3.1. Turbulent drag reduction

Figure 2 displays the time evolution of the spatially averaged skin-friction coefficient $\hat {C}_f$ for different SJS angles $\beta$. The coefficient is strongly influenced by the SJS; it displays intense fluctuations that become periodic after two forcing periods. The oscillating period of these wall-friction fluctuations is $T_{{osc},\hat {C}_f}^+=62.5$, half of the period $T_{{osc}}$ of the SJS. The fluctuations of $\hat {C}_f$ depend on the SJS velocity; they are smallest for $\beta = 0^{\circ }$, and largest for $\beta = 60^{\circ }$, which means that they increase as the streamwise component of the SJS velocity vector increases up $\beta = 60^{\circ }$. For larger angles, the fluctuating amplitude does not grow monotonically with the SJS angle; the case $\beta = 75^{\circ }$ has a smaller amplitude than the case $\beta = 60^{\circ }$. As $\beta$ approaches $90^{\circ }$, the SJS flow becomes more and more aligned along the streamwise direction, so that the area of influence of the SJS becomes smaller, leading to a small fluctuating amplitude of $\hat {C}_f$. The amplitude of the wall-shear stress therefore oscillates less for large $\beta$ values than for $\beta = 60^{\circ }$. When $\beta = 90^{\circ }$, no flow exhausts into the channel from the SJS exits because the SJS velocity vector is unrealistically parallel to the slots, thus being less able to influence the bulk flow.

Figure 2.

Time evolution of the skin-friction coefficient for different SJS angles.

Figure 3 shows the drag-reduction margin as a function of the SJS angle. The maximum value is $\mathcal {R}=10.5\,\%$ when $\beta = 75^{\circ }$. The SJS angle $\beta = 0^{\circ }$ gives the maximum drag increase, that is, the drag coefficient is 98.4 % larger than the coefficient of the uncontrolled channel. Interpolating the data in figure 3 leads to the estimate that SJS with $\beta = 54^{\circ }$ generate a flow with the same average drag as the uncontrolled flow. Appendix B discusses the effects of varying the slot height, the period of forcing, the peak SJS velocity and the distance between the slots.

Figure 3.

Drag reduction for different SJS angles.

3.2. Mean-flow and turbulence statistics

Figure 4(a) illustrates the time and spatially averaged streamwise velocity $\overline {u}^+$ near the slots for the reference case with the SJS off, the drag-increase case for $\beta = 0^{\circ }$, and the drag-reduction case for $\beta = 75^{\circ }$. The skin-friction coefficient of the case with SJS off is $\hat {C}_f=8.20 \times 10^{-3}$ for the parameters in table 1, i.e. $0.24\,\%$ larger than that of the smooth channel, $\hat {C}_f=8.18 \times 10^{-3}$ (Kim, Moin & Moser Reference Kim, Moin and Moser1987). As reported in Appendix A, this difference is within the uncertainty range, estimated to be less than 1 %, which proves that the steps have a negligible influence on the skin friction for $h_{{jet}}^+=2$. A region of negative mean streamwise velocity is found near the slots for the case $\beta = 75^{\circ }$, which is not present when the SJS eject at $\beta = 0^{\circ }$. Although the SJS are characterized by a zero-net-mass flux at the slots, the time averaging reveals that a significant near-wall counter flow opposite to the bulk streamwise flow occurs when $\beta = 75^{\circ }$. No net counter flow is detected for $\beta = 0^{\circ }$ because no SJS flow is imposed against the bulk flow. The generation of the counter flow near the SJS slots is similar to that induced by canonical synthetic jets exhausting perpendicularly to a cross flow. In their review, Glezer & Amitay (Reference Glezer and Amitay2002) discuss several cases where synthetic jets, although characterized by a zero-net-mass flux, modify the cross flow into which they discharge and produce a displacement of its streamlines, thereby engendering a virtual change in the surface shape. In our study, the distortion of the bulk turbulent flow also occurs along the wall-normal direction as in Glezer & Amitay (Reference Glezer and Amitay2002), but the change is due to the SJS forcing, which is parallel to the walls. Increasing the SJS angle from $\beta = 0 ^{\circ }$, the SJS flow is directed against the main flow, producing the counter flow near the exits. When the SJS angle is large enough, the counter flow becomes a dominant effect on altering the velocity profile to reduce the near-wall velocity gradient and therefore the friction drag.

Figure 4.

Time and spatially averaged flow near the slots on the $y$–$z$ plane. (a) Contours of the averaged streamwise-velocity. (b) Vectors of the induced time-averaged flow $(\overline {v}^+, \overline {w}^+)$. The magnitude of the velocity vectors is equal to $\sqrt {\overline {v}^{+2} + \overline {w}^{+2}}$. The arrow at the top of each graph denotes the unit length for the vectors in the graph. The maximum magnitudes in the cases with the SJS off, $\beta = 0^{\circ }$ and $\beta = 75^{\circ }$, are $0.08$, $3.57$ and $0.56$, respectively.

