3.1. Inputs and outputs

The control set-up consists of spanwise-periodic rows of actuators $u^i$ , estimation sensors $y^i$ and performance sensors $z_{\kern-1pt j}^i$ . In each case, the superscript $i$ indexes the spanwise location, with elements of identical type uniformly spaced by $\varDelta _{x_3} = 3.32 \delta _0^*$ and operating independently. While a single row is used for the actuators and estimation sensors, four streamwise rows of performance sensors are considered, indexed by $j = 1, \ldots, 4$ (see figure 1). In addition, to create instabilities in the flow, a spanwise-periodic row of exogenous disturbances $w^i$ is used. In the spanwise-direction $x_3$ , the computational domain comprises $n = 2m+1$ actuators and sensors ( $m=17$ so $n=35$ ); hence the flow field is periodic in the spanwise direction with period $L_{x_3}=n\varDelta _{x_3}$ . It should be noted that the number of spanwise exogenous disturbances $w^i$ need not match the number of sensors and actuators for the development of the method, as the controller synthesis is performed solely based on the available signals $y^i$ , $u^i$ and $z^i$ (see § 3.3). The number of sensors ( $y^i$ , $z^i$ ) within each row coincides with the number of actuators $u^i$ and this choice is justified § 3.5. The physical rationale for the input and output configuration (streamwise locations, types, spanwise spacing, number of rows, etc.) is detailed in § 5, the present section focusing mainly on establishing the control framework. Input and output variables may be gathered in column vectors of $\mathbb{R}^n$ ,

(3.1) \begin{align} \boldsymbol{w}&=[w^{-m},\ldots, w^0,\ldots w^m]^T,\quad &\boldsymbol{u}=[u^{-m},\ldots, u^0,\ldots u^m]^T, \\[-12pt] \nonumber \end{align}

(3.2) \begin{align} \boldsymbol{y}&=[y^{-m},\ldots, y^0,\ldots y^m]^T,\quad &\boldsymbol{z}_{\!j}=[z_{\kern-1pt j}^{-m},\ldots, z_{\kern-1pt j}^0,\ldots z_{\kern-1pt j}^m]^T. \\[10pt] \nonumber \end{align}

The forcing field $\boldsymbol{f}$ in (2.1) is taken as the sum of a noise term (acoustic perturbations, surface roughness, free stream turbulence, entropy spot, etc.) $\boldsymbol{f}_{\boldsymbol{w}}$ and an actuator term $\boldsymbol{f}_{\boldsymbol{u}}$ ,

(3.3) \begin{align} \boldsymbol{f}_{\boldsymbol{w}}(x_1,x_2,x_3,t)&=\epsilon \boldsymbol{g}(x_1-x_{1,\boldsymbol{w}}, \, x_2-\delta _0^*; \, \sigma _{x_1}, \, \sigma _{x_1}/10)\sum _{i=-m}^m B_{w^i}(x_3)w^i(t), \\[-28pt] \nonumber \end{align}

(3.4) \begin{align} \boldsymbol{f}_{\boldsymbol{u}}(x_1,x_2,x_3,t)&=\epsilon \boldsymbol{g}(x_1-x_{1,\boldsymbol{u}}, \, x_2-x_{2,\textit{GIP}}(x_{1,\boldsymbol{u}});\, \sigma _{x_1}, \, \sigma _{x_1}/3)\sum _{i=-m}^m B_{u^i}(x_3)u^i(t), \\[0pt] \nonumber \end{align}

where

(3.5) \begin{align} \boldsymbol{g}(x_1,x_2;\sigma _{x_1},\sigma _{x_2})&=[0,x_2\sigma _{x_1}/\sigma _{x_2},-x_1\sigma _{x_2}/\sigma _{x_1},0,0]^T\mathrm{exp} \!\left (-\dfrac {x_1^2}{2\sigma _{x_1}^2}-\dfrac {x_2^2}{2\sigma _{x_2}^2}\right ), \\[-28pt] \nonumber \end{align}

(3.6) \begin{align} B_{w^i}(x_3)&=\sum _{p=-\infty }^\infty b_w(x_3-i\varDelta _{x_3}+pn\varDelta _{x_3}),\quad b_w(x_3)=\delta _{\textit{Dirac}}(x_3), \\[-28pt] \nonumber \end{align}

(3.7) \begin{align} B_{u^i}(x_3)&=\sum _{p=-\infty }^\infty b_u(x_3-i\varDelta _{x_3}+pn\varDelta _{x_3}),\quad b_u(x_3)=\mathrm{exp} \!\left (-\dfrac {x_3^2}{2\sigma _{x_3}^2}\right ), \\[0pt] \nonumber \end{align}

