2.1. Simulation configuration

Turbulent channel flows rotating in the spanwise direction at Reynolds number ${Re_b := U_b H/\nu = 2500}$ (where $H$ is the channel half-height and $U_b$ is the bulk velocity) are simulated by DNS. The friction Reynolds number $Re_\tau := u_\tau H/\nu$ is 160 when the channel is not rotating. A 2-D bump defined by the parabolic formula (normalized by $H$)

(2.1)\begin{equation} y = \max[{-}a(x-4)^2 + h,0] - 1 \end{equation}

is placed on the bottom wall of the channel at $y=-H$ (see figure 2a). Depending on the sign of the rotation rate, this side is either anti-cyclonic (i.e. $Ro_b >0$) or cyclonic ($Ro_b<0$). The height of the bump is set to be $h=0.25H$. The parameter $a=0.15$ yields a bump with streamwise length $2.58H$ along the wall (i.e. $x/H=[2.71,5.29]$). These dimensions are chosen for two reasons. First, the blockage in the wall-normal direction is relatively low, and the rate of contraction/expansion is gradual. Compared with the Gaussian-shaped Boeing bump (Balin & Jansen Reference Balin and Jansen2021; Uzun & Malik Reference Uzun and Malik2022), for example, the relatively large length-to-height ratio of the current bump has several advantages for this study: (1) the favourable pressure gradient caused by the contraction at the windward side of the bump is mild such that flow is not relaminarized (Yuan & Piomelli Reference Yuan and Piomelli2011); (2) the APG at the aft part of the bump is mild (figure 2b) such that the separation point is non-fixed. The second reason for using $h=0.25H$ is that the flow up to the height of the bump will be in the destabilized (stabilized) region under anti-cyclonic (cyclonic) rotation. This will be shown in § 7.

Figure 2. (a) Bump profile and (b) inviscid streamwise pressure gradient along the surface. The profiles for the Gaussian-shaped bump used in Balin & Jansen (Reference Balin and Jansen2021) and Uzun & Malik (Reference Uzun and Malik2022) are scaled to the same height as the current bump for comparison. indicates current parabolic bump; indicates Gaussian bump.

Five cases corresponding to rotation numbers $Ro_b = 0$, $\pm 0.42$ and $\pm 1.0$ were performed. The cases are named ‘P/NXX’ where P or N denotes positive or negative $Ro_b$, while $\text {XX}=04$ denotes $|Ro_b|=0.42$, and $\text {XX}=10$ denotes $|Ro_b|=1.0$. The non-rotating case is denoted as case 00. Cases P04 and P10 will be referred to as the ‘positive-rotating’ or ‘anti-cyclonic’ cases interchangeably. Conversely, cases N04 and N10 will be referred to as ‘negative-rotating’ or ‘cyclonic’ cases. The simulation parameters for each case are summarized in table 1. Note that the description and discussion in this paper are limited to the selected rotation rates. Terms describing the trends with respect to the rotation rate, such as ‘(non-)monotonic’, are used with respect to the tested rotation rates. They have no implication on the trends occurring between the current rotation rates and the ultimate 2-D laminar dynamics (thus 2-D laminar separation) at sufficiently high rotation rates (Grundestam et al. Reference Grundestam, Wallin and Johansson2008a; Brethouwer Reference Brethouwer2017).

Table 1. Simulation parameters: $Re_{\tau,c}$ is the friction Reynolds number at the bump crest; $Re_{\tau,u}$ ($Re_{\tau,s}$) is the friction Reynolds number for the anti-cyclonic (cyclonic) side of the channel; $Re_{\tau }$ is the friction Reynolds number in the fully recovered channel section; $Ro_{\tau } := 2\varOmega H/u_\tau$ is the friction rotation number in the fully recovered channel section; $x_{sep}$ is the streamwise location of the mean separation point; $y_{sep}$ is the wall-normal location of the mean separation point (relative to the bottom wall); $L_{sep}$ is the streamwise length of the mean separation bubble; and $F_D$ is the total mean drag per unit span over the entire channel. Here, $x_{sep}$, $y_{sep}$ and $L_{sep}$ are normalized by $H$, and $F_D$ is normalized by $HU_b^2$. We write $\Delta y^+_{(1)}$ for the wall-normal grid spacing in wall units at the first point from the bottom wall.