Figure 4(b) shows the velocity vectors of the time-averaged cross-flow velocity components $(\overline {v}^+,\overline {w}^+)$. A mild cross flow with clockwise rotation occurs when the SJS are off, similar to the secondary flows reported by Hwang & Lee (Reference Hwang and Lee2018) numerically and Vanderwel & Ganapathisubramani (Reference Vanderwel and Ganapathisubramani2015) experimentally for turbulent flows over longitudinal rectangular roughness elements. Hwang & Lee (Reference Hwang and Lee2018) changed systematically the spanwise distance and the width of the steps, and reported that the strength of the secondary vortices increases when the spanwise distance increases or when the width decreases. Vanderwel & Ganapathisubramani (Reference Vanderwel and Ganapathisubramani2015) also revealed that the vortical flows exist next to the roughness elements, and their strength depends on their spanwise spacing. When the spacing is comparable with the boundary-layer thickness, the secondary vortices reach their maximum strength. The presence of instantaneous secondary flows was demonstrated further by Vanderwel et al. (Reference Vanderwel, Stroh, Kriegseis, Frohnapfel and Ganapathisubramani2019), using both experiments and numerical simulations.

When the SJS are on, although the time-averaged velocity components are null at the slots, a finite time-averaged cross flow is generated by the nonlinear interaction between the SJS and the bulk streamwise flow. The intensity of this cross flow is much larger than that in the reference case with SJS off, and the rotation is anticlockwise, that is, opposite to the rotation occurring when the SJS are off. When $\beta = 0^{\circ }$, the time-averaged SJS flow is more intense and more confined near the jet wall than in the case $\beta = 75^{\circ }$. The bulk of the cross flow is pushed upwards and away from the slots. The time-averaged cross flow is similar to the wall jets studied by Yao et al. (Reference Yao, Chen and Hussain2018), although in their case finite time-averaged jets are expected to form because they are generated by a steady spanwise body force.

Figure 5 compares further time and spatially averaged quantities for the cases with the SJS off, $\beta = 0^{\circ }$ and $\beta = 75^{\circ }$. Figure 5(a) shows the averaged scaled wall-shear stresses $\overline {C}_f$ along the spanwise direction. Consistently with the near-wall counter flow observed for $\beta = 75^{\circ }$ near the slots, the local wall-friction drag is reduced there. This result proves that the drag-reduction mechanism is different from other spanwise-forcing techniques, such as the oscillation wall (Quadrio & Ricco Reference Quadrio and Ricco2004) or the streamwise-travelling waves of spanwise wall velocity (Quadrio et al. Reference Quadrio, Ricco and Viotti2009), for which the wall-shear stress is never negative. As the case with $\beta = 75^{\circ }$ involves a significant oscillatory velocity component along the streamwise direction, the drag-reduction effect is akin to that reported by Zhou & Ball (Reference Zhou and Ball2008), who moved the wall obliquely with respect to the streamwise direction. They concluded that although forcing the flow purely along the spanwise direction led to the best performance, drag reduction was also found when most of the wall motion was along the streamwise direction. For $\beta = 0^{\circ }$, the wall-friction drag is more than six times larger than the uncontrolled value in the proximity of the slots, and the wall-shear stress is not reduced at any spanwise location. Along the central part of the jet wall, the trends of the wall-shear stress are flat and overlap in all three cases, indicating that the friction drag is unaffected along that portion of the wall because that region is too far from the SJS slots. Drag increase occurs over the step walls in both controlled cases.

Figure 5.

Comparisons of time and spatially averaged quantities at different spanwise positions. (a) Distributions of averaged skin friction along the spanwise direction. (b) Profile positions. (c) Profiles of averaged streamwise velocity, covering the height range of a half-channel.

The profiles of the time-averaged streamwise velocity are shown in figure 5(c), and the respective spanwise locations are indicated in figure 5(b). Over the step wall at $z_e^+ = 0.1$ and $17$, the large mean velocity in the controlled cases, which causes the local drag increase shown in figure 5(a), is limited only very close to the wall as the profiles show a good agreement with the uncontrolled profile at wall-normal distances $y^+>10$ from the step wall. The SJS with $\beta = 0^{\circ }$ create drag increase near the slot at $z_e^+ = 20$ by intensifying the mean streamwise velocity only up to $y^+=25$, while the reverse flow for $\beta = 75^{\circ }$ is confined up to $y^+=5$, i.e. in the viscous sublayer. At higher locations, the profiles for the controlled flows agree more closely to the uncontrolled profile, although a velocity deficit is found for $25< y^+<80$. In the middle of the jet wall at $z_e^+ = 62.8$, the three profiles overlap within the viscous sublayer, and the velocity deficit with respect to the uncontrolled case between $10< y^+<100$, is more pronounced than that at other spanwise locations, arguably because of the intense lift-up effect observed when the flow is time-averaged, as shown in figure 4(b). This deficit of mean streamwise velocity is indeed more intense for $\beta = 0^{\circ }$ than for $\beta = 75^{\circ }$ because the upward flow between the slots is more significant when the SJS are spanwise only, as the middle graph of figure 4(b) illustrates.