with $\sigma _{x_1}=1.06\delta _0^*$ , $\sigma _{x_3}=4.98\delta _0^*$ and streamwise positions of the rows of exogenous noises $x_{1,\boldsymbol{w}}$ and actuators $x_{1,\boldsymbol{u}}$ indicated in table 1. The amplitude $\epsilon$ is chosen in §§ 6 and 7.1 such that the dynamics is linear throughout the numerical domain. In § 7.2, the amplitude is increased to induce transition to turbulence within the domain. The form of the momentum forcing $\boldsymbol{g}$ is identical to that chosen in previous studies of transition to turbulence in incompressible and supersonic boundary layers (Bagheri et al. Reference Bagheri, Brandt and Henningson2009; Semeraro et al. Reference Semeraro, Bagheri, Brandt and Henningson2011; Belson et al. Reference Belson, Semeraro, Rowley and Henningson2013; Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024). The noise term $\boldsymbol{f_w}$ is localized close to the inlet, at a streamwise position $x_{1,\boldsymbol{w}}=4.1\delta _0^*$ , and centred about $x_2=\delta _0^*$ in the wall-normal direction. The independent-white-noise-in-time signals $w^i(t)$ modulate the amplitude of the spatial supports $B_{w^i}$ (which are based on a Dirac comb in the spanwise direction) of the noise sources; hence no spanwise wavenumber or frequency is initially privileged. On the other hand, actuation $\boldsymbol{f_u}$ is centred about the generalized inflection point $x_2=x_{2,\textit{GIP}}(x_{1,\boldsymbol{u}})$ at the streamwise position $x_{1,\boldsymbol{u}}=867.2\delta _0^*$ . The modelling of the actuation impact in the flow as a localized forcing term centred around the generalized inflection point may remain idealized with respect to a practical experimental implementation, although advanced actuation devices such as plasma actuators could potentially approximate such a forcing distribution (Dadfar et al. Reference Dadfar, Semeraro, Hanifi and Henningson2013; Shahriari et al. Reference Shahriari, Kollert and Hanifi2018; Samimy et al. Reference Samimy, Webb and Esfahani2019). The modelling of a realistic actuator nevertheless lies beyond the scope of the present study. We selected this wall-normal position to ease the control of the instabilities by maximizing the receptivity process (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023). The streamwise position of the actuators $x_{1,\boldsymbol{u}}$ is a compromise between controlling a sufficiently long domain representative of experimental configurations and restricting the complexity of the control problem by limiting the frequency–wavenumber range to control (see figure 3). The spatial supports $B_{u^i}$ of the actuators are modelled as a comb of Gaussian functions in the spanwise direction, while the signals $u^i(t)$ are not known a priori but are determined by the controller based on the estimation sensor signals $y^i(t)$ . For both $\boldsymbol{f_w}$ and $\boldsymbol{f_u}$ , only the $x_1$ and $x_2$ components of the momentum forcing terms are excited (via the function $\boldsymbol{g}$ ), which is optimal for the second mode, but suboptimal for the first (oblique) mode, as non-zero transverse momentum forcing would lead to greater growth according to Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019).

Table 1.

Streamwise position of rows of sources and sensors, in units of $\delta _0^*$ .

Table 1. Long description

A table with one row and seven columns. The columns are labeled as x1,w, x1,u, x1,y, x1,z1, x1,z2, and x1,z3 = x1,z4. The values in the row are 4.1, 867.2, 887.6, 1022.6, 1312.2, and 1775.0 respectively.

Next we introduce sensors. The row of $n=2m+1$ estimation sensors $y^i$ (which serve as inputs to the controller) is located at $x_{1,\boldsymbol{y}}=887.6\delta _0^*$ and is composed of fluctuation wall-pressure sensors. Following the conclusions of Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024), these estimation sensors are placed downstream of actuators to enforce performance robustness to free stream velocity variations. Therefore, this is a feedback set-up and not a feedforward one as typically found in the literature on noise-amplifier flows. The four rows of $n=2m+1$ performance sensors $z^i_j$ only serve to define an objective functional for optimizing the controller. Rows $1,2,3$ , respectively, located at $(x_{1,\boldsymbol{z}_1},\, x_{1,\boldsymbol{z}_2},\, x_{1,\boldsymbol{z}_3})=(1022.6,1612.2,1775.0)\delta _0^*$ , measure fluctuating wall-pressure $p'$ . Row $4$ is located at $x_{1,\boldsymbol{z}_4}=x_{1,\boldsymbol{z}_3}$ and measures longitudinal perturbation velocities $v_{x_1}'$ at the generalized inflection point. On each row, sensors are infinitely localized in $x_3$ , and regularly spaced in the transverse direction with $\varDelta _{x_3}$ . For $-m\leqslant i\leqslant m$ , we have

(3.8) \begin{align} y^i(t)&=p'(x_{1,\boldsymbol{y}},\, 0,\, i\varDelta _{x_3},\, t), \\[-28pt] \nonumber \end{align}

(3.9) \begin{align} z_{\kern-1pt j}^i(t)&=\begin{cases} p'(x_{1,\boldsymbol{z}_{\!j}},\, 0,\, i\varDelta _{x_3},\, t),\quad &j=1,2,3,\\ v_{x_1}'(x_{1,\boldsymbol{z}_{\!j}},\, x_{2,\textit{GIP}}(x_{1,\boldsymbol{z}_{\!j}}),\, i\varDelta _{x_3},\, t),\quad &j=4. \end{cases} \\[-12pt] \nonumber \end{align}

Again, these specific choices will be justified later in § 5.

The linear time-invariant dynamics of the supersonic boundary layer may be described using multiple-input–multiple-output (MIMO) transfer functions which are all functions of the Laplace variable $s$ . In the absence of control, the MIMO transfers $\boldsymbol{T}_{\!\boldsymbol{yw}}$ and $\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{w}}$ represent the impact of the disturbances $\boldsymbol{w}$ on the sensors $\boldsymbol{y}$ and $\boldsymbol{z}_{\!j}$ , respectively; i.e. $\boldsymbol{y}=\boldsymbol{T_{yw}}\boldsymbol{w}$ and $\boldsymbol{z}_{\!j}=\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{w}}\boldsymbol{w}$ . The impact of the actuator on the sensors, in the absence of disturbances, is represented by the MIMO transfers $\boldsymbol{T}_{\!\boldsymbol{yu}}$ and $\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{u}}$ ; i.e. $\boldsymbol{y}=\boldsymbol{T_{yu}}\boldsymbol{u}$ and $\boldsymbol{z}_{\!j}=\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{u}}\boldsymbol{u}$ . The MIMO transfer function of the controller is denoted $\boldsymbol{K}$ and it transforms the measurement signal $\boldsymbol{y}$ into the control signal $\boldsymbol{u}$ . A block-diagram representation is included in figure 4(a). We also introduce in this block-diagram a measurement noise $\boldsymbol{n}$ on the estimation sensors and an example of uncertainty block $\boldsymbol{{\varDelta }}$ representing a model error on $\boldsymbol{T}_{\!\boldsymbol{yu}}$ ; their role and importance will be detailed in § 4.

