Anisotropic turbulence and internal waves in stably stratified flows (QNSE theory (original) (raw)
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A quasinormal scale elimination model of turbulent flows with stable stratification
Physics of Fluids, 2005
Models of planetary, atmospheric and oceanic circulation involve eddy viscosity and eddy diffusivity, K M and K H , that account for unresolved turbulent mixing and diffusion. The most sophisticated turbulent closure models used today for geophysical applications belong in the family of the Reynolds stress models. These models are formulated for the physical space variables; they consider a hierarchy of turbulent correlations and employ a rational way of its truncation. In the process, unknown correlations are related to the known ones via "closure assumptions" that are based upon physical plausibility, preservation of tensorial properties, and the principle of the invariant modeling according to which the constants in the closure relationships are universal. Although a great deal of progress has been achieved with Reynolds stress closure models over the years, there are still situations in which these models fail. The most difficult flows for the Reynolds stress modeling are those with anisotropy and waves because these processes are scaledependent and cannot be included in the closure assumptions that pertain to ensemble-averaged quantities. Here, we develop an alternative approach of deriving expressions for K M and K H using the spectral space representation and employing a self-consistent, quasi-normal scale elimination (QNSE) algorithm. More specifically, the QNSE procedure is based upon the quasi-Gaussian mapping of the velocity and temperature fields using the Langevin equations. Turbulence and waves are treated as one entity and the effect of the internal waves is easily identifiable. This model implies partial averaging and, thus, is scale-dependent; it allows one to easily introduce into consideration such parameters as the grid resolution, the degree of the anisotropy, and spectral characteristics, among others.
A quasi-normal scale elimination model of turbulence and its application to stably stratified flows
Nonlinear Processes in Geophysics, 2006
Models of planetary, atmospheric and oceanic circulation involve eddy viscosity and eddy diffusivity, K M and K H , that account for unresolved turbulent mixing and diffusion. The most sophisticated turbulent closure models used today for geophysical applications belong in the family of the Reynolds stress models. These models are formulated for the physical space variables; they consider a hierarchy of turbulent correlations and employ a rational way of its truncation. In the process, unknown correlations are related to the known ones via "closure assumptions" that are based upon physical plausibility, preservation of tensorial properties, and the principle of the invariant modeling according to which the constants in the closure relationships are universal. Although a great deal of progress has been achieved with Reynolds stress closure models over the years, there are still situations in which these models fail. The most difficult flows for the Reynolds stress modeling are those with anisotropy and waves because these processes are scaledependent and cannot be included in the closure assumptions that pertain to ensemble-averaged quantities. Here, we develop an alternative approach of deriving expressions for K M and K H using the spectral space representation and employing a self-consistent, quasi-normal scale elimination (QNSE) algorithm. More specifically, the QNSE procedure is based upon the quasi-Gaussian mapping of the velocity and temperature fields using the Langevin equations. Turbulence and waves are treated as one entity and the effect of the internal waves is easily identifiable. This model implies partial averaging and, thus, is scale-dependent; it allows one to easily introduce into consideration such parameters as the grid resolution, the degree of the anisotropy, and spectral characteristics, among others. Applied to tur-
A Quasi-Normal Scale Elemination theory of turbulent flows with stable stratification
2007
A new spectral model of turbulent flows with stable stratification is presented. The theory is based upon a mapping of the actual velocity field to a quasi-Gaussian field using the Langevin equation. The parameters of the mapping are calculated using a systematic process of successive averaging over small shells of velocity and temperature modes that eliminates them from the equations of motion. This procedure does not differentiate between turbulence and internal waves and accounts for their combined effect. This approach offers a powerful mathematical tool for dealing with previously nearly intractable aspects of anisotropic turbulence; among these aspects are the threshold criterion for generation of internal waves and the modification of their dispersion relation by turbulence. The process of successive small scales elimination results in a model describing the largest scales of a flow. Partial scale elimination yields subgrid-scale viscosities and diffusivities that can be used in large eddy simulations. The elimination of all fluctuating scales results in RANS models. The model predicts various important characteristics of stably stratified flows, such as the dependence of the vertical turbulent Prandtl number on Froude and Richardson numbers, anisotropization of the flow filed, and decay of vertical diffusivity under strong stratification, all in good agreement with computational and observational data. The theory also yields analytical expressions for various 1D and 3D kinetic and potential energy spectra that reflect the effects of waves and anisotropy. The model's results are suitable for immediate use in practical applications.
Theoretical and Computational Fluid Dynamics, 2011
A Finite Volume-based large-eddy simulation method is proposed along with a suitable extension of the dynamic modelling procedure that takes into account for the integral formulation of the governing filtered equations. Discussion about the misleading interpretation of FV in some literature is addressed. Then, the classical Germano identity is congruently rewritten in such a way that the determination of the modelling parameters does not require any arbitrary averaging procedure and thus retains a fully local character. The numerical modelling of stratified turbulence is the specific problem considered in this study, as an archetypal of simple geophysical flows. The original scaling formulation of the dynamic sub-grid scale model proposed by Wong and Lilly (Phys. Fluids 6(6), 1994) is suitably extended to the present integral formulation. This approach is preferred with respect to traditional ones since the eddy coefficients can be independently computed by avoiding the addition of unjustified buoyancy production terms in the constitutive equations. Simple scaling arguments allow us not to use the equilibrium hypothesis according to which the dissipation rate should equal the sub-grid scale energy production. A careful a priori analysis of the relevance of the test filter shape as well as the filter-to-grid ratio is reported. Large-eddy simulation results are a posteriori compared with a reference pseudo-spectral direct numerical solution that is suitably post-filtered in order to have a meaningful comparison. In particular, the spectral distribution of kinetic and thermal energy as well as the viscosity and diffusivity sub-grid scale profiles are illustrated. The good performances of the proposed method, in terms of both evolutions of global quantities and statistics, are very promising for the future development and application of the method. supposed to be effective by means of an explicit/implicit filtering operation. Different SGS models for LES have been developed during the last two decades, i.e. the eddy-viscosity, scale-similarity and mixed models, as well as the more recent approximate deconvolution and Lagrangian models, . Moreover, the dynamic procedure proposed in allows for a wider feasibility as it can be applied to all modelling procedures, but it remains generally developed under the same equilibrium hypothesis of the Boussinesq sub-grid viscosity approach.