Figure 6 shows the contour plots of the root-mean-square (r.m.s.) values of the streamwise velocity fluctuations. Profiles in the top, middle and bottom rows refer to the total, periodic and turbulent fluctuations, respectively, as defined in (2.9). In figures 6(a,b,c) for the total fluctuations, two local peaks occur for $\beta = 0^{\circ }$, the most energetic one located in the proximity of the slot. Only one local peak is computed for $\beta = 75^{\circ }$, at the slot. The positions of these local peaks are the same as those of the periodic fluctuations caused by the SJS, shown in figures 6(d,e). The periodic fluctuations are most intense for $\beta = 75^{\circ }$ because part of the SJS velocity component is aligned along the streamwise direction. The periodic forcing becomes weaker at locations further away from the SJS slots because the streamwise velocity fluctuates the most near the SJS slots, causing only one local peak. For $\beta = 0^{\circ }$, the streamwise velocity of SJS is zero at the SJS slots, so no additional streamwise velocity fluctuations is found. During the blowing phase, a large near-wall velocity against the main channel flow is generated near the slot exit. However, during the suction phase near the slot exit, the streamwise velocity component is enhanced near the wall. These opposite behaviours lead to the local peak of the fluctuation that is closer to the slot. The other peak is induced by the blowing SJS only because the influenced region by the blowing SJS is larger than that by the suctioning SJS. As the locations of influence of blowing and suctioning are different, the fluid motions lead to two local peaks of the periodic fluctuations. Moreover, it is evident that the lift-up effect of the SJS occurs as soon as they discharge from the orifices. The periodic fluctuations are more energetic than the turbulent fluctuations, although the latter grow with respect to the uncontrolled case, as depicted in figures 6(g,h).

Figure 6.

Contours of root-mean-square values of the streamwise velocity fluctuations, $u'^+_{rms}$, $\tilde {u}^+_{rms}$ and $u''^+_{rms}$, in the $y$–$z$ view. The local peak positions are denoted by the white crosses. The panels show: (a) SJS off, $u'^+_{rms}$; (b) $\beta = 0^{\circ }$, $u'^+_{rms}$; (c) $\beta = 75^{\circ }$, $u'^+_{rms}$; (d) $\beta = 0^{\circ }$, $\tilde {u}^+_{rms}$; (e) $\beta = 75^{\circ }$, $\tilde {u}^+_{rms}$; ( f) SJS off, $u''^+_{rms}$; (g) $\beta = 0^{\circ }$, $u''^+_{rms}$; (h) $\beta = 75^{\circ }$, $u''^+_{rms}$.

Figure 7 shows that the Reynolds shear stresses $-\overline {u'v'}^+$, given by the total fluctuations, are increased by the SJS. In both controlled cases, $\beta = 0^{\circ }$ and $75^{\circ }$, larger Reynolds stress values than with the SJS off occur near the slots. The Reynolds stress values in the SJS cases are comparable near the slot, but for $\beta = 0^{\circ }$, large values are found above $y^+=15$ between the slots. The case for $\beta = 0^{\circ }$ also presents a region of Reynolds stress of opposite sign, centred at $y^+$=12 from the edge of the step wall. In the region where the Reynolds stress values are negative, the time and spatially averaged flow moves down towards the steps, as shown in figure 4. The low-speed region of streamwise velocity is generated by the blowing SJS and gives a negative $u'$, while when this part of flow is accelerated upwards, which has a positive $v'$, a region with negative $\overline {u'v'}$ is created. In the region where $\overline {u'v'}$ is positive, the flow motion in the wall-normal direction is opposite to that in the region of negative sign, although the flow motion is also in the streamwise direction.

Figure 7.

Contours of $-\overline {u'v'}^+$ in the $y$–$z$ plane. In this figure and in figures 9 and 10, the dashed lines denote negative values. (a) SJS off; (b) $\beta = 0^{\circ }$; (c) $\beta = 75^{\circ }$.

The phase and spatially ensemble averaged skin-friction coefficients are influenced significantly by the SJS, as shown in figure 8. Fluid is blown out from the left slot and is drawn in at the right slot from $\varphi = 0$ to $\varphi = {\rm \pi}$, while the opposite occurs in the other half-period. In both cases, $\beta = 0 ^{\circ }$ and $\beta = 75^{\circ }$, and near the slots, the wall-friction drag is reduced during blowing and increased during suction. The drag is never negative for $\beta = 0 ^{\circ }$, while it is decreased significantly and becomes negative for $\beta = 75 ^{\circ }$. The integral of the reduced drag offsets the integral of the raised drag for $\beta = 75 ^{\circ }$, so an overall drag reduction is obtained.

Figure 8.

Distributions of the phase and spatially ensemble averaged skin-friction coefficient. The blue lines denote the profiles at phases $\varphi ={\rm \pi} / 2$ and $3 {\rm \pi}/2$. (a) Three-dimensional, $\beta = 0 ^{\circ }$; (b) two-dimensional, $\beta = 0 ^{\circ }$; (c) three-dimensional, $\beta = 75 ^{\circ }$; (d) two-dimensional, $\beta = 75 ^{\circ }$.