Figure 4.

(a) System representation of the control set-up, including an inverse multiplicative input uncertainty block $\boldsymbol{\varDelta }$ and a measurement noise $\boldsymbol{n}$ . (b) The $n\times n$ centralized MIMO problem of panel (a), with $n=2m+1$ may be recast as $m+1$ independent single-input–single-output (SISO) problems in the transverse wavenumber space. (c) Introduction of the spectral factor $\hat {W}^k_{yw}$ and the Wiener filter $\hat {T}^k_{z_{\kern-1pt j}y_w}$ to deal with the unknown input $\hat {w}^k$ . This modifies the phase of the closed-loop transfer function, but preserves its module up to a constant scaling factor. Similarly, the performance outputs are redefined as $\hat {z}^k_j(t):=\hat {z}^k_j(t+\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}})$ with $\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}$ the convection delay from $\boldsymbol{y_w}$ to $\boldsymbol{z}_{\!j}$ , without impacting the module of the closed-loop transfer function.

Ignoring $\boldsymbol{n}$ and $\boldsymbol{\varDelta }$ for now, the closed-loop transfer function from $\boldsymbol{w}$ to $\boldsymbol{z}_{\!j}$ is

(3.10) \begin{align} \boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{w}}^c=\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{w}}+\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{u}}\boldsymbol{K}\underbrace {(\boldsymbol{I}-\boldsymbol{T}_{\!\boldsymbol{yu}}\boldsymbol{K})^{-1}}_{\boldsymbol{S}}\boldsymbol{T}_{\!\boldsymbol{yw}}, \end{align}

where $\boldsymbol{S}$ is the sensitivity function (Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005). When designing the controller, we will be concerned with two different norms of the closed-loop transfer function, the classical $H_2$ and $H_\infty$ norms, subsequently denoted $\|(.)\|_2$ and $\|(.)\|_\infty$ . The $H_2$ norm has the following interpretation in the stochastic framework: if $\boldsymbol{w}$ is a white-noise of unit intensity, i.e. of power spectral density (PSD) $\boldsymbol{\varPhi }_{\boldsymbol{w}}=\boldsymbol{I}$ , then the $H_2$ norm of $\boldsymbol{T}^c_{\boldsymbol{z}_{\!j}\boldsymbol{w}}$ provides the standard deviation of $\boldsymbol{z}_{\!j}$ at any point in time (in the statistically steady regime), i.e.

(3.11) \begin{align} \|\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{w}}^c\|_2=\sqrt {E[\|\boldsymbol{z}_{\!j}\|^2]}\quad \text{for }\boldsymbol{\varPhi }_{\boldsymbol{w}}=\boldsymbol{I}, \end{align}

where $E[.]$ denotes the expectation. On the other hand, the $H_\infty$  norm provides the worst-case amplification of finite-energy perturbations. Denote the time-integrated energy $\|\boldsymbol{z}_{\!j} (t)\|_2=\sqrt {\int _0^\infty \|\boldsymbol{z}_{\!j}\|^2\,\mathrm{d}t}$ , then

(3.12) \begin{align} \|\boldsymbol{T}_{\!\boldsymbol{z}_{\!j}\boldsymbol{w}}^c\|_\infty =\max _{0\lt \|\boldsymbol{w}(t)\|_2\lt \infty }\dfrac {\|\boldsymbol{z}_{\!j}(t)\|_2}{\|\boldsymbol{w}(t)\|_2}. \end{align}

For stable systems, these norms may be computed as follows:

(3.13) \begin{align} \|\boldsymbol{T}\|_2&=\sqrt {\dfrac {1}{2\pi }\int _{-\infty }^\infty \|\boldsymbol{T}(\mathrm{i}\omega )\|^2_{{F}}\,\mathrm{d}\omega },&\text{with }\quad \|\boldsymbol{T}(\mathrm{i}\omega )\|^2_{{F}}&=\mathrm{Tr}(\boldsymbol{T}^H(\mathrm{i}\omega )\boldsymbol{T}(\mathrm{i}\omega )),\nonumber \\ \|\boldsymbol{T}\|_\infty &=\sup _{\omega \in \mathbb{R}}\|\boldsymbol{T}(\mathrm{i}\omega )\|_2,&\text{with }\quad \|\boldsymbol{T}(\mathrm{i}\omega )\|_2&=\bar {\sigma }\left (\boldsymbol{T}(\mathrm{i}\omega )\right ), \end{align}

where $\|(.)\|_{{F}}$ denotes the Frobenius norm and $\bar {\sigma }(.)$ denotes the largest singular value.

The controller $\boldsymbol{K}$ is a full $n\times n$ matrix so the control set-up is called centralized. For large values of $n$ , designing a structured (i.e. of a given order) controller to minimize the $H_2$ / $H_\infty$ norms is a challenging task so in the next section, we will exploit the symmetries of the problem in order to decouple the problem into a fully decentralized one.

