3.1 Overview of one-point statistics

We select the channel flow at $Re_{\unicode[STIX]{x1D70F}}\approx 500$ with $\unicode[STIX]{x1D6F1}=60$ as a representative case to illustrate the non-equilibrium response of the flow succeeding the imposition of the lateral pressure gradient. The systematic analysis for various $Re_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D6F1}$ is presented in § 3.2. For $t\rightarrow \infty$, the system attains a new statistically steady state corresponding to a 2-D channel flow at higher $Re_{\unicode[STIX]{x1D70F}}$ and mean-flow direction parallel to the vector $(\text{d}P/\text{d}x_{1},0,\text{d}P/\text{d}x_{3})$. We focus on the initial transient dominated by 3-D non-equilibrium effects for $t^{\ast }<1$. The statistical quantities of interest are computed by averaging the flow in the homogeneous directions, over the top and bottom halves of the channel, and among different runs. The averaging operator is hereafter denoted by $\langle \cdot \rangle$, and velocity fluctuations are signified by $(\cdot )^{\prime }$. Fluctuating velocities are measured with respect to the time-evolving mean velocity profiles in the streamwise and spanwise direction, $\langle u_{1}\rangle (x_{2},t)$ and $\langle u_{3}\rangle (x_{2},t)$, respectively.

The mean velocity profiles are shown in figure 3 at several times. The streamwise mean velocity undergoes mild changes in shape (figure 3a), and the main outcome of the lateral pressure gradient is the development of a spanwise boundary layer of thickness $\unicode[STIX]{x1D6FF}_{3}$ (figure 3b). The growth of $\unicode[STIX]{x1D6FF}_{3}$ is initially governed by viscous diffusion, i.e. $\unicode[STIX]{x1D6FF}_{3}\sim \sqrt{\unicode[STIX]{x1D708}t}$ for $t<t_{\unicode[STIX]{x1D708}}$. A rough estimation of $t_{\unicode[STIX]{x1D708}}$ is given by $t_{\unicode[STIX]{x1D708}}^{+}\approx 70$ (Moin et al. Reference Moin, Shih, Driver and Mansour1990), such that the initial viscous growth period becomes a smaller fraction of $T$ as $Re_{\unicode[STIX]{x1D70F}}$ increases. For $t>t_{\unicode[STIX]{x1D708}}$, turbulent diffusion prevails and $\unicode[STIX]{x1D6FF}_{3}\sim \sqrt{\unicode[STIX]{x1D708}_{e}t}$, where $\unicode[STIX]{x1D708}_{e}$ is the turbulent eddy viscosity. Assuming the mixing-length hypothesis, $\unicode[STIX]{x1D708}_{e}\sim u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}_{3}$, then $\unicode[STIX]{x1D6FF}_{3}\sim u_{\unicode[STIX]{x1D70F}}t$, i.e. the spanwise boundary layer grows linearly in time regardless of $\text{d}P/\text{d}x_{3}$ in first-order approximation. The agreement of the approximation $\unicode[STIX]{x1D6FF}_{3}^{+}\approx 0.445t^{+}$, included in figure 3(b), with $\unicode[STIX]{x1D6FF}_{3}$ highlights the validity of the previous assumptions after the initial viscous phase. The inertial core of the channel, $\langle \cdot \rangle _{\infty }$, is accelerated by the mean spanwise pressure gradient such that $\unicode[STIX]{x1D70C}\langle u_{3}\rangle _{\infty }\approx (\text{d}P/\text{d}x_{3})t$, which controls the additional spanwise shear, $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}\sim \langle u_{3}\rangle _{\infty }/\unicode[STIX]{x1D6FF}_{3}\sim (\text{d}P/\text{d}x_{3})/(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}})$. In summary, the sudden imposition of $\text{d}P/\text{d}x_{3}$ results in the emergence of a spanwise boundary layer diffusing upwards from the wall linearly in time, $\unicode[STIX]{x1D6FF}_{3}\sim u_{\unicode[STIX]{x1D70F}}t$, accompanied by an additional mean shear proportional to $\text{d}P/\text{d}x_{3}$.

Figure 3. Mean velocity profile in (a) the streamwise direction and (b) the spanwise direction for $t^{+}=12$, 72, 132, 192, 252, 312, 372 and 432. Colours indicate time from $t^{+}=0$ (black) to $t^{+}=432$ (red). The vertical dotted lines are the boundary layer thickness $\unicode[STIX]{x1D6FF}_{3}$ defined by the wall-normal distance at which $\langle u_{3}\rangle =0.99\langle u_{3}\rangle _{\infty }=0.99\langle u_{3}\rangle (h,t)$, and the vertical dashed lines are the estimated boundary layer thickness given by $\unicode[STIX]{x1D6FF}_{3}^{+}=0.445t^{+}$.

Figure 4. Mean Reynolds stress for $\unicode[STIX]{x1D6F1}=60$ at $Re_{\unicode[STIX]{x1D70F}}\approx 500$. Different lines correspond to different times $t^{+}=12$, 72, 132, 192, 252, 312, 372 and 432. Colours indicate time from $t^{+}=0$ (black) to $t^{+}=432$ (red). The arrows indicate the direction of time.

The evolution of the mean Reynolds stresses is shown in figure 4. Considering that the flow is subjected to the additional strain $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$, the classic theory anticipates an increase of the Reynolds stresses under the equilibrium assumption $-\langle u_{i}^{\prime }u_{j}^{\prime }\rangle +(1/3)\langle u_{k}^{\prime }u_{k}^{\prime }\rangle \unicode[STIX]{x1D6FF}_{ij}\propto \unicode[STIX]{x1D708}_{e}\langle \unicode[STIX]{x1D64E}_{ij}\rangle$, where $\unicode[STIX]{x1D64E}_{ij}$ is the rate-of-strain tensor and $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker delta. Figure 4 shows that the behaviour of $\langle u_{i}^{\prime }u_{j}^{\prime }\rangle$ is consistent with the equilibrium prediction for large times. However, during the first stages of the transient, $\langle u_{1}^{\prime }u_{1}^{\prime }\rangle$ and $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$ experience a vigorous depletion, whereas $\langle u_{2}^{\prime }u_{2}^{\prime }\rangle$ and $\langle u_{3}^{\prime }u_{3}^{\prime }\rangle$ remain roughly constant. Thus, the initial transient exhibits a counter-intuitive behaviour of Reynolds stresses, inconsistent with the equilibrium assumption. The reduction in magnitude of those stresses comprising $u_{1}^{\prime }$ hints at a deficiency in the streak generation cycle triggered during the transient; the structural origin of such a deficiency is discussed in § 3.4. A similar equilibrium argument applies to the angle of Reynolds stress direction, $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70F}}=\text{atan}[\langle u_{2}^{\prime }u_{3}^{\prime }\rangle /\langle u_{1}^{\prime }u_{2}^{\prime }\rangle ]$, and mean shear direction, $\unicode[STIX]{x1D6FE}_{S}=\text{atan}[(\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2})/(\unicode[STIX]{x2202}\langle u_{1}\rangle /\unicode[STIX]{x2202}x_{2})]$, which are expected to satisfy $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70F}}\approx \unicode[STIX]{x1D6FE}_{S}$ in an equilibrium 2DTBL. As seen from figure 5(a), the equilibrium condition is not met for the angles; the Reynolds stress direction lags behind the mean direction closer to the wall and leads further away. We will focus most of our attention on the tangential Reynolds stress, $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$, because the initial non-equilibrium response is most vividly manifested on that component, although other coordinate-dependent metrics can be defined to measure non-equilibrium effects. In particular, it was assessed that the conclusions drawn below are also valid when non-equilibrium effects are quantified in terms of the classic Townsend (Reference Townsend1976) structure parameter as shown in appendix C.

It could be argued that the drop in $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$ in figure 4(d) is an artefact of the static frame of reference ${\mathcal{F}}:(x_{1},x_{2},x_{3})$. The direction given by ${\mathcal{F}}$ is no longer coplanar with the mean shear vector, which is the primal source responsible for the injection of kinetic energy into the turbulence intensities. To show that the depletion of $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$ is not the consequence of observing the flow from the point of view of ${\mathcal{F}}$, we define the wall-normal and time-dependent frame of reference $\tilde{{\mathcal{F}}}:(\tilde{x}_{1},x_{2},\tilde{x}_{3})$ such that $\tilde{x}_{1}$ points in the direction of the local mean shear vector $(\unicode[STIX]{x2202}\langle u_{1}\rangle /\unicode[STIX]{x2202}x_{2},0,\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2})$ at each wall-normal location and time instant. The angle between $\tilde{x}_{1}$ and $x_{1}$ is given by $\unicode[STIX]{x1D6FE}_{S}$ (figure 5a). The velocity components in the frame of reference $\tilde{{\mathcal{F}}}$ are denoted by $\tilde{u} _{1}$, $\tilde{u} _{2}~(\equiv u_{2})$ and $\tilde{u} _{3}$. Figure 5(b) demonstrates that the shear-aligned tangential Reynolds stress, $-\langle \tilde{u} _{1}^{\prime }\tilde{u} _{2}^{\prime }\rangle$, also experiences a strong reduction in magnitude. An alternative frame of reference is that aligned with the principal Reynolds stress direction defined by the angle $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70F}}$ (Moin et al. Reference Moin, Squires, Cabot and Lee1991). The difference between $\unicode[STIX]{x1D6FE}_{S}$ and $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70F}}$ is small (figure 5a), and the history of the Reynolds stresses in the frame of reference of the principal Reynolds stress direction (not shown) is similar to the results from figure 5(b).

