2.1. Direct numerical simulations

The fluid flow is simulated by the spectral element code Nek5000 (Fischer Reference Fischer2017). The code employs the spectral finite element method (SEM), based on the Galerkin discretization of the weak form of the Navier–Stokes equations. The fluid domain is split into hexahedral elements where each fluid quantity $q$ is expressed in terms of Lagrange polynomials with nodes following the distribution of Gauss–Lobatto–Legendre (GLL) points. Each element is discretized with $l_x l_y l_z=l_x^3$ GLL points, where $l_x=l_y=l_z$ are the number of points used for each edge of the element. The Navier–Stokes equations are discretized in time using an operator integration factor splitting scheme (Maday, Patera & Rønquist Reference Maday, Patera and Rønquist1990) that permits large time steps $\Delta t$ due to its semi-implicit nature. The time step is set to the constant value $\Delta t= 1.69\times 10^{-4}$ and the related Courant–Friedrichs–Lewy number is in the range 1.5–2.0. Figure 2 depicts the mesh around the lower bar. The mesh radiates from the bar to guarantee a high clustering of points especially around the left and the right top corners, where singular behaviours of the local pressure and the local wall-shear stress occur. A total number of $76\,496$ hexahedral elements are employed. The mesh is identical on each $x$–$y$ plane and 14 identical elements are used along $z$. A number of $6$ GLL points is adequate to compute the total drag accurately, as shown by the convergence study in appendix A. The first point above the cavity is at a distance $\Delta y=1.1\times 10^{-3}$ from the wall, while the last point along $y$ is at a distance $\Delta y=1.3\times 10^{-2}$ from the centreline. The maximum streamwise resolution is $\Delta x=1.5\times 10^{-2}$. The minimal $\Delta x=\Delta y =5.7\times 10^{-5}$ refer to the grid elements at the side edges and the crests of the bars, respectively. Even though the spectral elements along $z$ are identical, the spanwise resolution varies from $\Delta z_{min}=2.6\times 10^{-2}$ to $\Delta z_{max}=6.4\times 10^{-2}$. All the simulations are performed on the high performance computing facility of the A*STAR Computational Resource Centre in Singapore, employing 2000–8000 processors. The flow statistics and the drag coefficients are computed after the end of an initial transient time, evaluated by the inspection of the time history of total drag. This is crucial to assess that the flow has reached a statistically steady state.

Figure 2.

Mesh for the channel with bars. The grid is symmetrical around the axes $x^*=4\delta ^*$ and $y^*=1.2\delta ^*$ and only the quarter of the computational domain in the range $\left [ 0,4\delta ^* \right ] \times \left [ 0,1.2\delta ^* \right ]$ is shown. Only the hexahedral elements are shown.

2.2. Averaging operators and flow decomposition

The average of a quantity $q$ along the spanwise direction is

(2.2)\begin{equation} \left \langle q \right\rangle(x,y,t)=\frac{1}{L_z}\int_0^{L_z} q(x,y,\hat{z},t) \, \mathrm{d} \hat{z}, \end{equation}

where the integral is approximated via the Gauss–Legendre quadrature that takes into account the distribution of the GLL points. The time-ensemble average is

(2.3)\begin{equation} \hat{q}\ (x,y,z,\tau)=\frac{1}{N}\sum_{n=0}^{N-1} q \left( x,y,z,\tau+n \, T \right), \quad 0\leq \tau< T, \end{equation}

where $N$ is the number of periods, and the time average between times $t_i$ and $t_f$ is

(2.4)\begin{equation} \bar{q}(x,y,z)=\frac{1}{t_f-t_i}\int_{t_i}^{t_f} q\left(x,y,z,\hat{t}\ \right) \,\mathrm{d} \hat{t}. \end{equation}

A quantity $q$ of the flow field is decomposed as

(2.5)\begin{equation} q=q_m\left( x,y \right) +q_{osc}\left( x,y,\tau \right) +q_t\left( x,y,z,t \right), \end{equation}

where $q_m\left ( x,y \right ) =\left \langle \bar {q} \right \rangle$ is the mean flow, $q_{osc}\left ( x,y,\tau \right ) =\left \langle \hat {q} \right \rangle -q_m$ is the time-periodic flow and $q_t=q-\left \langle \hat {q} \right \rangle$ is the turbulent fluctuation.

2.3. Parameters defining the actuation performance

For a separated flow the total drag force is composed of a pressure-drag force $F_p^*$ and a skin-friction drag force $F_f^*$, defined as

(2.6a,b)\begin{equation} F_p^*=\int_{S^*} p^*\, \boldsymbol{i} \boldsymbol{\cdot} \boldsymbol{n} \, \mathrm{d} S^*, \quad F_f^*=-\int_{S^*} \left.\mu^* \frac{\partial u^*}{\partial \boldsymbol{n}}\right\vert_w \, \mathrm{d} S^*, \end{equation}

where $\boldsymbol {n}$ is the normal unit vector pointing out of the fluid domain, $\boldsymbol {i}$ is the unit vector along the $x$ direction, $S^*$ is the solid surface and the $w$ denotes the wall. For the vertical surfaces of the bar $\boldsymbol {n} \boldsymbol {\cdot } \boldsymbol {i}\neq 0$, and therefore the pressure drag force is

(2.7)\begin{equation} F_p^*=\int_{S^*_l} \overline{p^*} \, \mathrm{d} S^* -\int_{S^*_r} \overline{p^*} \, \mathrm{d} S^*, \end{equation}

where $S^*_l$ and $S^*_r$ are the left and right vertical surfaces of the bar of area $h^*L_z^*$. The skin-friction drag is

