Semi analytical method for calculating Dean's vortex in Torus of elliptical cross section

Abstract

Based on the work of Dean (1927 and 1928) [1-2] and Cuming (1952) [3], the stationary flow of an incompressible Newtonian fluid through a curved pipe of uniform curvature and with elliptic cross section is studied. The Navier-Stokes equations are expressed in toroïdal coordinates system (s,r,θ). Following Dean’s approach, the governing equations for the fluid motion through a curved elliptical channel are solved by using an original semi analytical method for the resolution of a biharmonic equation. The main interest in this study is to test and validate in the case of an elliptical cross section the proposed semi-analytical method. The latter can then be used for other geometries for which explicit solutions are not available.

Keywords

• Toroïdal coordinates
• Curvature
• Curved pipes
• Centrifugal force
• Dean vortex.