An adjoint method for control of liquid-gas flows using a sharp interface model

We present a computational, continuous adjoint framework for the control of liquid-gas flows using a sharp interface model. The two-phase Navier–Stokes equations are solved using a mass-conserving geometric Volume-of-Fluid method, while the adjoint equations consider a level set-based representation of the interface. To facilitate the accurate transport of a surface adjoint variable, a geometric surface transport method is formulated and applied. We verify our method by comparing adjoint-calculated gradients against analytical gradients or finite difference approximations. The method is then applied to a variety of benchmark two-phase flow problems, including the multi-dimensional inflow control of droplet position and optimal control of the initial velocity profile in a temporally evolving liquid-gas mixing layer.