3.1 Length scales and the classification into layers

The length scales of turbulence range from a small limit of the order of a few viscous Kolmogorov lengths, $\unicode[STIX]{x1D702}$ , to a large limit of the order of the integral length, $L_{\unicode[STIX]{x1D700}}$ , and it follows from the definitions in the previous section that $L_{\unicode[STIX]{x1D700}}/\unicode[STIX]{x1D702}=Re_{L}^{3/4}$ . This ratio is usually large, and there is an intermediate ‘inertial’ range in which neither of the two scales is important, and where eddies can only be self-similar.

Figure 3.

(a) The line contours are the premultiplied spectrum of the kinetic energy $\unicode[STIX]{x1D719}_{qq}/{u_{\unicode[STIX]{x1D70F}}}^{2}=0.5(1)4.5$ , as a function of the wall-parallel wavelength, $\unicode[STIX]{x1D706}_{I}$ , and of the distance from the wall. The shaded contours are the normalised premultiplied spectrum of the vorticity magnitude $\unicode[STIX]{x1D708}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}}/\unicode[STIX]{x1D700}=0.5(1)4.5$ . Channel CH5200. - - -, $\unicode[STIX]{x1D706}_{I}=5x_{2}$ ; – ⋅ – ⋅ –, $\unicode[STIX]{x1D706}_{I}=25\unicode[STIX]{x1D702}$ . (b) Turbulent boundary layer BL6600 at $h^{+}\approx 1800$ . The large central object is an isosurface of the streamwise velocity $(u_{1}^{+}=2)$ . It is approximately $2.5h$ long, and spans most of the thickness of the layer. The smaller objects are vortices visualised as isosurfaces of the discriminant of the velocity gradient. Flow is from left to right (picture credits, J. A. Sillero).

The best-known self-similar range is the Kolmogorov (Reference Kolmogorov1941) inertial energy cascade, but more relevant to our discussion is the logarithmic layer of wall-bounded flows. For an equilibrium shear flow that is statistically homogeneous in the streamwise and spanwise directions, the conservation of streamwise momentum can be written as (Tennekes & Lumley Reference Tennekes and Lumley1972),

where $P=\langle \widetilde{p}\rangle$ is the ensemble-averaged kinematic pressure. Far enough from the wall for viscosity to be unimportant, $x_{2}^{+}\gg 1$ , but close enough for pressure gradients and other streamwise non-uniformities to be negligible, $x_{2}\ll h$ , (3.1) requires that the tangential Reynolds stress satisfies $-\langle u_{1}u_{2}\rangle \approx u_{\unicode[STIX]{x1D70F}}^{2}(1-x_{2}/h)\approx u_{\unicode[STIX]{x1D70F}}^{2}$ . This implies that the ‘correlated’ parts of $u_{1}$ and $u_{2}$ scale with $u_{\unicode[STIX]{x1D70F}}$ , and suggests that the same should be true for the full intensities, $u_{j}^{\prime }\sim q\sim u_{\unicode[STIX]{x1D70F}}$ . In addition, neither the flow thickness nor viscosity can be relevant in this range of wall distances, and there is no length scale. The result is that structures in this intermediate region can only have a characteristic aspect ratio, but not a characteristic size, and that the size of the largest momentum- and energy-carrying eddies has to grow linearly with $x_{2}$ .

It follows from these considerations that the possible structures of wall-bounded flows are stratified in scale space by their size, and in position by their distance from the wall, and that the flow can be approximately classified into three layers: a viscous or buffer layer, where all eddy sizes scale in wall units; an outer layer, where the length scale of the energy and of the energy production is the flow thickness $h$ ; and a scale-less intermediate layer in which the length scale of the energy production is proportional to $x_{2}$ . Everywhere, the velocity scale is $u_{\unicode[STIX]{x1D70F}}$ , and the dissipation length is $\unicode[STIX]{x1D702}$ .

Spectra of the energy and enstrophy are presented in figure 3(a) in terms of an ‘isotropic’ wall-parallel wavelength

which represents the size of the eddy, but neglects for the moment the possible anisotropies of the flow. Each horizontal section of this figure is a spectral density at a given distance from the wall and, as expected, the peak of the enstrophy spectrum is everywhere near some small multiple of the Kolmogorov scale, $\unicode[STIX]{x1D706}_{I}^{+}\approx 25\unicode[STIX]{x1D702}^{+}\approx 20(x_{2}^{+})^{1/4}$ . For these small scales, most of the enstrophy is concentrated near $k_{1}\approx k_{3}$ , so that $\unicode[STIX]{x1D706}_{I}\approx \unicode[STIX]{x1D706}_{3}/\sqrt{2}$ and the above relation is equivalent to $\unicode[STIX]{x1D706}_{1}\approx \unicode[STIX]{x1D706}_{3}\approx 35\unicode[STIX]{x1D702}$ (Jiménez Reference Jiménez2013b ). On the other hand, the scale of the energy-containing eddies grows linearly away from the wall. We will see later that $k_{1}\ll k_{3}$ for these eddies, so that $\unicode[STIX]{x1D706}_{I}\approx \unicode[STIX]{x1D706}_{3}$ .

The boundary between the intermediate and outer layers, conventionally taken to be $x_{2}/h\approx 0.2$ , is defined by the end of the linear growth of the size of the energy-containing eddies. The transition between the intermediate and buffer layers is defined by the level, $x_{2}^{+}\approx 80$ ( $x_{2}\approx 0.015h$ in figure 3 a), at which the length scales of the energy and of the enstrophy become comparable. Below this point, viscosity is important for all eddies, including the energy-containing ones. Above it, the vortices contain very little energy, and we will see below that they do not participate in the energy production.

A flow snapshot displaying the separation between the energy and dissipation scales is figure 3(b), where a perturbation velocity isosurface is shown together with the much smaller vortices. Even at this moderate Reynolds number $(h^{+}=1800,\,Re_{\unicode[STIX]{x1D706}}\approx 110)$ , the range of lengths is $L_{\unicode[STIX]{x1D700}}/\unicode[STIX]{x1D702}\approx 150$ , and it is hard to avoid the conclusion that the two types of eddies can only interact indirectly across an intermediate range in which neither the small nor the large length scale are relevant. A more extreme example is the atmospheric surface layer, where $\unicode[STIX]{x1D702}\approx 30~\unicode[STIX]{x03BC}\text{m}$ , and $h=O(100~\text{m})$ . The two scales are then separated by a factor of $10^{6}$ .

It is shown in appendix A that the functional relation between two variables can often be derived from their scaling properties. For example, the power law in the Kolmogorov (Reference Kolmogorov1941) inertial energy spectrum is a consequence of the lack of both a velocity and a length scale. The intermediate layer in wall-bounded turbulence lacks a length scale but not a velocity scale, and its mean velocity profile is bound to be logarithmic (see appendix A),

from where the layer takes its common name. The constants $A$ and $\unicode[STIX]{x1D705}$ have to be determined experimentally. The Kármán constant, $\unicode[STIX]{x1D705}\approx 0.4$ , reflects the dynamics of turbulence in the logarithmic layer and is approximately universal, but $A$ is not.

The reason for the latter is that (3.3) is only a particular self-similar solution of the equations of motion, to which other solutions tend in the range of wall distances where boundary conditions can be approximately neglected. Typically, this self-similar range only exists in some limiting case ( $h^{+}\gg 1$ in wall-bounded turbulence), outside which (3.3) is only an approximation that requires additional adjustable parameters. For example, the assumptions leading to (3.3) do not hold near the wall, and $A$ substitutes for the missing no-slip boundary condition. It depends on the details of the wall and of the viscous layer, and is $A\approx 5$ for smooth walls. Additional boundary-condition surrogates have been proposed, such as a virtual origin for $x_{2}$ (Oberlack Reference Oberlack2001). They improve the agreement in experiments at moderate Reynolds numbers (Mizuno & Jiménez Reference Mizuno and Jiménez2011), but can be neglected as the Reynolds number increases. In the case of a truly asymptotic logarithmic layer, even $A$ becomes negligible compared to $\log x_{2}^{+}\gg 1$ .

3.2 The energy balance

Figure 4.

(a) Mean velocity profile for the channel CH2000. (b) Velocity fluctuation intensities: ——, $u_{1}^{\prime }$ ; - - - -, $u_{2}^{\prime }$ ; – ⋅ – ⋅ –, $u_{3}^{\prime }$ . The dashed vertical lines are conventional upper limits for the buffer $(x_{2}^{+}=80)$ , and logarithmic $(x_{2}/h=0.2)$ layers.

Even if they are relatively thin, the viscous and logarithmic layers are important for the flow as a whole. The thickness of the buffer layer with respect to the total thickness is $80/h^{+}$ , which ranges from 40 % for barely turbulent flows $(h^{+}\approx 200)$ to $10^{-4}$ for large water mains $(h^{+}\approx 5\times 10^{5})$ , but it follows from (3.3) that, even in the latter case, 40 % of the velocity drop takes place below $x_{2}^{+}=80$ (see figure 4 a). The maximum turbulence intensity is also found in the buffer layer, and the fluctuations decay away from the wall (see figure 4 b). The mean shear derived from (3.3), $S=u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D705}x_{2}$ , which is the energy source for the turbulence fluctuations, is also maximum near the wall.

Figure 5.

(a) Contribution to the energy dissipation below a given wall distance from the different terms in the right-hand side of (3.5). —  $\circ$  —, Dissipation due to the mean shear; ——, total dissipation; - - - -, total energy input on the left-hand side of (3.5). The dashed vertical lines are conventional upper limits for the buffer $(x_{2}^{+}=80)$ , and logarithmic $(x_{2}/h=0.2)$ layers. Channel CH2000. (b) Fraction of the dissipation due to different layers in channels, versus the Reynolds number. $\circ$ , from the mean shear; ▵, below $x_{2}^{+}=80$ ; ▿, below $x_{2}/h=0.2$ . The trend lines are the semiempirical fits discussed in the text: – ⋅ – ⋅ –, $9.1/U_{b}^{+}$ ; - - - -, $15.5/U_{b}^{+}$ ; ——, $1-2.5/U_{b}^{+}$ .

Consider the energy balance in a turbulent channel (Tennekes & Lumley Reference Tennekes and Lumley1972). Energy enters the system through the work of the pressure gradient $\unicode[STIX]{x2202}_{1}P=-u_{\unicode[STIX]{x1D70F}}^{2}/h$ on the volumetric flux $2hU_{b}$ , where $U_{b}=h^{-1}\int _{0}^{h}U_{1}\,\text{d}x_{2}$ is the bulk velocity. This energy input has to balance the total dissipation if the flow is to be statistically stationary. In wall units, this is expressed as

where $\unicode[STIX]{x1D700}$ in the last integral is the ‘turbulent’ dissipation due to the gradients of the velocity fluctuations, and ${S^{+}}^{2}$ is the dissipation due to the effect of the viscosity on the mean velocity profile. When the balance leading to (3.4) is applied to a layer stretching from the wall to $x_{2}$ it gives an idea of the contributions to the dissipation from the different parts of the flow. The equation takes the form,

where the extra term in the energy input in the left-hand side is the work of the tangential Reynolds stress against the mean profile. The two terms in the right-hand side of (3.5) are plotted in figure 5(a). The small difference between the energy input (dashed) and the dissipation (solid line) in this figure is the effect of the small internal energy fluxes represented by the trailing dots in (3.5). They are negligible at high Reynolds numbers.

In shear flows without walls, the dissipation due to the mean velocity profile is $O(\unicode[STIX]{x1D700}/Re)$ , and can be neglected. In the wall-bounded case, figure 5(a) shows that both contributions are of the same order, although the dissipation due to the mean profile resides almost exclusively below $x_{2}^{+}=20$ . Because the shear in that region scales well in wall units, this part of the dissipation is very nearly independent of the Reynolds number, $\int _{0}^{h}{S^{+}}^{2}\,\text{d}x_{2}\approx 9.1$ . Figure 5(a) also shows that a relatively large fraction of the turbulent dissipation, denoted as $\unicode[STIX]{x1D700}_{80}$ , takes place below $x_{2}^{+}=80$ . The velocity gradients in this part of the flow are also approximately independent of the Reynolds number, and $\unicode[STIX]{x1D700}_{80}^{+}\approx 6.4$ .

Most of the remaining dissipation takes place within the logarithmic layer, $80\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}<x_{2}<0.2h$ , and can be estimated from (3.3). The energy balance of the fluctuations, averaged over wall-parallel planes, takes the form (Tennekes & Lumley Reference Tennekes and Lumley1972)

where $\text{D}_{t}=\unicode[STIX]{x2202}_{t}+U_{1}\unicode[STIX]{x2202}_{1}$ is the mean advective derivative. The trailing dots stand for transfer terms that are not important in the logarithmic layer, where the dissipation is almost exclusively due to $\unicode[STIX]{x1D700}$ , and has to be approximately balanced by the local energy production, $-S\langle u_{1}u_{2}\rangle$ . This can be written as

which can be integrated to

A similar estimate for the total energy input shows that

also grows logarithmically with $h^{+}$ , so that the remaining dissipation above the logarithmic layer is approximately independent of the Reynolds number, $\unicode[STIX]{x1D700}_{out}^{+}\approx 2.5$ . As a consequence, the relative contributions of the buffer and outer layers to the dissipation decrease logarithmically as the Reynolds number increases. This is shown in figure 5(b), which includes simulation results from channels at several Reynolds numbers, and logarithmic fits based on the arguments above. In the asymptotic limit of an ‘infinite’ Reynolds number, most of the dissipation resides in the logarithmic layer, but figure 5(b) shows that the fraction of the dissipation due to the mean and fluctuating velocities in the buffer layer is still of the order of 50 % of the total at the largest ‘realistic’ Reynolds numbers, $h^{+}=O(10^{6})$ .

Because of this ‘singular’ nature, the near-wall layer is not only important for the rest of the flow, but it is also essentially independent from it. This was shown by Jiménez & Pinelli (Reference Jiménez and Pinelli1999) using ‘autonomous’ simulations in which the outer flow was artificially removed above a certain distance, $\unicode[STIX]{x1D6FF}$ , from the wall. The dynamics of the buffer layer was unaffected as long as $\unicode[STIX]{x1D6FF}^{+}\gtrsim 60$ . The same conclusion can be drawn from the minimal flow experiments by Jiménez & Moin (Reference Jiménez and Moin1991), who simulated channels in numerical boxes small enough for no large flow scales to be possible above $x_{2}^{+}\approx 100$ . Again, the buffer layer remained essentially unaffected. Minimal flows were extended to the logarithmic layer by Flores & Jiménez (Reference Flores and Jiménez2010) with similar results: the dynamics of the higher-shear region near the wall is essentially independent from outside influences. These observations should not be interpreted to mean that there are no interactions between the inner and outer layers. These interactions will be documented below, but they can mostly be expressed as modulations or superpositions, which are not required for the maintenance of the flow.

Understanding the structure of this near-wall region has practical implications. Energy dissipation by turbulence is the root cause of hydrodynamic friction drag, which is estimated to be responsible for 5 % of the total energy expenditure of humankind (Jiménez Reference Jiménez2013b ). The energy input, $U_{b}^{+}=(2/c_{f})^{1/2}$ , determines the friction coefficient $c_{f}$ , and it follows from figure 5 that any attempt to understand or to control wall friction has to take into account the buffer and logarithmic layers.

3.3 The scales of the energy production

The above discussion says little about how turbulence extracts energy from the mean flow. We saw in (3.6) that the average energy production in a parallel shear flow is $-S\langle u_{1}u_{2}\rangle$ , which depends on the coupling of the Reynolds stress with the mean shear.

Figure 6.

(a) Corrsin (Reference Corrsin1958) integral shear parameter $S^{\ast }$ . ——, Channels and boundary layers $h^{+}=2000{-}5200$ (various sources); $\circ$ , statistically stationary homogeneous shear turbulence, plotted at the equivalent $Re_{\unicode[STIX]{x1D706}}$ (Sekimoto et al. Reference Sekimoto, Dong and Jiménez2016). (b) The line contours are the spectral energy density of the kinetic energy, as in figure 3(a). The shaded contours are the spectral Corrsin shear parameter $s^{\ast }(\unicode[STIX]{x1D706}_{I})=2(\times 10)2000$ , from light to dark. Channel CH5200. ——, $\unicode[STIX]{x1D706}_{I}=x_{2}$ ; - - - -, $\unicode[STIX]{x1D706}_{I}=5x_{2}$ ; – ⋅ – ⋅ –, $\unicode[STIX]{x1D706}_{I}=25\unicode[STIX]{x1D702}$ . The horizontal dashed line is $x_{2}^{+}=80$ .

The coupling criterion was established by Corrsin (Reference Corrsin1958). The turnover time over which the nonlinear self-interaction of an eddy of size $\ell$ and characteristic velocity $u_{\ell }$ changes its energy is of order $\unicode[STIX]{x1D70F}_{to}=\ell /u_{\ell }$ , while its deformation by the shear takes place in $\unicode[STIX]{x1D70F}_{s}=S^{-1}$ . The Corrsin parameter is the ratio of these times, $s^{\ast }(\ell )=\unicode[STIX]{x1D70F}_{to}/\unicode[STIX]{x1D70F}_{s}=S\ell /u_{\ell }$ . If $s^{\ast }(\ell )\ll 1$ , the eddy evolves independently of the shear and there is little or no energy production. This is the regime of the inertial cascade. If $s^{\ast }(\ell )\gg 1$ , the eddy is controlled by the shear, and can extract (or lose) energy from (or to) it. This is the range of the energy production. Note that this implies that the energy-producing eddies of a shear flow are quasilinear, in the sense that they are controlled by the interaction of the fluctuations with the mean flow, with only slower nonlinear effects. The inertial cascade is fully nonlinear.

When this criterion is applied to the integral scales, where $\ell =L_{\unicode[STIX]{x1D700}}$ and $u_{\ell }=q$ , we obtain an integral Corrsin parameter, $S^{\ast }=Sq^{2}/\unicode[STIX]{x1D700}$ , which determines whether some part of the flow is involved in the production of turbulent energy (if $S^{\ast }\gg 1$ ), or just transfers or dissipates it (if $S^{\ast }\ll 1$ ).

Figure 6(a) shows that equilibrium shear flows tend to have $S^{\ast }\approx 10$ , at least above the buffer layer in the wall-bounded case. This is a moderately large number that implies a quasilinear interaction of the energy-containing eddies with the mean flow throughout the whole logarithmic layer. Linear processes do not have an intrinsic amplitude, and the fact that $S^{\ast }$ is similar for fairly different flows suggests that the ‘linear’ energy production eventually saturates by means of a relatively universal mechanism that drains its energy to the cascade, roughly equivalent to an eddy viscosity (del Álamo & Jiménez Reference del Álamo and Jiménez2006). This recalls the engineering rule of thumb that the Reynolds number based on the eddy viscosity of turbulent flows is always of order 10–30 (Tennekes & Lumley Reference Tennekes and Lumley1972). The very high value of $S^{\ast }$ in the buffer layer is not an indicator of extremely sheared flow in the sense just discussed. The shear in this region is the highest in the flow, but the argument of Corrsin (Reference Corrsin1958) assumes that the balance is between nonlinearity and shear, while the fastest evolution time near the wall is viscous. For eddies of size $\ell$ the viscous decay time is $T_{\unicode[STIX]{x1D708}}=\ell ^{2}/\unicode[STIX]{x1D708}$ , and the relevant shear parameter is $S_{\unicode[STIX]{x1D708}}^{\ast }=S\ell ^{2}/\unicode[STIX]{x1D708}$ . In fact, the transition of $S^{\ast }$ from its near-wall peak to the outer plateau can be used as a convenient definition of the upper edge of the buffer layer, and is the origin of the value used in this article, $x_{2}^{+}\approx 80$ .

