Navier-Stokes-alpha model: LES equations with nonlinear dispersion

We present a framework for discussing LES equations with nonlinear dispersion. In this framework, we discuss the properties of the nonlinearly dispersive Navier-Stokes-alpha model of incompressible fluid turbulence --- also called the viscous Camassa-Holm equations and the LANS equations in the literature --- in comparison with the corresponding properties of large eddy simulation (LES) equations obtained via the approximate-inverse approach. In this comparison, we identify the spatially filtered NS-alpha equations with a class of generalized LES similarity models. Applying a certain approximate inverse to this class of LES models restores the Kelvin circulation theorem for the defiltered velocity and shows that the NS-alpha model describes the dynamics of the defiltered velocity for this class of generalized LES similarity models. We also show that the subgrid scale forces in the NS-alpha model transform covariantly under Galilean transformations and under a change to a uniformly rotating reference frame. Finally, we discuss in the spectral formulation how the NS-alpha model retains the local interactions among the large scales, retains the nonlocal sweeping effects of large scales on small scales, yet attenuates the local interactions of the small scales amongst themselves.Comment: 15 pages, no figures, Special LES volume of ERCOFTAC bulletin, to appear in 200

Energy transfer in isotropic turbulence at low Reynolds numbers

Detailed measurements were made of energy transfer among the scales of motion in incompressible turbulent fields at low Reynolds numbers generated by direct numerical simulation. It was observed that although the transfer resulted from triad interactions that were non-local in k space, the energy always transferred locally. The results are consistent with the notion of non-uniform advection of small weak eddies by larger and stronger ones, similar to transfer processes in the far dissipation range at high Reynolds numbers

Variational Principles for Lagrangian Averaged Fluid Dynamics

The Lagrangian average (LA) of the ideal fluid equations preserves their transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its convection of potential vorticity and its conservation of helicity. Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational framework that implies the LA fluid equations. This is expressed in the Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure

Incompressible Turbulence as Nonlocal Field Theory

It is well known that incompressible turbulence is nonlocal in real space because sound speed is infinite in incompressible fluids. The equation in Fourier space indicates that it is nonlocal in Fourier space as well. Contrast this with Burgers equation which is local in real space. Note that the sound speed in Burgers equation is zero. In our presentation we will contrast these two equations using nonlocal field theory. Energy spectrum and renormalized parameters will be discussed.Comment: 7 pages; Talk presented in Conference on "Perspectives in Nonlinear Dynamics (PNLD 2004)" held in Chennai, 200

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 Chekhlov et 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 simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary djnlx.tex fil

Leray and LANS-α\alpha modeling of turbulent mixing

Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e., Leray and LANS$-\alpha$ regularisation. The Leray principle introduces a {\bfi smoothed transport velocity} as part of the regularised convective nonlinearity. The LANS$-\alpha$ principle extends the Leray formulation in a natural way in which a {\bfi filtered Kelvin circulation theorem}, incorporating the smoothed transport velocity, is explicitly satisfied. These regularisation principles give rise to implied subgrid closures which will be applied in large eddy simulation of turbulent mixing. Comparison with filtered direct numerical simulation data, and with predictions obtained from popular dynamic eddy-viscosity modelling, shows that these mathematical regularisation models are considerably more accurate, at a lower computational cost.Comment: 42 pages, 12 figure

Optical measurement of rates of dissipation of temperature variance due to oceanic turbulence

Inhomogeneities in the refractive index induced by temperature fluctuations in turbulent flows have the effect of scattering light in near-forward angles. We have developed a method that extracts the rate of Temperature Variance Dissipation (TVD) and its spectrum from the properties of light scattering and have built an instrument - Optical Turbulence Sensor (OTS) - that implements the method. OTS uses a linear wavefront sensing Hartmann array and allows for nearly instantaneous measurements of temperature variance in turbulent flows. The instrument has been tested in an situ experiment carried out from a drifting vessel at a site off the coast of Newport, Oregon. Here we compare the temperature variance measured by OTS and its spectra with both theoretical predictions and with spectra obtained from a fast thermistor sensor.This paper was published in Optics Express and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/oe/home.cfm. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.Keywords: Wave-front sensing, Turbulenc

Local shell-to-shell energy transfer via nonlocal Interactions in fluid turbulence

In this paper we analytically compute the strength of nonlinear interactions in a triad, and the energy exchanges between wavenumber shells in incompressible fluid turbulence. The computation has been done using first-order perturbative field theory. In three dimension, magnitude of triad interactions is large for nonlocal triads, and small for local triads. However, the shell-to-shell energy transfer rate is found to be local and forward. This result is due to the fact that the nonlocal triads occupy much less Fourier space volume than the local ones. The analytical results on three-dimensional shell-to-shell energy transfer match with their numerical counterparts. In two-dimensional turbulence, the energy transfer rates to the near-by shells are forward, but to the distant shells are backward; the cumulative effect is an inverse cascade of energy.Comment: 10 pages, Revtex