3.2. Exploiting symmetries

The flow configuration has key symmetries that we wish to exploit. The basic flow has no transverse velocity component, i.e. $\bar {v}_{x_3}=0$ , and is invariant in the transverse direction, i.e. $\partial _{x_3} \bar {v}_{x_1}=\partial _{x_3} \bar {v}_{x_2}=\partial _{x_3} \bar {p} = \partial _{x_3} \bar {T} = 0$ . The control set-up is periodic in the transverse direction, with period $L_{x_3}$ , and flow perturbations about the basic flow are also $ L_{x_3}$ -periodic. Furthermore, source and measurement fields consist in arrays of identical fields regularly distributed along the transverse direction, with the same number $n$ in one period $L_{x_3}$ . The noise–actuator–sensor set-up is therefore periodic with period $ L_{x_3}/n=\varDelta _{x_3}$ , even though the flow field is only $ L_{x_3}$ -periodic.

Under these hypotheses, all the previously introduced transfer matrices are circulant. For example, ${\boldsymbol{y}}={\boldsymbol{T}}_{\boldsymbol{yu}}{\boldsymbol{u}}$ may be written more explicitly,

(3.14) \begin{align} \left (\begin{array}{c} \vdots \\ y^{-1} \\ y^0 \\ y^{1} \\ \vdots \end{array}\right )&= \begin{pmatrix} \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\[5pt] T_{yu}^{-16} & T_{yu}^{-15} & {\cdots} & T_{yu}^{1} & {\cdots} & T_{yu}^{17} & T_{yu}^{-17} \\[5pt] T_{yu}^{-17} & T_{yu}^{-16} & {\cdots} & T_{yu}^{0} & {\cdots} & T_{yu}^{16} & T_{yu}^{17} \\[5pt] T_{yu}^{17} & T_{yu}^{-17} & {\cdots} & T_{yu}^{-1} & {\cdots} & T_{yu}^{15} & T_{yu}^{16} \\[5pt] \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \end{pmatrix}\left (\begin{array}{c} \vdots \\ u^{-1} \\ u^0 \\ u^{1} \\ \vdots \end{array}\right )\!, \end{align}

with $T_{yu}^{\,k}=T_{y^i u^j}$ ( $T_{y^i u^j}$ representing the SISO transfer from $u^j$ to $y^i$ ) and $k = \mathrm{mod} (j-i+m,\,n)-m$ for all $ |i| \leqslant m$ and $ |j| \leqslant m$ . Since circulant systems can be expressed as circular discrete convolutions, passing to Fourier space transforms the convolution into a componentwise multiplication, resulting in a diagonal representation of the transfer matrices,

(3.15) \begin{align} \underbrace {\left (\begin{array}{c} \hat {y}^0 \\ \hat {y}^{1} \\ \vdots \\ \hat {y}^{m} \\ \hat {y}^{-m} \\ \vdots \\ \hat {y}^{-1}\end{array}\right )}_{\boldsymbol{\hat {y}}}&=\underbrace {\left ( \begin{array}{ccccccc} \hat {T}_{yu}^{0} & 0 & {\cdots} & {\cdots} & {\cdots} & {\cdots} & 0 \\ 0 & \hat {T}_{yu}^{1} & 0 & {\cdots} & {\cdots} &{\cdots} & \vdots \\ \vdots & \ddots & \ddots & \ddots & {\cdots} & {\cdots} & \vdots \\ \vdots & {\cdots} & 0 &\hat {T}_{yu}^{m} & 0 & {\cdots} & \vdots \\ \vdots & {\cdots} & {\cdots} & 0 & \hat {T}_{yu}^{-m} & 0 & \vdots \\ \vdots & {\cdots} & {\cdots} & {\cdots} & \ddots & \ddots & \vdots \\ 0 & {\cdots} & {\cdots} & {\cdots} & {\cdots} & 0 & \hat {T}_{yu}^{-1} \end{array} \right )}_{\hat {\boldsymbol{T}}_{\boldsymbol{yu}}=\boldsymbol{F}_n\boldsymbol{T}_{\!\boldsymbol{yu}} \boldsymbol{F}_n^{-1}} \underbrace {\left (\begin{array}{c} \hat {u}^0 \\ \hat {u}^{1} \\ \vdots \\ \hat {u}^{m} \\ \hat {u}^{-m} \\ \vdots \\ \hat {u}^{-1}\end{array}\right )}_{\boldsymbol{\hat {u}}}, \end{align}

where $\boldsymbol{\hat {y}}=\boldsymbol{F}_n \boldsymbol{{y}}$ denotes the discrete Fourier transform (DFT) of vector $\boldsymbol{{y}}$ (and similarly for $\boldsymbol{w}$ , $\boldsymbol{u}$ and $\boldsymbol{z}$ ), the matrix $ \boldsymbol{F}_n$ being the unitary DFT matrix defined by $(F_n)_{ki}=n^{-1/2}\omega _n^{ki}$ for $k,i=0,\ldots, n-1$ and satisfying $\boldsymbol{F}_n\boldsymbol{F}_n^H=\boldsymbol{F}_n^H\boldsymbol{F}_n=\boldsymbol{I}$ . Here $(.)^H$ denotes the Hermitian conjugate, and $\omega _n=\mathrm{e}^{-{i}2\pi /n}$ is the primitive nth root of unity. Hence, each $\hat {T}_{yu}^{k}$ represents the transfer function associated with the $k^{th}$ spanwise Fourier mode of $\boldsymbol{{y}}$ and $\boldsymbol{{u}}$ . The kth spanwise mode corresponds to a non-dimensional transverse wavenumber $(k\beta /n)\delta _0^*$ , where $\beta = 2\pi /\varDelta _{x_3}$ is the wavenumber associated with the distance $\varDelta _{x_3}$ between two consecutive sensors/actuators in $x_3$ -direction.

