research: Computational Fluid Dynamics

Coarse Projective Integration for Model Turbulent Flows

By implementing and applying a class of recently-proposed methods (the so-called equation-free multiscale (EFM) methods), and, in particular, a technique called coarse projective integration (CPI), we hope to develop a better computer-assisted understanding of certain types of turbulence, and to achieve a significant savings in computational time for turbulence simulations.

In particular, we focus on scaling laws for the energy spectrum. An explicit equation for the evolution of the expected energy spectrum of a turbulent flow cannot, in general, be written down; yet we propose to numerically construct such an equation '"on the fly" by processing the results of short bursts of appropriately initialized ensembles of Navier-Stokes (or possibly LES or RANS) simulations.

Processing the results of these simulations allows us to estimate the right-hand side (the time derivative) of the unavailable-in-closed-form spectrum evolution equation, as well as the action of its linearization (Jacobian).

With these estimates we can implement both numerical integration (so-called projective integration) and fixed-point (e.g. Newton-Raphson) techniques to simulate the unavailable spectrum evolution in time, and find its stationary (and possibly even self-similar) states.

These methods have the potential to lead to a significant  savings in computational time compared to direct simulation.  Preliminary work with Burger's equation (see below) suggests that this savings could be on the order of 10² for a three-dimensional velocity field.

See a recent presentation on coarse projective integration for model turbulent flows here.

Go to the website of Graduate Student Anne Staples.

This work is in collaboration with Prof. Yannis Kevrekidis in the Department of Chemical Engineering at Princeton, and Prof.
Leslie Smith in the Department of Mathematics at the University of Wisconsin.

Burger's equation integrated in two ways: with a standard 3rd-order Runge-Kutta (RK3) time integration scheme (green curve), and with a hybrid Forward Euler-RK3 projective integration scheme (red curve). The blue curve is the initial condition (multiplied by 2000). The saving in CPU time for the hybrid FE-RK3 scheme was a factor of about 4.7.