2.1 Basic set-up

Figure 1. Integration domain and coordinate systems.

The DNS are performed with our compressible, high-order, finite-difference code NS3D; see e.g. Dörr & Kloker (Reference Dörr and Kloker2015b ) for details. The vector $\boldsymbol{u}=[u,v,w]^{\text{T}}$ denotes the velocity components in the chordwise, wall-normal and spanwise directions $x$ , $y$ and $z$ , respectively. Velocities and length scales are normalized by the chordwise reference velocity $\bar{U}_{\infty }$ and the reference length $\bar{L}$ , respectively, with the overbar denoting dimensional values; further, the reference density is $\bar{\unicode[STIX]{x1D70C}}_{\infty }$ and the reference temperature $\bar{T}_{\infty }$ . In the experiments the Mach number based on the chordwise free-stream velocity $\bar{U}_{\infty ,exp}=22.66~\text{m}~\text{s}^{-1}$ is $Ma_{\infty ,exp}=0.066$ . The reference length is defined as $\bar{L}_{exp}=0.1~\text{m}$ . For the simulations a computationally non-prohibitive Mach number $Ma_{\infty }=0.2$ is chosen, keeping the ambient conditions and the Reynolds number range identical. The reference values in the simulations are then $\bar{U}_{\infty }=68.871~\text{m}~\text{s}^{-1}$ , $\bar{L}=0.033~\text{m}$ , $\bar{\unicode[STIX]{x1D70C}}_{\infty }=1.181~\text{kg}~\text{m}^{-3}$ and $\bar{T}_{\infty }=\bar{T}_{wall}=295.0~\text{K}$ . For details of the base-flow generation for the DNS, see Dörr & Kloker (Reference Dörr and Kloker2017); note that the reference length $\bar{L}_{exp}$ is not the length of the plate $\bar{L}_{plate,exp}=0.6~\text{m}$ in the experiments.

A rectangular integration domain with a block-structured Cartesian grid is considered for the simulations (see figure 1). The inflow and outflow are treated with characteristic boundary conditions. In addition, sponge layers based on a volume-forcing term and a spatial compact tenth-order low-pass filter are employed to further minimize disturbances. To allow mass flow through the free-stream boundary for the simulations including the plasma actuators, the conditions there are as follows: the base-flow values for $\unicode[STIX]{x1D70C}$ and $T$ are kept and $\text{d}u/\text{d}y|_{e}=\text{d}w/\text{d}y|_{e}=0$ allows $u_{e}$ and $w_{e}$ to adapt, where the subscript $e$ denotes the values at the upper boundary of the domain; in addition, $v_{e}$ is calculated according to $\text{d}v/\text{d}y|_{e}=-(\text{d}(\unicode[STIX]{x1D70C}_{e}u_{e})/\text{d}x+\text{d}(\unicode[STIX]{x1D70C}_{e}w_{e})/\text{d}z)/\unicode[STIX]{x1D70C}_{e}$ , assuming $\text{d}\unicode[STIX]{x1D70C}/\text{d}y|_{e}=0$ . For more details, see Dörr & Kloker (Reference Dörr and Kloker2016). At the lateral boundaries periodicity conditions are prescribed. The fundamental spanwise wavelength is $\unicode[STIX]{x1D706}_{z,0}=0.180$ , corresponding to the fundamental spanwise wavenumber $\unicode[STIX]{x1D6FE}_{0}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z,0}=35.0$ . Explicit finite differences of eighth order are used for the discretization in the $x$ -direction and $y$ -direction, and a Fourier spectral ansatz is used for the $z$ -direction. In the chordwise direction $x$ , an equally spaced grid with 2604 points is used, covering $0.173\leqslant x\leqslant 4.078$ , with $\unicode[STIX]{x0394}x=1.5\times 10^{-3}$ . In the wall-normal direction $y$ , a stretched grid with 152 points is employed, $0.000\leqslant y\leqslant 0.121$ , with $\unicode[STIX]{x0394}y_{wall}=1.251\times 10^{-4}$ and $\unicode[STIX]{x0394}y_{max}=2.135\times 10^{-3}$ . The fundamental spanwise wavelength $\unicode[STIX]{x1D706}_{z,0}$ is discretized with 64 points ( $K=21$ de-aliased Fourier modes), yielding $\unicode[STIX]{x0394}z=2.805\times 10^{-3}$ . An explicit fourth-order Runge–Kutta scheme is employed for the time integration. The fundamental angular frequency is $\unicode[STIX]{x1D714}_{0}=6.0$ , and the time step is $\unicode[STIX]{x0394}t=8.727\times 10^{-6}$ .

A disturbance strip at the wall with synthetic blowing and suction, centred at $x=0.800$ , $0.766\leqslant x\leqslant 0.835$ , and alternating in spanwise direction $z$ , is employed to excite the primary CF vortex disturbances $(0,2)$ or $(1,+2)$ ; the double-spectral notation ( $h\unicode[STIX]{x1D714}_{0}$ , $k\unicode[STIX]{x1D6FE}_{0}$ ) is used. To initiate controlled laminar breakdown, two additional disturbance strips are positioned farther downstream to excite pulse-like (background) disturbances ( $h,\pm 1$ ), with $h=1{-}50$ . The disturbances are forced near $x=1.0$ , $x=2.0$ and $x=2.5$ with a strip extension of 0.045, respectively. Note that the modes $|k|\geqslant 2$ are nonlinearly generated at once by the primary modes.

2.2 Modelling of the plasma actuator

Figure 2. (a) Schematic of plasma actuator and planar volume-force distribution. (b) Angle $\unicode[STIX]{x1D6FD}_{PA}$ for rotation about the wall-normal axis $y$ ; see Dörr & Kloker (Reference Dörr and Kloker2017).