The spectral shear parameter for individual wavelengths, $s^{\ast }(\unicode[STIX]{x1D706}_{I},x_{2})$ , can be estimated by identifying the eddy size $\ell$ with the wavelength $\unicode[STIX]{x1D706}_{I}$ , and the eddy velocity scale with the spectral energy density $u_{\ell }=\unicode[STIX]{x1D719}_{qq}^{1/2}$ . The resulting $s^{\ast }=S\unicode[STIX]{x1D706}_{I}/\unicode[STIX]{x1D719}_{qq}^{1/2}$ is overlaid in figure 6(b) to the energy spectrum from figure 3(a). It increases sharply towards the longer wavelengths near the wall, and falls below $s^{\ast }=2$ to the left of $\unicode[STIX]{x1D706}_{I}\approx L_{c}=x_{2}$ . The Corrsin length $L_{c}$ defined in this way represents the scale of the smallest eddies that interact directly with the shear, and is typically a fixed small fraction of the integral length, $L_{c}/L_{\unicode[STIX]{x1D700}}\approx (S^{\ast })^{-3/2}$ . As in figure 6(a), the shear-dominated region below $x_{2}^{+}\approx 80$ should be interpreted as viscous, including the highest $s^{\ast }$ in the very long near-wall region in the lower right-hand corner of figure 6(b).

Note that, as expected, the length scale of the vorticity, $\unicode[STIX]{x1D706}_{I}\approx 25\unicode[STIX]{x1D702}$ , is below the Corrsin scale for $x_{2}^{+}\gtrsim 50$ , equivalent to $x_{2}/h\approx 10^{-2}$ in the case of figure 6(b). Except in the buffer layer, these viscous vortices do not interact with the shear, and do not participate in the turbulence-production process.

3.4 Anisotropy

Although figure 3(a) is drawn in terms of an isotropic wavelength and of the kinetic energy, shear flows are not isotropic. Figure 4(b) shows that the intensities of the three velocity components are different. The largest share of the kinetic energy is contained in $u_{1}^{\prime 2}$ . This is especially true in the buffer layer, but the contribution of this component is at least half of the total at all wall distances. The other two velocity components split the rest of the energy approximately evenly, at least far from the wall. This is most easily understood in terms of the energy equation for individual components (Tennekes & Lumley Reference Tennekes and Lumley1972), although we will see later how this difference is implemented in detail by the energy-production mechanisms. In a channel,

for the streamwise component, and

for each of the two transverse components, where the repeated indices in the pressure term do not imply summation. The streamwise component is special because, on average, it is the only one that receives energy directly from the shear through the production term, $-S\langle u_{1}u_{2}\rangle$ . Approximately half of this energy is dissipated to viscosity, and the rest is transferred to the other two velocity components by the pressure–strain term, $\langle p\,\unicode[STIX]{x2202}_{1}u_{1}\rangle$ . This is a redistribution term, because it follows from continuity that $\langle p\,\unicode[STIX]{x2202}_{j}u_{j}\rangle =0$ , so that the net effect of the pressure on the kinetic energy vanishes. Its effect on the velocities is approximately isotropic, and each of the two transverse components receives roughly equal amounts of energy (Hoyas & Jiménez Reference Hoyas and Jiménez2008). Roughly speaking, the kinetic energy of the streamwise velocity is twice that of the other two components because it receives twice as much energy as they do.

The details of the distribution of the kinetic energy among the three velocity components depend on the flow. The buffer layer is approximately universal, but the outer layers are not. The contribution of the transverse velocities to the kinetic energy is somewhat larger in boundary layers than in channels, most probably due to the intermittency at the turbulent–non-turbulent interface (Sillero et al. Reference Sillero, Jiménez and Moser2013). In contrast, the streamwise component is substantially stronger in Couette flow than in either boundary layers or channels, reflecting the presence of strong persistent streamwise ‘rollers’ which are not found in other flows (Pirozzoli, Bernardini & Orlandi Reference Pirozzoli, Bernardini and Orlandi2014).

Figure 7.

(a–c) Two-dimensional spectral densities of the velocity components at $x_{2}^{+}=15$ . The dashed diagonal is $\unicode[STIX]{x1D706}_{1}=8\unicode[STIX]{x1D706}_{3}$ , the hyperbolic curve is $\unicode[STIX]{x1D706}_{I}^{+}=1000$ , and the circles are $\unicode[STIX]{x1D706}_{1}=\unicode[STIX]{x1D706}_{3}$ . The grey patch is the integrated energy density, $\unicode[STIX]{x1D6F7}=(1/h)\int _{0}^{h}\unicode[STIX]{x1D719}\,\text{d}x_{2}$ for $h^{+}=5200$ . (a) $\unicode[STIX]{x1D719}_{11}^{+}=(0.25,0.5,1.0)$ . $\unicode[STIX]{x1D6F7}_{11}^{+}=0.1$ (b) $\unicode[STIX]{x1D719}_{22}^{+}=(0.006,0.012,0.025)$ . $\unicode[STIX]{x1D6F7}_{22}^{+}=0.02$ (c) $\unicode[STIX]{x1D719}_{33}^{+}=(0.05,0.1,0.2)$ . $\unicode[STIX]{x1D6F7}_{33}^{+}=0.05$ (d–f). One-dimensional streamwise spectral densities as functions of wall distance. The two diagonals are approximately: ——, $\unicode[STIX]{x1D706}_{I}=x_{2}$ ; - - - -, $\unicode[STIX]{x1D706}_{I}=5x_{2}$ , converted to $\unicode[STIX]{x1D706}_{1}$ by taking into account the aspect ratio of the outer-layer spectra. (d) $\unicode[STIX]{x1D719}_{11}^{+}=0.2(0.4)1.8$ . The grey patch is the vertically correlated region near the wall for $h^{+}=2003$ , from figure 11(e). (e) As in (d), for $\unicode[STIX]{x1D719}_{22}^{+}=0.1(0.2)0.9$ . (f) $\unicode[STIX]{x1D719}_{33}^{+}=0.1(0.2)0.9$ . Channels: – ⋅ – ⋅ –, CH950; - - - -, CH2000; ——, CH5200.

Figures 7(a)–7(c) display wall-parallel two-dimensional spectral densities for the three velocity components at the location of the buffer-layer peak of $u_{1}^{\prime }$ . The streamwise-velocity spectrum in figure 7(a) has two well differentiated parts (Hoyas & Jiménez Reference Hoyas and Jiménez2006): a near-wall one for $\unicode[STIX]{x1D706}_{I}^{+}\lesssim 10^{3}$ , which scales in wall units; and a ridge along $\unicode[STIX]{x1D706}_{1}\approx 8\unicode[STIX]{x1D706}_{3}$ , which gets longer as $h^{+}$ increases, and is therefore associated with the outer flow. Figures 7(a)–7(c) also contain an isocontour of the vertically integrated energy densities, $\unicode[STIX]{x1D6F7}=(1/h)\int _{0}^{h}\unicode[STIX]{x1D719}\,\text{d}x_{2}$ , showing that the large-scale energy in the buffer region is generally only associated with the long-wavelength edge of the integrated spectrum. Most of the kinetic energy, which resides in the outer part of the flow, does not reach near the wall.

The wall-normal structure of these spectra is displayed in figure 7(d–f) as spanwise-integrated one-dimensional spectral densities, plotted as functions of the distance from the wall and of the streamwise wavelength. The first conclusion from figure 7 is that the wall-normal velocity is damped in the neighbourhood of the wall, and that the damping is strongest for the largest eddies. There is no large-scale $\unicode[STIX]{x1D719}_{22}$ in figure 7(b), even if the contours in this figure are forty times weaker than those for $\unicode[STIX]{x1D719}_{11}$ in figure 7(a). Farther from the wall, the transverse velocities in figures 7(e) and 7(f) are weaker than $u_{1}$ , but only by a factor of two, which is also the ratio of their overall intensities. Moreover, the comparison of the different spectra, using as reference the various trend lines in the figures, shows that $u_{2}$ and $u_{3}$ tend to occur at similar scales far from the wall, suggesting that they may be part of a common structure in that region, while $u_{1}$ and $u_{3}$ are similarly paired near it. This is also suggested by the shaded grey patches in figure 7(d–f), which mark the depth of the near-wall vertically coherent layer for the three velocities. This information cannot be obtained from the spectra, and will be discussed in detail in § 3.5, but the results have been added to figure 7(d–f) to aid in their interpretation. There is a near-wall layer in which all the velocity components are vertically correlated, but the coherent layer of $u_{2}$ does not extend beyond $\unicode[STIX]{x1D706}_{1}^{+}\approx 5000$ and $x_{2}^{+}\approx 80$ , scaling in wall units. The coherent layer of $u_{1}$ and $u_{3}$ extends up to $\unicode[STIX]{x1D706}_{1}\approx 7h$ and $x_{2}/h\approx 0.1{-}0.2$ , scaling in outer units. Beyond $\unicode[STIX]{x1D706}_{1}\approx 7h$ , the near-wall coherent layer of $u_{3}$ disappears, but $u_{1}$ continues to get taller until it fills most of the channel at very long wavelengths.

The grey patches in figure 7(a–c) show that the spectra of the three velocities are very different above the buffer layer. The two transverse components are approximately equilateral, $\unicode[STIX]{x1D706}_{1}\approx \unicode[STIX]{x1D706}_{3}$ , but the spectrum of the streamwise velocity is longer. Its short-wavelength edge, $\unicode[STIX]{x1D706}_{1}\approx 2\unicode[STIX]{x1D706}_{3}$ , is only slightly more elongated than for the transverse velocities, but its longest wavelengths are at least ten times longer than for the two transverse velocities, and extend to the longest dimension of the computational box. This long-wavelength behaviour will be discussed in the next section, and suggests that, if there are coherent structures in the flow, there are at least two kinds: very elongated ‘streaks’ of $u_{1}$ , and shorter structures of $(u_{2},u_{3})$ .

3.5 Correlations

While spectra describe the size of eddies along homogeneous directions, non-homogeneous directions have to be analysed using two-point covariances, defined for $u_{i}$ as

where the repeated index does not imply summation. The covariance is symmetric in its two arguments, but we will define $\tilde{\boldsymbol{x}}$ as the reference point, and $\boldsymbol{x}$ as the variable argument. Along homogeneous directions, $R_{ii}$ depends only on the coordinate increment, $\unicode[STIX]{x0394}x_{j}=x_{j}-\tilde{x}_{j}$ , and forms a Fourier-transform pair with the power spectrum (see appendix B.2). For example, in channels, $R_{ii}=R_{ii}(\unicode[STIX]{x0394}x_{1},x_{2},\tilde{x}_{2},\unicode[STIX]{x0394}x_{3})$ . At the reference point, the covariance reduces to the variance $R_{ii}(\tilde{\boldsymbol{x}},\tilde{\boldsymbol{x}})=u_{i}^{\prime 2}(\tilde{\boldsymbol{x}})$ , and the dimensionless version of the covariance is the two-point correlation,

which is unity at $\boldsymbol{x}=\tilde{\boldsymbol{x}}$ . Correlations and covariances are high-dimensional quantities. In channels, the correlation is four-dimensional. In boundary layers, where the only homogeneous direction is the span, it is five-dimensional. This means that, except for relatively low Reynolds numbers, it is difficult to compute and store correlations or covariances for more than a few reference points. For a channel simulation using a moderately sized grid with $m=1000^{3}=10^{9}$ points, the correlation is an $(m\times m)$ matrix, which can be reduced, using homogeneity, to a block-diagonal form of $m^{2/3}=10^{6}$ submatrices of size $(m^{1/3}\times m^{1/3})$ . The total number of non-zero elements is $m^{4/3}=10^{12}$ , although the rank is only $m$ . The problem of using empirical correlations is more fundamental than a practical question of computer storage. It is shown in appendix B that the covariance can be represented as a matrix $\unicode[STIX]{x1D64D}=\unicode[STIX]{x1D650}\unicode[STIX]{x1D650}^{\ast }$ , where $\unicode[STIX]{x1D650}$ is an $(m\times n)$ matrix whose columns are the $n$ observations (‘snapshots’), and the asterisk denotes Hermitian transpose. It is clear from this definition that the rank of $\unicode[STIX]{x1D64D}$ constructed in this way is at most $n$ . The consequence is that the computation of the covariance requires as many independent snapshots as degrees of freedom, which increases with the cube of the grid diameter. Any smaller number of independent samples only provides an approximation to the covariance. In practice, this means that covariances are typically only used in the form of incomplete approximations of deficient rank, usually through some variant of the method of snapshots (Sirovich Reference Sirovich1987, see appendix B.1).

Figure 8.

Three-dimensional representation of the correlation function of the streamwise-velocity fluctuations, $C_{11}$ , with respect to the reference point $\tilde{x}_{2}/h=0.6$ . Boundary layer BL6600 at $h^{+}=1530$ . The flow goes from left to right. Several isosurfaces are shown, with $C_{11}=-0.09$ (white), $+0.09$ (turquoise), $+0.4$ (yellow) and $+0.8$ (blue). In the different planar cross-sections, the contour lines of positive and negative values are coloured red and white, respectively. Reproduced with permission from Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2014).

In spite of these limitations, correlations give useful information about the three-dimensional structure of the flow variables. Figure 8 is an example of $C_{11}$ for a boundary layer. It shows an inclined central positive lobe surrounded by smaller negative lobes. The cross-flow sections are approximately circular, while the streamwise sections are elongated. Most of the data in this subsection are drawn from Sillero et al. (Reference Sillero, Jiménez and Moser2014), which should be consulted for further details. As in spectra, it is striking that the correlations of the different velocity components have very different geometries. While the inclination angle of the streamwise-velocity correlation in figure 8 is approximately $10^{\circ }$  with respect to the wall, the correlation of the wall-normal velocity is vertical, and that of the spanwise velocity is tilted approximately $30^{\circ }$ . These values remain approximately constant away from the buffer and outer layers, apply to boundary layers and channels, and are independent of the Reynolds number within the range in which they have been studied. The correlation of the pressure fluctuations is also vertical, broadly similar to that of $u_{2}$ .

Figure 9.

Two-dimensional sections through the reference point $(\tilde{x}_{2}/h=0.35)$ of the two-point autocorrelation function of the three velocity components. Channel CH2000. (a) Spanwise velocity. (b) Wall-normal velocity. (c) Streamwise velocity. Contours are $C=[-0.1,-0.05,0.05,0.1,0.3]$ . Negative contours are dashed. The grey patches in (a–b) are the $C=0.05$ contour of the other transverse-velocity component. It has been shifted horizontally by $0.25h$ in the cross-flow sections to suggest a possible completion of the streamwise roller. The grey patch in (c) is $C_{33}$ .

Streamwise and cross-flow sections of the velocity autocorrelation function of the three velocity components are given in figure 9 for a reference point in the outer edge of the logarithmic region, $\tilde{x}_{2}/h=0.35$ . This is approximately where the correlations of $u_{1}$ are longest in channels. The correlations of the two transverse velocities in figure 9(a,b) have comparable dimensions. As mentioned above, the correlation of the spanwise velocity in figure 9(a) is tilted at approximately 30 $^{\circ }$  to the wall. It has relatively strong negative lobes which are tilted at approximately the same inclination, located above and below the main positive lobe. The cross-flow section in the right-hand part of figure 9(a) shows that the positive correlation lobe is flat and wide with respect to its height. Conversely, the correlation of the wall-normal velocity in figure 9(b) is relatively isotropic in the streamwise section, slightly tilted backwards, and tall and narrow in the cross-flow plane.

It should be noted that the centres of the two correlations do not necessarily correspond to the same location in the homogeneous wall-parallel directions. Correlations describe the relation between the same velocity component at two points. For example, a positive $C_{22}$ means that $u_{2}(x_{2})$ has, on average, the same sign as $u_{2}(\tilde{x}_{2})$ , but their common sign can be positive or negative. Similarly, $C_{22}$ and $C_{33}$ say little about the relative position of $u_{2}$ with respect to $u_{3}$ . However, if figures 9(a) and 9(b) are taken together, they are consistent with a ‘roller’, oriented approximately streamwise and inclined at 30 $^{\circ }$  to the wall. The positive lobe of $C_{33}$ would correspond to the top or to the bottom of the roller, where the transverse velocity is directed spanwise. Its negative lobe corresponds to the other edge of the roller, where $u_{3}$ has opposite sign. The positive lobe of $C_{22}$ would correspond to the two ‘sidewalls’ closing the circuit. Grey patches of $C_{22}$ have been added to the $C_{33}$ plot, and vice versa, to suggest the relation implied by this model between the two components. The two correlations have been shifted by $0.25h$ in the spanwise direction, to make $C_{22}$ coincide with a possible lateral edge of the roller. This offset would imply a roller diameter of approximately $0.5h$ , of the same order as the distance from the wall to the reference point, $\tilde{x}_{2}=0.35h$ , and consistent with the vertical distance between the positive and negative lobes of $C_{33}$ . The negative lobes in the transverse cross-section of $C_{22}$ in figure 9(b) are also at the right distance and position to represent the opposite lateral wall of the roller. Note that the symmetry of $C_{22}$ with respect to $x_{3}=0$ is statistical, and does not imply the symmetry of individual eddies. The presence of both an upper and a lower negative lobe in $C_{33}$ is also statistical. Some rollers are detected at their upper edge, and others at their lower edge.

Inclined ‘vortices’ have often been mentioned in descriptions of boundary-layer eddies (Adrian Reference Adrian2007), and detected in shear flows by stochastic estimation (Adrian & Moin Reference Adrian and Moin1988). We shall see later that they appear as parts of conditional Reynolds-stress structures (Dong et al. Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017), but it should be stressed that the dimensions of the correlations in figure 9(a,b) are much larger than those of individual vortices. The diameter of the roller implied by them is approximately 1000 wall units, or $300\unicode[STIX]{x1D702}$ , and it can be shown that these dimensions scale with $h$ as the Reynolds number changes. On the other hand, figure 3(a) shows that the size of the vorticity scales in viscous units, and is always approximately $30\unicode[STIX]{x1D702}$ .

The correlation of $u_{1}$ in figure 9(c) is much longer streamwise than those of the transverse velocities, although not much wider in the cross-plane. The shadow of $C_{33}$ included in figure 9(c) emphasises the relation of the respective sizes. It is interesting that, even at this relatively small distance from the wall, there is a negative lobe of $C_{11}$ near the opposite wall. The streamwise-velocity eddies are large enough to be ‘global’, involving the whole channel. Although the energy considerations in § 3.2 imply that the $u_{1}$ streak and the cross-flow rollers have to be related, and we shall see later that they appear together in conditional flow fields, the difference in their size makes it difficult to describe them as parts of a single eddy. At the very least, each streak must contain several rollers. It should also be noted at this point that, because the correlations are second-order quantities, they favour the locations and times where the velocity is strongest, which need not occur simultaneously for all the velocity components.

