2.1. Numerical method

Plane Couette flow is numerically simulated through discretization of the incompressible, constant viscosity Navier–Stokes (N–S) equations in a Cartesian coordinate box. In LES, a general filter with size $\varDelta _c$ is applied to the N–S equation, which results in

(2.1a,b)\begin{equation} \frac{\partial \widetilde{u_i}}{\partial t} + \frac{\partial \widetilde{u_i} \widetilde{u_j}}{\partial x_j} ={-} \frac{1}{\rho} \frac{\partial \tilde{p}}{ \partial x_i} + \nu \frac{\partial^2 \widetilde{u_i}}{\partial x_j^2} - \frac{\partial T_{ij}}{\partial x_j}, \quad \frac{\partial \widetilde{u_i}}{\partial x_i} = 0. \end{equation}

Here $\widetilde {u_i}$ is the filtered velocity, $\rho$ the density and $\tilde {p}$ the filtered pressure. Additionally, $T_{ij}= \widetilde {u_iu_j}- \widetilde {u_i}\widetilde {u_j}$ denotes the effect of unresolved scales on the resolved-scale motion, which is modelled by an SGS model.

The computational box is set up with dimensions $L_x \times L_y \times L_z$ with periodic boundary conditions in both the streamwise ($x$-) and spanwise ($z$-) directions, and corresponding mesh $N_x \times N_y \times N_z$. The mesh is uniform in the $x$- and $z$-directions. In the wall-normal $y$-direction, the mesh is stretched for WRLES but uniform for WMLES. The stretching function employed is that used by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014), namely an error function mapping $\textrm {erf}(\alpha \xi /2)/\textrm {erf}(\alpha /2)$ with $\alpha =4$ and $\xi$ varying in $[-1,1]$. Corresponding velocity components are $u$, $v$ and $w$ in the $x$-, $y$- and $z$-directions, respectively. Moving both walls in the streamwise direction with equal and opposite speeds $\pm U_c$ generates the flow.

The governing equations are solved using a third-order Runge–Kutta method for time evolution with a fractional-step method for marching Eulerian velocities. The code is an extension of that used by Inoue & Pullin (Reference Inoue and Pullin2011). In solving the velocity-update equation, only the wall-normal viscous term is treated implicitly and the wall-parallel viscous terms are treated explicitly. The skew-symmetric form of the nonlinear term is employed. For spatial discretization, a Fourier spectral method is used in the streamwise ($x$-) direction, while a fourth-order finite-difference method on a staggered grid is used in the other two directions. The pressure Poisson equation reduces to a sequence of two-dimensional modified Helmholtz equations in the $y$–$z$ plane and is further reduced to a one-dimensional linear system by applying a Fourier transform in the spanwise $z$-direction. A septa-diagonal matrix solver (from LAPACK) is used for solving the resulting equations in the wall-normal $y$-direction. For code parallelization, we use MPI in only the streamwise direction.

2.2. Subgrid-scale model

In the present LES, unresolved subgrid motions are represented by the stretched-vortex (SV) SGS model (Misra & Pullin Reference Misra and Pullin1997; Chung & Pullin Reference Chung and Pullin2009). More specifically, in a numerical cell, the unresolved subgrid motion is modelled by a stretched-spiral vortex (Lundgren Reference Lundgren1982) within each cell with direction vector $ {\boldsymbol e}^v$, resulting in the subgrid stress,

(2.2)\begin{equation} T_{ij} =(\delta_{ij} - {\boldsymbol e}_i^v {\boldsymbol e}_j^v) K, \end{equation}

where $K$ is the subgrid kinetic energy. This is computed from an integral of the stretched-spiral vortex energy spectrum $E(k)$ as

(2.3)\begin{equation} K=\int_{k_c}^\infty E(k) \,\textrm{d}k = \frac{\mathscr{K}'_0}{2} \varGamma\left[{-}1/3, \frac{2\nu k_c^2}{3|\tilde{a}|}\right]. \end{equation}

Here $k_c={\rm \pi} /\varDelta _c$ is the cutoff wavenumber, $\tilde {a}= {\boldsymbol e}_i^v {\boldsymbol e}_j^v \tilde {S}_{ij}$ is the stretching along the subgrid vortex with $\tilde {S}_{ij}$ the local resolved-scale, rate-of-strain tensor. In the present implementation, the ${\boldsymbol e}_j^v$ are aligned with the principal extensional eigenvector of $\tilde {S}_{ij}$. The grouped constant $\mathscr {K}'_0$ is calculated dynamically from the resolved-scale velocity. The model implementation is dynamic with no adjustable parameters except that $\varDelta _c$ is taken as the local cell size. For details, see Misra & Pullin (Reference Misra and Pullin1997), Voelkl et al. (Reference Voelkl, Pullin and Chan2000), Chung & Pullin (Reference Chung and Pullin2009) and Lundgren (Reference Lundgren1982). In DNS and WRLES, non-slip boundary conditions are applied on both walls.

2.3. Virtual wall model for WMLES

In WRLES, the skin-friction velocity is calculated directly from the wall-normal gradient of the wall-parallel velocity evaluated at the wall. A one-sided, four-point stencil is used as described by (2.12) of Cheng et al. (Reference Cheng, Pullin, Samtaney, Zhang and Gao2017). Because the near-wall flow is not resolved in WMLES, the local skin-friction velocity must be otherwise determined.

For WMLES, we use the two-tier, VWM (Chung & Pullin Reference Chung and Pullin2009). First, an assumption of local inner scaling on $u_\tau$ and $\delta _\nu$ is used in the wall-normal-average (over a thin layer $0< y< h_0$) of the wall-parallel, streamwise momentum equation. This gives an ordinary differential equation (ODE) describing $u_\tau (x,z,t)$ at each wall point. Each ODE is coupled to the outer-flow LES through the instantaneous flow state at the ‘virtual-wall’ interface at $y=h_0$ of the wall-modelled layer and the outer-flow LES. This locally determined $u_\tau$ is then used together with a log-linear-based approximation to provide a slip velocity at the virtual wall $y=h_0$, with dynamical evaluation of the local Kármán constant. The virtual-wall height parameter $h_0$ is subgrid and must be smaller than the wall-adjacent, wall-normal cell size $\Delta y$. Presently, we use $h_0 = 0.18\,\Delta y$ as suggested by Chung & Pullin (Reference Chung and Pullin2009) following a sensitivity study. Near-wall RANS is not required and $u_\tau (x,y,t)$ is determined dynamically. Hence $h_0^+ \equiv h_0\,\nu /u_\tau$ is determined dynamically.