Density stratification has a strong impact on turbulence in geophysical flows. Stratification changes the spatial turbulence spectrum and the energy transport and conversion within the spectrum. We analyze these effects based on a series of direct numerical simulations (DNS) of stratified turbulence. To facilitate simulations of real-world problems, which are usually beyond the reach of DNS, we propose a subgrid-scale turbulence model for large eddy simulations of stratified flows based on the Adaptive Local Deconvolution Method (ALDM). Flow spectra and integral quantities predicted by ALDM are in excellent agreement with direct numerical simulation. ALDM automatically adapts to strongly anisotropic turbulence and is thus a suitable tool for studying turbulent flow phenomena in atmosphere and ocean.
Spectral eddy viscosity of stratified turbulence
2014
The spectral eddy viscosity (SEV) concept is a handy tool for the derivation of large-eddy simulation (LES) turbulence models and for the evaluation of their performance in predicting the spectral energy transfer. We compute this quantity by filtering and truncating fully resolved turbulence data from direct numerical simulations (DNS) of neutrally and stably stratified homogeneous turbulence. The results qualitatively confirm the plateau–cusp shape, which is often assumed to be universal, but show a strong dependence on the test filter size. Increasing stable stratification not only breaks the isotropy of the SEV but also modifies its basic shape, which poses a great challenge for implicit and explicit LES methods. We find indications that for stably stratified turbulence it is necessary to use different subgrid-scale (SGS) models for the horizontal and vertical velocity components. Our data disprove models that assume a constant positive effective turbulent Prandtl number.
Direct and large eddy simulation of stratified turbulence
Simulations of geophysical turbulent flows require a robust and accurate subgrid-scale turbulence modeling. To evaluate turbulence models for stably stratified flows, we performed direct numerical simulations (DNSs) of the transition of the three-dimensional Taylor–Green vortex and of homogeneous stratified turbulence with large-scale horizontal forcing. In these simulations we found that energy dissipation is concentrated within thin layers of horizontal tagliatelle-like vortex sheets between large pancake-like structures. We propose a new implicit subgrid-scale model for stratified fluids, based on the Adaptive Local Deconvolution Method (ALDM). Our analysis proves that the implicit turbulence model ALDM correctly predicts the turbulence energy budget and the energy spectra of stratified turbulence, even though dissipative structures are not resolved on the computational grid.
Journal of Fluid Mechanics, 2003
Stably stratified freely decaying homogeneous turbulence is investigated by means of direct numerical simulations (DNS) and a two-point closure statistical model of the EDQNM type; a careful comparison with laboratory experiments is also made. Several aspects of anisotropy in the flow are studied, both at large and small scales. DNS and EDQNM approaches give very similar results up to the finest indicators of the flow, namely anisotropic spectra of velocity fields. Hence the statistical model predicts the structure of the flow at all scales.
A Quasi-Normal Scale Elimination Theory of Turbulent Flows With Stable Stratification
Volume 4: Fatigue and Fracture; Fluids Engineering; Heat Transfer; Mechatronics; Micro and Nano Technology; Optical Engineering; Robotics; Systems Engineering; Industrial Applications, 2008
The Quasi-Normal Scale Elimination (QNSE) theory is a second order spectral closure capable of dealing with host of complicated factors introduced by nonlinearity and stable stratification. The theory is based upon a mapping of the actual velocity field to a quasi-Gaussian field whose modes are governed by the Langevin equation. The parameters of the mapping are calculated using a systematic process of successive averaging over small shells of velocity and temperature modes that eliminates them from the momentum and temperature equations. Turbulence and waves are treated as one entity and the effect of waves is easily identifiable. The model shows that there exists a range of Richardson numbers, between, approximately, 0.1 and 1, in which turbulence and heat transfer undergo remarkable anisotropization; the vertical mixing becomes suppressed while the horizontal mixing is enhanced. The theory yields analytical expressions for various 1D and 3D kinetic and potential energy spectra th...
Quasi-normal scale elimination theory of turbulence
2008
We present an analytical theory of turbulence based upon the procedure of successive elimination of small-scale modes that leads to gradual coarsening of the flow field and accumulation of eddy viscosity. The Reynolds number based upon the eddy viscosity, velocity and length scale of the remaining smallest scales stays O(1). The main results of the theory are analytical expressions for eddy viscosity and kinetic energy spectrum. Partial scale elimination yields a subgrid-scale representation for large eddy simulations while the elimination of all fluctuating scales is analogous to the Reynolds averaging.