## Abstract

Field observations and laboratory experiments suggest that at high Reynolds numbers *Re* the outer region of turbulent boundary layers self-organizes into quasi-uniform momentum zones (UMZs) separated by internal shear layers termed ‘vortical fissures’ (VFs). Motivated by this emergent structure, a conceptual model is proposed with dynamical components that collectively have the potential to generate a self-sustaining interaction between a single VF and adjacent UMZs. A large-*Re* asymptotic analysis of the governing incompressible Navier–Stokes equation is performed to derive reduced equation sets for the streamwise-averaged and streamwise-fluctuating flow within the VF and UMZs. The simplified equations reveal the dominant physics within—and isolate possible coupling mechanisms among—these different regions of the flow.

This article is part of the themed issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

## 1. Introduction

Turbulent wall flows, including the canonical flat-plate turbulent boundary layer (BL), exhibit quasi-coherent flow structures on a wide range of spatio-temporal scales. In the near-wall region, i.e. the viscous sublayer and buffer layer, the interaction among space-filling 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, quasi-coherent flow structures also arise in the outer region of the BL (i.e. outside the buffer layer), where the long-time mean dynamics are inertially dominated. These structures, too, have been observed and studied in laboratory experiments and DNS. Moreover, Hwang *et al.* [11,12] recently have employed over-damped large-eddy simulations strategically designed to isolate large-scale structures away from the wall for *Re*_{τ}≈100–1000. To date, however, quantitative *theories*—systematically developed from the governing incompressible Navier–Stokes (NS) equation—that describe the essential nonlinear dynamics by which self-organized flow features in the inertial layer are maintained have not been developed to the extent that is true for their near-wall counterparts. One exception is the large Reynolds number asymptotic theory developed by Deguchi & Hall [13], who demonstrate that coherent vortical structures (nearly) in the free stream of laminar BL flows that approach their free-stream speeds exponentially with distance from the wall can be self-sustained independently of near-wall excitation. (In fact, these authors show that the free-stream structures can *drive* strong streaky flows adjacent to the wall.) Nevertheless, their theory does not specifically account for flow features observed within the inertial layer, the subject of interest here.

Such quantitative theories are desirable for at least two reasons. First, they provide a framework for understanding the mechanisms controlling the formation and evolution of flow structures within the inertial layer, which ultimately may facilitate high-fidelity numerical simulations of wall-bounded flows at extreme values of *Re*_{τ} (>10^{4}) by guiding the development of customized, e.g. multi-scale, numerical algorithms. Second, elucidation of the intrinsic nonlinear dynamics and structure of the inertial layer is a necessary prerequisite for increased understanding of ‘inner–outer’ interactions, understanding that may be leveraged for the design of improved flow control strategies. As an initial step towards the development of a first-principles quantitative theory, we propose in this investigation a simplified model for a self-sustaining process (SSP) that may support quasi-coherent structures *away* from the wall in turbulent shear flows at extreme values of the Reynolds number. The model, which provides a conceptual framework for interpreting the emergence of characteristic inertial layer structures (described below), is made quantitative in the physically relevant limit using multi-scale asymptotic analysis.

Various quasi-coherent flow structures have been observed in the outer region of turbulent wall flows. These structures have length scales much larger than 100 *ν*/*u*_{τ} penetrate into or exist wholly within the inertial region, and include so-called large-scale and very-large-scale motions (LSMs and VLSMs, respectively) and superstructures [14], which exhibit spatial coherence over increasing streamwise length scales. In particular, inertial-layer superstructures, emergent flow features carrying positive and negative streamwise velocity fluctuations, have been shown to extend for roughly 5–15 *h* in the streamwise direction, to carry a significant fraction of the total turbulent kinetic energy of the flow, and to impress a modulational signature in the near-wall region [15]. Despite their likely significant impact on turbulent transport and inner–outer interactions, the origin of these various large-scale structures is not wholly clear. For example, it has been argued that LSMs and VLSMs arise from the spontaneous organization of attached and/or detached hairpin vortices and hairpin vortex packets [14]. Here, we investigate a complementary possibility, namely that certain large-scale quasi-coherent flow structures in the outer region of turbulent wall flows arise *directly* from a self-sustaining, multiple space and time-scale process, in loose analogy with the way in which near-wall streaks and vortices have been argued to arise from a single-scale (uniformly space-filling) SSP [1]. It should be emphasized that a similar thesis has been advanced recently by Hwang and co-workers [11,12,16], but here we focus specifically on developing a semi-analytical, mechanistic description that has the potential to explain the formation and maintenance of uniform momentum zones (UMZs) and interlaced ‘vortical fissures’ (VFs), arguably the primal coherent structures in the outer part of turbulent wall flows at extreme Reynolds number.