The correlation functions for the cross-flow velocities are relatively independent of the Reynolds number, and vary little among the flows in which they have been studied (Sillero et al. Reference Sillero, Jiménez and Moser2014). They are good candidates for ‘universal’ features of shear flows. Those of $u_{1}$ are not, and are known with less certainty. The lengths implied by figure 9(c) for channels $(\unicode[STIX]{x1D706}_{1}\approx 20h)$ are, unfortunately, comparable to the length of the computational box $(L_{1}=25h)$ , and vary by $\pm 20\,\%$ among different simulations, including those performed within the same research group using similar codes (Sillero et al. Reference Sillero, Jiménez and Moser2014). Experiments are also ambiguous because it is difficult to deduce long spatial dimensions from temporal data, or to collect spatial measurements over very long distances. Nevertheless, the lengths in figure 9(a) are probably approximately correct. A simulation in a very large box $(L_{1}=190h)$ was performed by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014a ) at $h^{+}=550$ , and did not result in longer eddies. The length-to-width ratio, $\unicode[STIX]{x1D706}_{1}/\unicode[STIX]{x1D706}_{3}\approx 8$ , of the spectral ridge in figure 7(a) is consistent with $\unicode[STIX]{x1D706}_{1}=16h$ for streaks of width $\unicode[STIX]{x1D706}_{3}=2h$ , which is the maximum value implied by the cross-flow section in figure 9(c). This width is much narrower than the spanwise dimensions of the computational box, and there is no reason to believe that it is numerically constrained. The shorter end of the spectral ridge in figure 7(a) is also short with respect to the computational box, and its aspect ratio should not be numerically constrained. The fact that it does not change with the Reynolds number, which modifies the ratio between the length of these streaks and the size of the box, and that the aspect ratio of the shorter streaks is similar to that of the longer streaks, gives some confidence on the results for the longer eddies.

On the other hand, the correlation length of $u_{1}$ in boundary layers is shorter than in channels, $\unicode[STIX]{x1D706}_{1}\approx 4h$ (Sillero et al. Reference Sillero, Jiménez and Moser2014), while that in Couette flows appears to be longer than any of the experiments or simulations that have been performed; it might be infinite (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014). The reason for these differences is unclear, although reasonable models can be advanced in some cases. For example, very long streaks presumably require very long times to organise (see the discussion of figure 29 in § 6.1). Visually tracking the large scales in simulations of channels confirms that they evolve extremely slowly, with evolution times of the order of many turnovers, $tu_{\unicode[STIX]{x1D70F}}/h\gg 1$ . Assuming that the relevant deformation velocity is $u_{\unicode[STIX]{x1D70F}}$ , the time needed to organise an eddy of length $\unicode[STIX]{x1D706}_{1}$ is $\unicode[STIX]{x1D706}_{1}/u_{\unicode[STIX]{x1D70F}}$ , during which time the flow is advected approximately by $U_{b}\unicode[STIX]{x1D706}_{1}/u_{\unicode[STIX]{x1D70F}}\approx 30\unicode[STIX]{x1D706}_{1}$ . In a uniform channel, very long advection lengths are available, but, in a boundary layer, a streak of length $\unicode[STIX]{x1D706}_{1}=4h$ would be advected by approximately $100h$ during its formation. Over that distance, the thickness of the boundary layer grows by a factor of two. In essence, streaks in boundary layers do not have time to become very long before the mean velocity profile changes enough to require them to adapt to a new size. A similar argument could explain the scatter among the measured correlation lengths of different channel simulations, because not all of them are run for the same time. The streaks that we see in channels could still be slowly growing in some cases. This problem also applies to the development length of laboratory experiments.

Hutchins & Marusic (Reference Hutchins and Marusic2007) have proposed that real streamwise-velocity eddies may be very long, meandering on a shorter scale. They note that the meanders hide the real length of the streaks in spectra and correlations, and they present visual evidence of lengths of $20h$ in boundary layers, meandering at the spectrally measured length of $6h$ . Unfortunately, their argument does not explain why channels or Couette flows should meander less than boundary layers, and thus appear longer in the spectra. Neither do the estimates of the evolution time in the previous paragraph explain the differences between Couette flow and channels. The question of the real length of the large velocity streaks in wall-bounded turbulence, and the reasons for it, remain at the moment unsettled. This question will be revisited when discussing individual velocity structures in § 5.3.

3.6 Filtered correlations

Figure 10.

(a) One-dimensional two-point correlation, $C_{(2)}(x_{2},\tilde{x}_{2})$ , as a function of $x_{2}$ . Above the diagonal, $C_{(2)22}$ . Below the diagonal, $C_{(2)11}$ . Contours are $C_{(2)}=[0.3,0.5,0.7,1]$ . (b) Correlation depth at $C_{(2)}=0.3$ of the different velocity components, as a function of the distance from the wall to the reference point. ▵, $u_{1}$ ; $\circ$ , $u_{2}$ ; ▿, $u_{3}$ . Channel CH2000.

Consider next the question of defining the correlation depth of the velocities at different distances from the wall. In channels, we can define a one-dimensional vertical correlation profile,

where the value at $\unicode[STIX]{x0394}x_{1}=0$ is substituted by the maximum over $\unicode[STIX]{x0394}x_{1}$ to take into account the inclination of the correlations. This quantity is represented in figure 10(a) for $u_{1}$ and $u_{2}$ , using the symmetry of $C_{(2)}$ with respect to its two arguments to include the two velocity components in the same figure.

There are many ways of defining the correlation depth, $\unicode[STIX]{x1D6EC}_{j}(\tilde{x}_{2})$ , of $u_{j}$ at a given distance from the wall, but the simplest way is to measure the width at a given level of the correlation profile centred at $\tilde{x}_{2}$ . Figure 10(b) displays the correlation depth defined at $C_{(2)ii}=0.3$ for the three velocity components. It is evident that $u_{1}$ is deeper than $u_{2}$ or $u_{3}$ , and that all the velocities are more deeply correlated away from the wall than near it. But it is difficult to extract other conclusions from the figure. In particular, since the spectra in figure 7(d–f) show that all the velocity components get longer as they move away from the wall, it is unclear from figure 10(b) whether the growth in depth is due to the differences in wall distance, or to the longer eddies. An interesting related question, when the reference point $\tilde{x}_{2}$ is chosen in the buffer layer, is how far from the wall are the eddies responsible for the spectral ridge in figure 7(a).

Fortunately, the Fourier duality between covariances and spectra along homogeneous directions allows us to filter the covariance to only include a restricted range of wall-parallel wavelengths.

Consider a generic variable, $u$ , whose covariance is $R_{uu}(\unicode[STIX]{x0394}x_{1},x_{2},\tilde{x}_{2})$ . Its spectrum, $E_{uu}(k_{1},x_{2},\tilde{x}_{2})=\langle \widehat{u}(k_{1},x_{2})\widehat{u}^{\ast }(k_{1},\tilde{x}_{2})\rangle$ , is the Fourier transform of $R_{uu}$ (see appendix B.2). This is true for any spectrum. In particular, if we apply to $\widehat{u}$ a spectral filter $G(k_{1})$ that restricts it to some range of wavenumbers, $\overline{\widehat{u}}(k_{1})=\widehat{u}(k_{1})G(k_{1})$ , its spectrum is similarly filtered to $E_{\bar{u}\bar{u}}(k_{1},x_{2},\tilde{x}_{2})=E_{uu}(k_{1},x_{2},\tilde{x}_{2})|G(k_{1})|^{2}$ . The covariance of the filtered variable can then be obtained as the inverse Fourier transform of the filtered spectrum. A similar procedure can be applied to $k_{3}$ and to three-dimensional covariances.

Figure 11.

Two-dimensional sections through the reference point $(\tilde{x}_{2}/h=0.35)$ of the two-point filtered autocorrelation function of the streamwise velocity. (a) $\unicode[STIX]{x1D706}_{1}/h\in (1.4,2.2)$ , (b) $(5.0,8.2)$ and (c) $(13.6,22.0)$ . Contours are $C_{11}=\pm [1,2]$ , and negative contours are dashed. (d) One-dimensional filtered correlation, $C_{(2)11}(x_{2},\tilde{x}_{2};\unicode[STIX]{x1D706}_{1})$ , of the streamwise velocity, as a function of the distance from the wall. Above the diagonal, as in (a). Below the diagonal, as in (b). Contours are $C_{(2)11}=[0.3,0.5,1]$ . (e) Correlation depth of $u_{1}$ , as a function of the central filter wavelength. Each pair of lines represents the upper and lower limits within which $C_{(2)11}>0.4$ . $\tilde{x}_{2}=[15/h^{+},0.1,0.2,0.35,0.6]$ , as marked by their common origin at $\unicode[STIX]{x1D706}_{1}=0$ . The circles are $x_{2}=0.03\unicode[STIX]{x1D706}_{1}$ . Channel CH2000.

Filtered correlation functions can be defined by normalising the covariance with the standard deviation of the filtered variables, but it is more useful to normalise them with the total fluctuation intensities,

This has the advantage of keeping some spectral information about the relative intensity of the filtered variable in the different wavenumber ranges, but at the price of lacking a well-defined maximum value. The value of $C_{\bar{u}\bar{u}}$ at $\boldsymbol{x}=\tilde{\boldsymbol{x}}$ is the relative energy of the filtered variable with respect to the total. In (3.15), and in the figures in the rest of this section, we have assumed that the filter is used to separate the flow into $N_{bands}$ approximately equal logarithmic bands, and we have multiplied the filtered correlations by $N_{bands}$ to get maxima of order unity. Even so, the filtered correlations of the different variables reach different maximum levels, and the threshold used to determine correlation depths has to be adjusted to some fraction of the empirical maximum of each case.

Using these definitions, the effect of filtering the $C_{11}$ correlation in figure 9(c) is shown in figure 11(a–c). The streamwise coordinate is filtered with a family of self-similar sharp spectral box filters in which the maximum and minimum wavelengths differ by a factor of 1.6. The central filter wavelength increases from figure 11(a) to 11(c), and the longest filter spans roughly half of the length of the computational box. It can be shown that the streamwise average of any correlation which does not include $k_{1}=0$ has to vanish, and all the filtered correlations in figure 11 are oscillatory. As the central wavelength is made shorter, the depth of the correlation decreases, until it eventually separates from the wall.

This is made clearer by the correlation profile $C_{(2)11}(x_{2},\tilde{x}_{2};\unicode[STIX]{x1D706}_{1})$ in figure 11(d), built from the filtered correlations in figure 11(a,b), which should be compared with the unfiltered figure 10(a). The main difference between the two figures is that the depth of the filtered correlations in figure 11(d) is more independent of $\tilde{x}_{2}$ than it was in figure 10(a). Moreover, different filter wavelengths produce very different depths, answering the question of whether the correlation depth is linked to the length of the eddies or to their distance from the wall.

Figure 11(e) displays the minimum and maximum distances from the wall at which the filtered correlation profiles exceed the arbitrary threshold $C_{(2)11}=0.4$ . The depth at short wavelengths is very small, and the maximum and minimum heights coincide. As the central wavelength of the filter increases, so does the depth, and the lower limit reaches the wall at some wavelength that depends on the reference height $\tilde{x}_{2}$ . Eddies longer than that limit ‘attach’ to the wall and link directly the inner- and outer-flow regions. The solid line starting near the origin in figure 11(e) corresponds to the upper correlation limit of eddies centred at $\tilde{x}_{2}^{+}=15$ , and marks the depth of the correlated wall layer as a function of the streamwise wavelength. It grows approximately linearly as $x_{2}\approx \unicode[STIX]{x1D706}_{1}/30$ , until it saturates to $x_{2}\approx 0.3h$ for very long wavelengths. The slope of this line depends on the correlation level used to define the depth, but it is clear from figure 11(e) that other thresholds behave similarly. For short wavelengths, $\unicode[STIX]{x1D706}_{1}^{+}\lesssim 2000$ , the correlation depth settles to approximately $x_{2}^{+}\approx 30$ , and the vertically correlated layer does not extend above the buffer layer. Similarly defined coherence limits for the three velocity components have been incorporated as grey patches to the spectra in figure 7(d–f).

It is interesting to note that a large coherence depth is a property of long eddies, independently of whether they are attached to the wall or not. For example, the pair of lines corresponding to $\tilde{x}_{2}=0.6$ in figure 11(e) define a correlation depth spanning half of the channel for wavelengths that are much shorter than the wavelength at which this particular correlation attaches to the wall, $\unicode[STIX]{x1D706}_{1}\approx 15h$ . The message of figure 11(e) is that long eddies are also deep. They attach to the wall when they grow to be too deep to do otherwise, but they do not appear to originate from the wall.

3.7 The effect of the Reynolds number

Figure 12.

(a) Sketch of attached and detached eddies in a channel, created with independent random sizes and positions. The thicker (red) objects are attached. (b) Maximum value of the velocity fluctuation intensities. Various experimental and numerical flows: open symbols, $u_{1}^{\prime 2^{+}}$ ; closed, $u_{2}^{\prime 2^{+}}$ ; light blue fill, $u_{3}^{\prime 2^{+}}$ . ▿, Boundary layers; ▵, channels; $\circ$ , pipes. Each velocity component has been offset by a different constant, so that the trend lines cross zero at $h^{+}=100$ . The trend lines are: $u_{1}^{\prime 2^{+}}=3.58+0.65\log h^{+}$ , $u_{2}^{\prime 2^{+}}=0.51+0.1\log h^{+}$ , and $u_{3}^{\prime 2^{+}}=-0.60+0.4\log h^{+}$ . (c) Correlation limits for $\tilde{x}_{2}/h=0.6$ , as functions of the central filter wavelength. ▿, $C_{(2)11}=0.4$ ; ▵, $C_{(2)33}=0.25$ ; ▫, $C_{(2)pp}=0.25$ . Each pair of lines represents the upper and lower limits within which $C_{(2)}$ exceeds the specified level. (d) One-dimensional streamwise spectral density of the pressure, as a function of the wall distance. The two diagonals are approximately: ——, $\unicode[STIX]{x1D706}_{I}=x_{2}$ ; - - - -, $\unicode[STIX]{x1D706}_{I}=5x_{2}$ , reduced to $\unicode[STIX]{x1D706}_{1}$ assuming $\unicode[STIX]{x1D706}_{1}\approx \unicode[STIX]{x1D706}_{3}$ . The grey patch is the region vertically correlated with the wall. Channel CH2000.

The resulting organisation of the velocity eddies is sketched in figure 12(a). Eddies of all sizes can be found at all heights, but they cannot grow larger than their distance from the wall. Statistically, this means that the size of the largest eddies scales linearly with the wall distance.

The relevance of attached eddies was first stressed by Townsend (Reference Townsend1961) and developed by Perry, Henbest & Chong (Reference Perry, Henbest and Chong1986). They noted that, if the intensity of eddies centred within the logarithmic layer scales with $u_{\unicode[STIX]{x1D70F}}$ , and if those eddies retain their intensity down to the wall, the fluctuation energy near the wall should increase with the logarithm of the Reynolds number. The argument hinges on the approximately uniform long-wavelength spectral ‘skirt’ of $\unicode[STIX]{x1D719}_{11}$ and $\unicode[STIX]{x1D719}_{33}$ in figure 7(d,f). It does not apply to $\unicode[STIX]{x1D719}_{22}$ , where the damping of $u_{2}$ by the impermeability condition ensures that the near-wall coherent layer in figure 7(e) contains no energy at long wavelengths.

It can be shown that the longest wavelengths of the spectra in figure 7 scale in outer units, while their short-wavelength end scales in wall units. The range of energy-containing eddies near the wall thus scales like $\unicode[STIX]{x1D706}_{max}/\unicode[STIX]{x1D706}_{min}\sim h^{+}$ , and the total fluctuation energy is $\int \unicode[STIX]{x1D719}\,\text{d}(\log \unicode[STIX]{x1D706}_{1})\sim \log (h^{+})$ . Figure 12(b) shows that this is true for the maxima of $u_{1}^{\prime 2+}$ and $u_{3}^{\prime 2+}$ , but not (or much more weakly) for $u_{2}^{\prime 2+}$ .

Figure 12(c) shows the vertical correlation limits of $u_{1}$ and $u_{3}$ for eddies centred at one particular location in the outer layer $(\tilde{x}_{2}=0.6h)$ . The main difference between the two variables in this figure is that, while the depth of $u_{1}$ increases with the wavelength over the whole range of the figure, that of $u_{3}$ only grows up to $\unicode[STIX]{x1D706}_{1}\approx 3h$ , beyond which the energy of $u_{3}$ decreases, and the filtered correlation becomes too weak to show at the selected threshold. As a consequence, the wall-attached eddies of $u_{1}$ eventually fill the whole channel in figure 7(d), but those of $u_{3}$ never reach above $x_{2}/h\approx 0.2$ in figure 7(f).

More surprising is the correlation depth of the pressure, which is included in figure 12(c) and in the pressure spectrum in figure 12(d). The pressure is an attached variable, with deep eddies that span a large fraction of the channel, but only at the comparatively short wavelengths of the $(u_{2},u_{3})$ eddies discussed in figure 9. Given the irrotational character of the longest $u_{1}$ eddies, whose vorticity is very low, one would expect their dynamics to be mainly controlled by the pressure. Indeed, the pressure correlation and the pressure spectrum are long, but the correlation in figure 12(c) gets thinner for the longer wavelengths, and detaches from the wall. Figure 12(d) shows that only a very weak vertical correlation of the pressure reaches the wall beyond $\unicode[STIX]{x1D706}_{1}\approx 5h$ . The implication is that the near-wall long and wide eddies in the spectral ridge of figure 7(a) are mostly maintained by weak but persistent Reynolds stresses. The pressure appears to be mostly associated with the rollers discussed in figure 9.

Figures 7(d) and 7(f) show that arguments similar to those above imply that the fluctuation profile above the buffer layer should be $u_{j}^{\prime 2+}\sim \log (h/x_{2})$ , for $u_{1}$ and $u_{3}$ . Both predictions have been confirmed observationally: by numerical simulations in the case of $u_{3}$ (Jiménez & Hoyas Reference Jiménez and Hoyas2008), and experimentally in the case of $u_{1}$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013).

It should be mentioned at this point that there is some controversy about the behaviour of even basic flow statistics at very high Reynolds numbers $(h^{+}\gtrsim 10^{4})$ . This range is still inaccessible to simulations, and hard to measure experimentally. For example, the presence of an ‘outer’ kinetic energy peak in the middle of the logarithmic layer has been variously reported (Smits et al. Reference Smits, McKeon and Marusic2011) as an outer maximum of the total fluctuation intensity, or as a peak in the spectral energy density. The two claims are different, and both have proved hard to confirm by high-resolution experiments (Örlü et al. Reference Örlü, Fiorini, Segalini, Bellani, Talamelli and Alfredsson2017). The second claim, a peak in the spectral density $\unicode[STIX]{x1D719}_{11}$ , could be due either to a peak in the total energy (the first claim), or to the concentration of the same energy into a narrower range of wavelengths. The latter would be a natural consequence of the narrowing of the spectrum as the flow gets closer to a single scale far from the wall (see figure 7 e), presumably because, as discussed in § 3.1, this layer has a natural unit of length. However, in the absence of unambiguous experimental confirmation, it is possible that a different kind of eddies than those described here exists at high Reynolds numbers, and that they are not captured by the simulations used in this article.

Summarising the results up to now, we have shown that the one- and two-point statistics of wall-bounded turbulent flows suggest the existence of two types of eddies: a self-similar family of inclined large-scale rollers that mostly involve the transverse-velocity components, and which are restricted to the logarithmic layer; and the much longer streaks of the streamwise velocity, also self-similar in the spanwise direction, that exists at all heights from the wall. The largest of these streaks fill most of the width of the flow. For sizes $\unicode[STIX]{x1D706}_{I}\gtrsim x_{2}$ , both types of eddies interact directly with the shear, and are therefore presumably involved in the energy-generation cycle, but their very different sizes suggest that they are only indirectly related to one another. Long eddies are deep in the wall-normal direction, but only attach to the wall if they become so large that they do not fit into the flow otherwise. Examples of structures of the transverse velocities are given later in figures 20 and 21. Examples of streaks are figure 23.

