An asymptotic theory for the high-Reynolds-number flow past a shear-free circular cylinder

TL;DR

This paper develops an asymptotic theory for high-Reynolds-number flow past a shear-free circular cylinder, accurately describing viscous effects, wake dynamics, and drag behavior, with results validated against simulations.

Contribution

It introduces a novel asymptotic framework capturing viscous modifications and wake structure in high-Reynolds-number shear-free flow, including a new boundary layer and vorticity transport analysis.

Findings

Viscous effects are localized and behave as linear perturbations.

Derived a boundary layer equation and self-similar solutions.

Found a logarithmic Reynolds number dependence of drag.

Abstract

We present an asymptotic theory for analytical characterization of the high-Reynolds-number incompressible flow of a Newtonian fluid past a shear-free circular cylinder. The viscosity-induced modifications to this flow are localized and except in the neighborhood of the rear stagnation point, behave like a linear perturbation of the inviscid flow. Our theory gives a highly accurate description of these modifications by including the contribution from the most significant viscous term in a correctional perturbation expansion about an inviscid base state. We derive the boundary layer equation for the flow and deduce a similarity transformation that leads to a set of infinite, shear-free-condition-incompatible, self-similar solutions. By suitably combining members from this set, we construct an all-boundary-condition-compatible solution to the boundary layer equation. We derive the governing equation for vorticity transport through the narrow wake region and determine its closed-form solution. The near and far field forms of our wake solution are desirably consistent with the boundary layer solution and the well-known, self-similar planar wake solution, respectively. We analyze the flow in the rear stagnation region by formulating an elliptic partial integro-differential equation for the distortion streamfunction that specifically accounts for the fully nonlinear and inviscid dynamics of the viscous correctional terms. The drag force and its atypical logarithmic dependence on Reynolds number, deduced from our matched asymptotic analysis, are in remarkable agreement with the high-resolution simulation results. The logarithmic dependence gives rise to a critical Reynolds number below which the viscous correction term, counterintuitively, reduces the net dissipation in the flow field.