Computational Mathematics Seminar

September 21, 3-4 pm, Thackeray 704

Speaker: Prof. William Layton

Similarity theory and truncation of scales in continuum models of turbulence

ABSTRACT:
This talk is based upon joint work with Ms Monika Neda.
Direct numerical simulation of a 3d turbulent flow typically requires
O(Re^{+9/4}) mesh points in space per time step, and thus is often not
computationally economical or even feasible. On the other hand, the
largest structures in the flow (containing most of the flow's energy)
are responsible for much of the mixing and most of the flow's momentum
transport. Thus, various numerical regularizations for truncating the
small structures and turbulence models of the large structures are used
for simulations seeking to predict flow statistics or averages. The
resulting simulations are typically complex with many uncertainties and
fitting/tuning parameters whose effects upon the computed solutions are
often poorly understood. Thus, it is important to understand how these
regularizations and models (and their parameters) act to truncate the
scales in a simulated flow to be represent-able on a computationally
feasible grid.
Turbulent flows consist of three dimensional eddies of various sizes. In
1941, I. Kolmogorov gave a remarkable, universal description of the
eddies in turbulent flow by combining a judicious mix of physical
insight, conjecture, mathematical analysis and dimensional analysis. In
his description, the largest eddies are deterministic in nature. Those
below a critical size are dominated by viscous forces, and die very
quickly due to these forces. This critical length scale (the Kolmogorov
micro-scale) is η =O(Re^{-3/4})
in 3d. From this estimate, it follows that direct numerical simulation
of a 3d flow thus requires x = y = z = O(Re^{-3/4}) giving O(Re^{+9/4})
mesh points in space per time step, and thus is often not
computationally economical or even feasible. This estimate is based upon
existence of an energy cascade in turbulent flow problems and
Kolmogorov's above estimate of the micro-scale at the bottom of the
energy cascade.
This talk will first give a very clear and simple presentation of the
K41 theory of homogeneous isotropic turbulence (aiming to be
understandable to general applied mathematicians). Next, it will
consider a numerical regularization and a turbulence model. Similarity
theory of those two continuum (PDE) models will be developed. This will
give precise insight into the way scales are truncated by both and
quantitative information about parameter selection in both.