(2.8)\begin{equation} F_f^*=F^*_{ca}+F^*_{cr}=\int_{S^*_{ca}} \left.\mu^* \frac{\partial \overline{u^*}}{\partial y^*}\right|_w \, \mathrm{d} S^* +\int_{S^*_{cr}} \left.\mu^* \frac{\partial \overline{u^*}}{\partial y^*}\right|_w \, \mathrm{d} S^*, \end{equation}

where $S^*_{ca}$ and $S^*_{cr}$ are the cavity and crest surfaces, respectively. Note that the minus sign in (2.6a,b) is absent in (2.8) because, for the bottom horizontal wall, $\boldsymbol {n}=-\boldsymbol {j}$, where $\boldsymbol {j}$ is the unit vector along the $y$ axis. The forces $F^*_{ca}$, $F^*_{cr}$ and $F_p^*$ include contributions from the bottom and top walls. The drag coefficients of the skin friction along the cavity ($C_{ca}$), the skin friction along the crest ($C_{cr}$), the pressure ($C_p$) are defined by dividing $F^*_{ca}$, $F^*_{cr}$ and $F_p^*$ by $\rho ^*{U^*_p}^2 L^*_xL^*_z$, respectively. The skin-friction coefficient is composed of the crest and the cavity drag coefficients, i.e. $C_f=C_{cr}+C_{ca}$. We do not define the drag coefficients on the left and the right bar sides and their reductions because, as remarked by Banchetti et al. (Reference Banchetti, Luchini and Quadrio2020), it is not meaningful to define a local pressure coefficient for an incompressible flow as the pressure values can be shifted by any constant.

We define a percentage reduction for each drag coefficient with respect to its value in the uncontrolled case, denoting these reductions with the letter $\mathcal {R}$ and the same subscript of the drag coefficients. For instance, the percentage reduction of the skin-friction coefficient along the cavity is

(2.9)\begin{equation} \mathcal{R}_{ca}(\%)=100 \left( 1-\frac{{C_{ca}}_o}{C_{ca}} \right), \end{equation}

where $C_{ca}$ and ${C_{ca}}_o$ are the uncontrolled and controlled values, respectively (the subscript $o$ stands for oscillating). The total drag reduction $\mathcal {R}(\%)$ is defined as

(2.10)\begin{equation} \mathcal{R}(\%) = 100 \left( 1- \frac{{F_p}_{o}^*+{F_f}_{o}^*}{F_p^*+F_f^*}\right)= 100 \left( 1- \frac{{C_f}_{o}+{C_p}_{o}}{{C_f}+{C_p}} \right), \end{equation}

where the pressure-drag force $F_p^*$ is defined in (2.7) and the skin-friction drag force $F_f^*$ is defined in (2.8). Banchetti et al. (Reference Banchetti, Luchini and Quadrio2020) fixed the total pressure at the inflow location of their computational domain in order to quantify the effect of the pressure drop given by their control method. In all our cases, we instead maintain the pressure constant at the further end of our computational domain.

As we study an active technique, it is crucial to account for the power $\mathcal {P}_z(\%)$ supplied to the fluid for controlling the flow, defined as

(2.11)\begin{equation} \mathcal{P}_z(\%)=\frac{100}{\bar{\varPi}_x U_b (t_f-t_i)} \int_{t_i}^{t_f} \varPi_z\left( \hat{t} \right)\, W_b\left( \hat{t} \right) \, {\mathrm{d} \hat{t}}, \end{equation}

expressed as a percentage of the power used for driving the fluid along the streamwise direction. In (2.11) $\bar {\varPi }_x$ is the mean pressure gradient driving the flow along $x$ in the uncontrolled case, $U_b$ is the constant volume-averaged velocity, $U_b=V^{-1} \int _{V} \bar {u} \, \mathrm {d} V$, $W_b(t)$ is the spanwise volume-averaged velocity due to the control, $W_b=V^{-1} \int _{V} w(x,y,z,t) \, \mathrm {d} V$ and $V$ is the fluid volume. Appendix B presents the derivation for the power used to drive the fluid along $x$, ${P}_x=-\bar {\varPi }_x U_b>0$.

It is verified numerically that the force exerted by the pressure gradient to drive the fluid along the streamwise direction matches the sum of the drag forces, i.e. $-\bar {\varPi }_x L_y=C_f+C_p$ and $-{\bar {\varPi }_x}_o L_y={C_f}_o+{C_p}_o$. By using this force balance, the percentage of power saved is

(2.12)\begin{equation} \mathcal{P}_{sav}(\%) = 100 \left( 1 - \frac{\varPi_{x_o} U_b}{\bar{\varPi}_{x} U_b}\right) = 100 \left( 1- \frac{\bar{\varPi}_{x_o}}{\bar{\varPi}_x} \right) = 100 \left( 1 - \frac{C_{f_o} + C_{p_o}}{C_f + C_p} \right) =\mathcal{R}(\%), \tag{2.12}\end{equation}

because the volume-averaged velocity $U_b$ is constant.

The difference between the power $\mathcal {P}_{sav}(\%)$ saved through the oscillating pressure gradient and the power $\mathcal {P}_z(\%)$ spent for controlling the flow is the net power saved

(2.13)\begin{equation} \mathcal{P}_{net}(\%)=\mathcal{R}(\%)-\mathcal{P}_z(\%). \end{equation}

4.1. Uncontrolled flow

The mean-flow streamlines in the region surrounding the bar in the uncontrolled flow case are depicted in figure 5. The separation zone is occupied by three recirculating vortices: two of them, A and B, are adjacent to the vertical surfaces of the bar and have a size comparable with the bar height. Vortex C is located downstream and on top of vortex B, extending for a length of more than five bar heights along the streamwise direction and covering the crest where a mild separation occurs. Vortices A and C rotate clockwise, while vortex B rotates anticlockwise. Sufficiently far from the bar ($x<3$ and $x>6.5$) the flow is not influenced by the obstacle and the mean-flow streamlines become straight and aligned with the streamwise direction. We define the reattachment points at the positions where the time and spanwise-averaged wall-shear stress is zero. They separate the forward and backward mean flows near the wall and are found at $x=3.56, 4.39, 5.38$. The mean streamlines agree visually with those of Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003) and Ikeda & Durbin (Reference Ikeda and Durbin2007), although their Reynolds number is slightly larger.

Figure 5.

Spanwise- and time-averaged streamlines for the flow without control. The recirculation vortices are indicated by the letters A, B and C.

Figure 6 shows the mean pressure of the uncontrolled flow. The pressure reaches high values right upstream of the bar where vortex A occurs, with the maximum pressure occurring along the left vertical surface of the bar. The core of vortex C, located approximately at $x=4.6$, is characterized by a low pressure. The effect of the bar on the wall-shear stress is clearly shown in figure 7 by the distribution of the local friction Reynolds number $Re_\tau =u_\tau ^* (h^*+\delta ^*)/\nu ^*$ as a function of the streamwise direction, where $u_\tau ^*=\sqrt {\tau _w^*/\rho ^*}$ is the local wall-friction velocity and $\tau _w^*$ is the time- and spanwise-averaged wall-shear stress. The Reynolds number $Re_\tau$ is only computed where the wall-shear stress is positive. It varies significantly in the proximity of the bar, while it is almost constant at $Re_{\tau }=218$ away from the bar, for $x<2$ and $x>6$. It must be remarked that scaling quantities with the local wall-friction velocity is only meaningful in the region where the flow is fully attached and relatively undisturbed from the separation regions so that the skin friction approaches a positive constant. It is instead not appropriate to employ wall-friction scaling in the recirculating regions where the skin friction is either vanishingly small or negative. The constant wall-friction velocity in the fully attached region, $u_\tau =0.044$, is used to scale quantities in wall units, as follows

(4.1)\begin{gather} A^+ = \frac{A^* \nu^*}{\rho^* u_{\tau}^{*3}} = \frac{A(h+1)^3 Re_p^2}{h Re_\tau^3}, \end{gather}

(4.2)\begin{gather}T^+ = \frac{T^* u_{\tau}^{*2}}{\nu^*} = \frac{T Re_\tau^2}{(h+1)^2 Re_p}, \end{gather}

(4.3)\begin{gather}J^+ = \frac{J^*}{\rho^* u^*_{\tau}} = \frac{J (h+1) Re_p}{Re_\tau}. \end{gather}

Scaling the forcing parameters in wall units is useful to compare the performance of the actuation in reducing the skin-friction drag of the attached flow along the cavity with the widely studied spanwise wall-oscillation technique (Quadrio & Ricco Reference Quadrio and Ricco2004). We also find that the uncontrolled scaled wall-shear stress of the attached flow away from the bars, $Re_p^{-1}\mathrm {d}U/\mathrm {d}y\vert _{y=0}=0.0034$, is less than 1 % different from the coefficient found from the empirical correlation $Re_p^{-1}\mathrm {d}U/\mathrm {d}y\vert _{y=0}=0.0336 U_b^2 Re_\tau ^{-0.273}$ (Pope Reference Pope2000), valid for fully developed turbulent channel flows over flat walls.

Figure 6.

Colour map and contour lines of the mean pressure in the uncontrolled case. The isoline values are equispaced at intervals of $0.02$ in the range $[-0.25,0.1]$.

Figure 7.

Local friction Reynolds number along the channel in the uncontrolled case. The location of the bar is marked by the shaded region. The thin solid line denotes the friction Reynolds number $Re_{\tau }=218$, averaged in the constant-friction region, $x<2$ and $x>6.5$.

4.2. Skin-friction and pressure-drag coefficients

As the current control strategy aims at reducing both the viscous drag along the streamwise-parallel surfaces and the significant form drag experienced by the bar because of the recirculation regions, it is central to precisely quantify the drag coefficients and their changes in the controlled cases.

The effect of the control on the drag coefficients for a large drag-reduction case with $T=6.33$ and $A=1.08$ ($J^+=50$, $T^+=50$) is presented in figure 8. The flow responds quickly to the control activation at $t=250$ and the coefficients adjust to the new value monotonically. This transient response is shorter and smoother than in the case of spanwise wall oscillations, where the wall-shear stress may evolve non-monotonically and reach a very low instantaneous value before adjusting to the new fully developed value (Quadrio & Ricco Reference Quadrio and Ricco2003). The time-averaged coefficients are computed after the end of the transient phase, estimated at around $t=270$, i.e. when all the coefficients reach a new statistically steady state. The control guarantees a substantial reduction of total drag $C_p+C_f$ due to a reduction of both $C_p$ and $C_f$. This result proves that a spanwise oscillating pressure gradient can reduce the skin-friction drag of a cavity flow and the pressure drag given by the difference on the bar sides in the integrated pressures of a fully separated flow. The drop of total pressure drag may not be achieved by the traditional separation control methods, such as vortex generators and jets, which may introduce a viscous drag penalty due to the enhanced flow mixing. The time histories of the pressure coefficient is much more oscillatory than that of the skin-friction coefficient, which is expected because the pressure drag is affected by the intense fluctuations in the separated region downstream of the bar. The skin-friction reduction is mostly due to the decrease of the cavity drag. The friction drag on the crest is very small and negative because of the reversed flow in the small recirculation region occurring due to separation. The spanwise forcing has a marginal effect on the crest friction.

Figure 8.

(a) Time evolutions of the total drag coefficient $C_p+C_f$. (b) Time evolutions of the pressure-drag coefficients $C_{p}$ and the skin-friction coefficients $C_{cr}$ and $C_{ca}$. The control parameters are $T^+=50$ and $J^+=50$. The control is activated at $t=250$ and the circles denote the start of averaging. The horizontal dashed and solid lines indicate the mean uncontrolled and controlled coefficients, respectively.