A computational domain of $39H \times 2H \times 6H$ in the streamwise ($x$), wall-normal ($y$) and spanwise ($z$) directions is employed. The long computational domain in the streamwise direction is used to ensure that the channel is well recovered near the outlet, as a streamwise-periodic boundary condition is employed. This allows for a complete examination of the recovery of the wake flow and avoidance of auxiliary boundary conditions for generating physical rotating turbulence at the inflow. In previous studies, it is reported that a flow without spanwise rotation is well recovered after 30 times the height of an obstacle (Castro Reference Castro1979; Le, Moin & Kim Reference Le, Moin and Kim1997; Song, DeGraaff & Eaton Reference Song, DeGraaff and Eaton2000; Mollicone et al. Reference Mollicone, Battista, Gualtieri and Casicola2017). In the rotating BFS study performed by Barri & Andersson (Reference Barri and Andersson2010), they showed that the mean skin friction coefficient gradually approached a constant level but was not completely recovered by 32 step heights. The length of the domain in this study is greater than 130 bump heights. It is carefully justified that not only the first-order mean quantities but also the higher-order statistics are well developed when the flow approaches the outflow boundary. An auxiliary simulation of case N04 (the slowest case to recover from the bump wake, see figure 9) was performed with an extended streamwise length $L_x = 100H$. The mean separation point, reattachment point and separation length change by less than $0.01H$ compared with the current domain. In the rear part of the SSL where fluctuations are the most prominent, the Reynolds stresses differ by less than 3 %. These minor discrepancies further indicate that the current domain length is sufficient. A periodic boundary condition is also employed in the spanwise direction. For the remaining boundary conditions, the no-slip condition is enforced along the bottom and top walls, including the bump surface.

2.2. Numerical methods

Incompressible Navier–Stokes equations for a Newtonian fluid, non-dimensionalized by $U_b$ and $H$,

(2.2)$$\begin{gather} \frac{\partial u_i}{\partial x_i} = 0, \end{gather}$$

(2.3)$$\begin{gather}\frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j}(u_iu_j) ={-}\frac{\partial p}{\partial x_i} + \frac{1}{Re_b}\,\nabla^2u_i - Ro_b\,\epsilon_{i3k} u_k + f_i, \end{gather}$$

are solved by DNS. Here, $p$ is the modified pressure, and $Ro_b$ is the bulk rotation number. The term $f_i$ is used to enforce the no-slip boundary conditions on the bump, which is achieved by an immersed boundary method (IBM) based on a volume-of-fluid (VOF) approach (Peskin Reference Peskin1972; Scotti Reference Scotti2006). The fraction of cell volume that is occupied by the fluid, denoted as $\phi$, is calculated analytically in a pre-processing calculation using the simulation grid. During the simulation, the velocity in the cells that are occupied partially or fully by the solid is weighted by $\phi$ through term $f_i$. This method has been used extensively (Scotti Reference Scotti2006; Yuan & Piomelli Reference Yuan and Piomelli2014a,Reference Yuan and Piomellib; Wu, Banyassady & Piomelli Reference Wu, Banyassady and Piomelli2016; Wu & Piomelli Reference Wu and Piomelli2018; Wu et al. Reference Wu, Piomelli and Yuan2019; Savino, Yeom & Wu Reference Savino, Yeom and Wu2023b) in the investigations of flow around embedded objects.

A Cartesian grid is designed such that the bump surface and the wake of the bump are well resolved. The grid is uniform in the $x$ direction around the bump and in the far wake (i.e. $\Delta x/H = 0.011$ for $x/H=[2.09,6.68]$ and $\Delta x/H = 0.053$ for $x=[12, 39]$), while being stretched between $x/H = [0, 2.09]$ and $x/H =[6.68, 12]$ to transition from the two uniform spacings. The grid is also uniform in the $y$ direction below the crest with a hyperbolic tangent stretching towards the centreline. The grid is uniform in the $z$ direction. The maximum stretch ratio is less than 3 % in all directions. The grid in the $x$–$y$ plane is shown in figure 3 for reference.

Figure 3. Computational grid in the $x$–$y$ plane. Every fifth grid cell is shown in both directions for clarity. The main figure is limited to $x/H=[0, 10]$, and the inset is limited to $x/H=[2.5, 5.5]$, $y/H=[-1, -0.7]$ for clarity. The bump surface is shown by the red solid line.

Two grids are tested for grid convergence. The first grid, denoted as grid I, consists of $1196 \times 192 \times 184$ grid points in the streamwise, wall-normal and spanwise directions, respectively. For all the five cases using grid I, the maximum $\Delta x^+$ and $\Delta z^+$ near the two walls are 9.3 and 6.5; $\Delta y^+$ at the first cell from the bottom wall is reported in table 1. Because the $y$ grid is uniformly spaced to the bump crest, $\Delta y^+ < 1$ is maintained along the bump and in the wake. Away from the wall, grid I gives $\Delta h/\eta \le 6 \ (\Delta h = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2})$, which is much smaller than the length scale at which the maximum dissipation occurs, $24\eta$ (Pope Reference Pope2000). Thus grid I is expected to be capable of resolving a substantial portion of the dissipation spectrum. Nevertheless, a finer grid (grid II) is employed for the P04 case to verify the grid convergence. This case is chosen because the moderate rotation rate leads to the greatest increase in turbulence intensity on the anti-cyclonic wall. Grid II consists of $1584\times 239\times 288$ grid points in the $x$, $y$ and $z$ directions, which corresponds to a refinement factor of $132\,\% \times 125\,\% \times 157\,\%$. The mean streamwise velocity and TKE are compared at selected streamwise locations in figure 4. The agreement indicates that the solutions on both grids are grid-independent. In the following, only the results calculated using grid I are shown.

