A key challenge in the simulations of turbulent flows is modeling the effect of the smallest scales on the large scales of motion. The large-scale dynamics are really of interest, but in many cases it is difficult to obtain a closed form of the equations governing them.
For accelerating simulations of multi-scale dynamical systems, Yannis Kevrekidis, Bill Gear and his co-workers have developed an “equation-free” approach over the last decade or so, and have applied it to many different areas. In this approach, given a resolved or “fine” numerical solver, scientific computing tasks (such as bifurcation or stability analysis) are performed at the large or “coarse” scales, without ever deriving the governing equations in a closed form.
We applied the method to a simple model of turbulence, the 1D Burgers equation with near-vanishing viscosity and with random forces acting at the small scales. The evolution of its velocity field was stochastic, however that of the ensemble-averaged energy spectrum had a definite structure. We effectively solved for the evolution of the spectrum using appropriate calls to the direct simulator, and obtained a computational speed-up factor of around 4. The evolution of higher moments was shown to have much smaller relaxation time, and thus did not couple to the evolution of the spectrum. The evolution of the energy spectrum at the largest scales was shown to be self-similar, and we could also compute the self-similar shape and the corresponding exponents.
S. Ahuja, V. Yakhot and I. G. Kevrekidis. Computational coarse graining of a randomly forced 1-D Burgers equation. Physics of Fluids, 20, 035111, 2008; arxiv/0707.0510. (pdf)