▪ Abstract Relationships between small and large scales of motion in turbulent flows are of much interest in large-eddy simulation of turbulence, in which small scales are not explicitly resolved and must be modeled. This paper reviews models that are based on scale-invariance properties of high-Reynolds-number turbulence in the inertial range. The review starts with the Smagorinsky model, but the focus is on dynamic and similarity subgrid models and on evaluating how well these models reproduce the true impact of the small scales on large-scale physics and how they perform in numerical simulations. Various criteria to evaluate the model performance are discussed, including the so-called a posteriori and a priori studies based on direct numerical simulation and experimental data. Issues are addressed mainly in the context of canonical, incompressible flows, but extensions to scalar-transport, compressible, and reacting flows are also mentioned. Other recent modeling approaches are briefly introduced.

Scale-Invariance and Turbulence Models for Large-Eddy Simulation

The dynamic model for large-eddy simulation of turbulence samples information from the resolved velocity field in order to optimize subgrid-scale model coefficients. When the method is used in conjunction with the Smagorinsky eddy-viscosity model, and the sampling process is formulated in a spatially local fashion, the resulting coefficient field is highly variable and contains a significant fraction of negative values. Negative eddy viscosity leads to computational instability and as a result the model is always augmented with a stabilization mechanism. In most applications the model is stabilized by averaging the relevant equations over directions of statistical homogeneity. While this approach is effective, and is consistent with the statistical basis underlying the eddy-viscosity model, it is not applicable to complex-geometry inhomogeneous flows. Existing local formulations, intended for inhomogeneous flows, are most commonly stabilized by artificially constraining the coefficient to be positive. In this paper we introduce a new dynamic model formulation, that combines advantages of the statistical and local approaches. We propose to accumulate the required averages over flow pathlines rather than over directions of statistical homogeneity. This procedure allows the application of the dynamic model with averaging to in-homogeneous flows in complex geometries. We analyse direct numerical simulation data to document the effects of such averaging on the Smagorinsky coefficient. The characteristic Lagrangian time scale over which the averaging is performed is chosen based on measurements of the relevant Lagrangian autocorrelation functions, and on the requirement that the model be purely dissipative, guaranteeing numerical stability when coupled with the Smagorinsky model. The formulation is tested in forced and decaying isotropic turbulence and in fully developed and transitional channel flow. In homogeneous flows, the results are similar to those of the volume-averaged dynamic model, while in channel flow, the predictions are slightly superior to those of the spatially (planar) averaged dynamic model. The relationship between the model and vortical structures in isotropic turbulence, as well as ejection events in channel flow, is investigated. Computational overhead is kept small (about 10% above the CPU requirements of the spatially averaged dynamic model) by using an approximate scheme to advance the Lagrangian tracking through first-order Euler time integration and linear interpolation in space.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/783398B2D0BE53C120151E4E911CA833/S0022112096007379a.pdf/div-class-title-a-lagrangian-dynamic-subgrid-scale-model-of-turbulence-div.pdf

A Lagrangian dynamic subgrid-scale model of turbulence

The paper presents large-eddy simulation (LES) formalism, along with the various subgrid-scale models developed since Smagorinsky’s model. We show how Kraichnan’s spectral eddy viscosity may be implemented in physical space, yielding the structure-function model. Recent developments of this model that allow the eddy viscosity to be inhibited in transitional regions are discussed. We present a dynamic procedure, where a double filtering allows one to dynamically determine the subgrid-scale model constants. The importance of backscatter effects is discussed. Alternatives to the eddy-viscosity assumption, such as scale- similarity models, are considered. Pseudo-direct simulations in which numerical diffusion replaces subgrid transfers are mentioned. Various applications of LES to incompressible and compressible turbulent flows are given, with an emphasis on the generation of coherent vortices

