13 research outputs found
Anomalous scaling of passive scalar in turbulence and in equilibrium
We analyze multi-point correlation functions of a tracer in an incompressible
flow at scales far exceeding the scale at which fluctuations are generated
(quasi-equilibrium domain) and compare them with the correlation functions at
scales smaller than (turbulence domain). We demonstrate that the scale
invariance can be broken in the equilibrium domain and trace this breakdown to
the statistical integrals of motion (zero modes) as has been done before for
turbulence. Employing Kraichnan model of short-correlated velocity we identify
the new type of zero modes, which break scale invariance and determine an
anomalously slow decay of correlations at large scales
Lamb-type solution and properties of unsteady Stokes equations
We derive the general solution of the unsteady Stokes equations for an
unbounded fluid in spherical polar coordinates, in both time and frequency
domains. The solution is an expansion in vector spherical harmonics and given
as a sum of a particular solution, proportional to pressure gradient exhibiting
power-law spatial dependence, and a solution of vector Helmholtz equation
decaying exponentially in far field, the decomposition originally introduced by
Lamb. The solution can be applied to construct the transient exterior flow
induced by an arbitrary velocity distribution at the spherical boundary, such
as arising in the squirmer model of a microswimmer. It can be used to construct
solutions for transient flows driven by initial conditions, unbounded flows
driven by volume forces or disturbance to the unsteady flow due to a stationary
spherical particle. The long-time behavior of solution is controlled by the
flow component corresponding to average (or collective) motion of the boundary.
This conclusion is illustrated by the study of decay of transversal wave in the
presence of a fixed sphere. We further show that the general representation
reduces to the well-known solutions for unsteady flow around a sphere
undergoing oscillatory rigid-body (translation and rotation) motion. The
proposed solution representation provides an explicit form of the velocity
potential far from an oscillating body ("generalized" Darcy's law) and high-
and low-frequency expansions. The leading-order high-frequency expansion yields
the well-known ideal (inviscid) flow approximation, and the leading-order
low-frequency expansion yields the steady Stokes equations. We derive the
higher-order corrections to these approximations and discuss d'Alembert
paradox. Continuation of the general solution to imaginary frequency yields the
general solution of the Brinkman equations describing viscous flow in porous
medium.Comment: 30 pages, revised versio