In the second part, a streamwise slip-velocity boundary condition at $y=h_0$ is calculated, with the velocity relative to the wall given as

(2.4)\begin{equation} \tilde{u} (h_0^+) = u_\tau \left( \frac{1}{\kappa} \log \left( \frac{h_0^+}{h_\nu^+} \right) +A \right), \end{equation}

with

(2.5a,b)\begin{equation} A = h_v^+{-} \frac{ \log h_\nu^+} {\kappa} , \quad \kappa = \frac{\gamma K^{1/2}}{2 ({-}T_{xy}/ u_\tau)} . \end{equation}

Here $\gamma$ is a constant parameter which is estimated via Townsend's structure parameter (Chung & Pullin Reference Chung and Pullin2009). Additionally, $h_\nu ^+ = 10.2$ is an empirical parameter which is essentially the intersection point between the near-wall, linear and log profiles. Similar wall-normal averaging of the continuity equation provides a wall-normal velocity at $y=h_0$. These velocity components provide virtual-wall, Dirichlet boundary conditions for the outer-flow LES. See Chung & Pullin (Reference Chung and Pullin2009) and Cheng et al. (Reference Cheng, Pullin and Samtaney2015) for further details. The WMLES code has been verified in previous studies by comparison with DNS and experiments on flat-plate boundary layer flow (Inoue & Pullin Reference Inoue and Pullin2011) and separation/reattachment flow (Cheng et al. Reference Cheng, Pullin and Samtaney2015). In subsequent sections, we will provide further code verification for PC flow.

3.1. Cases implemented

For PC flow, the single driving dimensionless parameter is $Re_c$, and $Re_\tau (Re_c)$ must be determined. Other parameters quantify the size and aspect ratio of the simulation domain and the grid resolution. In figure 1(a), we show a schematic of PC flow with two parallel plates, separated by $L_y=2h$, moving in the $\pm x$-directions with equal speed $U_c$. A sketch of a $(y\text {--}z)$ cross-section of the colour-contoured, streamwise averaged, wall-normal velocity is shown in figure 1(a). A stream-wise velocity profile, averaged in both the streamwise and spanwise directions is shown in figure 1(c) to illustrate the distribution of mass flux for PC flow. The present code implementation guarantees that the mass flux in both wall-parallel directions is always zero, which implies a dynamic mean pressure gradient fluctuating around zero. A test case that fixes a zero mean pressure gradient while allowing fluctuating mass flux at $Re_\tau \approx 500$ was also carried out. Little difference was observed on the present diagnostic results between the two methods. We will discuss the LES results in terms of $Re_\tau$ variation as this provides a clearer contact with other canonical flows such as Poiseuille-channel and pipe flow.

Figure 1.

Sketch of PC flow. (a) Two parallel walls with separation distance $L_y = 2h$. Upper and lower walls move with velocity $\pm U_c$. (b) Visualizing rolls via contours of streamwise averaged spanwise velocity components on a $(y\text {--}z)$ plane. (c) Distribution of streamwise mean velocity (averaged in both streamwise and spanwise directions).

Verification begins with the DNS of Lee & Moser (Reference Lee and Moser2018) at low Reynolds number $Re_c = 4000$ ($Re_\tau \approx 220$) within a domain of $(L_x, L_y, L_z) = (20{\rm \pi} h, 2h, 5{\rm \pi} h)$. This domain is also used later for some LES. We then implement WRLES at $Re_c = 10\,000$ ($Re_\tau \approx 500$) and $Re_c = 22\,000$ ($Re_\tau \approx 1000$) to demonstrate the SGS model capability for capturing turbulence in both the resolved, near-wall and the outer modelled domain sectors. Wall-resolved LES of two other flows with moderately large $Re_c=44\,000$, 90 000 are then discussed.

Wall-modelled LES at $Re_c= 90\,000$, with different meshes, overlaps the WRLES and provides assessment of the VWM performance. At these $Re_c$, the nearest wall-normal grid point approaches the lower limit of the log region. Wall-modelled LES with $Re_c = 3\times 10^5$, $10^6$, $3\times 10^6$ and $10^7$ are then described. All cases are listed in table 1 for WRLES and table 2 for WMLES. We list prescribed parameters $Re_c$, mesh resolution $N_x \times N_y \times N_z$, the calculated $Re_\tau$ and the mesh size in near-wall scaling against $\delta _\nu$: $\Delta x^+$ and $\Delta z^+$. In the wall-normal direction, we show the minimal $\Delta y^+_{min}$ for WRLES and the uniform $\Delta _y^+$ for WMLES. The time duration used for averaging results is also shown, defined as $T u_\tau /h$, which represents a streamwise distance travelled with speed $u_\tau$. In table 2, for $Re_c=9\times 10^4$, two different resolutions are shown: $768 \times 96 \times 192$ and $3072 \times 96 \times 768$ referred to subsequently as coarse-mesh and fine-mesh WMLES, respectively. Results using both resolutions will be discussed later in relation to resolution requirements for the WMLES capture of spanwise roll structures.

Table 1.

Cases implemented in wall-resolved LES. Here, $N_x, N_y,N_z$ are mesh resolutions in the streamwise, wall-normal and spanwise directions, respectively. Additionally, $\Delta x^+, \Delta y^+,\Delta z^+$ are cell sizes scaled on $\delta _\nu$ and $T\,u_\tau /h$ is the time window for time averaging.

Table 2.

Cases implemented for wall-modelled LES. For the key, see table 1. Here, $Tu_\tau /h \approx 50.0$ for all cases. In the discussion, WMLES with $768 \times 96 \times 192$ will be referred as ‘coarse-mesh’ while WMLES with $3072 \times 96 \times 768$ will be denoted as ‘fine-mesh’.