Table 1 lists the values of the drag coefficients for different amplitudes $A^+$, periods of oscillations $T^+$ and impulses in the range $J^+=1\text {--}50$. In uncontrolled conditions, the contribution of the pressure drag to the total drag is more than four times the contribution of the skin-friction drag.

Table 1.

Control parameters and mean drag coefficients. In this table and in tables 2 and 3, the case highlighted in bold is studied in § 5.

Table 2 presents the computed percentage variations of the drag coefficients. Note that, while the partial skin-friction and pressure-drag coefficients in table 1 add to give the corresponding total coefficients, the partial reductions of the coefficients in table 2 do not because each reduction is defined with respect to the corresponding value in the uncontrolled case, as clear from the definitions (2.9) and (2.10).

Table 2.

Percentage variations of the drag coefficients. Positive values indicate drag reduction and negative values denote drag increase.

The control with small period $T=0.2$ ($T^+=1.57$) reduces the total drag marginally as the impulse $J$ is too small. The small drag reduction is due to the small reduction of $C_p$, while both the cavity and the total skin friction increase slightly. For such a low period, a variation of $A$ of two orders of magnitude in the range $0.1\text {--}10$ ($A^+=0.29\text {--}29$) does not cause any improvement. The drag increases slightly in some cases for $J^+=3.2$ and $4.4$. Both the skin-friction drag and the pressure drag decrease at higher impulse values, $J^+=15, 50$. The case with $T=10.1$ and $A=0.67$ ($T^+=80$ and $J^+=50$) shows the maximum total drag-reduction margin, $\mathcal {R}=25\,\%$, the maximum skin-friction drag-reduction margin, $\mathcal {R}_f=37\,\%$, and the maximum pressure-drag reduction margin, $\mathcal {R}_p=22\,\%$.

Figure 9 shows the total drag-reduction margin as a function of the actuation period at the impulse values $J^+=15$ and 50. For a constant $T^+$, the drag-reduction margin is larger as the impulse increases, while, for a constant $J^+$, it increases with the period up to a maximum at $T^+=80$. The existence of an optimum period of oscillation is consistent with similar control strategy for skin-friction drag reduction, such as the oscillating-wall technique in smooth channels (Ricco & Quadrio Reference Ricco and Quadrio2008). The optimal period of wall oscillation in the flat-wall case is $T^+=100$ (Touber & Leschziner Reference Touber and Leschziner2012), which gives maximum drag-reduction margins of 35 %–40 %, thus in good agreement with the maximum found in the channel with bars altered by a spanwise pressure gradient.

Figure 9.

Drag-reduction margin as a function of the oscillating period. The circles indicate the numerical results and the solid lines are the best fits at constant $J^+=A^+ T^+/{\rm \pi}$.

Figure 10 reveals that the reduction in viscous drag along the cavity contributes to most of the total viscous reduction because the skin friction on the bar crest is negligible. It is also shown that the total friction drag reduction $\mathcal {R}_f$ is almost double the total pressure-drag reduction $\mathcal {R}_p$. The viscous drag reduction reaches a maximum of approximately $37\,\%$ for $T^+=50\text {--}100$, while the pressure-drag reduction reaches its maximum value of approximately $22\,\%$ for $T^+=80\text {--}100$. The amount of skin-friction drag reduction is comparable with the maximum reduction brought about by spanwise wall oscillations in a fully developed channel flow for $T^+=100$, and similar wall-friction Reynolds numbers and maximum velocity amplitude of oscillation (Quadrio & Ricco Reference Quadrio and Ricco2004). While the scaled forces are additive, i.e. $-\bar {\varPi }_x L_y=C_f+C_p$, the reductions of viscous drag, $\mathcal {R}_f$, and of the pressure drag, $\mathcal {R}_p$, lead to $\mathcal {R}$ via

(4.4)\begin{equation} \mathcal{R} = 100 - \frac{1}{\bar{C}+1}\left[ 100 - \mathcal{R}_f + \bar{C} \left(100-\mathcal{R}_p \right) \right], \end{equation}

where $\bar {C}=C_p/C_f=4.3$.

Figure 10.

Drag-reduction margins as functions of the oscillating period for an impulse $J^+=50$.

4.3. Power spent and net power saved

Table 3 gives the values of the drag-reduction margin, $\mathcal {R}(\%)$ (which coincides with the power saved), the power spent to activate the control, $\mathcal {P}_z(\%)$, and the laminar power $\mathcal {P}_{z,l}(\%)$ for different parameters $T$ and $A$. Very marginal savings in terms of $\mathcal {P}_{net}(\%)$ are found when the drag-reduction effect is small and these values are within the uncertainty of the computed $\mathcal {R}(\%)$. For the cases with large drag reduction, the decrease in pressure gradient along the streamwise direction comes at the expense of a high consumption of energy, which leads to no savings in terms of $\mathcal {P}_{net}(\%)$. This finding opens new avenues of research to improve the net energy gain with a view to rendering this technique competitive with vortex generators which, at this stage, are more effective and guarantee estimated net drag reductions of the order of $10\,\%$ (Calarese et al. Reference Calarese, Crisler and Gustafson1985).

Table 3.

Power saved $\mathcal {P}_z(\%)$, net power $\mathcal {P}_{net}(\%)$ and laminar power $\mathcal {P}_{z,l}(\%)$ as functions of the control parameters $A$ and $T$.

Figure 11 shows the values of $\mathcal {P}_z$ obtained via the direct numerical simulations versus the laminar prediction $\mathcal {P}_{z,l}$. The laminar power $\mathcal {P}_{z,l}$ gives an excellent approximation of $\mathcal {P}_z$ for all the values of $A$ and $T$ considered. The laminar solution found in § 3 is therefore able to predict the turbulent power spent very well.

Figure 11.

Turbulent power spent $\mathcal {P}_z(\%)$ versus its laminar prediction $\mathcal {P}_{z,l}(\%)$. The solid line indicates the line $\mathcal {P}_z(\%)=\mathcal {P}_{z,l}(\%)$.

