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 two-dimensional 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., J. Fluid Mech., vol. 902, 2020, A16). Here, we determine the inflow and outflow channel geometries that minimise washout times. For a given flow field $\boldsymbol {u}$, vortices are characterised by regions where $\det \boldsymbol {\nabla }\boldsymbol {u} > 0$ (Jeong & Hussain, J. Fluid Mech., vol. 285, 1995, pp. 69–94). Integrating a smooth form of $\max (0, \det \boldsymbol {\nabla }\boldsymbol {u})$ over the domain provides an objective to minimise recirculation zones (Kasumba & Kunisch, Comput. Optim. Appl., vol. 52, 2012, pp. 691–717). 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 inflow/outflow channel geometry.