Subgrid modeling for convection-diffusion-reaction in one space dimension using a Haar Multiresolution analysis

In this paper we propose and study a subgrid model for linear convection-diffusion-reaction equations with fractal rough coefficients. The subgrid model is based on scale extrapolation of a modeling residual from coarser scales using a computed solution on a finest scale as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a scale less than h/4. We also present error estimates for the modeling error in terms of modeling residuals.     

The variational multiscale method for laminar and turbulent flow

Summary  The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.     

Investigating an advective approach to subgrid modeling in large-eddy simulations

The purpose of this paper is to investigate and validate an alternative subgrid model to be used in large-eddy simulations, based on an advective formulation. Rather than modeling the subgrid tensor that appears in the LES formulation as is commonly done, the subgrid force vector, which is the divergence of the subgrid tensor, is modeled directly. It is designed to comply with two basic principles. First, it is required to act only on the smallest scales that the mesh can represent. Second, it must be of an advective nature, which means it must have a preferred direction aligned with the fluid velocity. The results for two benchmark test cases, including Homogeneous Isotropic Turbulence and Turbulent Channel Flow, show that this approach can successfully represent the effect of the small scales on the resolved ones, while guaranteeing numerical stability and greater robustness in adverse mesh environments, when compared to some traditional eddy-viscosity based models, such as the Smagorinsky and the dynamic model from Germano.     

Large eddy simulation of pulsating flow over a circular cylinder at subcritical Reynolds number

The pulsating cross-flow over a single circular cylinder at the subcritical Reynolds number Re_D = 2580 is studied with the large eddy simulation (LES) technique using the standard Smagorinsky model as well as a dynamic model in which the test filtered quantities are evaluated through a truncated Taylor series expansion. The filtered equations are discretised using the finite volume method in an unstructured, collocated grid arrangement with a second-order accurate method, in both space and time. The predictions are compared against very detailed experiments for mean velocities and Reynolds stresses that were performed in a duct of cross-section 72 mm × 72 mm using the PIV technique. The effects of mesh refinement close to the cylinder as well as of subgrid scale model are also examined. The numerical predictions are in very good agreement with the measurements in terms of mean as well as turbulence quantities. The instantaneous flow patterns of the flow field are examined and the effect of the external flow pulsation on the wake characteristics such as vortex formation length, vortex strength, Strouhal number as well as the lift and drag coefficients is quantified. The vortex formation length is decreased while the mean drag, as well as the rms values of the drag and lift coefficients increase significantly under pulsating flow conditions. The performance of the LES technique is analysed in the light of the wake characteristics.     

Numerical modeling of the flow field and performance in cyclones of different cone-tip diameters

The effect of the cone tip-diameter on the flow field and performance of cyclone separator was investigated computationally and via mathematical models. Three cyclones with different cone tip diameters were studied using large eddy simulation (LES). The cyclone flow field pattern has been simulated and analyzed with the aid of velocity components and static pressure contour plots. In addition the cyclone collection efficiency based on one-way discrete phase modeling has been investigated. The results obtained demonstrate that LES is a suitable approach for modeling the effect of cyclone dimensions on the flow field and performance. The cone tip-diameter has an insignificant effect on the collection efficiency (the cut-off diameter) and the pressure drop. The simulation results agree well with the published experimental results and the mathematical models trend.     

Simulations of confined turbulent vortex flow

Large-eddy simulations (LES) of the turbulent flow in a swirl tube with a tangential inlet have been performed. The geometry, and flow conditions were chosen according to an experimental study by [Escudier MP, Bornstein J, Zehnder N. Observations and LDA measurements of confined turbulent vortex flow. J Fluid Mech 1980;98:49-63]. Lattice-Boltzmann discretization was used to numerically solve the Navier-Stokes equations in the incompressible limit. Effects of spatial resolution and choices in subgrid-scale modeling were explicitly investigated with the experimental data set as the testing ground. Experimentally observed flow features, such as vortex breakdown and laminarization of the vortex core were well represented by the LES. The simulations confirmed the experimental observations that the average velocity profiles in the entire vortex tube are extremely sensitivity to the exit pipe diameter. For the narrowest exit pipe considered in the simulations, very high average velocity gradients are encountered. In this situation, the LES shows the most pronounced effects of spatial resolution and subgrid-scale modeling.     