4.1 Advection velocities

We have not discussed up to now the temporal evolution of eddies, but whether they can be considered coherent or not depends on whether they are able to keep their shape for dynamically significant times. This depends, among other things, on how their propagation velocity changes with the wavenumber and with the location along inhomogeneous directions.

Consider a variable $\unicode[STIX]{x1D712}$ that we wish to approximate as a wave with phase velocity $c_{\unicode[STIX]{x1D712}}$ along the direction $x_{1}$ . A simple definition of the phase velocity was introduced by del Álamo & Jiménez (Reference del Álamo and Jiménez2009) by minimising $\langle (\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D712}+c_{\unicode[STIX]{x1D712}}\unicode[STIX]{x2202}_{1}\unicode[STIX]{x1D712})^{2}\rangle$ . The result is

which can be expressed as

for individual Fourier modes, where the asterisk stands for complex conjugation, and $\text{Im}$ for the imaginary part.

Denoting by $c_{j}$ the advection velocity of the $u_{j}$ velocity component, the spectral phase velocity of the streamwise-velocity component, $c_{1}$ , is shown in figure 14 at three heights in the channel, normalised with the mean flow velocity at each height. Del Álamo & Jiménez (Reference del Álamo and Jiménez2009) showed that the phase velocity agrees with the local velocity for most wavelengths, except for very large scales, where $c_{1}\approx U_{b}$ independently of $x_{2}$ , and in the buffer layer, where $c_{1}^{+}\approx 11$ . Figure 14 shows that this remains approximately true in the present case, at a higher Reynolds number than in del Álamo & Jiménez (Reference del Álamo and Jiménez2009), and that the scale dependence is reasonably well described as a function of the wall-parallel isotropic wavelength $\unicode[STIX]{x1D706}_{I}$ .

Figure 15(a) summarises the wall-normal dependence of $c_{1}$ , and shows that the transition to the large-scale behaviour in which the eddies do not follow the flow takes place at $\unicode[STIX]{x1D706}_{I}\approx 5x_{2}$ above $x_{2}^{+}\approx 100$ . Closer to the wall, the transition wavelength saturates at $\unicode[STIX]{x1D706}_{I}^{+}\approx 10^{3}$ , which is also the boundary in figure 7(a) between the near-wall spectral energy component and the self-similar outer ridge. A similar behaviour is found for the other two velocity components, although there is very little large-scale energy in the case of $u_{2}$ .

Figure 14.

Phase velocity for $u_{1}$ normalised with the mean velocity at each height, $c_{1}/U_{1}(x_{2})$ . Channel CH2000. Contours are spaced by 0.1, drawn only where the spectral density $\unicode[STIX]{x1D719}_{11}$ is greater than 2 % of its maximum. (a) $x_{2}^{+}=15$ . (b) $x_{2}^{+}=80$ ; (c) $x_{2}/h=0.15\,(x_{2}^{+}=300)$ . The hyperbola-like thicker lines are: ——, $\unicode[STIX]{x1D706}_{I}=x_{2}$ ; - - - -, $\unicode[STIX]{x1D706}_{I}=5x_{2}$ .

Figure 15.

Advection velocity parameters for $u_{1}$ in channels, as a function of $x_{2}$ , averaged over shells of constant $\unicode[STIX]{x1D706}_{I}$ . Shaded contours are CH2000. The thicker lines are: – ⋅ – ⋅ –, $h^{+}=550$ (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2004); - - - -, CH950; ——, CH2000. (a) Phase velocity normalised with the mean flow velocity. Shaded contours are $c_{1}/U_{1}(x_{2})=0.9(0.1)1.4$ , and thick lines are $c_{1}/U_{1}(x_{2})=1$ . (b) Wavenumber dispersion parameter, as defined in (4.8). Shaded contours are $c^{\ast }=1.0(0.3)2.2$ , and thick lines are $c_{1}^{\ast }=1$ . (c) Shearing parameter, as defined in (4.9). Shaded contours are $\unicode[STIX]{x1D6FE}_{s1}=0.2(0.2)1.0$ , and thick lines are $\unicode[STIX]{x1D6FE}_{s1}=0.2$ . The straight lines in (a–c) are: ——, $\unicode[STIX]{x1D706}_{I}=x_{2}$ ; - - - -, $\unicode[STIX]{x1D706}_{I}=5x_{2}$ and $x_{2}^{+}=80$ . (d) Instantaneous snapshots of an attached logarithmic-layer ejection in a channel at $h^{+}=4200$ . The flow (and time) goes from left to right, and the streamwise displacement of the structure has been shortened in order to fit its complete lifetime in the figure. The structure is coloured with the distance from the wall. Reproduced with permission from Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014b ).

The propagation velocity of wave packets is the group velocity (Whitham Reference Whitham1974),

The difference between the phase and group velocities is generally small, but significant. For example, it was shown by del Álamo & Jiménez (Reference del Álamo and Jiménez2009) that it influences the reduction of experimental frequency spectra to their wavenumber counterparts. In the present context, the dispersion lifetime of a wave packet depends on the difference between the two velocities. Assume that the wave packet is chosen such that $\unicode[STIX]{x0394}k_{1}\approx k_{1}$ , as discussed in the previous section. The packet disperses at a rate $\unicode[STIX]{x0394}c\approx |\unicode[STIX]{x2202}_{k_{1}}c|\unicode[STIX]{x0394}k_{1}\approx |\unicode[STIX]{x2202}_{k_{1}}c|k_{1}=|c_{g}-c|$ . We can then define an analogue of the Corrsin parameter discussed in § 3.3,

which compares the dispersion and turnover times. If $c_{j}^{\ast }\ll 1$ , the lifetime of the packet is determined by its nonlinear turnover; otherwise, it is limited by dispersion. Figure 15(b) shows that dispersive eddies of $u_{1}$ are only found at long wavelengths, $\unicode[STIX]{x1D706}_{I}\gtrsim h$ , and near the wall. Approximately the same limits apply to the other two velocity components. Figure 15(b) should be used in conjunction with the Corrsin parameter in figure 6(b), which measures the importance of the shear with respect to nonlinearity. A comparison of the contour levels of the two figures shows that deformation by the shear is faster than wavenumber dispersion almost everywhere.

This could suggest that all large eddies are sheared by the mean flow, but this turns out not to be the case. The actual effect of the shear on the deformation of the wave packets can be quantified by a ‘shear-deformation parameter’ that compares the shear with the wall-normal variation of the phase velocity,

When $\unicode[STIX]{x1D6FE}_{s}\approx 1$ , the (nonlinear) self-interaction sustains the shape of the structure against the effect of the mean flow, and $c_{j}$ is approximately independent of $x_{2}$ . When $\unicode[STIX]{x1D6FE}_{s}\ll 1$ , eddies are advected and deformed by the shear. This does not imply that the sheared eddies are not coherent, but suggests that their interaction with the mean flow is essentially linear, and that they are unlikely to survive much longer than the shearing time, $ST_{s}=O(1)$ .

The Corrsin parameters $s^{\ast }$ and $c^{\ast }$ quantify processes acting on the eddies, while $\unicode[STIX]{x1D6FE}_{s}$ measures the result of those processes. The difference between $s^{\ast }$ and $c^{\ast }$ on one side and $\unicode[STIX]{x1D6FE}_{s}$ on the other, encapsulates the difference between statistically defined eddies, and the internal dynamics of the structures. The shear-deformation parameter for $u_{1}$ is shown in figure 15(c). Only structures in the buffer layer, $x_{2}^{+}\lesssim 50$ , and those longer than $\unicode[STIX]{x1D706}_{I}/x_{2}\approx 5$ are nonlinearly coherent.

The dispersive behaviour implied by figure 15(b) is reasonably well understood in the case of the buffer layer, and illustrates the relation between coherence and dispersion. Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014b ) measured the propagation velocity of individual structures defined by intense isosurfaces of the tangential Reynolds stress, $-u_{1}u_{2}$ (see § 5). They measured the phase velocity at each wall distance by tracking the motion of small-scale features within individual structures, and the group velocity by tracking the motion of the rectangular box circumscribing each structure. All the structures in the sample were relatively large, and attached to the wall in the sense of having roots well within the buffer layer. They found that the two propagation velocities were similar above $x_{2}^{+}\approx 100$ , but different below that level. An example is given in figure 15(d), which shows the evolution of an ejection $(u_{1}<0,u_{2}>0)$ throughout its lifetime. The upper part of the ejection extends into the logarithmic layer and moves approximately as a unit. It drags underneath a dispersive viscous root that moves more slowly, and which keeps getting left behind (second to fourth attached frames), and reforming at its leading edge (frame five).

In essence, dispersion happens because eddies at a given distance from the wall are superpositions of eddies of different sizes and heights. Larger eddies, with longer wavelengths, tend to be centred at higher distances from the wall, and move with the faster velocity corresponding to their taller vertical range. This dependence of the phase velocity on wavelength defines dispersion.

Figure 16.

Sketch of the expected behaviour of the flow structures in channels, as a function of the wavelength and of the distance from the wall. All limits should be understood as approximate.

Figure 17.

Behaviour of the velocity fluctuations integrated over the spectral ranges sketched in figure 16, as functions of the distance from the wall. – ⋅ – ⋅ –, CH950; - - - -, CH2000; ——, CH5200. (a,d,g) Near-wall viscous waves; $\unicode[STIX]{x1D706}_{1}^{+}\leqslant 10^{4},\,\unicode[STIX]{x1D706}_{3}^{+}\leqslant 2\times 10^{3}$ . (b,e,h) Large inertial waves; $\unicode[STIX]{x1D706}_{I}\geqslant 2h$ . (c,f,i) Shear-driven eddies; $x_{2}\leqslant \unicode[STIX]{x1D706}_{I}<2h$ . (a–c) Total integrated fluctuation intensity, $q=\sqrt{u_{i}u_{i}/3}$ . The horizontal line in (a) is the wall distance at which $U_{1}^{+}(x_{2})=11$ ; the one in (b) is $U_{1}(x_{2})=U_{b}$ . (d–f) Spectral densities, $\unicode[STIX]{x1D719}_{qq}^{+}=0.1$ . The shaded regions are the ranges used in (a–c) to integrate the different intensity profiles. Drawn at: (d) $x_{2}^{+}=15$ , (e) $x_{2}/h=0.35$ , (f) $x_{2}/h=0.2$ . (g–i) - - - -, Profiles of the phase velocity of $u_{1}$ at the spectral points marked with a circle in (d–f); ——, mean velocity profile. Channel CH2000.

The conclusions from this first part of the paper can be summarised in the sketch in figure 16. There are two coherent parts of the $(x_{2},\unicode[STIX]{x1D706}_{I})$ parameter plane: the buffer layer, where structures are held together by viscosity, and the very large structures in the outer region, which presumably survive because they draw their energy from the high shear near the wall, but are only slowly deformed by the weaker shear in the outer part of the flow. This agrees with our previous argument that these are the only two parts of the flow that possess an intrinsic unit of length: the wall unit near the wall, and the flow thickness for the very large eddies. It also suggests that the structures in these two ranges could be approximately described as semi-permanent solutions of the Navier–Stokes equations. They draw their energy from the shear, but possess enough internal dynamics to maintain a uniform propagation velocity across their wall-normal extent.

The eddies in the trapezoidal region labelled as ‘shear-driven bursts’ in figure 16 possess no such unit of length, and are therefore difficult to describe as equilibrium solutions. Accordingly, this is also the region in which the propagation velocity is found to track the mean profile, and where the life of the structures is determined by their deformation by the shear. Finally, structures to the left of the Corrsin length are too small to couple with the shear, and constitute the quasi-isotropic Kolmogorov cascade. The question mark to the right of $\unicode[STIX]{x1D706}_{I}/h\approx 30$ reflects the uncertainty in the maximum length of the large-scale velocity streaks. As discussed in § 3.5, this limit probably depends on the flow involved.

The effect of the behaviour of the advection velocity on the kinetic energy profiles is shown in figure 17. Figures 17(a)–17(c) display the profiles of the kinetic energy integrated over spectral regions corresponding, respectively, to the near-wall viscous waves, the large-scale inertial waves, and the logarithmic-layer sheared structures. The corresponding spectral ranges are plotted in figure 17(d–f) at wall-parallel planes characteristic of the corresponding structural regimes, and the vertical profiles of the phase velocities for the three cases are compared with the mean velocity profile in figure 17(g–i). The point where the phase and mean velocity profiles intersect defines the critical layer at which a permanent wave resonates with the mean flow and is expected to reach its maximum amplitude. This resonance is both a linear result (McKeon & Sharma Reference McKeon and Sharma2010), and an approximate nonlinear result (Hall & Sherwin Reference Hall and Sherwin2010). It is clearly visible in figure 17(a,b), where the structures in the viscous and outer layers reach a maximum close to the level at which their approximately constant phase velocity crosses the mean velocity profile.

Figures 17(c), 17(f) and 17(i) contain results for the self-similar structures of the logarithmic layer. For them, figure 17(i) shows that the phase velocity follows very closely the mean profile, and a critical layer cannot be defined. We have already seen that these are not permanent waves. They have a finite lifetime due to their deformation by the shear, and display no amplitude maximum in figure 17(c).

5.1 Intermittency

Consider first the intensity as an indicator of coherence. Isolating individually connected regions of the flow by thresholding their intensity is a classical way of identifying coherent structures, but it implies the choice of a threshold. This is easiest for intermittent quantities, for which high intensity is localised. There is a well-developed theory of the intermittency associated with a singular behaviour in the limit of very high Reynolds numbers (Sreenivasan Reference Sreenivasan1991), but we will use a less rigorous definition. If, for a quantity $\unicode[STIX]{x1D712}$ with p.d.f. $p(\unicode[STIX]{x1D712})$ , we define the volume fraction of the data above a threshold $\unicode[STIX]{x1D712}_{0}$ as

and the fraction of $\unicode[STIX]{x1D712}$ above that threshold as

$\unicode[STIX]{x1D712}$ will be considered to be intermittent if a threshold can be found such that $F(\unicode[STIX]{x1D712}_{0})$ is relatively large for a relative small volume $V(\unicode[STIX]{x1D712}_{0})$ .

Figure 18.

(a) Probability density function of several quantities, illustrating the effect of different representations. (b) Cumulative property fraction (5.2) against cumulative probability (5.1). Channel CH2000. ——, $\unicode[STIX]{x1D712}=-u_{1}u_{2}$ ; - - - -, $|u_{2}|$ ; 

,  $u_{2}^{2}$ ; – ⋅ – ⋅ –, $|\unicode[STIX]{x1D714}|$ ; 

,  $\unicode[STIX]{x1D714}^{2}$ . The shaded patch in (b) is $V_{\unicode[STIX]{x1D712}}\in (0.05-0.1)$ .

Unfortunately, this definition is physically ambiguous. Consider the problem of isolating velocity structures in which the absolute value of $u_{2}$ is above a threshold. The p.d.f. of the velocity is known to be approximately Gaussian, as shown by the dashed curve in figure 18(a). Correspondingly, figure 18(b) shows that the velocity fraction (5.2) grows relatively slowly with the volume fraction (5.1).

However, the distribution of the same quantity becomes more intermittent if we consider $u_{2}^{2}$ (dashed line with symbols in figure 18 a). As a consequence, the fraction of the wall-normal component of the kinetic energy also grows more steeply with the volume than in the case of $|u_{2}|$ (figure 18 b). The same is true of the vorticity magnitude, which has been included in figure 18(b). Even for such a technically intermittent variable, the fraction of $|\unicode[STIX]{x1D714}|$ included in a given volume fraction of strong vorticity grows approximately as that of $|u_{2}|$ , while that of the enstrophy, $|\unicode[STIX]{x1D714}|^{2}$ , behaves approximately as $u_{2}^{2}$ . Any p.d.f. can be made more intermittent by representing it in terms of a higher power of its variable. Of particular interest is the product $-u_{1}u_{2}$ , which we will use in the next section to classify the flow into regions of active wall-normal transfer of the streamwise momentum. Its p.d.f has been added to figure 18(a,b) and behaves as a quadratic variable. In fact, it was shown by Antonia & Atkinson (Reference Antonia and Atkinson1973) and Lu & Willmarth (Reference Lu and Willmarth1973) that the probability distribution of $-u_{1}u_{2}$ is essentially that of the product of two Gaussian variables. The main reason why we can partition the flow into discrete intense regions that contain a relatively large fraction of quantities of interest within a small volume fraction, is that the interesting quantities in fluid mechanics (energy, enstrophy and Reynolds stresses) are often quadratic.

A systematic way of choosing a threshold to partition the flow into separate intense regions was introduced by Moisy & Jiménez (Reference Moisy and Jiménez2004). Consider the enstrophy. A very high threshold isolates a few very intense vortices which account for a very small fraction of the total enstrophy. Conversely, a very low threshold includes a larger enstrophy fraction, but at the price of linking all the vortices into a single large tangle. The ‘percolation’ transition between the two limits is typically abrupt (Stauffer & Aharony Reference Stauffer and Aharony1994), and can be used to define a threshold that includes as much enstrophy as possible while still segmenting the flow into individual structures. While the percolation threshold depends on the shape of the individual objects being thresholded, it typically takes place for volume fractions of the order of 5–10 % in three dimensions. This range has been added to figure 18(b), and shows that we can expect to identify intense structures accounting for approximately 40–60 % of the quadratic quantities of the flow, but for a much smaller fraction (15–25 %) of the linear ones. Details of the application of this technique to wall-bounded flows can be found in del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006) for the vorticity, in Lozano-Durán, Flores & Jiménez (Reference Lozano-Durán, Flores and Jiménez2012) and Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017) for the Reynolds stresses, and in Sillero (Reference Sillero2014) for individual velocity components. The resulting structures will be discussed in the rest of this section.

It is clear that the percolation threshold is only one among many possible threshold choices, and that conclusions derived from the structures thus obtained have to be tested against other methods of analysis. For example, individual structures should only be treated as indicative, while the statistical properties of classes of structures are more meaningful. Also, the percolation analysis described above typically results in a range of thresholds approximately spanning a decade. Choosing its midpoint as a nominal threshold is reasonable, but any conclusion should be tested against the results of thresholding above and below the nominal value. Finally, although it should be taken into account that different quantities and methods often result in different statistics, any conclusion derived from thresholded structures should be complemented by the analysis of the statistical eddies discussed in §§ 3 and 4.

5.2 The tangential Reynolds stress

Figure 19.

(a) Joint p.d.f. of $(u_{1},u_{2})$ . ——, Joint-Gaussian p.d.f. with correlation coefficient $\langle u_{1}u_{2}\rangle /u_{1}^{\prime }u_{2}^{\prime }=-0.4$ ; - - - -, homogeneous shear turbulence, HSF100; – ⋅ – ⋅ –, channel CH950 at $x_{2}/h=0.15$ . Contours contain 70 %, 30 %, and 5 % of the total mass of the p.d.f. The hyperbolic lines are $u_{1}u_{2}=\pm 1.75u_{1}^{\prime }u_{2}^{\prime }$ . (b) Sketch of a Reynolds-stress structure (Q2) within its circumscribed parallelepiped, coloured by the distance from the wall. The flow is from left to right, and the top of the structure is at $x_{2,max}=x_{2,min}+\unicode[STIX]{x1D6E5}_{2}$ . (c) Joint p.d.f. of the streamwise and wall-normal dimensions of the Q $^{-}$ structures. ——, Attached $\text{Q}^{-}\text{s}$ in channel CH2000; – ⋅ – ⋅ –, detached $\text{Q}^{-}\text{s}$ in CH2000; - - - -, homogeneous shear turbulence HSF250, with dimensions normalised by the span of the computational box. The diagonal is $\unicode[STIX]{x1D6E5}_{1}=2\unicode[STIX]{x1D6E5}_{2}$ , and contours contain 90 % of the p.d.f.s. (d) As in (c), for the spanwise dimension. The diagonal is $\unicode[STIX]{x1D6E5}_{3}=\unicode[STIX]{x1D6E5}_{2}$ .