New Trends in Large-Eddy Simulations of Turbulence

A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries

Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow

The dynamic subgrid‐scale eddy viscosity model of Germano et al. [Phys. Fluids A 3, 1760 (1991)] (DSM) is modified by employing the mixed model of Bardina et al. [Ph.D dissertation, Stanford University (1983)] as the base model. The new dynamic mixed model explicitly calculates the modified Leonard term and only models the cross term and the SGS Reynolds stress. It retains the favorable features of DSM and, at the same time, does not require that the principal axes of the stress tensor be aligned with those of the strain rate tensor. The model coefficient is computed using local flow variables. The new model is incorporated in a finite‐volume solution method and large‐eddy simulations of flows in a lid‐driven cavity at Reynolds numbers of 3200, 7500, and 10 000 show excellent agreement with the experimental data. Better agreement is achieved by using the new model compared to the DSM. The magnitude of the dynamically computed model coefficient of the new model is significantly smaller than that from DSM.

A dynamic mixed subgrid‐scale model and its application to turbulent recirculating flows

A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates

The evolution of the coastal upwelling and interfacial instability of a stratified and rotating fluid is studied numerically by using large-eddy simulation. Upwelling is generated near the sidewall of a rotating annulus by the shear at the top. The fluid initially consists of a stably stratified 'two-layer' structure with a narrow interface separating the two layers. The large-scale motion of the flow is simulated by solving the time-dependent non-hydrostatic incompressible Navier-Stokes and scalar transport equations while the small-scale motion is represented by a dynamic subgrid-scale model. The upwelling process contains both stable and unstable stratification. The vertical structure of upwelling consists of a persistent primary front, a trailing mixing zone on the shore side of the front, and a temporary secondary front which leads a top inversion layer. The longshore velocity profile has two maxima which occur at the edge of the sidewall boundary layer and at the density front. The upwelled density front is unstable to azimuthal perturbations and baroclinic waves develop and grow to large amplitude. Pairs of cyclonic and anticyclonic waves appear at the front which form 'jet-streams'. The secondary front is unstable to azimuthal perturbations. Its instability, and the associated drop of the top inversion layer, take the form of radial bands which subsequently break up into isolated patches and eventually sink. The computed values of various upwelling time and length scales are compared to and are in good agreement with past experimental data.

Numerical simulation of coastal upwelling and inerfacial instability of a rotaion and stratified fluid

The dynamic subgrid-scale model of Germano et al. is implemented in a finite volume formulation and applied to the simulation of turbulent flow in a three-dimensional lid-driven cavity at Reynolds number of 7500. The filtering operation is carried out in physical space, and the model coefficient is calculated locally. The computed mean and rms velocities as well as the Reynolds stress are compared with experimental data. It is shown that backscatter from small to large scales is necessary to sustain turbulent fluctuations. The model is being applied to the simulation of turbulent flows in a stratified and rotating environment in complex geometries.

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Application of a dynamic subgrid-scale model to turbulent recirculating flows

Presented is a numerical method for simulating free-surface flows through solution of the timedependent, incompressible, Navier-Stokes equations and the non-linear dynamic and kinematic boundary conditions. The numerical method uses boundary-fitted curvilinear coordinates with a finite-volume, time-splitting, approximate-factorization method formulated in primitive variables on a non-staggered grid. The pressure Poisson equation is solved using a multigrid technique. A new extension of domain decomposition methods is developed which involves splitting the free surface from the fluid volume for the iterative enforcement of the pressure equation and the dynamic boundary condition. A derivation in curvilinear coordinates of the Eulerian kinematic boundary condition is presented and is used for advancing the free surface in a Crank-Nicolson formulation. Development of the numerical method is presented for three dimensions; preliminary results are given from twodimensional non-linear simulations of standing waves.

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Yan Zang

Papers

A dynamic mixed subgrid‐scale model and its application to turbulent recirculating flows

A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates

Numerical simulation of coastal upwelling and inerfacial instability of a rotaion and stratified fluid

Application of a dynamic subgrid-scale model to turbulent recirculating flows

A method for simulation of viscous, non-linear, free-surface flows ∗