The phase and spatially ensemble averaged streamwise velocity $\langle u \rangle ^+$ in the near-wall region is shown in figure 9 for half the period. The blowing from the left slot influences the streamwise velocity to larger wall-normal distances than the suction from the right slot. Fluid with small streamwise velocity is blown upwards by the left SJS, while fluid with large streamwise velocity is brought downwards by the right SJS. The former reduces drag, while the latter increases it. The SJS blowing with $\beta = 0^{\circ }$ influence the flow to larger wall-normal locations than the SJS blowing with $\beta = 75^{\circ }$. The counter flow is generated by the SJS with $\beta = 75^{\circ }$ during blowing, as depicted at $\varphi = 3{\rm \pi} /5$.

Figure 9.

Contours of phase and spatially averaged streamwise velocities $\langle u \rangle ^+$ in the $y$–$z$ plane at different phases for the cases (a) $\beta = 0^{\circ }$, (b) $\beta = 75^{\circ }$.

Figure 10 shows the contour plots of the Reynolds shear stresses $-\langle u''v'' \rangle ^+$ given by the purely turbulent fluctuations for $\beta = 0^{\circ }$ and $\beta = 75^{\circ }$ from $\varphi = 0$ to $\varphi = {\rm \pi}$. The Reynolds shear stresses are enhanced by blowing, while the drag is decreased. Both positive and negative values of $-\langle u''v'' \rangle ^+$ are produced by the blowing region of the SJS for $\beta = 0^{\circ }$. The region with negative values of $-\langle u''v'' \rangle ^+$ is closer to the steps than the region with positive values. For $\beta = 75^{\circ }$, the positive values of $-\langle u''v'' \rangle ^+$ dominate the flow field, while the region with negative values is negligible. As shown by the contours of $\overline {u'v'}^+$ and $\langle u''v'' \rangle ^+$ in figures 7 and 10, large values of $\overline {u'v'}^+$ occur near the slots, while the values of $\langle u''v'' \rangle ^+$ are low, indicating that the periodic velocity fluctuations contribute the most to the total Reynolds shear stress there.

Figure 10.

Contours of the Reynolds shear stress $-\langle u''v'' \rangle ^+$ in the $y$–$z$ plane at different phases for the cases (a) $\beta = 0 ^{\circ }$, (b) $\beta = 75 ^{\circ }$.

The overall picture is therefore that the SJS cause drag reduction for $\beta = 75^{\circ }$ through the intense near-wall counter flow being larger than the forward flow occurring in the near-wall region in the proximity of the SJS slots, despite the intensified Reynolds shear stresses. The drag-increasing case for $\beta = 0^{\circ }$ also enhances the Reynolds shear stresses, but it does not benefit from the counter flow because the SJS are aligned along the spanwise direction only.

3.3. Turbulent-flow structures

Instantaneous flows for the cases with $\beta = 0^{\circ }$ and $75^{\circ }$ are discussed in this subsection. Figure 11 shows the isosurfaces of $\lambda _2^+ = -2$ in the bottom half-channel at different phases. The $\lambda _2$ technique to detect the vortex cores was developed by Jeong & Hussain (Reference Jeong and Hussain1995). The flows altered by the SJS display more intense vortical structures than the case without SJS forcing, irrespective of whether the drag reduces or increases. The response of the flow to the forcing is therefore different from other spanwise forcing methods, such as streamwise-travelling waves of spanwise wall velocity (Quadrio et al. Reference Quadrio, Ricco and Viotti2009; Quadrio & Ricco Reference Quadrio and Ricco2011), which lead to a less intense and more sporadic population of near-wall vortical structures accompanied by a reduction of wall-friction drag. Vortical structures, shaped like elongated tubes near the SJS slots, are generated during blowing, at the phase $\varphi = {\rm \pi}/2$. At the phase $\varphi = 0$ and during suction, the vortical structures are broken by the main flow and the vortices are more apart than when $\varphi = {\rm \pi}/2$. Figure 11 shows that the diameter and the intensity of the weaker vortices, coloured in green and located in the bulk of the flow, are only slightly influenced by the SJS angle. In the $\beta =0^\circ$ case, intense elongated tubular structures appear near the wall and in the proximity of the SJS exits where blowing occurs. These structures are much weaker in the $\beta =75^\circ$ case.

Figure 11.

Isosurfaces of $\lambda _2^+ = -2$ at different phases for the cases with (a) SJS off, (b) $\beta = 0^{\circ }$, and (c) $\beta = 75^{\circ }$. The isosurfaces are coloured by $u'^+$, and ‘B’ and ‘S’ stand for blowing and suctioning slots, respectively.

Two-point autocorrelations, defined as

(3.1)\begin{equation} R_{u''u''}(\Delta x_i) = \frac{\overline{u''(x_i+\Delta x_i)\,u''(x_i)}}{\overline{u''u''}}, \end{equation}

are computed for $x_i=x,z$. Figure 12 shows the distributions of $R_{u''u''}(\Delta x)$ and $R_{u''u''}(\Delta z)$ at different heights for cases with SJS off, $\beta = 0 ^{\circ }$ and $\beta = 75 ^{\circ }$. Figure 12(a) shows that at $y^+=10$, for $\beta = 75 ^{\circ }$, the spanwise turbulent length scales are smallest, while for $\beta = 0 ^{\circ }$, the scales are largest, confirming quantitatively what is observed in the flow visualizations of figure 11. In figure 12(b), the $R_{u''u''}(\Delta z^+)$ values at higher locations ($y^+=100$) are similar for both SJS cases, as also shown by the green vortices in figure 11. Figure 12(c) shows that at $y^+=10$, the streamwise turbulent length scales are reduced slightly by SJS, and the case $\beta = 0 ^{\circ }$ shows the shortest scales when the SJS velocity amplitude reaches its maximum. Figure 12(d) shows that at $y^+=100$, the SJS increase the length scales the most when $\beta = 75 ^{\circ }$. The SJS influence the spanwise length scales more than the streamwise length scales.

