TY - JOUR

T1 - Hampering Görtler vortices via optimal control in the framework of nonlinear boundary region equations

AU - Sescu, Adrian

AU - Afsar, Mohammed

PY - 2018/4/14

Y1 - 2018/4/14

N2 - The control of stream-wise vortices in high Reynolds number boundary layer flows often aims at reducing the vortex energy as a means of mitigating the growth of secondary instabilities, which eventually delay the transition from laminar to turbulent flow. In this paper, we aim at utilizing such an energy reduction strategy using optimal control theory to limit the growth of Görtler vortices developing in an incompressible laminar boundary layer flow over a concave wall, and excited by a row of roughness elements with span-wise separation in the same order of magnitude as the boundary layer thickness. Commensurate with control theory formalism, we transform a constrained optimization problem into an unconstrained one by applying the method of Lagrange multipliers. A high Reynolds number asymptotic framework is utilized, wherein the Navier-Stokes equations are reduced to the boundary region equations (BRE), in which wall deformations enter the problem through an appropriate Prandtl transformation. In the optimal control strategy, the wall displacement or the wall transpiration velocity serve as control variables, while the cost functional is defined in terms of the wall shear stress. Our numerical results indicate, among other things, that the optimal control algorithm is very effective in reducing the amplitude of the Görtler vortices, especially for the control based on wall displacement.

AB - The control of stream-wise vortices in high Reynolds number boundary layer flows often aims at reducing the vortex energy as a means of mitigating the growth of secondary instabilities, which eventually delay the transition from laminar to turbulent flow. In this paper, we aim at utilizing such an energy reduction strategy using optimal control theory to limit the growth of Görtler vortices developing in an incompressible laminar boundary layer flow over a concave wall, and excited by a row of roughness elements with span-wise separation in the same order of magnitude as the boundary layer thickness. Commensurate with control theory formalism, we transform a constrained optimization problem into an unconstrained one by applying the method of Lagrange multipliers. A high Reynolds number asymptotic framework is utilized, wherein the Navier-Stokes equations are reduced to the boundary region equations (BRE), in which wall deformations enter the problem through an appropriate Prandtl transformation. In the optimal control strategy, the wall displacement or the wall transpiration velocity serve as control variables, while the cost functional is defined in terms of the wall shear stress. Our numerical results indicate, among other things, that the optimal control algorithm is very effective in reducing the amplitude of the Görtler vortices, especially for the control based on wall displacement.

KW - Görtler vortices

KW - Rynolds number

KW - boundary layers

UR - https://www.cambridge.org/core/journals/journal-of-fluid-mechanics

M3 - Article

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -