2.1 Wing model, wind tunnel and reference systems

The model used in the current investigation is a $45^{\circ }$ swept wing of $1.25$  m span and $1.27$  m chord in the free stream direction. The airfoil used is a modified version of the NACA 66018 shape that was named 66018M3J (more details can be found in Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), the wing and the airfoil are presented in figures 1(b) and 5).

The wind tunnel where the experiments were performed is the closed-loop low-turbulence tunnel (LTT) installed at TU Delft Aerodynamics. The tunnel testing chamber dimensions are: $1.25~\text{m}\times 1.80~\text{m}\times 2.6~\text{m}$ in height, width and length, respectively. Because of seven anti-turbulence screens and a contraction ratio of 17 : 1, the flow has very low levels of free stream turbulence. The latter was measured with a single hot-wire sensor to be $Tu/U_{\infty }=0.07\,\%$ at the free stream velocity of $U_{\infty }=24~\text{m}~\text{s}^{-1}$ (bandpass filtered between 2 and 5000 Hz). The $Tu/U_{\infty }$ value is low enough to observe stationary cross-flow waves (see Bippes Reference Bippes1999) and indeed this was the case for the test case at $U_{\infty }=25.6~\text{m}~\text{s}^{-1}$ and free stream Reynolds number $Re=2.17\times 10^{6}$ (see also Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). The experiment was performed at incidence of $\unicode[STIX]{x1D6FC}=3^{\circ }$ and the flow over the wing pressure side was investigated.

In order to be used for studies on the cross-flow instability, the flow has to be representative of a semi-infinite swept wing thus invariant along the leading edge direction (Deyhle & Bippes Reference Deyhle and Bippes1996; Reibert et al. Reference Reibert, Saric, Carrillo and Chapman1996; Radeztsky et al. Reference Radeztsky, Reibert and Saric1999). The invariance of the flow along the leading edge direction is assessed by two arrays of 46 pressure taps placed near the two wing ends (300 mm from the tunnel walls). At $\unicode[STIX]{x1D6FC}=3^{\circ }$ , the flow shows a high degree of uniformity when checking the pressure distributions of the two arrays (see figure 1 a). Furthermore it is worth noticing that, for this flow configuration, the measured pressure minimum point is at $X/c_{X}=0.63$ thus ensuring a fully stable boundary layer with respect to Tollmien–Schlichting (TS) modes.

Figure 1.

(a) Pressure distribution on the wing pressure side measured by two arrays of 46 pressure taps placed 300 mm from the top and bottom wing tips respectively at $\unicode[STIX]{x1D6FC}=3^{\circ }$ and $Re=2.17\times 10^{6}$ . (b) Schematic of the model and definition of the $xyz$ , $XYZ$ and $x_{t}y_{t}z_{t}$ reference systems.

Due to the inherent complexity of the geometry and flow topology, several reference systems are introduced. These are presented in figure 1(b). Firstly, the unswept coordinate system is defined with the upper case letters $XYZ$ and is positioned such that the $X$ and $Y$ axes lie in a horizontal plane ( $X$ being in the chord plane and $Y$ orthogonal to that plane) while the $Z$ is the vertical direction. The velocity components along this coordinate system are indicated with upper case letters $UVW$ . The wing is rotated at incidence angle of $\unicode[STIX]{x1D6FC}=3^{\circ }$ and, as such, the wind tunnel reference system does not coincide with the unswept wing coordinates. The velocity components along the wind tunnel reference systems are denoted with upper case letters and the subscript: $_{\infty }$ .

The swept reference system is such that its $x$ axis is orthogonal to the wing leading edge, the $z$ axis is parallel to the leading edge (both these axes belong to the chord plane) and the $y$ axis coincides with the unswept $Y$ direction. By rotating the $XYZ$ system about the $Y$ axis by $\unicode[STIX]{x1D6EC}=45^{\circ }$ , one goes to the swept $xyz$ system. The velocity components in this system of coordinates are indicated with the lower case letters $uvw$ .

A last coordinate system needs to be introduced as the local tangential system (the definition of tangential is simplified here as it implies only the surface curvature along the $X$ direction). This system is positioned such that its $x_{t}$ axis is along the surface tangent, the $y_{t}$ is the wall-normal direction and $z_{t}$ coincides with the $Z$ axis.

2.2 AC-DBD plasma actuators

As identified earlier, for the needs of this study, actuator-induced roughness is of great importance. To this end, special actuators were designed, built and employed. The actuators were manufactured using a metal deposition technique developed in-house. The technique makes use of computer-controlled spraying of sub-micron conductive silver particles on the dielectric surface. The resulting electrode thickness is of the order of 1  $\unicode[STIX]{x03BC}\text{m}$ . For comparison, traditional hand-made actuators based on adhesive copper tape, typically have electrode thickness of 60  $\unicode[STIX]{x03BC}\text{m}$ . The plasma actuators were placed at $x/c=0.025$ , close to the primary mode critical station (see the LST analysis of § 3). The plasma actuators were designed towards forcing a single monochromatic primary cross-flow mode. To force the desired mode, the employed actuators feature one straight electrode, which is exposed to the flow and is powered by the high-voltage supply. The grounded electrode is encapsulated below the dielectric barrier and is made of a series of stems spaced equally from each other and oriented along $x$ (see figure 3 b). The spacing between the stems defines the spanwise wavelength of the plasma forcing, and as such the spanwise wavelength of the forced CF modes. Three different modes were forced in this study, namely $\unicode[STIX]{x1D706}_{z}=9$  mm, which corresponds to critical mode (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016), and $\unicode[STIX]{x1D706}_{z}=7$  mm and $\unicode[STIX]{x1D706}_{z}=12$  mm, corresponding to sub-critical and super-critical modes respectively.

A detail of the DBD plasma actuators used in this study is presented in figure 2. The covered electrode stems are $d=10$  mm long and $w=3$  mm wide. They are projected towards the straight electrode overlapping with it by $o=2$  mm, in order to ensure consistent plasma formation. Given the relative large length of these stems, the flow ionisation region corresponds only to the electrode overlapping area thus ensuring the desired spanwise modulation of the forced jet. It is important to note that the exposed electrode was placed downstream so as to have the induced jet aligned with $-x$ (see figure 3 b).

Kapton polyimide film of 50  $\unicode[STIX]{x03BC}\text{m}$ thickness was used as dielectric barrier. A second layer of Kapton was placed between the actuator and the model surface to better protect the latter. As such, the total thickness of the actuator was measured to be 125  $\unicode[STIX]{x03BC}\text{m}$ . In order to avoid exposure of the incoming flow to a step of such high thickness, the Kapton sheet was wrapped around the wing leading edge. This ensured that the only surface perturbation seen by the flow was a two-dimensional backward facing step of 125  $\unicode[STIX]{x03BC}\text{m}$ at approximately 10 % of the model chord. Preliminary investigations, using PIV and infrared thermography, indicated no adverse effects on the transition location by the physical presence of the actuator on the wing.

Figure 2.

Detail (picture) of a DBD plasma actuator used for selective forcing of monochromatic primary cross-flow instability modes. The Kapton sheet, the covered electrode (CE), the exposed electrode (EE), the stems wavelength ( $\unicode[STIX]{x1D706}_{z}$ ), the width ( $w$ ), the distance between the electrodes ( $d$ ) and the overlap ( $o$ ) are also shown. View of the underside of the actuator, CE on foreground, EE on background.

Figure 3.

(a) Schematic of the wind tunnel model with the actuators (Kapton sheet in orange, CE in dark grey, EE in light grey). Definition of the field of view of the IR thermography (blue dashed rectangle) and of the PIV (red dashed rectangle) experiments. (b) Schematic of the plasma actuator. The plasma jets (magenta), the inviscid streamline (cyan) and the related reference system (ISL) are shown together with the swept reference system ( $xz$ ). The schematics are not to scale.