Figure 12.

Spanwise and streamwise two-point correlations of streamwise velocity fluctuations $u''$, for: (a) $y^+ = 10$; (b) $y^+ = 100$; (c) $y^+ = 10$; (d) $y^+ = 100$.

3.4. Flow and power balance of the jet-sheet actuator

A model of the actuators that generate the SJS is presented in this subsection. The focus is on the flow inside the actuators, located underneath the channel walls, and on the power required to operate the SJS. As for the turbulent channel flow simulations, the channel half-height $h^*$ and the centreline velocity $U^*_p$ of the laminar parabolic Poiseuille flow of the reference smooth channel case are used for scaling.

3.4.1. Flow in the cavity chambers

Figure 13 shows a schematic of the system. A piston separates two chambers of a cavity and oscillates sinusoidally along the spanwise direction, forcing the fluid in the compression chamber to discharge through the SJS opening and out into the channel. Simultaneously, the pressure drops in the suction chamber, causing the fluid to enter the SJS opening and to fill the chamber. The flows in the chambers alternate their behaviour every half-cycle, according to the motion of the piston. The parameters of the chambers and the simulation details are listed in table 2.

Figure 13.

Schematic of the cavity chambers. The thickness of the piston is not to scale.

Table 2.

Simulation and chamber parameters, where $\langle w\rangle _{{jet,max}}$ is the spatially averaged SJS velocity, and $\Delta t$ is the time step. The other quantities are defined in figure 13.

The in-house code SHEFFlow is used to compute the flow in the chambers and outside the SJS openings. The domain of the simulation consists of the chambers and half of the channel above them. The flow is assumed to be two-dimensional. Mirror boundary conditions are imposed at the half-channel boundary, and periodic boundary conditions are enforced at the sides of the computational domain. Figure 13 represents the purely spanwise case ($\beta =0 ^{\circ }$). In the case of oblique SJS, the exits would require inclined vanes to drive the fluid out into the channel at an angle. The losses due to the three-dimensionality of the vanes are assumed small with respect to the rest of the losses, an approximation that is confirmed in § 3.4.4. The wall that separates the SJS and the channel has a finite thickness, which is smaller than the height of the exits. This finite thickness avoids a sharp wall end at the exit, which would create intense velocity gradients when the fluid leaves the chamber.

Figure 14 shows that the density and the temperature are not uniform inside the chambers. When the piston is in the neutral position ($\varphi = {\rm \pi}/2$), the piston velocity is maximum. The fluid is then compressed in the right chamber, while in the left chamber the fluid expands. At $\varphi = 0$, the displacement is maximum and the density in the right chamber decreases by releasing the fluid inside the chamber to the channel, while in the left chamber, the fluid enters the chamber and the density increases. These compressibility effects cause a phase lag of about $\varphi =0.03{\rm \pi}$ between the pressure experienced by the piston and the velocity of the piston, rendering the power spent lower than that without delay.

Figure 14.

Contours of (a) density, (b) temperature, and (c) spanwise velocity, at phases $\varphi = 0$ (left) and $\varphi = {\rm \pi}/2$ (right). The horizontal arrow indicates the direction of piston motion.

Figure 15 shows that the computed horizontal SJS velocity profiles are approximated well by parabolic profiles. The results of the two-dimensional simulation therefore validate the assumption of a parabolic SJS flow for the three-dimensional channel flow simulation, modelled by (2.4). The largest deviations from a symmetric profile occur when the mass flow rate is maximum during blowing or suction. When air exits a chamber and enters the channel, the peak of the maximum velocity is slightly larger than in the mid position at $y^+=1$, while the contrary happens in the suction phase, during which the maximum velocity peak is lower than in the mid position. In the blowing phase, this small effect is caused by a localized region of low pressure at the tip of the step wall. This low pressure causes the air exiting the SJS aperture to move upwards, away from the jet wall. In the suction phase, the curvature of the chamber forces the air in the lower half of the SJS aperture to turn downwards.

Figure 15.

Velocity profiles at the exit of the left cavity at different phases of the oscillation. The dashed lines represent the parabolic boundary condition given in (2.4), imposed in the three-dimensional channel flow simulations. The solid lines represent the velocity profiles obtained with the two-dimensional simulation.

3.4.2. Motion of the piston

The position of the piston is described by

(3.2)\begin{equation} z_{{piston}} = \mathcal{Z}_{{max}} \cos \left(\frac{2{\rm \pi}}{T_{{piston}}}\,t\right),\end{equation}

where $T_{{piston}}$ is the period of oscillation of the piston, and $\mathcal {Z}_{{max}}$ is the maximum displacement travelled by the piston with respect to the central position. The period of oscillation $T_{{piston}}$ is the same as the period of oscillation $T_{{osc}}$ of the SJS boundary condition in the full three-dimensional simulation.

