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In this paper, we have found two new nonlinear travelling wave solutions in pipe flows. We investigate possible asymptotic structures at large Reynolds number
when wavenumber is independent of
and identify numerically calculated solutions as finite
realizations of a nonlinear viscous core (NVC) state that collapses towards the pipe centre with increasing
at a rate
. We also identify previous numerically calculated states as finite
realizations of a vortex wave interacting (VWI) state with an asymptotic structure similar to the ones in channel flows studied earlier by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In addition, asymptotics suggests the possibility of a VWI state that collapses towards the pipe centre like
, though this remains to be confirmed numerically.
The robustness of localized states that transport energy and mass is assessed by a numerical study of the Euler equation in two space dimensions. The localized states are the translating ‘V-states’ discovered by Deem & Zabusky. These piecewise- constant dipolar (i.e. oppositely-signed ± or ±) vorticity regions are steady translating solutions of the Euler equations. A new adaptive contour-dynamical algorithm with curvature-controlled node insertion and removal is used. The evolution of one V-state, subject to a symmetric-plus-asymmetric perturbation is examined and stable (i.e. non-divergent) fluctuations are observed. For scattering interactions, coaxial head-on (or ± on ±) and head-tail (or & on ±) arrangements are studied. The temporal variation of contour curvature and perimeter after V-states separate indicate that internal degrees of freedom have been excited. For weak interactions we observe phase shifts and the near recurrence to initial states. When two similar, equal-circulation but unequal-area V-states have a head-on interaction a new asymmetric state is created by contour ‘exchange’. There is strong evidence that this is near to a V-state. For strong interactions we observe phase shifts, ‘breaking’ (filament formation) and, for head-tail interactions, merger of like-signed vorticity regions.
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