Based on the empirical model given by Maden et al. (Reference Maden, Maduta, Kriegseis, Jakirlić, Schwarz, Grundmann and Tropea2013), a non-dimensional wall-parallel volume-force distribution $f(x,y)$ in the plane perpendicular to the electrode axis is prescribed. The three-dimensional distribution of the volume force $f(x,y,z)$ produced by a plasma actuator with electrode length $l_{PA}$ is then modelled by extrusion of $f(x,y)$ along the electrode axis, as sketched in figure 2. At the lateral edges a fifth-order polynomial is imposed over a range of 10 % of the electrode length to smooth the changeover from zero to maximum forcing. The angle $\unicode[STIX]{x1D6FD}_{PA}$ defines the clockwise rotation of the actuator about the wall-normal axis through $(x_{PA},z_{PA})$ . According to previous investigations by Dörr & Kloker (Reference Dörr and Kloker2015b ) and Dörr & Kloker (Reference Dörr and Kloker2016), the effect of the wall-normal force component is negligible.

Based on the alternating-current operation of plasma actuators, an unsteady volume force with alternating direction (push and pull events with non-zero time mean) is inherently induced. If the frequency of the operating voltage is set clearly above the unstable frequency range of the boundary layer, the high-frequency unsteadiness can be neglected, and only the steady mean force resulting from the asymmetric push and pull events has to be taken into consideration. In the present study, in contrast, we simulate plasma actuators with low-frequency operation and make full use of the unsteady nature for the excitation of unsteady CF-vortex modes, resembling the direct-frequency mode for active wave cancellation investigated by Kurz et al. (Reference Kurz, Tropea, Grundmann, Forte, Vermeersch, Seraudie, Arnal, Goldin and King2012). To mimic the resulting unsteady volume force, $f(x,y,z)$ is multiplied by a sinusoidal modulating factor

(2.1) $$\begin{eqnarray}Z(t)=c_{s}+c_{u}\sin (\unicode[STIX]{x1D714}_{PA}t+\unicode[STIX]{x1D719}_{PA}),\end{eqnarray}$$

with angular frequency $\unicode[STIX]{x1D714}_{PA}$ and phase $\unicode[STIX]{x1D719}_{PA}$ , yielding $f(x,y,z,t)=f(x,y,z)Z(t)$ . The parameters $c_{s}$ and $c_{u}$ denote the amplitude of the steady and the unsteady component, respectively. In fact, the unsteady fluctuating part can be up to about ten times the steady mean value. See e.g. Benard & Moreau (Reference Benard and Moreau2014) for further details about the unsteady aspects of the plasma actuation.

The actuator parameters for the current investigations are provided in table 1. To give additional information on the actuation strength, the maximal actuation momentum coefficient $c_{\unicode[STIX]{x1D707},L_{exp}}$ based on the reference length $\bar{L}_{exp}$ is calculated using equations (3) and (4) in Dörr & Kloker (Reference Dörr and Kloker2016); $c_{\unicode[STIX]{x1D707},\unicode[STIX]{x1D703}_{s}}$ based on the local momentum thickness $\bar{\unicode[STIX]{x1D703}}_{s}$ in the streamline-oriented system is calculated as $c_{\unicode[STIX]{x1D707},\unicode[STIX]{x1D703}_{s}}=c_{\unicode[STIX]{x1D707},L_{exp}}\bar{L}_{exp}/\bar{\unicode[STIX]{x1D703}}_{s}$ . For further details of the volume-force model and a discussion on its limitations, see Dörr & Kloker (Reference Dörr and Kloker2016) and Dörr & Kloker (Reference Dörr and Kloker2015a ).

3.1 Base-flow characteristics

Figure 3. (a) Base-flow parameters in the streamline-oriented coordinate system (the $x_{s}$ -direction points in the local potential-flow direction; see figure 1). Reynolds number $Re_{\unicode[STIX]{x1D6FF}_{1,s}}$ is given on the right ordinate. (b) Spatial chordwise amplification rates $\unicode[STIX]{x1D6FC}_{i}$ (coloured), wave-vector angle $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}$ (spanned by the $x$ -axis and the wave vector $(\unicode[STIX]{x1D6FC}_{r},0,\unicode[STIX]{x1D6FE})^{\text{T}}$ , dotted white lines) and $n$ -factors (solid black lines) of unstable steady CF-instability modes. Spatial chordwise amplification rate $\unicode[STIX]{x1D6FC}_{i}=\text{d}(\ln A)/\text{d}x$ , where $A$ is the disturbance amplitude.

Table 1. Plasma-actuator volume-force parameters for the simulations presented; $\bar{f}=f(\bar{\unicode[STIX]{x1D70C}}_{\infty }\bar{U}_{\infty }^{2}/\bar{L})=f(\bar{\unicode[STIX]{x1D70C}}_{\infty }^{2}\bar{U}_{\infty }^{3}/Re\bar{\unicode[STIX]{x1D707}}_{\infty })$ , $\max \{\bar{f}\}=\max \{[\bar{f}_{x}^{2}+\bar{f}_{z}^{2}]^{1/2}\}$ .

Figure 3(a) shows the evolution of boundary-layer parameters. Note that the subscript $s$ denotes the variables in the streamline-oriented coordinate system. The flow is strongly accelerated near the leading edge, with weak but continuous flow acceleration over the whole length of the plate. The local angle of the external streamline $\unicode[STIX]{x1D719}_{e}$ varies from $70.0^{\circ }$ to $45.2^{\circ }$ . The Reynolds number $Re_{\unicode[STIX]{x1D6FF}_{1},s}$ based on the streamwise displacement thickness rises from 263 to 1180 and the shape factor $H_{12,s}$ is about 2.45. The stability properties with respect to the steady modes and low-frequency unsteady modes with $\unicode[STIX]{x1D714}=3$ and $\unicode[STIX]{x1D714}=6$ resulting from linear stability analysis are shown in figures 3(b) and 4, respectively. The investigated primary modes $(0,2)$ and $(1,+2)$ with $\unicode[STIX]{x1D6FE}=70$ are the integrally most amplified steady and unsteady modes, respectively. A higher $n$ -factor, $n=\int _{x_{0}}^{x}-\unicode[STIX]{x1D6FC}_{i}\,\text{d}x$ , is found for the unsteady mode $(1,+2)$ , travelling against the CF, i.e. travelling rightwards if looking downstream on a $yz$ -plane.