As the flow inside the chambers is compressible, an exact a priori relationship between the piston motion and the SJS velocity cannot be found without simulating the flow because the fluid density at the piston surfaces and the fluid density at the cavity exits are not known. Therefore, in order to obtain an estimate of the maximum displacement $\mathcal {Z}_{{max}}$ as a function of the average SJS velocity, the incompressible mass conservation equation is used:

(3.3)\begin{equation} \int_A \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d} A = 0, \end{equation}

where $A$ is the surface of the chamber including the SJS orifices, and $\boldsymbol {n}$ is the normal unit vector perpendicular to $A$, pointing outwards. Since the vectors $\boldsymbol {n}$ for the piston surface and for the SJS exit surface point in the spanwise direction, (3.3) becomes

(3.4)\begin{equation} {-}w_{{piston}} (t)\,A_{{piston}}+ \int_{A_{{fluid}}} w_{{jet}} (y,t)\,\mathrm{d} A_{fluid} = 0, \end{equation}

where $w_{{piston}}$ is the velocity of the piston, $w_{{jet}}$ is the velocity profile at the SJS exit, $A_{{piston}}$ is the surface of the piston, and $A_{\mathrm {fluid}}$ is the area of the SJS apertures. Writing (3.4) per unit depth, the surface integral becomes

(3.5)\begin{equation} {-}w_{{piston}} (t)\,h_{{piston}} + \int_0^{h_{{jet}}} w_{{jet}} (y,t)\, \mathrm{d}y = 0, \end{equation}

where $h_{{piston}}$ is the height of the piston. The spatial mean velocity of the SJS, defined as

(3.6)\begin{equation} \langle w\rangle_{{jet}}(t)=\frac{1}{h_{{jet}}} \int_{h_{{jet}}} w_{{jet}} (y,t)\, \mathrm{d}y, \end{equation}

and $w_{{piston}}$, found from differentiating (3.2), are substituted into (3.5) to find the maximum displacement of the piston:

(3.7)\begin{equation} \mathcal{Z}_{{max}} = \frac{T_{piston}}{2{\rm \pi}}\,\frac{h_{{jet}}}{h_{{piston}}}\, \langle W\rangle_{{jet,max}},\end{equation}

where $\langle W\rangle _{{jet,max}}$ is the maximum spatial mean velocity of the SJS within a period. The boundary conditions for the SJS velocity in the three-dimensional simulation with the parameters given in table 2 result in amplitude $\mathcal {Z}_{{max}}=0.079$ and maximum piston velocity $W_{{piston,max}}= 0.031$.

3.4.3. Power balance of the cavity flow for $\beta =0^\circ$

The first step to study the power balance of the SJS actuators is to calculate the power per unit depth required to move the piston, $\mathcal {W}_{{piston}}$, in the two-dimensional configuration ($\beta =0^\circ$). This power is exerted by the force that the piston has to overcome at any time to generate the SJS, given by the difference of the integrated pressures on the two sides of the piston as a function of time. The power $\mathcal {W}_{{piston}}$ is

(3.8)\begin{equation} \mathcal{W}_{{piston}}(t) = w_{{piston}}(t)\int_{h_{{piston}}} \Delta p_{{piston}}(y,t)\,\mathrm{d}y, \end{equation}

where $\Delta p_{{piston}}$ is the difference between the pressures on the two sides of the piston. Figure 16 shows the piston velocity $w_{{piston}}(t)$, the force per unit depth of the cavity acting on the piston,

(3.9)\begin{equation} F_{{piston}} (t) = \int_{h_{{piston}}} \Delta p_{{piston}} (y,t)\,\mathrm{d}y, \end{equation}

and the power $\mathcal {W}_{{piston}}$ during a period of oscillation. A phase lag of about $\varphi =0.1{\rm \pi}$, shown in figure 16, occurs between the integrated pressure $\Delta F_{{piston}}$ experienced by the piston and the velocity of the piston $w_{{piston}}(t)$. This phase lag causes the power spent to be lower than that if the maxima of the piston force and the piston velocity occurred at the same time. For the parameters given in table 2, the average power required to move the piston is $\overline {\mathcal {W}}_{{piston}}=8\times 10^{-3}$.

Figure 16.

Ensemble average of the velocity of the piston, pressure difference, and power required to move the piston.