As first documented by Meinhart & Adrian [17], the wall-normal structure of the *instantaneous* streamwise velocity in a turbulent BL exhibits a staircase-like variation, with relatively uniform regions (UMZs) segregated by VFs across which there are discernible jumps in the streamwise velocity; that is, the VFs essentially are internal shear layers. A spate of more recent investigations [18,19] has confirmed and further quantified the basic conception of UMZs advanced in the pioneering study by Meinhart & Adrian [17]; see figure 1*a*,*b*, which is adapted from de Silva *et al.* [19]. These latter authors also show that the number of UMZs (and VFs) increases logarithmically with increasing Reynolds number. Klewicki [20,21] has proposed a kinematic description of the spanwise vorticity associated with this staircase UMZ/VF structure that is consistent both with the mean momentum equation and with the intriguing notion that the turbulent BL comprises logarithmically many viscous internal layers containing most of the vorticity and, hence, dissipation. In a related study, Cuevas *et al.* [22] demonstrate that the logarithmic profile and other statistical features associated with the long-time mean streamwise flow can be recovered by ensemble averaging staircase-like streamwise velocity profiles with semi-empirically determined distributions for the spacing between and magnitude of the velocity jumps (or ‘steps’).

Motivated by observations of the preponderance of and likely fundamental dynamical role played by UMZ/VF structures in the inertial region of turbulent wall flows, we propose, in §2, a conceptual model of a single (geometrically) idealized VF and adjacent UMZs. A large Reynolds number asymptotic analysis is presented in §3 to render this model quantitative. The analysis exploits the emergence of distinct dominant balances of forces in different subdomains of the flow and yields simplified (‘reduced’) equations that govern the flow in each region, which then must be coupled via suitable matching conditions. This rather technical section is followed in §4 by a summary of the resulting reduced model and a description of a suitable iterative solution algorithm; here, the global structure of the nonlinear couplings characterizing the underlying SSP is laid bare. (Some readers may prefer initially to skim §3 and subsequently to return to this section after reading §4.) We conclude in §5 with a brief discussion of the mathematical and physical attributes of the present SSP model, including its connection to observed BL features.

## 2. Conceptual model

Perhaps the most fundamental question to be addressed is why, in the first instance, should regions of quasi-uniform momentum appear in a sheared flow? To this end, we note that UMZs are observed not only in flat-plate turbulent BLs (for which, in the absence of turbulent motions, the flow indeed would be quite uniform except within an exquisitely thin laminar BL adjacent to the wall) but also in plane Poiseuille and other canonical wall flows whose laminar states are sheared on *O*(*h*) wall-normal length scales. The innovative large Reynolds number analysis by Deguchi & Hall [13] discussed in §1 exploits a presumed pre-existing shear-layer or laminar-BL structure but does not yield insights into physical processes that would be capable of inducing sharply varying velocity profiles were those profiles not supported by external agencies. The hypothesis we propose in this investigation is based on our interpretation that large-scale structures in the outer region consist of streamwise (*x*-directed) vortices and streaks, much like the streamwise vortices and streaks that occur in the near-wall region, with the streaks being associated with spanwise anomalies in streamwise flow speed between laterally adjacent UMZs and the streamwise vortices accounting for a comparably weak rotary flow in wall-normal/spanwise (*y*/*z*) or ‘perpendicular’ (⊥) planes within the UMZs. However, we postulate a number of crucial differences between near-wall and inertial-layer streaks and vortices, as discussed next.

First, even at asymptotically large Reynolds number, the streaks associated with lower- and also upper-branch exact coherent states (ECS) arising in the SSP theory of Waleffe and co-workers [5,23] and in the large Reynolds number limit of this theory independently developed by Hall and co-workers [6,7] (who employ the terminology ‘vortex–wave interaction’ or VWI theory; see Chini [24]), the *x*-averaged streaky streamwise flow varies smoothly throughout the entire spatial domain. 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*.

Consistent with this requirement, we demonstrate in §3a that the large-*Re*_{τ} limiting form of the *x*-averaged streamwise momentum equation reduces to a two-dimensional advection–diffusion equation (i.e. in the ⊥-plane) in the weak diffusion limit for the averaged streamwise flow, with a prescribed advecting velocity field generated by the streamwise roll motions, i.e. the streaky flow essentially behaves as a passive scalar field at large effective ‘Peclet’ number. Crucially, numerous studies of two-dimensional scalar advection by a counter-rotating cellular velocity field in the weak diffusion (large Peclet number) limit confirm that the scalar field is strongly homogenized [25,26,27], *thereby providing a possible mechanism for the observed quasi-uniformity of the streamwise momentum within UMZs*. Empirical support for this proposition is provided by the DNS results of Papavassiliou & Hanratty [28], which confirm that low-momentum regions of large-scale (inertial layer) structures in turbulent plane Couette flow are separated by the vortex cores of nearly inviscid streamwise roll modes, implying a large effective Peclet number for the advected streamwise flow.