Figure 4. Comparison of mean streamwise velocity and TKE at $x/H=2$, 4, 5.5, 8 and 20, case P04. indicates grid I (coarser one); indicates grid II (finer one). Each profile is shifted to the right by 2 units for $U$, and by 0.1 units for TKE, for clarity.

The equations of motion are solved using a well-validated finite difference code (Keating et al. Reference Keating, Piomelli, Bremhorst and Nešić2004; Wu et al. Reference Wu, Banyassady and Piomelli2016, Reference Wu, Piomelli and Yuan2019; Wu & Piomelli Reference Wu and Piomelli2018; Savino, Patel & Wu Reference Savino, Patel and Wu2023a; Wu & Savino Reference Wu and Savino2023) which solves (2.2) and (2.3) on a staggered grid. The code is second-order accurate in both time and space: a second-order-accurate central differencing scheme is used for all spatial derivatives. A second-order-accurate semi-implicit time advancement method is employed in which the Crank–Nicolson scheme is used for the wall-normal diffusion terms, while the Adams–Bashforth scheme is applied to all remaining terms. The Poisson equation is solved directly via a pseudo-spectral method (Moin Reference Moin2010) using the blktri matrix solver from the FISH-PACK software library in which a generalized cyclic reduction algorithm is employed (Sweet Reference Sweet1974; Swarztrauber & Sweet Reference Swarztrauber and Sweet1979). The code is parallelized using the message-passing interface (MPI) protocol. After each case reaches its statistically steady state, the 3-D flow field data is saved at time interval $\delta t = 5H/U_b$ over a total $400H/U_b$ (1600$h/U_b$) for statistical averaging. The 2-D slides of the flow field are also sampled at several planes every 115 time steps (${\sim }0.17H/U_b$), and history profiles of the velocity are monitored at every time step at selected locations. Statistical averages are performed in time and over the homogeneous spanwise direction. The mean quantities are denoted by capital letters or by operator $\bar {(\, \, )}$. The superscript $+$ denotes quantities scaled with the wall units. Results in the near wake of the bump will be focused on, and the region far downstream will not be shown unless necessary.

2.3. Validation

The convergence of the statistics is ensured by checking that the mean velocity and Reynolds stresses obtained using only half of the sample are within 1 % of the values calculated using the entire sample (analysis not shown for brevity). The mean velocity profiles in the fully recovered region ($x/H=[28,31]$) are shown in figure 5. Figure 5(a) shows the mean streamwise velocity scaled in outer units. As shown by the dash-dotted lines, the expected region of $\partial U/\partial y = 2\varOmega$ is captured for both moderate and high rotation rates. Figure 5(b) shows the mean streamwise velocity scaled with wall units. Case 00 is compared with the DNS of Lee & Moser (Reference Lee and Moser2015) at $Re_{\tau } = 180$. Despite slight differences in Reynolds number, the data collapse well. Note that $Re_{\tau }$ differs on the anti-cyclonic and cyclonic walls for the rotating cases (see table 1). Thus the top and bottom portions of the channel are normalized with $Re_{\tau,s}$ and $Re_{\tau,u}$, respectively. The near-wall flow collapses on the linear law of the wall for the viscous sub-layer, indicating the sufficient resolution of near-wall turbulence regardless of modulation in turbulence intensity. Additionally, in the outer layer, the anti-cyclonic side displays a downshift compared to the canonical logarithmic relation, while the cyclonic side displays a parabolic laminar profile. This is consistent with the behaviour observed in the studies of Watmuff, Witt & Joubert (Reference Watmuff, Witt and Joubert1985), Tafti & Vanka (Reference Tafti and Vanka1991), Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993) and Wu et al. (Reference Wu, Piomelli and Yuan2019).

Figure 5. (a) Mean streamwise velocity scaled with outer units. Dot-dashed lines representing $\partial U/\partial y = 2\varOmega$ are shown for each case. (b) Mean streamwise velocity scaled with wall units. Note that for the rotating cases, the anti-cyclonic and cyclonic walls are scaled with their respective friction velocity. Case 00 is compared with DNS data from Lee & Moser (Reference Lee and Moser2015). In both plots, profiles are averaged in the streamwise direction in the range $x/H = [28,31]$.

4.1. Total drag

Using the VOF IBM, the force exerted on the fluid to enforce the no-slip condition by the bump is calculated during every time iteration. This force contains both the frictional and form drag components produced by the bump. Integrating this force along with the wall-shear stress on the planar walls provides the total drag of the channel. The conventional notion regarding the relation between flow separation and form drag is that larger separation implies larger form drag, and vice versa. The results below show that this intuition can be misleading.

