2.1. DNS solver and computational set-up

The DNS solver used in this study is the same as that employed by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022), and was extensively described and validated earlier by Memmolo, Bernardini & Pirozzoli (Reference Memmolo, Bernardini and Pirozzoli2018). The compressible Navier–Stokes equations are solved for an ideal gas, with the heat flux vector and the viscous stress tensor modelled by Fourier’s law and the Newtonian hypothesis, respectively. The dependence of dynamic viscosity $\mu$ on temperature $T$ is described by the power law $\mu / \mu _\infty = ( T / T_{\infty } )^{0.76}$ (Smits & Dussauge Reference Smits and Dussauge2006). Here and throughout the paper, the subscript ‘ $\infty$ ’ refers to the free stream undisturbed flow. The equations, cast in integral form, are discretised using a second-order, energy-consistent finite-volumes method (Pirozzoli Reference Pirozzoli2011). In the presence of shock waves, detected by a modified Ducros sensor (Ducros et al. Reference Ducros, Ferrand, Nicoud, Weber, Darracq, Gacherieu and Poinsot1999), the code switches locally to a third-order weighted essentially non-oscillatory scheme (Liu, Osher & Chan Reference Liu, Osher and Chan1994).

The case of interest is the same as that considered by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022), i.e. an aerofoil immersed in a uniform flow. The sectional profile is the supercritical V2C aerofoil, designed within the European project TFAST (Doerffer et al. Reference Doerffer, Flaszynski, Dussauge, Babinsky, Grothe, Petersen and Billard2021), and already studied experimentally (Placek & Ruchala Reference Placek and Ruchala2018) and numerically (Szubert et al. Reference Szubert, Asproulias, Grossi, Duvigneau, Hoarau and Braza2016; Zauner, De Tullio & Sandham Reference Zauner, De Tullio and Sandham2019). The angle of attack is $\alpha = 4^\circ$ , which provides the maximum lift-to-drag ratio. The airflow has inflow conditions of Mach number $M_\infty \equiv U_\infty /a_\infty = 0.7$ (where $U_\infty$ is the free stream velocity and $a_\infty$ is the speed of sound in the free stream) and Reynolds number $Re_\infty \equiv U_\infty c / \nu _\infty = 3 \times 10^5$ (where $c$ is the profile chord and $\nu _\infty$ is the kinematic viscosity in the free stream). Periodicity is enforced in the spanwise direction and the inflow is laminar: to avoid the large excursions of the transition location (Zauner et al. Reference Zauner, De Tullio and Sandham2019), the boundary layer is tripped, as in experiments. A small Gaussian blob of wall-normal random volume force (Schlatter & Örlü Reference Schlatter and Örlü2012) acts on both sides of the aerofoil $0.1 c$ downstream of the leading edge. The intensity of the tripping force ( $100 U_\infty ^2/c$ ) was selected as the minimum value granting healthy development of the turbulent boundary layer.

Unless otherwise indicated, the length scale $c$ and velocity scale $U_\infty$ are used to form dimensionless quantities, and are omitted hereinafter. Alternatively, viscous units will also be adopted, where the reference velocity is the friction velocity ( $u_\tau =\sqrt {\tau _w/\rho }$ ) and the reference length is the viscous length ( $\nu /u_\tau$ ). In these definitions, $\tau _w, \rho , \nu$ are the wall-shear stress, density and kinematic viscosity, respectively. Quantities made non-dimensional with the viscous units extracted from the reference uncontrolled simulation are denoted with a ‘+’ superscript, while the ‘ $\ast$ ’ scaling denotes quantites made non-dimensional with the friction velocity of the current simulation. Note, however, that the wall-shear stress varies significantly along the profile. In this paper, friction is evaluated on the suction side at $x=0.4$ , a position that is downstream enough for the actuation to actually develop its effects, but also sufficiently upstream of the shock such that the wall pressure gradient is almost constant across the various simulations. This aspect is further discussed in § 3.2 and in Appendix A.

The computational set-up replicates the baseline case described by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022), who validated it in terms of resolution requirements. The mesh features a C-type topology with $4096 \times 512 \times 256$ cells. It has a radius $25$ chords and a spanwise extension of $L_z = 0.1$ , which is sufficiently large to grant that the flow statistics are independent from this length. For verification, an additional simulation was carried out for the reference flow with a spanwise extension of $L_z = 0.4$ , only to find overlapping results. The cells are uniformly spaced in the spanwise direction, while a hyperbolic-tangent clustering is adopted in the wall-normal direction. The mesh meets the requirements for a fully resolved DNS, with $\Delta x^+ \lt 10$ , $\Delta y^+ \lt 0.5$ and $\Delta z^+ \lt 5$ at the wall, where $\Delta x$ , $\Delta y$ and $\Delta z$ are the grid spacings in the $x$ , $y$ and $z$ directions, respectively. Curvilinear coordinates $\xi$ and $\eta$ will be used in the following to denote directions tangent and normal to the wall, respectively. Time integration is performed using a low-storage, third-order Runge–Kutta scheme, with a fixed time step of $\Delta t = 1.2 \times 10^{-4}$ , chosen to keep the maximum Courant–Friedrichs–Lewy number below unity. The quantities of interest are averaged in the spanwise direction and over time for a minimum duration of $\Delta T = 40$ ; the most significant cases, discussed in the next section, are run for a maximum $\Delta T = 130$ . The initial transient, determined on a case-by-case basis, is excluded from the time-averaging process. An a posteriori analysis of the lift and drag time histories using the algorithm developed by Russo & Luchini (Reference Russo and Luchini2017) confirms that the ratio between the standard deviation $\sigma$ of the estimated mean value of the lift coefficient and the coefficient’s value remains below $0.4\,\%$ across all cases; the same quantity for the drag coefficient is below $0.7\,\%$ . These metrics are presented as $\sigma _L$ and $\sigma _D$ in table 1.

Table 1.