The actuator was powered using a Trek 20/20C HS high-speed high-voltage amplifier. Depending on the case, the actuation signal was sinusoidal with amplitudes (E) between 2 and 3 kV. The scope of this study is to prove the applicability of DBD plasma actuators as control devices to condition the spanwise wavelength of primary cross-flow instability modes. As such, the high-voltage signal frequency (carrier frequency) was chosen considering the development of unsteady instability modes in the boundary layer. Preliminary linear stability theory analysis (presented in § 3) showed that the considered boundary layer is unstable to travelling primary instability modes comprised in the band 0–1 kHz. Therefore for the main tested cases, the actuators were operated at 2 kHz. To decouple the effects of unsteady forcing from the effects of forcing monochromatic CFI modes, a two-dimensional spanwise-uniform actuator was also considered.

2.3 Infrared thermography

The boundary layer transition evolution was inspected by IR thermography. The model was irradiated by two 1 kW halogen lamps placed approximately one metre from the model. The convective heat flux from the wing surface to the flow is a function of the local wall shear. As such, regions of high shear cool more rapidly than respective low shear regions. Differences in wall temperature captured by the thermal camera can therefore be used for visualisation between laminar and turbulent regions (see also Saric, Carpenter & Reed Reference Saric, Carpenter and Reed2011). In addition, using highly sensitive, actively cooled thermal imagers, details in the shear distribution of laminar boundary layers, such as the characteristic streaky footprint of CFVs can be retrieved.

The set-up for the IR thermography experiment is presented in the schematic of figure 4(a). The camera was mounted at about 1 m distance from the model surface and placed outside the wind tunnel, looking through an opportune germanium window. The imaged field of view was $480\times 360~\text{mm}^{2}$ and centred at $X/c_{X}=0.25$ , $Z=0$ (see the schematic of figure 3(a), where the IR domain is shown with a dashed (blue) rectangle). The IR camera used for the current experiment was a CEDIP Titanium 530L, featuring a mercury–cadmium–telluride quantum sensor with $320\times 256~\text{px}^{2}$ . The sensor is actively cooled by means of a Stirling cycle to the working temperature of 77 K resulting in a noise equivalent temperature difference of less than 18 mK. The integration time and the operated frame rates were 200  $\unicode[STIX]{x03BC}\text{s}$ and 100 Hz respectively. The camera was equipped with a 25 mm focal length germanium lens operated at $f_{\sharp }=2.0$ .

Figure 4.

Schematics (not to scale) of the experimental set-ups. The flow comes from right. The model (in black), the testing chamber (semi-transparent) and the opportune diagnostics windows are shown. (a) IR-thermography experiment with the IR camera (in brown). (b) High-speed PIV experiment with the PIV camera (in blue), the laser head (in dark green) and the light sheet from downstream (light green).

2.4 Particle image velocimetry

Particle image velocimetry was performed towards quantification of the boundary layer velocity field. Planar high-speed PIV was chosen for this experiment. The inspected planes are along the $X-Z$ directions at a distance of 1.5 mm from the model surface. The planes are centred with the station $X/c_{X}=0.25-Z=0$ of the model (see the schematic of figure 3(a), where the PIV domain is shown with a dashed (red) rectangle).

The PIV experimental set-up is presented in figure 4(b). As shown, the camera was imaging the field of view (FOV) from outside the wind tunnel through available optical windows and was placed at approximately 1 m distance from the measured plane. The camera used in this experiment is a LaVision Imager HS featuring a sensor of $2016\times 2016~\text{px}^{2}$ and pixel size of 11  $\unicode[STIX]{x03BC}\text{m}$ and 12 bits of digital resolution. The active image sensor was reduced to $1632\times 496~\text{px}^{2}$ for the current experiment. The imager was equipped with a Nikon Nikkor 200 mm micro lens operated at $f_{\sharp }=5.6$ . The resulting magnification factor of the PIV experiment is $M=0.24$ . Illumination was provided from downstream as shown in the schematic of figure 4(b). A Nd:YAG high-speed Continuum Mesa PIV laser (18 mJ pulse $^{-1}$ ) was employed for particle illumination. The light beam was opportunely shaped into a sheet of approximately 1 mm thickness. The light sheet was oriented so as to be parallel to the model surface for all the imaged domain as shown in figure 5. In the schematic the AC-DBD plasma actuator is also shown. The Kapton sheet is in light grey and the electrodes in dark grey. Note that this is the wing cross-section orthogonal to the leading edge direction.

Figure 5.

Wing section and schematic (not to scale) of the laser light (light green) and of the boundary layer (cyan). The AC-DBD plasma actuator is also shown: Kapton sheet in light grey; electrodes in dark grey. Inset: magnified view of the actuator. The wing section is orthogonal to the leading edge.

The flow was seeded using Safex water glycol injected downstream of the testing chamber. The average particle diameter is 1  $\unicode[STIX]{x03BC}\text{m}$ . Synchronisation of the system was guaranteed by using the LaVision Davis 8.2 suite and a LaVision High Speed Controller unit. Image acquisition, pre-processing and correlation were performed with the same software. Cross-correlation was performed with final interrogation windows of $24\times 24~\text{px}^{2}$ with relative overlap set to 75 %. As such the final vector spacing was approximately 0.27 mm in both $X$ and $Z$ . The time interval between the two laser pulses was set to 15  $\unicode[STIX]{x03BC}\text{s}$ so as to give an average particle displacement in the free stream of 15 px. The measurement frame rate (FR) was 3000 Hz and sequences of 6000 image pairs were acquired, for every tested flow case. The resulting measurement time was 2 s.

The correlation random error in planar PIV is estimated to be 0.1 px (Raffel et al. Reference Raffel, Willert, Wereley and Kompenhans2007). Therefore the random error, relative to the free stream velocity, is $\unicode[STIX]{x1D700}_{(U,V)}=0.1/15=0.007$ . This is also the measurement error for the statistic velocity fluctuations fields ( $\{U_{t}^{\prime },V_{t}^{\prime }\}_{RMS}$ ). Here $U_{t}^{\prime }$ and $V_{t}^{\prime }$ are the time velocity fluctuations (i.e. after subtracting the time average). The uncertainty on the time-averaged velocity fields is retrieved from the flow fluctuations magnitude, estimated to be less than 10 % of the free stream velocity, and from the measurement ensemble size ( $ES$ ): $\unicode[STIX]{x1D700}_{(\bar{U},\bar{V})}=0.1/\sqrt{ES}$ , where the overbar denotes time averaging. The ensemble size to be considered here does not encompass all the 6000 captured fields. These are correlated in time with respect to the travelling instability modes, which account for most of the fluctuation energy at the measurement location (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). The fluctuations related to this instability feature high energy content in the frequency band $100~\text{Hz}\leqslant f\leqslant 400~\text{Hz}$ (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). Therefore a conservative time sub-sampling of the measured fields is required in order to formally render them uncorrelated in time. This factor can be estimated as FR $/\text{min}(f_{(U_{t}^{\prime },V_{t}^{\prime })})=3000~\text{Hz}/100~\text{Hz}=30$ and is used to estimate the uncorrelated ensemble size as a fraction of the total of the captured samples: $ES_{u}=ES/30=200$ . Finally, the uncertainty on the time-averaged velocity fields can be evaluated as $\unicode[STIX]{x1D700}_{(\bar{U},\bar{V})}=0.1/\sqrt{ES_{u}}\approx 0.018$ .

2.5 Non-dimensionalisation and normalisation

Several reference systems and different experimental and numerical techniques were used and described in the present study. Part of the results is therefore presented in non-dimensional units. The airfoil chords along the $X$ and $x$ directions are used as reference length scales for the respective coordinate systems. However, some results are presented in dimensional units to ease comparison with dimensional quantities and avoid introduction of elaborated nomenclature. This consideration follows from the actuators employed in this study. These devices can be described by non-dimensional quantities as e.g. the momentum coefficient (see § 5). However, the main dimensional parameters, namely the frequency of the driving AC voltage and the wavelength of the electrodes, are crucial for the description and understanding of the presented analyses. The velocity scale used in the remainder is the free stream velocity ( $U_{\infty }$ ). In addition, $U_{\infty }^{2}$ is used to non-dimensionalise the velocity power spectra.

