How linear is wall-bounded turbulence?

Abstract

The relevance of Orr's inviscid mechanism to the transient amplification of disturbances in shear flows is explored in the context of bursting in the logarithmic layer of wall-bounded turbulence. The linearized problem for the wall normal velocity is first solved in the limit of small viscosity for a uniform shear and for a channel with turbulent-like profile, and compared with the quasiperiodic bursting of fully turbulent simulations in boxes designed to be minimal for the logarithmic layer. Many properties, such as time and length scales, energy fluxes between components, and inclination angles, agree well between the two systems. However, once advection by the mean flow is subtracted, the directly computed linear component of the turbulent acceleration is found to be a small part of the total. The temporal correlations of the different quantities in turbulent bursts imply that the classical model, in which the wall-normal velocities are generated by the breakdown of the streamwise-velocity streaks, is a better explanation than the purely autonomous growth of linearized bursts. It is argued that the best way to reconcile both lines of evidence is that the disturbances produced by the streak breakdown are amplified by an Orr-like transient process drawing energy directly from the mean shear, rather than from the velocity gradients of the nonlinear streak. This, for example, obviates the problem of why the cross-stream velocities do not decay once the streak has broken down. C

Effect of the computational domain on direct simulations of turbulent channels up to Reτ = 4200

“…9, and is known to have incorrect statistics above y ≈ L z /3 ≈ 0.25h. 2 The new small case S950 has a somewhat larger computational domain with streamwise and spanwise periodicities L x = π h and L z = π h/2 at Re τ = 939, and Ref.…”

Coherent structures in statistically-stationary homogeneous shear turbulence

“…They are followed some time later by a somewhat weaker rise of v 2 and w 2 . This relation between velocity components is reminiscent of the bursting in minimal channels analyzed in Jiménez (2013a), and the bursting time scale history in figure 2.7(a).…”

“…Orr (1907) The overtaking of shear-periodic copies advected by the mean velocities in neighboring boxes is not very different from the shearing mechanism in Orr's. For example, the characteristics dip in u 2 during the burst of v 2 is the same in both cases (Umurhan & Regev, 2004;Jiménez, 2007Jiménez, , 2013a, and the essence of the interaction is the tilting of the structures due to the shear. However, the details are different.…”

“…Because of the streamwise periodicity of the domain, the shift SL y t of the upper boundary with respect to the bottom induces a characteristic time period, ST s = L x /L y (hereafter, box period), in which the flow is forced by the passing of the shifted-periodic boxes immediately above and below (Jiménez, 2013a). Since this forcing acts at the box scale, small turbulent structures can be expected to be roughly independent of the box geometry, but care should be taken to account for resonances between the largest flow scales and T s (see § 2.3.2).…”

“…The average length T b of individual bursts is given in figure 2.8(b), computed as the width of the temporal autocorrelation function of v 2 , measured at C v 2 v 2 = 0.5 (Jiménez, 2013a). For short boxes (ST s 2), the period stays relatively constant, ST b ≈ 15.…”