Figure 5. (a) Angle of the mean Reynolds stress direction $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D70F}}$ (solid) and mean shear direction $\unicode[STIX]{x1D6FE}_{S}$ (dotted) with respect to $x_{1}$. (b) Mean tangential Reynolds stress in the wall-normal and time-dependent frame of reference $\tilde{{\mathcal{F}}}$ aligned with the mean shear direction $\unicode[STIX]{x1D6FE}_{S}$. The arrow in panel (b) indicates the direction of time. Different lines correspond to different times $t^{+}=12$, 72, 132, 192, 252, 312, 372 and 432. In both panels, the colours denote time from $t^{+}=0$ (black) to $t^{+}=432$ (red). The data are for $\unicode[STIX]{x1D6F1}=60$ at $Re_{\unicode[STIX]{x1D70F}}\approx 500$.

3.2 Flow regimes

We quantify the flow regimes of the transient response of the momentum-carrying eddies (responsible for $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$) subjected to non-equilibrium effects. It is assumed that the Reynolds number of the channel flow is sufficiently high to develop a multi-scale collection of randomly distributed momentum-carrying motions with their roots attached to the wall, as first conjectured by Townsend (Reference Townsend1976) and currently supported by a growing number of studies (e.g. Davidson, Nickels & Krogstad Reference Davidson, Nickels and Krogstad2006; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Hwang Reference Hwang2015; Hellström, Marusic & Smits Reference Hellström, Marusic and Smits2016; Hwang & Bengana Reference Hwang and Bengana2016; Baars, Hutchins & Marusic Reference Baars, Hutchins and Marusic2017; Dong et al. Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017; Hwang & Sung Reference Hwang and Sung2018; Cheng et al. Reference Cheng, Li, Lozano-Durán and Liu2019; Yang, Willis & Hwang Reference Yang, Willis and Hwang2019). We can anticipate that, for low values of $\unicode[STIX]{x1D6F1}$, the perturbation introduced by the lateral forcing is very gentle and eddies evolve in a quasi-equilibrium state irrespective of their size and lifespan. Conversely, large values of $\unicode[STIX]{x1D6F1}$ are expected to drive the entire population of eddies at all scales across the boundary out of equilibrium. The non-dimensional parameters governing these flow regimes are $Re_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D6F1}$.

The level of non-equilibrium endured by the momentum-carrying eddies can be estimated by assuming that, prior to the application of $\unicode[STIX]{x1D6F1}$, the boundary layer is populated by a collection of wall-attached self-similar eddies with sizes $l_{e}$ proportional to the distance to the wall, $l_{e}\sim x_{2}$, and characteristic velocity $u_{\unicode[STIX]{x1D70F}}$ (Townsend Reference Townsend1976). Consequently, the characteristic lifetime of eddies of size $l_{e}$ is $t_{e}\sim x_{2}/u_{\unicode[STIX]{x1D70F}}$. The smallest momentum-carrying eddies are found close to the wall at $x_{2}\sim \unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ due to the limiting effect of viscosity, and their lifetimes reduce to $t_{e}\sim \unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}^{2}$. The largest eddies are constrained by the channel height $x_{2}\sim h$, with lifetimes $t_{e}\sim h/u_{\unicode[STIX]{x1D70F}}$. The lateral mean pressure gradient introduces an additional time scale associated with the spanwise acceleration of the flow, $t_{p}\sim \unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}/(\text{d}P/\text{d}x_{3})$. The condition for non-equilibrium is $t_{p}<t_{e}$, i.e. the characteristic time to accelerate the flow in the spanwise direction is shorter than the lifetime of the momentum-carrying eddies in order to shove the latter out of the equilibrium state. A similar conclusion is drawn by reasoning in terms of the minimum strength of the lateral shear layer $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$ necessary to disturb the local-in-$x_{2}$ mean shear of the wall-attached eddies $\unicode[STIX]{x2202}\langle u_{1}\rangle /\unicode[STIX]{x2202}x_{2}$. The former was shown to be $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}\sim (\text{d}P/\text{d}x_{3})/(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}})$ in § 3.1, while the latter can be approximated by assuming a logarithmic mean velocity profile of the form $\langle u_{1}\rangle \sim (u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D705})\log (x_{2}^{+})$ such that $\unicode[STIX]{x2202}\langle u_{1}\rangle /\unicode[STIX]{x2202}x_{2}\sim u_{\unicode[STIX]{x1D70F}}/x_{2}$. Then, the lateral mean shear required to overcome the mean streamwise shear is $u_{\unicode[STIX]{x1D70F}}/x_{2}<(\text{d}P/\text{d}x_{3})/(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}})$, which is equivalent to time-scale argument, $t_{p}<t_{e}$, discussed above.

Based on the flow scales discussed above, we differentiate three flow regimes as sketched in figure 6(a). For $\unicode[STIX]{x1D6F1}<O(1)$ ($t_{p}>t_{e}$), the spanwise pressure gradient is categorised as weak, and all flow scales relax instantly to a quasi-equilibrium state during the transient period. Conversely, for $\unicode[STIX]{x1D6F1}>O(Re_{\unicode[STIX]{x1D70F}})$ ($t_{p}<t_{e}$), the momentum-carrying eddies are unable to adjust to the prompt imposition of the shear regardless of their size. For intermediate values of $\unicode[STIX]{x1D6F1}$, eddies coexist in both quasi-equilibrium and non-equilibrium states, the former being the eddies located in the region closer to the wall.

Figure 6. (a) Schematic of self-similar, wall-attached, momentum-carrying eddies, and different flow regimes as a function of the spanwise-to-streamwise mean pressure gradient ratio $\unicode[STIX]{x1D6F1}$. The eddies coloured in green are in a quasi-equilibrium state, whereas the eddies coloured in red are out of equilibrium. (b,c) The percentage drop of tangential Reynolds stress, $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}$, in the frame of reference of the mean shear $\tilde{{\mathcal{F}}}$ as a function of the spanwise-to-streamwise mean pressure gradient ratio $\unicode[STIX]{x1D6F1}$ and wall-normal distance $x_{2}^{\ast }$ for (b) $Re_{\unicode[STIX]{x1D70F}}\approx 500$ and (c) $Re_{\unicode[STIX]{x1D70F}}\approx 1000$. The vertical lines in panel (b) represent flow states ranging from the equilibrium regime (green) to non-equilibrium regime (red).

The analysis above is corroborated in figure 6(b,c), which shows the maximum percentage drop of the tangential Reynolds stress during the transient period after the imposition of the lateral mean pressure gradient, $\min _{t}\{D_{\unicode[STIX]{x1D70F}}(x_{2},t)\}$, where $D_{\unicode[STIX]{x1D70F}}$ is defined as

(3.1)$$\begin{eqnarray}D_{\unicode[STIX]{x1D70F}}(x_{2},t)=\frac{\langle \tilde{u} _{1}\tilde{u} _{2}\rangle (x_{2},t)-\langle \tilde{u} _{1}\tilde{u} _{2}\rangle (x_{2},0)}{\langle \tilde{u} _{1}\tilde{u} _{2}\rangle (x_{2},0)}\times 100.\end{eqnarray}$$

Note that the Reynolds stress in (3.1) is referred to the frame of reference $\tilde{{\mathcal{F}}}$ aligned with the mean shear. Similar conclusions are drawn when the stress is referred to ${\mathcal{F}}$. The results in figure 6(b) reveal that the relative reduction in the Reynolds stress attains up to 30 %, and that the drop accentuates for increasing $\unicode[STIX]{x1D6F1}$ and $x_{2}^{\ast }$. Figure 6(c) confirms that the trend holds at higher $Re_{\unicode[STIX]{x1D70F}}$.

Figure 7. Maximum percentage drop of the tangential Reynolds stress $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}$ in the frame of reference of the mean shear. Colours are black for cases at $Re_{\unicode[STIX]{x1D70F}}\approx 500$ and red for cases at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$. In panel (a), solid lines with circles are $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}$ at $x_{2}^{+}=30$, and the dashed line is $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}\approx -160\unicode[STIX]{x1D6F1}/Re_{\unicode[STIX]{x1D70F}}$. In panel (b), lines with symbols are $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}$ at $x_{2}^{\ast }=0.2$ (○) and $x_{2}^{\ast }=0.4$ (▿), and the dashed line corresponds to $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}\approx -\unicode[STIX]{x1D6F1}x_{2}^{\ast }$.

The scaling of $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}$ is inspected in figure 7, which contains various cuts of the $(\unicode[STIX]{x1D6F1},x_{2}^{\ast })$ maps shown in figure 6(b,c). Within the buffer region (figure 7a), the response of the flow is controlled by the viscous scales. The momentum equation in inner units is given by

(3.2)$$\begin{eqnarray}\frac{{\mathcal{D}}u_{i}^{+}}{{\mathcal{D}}t^{+}}=-\frac{\unicode[STIX]{x2202}p^{\prime +}}{\unicode[STIX]{x2202}x_{i}^{+}}-\frac{\text{d}P^{+}}{\text{d}x_{i}^{+}}\unicode[STIX]{x1D6FF}_{i3}+\frac{\unicode[STIX]{x2202}^{2}u_{i}^{+}}{\unicode[STIX]{x2202}x_{k}^{+}\unicode[STIX]{x2202}x_{k}^{+}},\end{eqnarray}$$

where ${\mathcal{D}}$ denotes material derivative and $\text{d}P^{+}/\text{d}x_{1}^{+}=O(1/Re_{\unicode[STIX]{x1D70F}})$ has been neglected. From (3.2), we conclude that a similar reduction in the Reynolds stress is obtained across different $Re_{\unicode[STIX]{x1D70F}}$ for identical values of $\text{d}P^{+}/\text{d}x_{3}^{+}=\unicode[STIX]{x1D6F1}/Re_{\unicode[STIX]{x1D70F}}$, which is the relevant spanwise-to-streamwise mean pressure gradient for the buffer region.