The first coherent structures to be treated quantitatively in wall-bounded flows were those of the tangential Reynolds stress, $-u_{1}u_{2}$ , which is the quantity associated with mean momentum transfer in (3.1), and with the mean production of turbulent energy in (3.6). Consider the joint p.d.f. of the two velocity components in figure 19(a), which can be classified into the four quadrants labelled Q1 to Q4. Most points for which $|u_{1}u_{2}|$ is large either belong to Q2 (ejections), where positive $u_{2}$ carries low streamwise velocity from the wall upwards, or to Q4 (sweeps), where the opposite is true. Wallace, Eckelman & Brodkey (Reference Wallace, Eckelman and Brodkey1972) and Lu & Willmarth (Reference Lu and Willmarth1973) argued that these events are the dominant contributors to the exchange of streamwise momentum among different layers of the flow, and ultimately to the generation of turbulent drag. They introduced the analysis of the flow in terms of strong ‘quadrant’ events, defined by an empirically determined threshold for $|u_{1}u_{2}|$ . These early single-point velocity measurements eventually evolved into fully three-dimensional structures in direct simulations, denoted here as Qs. In agreement with our discussion of figure 18(b), the percolation analysis described in the previous section isolates Qs that fill approximately 7 % of the volume, and 60 % of the total momentum transfer. It is encouraging that the threshold defined in this way agrees approximately with those found by other groups from single-point temporal signals, using very different methods. A typical value $|u_{1}u_{2}|>1.75\,u_{1}^{\prime }u_{2}^{\prime }$ has been added to figure 19(a) as the four hyperbolic lines bounding the Qs.

Qs have been studied for wall turbulence by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012), and for homogeneous shear turbulent flows (HSF) by Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017). Their temporal evolution, including mergings and splits, has been documented in Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014b ). The interested reader should consult these papers for details. We only summarise here their most salient features.

When structures are circumscribed in a parallelepiped aligned with the coordinate directions, as in the sketch in figure 19(b), they can be classified according to their position and dimensions. Structures separate into two clearly distinct classes: those which are attached to the wall, and those which are not. This distinction appears clearly in the joint p.d.f. of minimum and maximum distances from the wall, and applies to all the structures that have been studied up to now in wall-bounded turbulence, not only to the Qs. It is determined by whether the minimum distance from the wall is below or above $x_{2}^{+}\approx 20$ (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006). In the example in figure 15(d), this corresponds to whether the structures have developed a viscous dispersive root near the wall. As seen in that figure, a given structure need not be attached or detached over its full lifetime. Ejections, which tend to move away from the wall, start their lives as attached, and eventually detach. Sweeps behave the other way around. Notwithstanding these differences, the properties of Q2s and Q4s are fairly similar, and we will treat them together from now on as $\text{Q}^{-}\text{s}$ . Although detached structures are numerically much more common than attached ones, they are also smaller, and most of the volume associated with intense Reynolds stress is in attached structures. In the case of Qs in channels, approximately 60 % of all the intense structures are $\text{Q}^{-}\text{s}$ , but only 25 % of them are attached to the wall. In spite of this, attached structures account for approximately 70 % of the volume of intense structures, and carry approximately 60 % of the total tangential Reynolds stress (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012).

Figure 19(c) shows p.d.f.s of the streamwise and wall-normal dimensions of $\text{Q}^{-}\text{s}$ in three classes of structures: attached and detached $\text{Q}^{-}\text{s}$ in a channel, and $\text{Q}^{-}\text{s}$ in HSF, where the absence of walls makes attachment irrelevant. All of them describe self-similar families, at least above the buffer layer. Detached and HSF $\text{Q}^{-}\text{s}$ have similar aspect ratios $(\unicode[STIX]{x1D6E5}_{1}/\unicode[STIX]{x1D6E5}_{2}\approx 2)$ , while attached $\text{Q}^{-}\text{s}$ are slightly more elongated $(\unicode[STIX]{x1D6E5}_{1}/\unicode[STIX]{x1D6E5}_{2}\approx 3)$ . This discrepancy was investigated by Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017), who showed that it is due to ‘spurious’ connections between neighbouring attached $\text{Q}^{-}\text{s}$ through their viscous roots. If the points below $x_{2}^{+}=100$ are removed from the identification of connected structures, even objects that would otherwise be attached have the same aspect ratio as the detached ones. Note that these aspect ratios are approximately consistent with those in figure 9(a,b), where the correlations of $u_{2}$ and $u_{3}$ have heights of order $h$ , and streamwise lengths of order $2h$ . They also agree with the ‘short’ POD reconstructions of $u_{2}$ and $u_{3}$ in figure 13(f), but they are much shorter than the ‘long’ correlations of $u_{1}$ in figures 9(c) and 13(f). The spanwise aspect ratios in 19(d) are identical for the three families $(\unicode[STIX]{x1D6E5}_{3}/\unicode[STIX]{x1D6E5}_{2}\approx 1)$ , in agreement with the cross-flow correlations in figure 9. In all these cases, there is essentially no difference between the channel and the wall-less HSF, reinforcing the conclusion in §§ 3 and 4 that coherent structures and eddies are a consequence of the shear, not of the wall. Structures attach to the wall when they become too large to fit in the channel otherwise.

The fact that the attached $\text{Q}^{-}\text{s}$ in channels are longer than the detached ones, but not wider, suggests that neighbouring $\text{Q}^{-}\text{s}$ are arranged streamwise from one another. This was confirmed by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012), who showed that attached $\text{Q}^{-}\text{s}$ of the same kind (e.g., a Q2 and its nearest Q2) are preferentially located streamwise from each other, while those of different kind (i.e., Q2 and Q4) form spanwise pairs. Interestingly, the same is only true in HSF for Qs whose diagonal dimension is larger than the Corrsin scale. Smaller structures, whose internal turnover time is too fast to couple with the shear, do not have a frame of reference from which to determine their orientation, and are statistically isotropic (Dong et al. Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017). For these small structures, it is possible to define Q $_{ij}$ ’s based on the intensity of the Reynolds stress $-u_{i}u_{j}$ . Many of their characteristics are the same as the more classical Qs based on $-u_{1}u_{2}$ , but the statistics of their relative positions are oriented in each case with respect to the velocity components with which they are defined. In this sense, only larger-than-Corrsin Qs are physically relevant for the energy and momentum balance of shear turbulence. As discussed in § 3, smaller structures are part of the isotropic Kolmogorov cascade. Note that, since the Corrsin scale in channels is $L_{c}\approx x_{2}$ , all attached structures are larger than $L_{c}$ , and most structures larger than $L_{c}$ are attached.

It is interesting in this respect that there are essentially no large Q1 or Q3 structures in HSF (Dong et al. Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017), in the same way that there are very few attached Q1 and Q3 in channels (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). Large eddies with $u_{1}u_{2}<0$ draw energy from the shear, while those with $u_{1}u_{2}>0$ lose it. As a consequence, the former grow while the latter dwindle, and the only Qs which are approximately evenly distributed among the four quadrants are the smaller ones uncoupled from the shear. In this sense, attached eddies in wall-bounded turbulence are equivalent to larger-than-Corrsin eddies in HSF.

Figure 20.

Flow fields conditioned to attached Q2–Q4 pairs reaching into the logarithmic layer. Channel CH2000. Flow is from top-left to bottom-right, and axes are scaled with the distance from the centre of gravity of the pair to the wall. (a) P.d.f.s of the points belonging to the Q2 (yellow at left) and Q4 (blue at right). The isosurfaces plotted are $0.75$ times the maximum value of each p.d.f. (b) Conditional streamwise perturbation velocity. The yellow object at left is the low-speed isosurface, $u_{1}^{+}=-0.5$ . The blue one is $u_{1}^{+}=0.5$ . The arrows in (a,b) are transverse velocities in the cross-flow plane, with the longest one approximately $0.5u_{\unicode[STIX]{x1D70F}}$ . (c) Instantaneous Q2–Q4 pair, with its associated vortex tangle. Yellow is the ejection, blue is the sweep, and red are the vortices. Flow is from left to right.

Conditional flow fields around pairs of attached $\text{Q}^{-}\text{s}$ in a channel are shown in figure 20(a,b). Since we saw in figure 19(c,d) that pairs come in all sizes with self-similar aspect ratios, the conditional average is compiled after rescaling each pair to a common size and centring it on the centre of gravity of the pair. Therefore, the coordinates in figure 20(a,b), $\unicode[STIX]{x1D6FF}_{i}=x_{i}/\bar{x}_{2}$ , are multiples of the distance, $\bar{x}_{2}$ , from the wall to the centre of gravity of each individual pair. Also, because the equations are symmetric with respect to reflections along $x_{3}$ , the ejection is always defined to lie to the left of the picture. Figure 20(a) displays the mean conditional geometry of the Q-pair, and is actually an isosurface of the probability of finding a point belonging to a structure in the similarity coordinates. The object to the left is the ejection, and the one to the right is the sweep. The arrows in the central cross-section are conditional velocities in that plane. They reveal an approximately streamwise roller, rising out of the ejection and sinking into the sweep. There are secondary counter-rollers to both sides of the pair, but they are much weaker than the primary central one, and there is little evidence of symmetric ‘hairpin legs’. The Qs in figure 20(a) are in the logarithmic layer $(x_{2,max}^{+}>100)$ . Their rollers can only be identified as conditional objects, and it is difficult to relate different rollers to one another, but the arrangement of streamwise vortices in the buffer layer has been studied extensively. They are arranged antisymmetrically, with vortices of opposite sign alternatively staggered along each side of a streak (Stretch Reference Stretch1990; Schoppa & Hussain Reference Schoppa and Hussain2002). The lack of strong counter-rollers in figure 20(a) suggests that the average arrangement in the logarithmic layer is similar. The aspect ratio of the Q-pair in figure 20(a) is approximately $4:1:1.5$ along the three coordinate directions (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012).

Because $u_{1}<0$ in ejections, and $u_{1}>0$ in sweeps, the ejection sits in a low-velocity streak, and the sweep in a high-velocity streak. The pair sits at the border between the two, and the intervening roller has the direction of rotation required to reinforce both streaks. Figure 20(b) shows the conditional streaks associated with the pair in figure 20(a). The high-velocity streak is larger than the low-velocity streak. The early experimental perception was the opposite, probably because ejections collect tracers from the wall, and are easier to visualise, but high-speed regions are fed from above, while low-speed regions are blocked by the wall. This intrinsic asymmetry was shown by Orlandi & Jiménez (Reference Orlandi and Jiménez1994) to be at the root of why the skin friction is higher in turbulent flows than in laminar flows. The size relation between the two streaks in figure 20(b) is statistically robust, and is evident both in instantaneous pictures and in conditional ones. Both conditional streaks are longer than the Qs, especially the high-velocity streak, but not as much as the very long structures implied by the correlations of $u_{1}$ in figure 11(e). In fact, the dimensions of the low-speed streak in figure 20(b) are very similar to those of the conditional ejection in figure 20(a). The high-speed streak is longer, but also taller, and its length-to-height ratio, approximately $5:1$ , is also not very different from either the Qs or from the low-velocity streak, although shorter than the $20:1$ ratios for attached eddies implied by figure 11(e). This point will be further discussed in § 5.3, after we see examples of instantaneous streaks.

It should be made clear that the rollers in figure 20(a,b) are not vortices, and that individual structures are not as smooth as the conditional ones. A typical instantaneous sweep–ejection pair is shown in figure 20(c). Its width is $0.5h$ , or 1000 wall units. The much smaller vortices in the vorticity cluster associated with the pair have been added to the figure, for comparison. The dimensions of the conditional rollers in figure 20(a,b) are similar to those of the correlation rollers discussed in connection with figure 9(a,b).

Figure 21.

Effect of the inhomogeneity of the mean velocity profile on the flow field conditioned to Q2–Q4 pairs. The central opaque S-shaped object is the isosurface of the magnitude of the conditional perturbation vorticity, at 25 % of its maximum. The two translucent objects are isosurfaces of the conditional perturbation streamwise velocity, $u_{1}^{+}=\pm 0.6$ . (a) Homogeneous shear turbulence, HSF100. (b) Detached Qs in channel CH950. (c) Attached Qs in channel CH950. Adapted with permission from Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017).

The three-dimensional geometry of the conditional rollers associated with the $\text{Q}^{-}$ pairs is displayed in figure 21. The three figures correspond to different flow configurations. They are reproduced from Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017), which should be consulted for details. Figure 21(a) corresponds to homogeneous shear turbulence, and is the clearest. Because of the statistical symmetries of this flow, the conditional sweep and ejection (not shown in the figure, for clarity) are equivalent, and are arranged antisymmetrically with respect to the centre of gravity of the pair. The roller is located between them, approximately aligned to the $(x_{1},x_{2})$ plane, and inclined at 45 $^{\circ }$  with respect to the mean flow velocity. This is the direction of maximum extension by the shear (Rogers & Moin Reference Rogers and Moin1987). At both sides of the roller are the high- and low-velocity streaks, each one extending downstream from the roller: the high-speed streak extends towards the right of the figure, and the low-speed streak extends to the left. The implication is that the roller moves at an intermediate velocity from that of the two streaks, so that the faster high-speed streak moves ahead of it while the slower low-speed streak is left behind. It is known that the streamwise advection velocities of the sweeps, which are located in high-speed streaks, are approximately $3u_{\unicode[STIX]{x1D70F}}$ faster than those of the ejections, which are located in low-speed streaks (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b ). The roller in figure 21(a) is capped at its top and bottom by two ‘hooks’ resembling incomplete hairpins. These are also conditional structures.

Figure 21(b) is plotted with the same parameters as figure 21(a), but for detached pairs in the logarithmic layer of a channel. The arrangement is similar, but the roller is less symmetric, and less inclined to the free stream (20 $^{\circ }$ ). The high-speed streak, which is now farther from the wall than the low-speed streak, is also substantially larger. The conditional geometry of wall-attached pairs is displayed in figure 21(c). It can best be understood as a further evolution of the transition from figures 21(a) to 21(b). The roller is almost parallel to the wall, and the low-speed streak has almost disappeared underneath the surviving upper hook of the roller, which now looks as an essentially full asymmetric hairpin. Note that the lower hook is still visible in the figure, although almost completely truncated by the wall and hidden underneath the high-speed streak.

As we have already seen several times in this article, the conclusion has to be that the structures depicted in figure 21(a–c) are aspects of the same process, although both the low-speed streak and the lower hook of the roller get increasingly modified by the non-uniformity of the mean shear. They are also damped by the impermeability of the wall as they approach it.

5.3 The velocity structures

Figure 22.

(a) One-dimensional p.d.f.s of the logarithm of the streamwise length, $\unicode[STIX]{x1D6E5}_{1}$ , of intense attached structures of $u_{1}$ , as functions of $\unicode[STIX]{x1D6E5}_{2}$ . (b) Same for the transverse velocities. (c) Same for the spanwise length, $\unicode[STIX]{x1D6E5}_{3}$ . In all cases: ——, $|u_{1}|^{+}>2$ in BL6600 at $h^{+}\approx 1600$ (Sillero Reference Sillero2014);  $\cdots \cdots$ , same for CH2000 (Lozano–Durán, private communication); - - - -, $|u_{2}|^{+}>1.3$ in BL6600; – ⋅ – ⋅ –, $|u_{3}|^{+}>1.6$ in BL6600. The diagonals in (a,b) are: ——, $\unicode[STIX]{x1D6E5}_{1}=2\unicode[STIX]{x1D6E5}_{2}$ ; - - - -, $\unicode[STIX]{x1D6E5}_{1}=5\unicode[STIX]{x1D6E5}_{2}$ . The diagonal in (c) is $\unicode[STIX]{x1D6E5}_{3}=\unicode[STIX]{x1D6E5}_{2}$ . (d) Two-dimensional streamwise section through the reference point $(\tilde{x}_{2}/h=0.4)$ of the two-point autocorrelation function of $u_{1}$ . Contours are $C_{11}=[0.05,0.1,0.3]$ . ——, Channel CH2000; - - - -, boundary layer BL6600 at $h^{+}=1530$ ; the grey patch is the $C_{33}=0.05$ contour of the $u_{3}$ correlation in the channel.

The intense structures of individual velocity components were studied by Sillero (Reference Sillero2014). They also separate into attached and detached families, and are self-similar in the logarithmic layer. Figure 22 displays the p.d.f.s of their aspect ratios. The elongation of the structures of $u_{1}$ in figure 22(a), $\unicode[STIX]{x1D6E5}_{1}/\unicode[STIX]{x1D6E5}_{2}\approx 5$ , is higher than for the transverse velocities in figure 22(b), $\unicode[STIX]{x1D6E5}_{1}/\unicode[STIX]{x1D6E5}_{2}\approx 2$ , although, somewhat surprisingly, not much more so. The latter value is also the aspect ratio of the Qs discussed in figure 19(b), showing that the elongation of the Qs is that of the transverse-velocity components. The spanwise aspect ratio is shown in figure 22(c), and approximately agrees for all variables, including $u_{1}$ and the Qs.

More interesting is the comparison in figure 22(a) between the $u_{1}$ structures of channels and of boundary layers. Figure 22(d) shows that, as already discussed in § 3.5, the correlations of $u_{1}$ in boundary layers are considerably shorter than those in the channels, but it turns out that the intense $u_{1}$ structures in the logarithmic layer are very similar in the two flows. If we interpret the reference height of the two-point correlation as the centre of gravity of the structures, the p.d.f.s in figure 22(a) would imply that the structures of $u_{1}$ responsible for the correlation in figure 22(d) would have $\unicode[STIX]{x1D6E5}_{1}/h\approx 2$ –6. This is in reasonable agreement with the correlations of the boundary layer, and with the approximate size of the individual velocity substructures in figures 3(b) and 23(b), but is too short for the correlations of the channel. Figure 22(a) suggests that the difference between the correlations of boundary layers and channels is mostly due to the very long and tall ‘global’ modes that dominate the outer part of the flow, and which appear in the p.d.f.s as ‘overhangs’ at the upper end of the self-similar region. These structures contribute strongly to the correlations in the channel because they are large, but figure 22(a) shows that they are not particularly intense, or at least that they do not strongly affect the smaller structures in the logarithmic layer. Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) mention that there is at least one very large Q4 that crosses the whole computational box in most flow fields, but that those objects are concatenations of smaller ones. The suggestion is that the difference between the correlation of $u_{1}$ in boundary layers and channels lies in the details of the concatenation mechanism, rather in the substructures themselves. Given our discussion in § 3.5, it would be interesting to repeat the present analysis for Couette flow.

Figure 23.

Perturbation velocity isosurfaces, with the flow from lower left to upper right. (a) Buffer-layer low-velocity streaks, $u_{1}^{+}=-2$ (light grey). Channel CH2000. Structures higher than $x_{2}^{+}=150$ and smaller than $10^{6}$ cubic wall units have been removed for clarity. The plot spans $(L_{1}^{+}\times L_{3}^{+}=2300\times 1000)$ . The darker (red) objects are $|u_{2}|^{+}=2$ . (b) Outer-flow high-velocity streaks, $u_{1}^{+}=2$ . Boundary layer BL6600 at $h^{+}\approx 1200$ –1400. Structures shorter than $\unicode[STIX]{x1D6E5}_{1}=h/2$ have been removed. $(L_{1}^{+}\times L_{3}^{+}=21\,000\times 9000)$ (picture credits, J. A. Sillero).

