Boundary-layer analysis for effects of viscosity of the irrotational flow on the flow induced by a rapidly rotating cylinder in a uniform stream

Abstract

We study the streaming flow past a rapidly rotating circular cylinder. The starting point is the full continuity and momentum equations without any approximations. We assume that the solution is a boundary-layer flow near the cylinder surface with the potential flow outside the boundary layer. The order of magnitude of the terms in the continuity and momentum equations can be estimated inside the boundary layer. When terms of the order of $\delta/a$ and higher are dropped, where $\delta$ is the boundary-layer thickness and $a$ is the radius of the cylinder, the equations used by M. B. Glauert (Proc. R. Soc. Lond. A, vol. 242, 1957, p. 108) are recovered. Glauert's solution ignores the irrotational rotary component of the flow inside the boundary layer, which is consistent with dropping $\delta/a$ terms in the governing equations.

We propose a new solution to this problem, in which the velocity field is decomposed into two parts. Outside the boundary layer, the flow is irrotational and can be decomposed into a purely rotary flow and a potential flow past a fixed cylinder. Inside the boundary layer, the velocity is decomposed into an irrotational purely rotary flow and a boundary-layer flow. Inserting this decomposition of the velocity field inside the boundary layer into the governing equations, we obtain a new set of equations for the boundary-layer flow, in which we do not drop the terms of the order of $\delta/a$ or higher. The pressure can no longer be assumed to be a constant across the boundary layer, and the continuity of shear stress at the outer edge of the boundary layer is enforced. We solve this new set of equations using Glauert's method, i.e. to expand the solutions as a power series of $\alpha \,{=}\, 2 U_0 /Q$, where $U_0$ is the uniform stream velocity and $Q$ is the circulatory velocity at the outer edge of the boundary layer. The pressure from this boundary-layer solution has two parts, an inertial part and a viscous part. The inertial part comes from the inertia terms in the momentum equations and is in agreement with the irrotational pressure; the viscous part comes from the viscous stress terms in the momentum equations and may be viewed as a viscous pressure correction, which contributes to both drag and lift. Our boundary-layer solution is in reasonable to excellent agreement with the numerical simulation in the companion paper by Padrino & Joseph (2006).