For the logarithmic layer, for which the high-$Re_{\unicode[STIX]{x1D70F}}$ analysis holds (figure 7b), wall-attached eddies of a given size $l_{e}\sim x_{2}$ experience a similar drop in the Reynolds stress when the mean spanwise pressure gradient is normalised by the characteristic scales, $x_{2}$ and $u_{\unicode[STIX]{x1D70F}}$, controlling the eddies. Analysis of the non-dimensional equations obtained by introducing the similarity variable $\unicode[STIX]{x1D702}=t/t_{e}=tu_{\unicode[STIX]{x1D70F}}/x_{2}$ reveals that the condition for self-similar Reynolds stress depletion at a given wall-normal distance is obtained by a common value of the compensated spanwise-to-streamwise mean pressure gradient ratio, $\unicode[STIX]{x1D6F1}x_{2}^{\ast }$, consistent with the results from figure 7(b).

From the scaling analysis above and the numerical results in figure 7, the quantitative drop in Reynolds stress for the flow motions free of viscous effects at a given $x_{2}$ location is well approximated by

(3.3)$$\begin{eqnarray}\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}\approx -\unicode[STIX]{x1D6F1}x_{2}^{\ast }.\end{eqnarray}$$

If we further assume that the self-similar scaling of the flow motions with $x_{2}$ does not hold below $x_{2}^{+}\approx 160$, the inner-layer scaling law for the Reynolds stress decrease implied by (3.3) is

(3.4)$$\begin{eqnarray}\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}\approx -160\frac{\unicode[STIX]{x1D6F1}}{Re_{\unicode[STIX]{x1D70F}}},\end{eqnarray}$$

which is valid for the buffer region and serves as an approximation to the trends observed in figure 7(a).

Finally, a tentative relation delimiting the necessary spanwise forcing to achieve the fully non-equilibrium regime (eddies out of equilibrium across almost the entire boundary layer), arbitrarily delimited by $\min _{t}\{D_{\unicode[STIX]{x1D70F}}\}<-5\,\%$, is given by

(3.5)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}>0.03Re_{\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

Equation (3.5) shows that the lateral mean pressure gradient required to attain the fully non-equilibrium regime increases proportionally to the Reynolds number. Note that (3.5) is a non-equilibrium condition for $x_{2}^{+}>30$. Prescribing a lower wall-normal limit would result in an even more stringent condition than (3.5). The meaning of $\unicode[STIX]{x1D6F1}$ in this particular flow cannot be unambiguously extrapolated to more general flow configurations. Nonetheless, the time-scale argument used to derive (3.5) suggests that, in external aerodynamic applications, the inner layer is most likely to be found in a quasi-equilibrium state given the high Reynolds numbers typically encountered in these situations.

3.3 Evolution of the tangential Reynolds stress

In the previous subsection we were concerned with the maximum drop in the tangential Reynolds stress without consideration of its time response. Here, we discuss the scaling of the evolution of $D_{\unicode[STIX]{x1D70F}}$ for 3-D channels in the fully non-equilibrium regime, i.e. $\unicode[STIX]{x1D6F1}>0.03Re_{\unicode[STIX]{x1D70F}}$, which is the most intriguing case from the physical viewpoint. As in § 3.2, we perform the analysis separately for the buffer region and logarithmic layer, although the former can be thought of as the near-wall limit of the latter.

The evolution of $D_{\unicode[STIX]{x1D70F}}$ in the buffer layer is plotted in figure 8 for various pairs of $(Re_{\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D6F1})$. Three scalings are inspected. Figure 8(a) shows the evolution of $D_{\unicode[STIX]{x1D70F}}$ as a function of time normalised in outer units. Unsurprisingly, both the intensity of $D_{\unicode[STIX]{x1D70F}}$ and the time instant for the maximum drop vary considerably among distinct combinations of $(Re_{\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D6F1})$. Inasmuch as the near-wall eddies do not scale in outer units, the results in figure 8(a) are included only to expose the lack of collapse among cases under an inadequate normalisation. The time scaling using wall units is tested in figure 8(b). It was argued in § 3.2 that the depletion of Reynolds stress within the inner layer is proportional to $\unicode[STIX]{x1D6F1}/Re_{\unicode[STIX]{x1D70F}}$. Consequently, the results in figure 8(b) are plotted against the compensated Reynolds stress drop, $D_{\unicode[STIX]{x1D70F}}Re_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D6F1}$. The new scaling improves the collapse of the results, especially for $t^{+}<150$, above which the evolution of $D_{\unicode[STIX]{x1D70F}}Re_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D6F1}$ diverges among cases. The absence of collapse for $t^{+}>150$ coincides with the typical lifetime of the momentum-carrying eddies in the buffer layer (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b). Thus, $u_{\unicode[STIX]{x1D70F}}$ (defined at $t=0$) is representative of the originally-in-equilibrium near-wall eddies until the generation cycle is restarted and newborn eddies emerge under different flow conditions. Following the previous reasoning, the collapse can be further improved under the assumption that the length and time scales of the newly created eddies are controlled by the local-in-time friction velocity

(3.6)$$\begin{eqnarray}u_{\unicode[STIX]{x1D70F}}^{\star \,2}(t)=\left.\sqrt{\left(\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\langle u_{1}\rangle }{\unicode[STIX]{x2202}x_{2}}\right)^{2}+\left(\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\langle u_{3}\rangle }{\unicode[STIX]{x2202}x_{2}}\right)^{2}}\right|_{x_{2}=0}.\end{eqnarray}$$

The local wall units, denoted by $(\cdot )^{\star }$, are analogously defined in terms of $\unicode[STIX]{x1D708}$ and $u_{\unicode[STIX]{x1D70F}}^{\star }(t)$, and the local friction Reynolds number is $Re_{\unicode[STIX]{x1D70F}}^{\star }(t)=u_{\unicode[STIX]{x1D70F}}^{\star }(t)h/\unicode[STIX]{x1D708}$. The results in figure 8(c) confirm that the local scaling ($t^{\star }$ versus $D_{\unicode[STIX]{x1D70F}}Re_{\unicode[STIX]{x1D70F}}^{\star }/\unicode[STIX]{x1D6F1}$) holds for longer times.

Figure 8. Evolution of the percentage change of tangential Reynolds stress $D_{\unicode[STIX]{x1D70F}}$ in the buffer layer for $x_{2}^{+}=30$ in panels (a) and (b), and for $x_{2}^{\star }=30$ in panel (c). The lines are for $Re_{\unicode[STIX]{x1D70F}}\approx 500$ (——) and for $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ (– – –). For cases at $Re_{\unicode[STIX]{x1D70F}}\approx 500$, colours are $\unicode[STIX]{x1D6F1}=20$, 40 and 60 from dark red to light red. For cases at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$, colours are dark red for $\unicode[STIX]{x1D6F1}=60$ and light red for $\unicode[STIX]{x1D6F1}=100$.

The evolution of $D_{\unicode[STIX]{x1D70F}}$ for the momentum-carrying eddies across the logarithmic layer is shown in figure 9, where three scaling laws are investigated. The evolution of $D_{\unicode[STIX]{x1D70F}}$ in outer units is included in figure 9(a). Wall-attached eddies follow an ordered response in time after the sudden imposition of the transverse pressure gradient: eddies closer to the wall react earlier and are the least perturbed, while larger eddies experience a more acute Reynolds stress reduction at later times. The preceding analysis for the buffer region is extended to the logarithmic layer by taking into consideration that the lifetimes of the wall-attached eddies scale as ${\sim}x_{2}/u_{\unicode[STIX]{x1D70F}}$, with a consistent drop in the Reynolds stress proportional to $\unicode[STIX]{x1D6F1}x_{2}^{\ast }$. The self-similar response of wall-attached eddies under the lateral force is evidenced by the improved collapse in figure 8(b), at least for $tu_{\unicode[STIX]{x1D70F}}/x_{2}\lesssim 1$. Analogously to the inner layer, $u_{\unicode[STIX]{x1D70F}}$ stands as the characteristic velocity scale of the original eddies in the equilibrium state, but does not hold as such for times longer than the lifespan of individual wall-attached eddies, $tu_{\unicode[STIX]{x1D70F}}/x_{2}\approx 1$ (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b). The collapse among cases is perfected by using the local time scale $tu_{\unicode[STIX]{x1D70F}}^{\star }/x_{2}$ (figure 8c), which accounts for variations in the momentum transfer controlling the eddies during the transient. We close this subsection by highlighting that the scaling properties of the flow studied above are conspicuous only at moderate to high Reynolds numbers. The response of the flow at lower Reynolds numbers is discussed in appendix A, where it is shown that 3-D channels at $Re_{\unicode[STIX]{x1D70F}}\approx 180$ lack the necessary scale separation to exhibit a multiscale depletion of the Reynolds stress.

Figure 9. Evolution of the percentage change of tangential Reynolds stress $D_{\unicode[STIX]{x1D70F}}$ in the logarithmic layer for $x_{2}^{\ast }=0.15$, 0.2, 0.3 and 0.4 represented by lines coloured from dark red to light red. The lines are for cases at $Re_{\unicode[STIX]{x1D70F}}\approx 500$ (——) and at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ (– – –), both for $\unicode[STIX]{x1D6F1}=60$. The arrow in panel (a) indicates increasing wall-normal distance.

