# Optimal transient growth in an incompressible flow past a backward-slanted step

## TL;DR

This paper investigates the transient growth of disturbances in incompressible flow past a backward-slanted step, combining linear stability analysis and optimal control to identify maximal energy amplification.

## Contribution

It introduces a novel approach to analyze and control transient growth phenomena in flow separation over a backward-slanted step, relevant for aerodynamic drag reduction.

## Key findings

- Achieved a kinetic-energy gain of about one million.
- Developed a procedure for physically-realizable disturbances.
- Performed linear stability and optimal control analysis.

## Abstract

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 focusing on a specific aerodynamic geometry, namely a backward-slanted step at 25 degrees 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-Helmholtz instabilities. A linear stability analysis is performed, and an optimal control problem with a steady volumic forcing is tackled by means of variational formulation, adjoint method, penalization scheme and 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 of the order of one million with respect to the power introduced by the external forcing.