Figure 4. Like figure 3(b) but for unsteady CF-instability modes with (a)  $\unicode[STIX]{x1D714}=3$ and (b)  $\unicode[STIX]{x1D714}=6$ .

Figure 5. Spatial chordwise amplification rates $\unicode[STIX]{x1D6FC}_{i}$ (coloured) at (a)  $x=1.0$ and (b)  $x=2.0$ of unstable CF-instability modes as function of the spanwise wave number $\unicode[STIX]{x1D6FE}$ and the angular frequency $\unicode[STIX]{x1D714}$ .

For a successful UFD, the control CF vortices have to be (i) narrowly spaced so that they possibly saturate with lower amplitude without invoking significant secondary instability and (ii) strongly amplified at first so that they can dominate the primary state and generate a beneficial mean-flow distortion. Among the candidates, the narrowly spaced, rightward-travelling CF-vortex modes $(0.5,+3)$ and $(1,+3)$ are strongly amplified in the leading-edge region but damped for $x>2.9$ and $x>2.4$ , respectively. The leftward-travelling modes $(0.5,-3)$ and $(1,-3)$ are amplified over a longer distance but their maximal amplification rates are distinctly lower. For an overview of the stability properties of the CF-vortex modes with higher frequencies, the dependence of the spatial amplification rate $\unicode[STIX]{x1D6FC}_{i}$ on $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D714}$ at two chordwise positions is shown in figure 5. The rightward-travelling modes are generally more unstable. At $x=1.0$ the highest amplification is found for $\unicode[STIX]{x1D714}\approx 12$ and $\unicode[STIX]{x1D6FE}\approx 80$ . Farther downstream at $x=2.0$ , the unstable wavenumber range shrinks stronger with increasing frequency. As also proven by preliminary investigations using DNS, the CF-vortex modes with higher frequencies are amplified only in a short region and strongly damped downstream. Hence, the mean-flow distortion generated is not sufficient to effectively stabilize the flow. Further findings of our preliminary investigations of the influence of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D714}$ are summarized at the end of § 4.1.

3.2 Reference cases

Figure 6. Downstream development of modal $\tilde{u} _{s,(h,k)}^{\prime }$ and $\tilde{u} _{s,(h)}^{\prime }$ amplitudes for (a) case REF-S and (b) case REF-U from Fourier analysis (maximum over $y$ or $y$ and $z$ , $6\leqslant \unicode[STIX]{x1D714}\leqslant 180$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}=6$ ).

Table 2. Definition of investigated cases.

We provide two reference cases with a single steady or unsteady mode as primary disturbance input, respectively. In both cases the plasma actuators are inactive. Table 2 summarizes the disturbance inputs for the various cases presented in this paper. For case REF-S the most amplified steady mode $(0,2)$ is excited by the primary disturbance strip. In figure 6, the modal amplitude development of the streamline-oriented disturbance velocity component $\tilde{u} _{s}^{\prime }=u_{s}^{\prime }/u_{B,s,e}=(u_{s}-u_{B,s})/u_{B,s,e}$ is shown. The amplitude of the mode $(0,2)$ surpasses $10\,\%$ of $u_{B,s,e}$ at $x\approx 2.2$ while the amplification rate progressively decreases proceeding further into the saturation state. Convective secondary instability is triggered by the large-amplitude CF vortices for $x>2.5$ . The strong secondary growth of the high-frequency mode $\unicode[STIX]{x1D714}=90$ is followed by further amplification of higher-frequency unsteady components, leading to the transition to turbulence. Figure 7(a) shows the vortical structures in the rotated reference system

(3.1) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}x_{r}\\ z_{r}\end{array}\right)=\left(\begin{array}{@{}cc@{}}\cos \unicode[STIX]{x1D6F7}_{r} & \sin \unicode[STIX]{x1D6F7}_{r}\\ -\sin \,\unicode[STIX]{x1D6F7}_{r} & \cos \unicode[STIX]{x1D6F7}_{r}\end{array}\right)\left(\begin{array}{@{}c@{}}x-x_{0}\\ z-z_{0}\end{array}\right), & & \displaystyle\end{eqnarray}$$

with $x_{0}=0.4$ , $z_{0}=0$ and $\unicode[STIX]{x1D6F7}=45^{\circ }$ . In the physical space, steady CF vortices corresponding to the mode $(0,2)$ appear, with axes nearly parallel to the potential-flow streamlines. The finger-like high-frequency secondary structures emerge on the main CF vortices and convect downstream, finally developing into turbulence spots.

Figure 7. Vortex visualization (snapshots, $\unicode[STIX]{x1D706}_{2}=-4$ , colour indicates $y$ ) for case REF-S at $t/T_{0}=18$ (a) and case REF-U at $t/T_{0}=17.25$  (b), 17.5 (c), 17.75 (d) and 18 (e). The rotated reference system according to (3.1) is used. Note the compression of the $x_{r}$ -axis ( $z_{r}:x_{r}=2:1$ ).

In case REF-U the most amplified unsteady mode $(1,+2)$ is excited as primary disturbance. The forcing amplitude of the primary disturbance strip is chosen such that the amplitude of the primary mode $(1,+2)$ reaches the same level at $x=2.2$ as that of mode $(0,2)$ in REF-S; see figure 6(b). In accordance with results of linear stability analysis, the amplification of the unsteady mode $(1,+2)$ is significantly stronger, i.e. only a smaller initial amplitude of the mode $(1,+2)$ is required; $(1,+2)$ saturates earlier than $(0,2)$ in REF-S and triggers somewhat stronger secondary instability at about the same $x$ -position. The background disturbances rise explosively downstream of their forcing, leading to rapid transition. In figure 7(b–e), snapshots of the vortical structures at four time instances within a fundamental period $T_{0}$ are presented. The CF vortices are travelling in positive spanwise and $x_{r}$ direction. The finger-like structures emerge on the main CF vortices and ride along them.