3.4 Structural changes in the conditionally averaged flow field

We examine the structural evolution of the flow in the surroundings of the momentum-carrying eddies. To that end, we identify 3-D structures of the intense momentum transfer using the methodology introduced by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) (see also Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b; Lozano-Durán & Borrell Reference Lozano-Durán and Borrell2016). An individual structure (or object) of intense momentum transfer at time $t$ is defined as a spatially connected region in the flow satisfying

(3.7)$$\begin{eqnarray}-u_{1}^{\prime }(x_{1},x_{2},x_{3},t)u_{2}^{\prime }(x_{1},x_{2},x_{3},t)>H\langle u_{1}^{\prime \,2}\rangle ^{1/2}(x_{2},t)\langle u_{2}^{\prime \,2}\rangle ^{1/2}(x_{2},t),\end{eqnarray}$$

where $H$ is a thresholding parameter (hyperbolic hole size (Bogard & Tiederman Reference Bogard and Tiederman1986)) equal to 1.75 obtained following the analysis by Moisy & Jiménez (Reference Moisy and Jiménez2004). It was tested that varying $H$ within the range $0.5<H<3$ does not change the conclusions below. The original frame of reference defined by ${\mathcal{F}}$ is preferred to $\tilde{{\mathcal{F}}}$ in order to avoid artificial distortions in the flow due to the time and space variations in $\tilde{{\mathcal{F}}}$. Hereafter, we refer to individual structures of intense $-u_{1}^{\prime }u_{2}^{\prime }$ events as $-u_{1}^{\prime }u_{2}^{\prime }$ structures. Numerically, 3-D structures are constructed by connecting neighbouring grid points fulfilling (3.7) and using the 6-connectivity criteria (Rosenfeld & Kak Reference Rosenfeld and Kak1982). Figure 10 shows the wall-attached $-u_{1}^{\prime }u_{2}^{\prime }$ structures identified before and after the imposition of the spanwise pressure gradient. Figure 10 also includes one individual $-u_{1}^{\prime }u_{2}^{\prime }$ structure highlighted by the box with red edges.

Figure 10. Instantaneous $-u_{1}^{\prime }u_{2}^{\prime }$ structures defined by (3.7) for $\unicode[STIX]{x1D6F1}=60$ and $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ at (a) $t^{\ast }=0$ and (b) $t^{\ast }=0.5$. Only $-u_{1}^{\prime }u_{2}^{\prime }$ structures attached to the bottom wall are shown. The colours represent the distance to the wall, from yellow (closer to the wall) to blue (farther from the wall). The box with edges coloured in red is the bounding box of one individual $-u_{1}^{\prime }u_{2}^{\prime }$ structure with streamwise, wall-normal and spanwise sizes equal to $l_{1}$, $l_{2}$ and $l_{3}$, respectively.

We focus our attention on the channel at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ and $\unicode[STIX]{x1D6F1}=60$, but similar results are obtained consistently across different $Re_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D6F1}$, provided that the latter is large enough to attain the fully non-equilibrium regime. We select three time instants to assess the structural changes in the flow, namely, $t^{\ast }=0$, $t^{\ast }=0.25$ and $t^{\ast }=0.50$. The evolution of $D_{\unicode[STIX]{x1D70F}}$ is plotted in figure 11(a), which shows that the maximum drop in the tangential Reynolds stress occurs at $t^{\ast }\approx 0.50$.

Figure 11. (a) Evolution of the percentage change of tangential Reynolds stress $D_{\unicode[STIX]{x1D70F}}$ at $x_{2}^{\ast }=0.25$. The vertical lines are the times selected to study the flow structure in addition to $t^{\ast }=0$, namely $t^{\ast }=0.25$ (– – –) and $t^{\ast }=0.5$ ($\cdots \cdots$). (b) Joint p.d.f.s of the logarithms of the streamwise $l_{1}$ and wall-normal $l_{2}$ sizes of wall-attached $-u_{1}^{\prime }u_{2}^{\prime }$ structures, $p(l_{1}^{+},l_{2}^{+})$. The contours plotted contain 50 % and 99.8 % of the probability. The lines are for $t^{\ast }=0$ (

), $t^{\ast }=0.25$ (

) and $t^{\ast }=0.5$ (

). The straight dashed line is $l_{1}^{+}=3l_{2}^{+}$ and the arrow indicates the direction of time. The results are for $Re_{\unicode[STIX]{x1D70F}}\approx 1000$ and $\unicode[STIX]{x1D6F1}=60$.

The identification procedure above yields approximately $10^{5}$ structures at each time instant after discarding those objects with volumes smaller than $30^{3}$ wall units. The sizes of the objects are measured by circumscribing each structure within a box aligned to the Cartesian axes, whose streamwise, wall-normal and spanwise sizes are denoted by $l_{1}$, $l_{2}$ and $l_{3}$, respectively. The minimum and maximum distances of each object to the closest wall are $x_{2,min}$ and $x_{2,max}$, respectively, and such that $l_{2}=x_{2,max}-x_{2,min}$. An example of an individual $-u_{1}^{\prime }u_{2}^{\prime }$ structure and its bounding box is included in figure 10(a). The bounding boxes of the structures are aligned with the original flow direction. At $t^{\ast }=0.5$, the mean shear at the centre of gravity of a $-u_{1}^{\prime }u_{2}^{\prime }$ structure of size $x_{2}^{\ast }\approx 0.4$ is rotated by $\unicode[STIX]{x1D6FE}_{S}\approx 15^{\circ }$ (see also figure 10b). This low turning angle justifies the selection of ${\mathcal{F}}$ to study the flow for short times, as is the case here, but the investigation of structural changes for longer times would require a properly chosen rotating frame of reference. We centre our attention on wall-attached $-u_{1}^{\prime }u_{2}^{\prime }$ structures, defined as those with $x_{2,min}^{+}<25$ (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006). For the value of $H$ selected, wall-attached structures are responsible for more than 60 % of the tangential Reynolds stress at all three times considered. Figure 11(b) shows the joint probability density function (p.d.f.) of the sizes of the wall-attached structures, $p(l_{1}^{+},l_{2}^{+})$. At $t^{\ast }=0$, the distribution of sizes is consistent with a geometrically self-similar population of structures akin to the wall-attached eddies envisioned by Townsend (Reference Townsend1976) at high Reynolds numbers. The mode of the p.d.f. follows a reasonably well-defined linear law, $l_{1}\sim 3l_{2}$, consistent with previous studies (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). From $t^{\ast }=0$ to $t^{\ast }=0.50$, the most pronounced modification in the geometry of the structures is a gradual shortening of their streamwise length, while their wall-normal heights are barely affected.

Each $-u_{1}^{\prime }u_{2}^{\prime }$ structure can be classified either as an ejection, when the average wall-normal velocity within its enclosed volume is positive, or as a sweep otherwise. Sweeps and ejections are known to be spatially organised in pairs side-by-side along the spanwise direction (Ganapathisubramani Reference Ganapathisubramani2008; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Wallace Reference Wallace2016; Osawa & Jiménez Reference Osawa and Jiménez2018). This sweep–ejection group, representative of a streamwise roll, is the predominant logarithmic-layer flow structure responsible for the generation of tangential Reynolds stress. Consequently, we are interested in examining the modification of the flow around sweep–ejection pairs during the transient period. We denote the centre of gravity of the bounding boxes of the $n$th sweep and its paring ejection as $\boldsymbol{x}_{s}^{n}$ and $\boldsymbol{x}_{e}^{n}$, respectively. The wall-normal size of the sweep is $l_{2,s}^{n}$ and that of the ejection is $l_{2,e}^{n}$. The averaged flow field conditioned to the presence of a sweep–ejection pair is computed by averaging the velocity vector in a rectangular domain along different $n$th pairs, whose centre coincides with $\boldsymbol{x}_{p}^{n}=(\boldsymbol{x}_{e}^{n}+\boldsymbol{x}_{s}^{n})/2$, and it edges are $\boldsymbol{r}$ times the average wall-normal height $l_{p}^{n}=(l_{2,e}^{n}+l_{2,s}^{n})/2$. Then, the conditionally averaged flow around sweep–ejection pairs is given by

(3.8)$$\begin{eqnarray}\{u_{i}^{\prime }\}(\boldsymbol{r})=\mathop{\sum }_{n=1}^{N}\frac{u_{i}^{\prime }(\boldsymbol{x}_{p}^{n}+l_{p}^{n}\boldsymbol{r})}{N},\end{eqnarray}$$

where $n=1,\ldots ,N$ is the set of sweep–ejection pairs selected to perform the conditional average, and $\boldsymbol{r}=(r_{1},r_{2},r_{3})$. We also take advantage of the spanwise symmetry of the flow, and $r_{3}$ is always chosen to be positive towards the sweep. The reader is referred to Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Dong et al. (Reference Dong, Lozano-Durán, Sekimoto and Jiménez2017) for additional details on the procedure to obtain conditional flow fields.

Figure 12. Averaged flow fields conditioned to wall-attached pairs of sweeps and ejections with wall-normal sizes in the range $0.2<l_{p}^{n\ast }<0.3$ at (a) $t^{\ast }=0$, (b) $t^{\ast }=0.25$ and (c) $t^{\ast }=0.50$. The panels on the left contain isosurfaces of the low-velocity (blue) and high-velocity (red) streaks defined by $\pm \unicode[STIX]{x1D6FC}$ of the maximum positive and negative, respectively, fluctuating streamwise velocity of the average flow with (a) $\unicode[STIX]{x1D6FC}=0.6$, (b) $\unicode[STIX]{x1D6FC}=0.55$ and (c) $\unicode[STIX]{x1D6FC}=0.43$. The arrows indicate the mean flow direction. The panels on the right display the cross-flow velocity vector field $(\{u_{2}^{\prime }\},\{u_{3}^{\prime }\})$ (arrows) and the fluctuating velocity $\{u_{1}^{\prime }\}$ (colours). The dashed white line shows the wall-normal extension from the wall of the incoherent streamwise velocity field represented by low values of $\{u_{1}^{\prime }\}$ in green. Velocities are normalised by $u_{\unicode[STIX]{x1D70F}}$. Results are for $\unicode[STIX]{x1D6F1}=60$ at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$.