Second, the kinematic configuration of vortices and streaks in the near-wall region differs from that which we propose in our simplified model of inertial-layer dynamics. Motivated by the conceptual picture of UMZs and VFs illustrated in the schematic (adapted from Priyadarshana *et al.* [29]) shown in figure 2*a*, we postulate that away from the wall a spanwise array of streamwise-aligned, counter-rotating roll vortices stacked in the wall-normal direction and exposed to a background (e.g. laminar) wall-normal shear flow will—at large effective Reynolds number—naturally induce UMZs separated by VFs (figure 2*b*). If realized, this configuration would thereby provide a mechanism for the formation and sustenance of the internal shear layers exhibited by the streamwise velocity field. In the absence of the straining provided by the vortices, the shear layers (and, hence, VFs) ultimately would disintegrate owing to instability and viscous diffusion. To accord with observations, the stacked vortices must maintain the VFs through an advective/diffusive balance in which the jump in the streamwise-averaged streamwise velocity is *O*(*u*_{τ}) (figure 1). Owing to this kinematic configuration and the dynamics by which it is sustained, the SSP underlying our simplified model of inertial layer motions inherently involves *multiple spatial scales*, also in contrast with the near-wall SSP.

Finally, the feedback mechanism by which the streamwise rolls are sustained in the near-wall SSP differs from that conceived here for their inertial-layer counterparts. For the SSP operative in the near-wall region, the rolls are maintained by nonlinear interactions among *x*-varying Rayleigh instabilities arising from spanwise inflections of the streaky streamwise flow [5,7]. In contrast, in the model contemplated here, the rolls within the UMZs are triggered by instabilities within the VFs, i.e. outside the UMZs, because the fissures are the sites of strong wall-normal inflections and thus are prone to Rayleigh instabilities with disturbances predominantly varying in *x*/*y* planes. Although linear stability considerations indicate that the fastest-growing disturbances have streamwise wavelengths comparable to the (dimensional) VF thickness Δ_{f}, these modes cannot directly couple with motions in the UMZs because their wall-normal extent similarly scales with Δ_{f}, implying that such disturbances are confined to the fissure. In contrast, disturbances with streamwise wavelengths long compared with the fissure thickness induce pressure fluctuations that penetrate well outside the VF. Moreover, the near-marginal instabilities most relevant to an SSP are, in fact, long-wavelength modes. As shown in §3, the *x*-varying streamwise velocity component associated with these long-wavelength near-marginal modes is amplified within the VF, enabling the fluctuation dynamics to nonlinearly couple with the streamwise-averaged flow inside the fissure.

## 3. Large Reynolds number analysis

In this section, we perform a large Reynolds number asymptotic analysis of the NS equation to derive simplified equations governing the flow configuration shown in figure 2*b*. In the idealization considered here, the domain is imagined to have been excised from a large-*Re*_{τ} flow sufficiently far from the wall that the *x*-averaged streamwise velocity component varies by only *O*(*u*_{τ}) across an outer wall-normal distance *l*_{y}<*h* comparable to the wall-normal spacing between adjacent VFs, except near the fissures, where the *x*-averaged streamwise velocity also varies by *O*(*u*_{τ}) but across a distance comparable to the fissure width Δ_{f}.

Scaling all velocity components by *u*_{τ} and lengths by *l*_{y}, the dimensionless NS equation can be expressed as
3.1
which together with the incompressibility constraint ∇⋅**u**=0 constitute the ‘master’ equations. In equation (3.1), **u**=(*u*,*v*,*w*) and *p* are the velocity vector and pressure, respectively, and *Re*≡*u*_{τ}*l*_{y}/*ν* is a Reynolds number defined using the mean wall-normal spacing *l*_{y} between adjacent fissures rather than the outer length scale *h*, but which nevertheless is numerically of the same order of magnitude as *Re*_{τ}. The body force per unit mass , where is a unit vector in the *x*-direction, is included to drive the required background (laminar) shear flow that would exist in the absence of the turbulent motions.

We choose a local Cartesian coordinate system moving with the presumed planar VF (a geometrical configuration perhaps more likely to be relevant to VFs in fully developed internal flows, such as plane Poiseuille flow; see Eisma *et al.* [18]), which is taken to be coincident with the plane *y*=0. The fissure, or internal shear layer, has *O*(Δ) dimensionless thickness, where Δ(*Re*)→0 as . It transpires that in addition to the internal shear layer an even thinner ‘inner’ region, a critical layer (CL), emerges with thickness *O*(*δ*), where *δ*=*o*(Δ) as (figure 3). We separately analyse the flow within the UMZs above (*y*>0, denoted ‘+’) and below (*y*<0, denoted ‘−’) the fissure, within the internal shear layer and within the CL in the limit of large Reynolds number.