The total drag per unit span is compared in figure 8, decomposed into four sources: the friction drag produced by the bottom wall (excluding the bump), $F_{D,{bot}} = \int \tau _w(x,-H) \,{{\rm d}\kern 0.06em x}$; the friction drag produced by the top wall, $F_{D,{top}} = \int \tau _w(x,H) \,{{\rm d}\kern 0.06em x}$; the drag produced on the wind side of the bump, $F_{D,{wind}} = \int _{-H}^{H}\int _{{LE}}^{x_c} F_1 \,{{\rm d}\kern 0.06em x}\, {{\rm d}\kern 0.05em y}$; and the drag produced on the lee side of the bump, $F_{D,{lee}} = \int _{-H}^{H}\int _{x_c}^{{TE}} F_1\, {{\rm d}\kern 0.06em x}\, {{\rm d}\kern 0.05em y}$. Here, LE, TE and $x_c$ correspond to the streamwise locations of the bump leading edge, trailing edge and crest, respectively, and $F_1$ is the mean IBM force from (2.3) in the streamwise direction. For cases P04 and P10, the bottom wall skin friction (blue bar) and the drag produced by the bump (green and red bars) sum to the total drag on the anti-cyclonic side (hatched regions). In cases N04 and N10, the friction on the top wall (orange, hatched bar) is solely responsible for anti-cyclonic side drag production. Note that the total drag (sum of these four terms) is given in table 1.

Figure 8. Total drag per unit span. The bars are split into drag produced on the bottom wall ($F_{D,{bot}}$), top wall ($F_{D,{top}}$), wind side ($F_{D,{wind}}$) and lee side ($F_{D,{lee}}$) of the bump, as shown in the legend. The inset compares the drag produced by the wind and lee sides of the bump. In all plots, hatched regions represent drag produced on the anti-cyclonic side of the channel.

All rotating cases exhibit a decrease in total drag compared to case 00. In the current configuration, the modulation of skin friction along the walls appears to contribute most significantly to the change in total drag, given that the length of the channel is considerably longer than the bump (which occupies $7\,\%$ of a single wall). The primary contributor to the total drag decrease is the reduction in skin friction along the cyclonic wall (orange bars for cases P04 and P10, blue bars for cases N04 and N10), a consequence of diminished turbulence intensity due to relaminarization by the cyclonic rotation. Conversely, the skin friction on the anti-cyclonic wall changes non-monotonically with $Ro_b$ (hatched bars). At the moderate rotation rate (P04 and N04), skin friction on the anti-cyclonic wall is increased, counteracting the drag reduction on the cyclonic side and thus leading to a minor net reduction in total drag. At the high rotation rate (P10 and N10), the skin friction on the anti-cyclonic wall is lower than in the non-rotating case, resulting in a drastic total drag decrease together with the laminar opposite side. The suppression of turbulence and the associated drag over the anti-cyclonic wall at high rotation rates is consistent with the observations in the attached rotating flow studies of Johnston et al. (Reference Johnston, Halleen and Lezius1972) and Brethouwer (Reference Brethouwer2017).

Compared to the total drag differences, the drag produced solely by the bump differs little between all cases despite the significant change in the size of the separation region. The bump-produced drag is dominated by the positive wind-side force (drag, green bars) compared to the negative lee-side force (thrust, red bars). This behaviour is better illustrated by the inset of figure 8, where the bump-produced forces are compared directly. The predominant change by rotation is the decrease in drag on the wind side of the bump rather than the thrust on the lee side. It is found that the decrease in wind-side drag follows the same trend as the incoming MMD. Phenomenologically, this is expected, as a higher (lower) incoming velocity will result in a larger (smaller) increase of the stagnation pressure at the wind side of the bump. Case 00 displays the largest incoming velocity, and consequently the largest wind-side drag. Case P10, with the lowest incoming velocity, produces the least wind-side drag. The negative force on the lee side is often considered as a ‘back-pressure’ whose recovery depends on the size of the separation region. That is, a smaller recirculation region is expected to result in a greater back-pressure (i.e. more negative). Our results support this trend in general, with the exception being that case P04 has a similar $F_{D,{lee}}$ to case 00, yet a smaller separation region. The variation of back-pressure between the four rotating cases, nevertheless, is small compared with that of the wind-side drag, and the latter remains the dominant contributing force. Therefore, the variation of separation size contributes little to the change in bump-produced drag. These observations indicate that the size of the separation region is not a proper indicator of the drag, even locally around the bump, at least for the current rotating configuration in question.

4.2. Skin friction on the channel walls

Because the skin friction on the channel walls is found to be the dominant term contributing to total drag and its variation with rotation, we now further examine its streamwise evolution associated with the separation region, wake and flow recovery. The mean skin friction coefficients ($C_f := 2\tau _w/U_b^2$) along the bottom and top walls are shown in figure 9. Skin friction on the bump is excluded from the bottom wall drag as in the previous subsection. Integrating $C_f$ along $x$ on both walls shows that the accumulated skin friction exceeds the drag produced by the bump 6$H$–7$H$ downstream of the bump (or $\sim$25 bump heights). Therefore, unless the wall extends only over such a short distance downstream of the protrusion in physical applications, the skin friction along it during the prolonged recovery of the wake would remain the main source of total drag. This is typical, for example, when the protrusion is located near the leading edge of a turbine.

