Optimal Flow Control and Topology Optimization Using the Continuous Adjoint Method in Unsteady Flows

Abstract

This paper presents the development and application of the unsteady continuous adjoint method to the incompressible Navier-Stokes equations and its use in two different optimization problems. The first is the computation of the optimal setting of a flow control system, based on pulsating jets located along the surface of a square cylinder, in order to minimize the time-averaged drag. The second is dealing with unsteady topology optimization of a duct system with four fixed inlets and a single outlet, with periodic in time inlet velocity profiles, where the target is to minimize the time-averaged viscous losses. The presentation of the adjoint formulation is kept as general as possible and can thus be used to other optimization problems governed by the unsteady Navier-Stokes equations. Though in the examined problems the flow is laminar, the extension to turbulent flows is doable.

Aerodynamic Shape Optimization Using “Turbulent” Adjoint And Robust Design in Fluid Mechanics

“…In unsteady flows, the adjoint method, [17], can also be used to compute the optimal characteristics of unsteady jets, such as pulsating or oscillating ones. Such an application, where the optimal amplitudes of pulsating jets have been computed using the unsteady continuous adjoint method is presented in Fig.…”

On the incremental singular value decomposition method to support unsteady adjoint‐based optimization

“…In the present work, all jets have the same frequency and phase, thus, only the jet amplitudes A m are considered as design variables. Positive and negative A m correspond to blowing and suction, respectively …”

“…Positive and negative A m correspond to blowing and suction, respectively. 39,40 Following the derivation described in Section 2 the sensitivity derivatives expression reads…”

On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization

“…The problem becomes much more pronounced in unsteady adjoint solvers, [5,6], 30 in which, for a time-averaged objective function, the SD computation must be repeated for each and every time-step. In the same problem, the cost of SI is still negligible.…”

“…Both result to the same adjoint field equations and boundary conditions, with, however, two distinct expressions for the sensitivity derivatives (SD) of the objective function J with respect to (w.r.t.) 5 the design variables b. The first published formulation, [1], led to SD expressions including field integrals (FI) of the variations in the coordinates x w.r.t.…”