5.1. Controlled instantaneous fields and average flow

Instantaneous flow fields are shown in figures 12–14 to illustrate the effects of the square bar and of the spanwise-forcing control on the flow dynamics. Figure 12 shows isosurfaces of the invariant $\lambda _2$, proposed by Jeong & Hussain (Reference Jeong and Hussain1995) to visualize vortical structures. Vortices are formed on top and downstream of the bar, especially where the flow fully separates, and are mostly attenuated upstream of the bar. Figures 13 and 14 depict instantaneous streamwise and spanwise velocity fields in the central $x$–$y$ plane at $z={\rm \pi} /2$. The controlled streamwise velocity is much more uniform than in the reference case, both upstream and downstream of the bar. In the controlled case, the spanwise velocity is positive everywhere in the visualization plane and much larger than in the reference case because of the large spanwise pressure gradient at this phase of the oscillation.

Figure 12.

Isosurfaces of $\lambda _2=-1.0$ in the channel for the uncontrolled case (a) and the controlled case (b). The isosurface contours are coloured by the streamwise velocity.

Figure 13.

Contour plots of instantaneous streamline velocity $u$ in the channel for the uncontrolled case (a) and the controlled case (b).

Figure 14.

Contour plots of instantaneous spanwise velocity, $w$, in the channel for uncontrolled (a) and controlled case (b).

The contours of the mean streamwise velocity, shown in figure 15, confirm that the main flow features around the bar are qualitatively unvaried in the controlled case. The three recirculation regions are still present; the large recirculation area behind the bar is slightly compressed towards the wall and the one upstream of the bar is weakened. Figure 16 compares the mean streamlines in the uncontrolled and controlled cases. The forcing reduces the size of vortices B and C and brings them closer to the wall, thus rendering the flow more attached downstream of the bar. The reattachment point upstream of the vortex A displaces slightly and a small vortex forms upstream of vortex A with its core at $x=3.72$. After the bar, although the control shifts the second reattachment point downstream, this change only marginally affects the pressure-drag balance, which is instead mainly influenced by the dynamics in the whole separated region and by the flow modifications upstream of the bar. Similar changes of the separation area as an effect of the control were also found in previous separation drag studies (Minelli et al. Reference Minelli, Tokarev, Zhang, Liu, Chernoray, Basara and Krajnović2019; Banchetti et al. Reference Banchetti, Luchini and Quadrio2020).

Figure 15.

Contour plots of time- and spanwise-averaged velocity $\langle \bar {u} \rangle$ in the channel for uncontrolled (a) and controlled case (b). The contour lines are for the negative velocity range of $\langle \bar {u} \rangle$ in $(-0.2,-0.15,-0.1,-0.05,-0.025)$.

Figure 16.

Effect of the control on the separation areas. Dashed and solid lines indicate the time- and spanwise-averaged streamlines for the uncontrolled and controlled flow, respectively.

5.2. Distribution of pressure and skin-friction drag coefficients

The difference of the time- and spanwise-averaged pressure between the controlled case and the uncontrolled case is shown in the contour plot of figure 17. The pressure is unvaried in a region that extends from the bar upward toward the channel centreline. The pressure upstream of the bar decreases, while the pressure instead increases in the region downstream the bar with the maximum increase inside the area delimited by the isoline around the point $(x,y)=(4.75,0.1)$. This dual effect on the pressure distribution is the main cause of the reduced pressure difference on the two sides of the bar, which gives a lower overall pressure drag.

Figure 17.

Colour map and contours of the difference of the mean pressure between the uncontrolled case and the controlled case. The positive values indicate a pressure increase and the negative values denote a pressure decrease. The isoline values are equispaced at intervals of $0.002$ in the range $[-0.02,0.02]$.

Figure 18 depicts the mean pressures in the controlled and uncontrolled cases (18a) and their difference (18b). The change of pressure is much smaller away from the bar, especially downstream of the large separation region, in the fully attached region for $s>6.5$, where $s$ is the streamwise coordinate running along the solid wall. The pressures are kept equal to zero at the further end of the domain, as discussed in § 2.3. Excellent agreement is found between our data and the data behind the bar from the direct numerical simulations of Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003) (red circles), obtained for a flow over a squared bar with the same height as ours, a slightly different Reynolds number and a distance between bars which is approximately half of ours ($h/L_x=19$). This match confirms that the recirculation area is largely unaffected once the bars are sufficiently spaced apart. The distribution of the pressure difference in the separated region is qualitatively similar to that found by Banchetti et al. (Reference Banchetti, Luchini and Quadrio2020) for a channel flow modified by a smooth bump and altered by wall travelling waves. In their flow, the total pressure sufficiently downstream of the obstacle recovers the negative constant slope behaviour proper of channel flows over smooth walls. In our case, figure 18(a) instead shows that the pressure further downstream of the bar tends to a constant value because the effect of the intense separation is still present. The same trend was found by Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003). This finding is expected because Banchetti et al.'s (Reference Banchetti, Luchini and Quadrio2020) flow is characterized by a mild separation being their obstacles smoother and thinner than our bars and more spaced apart along the streamwise direction.

Figure 18.

(a) Time- and spanwise-averaged total pressure in the uncontrolled case (dashed line) and the controlled case (solid line) as a function of the coordinate $s$ running along the wall surface. The sides of the bar are shown in grey and the crest of the bar is marked in light blue. The red circles are data from the direct numerical simulations of Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003). (b) Difference between the time- and spanwise-averaged pressure in the controlled and uncontrolled cases.