### (a) Uniform momentum zones

First, we analyse the flow within UMZ^{+} and UMZ^{−}. As for near-wall streaks and vortices, we anticipate that at large *Re* a comparably weak vortex flow of size , where as , exposed to an *O*(1) dimensionless wall-normal shear can induce an *O*(1) streaky flow: a streamwise-averaged streamwise flow that varies in the spanwise (*z*) direction. Thus, we posit the following expansions for the velocity and pressure fields:
3.2
where we have introduced a slow time scale , so that , and we have decomposed all fields into a streamwise (*x*) and ‘fast-time’ *t* average, denoted with an overbar, and a fluctuation with zero streamwise/fast-time mean, denoted with a prime. The expansion parameter for the fluctuation fields *a*′(*Re*)→0 as , and the numerical subscripts indicate the *a posteriori* determined scaling of the field variables with respect to *Re*^{−1} (table 1). As noted in §2 and suggested by figure 3*b*, the fluctuating streamwise velocity component induced by long-wavelength disturbances on an internal shear layer characterized by an *O*(1) jump in dimensionless mean streamwise flow speed is amplified by the factor 1/Δ within the region of strong shear; therefore, we require the leading-order fluctuating streamwise velocity component to be *O*(*a*′), where *a*′≤Δ, within the UMZs. The three-dimensional incompressibility condition then constrains the size of the perpendicular fluctuation velocity components and the fluctuation pressure also to be *O*(*a*′).

Substituting these expansions into the NS equation (3.1) and the incompressibility condition and performing an *x*/*t* average yields the leading-order equations for the streamwise-averaged flow within the UMZs:
3.3
where is the component of the mean body force. Presuming as , as confirmed *a posteriori* in §3c, viscous diffusion formally is subdominant in, and hence absent from, the leading-order mean momentum equations. The expected Reynolds stress divergence (RSD) terms also are subdominant in the mean momentum equations. Together these considerations confirm the assertion made in §2 that, given the roll velocity field [,] within the UMZs, the streak velocity component behaves as a passive scalar field at large effective ‘Peclet’ number. The counter-rotating cellular velocity field associated with the rolls then may be expected to largely homogenize the streamwise-averaged streamwise flow, thereby creating a UMZ. For this same reason, however, the leading-order mean momentum equations within the UMZs must be regularized by retaining ⊥-Laplacian diffusion: through a shear dispersion mechanism, large gradients in the *y*–*z* plane are generated, first along mean ⊥-streamlines and, ultimately, across them [26]. To properly capture this phenomenon, the regularization terms , respectively, should be included on the right-hand sides of the first three equations in system (3.3), where the perpendicular Laplacian operator . Accordingly, the effective Reynolds number 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 shows a numerical solution of the regularized -equation for specified roll motions ]; note the emergence of UMZs and an interlaced VF.

Subtracting equations (3.3) from the instantaneous NS and continuity equations yields the leading-order fluctuating momentum and continuity equations within the UMZs:
3.4
We note that this system, which is *quasi-linear* about the comparably slowly temporally varying streak velocity , is identical to that first derived by Hall & Horseman [30]. Because equations (3.4) are autonomous in *x* and *t*, a normal-mode ansatz can be made
3.5
where ‘c.c.’ denotes the complex conjugate, 2*π*/*α* is the streamwise wavelength of the mode and *c* is its complex phase speed. Analogous representations are employed for the other fluctuation fields. Substituting this ansatz into system (3.4) yields a generalized Rayleigh equation [30] for the Fourier-transformed pressure fluctuation ,
3.6
For the envisaged flow configuration, we can, without further loss of generality, choose our coordinate system and forcing function so that within UMZ^{+} or UMZ^{−}; thus, equation (3.6) is not singular within the UMZs.

### (b) Internal shear layer (VF)

Given smoothly varying roll motions within the UMZs, equation (3.1) indicates that a simple advective/wall-normal-diffusive balance is attained when wall-normal gradients in the *x*-averaged streamwise velocity occur over a wall-normal length scale . To analyse the flow within this internal shear layer (or VF), we introduce a rescaled wall-normal coordinate , so that , and posit the following asymptotic expansions for the various fields:
3.7
The scalings used for the leading-order mean fields can be rationalized as follows. Presuming that within the UMZs linearly in *y* as *y*→0, then within the internal shear layer. By continuity, within this layer, while for matching with the mean flow in the UMZs and . Owing to a Rayleigh instability of the shear layer, the leading-order fluctuating streamwise velocity component *u*′=*O*(*a*′)×*O*(1/Δ)=*O*(*a*′/Δ); i.e. this velocity component is amplified relative to its size within the UMZs (figure 3*b*). Continuity and matching with the fluctuating fields in the UMZs then requires *v*′=*O*(*a*′) and *p*′=*O*(*a*′). Similarly, matching and analysis of the three-dimensional Rayleigh equation indicates that *w*′=*O*(*a*′), implying that the size of *w*′ within the VF is *not* constrained by incompressibility (again, see figure 3*b*).

