetic energy of turbulence from the energy of the average flow. Two models based on the distributions of the kinetic turbulence energy and the intensity of scalar-field pulsations by wave numbers and length scales are used for calculation of the statistical characteristics of turbulent velocity and scalar fields. The calculation results are compared to the data of the direct numerical modeling performed under the same initial conditions. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 79, No. 1, pp. 148–154, January–February, 2006.     

Dynamic and scalar turbulent fluctuation in a diffusion flame of an-axisymmetric methane jet into air

 A study of turbulence/combustion interactions in a relatively large turbulent diffusion flame of an axisymmetric methane jet into air is presented. A first order k–ɛ turbulence closure model is used along with two different models (equal scales and non-equal scales) for the submodel describing the scalar dissipation rate. The flamelet concept is used to model the turbulent combustion along with a joint mixture fraction/strain rate probability density function (PDF) for the prediction of the average parameters of the turbulent diffusion flame. The numerical approach is that of Patankar and Spalding, while the flamelet simulations are obtained from the RUN-1DL code of Rogg and co-workers based on a 17 species detailed reaction mechanism. The chosen configuration is that of the experimentally studied turbulent diffusion flame of Streb [1]. A comparison between these experimental results and the obtained numerical ones is thus presented. Relatively good agreements are obtained which show the usefulness of the two-scale model compared to the classical one-scale model for predicting turbulent diffusion flames. Nonetheless some discrepancies are obtained in the outer and downstream regions of the jet, especially in comparison with the experimental data. These are attributed to short coming of the considered turbulence model and soot radiation which is not accounted for. Received: 2 May 2002 / Accepted: 31 January 2003     

Wiener filtering of interferometry measurements through turbulent air using an exponential forgetting factor

The problem of imaging through turbulent media has been studied frequently in connection with astronomical imaging and airborne radars. Therefore most image restoration methods encountered in the literature assume a stationary object, e.g., a star or a piece of land. In this paper the problem of interferometric measurements of slowly moving or deforming objects in the presence of air disturbances and vibrations is discussed. Measurement noise is reduced by postprocessing the data with a digital noise suppression filter that uses a reference noise signal measured on a small stationary plate inserted in the field of view. The method has proven successful in reducing noise in the vicinity of the reference point where the size of the usable area depends on the degree of spatial correlation in the noise, which in turn depends on the spatial scales present in the air turbulence. Vibrations among the optical components in the setup tend to produce noise that is highly correlated across the field of view and is thus efficiently reduced by the filter.     

High-performance computing in computational fluid dynamics: progress and challenges

Computational fluid dynamics (CFD) is by far the largest user of high-performance computing (HPC) in engineering. The main scientific challenge is the need to gain a greater understanding of turbulence and its consequences for the transfer of momentum, heat and mass in engineering applications, including aerodynamics, industrial flows and combustion systems. Availability of HPC has led to significant advances in direct numerical simulation (DNS) of turbulence and turbulent combustion, and has encouraged the development of large-eddy simulation (LES) for engineering flows. The statistical data generated by DNS have provided valuable insight into the physics of many turbulent flows and have led to rapid improvements in turbulence and combustion modelling for industry. Nevertheless, major challenges remain and the computational requirements for turbulence research, driven by well-established physical scaling laws, are likely to remain at the limit of the available HPC provision for some time to come.     

A multidomain boundary element method for two equation turbulence models

The paper deals with the multidomain Boundary Element Method (BEM) for modelling 2D complex turbulent flow using low Reynolds two equation turbulence models. While the BEM is widely accepted for laminar flow this is the first case, where this method is applied for a complex flow problems using k–ε turbulence model. The integral boundary domain equations are discretised using mixed boundary elements and a multidomain method also known as subdomain technique. The resulting system matrix is overdetermined, sparse, block banded and solved using fast iterative linear least squares solver. The simulation of turbulent flow over a backward step is in excellent agreement with the finite volume method using the same turbulent model.     

Entropic lattice Boltzmann model for Burgers's equation

Entropic lattice Boltzmann models are discrete-velocity models of hydrodynamics that possess a Lyapunov function. This feature makes them useful as nonlinearly stable numerical methods for integrating hydrodynamic equations. Over the last few years, such models have been successfully developed for the Navier-Stokes equations in two and three dimensions, and have been proposed as a new category of subgrid model of turbulence. In the present work we develop an entropic lattice Boltzmann model for Burgers's equation in one spatial dimension. In addition to its pedagogical value as a simple example of such a model, our result is actually a very effective way to simulate Burgers's equation in one dimension. At moderate to high values of viscosity, we confirm that it exhibits no trace of instability. At very small values of viscosity, however, we report the existence of oscillations of bounded amplitude in the vicinity of the shock, where gradient scale lengths become comparable with the grid size. As the viscosity decreases, the amplitude at which these oscillations saturate tends to increase. This indicates that, in spite of their nonlinear stability, entropic lattice Boltzmann models may become inaccurate when the ratio of gradient scale length to grid spacing becomes too small. Similar inaccuracies may limit the utility of the entropic lattice Boltzmann paradigm as a subgrid model of Navier-Stokes turbulence.     

Turbulence Investigation at Small Scale by Angle-of-arrival Fluctuations of a Laser Beam

In this paper the possibility is investigated of deriving information on the shape of the structure function of the refractive index of a turbulent medium at small scales (or, which amounts to the same thing, on the shape of the power spectrum at a high-wavenumber) from measurements of angle-of-arrival fluctuations of an optical wave propagated through it. As a result, a method is also devised for the determination of the inner scale of the turbulence.     

Large deformation analysis of rubber based on a reproducing kernel particle method

 A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal shape functions derived on␣the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation analysis. Since the current approach employs global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing with large material distortions.     

一种改进的大涡模拟入口湍流生成方法研究

大涡模拟中的入口湍流的生成方法研究,是当前计算风工程领域国内外研究的热点问题。该文在NSRFG(narrowband synthesis random flow generation)方法的基础上,对其中重要参数无量纲长度尺度\begin{document}$\beta$\end{document}、空间相关性\begin{document}$R$\end{document}和调谐因子\begin{document}${\gamma _j}$\end{document}进行深入理论分析,推导了调谐因子\begin{document}${\gamma _j}$\end{document}与无量纲长度尺度\begin{document}$\beta $\end{document}的函数关系,建议了一种改进的入口湍流合成技术——INSRFG(improved NSRFG)方法。利用该方法进行了与规范相对应的4类标准地貌湍流风场的大涡模拟数值仿真;通过对比分析,表明INSRFG方法模拟的大气边界层湍流风场,能较好满足脉动风速功率谱、空间相关性等湍流风场基本特性,并较好实现大气边界层风场模拟中的平衡态基本要求。研究表明,这种新的INSRFG湍流合成方法具有参数取值明确、数学模型简洁、计算效率相对较高的优点,是一种进行建筑结构大涡模拟研究的具有较好前景的通用入口湍流生成方法。