Vortexons in axisymmetric Poiseuille pipe flows

Abstract

Abstract. We present a study on the nonlinear dynamics of small long-wave disturbances to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. At high Reynolds numbers, the associated Navier-Stokes equations can be reduced to a set of coupled Korteweg-de Vries-type (KdV) equations that support inviscid and smooth travelling waves numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices concentrated near the pipe boundaries (wall vortexons) or that wrap around the pipe axis (centre vortexons), in agreement with the analytical soliton solutions derived by Fedele (2012) [12]. The KdV dynamics of a perturbation is also investigated by means of an high accurate Fourier-based numerical scheme. We observe that an initial vortical patch splits into a centre vortexon radiating patches of vorticity near the wall. These can undergo further splitting leading to a proliferation of centre vortexons that eventually decay due to viscous effects. The splitting process originates from a radial flux of azimuthal vorticity from the wall to the pipe axis in agreement with the inverse cascade of cross-stream vorticity identified in channel flows by Eyink (2008) [10]. The inviscid vortexon most likely is unstable to non-axisymmetric disturbances and may be a precursor to puffs and slug flow formation.

Camassa–Holm equations and vortexons for axisymmetric pipe flows

“…This is the mechanism of inverse cascade of cross-stream vorticity in channel flows identified by Eyink (2008). Similar dynamics is also observed for long-wave disturbances associated to the KdV equations (4.2) (Fedele & Dutykh 2013).…”

Wave dispersion in moderate channel turbulence

“…The observed deviations from Taylor’s hypothesis appear to be the signature of the dispersive nature of turbulence. Moreover, the observed wave dispersion fairly agrees with the theoretical dispersion predicted for axisymmetric flows 18 , 19 , 21 .…”

“…Fedele and Dutykh 18 , 19 investigated the dynamics of non-rotating axisymmetric pipe flows in terms of nonlinear soliton bearing equations. They showed that at high Reynolds numbers, the dynamics of perturbations to the laminar flow obey a coupled system of nonlinear Camassa–Holm (CH) equations 20 .…”

“…Drawing on Fedele and Dutykh 18 , 19 , 21 consider the axisymmetric perturbation to a laminar pipe fluid flowing in the streamwise direction x , with r the radial direction. The associated streamfunction is , where is the least stable eigenmode of the linearized axisymmetric Navier–Stokes equations around the laminar flow solution 21 and the amplitude A ( x , t ) satisfies a CH type equation 18 where is a mean advection velocity, are free parameters, ( F , H ) are the coefficients of the nonlinear terms and r is the radial direction.…”

On time relaxed schemes and formulations for dispersive wave equations

“…However, the KdV and BBM-type equations appear in many other physical settings as well (see e.g. [11,17,18,20,30]).…”