The power balance inside the cavity chambers is studied through the balance equation for the total power integrated over the control volume. The balance equation, derived in Appendix C following Panton (Reference Panton2013), reads

(3.10)\begin{align} & \underbrace{\frac{\mathrm{d}}{\mathrm{d} t} \int_V \rho \left(e + \frac{\vert \boldsymbol{u} \vert^2}{2} \right) \mathrm{d}V }_{\mathrm{d}\mathcal{E}/\mathrm{d}t}={-} \underbrace{\int_{A_{fluid}} p \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}A}_{\mathcal{P}_{fluid}} -\underbrace{\int_{A_{fluid}} \rho e \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}A}_{\mathcal{F}_{i-fluid}} \nonumber\\ &\quad -\underbrace{\int_{A_{fluid}} \rho\, \frac{\vert \boldsymbol{u} \vert^2}{2}\,\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}A}_{\mathcal{F}_{m-fluid}} -\underbrace{w_{{piston}}(t)\int_{A_{piston}} \Delta p_{{piston}}\,\mathrm{d}A}_{\mathcal{W}_{{piston}}} + \underbrace{\frac{1}{Re_p}\int_{A_{fluid}} (\boldsymbol{\underline{\tau}} \boldsymbol{\cdot} \boldsymbol{u}) \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}A}_{\mathcal{T}_{fluid}} \nonumber\\ &\quad +\underbrace{\frac{1}{Re_p} \int_A k\,\boldsymbol{\nabla} T \boldsymbol{\cdot} \boldsymbol{n}\,\mathrm{d}A}_{\mathcal{Q}}. \end{align}

Equation (3.10) expresses the following physical mechanisms. The power injected into the cavity chambers via the pressure work of the piston $\mathcal {W}_{{piston}}$ partly generates the time rate of change of the integrated total energy, $\mathrm {d}\mathcal {E}/\mathrm {d}t$, is partly transferred to the channel as the fluid exhausts through the SJS apertures, via the flux of internal energy per unit time $\mathcal {F}_{i-fluid}$, the mechanical power flux $\mathcal {F}_{m-fluid}$, the pressure work $\mathcal {P}_{fluid}$ and the shear stress work $\mathcal {T}_{fluid}$ of the SJS, and is lost to the outside of the cavity chambers via the heat transfer $\mathcal {Q}$.

Figure 17 shows the time evolution of the terms of (3.10) during one period of oscillation. The kinetic energy term $\mathrm {d}\mathcal {E}/\mathrm {d}t$ shows the most intense oscillations, while the power transferred via the shear stresses at the SJS apertures, $\mathcal {T}_{fluid}$, is found to be negligible. At any phase of the oscillation, the piston always injects power into the chambers, while heat is always extracted from the chambers. The flux terms and the pressure term related to the SJS instead oscillate between positive and negative values, being directly related to the SJS.

Figure 17.

Time evolution of terms in the power equation (3.10). Positive values indicate power into the chambers, while negative values indicate power lost from the chambers.

Figure 18 displays the time evolution of the flux terms $\mathcal {F}_{m-fluid}$ and $\mathcal {F}_{i-fluid}$. The flux of internal energy per unit time $\mathcal {F}_{i-fluid}$ is much larger than the mechanical power flux $\mathcal {F}_{m-fluid}$ because the difference in temperature at the SJS exits is larger than the difference in the averaged velocities. Mechanical power is passed from the chambers to the channel for $47\,\%$ of an oscillating period, during $(0.31 {\rm \pi}, 0.86 {\rm \pi})$ and $(1.31 {\rm \pi}, 1.86 {\rm \pi})$. The time-averaged values of the fluxes of mechanical power and internal energy per unit time are $3.04 \times 10^{-4}$ and $1.84 \times 10^{-4}$, i.e. the mechanical part takes $62.3\,\%$ of the convective flux and is $3.8\,\%$ of the time average of $\mathcal {W}_{{piston}}$.

Figure 18.

Time evolution of mechanical power flux $\mathcal {F}_{m-fluid}$ and the flux of internal energy per unit time $\mathcal {F}_{i-fluid}$.

The terms in (3.10) are time-averaged to quantify further the balance of the SJS actuator. The averaged values are listed in table 3. For the convective flux and the pressure work, power flows into the two chambers when the value is negative and out of them when the value is positive. The channel flow thus receives power from the SJS in the three ways, i.e. through the flux of internal energy per unit time $\overline {\mathcal {F}}_{i-fluid}$, the mechanical power flux $\overline {\mathcal {F}}_{m-fluid}$, and the pressure work term $\overline {\mathcal {P}}_{fluid}$. The last two types of power are relevant for flow control: $\overline {\mathcal {F}}_{m-fluid}$ and $\overline {\mathcal {P}}_{fluid}$ are $3.8\,\%$ and $1.7\,\%$ of $\overline {\mathcal {W}}_{{piston}}$, respectively. The effective power $\overline {\mathcal {P}}_{\mathrm {jet-sheet}}$ is

(3.11)\begin{equation} \overline{\mathcal{P}}_{{jet-sheet}} = \overline{\mathcal{F}}_{m-fluid} + \overline{\mathcal{P}}_{fluid}= 0.038 \times \overline{\mathcal{W}}_{{piston}} + 0.017 \times \overline{\mathcal{W}}_{{piston}}=0.054 \times \overline{\mathcal{W}}_{piston}. \end{equation}

The efficiency of the SJS actuator is therefore $\overline {\mathcal {P}}_{{jet-sheet}}/\overline {\mathcal {W}}_{{piston}}=5.4\,\%$. A more conservative estimation of the power employed by distributed suction and blowing was utilized by Bewley, Moin & Temam (Reference Bewley, Moin and Temam2001), Chung & Talha (Reference Chung and Talha2011) and Stroh et al. (Reference Stroh, Frohnapfel, Schlatter and Hasegawa2015). They used the absolute values of the integrands that define $\overline {\mathcal {F}}_{m-fluid}$ and $\overline {\mathcal {P}}_{fluid}$, as defined in (3.12) and (3.13) below. If we adopt those authors’ definitions of power consumption,