Large eddy simulation and ALE mesh motion in Rayleigh-Taylor instability simulation

Many large eddy simulation (LES) techniques have been developed for stationary computational meshes. This study applies a single equation LES to Arbitrary Lagrangian-Eulerian (ALE) simulations of Rayleigh-Taylor instability and investigates its effects. Behavior of LES is similar for Eulerian and ALE simulations for the test problem studied. However, the motion of the mesh can be tied to the subgrid scale model in the form of a relaxation weight based on subgrid scale energy. This increases mesh resolution in areas of high subgrid scale energy.     

Numerical simulation of phase separation and a priori two-phase LES filtering

This work reports on the potential application of Large Eddy Simulation (LES) in the calculation of turbulent isothermal two-phase flows, in the case where the large scales of each phase are resolved and small interface structures can be smaller than the mesh size. In comparison with single phase flows, application of LES to two-phase flow problems should account for the complex interaction between the interface and the turbulent motion. The complete filtered two-phase flow equations are formulated to deal with turbulence at the interface. Explicit filtering of 3D direct numerical simulations of a phase separation problem has been employed to evaluate the order of magnitude of the specific subgrid contributions. Analyses of the numerical results have been conducted to derive conclusions on the relative importance of the different subgrid scale contributions. Modeling issues and turbulent energy transfer across the interface are discussed.     

Discontinuous subgrid formulations for transport problems

In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.     

Toward advanced subgrid models for Lattice-Boltzmann-based Large-eddy simulation: Theoretical formulations

This paper addresses the issue of developing advanced subgrid model for large-eddy simulations (LES) of turbulent flows based on Lattice Boltzmann methods (LBM). Most of already existing subgrid closures used in LES-LBM are straightforward extensions of the most crude model developed within the Navier–Stokes equations, namely the Smagorinsky eddy-viscosity model. In a first part, it is shown how to obtain an improved eddy-viscosity subgrid model for LBM. The original implementation of the Inertial-Range Consistent Smagorinsky model proposed by Dong and Sagaut for the D3Q19 scheme is used as an illustration. In a second step, an original extension of the Approximate Deconvolution Method proposed by Adams and Stolz for Navier–Stokes simulation is proposed. This new LBM-LES approach does not rely on the eddy-viscosity concept and is written directly within the LBM framework. It is shown that it can be implemented thanks to a trivial modification of the existing LBM solvers for Direct Numerical Simulation.     

Advance in RANS-LES coupling,a review and an insight on the NLDE approach

Summary The aim of this article is twofold. The first purpose is to propose a review of existing RANS-LES methods and is addressed in the first part in a comprehensive way, detailing the advantages and the drawbacks of the different techniques. In a second time, a hybrid RANS-LES approach is presented, which can be interpreted as the most general case of the NLDE approach as defined by Morriset al. A decomposition into three parts of the exact solution of the Navier-Stokes equations is considered: mean flow, resolved fluctuations and unresolved (subgrid) fluctuations. The mean flow is computed using a classical RANS method, while resolved fluctuations are derived from a LES method. Several features on this approach are at first discussed in this paper, that are: the development of a non-zero mean for the resolved fluctuations (also called hereafter details), the computational problems due to the use of different schemes and meshes for the RANS and LES calculations, and the use of a boundary condition suited to the fluctuating part of the field. This approach is then used to simulate the acoustic sources of the flow around the slat of a high-lift system in landing configuration. The mean instabilities of the flow are studied and the resulting acoustic near field is carefully investigated.     