Figure 9. Skin friction coefficient along (a) the bottom wall and (b) the top wall. The skin friction over the surface of the bump is excluded from the plot.

When the flow is non-rotating (case 00), the skin friction along the bottom wall displays the expected behaviour of a separation bubble and reattaching flow. Specifically, a negative peak prior to reattachment signifies strong reverse flow in the recirculation region, while a positive peak post reattachment indicates strong forward flow. The latter is characteristic of impinging-type reattachment in which a strong mean downwash and/or rapidly decaying roller vortices strike the surface, as discussed in Le et al. (Reference Le, Moin and Kim1997) and Na & Moin (Reference Na and Moin1998). The steep separating streamline of case 00 near the reattachment point (refer to figure 7b) supports this claim. The recovery of $C_f$ on the bottom wall takes ${\sim }20 H$ (80 bump heights). On the top wall, $C_f$ shows a variation due to the change of flow area by the bump. It reaches its peak at $x=4H$, where the crest of the bump is, and its minimum at $x \approx 7H$, slightly downstream of the reattachment point, indicating that the ‘dead fluid zone’ within the separation bubble (Na & Moin Reference Na and Moin1998) reduces the effective flow area of the channel. Following the minimum, the flow gradually recovers, attaining the recovered $C_f$ value ${\sim }15H$ (60 bump heights) downstream of the reattachment point.

When subjected to rotation, the wake of the bump on the bottom wall exhibits a negative–positive–peak pattern near the reattachment region similar to the non-rotating case. Positive rotation leads to a shorter transition between the two peaks due to the reduced separation region, and rapid recovery to a constant $C_f$ within a few $H$ of the bump trailing edge. Note that the reduction of the recovery length is much more significant than that of the separation region. This indicates that rotation modulates not only the mean separation region but also the reattached flow. The non-monotonic change of the skin friction on the anti-cyclonic side in figure 8 is quantified in figure 9 as a significant plateau that is attained at a higher (lower) $C_f$ for case P04 (P10) than for case 00. This is consistent with Brethouwer (Reference Brethouwer2017) (among others), who notes a reduction in skin friction on the anti-cyclonic wall beyond $Ro\approx 0.45$. Negative rotation, on the other hand, attenuates the peaks near reattachment. This indicates that the reattachment at the end of the long separation bubbles of cases N04 and N10 is characterized by diffusion and mild impingement compared to the former cases. As shown in figure 7(b), the separating streamlines indeed exhibit more mild curvature than the positive-rotating cases. Comparing cases N04 and N10, the positive peak after reattachment is negligible at the higher rotation rate, thus $C_f$ reaches its asymptomatic value shortly downstream of reattachment. In case N04, conversely, the recovery of $C_f$ is significantly slower, such that the wake persists downstream until $x\approx 32H$. Along the top, planar wall, the streamwise variation of $C_f$ due to the bump blockage remains similar among most cases, with a shift in the asymptotic value to which $C_f$ recovers in each case. Again, $C_f$ is reduced (increased) when the top wall is the cyclonic (anti-cyclonic) wall during rotation. In the study of a rotating BFS by Barri & Andersson (Reference Barri and Andersson2010), an additional laminar separation bubble is observed on the cyclonic, planar wall opposite the step. This does not occur in the positive-rotating cases with the current configuration. Rather, $C_f$ remains positive at its minimum when the flow mildly expands over the bump and the SSL.

9.1. Instantaneous flow field

The small-scale structures are shown by the isosurfaces of the second invariant of the velocity gradient tensor ($Q:= - (1/2)(\partial u_j/\partial x_i)(\partial u_i/\partial x_j)$) in figure 17. In figure 18, spanwise vorticity ($\omega _z$) and $Q$ are extracted in the streamwise–wall-normal ($x\unicode{x2013}y$) plane at the channel midspan to visualize the shear layer and the wall-normal distribution of the vortices. Notably, TG vortices are not able to be visualized by these quantities due to their large-scale coherence. Thus instantaneous flow fields of streamwise vorticity ($\omega _x$) and wall-normal velocity in the cross-flow ($z$–$y$) plane in the wake of the bump are provided in figure 19 to visualize them. They are observed by the large-scale counter-rotating swirls, indicating alternating regions of upwash and downwash. The absence of vortices in the streamwise-elongated regions in the top view of the $Q$ isosurfaces (figure 17) serves as an additional depiction of TG vortices.

Figure 17. Isosurfaces of the second invariant of the velocity gradient tensor visualized at a level ${Q=2.0 U_b^2/H^2}$. (a,c,e,g,i) isometric view coloured by wall-normal distance to the bottom wall ($y/H$). (b,d,f,h,j) top view coloured by instantaneous streamwise velocity fluctuation ($u'/U_b$). From top to bottom, rows correspond to cases 00, P04, P10, N04, N10, as in preceding figures. Note that for the anti-cyclonic cases, the circled packets of ejected hairpin vortices correspond to the TG vortex-induced upwash indicated by arrows in figure 19.