When specified, the presented spectra are normalised as shown in (2.1) (in agreement with Parseval’s theorem),

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FE}}^{\ast }\cdot \unicode[STIX]{x1D6FF}_{f}=(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FE}}\cdot \unicode[STIX]{x1D6FF}_{f})\frac{\displaystyle \frac{1}{ES-1}\unicode[STIX]{x1D6F4}_{t}(\unicode[STIX]{x1D6FE}^{\prime })^{2}}{\unicode[STIX]{x1D6F4}_{f}(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FE}}\cdot \unicode[STIX]{x1D6FF}_{f})},\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}$ is the power spectral density (PSD), $\unicode[STIX]{x1D6FF}_{f}$ is the spectra frequency resolution, $\unicode[STIX]{x1D6FE}$ is the spectra variable and $ES$ the temporal ensemble size. The sums are performed with respect to the subscript quantity (i.e. time ( $t$ ) at the numerator and frequency ( $f$ ) at the denominator).

3.1 Formulation

The linear stability problem is presently solved using the Chebyshev spectral collocation method, with 100 polynomials. The spatial formulation is chosen where the flow disturbances are expressed in the waveform of (3.1),

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D713}(x,y,z,t)=\unicode[STIX]{x1D719}(y)\text{e}^{\text{i}(\unicode[STIX]{x1D6FC}x+\unicode[STIX]{x1D6FD}z-\unicode[STIX]{x1D714}t)},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}$ (angular frequency) is real, $\unicode[STIX]{x1D6FC}$ (wavenumber along $x$ ) is complex and $\unicode[STIX]{x1D6FD}$ (wavenumber along $z$ ) is real as no growth is assumed along the $z$ direction based on the assumption of spanwise invariance.

Details on the derivation of the Orr–Sommerfeld (OS) equation for 2.5-dimensional flows are given by Mack (Reference Mack1984) (pp. 3–13). The companion matrix technique of Bridges & Morris (Reference Bridges and Morris1984) is employed to overcome the eigenvalue nonlinearity. N factors ( $N$ ) are computed as the integral of growth rates (i.e. the imaginary part of the chordwise wavenumbers: $-\unicode[STIX]{x1D6FC}_{i}$ ) along $x$ . Envelope N factors ( $N^{env}$ ) are the $x$ -integral of the local maximum values, per chord station, among all the considered modes (i.e. wavenumbers) (Arnal et al. Reference Arnal, Gasparian and Salinas1998; Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). Equations (3.2a,b ) give the mathematical formulations of these two quantities,

(3.2a,b ) $$\begin{eqnarray}N(x,\unicode[STIX]{x1D706}_{z},\unicode[STIX]{x1D714})=\int _{x_{0}(\unicode[STIX]{x1D706}_{z},\unicode[STIX]{x1D714})}^{x}-\unicode[STIX]{x1D6FC}_{i}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D706}_{z},\unicode[STIX]{x1D714})\,\text{d}\unicode[STIX]{x1D701};\quad N^{env}(x,\unicode[STIX]{x1D714})=\max |_{\unicode[STIX]{x1D706}_{z}}(N(x,\unicode[STIX]{x1D706}_{z},\unicode[STIX]{x1D714})).\end{eqnarray}$$

5.1 Experimental set-up

The camera, laser and synchronisation equipment used for this experiment are the same as described in § 2.4. For this experiment, the camera’s active sensor was reduced to $960\times 440~\text{px}^{2}$ to increase its maximum frame rate. The camera was equipped with a Nikon Nikkor 200 mm macro lens operated at $f_{\#}=8$ . The magnification factor was $M=1.14$ . The air in the closed box was seeded with droplets of paraffin oil, $1~\unicode[STIX]{x03BC}\text{m}$ in average diameter, generated by a TSI Atomizer nozzle. The coordinate system used in this experiment is represented by the symbols $x_{p}y_{p}z_{p}$ (see figure 3 b). The system is arranged such that the $x_{p}$ axis is in the direction of the induced jet (i.e. orthogonal to the electrodes length), the $y_{p}$ is perpendicular to the plane of the actuator and $z_{p}$ is along the length of the electrodes. The plasma actuator was installed in a transparent acrylic glass box enclosing a volume of $0.25~\text{m}^{3}$ (1 m long along $x_{p}$ ). The plane inspected with PIV is aligned with $x_{p}-y_{p}$ , given the uniformity of the induced flow field along $z_{p}$ . The laser, camera and transparent box were installed on a pneumatically stabilised table (i.e. an optical table) to avoid vibrations on the set-up which could be detrimental for such high magnification planar PIV experiment. Cross-correlation was carried out with final interrogation windows of $24\times 24~\text{px}^{2}$ with relative overlap set to $75\,\%$ . As such, the final vector spacing was approximately 0.053 mm in both $x_{p}$ and $y_{p}$ directions. The system was operated at acquisition rate of 10 kHz in single-frame mode thus the pulse separation between the correlated frames was $100~\unicode[STIX]{x03BC}\text{s}$ . Sequences of 25 000 images were acquired for each test case for a total measurement time of 2.5 s. The measurement started prior to plasma actuation thus capturing the initial flow acceleration phase.

5.2 Performance

In quiescent flow environments, when the actuator is supplied with the voltage signal, the strong electric field established between the electrodes ionises and accelerates the air molecules. The incompressible Navier–Stokes momentum equation is used to retrieve the induced body force. In this very initial regime, advection, diffusion and pressure terms are negligible with respect to the unsteady acceleration and body-force terms. Following Kotsonis et al. (Reference Kotsonis, Ghaemi, Veldhuis and Scarano2011), under the specified assumptions, the momentum equation reduces to $\boldsymbol{F}=\unicode[STIX]{x1D71A}$ $\unicode[STIX]{x2202}\boldsymbol{U}/\unicode[STIX]{x2202}t$ , where $\boldsymbol{F}$ is the body force and $\boldsymbol{U}$ the velocity. Low-pass filtering of the measured velocity signal was applied in order to eliminate high frequency oscillations associated with the AC voltage frequency. In addition to the body forces, the time-averaged velocity field is computed, after the induced jet has fully developed, from the measured time series and presented in figure 7 for the case of $f_{f}=2~\text{kHz}-E=3~\text{kV}$ .

The induced jet originates in the vicinity of the inner edge of the exposed electrode and is mainly aligned with the $x_{p}$ direction. The wall-normal velocity component, at the electrodes interface, supplies the mass for the wall-parallel jet (Kotsonis et al. Reference Kotsonis, Ghaemi, Veldhuis and Scarano2011). A secondary wall-parallel jet, oriented opposite to the main velocity, is observed on top of the exposed electrode. This secondary jet is not observed with the usual copper electrode actuators and can be caused by the extremely low thickness of the employed actuators building up a competing electrical field. A confined region of negative $x_{p}$ -body force is also observed. The body-force field is overall rather weak, for the tested conditions, and predominant along the $x_{p}$ direction (Kotsonis Reference Kotsonis2015).

Figure 7.

(a) Time-averaged velocity magnitude field (grey solid lines: 5 levels, from 0 to $0.6~\text{m}~\text{s}^{-1}$ ). (b) Body-force magnitude field (grey solid lines: 5 levels, from 50 to $1000~\text{N}~\text{m}^{-2}$ ). Vectors: $x_{p}-y_{p}$ components of the respective contour variables (reference vector in the upper left corner). AC-DBD actuator operated at $f_{f}=2~\text{kHz}-E=3~\text{kV}$ . The two horizontal grey lines below $y_{p}=0$ represent (to scale) the electrodes of the actuator. The markers at $x_{p}=1.6~\text{mm}-y_{p}=0.25$ (grey) and 0.75 mm (black) refer to figure 8.