On a numerical subgrid upscaling algorithm for Stokes–Brinkman equations

This paper discusses a numerical subgrid resolution approach for solving the Stokes–Brinkman system of equations, which is describing coupled flow in plain and in highly porous media. Various scientific and industrial problems are described by this system, and often the geometry and/or the permeability vary on several scales. A particular target is the process of oil filtration. In many complicated filters, the filter medium or the filter element geometry are too fine to be resolved by a feasible computational grid. The subgrid approach presented in this paper is aimed at describing how these fine details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid. This is done via a systematic and careful procedure of modifying and updating the coefficients of the Stokes–Brinkman system in chosen cells. This numerical subgrid approach is motivated from one side from homogenization theory, from which we borrow the formulations for the so-called cell problem, and from the other side from the numerical upscaling approaches, such as Multiscale Finite Volume, Multiscale Finite Element, etc. Results on the algorithm’s efficiency, both in terms of computational time and memory usage, are presented. Comparison of the full fine grid solution (when possible) of the Stokes–Brinkman system with the subgrid solution of the upscaled Stokes–Brinkman system (including effective permeabilities for the quasi-porous cells), are presented in order to evaluate the accuracy and the efficiency. Advantages and limitations of the considered subgrid approach are discussed.     

Large eddy simulation of two-dimensional isotropic turbulence

Large eddy simulation (LES) of forced, homogeneous, isotropic two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlovet al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplified SGS representation was designed, referred to as the stabilized negative viscosity (SNV) representation, which was based on two algebraic terms only, negative Laplacian and positive biharmonic ones. It was found that the SNV scheme performed in a fashion very similar to the full equation and it was argued that this scheme and its derivatives should be applied for SGS representation in LES of quasi-2D flows.     

Modeling of delta-wing type vortex generators

A numerical model of delta-wing type vortex generator was developed in two steps.The first step was to obtain a parameterized model of the shedding vortex based on delta-wing theory,which relates the geometry parameters and flow field parameters to the strength of shedding vortex which directly decides the source term.In the second step,a method was proposed to add source terms into the flow control equations so that the shedding vortex could be simulated numerically.As soon as the numerical model was compl...     

Adaptive large eddy simulation

Several a posteriori indicators in the framework of local grid adaptation and large eddy simulation (LES) are evaluated. In LES indicators must be capable to bound not only the discretisation error, but also the modeling error. Moreover, the numerical method must be able to adapt the computational grid dynamically, as the regions requiring different resolution are not static. The performance of different indicators is evaluated in two flow configurations. It turns out that the classic residual based error indicator and the newly introduced heuristic indicator perform best.     

Performance analysis of parallel Schwarz preconditioners in the LES of turbulent channel flows

We present a comparative study of parallel Schwarz preconditioners in the solution of linear systems arising in a Large Eddy Simulation (LES) procedure for turbulent plane channel flows. This procedure applies a time-splitting technique to suitably filtered Navier–Stokes equations, in order to decouple the continuity and momentum equations, and uses a semi-implicit scheme for time integration and finite volumes for space discretisation. This approach requires the solution of four sparse linear systems at each time step, accounting for a large part of the overall simulation; hence the linear system solvers are a crucial component in the whole procedure. Several preconditioners are applied in the simulation of a reference test case for the LES community, using discretisation grids of different sizes, with the aim of analysing the effects of different algorithmic choices defining the preconditioners, and identifying the most effective ones for the selected problem. The preconditioners, coupled with the GMRES method, are run within SParC-LES, a recently developed LES code based on the PSBLAS and MLD2P4 libraries for parallel sparse matrix computations and preconditioning.     

