Flow and passive transport in planar multipolar flows

Abstract

We study the flow and transport of heat or mass, modelled as passive scalars, within a basic geometrical unit of a three-dimensional multipolar flow – a triangular prism – characterised by a side length $L$, a normalised thickness $0.01\leqslant \unicode[STIX]{x1D700}\leqslant 0.1$ and an apex angle $0<\unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}$, and connected to inlet and outlet pipes of equal normalised radius $0.01\leqslant \unicode[STIX]{x1D6FF}\leqslant 0.1$ perpendicularly to the plane of the flow. The flow and scalar fields are investigated over the range $0.1\leqslant Re_{p}\leqslant 10$ and $0.1\leqslant Pe_{p}\leqslant 1000$, where $Re_{p}$ and $Pe_{p}$ are respectively the Reynolds and Péclet numbers imposed at the inlet pipe when either a Dirichlet ($\text{D}$) or a Neumann ($\text{N}$) scalar boundary condition is imposed at the wall unattached to the inlets and outlets. A scalar no-flux boundary condition is imposed at all the other walls. An axisymmetric model is applied to understand the flow and scalar transport in the inlet and outlet regions, which consist of a turning region close to the pipe centreline and a channel region away from it. A separate two-dimensional model is then developed for the channel region by solving the integral form of the momentum and scalar advection–diffusion equations. Analytical relations between geometrical, flow and scalar transport parameters based on similarity and integral methods are generated and agree closely with numerical solutions. Finally, three-dimensional numerical calculations are undertaken to test the validity of the axisymmetric and depth-averaged analyses. Dominant flow and scalar transport features vary dramatically across the flow domain. In the turning region, the flow is a largely irrotational straining flow when $\unicode[STIX]{x1D700}\geqslant \unicode[STIX]{x1D6FF}$ and a dominantly viscous straining flow when $\unicode[STIX]{x1D700}\ll \unicode[STIX]{x1D6FF}$. The thickness of the scalar boundary layer scales to the local Péclet number to the power $1/3$. The diffusive flux $j_{d}$ and the scalar $C_{s}$ at the wall where ($\text{D}$) or ($\text{N}$) is imposed, respectively, are constant. In the channel region, the flow is parabolic and dominated by a source flow near the inlet and an irrotational straining flow away from it. When $(\text{D})$ is imposed the scalar decreases exponentially with distance from the inlet and the normalised scalar transfer coefficient converges to $\unicode[STIX]{x1D6EC}_{\infty }=2.5694$. When $(\text{N})$ is imposed, $C_{s}$ varies proportionally to surface area. Transport in the straining region downstream of the inlet is diffusion-limited, and $j_{d}$ and $C_{s}$ are functions of the geometrical parameters $L$, $\unicode[STIX]{x1D700}$, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$. In addition to describing the fundamental properties of the flow and passive transport in multipolar configurations, the present work demonstrates how geometrical and flow parameters should be set to control transfers in the different regions of the flow domain.