The long and narrow structures of $u_{1}$ are the well-known streamwise streaks, which are found at all sizes and at all distances from the wall. Illustrating their range of scales, figure 23(a) displays isosurfaces of the perturbation velocity in the buffer layer, and figure 23(b) displays the large outer streaks spanning most of the thickness of the flow. The geometry of $u_{1}$ is similar in the two figures, but the sizes are very different. While the streaks in figure 23(a) have widths of the order of $\unicode[STIX]{x1D6E5}_{3}^{+}\approx 100$ , those in figure 23(b), have $\unicode[STIX]{x1D6E5}_{3}^{+}\approx h^{+}\approx 1500$ . The main difference between them is that the larger structures have a higher intrinsic Reynolds number, which results in a wider range of scales and in a rougher appearance. Another difference is that the near-wall structures are less uniformly distributed than the outer structures. This reflects the modulation of the near-wall layer by the outer flow. The isosurfaces in figure 23(a,b) are defined in wall units that use a uniform friction velocity, while different parts of the wall ‘live’ in the local environment created by the larger outer eddies. It was shown by Jiménez (Reference Jiménez2012) that some of the non-uniformity of the wall can be absorbed by scaling the fluctuations with a ‘local’ friction velocity. Similar models have been used as boundary conditions in large-eddy simulations for a long time (Deardorff Reference Deardorff1970; Piomelli & Balaras Reference Piomelli and Balaras2002), and a related idea has been expressed as a modulation of the inner flow by the outer flow in Marusic, Mathis & Hutchins (Reference Marusic, Mathis and Hutchins2010). This works well for the intensity of the fluctuations, but it does not fully describe the effect of the outer flow on the wall structures. Continuity requires that large-scale ejections $(u_{2}>0)$ should be regions of local lateral convergence, which tend to concentrate the streaks underneath. The opposite is true underneath large-scale sweeps for which $u_{2}<0$ (Toh & Itano Reference Toh and Itano2005). Although figure 23(a,b) correspond to different flows, the spanwise width of the emptier regions with few streaks in figure 23(a) is approximately the same, $O(h)$ , as the width of the large streaks in figure 23(b). We saw in figure 22(c) that the width of all the velocity structures is $\unicode[STIX]{x1D6E5}_{3}\approx h$ when $\unicode[STIX]{x1D6E5}_{2}\approx h$ .

It is interesting that the outer flow in figure 23(b) is relatively well organised, suggesting that, in the absence of exogenous modulation by still larger structures, and when stripped of small-scale instabilities, the largest scales of the channel tend to form a relatively organised pattern which could perhaps be described by some simple theoretical model of coherence. This agrees with the narrowing of the energy spectra in figure 7 away from the wall, and with the evidence from the advection velocity discussed in § 4.1. It is also interesting that, independently of the differences between small and large streaks, the absolute magnitude of the $u_{1}$ isocontours in figure 23(a,b) is the same in wall units, in agreement with the role of $u_{\unicode[STIX]{x1D70F}}$ as a universal velocity scale.

Figure 23(a) includes isosurfaces of the wall-normal velocity. They are located near the streaks, but they are much shorter than them, as already implied by the correlations in § 3.5. The spanwise velocity and the strong vortices near the wall (not shown) have sizes comparable to $u_{2}$ , but the structures of $u_{3}$ tend to be flat ‘flakes’ parallel to the wall, in agreement with the transverse correlation in figure 9(a). The vortices are quasi-streamwise ‘worms’ with sizes comparable to the width of the streaks. This suggest that the vortices are responsible for the wall-normal velocities in the buffer layer, at least for short stretches comparable to the length of the $u_{2}$ structures in figure 23(a), but the same is not true farther from the wall. The large central structure in figure 3(b) is a substructure of the velocity streaks in the flow in figure 23(b). Figure 3(b) also includes vortices, but they are much smaller than the velocity structures, and cannot create wall-normal velocities on a scale appropriate to modify $u_{1}$ (see also figure 20 c). It was shown by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006) that the ejections of the logarithmic layer are associated with large vortex clusters, rather than with individual vortices, and by Jiménez (Reference Jiménez2013b ) that the vortices away from the wall are essentially isotropically oriented, and generate little net large-scale velocity. The counterpart of the near-wall vortices far from the wall are the conditional rollers discussed in § 5.2. They are the collective effect of the residual anisotropy of the vorticity organisation in the presence of the shear of the mean velocity profile.

On the other hand, examples of well-organised hairpin forests have been shown to exist in turbulent flows (Adrian Reference Adrian2007), although they tend to be more common at relatively low Reynolds numbers, and to become disorganised when the Reynolds number increases. Somewhat confusingly, this disorganised vorticity is sometimes referred to as hairpin ‘fragments’, which makes it difficult to distinguish it from regular vortices. An exception may be the structures reported in Wu et al. (Reference Wu, Moin, Wallace, Skarda, Lozano-Durán and Hickey2017). Hairpin forests have always been known to form in the turbulent spots that mediate bypass laminar–turbulent transition. In this sense they are low-Reynolds-number structures. The observation in Wu et al. (Reference Wu, Moin, Wallace, Skarda, Lozano-Durán and Hickey2017) is that very similar spots and hairpins form intermittently underneath fully turbulent boundary layers ( $x_{2}^{+}\lesssim 100$ and $h^{+}\lesssim 1000$ ), although they also tend to disorganise as they move farther from the wall. In this sense they may be important contributors to the regeneration of turbulence in the buffer layer.

It is striking that neither far nor near the wall are there indications of the long streamwise rollers that are often important ingredients of low-order models of the turbulence energy cycle. The observations support shorter structures that would not, by themselves, create a long streak. On the other hand, we saw in § 5.2 that the conditional rollers are arranged with respect to the streaks in such a way as to always help to sustain them: clockwise rollers are located on the left-hand edge of a high-velocity streaks, and counter-clockwise streaks are located on the right hand. Thus, the streamwise average of all the shorter rollers can be modelled as a long average roller that creates a long average streak, even if the flow is better described locally as a concatenation of the smaller aligned units in figure 23(b). The mechanism for this alignment is unclear, and we will come back to it in § 7.1 as one of the open problems in modelling wall-bounded turbulence.

Note that the streaks are different from the ‘hairpin packets’ often proposed as building blocks for the logarithmic layer (Adrian Reference Adrian2007). Even if hairpins, rather than just isolated inclined vortices or conditional rollers, could be shown to exist at high Reynolds numbers, the individual hairpins would be equivalent to the $u_{2}$ – $u_{3}$ structures whose aspect ratio is displayed in figure 22(b). The packets would be the somewhat longer $u_{1}$ structures in figure 22(a), whose dimensions suggest that each of them contain at most three or four shorter structures. The streaks themselves are the overhang at the top of the p.d.f.s in figure 22(a), and correspond to the approximately linear arrangement of substructures spanning the full thickness of the boundary layer in figure 23(b).

5.4 Other variables

The main arbitrariness in the definition of the structures described in the previous sections is the choice of the variable being thresholded. For example, while the streamwise velocity can be considered a surrogate for kinetic energy, it is unclear why the velocity fluctuations, $u_{1}$ , rather than the full velocity, $\widetilde{u}_{1}$ , should be considered as the variable of interest. In terms of the total energy equation, the fluctuating $u_{1}$ is negligible, and the transverse-velocity fluctuations even more so. On the other hand, the total kinetic energy is not a Galilean invariant quantity. Unfortunately, the geometry of the two fields is quite different. While the streaks of $u_{1}$ can be attached or detached from the wall (Sillero Reference Sillero2014), the instantaneous profile of the full velocity is almost always monotonic. Thresholding $\widetilde{u}_{1}$ always results in an irregular layer attached to the wall, and the streaks become uniform-momentum regions in which the mean velocity profile has been mixed by the tangential stresses. These are the very large uniform-momentum zones mentioned by Adrian, Meinhart & Tomkins (Reference Adrian, Meinhart and Tomkins2000). They are not part of the self-similar organisation of the logarithmic layer, and are at least ten times longer than individual $u_{1}$ structures.

The Reynolds stresses discussed in § 5.2 appear to be a more objective choice, because they are in the equation for momentum transfer, and ultimately determine drag. However, it was noted by Jiménez (Reference Jiménez2016) that the quantity in the momentum equation is the divergence of the stresses, and that the stresses themselves can be modified by adding any symmetric tensor of zero divergence without changing anything. As an example, he introduced an ‘optimal’ stress tensor, $\unicode[STIX]{x1D719}_{ij}$ , which minimises the integrated squared magnitude of the stresses over the channel. This tensor field is an arbitrary modification of the classical stresses, but not more arbitrary than the classical one, and it turns out that its statistics are rather different from those of $u_{i}u_{j}$ . In particular, the p.d.f. of $\unicode[STIX]{x1D719}_{12}$ in a channel is much closer to Gaussian than that of $u_{1}u_{2}$ , and the amount of ‘counter-gradient’ momentum flux in figure 18(b) is much reduced. Conversely, the product $\widetilde{u}_{1}u_{2}$ , which uses the full streamwise velocity rather than the perturbations, and which correspond to a different rearrangement of the momentum equation, is much more intermittent than the classical Reynolds stress, and involves much more counter-gradient backscatter (Jiménez Reference Jiménez2016).

None of this should be fundamentally troubling, because these alternative fluxes are just different choices of the gauge used to represent the Reynolds-stress field, and they leave the equations invariant. But it raises the question of what is the real significance of the quadrant analysis in § 5.2, and of whether the properties that we have discussed for the Qs might be artefacts linked to a particular form of the equations.

There are two options. The first one is to renounce to the Reynolds stresses, and to consider the geometry of its divergence. This ‘Lamb vector’ has been studied, for example, by Wu et al. (Reference Wu, Zhou, Lu and Fan1999) in the context of modelling, and it changes the character of the stress field completely. The stresses are large-scale quantities, while their divergence is associated with much smaller scales. The question is akin to whether the quantity of interest in a flow is the pressure or the pressure gradient. The gradient is the only term in the equations of motion, and determines the acceleration of the fluid; but the pressure, which is the integral of the gradient, is more directly related to the velocities. Think of Bernoulli’s equation.

Similarly, while the Lamb vector is the quantity directly related to the acceleration of the fluid particles, its integral determines the overall velocity profile. However, this leaves open the question of which gauge to use, and of whether we should study the properties of the Reynolds-stress Qs, as in § 5.2, the thresholded regions of strong $\unicode[STIX]{x1D719}_{12}$ , or something else. The most direct way of answering this question is to repeat the analysis in § 5.2 for $\unicode[STIX]{x1D719}_{12}$ . This was done recently by Osawa & Jiménez (Reference Osawa and Jiménez2018), and the tentative result is that the streamwise elongation of the optimal flux structures (‘op-sters’) is similar to that of the Qs, while their spanwise aspect ratio is twice as wide. Other correlations suggest that, since op-sters do not differentiate between sweeps and ejections, they approximately correspond to the combined Q2–Q4 pairs.

Note that, because the difference between $\unicode[STIX]{x1D719}_{ij}$ and $u_{i}u_{j}$ is divergence-free, the integral of the two quantities over any sufficiently large volume only differs by a small boundary term. In particular, the ensemble-averaged stresses of the two representations are identical. The preliminary analysis just discussed suggests that their structures are also essentially similar, giving some hope that they are not artefacts, and that they are physically significant in both cases.

5.5 Small and minimal flow units

Minimal simulations began to appear in the 1980s as devices to isolate invariant (initially meaning ‘simple’) solutions in two-dimensional channels (Herbert Reference Herbert, de Vooren and Zandbergen1976; Jiménez Reference Jiménez1987a ). Turbulent channels are typically simulated in numerical domains that are periodic in the two wall-parallel directions, so that a fundamental flow unit repeats itself in a doubly infinite sequence of identical copies. Usually, simulation boxes are chosen large enough for the periodicity not to interfere with the solution, making sure that copies are ‘far enough not to see each other’.

A different approach is to do the opposite, choosing a periodicity so short that, if individual structures exist, the fundamental cell contains a single structure with just enough dynamics to sustain itself. The original hope was that, if the flow was constrained enough, individual structures could be made to be steady, or otherwise simpler to describe than the chaotic structures in real flows. This, for example, was found to be the case in the two-dimensional turbulent channels mentioned above. Even if that flow is different enough from three-dimensional turbulence to be irrelevant to the present article, those results proved that single nonlinear structures with a well-defined characteristic size could survive in a sheared environment, and that incrementally releasing the computational constraints could lead to bifurcations into temporally periodic orbits, chaos, and spatially localised states.

Figure 24.

(a) Velocity fluctuation profiles of: ——, a minimal channel at $h^{+}=181$ , $L_{1}^{+}=460$ , $L_{3}^{+}=127$ ; - - - -, a full-sized channel at $h^{+}=180$ . (b) Evolution during a burst in the minimal channel in (a) of the instantaneous: – ⋅ – ⋅ –, box-averaged velocity gradient at the wall; ——, root-mean-squared intensity of the streamwise-averaged value of $u_{1}$ in $x_{2}^{+}\in (25{-}50)$ , representing the strength of the streak; - - - -, same for $\unicode[STIX]{x1D714}_{1}$ , representing the strength of the streamwise roller. All quantities are normalised with their long-time average. The symbols correspond to the snapshots in (c), which show the evolution of the buffer layer before and during the burst. The grey central object is the low-velocity streak, $u_{1}^{+}=-4$ , and the shorter coloured objects are the vortices, $\unicode[STIX]{x1D714}_{1}^{+}=\pm 0.25$ . Objects taller than $x_{2}^{+}=80$ have been removed for clarity. The view looks towards the wall, with the flow from top to bottom; time moves from left to right, with an interval between frames $\unicode[STIX]{x0394}t^{+}\approx 8$ . Axes move downstream with a velocity $c^{+}=7.6$ , to keep the wave in the streak approximately fixed.

Minimal simulations of three-dimensional channels appeared soon after (Jiménez & Moin Reference Jiménez and Moin1991). They were at first restricted to the viscous layer near the wall, where they provided the first evidence that the empirical spanwise streak separation, $\unicode[STIX]{x1D6E5}_{3}^{+}\approx 100$ , is essentially a critical Reynolds number below which turbulence cannot be sustained. These solutions were not steady, but they contained a wavy streamwise-velocity streak with two flanking staggered quasi-streamwise vortices (see the individual frames in figure 24 c), strongly reminiscent of the structures that had been educed from visual inspection of boundary-layer simulations by Robinson (Reference Robinson1991), or by machine processing of large-box channels by Stretch (Reference Stretch1990).

An intuitive interpretation of minimal flow simulations is that they substitute the disordered arrangement of the structures in real turbulence by a periodic ‘crystal’ in which the structures can be studied more easily. The surprising observation is that, even after what is clearly a major change in the dynamics, the low-order statistics of the minimal flow are essentially correct (figure 24 a). This strongly suggests that the structures isolated by a minimal cell are fundamentally autonomous, with dynamics that depend only weakly on the interaction with their neighbours, or at least on the detailed geometry of those interactions. They therefore satisfy the basic criterion for a coherent structure, as defined in the introduction to this article.

A striking characteristic of minimal turbulent solutions is that they burst intermittently and irregularly. The streak is approximately straight and steady most of the time, and occasionally meanders and breaks down into a burst of vorticity and wall-normal velocity that recalls the observations of tracers in early experiments with boundary layers (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). The cycle then restarts (Jiménez & Moin Reference Jiménez and Moin1991). The evolution of several box-integrated quantities during a burst is presented in figure 24(b). Different quantities peak at different moments during the burst. The streamwise-averaged streamwise velocity grows first, followed by the formation of a roller indicated by the growth of the streamwise-averaged $\unicode[STIX]{x1D714}_{1}$ , and finally by the velocity gradient at the wall. These quantities are compiled in figure 24(b) over a flow slab near $x_{2}^{+}=40$ , which is where the flow in figure 24(c) suggests that most of the bursting activity is concentrated. Averages at different distances from the wall result in slightly different timing relations, and it is risky to draw conclusions from a single flow realisation, but the general sequence of events is consistent with the picture derived by Jiménez (Reference Jiménez2013a ) using the temporal cross-correlation of different variables in bursts farther from the wall. It is interesting that the wall friction is a trailing indicator of bursting, and that the velocity gradient at the wall is still increasing when most of the burst appears to have subsided. This suggests that the burst originates far from the wall and migrates downwards. The evolution of the corresponding flow fields is given in figure 24(c), which only includes structures fully contained below $x_{2}^{+}=80$ . Soon after the last frame in figure 24(c), most of the vorticity fluctuations migrate above the buffer layer, and disappear from our visualisation box. The streak disorganises for a while, to reform later.

Refinements of the idea that an instability of the streak originates streamwise vortices, which in turn reinforce the streak, led to the codification of a self-sustaining process (SSP), first in minimal low-Reynolds-number flows by Hamilton, Kim & Waleffe (Reference Hamilton, Kim and Waleffe1995), and eventually in the near-wall layer of larger-scale channels by Schoppa & Hussain (Reference Schoppa and Hussain2002).

At about the same time, numerically exact three-dimensional permanent-wave solutions of the Navier–Stokes equations were found for minimal Couette flow by Nagata (Reference Nagata1990). They also contain a streak flanked by staggered vortices, and have dimensions that, when expressed in wall units, agree with those of minimal turbulent flows. These solutions are unstable to perturbations, and can only be computed by numerically imposing stationarity or temporal periodicity, although enough numerical constraints stabilises them in the same way as in the two-dimensional experiments mentioned at the beginning of this section. As in that case, progressively releasing the constraints leads to bifurcations into limit cycles and chaos (Jiménez & Simens Reference Jiménez and Simens2001). Two examples are given in figure 25. The example in figure 25(a) is a permanent wave, and the example in figure 25(b) is an oscillatory solution in which the two wavelengths in the box exchange amplitudes periodically. The geometry of the two solutions is similar, and also similar to the chaotic minimal channel in figure 24(c), but the unsteadiness in figure 25(b) is relatively weak. The temporal oscillation of its friction coefficient is approximately 5 %, compared to approximately 20 % in the minimal channel in figure 24(c). Most known exact periodic solutions share this weakness, but some of the homoclinic orbits that have also been found do not. A quantitative comparison of the properties of simple flow solutions, chaotic minimal flows, and large-scale wall turbulence can be found in Jiménez et al. (Reference Jiménez, Kawahara, Simens, Nagata and Shiba2005).

Figure 25.

Two examples of simple solutions in a minimal box that has been stabilised by filtering the simulation above a distance $\unicode[STIX]{x1D6FF}$ from the wall (Jiménez & Simens Reference Jiménez and Simens2001). View and notation, as in figure 24(c), but the flow is from left to right, and the isolevels are, $u_{1}^{+}=-3.5$ and $\unicode[STIX]{x1D714}_{1}^{+}=\pm 0.18$ . (a) A steady wave with $h^{+}=118$ , $L_{1}^{+}=146$ , $L_{3}^{+}=177$ , $\unicode[STIX]{x1D6FF}^{+}=49$ . (b) An oscillatory wave. The two wavelengths exchange amplitudes with a period of $T^{+}\approx 370$ and an amplitude of the friction coefficient, $c_{f}^{\prime }/\langle c_{f}\rangle \approx 0.05$ . $h^{+}=124$ , $L_{1}^{+}=520$ , $L_{3}^{+}=185$ , $\unicode[STIX]{x1D6FF}^{+}=51$ .

