Designing vortices in pipe flow with topography-driven Langmuir circulation

We present direct numerical simulation of a mechanism for creating longitudinal vortices in pipe flow, compared with a simple model theory. By furnishing the pipe wall with a pattern of crossing waves secondary flow in the form of spanwise 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 accute angle with the axis, $\phi ={10}^{\circ }$ - ${20}^{\circ }$ (a contracted eggcarton'), changes sign near $45^\circ$ and is weak and opposite 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 $\phi \gtrsim {45}^{\circ }$ reversing the flow. Curiously, circulation strength is a faster-than-linearly increasing function of Reynolds number for the contracted case. 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, $\varphi=18.6^\circ, 45^\circ$and$60^\circ$. Mean-vorticity production is organised in a ring-like structure with the two rings contributing to rotating flow in opposite senses. For the larger $\phi$ the inner ring decides the main swirling motion, whereas for $\phi ={18.6}^{\circ }$ 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 $\phi ={18.6}^{\circ }$ both contribute approximately equally to production in the outer ring.