Complete set of simulations. The control parameters are shown in the left part of the table. The middle part reports the aerodynamic coefficients ( $C_L$ for lift, $C_D$ for drag and $C_M$ for moment) with the percentage relative standard deviation $\sigma$ , the friction and pressure components of drag, $C_{D,f}$ and $C_{D,p}$ , and aerodynamic efficiency $E$ , shock position $x_s$ and shock intensity $I$ . The rightmost part shows the performance indicators discussed in the text. The relative change of an indicator $\Phi$ is defined as the percentage increase with respect to the uncontrolled case (denoted with the ‘0’ subscript), i.e. $\Delta \Phi = 100(\Phi /\Phi _0-1)$ ; in case R10, variations are computed against RREF.

Figure 2.

Visualisation of the mean modified Ducros shock sensor over the suction side of the aerofoil (reference simulation), highlighting regions where $\Theta \gt 0.99$ . The blue line is the isoline at $M=1$ and the black dots denote the cells used to compute the shock intensity. The inset shows the upper edge of the shock superimposed to the grid, with the red isoline corresponding to $\Theta =0.99$ .

The discussion that follows will consider the position and strength of the shock wave. To identify the shock, we use the modified Ducros sensor (Pirozzoli Reference Pirozzoli2011)

(2.1) \begin{equation} \Theta = \max \left ( - \frac {\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {u}} {\sqrt {\left (\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {u} \right )^2 + \left (\boldsymbol {\nabla } \times \boldsymbol {u}\right )^2 + \left ( U_\infty / c \right )^2}}; 0 \right ) \end{equation}

where $\boldsymbol {u}$ is the local velocity vector. Figure 2 displays the mean value of $\Theta$ near the shock wave for the reference case. We define a point to be within the shock if $\Theta \gt 0.99$ . The shock position, denoted with $x_s$ in table 1, is computed as the $x$ coordinate of the point closest to the wall where $\Theta \gt 0.99$ . The shock intensity, denoted as $I$ in table 1, is determined as the wall-normal average of the ratio between the pre- and post-shock Mach numbers. At a given wall distance, the pair of Mach number values is extracted at the two sides of the $\Theta = 0.99$ iso-line, i.e. at the black dots of figure 2.

2.2. Description of the numerical experiments

The streamwise-travelling waves (1.1) are characterised by five parameters. Three of them also apply to the canonical case of the plane channel, namely the spatial wavenumber $\kappa _x$ , the pulsation $\omega$ and the amplitude $A$ . Additionally, here, the finite controlled region on the suction side of the aerofoil begins at the position $x_b$ and ends at the position $x_e$ . Following Yudhistira & Skote (Reference Yudhistira and Skote2011), an exponential smoothing function acting on a length scale of $0.05$ chord units is used at the boundary between active and inactive regions. The space of these five parameters is explored with twenty-nine DNS, listed in table 1, where the control parameters are reported along with some global results to be discussed later.

Figure 3.

Control parameters for flow cases C1–C20, superposed to the drag reduction map for incompressible plane channel flow (Gatti & Quadrio Reference Gatti and Quadrio2016). The diamond corresponds to case C2 of Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022). The isolines indicate percentage reduction of friction drag.

Case REF corresponds to the baseline unmanipulated flow at incidence angle of $4^\circ$ . Cases C1 to C20 have a fixed amplitude ( $A=0.684$ , equivalent to $A^+=11$ ), and fixed starting point ( $x_b=0.2$ ) and end point ( $x_e=0.78$ ) of the forcing, as in Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022). These cases are depicted by dots in figure 3, where the drag reduction map for the incompressible channel flow at $Re_\tau =200$ (Gatti & Quadrio Reference Gatti and Quadrio2016) is reported. The first subset of cases lies on the vertical line L1 at $\omega =0$ , corresponding to steady forcing; the second subset lies on the horizontal line L2 at constant wavelength, crossing both drag-reducing and drag-increasing regions of the parameter space; the third subset lies along the oblique line L3, corresponding to the ridge of maximum drag reduction in incompressible channel flow. Additional cases from C21 to C26 are designed to understand the effects of the remaining control parameters ( $x_b$ , $x_e$ and $A$ ). One simulation (labelled as R10 in table 1) modifies one of the best-performing cases (namely C10) by reducing the incidence angle from $4^\circ$ to $3.45^\circ$ , in such a way that the controlled aerofoil yields the same lift of the original uncontrolled case. Case RREF is the corresponding unmanipulated simulation at reduced incidence.

In the remainder of the paper, special attention will be paid to cases C10 (one of the best performers), C12 (the only case yielding drag increase) and R10 (same as C10 but with reduced incidence). For readability, they will be renamed DR (drag reduction), DI (drag-increase) and DRCL (drag reduction at constant lift).

3.1. Force and moment coefficients

Aerodynamic forces and moments derive from the integrated distributions of the projected wall friction and pressure, whose coefficients are defined as

(3.1) \begin{equation} C_f(x) = \frac {2 \tau _w(x)}{\rho _\infty U_\infty ^2}, \qquad \qquad C_p(x) = \frac {2\left [p_w(x) - p_\infty \right ]}{\rho _\infty U_\infty ^2}, \end{equation}

where $\tau _w$ and $p_w$ are the mean wall-shear stress and the mean wall pressure, respectively, obtained after averaging in time and along the spanwise direction.

