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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 high-precision 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 wave-numbers 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., Comment: 23 pages; 6 figures

We study flocking of self-propelled, interacting microorganisms with finite sizes and mass immersed in a turbulent flow. In the presence of the competing interactions of self-propulsion and the carrier turbulent flow, as is typical in nature, we show that including the effect of inertia is essential for the stability of flocks. We examine the problem from the point of view of global as well as local order and the statistics of the velocity of the microorganisms as a function of the inertia, the interaction radius, the level of self-propulsion as well as noise.

We systematize the study of dynamic multiscaling of time-dependent structure functions in different models of passive-scalar and fluid turbulence. We show that, by suitably normalizing these structure functions, we can eliminate their dependence on the origin of time at which we start our measurements and that these normalized structure functions yield the same linear bridge relations that relate the dynamic-multiscaling and equal-time exponents for statistically steady turbulence. We show analytically, for both the Kraichnan model of passive-scalar turbulence and its shell model analogue, and numerically, for the Gledzer–Ohkitani–Yamada (GOY) shell model of fluid turbulence and a shell model for passive-scalar turbulence, that these exponents and bridge relations are the same for statistically steady and decaying turbulence. Thus, we provide strong evidence for dynamic universality, i.e. dynamic-multiscaling exponents do not depend on whether the turbulence decays or is statistically steady.