(3.12)$$\begin{gather} \lvert \overline{\mathcal{F}}_{m-fluid} \rvert =\int_{T_{osc}} \int_{A_{fluid}} \left\lvert \rho\,\frac{\vert \boldsymbol{u} \vert^2}{2}\,\boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{n} \right\rvert\mathrm{d}A\,\mathrm{d}t, \end{gather}$$

(3.13)$$\begin{gather}\lvert \overline{\mathcal{P}}_{fluid} \rvert = \int_{T_{osc}} \int_{A_{fluid}} \lvert p \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \rvert\,\mathrm{d}A\, \mathrm{d}t, \end{gather}$$

then we find $\lvert \overline {\mathcal {F}}_{m-fluid} \rvert =0.42 \times 10^{-3} = 0.052 \times \overline {\mathcal {W}}_{{piston}}$ and $\lvert \overline {\mathcal {P}}_{fluid} \rvert =6.22 \times 10^{-3} = 0.774 \times \overline {\mathcal {W}}_{{piston}}$.

Table 3.

Time-averaged terms in (3.10).

3.4.4. Power balance of the cavity flow for $\beta \neq 0^\circ$

The analysis of the power efficiency of the cavity flow presented in § 3.4.3 is limited to the two-dimensional case with injection angle $\beta =0^\circ$. To compute the power efficiency of the cavity flow for finite angles $\beta$, we estimate the power loss by considering the flow through a series of guide vanes placed between the cavity and the SJS exits, as shown in figure 19. For the power-loss estimation, it is useful to refer to studies on flows through guide vanes in low-speed wind tunnels. The Reynolds number $\mbox {{Re}}_c$ used in the analysis of guide vanes is typically based on the mean inlet velocity and the chord of the guide vanes; in our case, a sound estimate based on the maximum inlet velocity is $\mbox {{Re}}_c=25\,000$. According to Sahlin & Johansson (Reference Sahlin and Johansson1991) and Lindgren, Österlund & Johansson (Reference Lindgren, Österlund and Johansson1998), the guide-vane pressure-loss coefficient $\mathcal {K}$ at this chord Reynolds number for a $90^{\circ }$ turn and expansion ratios close to unity varies in the range

(3.14)\begin{equation} \mathcal{K} = \frac{\Delta H^*}{q^*_{{in}}}=0.08-0.12, \end{equation}

where $\Delta H^* = (p^*_{{in}}+q^*_{{in}})- (p^*_{{out}}+q^*_{{out}})$; the subscripts ‘in’ and ‘out’ stand for the flow entering and exhausting from the vane, respectively, and $q^*_{{in}}=\rho _c^*\, \langle W\rangle _{{jet,max}}^{*2}/2=q^*_{{out}}$ because the expansion ratio is assumed to be unity. A $90^{\circ }$ turning vane represents the worst possible scenario for the power loss, and adequately models the SJS case with $\beta =75^{\circ }$, which leads to the maximum drag reduction. The estimate of the pressure coefficient translates to a 10 % decrease of the mechanical power flux $\overline {\mathcal {F}}_{m-fluid}$ injected into the channel. The overall efficiency of the device reduces to 5.2 %. Moreover, it means that the case with $\beta =75^{\circ }$ needs $1.11 \times \overline {\mathcal {W}}_{{piston}}$ to have the same amount of power at the SJS exits as the case with $\beta =0^{\circ }$.

Figure 19.

Schematic of the guide vanes used to change the direction of the flow to angle $\beta$.

It is also instructive to compare the power used to activate the piston with the power saved by drag reduction. From (2.13), the power spent per streamwise length used to drive the fluid in the channel is

(3.15)\begin{equation} \overline{\mathcal{P}}_{{channel}}^*=2\tau_w^*U_b^* (L^*_{jet} + L^*_{step})= [C_f] \rho_c^* U_b^{*3}(L^*_{jet} + L^*_{step}). \end{equation}

Using the channel half-height $h^*$ and the centreline velocity $U_p^*$ for scaling, one finds

(3.16)\begin{equation} \overline{\mathcal{P}}_{{channel}}=[C_f] \left( \frac{U_b^*}{U_p^*} \right)^3 \frac{L^*_{jet} + L^*_{step}}{h^*}=16.97 \times 10^{{-}4}. \end{equation}

For $\beta = 75^{\circ }$, the skin-friction coefficient is reduced by 10.5 %, which leads to the maximum power saved by the SJS actuation, $\overline {\mathcal {P}}_{{channel,SJSoff}} - \overline {\mathcal {P}}_{{channel,SJS75}}= 17.82 \times 10^{-5}$. The net power saved in this case is $\overline {\mathcal {P}}_{{channel,SJSoff}} - \overline {\mathcal {P}}_{{channel, SJS75}} - 1.11 \times \overline {\mathcal {W}}_{{piston}}= -8.75 \times 10^{-3}$, which means that the actuation power is larger than the saved power due to drag reduction.