The averaged flow field conditioned to sweep–ejection pairs with $0.2<l_{p}^{n\,\ast }<0.3$ is plotted in figure 12. At $t^{\ast }=0$ (figure 12a), the characteristic flow structure consistent with the statistically-in-equilibrium flow state is a streamwise roll flanked by one low-velocity streak and one high-velocity streak. At succeeding times (figure 12b,c), the roll persists, while the intensity and size of the low-velocity streak have already decreased at $t^{\ast }=0.5$. The high-velocity streak and roll are also weakened, but the variations are less pronounced. Another observation is the loss of coherence in a developing layer underneath the low-velocity streak. This effect is demonstrated by the low values of $\{u_{1}^{\prime }\}$ represented by the green regions below the white dashed lines in figure 12 (right panels). The nearly zero averaged streamwise velocity for $t^{\ast }>0$ is an indication that, from the viewpoint of the larger rolls, the fluid motions below cancel out and are perceived as incoherent. During the transient, both low- and high-velocity streaks shorten in the streamwise direction in accordance with the geometric analysis in figure 11(b). Although not shown, the results above are also applicable to sweep–ejection pairs across different ranges of $l_{p}^{n}$ when the times are appropriately scaled by $l_{p}^{n}/u_{\unicode[STIX]{x1D70F}}^{\star }$, implying that the modifications in the flow are self-similar in space and time.

The message from figure 13 is that the main structural alteration during the transient is the weakening of the low-velocity streaks, which is in turn associated with the loss of coherence of the flow within a growing layer underneath the streamwise rolls. The aforementioned loss of coherence may be attributed to (i) the relative displacement of wall-parallel layers at different heights and (ii) the additional mean spanwise shear which enhances the generation of smaller-scale momentum-carrying eddies (as shown by Mizuno & Jiménez (Reference Mizuno and Jiménez2011), Jiménez (Reference Jiménez2018) and Lozano-Durán & Bae (Reference Lozano-Durán and Bae2019)). This is illustrated in figure 13, which contains the instantaneous streamwise velocity at two wall-normal distances: one closer to the wall at $x_{2}^{\ast }=0.1$ influenced by the additional shear from the lateral boundary layer, and another farther from the wall at $x_{2}^{\ast }=0.3$ still unaffected.

Figure 13. (a) Schematic of the channel flow domain with the wall-normal planes shown in panels (b) and (c). Panels (b) and (c) contain the streamwise velocity at (b) $x_{2}^{\ast }=0.1$ and (c) $x_{2}^{\ast }=0.3$ at the same time instant $t^{\ast }=0.5$ for $\unicode[STIX]{x1D6F1}=60$ at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$. The red straight lines in (b) and (c) indicate the mean flow direction. Velocities are normalised by $u_{\unicode[STIX]{x1D70F}}$.

3.5 Structural model

On the basis of the above observations, we propose a conceptual model that accounts for the changes undergone by the flow. The model is sketched in figure 14. The key elements are the low- and high-velocity streaks and their relative alignment with respect to the streamwise roll. Although the flow can be conceptually divided into streaks and rolls, both are interdependent flow entities that interact in a self-sustaining cycle (Waleffe Reference Waleffe1997; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Hwang & Cossu Reference Hwang and Cossu2011; Cossu & Hwang Reference Cossu and Hwang2017; Lozano-Durán, Bae & Encinar Reference Lozano-Durán, Bae and Encinar2020). At a given wall-normal distance $x_{2}$ and $t=0$, the flow is configured in an equilibrium array of rolls and streaks with their centres at $x_{2}$, sizes $2x_{2}$ and lifetimes $2x_{2}/u_{\unicode[STIX]{x1D70F}}$ (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014b). The tangential Reynolds stress $\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$ at $x_{2}$ is the result of the wall-normal momentum transport conducted by the rolls and the arrangement of streaks in the equilibrium state. The momentum transfer at $t=0$ can be modelled as the sum of two contributions,

(3.9)$$\begin{eqnarray}\displaystyle \langle u_{1}^{\prime }u_{2}^{\prime }\rangle (x_{2},t=0)^{model} & {\approx} & \displaystyle (u_{1}^{\prime S}u_{2}^{\prime R})_{top}+(u_{1}^{\prime S}u_{2}^{\prime R})_{bot}\end{eqnarray}$$

(3.10)$$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle (u_{\unicode[STIX]{x1D70F}})(-u_{\unicode[STIX]{x1D70F}}/2)+(-u_{\unicode[STIX]{x1D70F}})(u_{\unicode[STIX]{x1D70F}}/2)\approx -u_{\unicode[STIX]{x1D70F}}^{2},\end{eqnarray}$$

where $(u_{1}^{\prime S}u_{2}^{\prime R})_{top}$ represents the wall-normal transport of the high-velocity streak, $u_{1}^{\prime S}\approx u_{\unicode[STIX]{x1D70F}}$, by the downward motion of the roll, $u_{2}^{\prime R}\approx -u_{\unicode[STIX]{x1D70F}}/2$, above $x_{2}$. Conversely, $(u_{1}^{\prime S}u_{2}^{\prime R})_{bot}$ is the wall-normal transport of the low-velocity streak, $u_{1}^{\prime R}\approx -u_{\unicode[STIX]{x1D70F}}/2$, by the upward motion of the roll, $u_{2}^{\prime R}\approx u_{\unicode[STIX]{x1D70F}}$, below $x_{2}$. The intensities of $u_{1}^{\prime S}$ and $u_{2}^{\prime R}$ are adjusted to produce a total momentum transfer equal to $-u_{\unicode[STIX]{x1D70F}}^{2}$, although the discussion is extensive to other values.

Figure 14. Structural model of self-similar wall-attached eddies subjected to a sudden mean spanwise pressure gradient. The figure shows one building block structure that comprises a streamwise roll flanked by one low-velocity streak and one high-velocity streak. Note that the complete flow field consists of the superposition of multiple building block structures of different sizes. (a) Statistically-in-equilibrium wall-attached momentum-carrying eddies of size $2x_{2}$ at $t=0$ generating a momentum transfer ${\approx}-u_{\unicode[STIX]{x1D70F}}^{2}$. (b) Non-equilibrium wall-attached momentum-carrying eddies at $t=x_{2}/u_{\unicode[STIX]{x1D70F}}$ after the imposition of a transverse mean pressure gradient generating a momentum transfer ${\approx}-u_{\unicode[STIX]{x1D70F}}^{2}/2(1+\unicode[STIX]{x1D706})$. In panel (a), $(u_{1}^{\prime S}u_{2}^{\prime R})_{top}$ and $(u_{1}^{\prime S}u_{2}^{\prime R})_{bot}$ represent the downward and upward, respectively, wall-normal momentum transfer by the streamwise roll. In panel (b), $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ is the spanwise boundary layer thickness and $\unicode[STIX]{x1D6E5}_{r}$ is the lateral displacement of the flow above $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ due to the uniform acceleration $(1/\unicode[STIX]{x1D70C})\,\text{d}P/\text{d}x_{3}$ imposed by the mean spanwise pressure gradient. The smaller-scale low- and high-velocity streaks underneath the larger roll in panel (b) represent the incoherent fluid motions discussed in § 3.4 (figure 12b,c) and visualised in figure 13(b). At $t=0$, the flow also contains smaller streamwise streaks underneath the larger roll; although in this case they are aligned with the $x_{1}$ direction. However, smaller streaks are not represented in panel (a), as they do not contribute to the destruction of the near-wall coherence of the rolls above (figure 12a).

At half the lifespan of the eddies $t\approx x_{2}/u_{\unicode[STIX]{x1D70F}}$, the spanwise boundary layer extends up to $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}\approx 0.445x_{2}$, based on the estimations in § 3.1, and remains below the centre of the rolls located at $x_{2}$. Simultaneously, the upper flow is laterally displaced by $\unicode[STIX]{x1D6E5}_{r}\approx (1/\unicode[STIX]{x1D70C})(\text{d}P/\text{d}x_{3})(x_{2}/u_{\unicode[STIX]{x1D70F}})^{2}$. For values of $\unicode[STIX]{x1D6E5}_{r}$ larger than the spanwise coherence of the roll–streak structure, namely $\unicode[STIX]{x1D6E5}_{r}>2x_{2}$, the centre of the rolls is misaligned with the underneath streaks within the lateral boundary layer. The latter streaks are also altered by $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$, which increases the local Reynolds number and triggers the emergence of smaller scales (as discussed in § 3.4, figure 13). These changes originate a new flow configuration that is less efficient in producing $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$ compared to the equilibrium state. The rationale behind such a reduction is a loss of flow coherence underneath the rolls, as supported by figure 12, which weakens the vertical momentum transported by the rolls from the layers closer to the wall. The new momentum transfer at $t\approx x_{2}/u_{\unicode[STIX]{x1D70F}}$ can be modelled similarly to (3.10) by assuming that $(u_{1}^{\prime S}u_{2}^{\prime R})_{top}$ is barely affected, whereas $(u_{1}^{\prime S}u_{2}^{\prime R})_{bot}$ provides a deficient momentum transfer such that

(3.11)$$\begin{eqnarray}\displaystyle \langle u_{1}^{\prime }u_{2}^{\prime }\rangle (x_{2},t=x_{2}/u_{\unicode[STIX]{x1D70F}})^{model} & {\approx} & \displaystyle (u_{1}^{\prime S}u_{2}^{\prime R})_{top}+(u_{1}^{\prime S}u_{2}^{\prime R})_{bot}\end{eqnarray}$$