3.2. Diagnostics

Owing to the existence of spanwise rolls, the spanwise direction is not homogeneous, which leads to distinct averaging methods. Two averages are defined, the first including streamwise, spanwise and time averaging, and the second that excludes the spanwise average to recognize the roll structure. We define the following two averaging operators for any instantaneous field $\phi (x,y,z,t)$:

(3.1a,b)\begin{equation} \bar{\phi}(y) = \frac{1}{N_t} \frac{1}{N_x} \frac{1}{N_z} \sum_{z,x,t} \phi(x,y,z,t), \quad \tilde{\phi}(y,z) = \frac{1}{N_t} \frac{1}{N_x} \sum_{x,t} \phi(x,y,z,t), \end{equation}

where $N_t$ is the number of time steps. From instantaneous flow fields $u(x,y,z,t)$, $v(x,y,z,t)$ and $w(x,y,z,t)$, noting that $\bar {v}(y)=0$ and $\bar {w}(y)=0$, this allows computation of one-dimensional profiles $\bar {u}(y)$ and two-dimensional fields $\tilde {u}(y,z)$, $\tilde {v}(y,z)$ and $\tilde {w}(y,z)$.

The fluctuation part of the field can also be defined in two ways:

(3.2a,b)\begin{equation} \phi'(x,y,z,t) = \phi(x,y,z,t) - \bar{\phi}(y), \quad \phi^\bullet (x,y,z,t) = \phi(x,y,z,t) - \tilde{\phi}(y,z). \end{equation}

Further applying the averaging operators to velocity fluctuation components results in distinct Reynolds-stress distributions. The ‘$\bar {\phi }$’ operator produces functions of $y$ only, $\overline {u_i'u_j'}(y)$ and $\overline {u_i^\bullet u_j^\bullet }(y)$, while the ‘$\tilde {\phi }$’ operator generates fluctuation fields that are functions of $(y,z)$, namely $\widetilde {u_i'u_j'}(y,z)$ and $\widetilde {u_i^\bullet u_j^\bullet }(y,z)$ where $u_i$ and $u_j$ with $i,j=1,2,3$ could be velocity components $u$, $v$ or $w$.

We refer to both inner and outer length scaling $y^+ = y/\delta _\nu$ and $y/h$, respectively, and corresponding velocity scales $u_\tau$ and $U_c$. We use the following notation for inner scaling:

(3.3a–c)\begin{equation} U^+{=} \frac{ \bar{u}} {u_\tau}, \quad \left( u'u'\right)^+{=} \frac{ \overline{u'u'}}{u_\tau^2}, \quad \left( u^\bullet u^\bullet \right)^+{=} \frac{ \overline{u^\bullet u^\bullet }}{u_\tau^2}, \end{equation}

which is also applicable to other Reynolds stresses. Additional scalings, such as $\widetilde {u_i'u_j'}(y,z) / u_\tau ^2$ and $\widetilde {u_i^\bullet u_j^\bullet }(y,z)/ u_\tau ^2$, will be described as required. In what follows, profiles of the mean velocity will refer to the mean-velocity magnitude relative to a moving wall.

3.3. Verification of DNS and wall-resolved LES

Figure 2 shows DNS verification at $Re_c=4400$, compared with DNS at the same $Re_c$ by Lee & Moser (Reference Lee and Moser2018). Streamwise mean-velocity $U^+$ and scaled velocity gradient $\varXi = y^+ \textrm {d}U^+/{\textrm {d} y}^+$ versus $y^+$ are shown in figure 2(a) and scaled turbulent intensities $(u'u')^+$, $(v'v')^+$ and $(w'w')^+$ are shown in figure 2(b). We find that the two DNS match reasonably well, notably on $\varXi$ which clearly shows an uptilt near the centreline. This tendency is characteristic of medium-Reynolds-number PC flows as described by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014).

Figure 2.

Verification with DNS at $Re_\tau \approx 220$. (a) Mean velocity profile $U^+$ and scaled velocity gradient $\varXi$. (b) Turbulence intensities $(u'u')^+$, $(v'v')^+$ and $(w'w')^+$. Lines, data from Lee & Moser (Reference Lee and Moser2018). Lines with symbols, present DNS results.

Beyond DNS, we compare in figure 3 our LES at $Re_\tau \approx 500$ and $1000$ with the DNS data from Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014). DNS data at $Re_\tau =502.6$ from Lee & Moser (Reference Lee and Moser2018) are also included. Figure 3(a) shows streamwise velocity profiles of $U^+$ versus $y^+$. The LES generally agrees with DNS at both $Re_\tau$ values. For $y^+\sim Re_\tau$ with $Re_\tau \approx 500$, differences between our LES and the two DNS are similar to the differences between the two reference DNS results. At $Re_\tau \approx 1000$, our LES matches DNS data (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014), especially near the centreline as shown in the lower-right close-up inset. Near $y^+$ around $20\sim 40$, a small but systematic difference is apparent between DNS and LES, as shown in the upper-left close-up inset. This can be ascribed to the SGS model effect on the basic mean velocity for PC flow. The wall-normal averaged Navier–Stokes equations for statistically stationary PC flow show that $\textrm {d} U^+/ {\textrm {d} y}^+ - (u'v')^+ = 1$. In LES, the SGS model contributes to a part of $(u'v')^+$ and so directly influences the gradient of the mean velocity. This effect can be found in a plot of $\varXi$ in figure 3(b), where $\varXi$ from the LES is generally lower than that obtained by DNS at around $y^+ = 10$. In other regions of the $\varXi$ profile, the WRLES follows the trend of the corresponding reference DNS fairly well.

Figure 3.

Verification of LES at $Re_\tau \approx 500$ and $1000$. (a) Velocity profiles $U^+$. (b) Scaled velocity gradient $\varXi$. Dashed, DNS of Lee & Moser (Reference Lee and Moser2018). Dash–dotted and dash–double dotted, DNS of Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014). Line with $\blacktriangledown$ (red), LES of $Re_\tau = 500$. Line with $\blacktriangle$, LES of $Re_\tau = 1000$.