The graphs of figure 19 show that the wall-shear stress decreases along the major part of the cavity area where the flow is attached ($x<3.5$ and $x>5.4$), while there is no significant change of skin friction upstream of the bar ($3.5<x<3.9$). The wall-shear stress is mostly affected by the forcing when it is positive, i.e. where the flow is attached along the cavity, reaching a reduction of $31\,\%$. This value is consistent with the drag-reduction margin of $\mathcal {R}=35\,\%$ obtained by forcing a flat-wall turbulent channel flow by spanwise wall oscillations at a similar friction Reynolds number, period $T^+=50$ and maximum spanwise velocity of approximately $27u_\tau ^*$ (Quadrio & Ricco Reference Quadrio and Ricco2004). As for the pressure distribution, the red circles show that the skin-friction distribution in the recirculation area agrees very well with that of Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003).

Figure 19.

(a) Time- and spanwise-averaged wall-shear stress in the uncontrolled case (dashed line) and the controlled case (solid line). The red circles are data from the direct numerical simulations of Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003). (b) Difference between the time- and spanwise-averaged wall-shear stress caused by the control. The prime indicates derivation with respect to $y$.

The negative skin friction in the region behind the bar is instead largely unperturbed. The wall-shear stress in the separation area B between $x=4.1$ and $x=4.4$, defined in figure 16, is unaffected by the forcing, although the near-wall flow is along the streamwise direction. This finding is likely to be due to the stress being small and the flow not being in the fully developed turbulent regime there. This result is consistent with the spanwise forcing not having an effect on a streamwise laminar flow (Ricco & Hicks Reference Ricco and Hicks2018). Figure 19(a) also shows that the reattachment points upstream of the bar and immediately downstream of the bar are unaffected by the control, while the reattachment point further downstream is shifted only marginally in the streamwise direction.

5.3. Turbulent-flow statistics

Figures 20 and 21 show the time- and spanwise-averaged profiles of the streamwise and wall-normal mean velocities along the channel for the uncontrolled and controlled cases. Close to the wall, the streamwise velocity is smaller in the controlled case resulting in the lower skin friction. The conservation of mass flow rate manifests itself in the wall-normal location increasing from the wall. Away from the wall, the streamwise velocity in the controlled case becomes first larger and then smaller than in the uncontrolled case as the channel centreline is approached when the flow is streamwise only at every wall-normal location. The opposite occurs when the wall-shear stress is instead negative. The decrease of pressure on the left-hand side of the bar could be due to the reduced dynamic pressure of the flow encountering the bar, due to the lower mean velocity, shown in the first three profiles of figure 20(a). The wall-normal velocity is unaffected by the control when the flow is fully attached, while it slightly increases in the recirculation region.

Figure 20.

Profiles of streamwise mean velocity $\langle \bar {u} \rangle$ at different $x$ locations along the channel. The velocity profiles are shown up to the height of the bar (a) and to the channel centreline (b). In this figure and in figures 21 and 22, solid and dashed lines show profiles in the controlled and uncontrolled cases, respectively. The top horizontal axis denotes the velocity magnitude. The bars are not drawn to scale.

Figure 21.

Profiles of the wall-normal mean velocity $\langle \bar {v} \rangle$ at different $x$ locations along the channel. The bar is not drawn to scale.

Figure 22.

Mean streamwise velocity (a) and wall-normal velocity (b) profiles scaled in viscous units at different locations along the channel. The wall-friction velocity for each case, defined in § 4, is used for scaling.

Figure 22 shows the velocity profiles scaled in viscous units and at locations where the streamwise flow is attached. The wall-friction velocity of each case is used. The streamwise velocity profile shifts upwards in the controlled case, thus showing a thicker viscous layer and buffer region, a well-known feature of drag-reduced attached flows. The wall-normal velocity is instead only marginally affected.

Figure 23 depicts the spanwise- and phase-averaged velocity components at different streamwise locations and phase of oscillation $\tau$. The velocities $\left \langle \hat {u} \right \rangle$ and $\left \langle \hat {v} \right \rangle$ do not depend on $\tau$ and overlap with the time-averaged velocities $\left \langle \bar {u} \right \rangle$ and $\left \langle \bar {v} \right \rangle$ when the flow is fully attached and show only a marginal modulation with the phase in the recirculation region downstream of the bar. This result is in accordance with the behaviour of turbulent channel flows with spanwise oscillating walls (Jung et al. Reference Jung, Mangiavacchi and Akhavan1992) and validates the assumption of the steadiness of $\left \langle \hat {u} \right \rangle$ and $\left \langle \hat {v} \right \rangle$, adopted for the laminar-flow analysis of § 3. The velocity $\left \langle \hat {v} \right \rangle$ is smaller than $\left \langle \hat {u} \right \rangle$ for every $x$ and $\tau$, and is negligible outside the recirculation areas ($x=7$). The spanwise profiles $\left \langle \hat {w} \right \rangle$ at $\tau =0$ and $\tau =T/4$ are $90^{\circ }$ out of phase with the $\left \langle \hat {w} \right \rangle$ profiles at $\tau =T/2$ and $\tau =3/4 T$. We also note that the spanwise velocity $\left \langle \hat {w} \right \rangle$ is comparable with $\left \langle \bar {u} \right \rangle$ in the bulk of the channel.

Figure 23.

Profiles of spanwise- and phase-averaged velocities $\left \langle \hat {u}\right \rangle$, $\left \langle \hat {v}\right \rangle$ and $\left \langle \hat {w}\right \rangle$ at different $x$ locations. Thin solid lines with symbols are spanwise- and phase-averaged profiles at different phases (circle: $\tau =0$; square: $\tau =T/4$; diamond: $\tau = T/2$ and triangle: $\tau =3T/4$). The lines indicate the time- and spanwise-averaged velocities $\left \langle \bar {u}\right \rangle$, $\left \langle \bar {v}\right \rangle$ and $\left \langle \bar {w}\right \rangle$ for the controlled case (solid line) and the uncontrolled case (dashed line). The bars are not drawn to scale.