Figure 18. Contours of instantaneous flow in the channel midspan: (a) spanwise vorticity ($\omega _z$); (b) second invariant of the velocity gradient tensor ($Q$, see text).

Figure 19. Contours of instantaneous flow in the cross-flow plane at $x/H=7.0$: (a) streamwise vorticity $\omega _x$; (b) wall-normal velocity and velocity vectors. The vectors are shown every $0.3H$ in $z$, and $0.1H$ in $y$, normalized by their magnitude for clarity. Horizontal lines represent (if applicable):  $y\vert _{S=-0.5}$; $y\vert _{{max}(\partial U/\partial y)}$; $y\vert _{U=0}$. Note that for the anti-cyclonic cases, the regions of upwash indicated by arrows correspond to the regions of ejected hairpin vortices circled in figure 17.

When the channel is not rotating, a roller vortex can be seen underneath a nest of hairpin vortices in the SSL. The latter dominates the downstream region with several well-defined arched heads appearing randomly across the domain. In the positive-rotating cases, the SSL is strongly disrupted. Weakly coherent spanwise-oriented vortices in the shear layer for case P04 are highlighted in figure 17 and the spanwise vorticity contour of figure 18. It is observed that TG vortices induce spanwise variation of the separation point for the anti-cyclonic cases, as shown through instantaneous streamwise velocity on the bump along with $z$–$y$ planes of instantaneous wall-normal velocity in figure 20. Regions of upwash and downwash induced by the TG vortices are clearly correlated with locally separated and attached flow, respectively. The hairpin vortices in the incoming flow are elongated in the streamwise direction as shown by the $Q$ isosurfaces. They are longer and more parallel to the wall in case P10 than in P04. This agrees with the findings in the literature that rotational instability favours the growth of streamwise vortices (Lesieur, Yanase & Métais Reference Lesieur, Yanase and Métais1991; Yanase et al. Reference Yanase, Flores, Métais and Riley1993; Métais et al. Reference Métais, Flores, Yanase, Riley and Lesieur1995; Lamballais et al. Reference Lamballais, Lesieur and Métais1996), and that hairpin legs are enhanced in rotating flows (Iida, Tsukamoto & Nagano Reference Iida, Tsukamoto and Nagano2008; Yang & Wu Reference Yang and Wu2012; Brethouwer Reference Brethouwer2017).

Figure 20. Contours of instantaneous streamwise velocity on the bump surface and wall-normal velocity in the $z$–$y$ plane. Velocity vectors are superposed onto the $z$–$y$ plane. Only the non-rotating and anti-cyclonic cases are displayed since the TG vortices interact with the SSL only on the anti-cyclonic side.

The wake of the bump exhibits rich dynamics represented by the evolution of the hairpin vortices. In the spanwise vorticity and $Q$ contours at the channel midspan (figure 18), it can be seen that the series of hairpin vortices is distributed along a path that extends from the SSL towards the channel centreline. This trajectory aligns with the extended $-1< S<0$ region and where $\overline {v'v'}$ and $\overline {w'w'}$ exhibit enhancement. This trend can also be observed in the $Q$ isosurfaces (figure 17) and the cross-flow contours (figure 19), where a clear correlation of the region between TG vortex pairs (i.e. strong upwash) and the ejection of hairpin vortices can be identified. At the time instants shown in these figures, four ejection gaps can be seen in case P04, and six are visible in case P10. This agrees with the literature that TG vortices are smaller and closer to the anti-cyclonic wall at high rotation rates. Correlating the cross-flow contours, the $Q$ isosurfaces and the Reynolds stresses contours (figures 11–14), the eddies being ejected have large streamwise vorticity near the wall, thus corresponding to the legs of the hairpin vortices. The bottom of these legs creates the high $\overline {w'w'}$ in the vicinity of the wall, while their rotation appears as the strong $\overline {v'v'}$ and $\overline {u'u'}$. Beyond $y/H\sim -0.6$ to $-$0.7, the streamwise vorticity decreases, forming arched structures in the cross-flow contours. This is where the head of the hairpin is formed. The vortex filaments generated by the breakdown of the hairpin heads can be ejected up to the cyclonic side of the channel at $y/H\sim 0.5$ (refer to the 2-D $Q$ contour in figure 18). Due to the smaller TG vortices at the higher rotation rate, the width and ejection height of the hairpin vortices in the wake of the bump are smaller in case P10 than in P04.

