Steady flow in a curved tube of triangular cross section

Abstract

The steady flow of a viscous fluid moving under a constant pressure gradient in a curved tube with a uniform triangular cross section is investigated. Numerical solutions of the equations of motion have been found for the range 100-12 000 of the Dean numberD=Ga3√(2a/L)/μv, whereGis the constant pressure gradient,ais a dimension of the triangle,Lthe radius of the circle in which the tube is coiled,μthe viscosity andvthe coefficient of kinematic viscosity of the fluid. The results for lowDhave been checked by an independent numerical method in which the stream function is expanded in a series of powers ofDfollowing the method of Dean (1928). All the results have been checked for accuracy by varying the grid size used in the numerical computations. The trend of the results asDincreases is examined for evidence of the development of a boundary-layer structure asD→ ∞. Some indication is found of the formation of a boundary layer of thickness proportional toD–1/3near the side walls of the tube with an associated inviscid core region in the centre of the tube. In particular, comparison is made with details of an asymptotic model asD→ ∞ proposed by Smith (1976). A measure of agreement with the general characteristics of this model is obtained, although there are some discrepancies in the precise details. It is possible that the range ofDconsidered in the present work is not great enough to form any definite conclusions regarding the precise nature of the flow asD→ ∞. A feature of the present results which develops forD> 3000 is that the maximum axial velocity in the tube ceases to occur on the axis of symmetry of the cross section. This feature appears to be generally consistent with numerical results obtained by Cheng & Akiyama (1970) and Hocking (unpublished) for a tube of rectangular cross section. The sequence of corner vortices of the type identified by Moffatt (1964) is found to occur in the numerical solutions. A detailed study of the vortices has already been published (Collins & Dennis 1976).