Hampering G\"{o}rtler vortices via optimal control in the framework of nonlinear boundary region equations

TL;DR

This paper develops an optimal control method within a boundary layer framework to effectively reduce G"{o}rtler vortices in high Reynolds number flows over concave surfaces, aiming to delay transition to turbulence.

Contribution

It introduces a novel optimal control approach using boundary region equations to suppress G"{o}rtler vortices via wall displacement or transpiration control.

Findings

Optimal control significantly reduces vortex amplitude.

Wall displacement control is particularly effective.

Method delays laminar-turbulent transition.

Abstract

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 G\"{o}rtler 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 G\"{o}rtler vortices, especially for the control based on wall displacement.