4.1 Single travelling UFD modes excited by blowing/suction

In cases BS-UFD-R and BS-UFD-L, the actuator row for UFD is set at $x=0.5$ by an additional blowing/suction strip extending over $0.481\leqslant x\leqslant 0.520$ , exciting the single control mode $(0.5,+3)$ or $(0.5,-3)$ , respectively. The amplitude of the blowing/suction is kept identical for both cases. To clarify the pure effect of the travelling UFD modes, the strip for primary (‘test-mode’) disturbance input at $x=0.8$ is deactivated. For indication of secondary instabilities that might possibly arise farther upstream, (background) pulsing at $x=1.0$ and $x=2.0$ is introduced for both cases.

4.1.1 Development of disturbance amplitudes

Figure 8. Like figure 6 but for (a) case BS-UFD-R and (b) case BS-UFD-L.

In case BS-UFD-R the rightward-travelling mode $(0.5,+3)$ saturates at $x\approx 1.6$ with an amplitude level of 21 %; see figure 8(a). Upstream of the secondary pulsing, the superharmonics of $(0.5,+3)$ are represented by the lower-frequency curves resulting from the $t$ -modal decomposition. In addition, a strong mean-flow distortion $(0,0)$ is nonlinearly generated. According to linear stability analysis, the mode $(0.5,+3)$ is damped only downstream of $x\approx 2.9$ . However, due to the stabilizing effect of the mean-flow distortion $(0,0)$ as discussed in § 4.1.3, the damping of $(0.5,+3)$ occurs earlier. It decays monotonically to an amplitude level of 0.3 % at the end of the considered domain, whereas the mean-flow distortion $(0,0)$ is more persistent. Unlike the reference cases REF-S and REF-U, no strong, persistent secondary instability is triggered. At the end of the domain all unsteady modes fall far below 1 %. In case BS-UFD-L, as shown in figure 8(b), the leftward-travelling mode $(0.5,-3)$ is amplified more weakly than $(0.5,+3)$ and attains saturation with an amplitude of 25 % at $x\approx 2.0$ , farther downstream than $(0.5,+3)$ . Nevertheless, the maximal amplitude of the nonlinearly generated mean-flow distortion $(0,0)$ is significantly lower than in case BS-UFD-R. Farther downstream, a significant growth of all unsteady disturbance components is observed. Since both the superharmonics of the unsteady primary mode and the secondarily unstable modes may contribute to this growth, secondary instabilities cannot be clearly identified.

Figure 9. Downstream development of modal $\tilde{u} _{s,m(l)}^{\prime }$ amplitudes in moving systems for (a) case BS-UFD-R and (b) case BS-UFD-L from Fourier analysis (maximum over $y$ and $z$ , $\unicode[STIX]{x1D714}_{m}=0$ and $5\leqslant \unicode[STIX]{x1D714}_{m}\leqslant 179$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}_{m}=6$ ). (c) Illustration of the connection between the ( $t$ – $z$ )-modal decomposition in the fixed system and the $t$ -modal decomposition in the Galilean-transformed system moving with the mode $(0.5,+3)$ (left) and $(0.5,-3)$ (right), respectively. The frequency factor in the moving system is denoted by $l$ instead of $h$ with $\unicode[STIX]{x1D714}_{m}=l\unicode[STIX]{x1D714}_{0}$ .

In order to unambiguously clarify the secondary instabilities triggered, we analyse the amplitude development referring to a Galilean-transformed system $(x,y,z_{m})$ , see Wassermann & Kloker (Reference Wassermann and Kloker2003), $z_{m}=z-c_{(0.5,\pm 3)}t$ , where $c_{(0.5,\pm 3)}=\pm 0.5\unicode[STIX]{x1D714}_{0}/(3\unicode[STIX]{x1D6FE}_{0})$ is the spanwise phase velocity of the primary mode $(0.5,\pm 3)$ . With respect to the transformed moving system, the primary state becomes steady. The modal component $\unicode[STIX]{x1D714}_{m}=0$ consists of a spanwise mean $(0,0)$ and a purely three-dimensional component $(\unicode[STIX]{x1D714}_{m}=0)-(0,0)$ that can be recomposed by the primary mode and its superharmonics; see figure 9(c). Furthermore, other ( $t$ – $z$ )-modal components belonging to the same secondary instability mode carried by the primary wave, i.e. on the same diagonal as sketched in figure 9(c), are recombined. The secondary-instability behaviour is directly indicated by the modal components $\unicode[STIX]{x1D714}_{m}\neq 0$ .

For case BS-UFD-R (figure 9 a), secondary instabilities are observed in a short region downstream of the pulsing at $x=1.0$ . The leading mode $\unicode[STIX]{x1D714}_{m}=89$ reaches an amplitude of 0.4 % at $x=2.1$ and becomes virtually stable farther downstream. For $2.0<x<3.0$ a distinct secondary growth is found for the modes with higher frequencies. At the end of the domain, the three-dimensional flow deformation dies out, and almost all unsteady components are stable. Within the considered domain, none of the secondary instability modes exceeds an amplitude of 0.5 %. For case BS-UFD-L (figure 9 b), some low-frequency modes undergo a strong amplification. Farther downstream, the leading mode with $\unicode[STIX]{x1D714}_{m}=17$ becomes nonlinear, significantly alters the primary state downstream and possibly drives the filling up of the perturbation spectrum. Compared to the rightward-travelling UFD mode $(0.5,+3)$ or the steady mode $(0,3)$ , the mode $(0.5,-3)$ seems more susceptible to low-frequency secondary instability.

