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Shape optimisation for faster washout in recirculating flows. (English) Zbl 1461.76583

Summary: 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 [J. G. Williams et al., ibid. 902, Paper No. A16, 26 p. (2020; Zbl 1460.76960)]. 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 \(\det \nabla \boldsymbol{u} >0\) [J. Jeong and F. Hussain, ibid. 285, 69–94 (1995; Zbl 0847.76007)]. Integrating a smooth form of \(\max (0, \det \boldsymbol{\nabla }\boldsymbol{u})\) over the domain provides an objective to minimise recirculation zones [H. Kasumba and K. Kunisch, Comput. Optim. Appl. 52, No. 3, 691–717 (2012; Zbl 1258.49070)]. 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.

MSC:

76Z05 Physiological flows
76D55 Flow control and optimization for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics

Software:

Fireshape; COFFEE; TSFC; PETSc
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