The mean aerodynamic force $\boldsymbol {F}$ is

(3.2) \begin{equation} \boldsymbol {F} = \oint _S \left [\tau _w \boldsymbol {\hat {t}} - p_w \boldsymbol {\hat {n}} \right ] {\textrm{d}}S = \frac {1}{2} \rho _\infty U_\infty ^2 \oint _S \left [C_f \boldsymbol {\hat {t}} - C_p \boldsymbol {\hat {n}}\right ] {\textrm{d}}S ,\end{equation}

where $\boldsymbol {\hat {t}}$ and $\boldsymbol {\hat {n}}$ are the local unit vectors respectively tangent and normal to the wall, and $S$ is the wing surface. Here, $\boldsymbol {F}$ is conventionally decomposed into lift $L$ and drag $D$ components, respectively normal and parallel to the free stream velocity. The aerodynamic pitching moment $M_{TE}$ with respect to the trailing edge is determined as

(3.3) \begin{equation} M_{TE} = \boldsymbol {\hat {z}} \boldsymbol {\cdot } \oint _S \boldsymbol {r} \times \left [\tau _w \boldsymbol {\hat {t}} - p_w \boldsymbol {\hat {n}} \right ] {\textrm{d}}S ,\end{equation}

where $\boldsymbol {\hat {z}}$ is the unit vector in the spanwise direction and $\boldsymbol {r}$ is the distance vector from the trailing edge. The values of the force and momentum coefficients $C_L$ , $C_D$ and $C_M$ are reported in table 1.

Figure 4.

Distributions of $(a)$ friction coefficient ( $C_f$ ) and $(b)$ pressure coefficient ( $C_p$ , along the aerofoil. Dots are from the reference simulation by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022) and overlap with the present REF case. The vertical lines mark the shock position for each case (note that DR and DRCL overlap); the grey area indicates the active region (on the suction side only).

Figure 4(a) shows the distribution of the friction coefficient along the chord. The reference case (black line) overlaps with the nominally identical one computed by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022), with minor discrepancies near the leading edge due to differences in the numerical procedure adopted. The plot highlights some important flow features, as the local peak of $C_f$ at $x=0.1$ due to tripping, the local reduction under the shock wave (whose position is indicated with the vertical dashed line) and the near zero values at the trailing edge on the suction side. To analyse the effects of wall actuation on friction, the reference case is compared with cases DR, DRCL and DI. Direct effects of control are observed on the suction side only, where it is applied. In terms of friction, the effects of StTWs are as expected: after the forcing is applied (i.e. for $x\gt 0.2$ ), friction decreases (or increases, for DI), becoming mildly negative past the shock for cases DR and DRCL. Simulation DRCL with modified incidence is very similar to DR on either side of the aerofoil, with exception of the region right upstream of the shock, where friction is slightly higher. The shock moves downstream in the drag-reducing cases, whereas it moves upstream for DI. Figure 4(b) shows the distribution of the pressure coefficient. In the REF case, the sudden flow expansion at the leading edge is followed by a plateau where, by design, the aerofoil achieves nearly zero pressure gradient. Further downstream, the mildly negative slope of $C_p$ reveals the presence of a shock. The wall pressure distributions for DR and DRCL do not coincide as a consequence of the different incidence, while the position of the shock across the two cases is unchanged. Case DRCL provides the same lift as the reference case, although its wall pressure remains higher (hence with a lower $C_p$ ) than the reference simulation in the first 40 % of the chord: StTWs displace the shock downstream and the supersonic low-pressure region is wider.

Figure 5.

(a,b) Local friction ( $C_{d,f}$ ) and (c,d) pressure ( $C_{d,p}$ ) contributions to $C_D$ on the (a,c) suction side and (b,d) pressure side of the aerofoil. The grey area highlights the active region (on the suction side only).

Figure 5 shows the local friction and pressure contributions to the drag coefficient ( $C_{d,f}$ and $C_{d,p}$ , respectively, whose integrals are presented in table 1 as $C_{D,f}$ and $C_{D,p}$ ). The distribution of $C_{d,f}$ is very similar to the friction coefficient (depicted in figure 4) on the suction side (panel a), as in this region, the wall is rather flat and aligned to the free stream. The effect of wall actuation on the pressure side, shown in panel $(b)$ , is instead rather small, with minor differences close to the trailing edge associated with different values of the angle of attack. As for $C_{d,p}$ (panels c and d), all the distributions essentially overlap. However, one should bear in mind the different magnitude of the contributions of pressure and friction to the overall drag, which renders pressure changes significant.

Figure 6.

Local difference of (a,b) frictional ( $\Delta C_{d,f}$ ) and (c,d) pressure ( $\Delta C_{d,p}$ ) contributions to the drag coefficient on the suction and pressure sides of the aerofoil. The black lines express the differences between DR and REF cases, whereas the blue lines are obtained comparing cases at lower incidence DRCL and RREF.

This is emphasised in figure 6, where differences between the controlled and uncontrolled cases at the two incidence angles of cases DR and DRCL are shown. On the suction side (panels a and c), the frictional contribution only shows small variations, whereas the pressure contribution exhibits significant changes close to the shock, around the mid-chord. In that region, control yields positive variations of $C_{d,p}$ and contributes to drag increase. This positive contribution is smaller at the lower incidence $\alpha =3.45^\circ$ , as the shock is weaker (see table 1). Panels $(b)$ and $(d)$ show that changes on the pressure side are instead smaller, and do not depend on incidence.

Having isolated the effects of the reduced incidence angle from those directly ascribed to StTWs, some conclusions can be drawn: (i) StTWs always reduce friction drag, except for those combinations of control parameters for which drag increase is expected from channel flow information; (ii) StTWs tend to increase pressure drag, as a result of shock strengthening; (iii) a smaller angle of attack reduces the pressure contribution to the drag force, at least within the range of values considered here, with a less significant effect on the friction contribution; (iv) although the friction and pressure contributions to drag have different magnitudes, the relative control-induced variations are comparable.

3.2. Effects of the control parameters

The performance of StTWs as a function of the control parameters is well understood in the canonical plane channel flow, where drag is entirely due to friction, and the mean friction is constant along the channel; comprehensive information exists which maps the control parameters into drag reduction figures. Furthermore, even if compressibility is taken into account, the reduction of the skin-friction coefficient (and the net power gain) is essentially unchanged as long as the framework remains that of the plane channel flow, as demonstrated by Gattere et al. (Reference Gattere, Zanolini, Gatti, Bernardini and Quadrio2024). Hence, at least locally, the effects of StTWs on the skin-friction over the transonic wing could be derived directly from the canonical channel flow. However, the same does not apply on a global level, since the mean flow properties vary strongly along the streamwise direction due to wall curvature, adverse pressure gradient and the shock wave.

