Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 469, Issue 2153, Pages -Publisher
ROYAL SOC
DOI: 10.1098/rspa.2012.0550
Keywords
finite-dimensional attractors; transitional flows; stochastic reduction; nonlinear dimensionality; nonlinear energy transfer; dynamical orthogonality
Categories
Ask authors/readers for more resources
We examine the geometry of the finite-dimensional attractor associated with fluid flows described by Navier-Stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components (modes) of the flow. Specifically, we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order-reduction and describe the stochastic attractor in the reduced-order phase space in terms of the associated probability measure. We introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite-time Lyapunov exponents. Subsequently, we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms. In particular, we illustrate the connection of the low-dimensionality of the attractor with the circulation of energy: (i) from the mean flow to the unstable modes (due to their linearly unstable character), (ii) from the unstable modes to the stable ones (due to a nonlinear energy transfer mechanism) and (iii) from the stable modes back to the mean (due to the linearly stable character of these modes).
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available