These separate lines of research eventually merged into a model in which the steady solutions are understood as ‘frozen’ versions of the SSP (Waleffe Reference Waleffe1997), while related periodic and homoclinic or heteroclinic orbits represent the bursts. A review can be found in Kawahara et al. (Reference Kawahara, Uhlmann and van Veen2012), but this is a rapidly growing field, and interested readers are encouraged to check recent individual research papers.

Figure 26.

(a) Lifetime of sweeps and ejections, as a function of the average distance from the top of the structure to the wall (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b ). ——, CH950; - - - -, CH2000. The contours contain 50 % and 75 % of the data. The chain-dotted diagonal is $T^{+}=0.6x_{2,max}^{+}$ ; the dashed one is $T^{+}=3x_{2,max}^{+}$ . (b) Evolution of the kinetic energy of the streamwise velocity during a burst in a small channel $(h^{+}=1850,\,L_{1}/h=\unicode[STIX]{x03C0}/2,\,L_{3}/h=\unicode[STIX]{x03C0}/4)$ which is minimal at $x_{2}/h\approx 0.25$ (Flores & Jiménez Reference Flores and Jiménez2010). - - - -, Averaged over $x_{2}/h\in (0.05,0.1)$ ; ——, $x_{2}/h\in (0.1,0.25)$ . Each curve is normalised with its long-time average, and time is scaled with the shear of the mean velocity profile at $x_{2}/h=0.2$ . The symbols correspond to the snapshots in (c), which show the evolution of the low-velocity streak, $u_{1}^{+}=-2$ , leading to the burst. Only the layer $x_{2}/h\in (0.1,0.25)$ is represented. The view looks towards the wall, with the flow from top to bottom. Time moves from left to right, with an interval between frames $S\unicode[STIX]{x0394}t\approx 0.7$ $(\unicode[STIX]{x0394}t^{+}\approx 106)$ . Axes move streamwise with $c^{+}=20$ , approximately equal to $U_{1}$ at $x_{2}/h=0.2$ .

Although the first minimal simulations were truly minimal, in the sense of having the smallest dimensions that can sustain turbulence, their main advantage from the point of view of turbulence research is that they contain a single structure that can be easily studied and followed in time without the need for tracking it in space. When the structure crosses the downstream end of the computational box, an identical copy enters upstream. Minimal boxes also allow low-order quantities, such as the intensity of the bursting structures, to be studied by integration over the whole computational box, as in figure 24(b). It was eventually realised that what could be called ‘small-box’ simulations can be used in similar ways to study the larger structures of the logarithmic layer. If the box size is large enough to contain several buffer-layer structures, but only one of the larger structures that reach up to, say, $x_{2}=0.2h$ , that simulation can be used as a tool to study the logarithmic-layer structure in isolation. Flores & Jiménez (Reference Flores and Jiménez2010) found that the critical box dimension is the spanwise period, $L_{3}$ , and that boxes are minimal with respect to the flow at $x_{2}\approx L_{3}/3$ . If we identify this height with the centre of gravity of the largest structure fitting in the computational box, the resulting aspect ratio, $\unicode[STIX]{x1D6E5}_{3}=L_{3}\approx 3x_{2}\approx 1.5\unicode[STIX]{x1D6E5}_{2}$ , is consistent with the p.d.f.s in figure 22(c) and with the cross-plane correlations in figure 9. These maximal structures are not as smooth as the buffer-layer ones in figure 24(c), and the geometry of the burst is correspondingly more complex, but they can be identified as coherent by neglecting the superimposed small scales, in the same sense as for the streaks in figure 23(b). In particular, Flores & Jiménez (Reference Flores and Jiménez2010) showed that logarithmic-layer structures also burst intermittently, and that the process is a larger-scale version of the one in the buffer layer. The largest structure that fits in the box is a streamwise streak flanked by a sweep and an ejection. The streak occasionally meanders and destroys itself while creating a pulse of wall-normal velocity, Reynolds stresses, and dissipation. It eventually reforms.

These logarithmic-layer bursts were studied in large-box simulations by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014b ) as part of their detailed study of wall-attached sweeps and ejections. The dependence of their lifetimes on the maximum height of the structure is given in figure 26(a). There is a self-similar regime in which the lifetime is proportional to the height, $T^{+}\approx 0.6x_{2,max}^{+}$ , and a lower range in which the structures stay within the buffer layer, and their lifetime is $T^{+}\approx 30$ . The two regimes appear as continuations of each other, giving credence to the idea that they are different aspects of the same phenomenon. If we empirically identify $x_{2,max}\approx 2x_{2,c}$ , where $x_{2,c}$ is the average height of the centre of gravity of the structures during their lifetime, the self-similar relation in figure 26(a) is equivalent to $T^{+}\approx 1.5x_{2,c}^{+}$ , or $S_{c}T\approx 4$ , where $S_{c}^{+}=1/\unicode[STIX]{x1D705}x_{2,c}^{+}$ is the mean shear. This is the order of magnitude of the lifetimes deduced by Jiménez (Reference Jiménez2013a ) for any shear flow, on the basis of the time it takes for a structure to be sheared. The history of the box-averaged intensities of one such logarithmic-layer burst is given in figure 26(b), and the width of the burst approximately agrees with the above estimate. The duration of the buffer-layer structures in figure 26(a) also agrees approximately with the width of the peaks in figure 24(b).

A different definition of the bursting period was used by Flores & Jiménez (Reference Flores and Jiménez2010), as the spectral peak of the frequency spectrum of box-integrated quantities in small-box simulations. Their result, $T^{+}=6x_{2}^{+}\approx 3x_{2,max}^{+}$ is approximately four times longer than the tracking result in figure 26(a). The two trend lines have been included in that figure. They refer to different quantities: the Fourier result measures the time interval between bursts, while the tracking one measures the duration of its intense phase. The difference between the two periods measures the fraction of time (20 %) that the flow is bursting. This minimal-box result probably overestimates the prevalence of bursting in large-scale flows; Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) find that attached $\text{Q}^{-}\text{s}$ only cover approximately 8 % of the wall.

The meandering streak corresponding to figure 26(b), filtered to the same relative resolution as the buffer-layer figures 24 and 25, is shown in figure 26(c).

5.6 Causality

We have mentioned in several occasions that turbulent structures can form far from the wall, and that they appear to only require the presence of shear. Of course, this does not rule out the possibility that in wall-bounded turbulence, where a wall is present, structures may predominantly form near it, and it is interesting to ascertain whether, under those circumstances, the structures of wall-bounded flows originate at the wall, or away from it. At first sight, it looks that the wall should be the origin. We mentioned in § 3.2 that the highest turbulence intensities are in the buffer layer, and that so is the strongest shear, which is the source from which turbulence draws its energy. Equation (3.7) shows that the balance between dissipation and the production, both of which are proportional to the shear, requires that ${\unicode[STIX]{x1D714}^{\prime }}^{2+}\sim 1/x_{2}^{+}$ in the logarithmic layer, so that vorticity is also concentrated near the wall. We discussed in § 3.2 experiments that show that the dynamics of the near-wall region wall is essentially independent of the outer flow, and it is tempting to hypothesise, from this evidence, that turbulent fluctuations are created in the buffer layer and diffuse outwards in some unspecified manner (see, for example, Adrian Reference Adrian2007).

However, none of these results prove that the converse cannot also be true, and that the outer flow can survive independently of the wall. It is clear that the wall is required to create a shear, but it is possible that this is its main role, and that structures are created everywhere, with each wall distance basically independent from others. In this alternative model, turbulence draws its energy from the local shear, and either decays locally or diffuses away from its origin, including towards the wall. The arguments for this view have also been repeated often. The earliest one is the lack of influence of wall roughness on the properties of the logarithmic and outer flows (Townsend Reference Townsend1976). Roughness destroys the details of the near-wall region, including the near-wall peak of the turbulence intensity, but it only has minor effects above a layer of the order of a few roughness heights (Jiménez Reference Jiménez2004). Similarly, large-eddy simulations, which tend to represent poorly the near-wall layer, reproduce features of the outer flow relatively well (Moin & Kim Reference Moin and Kim1982; Kravchenko, Moin & Moser Reference Kravchenko, Moin and Moser1996; Piomelli & Balaras Reference Piomelli and Balaras2002). Mizuno & Jiménez (Reference Mizuno and Jiménez2013) present computations of a turbulent channel in which the wall is substituted by an off-wall boundary condition that mimics the logarithmic, rather than the buffer layer, with relatively few deleterious effects, and we saw in figure 21 that there is a smooth transition between the attached eddies of wall-bounded flows and the structures of homogeneous shear turbulence, in which there is shear but no walls.

Figure 27.

Temporal evolution of the Reynolds stresses in the small channel in figure 26(b,c). (a) Evolution in the $(x_{2},\,t)$ plane of the streamwise-velocity fluctuations, averaged over wall-parallel planes. The colour range (dark to light) is $\langle u_{1}^{2}\rangle _{13}^{+}\in (0,11)$ . The two dashed diagonals have slope $\text{d}x_{2}/\text{d}t=\pm u_{\unicode[STIX]{x1D70F}}$ . (b) As in (a), for $-\langle u_{1}u_{2}\rangle _{13}^{+}\in (0,1.8)$ . (c) Temporal cross-correlation coefficient between $\langle u_{1}^{2}\rangle _{13}$ at $\tilde{x}_{2}^{+}=25$ and: ——, $x_{2}^{+}=25$ ; ▫, 70; ▵, 140; $\circ$ , 275; ▿, 460 or $x_{2}/h=0.25$ . (d) As in (c), for $\langle u_{1}u_{2}\rangle _{13}$ .

All these experiments strongly suggests that the dominant root cause of the structures of wall turbulence is the shear rather than the viscous wall, and it was indeed shown by Tuerke & Jiménez (Reference Tuerke and Jiménez2013) that even minor artificial changes of the mean shear produce major effects in the fluctuations. However, the most direct evidence for the direction of causality comes from simulations in the small channels discussed in the previous section, in which individual structures manifest themselves in the global averages. The channels in Flores & Jiménez (Reference Flores and Jiménez2010) are minimal with respect to structures of height $x_{2}/h\approx 0.2$ –0.3, within the logarithmic layer, and the corresponding structures can be detected by averaging the intensities over wall-parallel planes. These planar averages are only functions of the wall distance and of time, and are represented in the $x_{2}$ – $t$ maps in figure 27(a,b). The first of these two figures displays the evolution of the plane-averaged intensity of the streamwise velocity, and figure 27(b) displays the tangential Reynolds stress. In both cases, it is evident that eddies move both towards and away from the wall, with wall-normal velocities of the order of $\pm u_{\unicode[STIX]{x1D70F}}$ (Flores & Jiménez Reference Flores and Jiménez2010). The dominant direction of causality is tested in figures 27(c) and 27(d), which plot the cross-correlation of the planar averages of the fluctuations in the buffer layer with those at different distances from the wall. Within the distances plotted in the figure, the peak of the cross-correlation of the planar average $\langle u_{1}^{2}\rangle _{13}$ stays centred near $t=0$ , but it gets wider and develops a flat top as $x_{2}$ moves away from the wall, probably reflecting that the taller eddies are related to those at the wall either at earlier or at later times. On the other hand, the correlations of the mean Reynolds stress, $\langle u_{1}u_{2}\rangle _{13}$ , drift towards earlier times as $x_{2}$ increases, implying that the dominant evolution of these eddies in the logarithmic layer progresses from the outside towards the wall, rather than the other way around. Different variables behave differently, although most tend to behave like $u_{1}$ in figure 27(c). We could not find any case in which correlations moved outwards on average at a rate similar to the inwards velocity of the Reynolds stress in figure 27(d).

It should be cautioned that these results could be affected by the small computational box. It can indeed be shown that they are corrupted by the box above $x_{2}/h\approx 0.5$ , where causality reverses, but the wall-normal propagation velocity of individual strong $(u_{1}u_{2})$ structures was measured by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014b ) in large channels, free of minimal-box effects. They found that ejections $(u_{2}>0)$ move away from the wall with a distribution of velocities centred around $u_{\unicode[STIX]{x1D70F}}$ , while sweeps $(u_{2}<0)$ move towards the wall at approximately the same rate. Since the two types of structures tend to forms pairs of one sweep and one ejection, the net wall-normal velocity of the pair approximately vanishes. Note that, in this description, the wall only plays an essential role in creating the shear, but not in the generation of either sweeps or ejections.

6.1 Linear approximations

The key dichotomy in the linear behaviour of shear flows is between the modal instabilities characteristic of self-adjoint operators, and the transient growth in non-normal ones. In the former, fluctuations can be expanded in a set of orthogonal eigenfunctions, and stability analysis reduces to determining the properties of the corresponding eigenvalues. If a temporal eigenvalue has a positive real part, the associated eigenfunction grows exponentially and eventually dominates. Stable eigenfunctions die exponentially, and orthogonality ensures that individual eigenfunctions can be treated as essentially independent of one another as long as the system remains linear and autonomous.

On the other hand, the eigenfunctions of non-normal operators are not necessarily orthogonal. In general, there will be cancellations among some of the eigenfunctions contributing to a given initial condition and, even if the evolution of the system is such that all the eigenvalues are stable and decay exponentially, the balance of those cancellations may change during the decay. The result is that some initial conditions grow for a while, typically algebraically, even if they eventually decay exponentially when all the involved eigenfunctions do so. If a group of eigenfunctions are almost parallel to each other, the effect can be large. A recent survey of applications to hydrodynamic transition is Schmid (Reference Schmid2007), and a textbook account is Schmid & Henningson (Reference Schmid and Henningson2001).

For parallel flows, the linearised version of the Navier–Stokes equations is

where primed capitals denote derivatives with respect to $x_{2}$ , and $\unicode[STIX]{x1D6FF}_{i}^{j}$ is Kronecker’s delta. It can be reworked into the Orr–Sommerfeld equation for $u_{2}$ (Drazin & Reid Reference Drazin and Reid1981),

and the Squires equation for the wall-normal component of the vorticity,

Several terms in these equations can amplify fluctuations. The leading operator in the left-hand side of (6.1)–(6.3) is advection by the mean profile. It does not amplify fluctuations by itself, but, because $U_{1}$ is a function of $x_{2}$ , it deforms them, generally tilting them forward. Because (6.2) is autonomous in $u_{2}$ , it determines whether the flow is stable or not. The term marked ‘K–H’ on its right-hand side is the deformation of the mean shear by the cross-shear velocity fluctuations. It rearranges the distribution of vorticity without amplifying it, but it is responsible for the Kelvin–Helmholtz modal instability when the mean velocity profile has a shear maximum (i.e., an inflection point) because it moves vorticity fluctuations to where originally there were none. Because wall-bounded flows are not inflectional in the absence of strong pressure gradients, this term is generally not important for them.

Figure 28.

(a) Sketch of the evolution of the lift-up mechanism in the cross-plane $(x_{3},x_{2})$ . A pair of stationary vortices with zero net circulation, represented as red circles, deforms the background shear, represented by the solid isolines. Time increases from left to right. (b) Evolution of the perturbation energy $\langle u_{1}^{2}\rangle$ during (a). Time is normalised with the maximum of $u_{2}$ and with the distance from the wall to the centre of gravity of the vortex pair. (c) Sketch of the evolution of the Orr mechanism in the streamwise plane $(x_{1},x_{2})$ . Time increases from left to right. As the upper vortex is advected by the ambient shear above the lower one, their wall-normal velocities reinforce each other. When the vortices are not vertically aligned, their velocities tend to cancel. (d) Evolution of the integrated energy of the wall-normal velocity during the evolution in (c). Time is normalised with the shear advecting the vortices. The symbols in (b,d) are the times of the snapshots in (a,c), respectively. The units for the amplitudes are arbitrary.

The advection of $\unicode[STIX]{x1D714}_{2}$ in Squires equation (6.3) is forced by the wall-normal fluctuations generated by (6.2). This does not create instability by itself, but it can amplify fluctuations considerably. The relevant term is marked as ‘lift-up’, and represents the deformation of the mean velocity profile by the spanwise variations of $u_{2}$ . This is one of the terms responsible for non-orthogonal eigenvectors of the evolution operator (6.2)–(6.3). It leads to the formation of the streaks of $u_{1}$ , because it acts most strongly on long narrow features for which $\unicode[STIX]{x1D714}_{2}\approx \unicode[STIX]{x2202}_{3}u_{1}$ . A sketch is given in figure 28(a), drawn in the $(x_{3},x_{2})$ cross-flow plane for simplicity. The solid contours are $U_{1}$ , which in this case increases from bottom to top. The arrows are the field of transverse velocities due to a pair of streamwise vortices, shown here as stationary. Lift-up works by moving low-velocity fluid from the wall upwards and vice versa. In the particular case of figure 28(a), it is creating a low-velocity streak in the centre of the sketch, where the $U_{1}$ contours move away from the wall. Figure 28(b) shows the growth of the perturbation energy, which is typically algebraic. Unless the mean velocity profile stops growing far from the wall, there is no obvious way to limit the growth of the lifted streak, because high-velocity fluid keeps being moved towards the wall, and vice versa. However, the growth rate typically slows after some time, because fast fluid has to be drawn from the upper layers where the transverse velocities created by vortices located near the wall are weaker. Nonlinearity also limits the rate of growth by mixing laterally the fluid in the streak, as is beginning to happen in the right-most snapshot in figure 28(a). The net effect is to rearrange the mean velocity profile. In the case of the figure, it flattens the profile near the wall, and creates a sharper shear layer above the vortex pair.

The other non-normal part of the evolution is the Laplacian marked as ‘Orr’ in the left-hand side of (6.2). If the advection in (6.2) acted on $u_{2}$ instead of on $\unicode[STIX]{x1D6FB}^{2}u_{2}$ , its effect would just be to reorganise the velocity fluctuations without changing their amplitude. However, this deforms their shape, violating continuity. The Laplacian represents the reaction of the pressure to restore continuity. The effect of this term, first discussed by Orr (Reference Orr1907), is to amplify backwards-tilting packets of $u_{2}$ as they are carried towards the vertical by the shear. The effect is transient, with a time scale $O(S^{-1})$ , because $u_{2}$ is eventually damped again as the packets are tilted past the vertical. However, the net effect of a transient burst of $u_{2}$ is to create a streak of $u_{1}$ through the lift-up mechanism mentioned above. The damping effect of the tilting on $u_{2}$ is due to the change of $\unicode[STIX]{x2202}_{2}u_{2}$ as the structure becomes vertically stacked. The streaks created by the lift-up are also tilted and vertically thinned by the shear, but $\unicode[STIX]{x2202}_{2}u_{1}$ is not in the continuity equation, and pressure does not damp them. As a consequence, the effect of the Orr bursts of $u_{2}$ is long-lasting.

The intuitive mechanism behind the Orr amplification is sketched in figure 28(c), which portraits, for simplicity, a two-dimensional case in the longitudinal $(x_{1},x_{2})$ plane. The figure shows two co-rotating vortices being advected by the mean flow in such a way that vortex ‘A’ overtakes ‘B’. When the two vortices are far apart, their velocity fields are independent. As they get closer, the downwash of ‘A’ initially cancels the upwash of ‘B’, and the total energy of the pair decreases slightly. Later, when they pass above each other, their velocities mutually reinforce, the maximum wall-normal velocity doubles and, even if the integral of $u_{2}^{2}$ is concentrated over a smaller area, the integrated energy doubles. As the vortices separate again, the reinforcement decreases and the energy amplification is lost. The evolution of the integrated energy during the events in figure 28(c) is plotted in figure 28(d). It should be clear from this explanation that Orr’s is a robust mechanism which does not depend on the details of the velocity profile responsible for the overtaking of the two structures, and that similar results would be obtained if the two vortices were substituted by a tilting elliptical vorticity patch, or by a three-dimensional vorticity distribution. Note that pressure is not explicitly invoked by this second explanation, but that the expression for the velocity in terms of the vorticity distribution implies the Biot–Savart law, which incorporates continuity.