The DFT matrix being unitary, we have (Parseval)

(3.16) \begin{align} \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}(\mathrm{i}\omega )\|_{{F}}=\|\boldsymbol{T}_{\!\boldsymbol{y}\boldsymbol{u}}(\mathrm{i}\omega )\|_{{F}}&\quad \Rightarrow \quad \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}\|_2=\|\boldsymbol{T}_{\!\boldsymbol{y}\boldsymbol{u}}\|_2,\nonumber \\ \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}(\mathrm{i}\omega )\|_{2}=\|\boldsymbol{T}_{\!\boldsymbol{y}\boldsymbol{u}}(\mathrm{i}\omega )\|_{2}&\quad \Rightarrow \quad \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}\|_\infty =\|\boldsymbol{T}_{\!\boldsymbol{y}\boldsymbol{u}}\|_\infty . \end{align}

Finally, $\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}$ being diagonal, we have

(3.17) \begin{align} \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}(\mathrm{i}\omega )\|^2_{{F}}=\sum _k |\hat {T}^k_{yu}(\mathrm{i}\omega )|^2 & \quad \Rightarrow \quad \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}\|^2_2=\sum _k \|\hat {T}^k_{yu}\|_2^2,\nonumber \\ \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}(\mathrm{i}\omega )\|_2=\max _k |\hat {T}^k_{yu}(\mathrm{i}\omega )| & \quad \Rightarrow \quad \|\hat {\boldsymbol{T}}_{\boldsymbol{y}\boldsymbol{u}}\|_{\infty }=\max _k \|\hat {T}^k_{yu}\|_\infty . \end{align}

Therefore, minimizing the $H_2$ or the $H_\infty$ norm for each integer $k$ independently results in the minimization of the $H_2$ or $H_\infty$ norm of the full system. By exploiting the symmetries of our set-up, we convert a $n\times n$ centralized MIMO problem into a decentralized MIMO problem made of $n$ SISO subproblems, represented in figure 4(b). The closed-loop transfer function for each integer $-m\leqslant k\leqslant m$ is

(3.18) \begin{align} \hat {T}^{k,c}_{z_{\kern-1pt j}w}=\hat {T}^k_{z_{\kern-1pt j}w}+\hat {T}^k_{z_{\kern-1pt j}u}\hat {K}^k\underbrace {\big(1-\hat {T}^k_{yu}\hat {K}^k \big)^{-1}}_{\hat {S}^k}\hat {T}^k_{yw}. \end{align}

The SISO block diagram is similar to the one introduced in Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023) for the case of spanwise invariant perturbations, so we may use the same approach for synthesis. One difference with the 2-D case though is that the input and output variables are now complex in the time-domain. This is not an issue here because the impulse responses associated with the complex input–output pairs are real in our case (see Appendix B), therefore system identification of controller synthesis may be performed in Fourier space, using real algebra, exactly as in Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023). Moreover, all transfer functions satisfy the symmetry

(3.19) \begin{align} \hat {T}^{-k}=\hat {T}^k \end{align}

(see Appendix B), therefore we only need to perform identification and synthesis for the $m+1$ transverse wavenumbers $k=0,\ldots, m$ .

Each controller $\hat {K}^k$ is synthesized independently in state-space form $(\hat {\boldsymbol{A}}_K^k,\hat {\boldsymbol{B}}_K^k,\hat {\boldsymbol{C}}_K^k,0)$ where $\hat {\boldsymbol{A}}_K^k$ , $\hat {\boldsymbol{B}}_K^k$ and $\hat {\boldsymbol{C}}_K^k$ denote the state, input and output matrices (Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005) of the controller associated with mode $k$ . As long as the perturbations remain within the linear regime, each spanwise mode $k$ evolve independently and consequently, each controller $\hat {K}^k$ operates independently. A real state-space representation of the total controller $\boldsymbol{K}$ is then retrieved as $(\hat {\boldsymbol{A}}_K,\hat {\boldsymbol{B}}_K\boldsymbol{F}_n,\boldsymbol{F}_n^{-1}\hat {\boldsymbol{C}}_K,0)$ , where $\hat {\boldsymbol{A}}_K=\mathrm{blkdiag}(\hat {\boldsymbol{A}}_K^0,\ldots, \hat {\boldsymbol{A}}_K^m,\hat {\boldsymbol{A}}_K^m,\ldots, \hat {\boldsymbol{A}}_K^1)$ , $\hat {\boldsymbol{B}}_K=\mathrm{blkdiag}(\hat {\boldsymbol{B}}_K^0,\ldots, \hat {\boldsymbol{B}}_K^m,\hat {\boldsymbol{B}}_K^m,\ldots, \hat {\boldsymbol{B}}_K^1)$ and $\hat {\boldsymbol{C}}_K=\mathrm{blkdiag}(\hat {\boldsymbol{C}}_K^0,\ldots, \hat {\boldsymbol{C}}_K^m,\hat {\boldsymbol{C}}_K^m,\ldots, \hat {\boldsymbol{C}}_K^1)$ . The matrices $\hat {\boldsymbol{B}}_K\boldsymbol{F}_n$ and $\boldsymbol{F}_n^{-1}\hat {\boldsymbol{C}}_K$ are real indeed because of the symmetry $\hat {\boldsymbol{B}}^{-k}_K=\hat {\boldsymbol{B}}^k_K$ and $\hat {\boldsymbol{C}}^{-k}_K=\hat {\boldsymbol{C}}^k_K$ . The total controller $\boldsymbol{K}$ is the one implemented in the computational fluid dynamics (CFD) solver FastS for the closed-loop simulations: the control signal vector $\boldsymbol{u}$ is updated from the estimation signal vector $\boldsymbol{y}$ by solving the first-order differential equations of the controller state-space representation (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024).

