Reduced description of exact coherent states in parallel shear flows

Abstract

A reduced description of exact coherent structures in the transition regime of plane parallel shear flows is developed, based on the Reynolds number scaling of streamwise-averaged (mean) and streamwise-varying (fluctuation) velocities observed in numerical simulations. The resulting system is characterized by an effective unit Reynolds number mean equation coupled to linear equations for the fluctuations, regularized by formally higher-order diffusion. Stationary coherent states are computed by solving the resulting equations simultaneously using a robust numerical algorithm developed for this purpose. The algorithm determines self-consistently the amplitude of the fluctuations for which the associated mean flow is just such that the fluctuations neither grow nor decay. The procedure is used to compute exact coherent states of a flow introduced by Drazin and Reid [Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 1981)] and studied by Waleffe [Phys. Fluids 9, 883 (1997)]: a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force. Numerical continuation of the lower-branch states to lower Reynolds numbers reveals the presence of a saddle node; the saddle node allows access to upper-branch states that are, like the lower-branch states, self-consistently described by the reduced equations. Both lower-and upper-branch states are characterized in detail.

A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows

“…The absence of anything akin to a UMZ is consistent with the effective Reynolds number for this x-mean flow being O(1) [7,8]. Although it has long been understood that comparably weak roll motions in the presence of O(1) dimensionless shear can induce O(1) streaks through the lift-up effect associated with roll advection (indeed, this is a cornerstone of VWI theory), a primary facet of the model we propose is that the streamwise vortices-while still weaker than the streaks-must be sufficiently strong that the effective Reynolds number for the x-averaged flow is large.…”

“…Accordingly, the effective Reynolds numberāRe felt by the streamwise-averaged streamwise flow within the UMZs is small relative to the Reynolds number of the instantaneous flow (Re) but large compared with unity. This asymptotic ordering renders system (3.3) distinct from related asymptotically reduced descriptions of SSPs in shear flows [7,8]. Figure 4 and ∂ x u 7/10 + ∂ y v 7/10 + ∂ z w 7/10 = 0.…”

“…As in related studies [7,8,33], this scalar can be determined by employing an iterative algorithm. Following Dempsey et al [34], we instead specify A and treat the streamwise wavenumber α as a scalar unknown that must be determined via the solution of a nonlinear eigenvalue problem.…”

“…the viscous sublayer and buffer layer, the interaction among spacefilling streamwise vortices and streaks has been shown to give rise to a regeneration cycle capable of locally sustaining the turbulence on scales of approximately 100 viscous units ν/u τ in extent, where ν is the kinematic viscosity and u τ is the wall friction velocity [1]. There is a large body of literature documenting the occurrence and properties of these near-wall coherent structures in laboratory experiments [2] and direct numerical simulations (DNS) [3] and establishing quantitative theories for their self-sustenance [4][5][6][7][8][9][10]. At large values of the friction Reynolds number Re τ ≡ u τ h/ν, where h is an appropriate outer length scale such as the BL thickness, quasicoherent flow structures also arise in the outer region of the BL (i.e.…”

Prospectus: towards the development of high-fidelity models of wall turbulence at large Reynolds number

“…Restricted nonlinear models (RNL) of wall turbulence generically invoke some level of streamwise averaging, as they were: (i) at least partially motivated by the slowly varying streamwise structure of the roll-streak motions in near-wall turbulence and the associated self-sustaining (exact coherent structure) process model [36,37], and (ii) were partly justified by single streamwise-wavenumber systematic asymptotic reductions of the full Navier-Stokes equations [38][39][40]. Given a streamwise-averaged version of the Navier-Stokes equations (or more generally, a representation involving a small number of streamwise modes), equations for the fluctuating fields are developed, and model reduction is attained by neglecting the nonlinear terms involving fluctuation-fluctuation products (e.g.…”