Turbulence Regeneration in Pipe Flow at Moderate Reynolds Numbers

Abstract

We present the results of an experimental investigation into the nature and structure of turbulent pipe flow at moderate Reynolds numbers. A turbulence regeneration mechanism is identified which sustains a symmetric traveling wave within the flow. The periodicity of the mechanism allows comparison to the wavelength of numerically observed exact traveling wave solutions and close agreement is found. The advection speed of the upstream turbulence laminar interface in the experimental flow is observed to form a lower bound on the phase velocities of the exact traveling wave solutions. Overall our observations suggest that the dynamics of the turbulent flow at moderate Reynolds numbers are governed by unstable nonlinear traveling waves. A process which is believed to be of relevance for transition in shear flows is the so called lift-up mechanism [6]. Here streamwise vortices of relatively small magnitude transport low momentum fluid away from the wall and high momentum fluid towards the wall creating strong anomalies in the velocity profile, so-called low-and high-speed streaks (elongated regions of low or high streamwise velocity with respect to their surroundings). During this process the initial perturbation amplitude can grow substantially. Since the mechanism is of a linear nature, in linearly stable flows eventually all perturbations will decay. However it is assumed that once the perturbation has reached a sufficiently large amplitude nonlinear effects will become important and lead to secondary instabilities. The mathematical reason for the transient growth of initial perturbations is the non-normality of the linearized NavierStokes operator and this mechanism has attracted considerable attention [7][8][9].In more recent studies the relevance of nonlinear effects and, in particular, the role of nonlinear solutions has been discussed [10,11]. Such solutions have been calculated for Couette and Poiseuille flow [12 -16] and more recently also for pipe flow [17,18]. These new flow states arise at finite values of the control parameter (Re) and in the case of pipe flow they take the form of traveling waves (TWs), which are coherent flow structures propagating at a constant phase speed in the streamwise direction. Typically these TW solutions are unstable so that the laminar flow state remains the only stable solution. In analogy to observations in low dimensional models [19,20] it has been suggested that as the Reynolds number is increased further the unstable states undergo secondary bifurcations and form an attracting region in phase space which gives rise to turbulent dynamics. The basin of attraction of the turbulent state grows with increasing Reynolds number, so that smaller and smaller perturbations are sufficient to destabilize the laminar flow.Confirmation of this picture has been provided by recent experiments in pipe flow where transients of unstable TWs have been observed that are in close agreement with the numerical solutions [21]. Further experimental investigations have demonstrated that the sta...

Repeller or Attractor? Selecting the Dynamical Model for the Onset of Turbulence in Pipe Flow

“…Underlying such a structure are unstable states and for pipe flow unstable solutions to the governing equations have been identified in the form of traveling waves [14,15]. Surprisingly clear transients of such traveling waves were observed in experiments [16,17] confirming their relevance to the turbulent dynamics. More recently traveling wave transients were also reported in numerical studies [18,19].…”

Dynamical systems and the transition to turbulence in linearly stable shear flows

“…Plane Couette flow is the historically first example, where lifetimes have been studied (Bottin & Chaté 1998;Bottin et al 1998b) and coherent structures have been identified (Nagata 1990(Nagata , 1996Clever & Busse 1992, 1997Eckhardt et al 2002). Similar investigations of pipe flow came later: lifetimes are studied in the papers by , Hof et al (2005Hof et al ( , 2006, Mullin & Peixinho (2006a,b) and Peixinho & Mullin (2006), and coherent states in the papers by Faisst & Eckhardt (2003), Hof et al (2004), and Schneider et al (2007b). The status of pipe flow has been reviewed by Kerswell (2005) and Eckhardt et al (2007) and the related studies of other flows may be traced from Mullin & Kerswell (2004).…”

Highly symmetric travelling waves in pipe flow

“…Faisst & Eckhardt (2003) and employed this technique within the geometry of pipe flow to find the expected TWs, although all were without swirl and therefore unrelated to the predictions of Smith & Bodonyi (1982). Experimental observations (Hof et al 2004(Hof et al , 2005 and numerical computations (Kerswell & Tutty 2007;Schneider et al 2007a;) then followed, which indicated that these waves are realized, albeit transiently, as coherent structures in turbulent pipe flow. These waves are all born in saddle-node bifurcations with the lowest at ReZ1251 for TWs with a discrete rotational symmetry.…”