  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,
$\varphi=10^\circ$ - $20^\circ$ (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 $\varphi\gtrsim 45^\circ$ 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 $\varphi$ the inner ring decides the main swirling motion,
whereas for $\varphi=18.6^\circ$ 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 $\varphi=18.6^\circ$ both
contribute approximately equally to production in the outer ring.