(3.12)$$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle (u_{\unicode[STIX]{x1D70F}})(-u_{\unicode[STIX]{x1D70F}}/2)+(-u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D706})(u_{\unicode[STIX]{x1D70F}}/2)\approx -\frac{u_{\unicode[STIX]{x1D70F}}^{2}}{2}(1+\unicode[STIX]{x1D706}),\qquad\end{eqnarray}$$

where $\unicode[STIX]{x1D706}$ is a damping factor accounting for the reduction in the Reynolds stress generation due to the loss of streak coherence within the lateral boundary layer. The functional form of $\unicode[STIX]{x1D706}$ is modelled by assuming that the loss of streak coherence is, in first-order approximation, linearly proportional to the relative spanwise mean shear,

(3.13)$$\begin{eqnarray}\unicode[STIX]{x1D706}=1-\frac{\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}}{\unicode[STIX]{x2202}\langle u_{1}\rangle /\unicode[STIX]{x2202}x_{2}},\end{eqnarray}$$

such that $\langle u_{1}^{\prime }u_{2}^{\prime }\rangle =-u_{\unicode[STIX]{x1D70F}}^{2}$ for $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}=0$. If we consider the approximations $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}\approx (\text{d}P/\text{d}x_{3})/(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}})$ and $\unicode[STIX]{x2202}\langle u_{1}\rangle /\unicode[STIX]{x2202}x_{2}\approx u_{\unicode[STIX]{x1D70F}}/(2x_{2})$ (see § 3.1), then

(3.14)$$\begin{eqnarray}\langle u_{1}^{\prime }u_{2}^{\prime }\rangle (x_{2},t=x_{2}/u_{\unicode[STIX]{x1D70F}})^{model}\approx -u_{\unicode[STIX]{x1D70F}}^{2}\left(1-\frac{x_{2}\,\text{d}P/\text{d}x_{3}}{\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2}}\right).\end{eqnarray}$$

Equation (3.14) can be rearranged as $\min _{t}\{D_{\unicode[STIX]{x1D70F}}^{model}\}\approx -\unicode[STIX]{x1D6F1}x_{2}^{\ast }$, which coincides with the maximum Reynolds stress depletion from (3.3).

Additionally, the model above predicts that the condition for non-equilibrium of flow structures at height $x_{2}$ is given by $\unicode[STIX]{x1D6E5}_{r}>2x_{2}$, which in non-dimensional form yields $\unicode[STIX]{x1D6F1}x_{2}^{\ast }>2$. In order to disturb the wall-parallel layers at all heights across the boundary layer, $x_{2}^{\ast }$ should be fixed in wall units and such that $\unicode[STIX]{x1D6F1}>O(Re_{\unicode[STIX]{x1D70F}})$, also consistent with the estimation of $\unicode[STIX]{x1D6F1}>0.03Re_{\unicode[STIX]{x1D70F}}$ provided in § 3.2.

The scenario promoted above is self-similar: the continuous depletion in time of the Reynolds stress in figure 5(b) is the result of the time-ordered disruption of streaks and rolls from their natural equilibrium by the growth of the spanwise boundary layer. The present model complements and generalises previous studies formulated in terms of single-scale near-wall streaks and quasi-streamwise vortices to the analogous log-layer streaks and rolls of arbitrary size $x_{2}$. The mechanism above also shares some similarities with the physical arguments pertaining to the modification of near-wall turbulence in the presence of oscillating walls characteristic of drag reduction studies (Jung et al. Reference Jung, Mangiavacchi and Akhavan1992; Laadhari, Skandaji & Morel Reference Laadhari, Skandaji and Morel1994; Choi & Clayton Reference Choi and Clayton2001; Choi, Xu & Sung Reference Choi, Xu and Sung2002; Ricco & Quadrio Reference Ricco and Quadrio2008), although our model is tailored for multiscale flows and uniform accelerations.

It is worth noting that the reduction of the Reynolds stress has been mainly modelled on the basis of non-equilibrium effects rather than on the three-dimensionality of the mean flow and, therefore, is not constrained to the application of additional mean pressure gradients only in the spanwise direction. Accordingly, the model also predicts that a sudden forcing in the streamwise direction would encompass a decrease of the Reynolds stress as long as the relative shift between wall-parallel layers is capable of misaligning the cores of the roll and the streaks underneath. As it is well known that streaks are longer than wider across the logarithmic layer by a factor of 3–6, we can anticipate that suddenly imposed streamwise mean pressure gradients are less efficient in decreasing the Reynolds stress than their spanwise counterparts. This view is supported by the studies by He & Seddighi (Reference He and Seddighi2013, Reference He and Seddighi2015), Seddighi et al. (Reference Seddighi, He, Pokrajac, ODonoghue and Vardy2015) and Mathur et al. (Reference Mathur, Gorji, He, Seddighi, Vardy, ODonoghue and Pokrajac2018), who showed that channel flows subjected to streamwise mean pressure gradients exhibit a similar, but less exacerbated, counter-intuitive response of flow consistent with the model presented here.

To conclude, we comment on the similarities and differences of the present structural model with respect to previous models in the literature. We should stress first that our model relies on the multiscale organisation of the flow, while past models are usually formulated under the assumption of a single-scale flow at low Reynolds number. From figure 13, it was argued that the flow structures produced under the additional mean shear are smaller in size, and that this contributes to the loss of coherence from the viewpoint of the upper flow layers. The generation of smaller eddies is supported by the work of Lozano-Durán & Bae (Reference Lozano-Durán and Bae2019), who showed that the characteristic length scale of the flow structures decreases proportionally to the mean shear. Our model is also consistent with the generation of smaller flow features postulated by Lohmann (Reference Lohmann1976) and corroborated experimentally by Kiesow & Plesniak (Reference Kiesow and Plesniak2002) for the near-wall region. Bradshaw & Pontikos (Reference Bradshaw and Pontikos1985) suggested that the mechanism responsible for the reduction in the Reynolds stress is the tilting about the streamwise direction of pre-existing flow structures caused by the ‘instant’ introduction of $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$ by inviscid skewing. In the present flow configuration, $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$ diffuses upwards from the wall in a finite time. Yet, the aforementioned tilting mechanism is not completely incompatible with the model depicted in figure 14, as the relative displacement between wall-parallel flow layers still entails some degree of inclination of the eddies in the $x_{1}$–$x_{2}$ plane. Coleman et al. (Reference Coleman, Kim and Spalart1996b) noted that the toppling of eddies dominates when the three-dimensionality is introduced by inviscid skewing, but its role might be secondary in cases with uniform lateral acceleration as in the present study. Comparisons with previous models framed in terms of the evolution of streaks, ejections and sweeps are complicated due to the lack of interaction between flow structures at different scales. Eaton (Reference Eaton1991) and Kannepalli & Piomelli (Reference Kannepalli and Piomelli2000) hypothesised that the cross-flow could inhibit the intensity of the low-velocity streaks and, consequently, the associated Reynolds stress from the streak breakdown. Other works advocate for an asymmetric distribution of sweeps and ejections as a key requirement for the Reynolds stress reduction (Anderson & Eaton Reference Anderson and Eaton1989; Sendstad Reference Sendstad1992; Littell & Eaton Reference Littell and Eaton1994; Eaton Reference Eaton1995; Chiang & Eaton Reference Chiang and Eaton1996; Wu & Squires Reference Wu and Squires1997). Although the weakening of the streaks and the asymmetry of sweeps and ejections are not necessarily incompatible with our model, they are not essential components to attain a Reynolds stress reduction.

3.6 Evolution of the tangential Reynolds stress budget

We examine the reduction of $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$ from the Reynolds stress budget viewpoint to complement the physical insight gained from the structural analysis in § 3.4. We use the static frame of reference ${\mathcal{F}}$ to avoid the complexity of additional terms of the form $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$ in the budget equation. Following Mansour, Kim & Moin (Reference Mansour, Kim and Moin1988), the dynamic equation for the component $\langle u_{i}^{\prime }u_{j}^{\prime }\rangle$ is given by

(3.15)$$\begin{eqnarray}\frac{{\mathcal{D}}\langle u_{i}^{\prime }u_{j}^{\prime }\rangle }{{\mathcal{D}}t}=P_{ij}+\unicode[STIX]{x1D700}_{ij}+T_{ij}+PS_{ij}+PD_{ij}+V_{ij},\end{eqnarray}$$

where the terms on the right-hand side are the Reynolds stress production ($P_{ij}$), dissipation ($\unicode[STIX]{x1D700}_{ij}$), turbulent diffusion ($T_{ij}$), pressure strain ($PS_{ij}$), pressure diffusion ($PD_{ij}$) and viscous diffusion ($V_{ij}$) defined as

(3.16)$$\begin{eqnarray}\displaystyle \displaystyle P_{ij} & = & \displaystyle -\langle u_{i}^{\prime }u_{k}^{\prime }\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{k}}\right\rangle -\langle u_{j}^{\prime }u_{k}^{\prime }\rangle \left\langle \frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{k}}\right\rangle ,\end{eqnarray}$$

(3.17)$$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D700}_{ij} & = & \displaystyle -2\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\right\rangle ,\end{eqnarray}$$

(3.18)$$\begin{eqnarray}\displaystyle \displaystyle T_{ij} & = & \displaystyle -\left\langle \frac{\unicode[STIX]{x2202}u_{i}^{\prime }u_{j}^{\prime }u_{k}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\right\rangle ,\end{eqnarray}$$

(3.19)$$\begin{eqnarray}\displaystyle \displaystyle PS_{ij} & = & \displaystyle -\left\langle u_{i}^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{j}}+u_{j}^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{i}}\right\rangle ,\end{eqnarray}$$

