On the structure of wave fronts in nonlinear dissipative media

Баренблатт, Г. И.; Ivanov, M. Ya.; Shapiro, G. I.

doi:10.1007/bf00250915

Cited by 16 publications

(7 citation statements)

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“…We create a steady traveling wave at long times by the boundary conditions u(Ϫϱ) ϭ u 0 , uЈ(Ϫϱ) ϭ 0, u(ϩϱ) ϭ 0 (see analyses in refs. [17][18][19]. The translation of the wave does not affect the spectrum.…”

Section: The Inertial Range Of the Kdvb Equationmentioning

confidence: 99%

Viscosity-dependent inertial spectra of the Burgers and Korteweg–deVries–Burgers equations

Chorin

Hald

2005

Proc. Natl. Acad. Sci. U.S.A.

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We show that the inertial range spectrum of the Burgers equation has a viscosity-dependent correction at any wave number when the viscosity is small but not zero. We also calculate the spectrum of the Korteweg-deVries-Burgers equation and show that it can be partially mapped onto the inertial spectrum of a Burgers equation with a suitable effective diffusion coefficient. These results are significant for the understanding of turbulence.I n a series of papers we, with G. I. Barenblatt and V. M.Prostokishin (1-7), presented a theory based on similarity considerations that derived Reynolds number-dependent scaling laws for the mean velocity profiles in turbulent boundary layers, as well as related Reynolds number-dependent anomalous corrections for the inertial range of turbulence. A mathematical model for the turbulent boundary layer that exhibits this scaling was presented in ref.8. This scaling theory agrees very well with the experimental data.It is a basic postulate of turbulence theory that fully developed turbulence far from walls has, on intermediate scales, small compared with the energy-containing scales but large compared with the dissipation scales, a universal ''inertial'' energy spectrum E(k) of the form E(k) ϭ CD 2/3 k Ϫ␥ , where k is the wave number and D is the rate of energy dissipation (9). In their original analysis Kolmogorov and Obukhov (see ref. 9) deduced that ␥ ϭ 5͞3; in recent years it has been claimed that this ␥ must be corrected by a positive ''intermittency correction'' that is independent of Reynolds number. The prefactor C has always been assumed to be independent of the Reynolds number. To the contrary, we have found from our theory and from analysis of the data that C was a function of the Reynolds number R, and that there was indeed a correction to the Kolmogorov-Obukhov exponent but that it was Reynolds number-dependent and tended to zero as the Reynolds number increased; thus the effects of viscosity are felt even in the range of intermediate, inertial scales.It would, of course, be desirable to deduce the scaling laws from the Navier-Stokes equations, but until this can be done it is of interest to examine the spectral properties of simpler model problems that exhibit an inertial range. This is done in the present article, where we rigorously show that the Burgers equation, one of the simplest models in fluid dynamics, exhibits the phenomena we see in the more complex situation. We also produce a closely related analysis of the spectrum of the Korteweg-deVries-Burgers (KdVB) equation that provides a useful glimpse of what has to be done in turbulence calculations. The Inertial Range of the Burgers EquationAs is well known, the Burgers equationwhere x is a physical space variable, t is the time, is a viscosity and subscripts denote differentiation, has in the limit 3 0 a Fourier transform û(k) with an inertial range spectrum E(k) ϳ k Ϫ2 , as can be readily seen from the fact that the solution u develops shocks whose spectral signature is k Ϫ2 . But what happens to that spectrum ...

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Section: The Inertial Range Of the Kdvb Equationmentioning

confidence: 99%

Viscosity-dependent inertial spectra of the Burgers and Korteweg–deVries–Burgers equations

Chorin

Hald

2005

Proc. Natl. Acad. Sci. U.S.A.