Noise prediction for a turbulent jet using different hybrid methods

The acoustic field of a cold single stream jet at Mach number 0.9 and Reynolds number 3600 is determined via computational aeroacoustics (CAA) methods. The jet computation of the acoustical field is performed by two hybrid approaches using a large-eddy simulation (LES) for the flow field and various systems of equations for the acoustical field to construct a robust, efficient, and reliable LES/CAA solver. The acoustic equations are the Ffowcs Williams-Hawkings equation (FWH) in the frequency domain and the acoustic perturbation equations (APE). The pronounced impact of the data windowing and the radial and streamwise extension of the integration surface on the directivity of the FWH solution is discussed at length. The comparison with available experimental and numerical results at similar flow conditions based on the noise characteristics in the near field shows the solution of the APE system to match the results of the direct LES more accurately than the FWH approach. The APE solution is less susceptible to the size of the source term region than the FWH approach to the location of the source surface. In conjunction with the APE formulation the LES domain can be chosen smaller than for the FWH ansatz resulting in less computational cost for the jet flow. The dominant source term in the APE system for cold jet noise is shown to be the Lamb vector.     

Study of jet precession, recirculation and vortex breakdown in turbulent swirling jets using LES

Large eddy simulations (LES) are used to investigate turbulent isothermal swirling flows with a strong emphasis on vortex breakdown, recirculation and instability behaviour. The Sydney swirl burner configuration is used for all simulated test cases from low to high swirl and Reynolds numbers. The governing equations for continuity and momentum are solved on a structured Cartesian grid, and a Smagorinsky eddy viscosity model with the localised dynamic procedure is used as the sub-grid scale turbulence model. The LES successfully predicts both the upstream first recirculation zone generated by the bluff body and the downstream vortex breakdown bubble. The frequency spectrum indicates the presence of low frequency oscillations and the existence of a central jet precession as observed in experiments. The LES calculations well captured the distinct precession frequencies. The results also highlight the precession mode of instability in the center jet and the oscillations of the central jet precession, which forms a precessing vortex core. The study further highlights the predictive capabilities of LES on unsteady oscillations of turbulent swirling flow fields and provides a good framework for complex instability investigations.     

Optimal Trajectory Planning for Flexible Link Manipulators with Large Deflection Using a New Displacements Approach

The main objective of the present paper is to determine the optimal trajectory of very flexible link manipulators in point-to-point motion using a new displacement approach. A new nonlinear finite element model for the dynamic analysis is employed to describe nonlinear modeling for three-dimensional flexible link manipulators, in which both the geometric elastic nonlinearity and the foreshortening effects are considered. In comparison to other large deformation formulations, the motion equations contain constant stiffness matrix because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms. In this investigation, the computational method to solve the trajectory planning problem is based on the indirect solution of open-loop optimal control problem. The Pontryagin’s minimum principle is used to obtain the optimality conditions, which is lead to a standard form of a two-point boundary value problem. The proposed approach has been implemented and tested on a single-link very flexible arm and optimal paths with minimum effort and minimum vibration are obtained. The results illustrate the power and efficiency of the method to overcome the high nonlinearity nature of the problem.     

The computation of massively separated flows using compressible vorticity confinement methods

Vorticity confinement methods have been shown to be very effective in the computation of flows involving the convection of thin vortical layers. These are the only Eulerian methods whereby simulations of these layers remain very thin and persist long distances without significant dissipation. Initially developed by Steinhoff and co-workers for incompressible flow, these methods have been used successfully to predict complex flows, particularly involving helicopter rotors. Recently, the method has been extended to a compressible finite-volume form, which will enable the methods to be used for a much broader class of problems. In this paper, we examine the ability of the compressible vortex confinement methodology to handle an important class of vortex-dominated flows involving massive separation from bluff bodies. We evaluate the effectiveness of the method by comparisons with experimental data and available state-of-the-art computations. An important conclusion of the present work is that vortex confinement applied to massively separated flows, without modeling the viscous terms and on an essentially inviscid grid, can result in a reasonable approximation to turbulent separated flows. The computed flow structures and velocity profiles were in good agreement with time-averaged values of the data and with LES simulations even though the confinement approach utilized more than a factor of 50 fewer cells in the computation (20,000 compare to more the 1 million). The success of the method for these classes of flows may be attributed to the accurate calculation of the rotational inviscid flow which dominates the convection of the large-scale flow structures.