We interpret the correlation between the $-1< S<0$ region and the trajectory of the hairpin vortices as follows. Recall that the lee side of the bump is in the destabilized regime, as discussed in § 7. Therefore, hairpins here are ejected into a region where they are augmented and able to extract more significant energy from the mean flow (versus being ejected into a neutrally stable regime, as occurs in a planar rotating channel). Furthermore, the ejection regions between TG vortices are found to have negative $u^\prime$, as is clearly visible in the top view of the $Q$ isosurfaces. Thus the ejection mechanism is similar to the ejection events between low-speed streaks in TBLs: a local low streamwise velocity will increase the wall-normal velocity by mass conservation. However, the causation is the opposite here since the upwash generated by the TG vortex pair initiates the local deceleration. The reduced streamwise velocity increases $\partial u/\partial y$ in the wake of the bump (i.e. away from the wall). As discussed in the stability regime section (§ 7), an increased velocity gradient leads to a less negative $S$ that makes such regions unstable. The extraction of kinetic energy from the mean flow to $\overline {u'u'}$ and redistributed to $\overline {v'v'}$ via the Coriolis terms is the statistical description of the augmentation of the hairpin vortices mentioned above. Moreover, we attribute this process to the inclined extension of the destabilization region that reaches up to the channel centreline. On one hand, it can be viewed as a faster recovery of the near-wall region to the neutral $S=-1$ regime, compared to the channel centreline. On the other hand, it also represents the consequence of the continuous mean momentum extraction as the enhanced hairpin vortices are ejected. Both of these two processes are characterized by the Reynolds stress divergence in the mean momentum budget, and the production and turbulent diffusion in Reynolds stress budgets. The physical picture corresponding to this process is that as the hairpin vortices are lifted in the ejection regions, their (enhanced) legs bring high momentum to the region underneath by their rotation to facilitate the recovery of the near-wall flow. Yet they keep draining the mean momentum at their height through the Coriolis terms. Thus the region away from the wall attains a prolonged modulation. This process continues until the TG vortices cannot further sustain the ejection of the enhanced hairpin vortices, and/or the hairpin vortices break down such that they no longer extract significant energy from the mean flow.

In the negative-rotating cases, the hairpin and TG vortices are on the opposite side of the bump and do not directly impact the SSL. The $Q$ isosurfaces show the footprints of the oblique waves on the roller vortices formed far downstream of the bump. A spanwise wavelength about half of the spanwise domain size can be seen. Case N04 exhibits significantly more distorted vortices than N10. This is consistent with the literature that turbulent spots on the stable side diminish as the rotation rate increases (Grundestam et al. Reference Grundestam, Wallin and Johansson2008a; Brethouwer Reference Brethouwer2017). Note that there are diverse opinions about the origin of these turbulent spots. Local cyclic turbulent bursts by the growth and breakdown of Tollmien–Schlichting (TS) waves, and turbulent fluctuations penetrating into the stable region, are two main mechanisms proposed (Kristoffersen & Andersson Reference Kristoffersen and Andersson1993; Brethouwer et al. Reference Brethouwer, Schlatter, Duguet, Henningson and Johansson2014; Brethouwer Reference Brethouwer2016; Dai et al. Reference Dai, Huang and Xu2016). For the flow in the current study, we are inclined to the latter because: (1) there is a prominent ejection of the hairpin vortices excited by the extended unstable region in the wake of the bump; and (2) the relatively low $Re_\tau$ and $Ro_\tau$ prevent the TS waves from reaching very large amplitude (Brethouwer et al. Reference Brethouwer, Schlatter, Duguet, Henningson and Johansson2014). The cross-flow contours of these two cases (figure 19) exhibit that the head of the hairpin vortices ejected from the opposite side indeed reaches the SSL in case N04, while it does not seem so in case N10. This is again because the TG vortices are smaller and nearer the anti-cyclonic wall as the rotation rate increases.

The dynamics of the structures elaborated above indicates that any increase in shear, whether by deceleration of near-wall flow as observed in this study, or acceleration of outer flow, will result in an increased $\partial U/\partial y$, which increases the absolute vorticity ratio from $S\sim -1$ to $-1< S<0$. In such a region, the hairpin vortices in the boundary layer will be augmented and sustained longer as they are ejected by the TG vortices, reaching further into the core of the flow (compared with in the same rotating system under a zero pressure gradient, in which the hairpins also get ejected by the TG vortices). Therefore, this process could also occur in attached flows, including on the anti-cyclonic side of our negative-rotating cases. Indeed, we observed an increased $S$ near the top anti-cyclonic wall, and enhanced $\overline {v'v'}$ and $\overline {w'w'}$ in cases N04 and N10 (figures 12 and 13). Therefore, the turbulent patches superposed to the oblique wave and disturbing the laminar SSL in case N04 are likely energized filaments of the ejected hairpins from the opposite (attached) side. In particular, the increased $\overline {v'v'}$ and $\overline {w'w'}$ near the anti-cyclonic wall in cases N04 and N10 are most prominent not immediately downstream of the bump, but where the wall-parallel laminar shear layer starts to curve towards the wall, creating an expansion of the nominal cross-sectional area and thus an APG. This supports our claim that near-wall deceleration is what causes destabilization of anti-cyclonic flow, leading to the observed augmentation of hairpin vortices and turbulence.