One can define the relative local reduction of the friction coefficient as

(3.4) \begin{equation} \Delta C_f(x) = 100 \frac {C_f(x)-C_{f,0}(x)}{C_{f,0}(x)} ; \end{equation}

this quantity varies because of changes of $C_f$ itself, and because of the presence of the gradual spatial evolution after $x_b$ of the control effect (Skote Reference Skote2012). Figure 7 plots $\Delta C_f(x)$ for flow cases DR and DRCL, limited to the portion of the suction side where forcing is active. The variations of $C_f$ along the chord are significant, thus making it difficult to distil the function $\Delta C_f(x)$ into a single number to compare with the case of the channel flow. However, one can identify a portion of the curve (for example, $x \leqslant 0.4$ ) over which changes in friction can be attributed mainly to wall actuation.

Figure 7.

Relative local change of the friction coefficient ( $\Delta C_f(x)$ ) on the portion of the suction side where control is active, for cases DR and DRCL compared with case REF. The vertical blue line at $x=0.4$ shows where frictional drag reduction is evaluated, whereas the two (nearly coincident) vertical dashed lines display the positions of the shock.

Hence, we consider the value of $\Delta C_f$ at $x=0.4$ as a surrogate measure of frictional drag reduction and define the viscous units with the friction velocity measured at $x=0.4$ . A detailed analysis of how this choice affects the comparison with the case of channel flow is presented in Appendix A, together with law-of-the-wall plots of the mean streamwise velocity profiles extracted at $x=0.4$ .

Figure 8.

Comparison of local reduction of friction coefficient ( $- \Delta C_f$ ) for flow cases C1–C20, across lines L1, L2 and L3 in figure 3 and incompressible channel flow at $Re_\tau = 200$ (Gatti & Quadrio Reference Gatti and Quadrio2016).

In figure 8, we compare the values of friction drag reduction $-\Delta C_f$ at $x=0.4$ for the simulations C1–C20 at constant forcing intensity $A$ with data for the incompressible channel flow at $Re_\tau = 200$ (Gatti & Quadrio Reference Gatti and Quadrio2016), properly interpolated to precisely match the control parameters of each case. While the magnitudes of $\Delta C_f$ should be regarded qualitatively for the reason discussed above, the trends are remarkably similar. In particular, in both cases, the global maximum of drag reduction is found for slow, forward-travelling waves.

Figure 9.

Relative variations of lift ( $\Delta C_L$ ), drag ( $\Delta C_D$ ) and pitching moment ( $\Delta C_M$ ) coefficients, for the cases C1–C20 across lines L1, L2 and L3 of figure 3.

Relative variations of the force and momentum coefficients of the aerofoil for flow cases C1–C20 are reported in figure 9. Perhaps surprisingly, the drag coefficient is found to undergo little changes, with at most a reduction of approximately $1\,\%$ , as a consequence of wall actuation; in addition, in some cases, it even increases slightly. As expected, changes in the pitching moment, which is mainly controlled by pressure, are analogous to changes in $C_L$ .

Figure 10.

Percentage friction contribution to drag ( $100 C_{D,f}/C_D$ ), as a function of the friction drag changes ( $\Delta C_f$ ) for cases C1–C20 across lines L1, L2 and L3 of figure 3. The colour of each data point encodes the shock position. The cross indicates the reference case, while the dashed line is a linear regression of the data for $\Delta C_f \lt 0$ .

Figure 10 displays the friction contribution to the total drag combined with the friction drag reduction. StTWs generally decrease the friction contribution to the overall drag, with the exception of the friction-drag-increasing case DI. Whereas in the reference simulation friction is responsible for $34\,\%$ of the entire drag, this fraction reduces to $29\,\%$ in the presence of wall actuation. The minor changes observed in $C_D$ imply that the relationship between $\Delta C_f$ and $C_{D,f}$ is close to linear. The shock wave is located at $x_s=0.47$ in the reference case, and a connection between a large local friction reduction and a downstream displacement of the shock is clearly visible. The further the shock is delayed towards the trailing edge, the stronger the expansion on the suction side of the aerofoil. Since the post-shock conditions to be matched are approximately the same (see $C_p$ in figure 4), we can conclude that the shock intensity does depend on the shock position, as can be seen from the parameters listed in table 1. Hence, a stronger shock leads to a larger pressure contribution to the aerodynamic coefficients.

3.3. Net savings

As discussed in § 1, assessing the effectiveness of active flow control strategies requires to estimate the energy expenditure for the actuation. In canonical flows, in which drag consists only of friction, one may want to exploit the positive effect of friction-reducing control to improve the balance between performance (e.g. the mass flow rate in channel flow or the pressure losses through the duct) and the related cost, the pumping power. However, infinite ways exist to modify such balance, the two extreme cases being maximisation of performance for given cost or minimisation of cost for given performance. In a plane channel flow (Frohnapfel, Hasegawa & Quadrio Reference Frohnapfel, Hasegawa and Quadrio2012) the two approaches are referred to as constant-pressure-gradient comparison (CPG, whereby the pumping force per unit volume is unchanged and the flow rate can change) and the constant-flow-rate comparison (CFR, whereby the flow rate is unchanged and the pumping force per unit volume can change). A similar line of reasoning applies to the present case, but the definition of costs and benefits is less obvious. As we have seen, the main effect of wall actuation is to increase lift; however, lift is a poor indicator of performance. In fact, in cruise flight, the wing is bound to always generate the same lift force for balancing the aircraft weight.