Spatial integration of the body-force field retrieves the exerted specific (i.e. per actuator unit length) thrust. This is eventually used to compute the momentum coefficient, according to (5.1), which is used to compare the actuators authority for the different forcing conditions,

(5.1) $$\begin{eqnarray}C_{\unicode[STIX]{x1D707}}=\frac{T_{x_{p}}}{\frac{1}{2}\unicode[STIX]{x1D71A}{u_{e}}^{2}\unicode[STIX]{x1D717}_{u}},\end{eqnarray}$$

where the local (at the actuator position) boundary layer edge velocity $u_{e}=14~\text{m}~\text{s}^{-1}$ and momentum thickness $\unicode[STIX]{x1D717}_{u}=7.81\times 10^{-2}$  mm are extracted from the boundary layer numerical solution for the investigated case (see appendix A).

The thrust generated by the plasma actuator along $x_{p}$ and the respective momentum coefficient for the different test cases considered in this study are presented in table 2. When the actuator is operated at $f_{f}=2~\text{kHz}-E=3$  kV, the horizontal component of the thrust generated is $T_{x_{p}}=0.432~\text{mN}/m_{act}$ and the momentum coefficient is $C_{\unicode[STIX]{x1D707}}=0.046$ . This is the combination of supplied voltage and frequency that leads to the strongest induced jet. When the frequency is reduced to 400 Hz, the performance of the actuator is reduced. Similarly, decreasing the voltage amplitude to 2 kV at the same operated frequency of 2 kHz, lowers $C_{\unicode[STIX]{x1D707}}$ .

For all the investigated configurations of table 2, the generated thrust is comparatively less than what reported by Chernyshev et al. (Reference Chernyshev, Kuryachii, Manuilovich, Rusyanov and Skvortsov2013) and Dörr & Kloker (Reference Dörr and Kloker2015b ) in their studies. However, in the current study, the AC-DBD plasma jets are used in a respective fashion to isolated roughness elements for DRE/UFD flow control. In such a configuration, the actuators are used to induce a weak localised perturbation of the incoming flow thus requiring only moderate forcing amplitudes. The specific thrust applied by the actuators used by Dörr & Kloker (Reference Dörr and Kloker2017) for the same purpose is comparable to the values reported here.

Table 2.

AC-DBD actuator-induced thrust and momentum coefficient.

Capitalising on the high sampling rate of the employed PIV method, access to the unsteady and spectral features of the plasma-induced flow field is gained. Spectral analysis of the velocity fluctuations, at two points at $x_{p}=1.6~\text{mm}-y_{p}=0.25$ , 0.75 mm (markers in figure 7 a), was carried out for the two different frequencies at which the actuators were operated (0.4 and 2 kHz). The results are shown in figure 8. The spectra are normalised according to (2.1) and non-dimensionalised with $U_{\infty }^{2}$ thus they directly show the relative kinetic energy per frequency band. They reveal strong unsteady fluctuations at the respective carrier frequency as well as higher harmonics. Moreover, low frequency fluctuations close to the wall are observed. These can be attributed to a range of factors including plasma-induced thermal disturbances, measurement noise and unstable plasma formations. The exact origin of these fluctuations should be taken into account in the future for a complete characterisation of the plasma effect. Moreover, it is currently unclear to what extent these low frequency fluctuations would be present also on the wing mounted configuration under external flow condition. Such study will require a dedicated effort that goes beyond the scope of this article.

7.1 Time-averaged flow topology

The time-averaged $U$ velocity fields for the different test cases are presented in figure 13. Figure 13(a) shows the unforced flow. The actuator is mounted on the wing but is not supplied with voltage. The boundary layer, within the imaged domain, is still laminar and slightly modified by natural CFI mechanisms as seen in figure 9(a). The appearance of stationary vortices is observed in the presented velocity field as a weak velocity modulation along the span.

Figure 13.

Time-averaged $U$ velocity fields (10 levels, from 0.84 (black) to $1U_{\infty }$ (white)). The flow comes from left. (a) Actuator non-operated; (b) spanwise-uniform forcing; (c) sub-critical forcing ( $\unicode[STIX]{x1D706}_{f}=7$  mm); (d) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm); (e) super-critical forcing ( $\unicode[STIX]{x1D706}_{f}=12$  mm). Actuator operated at $f_{f}=2~\text{kHz}-E=3$  kV ( $C_{\unicode[STIX]{x1D707}}=0.046$ ). The (red) lines and labels refer to analyses presented in the remainder. Inset: spanwise fluctuation of the $\bar{U}_{z}^{\prime }$ velocity signal sampled along the line AB in fields: (c) black, (d) dark grey, (e) light grey line.

When two-dimensional forcing is applied (figure 13 b), the flow drastically rearranges. The overall magnitude of velocity is reduced, showing a thickening of the boundary layer. Moreover, the velocity local minima are now more distinguishable, suggesting the enhancement of the stationary primary vortices. When spanwise-modulated forcing is applied (figure 13 c–e), separate and more uniform stationary vortices become clearly visible. The spanwise spacing, along a line parallel to the leading edge (segment AB in the contours) can be compared to the actuated wavelength. The velocity spanwise fluctuations ( $\bar{U}_{z}^{\prime }$ ) for the spanwise-modulated flow cases are presented in the inset of figure 13, further reconciling with the outcomes of the flow visualisation experiment of § 6. When the super-critical mode ( $\unicode[STIX]{x1D706}_{f}=12$  mm) is forced, the superharmonic mode seen before in figure 11(c) becomes evident, downstream of $x/c=0.25$ , in the regions in between the stronger CFVs. The lower side of the presented flow fields always shows lower velocities in comparison to the upper region. This effect is attributed to a possible small inclination of the laser sheet (i.e. rotation about the $X$ axis), such that the lower illuminated region is slightly closer to the wall.

Inspecting the fields pertaining to the three different forced wavelengths shows that, when critical forcing is applied, the vortices appear more developed than the sub-critical and super-critical forcing cases. In fact it can be seen that the velocity minima corresponding to the low-momentum region attain the lowest values for the $\unicode[STIX]{x1D706}_{z}=9$  mm vortices. In addition, the spatial evolution of the stationary waves can be quantified by computing their amplitude as defined by equation (7.1) (White & Saric Reference White and Saric2005 and Serpieri & Kotsonis Reference Serpieri and Kotsonis2016),

(7.1) $$\begin{eqnarray}A^{CFV}(x)=z_{rms}\{\bar{U_{z}^{\prime }}(x)/U_{\infty }\}|_{y_{t}=y_{t}^{PIV}}.\end{eqnarray}$$

Figure 14.

Spatial evolution of the individual vortices of figure 13(c–e) computed from (7.1). (a) Sub-critical forcing ( $\unicode[STIX]{x1D706}_{f}=7$  mm); (b) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm); (c) super-critical forcing ( $\unicode[STIX]{x1D706}_{f}=12$  mm).

As evident in figure 13, ensembles of two to three individual vortices are available through the PIV measurements. Due to the fixed FOV, the number of fully resolved vortices is dependent on the forcing wavelength. For the calculation of the amplitudes, the individual vortices are tracked in their spatial evolution. The estimated amplitudes for the individual vortices in the spanwise-modulated forcing cases (figure 13 c–e) are shown in figure 14(a–c) respectively.