3.3. Dealing with the unknown input $\boldsymbol{w}$

The SISO transfer functions $\hat {T}^k_{yu}$ and $\hat {T}^k_{z_{\kern-1pt j}u}$ may be identified from input–output data as we shall see in § 6.1, however, this is not possible for $\hat {T}^k_{yw}$ and $\hat {T}^k_{z_{\kern-1pt j}w}$ , because the exogenous noise $\boldsymbol{w}$ cannot be measured in a realistic set-up. First we introduce a Wiener filter $\hat {T}^k_{z_{\kern-1pt j} y_w}$ (Sasaki et al. Reference Sasaki, Morra, Fabbiane, Cavalieri, Hanifi and Henningson2018, Reference Sasaki, Morra, Cavalieri, Hanifi and Henningson2020; Morra et al. Reference Morra, Sasaki, Hanifi, Cavalieri and Henningson2020) to optimally estimate $\hat {z}_{j,w}^k$ from $\hat {y}_w^k$ (estimation measurements in absence of control). As a result

(3.20) \begin{align} \hat {z}^k_{j,w}=\hat {T}^k_{z_{\kern-1pt j} y_w}\hat {y}_w^k+\hat {r}^k, \end{align}

where $\hat {r}^k$ is a residual due to the fact that some information measured by $\hat {z}_{j,w}^k$ is missing in $\hat {y}_w^k$ (e.g. nonlinearities, noise on performance sensors, spectral leakage, linear contributions from other unmeasured inputs, etc.) (Bendat & Piersol Reference Bendat and Piersol1971; Morra et al. Reference Morra, Sasaki, Hanifi, Cavalieri and Henningson2020). Because sensors $\boldsymbol{y}$ are far upstream of $\boldsymbol{z}_{\!j}$ , there are no specific precautions to take with respect to causality and the simple formula

(3.21) \begin{align} \hat {T}^k_{z_{\kern-1pt j}y_w}(\mathrm{i}\omega )=\dfrac {\hat {\varPhi }_{z_{\kern-1pt j}y_w}^k(\omega )}{\hat {\varPhi }_{y_w}^k(\omega )} \end{align}

provides the frequency response of the optimal causal filter (Glad & Ljung Reference Glad and Ljung2000). Both spectral and cross-spectral power densities $\hat {\varPhi }^k(\omega )$ may be estimated from data with actuation switched off. Using the filter, we may write

(3.22) \begin{align} \hat {T}^k_{z_{\kern-1pt j}w}\approx \hat {T}^k_{z_{\kern-1pt j}y_w}\hat {T}^k_{yw} \end{align}

which leads, using (3.18), to

(3.23) \begin{align} \hat {T}^{k,c}_{z_{\kern-1pt j}w}\approx \big(\hat {T}^k_{z_{\kern-1pt j}y_w}+\hat {T}^k_{z_{\kern-1pt j}u}\hat {K}^k\hat {S}^k \big)\hat {T}^k_{yw}. \end{align}

Optimizing the $H_2$ or $H_\infty$ norm only requires the computation of the module of $\hat {T}^{k,c}_{z_{\kern-1pt j}w}$ , therefore, the phase information of $\hat {T}^k_{yw}$ is irrelevant. The module of $\hat {T}^k_{yw}$ satisfies

(3.24) \begin{align} \hat {\varPhi }^k_{y_w}=|\hat {T}^k_{yw}|^2\hat {\varPhi }^k_w, \end{align}

where, again, the spectral density $\hat {\varPhi }^k_{y_w}$ of $\hat {y}^k_w$ can be measured. Without loss of generality, we may assume that $\boldsymbol{w}$ is a white-noise, and we introduce $\hat {W}^k_{yw}$ as a spectral factor of $\hat {\varPhi }^k_{y_w}$ . Hence $\hat {W}^k_{yw}$ is just a stable and causal filter, whose inverse is also a stable and causal filter, such that $|\hat {W}^k_{yw}(\mathrm{i}\omega )|\propto \sqrt {\hat {\varPhi }^k_{y_w}(\omega )}$ for all frequencies. The weighting function $\hat {W}^k_{yw}$ is not unique, being defined up to an arbitrary phase and scaling factor in magnitude, which do not affect the synthesis step due to $H_2/H_{\infty }$ minimization. This leads to the final equation,

(3.25) \begin{align} \hat {T}^{k,c}_{z_{\kern-1pt j}w}\approx \big(\hat {T}^k_{z_{\kern-1pt j}y_w}+\hat {T}^k_{z_{\kern-1pt j}u}\hat {K}^k\hat {S}^k \big)\hat {W}^k_{yw}, \end{align}

where all the SISO transfer functions may be identified from input–output data in an experimental configuration. The block diagram involving the Wiener filter and the spectral factor is represented in figure 4(c): it represents the closed-loop transfer function, up to a negligible phase error and magnitude scaling factor both due to $\hat {W}_{yw}^k$ , and up to the residual error $\hat {r}^k$ of (3.20) from the Wiener filter.