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“…In each subset pick one variable, in a fixed location within the shape of the subset, to be a member ofφ and the remainder to be members ofφ (for more general choices see [29]) . Then write H (0) = H and H (1) =Ĥ , eliminate the members ofφ and renumber the members ofφ so that they occupy the original lattice. We know (see Sec.…”

Section: Connection With Renormalizationmentioning

confidence: 99%

“…2) that the mapping from the old system to the new system preserves probability and equivalently, the partition function, and constitutes thus by definition a Kadanoff renormalization group transformation [21]. This transformation can be repeated and generates a sequence of Hamiltonians H (0) , H (1) , H (2) , . .…”

Section: Connection With Renormalizationmentioning

confidence: 99%

Prediction from Partial Data, Renormalization, and Averaging

2006

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We summarize and compare our recent methods for reducing the complexity of computational problems, in particular dimensional reduction methods based on the Mori-Zwanzig formalism of statistical physics, block Monte-Carlo methods, and an averaging method for deriving an effective equation for a nonlinear wave propagation problem. We show that their common thread is scale change and renormalization.

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“…1). This problem was chosen because of its apparent simplicity, because previous work by Barenblatt et al (8,9) has suggested interesting conjectures about its solution and also because of interesting connections with fluid mechanics.In the following sections I describe the problem, review the equilibrium averaging theory for Hamiltonian systems, set up the analogy for the KdVB equation, and provide a numerical analysis of its validity. I also compare the numerical results to earlier analytical work and point out interesting scaling relations that emerge from the calculations.…”

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confidence: 99%

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confidence: 99%

Averaging and renormalization for the Korteveg–deVries–Burgers equation

Chorin

2003

Proc. Natl. Acad. Sci. U.S.A.

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We consider traveling wave solutions of the Korteveg-deVriesBurgers equation and set up an analogy between the spatial averaging of these traveling waves and real-space renormalization for Hamiltonian systems. The result is an effective equation that reproduces means of the unaveraged, highly oscillatory, solution. The averaging enhances the apparent diffusion, creating an ''eddy'' (or renormalized) diffusion coefficient; the relation between the eddy diffusion coefficient and the original diffusion coefficient is found numerically to be one of incomplete similarity, setting up an instance of Barenblatt's renormalization group. The results suggest a relation between self-similar solutions of differential equations on one hand and renormalization groups and optimal prediction algorithms on the other. An analogy with hydrodynamics is pointed out.T here are many equations of interest in science whose solution is too complicated to be computed with sufficient accuracy, and one may be interested in finding an effective equation, easier to work with, whose solution is an average of the real solution. Recently, there has been a major effort to find rational ways to derive such equations for nonlinear problems (see, e.g., refs. 1 and 2).Optimal prediction methods (3) have been developed to find effective equations where the average is over a probability density for unresolved degrees of freedom. In the case of systems in thermal equilibrium the derivation of the reduced equations simplifies considerably and turns out to be closely related to renormalization group (RNG) methods (4).It is of great interest to extend these simpler optimal prediction methods to non-Hamiltonian systems, and we attempt to do so here by analogy; a more ambitious extension of RNG ideas to systems far from equilibrium, including diffusive systems, has been presented in refs. 5-7. We work here with the specific example of the Korteveg-deVries-Burgers (KdVB) equation, with boundary conditions that give rise to traveling waves. The problem has a single dimensionless parameter that we call R (to highlight an analogy with the Reynolds number of fluid mechanics). For large values of R the traveling waves are highly oscillatory; we consider a spatial (not ensemble) average of these traveling waves and look for an equation whose solution approximates this average (see Fig. 1). This problem was chosen because of its apparent simplicity, because previous work by Barenblatt et al. (8,9) has suggested interesting conjectures about its solution and also because of interesting connections with fluid mechanics.In the following sections I describe the problem, review the equilibrium averaging theory for Hamiltonian systems, set up the analogy for the KdVB equation, and provide a numerical analysis of its validity. I also compare the numerical results to earlier analytical work and point out interesting scaling relations that emerge from the calculations. Note that the averaging͞ renormalization presented here is applied to stationary solutions; the extension to...

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On the structure of wave fronts in nonlinear dissipative media

Cited by 16 publications

References 5 publications

Viscosity-dependent inertial spectra of the Burgers and Korteweg–deVries–Burgers equations

Viscosity-dependent inertial spectra of the Burgers and Korteweg–deVries–Burgers equations

Prediction from Partial Data, Renormalization, and Averaging

Averaging and renormalization for the Korteveg–deVries–Burgers equation

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