In analogy with the CFR comparison procedure, an increase of lift can be exploited to reduce the angle of attack, thus aiming at cost minimisation for unchanged performance. Let $\Pi _0 = D_0 U_\infty$ be the nominal power required for flying the aircraft at constant speed $U_\infty$ , with $D_0$ the drag force in the uncontrolled configuration. One can define the net rate of change of the cruise flight power requirement ( $\Delta P_n$ ) as the ratio between the reduction of flight power (incremented by the control power) and the reference power:

(3.5) \begin{equation} \Delta P_n = \frac {\left (\Pi + \Pi _c\right ) - \Pi _0}{\Pi _0}, \end{equation}

where $\Pi = D \, U_\infty$ is the power required to fly the wing at the same speed $U_\infty$ in the controlled configuration. The subscript $n$ in $\Delta P_n$ emphasises that the power change is a net balance that accounts for the extra power $\Pi _c$ required by the active forcing. At the numerator of (3.5), the power reduction $- (\Pi - \Pi _0 )$ is obviously linked to the drag reduction $- (D - D_0 )$ , where the drag $D$ is the drag obtained in a configuration yielding the same lift of the reference case. Finally, the control power $\Pi _c$ is the power transferred from the StTWs to the fluid, assuming unitary efficiency $\gamma =1$ of an ideal actuator, as alway done in numerical studies. Values for $\gamma$ are typically not provided by experimental studies (e.g. Bird et al. Reference Bird, Santer and Morrison2018; Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021; Fumarola, Santer & Morrison Reference Fumarola, Santer and Morrison2024; Knoop et al. Reference Knoop, Hartog, Schrijer, van Campenhout, van Nesselrooij and van Oudheusden2024) where StTWs were tested; these studies either overlook the net savings or simply assume ideal efficiency. While Gatti et al. (Reference Gatti, Güttler, Frohnapfel and Tropea2015) mention a large value of $\gamma =0.7$ for their electro-mechanical actuator for the spanwise oscillating wall, the typical proof-of-principle experiment employing simple mechanical actuators has low efficiencies: for example, actuation power reported by Auteri et al. (Reference Auteri, Baron, Belan, Campanardi and Quadrio2010) for their mechanical implemention of StTWs is 1528 mW, against an ideal control power of 2.1 mW ( $\gamma =0.0014$ ).

The drag $D$ required to evaluate (3.5) is derived from the lift increase $\Delta L$ brought about by wall actuation via the following procedure: (1) the lift curve $C_L(\alpha )$ of the uncontrolled wing is used to determine the reduced incidence angle required to recover the original lift when the control is applied (hence, causing the variation $\Delta C_L$ available in table 1); (2) the drag curve $C_D(\alpha )$ of the uncontrolled wing is used to interpolate drag at the reduced incidence (the presence of forcing is not accounted for yet); (3) the drag coefficient at the reduced incidence is modified to account for the variation $\Delta C_D$ (available in table 1) caused by the direct action of StTWs.

Points (1) and (2) above require the polar curve of the V2C aerofoil, which is impractical to obtain via DNS, and has been computed with RANS instead. In particular, the open-source compressible RANS solver SU2 v.7.5.1 is used, with differential operators at second-order accuracy, and free stream conditions matching those of the DNS. The central Jameson–Schmidt–Turkel scheme (Jameson, Schmidt & Turkel Reference Jameson, Schmidt and Turkel1981) has been employed for the convection terms, with the default Venkatakrishnan slope limiter (Venkatakrishnan Reference Venkatakrishnan1995) employed to preserve monotonicity near shocks. The weighted least squares method is employed for discretising the gradient operator involved in the viscous terms. Turbulence is modelled through the wall-resolved Spalart–Allmaras model (Spalart & Allmaras Reference Spalart and Allmaras1992). The grid is an unstructured, two-dimensional circular mesh with radius of $100$ chords, consisting of 429 100 volume elements. It features an orthogonal prism layer refinement at the aerofoil surface, which guarantees element orthogonality and a wall resolution of approximately one viscous unit just ahead of the shock. To avoid additional uncertainty, no transition modelling has been employed. The simulations are run until the relative change of drag coefficient is smaller than $10^{-8}$ . The pseudo time-stepping required to reach convergence of the steady simulation is performed with implicit Euler. The RANS-based polar is shown in figure 12, together with the datapoints computed with DNS.

Figure 11.

Changes in aerodynamic efficiency ( $\Delta E$ ) and net power ( $\Delta P_n$ ) for cases C1–C20 across lines L1 (standing wave), L2 (fixed $\kappa _x$ ) and L3 (ridge of maximum drag reduction) of figure 3. The cyan markers highlight case DI.

Figure 12.

Polar curve of the V2C aerofoil.

Results of the procedure in terms of $\Delta P_n$ are presented in figure 11: we recall that under the convention employed in this paper, positive $\Delta P_n$ implies more power required in the controlled case, hence power loss. Figure 11 shows that changes of aerodynamic efficiency are generally positive with $\Delta E$ typically increasing by $10\,\%$ on average and up to 30 %. An exception is case DI, highlighted by the cyan markers, which yields drag increase and consequently reduced efficiency. Against the improved efficiency, the required power does not always decrease, being basically unchanged in a few cases and increased in the remaining ones. This is an expected result that parallels the incompressible channel flow, where the power balance at this rather large forcing amplitude is unfavourable. As in the channel, the net balance is found to be best for slow, forward-travelling waves (Quadrio et al. Reference Quadrio, Ricco and Viotti2009). These figures are minimally altered when using the RANS datapoint for the reference state, which yields a slightly better (up to 1.4 % extra) power saving.

Extent and position of the active region are two control parameters with no counterpart in the homogeneous channel flow. Simulations C21–C24 have been performed to quantify how the baseline values of $x_b=0.2$ and $x_e=0.78$ considered by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022) affect the results. Specifically, in flow cases C21 and C22, the end point of the actuated region is moved upstream for fixed $x_b$ , whereas in cases C23 and C24, the origin is shifted downstream for a fixed $x_e$ . In all cases, the active area is reduced, yet the results listed in table 1 show that $\Delta C_f$ is minimally altered when $x_e$ is shifted upstream; in contrast, moving $x_b$ downstream adversely affects $\Delta C_f$ , $x_b=0.4$ being the worst case. The best choice for increasing the aerodynamic efficiency seems thus to be $x_b=0.2$ and $x_e=0.6$ , yielding $\Delta P_n = -9.7\,\%$ , which is the best value obtained among all the DNS performed at $\alpha =4^\circ$ .