Figure 4 shows a comparison of the three diagonal and one off-diagonal components of the Reynolds stresses with DNS. All three turbulent intensities between two cases show satisfactory agreement. The behaviour of $(u'v')^+$ agrees with the DNS data, starting from zero and increasing asymptotically to unity in the velocity-log region.

Figure 4.

Comparison of LES with DNS at $Re_\tau \approx 500$ and $1000$: (a) $(u'u')^+$; (b) $(v'v')^+$; (c) $(w'w')^+$; (d) $(u'v')^+$. For the key, see figure 3.

A kinetic-energy diagnostic can be defined for a velocity component $\phi$ by

(3.4)\begin{equation} e_\phi (y) = \frac{1}{N_z} \sum_{z} \left( \frac{ \tilde{\phi}(y,z) }{U_c} \right) ^2 \end{equation}

leading to $e_u(y)$, $e_v(y)$ and $e_w(y)$, which are energy densities in the wall-normal direction per unit span, scaled using the moving-wall velocity $U_c$. Both $e_v$ and $e_w$ are measures of the roll energy magnitude. We compare wall-normal profiles of $e_v$ and $e_w$ from the WRLES with DNS data (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014). In figure 5(a), both $e_v$ and $e_w$ scaled on $U_c^2$ show similar trends to DNS data with maximum discrepancy near the centreline for $e_v$ of approximately $10\,\%$. We also plot $e_v + e_w$, which can be interpreted as a measure of the energy of the rolls that exhibit structured fluid motion in the plane perpendicular to the streamwise direction. Figure 5(b) shows profiles of the four different quantities for WRLES and DNS. At the flow centreline, the turbulence intensities are approximately $1\,\%$ of $U_c^2$, while the roll energy, $e_v+e_w$, is approximately $0.01\,\%$ of $U_c^2$. The WRLES represented by lines agrees quite well with the reference DNS data, which demonstrates its capability of capturing subtle physical properties of the spanwise rolls.

Figure 5.

Verification of roll energy at $Re_\tau \approx 1000$. (a) Roll kinetic energy components $e_v$, $e_w$. (b) Comparison of four quantities: $e_u$, $(u'u')^+$, $(v'v')^+ +(w'w')^+$ and $e_v+e_w$. Lines, present LES results; symbols, reference DNS by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014).

4.1. Velocity profiles

Figure 6(a) shows the scaled streamwise velocity $U^+$ of four WRLES. Also plotted is a log law $U^+ = \ln (y^+)/\kappa + B$ with $\kappa = 0.4$ and $A=5.0$. The WRLES shows a consistent log-region across the $Re_c$ range. This is also observed in the $\varXi -y^+$ plot of figure 6(b).

Figure 6.

Mean velocity profile of wall-resolved LES data. (a) Velocity profiles $U^+$. (b) Scaled velocity gradient $\varXi$. Here, $\blacktriangle$ (red), $Re_\tau \approx 500$; $\blacktriangledown$ (blue), $Re_\tau \approx 1000$; $\blacktriangleleft$, $Re_\tau \approx 1800$; $\blacktriangleright$ (red), $Re_\tau \approx 3600$.

4.2. Fluctuations

Figure 7 shows the averaged Reynolds stress $(u'u')^+$ and $(w'w')^+$ in inner scaling. The four WRLES up to $Re_\tau \approx 3,600$ show consistent trends. The inner scaling of the two intensities show a lack of universality in the general shape, in agreement with the observation of Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014). The wall-normal position of the first peak, at $y^+ \approx 15$ for all cases, is in general agreement with other canonical flows including channel flow data by Lee & Moser (Reference Lee and Moser2015), pipe flow data by Willert et al. (Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017) and also Taylor–Couette flow (Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016; Cheng, Pullin & Samtaney Reference Cheng, Pullin and Samtaney2020). Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014) emphasize the presence of a secondary peak in $(u'u')^+$ at $y/h\approx 0.33$ in DNS-PC data at $Re_\tau \approx 1000$. For WRLES, this secondary peak is presently observed at similar locations for all higher $Re_\tau$ flows, as shown in figure 8(a,b) for outer scaling plots of $(u'u')^+$ and $(w'w')^+$. For PC flow, this secondary peak clearly emerges at $Re_\tau \approx 1000$, which is similar to the range of $Re_\tau \approx 2000$ found in Taylor–Couette flow (Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016). These $Re_\tau$ values are generally much smaller than those observed for the first appearance of a secondary peak in canonical flows, for example $Re_\tau =19\,918$ in the pipe flow experiments by Willert et al. (Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017). A plausible hypothesis is that the secondary peak in Couette flow is probably related to the existence of large-scale rolls which effectively enhance the turbulent mixing and thus the energy transport.

Figure 7.

Turbulent intensities in inner scale $y^+$ for wall-resolved LES cases: (a) $(u'u')^+$; (b) $(w'w')^+$. Here, $\blacktriangle$ (red), $Re_\tau \approx 500$; $\blacktriangledown$ (blue), $Re_\tau \approx 1000$; $\blacktriangleleft$, $Re_\tau \approx 1800$; $\blacktriangleright$ (red), $Re_\tau \approx 3600$.

Figure 8.

Turbulent intensities in outer scaling $y/h$ for wall-resolved LES: (a) $(u'u')^+$; (b) $(w'w')^+$; (c) $(u^\bullet u^\bullet )^+$; (d) $(w^\bullet w^\bullet )^+$. Here, $\blacktriangle$ (red), $Re_\tau \approx 500$; $\blacktriangledown$ (blue), $Re_\tau \approx 1000$; $\blacktriangleleft$, $Re_\tau \approx 1800$; $\blacktriangleright$ (red), $Re_\tau \approx 3600$.

To understand fluctuations generated by the small-scale turbulence only, we plot $\overline {u^\bullet u^\bullet }$ in the form $(u^\bullet u^\bullet )^+$ in figure 8(c) and $(w^\bullet w^\bullet )^+$ in figure 8(d). Figure 5(b) indicates that for the spanwise component, $e_w \approx 1\,\%\, \overline {w'w'}$. Because $\overline {w^\bullet w^\bullet } = \overline {w'w'} - e_w$, it is clear that wall-normal profiles of $\overline {w^\bullet w^\bullet }$ will be similar to those for $\overline {w'w'}$, in both inner and outer scaling. In contrast, the streamwise rolls do impose a strong imprint on streamwise turbulent intensities. The profiles of $(u^\bullet u^\bullet )^+$ are generally rather smaller in magnitude than $(u'u')^+$ and do not show a monotonic increasing tendency at fixed $y/h$ with increasing $Re_\tau$ in the outer flow region.

