Optimal Control of Bypass Transition

Authors: Högberg, M., Henningson, D.S.H., Berggren, M.
Document Type: Conference
Pubstate: Published
Journal: Advances in Turbulence VIII. (2000) Proc. 8th European Turbulence Conference, Barcelona, Spain
Volume:    205-208
Year: 2000


Transition to turbulence is an ambiguous process, and since knowledge about the underlying non-linear mechanism is limited, it is difficult to design an efficient control. Successful attempts have been made to control the linear mechanisms involved in transition scenarios such as the growth of Tollmien-Schlichting waves. More complicated processes leading to transition such as the bypass transition scenarios with non-linear effects are not as intuitive and therefore more difficult to control. The goal of the current work is to determine how to control such processes in the optimal way given the method of controlling the flow, and an objective function describing the features of the flow to be controlled. There are several different possibilities to affect the flow. The method chosen here is blowing and suction at the walls, since it is a fairly simple way of acting on the flow, and also because it is a technique that is widely used. Blowing and suction has successfully been used for similar problems, namely control of turbulence, where complete relaminarization was obtained Bewley et al. JFM to appear. The blowing and suction is applied to flow in a channel, where we can find many of the interesting bypass transition scenarios. Optimization of the control is not a trivial issue. Direct search methods are costly, requiring a lot of computations, and often inefficient for non-linear processes. Methods where one can access the gradient of the objective function are more efficient, and there are several algorithms developed for this type of optimization. To compute gradients of the objective function the adjoint equation approach is used. It is an efficient method in the sense that only two computations are required for each optimization iteration independent of the number of degrees of freedom of the control. First the state equation ( Navier--Stokes ) is solved and then this solution is used as input to the adjoint equation that is solved next and gives the gradient of the objective function. There is no need to have any knowledge of the underlying physics nor of the functional behavior of the control. After computing the optimal control for a specific transition scenario, it is possible to analyze the computed control and perhaps gain some insight to the physics of the transition process. The transition scenario we study here is oblique transition in channel flow, since it contains many different stages, and is highly nonlinear. The transition threshold for this scenario has been thoroughly studied in Reddy et al. JFM 1998 Optimization is performed with a limited memory quasi Newton method described in Byrd,Lu and Nocedal (1994)