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Direct control of the small-scale energy balance in two-dimensional fluid dynamics

Published online by Cambridge University Press:  07 October 2015

Jason Frank ,
Benedict Leimkuhler  and
Keith W. Myerscough
Jason Frank
Affiliation:
Mathematical Institute, Utrecht University, PO Box 80010, 3508 TA Utrecht, The Netherlands
Benedict Leimkuhler
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Edinburgh EH9 3JZ, UK
Keith W. Myerscough*
Affiliation:
Centrum Wiskunde and Informatica, PO Box 94079, 1090 GB Amsterdam, The Netherlands
*
†Email address for correspondence: keith@myerscough.nl
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Abstract

We explore the direct modification of the pseudo-spectral truncation of two-dimensional, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theoretical prediction or high-fidelity simulation. In the scheme outlined here, Nosé–Hoover thermostats, commonly used in molecular dynamics, are introduced as feedback controls applied to energy shells of the Fourier-discretized Navier–Stokes equations. As we demonstrate in numerical experiments, the dynamical properties (quantified using autocorrelation functions) are only modestly perturbed by our device, while ensemble dispersion is significantly enhanced compared with simulations of a corresponding truncation incorporating hyperviscosity.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Mathematical Foundations: Computational methods Turbulent Flows: Homogeneous turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 782 , 10 November 2015 , pp. 240 - 259
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Copyright
© 2015 Cambridge University Press 

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Frank et al. supplementary movie

A movie showing the evolution and chaotic divergence of the turbulent simulations. Both rows feature simulation results using (left) the model with added Hyperviscosity, (centre) the Reference simulation, and (right) the Nos ́e-Hoover method as indicated with † in table 1. The initial conditions are the same for all of the simulations in one row, but slightly different between top and bottom rows. Compare with Figure 4 in the paper.

Download Frank et al. supplementary movie(Video)
Video 4.4 MB