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تسجيل مستخدم جديدThe decay of multiscale signals - deterministic model of the Burgers turbulence
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This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers equation in the limit of vanishing viscosity. A deterministic model of turbulence-like evolution is considered. We con- struct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wavenumbers of this function form a Weierstrass spectrum, which accumulates at the origin in geometric progression.Reverse sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in Burgers turbulence. Combining these two ideas allows us to obtain an exact analytical solution for the velocity field. We also notice that such multiscale waves may be constructed for multidimensional Burgers equation. This solution has scaling exponent h=-(1+n)/2 and its evolution in time is self-similar with logarithmic periodicity and with the same average law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-like struc- tures. This model also describes important properties of the Burgers turbulence such as the self-preservation of the evolution of large scale structures in the presence of small scales perturbations.
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This work is devoted to the decay ofrandom solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at small wavenumbers $k
$ and falling off quickly at large wavenumbers. In physical space, at sufficiently large distances, there is an ``outer region, where the velocity correlation function preserves exactly its initial form (a power law) when $n$ is not an even integer. When $1<n<2$ the spectrum, at long times, has three scaling regions : first, a $|k|^n$ region at very small $k$ms1 with a time-independent constant, stemming from this outer region, in which the initial conditions are essentially frozen; second, a $k^2$ region at intermediate wavenumbers, related to a self-similarly evolving ``inner region in physical space and, finally, the usual $k^{-2}$ region, associated to the shocks. The switching from the $|k|^n$ to the $k^2$ region occurs around a wave number $k_s(t) propto t^{-1/[2(2-n)]}$, while the switching from $k^2$ to $k^{-2}$ occurs around $k_L(t)propto t^{-1/2}$ (ignoring logarithmic corrections in both instances). The key element in the derivation of the results is an extension of the Kida (1979) log-corrected $1/t$ law for the energy decay when $n=2$ to the case of arbitrary integer or non-integer $n>1$. A systematic derivation is given in which both the leading term and estimates of higher order corrections can be obtained. High-resolution numerical simulations are presented which support our findings.
In a seminal article, citet[J. Fluid Mech., 174:441-465]{maxey87} presented a theoretical analysis showing that enhanced particle settling speeds in turbulence occur through the preferential sweeping mechanism, which depends on the preferential sampl
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