Figure 13.

Performance metrics for flow cases with different forcing amplitude $A^+$ : C25 ( $A^+ \approx 3$ ), C26 ( $A^+ \approx 6$ ) and DR ( $A^+ \approx 11$ ). $(a)$ Changes in $C_f$ , compared with incompressible channel flow data; $(b)$ changes in the aerodynamic efficiency and net savings.

It is well known that the energetic efficiency of StTWs varies with the forcing amplitude $A$ in the case of flow over plane walls. This dependence is here verified in flow cases C25 and C26, which replicate case DR with lower forcing amplitudes. The results are presented in figure 13, in which panel $(a)$ shows the resulting changes of $\Delta C_f$ compared with the incompressible channel flow data, interpolated from the database by Gatti & Quadrio (Reference Gatti and Quadrio2016), while panel $(b)$ displays $\Delta E$ and $\Delta P_n$ . Just as for incompressible channels, the friction drag reduction decreases for smaller forcing amplitudes. The agreement, both in qualitative and quantitative terms, between the curves of wing and plane channel flow is remarkable. Interestingly, the improvement of the aerodynamic efficiency (panel b) does not follow $\Delta C_f$ , and halving the control intensity from $A^+ \approx 11$ to $A^+ \approx 6$ does not decrease $\Delta E$ . We find a minumum value of $\Delta P_n$ for the case C27, corresponding to $A^+=6$ . This is a significant analogy with the incompressible channel flow, in which maximum efficiency is obtained for $A^+ \approx 5$ (Quadrio et al. Reference Quadrio, Ricco and Viotti2009), as the power cost grows quadratically with $A$ .

Finally, it is important to stress that in this specific application, the true benefits of StTWs hinge upon the increase of aerodynamic efficiency, and become more significant when the entire aircraft is considered instead of the wing alone. Therefore, for case DR, we replicate the line of reasoning followed by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022) and extrapolate the potential saving to a complete aircraft. Specifically, we consider the same wing-body configuration, i.e. the ‘DLR-F6’ (Laflin et al. Reference Laflin, Klausmeyer, Zickuhr, Vassberg, Wahls, Morrison, Brodersen, Rakowitz, Tinoco and Godard2005), at the reference uncontrolled conditions of $M=0.75$ , $Re_\infty =3 \times 10^6$ and $\alpha _0=0.52^\circ$ . Aircraft information is extracted from the dataset available online at https://aiaa-dpw.larc.nasa.gov/Workshop2/DPW_forces_WB_375.

The simplifying hypotheses adopted are the following:

1. (i) the wing-generated lift is the only contributor to the overall lift of the aircraft: the fuselage and the tail do not contribute (hence, no additional lift sources are considered);

2. (ii) the non-lift-induced drag (i.e. the drag that is computed via this procedure) corresponds to one-third of the overall drag;

3. (iii) the aerodynamic coefficients and the control effects are constant along the wingspan, and do not change with $Re_\infty$ and $M_\infty$ , so that results obtained for the wing section are extended to the whole wing without finite-wing corrections;

4. (iv) the slopes of the curves $C_L-\alpha$ and $C_D-\alpha$ are not affected by the control;

5. (v) the actuated surface is one-fourth of the wing surface and one-twelfth of the whole aircraft surface.

Under these assumptions, one can estimate how StTWs as in case DR would improve the drag coefficient of the entire aircraft. The procedure adopted, which resembles that described above to obtain $\Delta P_n$ , has already been presented by Quadrio et al. (Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022) and therefore is not reported here in detail. The smaller angle of attack in cruise flight, made possible by StTWs, translates into a decrease of the aircraft drag coefficient of $\Delta C_D=-8\,\%$ . This reduction is then augmented by the effect of the StTWs at the reduced angle of attack (see table 1), which yields the additional benefit of $\Delta C_D = -4.4\,\%$ . From the energetic viewpoint, the theoretical $\gamma =1$ control power $\Pi _c$ required by the actuation at unitary efficiency is $0.4\,\%$ of the overall flight power, yielding a nominal net gain of 12 %. While a suitable actuator for StTWs is missing, it may be interesting to mention that this combination of net savings and required power translates into the minimum value of efficiency required by an actuator to provide a positive power gain. The mimimum efficiency is simply the inverse of the maximum gain discussed by Kasagi et al. (Reference Kasagi, Hasegawa, Fukagata and Eckhardt2009). If the requirement for net savings is reformulated via (3.5) for the complete aircraft and for non-unitary control efficiency as

(3.6) \begin{equation} \Delta P_n = \frac {\left (\Pi + \Pi _c / \gamma \right ) - \Pi _0}{\Pi _0} \leqslant 0, \end{equation}

the control performance is such that an actuator efficiency as low as $\gamma = 0.045$ is enough to yield a net power gain.

4.1. Boundary layer

Figure 14 shows a comparative view of the instantaneous wall-shear stress ( $\tau _w$ ) on the suction side of the wing for the REF, DR and DI flow cases. Close to the leading edge, the flow is laminar up to the tripping location, where streaks of high- and low-speed fluid, whose footprints are visible at the wall, begin to form due to the imposed transition. In the reference case, turbulent structures are convected downstream towards the shock, where the strong adverse pressure gradient slows the flow and changes their shape. The wall-shear stress remains positive on average around the shock, while showing instantaneous, locally negative spots. In case DR, the streaks are similar to the reference case right past the tripping point, but the forcing affects their dowstream development past $x_b=0.2$ . Further downstream, where the influence of the shock is stronger, mild flow reversal develops upstream of the shock position, as made evident by the large region with negative wall shear. The observed flow reversal however is likely connected with the limited Reynolds number of the present configuration. In flow case DI, in which wall actuation is ineffective and even yields local increase of $\tau _w$ , the near-wall cycle is strengthened, yielding turbulence intensification and higher friction. Here, the streamwise modulation of the flow induced by wall actuation is clearly visible, both upstream and downstream of the shock. According to Quadrio et al. (Reference Quadrio, Ricco and Viotti2009), this is an indicator that spanwise forcing is operating far from the optimal conditions.

