Shape optimisation for faster washout in recirculating flows

How to design an optimal biomedical flow device to minimise trapping of undesirable biological solutes/debris and/or enhance their washout is a pertinent but complex question.
While biomedical devices often utilise externally-driven flows to enhance washout, the
presence of vortices – arising as a result of fluid flows within cavities – hinder washout
by trapping debris. Motivated by this, we solve the steady, incompressible Navier–Stokes
equations for flow through channels into and out of a 2D cavity. In endourology, the
presence of vortices – enhanced by flow symmetry breaking – has been linked to long
washout times of kidney stone dust in the renal pelvis cavity, with dust transport
modelled via advection and diffusion of a passive tracer (Williams et al. 2020). Here
we determine the inflow and outflow channel geometries that minimise washout times.
For a given flow field u, vortices are characterised by regions where det ∇u > 0 (Jeong
& Hussain 1995). Integrating a smooth form of max(0, det ∇u) over the domain provides
an objective to minimise recirculation zones (Kasumba & Kunisch 2012). We employ
adjoint-based shape optimisation to identify inflow and outflow channel geometries that
reduce this objective. We show that a reduction in the vortex objective correlates with
reduced washout times. We additionally show how multiple solutions to the flow equations
lead to solution branch-switching during the optimisation routine by characterising the
change in solution bifurcation structure with the change in scope tip geometry.