Nonlinear Dynamics of the Lid-Driven Cavity

 

This famous flow problem acts as an ideal testing environment for 1) a researcher to become familiar with turbulent flows and 2) novel model-order reduction techniques to be tested. The problem itself if physically quite simple: a box with the top open. The unambiguous boundary conditions


of this control volume allow for extensive comparisons between computational and experimental data. It is also a flow regime that can exhibit virtually all naturally occurring incompressible phenomena, given the right initial conditions. The reader is directed to this Annual Review by Shankar and Deshpande for a more comprehensive summary of the problem.

The above video is of a 2D lid-driven cavity, simulated using different methods, at a Reynolds number of 6,000: a laminar flow regime. The leftmost animation is constructed using direct numerical simulation: an “exact” solution of the Navier-Stokes equations for this problem. This flow data is then used to create a series of POD modes which represent the dominant coherent structures in the flow (used to generate the middle animation). These POD modes are then used to create an integrated reduced-order model of the flow, as seen in the animation on the right. The progression from the middle to the right animation can happen in a number of ways. Currently, this research applies an energy balancing algorithm to ensure that the modes in the reduced-order model contain as much energy as the original flow.

The below video shows how sensitive the reduced-order model creation can be. It was generated using the same procedure as described above, but at a Reynolds number of 20,000: a chaotic flow regime. Although the POD modes seem to adequately capture the coherent structures of this more chaotic flow, creating a reduced-order model from these modes can be a challenging step.

Current research is using step and impulse perturbations in converged flows (including limit-cycle oscillations at higher Reynolds numbers) to study the damping behavior of the responses at different Reynolds numbers. It is hypothesized that trends in the flow’s damping behavior can be used to predict a flow’s transition from laminar to turbulent. If this can be substantiated for the lid-driven cavity, then the experiment can be run with other flow scenarios, including airfoils, to see if turbulence can be predicted with only laminar flow data.

 

Appreciation is extended to Dr. Maciej Balajewicz for developing the software applied in this research.