Camassa–Holm equations and vortexons for axisymmetric pipe flows

Abstract

Abstract. In this paper, we study the nonlinear dynamics of an axisymmetric disturbance to the laminar state in non-rotating Poiseuille pipe flows. In particular, we show that the associated Navier-Stokes equations can be reduced to a set of coupled Camassa-Holm type equations. These support inviscid and smooth localized travelling waves, which are numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices that concentrate near the pipe boundaries (wall vortexons) or wrap around the pipe axis (centre vortexons) in agreement with the analytical soliton solutions derived by Fedele (2012) for small and long-wave disturbances. Inviscid singular vortexons with discontinuous radial velocities are also numerically discovered as associated to special traveling waves with a wedge-type singularity, viz. peakons. Their existence is confirmed by an analytical solution of exponentially-shaped peakons that is obtained for the particular case of the uncoupled Camassa-Holm equations. The evolution of a perturbation is also investigated using an accurate Fourier-type spectral scheme. We observe that an initial vortical patch splits into a centre vortexon radiating vorticity in the form of wall vortexons. These can under go further splitting before viscosity dissipates them, leading to a slug of centre vortexons. The splitting process originates from a radial flux of azimuthal vorticity from the wall to the pipe axis in agreement with Eyink (2008). The inviscid and smooth vortexon is similar to the nonlinear neutral structures derived by Walton (2011) and it may be a precursor to puffs and slugs observed at transition, since most likely it is unstable to non-axisymmetric disturbances.

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. Here, we added a fifth-order dispersion term ruled by to model dispersive effects beyond axisymmetry.…”

“…Such a wave dispersion is associated to a hidden elastic energy that has a counterpart in axisymmetric Navier–Stokes flows, which are essentially two-dimensional (2-D) since vortex stretching is absent. To understand the physical origin of such a wave dispersion, Fedele and Dutykh 18 consider the 2-D Euler equations for an inviscid fluid over the domain V in a Cartesian frame. Given the streamfunction , the divergent-free velocity field and the vorticity .…”

“…The paper is structured as follows. We first briefly overview the theoretical results on the wave dispersion of axisymmetric turbulent flows 18 . Then, we introduce the case study of DNS turbulent channel flows simulations, and then overview Taylor’s hypothesis 1 in relation to our work.…”