It is important to stress that neither lift-up nor Orr are intrinsically linear processes, although they subsist in the linear approximation, and are most easily analysed in that limit. The effects depicted in figure 28 work equally well for strong perturbations as for weak ones. In particular, the Orr superposition of flow fields in figure 28(c) only depends on having eddies moving on average at different speeds at different heights, and it does not require that the mean velocity profile exists as such at any moment. Of course, if the overtaking eddies interact with other perturbations of comparable intensity, the effect of Orr superposition might be difficult to isolate. Linearisation does not add anything to the equations, and the effects surviving the removal of the nonlinear terms constitute the core of the evolution operator under the conditions in which linearisation makes sense. The linearity to which we allude in this subsection is a simplification that recognises that, for the large scales in the presence of shear, the mechanisms in the linearised equations are fast enough to remain relevant even in the presence of nonlinearity, and strong enough to be responsible for many of the locally intense structures.

The viscous terms on the right-hand side of (6.2) and (6.3) act to damp the fluctuations, specially when the eddies are thinned by tilting at the beginning or at the end of a burst. Viscosity can be responsible for instabilities that are important in transition, but the times involved are much longer than the shear time, and they are not usually relevant in turbulence.

Because we are considering linear dynamics in this section, the evolution of any initial condition subject to (6.2)–(6.3) is most easily analysed for individual spatial Fourier modes. An appendix in Schmid & Henningson (Reference Schmid and Henningson2001) provides a practical computer code to optimise the amplification history of some chosen norm,

where $\Vert \unicode[STIX]{x1D712}\Vert ^{2}=\int |\unicode[STIX]{x1D712}|^{2}\,\text{d}V$ is integrated over the whole flow as a function of time. The analysis provides the most amplified initial condition and its temporal evolution, including the time for maximum amplification. The only input required is the mean velocity profile and the viscosity. The most amplified initial conditions play the same role as the most unstable eigenfunctions of modal instabilities. For any random initial condition, the system tends to select the projection over the most amplified direction, whose evolution eventually dominates the flow. The difference is that, while modal eigenfunctions grow while retaining their shape during their evolution, non-normal initial conditions change as they evolve. The shape of the most dangerous initial condition is in general very different from the final amplified perturbation that dominates the statistics. Also, because the growth in the amplitude of each initial condition is not a simple exponential, the dominant perturbation at different times may correspond to different initial conditions, as each perturbation grows and decays, to be substituted by another one with a slower evolution time.

This analysis was first applied to the velocity profile of a turbulent channel by Butler & Farrell (Reference Butler and Farrell1993), who found that the most amplified initial condition is a backwards-leaning set of oblique rollers with weak $u_{1}$ , which evolve into strong streaky structures of the streamwise velocity. The aspect ratio of the different perturbations is fixed by the wavelengths chosen for the analysis, but the most amplified initial conditions are those which evolve into infinitely long structures with $\unicode[STIX]{x1D706}_{3}/h\approx 3$ , filling most of the height of the channel. The buffer-layer streaks, with $\unicode[STIX]{x1D706}_{3}^{+}\approx 100$ , only become prominent if the time during which the system is allowed to evolve is artificially restricted.

Figure 29.

Maximum amplification under linearised dynamics of the energy integrated over the whole channel height, as a function of the wall-parallel wavelengths. Full eddy viscosity and $h^{+}=2000$ . (a) ——, Kinetic energy amplification, $A_{q,max}=\sqrt{2},2,2\sqrt{2},\ldots$ ; - -  $\circ$  - -, time for maximum amplification, $u_{\unicode[STIX]{x1D70F}}t_{max}/h=1$ and 2; the shaded contours are the vertically integrated energy spectrum, $\unicode[STIX]{x1D6F7}_{qq}$ , for CH2000. The dotted diagonal is $\unicode[STIX]{x1D706}_{1}=\unicode[STIX]{x1D706}_{3}$ . (b) Amplification of $u_{2}$ for the optimal histories in (a), and $\unicode[STIX]{x1D6F7}_{22}.$ (c) For $u_{3}$ and $\unicode[STIX]{x1D6F7}_{33}$ . (d) Amplification history of the three velocity components in the linearised case marked by a symbol in (a–c): $\unicode[STIX]{x1D706}_{1}/h=3.9$ , $\unicode[STIX]{x1D706}_{3}/h=5.55$ . ——, $u_{1}$ ; - - - -, $u_{2}$ ; – ⋅ – ⋅ –, $u_{3}$ . Symbols are the snapshots in (f–h). (e) Leading PODs of the three velocity components for CH2000 and $\unicode[STIX]{x1D706}_{1}/h=3.2$ . From top to bottom, $u_{1}$ , $u_{2}$ and $u_{3}$ . Black patches are $u_{j}>0.5$ , and grey ones are $u_{j}<-0.5$ . (f) Evolution of the most amplified linearised solution for the case in (d). Time and flow run from left to right, with $\unicode[STIX]{x0394}tu_{\unicode[STIX]{x1D70F}}/h\approx 0.15$ . The advection velocity is adjusted arbitrarily to fit all the snapshots within the plot. From top to bottom, the contours are $u_{1}=4$ , $u_{2}=1$ and $u_{3}=3$ , in arbitrary units.

The analysis was repeated by del Álamo & Jiménez (Reference del Álamo and Jiménez2006) by adding to the right-hand side of (6.2)–(6.3) the eddy viscosity required to maintain the mean velocity profile (for the extra terms arising in the equations, see Pujals et al. Reference Pujals, García-Villalba, Cossu and Depardon2009). Because the eddy viscosity depends on $x_{2}$ , the results are roughly comparable to introducing a damping time that varies with the distance from the wall, and both the outer and the buffer-layer streaks appear naturally. An example of the results is given in figure 29(a–c), where the maximum amplification of each velocity component is compared to the vertically integrated spectral density of a turbulent channel at the same Reynolds number. The three figures refer to the histories and initial conditions that maximise the amplification, $A_{q}$ , of the kinetic energy. The maximum energy amplification is given in figure 29(a) as a function of the wall-parallel wavelengths, and reflects mostly the growth of $u_{1}$ . The outer streaks appear at long wavelengths $(\unicode[STIX]{x1D706}_{1}\gg h)$ and $\unicode[STIX]{x1D706}_{3}\approx 3h$ , and it is interesting that they approximately agree with the results of Butler & Farrell (Reference Butler and Farrell1993) using a very different viscosity model. This supports our previous observation that viscosity is secondary to the mechanism of energy production. In fact, the analysis of the evolution of $u_{2}$ can be done using an inviscid code, because the impenetrability boundary condition is inviscid, but $u_{1}$ and $u_{3}$ require some viscosity to enforce no slip at the wall (Jiménez Reference Jiménez2013a ).

As in Butler & Farrell (Reference Butler and Farrell1993) the intensity profiles of the large amplified structures in del Álamo & Jiménez (Reference del Álamo and Jiménez2006) fill most of the channel height. They have dimensions that scale in outer units for different Reynolds numbers, and appear to be fairly universal features of equilibrium shear flows. The spectrum in figure 29(a) shows that the spanwise wavelength of this outer amplification peak agrees well with that of the longest turbulent energy in channels.

There is a secondary, weaker, amplification ridge near $\unicode[STIX]{x1D706}_{3}^{+}=100$ , which corresponds to the buffer-layer streaks. It has intensity profiles concentrated near the wall, and scales in wall units with the Reynolds number. A curious feature of both ridges is that the maximum amplification occurs for infinitely long structures, while we have seen that the streaks in real flows are long, but probably not infinite. This discrepancy should not be taken too seriously. The use of a wavelength-independent eddy viscosity for all the fluctuations should only be understood as a rough approximation to the energy dissipation by smaller-scale turbulence. Actual nonlinear processes can be expected to act over times of the order of the eddy turnover, which is $O(h/u_{\unicode[STIX]{x1D70F}})$ for the scales represented by the outer ridge. It is intuitively clear that the higher amplification of the longest structures can only be achieved at the expense of longer amplification times. This is confirmed by the analysis. Isolines of the time required to reach maximum amplification are included in figure 29(a), and suggest that the length of the streaks is probably limited by evolution times longer than a few turnovers. We gave a similar argument in § 3.5 to justify the shorter streamwise correlations of boundary layers, and we mentioned above that Butler & Farrell (Reference Butler and Farrell1993) used a time limit to retrieve the inner amplification ridge. Note that the turnover time, as well as the eddy viscosity, are nonlinear effects that depend on the finite amplitude of the fluctuations, and have been artificially superimposed on the linear analysis.

Transient-growth analysis selects the maximum singular value from the spectrum of the evolution operator, but there often are other singular values which are close to the first one. Because of the symmetry of the velocity profile in channels, the spectrum is formed by pairs of very similar magnitude, corresponding to perturbations that are symmetric or antisymmetric with respect to the centreline. For short wavelengths, the amplification of the two members of the pair is almost the same, because they represent essentially independent solutions evolving near one of the walls. This symmetry breaks for longer or wider wavelengths, and solutions with a symmetric $u_{2}$ spanning the whole channel tend to be slightly more amplified. The optimal solutions represented in figure 29(a) mostly have this symmetry, but both symmetric and antisymmetric solutions can be expected to occur in real cases.

This mix of symmetry classes is found both in the correlations and in the PODs. For example, the correlation of $u_{1}$ in figure 9(c) has a negative lobe on the opposite half of the channel, even if it is compiled relatively near the lower wall, suggesting an antisymmetric $u_{1}$ . The issue was avoided for the filtered PODs in § 4 by restricting the analysis to the lower half of the channel, because using the whole channel interferes with the compactification of asymmetric variables whose integral vanishes. This is why the PODs of all variables in figure 13 appear to be restricted to a half-channel, but we will see below that full-channel structures reappear when the correlation of long wavelengths is analysed over the whole flow.

Figures 29(b) and 29(c) display the maximum amplification of the two transverse-velocity components for the same optimal histories used in figure 29(a). The strongest amplification of the different components does not take place for the same wavelengths, nor at the same times. For example, the maximum amplification of $u_{2}$ occurs at the two ends of the $\unicode[STIX]{x1D719}_{22}$ spectrum, but it is also large at very wide wavelengths for which the kinetic energy is not amplified, and where the spectrum of turbulence has no energy. The linear evolution of $u_{2}$ is controlled by the Orr mechanism, which is most efficient for wide $(\unicode[STIX]{x1D706}_{3}\gg \unicode[STIX]{x1D706}_{1})$ waves. But the lift-up depends on the spanwise derivative in the right-hand side of (6.3). Very wide $u_{2}$ perturbations, even if they are amplified, cannot create a streak and do not result in long-lasting energy amplification. This also means that these perturbations are not found in turbulence, because there are no initial conditions in that range of wavelengths to serve as seeds for two-dimensional waves of $u_{2}$ . A similar analysis applies to $u_{3}$ in figure 29(c).

Figure 29(d) shows the amplification history of the wavelength combination marked by a symbol in figure 29(a–c). The different evolution of the three velocity components is clear, as is the transient nature of their amplification. Six snapshots of the evolution of the perturbation field of each of the three velocity components are shown in figure 29(f). Note the difference between the initial and final shape of the perturbations, as mentioned when discussing transient growth at the beginning of this subsection. All the structures are progressively tilted forward during their evolution, and it is interesting to note the similarity between the shape of the structures at their most amplified stage (between snapshots three and four) and the PODs of a real channel at a similar $\unicode[STIX]{x1D706}_{1}$ , displayed in figure 29(e).

A quantitative comparison of the correspondence between bursting in fully nonlinear small-box channel simulations and linear analysis is given in Jiménez (Reference Jiménez2015). While that reference should be consulted for details, a short summary of its conclusions is that the evolution of intense ‘minimal’ Fourier modes of the wall-normal velocity (i.e. those with wavelengths similar to the box dimensions) can be described well as a linearised transient Orr burst. For the simulations in that paper, which are minimal around $x_{2}/h=0.25$ , linear prediction works for time intervals of the order of $tu_{\unicode[STIX]{x1D70F}}/h\approx 0.15$ , which is approximately half the bursting lifetime. The time fraction of the flow history that can thus be described is approximately 65–70 %, and accounts for an even higher fraction of the total Reynolds stress. Bursts are essentially inviscid, and the addition of an eddy viscosity does not improve predictions.

Linearisation fails during periods of weak perturbations because the flow cannot be described by a few wall-normal eigenfunctions, but nonlinearity is overwhelmed by the linear process while bursting is active. This recalls the observation at the beginning of this section that nonlinearity in free-shear layers only matters when the thickening of the layer pushes the prevailing wavelength outside the instability range of the linear Kelvin–Helmholtz mechanism. In both cases, nonlinearity only becomes important for the large scales after the flow decouples from the shear, and it is striking and contrary to common intuition, that the strongest events are those best described by linearised models.

On the other hand, it should be emphasised that these conclusions refer to the dynamics of a few Fourier modes chosen to represents coherent structures. For example, each snapshot in figure 29(f) represents a single period of a uniform wave train, and the PODs in figure 29(e) are spatially periodic. Any discussion of the evolution of realistic initial conditions should be able to deal with wave packets that include a range of wavelengths, in the same spirit as the compact eddies discussed in § 4.

A summary of the ongoing work on the generalisation to full-sized simulations, in which several large scales coexist, is Encinar & Jiménez (Reference Encinar and Jiménez2016). Early indications are that the above results continue to hold for individual wavelengths, with relatively little interaction among sufficiently different scales.

6.2 Nonlinearity

7.1 Some open problems

It could be concluded from the previous summary that almost everything is known about the mechanics of individual structures in turbulent wall-bounded flows. This is probably true, although there are bound to be differences in interpretation. But this does not mean that we understand everything about wall-bounded turbulence, and it is the nature of articles like the present one to consider some of the open problems left in the field. The following are a few representative examples.

1. (1) The simplest question left open by the discussions above is how to precisely characterise what is the nonlinear event that restarts the self-sustaining cycle. We listed in § 6.2 some of the possibilities that have been discussed in the literature, but none of them is definitive enough to provide a predictive criterion for the location and time where a new cycle is about to begin. Since one of the most important applications of such a criterion is to inform active control strategies, it would be most useful if it could be reduced to a few variables, preferably at the wall. Given how much information we have on individual structures, this is the sort of problem that can probably be solved from existing data and in the next few years.

2. (2) How does a short roller become a long streak? We have already discussed this question in § 6.2.1, where we distinguish between the factor of two or three between the length of the intense structures of the streamwise and of the transverse velocities in figure 22, and the much larger difference among the correlation functions in figure 9. We have seen that the geometry of the structures of $u_{1}$ is consistent with the concatenation of shorter units. The p.d.f. of the length of the sublayer streaks was examined by Jiménez & Kawahara (Reference Jiménez, Kawahara, Davidson, Kaneda and Sreenivasan2013), who showed that it has an exponential tail, suggesting the concatenation of individual units of length $L_{c}^{+}\approx 500$ –1000. This is consistent with the length of minimal boxes (Jiménez & Moin Reference Jiménez and Moin1991), but the study was limited to $x_{2}^{+}\leqslant 60$ , and we are not aware of similar studies in the logarithmic layer. Minimal simulations are not very informative in this respect, because their streaks are always found to cross the box, and are therefore infinitely long. Simulations in longer boxes of minimal span do not result in longer rollers, but in several rollers along a longer streak (see figure 25 b).

   The question is how the concatenation takes place. Equivalently, why sweeps and ejections tend to form on the correct side of the streak to reinforce it, as in figure 21. We argued above that one of the effects of pre-existing streaks is to break the spanwise translational symmetry for the generation of new bursts, but the precise mechanism is unclear, as are its quantitative aspects. For example, how often are new bursts created in the ‘correct’ location? It could be the case that the long streaks that we observe are an example of observational bias: streaks grow by random superposition, and we are distracted by the streaks that happen to be long. The exponential p.d.f.s mentioned above would be consistent with this accretion model. The infinitely long streaks of minimal boxes would not be, but they could be an artefact of minimality.

3. (3) Perhaps the problem farthest from solution, and arguably the most important one, is how the multiscale nature of the flow is organised. We have a reasonable understanding of individual structures, but we know very little about their multiscale interactions. For example, each of the conditional structures in figures 20 and 21 is associated with a given length scale and with a given distance from the wall, but we saw in figure 19(c,d) that these structures exist at all scales in the logarithmic layer, and that they form a self-similar hierarchy. Geometrically, structures of different sizes coexist everywhere overlapping each other, and we have mentioned that they carry approximately 60 % of the tangential Reynolds stress.

   Presumably, the requirement that they should provide a given mean stress sets their velocity scale. Momentum conservation requires that the average of the Reynolds stress of all the structures that intersect a given plane has to be $u_{\unicode[STIX]{x1D70F}}^{2}$ , but how is this information communicated to individual structures? What happens to structures which are too weak or too strong? Do they interact with each other to ‘reach a consensus’ on the right intensity, or do they interact directly with the mean flow?

   It is easy to construct feed-back models in which the shear of the mean velocity profile controls the intensity of the bursts, especially if we admit, as we have argued, that bursts form at all heights instead of growing from the wall. Qualitatively, if the mean intensity of the structures in some layer is too weak, the shear increases, and the structures intensify. However, such ‘one-stage’ models are unlikely to be the whole story. Consider the structures reaching up to a given $x_{2}$ (e.g. one metre) within the logarithmic region of the atmospheric surface layer, whose thickness $h$ is $O(100)$ metres. How do these structures receive the information of the mean profile, which can only be defined by averaging over distances of several boundary-layer thicknesses? In this example, the problem of setting the velocity scale is not restricted to structures whose size is one metre, but also to everything bigger, because momentum conservation requires that the mean Reynolds stress of all the structures intersecting that plane has the right value. But the ratio of the turnover times of the largest and the smallest structures intersecting our plane is $h/x_{2}=O(100)$ . If the small scales adjust to the stress missing from the largest scales, how do they measure it? If the large scales adjust to the small scales, how do they do the average?

   The problem is not only that we do not know the answer to this question, but that we do not have the right tool to investigate it, and that we need to develop a system equivalent to minimal simulations for multiscale problems.

4. (4) A final unsolved problem is the one with which we opened § 6: what determines the mean velocity profile in wall-bounded turbulence? Motivated by it, a lot of progress has been made on the mechanisms of elemental structures, but very few things in § 6 refer to how the mean profile is put together. We mentioned in (3) that part of the problem is that we lack a multiscale theory, but it is not immediately clear how would we use such a theory if we had it. It may be time to revisit Malkus (Reference Malkus1956) idea of criticality. The quasilinear model in § 6.2.4 can be interpreted as a requirement of marginal stability for a base flow that now includes the infinitely long streak (Ioannou, private communication). The null Lyapunov exponent in that model takes the place of the neutral eigenvalue in Malkus (Reference Malkus1956).

   In a simpler modal setting, we have mentioned that linear theory can be used to quantitatively describe unstable free-shear layers, and that nonlinearity only takes effect when the linear modes become neutral. But that is not the whole story. The Reynolds stress is a nonlinear effect that depends on the amplitude of the fluctuations, and it controls the growth of the shear-layer thickness that eventually drives the Kelvin–Helmholtz eigenvalue to become neutral. A Malkusian interpretation would be that nonlinearity adjusts itself to keep the mean velocity profile at a state of (self-organised) criticality.

There are many more open questions that do not fit in the present article, and probably still more that I am not able to see at the present time, but, if I may finish the article on a personal note, this is what still makes wall-bounded turbulence fun.