Ohkitani, K. (2020) Study of the Hopf functional equation for turbulence: Duhamel principle and dynamical scaling. Physical Review E, 101. ISSN 1539-3755
Abstract
We consider a formulation for the Hopf functional differential equation which governs statistical solutions of the Navier-Stokes equations. By introducing an exponential operator with a functional derivative, we recast the Hopf equation as an integro-differential functional equation by the Duhamel principle. On this basis we introduce a successive approximation to the Hopf equation. As an illustration we take the Burgers equation and carry out the approximations to the leading order. Scale-invariance of the statistical Navier-Stokes equations in d-dimensions is formulated and contrasted with that of the deterministic Navier-Stokes equations. For the statistical Navier-Stokes equations, critical scale-invariance is achieved for the characteristic functional of the d-th derivative of the vector potential in d-dimensions. The deterministic equations corresponding to this choice of the dependent variable acquire the linear Fokker-Planck operator under dynamic scaling. In three dimensions it is the vorticity gradient that behaves like a fundamental solution (more precisely, source-type solution) of deterministic Navier-Stokes equations in the long-time limit. Physical applications of these ideas include study of a self-similar decaying profile of fluid flows. Moreover, we reveal typical physical properties in the late stage evolution by combining statistical scale-invariance and the source-type solution. This yields an asymptotic form of the Hopf functional in the long-time limit, improving the well-known Hopf-Titt solution. In particular, we present analyses for the Burgers equations to illustrate the main ideas and indicate a similar analysis for the Navier-Stokes equations.
Metadata
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Copyright, Publisher and Additional Information: | © 2020 American Physical Society. This is an author-produced version of a paper subsequently published in Physical Review E. Uploaded in accordance with the publisher's self-archiving policy. | ||||
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Institution: | The University of Sheffield | ||||
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) | ||||
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Depositing User: | Symplectic Sheffield | ||||
Date Deposited: | 18 Dec 2019 14:12 | ||||
Last Modified: | 09 Jan 2020 08:42 | ||||
Status: | Published | ||||
Publisher: | American Physical Society | ||||
Refereed: | Yes | ||||
Identification Number: | https://doi.org/10.1103/PhysRevE.101.013104 |