Transient progress mechanisms operating on streaky shear flows are believed necessary for sustaining close to-wall turbulence. Of the three particular person mechanisms current - Orr, carry-up and ‘push-over’ - Lozano-Duran et al. J. Fluid Mech. 914, A8, 2021) have not too long ago noticed that each Orr and push-over have to be present to maintain turbulent fluctuations given streaky (streamwise-unbiased) base fields whereas lift-up does not. We present here, using Kelvin’s mannequin of unbounded constant shear augmented by spanwise streaks, that this is because the push-over mechanism can act in live performance with a ‘spanwise’ Orr mechanism to provide much-enhanced transient growth. Rey) occasions. Our outcomes therefore help the view that whereas raise-up is believed central for the roll-to-streak regenerative process, it's Orr and push-over mechanisms that are both key for 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 (not less than) a close to-wall sustaining cycle (Hamilton et al., 1995; Waleffe, Wood Ranger shears 1997; Jimenez & Pinelli, 1999) involving predominantly streaks and streamwise rolls (or vortices) which helps maintain the turbulence (e.g. see the critiques Robinson, 1991; Panton, 2001; Smits et al., 2011; Jimenez, 2012, 2018). The technology of these streaks from the rolls is commonly explained by the (linear) transient growth ‘lift-up’ mechanism (Ellingsen & Palm, 1975; Landahl, Wood Ranger shears 1980), but how rolls are regenerated from the streaks has proven rather less clear resulting from the need to invoke nonlinearity sooner or Wood Ranger shears later.
Just specializing in the initial linear half, Schoppa & Hussain (2002) urged that transient development mechanisms on the streaks were truly extra important than (linear) streak instabilities, and that it was these transiently growing perturbations which fed again to create streaks through their nonlinear interaction. While this view has been contested (e.g. Hoepffner et al., 1995; Cassinelli et al., 2017; Jimenez, 2018), it's supported by latest cause-and-impact numerical experiments by Lozano-Durán et al. 2021) who seemed more intently at all of the linear processes current. Specifically, Lozano-Durán et al. 2021) remoted the influence of the three different transient growth mechanisms: the acquainted Orr (Orr, 1907) and lift-up (Ellingsen & Palm, 1975) mechanisms current for a 1D shear profile UU
and Wood Ranger Power Shears for sale Wood Ranger Power Shears manual Wood Ranger Power Shears shop Wood Ranger Power Shears website order now a far much less-studied ‘push-over’ mechanism which may solely function 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 growth possible on a wall-regular shear plus monochromatic streak field in line with the buffer area at the wall.
Over appropriately brief occasions (e.g. one eddy turnover time as proposed by Butler & Farrell (1993)), they found a similarly clear sign that raise-up is unimportant whereas the elimination of push-over dramatically reduced the growth: see their determine 7. The necessity to have push-over operating with the Orr mechanism indicates they're working symbiotically. How this happens, nonetheless, is puzzling from the timescale perspective as Orr is considered 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 elevate-up in which viscously-decaying wall-regular velocities (as present in streamwise rolls) advect the bottom shear to supply streaks. Push-over (a time period coined by Lozano-Durán et al. Understanding exactly how these two mechanisms constructively interact is subsequently an interesting difficulty. 1) - was used by Orr (1907) for his seminal work and has been essential in clarifying the characteristics of each Orr and carry-up mechanisms subsequently (e.g. Farrell & Ioannou, 1993; Jimenez, 2013; Jiao et al., 2021) and as a shear-move testbed otherwise (e.g. Moffatt, durable garden trimmer 1967; Marcus & Press, 1977). The key options of the model are that the base circulation is: 1. unbounded and so not restricted by any boundary situations; and 2. a linear function of area.
These together permit plane wave options to the perturbation evolution equations where the spatially-varying base advection can be accounted for by time-dependent wavenumbers. This leaves simply 2 odd differential equations (ODEs) for the cross-shear velocity and cross-shear vorticity to be integrated forward in time. These ‘Kelvin’ modes kind a whole set but, unusually, are usually not individually separable in space and time and so the representation differs from the same old plane wave method with constant wavenumbers. The augmented base stream considered here - proven in Figure 1 and equation (1) under - builds in a streak field which introduces spatially-periodic spanwise shear. This is now not purely linear in house and so a Kelvin mode is now not an answer of the linearised perturbation equations. Instead, a single sum of Kelvin modes over spanwise wavenumbers is needed, but, importantly, Wood Ranger shears the wall-regular shear could be handled as ordinary, removing the unbounded advective term from the system.
This means the model system is still a very accessible ‘sandbox’ by which to check the transient progress mechanisms of Orr, Wood Ranger shears elevate-up and now, crucially, additionally ‘push-over’. The worth to be paid for introducing the streak subject is an order of magnitude increase in the variety of ODEs to be solved, but, since this is increased from 2 to O(20)O(20), it's trivial by today’s requirements. The plan of the paper is as follows. Section 2 introduces the mannequin, the evolution equations and Wood Ranger shears discusses applicable parameter values. Rey asymptotic scaling legal guidelines and discussing the timescales for Orr and lift-up development mechanisms. The presence of streaks is launched in §4, with the 2D limit of no streamwise variation utilized in §4.1 for example how the push-over mechanism behaves when it acts alone. That is followed by a normal evaluation of the transient progress possible for the full 3D system in §4.2 which is discovered to clearly capture the symbiotic relationship between Orr and push-over.