Analytic structure of solutions of the one-dimensional Burgers equation with modified dissipation

Abstract

We use the one-dimensional Burgers equation to illustrate the effect of replacing the standard Laplacian dissipation term by a more general function of the Laplacian -of which hyperviscosity is the best known example -in equations of hydrodynamics. We analyze the asymptotic structure of solutions in the Fourier space at very high wave-numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing Reynolds numbers from algebraic recursion relations involving iterated integrations. We give a detailed analysis of their analytic structure for two different types of dissipation: a hyperviscous and an exponentially growing dissipation term. Our results, obtained in the limit of vanishing Reynolds numbers, are validated by highprecision numerical simulations at non-zero Reynolds numbers. We then study the bottleneck problem, an intermediate asymptotics phenomenon, which in the case of the Burgers equation arises when ones uses dissipation terms (such as hyperviscosity) growing faster at high wavenumbers than the standard Laplacian dissipation term. A linearized solution of the well-known boundary layer limit of the Burgers equation involving two numerically determined parameters gives a good description of the bottleneck region.PACS numbers: 47.27.-i, 82.20.-w, 47.51.+a, 47.55.df IntroductionThe physics of a fluid in a turbulent state is multiscale. Hence, it is convenient to study turbulence by separating the scales into energy injection L, inertial r, and dissipation η ranges [1]. Such a classification has proved useful, both theoretically and numerically, to develop models which mimic such scales. These models have the advantage of being less complex than the original system and hence, more tractable. Indeed, we owe much of our understanding of the physics and mathematics of turbulent flows, validated by experiments, observations and detailed simulations, to such reduced models. The most celebrated example of this is the tremendous advance made within the framework of three-dimensional, homogeneous, isotropic turbulence (in the limit of vanishing kinematic viscosity ν). Such a ‡ Author to whom all correspondence should be addressed.Analytic structure of solutions of the one-dimensional Burgers equation with modified dissipation2 framework, which ignores the specific details of the forcing and dissipation mechanisms, has yielded several important and universal results [2] for the inertial scale and forms the basis of Kolmogorov's seminal work in 1941 [3]. In particular, the most important results stemming from such a model are those related to 2-point correlation functions, multiscaling, and their universality [2,4,5].Despite the success in understanding the physics of the inertial range, the theoretical underpinnings of the model of homogeneous, isotropic turbulence are largely irrelevant for questions related to the regularity of such flows. This is because these questions -which still rank amongst the most profound and fundament...

Thermal noise competes with turbulent fluctuations below millimeter scales

“…This neglect is commonly justified by the idea that there is a strong separation between hydrodynamic scales dominated by turbulent fluctuations and extremely small scales of order the mean-free path length where thermal fluctuations begin to play a role [17]. The smallest turbulent fluid scales below which eddies are damped by viscosity, known as the dissipation range, typically occurs at millimeter scales and there is currently intensive effort, by computation [18][19][20], theory [21][22][23] and experiment [19,24,25], to understand the dynamics and statistics at these scales. The motivation ranges from unraveling the nature of turbulence itself, and the associated unsolved issue of formation of singularities [26], to the phenomena and practical applications mentioned above and others.…”