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.     

Numerical simulation of sound generation in a mixing layer by the finite difference lattice Boltzmann method

We simulated aerodynamic sound in a two-dimensional mixing layer using the finite difference lattice Boltzmann method (FDLBM). We introduced a finite difference scheme, called the dispersion relation preserving (DRP) scheme, into the FDLBM to carry out an accurate simulation of aerodynamic problems. The scheme is designed such that the dispersion relation of the finite difference scheme is the same as that of the original partial differential equations and is useful for acoustic simulations. A turbulent flow field was simulated by large-eddy simulation (LES), using the Smagorinsky model, and the results were compared with those from a direct simulation based on the Navier–Stokes equations to confirm the usefulness of this method. The combination of the FDLBM and the DRP scheme was shown to be very effective for direct simulations of aerodynamic sound.     

Analysis of a variational multiscale method for Large–Eddy simulation and its application to homogeneous isotropic turbulence

In this paper a variational multiscale method based on local projection and grad–div stabilization for Large–Eddy simulation for the incompressible Navier–Stokes problem is considered. An a priori error estimate is given for a case with rather general nonlinear (piecewise constant) coefficients of the subgrid models for the unresolved scales of velocity and pressure. Then the design of the subgrid scale models is specified for the case of homogeneous isotropic turbulence and studied for the standard benchmark problem of decaying homogeneous isotropic turbulence.     

A Parallel Subgrid Stabilized Finite Element Method Based on Two-Grid Discretization for Simulation of 2D/3D Steady Incompressible Flows

Based on domain decomposition and two-grid discretization, a parallel subgrid stabilized finite element method for simulation of 2D/3D steady convection dominated incompressible flows is proposed and analyzed. In this method, a subgrid stabilized nonlinear Navier–Stokes problem is first solved on a coarse grid where the stabilization term is based on an elliptic projection defined on the same coarse grid, and then corrections are calculated in overlapped fine grid subdomains by solving a linearized problem. By the technical tool of local a priori estimate for finite element solution, error bounds of the approximate solution are estimated. Algorithmic parameter scalings of the method are derived. Numerical results are also given to demonstrate the effectiveness of the method.     

Computational error-analysis of a discontinuous Galerkin discretization applied to large-eddy simulation of homogeneous turbulence

A computational error-assessment of large-eddy simulation (LES) in combination with a discontinuous Galerkin finite element method is presented for homogeneous, isotropic, decaying turbulence. The error-landscape database approach is used to quantify the total simulation error that arises from the use of the Smagorinsky eddy-viscosity model in combination with the Galerkin discretization. We adopt a modified HLLC flux, allowing an explicit control over the dissipative component of the numerical flux. The optimal dependence of the Smagorinsky parameter on the spatial resolution is determined for second and third order accurate Galerkin methods. In particular, the role of the numerical dissipation relative to the contribution from the Smagorinsky dissipation is investigated. We observed an ‘exchange of dissipation’ principle in the sense that an increased numerical dissipation implied a reduction in the optimal Smagorinsky parameter. The predictions based on Galerkin discretization with fully stabilized HLLC flux were found to be less accurate than when a central discretization with (mainly) Smagorinsky dissipation was used. This was observed for both the second and third order Galerkin discretization, suggesting to emphasize central discretization of the convective nonlinearity and stabilization that mimics eddy-viscosity as sub-filter dissipation.     

New subgrid artificial viscosity Galerkin methods for the Navier–Stokes equations

We study subgrid artificial viscosity methods for approximating solutions to the Navier–Stokes equations. Two methods are introduced that add viscous stabilization via an artificial viscosity, then remove it only on a coarse mesh. These methods can be considered as conforming, mixed methods, the first for velocity and vorticity, and the second for velocity and its gradient, the former incorporating a naturally arising grad–div stabilization term. In this paper, we rigorously study the first scheme analytically, showing that it is unconditionally stable and optimally convergent, as well as both schemes computationally. Numerical experiments demonstrate the advantages of both of these methods.     