Figure 24 displays the profiles of the viscous-scaled phase-averaged velocities $\left \langle \hat {u} \right \rangle ^+$ and $\left \langle \hat {v} \right \rangle ^+$ at two locations where the flow is attached. Both velocity components fluctuate more downstream than upstream of the bar. The velocity $\left \langle \hat {v} \right \rangle$ increases up to an absolute maximum and then decreases monotonically to zero as the channel centreline is approached.

Figure 24.

Profiles of spanwise- and phase-averaged velocities $\left \langle \hat {u}\right \rangle$, $\left \langle \hat {v}\right \rangle$ scaled in viscous units at two $x$ locations. Symbols and lines are the same in figure 23.

Figure 25 depicts the profiles of $\left \langle \hat {w} \right \rangle$ and $w_l$ for $\tau =0, T/4, T/2,3/4 T$ at different $x$ stations. The laminar solution $w_l$ agrees well with $\left \langle \hat {w} \right \rangle$ outside the separation area ($x=2$ and $x=7$) for $\tau =T/4, 3/4 T$. At $\tau =0, T/2$, a large discrepancy occurs between the laminar and the turbulent profiles. The laminar $w_l$ tends to be null in the outer region of $y^+>10$, contrary to the finite values of the turbulent $\left \langle \hat {w} \right \rangle$. The differences between the laminar and turbulent flows are due to the transverse Reynolds stresses $-\partial \left \langle \widehat {u_t w_t} \right \rangle /\partial x$ and $-\partial \left \langle \widehat {v_t w_t} \right \rangle /\partial y$ in (3.4). Overall, the turbulent velocity $\left \langle \hat {w} \right \rangle$ shows a satisfactory agreement with the laminar $w_l$, proving that the analysis of § 3 gives a relevant physical model for the transverse oscillating mean flow obtained from the direct numerical simulations. This result is confirmed by the excellent laminar prediction of the spanwise power $\mathcal {P}_z$, shown in figure 11.

Figure 25.

Comparisons between the profiles of $\left \langle \hat {w} \right \rangle$ (solid lines) and the laminar $w_l$ obtained via (3.5) the last term on the right-hand side (dashed lines). Profiles are at $\tau =0$ (circles), $\tau =T/4$ (squares), $\tau = T/2$ (diamonds) and $\tau =3T/4$ (triangles); (a) $x = 2$, (b) $x = 5$, (c) $x=6$, (d) $x=7$.

The turbulence intensities, $\left \langle \widehat {u_t u_t}\right \rangle$, $\left \langle \widehat {v_t v_t}\right \rangle$, $\left \langle \widehat {w_t w_t}\right \rangle$ and $\left \langle \overline {u_t u_t}\right \rangle$, $\left \langle \overline {v_t v_t}\right \rangle$, $\left \langle \overline {w_t w_t}\right \rangle$, are shown in figure 26. They are highest at the $x$ locations corresponding to the bar sides where peaks of $\left \langle \widehat {u_t u_t}\right \rangle$ and $\left \langle \overline {u_t u_t}\right \rangle$ reach values two and four times higher than the values far from the separation regions ($x=3$ and $7$). The profiles of $\left \langle \widehat {u_t u_t}\right \rangle$ and $\left \langle \overline {u_t u_t}\right \rangle$ show two local maxima far from the bar with the global maxima occurring at $y=0.1$ for $x=3$ and $y=0.5$ for $x=7$. Similar to the mean-flow quantities, the phase-averaged quantities fluctuate more downstream ($4 < x < 7$) than upstream of the bar. The streamwise and spanwise correlations are more dominant than the wall-normal correlation in both the controlled and uncontrolled cases. Both $\left \langle \widehat {w_t w_t}\right \rangle$ and $\left \langle \overline {w_t w_t}\right \rangle$ show a peak around the bar height $y=0.2$ for every $\tau$. The quantities are all reduced for any $\tau$ and at any location along the channel. Figure 27 confirms that the turbulent kinetic energy $0.5(\left \langle \overline {u_t u_t}\right \rangle +\left \langle \overline {v_t v_t}\right \rangle +\left \langle \overline {w_t w_t}\right \rangle )$ decreases in the detached region downstream of the bar.

Figure 26.

Profiles of $\left \langle \widehat {u_t u_t}\right \rangle$, $\left \langle \widehat {v_t v_t}\right \rangle$, $\left \langle \widehat {w_t w_t}\right \rangle$ at different $x$ stations and $\tau$ (thin solid lines with symbols). Profiles are at $\tau =0$ (circle), $\tau =T/4$ (square), $\tau = T/2$ (diamond) and $\tau =3T/4$ (triangle). The phase-averaged quantities are compared with time-averaged flow statistics $\left \langle \overline {u_t u_t}\right \rangle$, $\left \langle \overline {v_t v_t}\right \rangle$, $\left \langle \overline {w_t w_t}\right \rangle$ for the control (solid lines) and uncontrolled (dashed lines) case. The bars are not drawn to scale.

Figure 27.

Contour plots of turbulent kinetic energy in the region around the bar for uncontrolled (a) and controlled (b) cases.

Figure 28 compares the Reynolds stresses of the controlled and uncontrolled flows. The Reynolds stress $\left \langle \widehat {u_t v_t}\right \rangle$ is dominant over $\left \langle \widehat {u_t w_t}\right \rangle$ and $\left \langle \widehat {v_t w_t}\right \rangle$ and decreases with respect to the uncontrolled case for almost every $y$ at any $x$ and $\tau$, similar to Jung et al. (Reference Jung, Mangiavacchi and Akhavan1992). The attenuating effect on $\left \langle \widehat {u_t v_t}\right \rangle$ is most intense downstream of the bar. The stresses $\left \langle \overline {v_t w_t}\right \rangle$ and $\left \langle \overline {u_t w_t}\right \rangle$ remain smaller than $\left \langle \widehat {u_t v_t}\right \rangle$ in the controlled case. (Note that $\left \langle \overline {u_t w_t}\right \rangle$ vanishes in the averaged $z$-momentum equation of a flat-wall turbulent channel flow because of streamwise averaging, while in the channel with bars the streamwise derivative of this term is present in the balance, as shown in (3.4).) The small amplitudes of these Reynolds stresses give support to the assumption adopted in the laminar analysis of (3.4), where these quantities are neglected.