Figure 14.

Instantaneous wall-shear stress on the suction side of the aerofoil. The black lines indicate the position of the tripping and the green lines denote the position of the shock in the three cases; the white contours correspond to $\tau _w = 0$ .

Figure 15.

Shock position (dots) and extent of the reverse-flow region (horizontal bars). The vertical dashed line marks the shock position in the reference case. The dots are coloured according to the skin-friction reduction rate $-\Delta C_f$ . The white diamond corresponds to the unmanipulated flow.

The reversed flow region around the shock is further explored in figure 15, where its length and position, colour-coded with $-\Delta C_f$ , are plotted against the position of the shock wave $x_s$ for cases with constant control extension (i.e. cases C1–C20 and C25–C26). Figure 15 highlights a dependence between the length of the separated region and the position of the shock wave, further influenced by the skin-friction reduction rate; note that this relationship is quite robust, as the plot includes also results with different control amplitude (cases C25–C26). In fact, we observe that the reversed flow area consistently includes the shock and all cases with large $\Delta C_f$ feature long separation bubbles. The longest reversed flow region (approximately 9 % of the chord) is found in case C2, which also achieves the largest friction drag reduction ( $\Delta C_f=-46\,\%$ ). Conversely, case DI (drag-increase) and case C15 (which has the smallest positive $-\Delta C_f$ ) exhibit no flow separation and the shock resides further downstream, similarly to the unactuated case.

Figure 16.

Distribution of integral boundary layer parameters and pressure gradient across the actuated region on the suction side for the REF, DR and DI flow cases: (a) displacement thickness ( $\delta ^\ast$ ); (b) momentum thickness ( $\theta$ ); (c) shape factor ( $H$ ) and (d) longitudinal component of the pressure gradient ( $\partial {C_p}/\partial {x}$ ).

The compressible boundary layer thicknesses $\delta ^*$ , $\theta$ and shape factor $H$ are are shown in figure 16. It is well known that the local wall-shear stress is linked to streamwise variations of the boundary layer thicknesses $\delta ^\ast$ and $\theta$ via the von Kármán integral equation. They are computed as done by Xu, Ricco & Duan (Reference Xu, Ricco and Duan2023):

(4.1) \begin{equation} \delta ^\ast =\int _0^\infty \left (1-\frac {\overline {\rho }\,\widetilde {u}}{\rho _e U_e}\right ) {\textrm{d}}\eta, \qquad \theta = \int _0^\infty \left (1-\frac {\widetilde {u}}{U_e}\right ) \frac {\overline {\rho }\,\widetilde {u}}{\rho _e U_e}{\textrm{d}}\eta, \qquad H=\delta ^\ast /\theta, \end{equation}

where $\overline {\cdot }$ and $\widetilde { (\cdot )}$ denote the temporal and Favre-averaging operators, and the subscript ‘e’ denotes a quantity at the edge of the boundary layer. The edge of the boundary layer is identified with the wall-normal position at which the tangential velocity reaches 99 % of the maximum local tangential velocity.

As long as the wall curvature is limited, a reduction in the wall-shear stress should be associated with a reduction in the growth rate of the momentum thickness. This is indeed observed in the curve corresponding to case DR infigure 16(b). Moreover, the pre-shock adverse pressure gradient region in case DR is wider, the shock moves downstream significantly and the deceleration required to bring the flow back to the trailing edge conditions is stronger: the larger velocity defect is reflected in a larger displacement thickness, which is indeed seen to grow significantly at $x \simeq 0.5$ . The combination of these two effects yields a higher shape factor in flow case DR, which indicates that the boundary layer is more prone to flow reversal. The drag-increasing case DI differs in that near the shock, both $\delta ^*$ and $\theta$ (panels a and b, respectively) slightly grow due to the adverse pressure gradient, hence the shape factor, panel $(c)$ , does not change significantly. The streamwise pressure gradient is shown in figure 16(d). For the reference and drag-increasing cases, the peak of $\partial {C_p}/\partial {x}$ corresponds to the shock position, whereas for flow case DR, the peak lies distinctly downstream of the shock, with approximately the same magnitude. A barely discernible secondary peak is also found upstream of the shock ( $x \approx 0.47$ ), the two peaks approximately marking the start and the end of the flow reversal.

Figure 17.

Profiles of tangential turbulent stress ( $\widetilde {u_t^{\prime\prime}u_t^{\prime\prime}}$ ) along the wall-normal coordinate $\eta$ , at five positions along the chord (whose streamwise coordinate is reported on top of each panel) for the REF, DR and DI flow cases. In the insets, the red lines denote the position along the chord and the orientation of the normal, while the vertical coloured lines show the positions of the shocks.