Residual based VMS subgrid modeling for vortex flows

This paper presents a residual based subgrid modeling approach for Large Eddy Simulations (LES) based on the variational multiscale method as a cure for the problem of preservation of vortices in numerical flow simulation. This approach combines a splitting of the non-linear term in the Navier–Stokes equations into strain and vorticity with a residual based modeling of the subgrid problems. The benefit is that certain driving phenomena, normally not present in subgrid modeling, e.g. vortex stretching, can be seen in the equations.Here, we focus on two of the subgrid terms arising from the subgrid scale problem. The effect of the two terms are illustrated in an LES of a three dimensional flow around a wing where the main feature is the formation and preservation of a tip vortex, an important phenomenon in many aerodynamic and hydrodynamical applications. We see that the addition of the new subgrid terms correctly counteracts the dissipative effect, arising from numerics and turbulence modeling, on the vortex and thus strongly improves prediction of the tip vortex.     

Depth-averaged Lattice Boltzmann and Finite Element methods for single-phase flows in fractures with obstacles

We use Lattice Boltzmann Method (LBM) MRT and Cumulant schemes to study the performance and accuracy of single-phase flow modeling for propped fractures. The simulations are run using both the two- and three-dimensional Stokes equations, and a 2.5D Stokes–Brinkman approximate model. The LBM results are validated against Finite Element Method (FEM) simulations and an analytical solution to the Stokes–Brinkman flow around an isolated circular obstacle. Both LBM and FEM 2.5D Stokes–Brinkman models are able to reproduce the analytical solution for an isolated circular obstacle. In the case of 2D Stokes and 2.5D Stokes–Brinkman models, the differences between the extrapolated fracture permeabilities obtained with LBM and FEM simulations for fractures with multiple obstacles are below 1%. The differences between the fracture permeabilities computed using 3D Stokes LBM and FEM simulations are below 8% . The differences between the 3D Stokes and 2.5 Stokes–Brinkman results are less than 7% for FEM study, and 8% for the LBM case. The velocity perturbations that are introduced around the obstacles are not fully captured by the parabolic velocity profile inherent to the 2.5D Stokes–Brinkman model.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Velocity inversion of micro cylindrical Couette flow: A lattice Boltzmann study

In this work the micro gas flow between two concentric cylinders is investigated by a lattice Boltzmann equation (LBE) model with multiple relaxation times. A local kinetic boundary condition is proposed for the LBE to model the gas–wall interaction. Numerical simulations are conducted to examine the tangential velocity distribution under different flow conditions. It is shown that the proposed LBE can capture the velocity inversion phenomenon successfully. Comparisons with the Navier–Stokes solutions and DSMC results are also made and it is shown that the LBE yields better predictions.     

Lattice Boltzmann modeling of dendritic growth in forced and natural convection

A two-dimensional (2D) coupled model is developed for the simulation of dendritic growth during alloy solidification in the presence of forced and natural convection. Instead of conventional continuum-based Navier–Stokes (NS) solvers, the present model adopts a kinetic-based lattice Boltzmann method (LBM), which describes flow dynamics by the evolution of distribution functions of moving pseudo-particles, for the numerical computations of flow dynamics as well as thermal and solutal transport. The dendritic growth is modeled using a solutal equilibrium approach previously proposed by Zhu and Stefanescu (ZS), in which the evolution of the solid/liquid interface is driven by the difference between the local equilibrium composition and the local actual liquid composition. The local equilibrium composition is calculated from the local temperature and curvature. The local temperature and actual liquid composition, controlled by both diffusion and convection, are obtained by solving the LB equations using the lattice Bhatnagar–Gross–Krook (LBGK) scheme. Detailed model validation is performed by comparing the simulations with analytical predictions, which demonstrates the quantitative capability of the proposed model. Furthermore, the convective dendritic growth features predicted by the present model are compared with those obtained from the Zhu–Stefanescu and Navier–Stokes (ZS–NS) model, in which the fluid flow is calculated using an NS solver. It is found that the evolution of the solid fraction of dendritic growth calculated by both models coincides well. However, the present model has the significant advantages of numerical stability and computational efficiency for the simulation of dendritic growth with melt convection.     