3.4. Dealing with delays

When introducing the spectral factor $\hat {W}^k_{yw}$ , we discussed the fact that introducing an error in the phase of the closed-loop transfer function has no impact on controller synthesis, because system norms are only concerned with $|\hat {T}^{k,c}_{z_{\kern-1pt j} w}|$ . We may exploit this property once again by swapping the performance sensors $\boldsymbol{z}_{\!j}(t)$ with the non-causal sensors

(3.26) \begin{align} \boldsymbol{z}_{\!j}(t):=\boldsymbol{z}_{\!j}(t+\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}), \end{align}

where $\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}\gt 0$ is the convection delay from $\boldsymbol{y_w}$ to $\boldsymbol{z}_{\!j}$ . This procedure has already been used in Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024). The fact that the new sensors are not causal is not problematic because performance sensors do not need to implemented in real-time, they only serve for the controller synthesis step, which is performed offline. Translating the $\boldsymbol{z}_{\!j}$ signals backward in time amounts to multiplying the transfer functions $\hat {T}^k_{z_{\kern-1pt j} y_w}$ and $\hat {T}^k_{z_{\kern-1pt j} u}$ by the non-causal filter $\mathrm{e}^{s\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}}$ . By examining (3.25), we see that this filter factors out and therefore applies to the full closed-loop transfer function. Since the frequency response of the filter has a constant module equal to unity, the module of the frequency response of the closed-loop transfer function remains unchanged. This procedure leads to no loss of information because even though the filter $\mathrm{e}^{s\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}}$ is non-causal, the shifted frequency responses from $\boldsymbol{u}$ and $\boldsymbol{y_w}$ to $\boldsymbol{z}_{\!j}$ remain causal, so no information is ‘lost in the past’ during the system identification step. Indeed, between $0\leqslant t\leqslant \tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}$ , both impulse responses are zero, because perturbations did not have time to reach $\boldsymbol{z}_{\!j}$ yet. We may therefore safely discard the initial deadtime $\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}$ from both impulse responses without affecting the closed-loop system norm. This procedure has great benefits for both system identification and controller synthesis, because the lower the time-delay, the lower the order of the corresponding state-space representation and the cheaper controller synthesis. Note that we may not remove the deadtime from $\boldsymbol{u}$ to $\boldsymbol{y}$ without artificially changing the closed-loop dynamics, so this deadtime must remain.

3.5. Defining the number of sensors and actuators

In this section, we start by assuming that the number of sensors $n_y$ and $n_z$ and the number of actuators $n_u$ , per unit length $L_{x_3}$ , may take different values. Noise sources are infinitely localized along the transverse direction $x_3$ so they excite all transverse Fourier modes equally. The smallest wavelengths are filtered by viscosity and at the sensor positions there are minimum wavelengths $\lambda _{\textit{min}}^{\boldsymbol{y_w}}$ and $\lambda _{\textit{min}}^{\boldsymbol{z}_{j,\boldsymbol{w}}}$ containing energy in the signals $\boldsymbol{y_w}$ and $\boldsymbol{z}_{j,\boldsymbol{w}}$ (the signals of $\boldsymbol{y}$ and $\boldsymbol{z}_{\!j}$ when excited only by the exogenous noises $\boldsymbol{w}$ ), respectively. The minimum wavelengths containing energy in the signals $\boldsymbol{y_u}$ and $\boldsymbol{z}_{j,\boldsymbol{u}}$ (the signals of $\boldsymbol{y}$ and $\boldsymbol{z}_{\!j}$ when excited only by the actuators $\boldsymbol{u}$ ) are, in the linear regime, greater than $\lambda _{\textit{min}}^{b_u}$ , the minimum wavelength in the spectrum (in space) of the actuator field $b_u(x_3)$ . To avoid aliasing, the number of sensors must satisfy

(3.27) \begin{align} n_y\geqslant 2\dfrac {L_{x_3}}{\min \big(\lambda _{\textit{min}}^{\boldsymbol{y_w}},\lambda _{\textit{min}}^{b_u} \big)},\quad n_z\geqslant 2\dfrac {L_{x_3}}{\min \big(\lambda _{\textit{min}}^{\boldsymbol{z}_{j,\boldsymbol{w}}},\lambda _{\textit{min}}^{b_u} \big)}. \end{align}

As long as the dynamics is linear, it is also expected that $\lambda _{\textit{min}}^{\boldsymbol{z}_{j,\boldsymbol{w}}}\geqslant \lambda _{\textit{min}}^{\boldsymbol{y_w}}$ , as small structures are damped as they propagate downstream. On the other hand, if transition begins upstream of both sets of sensors, the opposite is true, but we exclude that scenario in the present paper.

The number of observable transverse Fourier modes is equal to the number of sensors, i.e. $n_y$ and $n_z$ . Similarly, the number of controllable Fourier modes is equal to the number of actuators $n_u$ . If $n_y$ is too small, then aliasing occurs and results in measurement noise on the observable modes. Similarly, we need

(3.28) \begin{align} n_u\geqslant 2L_{x_3}/\lambda _{\textit{min}}^{b_u}, \end{align}

otherwise actuation noise related to uncontrollable modes is introduced. In the linear limit, this actuation noise can be completely filtered out by the measurement if inequalities (3.27) are satisfied, so the closed-loop is not affected for the $n_u$ controllable modes (it goes without saying that $n_u\leqslant \min (n_y,n_z)$ ). However, this actuation noise can affect the nonlinear dynamics and negatively impact transition to turbulence. Thus, to minimize the number of sensors and actuators while avoiding the problems of measurement and actuation noise, it is desirable to choose actuators whose support is such that $\lambda _{\textit{min}}^{b_u}\geqslant \lambda _{\textit{min}}^{\boldsymbol{y_w}}$ , to choose a common number $n=n_u=n_y=n_z$ of actuators/sensors, and verify the condition $ 2L_{x_3}/\lambda _{\textit{min}}^{\boldsymbol{y_w}}\leqslant n$ . In this paper we choose $n=35$ , which, as we shall see, is more than enough.