9.2. Premultiplied spanwise spectrum

The spanwise wavelength and wall-normal location of the embedded structures are quantified by the premultiplied energy spectrum of velocity fluctuations in figure 21. Note that each spectrum is normalized by its peak magnitude. Without rotation, the peaks of the $\overline {u'u'}$ and $\overline {v'v'}$ spectra collapse at the location of the inflection point, representing the generation of roller vortices. The primary wavelength is approximately $0.7H$. In wall units, it corresponds to $\lambda _z^+\sim 110$, the spanwise spacing of near-wall streaks in canonical channel flows. This indicates the interaction between low-speed streaks and the SSL. After the flow reattaches, the spanwise wavelength is reduced as three-dimensionality develops.

Figure 21. Spanwise premultiplied energy spectra of the wall-normal ($k_z\varPhi _{vv}$) and streamwise ($k_z\varPhi _{uu}$) fluctuating velocity at (from left to right) $x/H = 4$, 5, 5.5, 6, 7 and 8, for (a–e) cases 00, P04, P10, N04 and N10, respectively. Contours represent $k_z\varPhi _{vv}$, while the dashed contour lines represent $k_z\varPhi _{uu}$. Spectra are normalized by their maximum values. Contour levels 0.05, 0.08, 0.1, 0.2, 0.4, 0.6 and 0.8 are displayed for $k_z\varPhi _{uu}$. The horizontal solid black line represents the wall-normal location of maximum velocity, the dashed blue line represents the wall-normal location of $S=-0.5$, and the dash-dotted green line represents the wall-normal location of the maximum velocity gradient.

For the four rotating cases, the $\overline {v'v'}$ spectrum shows a peak away from the anti-cyclonic wall, which represents the TG vortices. The top and bottom limits of the TG vortices can be determined by the wall-normal locations where the outer peak of the $\overline {v'v'}$ spectrum diminishes. Our results show that the former is represented by the location of the peak streamwise velocity, and the latter is correlated to where $S=-0.5$. The spectrum indicates that the TG vortices persist across the entire channel, including over the bump where they get displaced. They have characteristic spanwise wavelength $1.5H$ at the lower rotation rate (i.e. four pairs over the spanwise extent of the domain), and $1.0H$ at the higher rotation rate (i.e. six pairs). The core of the TG vortices, denoted by the location of the peak of the spectrum, moves closer to the anti-cyclonic wall, and the wall-normal extent reduces as the rotation number increases. These results are consistent with previous studies (Kristoffersen & Andersson Reference Kristoffersen and Andersson1993; Dai et al. Reference Dai, Huang and Xu2016; Brethouwer Reference Brethouwer2017).

The augmentation and streamwise evolution of the hairpin vortices associated with the ejection mechanism described above are quantified by streamwise development of the spectra. Focusing initially on case P04, near-wall peaks of $\overline {u'u'}$ and $\overline {v'v'}$ spectra develop at $x/H=5$ and $x/H = 5.5$, respectively. These peaks lie between the inflection point of the shear layer and beneath the TG vortices, indicating that they represent the hairpin vortices and not rollers associated with the separation. The peaks occur in the self-exciting regime ($S>-0.5$), indicating that $\overline {u'u'}$ and $\overline {v'v'}$ will be enhanced. At this streamwise location, the characteristic wavelength of the inner peaks is lower than that of the TG vortices. Downstream of reattachment ($x/H= 6$), the inner peaks of $\overline {u'u'}$ and $\overline {v'v'}$ shift away from the wall. Further downstream, the $\overline {v'v'}$ spectrum merges into a single peak; the $\overline {u'u'}$ spectrum overlies that of $\overline {v'v'}$, indicating a strong correlation between the two. The characteristic wavelengths of the $\overline {u'u'}$ and $\overline {v'v'}$ spectra are now corresponding to that of the TG vortices. This continuous trajectory of the $\overline {u'u'}$ and $\overline {v'v'}$ peaks away from the wall moving downstream from the bump corresponds to the ejection of the hairpin filaments by the TG vortices. For case P10, the dynamics are similar, yet the wall-normal extent of the ejection process is less than for P04 (as expected due to the smaller, more confined TG vortices at the high rotation rate). Due to the lower characteristic wavelength of the TG vortices in case P10, there is no scale separation between them and the hairpin vortices. However, the ejection process described by spreading and upward trajectory of the $\overline {u'u'}$ remains qualitatively consistent with case P04, indicating that the same ejection process is occurring.

Also noteworthy is that on the cyclonic side, the $\overline {u'u'}$ spectrum shows a peak at $\lambda _z\sim 3H$, representing the oblique wave. The centre of this wave is near the peak streamwise velocity at approximately $y/H=0.5$. The characteristic wavelength of the oblique wave is consistent at the high rotation rate.

When subject to negative rotation, the TG vortices appear on the top (anti-cyclonic) side of the channel, and therefore do not interact with the SSL. The SSL is also not at the same wall distance as the oblique waves. In case N04, the two are slightly nearer since peak velocity occurs closer to the cyclonic wall. Yet the spanwise wavelength of the roller vortices in the laminar SSL matches that of the oblique waves. This suggests that the latter is the primary source of disturbance for the three-dimensionality of the shear layer.