Numerical simulations of solutions of a two-level averaged Navier–Stokes system

An averaging procedure for the Navier–Stokes equations has been proposed in an earlier article [I. Moise, R.M. Temam, Renormalization group method. Application to Navier–Stokes Equation, Discrete Contin. Dyn. Syst. 6 (1) (2000) 191–210]. This averaging procedure is based on a two-level decomposition of the solution into low and high frequencies. The aim of the present article is to investigate, with the help of numerical simulations, the behavior of the small scales of the corresponding system. Space-periodic solutions with a non-resonant period are considered. The time evolution of the averaged and standard (non-averaged) small scales are compared at different Reynolds numbers and for different values of the cut-off level used to separate large and small scales of the flow variables. The numerical results illustrate the efficiency of the proposed averaging procedure for the Navier–Stokes equations. The averaged small scales provide an accurate prediction of the time-averaged small scales of the Navier–Stokes solutions. As the computational cost is reduced for the averaged equations, long time integrations on more than 50 eddy-turnover times have been performed for cut-off levels ensuring a proper resolution of the large scales. In these cases, development of instabilities in the averaged small scale equation is observed.     

An immersed boundary method based on the lattice Boltzmann approach in three dimensions,with application

The immersed boundary (IB) method originated by Peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid. The Navier–Stokes (N–S) equations in the IB method are usually solved using numerical methods such as FFT and projection methods. Here in our work, the N–S equations are solved by an alternative approach, the lattice Boltzmann method (LBM). Compared to many conventional N–S solvers, the LBM can be easier to implement and more convenient for modeling additional physics in a problem. This alternative approach adds extra versatility to the immersed boundary method. In this paper we discuss the use of a 3D lattice Boltzmann model (D3Q19) within the IB method. We use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3D channel and determine a drag scaling law for the sheet. Our main conclusions are: (1) the hybrid method is convergent with first-order accuracy which is consistent with the immersed boundary method in general; (2) the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow.     

Micro-macro Kolmogorov–Fokker–Planck models for a hard-sphere gas

Using a stochastic microscopic model of a rigid-sphere gas in a phase space, which is diffusive in the velocity space and valid at moderate Knudsen numbers, macroscopic equations of gas dynamics are derived, which are different from the system of Navier–Stokes equations or quasi-gasdynamic systems. The main pecularity of our derivation is more accurate velocity averaging due to the analytical solution of stochastic differential equations with respect to the Wiener measure, which describes our original meso model. The problem of a shock-wave front is used as an example showing that such an approach yields a greater and thus more realistic diffusion of the front than the one based on the Navier–Stokes equation. The numerical solution is based on a “discontinuous” particle method well suited for supercomputer applications.     

Verification of RANS solvers with manufactured solutions

This paper discusses code verification of Reynolds-Averaged Navier Stokes (RANS) solvers with the method of manufactured solutions (MMS). Examples of manufactured solutions (MSs) for a two-dimensional, steady, wall-bounded, incompressible, turbulent flow are presented including the specification of the turbulence quantities incorporated in several popular eddy-viscosity turbulence models. A wall-function approach for the MMS is also described. The flexiblity and usefulness of the MS is illustrated with calculations performed in three different exercises: the calculation of the flow field using the manufactured eddy-viscosity; the calculation of the eddy-viscosity using the manufactured velocity field; the calculation of the complete flow field coupling flow and turbulence variables. The results show that the numerical performance of the flow solvers is model dependent and that the solution of the complete problem may exhibit different orders of accuracy than in the exercises with no coupling between the flow and turbulence variables.     

Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.     

Large eddy simulation of turbulent open duct flow using a lattice Boltzmann approach

Large eddy simulations of turbulent open duct flow are performed using the lattice Boltzmann method (LBM) in conjunction with the Smagorinsky sub-grid scale (SGS) model. A smaller value of the Smagorinsky constant than the usually used one in plain channel flow simulations is used. Results for the mean flow and turbulent fluctuations are compared to experimental data obtained in an open duct of similar dimensions. It is found that the LBM simulation results are in good qualitative agreement with the experiments.     