Substituting expansions (3.7) into the *x*-component of the NS equation (3.1), collecting terms at leading order and averaging yields
3.8
where . Comparing equation (3.8) with the first equation of system (3.3) confirms that, in addition to unsteady advection, wall-normal diffusion arises at leading order within the internal shear layer (where the effective Reynolds number is unity). The leading-order mean perpendicular momentum equations require and
3.9
The mean wall-normal velocity then is self-consistently obtained from the incompressibility constraint: .

The leading-order fluctuation equations within the internal shear layer are
3.10
along with and the leading-order incompressibility condition ; again, recall that, although ∂_{x}∼∂_{z}=*O*(1), the size of *w*′ within the VF is not constrained by three-dimensional incompressibility. This set of fluctuation equations comprises a form of Rayleigh’s equation in the limit of long-wavelength disturbances. After Fourier transforming the fluctuation fields, using notation analogous to that used in equation (3.5), we can identify a simple quasi-steady solution to these equations, namely
3.11
where the amplitude function *A*=*A*(*z*,*T*) is undetermined at this stage of the analysis. This solution is of particular relevance to *equilibrium* ECS (in the chosen convecting reference frame). Because is independent of , matching with the fluctuation pressure field in the UMZ yields . Using equation (3.11), it is readily deduced that the Reynolds stress component , where the asterisk denotes complex conjugation. Note that this expression is independent of ; consequently, the RSD term in equation (3.9) vanishes. The expression for , however, shows that the spanwise fluctuating velocity component diverges as , because must pass through zero at by the symmetry of the envisaged flow configuration. This CL singularity is healed within an even thinner layer, where physical effects not retained in the leading-order fluctuation equations in the internal shear layer become significant.

### (c) Critical layer

Presuming as (as shown to be self-consistent below), then , where *δ*=*o*(Δ) is the thickness of the CL. Matching with the mean fields in the internal shear layer requires and , while incompressibility then yields . Similarly, given the hypothesized behaviour of as , inspection of equations (3.11) shows that within the CL the fluctuation fields must have the following magnitudes: *u*′=*O*(*a*′/Δ), *v*′=*O*(*δa*′/Δ) and *w*′=*O*(*Δa*′/*δ*), while *p*′=*O*(*a*′). Because *w*′ is amplified within the CL, whereas *u*′ is not, three-dimensional incompressibility can be restored within this layer by requiring *δ*=Δ^{2}.

The thicknesses of both the critical and internal shear layers can now be determined by considering the dominant balance of terms in the fluctuating spanwise momentum equation. To heal the singularity in as , wall-normal diffusion must be comparable to *x*-advection within the CL, yielding the relation Δ=*δ*^{3} *Re*. Substituting *δ*=Δ^{2} then gives the thickness of the internal shear layer Δ=*Re*^{−1/5} and, thence, the thickness of the CL *δ*=*Re*^{−2/5}. Recalling that the mean advective/diffusive balance in the internal shear layer requires , we deduce that the roll amplitude . Finally, the amplification of *w*′ within the CL renders the RSD component sufficiently large to drive an *x*-averaged flow in the spanwise direction provided that the size *a*′ of the fluctuations is tuned appropriately. The key observation is that because the CL is internal to the flow (i.e. away from any walls) matching of the *x*-mean *x*-vorticity between the VF and the CL must be enforced. Consequently, within the CL, is a factor of *δ*/Δ smaller than the expected estimate of . Using this refined scaling, it is then straightforward to show that the RSD component balances wall-normal diffusion of mean spanwise momentum within the CL if *a*′=*Re*^{−7/10}. Thus, , confirming that the RSD driving the rolls is concentrated within the CL. Table 1 summarizes the implied scalings of the various fields in each of the three subdomains.

Denoting the inner fields with capital letters, e.g. *u*=*U*(*x*,*Y*,*z*,*τ*,*T*;*Re*), where *Y* ≡*Re*^{2/5}*y* and the intermediate time scale *τ*≡*Re*^{−1/5}*t*, the leading-order mean equations within the CL reduce to
3.12
The solution to the first of equations (3.12) is seen to be consistent with the assumed form of the mean streamwise velocity component in the internal shear layer as ; this asymptotic matching procedure determines the mean shear λ(*z*,*T*) within the CL. The factor of *Re*^{−1/5} in the third equation, arising from the refined scaling estimate discussed above, ensures the smooth matching of between the outer part of the CL and inner part of the VF. As in the VWI analysis of Hall & Sherwin [7], integration of this equation shows that there is a jump in across the CL, namely
3.13
where the decay of as has been employed, and the superscript plus and minus symbols have been used to explicitly distinguish VF fields above and below the CL, respectively.