Figure 28.

Profiles of $\left \langle \widehat {u_t v_t}\right \rangle$, $\left \langle \widehat {v_t w_t}\right \rangle$, $\left \langle \widehat {w_t u_t}\right \rangle$ at different $x$ stations and $\tau$ (thin solid lines with symbols). Lines and symbols are the same as in figure 26. The bars are not drawn to scale.

5.4. Role of the Reynolds stresses between bars

In order to gain further insight into the role of the Reynolds stresses in the controlled channel flow, we consider a momentum balance relating the total drag to the Reynolds stresses $-\left \langle \widehat {u_tv_t} \right \rangle$ along the horizontal line $l_m$ connecting the crests of two consecutive bars. The sum of the drag coefficients $C_p(\tau )+C_f(\tau )$ for the controlled flow is related to the product $\left \langle \widehat {u} \right \rangle \left \langle \widehat {v} \right \rangle$, the Reynolds stress $-\left \langle \widehat {u_tv_t} \right \rangle$ and the friction term $Re_p^{-1}\, \partial \left \langle \widehat {u} \right \rangle /\partial y|_h$ at $y=h$ via

(5.1)\begin{equation} C_p(\tau)+C_f(\tau) \approx -\frac{L_y}{L_x}\int_{l_m} \left \langle \widehat{u} \right\rangle\left \langle \widehat{v} \right\rangle \,{\textrm{d}x}-\frac{L_y}{L_x}\int_{l_m} \left \langle \widehat{u_tv_t} \right\rangle \,{\textrm{d}x} +\frac{L_y}{L_x Re_p}\int_{l_m} \frac{\partial \left \langle \widehat{u} \right\rangle}{\partial y} \,{\textrm{d}x}, \end{equation}

where the phase dependence of the quantities is kept herein.

Appendix E presents the derivation of (5.1). Note that in (5.1) the total drag $C_p(\tau )+C_f(\tau )$ is only approximated by the terms on the right-hand side because $C_{cr}$, which is much smaller than $C_{ca}$, is neglected. Equation (5.1) differs from (5.3) in Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003) as they did not include the term due to $\left \langle \widehat {u} \right \rangle \left \langle \widehat {v} \right \rangle$ and we use the time-ensemble average instead of the time average. Figure 29 depicts the integrands of the terms in (5.1) for the controlled and uncontrolled flows. The Reynolds stresses $-\left \langle \widehat {u_tv_t} \right \rangle$ and shear stresses $\partial \left \langle \widehat {u} \right \rangle /\partial y$ are positive at every $x$ along the cavity mouth, giving a positive contribution to the overall drag for both flows. The term involving $\partial \left \langle \widehat {u} \right \rangle /\partial y$ is, however, negligible compared to the other terms. The term $-\left \langle \widehat {u} \right \rangle \left \langle \widehat {v} \right \rangle$ is positive and streamwise oscillatory downstream of the bar in the controlled case, while it is negative upstream of the bar ($2<x<4$) because the wall-normal velocity $\left \langle \widehat {v} \right \rangle$ is large and positive. Due to conservation of mass, this upward velocity balances the decreasing streamwise mean velocity caused by the presence of the bar. Figure 29 shows that the reduction of the total drag mainly occurs because the control attenuates the Reynolds stresses $-\left \langle \widehat {u_tv_t} \right \rangle$, while the control effect of the time average of $-\left \langle \widehat {u} \right \rangle \left \langle \widehat {v} \right \rangle$ and $\partial \left \langle \widehat {u} \right \rangle /\partial y$ is marginal. The attenuation of $-\left \langle \widehat {u_tv_t} \right \rangle$ occurs not only in the region where the flow is attached ($x<3.5$ and $x>5.5$), but also along $l_m$ above the vortex A ($3.5<x<4$). Good agreement is obtained behind the bar between our $\partial \left \langle \widehat {u} \right \rangle /\partial y$ data along the channel mouth and the data from the direct numerical simulations of Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003) (red circles), while the comparison between the $-\left \langle \widehat {u_tv_t} \right \rangle$ data is only satisfactory downstream of the peak. The mismatch of the maximum values could be due to the large uncertainty caused by the large velocity fluctuations in the recirculation region.

Figure 29.

Comparisons between the profiles of $-\left \langle \widehat {u} \right \rangle \left \langle \widehat {v} \right \rangle$, $-\left \langle \widehat {u_tv_t} \right \rangle$, $Re_p^{-1} \partial \left \langle \widehat {u} \right \rangle /\partial y |_h$ of the controlled flow at phases $\tau =0, T/4, T/2, 3/4 T$ (thin solid lines) and the uncontrolled terms $-\left \langle \bar {u} \right \rangle \left \langle \bar {v} \right \rangle$, $-\left \langle {\overline {u_tv_t}}\right \rangle$, $Re_p^{-1} \partial \left \langle \bar {u} \right \rangle /\partial y |_h$ (thick solid lines). The red circles are data from the direct numerical simulations of Leonardi et al. (Reference Leonardi, Orlandi, Smalley, Djenidi and Antonia2003).

We can interpret the reduction of the Reynolds stresses as the effect of the oscillation on the bulk turbulent flow that in turn manifests itself at the wall surfaces as the attenuation of the pressure difference on the vertical surfaces of the bars and of the wall-shear stress over the cavity wall. Further research work is needed to establish the link between the bulk Reynolds-stress reduction and the pressure changes at the bar sides.