5.1. Effect of spanwise variation: $Re_\tau \approx 500$

In figure 9, we show the averaged flow field in the $y\text {--}z$ plane. Colour-coded contours of the streamwise velocity $\tilde {u}$ in figure 9(a), with mean $(y\text {--}z)$-plane streamlines added to show the averaged circulatory motion within rolls, clearly indicate a strong spanwise variation associated with roll motion. Figure 9(b) shows the variation of the local skin-friction velocity along the spanwise $z$ direction. We define local skin-friction velocities $u_{\tau,b}(z)$ and $u_{\tau,t}(z)$ on the bottom and top walls respectively as $u_{\tau,b}^2 = | \textrm {d} \tilde {u} / {\textrm {d}y}|_{y=-h}$ and $u_{\tau,t}^2 = | \textrm {d} \tilde {u} / {\textrm {d} y}|_{y=h}$. Both $u_{\tau,b}(z)$ and $u_{\tau,t}(z)$ show a wavy variation in $z$ produced by the cross-stream velocity field of the rolls. We remark that there may also exist a spanwise component of the skin-friction vector at the wall, which leads to an effective spanwise $u_{\tau }$, but this is small and is neglected.

Figure 9.

Span-wise variation of PC flow, $Re_\tau \approx 500$ (a) Contours of $\tilde {u}$ with streamlines of the mean cross-flow $(\tilde {v}-\tilde {w})$ velocity field. (b) Local skin-friction velocity: red, $u_{\tau,b}$ and dashed, $u_{\tau,t}$. (c) Contours of $\widetilde {u^\bullet u^\bullet }$. (d) Contours of $\widetilde {u' u'}$.

In the plots of $\widetilde {(u^\bullet u^\bullet )}$ in figure 9(c), we observe six regions in the domain shown, illustrated as near-white zones, which depict minimum values. Each region denotes momentum transport induced by a quite strong vertical flow. For example, at position ‘a’ of figure 9(a), a strong vertical flow from bottom to top corresponds to the minimum region of $\widetilde {(u^\bullet u^\bullet )}$ near the bottom wall at the same position in figure 9(c). Similarly, at position ‘c’, a strong vertical flow from top to bottom corresponds to the minimum region of $\widetilde {(u^\bullet u^\bullet )}$ near the top wall at the same position. Owing to the different averaging methods, there exist other possible diagnostics, for example $\widetilde {(u' u')}$, as plotted in figure 9(d). Because this does not generally exhibit any interesting structures, this is noted but not discussed subsequently.

5.2. Scaling of $\tilde {u}$ and $\widetilde {u^\bullet u^\bullet }$ at $Re_\tau \approx 500$

We examine the local velocity profiles at specific spanwise locations, for example, at three locations ‘$a$’, ‘$b$’ and ‘$c$’ as labelled in figure 9. In figure 10, we plot the scaled streamwise velocity profiles $\tilde {u}^+ = \tilde {u}/u_{\tau,b}$ and related turbulence intensities $\widetilde {u^\bullet u^\bullet }^+ = \widetilde {u^\bullet u^\bullet }/u^2_{\tau,b}$. When scaled using local skin-friction velocity, streamwise profiles at different locations show similar log-like variation, not much removed from the scaled mean flow $U^+$ which is also plotted. The behaviour of turbulence intensities is different. In near-wall regions, say $y^+<50$, turbulence intensities show similar behaviour to span-wise averaged intensities, while for regions near the centreline, the differences between turbulence intensities at different spanwise locations are substantial, and imply turbulence momentum transport by the spanwise circulatory flow.

Figure 10.

Comparisons at selected locations ‘$a$’, ‘$b$’ and ‘$c$’, $Re_\tau \approx 500$: (a) Streamwise velocity $\tilde {u}$; (b) turbulent intensities $(u^\bullet u^\bullet )^+$ dashed line, location ‘$a$’; dash–dotted line, location ‘$b$’; solid line, location ‘$c$’. Line with filled symbols, $U^+$ in panel (a) and $(u'u')^+$ in panel (b).

To give a full picture of velocity scaling, it is useful to examine a two-dimensional $y\text {--}z$ plane contour plot of the scaled streamwise velocity. We focus on the velocity near the bottom wall which is chosen as the reference coordinate. In figure 11, we show plots with different scalings. The spanwise coordinate is scaled as $z/h$. Figure 11(a) shows $U^+$ with normal distance using $y^+$. The spanwise velocity variation is apparent because this scaling uses the global $u_\tau$. In figure 11(b), both $\tilde {u}$ and $y$ are scaled using the local $u_{\tau,b}(z/h)$. Here we define $y_l^+(y,z) = y u_{\tau,b}/\nu$. Using the local $u_{\tau,b}$, the spanwise velocity variation is mostly flattened.

Figure 11.

Scaling of velocity component $\tilde {u}$ and streamwise turbulent intensities $\widetilde {u^\bullet u^\bullet }$, $Re_\tau \approx 500$: (a) $\tilde {u}/u_\tau$ in the $(y^+, z/h)$ plane; (b) $\tilde {u}/u_{\tau,b}$ in the $(y_l^+, z/h)$ plane; (c) $\widetilde {u^\bullet u^\bullet }/u^2_\tau$ in the ($y^+,z/h$) plane; (d) $\widetilde {u^\bullet u^\bullet }/u^2_{\tau,b}$ in the ($y_l^+,z/h$) plane. Here $y_l^+ = y u_{\tau,b}/\nu$.

For the scaling of streamwise turbulence intensities, we focus on the near-wall region in $y_l^+ < 30$. We compare two scalings, one is using $u_\tau$ in figure 11(c), the second using $u_{\tau,b}$ in figure 11(d), for both turbulence intensity and distance. Again, using local $u_{\tau,b}$ removes much but not all of the $z/h$ variation.