(3.20)$$\begin{eqnarray}\displaystyle \displaystyle PD_{ij} & = & \displaystyle -\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{k}}\left\langle p^{\prime }u_{i}^{\prime }\unicode[STIX]{x1D6FF}_{jk}+p^{\prime }u_{j}^{\prime }\unicode[STIX]{x1D6FF}_{ik}\right\rangle ,\end{eqnarray}$$

(3.21)$$\begin{eqnarray}\displaystyle \displaystyle V_{ij} & = & \displaystyle \left\langle \frac{\unicode[STIX]{x2202}^{2}u_{i}^{\prime }u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}\unicode[STIX]{x2202}x_{k}}\right\rangle .\end{eqnarray}$$

In order to obtain quantities that are only a function of time, we introduce the average along $x_{2}$ bands, which is indicated by $\overline{(\cdot )}$. The wall-normal bands inspected are $x_{2}^{+}\in [5,50]$ and $x_{2}^{\ast }\in [0.2,0.3]$, which lie within the buffer region and logarithmic layer, respectively. The gains produced by the budget components $\bar{\unicode[STIX]{x1D719}}_{ij}$ for $(i,j)=(1,2)$, $(1,1)$ and $(2,2)$ are defined as

(3.22)$$\begin{eqnarray}\displaystyle & \displaystyle \text{Gain}_{-12}=\frac{-\bar{\unicode[STIX]{x1D719}}_{12}(t)+\bar{\unicode[STIX]{x1D719}}_{12}(0)}{-\overline{P}_{12}(0)}, & \displaystyle\end{eqnarray}$$

(3.23)$$\begin{eqnarray}\displaystyle & \displaystyle \text{Gain}_{11}=\frac{\bar{\unicode[STIX]{x1D719}}_{11}(t)-\bar{\unicode[STIX]{x1D719}}_{11}(0)}{\overline{P}_{11}(0)}, & \displaystyle\end{eqnarray}$$

(3.24)$$\begin{eqnarray}\displaystyle & \displaystyle \text{Gain}_{22}=\frac{\bar{\unicode[STIX]{x1D719}}_{22}(t)-\bar{\unicode[STIX]{x1D719}}_{22}(0)}{\overline{PS}_{22}(0)}, & \displaystyle\end{eqnarray}$$

where $\text{Gain}_{-12}$, $\text{Gain}_{11}$ and $\text{Gain}_{22}$ represent the gain in the Reynolds stress budget equation for $-\overline{\langle u_{1}^{\prime }u_{2}^{\prime }\rangle }$, $\overline{\langle u_{1}^{\prime }u_{1}^{\prime }\rangle }$ and $\overline{\langle u_{2}^{\prime }u_{2}^{\prime }\rangle }$, respectively. Note that $\text{Gain}_{-12}$ is defined such that $-\overline{P}_{12}>0$ contributes to increasing the magnitude of $-\overline{\langle u_{1}^{\prime }u_{2}^{\prime }\rangle }$. We analyse the channel flow at $Re_{\unicode[STIX]{x1D70F}}\approx 500$ and $\unicode[STIX]{x1D6F1}=60$, in which case the maximum drop in $-\overline{\langle u_{1}^{\prime }u_{2}^{\prime }\rangle }$ occurs at $t^{+}\approx 170$ and $t^{\ast }\approx 0.7$ for the bands in the buffer region and logarithmic layer, respectively.

Figure 15. Evolution of the gain produced by the different terms of the Reynolds stress budget for (a,b) $-\overline{\langle u_{1}^{\prime }u_{2}^{\prime }\rangle }$, (c,d) $\overline{\langle u_{2}^{\prime }u_{2}^{\prime }\rangle }$ and (e,f) $\overline{\langle u_{1}^{\prime }u_{1}^{\prime }\rangle }$. Panels (a), (c) and (e) are for $x_{2}^{+}\in [5,50]$, and panels (b), (d) and (f) are for $x_{2}^{\ast }\in [0.2,0.3]$. The results are for $\unicode[STIX]{x1D6F1}=60$ at $Re_{\unicode[STIX]{x1D70F}}\approx 500$. Zero gain is represented by the horizontal dotted line.

The gains are reported in figure 15 as a function of time. We discuss first the results for the buffer layer region $x_{2}^{+}\in [5,50]$. Figure 15(a) shows the evolution of $-\overline{P}_{12}$, $-\overline{T}_{12}$ and $-\overline{PS}_{12}$. The remaining terms are not significant in magnitude, nor do they play any relevant role in the discussion below, and so they are omitted for the sake of simplicity. The main contributor to the destruction of $-\overline{\langle u_{1}^{\prime }u_{2}^{\prime }\rangle }$ is the drop in production $-\overline{P}_{12}$, which can be traced back to a deficit on the pressure–strain correlation in the budget equation for $\overline{\langle u_{2}^{\prime }u_{2}^{\prime }\rangle }$ (figure 15c). Similarly, the decline of the streaks is the consequence of a lower production $\overline{P}_{11}$ (figure 15e), also caused by the drop in $-\overline{\langle u_{1}^{\prime }u_{2}^{\prime }\rangle }$.

The sequence of events is similar farther away from the wall, as seen in figure 15(b,d,f) for $x_{2}^{\ast }\in [0.2,0.3]$. The main sink of tangential Reynolds stress arises from the turbulent production $-\overline{P}_{12}$. The reduction in $-\overline{P}_{12}$ is connected to the lower pressure–strain correlation $\overline{PS}_{22}$ in the budget equation of $\overline{\langle u_{2}^{\prime }u_{2}^{\prime }\rangle }$ akin to the situation described for the buffer region. The decay of the streaks is similarly governed by the drop in the production of streamwise Reynolds stress $\overline{P}_{11}$, with some additional contribution by the turbulent diffusion $\overline{T}_{11}$.

The process of Reynolds stress reduction is then sequentially described by (i) the growth of the spanwise boundary layer $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$, which (ii) inhibits the redistribution of energy to $\langle u_{2}^{\prime }u_{2}^{\prime }\rangle$ via pressure–strain correlation, followed by (iii) the weakening of the production of tangential Reynolds stress, which (iv) eventually causes the drop in $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$. The terms involved at each step of the process are summarised in figure 16. A similar effect has been observed in transitional boundary layer flows subjected to spanwise wall oscillations (Hack & Zaki Reference Hack and Zaki2015). Our findings are consistent with the previous theory on transversely strained boundary layer flows by Moin et al. (Reference Moin, Shih, Driver and Mansour1990) and Coleman et al. (Reference Coleman, Kim and Le1996a) and extends the results to the outer layer of wall-bounded turbulence. The leading role of $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$ in the drop of $-\langle u_{1}^{\prime }u_{2}^{\prime }\rangle$ is also consistent with the structural model promoted in § 3.4, where it was argued that the deficient transport of momentum by the streamwise rolls has its origin in the displacement among fluid layers induced by $\unicode[STIX]{x2202}\langle u_{3}\rangle /\unicode[STIX]{x2202}x_{2}$.

Figure 16. Summary of the sequential process of Reynolds stress reduction from the sudden imposition of a spanwise pressure gradient up to the final decrease in tangential Reynolds stress. Time goes from left to right. See text for details.

4.1 Wall models

At the coarse near-wall grid resolutions of WMLES, the usual no-slip condition ceases to produce an accurate estimate of the momentum drain at the wall. Hence, wall models are responsible for estimating the wall-shear stress. The LES equations are integrated in time using the wall-shear stress provided by the wall model as a Neumann boundary condition instead of the no-slip condition. The kinematic no-penetration condition is maintained for the impermeable walls of the channel. Three wall models are investigated in the present work: the equilibrium wall model by Kawai & Larsson (Reference Kawai and Larsson2012), and the non-equilibrium wall models by Park & Moin (Reference Park and Moin2014) and Yang et al. (Reference Yang, Sadique, Mittal and Meneveau2015). We briefly summarise the main characteristics of each model and the modifications performed in the present work with respect to their original formulations.

The model by Yang et al. (Reference Yang, Sadique, Mittal and Meneveau2015) accounts for non-equilibrium effects while retaining a moderate complexity. This model assumes a parametric velocity profile in the near-wall region, where the coefficients are determined by enforcing a set of physical constraints. These include the continuity of the profile, the LES matching condition at a specified wall distance, and the compliance with the vertically integrated momentum equation, among others. The model is usually referred to as the integral wall model (IWM), since the momentum integral constraint is crucial in accounting for non-equilibrium effects. In the original formulation, the wall-model input is averaged in time to regularise the wall-shear stress, which otherwise was found to cause numerical instabilities. In the present study, given the statistically unsteady nature of the flow, the time averaging is replaced by spatial averaging along wall-parallel planes. To comply with the outer LES equations, we modify the original formulation by Yang et al. (Reference Yang, Sadique, Mittal and Meneveau2015) to account for the spanwise pressure forcing.

The non-equilibrium wall model by Park & Moin (Reference Park and Moin2014, Reference Park and Moin2016) solves the full Reynolds-averaged Navier–Stokes equations on a separate near-wall mesh with a mixing-length-type eddy-viscosity closure that dynamically accounts for the resolved portion of the turbulence in the wall-model domain. This formulation is the most comprehensive amongst the considered wall-stress models, and accounts for non-equilibrium effects embedded into the original Navier–Stokes equations. Herein, this model is termed NEQWM. In order to avoid an overprediction of the skin friction, the resolved turbulent stress is evaluated on the fly, and it is then subtracted from the modelled stress. Similarly to the IWM, the formulation by Park & Moin (Reference Park and Moin2014) was adjusted to account for the spanwise pressure forcing. This turned out to be particularly important in order to provide the required dominant balance in the momentum conservation equation for the initial times of the transient.