On the Development of a Nonprimitive Navier–Stokes Formulation Subject to Rigorous Implementation of a New Vorticity Integral Condition

In this paper, a new integral vorticity boundary condition has been developed and implemented to compute solution of nonprimitive Navier–Stokes equation. Global integral vorticity condition which is of primitive character can be considered to be of entirely different kind compared to other vorticity conditions that are used for computation in literature. The procedure realized as explicit boundary vorticity conditions imitates the original integral equation. The main purpose of this paper is to design an algorithm which is easy to implement and versatile. This algorithm based on the new vorticity integral condition captures accurate vorticity distribution on the boundary of computational flow field and can be used for both wall bounded flows as well as flows in open domain. The approach has been arrived at without utilizing any ghost grid point outside of the computational domain. Convergence analysis of this alternative vorticity integral condition in combination with semi-discrete centered difference approximation of linear Stokes equation has been carried out. We have also computed correct pressure field near the wall, for both attached and separated boundary layer flows, by using streamfunction and vorticity field variables. The competency of the proposed boundary methodology vis-a-vis other popular vorticity boundary conditions has been amply appraised by its use in a model problem that embodies the essential features of the incompressibility and viscosity. Subsequently the proposed methodology has been further validated by computing analytical solution of steady Stokes equation. Finally, it has been applied to three benchmark problems governed by the incompressible Navier–Stokes equations, viz. lid driven cavity, backward facing step and flow past a circular cylinder. The results obtained are in excellent agreement with computational and experimental results available in literature, thereby establishing efficiency and accuracy of the proposed algorithm. We were able to accurately predict both vorticity and pressure fields.     

Viscous flow and colloid transport near air–water interface in a microchannel

In order to understand the transport behavior of colloids near an air–water interface (AWI), two computational methods are applied to simulate the local water flow field near a moving AWI in a 2D microfluidic channel. The first method is a mesoscopic multicomponent and multiphase lattice Boltzmann (LBM) model and the second is the macroscopic, Navier–Stokes based, volume-of-fluid interface tracking method. In the LBM, it is possible to predict the dynamic contact angles after the static contact angle is correctly set, and the predicted dynamic contact angles are in good agreement with previous observations. It is demonstrated that the two methods can yield a similar flow velocity field if they are applied properly. The flow field relative to AWI depends on the direction of the flow, and exhibits curved streamlines that transport fluid between the center of the channel and the wall region. Using the obtained flow, the motion of sub-micron colloids in a de-ionized water solution is then studied by a Lagrangian approach. The observed colloid trajectories are in qualitative agreement with our visualizations using a confocal microscope.     

Adjoint-based fluid flow control and optimisation with lattice Boltzmann methods

A lattice Boltzmann (LB) framework to solve fluid flow control and optimisation problems numerically is presented. Problems are formulated on a mesoscopic basis. In a side condition, the dynamics of a Newtonian fluid is described by a family of simplified Boltzmann-like equations, namely BGK–Boltzmann equations, which are linked to an incompressible Navier–Stokes equation. It is proposed to solve the non-linear optimisation problem by a line search algorithm. The needed derivatives are obtained by deriving the adjoint equations, referred to as adjoint BGK–Boltzmann equations. The primal equations are discretised by standard lattice Boltzmann methods (LBM) while for the adjoint equations a novel discretisation strategy is introduced. The approach follows the main ideas behind LBM and is therefore referred to as adjoint lattice Boltzmann methods (ALBM). The corresponding algorithm retains most of the basic features of LB algorithms. In particular, it enables a highly-efficient parallel implementation and thus solving large-scale fluid flow control and optimisation problems. The overall solution strategy, the derivation of a prototype adjoint BGK–Boltzmann equation, the novel ALBM and its parallel realisation as well as its validation are discussed in detail in this article. Numerical and performance results are presented for a series of steady-state distributed control problems with up to approximately 1.6 million unknown control parameters obtained on a high performance computer with up to 256 processing units.