When examining the normalized modal $u_{r}^{\prime }$ amplitude distribution in crosscuts in the rotated moving system

(4.1) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}x_{r,m}\\ z_{r,m}\end{array}\right)=\left(\begin{array}{@{}cc@{}}\cos \unicode[STIX]{x1D6F7}_{r} & \sin \unicode[STIX]{x1D6F7}_{r}\\ -\sin \,\unicode[STIX]{x1D6F7}_{r} & \cos \unicode[STIX]{x1D6F7}_{r}\end{array}\right)\left(\begin{array}{@{}c@{}}x-x_{0}\\ z-c_{(0.5,\pm 3)}t-z_{0}\end{array}\right), & & \displaystyle\end{eqnarray}$$

with $x_{0}=0.4$ , $z_{0}=0$ and $\unicode[STIX]{x1D6F7}=45^{\circ }$ , the origin of the secondary instabilities becomes transparent. For case BS-UFD-R at $x_{r,m}=1.5$ ( $x\approx 1.4$ ), as shown in figure 10(a), a type-I or $z$ -mode is found for $\unicode[STIX]{x1D714}_{m}=89$ which attains the largest amplitude. Farther downstream, this mode spreads upwards to the top region of the shear layer, likely reflecting both $z$ - and $y$ -modes; see crosscut at $x_{r,m}=3.0$ ( $x\approx 2.5$ ) in figure 10(b). The high-frequency component $\unicode[STIX]{x1D714}_{m}=149$ , being one of the most amplified secondary instability modes at this position, reveals a typical amplitude distribution of a type-II or $y$ -mode; see figure 10(d). Once the three-dimensional deformation fades out, the originally localized secondary modes are gradually smeared; see figure 10(c,e). In case BS-UFD-L, the mode $\unicode[STIX]{x1D714}_{m}=17$ is a typical type-III mode which is located in the near-wall region at the updraft side of the main CF vortices (figure 10 f,g). At $x_{r,m}=3.0$ , the vortical structures in the near-wall region are strongly modified, leading to an increase of the complexity of the shear layers.

Figure 10. Crosscuts for (a–e) case BS-UFD-R and (f,g) case BS-UFD-L at various downstream positions in the rotated reference system moving spanwise with the respective primary CF vortices. Dashed lines: $\unicode[STIX]{x1D706}_{2}$ isocontours ( $-12$ to $-2$ , $\unicode[STIX]{x1D6E5}=2$ ); solid lines: $\tilde{u} _{r}$ isocontours (0.05 to 0.95, $\unicode[STIX]{x1D6E5}=0.10$ ); colour: normalized modal $u_{r}^{\prime }$ amplitude distribution.

4.1.2 Vortical structures

Figure 11. Vortex visualization (snapshots, $\unicode[STIX]{x1D706}_{2}=-2.5$ , colour indicates  $y$ ) for (a) case BS-UFD-R and (b) case BS-UFD-L at $t/T_{0}=18$ . Rotated (fixed) reference system according to (3.1) is used. Note the compression of the $x_{r}$ -axis ( $z_{r}:x_{r}=2:1$ ). Arrows indicate the travelling directions of the CF vortices.

The snapshots of the vortical structures arising in cases BS-UFD-R and BS-UFD-L are shown in figures 11(a) and 11(b), respectively. In both cases, the main CF vortices triggered by blowing and suction are spaced more narrowly than those arising in cases REF-S and REF-U. We note the misalignment between the orientation of the main vortices in both cases. In accordance with the modal amplitude development, the primary CF vortices in case BS-UFD-R die out downstream of $x_{r}=3.9$ ( $x\approx 3.1$ ). The finger-like secondary structures, left over from the high-frequency type-I and type-II modes upstream, are distorted, and stretched in the spanwise direction. Unlike the turbulent spots appearing in the reference cases, these structures are situated farther away from the wall and cause no skin-friction increase (not shown). In case BS-UFD-L, the main CF vortices are visible up to $x_{r}=4.5$ ( $x\approx 3.5$ ). Each main CF vortex is accompanied by a counter-rotating CF vortex at the updraft side, which is strongly modulated by the secondarily amplified modes downstream of the pulsing. Since the secondary structures are mainly evolved from the low-frequency type-III mode $\unicode[STIX]{x1D714}_{m}=17$ , their shapes are significantly different from the finger-like structures arising on the top side of the main CF vortices in case BS-UFD-R. See Bonfigli & Kloker (Reference Bonfigli and Kloker2007) for a detailed comparison between the vortical structures resulting from different secondary instability modes.

4.1.3 Modification of mean-flow profiles and stability properties

For the classical UFD technique using a steady control mode it was demonstrated that the growth attenuation of the primary modes is mainly based on the mean-flow distortion $(0,0)$ , which has a stabilizing effect similar to homogeneous suction; see Wassermann & Kloker (Reference Wassermann and Kloker2002). In order to verify whether the $(0,0)$ generated by unsteady control modes also has the same stabilizing effect, the streamwise and crosswise velocity profiles are averaged in time and spanwise direction, and then analysed in terms of the stability properties using linear stability calculations. Since case BS-UFD-L shows transition to turbulence it is ruled out and in the following we concentrate on case BS-UFD-R.

Figure 12. Time- and spanwise-averaged (a) $\tilde{u} _{s,tzm}$ and (b) $\tilde{w}_{s,tzm}$ profiles for case BS-UFD-R in comparison to the corresponding base flow at various downstream positions ( $x=0.8$ , 1.2, 1.8, 2.4, 3.0, 3.6 from left to right; the abscissa shift is 0.5 and 0.1, respectively).

Figure 12 shows that the $\tilde{u} _{s,tzm}$ profiles are S-shaped with increase in the near-wall region and decrease farther away from the wall. The CF component $\tilde{w}_{s,tzm}$ is only slightly reduced at $x=1.2$ and 1.8, and even increased farther downstream, questioning a palpable stabilizing effect.

Figure 13. Spatial chordwise amplification rates $\unicode[STIX]{x1D6FC}_{i}$ of unstable CF-vortex modes for case BS-UFD-R in comparison to the corresponding base-flow data (lines with levels 0.0 to $-4.2$ , $\unicode[STIX]{x1D6E5}=0.3$ ). (a) Steady modes; the dashed lines mark the modes $(0,2)$ (lower) and $(0,3)$ (upper). (b) Unsteady modes with $\unicode[STIX]{x1D714}=6$ ; the dashed lines mark the modes $(1,+2)$ (lower) and $(1,+3)$ (upper).

