Transient progress mechanisms working on streaky shear flows are believed essential for sustaining near-wall turbulence. Of the three individual mechanisms current – Orr, lift-up and ‘push-over’ – Lozano-Duran et al. J. Fluid Mech. 914, A8, 2021) have recently observed that each Orr and push-over have to be current to sustain turbulent fluctuations given streaky (streamwise-unbiased) base fields whereas elevate-up doesn’t. We present right here, utilizing Kelvin’s model of unbounded constant shear augmented by spanwise streaks, that this is because the push-over mechanism can act in concert with a ‘spanwise’ Orr mechanism to provide much-enhanced transient development. Rey) times. Our results subsequently support the view that while raise-up is believed central for portable cutting shears the roll-to-streak regenerative course of, it is Orr and push-over mechanisms that are both key for Wood Ranger cordless power shears Shears manual the streak-to-roll regenerative process in close to-wall turbulence. Efforts to understand wall-bounded turbulence have naturally focussed on the wall and the (coherent) structures which kind there (Richardson, 1922). The consensus is that there is (a minimum of) a near-wall sustaining cycle (Hamilton et al., 1995; Waleffe, 1997; Jimenez & Pinelli, 1999) involving predominantly streaks and streamwise rolls (or vortices) which helps maintain the turbulence (e.g. see the opinions Robinson, 1991; Panton, 2001; Smits et al., 2011; Jimenez, 2012, 2018). The technology of those streaks from the rolls is usually explained by the (linear) transient development ‘lift-up’ mechanism (Ellingsen & Palm, 1975; Landahl, 1980), however how rolls are regenerated from the streaks has confirmed a little less clear as a result of the need to invoke nonlinearity sooner or later.

Just focusing on the preliminary linear part, Schoppa & Hussain (2002) suggested that transient growth mechanisms on the streaks had been actually more important than (linear) streak instabilities, and that it was these transiently growing perturbations which fed again to create streaks by way of their nonlinear interplay. While this view has been contested (e.g. Hoepffner et al., 1995; Cassinelli et al., 2017; Jimenez, 2018), it’s supported by current cause-and-impact numerical experiments by Lozano-Durán et al. 2021) who looked more intently at all the linear processes current. Particularly, Lozano-Durán et al. 2021) remoted the influence of the three totally different transient growth mechanisms: the familiar Orr (Orr, 1907) and portable cutting shears carry-up (Ellingsen & Palm, 1975) mechanisms present for a 1D shear profile U(y)U(y) and a far much less-studied ‘push-over’ mechanism which may only operate when the bottom profile has spanwise shear i.e. U(y,z)U(y,z). Markeviciute & Kerswell (2024) investigated this additional by looking at the transient development potential on a wall-normal shear plus monochromatic streak field per the buffer area on the wall.

Over appropriately quick instances (e.g. one eddy turnover time as proposed by Butler & Farrell (1993)), they found a equally clear signal that carry-up is unimportant whereas the removing of push-over dramatically diminished the growth: see their figure 7. The necessity to have push-over working with the Orr mechanism indicates they are working symbiotically. How this happens, nevertheless, is puzzling from the timescale perspective as Orr is taken into account a ‘fast’ mechanism which operates over inertial timescales whereas push-over seems a ‘slow’ mechanism working over viscous timescales. This latter characterisation comes from an analogy with raise-up during which viscously-decaying wall-normal velocities (as present in streamwise rolls) advect the base shear to provide streaks. Push-over (a time period coined by Lozano-Durán et al. Understanding precisely how these two mechanisms constructively interact is subsequently an fascinating situation. 1) – was used by Orr (1907) for his seminal work and has been essential in clarifying the characteristics of both Orr and lift-up mechanisms subsequently (e.g. Farrell & Ioannou, 1993; Jimenez, 2013; Jiao et al., 2021) and as a shear-move testbed in any other case (e.g. Moffatt, 1967; Marcus & Press, 1977). The important thing features of the model are that the bottom flow is: 1. unbounded and so not restricted by any boundary circumstances; and 2. a linear perform of area.

These collectively permit plane wave solutions to the perturbation evolution equations the place the spatially-various base advection can be accounted for by time-dependent wavenumbers. This leaves just 2 ordinary differential equations (ODEs) for the cross-shear velocity and cross-shear vorticity to be built-in forward in time. These ‘Kelvin’ modes type an entire set however, unusually, should not individually separable in area and time and so the representation differs from the same old plane wave strategy with fixed wavenumbers. The augmented base move thought of right here – shown in Figure 1 and equation (1) below – builds in a streak field which introduces spatially-periodic spanwise shear. That is now not purely linear in space and so a Kelvin mode is not an answer of the linearised perturbation equations. Instead, a single sum of Kelvin modes over spanwise wavenumbers is needed, but, importantly, the wall-regular shear might be handled as standard, removing the unbounded advective term from the system.

This means the model system remains to be a really accessible ‘sandbox’ in which to study the transient progress mechanisms of Orr, carry-up and portable cutting shears now, crucially, additionally ‘push-over’. The price to be paid for introducing the streak subject is an order of magnitude improve within the number of ODEs to be solved, however, since this is elevated from 2 to O(20)O(20), portable cutting shears it’s trivial by today’s requirements. The plan of the paper is as follows. Section 2 introduces the model, the evolution equations and discusses applicable parameter values. Rey asymptotic scaling laws and discussing the timescales for portable cutting shears Orr and raise-up growth mechanisms. The presence of streaks is introduced in §4, with the 2D limit of no streamwise variation utilized in §4.1 for instance how the push-over mechanism behaves when it acts alone. That is adopted by a normal evaluation of the transient growth attainable for the full 3D system in §4.2 which is discovered to clearly capture the symbiotic relationship between Orr and push-over.