Within the imaged field, the stationary vortices show similar growths despite the different amplitudes. Forcing the critical mode at $\unicode[STIX]{x1D706}_{f}=9$  mm leads to more developed waves compared to the other two modes, especially with respect to the sub-critical mode, in both initial amplitude and spatial amplification. These results, which match with the stronger energy shown for this mode in the IR spectra (figure 11), confirm that forcing sub-critical modes leads to less amplified stationary vortices (see e.g. Reibert et al. Reference Reibert, Saric, Carrillo and Chapman1996; Malik et al. Reference Malik, Li, Choudhari and Chang1999 and Wassermann & Kloker Reference Wassermann and Kloker2002). All the fields show monotonic growth of the stationary vortices. However, the amplitudes appear to saturate at the most downstream portion of the fields. This is expected from boundary layers dominated by stationary CFVs undergoing laminar-to-turbulent transition (Haynes & Reed Reference Haynes and Reed2000; Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). However, it has to be clarified here that performing this analysis on planes at constant distance from the model wall has some limitations. In fact, moving downstream, the stationary waves grow and so does the boundary layer (see the schematic of figure 5). Hence, moving downstream at a constant wall-normal position, the measurement plane intersects the stationary vortices in a lower region where the velocity modulation effect is milder. This results, when using the metrics of (7.1), in lower amplitudes. Therefore the growth rates of the stationary CF vortices should be retrieved from full three-dimensional fields (Reibert et al. Reference Reibert, Saric, Carrillo and Chapman1996; Haynes & Reed Reference Haynes and Reed2000; Wassermann & Kloker Reference Wassermann and Kloker2002; White & Saric Reference White and Saric2005; Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). Despite these observations, the effect is rather mild within small chordwise domains, such as the one considered here.

Figure 15.

Comparison between the measured spanwise-mean velocity magnitude ( $z_{mean}\{\bar{V}^{tot}\}|_{y_{t}=y_{t}^{PIV}}$ ) from the velocity fields of figure 13 (symbols) and the boundary layer numerical solution (solid line) at $x/c=0.25$ . (○) Actuator non-operated; (▫) spanwise-uniform forcing; (*) sub-critical forcing ( $\unicode[STIX]{x1D706}_{f}=7$  mm); ( $+$ ) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm); (♢) super-critical forcing ( $\unicode[STIX]{x1D706}_{f}=12$  mm). Inset: magnified view.

An additional consideration in the analysis of figure 14 relates to the fact that the mean-flow distortion of the considered fields caused by the primary CFVs is not even. In fact, more developed vortices modify the mean flow more significantly. A way to inspect this effect is proposed here by comparing the measured spanwise-mean velocity magnitude ( $z_{mean}\{\bar{V}^{tot}\}|_{y_{t}=y_{t}^{PIV}}$ ) from the velocity fields of figure 13 (sampled along the AB segment) and the undisturbed boundary layer numerical solution (described in appendix A). The outcomes of this comparison are presented in figure 15. Evidently, the boundary layer corresponding to the spanwise-uniform and spanwise-modulated critically forced ( $\unicode[STIX]{x1D706}_{f}=9$  mm) flows underwent significant mean-flow distortion. The unforced flow case as well as the super-critical ( $\unicode[STIX]{x1D706}_{f}=12$  mm) and sub-critical ( $\unicode[STIX]{x1D706}_{f}=7$  mm) forcing cases are distorted more mildly. The measured unforced boundary layer agrees relatively well with the numerical prediction. The slight discrepancy is attributed to modelling assumptions in the boundary layer solution as well as measurement uncertainties in the PIV fields.

Figure 16.

Time-averaged $U$ velocity fields (10 levels, from 0.84 (black) to $1U_{\infty }$ (white)). (a)  $f_{f}=0.4~\text{kHz}-E=3$  kV ( $C_{\unicode[STIX]{x1D707}}=0.013$ ); (b)  $f_{f}=2~\text{kHz}-E=2~\text{kV}$ ( $C_{\unicode[STIX]{x1D707}}=0.037$ ).

The time-averaged velocity field for the additional forcing case of the critical mode actuator operated at the lower frequency (400 Hz) and same voltage amplitude (3 kV, $C_{\unicode[STIX]{x1D707}}=0.013$ ) is presented in figure 16(a). The velocity field shows a spanwise modulation with amplified stationary vortices at different wavelengths. When the actuator is operated at $f_{f}=2~\text{kHz}-E=2~\text{kV}$ ( $C_{\unicode[STIX]{x1D707}}=0.037$ ) (figure 16 b), the boundary layer features weaker stationary vortices when compared to the $f_{f}=2~\text{kHz}-E=3$  kV case (figure 13 d). This further confirms the dependency of forced cross-flow vortex strength to the momentum coefficient of the actuator.

A comparison of the present study to the passive control investigation of Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016), is shown in figure 17. In this previous work, the critical cross-flow instability mode was conditioned with leading edge micron-sized roughness elements (diameter: $d_{r}=2.8$  mm, height: $k_{r}=10~\unicode[STIX]{x03BC}\text{m}$ ) installed at the same chordwise position as the plasma actuators in the present study ( $x/c=0.025$ ). The velocity profile was acquired with hot-wire scans. Note that the velocity measured in that experiment, coincides mainly with $\sqrt{U^{2}+V^{2}}$ , however, given the strong predominance of the $U$ component (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016), this comparison can still be valid. The comparison demonstrates that, when the actuator is operated at $f_{f}=2~\text{kHz}-E=3~\text{kV}$ ( $C_{\unicode[STIX]{x1D707}}=0.046$ ), the spanwise-averaged value of the mean velocity attains the lowest value among all cases. For the lower frequency case, the lower velocity value compared to the laminar solution is explained by the more advanced transitional status of the forced boundary layer. When the actuator was operated at 400 Hz, the boundary layer showed multiple stationary modes (figure 12 a) and enhanced fluctuations (see the discussion of figure 20 a). Moving to the other flow cases, the comparison with the passive roughness elements used by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) is more instructive given the CFI monochromatic state of the compared boundary layers. The hot-wire profile shows already at this chordwise station a small deviation from the laminar profile, caused by the mean-flow distortion (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). Moreover, the profile crosses the station of the current PIV measurements ( $y_{t}=1.5$  mm) between the velocity values of the tested voltages thus showing that the $d_{r}=2.8~\text{mm}-k_{r}=10~\unicode[STIX]{x03BC}\text{m}$ passive roughness has a milder effect on the boundary layer than the $f_{f}=2~\text{kHz}-E=3~\text{kV}$ ( $C_{\unicode[STIX]{x1D707}}=0.046$ ) active roughness but stronger compared to the $f_{f}=2~\text{kHz}-E=2~\text{kV}$ ( $C_{\unicode[STIX]{x1D707}}=0.037$ ) case.

Figure 17.

Comparison between the measured spanwise-mean velocity magnitude ( $z_{mean}\{\bar{V}^{tot}\}|_{y_{t}=y_{t}^{PIV}}$ ) from the velocity fields of figure 16 (symbols), the boundary layer numerical solution (solid line) at $x/c=0.25$ . Critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm): (○)  $f_{f}=0.4~\text{kHz}-E=3$  kV ( $C_{\unicode[STIX]{x1D707}}=0.013$ ); (▫)  $f_{f}=2~\text{kHz}-E=2$  kV ( $C_{\unicode[STIX]{x1D707}}=0.037$ ); (*)  $f_{f}=2~\text{kHz}-E=3$  kV ( $C_{\unicode[STIX]{x1D707}}=0.046$ ). Dotted line: $d_{r}=2.8~\text{mm}-k_{r}=10~\unicode[STIX]{x03BC}\text{m}$ passive roughness (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). Inset: magnified view.

7.2 Flow fluctuations

As demonstrated in § 5.2, the employed plasma actuators produce an inherently fluctuating forcing effect, at a frequency aligned to the oscillation of the AC high-voltage signal. The effect of this unsteady forcing on the boundary layer is investigated in this section. Dörr & Kloker, in their studies (Dörr & Kloker Reference Dörr and Kloker2015b , Reference Dörr and Kloker2016) (Reynolds number based on plate length and free stream velocity orthogonal to the leading edge $Re=92\,000$ ), showed very little direct effect of unsteady forcing on the unsteady instability modes. The actuators in the aforementioned studies were modelled with stationary or periodic high frequency body-force inputs in their numerical set-up. However, in the case of unsteady force, the actuators were operated at a frequency more than four times higher than the frequency of the pertinent type-I secondary instability mode. The present study is conducted at a higher Reynolds number (chord Reynolds number $Re=1.08\times 10^{6}$ ), at which the respective mode peaks at a frequency of 5.5 kHz (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016). In the current experimental framework, achieving frequencies four times larger than the type-I mode frequency was technically unfeasible. This restriction was already envisioned by Dörr & Kloker as a possible limitation for UFD-LFC with AC-DBD actuators for high Reynolds numbers flows. Based on the above considerations, it was chosen to operate the actuators of the present study at a frequency of 2000 Hz. Such frequency is beyond the primary travelling modes band ( $0<f<1000$  Hz), yet below the frequency range corresponding to secondary instabilities ( $f>4000$  Hz) (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016).