Figure 14. (a) Wall-normal gradient of $\tilde{w}_{s,tzm}$ for case BS-UFD-R in comparison to the corresponding base flow at various downstream positions ( $x=0.8$ , 1.2, 1.8, 2.4, 3.0, 3.6 from left to right; the abscissa shift is 20). (b) Deformation of the velocity profile $\tilde{w}_{eff,tzm}$ in the wave-vector direction of the travelling mode $(1,+2)$ at $x=2.4$ and (c) the corresponding wall-normal gradients. Here $\tilde{w}_{eff,tzm}=\tilde{w}_{s,tzm}\cos \unicode[STIX]{x1D719}_{s}+\tilde{u} _{s,tzm}\sin \unicode[STIX]{x1D719}_{s}$ , where $\unicode[STIX]{x1D719}_{s}$ is the angle between the wave vector and the $z_{s}$ -direction.

In figure 13, the stability diagrams for case BS-UFD-R are shown based on the time-averaged flow and one-dimensional eigenfunction linear stability theory. Because the wave vectors of the steady modes are nearly aligned with the $z_{s}$ -direction, the $\tilde{w}_{s,tzm}$ profile plays a decisive role for the instability. In both Wassermann & Kloker (Reference Wassermann and Kloker2002) and Dörr & Kloker (Reference Dörr and Kloker2017), the stabilizing effect of the mean-flow distortion for steady modes was explained as a consequence of the mean CF reduction. In fact, the possible reduction of the maximum wall-normal gradient $\text{d}\tilde{w}_{s,tzm}/\text{d}y$ at the inflection point is a cause for the attenuation of the inviscid instability. In figure 14(a) the wall-normal gradients $\text{d}\tilde{w}_{s,tzm}/\text{d}y$ are shown at various $x$ -positions.

The amplification of the unsteady modes with $\unicode[STIX]{x1D714}=6$ is more strongly reduced than that of the steady modes. As the wave vectors of the unsteady modes are misaligned with the $z_{s}$ -direction, the $\tilde{u} _{s,tzm}$ profile comes into play. As shown by Gregory, Stuart & Walker (Reference Gregory, Stuart and Walker1955), the stability properties of three-dimensional modes with a known wave-vector orientation are linked to the velocity profile $\tilde{w}_{eff,tzm}=\tilde{w}_{s,tzm}\cos \unicode[STIX]{x1D719}_{s}+\tilde{u} _{s,tzm}\sin \unicode[STIX]{x1D719}_{s}$ , where $\unicode[STIX]{x1D719}_{s}$ is the angle between the wave vector and the $z_{s}$ -direction. In figure 14(b,c) the deformation of the velocity profile $\tilde{w}_{eff,tzm}$ regarding the mode $(1,+2)$ at $x=2.4$ and the corresponding wall-normal gradients are illustrated. Due to the S-formed $\tilde{u} _{s,tzm}$ profiles, the maximal gradient of $\tilde{w}_{eff,tzm}$ is strongly reduced and the inflection point is shifted distinctly closer to the wall. Hence, the unsteady mode $(1,+2)$ is greatly stabilized.

Note that based on the inflection point at the upper part of the $\tilde{u} _{s,tzm}$ profiles arising from the UFD, a strong amplification of streamwise-travelling Tollmien–Schlichting waves is predicted by one-dimensional linear stability theory. However, they cannot be observed in DNS results because their receptivity and growth are possibly significantly attenuated due to the additional spanwise modulation by the UFD control mode.

In addition, various simulations were performed to examine the influence of the frequency and the spanwise wavenumber of the UFD control mode. If the angular frequency $\unicode[STIX]{x1D714}$ is increased from 3 to 6 or 9, both the leftward- and rightward-travelling modes propagate faster in the spanwise direction. The shear layers induced inside the boundary layer are more pronounced and the background disturbances are more strongly amplified, counteracting the stabilizing effect of $(0,0)$ . The unwanted secondary amplification can be reduced by increasing the spanwise wavenumber of the UFD mode. However, the CF vortices are then spaced closer so that they hinder the vortical motion of each other and decay earlier. As a result, the nonlinearly generated mean-flow distortion with frequency higher than $\unicode[STIX]{x1D714}=3$ is too weak to stabilize the flow effectively.

4.2 Excitation of travelling UFD modes by unsteady volume forcing

Figure 15. (a) Plasma-actuator volume-force distribution for case PA-UFD ( $f_{10\,\%}$ isosurfaces at the time of the maximal forcing, where $f_{10\,\%}=\max \{(f_{x}^{2}+f_{z}^{2})^{1/2}\}/10=0.017$ ; the colour indicates the wall-normal distance $y$ ). The dashed lines show the local orientation of the wave fronts of the CF-vortex modes $(0.5,+3)$ (blue) and $(0.5,-3)$ (red). (b) Crosscut along the dash-dotted line perpendicular to the electrode axes shown in (a). The colour indicates $f$ and the solid lines mark the $f_{10\,\%}$ isosurfaces at the time of the maximal forcing. The inset shows the physical time signal within a fundamental period. (c) Downstream development of modal $\tilde{u} _{s,(h,k)}^{\prime }$ and $\tilde{u} _{s,(h)}^{\prime }$ amplitudes from Fourier analysis (maximum over $y$ or $y$ and $z$ , $6\leqslant \unicode[STIX]{x1D714}\leqslant 180$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}=6$ ) for case PA-UFD. The green rectangle indicates the chordwise position of the volume forcing.

