Fractional hyperviscosity induced growth of bottlenecks in energy spectrum of Burgers equation solutions

Banerjee, Debarghya

doi:10.1140/epjb/e2019-90751-4

Cited by 3 publications

(2 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…x and u 0 (x) = Ae ix obtained by using Faà di Bruno's formula (28) where the second sum is taken over k − l + 1 nonnegative integers j 1 , ..., j k−l+1 such that…”

Section: Solutions Of the One-dimensional Burgers Equation For Comple...mentioning

confidence: 99%

“…The bottleneck has been shown to become more pronounced with increasing α, see, e.g. [24][25][26][27][28] as well as [29] for a review, allowing for theoretical calculations, in the large α limit to be checked against numerical simulations. It is also important to remember that the use of hyperviscosity has shed light on the problem of finite-time blow-up: it was shown [12,30] that there is no finite-time blow-up for α > 5/4 despite the existence of complex singularities.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Pauls

Ray

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

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...

show abstract

“…x and u 0 (x) = Ae ix obtained by using Faà di Bruno's formula (28) where the second sum is taken over k − l + 1 nonnegative integers j 1 , ..., j k−l+1 such that…”

Section: Solutions Of the One-dimensional Burgers Equation For Comple...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Pauls

Ray

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

The turbulent cascade of inertia-gravity waves in rotating shallow water

Thomas,

Rajpoot,

Gupta

2024

J. Fluid Mech.

View full text Add to dashboard Cite

In this work we study features of inertia-gravity wave turbulence in the rotating shallow water equations. On examining the dynamics of waves with varying rotation rates, we find that the turbulent cascade of waves is strongest at low rotation rates, forming a $k^{-2}$ energy spectrum, and a rich distribution of shocks in physical space. At high rotation rates, the forward cascade of waves weakens along with a steeper energy spectra and vanishing of shocks in physical space. The wave cascade is seen to be scale-local, resulting in a noticeable time interval for energy to get transferred from domain scale to dissipative scale. Furthermore, we find that the vortical flow has a non-negligible effect on the wave cascade, especially at high rotation rates. The vortical flow assists in the forward cascade of waves and shock formation at high rotation rates, while the waves by themselves in the absence of the vortical flow lack a forward cascade and shock formation at such high rotation rates. On investigating the physical space structures in the vortical flow and their connections to the wave cascade, we find that strain-dominant regions, that are located around the boundaries of coherent vortices, are the physical space regions that contribute majorly to the forward cascade of waves. Our results in general highlight intriguing features of dispersive inertia-gravity wave turbulence that are qualitatively similar to those seen in three-dimensional homogeneous isotropic turbulence and are beyond the predictions of asymptotic resonant wave interaction theory.

show abstract