Steady advection-diffusion in multiply-connected potential flows

Abstract

We consider the steady heat transfer between a collection of impermeable obstacles immersed in an incompressible 2D potential flow, when each obstacle has a prescribed boundary temperature distribution. Inside the fluid, the temperature satisfies a variable-coefficient elliptic partial differential equation (PDE), the solution of which usually requires expensive techniques. To solve this problem efficiently, we construct multiply-connected conformal maps under which both the domain and governing equation are greatly simplified. In particular, each obstacle is mapped to a horizontal slit and the governing equation becomes a constant-coefficient elliptic PDE. We then develop a boundary integral approach in the mapped domain to solve for the temperature field when arbitrary Dirichlet temperature data is specified on the obstacles. The inverse conformal map is then used to compute the temperature field in the physical domain. We construct our multiply-connected conformal maps by exploiting the flexible and highly accurate AAA-LS algorithm. In multiply-connected domains and domains with non-constant boundary temperature data, we note similarities and key differences in the temperature fields and Nusselt number scalings as compared to the isothermal simply-connected problem analyzed by Choi et al. (2005). In particular, we derive new asymptotic expressions for the Nusselt number in the case of arbitrary non-constant temperature data in singly connected domains at low P\'eclet number, and verify these scalings numerically. While our language focuses on the problem of conjugate heat transfer, our methods and findings are equally applicable to the advection-diffusion of any passive scalar in a potential flow.