Density Fields in Burgers and KdV-Burgers Turbulence

A model analytical description of the density field advected in a velocity field governed by the multidimensional Burgers equation is suggested. This model field satisfies the mass conservation law and, in the zero viscosity limit, coincides with the generalized solution of the continuity equation....

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Bibliographic Details
Published in:SIAM journal on applied mathematics 1996-08, Vol.56 (4), p.1008-1038
Main Authors: Saichev, A. I., Woyczynski, W. A.
Format: Article
Language:English
Subjects:
Average linear density
Burger equation
Coordinate systems
Density
Evolution
Linear equations
Modeling
Nonlinear equations
Probability distribution
Probability distributions
Spectral energy distribution
Turbulence
Turbulence models
Velocity
Velocity distribution
Viscosity
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Summary:A model analytical description of the density field advected in a velocity field governed by the multidimensional Burgers equation is suggested. This model field satisfies the mass conservation law and, in the zero viscosity limit, coincides with the generalized solution of the continuity equation. A numerical and analytical study of the evolution of such a model density field is much more convenient than the standard method of simulation of transport of passive tracer particles in the fluid. In the 1-dimensional case, a more general Korteweg-deVries (KdV)-Burgers equation is suggested as a model which permits an analytical treatment of the density field in a strongly nonlinear model of compressible gas which takes into account dissipative and dispersive effects as well as pressure forces, the former not being accounted for in the standard Burgers framework. The dynamical and statistical properties of the density field are studied. In particular, utilizing the above model in the 2-dimensional case and the (most interesting for us) situation of small viscosity, we can follow the creation and evolution of the cellular structures in the density field and the subsequent creation of the "quasi-particle" clusters of matter of enormous density. In addition, it is shown that in the zero viscosity limit, the density field spectrum has a power tail$\propto k^{-n}$, with different exponents in different regimes.
ISSN:0036-1399
1095-712X
DOI:10.1137/S0036139994266475