Spectral Eddy Viscosity Research Articles

Spectral and hyper eddy viscosity in high-Reynolds-number turbulence

For the purpose of studying the spectral properties of energy transfer between large and small scales in high-Reynolds-number turbulence, we measure the longitudinal subgrid-scale (SGS) dissipation spectrum, defined as the co-spectrum of the SGS stress and filtered strain-rate tensors. An array of four closely spaced X-wire probes enables us to approximate a two-dimensional box filter by averaging over different probe locations (cross-stream filtering) and in time (streamwise filtering using Taylor's hypothesis). We analyse data taken at the centreline of a cylinder wake at Reynolds numbers up to Rλ ∼ 450. Using the assumption of local isotropy, the longitudinal SGS stress and filtered strain-rate co-spectrum is transformed into a radial co-spectrum, which allows us to evaluate the spectral eddy viscosity, v(k, kΔ). In agreement with classical two-point closure predictions, for graded filters, the spectral eddy viscosity deduced from the box-filtered data decreases near the filter wavenumber kΔ. When using a spectral cutoff filter in the streamwise direction (with a box-filter in the cross-stream direction) a cusp behaviour near the filter scale is observed. In physical space, certain features of a wavenumber-dependent eddy viscosity can be approximated by a combination of a regular and a hyper-viscosity term. A hyper-viscous term is also suggested from considering equilibrium between production and SGS dissipation of resolved enstrophy. Assuming local isotropy, the dimensionless coefficient of the hyper-viscous term can be related to the skewness coefficient of filtered velocity gradients. The skewness is measured from the X-wire array and from direct numerical simulation of isotropic turbulence. The results show that the hyper-viscosity coefficient is negative for graded filters and positive for spectral filters. These trends are in agreement with the spectral eddy viscosity measured directly from the SGS stress–strain rate co-spectrum. The results provide significant support, now at high Reynolds numbers, for the ability of classical two-point closures to predict general trends of mean energy transfer in locally isotropic turbulence.

From Two-Point Closures of Isotropic Turbulence to LES of Shear Flows

We first recall the EDQNM two-point closure approach of three-dimensional isotropic turbulence. It allows in particular prediction of the infrared kinetic-energy dynamics (with ak4 backscatter) and the associated time-decay law of kinetic-energy, useful in particular for one-point closure modelling. Afterwards, we show how the spectral eddy viscosity concept may be used for large-eddy simulations: we introduce the plateau-peak model and the spectral-dynamic models. They are applied to decaying isotropic turbulence, and allow recovery of the EDQNM infrared energy dynamics. Anew infrared k2 law for the pressure spectrum, predicted by the closure, is also well verified.

Intermittency and relative scaling of subgrid-scale energy dissipation in isotropic turbulence

The rate at which large-scale kinetic energy in turbulent flows is transferred to, or from, unresolved scales (smaller than a filter scale Δ) is given by Π(x,t)=−τijS̃ij, where τij is the subgrid stress, and S̃ij is the resolved strain-rate tensor. The spatial distribution of Π(x,t) is computed from DNS of isotropic turbulence, and is found to be highly intermittent with increasing levels of intermittency as the filter size decreases. Relative scaling exponents of high-order moments of Π are measured using extended self-similarity, and are compared to those of longitudinal velocity structure functions. Reasonably good agreement is found, both sets of exponents clearly departing from the Kolmogorov (1941) theory. Relative scaling exponents of the SGS dissipation as predicted by several models are measured a priori from the DNS, and are compared to those of the true dissipation. We find the constant and spectral eddy viscosity models to be significantly less intermittent, and the local dynamic model to be much more intermittent than the true SGS dissipation field. The traditional and volume-averaged dynamic Smagorinsky models, together with the similarity model, yield more realistic levels of intermittency.

On the Asymptotic Regimes and the Strongly Stratified Limit of Rotating Boussinesq Equations

Asymptotic regimes of geophysical dynamics are described for different Burger number limits. Rotating Boussinesq equations are analyzed in the asymptotic limit of strong stratification in the Burger number of order one situation as well as in the asymptotic regime of strong stratification and weak rotation. It is shown that in both regimes the horizontally averaged buoyancy variable is an adiabatic invariant (approximate conservation law) for the full Boussinesq system. Spectral phase shift corrections to the buoyancy time scale associated with vertical shearing of this invariant are deduced. Statistical dephasing effects induced by turbulent processes on inertial-gravity waves are evidenced. The “split” of the energy transfer of the vortical and the wave components is established in the Craya–Herring cyclic basis. As the Burger number increases from zero to infinity, we demonstrate gradual unfreezing of energy cascades for ageostrophic dynamics. This property is related to the nonlinear geostrophic adjustment mechanism which is the capacity of ageostrophic dynamics to transfer energy to small scales. The energy spectrum and the anisotropic spectral eddy viscosity are deduced with an explicit dependence on the anisotropic rotation/stratification time scale which depends on the vertical aspect ratio parameter. Intermediate asymptotic regime corresponding to strong stratification and weak rotation is analyzed where the effects of weak rotation are accounted for by an asymptotic expansion with full control (saturation) of vertical shearing. The regularizing effect of weak rotation differs from regularizations based on vertical viscosity. Two scalar prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure) are obtained.

