Our concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier-Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex-wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641-666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier-Stokes equations in a layer located a distance of O(ln Re) from the wall. Here Re is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

The recently understood relationship between high Reynolds number vortex-wave interaction theory and computationally-generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex-wave interaction theory, which we now refer to as VWI, is used in the long streamwise wavelength limit to continue the development found at order one wavelengths by Hall and Sherwin (2010). The asymptotic description given reduces the Navier-Stokes equations to the so-called boundary-region equations for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from O(h) to O(Rh) where R is the Reynolds number and 2h is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave-vortex states. The solutions we calculate have an asymptotic error proportional to R −2 when compared to the full Navier-Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel so that the roll part of the flow is driven throughout the flow rather than as in Hall and Sherwin (2010) as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling which when approached causes the flow to localise and take on similarities with computationally-generated or experimentally-observed turbulent spots. In effect the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier-Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

Rational large Reynolds number matched asymptotic expansions of three-dimensional nonlinear magneto-hydrodynamic (MHD) states are concerned. The nonlinear MHD states, assumed to be predominantly driven by a unidirectional shear, can be sustained without any linear instability of the base flow and hence are responsible for subcritical transition to turbulence. Two classes of nonlinear MHD states are found. The first class of nonlinear states emerged out of a nice combination of the purely hydrodynamic vortex/wave interaction theory by and the resonant absorption theories on Alfvén waves, developed in the solar physics community (e.g. Sakurai et al. 1991;Goossens et al. 1995). Similar to the hydrodynamic theory, the mechanism of the MHD states can be explained by the successive interaction of the roll, streak, and wave fields, which are now defined both for the hydrodynamic and magnetic fields. The derivation of this 'vortex/Alfvén wave interaction' state is rather straightforward as the scalings for both of the hydrodynamic and magnetic fields are identical. It turns out that the leading order magnetic field of the asymptotic states appears only when a small external magnetic field is present. However, it does not mean that purely shear-driven dynamos are not possible. In fact, the second class of 'self-sustained shear driven dynamo theory' shows the magnetic generation that is slightly smaller size in the absence of any external field. Despite small size, the magnetic field causes the novel feedback mechanism in the velocity field through resonant absorption, wherein the magnetic wave becomes more strongly amplified than the hydrodynamic counterpart. arXiv:1809.03853v2 [physics.flu-dyn]

The relationship between nonlinear equilibrium solutions of the full Navier-Stokes equations and the high-Reynolds-number asymptotic vortex-wave interaction (VWI) theory developed for general shear flows by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641-666) is investigated. Using plane Couette flow as a prototype shear flow, we show that all solutions having O(1) wavenumbers converge to VWI states with increasing Reynolds number. The converged results here uncover an upper branch of VWI solutions missing from the calculations of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178-205). For small values of the streamwise wavenumber, the converged lower-branch solutions take on the long-wavelength state of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58-85) while the upper-branch solutions are found to be quite distinct, with new states associated with instabilities of jet-like structures playing the dominant role. Between these long-wavelength states, a complex 'snaking' behaviour of solution branches is observed. The snaking behaviour leads to complex 'entangled' states involving the long-wavelength states and the VWI states. The entangled states exhibit different-scale fluid motions typical of those found in shear flows.

Two new families of exact coherent states are found in plane Poiseuille flow. They are obtained from the stationary and the travelling-wave mirror-symmetric solutions in plane Couette flow by a homotopy continuation. They are characterized by the mirror symmetry inherited from those continued solutions in plane Couette flow. The first family arises from a saddle-node bifurcation and the second family bifurcates by breaking the top-bottom symmetry of the first family. We find that both families exist below the minimum saddle-node-point Reynolds number known to date (Waleffe, Phys. Fluids, vol. 15, 2003, pp. 1517-1534.

In a recent paper, Deguchi & Hall (J. Fluid Mech., vol. 752, 2014a, pp. 602-625) described a new kind of exact coherent structure which sits at the edge of an asymptotic suction boundary layer at high values of the Reynolds number Re. At a distance ln Re from the wall, the structure is driven by the fully nonlinear interaction of tiny rolls, waves and streaks convected downstream at almost the free-stream speed. The interaction problem satisfies the unit-Reynolds-number three-dimensional Navier-Stokes equations and is localized in a layer of the same depth as the unperturbed boundary layer. Here, we show that the interaction problem is generic to any boundary layer that approaches its free-stream form through an exponentially small correction. It is shown that away from the layer where it is generated the induced roll-streak flow is dominated by non-parallel effects which now play a major role in the streamwise evolution of the structure. The similarity with the parallel boundary layer case is restricted only to the layer where it is generated. It is shown that non-parallel effects cause the structure to persist only over intervals of finite length in any growing boundary layer and lead to a flow structure reminiscent of turbulent boundary layer simulations. The results found shed light on a possible mechanism to couple near-wall streaks with coherent structures located towards the edge of a turbulent boundary layer. Some discussion of how the mechanism adapts to a three-dimensional base flow is given.

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (2019). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous-resistive magneto-hydrodynamic equations in plane Couette flow have been utilized. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of the plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll-streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in Deguchi (2019). When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of 'self-sustained shear driven dynamos' at the zero-external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in Deguchi (2019). Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions. arXiv:1809.03855v2 [physics.flu-dyn]

The applications and implications of two recently addressed asymptotic descriptions of exact coherent structures in shear flows are discussed. The first type of asymptotic framework to be discussed was introduced in a series of papers by Hall & Smith in the 1990s and was referred to as vortex–wave interaction theory (VWI). New results are given here for the canonical VWI problem in an infinite region; the results confirm and extend the results for the infinite problem inferred the recent VWI computation of plane Couette flow. The results given define for the first time exact coherent structures in unbounded flows. The second type of canonical structure described here is that recently found for asymptomatic suction boundary layer and corresponds to freestream coherent structures (FCS), in boundary layer flows. Here, it is shown that the FCS can also occur in flows such as Burgers vortex sheet. It is concluded that both canonical problems can be locally embedded in general shear flows and thus have widespread applicability.