5.3. Scaling of $\tilde {u}$ and $\widetilde {u^\bullet u^\bullet }$ at $Re_\tau \approx 3600$

We repeat this scaling of $\tilde {u}$ and $\widetilde {u^\bullet u^\bullet }$ for higher Reynolds number flow. In figure 12(a,b), we show two plots, one is $\tilde {u}/u_\tau$ in the ($y^+,z/h$) plane and the other is $\tilde {u}/u_{\tau,b}$ in the ($y_l^+,z/h$) plane. Again, using local scaling $u_{\tau,b}$ does show a flattened variation in the spanwise direction.

Figure 12.

Scaling of streamwise velocity $\tilde {u}$ and streamwise turbulence intensities $\widetilde {u^\bullet u^\bullet }$, $Re_\tau \approx 3600$: (a) $\tilde {u}/u_\tau$ in the ($y^+,z/h$) plane; (b) $\tilde {u}/u_{\tau,b}$ in the ($y_l^+,z/h$) plane; (c) $\widetilde {u^\bullet u^\bullet }/u^2_\tau$ in the ($y^+,z/h$) plane; (d) $\widetilde {u^\bullet u^\bullet }/u^2_{\tau,b}$ in the ($y_l^+,z/h$) plane.

In figure 12(c,d), we again focus on the bottom near-wall region and display the scaling effect using the local $u_{\tau,b}(z/h)$. Compared with using $u_\tau$, which clearly shows the roll effect, employing $u_{\tau,b}$ smooths the effect of the large-scale roll structure. Using $u_{\tau,b}(z/h)$ does show the presence of small-scale spanwise fluctuations. We expect these to be even smoother using longer timewise averaging.

We also discuss scaled profiles at a specific spanwise location. In figure 13, we show the scaled streamwise velocity profiles $\tilde {u}^+$ and related turbulence intensities $\widetilde {u^\bullet u^\bullet }^+$. Streamwise mean-velocity profiles at different spanwise locations show a log-like region when scaled on the local skin-friction velocity. The behaviour of turbulence intensities is similar to the results at $Re_\tau \approx 500$. In the near-wall region, inner scaling using $u_\tau$ gives reasonable collapse, while outer-region structures are generally substantially different. The presence, in PC flow, of spanwise local inner scaling including a near-wall log layer in the mean-velocity profile is consistent with the present wall model, as described in § 2.3, which uses both of these features.

Figure 13.

Comparisons at selected locations ‘$a$’, ‘$b$’ and ‘$c$’, $Re_\tau \approx 3600$: (a) streamwise velocity $\tilde {u}$; (b) turbulence intensities $(u^\bullet u^\bullet )^+$. Symbols are the same as in figure 10.

7.1. Skin-friction relation and the centreline, mean-velocity gradient

The asymptotic behaviour of the mean-velocity gradient, both near the wall and at the centreline, is the key feature of the wall-bounded Couette-flow turbulence. In a previous study of Taylor–Couette flow with the inner cylinder rotating, a three-zone model was established that used the existence of a constant-angular-momentum zone in the bulk of the flow (Cheng et al. Reference Cheng, Pullin and Samtaney2020). For a PC flow, a simpler approach is to obtain an $Re_{\tau }\text {--}Re_c$ through a two-log match at the flow centreline (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014). Following their model by assuming the velocity profiles in the form of log laws without wakes, and with constants $\kappa$ and $A$, we have the following mean velocity profiles.

I: $-h< y \leqslant 0$ with

(7.1)\begin{equation} U = U_c - u_{\tau}\left( \frac{1}{\kappa}\ln\left(\frac{( h + y )\,u_{\tau}}{\nu}\right) +A \right). \end{equation}

II: $0 \le y < h$ with

(7.2)\begin{equation} U ={-}U_c + u_{\tau}\left( \frac{1}{\kappa}\ln\left(\frac{( h - y)\,u_{\tau}}{\nu}\right) +A \right).\end{equation}

Matching of these two profiles at the centreline $y=0$ gives the following relation:

(7.3)\begin{equation} \frac{Re_c}{Re_{\tau}} = \frac{1}{\kappa} \ln Re_\tau + A, \end{equation}

which provides the skin-friction relation for PC flow. This equation can be solved for $Re_\tau$ to give

(7.4)\begin{equation} Re_\tau = \frac{\kappa Re_c}{ W (\textrm{e}^{A\kappa} \kappa Re_c )}, \end{equation}

where $W(Z)$ is the ProductLog function or Lambert function, which is the solution of $Z = W\ln (W)$. Plots of $Re_{\tau }\text {--}Re_c$ are shown in figure 17, which includes existing DNS reference data and also the present WR and WMLES and (7.3). The constants $\kappa$ and $A$ are chosen using classical values $\kappa =0.4$ and $A=5.0$. For the $Re_{\tau }\text {--}Re_c$ relation, all available DNS data reasonably agree with the theoretical estimate. Our present LES results are plotted in figure 17(b). The wall-modelled LES reasonably follow the modelled equation in this $Re_c$ range.

Figure 17.

Skin-friction relation $Re_{\tau }\text {--}Re_c$: (a) wall-resolved LES data comparison with DNS; (b) all present LES results. Here, $\triangle$ (red), DNS by Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014); $\square$ (green), DNS by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014); $\diamond$ (blue), DNS by Lee & Moser (Reference Lee and Moser2018); $\bullet$, wall-resolved LES; $\blacktriangle$ (red), wall-modelled LES. Dashed line (blue), (7.3) with $\kappa =0.4$ and $A=5.0$.

The model can be extended to calculate the mean-flow velocity gradient at the centreline. Because the above two velocity profiles also have a continuous velocity gradient at the centreline, we can obtain

(7.5)\begin{equation} S \equiv \frac{h}{U_c} \frac{\partial U}{\partial y} ={-} \frac{1}{\kappa } \frac{Re_\tau}{Re_c}. \end{equation}

Using (7.4) to eliminate $Re_\tau$, we obtain

(7.6)\begin{equation} S = \frac{1}{ W (\textrm{e}^{A\kappa} \kappa Re_c ) }. \end{equation}

Plots of $S(Re_c)$ are given in figure 18. Lower $Re_c$ cases, which include existing DNS reference data and the present WRLES results, are also shown in figure 18(a). The theory generally underestimates the numerical data for relatively low Reynolds number but our largest $Re_c$ WRLES data, at $Re_\tau \approx 3600$, reasonably matches the model estimate. Figure 18(b) shows a higher $Re_c$ comparison where agreement between theory and the WMLES is acceptable.