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.

Analytical and phenomenological studies of rotating turbulence

A framework, which combines mathematical analysis, closure theory, and phenomenological treatment, is developed to study the spectral transfer process in turbulent flows that are subject to rotation. First, we outline a mathematical procedure that is particularly appropriate for problems with two disparate time scales. The approach that is based on the Green’s method leads to the Poincaré velocity variables and the Poincaré transformation when applied to rotating turbulence. The effects of the rotation are now conveniently included in the momentum equation as the modifications to the convolution of nonlinear term. The Poincaré transformed equations are used to obtain a time-dependent Taylor–Proudman theorem valid in the asymptotic limit when the nondimensional parameter μ≡Ωt→∞ (Ω is the rotation rate and t is the time). The ‘‘split’’ of the energy transfer in both direct and inverse directions is established. Second, we apply the Eddy-Damped-Quasinormal-Markovian (EDQNM) closure to the Poincaré transformed Euler/ Navier–Stokes equations. This closure leads to expressions for the spectral energy transfer. In particular, a unique triple velocity decorrelation time is derived with an explicit dependence on the rotation rate. This provides an important input for applying the phenomenological treatment of Zhou [Phys. Fluids 7, 2092 (1995)]. In order to characterize the relative strength of rotation, another nondimensional number, a spectral Rossby number, which is defined as the ratio of rotation, and turbulence time scales, is introduced. Finally, the energy spectrum and the spectral eddy viscosity are deduced.

Large-eddy simulations of shear layers

The main characteristics of coherent vortices with some basic elementary vortex interactions are recalled and the formalism of large-eddy simulations of turbulence are then presented, with Smagorinsky's model. A model based upon Kraichnan's spectral eddy viscosity and its counterpart in physical space, the structure-function model, are introduced. Applications of these models to 3D mixing layers, wakes, and separated flows are presented, with emphasis on coherent-vortex topology. The influence of compressibility is considered, and it is shown how the structure-function model can be selectively applied to simulate transition in a boundary layer.

A cascade model for neutrally buoyant dispersed two-phase homogeneous turbulence—I. Model formulation

This work proposes a cascade model approach to describe two-phase turbulent flows of neutrally buoyant liquid dispersions for high dispersed phase fractions. A continuum framework is selected for the representation of the mean flow momentum and continuity equations. The cascade model of Desnyansky & Novikov is extended to describe the energy spectrum and eddy intermittency in the presence of a dispersed phase. The Reynolds stress in the mean balance equation is related to the cascade variables by a spectral eddy viscosity model. A population balance is formulated for each eddy size. This approach permits the specification of the immediate environment of the drop processes modeled in the population balance. Specific drop-eddy events such as grazing collisions, drop entrapment and eddy shattering are suggested and their effects on the turbulent spectrum, eddy intermittencies and drop size distributions are examined. The energy, intermittency and population balance equations modified to include the effects of drop-eddy interactions form the proposed two-phase cascade model for homogeneous, neutrally buoyant, turbulent dispersions.

Spectral large-eddy simulation of isotropic and stably stratified turbulence

We first recall the concepts of spectral eddy viscosity and diffusivity, derived from the two-point closures of turbulence, in the framework of large-eddy simulations in Fourier space. The case of a spectrum which does not decrease as (Ck is Kolmogorov's constant), with α ≈ 1.32.

Large-eddy simulation of the turbulent flow in the downstream region of a backward-facing step.

A numerical simulation of a complex turbulent shear flow using large-eddy simulation techniques is carried out. The filtered Navier-Stokes equations are solved with a finite-volume method. The subgrid model is a local adaptation to the physical space of isotropic spectral eddy-viscosity models. The statistics of the mean field are in good agreement with the experimental data available, corresponding to low step. Calculations in a high-step case show that the eddy structure of the flow presents striking analogies with the plane shear layers, with large billows shed behind the step, and longitudinal hairpin vortices strained between these billows.

Some remarks on the spectra of stably stratified turbulent shear flow

In this paper, a simple model is proposed to solve the system of spectral equations of turbulent kinetic energy and temperature variance in a thermally stratified turbulent shear flow. The buoyancy (or heat flux) term and the production term are modelled by means of turbulent exchange coefficient (or spectral eddy viscosity). For the terms representing inertial transfer of turbulent kinetic energy and temperature variance, Pao's (1965, 1968) formulations are employed. Analytical expressions for the non-dimensional kinetic energy spectrum φ ( k ), temperature variance spectrum φ T ( k ), the shear stress spectrum φ uw ( k ) and the heat-flux spectrum φ wT ( k ) are obtained within the framwork of similarity theory. The results are computed numerically and discussed.