Lastly, the equilibrium wall model (EQWM) of Kawai & Larsson (Reference Kawai and Larsson2012) is derived from the NEQWM by retaining only the wall-normal diffusion terms. The model involves a simple ordinary differential equation, which is solved along the wall-normal direction on each wall face (Wang & Moin Reference Wang and Moin2002). Consistent with the one-dimensional nature of the model, the spanwise mean pressure gradient vector was projected to the local flow direction at the matching location, and this was added to the EQWM equation as a momentum source term. A similar term was added to the energy equation for consistency.

4.2 Numerical set-up

The codes used for wall-modelled calculations are different from the solver presented for DNS, mainly because the wall models were conveniently available in other well-validated LES codes. The calculations using the NEQWM and EQWM are conducted using the code CharLES, which is an unstructured-grid finite-volume LES code for compressible flows developed at the Center for Turbulence Research and currently maintained by Cascade Technologies, Inc. The nominal spatial accuracy of the code is second order, but the reconstruction scheme upgrades to a fourth-order accuracy on uniform Cartesian grids (Herrmann Reference Herrmann2010; Khalighi et al. Reference Khalighi, Ham, Nichols, Lele and Moin2011). The dynamic Smagorinsky model (Moin et al. Reference Moin, Squires, Cabot and Lee1991; Lilly Reference Lilly1992) is used as subgrid-scale (SGS) model in the filtered conservation equations. The bulk Mach number is fixed at 0.2 for comparison with the incompressible DNS solution.

For the IWM, we use the LESGO solver (LESGO 2019). The code solves the incompressible filtered Navier–Stokes equations in a half-channel with a staggered grid, using a pseudo-spectral approach in the wall-parallel directions and a second-order central finite-difference scheme in the wall-normal direction. The scale-dependent Lagrangian-dynamic Smagorinsky model is used as SGS model (Bou-Zeid, Meneveau & Parlange Reference Bou-Zeid, Meneveau and Parlange2005). One of the most important impediments of the wall models considered above is that they rely explicitly or implicitly on a Reynolds-averaged parametrisation of the flow, usually through an eddy viscosity, and their coefficients are adjusted by assuming fully developed turbulence that is statistically in equilibrium. By construction of these models, the imposition of additional mean shear is accompanied by an increment in the magnitude of the Reynolds stresses (see the reviews by Piomelli & Balaras (Reference Piomelli and Balaras2002) and Bose & Park (Reference Bose and Park2018)), which might not be case under non-equilibrium conditions.

The LES grid resolution is uniform in the three spatial directions and equal to $(\unicode[STIX]{x1D6E5}_{1}^{+},\unicode[STIX]{x1D6E5}_{2}^{+},\unicode[STIX]{x1D6E5}_{3}^{+})=(180,60,133)$ or $(\unicode[STIX]{x1D6E5}_{1}^{\ast },\unicode[STIX]{x1D6E5}_{2}^{\ast },\unicode[STIX]{x1D6E5}_{3}^{\ast })=(0.2,0.06,0.14)$. The size of the computational domain is $(L_{1}^{\ast },L_{2}^{\ast },L_{3}^{\ast })=(8\unicode[STIX]{x03C0},2,3\unicode[STIX]{x03C0})$, which yields a total of 265 980 grid cells distributed as $(N_{1},N_{2},N_{3})=(130,31,66)$, in the streamwise, wall-normal and spanwise directions, respectively. The internal grids for EQWM and NEQWM have 30–40 cells stretched along the wall-normal direction. Additionally, the NEQWM shares the same wall-parallel resolution as the LES grid. The wall-normal exchange between the wall model and the LES is located at the centroids of the third grid cell away from the wall, $x_{2}^{\ast }\approx 0.16$.

The calculations are initialised with a 2-D channel flow in a statistically steady state at $Re_{\unicode[STIX]{x1D70F}}\approx 1000$. Then, a spanwise pressure gradient of $\unicode[STIX]{x1D6F1}=10$ is applied to induce a cross-stream shear layer, as in § 2. The transverse mean pressure gradient selected is relatively low in order to mimic the fact that, at high Reynolds numbers, the near-wall layer is in a quasi-equilibrium state, as discussed in § 3.2. The simulations are run for one eddy turnover time based on the streamwise friction velocity and channel half-height, $t^{\ast }\approx 1$. The results are averaged in the homogeneous directions and among runs starting from 10 uncorrelated initial conditions.

4.3 Results and discussion

Figure 18 shows the evolution of the streamwise and spanwise mean wall-stress components denoted by $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ and $\langle \unicode[STIX]{x1D70F}_{3}\rangle$, respectively. We discuss first the predictions for $\langle \unicode[STIX]{x1D70F}_{3}\rangle$. A general observation from figure 18(a) is that the NEQWM produces a fairly accurate prediction of $\langle \unicode[STIX]{x1D70F}_{3}\rangle$ throughout the transient. For short times ($t^{\ast }<0.1$), the NEQWM predictions are closely followed by those from IWM, while the EQWM results in 50 % to 25 % underprediction of $\langle \unicode[STIX]{x1D70F}_{3}\rangle$ throughout the initial transient. The EQWM and the IWM still capture correctly the growth rate of $\langle \unicode[STIX]{x1D70F}_{3}\rangle$ for $t^{\ast }\gtrsim 0.1$. For $t^{\ast }\approx 1$, the errors by NEQWM and IWM are roughly 2 %, whereas the error for the EQWM is 10 %. As a reference, the laminar response of the flow is also included in figure 18(a), which shows that the spanwise wall stress agrees with the laminar solution for $t^{\ast }<0.1$.

Figure 18. Evolution of (a) the mean spanwise wall stress $\langle \unicode[STIX]{x1D70F}_{3}\rangle$ and (b) the mean streamwise wall stress $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ for WMLES and DNS. The stresses are normalised by the value of the initial 2-D streamwise wall stress $\langle \unicode[STIX]{x1D70F}_{1}\rangle _{,2D}$. The lines and symbols are:

, DNS; ▫, WMLES (NEQWM); ▵, WMLES (IWM); ○, WMLES (EWQM); – ⋅ – ⋅ –, laminar solution. Note that the variations in the vertical axis of panel (a) are up to 70 % of $\langle \unicode[STIX]{x1D70F}_{1}\rangle _{,2D}$, while those in panel (b) are only up to 15 %.

The evolution of $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ is plotted in figure 18(b). Note that the variations in $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ are only up to 10 % and well below the changes undergone by $\langle \unicode[STIX]{x1D70F}_{3}\rangle$, which are up to 70 %. The EQWM and the IWM predict the wall stress throughout the transient within 5 % and 2 % error, respectively. The NEQWM predicts a relatively faster variation in $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ for $t^{\ast }\lesssim 0.4$ compared to IWM and EQWM, with deviations from the DNS up to 7 %. In all cases, the errors decay as time advances. As expected, none of the wall models is able to reproduce the initial reduction in $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ for $t^{\ast }\lesssim 0.4$. Such a decrease in the streamwise wall-stress component is the result of the complex flow dynamics discussed in § 3. The wall models investigated are based on the eddy-viscosity assumption; increasing shear rates in the flow result in additional strain rates. Hence, it comes as no surprise that WMLES consistently exhibits an approximately monotonic increase in $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ after the sudden spanwise pressure gradient is applied due to the additional transverse straining of the flow in the near-wall region.

Evolution of the wall-shear angle, defined as $\unicode[STIX]{x1D6FE}_{w}=\tan ^{-1}(\langle \unicode[STIX]{x1D70F}_{3}\rangle /\langle \unicode[STIX]{x1D70F}_{1}\rangle )$, is shown in figure 19(a). The performance of the wall models resembles the trends reported for $\langle \unicode[STIX]{x1D70F}_{3}\rangle$. This similarity is easily understood by noting that the relative time variations in $\langle \unicode[STIX]{x1D70F}_{1}\rangle$ are modest compared to the variations in $\langle \unicode[STIX]{x1D70F}_{3}\rangle$. The development of the mean spanwise velocity over one eddy turnover time is shown in figure 19(b). All the WMLES considered provide an excellent prediction of the boundary layer growth. The spanwise velocity profile develops its own logarithmic region for $t^{\ast }>0.6$, although the slope is substantially smaller than that of equilibrium channel flows. The agreement in the spanwise profile is observed in the turbulent flow region, where contributions from the SGS models and the wall models are expected to play a role in the mean spanwise momentum balance. These findings highlight the capability of current WMLES and SGS models to predict the mean spanwise velocity profile that arises in response to mild transverse pressure perturbations. Although not shown, the mean streamwise velocity undergoes only minor changes in time from its initial 2-D state, and good agreement is also found between DNS and WMLES.

Figure 19. (a) Evolution of the wall shear-stress angle, $\unicode[STIX]{x1D6FE}_{w}=\text{tan}^{-1}(\langle \unicode[STIX]{x1D70F}_{3}\rangle /\langle \unicode[STIX]{x1D70F}_{1}\rangle )$. (b) Mean spanwise velocity at $t^{\ast }=0$, 0.21, 0.405, 0.615, 0.825 and 1.02 (from bottom to top). The lines and symbols are:

, DNS; ▫, WMLES (NEQWM); ▵, WMLES (IWM); ○, WMLES (EWQM); – ⋅ – ⋅ –, laminar solution; – – –, standard logarithmic law, $\langle u_{3}\rangle ^{+}=(1/0.41)\ln (x_{2}^{+})+5.2$.

In summary, our results show that the expected errors in WMLES under moderate non-equilibrium 3-D effects are reduced for increasing degree of modelling complexity. However, factors such as intricacy in model implementation or computational cost can favour the adoption of the simplest wall models for some flow configurations. Future efforts should be devoted to enhance the capabilities of wall models to accurately capture the flow physics in the presence of strong non-equilibrium effects.