Figure 18.

Scaled velocity gradient at centreline, $S$. (a) Wall-resolved LES data comparison with DNS. (b) All present LES results. Here, $\triangle$ (red), DNS by Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014); $\square$ (green), DNS by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014); $\triangleleft$ (blue), short domain DNS by Lee & Moser (Reference Lee and Moser2018); $\triangleright$ (blue), long domain DNS by Lee & Moser (Reference Lee and Moser2018); $\bullet$, wall-resolved LES; $\blacktriangle$ (red), wall-modelled LES. Line (blue), (7.6) with $\kappa =0.4$ and $A=5.0$.

7.2. Large $Re_\tau$ limit for streamwise mean flow velocity

Using (7.2) and (7.3) gives, for $0 \le y < h$,

(7.7)\begin{equation} \frac{U(y)}{U_c} = \frac{ \ln \left(1-\dfrac{y}{h} \right) }{ \ln Re_\tau + \kappa A }. \end{equation}

According to the model, when $Re_\tau \to \infty$ at any finite $y/h$, then $U(y)/U_c\to 0$, while from (7.6), $S\to 0$. This will not extend to the upper wall $y\to h$ where a thin viscous-dominated layer can be expected to remain at any $Re_\tau$. The velocity in the lower-half domain will show similar behaviour.

7.3. Spanwise skin-friction variation

In figure 19(a), we show the spanwise variation of the skin-friction velocity $u_{\tau }(z/h)/U_c$ at different $Re_\tau$. The periodic-like variation corresponds to the presence of spanwise rolls. It can be seen that, in addition to the decrease of the spanwise mean with increasing $Re_\tau$, the amplitude of the spanwise variation also decreases. Figure 19(b) shows the spanwise root-mean-squared variation $u'_\tau$ versus $Re_\tau$. When $u'_\tau$ is scaled using $U_c$, it generally shows a monotonic decreasing tendency. The plot of $u'_\tau /u _\tau$ shows an initial increase with increasing $Re_\tau$ followed by a slow decline.

Figure 19.

Spanwise variation of skin-friction velocity. (a) Comparison of $u_{\tau }$ at different $Re_c$. Here, $\blacktriangle$ (red), $Re_\tau \approx 500$; $\blacktriangledown$ (blue), $Re_\tau \approx 1000$; $\blacktriangleleft$, $Re_\tau \approx 1800$; $\blacktriangleright$ (red), $Re_\tau \approx 3600$; $\blacksquare$ (blue), $Re_\tau \approx 3.3\times 10^4$; $\bullet$, $Re_\tau \approx 2.8\times 10^5$. (b) Measure of root mean square of $u_{\tau }$: $u'_\tau / u_\tau$ and $u'_\tau / U_c$.

8.1. Instantaneous velocity fields

Visualization of component velocity fields provides insight into large-scale, steamwise roll behaviour. In figure 20, we show velocity contours on several two-dimensional planes in a front (streamwise)-top (wall-normal) view of the three-dimensional flow. The planes shown include one wall-parallel plane and seven cross-flow planes. These cross-flow planes are located at intervals of $L_x/8$. For a better view of the inflow, planes at inflow and $L_x/8$ are hidden. In each figure, we also show two close-up frames to observe detailed flow features. One is from the cross-flow plane at $x=3L_x/4$, while the other on the wall-parallel plane starting at $x= 3L_x/4$.

Figure 20.

Instantaneous velocity fields, $Re_\tau \approx 500$: (a) streamwise velocity u; (b) vertical velocity $v$; (c) spanwise velocity $w$.

Figure 20(a) shows a colour-coded image of the streamwise velocity $u$ on the wall-parallel plane $y=0$. The mean velocity is zero allowing clear observation of roll structures. In figures 20(b) and 20(c), where the wall-normal and span-wise velocities $v,w$, respectively, are shown, the parallel plane at $y=0.85h$ is chosen to enable clear visualization of both cross-flow and wall-parallel flow features. Instantaneous roll signatures are more apparent in the $u$-velocity image, including both the three pair rolls on the wall-parallel plane and their intrusion flow counterparts into opposite walls on cross-flow planes. In plots of $v$ and $w$, detectable flow structures seem relatively small and large-scale rolls are hard to discern. An interesting feature is that the near-wall flow structure is clearly smaller than that in the flow within the bulk region, especially in the $v$ and $w$ images.

To show $Re_\tau$ effects, similar images of $u$, $v$ and $w$ WRLES at $Re_\tau \approx 3600$ are plotted in figure 21. Compared with $Re_\tau \approx 500$, the $u$-visualization shows little apparent $Re_\tau$ effect in part because the colour bar is designed to show large-scale rolls rather than small-scale turbulence. The small-scale flows near the two walls in this higher $Re_\tau$ case are clearly smaller in size and closer to the wall compared with $Re_\tau \approx 500$, as observed in the cross-flow close-ups in figure 21(b,c).

Figure 21.

Instantaneous velocity fields, $Re_\tau \approx 3600$: (a) streamwise velocity u; (b) vertical velocity $v$; (c) spanwise velocity $w$.

It is interesting to investigate whether our WMLES at $Re_c=10^7$ can also capture the above described features. In figure 22, generally all three components $u$, $v$ and $w$ are ‘lighter’ compared with figures 20 and 21. This is expected because when using the same colour bar, a high-Reynolds-number flow indicates a smaller magnitude of velocity fluctuations. The two main features of figures 20 and 21, including the large-scale rolls on the wall-parallel plane of $u$ (figure 22a) and the smaller-scale features on the cross-flow plane images of $v$ and $w$ (figure 22b,c), are all found in this WMLES at $Re_c=10^7$.

Figure 22.

Instantaneous velocity fields, $Re_\tau \approx 2.8 \times 10^5$: (a) streamwise velocity $u$; (b) vertical velocity $v$; (c) spanwise velocity $w$.