Figure 18.

Standard deviation of $U$ velocity fields (10 levels, from 0 (white) to $0.0784U_{\infty }$ (black)). (a) Actuator non-operated; (b) spanwise-uniform forcing; (c) sub-critical forcing ( $\unicode[STIX]{x1D706}_{f}=7$  mm); (d) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm); (e) super-critical forcing ( $\unicode[STIX]{x1D706}_{f}=12$  mm). Upper region masked for higher measurement noise.

7.2.1 Statistical analysis

The statistic fluctuation fields, for the test cases presented in figure 13, are presented in figure 18. The overall level of velocity fluctuations for the unforced flow (figure 18 a) is very low. In contrast, spanwise-uniform forcing dramatically enhances the unsteady field (figure 18 b). This flow case was chosen to investigate the effects that the employed unsteady forcing has on the boundary layer. The higher level of fluctuations in conjunction to the thickening of the boundary layer discussed before suggest that this flow is in a more advanced transitional state compared to the unforced case.

When spanwise-modulated forcing is applied (figure 18 c,d and e for sub-critical, critical and super-critical wavelengths, respectively), the flow fluctuations appear enhanced only in the regions of the conditioned stationary vortices. This is expected in boundary layers dominated by stationary cross-flow vortices as the latter introduce strong velocity shears (Kawakami, Kohama & Okutsu Reference Kawakami, Kohama and Okutsu1999; Malik et al. Reference Malik, Li, Choudhari and Chang1999; White & Saric Reference White and Saric2005; Bonfigli & Kloker Reference Bonfigli and Kloker2007). The lower level of fluctuations for the sub-critical mode suggests a more stable flow in line with the DRE/UFD control strategies (this aspect will be addressed later in § 7.2.4).

Figure 19.

Non-dimensional normalised power spectra ( $\unicode[STIX]{x1D6F7}_{U_{t}^{\prime }}^{\ast }\cdot \unicode[STIX]{x1D6FF}_{f}/U_{\infty }^{2}$ , see (2.1),  $\unicode[STIX]{x1D6FF}_{f}=10$  Hz) of the $U_{t}^{\prime }$ velocity signal sampled along the AB line at $x/c=0.25$ of figures 13 and 18 (10 levels, from $-60$ (white) to $-30$  dB (black)). (a) Actuator non-operated; (b) spanwise-uniform forcing; (c) sub-critical forcing ( $\unicode[STIX]{x1D706}_{f}=7$  mm); (d) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm); (e) super-critical forcing ( $\unicode[STIX]{x1D706}_{f}=12$  mm).

7.2.2 Spectral characteristics

Time series of the $U_{t}^{\prime }$ velocity signal sampled along the AB segment at $x/c=0.25$ of figures 13 and 18 are analysed in the Fourier domain. The power spectra are computed using Welch’s method (Welch Reference Welch1967) with a final frequency resolution of $\unicode[STIX]{x1D6FF}_{f}=10$  Hz. The spectra are normalised according to (2.1) and non-dimensionalised with $U_{\infty }^{2}$ . The results are presented in figure 19 for the five flow cases of figures 13 and 18. While the unforced flow shows very low spectral energy in all the resolved band ( $0~\text{Hz}\leqslant f\leqslant 1500~\text{Hz}$ ), the spanwise-uniform actuation results in very pronounced fluctuations in the band $0~\text{Hz}\leqslant f\leqslant 350~\text{Hz}$ . This is somewhat unexpected as the operated AC voltage is fixed at the frequency of $f_{f}=2000$  Hz which is well beyond the corresponding band of travelling modes at these flow conditions (cf. § 3). While at this point the analysis is inconclusive, it is likely that either low frequency fluctuations directly introduced by the actuator (recall the results of § 5.2) or nonlinear interactions between the plasma-induced jets and the boundary layer modes take place in the unsteady velocity field, thus enhancing the fluctuations energy in the travelling CF waves band. Moreover, the spanwise uniformity of the unsteady forcing is also present in the spectra. This is a direct consequence of the absence of amplified steady vortices in this flow field as seen in the statistics fields (figure 13 b).

Analysis of the spanwise-modulated actuated flows shows that the fluctuations are modulated along the span directly corresponding to the position of the conditioned cross flow vortices. They attain considerable energy throughout the resolved spectrum, albeit stronger at the lower frequencies. Among the different spanwise-modulated actuators, the 9 and 12 mm wavelength actuators are responsible for the strongest fluctuations. The Nyquist frequency of the implemented PIV measurements is lower than the secondary instability band (Serpieri & Kotsonis Reference Serpieri and Kotsonis2016) and hence the secondary instability mechanisms cannot be directly inspected in the frequency domain. This analysis is presented in the remainder, capitalising on the measurements spatial resolution.

The velocity spectra of the velocity fields of figure 16 are shown in figure 20. Forcing the flow at $f_{f}=0.4~\text{kHz}-E=3~\text{kV}$ ( $C_{\unicode[STIX]{x1D707}}=0.013$ ), enhances the low frequency modes considerably, as expected. A narrow band of strong fluctuations at the forced frequency is clearly distinguishable thus suggesting the receptivity of the particular flow to this range of frequencies. The fluctuations are less modulated along the span than the energy comprised in the band $0~\text{Hz}\leqslant f\leqslant 350~\text{Hz}$ which occur at the locations of local velocity shears in the time-averaged fields. These two bands appear rather distinguished suggesting a different origin. While the 400 Hz fluctuations are directly introduced by the actuator, the lower frequency, wider-band fluctuations resemble those reported in figure 19(b–e).

For the case of $f_{f}=2~\text{kHz}-E=2~\text{kV}$ ( $C_{\unicode[STIX]{x1D707}}=0.037$ ), the actuation introduces rather weak velocity fluctuations. The spectrum is similar to the unforced field of figure 19(a) as for this voltage the actuator has reduced strength.

Figure 20.

Same notation as figure 19 for the fields of figure 16 (10 levels, from $-60$ (white) to $-30$  dB (black)). (a)  $f_{f}=0.4~\text{kHz}-E=3$  kV ( $C_{\unicode[STIX]{x1D707}}=0.013$ ); (b)  $f_{f}=2~\text{kHz}-E=2$  kV ( $C_{\unicode[STIX]{x1D707}}=0.037$ ).

7.2.3 Spatial organisation

The results discussed hitherto confirm the initial estimations towards the applicability of AC-DBD plasma actuators for control of cross-flow instabilities based on the UFD approach. More specifically, the capability of conditioning the amplified stationary mode with spanwise-modulated actuators is experimentally demonstrated for the swept wing case, analogous to the work of Schuele et al. (Reference Schuele, Corke and Matlis2013) in the supersonic cone case. However, the effects of forcing on the unsteady field result in strong amplification of travelling modes, which potentially can have detrimental effects for future LFC purposes.

Towards gaining insight on the low frequency fluctuations enhanced by the unsteady actuation (§ 7.2.2), description of the spatial organisation of the fluctuating velocity fields is sought. The unsteady flow field pertaining to a single stationary wave is analysed. Unsteady fluctuations are stronger when the stationary vortex is more amplified due to elevated shears. As such, the flow field encompassing the most amplified stationary vortex (CFV1, enclosed by the (red) rectangle in figures 13 and 18) is henceforth considered. Two forcing cases pertaining to the critical mode forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm) and the spanwise-uniform forcing are selected for comparison in the following analysis. The choice of these cases is dictated by their inherently different spectrum. While the $\unicode[STIX]{x1D706}_{f}=9$  mm forces the most critical stationary mode, the uniform forcing case is spanwise invariant and, as such, does not favour a certain wavelength.