The leading-order fluctuation equations within the CL simplify to
3.14
where the constancy of the leading-order fluctuation pressure across the CL has been used in the first and third equations. The solution of the fluctuation *z*-momentum equation is of particular interest, because the self-correlation (in *x*) of *W*′_{1/2} drives a mean spanwise flow within the internal shear layer (VF) through the jump condition (3.13). In turn, the mean spanwise flow ultimately induces roll motions outside the VF via incompressibility (and viscous momentum transport).

Restricting attention to equilibrium ECS, we Fourier transform in *x* the steady version of the third of equations (3.14) to obtain an inhomogeneous Airy-like equation for the Fourier-transformed spanwise fluctuation velocity component ,
3.15
Defining *s*≡(*αλ*)^{1/3}*Y* , the solution to equation (3.15) can be expressed in terms of the function Yi(*s*) introduced by Balmforth *et al.* [31], where
3.16
satisfies the ordinary differential equation Yi′′−i*s*Yi=1/*π* (see also Hall & Sherwin [7]). Thus,
3.17
Because Yi(*s*)∼i/(*πs*) as [31], the fluctuating spanwise velocity component decays in the far field of the CL to smoothly match with its functional form in the internal shear layer. Figure 5, adapted from Balmforth *et al.* [31], shows the real and imaginary parts of Yi(*s*).

## 4. Summary of reduced model

The large-*Re* analysis described in §3 suggests the possibility of an inertial-layer SSP in which rolls within the UMZs are driven by the RSD acting in a CL of thickness *δ*=*Re*^{−2/5} that itself is embedded within an internal shear layer (i.e. a VF) of thickness Δ=*Re*^{−1/5}. The *O*(*Re*^{−3/5}) counter-rotating rolls, stacked in the wall-normal direction, are sufficiently strong to differentially homogenize the *O*(1) ambient shear and maintain the VF. Once the mean streamwise momentum is homogenized within the UMZs, the *x*-varying fluctuations may be expected to be largely irrotational (as the generalized Rayleigh equation, equation (3.6), reduces to Laplace’s equation for spatially uniform ). It should also be noted that, in accord with the Prandtl–Batchelor theorem for steady two-dimensional cellular flows at large (effective) Reynolds number [32], the *x*-mean *x*-directed vorticity also may be expected to be largely uniform within the UMZs. The key elements of the proposed SSP are summarized in figure 6.

To demonstrate the closure of this feedback loop, we next outline an algorithm to compute the global flow structure. We begin by discussing the determination of the (*x*-Fourier-transformed) fluctuation pressure within UMZ^{+} (say) via the solution of the generalized Rayleigh equation, equation (3.6), subject to appropriate boundary conditions. We seek solutions for the fluctuating pressure in the UMZ that are 2*π*/*β*-periodic in the *z* direction and that match with the fluctuating pressure in the internal shear layer. The matching condition on the pressure fluctuation can be enforced indirectly by matching the *O*(*Re*^{−7/10}) fluctuating wall-normal velocity components. Evaluating the *y*-component of the fluctuation momentum equations in UMZ^{+} as *y*→0^{+}, we obtain (in steady state)
4.1
upon using the second of equations (3.11). An appropriate boundary condition on is also required at the upper boundary of UMZ^{+} (i.e. as *y*→1^{−}). Specification of this condition depends upon the physical model being contemplated. If, as suggested in figure 2, the fluid domain is imagined to have been excised from the inertial layer of a turbulent wall flow, then the location *y*=1 would correspond to a second fissure and so a boundary condition analogous to equation (4.1) would seem appropriate. Indeed, enforcing a symmetry constraint suggests that . Moreover, owing to the homogenizing action of the rolls in UMZ^{+}, as *y*→0^{+} and *y*→1^{−}, where symmetry suggests the constant as (figure 4*b*). We leave more detailed consideration of these boundary conditions and their modelling implications to a future investigation.

From boundary condition (4.1), the magnitude of the Fourier-transformed pressure fluctuation is proportional to some scalar (i.e. *z*-independent) measure of the amplitude function *A*(*z*) arising in equations (3.11), which we take to be
4.2
Thus, controls the strength of the RSD driving the mean spanwise flow within the critical and internal shear layers and, by incompressibility, the intensity of the roll motions induced within the UMZs. 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* and treat the streamwise wavenumber *α* as a scalar unknown that must be determined via the solution of a nonlinear eigenvalue problem. This eigenvalue problem is formulated from the steady version of equations (3.14) governing the fluctuation dynamics within the CL. Taking the *Y* -derivative of the first of these equations, using the fourth equation to eliminate ∂_{Y}*V* ′_{9/10}, and Fourier transforming the resulting equation yields
4.3
where and equation (3.17) was used to eliminate . Equation (4.3) is solved subject to the matching conditions as . By integrating in *Y* and ensuring a smooth match with in VF^{+}, an ordinary differential eigenvalue problem for *A*(*z*) and *α* is obtained (see below).

