20,561 research outputs found
Local transfer and spectra of a diffusive field advected by large-scale incompressible flows
This study revisits the problem of advective transfer and spectra of a
diffusive scalar field in large-scale incompressible flows in the presence of a
(large-scale) source. By ``large-scale'' it is meant that the spectral support
of the flows is confined to the wave-number region , where is
relatively small compared with the diffusion wave number . Such flows
mediate couplings between neighbouring wave numbers within of each other
only. It is found that the spectral rate of transport (flux) of scalar variance
across a high wave number is bounded from above by ,
where denotes the maximum fluid velocity and is the spectrum
of the scalar variance, defined as its average over the shell .
For a given flux, say , across , this bound requires
This is consistent with recent
numerical studies and with Batchelor's theory that predicts a spectrum
(with a slightly different proportionality constant) for the viscous-convective
range, which could be identified with . Thus, Batchelor's
formula for the variance spectrum is recovered by the present method in the
form of a critical lower bound. The present result applies to a broad range of
large-scale advection problems in space dimensions , including some
filter models of turbulence, for which the turbulent velocity field is advected
by a smoothed version of itself. For this case, and
are the kinetic energy spectrum and flux, respectively.Comment: 6 journal pages, 1 "cartoon" figure, to appear in PR
Enstrophy dissipation in freely evolving two-dimensional turbulence
Freely decaying two-dimensional Navier--Stokes turbulence is studied. The
conservation of vorticity by advective nonlinearities renders a class of
Casimirs that decays under viscous effects. A rigorous constraint on the
palinstrophy production by nonlinear transfer is derived, and an upper bound
for the enstrophy dissipation is obtained. This bound depends only on the
decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from
above in terms of initial data of the flows. An upper bound for the enstrophy
dissipation wavenumber is derived and the new result is compared with the
classical dissipation wavenumber.Comment: No figures, Letter to appear in Phys. Fluid
- β¦