Irrespective of the chosen forcing case, the velocity fluctuations occurring at this stage of the boundary layer development are still rather weak and therefore challenging to identify considering the experiment dynamic range. Technically, the latter could be increased by augmenting the PIV laser pulses separation, however this approach would compromise the effective resolution of the velocity shears (Scarano Reference Scarano2002). As such, statistical decomposition based on spatial coherence is employed. More specifically, proper orthogonal decomposition following the snapshot formulation (Sirovich Reference Sirovich1987) is used, similar to previous analyses on tomo-PIV data by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016).

Figure 21.

Power spectra ( $\unicode[STIX]{x1D6FF}_{f}=10$  Hz) of the first 30 POD modes’ time coefficients (5 levels, from $-20$ (white) to 10 dB) (black). The DBD plasma actuator is operated at $f_{f}=2~\text{kHz}-E=3$  kV ( $C_{\unicode[STIX]{x1D707}}=0.046$ ). (a) Spanwise-uniform forcing; (b) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm).

As mentioned, Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) investigated the same flow case as the one considered in the present study, albeit using DREs for conditioning the cross-flow vortices. They reported that a significant portion of kinetic energy, captured by the first POD mode, was related to a low frequency oscillation of the stationary vortices. Further analyses on less energetic POD modes allowed the identification of flow structures related to the type-I and type-III instabilities. The type-I mode, corresponding to a secondary instability mode of the strong spanwise shears (Högberg & Henningson Reference Högberg and Henningson1998; Kawakami et al. Reference Kawakami, Kohama and Okutsu1999; Malik et al. Reference Malik, Li, Choudhari and Chang1999; Wassermann & Kloker Reference Wassermann and Kloker2002) was shown to be of Kelvin–Helmholtz nature (Bonfigli & Kloker Reference Bonfigli and Kloker2007). Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) reported the type-I mode to have a wavelength of about 4.6 mm along the stationary vortices axes and occur in a range of frequencies centred at 5 kHz. The type-III mode is instead caused by the interaction of primary stationary and travelling waves (Fischer & Dallmann Reference Fischer and Dallmann1991; Högberg & Henningson Reference Högberg and Henningson1998; Kawakami et al. Reference Kawakami, Kohama and Okutsu1999; Malik et al. Reference Malik, Li, Choudhari and Chang1999). Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) found these vortices to have a wavelength of 25 mm and a frequency of ${\approx}500$  Hz.

The high sampling frequency of the current experiment allows direct investigation of the temporal evolution of the low frequency fluctuations (i.e. the type-III mode for the flow cases where stationary vortices have amplified). The secondary instability modes are instead occurring at higher frequencies, which lie beyond the measured band. Analysis of the temporal coefficients of the POD modes can infer the nature of the related structures. The spectra of the first 30 POD modes temporal coefficients is shown in figure 21 for the cases of spanwise-uniform forcing and critical forcing at $\unicode[STIX]{x1D706}_{f}=9$  mm. The spectra frequency resolution is $\unicode[STIX]{x1D6FF}_{f}=10$  Hz. When the spanwise-uniform actuator is employed, high energy levels are observed in the presented POD modes–frequency plane. In particular almost 50 % of the total turbulent kinetic energy (TKE) pertains to the first ten POD modes (not shown). These first ten modes feature most of their energy in the $0~\text{Hz}\leqslant f\leqslant 500$  Hz band, further corroborating the outcomes of figure 19(b). For the spanwise-uniform forced flow, the absence of strong stationary vortices, and related shears, leads to a different instability evolution without the growth of secondary high frequency instability modes. A continuous energy cascade towards higher frequency eddies and higher POD modes is therefore characterising the flow.

In the presence of strong stationary waves (figure 21 b), the spectral energy is less spread through the POD modes. POD mode 1 shows very high energy at the lowest frequencies and is related to the POD mode 1 observed by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016). High energy concentrations occur in the first ten POD modes, featuring strong fluctuations within the band $0~\text{Hz}\leqslant f\leqslant 500~\text{Hz}$ and reaching the relative cumulative TKE level of 23 % (not shown). This is the frequency band where the primary travelling waves are expected as predicted by LST (§ 3), as well as experimentally observed by Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016). Proceeding towards higher index modes, the associated energy shifts towards higher frequencies beyond the resolved band. In fact, inspection of the POD eigenmodes, suggests that these POD modes pertain to type-I secondary instability mechanisms along the stationary vortex shears (as will be shown in § 7.2.4). These modes feature frequencies in the range of 4–9 kHz for the same flow case, as reported in Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016). Due to the limited PIV sampling rate in the present study, frequencies higher than 1500 Hz are not resolved. This limits the following analysis to the low frequency/high energy fluctuations.

A reduced-order reconstruction of the instantaneous velocity field based on the first 10 POD modes, which were shown to be resolved in time by the performed measurement, is presented in figure 22, for the same flow fields of figure 21. Three subsequent instantaneous flow fields, separated from each other by $\unicode[STIX]{x0394}_{t}=1/3000~\text{Hz}=3.\bar{3}\times 10^{-4}$  s, are plotted by means of $U_{t}^{\prime }$ velocity contours. When spanwise-uniform forcing is supplied, strong velocity fluctuations take place compared to the unforced flow field shown in figure 18(a). These fluctuations are captured in the reconstructed field of figure 22(a). They appear as a sequence of broad structures, spanning the entire vertical extent of the field and advected mainly along $X$ with a smaller oscillating component along the $Z$ direction. The reconstructed time series for the case of critical spanwise-modulated forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm) is presented in figure 22(b). In this case, the flow structures appear significantly smaller and localised in space compared to the spanwise-uniform forced flow. The structures are advected downstream, moving from the downwelling flow region towards the upwelling region of the stationary waves similarly to what reported in previous studies for the type-III modes (Högberg & Henningson Reference Högberg and Henningson1998; Kawakami et al. Reference Kawakami, Kohama and Okutsu1999; Malik et al. Reference Malik, Li, Choudhari and Chang1999; Wassermann & Kloker Reference Wassermann and Kloker2002; Bonfigli & Kloker Reference Bonfigli and Kloker2007; Serpieri & Kotsonis Reference Serpieri and Kotsonis2016).

The presented time sequence encompasses a total period of ${\approx}1$  ms hence it is not representative of the whole fluctuations evolution. To better represent the latter, a spectral analysis is sought. The instantaneous velocity signal for the entire measurement time (2 s) is sampled along a line parallel to $X$ and centred in the domain of figure 22 (along the (red) solid line in the leftmost contours). The signals are therefore function of $X$ and time: $g=g(X,t)$ . The $g$ signals are divided in segments of 600 time samples with 50 % overlap between neighbouring segments, weighted with a two-dimensional Hanning function and decomposed in their spatio-temporal Fourier modes. The resulting frequency–wavenumber spectra are then averaged together (Welch Reference Welch1967). The spectra have a frequency resolution of $\unicode[STIX]{x1D6FF}_{f}=10$  Hz and a spatial resolution of $\unicode[STIX]{x1D706}_{X}^{min}=2.16$  mm (two times the size of the interrogation windows (Schrijer & Scarano Reference Schrijer and Scarano2008)).

