Travelling wave states in pipe flow

In this paper, we report numerical calculations of two new travelling wave states with shift and reflect two-fold azimuthal symmetry (S2 symmetry of Wedin and Kerswell (2004)), characterized by a shrinking core towards the pipe center with increasing R. Further, one of these solutions, termed C2, has additional shift and rotate symmetry. We also present arguments to identify all possible scaling solutions as R → ∞. Within the solution class where the axial wave length is independent of R, we identify two families. One is a nonlinear viscous core (NVC) solution characterized by a fully nonlinear interaction between rolls, streaks and waves in a shrinking core of radius δ = R−1/4 where all axial velocity components scale asR−1/2, while perpendicular plane components of velocity scale at R−3/4. In this case, the wave speed satisfies (1 − c) = O(R−1/2). These states are similar to the states calculated by Deguchi and Hall (2014) in boundary layer flows, except that the wave and roll components decay algebraically rather than exponentially in the far-field as one exits the core. The streak for two fold azimuthally symmetric states, with or without additonal shift and rotate symmetry, does not decay as one exits the core until wall effects become important. The asymptotics also suggests possibility of a second class of solutions that we call vortex wave interaction (VWI) solutions: For δ = 1 or δ = R−1/6, a linear wave of size δ−4/3R−5/6 travelling with wave speed c where 1− c = O(δ) concentrated in a critical layer of thickness δ(δ4R)−1/3 around the critical curve drives rolls of size R−1δ−1/3 localized in a core of radius δ, which in turn drives a streak of size δ. We identify previously calculated states by Viswanath (2009a) as a VWI state with δ = 1.