Homotopy of exact coherent structures in plane shear flows

Abstract

Three-dimensional steady states and traveling wave solutions of the Navier-Stokes equations are computed in plane Couette and Poiseuille flows with both free-slip and no-slip boundary conditions. They are calculated using Newton's method by continuation of solutions that bifurcate from a two-dimensional streaky flow then by smooth transformation ͑homotopy͒ from Couette to Poiseuille flow and from free-slip to no-slip boundary conditions. The structural and statistical connections between these solutions and turbulent flows are illustrated. Parametric studies are performed and the parameters leading to the lowest onset Reynolds numbers are determined. In all cases, the lowest onset Reynolds number corresponds to spanwise periods of about 100 wall units. In particular, the rigid-free plane Poiseuille flow traveling wave arises at Re ϭ44.2 for L x ϩ ϭ273.7 and L z ϩ ϭ105.5, in excellent agreement with observations of the streak spacing. A simple one-dimensional map is proposed to illustrate the possible nature of the ''hard'' transition to shear turbulence and connections with the unstable exact coherent structures.

Travelling waves in a straight square duct

“…Similar to what has been proposed in [9,11] we construct the initial conditions for our non-linear procedure as a superposition of various ingredients: (i) the laminar base flow, (ii) streamwise rolls (taken as one of the least decaying eigenmodes of the Stokes operator on the square), (iii) the streaks induced by the rolls, and (iv) neutrally stable linear perturbations (with specific parities I to IV, according to the nomenclature of [1]) of the roll/streak flow. Contrary to [12] we have focused upon initial conditions constructed from roll/streakinstabilites with type-II and type-III parities.…”

“…Here we will present results obtained by applying an iterative solution strategy to the steady Navier-Stokes equations in a moving frame of reference. In the absence of a "natural" primary bifurcation point, we resort to the method proposed by Waleffe [9], where streamwise vortices are artificially added to the base flow and forced against viscous decay, leading to streaks, which are in turn linearly unstable, feeding back into the original vortices. The non-linear solution is then continued back to the original problem, i.e.…”

The emergence of localized vortex–wave interaction states in plane Couette flow

“…These structures are often referred to as self-sustaining processes, coherent structures or vortex-wave interactions, and their common feature is that they apparently form at least part of the 'backbone' on which turbulent flows evolve. In fact it turns out that vortexwave interaction theory involving inviscid waves is simply the high Reynolds number description of the computationally-generated self-sustaining processes at finite Reynolds numbers; see for example Hall and Smith (1991), Hall and Sherwin (2010), Hall (2012a,b) for a discussion of vortex-wave interaction theory and Waleffe (1997Waleffe ( , 2001Waleffe ( , 2003 for discussions of self-sustaining processes. Henceforth we refer to self-sustaining processes as SSP.…”

“…For channel flows, Waleffe and co-workers, guided by the early ideas of Benney and colleagues, looked into the possibility that the Navier-Stokes equations might support wave systems occurring as instabilities of streamwise vortex flows but sufficiently large to drive the vortex flows; see for example Waleffe (1995Waleffe ( , 1997Waleffe ( , 1998Waleffe ( , 2001Waleffe ( , 2003, Wang et al (2007). The approach used was to find fully nonlinear solutions of the Navier-Stokes equations using various fictitious forces to identify equilibrium states and then continue them when the forcing was switched off.…”

Symmetry of Coherent Vortices in Plane Couette Flow

“…By contrast, only two of the three symmetries are satisfied by the streamwise vortex solution (SVS), which was previously obtained as a coherent structure typical of wall turbulence in Refs. [1,2,3,4]. Thus, the SVS is expected to bifurcate from the HVS via breaking of the spanwise reflection symmetry depicted in Fig.1.…”