Designing vortices in pipe flow with topography-driven Langmuir circulation

Abstract

We present direct numerical simulation of a mechanism for creating longitudinal vortices in pipe flow, compared with a model theory. By furnishing the pipe wall with a pattern of crossing waves, secondary flow in the form of streamwise vortex pairs is created. The mechanism, ‘CL1’, is kinematic and known from oceanography as a driver of Langmuir circulation. CL1 is strongest when the ‘wall wave’ vectors make an acute angle with the axis, φ=10∘–20∘, changes sign near 45∘ and is weak and of opposite sign beyond this angle. A competing, dynamic mechanism driving secondary flow in the opposite sense is also observed, created by the azimuthally varying friction. Whereas at smaller angles ‘CL1’ prevails, the dynamic effect dominates when φ≳45∘, reversing the flow. Curiously, the circulation strength is a faster-than-linearly increasing function of Reynolds number for small φ. We explore an analogy with Prandtl's secondary motion of the second kind in turbulence. A transport equation for average streamwise vorticity is derived, and we analyse it for three different crossing angles, φ=18.6∘,45∘ and 60∘. Mean-vorticity production is organised in a ring-like structure with the two rings contributing to rotating flow in opposite senses. For the larger φ, the inner ring decides the main swirling motion, whereas for φ=18.6∘, outer-ring production dominates. For the larger angles, the outer ring is mainly driven by advection of vorticity and the inner by deformation (stretching) whereas, for φ=18.6∘, both contribute approximately equally to production in the outer ring.