Based on the results in § 4.1, the travelling mode $(0.5,+3)$ is chosen as UFD control mode since it causes no impeding secondary growth by itself. In the following, three actuators per fundamental wavelength are located at $x=0.5$ , instead of the blowing/suction strip for UFD as used in the former cases. For an optimal excitation of the unsteady UFD mode, the most effective spatial distribution of the volume force used by Dörr & Kloker (Reference Dörr and Kloker2017) for exciting steady CF-vortex modes is followed. The force extends in the wall-normal direction nearly up to the boundary-layer edge. The sinusoidal time signal $Z(t)=0.04+0.3\sin (3t)$ is used, i.e. the steady volume force is modulated. Figure 15(a,b) shows the volume-force configuration in top view and in a crosscut. Analogous to the steady forcing, the steady mean of the volume force $c_{s}f(x,y,z)$ excites steady CF-vortex modes. Meanwhile, the unsteady component $c_{u}f(x,y,z)\sin (\unicode[STIX]{x1D714}_{PA}t)$ imparts an oscillatory perturbation (push and pull events), which excites pairs of leftward- and rightward-travelling waves with the same spanwise wavenumber. Here, using a row of three actuators per fundamental wavelength, operated with angular frequency $\unicode[STIX]{x1D714}_{PA}=3$ , a pair of travelling CF-vortex modes $(0.5,\pm 3)$ and a weak steady mode $(0,3)$ are simultaneously excited. Furthermore, the orientation of the electrode axes of the plasma actuators has a significant effect on the receptivity of each individual CF-vortex mode; see also Dörr & Kloker (Reference Dörr and Kloker2017), case CMF. Employing actuators with sufficient length and aligned with the wave front of the desired control mode, the initial amplitude of the corresponding mode can be maximized and the formation of other misaligned modes can be hindered by destructive interference. Therefore, the angle of the actuators, $\unicode[STIX]{x1D6FD}_{PA}$ , is adapted such that the electrode axis is aligned with the axes of the travelling CF vortices arising in case BS-UFD-R at $x=0.5$ to ensure the best receptivity. For the most important actuator parameters, see table 1. The actuators are equidistantly distributed in $z$ with identical volume-force set-up. The maximal momentum coefficient $c_{\unicode[STIX]{x1D707},\unicode[STIX]{x1D703}_{s}}$ based on the momentum thickness $\unicode[STIX]{x1D703}_{s}$ at $x=0.5$ is $3.5\times 10^{-3}$ , corresponding to 34 % of the most efficient case ACF in Dörr & Kloker (Reference Dörr and Kloker2017). The time-averaged volume force required for the unsteady UFD configuration is reduced to 4 % with respect to case ACF.

To analyse the pure effect of the unsteady volume forcing, the primary disturbance strip at $x=0.8$ is inactive for case PA-UFD. Two secondary disturbance strips at $x=1.0$ and $x=2.0$ are used to introduce unsteady (background) disturbances. The modal amplitude development is shown in figure 15(c). As expected, the modes $(0.5,\pm 3)$ are simultaneously excited; however, the difference of their initial amplitudes clearly reveals the receptivity discrepancy. The targeted rightward-travelling mode $(0.5,+3)$ dominates due to its higher receptivity and amplification rate. The downstream development of $(0.5,+3)$ and $(0,0)$ is qualitatively similar to that of case BS-UFD-R. Also the behaviour of the secondary instability modes is virtually identical. The initial amplitude of the steady mode $(0,3)$ is one order of magnitude lower than that of the dominating mode $(0.5,+3)$ . The downstream growth of both $(0,3)$ and $(0.5,-3)$ is suppressed due to the stabilized mean flow. The weak amplification of $(0.5,-3)$ for $x>1.7$ is most probably caused by a nonlinear interaction: modes $(0.5,+3)$ and $(0,3)$ lead to a contribution to $(0.5,-3)$ for example. Due to the low initial amplitude and growth rate, the leftward-travelling mode plays however only a minor role. The interaction between the steady and travelling modes seems not to be able to fill up the perturbation spectrum.

Figure 16. Vortex visualization (snapshots, $\unicode[STIX]{x1D706}_{2}=-4$ , colour indicates $y$ ) and the (active) volume-force set-up ( $f_{10\,\%}$ isosurface, dark) for case PA-UFD at $t/T_{0}=$ (a) 16.5, (b) 17, (c) 17.5 and (d) 18 in the rotated (fixed) reference system according to (3.1). Note the compression of the $x_{r}$ -axis ( $z_{r}:x_{r}=2:1$ ). Here $T_{0}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}_{0}=1.05$ , $\unicode[STIX]{x1D714}_{0}=6$ .

Figure 16(a–d) shows the vortical structures arising from the actuation at four time instances within one full excitation cycle ( $\unicode[STIX]{x1D714}=3$ ). The volume force attains the maximum at $t=0.5T_{0}$ . In the vicinity of the actuators, unsteady CF vortices travelling in the positive $z$ -direction are excited continuously. The vortex axes are mainly parallel to the electrode axis of the actuators and hence aligned with the wave front of the mode $(0.5,+3)$ . Downstream of the actuator row up to $x_{r}\approx 2.5$ , the vortical structures are quite similar to those in case BS-UFD-R due to the dominance of $(0.5,+3)$ . Due to the growing steady mode $(0,2)$ superposed onto the travelling mode, the main CF vortices break into isolated segments; see figure 16(a). Farther downstream, the primary CF vortices disappear and only some secondary structures far from the wall are visible.

Figure 17. Spatial chordwise amplification rates $\unicode[STIX]{x1D6FC}_{i}$ of unstable CF-vortex modes for case PA-UFD in comparison to the corresponding base-flow data (lines with levels 0.0 to $-4.2$ , $\unicode[STIX]{x1D6E5}=0.3$ ). (a) Steady modes; the dashed lines mark the modes $(0,2)$ (lower) and $(0,3)$ (upper). (b) Unsteady modes with $\unicode[STIX]{x1D714}=6$ ; the dashed lines mark the modes $(1,+2)$ (lower) and $(1,+3)$ (upper).