As noted previously, the mean streamwise flow will be strongly homogenized (see the discussion in §2) except within asymptotically thin, spanwise-localized jet-like regions of positive and negative anomolies in the mean streamwise speed (and, of course, within the fissures). Thus, we simplify the following discussion of the global solution algorithm by setting (in UMZ^{+}), i.e. a constant known from symmetry considerations—a more refined analysis would properly treat the variability of within the UMZs. For brevity of notation, we omit the subscripts from the field variables in the following algorithm.

For a given spanwise domain length 2*π*/*β* and scalar fluctuation amplitude :

(i) Generate an initial iterate for λ(

*z*), the streak-induced shear within the CL.(ii) Obtain an integral representation of the solution to Laplace’s equation to which the generalized Rayleigh equation (equation (3.6)) reduces for uniform , for within the UMZ

^{+}domain*y*∈(0^{+},1^{−}) and*z*∈[0,2*π*/*β*], subject to the boundary conditions (see the discussion surrounding equation (4.1)) Denote this solution, which is a functional of*A*(*z*) and depends on*α*, by .(iii) Solve the ordinary differential eigenvalue problem for

*A*(*z*) and*α*obtained by setting and by imposing the normalization constraint given by equation (4.2), where is determined via solution of equation (4.3) and is a functional of*A*(*z*) through , subject to a 2*π*/*β*-periodicity requirement in*z*. ( can be related to*A*(*z*) via further manipulation and partial solution of equations (3.14).)(iv) Time-advance to steady state the mean equations for the ⊥-flow within the VF. These equations (equation (3.9), noting that the RSD term vanishes; incompressibility; and the -independence of the mean pressure) can be cast in stream function/vorticity form, where and , and must be solved subject to the symmetry and matching conditions respectively. Note that the solution for in UMZ

^{+}is readily obtained up to a multiplicative constant via the solution of a Poisson equation for the UMZ stream function*ψ*(*y*,*z*), , where the right-hand side is constant owing to the Prandtl–Batchelor constraint, subject to*ψ*→0 around the periphery of each cell. We leave for future investigation the determination of this constant value of the core vorticity but anticipate that the imposition of global constraints, including a global energy balance and/or constancy of the wall-normal flux of mean streamwise momentum, may be useful [27].(v) Time-advance to steady state the mean equation for the streaky flow within the VF (equation (3.8)), subject to the symmetry and matching conditions at and as .

(vi) Evaluate the streak shear within the CL, , and return to step (ii) until convergence is achieved to within a desired tolerance.

## 5. Conclusion

Motivated by the occurrence of zones of quasi-uniform streamwise momentum (UMZs) separated by thin layers of concentrated spanwise vorticity (VFs) in the outer region of turbulent wall flows at extreme values of *Re*_{τ}, we have performed a large Reynolds number asymptotic analysis of the incompressible NS equation to investigate whether an SSP can *directly* support such flow structures. This possibility should be contrasted with the notion that large-scale, outer-region structures result from the spontaneous concatenation of smaller-scale flow features. In this work, we have taken an important step towards confirming the former scenario by deriving from the instantaneous NS equation a self-consistent reduced model of a single planar VF coupled to adjacent UMZs above and below the fissure. The analysis suggests a three-region asymptotic structure in the wall-normal direction, with: outer UMZs in which the streamwise-averaged (‘mean’) and streamwise-varying (‘fluctuation’) dynamics are largely inviscid; an internal shear layer across which the mean streamwise flow exhibits a sharp, step-like variation; and an even thinner CL (within the shear layer) in which the spanwise fluctuating velocity component is amplified. The wall-parallel gradient of the streamwise self-correlation of this velocity component drives a mean spanwise flow within the CL; this spanwise flow is viscously transmitted to the internal shear layer and, by continuity (and further viscous action), ultimately transformed into roll motions outside the shear layer. The induced rolls are weak compared with the background ambient shear (i.e. in dimensional terms, roll velocities are small compared with the friction velocity *u*_{τ}) but, unlike prior large Reynolds number asymptotic studies of ECS and the associated SSPs, the rolls are sufficiently strong to differentially homogenize the resulting streaky streamwise flow. The straining provided by the counter-rotating rolls, which are stacked in the wall-normal direction, reinforces the VF and prevents its diffusive disintegration. Of course, to verify the proposed SSP scenario, solutions of the reduced system of equations must be numerically constructed following the algorithm outlined in §4 (and, ideally, compared with full NS solutions). We hope to report on these computations in a subsequent publication.