8.2. Roll structure

To demonstrate the present interpretation of rolls, we show colour contours of the averaged mean streamwise velocity field for five WRLES in figure 23 and four WMLES in figure 24. The streamwise velocity $\tilde {u}$ is plotted to show its large fluctuations in each roll. A notable difference can be observed between $Re_\tau \approx 500$ and $Re_\tau \approx 2.8\times 10^5$. In figure 23(a), the zero line of $\tilde {u}$ shows a wavy behaviour which is similar to prior DNS (Lee & Moser Reference Lee and Moser2018). However, the zero line for $\tilde {u}$ for $Re_\tau \approx 2.8\times 10^5$ in figure 24(d) shows a more pronounced mushroom structure, which clearly enhances the mixing of the flows generated from two moving walls. It is plausible that this mushroom structure is unstable and that the rolls could potentially disappear in even higher $Re_c$ flows.

Figure 23.

Contours of streamwise velocity $\tilde {u}$. Wall-resolved LES: (a) $Re_\tau \approx 500$; (b) $Re_\tau \approx 1000$; (c) $Re_\tau \approx 1800$; (d) $Re_\tau \approx 3600$.

Figure 24.

Contours of streamwise velocity $\tilde {u}$. Wall-modelled LES: (a) $Re_\tau \approx 3600$; (b) $Re_\tau \approx 3.3 \times 10^4$; (c) $Re_\tau \approx 2.8 \times 10^5$.

8.3. Roll energy

Diagnostics suitable for characterizing roll behaviour are the kinetic-energy densities, which are defined in (3.4) as $e_u(y)$, $e_v(y)$ and $e_w(y)$. Integration in the wall-normal direction provides an overall measure of the energy associated with the mean roll motion. We define $E_\phi$ as the integral of the corresponding $e_\phi (y)$ and $E_r$ as a combination of the $v$ and $w$ components. Specifically,

(8.1a,b)\begin{equation} E_\phi = \frac{1}{2\,h} \int_{{-}h}^{h} e_\phi\, {\textrm{d} y} , \quad E_r = E_v + E_w, \end{equation}

where $E_r$ can be interpreted as the kinetic energy of the mean motion in the large-scale rolls, while $E_u$ is the corresponding streamwise energy. Both are functions of $Re_\tau$.

In figure 25(a), we plot $E_r$ versus $Re_\tau$ from the present LES and DNS results from Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014). The two DNS data at low $Re_\tau$ are not discussed because these $Re_\tau$ are within the range of incipient roll generation. The LES data agrees fairly well with DNS at $Re_\tau \approx 500$ and $1000$. Wall-resolved LES at these two $Re_\tau$ do not clearly reveal a trend while the wall-modelled LES do show a sharp decline of $E_r$ with increasing $Re_\tau$, which implies a relative weakening of roll strength as $Re_\tau$ increases. The ratio $E_v/E_r$, which measures the wall-normal component contribution to the total roll energy, is shown in figure 25(b). With the exception of the DNS of $Re_\tau =171$ (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014), all other cases indicate a near-constant a value around 0.30–0.35. In figure 25(c), we plot $E_u/U_c^2$ for WRLES and WMLES while in figure 25(d), the corresponding plots for $E_r/E_u$ are shown. It is clear that for all LES, $E_r/E_u$ maintains a nearly constant level of approximately $1\,\%$, which is consistent with figure 5(b).

Figure 25.

Roll energy diagnostics: (a) plot of $E_r/U_c^2$ versus $Re_\tau$; (b) plot of $E_v/E_r$; (c) plot of $E_u/U_c^2$; (d) plot of $E_r/E_r$. Here, $\square$, reference DNS by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014); $\blacktriangle$, wall-resolved LES; $\blacktriangledown$ (blue), coarse wall-modelled LES; $\bullet$ (red), wall-modelled LES.

In addition to the above scaling of $E_u$ using $U_c$, we further show scaling using $u_\tau$. In figure 26(a), we find that $E_u/u_\tau ^2$ does not decrease over two decades of $Re_\tau$. This may imply that while streamwise rolls may continue to weaken on outer scaling at quite high $Re_\tau$, they may survive on inner scaling. Figure 26(b) shows the plot of $E_u/(U_c^2 S)$. Its variation with $Re_\tau$ is generally around unity and does not show a clear tendency with increasing $Re_\tau$.

Figure 26.

Roll energy scaling using $u_\tau$ and $S$: (a) plot of $E_u/U_\tau ^2$ versus $Re_\tau$; (b) plot of $E_u/(U_c^2S)$. Here, $\square$, reference DNS by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2014); $\blacktriangle$, wall-resolved LES; $\bullet$ (red), wall-modelled LES.

8.4. Circulation

The roll circulation provides a further measure of the roll activity. This is implemented using the mean velocity field as shown in figures 23 and 24. A control area in the $y\text {--}z$ plane is constructed comprising the top and bottom wall boundaries, a fixed left boundary $z/h = \textrm {const.}$ and a right boundary also with $z/h= \textrm {const}$. The circulation around this closed area can then be calculated as $\varGamma = \oint {\boldsymbol {u}}\boldsymbol {\cdot } \textrm {d}\boldsymbol {l}$. The normalized circulation $C = \varGamma /(U_c\,h)$ is plotted against the distance between the left and right boundaries in figure 27(a) for $Re_\tau \approx 500$. This shows oscillatory behaviour as the right-hand boundary passes through full rolls. We define the circulation per roll, $C_r$, as the average between maxima and minima of $C(z/h)$ across the spanwise flow domain. This is shown for both WRLES and WMLES as a function of $Re_\tau$ in figure 27(b). The trend with increasing $Re$ is similar to that for $E_r(Re_\tau )$. Both are consistent with diminishing relative roll strength as $Re_\tau$ increases. This is consistent with the hypothesis that the rolls may eventually dissipate and disappear at sufficiently large $Re_\tau$.

Figure 27.

Circulation: (a) measure of circulation $C$ for case at $Re_\tau \approx 500$; (b) roll circulation $C_r$ for all cases. Here, $\blacktriangle$, wall-resolved LES; $\bullet$ (red), wall-modelled LES.