Flow and passive transport in planar multipolar flows
2019
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 6 e 6 0.1 and an
apex angle 0 < α < π, and connected to inlet and outlet pipes of equal normalised
radius 0.01 6 δ 6 0.1 perpendicularly to the plane of the flow. The flow and scalar
fields are investigated over the range 0.1 6 Rep 6 10 and 0.1 6 Pep 6 1000, where
Rep and Pep are respectively the Reynolds and Peclet numbers imposed at the inlet
pipe when either a Dirichlet (D) or a Neumann (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 e > δ and a dominantly viscous
straining flow when e � δ. The thickness of the scalar boundary layer scales to the
local Peclet number to the power 1/3. The diffusive flux jd and the scalar Cs at the
wall where (D) or (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 (D) is imposed the scalar decreases exponentially
with distance from the inlet and the normalised scalar transfer coefficient converges
to Λ∞ = 2.5694. When (N) is imposed, Cs varies proportionally to surface area.
Transport in the straining region downstream of the inlet is diffusion-limited, and
jd and Cs are functions of the geometrical parameters L, e, α and δ. 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.
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