The spectra are shown in figure 23 for the two considered cases. The relation between the wavenumber and the frequency, directly corresponding to the structures phase velocity (referred to as $U_{c}=f\unicode[STIX]{x1D706}_{X}$ ), can be retrieved from the presented plots. A least-squares linear fit (dashed (red) lines in figure 23) through the energy maxima per wavelength was applied on the spectra giving the values of $U_{c}=17.60~\text{m}~\text{s}^{-1}$ ( $0.69U_{\infty }$ ) for the spanwise-uniform actuator and $U_{c}=27.50~\text{m}~\text{s}^{-1}$ ( $1.08U_{\infty }$ ) for the critical mode flow. The different values of advection velocity correlate with the lower values of the local velocity seen in the time-averaged fields (figure 13 b,d). The phase velocity component in the $Z$ direction could be estimated in a similar fashion (sampling the velocity time series along the $Z$ direction). However, given the small magnitude of this velocity component, the results are not deemed reliable.

Figure 22.

Reconstructed time sequence of the $U_{t}^{\prime }$ velocity fields (mean subtracted) from the first 10 POD modes (black lines: solid (positive) and dashed-dotted (negative), 5 levels, from $\pm 0.058$ to $\pm 0.004U_{\infty }$ ) and time-averaged velocity field (solid grey lines: 5 levels, from 0.84 to $1U_{\infty }$ ). The flow fields presented here are shown with the (red) rectangle in figures 13 and 18(b,d). (a) Spanwise-uniform forcing; (b) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm). The solid (red) lines in the leftmost contour refer to figure 23.

Figure 23.

Wavenumber–frequency power spectra for $g=g(X,t)$ of the full time series presented in figure 22 (6 levels, from $-15$ (white) to 35 dB (black)). (a) Spanwise-uniform forcing; (b) critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm). The velocity signal were extracted along the (red) solid lines shown in the leftmost contours of figure 22. The (red) dashed lines indicate the phase velocities ( $U_{c}$ ).

Summarising, while the instantaneous fields for the critical stationary mode case are dominated by type-III structures, strong velocity fluctuations mainly advecting along the free stream direction are amplified by the unsteady 2-D forcing. These observations indicate that, for the current flow cases, the unsteady flow field is strongly modified by the used plasma actuators. In § 8 a further effort in understanding this phenomenon is presented. Despite this undesired effect, the actuators showed authority in conditioning the developing CFI to the forced modes. Therefore a comparative analysis of the unsteady fields pertaining to the different forced modes can be sought.

Figure 24.

(a,c) $U$ velocity fields of normalised POD modes (mode index indicated in the lower left corner) (shaded: 11 levels comprised between $\pm 0.05$ ) and time-averaged velocity field (solid black lines: 3 levels, from 0.84 to $1U_{\infty }$ ). The flow fields presented here are shown with the (red) rectangle in figures 13(d) and 18(d) (critical forcing ( $\unicode[STIX]{x1D706}_{f}=9$  mm)). (b,d) Wavenumber power spectral density of the velocity signal sampled along the (red) line inclined along the stationary vortex, of (a) and (c) respectively.

7.2.4 UFD flow control using plasma actuators

In this section, the unsteady fields of the spanwise-modulated forcing cases are further examined within the perspective of UFD flow control. To this goal, the unsteady fields pertaining to the three forced wavelengths cases (same forcing amplitude) are reconstructed from selected POD modes. The POD modes were first analysed in their wavenumber spectral content along lines parallel to the CFVs axes (inclined $-2.12^{\circ }$ at the measurement location with respect to $X$ ). Note that, given this small angle, approximately the same wavelength limits described in the previous section are resolved. In figure 24, two POD modes with longer and shorter-wavelength structures, respectively, are shown together with the wavenumber velocity spectra along the stationary vortices to illustrate the followed procedure.

Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) report, for the same flow field, wavelengths of $\unicode[STIX]{x1D706}_{X_{W}}=25$  mm for type-III modes and of $\unicode[STIX]{x1D706}_{X_{W}}=4.6$  mm for type-I modes. Following their results, the wavelength $\unicode[STIX]{x1D706}_{X_{W}}=20$  mm was used here as a dividing threshold to reconstruct the $U$ fluctuations in two bands. Similarly, White & Saric (Reference White and Saric2005) and Serpieri & Kotsonis (Reference Serpieri and Kotsonis2016) used time filters to separate the different frequency bands from hot-wire measurements. The results are presented in figure 25. It can be seen that, for both the longer and shorter-wavelength bands, the sub-critical control mode ( $\unicode[STIX]{x1D706}_{f}=7$  mm) demonstrates lower fluctuations level. The other stationary modes show enhanced disturbances. This is especially evident for the shorter-wavelength disturbances, which are related to secondary mechanisms (note that these modes are not resolved in time, given the high frequency as shown in figure 21). These are located on the outer upwelling flow side of the stationary waves (Malik et al. Reference Malik, Li, Choudhari and Chang1999; Wassermann & Kloker Reference Wassermann and Kloker2002), as shown in figure 24(c). This analysis further corroborates the known beneficial aspects of the UFD flow control approach on the evolution of secondary instability mechanisms.

Figure 25.

Standard deviation of the reconstructed $U$ field from POD: (a) modes with $\unicode[STIX]{x1D706}_{X_{W}}>20$  mm; (b) modes with $\unicode[STIX]{x1D706}_{X_{W}}\leqslant 20$  mm (shaded: 5 levels, from 0 (white) to $0.04U_{\infty }$ (black)) and time-averaged velocity field (solid black lines: 3 levels, from 0.84 to $1U_{\infty }$ ). The flow fields presented here are shown with the (red) rectangle in figures 13 and 18(c,e).

8.1 Model and flow cases

The suggested model is based on the addition of the actuator body force as a source term in the stationary, incompressible laminar boundary layer equations for the case of spanwise invariant flow (2.5-D flow) presented in appendix A. Therefore, the analysis will focus on the generalised case of spanwise-uniform forcing. The inclusion of the plasma effect can be conveniently modelled as a volume distributed body force that can be extracted with numerical or experimental means (Kotsonis Reference Kotsonis2015). It must be noted that certain assumptions are necessarily attached to such an approach. It is assumed that the external flow does not affect the plasma actuator body force (Pereira, Ragni & Kotsonis Reference Pereira, Ragni and Kotsonis2014; Dörr & Kloker Reference Dörr and Kloker2015b , Reference Dörr and Kloker2016), and thermal effects due to plasma are weak enough to assume incompressible conditions (Kotsonis Reference Kotsonis2015). Additionally, the force is assumed to be time invariant (recall the definition of steady effect given in the previous paragraph) and of negligible magnitude in the wall-normal direction ( $F_{y}\approx 0$ ). The latter is a common assumption (Dörr & Kloker Reference Dörr and Kloker2015a ) and is essential in the boundary layer formulation of the Navier–Stokes equations as wall-normal pressure invariance ( $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}y=0$ ), upon which the system of equations (A 1)–(A 3) is based, will otherwise not hold (Schlichting & Gersten Reference Schlichting and Gersten2000). Finally, $x$ diffusion terms are also neglected. The stated set of assumptions allows the modification of only the $x$ -momentum equation as shown in (8.1),

(8.1) $$\begin{eqnarray}u\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}=-\frac{1}{\unicode[STIX]{x1D71A}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}+\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}y^{2}}+\frac{F_{x}}{\unicode[STIX]{x1D71A}}.\end{eqnarray}$$

The boundary layer equations (A 1), (8.1), (A 3) with the boundary conditions described in appendix A are integrated with the same numerical solver used for the stability analyses of § 3. The body-force field ( $F_{x}$ ), measured in the dedicated experiment of § 5, was smoothed and interpolated at the actuator’s chordwise position on the solver grid.

To investigate the effects of both forcing direction and magnitude, five cases are presently considered. Additional to the unforced baseline case, two cases consider the use of the body-force field ( $C_{\unicode[STIX]{x1D707}}=0.046$ ) of figure 7(b) oriented along and against the boundary layer flow in $x$ respectively, thus respecting the semi-infinite (spanwise invariant) swept wing assumption. Finally, two cases consider a fivefold body force ( $C_{\unicode[STIX]{x1D707}}=0.230$ ) oriented again along and against the flow in $x$ . An overview of the modelled cases, and respective notation, is given in table 3.