Optimal Perturbations

Classical hydrodynamic linear stability theory is based on an eigenvalue (normal mode) analysis of the linearized equations of motion, (including the continuity equation) and centers on the characteristics of the fastest growing normal mode. A flow is said unstable if there exists a normal mode with a positive growth rate for the given mean conditions. Nonlinear effects will either instigate a transition of the flow to disorganized turbulence or allow the perturbation to modify the mean flow enough to generate an equilibrium secondary state, which can undergo a further, second instability and successive breakdown to turbulence.

Recently, nonmodal perturbations have been shown to have the potential for large transient energy growth compared to normal modes for a variety of mean flows. The nonmodal stability analysis is based on the fact that for most flows the linear stability equations are not selfadjoint (the eigenfunctions are not orthogonal). We can consider expanding an arbitrary initial perturbation of unit energy using the normalized eigenfunctions as a basis set. Because the basis set is not orthonormal, we will in general have an initial cancellation of contributions between certain combinations of eigenfunctions. However, because the contributions from the different eigenfunctions grow or decay at different rates and have different phase velocities, the initial cancellation of the contributions from nearly parallel eigenfunctions will be removed as space (or time) increases. This could (and typically does) allow transient growth in the energy of the perturbation. Transient growth can occur in subcritical conditions for which no eigenvalue has a positive growth rate. In supercritical conditions, this transient growth can greatly exceed the exponential growth of the unstable normal mode over a finite time interval.

Optimal perturbations provide an upper bound for this energy growth, as they are defined as the the initial conditions for the boundary layer equautions, that cause the highest energy growth (from which the term "optimal"). These initial conditions are optimal in the sense that at the end of any chosen space (or time) domain, no other perturbation will achieve a larger increase in energy density.

For more details, see the following related publications.

Zuccher, S., Tumin, A. & Reshotko, E. 2006 Parabolic Approach to Optimal Perturbations in Compressible Boundary Layers. Journal of Fluid Mechanics, 556, 189-216.

Zuccher, S., Shalaev, I., Tumin, A. & Reshotko, E. 2006 Optimal Disturbances in the Supersonic Boundary Layer Past a Sharp Cone. AIAA Paper 2006-1113.

Zuccher, S., Tumin, A. & Reshotko, E. 2005 Optimal Disturbances in Compressible Boundary Layers - Complete Energy Norm Analysis. AIAA Paper 2005-5314.

Zuccher, S., Luchini, P. & Bottaro A. 2004 Algebraic Growth of Blasius Boundary Layer: Optimal and Robust Control in the Nonlinear Regime. Journal of Fluid Mechanics 513, 135-160.

Zuccher, S., Bottaro A., & Luchini, P. 2006 Algebraic growth in a Blasius boundary layer: Nonlinear optimal disturbances. European Journal of Mechanics B 25, 1-17.

Zuccher, S. & Luchini, P. 2002 Time-Dependent Optimal Perturbations for the Algebraic Instability in the Nonlinear Regime. In Proceedings of the 2002 ASME Fluids Engineering Division Summer Meeting, American Society of Mechanical Engineers, Fluids Engineering Division FED, Volume 257, Issue 1 B, 2002, Pages 1387-1393. Montreal, Quebec, Canada.

Zuccher, S. 2002 Receptivity and control of flow instabilities in a boundary layer, Ph.D thesis, Politecnico di Milano, Milano.