The stability diagrams gained by the time- and spanwise-averaged mean flow are shown in figure 17. Compared to case BS-UFD-R, the onset of the stabilizing effect occurs somewhat farther downstream due to the slightly lower initial amplitude of the UFD mode $(0.5,+3)$ . Different from the steady actuation investigated by Dörr & Kloker (Reference Dörr and Kloker2017), the unsteady volume force does not permanently counteract the CF. Hence, a direct modification of the stability properties in the vicinity of the actuators is not observed. Farther downstream, the stability properties basically coincide with those for case BS-UFD-R. Additional simulations with higher actuation strength but identical spatial force distribution (not shown) reveal enhanced stabilization for $x<1.7$ but also higher amplitude level of the modes $(0,3)$ and $(0.5,-3)$ downstream, which possibly counteracts the UFD effect because of the stronger nonlinear interaction between steady and unsteady modes. Therefore, the sensitivity of the unsteady technique to detrimental ‘over’-actuation seems somewhat higher than that of the steady approach.

4.3 Transition delay using travelling UFD modes excited by unsteady volume forcing

Figure 18. (a) Downstream development of modal $\tilde{u} _{s,(h,k)}^{\prime }$ and $\tilde{u} _{s,(h)}^{\prime }$ amplitudes from Fourier analysis (maximum over $y$ or $y$ and $z$ , $6\leqslant \unicode[STIX]{x1D714}\leqslant 180$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}=6$ ) for case PA-UFD-S. The green rectangle indicates the chordwise position of the volume forcing. Open symbols denote the reference case. (b–e) Vortex visualization (snapshots, $\unicode[STIX]{x1D706}_{2}=-4$ , colour indicates $y$ ) for case PA-UFD-S at $t/T_{0}=$ (b) 16.5, (c) 17, (d) 17.5 and (e) 18 in the rotated (fixed) reference system according to (3.1). Note the compression of the $x_{r}$ -axis ( $z_{r}:x_{r}=2:1$ ).

Figure 19. Like figure 18 but for case PA-UFD-U.

In cases PA-UFD-S and PA-UFD-U, the same volume-force actuation set-up as used for case PA-UFD is now employed to cases REF-S and REF-U, respectively. As shown in figures 18(a) and 19(a), for both cases, the downstream development of the UFD modes $(0.5,\pm 3)$ and $(0,3)$ and the mean-flow distortion $(0,0)$ agrees exactly with that of case PA-UFD up to $x\approx 2.3$ . In case PA-UFD-S, the test mode $(0,2)$ reaches an amplitude of 10 % only at $x=3.9$ , whereas this is at $x=2.2$ in the reference case REF-S. However, compared to the steady UFD approach investigated by Dörr & Kloker (Reference Dörr and Kloker2017) for the identical base flow, the growth attenuation is somewhat less effective. For $x>3.2$ , the low-frequency components of the perturbation spectrum are filled up progressively due to the interaction between the actuation-induced modes and the growing test mode $(0,2)$ which attains a nonlinear amplitude. Compared to the reference case REF-S, the strong growth of the high-frequency secondary instability modes $\unicode[STIX]{x1D714}\approx 90$ induced by the mode $(0,2)$ is distinctly attenuated, and the transition does not occur in the considered domain. In case PA-UFD-U, the amplification of the test mode $(1,+2)$ is virtually completely suppressed, and consequently the secondary instability.

In physical space, the vortical structures arising for case PA-UFD-S for $x_{r}<2.5$ are virtually identical to those appearing in case PA-UFD without test mode (not shown). Farther downstream, as shown in figure 18(b,e), the oncoming CF vortices, mainly associated with the dominating UFD mode $(0.5,+3)$ , are strongly detuned due to the modulation by the nonlinear steady mode $(0,2)$ . For $x_{r}>3.5$ , a distinct CF vortex $A$ detaches from the oncoming CF vortex and keeps growing whereas other primary structures fade out. Being supported by $A$ , the complex secondary structures in figure 18(c) at the end of the visualized region may finally evolve into turbulent spots farther downstream. For case PA-UFD-U, as expected, the vortical structures are identical to those of case PA-UFD due to the completely suppressed test mode; see figure 19(b–e).

As for a combined steady/unsteady test-modes scenario the following holds. Once a dominating mode becomes nonlinear ( ${>}$ 5 %), the amplification of other primary modes is suppressed; see Bonfigli & Kloker (Reference Bonfigli and Kloker1999). If both steady and unsteady modes reach nonlinear amplitudes simultaneously, the transition is promoted by strong nonlinear interaction. This complex scenario was investigated in detail by Bonfigli & Kloker (Reference Bonfigli and Kloker2005). An additional simulation with the control actuator (case ‘PA-UFD-S $+$ U’, not shown) reveals that also in this case the travelling mode $(1,+2)$ is damped out by the unsteady UFD and the resulting flow field is quite similar to that of case PA-UFD-S.

To quantify the reduction of the skin-friction drag achieved, the wall-normal gradient of the mean velocity component in the direction of the oncoming flow at the wall is calculated, which is proportional to the local skin-friction coefficient. The mean velocity in the direction of the oncoming flow is denoted as $u_{41^{\circ }}$ , where $41^{\circ }$ is the effective sweep angle of the plate. Results of the controlled cases are compared with the corresponding steady or unsteady reference cases; see figures 20(a) and 20(c), respectively. Due to the presence of the nonlinear UFD mode in the controlled cases, the wall-normal gradient is increased in the upstream region. However, due to the delayed transition, the sharp rise of the wall-normal gradient as found for case REF-S at $x\approx 3.5$ and for case REF-U at $x\approx 3.2$ does not occur. The chordwise integrated value at the end of the domain in both controlled cases is significantly lowered; see figures 20(b) and 20(d), respectively.

Figure 20. Downstream development of the (a,c) wall-normal gradient and (b,d) chordwise direction integrated wall-normal gradient of the spanwise mean velocity component in the direction of the oncoming flow at the wall. For the integration, the maximum value over one fundamental period is used to account for a fully turbulent flow.