Traveling wave instability in helical coil flow

Abstract

Complementary flow visualization photographs and numerical calculations are presented for the transitional state between the laminar and turbulent flow regimes in a helically coiled pipe. The flow visualization covers a Reynolds number range from 3800 to 8650 (890<De<2030, where De is the Dean number). Estimates of the wavelength and wave speed of a traveling wave instability are made from photographs and video recordings at Re=5060 and 5480 (De=1190 and 1280). The unsteady three-dimensional finite difference approximations of the Navier–Stokes equations formulated for the toroidal coordinate system are solved numerically. The calculations are performed in a curved pipe with a radius of curvature to pipe radius ratio equal to 18.2 and Re=5480 (De=1280). These test conditions match the flow visualization and previously reported laser Doppler velocimetry measurements. The calculations reveal a complex interaction between the centrifugal force and the cross-stream velocity, hence explaining the mechanism for maintaining the traveling wave. An analogy is made with known centrifugal instabilities to explain the character of the motion observed in the inner half of the pipe along planes defined by the radial and streamwise coordinate directions. Simple considerations show that the cross-stream flow has the potential for a centrifugal instability.