Last, we have inspected the Reynolds stress tensor to obtain further insight into the turbulence activity within the boundary layer. The components of the Favre-averaged Reynolds stress tensor ( $R_{ij}=\widetilde {u_i^{\prime\prime}u_j^{\prime\prime}}$ , where $ (\cdot ) ''$ denotes the Favre fluctuating component) have been determined by locally decomposing the velocity field into tangential ( $t$ ), normal ( $n$ ) and spanwise ( $z$ ) velocity components. Figure 17 shows the profiles of the tangential stress $R_{tt}$ scaled in reference viscous units at five stations along the suction side. In the reference case, the stress is maximum upstream of the shock, ( $x \approx 0.45$ , second panel), the amplification being related to the formation of a mixing layer (Pirozzoli, Bernardini & Grasso Reference Pirozzoli, Bernardini and Grasso2010). Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020) have shown that in the decelerating region upstream of the foot of the shock, the adverse pressure gradient decelerates the boundary layer, thus increasing the production of turbulence kinetic energy via the longitudinal production term $-\widetilde {u_t^{\prime\prime}u_t^{\prime\prime}}\partial {\widetilde {u}}/\partial {\xi }$ . Hence, a positive feedback loop is established for the turbulent kinetic energy and, in particular, for $\widetilde {u_t^{\prime\prime}u_t^{\prime\prime}}$ , until the flow stops decelerating downstream of the shock. For simulation DR, the local maximum of $R_{tt}$ takes place further downstream, at $x=0.51$ (third panel). Upstream of the shock, case DR exhibits nearly halved fluctuations, whose peak is shifted away from the wall (from $\eta ^+ \approx 10$ of the reference case to $\eta ^+ \approx 30$ ). Flow case DI instead exhibits the opposite behaviour, with slight reduction of the peak, which also moves closer to the wall. Differences are striking at $x=0.51$ , which is upstream of the shock in case DR, but downstream of it in cases REF and DI. Looking at the two rightmost panels, the growth of the boundary layer is reflected in the larger wall distance of the peaks, which occurs at $\eta ^+ \approx 90$ . The monotonic decrease of $\widetilde {u_t^{\prime\prime}u_t^{\prime\prime}}$ confirms the gradual spreading of the mixing layer (Pirozzoli et al. Reference Pirozzoli, Bernardini and Grasso2010).

4.2. Transient analysis

To further elucidate the relationship between StTWs and flow reversal, we have analysed the transient following the sudden application of StTWs at $t=0$ . Specifically, flow samples separated in time by $\Delta t = 6.2 \times 10^{-3}$ have been collected over a time span of $16.6$ convective time units after wall actuation is turned on. Figure 18 describes flow cases C10 (denoted as ‘DR’ so far) and C13; the former has a large reversed flow region, whereas in the latter, the region of reversal is significantly smaller. The left panels show the time history of spanwise-averaged quantities, additionally smoothed out by a running average over six time samples. The quantities are, from top to bottom, the local suction-side friction coefficient ( $C_{f,l}$ ), defined here as the average over the streamwise interval $0.3 \leqslant x \leqslant 0.4$ , the shock position $x_s$ and the flow reversal length ( $L_r$ ). The right panels plot two snapshots of the magnitude of the instantaneous field of the pressure gradient, $\| \boldsymbol {\nabla } p \|$ , plotted on an $(x,y)$ plane at $t=1.7$ and at $t=16.6$ . Movies for the evolution of the pressure gradient field over the transient are available online.

Figure 18.

Transient of flow cases C10 (top) and C13 (bottom). On the left: time histories of the spanwise-averaged mean friction coefficient between $x=0.3$ and $x=0.4$ ( $C_{f,l}(t)$ , top), the shock position ( $x_s(t)$ , centre) and the length of the recirculation bubble ( $L_r(t)$ , bottom), after sudden activation of StTWs. The horizontal lines show the mean asymptotic value (coloured) and the mean value of the REF flow case (black). On the right: magnitude of the pressure gradient $\| \boldsymbol {\nabla } p\|$ on an $(x,y)$ plane, shortly after the control activation ( $t = 1.7$ ) and at a later time ( $t=16.6$ ). Movies are available online.

In both simulations, after enabling control at $t=0$ , the friction coefficient approaches its asymptotic value on a time scale that is short enough to be barely visible in the plots (top-left panels). A non-equilibrium state then ensues, as the reduced friction causes the shock to move downstream. This takes place on a much longer time scale, corresponding to approximately eight convective time units for flow case C10 and four time units for flow case C13. Flow reversal is present in flow case C10 since the earliest stages, intermittently at first, then growing as the shock moves downstream, and becoming nearly stationary for $t\gtrsim 8.2$ . Flow case C13 initially features intermittent flow reversal, with relatively long time intervals with fully attached flow. Approximately five time units are needed for flow reversal to fully establish. Since the shock is located more upstream in C13, the intensity and extent of the adverse pressure gradient region is less, hence the final length of the reversed flow region is significantly smaller than for case C10.

Flow reversal past the shock is only an aspect of transonic shock wave / turbulent boundary layer interaction, which was analysed via DNS by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Grasso2010). Turbulent boundary layers hit by a shock tend to thicken upstream of the interaction owing to the strong pressure rise caused by the shock. This creates compression waves, and, if the Mach number is large enough, induces a separation, ultimately resulting in a $\lambda$ -shock configuration (Délery & Marvin Reference Délery and Marvin1986). The right panels of figure 18 display two different shock patterns. For flow case C10, at the short time $t = 1.7$ , the lower part of the shock tilts, generating compression waves travelling downstream towards the trailing edge. During this motion, they encounter and merge with pressure waves from the wake at the trailing edge, resulting in a secondary, smaller shock, which stabilises quickly. Exchange of information via pressure waves takes place between the two shocks through both the supersonic and subsonic regions, until the system reaches a stable configuration around $t = 16$ . It is worth noting that the final shock configuration resembles the typical topology of neither a weak shock/boundary layer interaction (i.e. with a single shock), nor a strong one (in which a $\lambda$ -shock is formed). In flow case C13, we observe intensification of the shock with respect to its initial configuration, without the secondary shock. The time evolution from $t=1.7$ onwards shows that pressure waves depart from the main shock and merge with those generated at the trailing edge, but they do vanish at $t \approx 8$ .

Therefore, we conclude that StTWs affect first the turbulent boundary layer directly, on a very short time scale, by abating the friction coefficient locally. On a longer time scale, the reduced friction indirectly causes the shock to move towards the trailing edge; this induces global modifications of the wall pressure distribution and then of the entire boundary layer. Flow reversal, which seems to be more a consequence of the modification of the shock position rather than a direct effect of wall actuation, causes the development of a secondary supersonic region. Whenever the recirculating bubble is large enough, a second, weaker shock develops.