Optimal Transient Growth in an Incompressible Flow past a Backward-Slanted Step

With the aim of providing a first step in the quest for a reduction of the aerodynamic drag on the rear-end of a car, we study the phenomena of separation and reattachment of an incompressible flow by focusing on a specific aerodynamic geometry, namely a backward-slanted step at 25<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>∘</mo> </msup> </semantics> </math> </inline-formula> of inclination. The ensuing recirculation bubble provides the basis for an analytical and numerical investigation of streamwise-streak generation, lift-up effect, and turbulent-wake and Kelvin&#8315;Helmholtz instabilities. A linear stability analysis is performed, and an optimal control problem with a steady volumic forcing is tackled by means of a variational formulation, adjoint methods, penalization schemes, and an orthogonalization algorithm. Dealing with the transient growth of spanwise-periodic perturbations, and inspired by the need of physically-realizable disturbances, we finally provide a procedure attaining a kinetic-energy maximal gain on the order of <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>6</mn> </msup> </semantics> </math> </inline-formula>, with respect to the power introduced by the external forcing.