Topology optimization of roughness elements to delay modal transition in boundary layers

It is well understood that spanwise arrays of roughness elements can be used to generate steady streaks in boundary layers. This modulation of the boundary layer has the potential to attenuate the growth of Tollmien-Schlichting (TS) waves which can lead to the transition to turbulence in low turbulence intensity environments, such as those experienced by an aircraft's fuselage in atmospheric flight. This article applies density based topology optimization in order to design roughness elements capable of exploiting the aforementioned stabilizing effect as a means of passive flow control. The geometry of the roughness elements are represented using a Brinkman penalization when conducting Direct Numerical Simulations (DNS) to simulate the streaky boundary layer flow. Similarly, the unsteady linearized Navier-Stokes equations are evolved to assess the spatial growth of the TS waves across the flat plate. The optimization procedure aims to minimize the TS wave amplitude at a given downstream position while a novel constraint is used promoting a stable baseflow. The optimization problem is solved with gradient descent algorithms where the adjoint-variable method is used to compute gradients. This method has been applied to three initial material distributions yielding three distinct and novel designs capable of damping the downstream growth of the TS wave significantly more than a reference Minature Vortex Generator (MVG) of comparable size. The optimized designs and streaky baseflows they induce are then studied using an energy budget analysis and local stability analysis.