Detailed comparisons of the proposed theory with measurements of turbulent flows from laboratory experiments are not yet feasible. Nevertheless, we can compare the predicted scaling of the VF thickness with that extracted from laboratory and field data. In particular, Klewicki [20] estimates that the dimensional fissure thickness Δ_{f} normalized by the outer length scale (labelled *h* in this study) varies in direct proportion to using data that span more than two decades in *Re*_{τ}. By contrast, our theory indicates that the corresponding dimensionless thickness of the internal shear layer Δ=*O*(*Re*^{−1/5}), where *Re*=(*l*_{y}/*h*) *Re*_{τ}; i.e. the theory evidently predicts a thicker fissure than has been observed. One potential explanation for this discrepancy is the necessity of arbitrarily choosing thresholds for the quantification of Δ_{f} from raw empirical data. It is also plausible that distinct large-*Re* reduced models, based on different scalings, may yield a fissure thickness that accords with the measured Δ_{f}/*h*. Even in that circumstance, we conjecture that elements of the global flow structure predicted by the present analysis still may be realized (e.g. the homogenizing action of the rolls). It is also worth noting that the present theory does, in fact, predict an inner region of thickness *O*(*Re*^{−2/5}), but the streamwise-averaged streamwise velocity is not sheared over this length scale. Finally, the conclusion reached above regarding the apparent discrepancy in the predicted and measured fissure thicknesses tacitly presumes the constancy of the length-scale ratio *l*_{y}/*h* when, in fact, this ratio appears to be a weak function of *Re*_{τ}.

More qualitatively, we comment that the UMZ/VF flow structure analysed here may constitute a ‘straight’ section of a longer superstructure that slowly meanders in the streamwise direction, and that the coupling mechanisms elucidated herein have the potential both to complement those described by the modelling framework of Sharma *et al.* [35] and to connect to the features observed in the numerical and physical experiments of Cossu & Hwang [36] and Baars *et al.* [37], respectively. In contrast to the recent work by Hwang *et al.* [11,12,16] discussed in §1, this study captures only a single interacting UMZ/VF ‘unit’. Nevertheless, based on the self-similar hierarchical rescaling of VWI states performed by Blackburn *et al.* [38] and of ‘resolvent modes’ by Moarref *et al.* [39], we speculate that the flow structures investigated here may constitute one member of a family of similar motions on a hierarchy—the varying spacing between VFs with distance from the wall will naturally generate a hierarchy of scales.

We conclude by commenting again on the relationship between the more well-studied near-wall and the newly proposed outer-region SSPs and associated ECS. In the near-wall region, the streamwise-averaged flow structures, i.e. rolls and streaks, are space filling and single scale even as . By contrast, the proposed SSP supports a multi-scale flow structure away from the wall in that limit. Continuing with this dichotomy, it is helpful to recall that Hall & Sherwin [7] describe the near-wall SSP they study as arising from a novel variant of ‘steady streaming’ within a CL, i.e. the single-wavenumber fluctuation/fluctuation nonlinearity within the CL drives a mean spanwise component of the roll flow. Because the *effective* Reynolds number felt by the mean flow is *O*(1), the streaming could be described more precisely as being of the Rayleigh–Nyborg–Westervelt variety [40,41,42]. The proposed SSP also may be viewed as arising from a novel form of steady streaming—but in the limit in which the effective Reynolds number is large. This so-called ‘Stuart’ streaming [43] necessarily involves the emergence of *nested* boundary or, here, internal layers. In summary, the self-sustaining interaction investigated here clearly has important dynamical connections both with the VWI exhibited by specific ECS in Couette flow and analysed by Hall & Sherwin [7] and with a vortex/Tollmien–Schlichting wave interaction arising in plane Poiseuille flow and analysed by Dempsey *et al.* [34]. Indeed, the present self-sustaining process theory blends in a novel fashion elements of each of these analyses, while also including an irreducible and essentially inviscid ‘core’ dynamics that accords with the behaviour observed in the outer region of turbulent wall flows at extreme values of *Re*_{τ}.

## Authors' contributions

G.P.C. conceived of and with B.M. carried out the asymptotic analysis and computations and drafted the manuscript. J.K. conceived of the study and with C.M.W. participated in the model development. All authors read and approved the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

This material is based upon work supported by the National Science Foundation under grant no. CBET-1545564. J.K. also acknowledges the support of the Australian Research Council under grant no. DP150102593.

## Acknowledgements

G.P.C. acknowledges useful discussions with W. Young and N. Balmforth at an early stage of this work.

## Footnotes

One contribution of 14 to a theme issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

- Accepted December 21, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.