Nonlinear exact coherent structures in pipe flow and their instabilities

In this paper, we present computational results of some two-fold azimuthally symmetric travelling waves and their stability. Calculations over a range of Reynolds numbers ( $Re$ ) reveal connections between a class of solutions computed by Wedin & Kerswell (J. Fluid Mech., vol. 508, 2004, pp. 333-371) (henceforth called the WK solution) and the $Re\rightarrow \infty$ vortex-wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641-666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178-205). In particular, the continuation of the WK solutions to larger values of $Re$ shows that the WK solution bifurcates from a shift-and-rotate symmetric solution, which we call the WK2 state. The WK2 solution computed for $Re\leqslant 1.19\times 10<^>{6}$ shows excellent agreement with the theoretical $Re<^>{-5/6}$ , $Re<^>{-1}$ and $O(1)$ scalings of the waves, rolls and streaks respectively. Furthermore, these states are found to have only two unstable modes in the large $Re$ regime, with growth rates estimated to be $O(Re<^>{-0.42})$ and $O(Re<^>{-0.92})$ , close to the theoretical $O(Re<^>{-1/2})$ and $O(Re<^>{-1})$ asymptotic results for edge and sinuous instability modes of vortex-wave interaction states (Deguchi & Hall, J. Fluid Mech., vol. 802, 2016, pp. 634-666) in plane Couette flow. For the nonlinear viscous core states (Ozcakir et al., J. Fluid Mech., vol. 791, 2016, pp. 284-328), characterized by spatial a shrinking of the wave and roll structure towards the pipe centre with increasing $Re$ , we continued the solution to $Re\leqslant 8\times 10<^>{6}$ and we find only one unstable mode in the large Reynolds number regime, with growth rate scaling as $Re<^>{-0.46}$ within the class of symmetry-preserving disturbances.