5,713 research outputs found
On the elimination of the sweeping interactions from theories of hydrodynamic turbulence
In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian
same time correlation tensors are equal. This statement allows us to transform
the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic
turbulence back to the Eulerian representation. We define a hierarchy of
homogeneity symmetries between incremental homogeneity and global homogeneity.
It is shown that both the elimination of the sweeping interactions and the
derivation of the 4/5-law require a homogeneity assumption stronger than
incremental homogeneity but weaker than global homogeneity. The
quasi-Lagrangian transformation, on the other hand, requires an even stronger
homogeneity assumption which is many-time rather than one-time but still weaker
than many-time global homogeneity. We argue that it is possible to relax this
stronger assumption and still preserve the conclusions derived from theoretical
work based on the quasi-Lagrangian transformation.Comment: v1: submitted to Physica D. v2: major revisions; resubmitted to
Physica D. v3: minor revisions requested by referee
Turbulence for (and by) amateurs
Series of lectures on statistical turbulence written for amateurs but not
experts. Elementary aspects and problems of turbulence in two and three
dimensional Navier-Stokes equation are introduced. A few properties of scalar
turbulence and transport phenomena in turbulent flows are described.
Kraichnan's model of passive advection is discussed a bit more precisely.
{Part 1: Approaching turbulent flows.} Navier-Stokes equation. Cascades and
Kolmogorov theory. Modeling statistical turbulence. Correlation functions and
scaling.
{Part 2: Deeper in turbulent flows.} Turbulence in two dimensions.
Dissipation and dissipative anomalies. Fokker-Planck equations. Multifractal
models.
{Part 3: Scalar turbulence.} Transport and Lagrangian trajectories.
Kraichnan's passive scalar model. Anomalous scalings and universality.
{Part 4: Lagrangian trajectories.} Richardson's law. Lagrangian flows in
Kraichnan's model. Slow modes. Breakdown of Lagrangian flows. Batchelor limit.
Generalized Lagrangian flows and trajectory bundles.Comment: 37 pages, 6 figures, lecture note
Exact Solutions of a Remarkable Fin Equation
A model "remarkable" fin equation is singled out from a class of nonlinear
(1+1)-dimensional fin equations. For this equation a number of exact solutions
are constructed by means of using both classical Lie algorithm and different
modern techniques (functional separation of variables, generalized conditional
symmetries, hidden symmetries etc).Comment: 6 page
- …