an:04106463
Zbl 0675.76040
Germano, M.
The Dean equations extended to a helical pipe flow
EN
J. Fluid Mech. 203, 289-305 (1989).
00154325
1989
j
76D10 35Q99
Dean equations; secondary flow; flow in a helical pipe; extended Dean equations
Summary: The Dean equations are extended to the case of a helical pipe flow, and it is shown that they depend not only on the Dean number K but also on a new parameter \(\lambda\) /\({\mathcal R}\), where \(\lambda\) is the ratio of the torsion \(\tau\) to the curvature \(\kappa\) of the pipe axis and \({\mathcal R}\) the Reynolds number referred in the usual way to the pipe radius a and to the equivalent maximum speed in a straight pipe under the same axial pressure gradient. The fact that the torsion has no first-order effect on the flow is confirmed, but it is shown that this is peculiar to a circular cross-section. In the case of an elliptical cross-section there is a first-order effect of the torsion on the secondary flow, and in the limit \(\lambda\) /\({\mathcal R}\to \infty\) (twisted pipes, provided only with torsion), the first-order `displacement' effect of the walls on the secondary flow is recovered.
Different systems of coordinates and different orders of approximations have recently been adopted in the study of the flow in a helical pipe. Thus comparisons between the equations and the results presented in different reports are in some cases difficult and uneasy. In this paper the extended Dean equations for a helical pipe flow recently derived by \textit{H. C. Kao} [ibid. 184, 335-356 (1987; Zbl 0645.76045)] are converted to a simpler form by introducing an appropriate modified stream function, and their equivalence with the present set of equations is recovered. Finally, the first-order equivalence of this set of equations with the equations obtained by \textit{S. Murata}, \textit{Y. Miyake}, \textit{T. Inaba} and \textit{H. Ogawa} [Bull. JSME 24, 355 ff. (1981)] is discussed.
Zbl 0645.76045