Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence

Inviscid regularization modeling of turbulent flow is investigated. Homogeneous, isotropic, decaying turbulence is simulated at a range of filter widths. A coarse-graining of turbulent flow arises from the direct regularization of the convective nonlinearity in the Navier–Stokes equations. The regularization is translated into its corresponding sub-filter model to close the equations for large-eddy simulation (LES). The accuracy with which primary turbulent flow features are captured by this modeling is investigated for the Leray regularization, the Navier–Stokes-α formulation (NS-α), the simplified Bardina model and a modified Leray approach. On a PDE level, each regularization principle is known to possess a unique, strong solution with known regularity properties. When used as turbulence closure for numerical simulations, significant differences between these models are observed. Through a comparison with direct numerical simulation (DNS) results, a detailed assessment of these regularization principles is made. The regularization models retain much of the small-scale variability in the solution. The smaller resolved scales are dominated by the specific sub-filter model adopted. We find that the Leray model is in general closest to the filtered DNS results, the modified Leray model is found least accurate and the simplified Bardina and NS-α models are in between, as far as accuracy is concerned. This rough ordering is based on the energy decay, the Taylor Reynolds number and the velocity skewness, and on detailed characteristics of the energy dynamics, including spectra of the energy, the energy transfer and the transfer power. At filter widths up to about 10% of the computational domain-size, the Leray and NS-α predictions were found to correlate well with the filtered DNS data. Each of the regularization models underestimates the energy decay rate and overestimates the tail of the energy spectrum. The correspondence with unfiltered DNS spectra was observed often to be closer than with filtered DNS for several of the regularization models

A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form $dv/dt = F(v)$, in the Banach space, $X$, of all bounded continuous functions of the variable $s\in\mathbb{R}$ with values in certain finite-dimensional linear space. This new evolution ODE, named determining form, induces an infinite-dimensional dynamical system in the space $X$ which is noteworthy for two reasons. One is that $F$ is globally Lipschitz from $X$ into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form $v(t, s) = v_{0}(t + s)$, correspond exactly to initial data $v_{0}$ that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter)

A data assimilation algorithm for the subcritical surface quasi-geostrophic equation

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood–Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators

Analyticity of Essentially Bounded Solutions to Semilinear Parabolic Systems and Validity of the Ginzburg-Landau Equation

Some analytic smoothing properties of a general strongly coupled, strongly parabolic semilinear system of order $2m$ in $realnos^D times (0,T)$ with analytic entries are investigated. These properties are expressed in terms of holomorphic continuation in space and time of essentially bounded global solutions to the system. Given $0 < T' < T le infty$, it is proved that any weak, essentially bounded solution ${bold u} = (u_1,dots,u_N)$ in $realnos^Dtimes (0,T)$ possesses a bounded holomorphic continuation $bold u (x+iy,sigma + itau )$ into a region in $complexnos^Dtimescomplexnos$ defined by $(x,sigma )in realnos^Dtimes (T',T)$, $|y| < A'$ and $|tau | < B'$, where $A'$ and $B'$ are some positive constants depending upon $T'$. The proof is based on analytic smoothing properties of a parabolic Green function combined with a contraction mapping argument in a Hardy space $H^infty$. Applications include weakly coupled semilinear systems of complex reaction-diffusion equations such as the complex Ginzburg-Landau equations. Special attention is given to the problem concerning the validity of the derivation of amplitude equations which describe various instability phenomena in hydrodynamics