4.1. Nominal performance/stability

In this study, the concept of nominal performance focuses to noise cancellation, in order to delay the laminar-to-turbulent transition, which occurs once a critical amplitude threshold is exceeded. Hence, the lower the ratio $\|\hat {T}^{k,c}_{z_{\kern-1pt j}w}\|_2/\|\hat {T}^{k}_{z_{\kern-1pt j}w}\|_2$ in nominal point condition (i.e. no uncertainties, errors or variations in the modelled transfer functions are considered), the better the nominal performance. One might also ask what is the most effective set-up for delaying transition between minimizing $\|\hat {T}^{k,c}_{z_{\kern-1pt j}w}\|_2/\|\hat {T}^{k}_{z_{\kern-1pt j}w}\|_2$ or $\|\hat {T}^{k,c}_{z_{\kern-1pt j}w}\|_\infty /\|\hat {T}^{k}_{z_{\kern-1pt j}w}\|_\infty$ but this is considered beyond the scope of the current study.

Concerning the concept of nominal stability, it refers to the requirement that the closed-loop system, built from the nominal model $\hat {T}^k_{yu}$ and the designed controller $\hat {K}^k$ , is stable in the absence of uncertainty. Since $-\hat {T}^k_{yu}\hat {K}^k$ does not exhibit any unstable pole ( $\hat {T}^k_{yu}$ is stable because the boundary layer flow is globally stable, while $\hat {K}^k$ is stable by design), the closed-loop system is stable if and only if the Nyquist plot of $-\hat {T}^k_{yu}\hat {K}^k$ does not encircle the critical point (−1,0). This notion of nominal stability (Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005) is meaningful only in a feedback configuration (feedforward control is unconditionally stable as $\hat {T}^k_{yu}=0$ ) and serves as the foundational prerequisite for subsequent analyses of robust stability.

4.2. Robust stability

Robust stability concerns the preservation of closed-loop stability when the true plant $\hat {T}^{k,\textit{real}}_{yu}$ deviates from its nominal model $\hat {T}^k_{yu}$ within a class of uncertainty $\hat {\varDelta }^k$ . The uncertain plant can be expressed as the feedback interconnection of a stable closed-loop transfer operator $\hat {M}^k$ with the norm-bounded stable uncertainty $\hat {\varDelta }^k$ . The small gain theorem (Zames Reference Zames1966; Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005) then yields a necessary and sufficient condition for robust stability,

(4.1) \begin{align} \|\hat {M}^k\hat {\varDelta }^k\|_\infty \lt 1. \end{align}

Indeed, ensuring this $H_\infty$ criterion guarantees that the Nyquist curve of the uncertain loop cannot be displaced so as to encircle the critical point, thereby preserving stability (Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005).

In the case where the real system can be expressed as

(4.2) \begin{align} \hat {T}_{yu}^{k,\textit{real}}=\frac {\hat {T}^k_{yu}}{1-\hat {\varDelta }^k} \quad \Leftrightarrow \quad \hat {\varDelta }^k=\frac {\hat {T}_{yu}^{k,\textit{real}}-\hat {T}^k_{yu}}{\hat {T}_{yu}^{k,\textit{real}}}, \end{align}

the operator $\hat {\varDelta }^k$ is referred to as an inverse multiplicative uncertainty (see in figure 4). This type of uncertainty is often preferred, as it can be naturally interpreted as a relative modelling error. The requirement that $-\hat {T}_{yu}^{k,\textit{real}}K$ does not encircle the critical point (−1,0) is then equivalent to satisfying the condition (Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005)

(4.3) \begin{align} |1-\hat {T}^k_{yu}\hat {K}^k|&\gt |\hat {\varDelta }^k| \quad \forall \omega, \nonumber \\ &\Leftrightarrow \nonumber \\ \|\hat {S}^k\hat {\varDelta }^k\|_{\infty }&\lt 1. \end{align}

Therefore, from a control point of view, imposing this $H_{ \infty }$ constraint guarantees the internal stability of the closed-loop up to a value $\hat {\varDelta }^k$ of relative model error on $\hat {T}^k_{yu}$ . Nevertheless, this type of multiplicative uncertainty has two limitations. The first is that it is only meaningful within the system’s bandwidth; outside this range, the very low values of $\hat {T}^{k,\textit{real}}_{yu}$ render the multiplicative uncertainty ill-conditioned. The second limitation is that for non-minimum phase systems (Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005) such as noise-amplifier flows, $\hat {\varDelta }^k$ is often unstable or non-causal unless special precautions are taken during its construction, making the classical small-gain criterion inapplicable. Consequently, multiplicative uncertainties account only for uncertainties that can be represented, for example, such as the worst-case combination of pure gain and pure phase shifts (so the most generic measure of available stability margin), representing in our case growth rate and convection speed estimation errors, respectively.

In order to address the shortcomings of multiplicative uncertainty, it is often convenient to also introduce an additive uncertainty,

(4.4) \begin{align} \hat {\varDelta }^k=\hat {T}_{yu}^{k,\textit{real}}-\hat {T}^k_{yu}. \end{align}

Indeed, additive uncertainty allows one to capture a broad range of model discrepancies while ensuring that $\hat {\varDelta }^k$ remains stable for boundary layer flows. Moreover, additive uncertainty is particularly well-suited for modelling discrepancies outside the system’s bandwidth, where multiplicative uncertainty would otherwise result in an ill-conditioned ratio. Hence, this additive uncertainty framework is therefore more general than the multiplicative framework, albeit at the cost of being more conservative within the bandwidth of interest (Skogestad & Postlethwaite Reference Skogestad and Postlethwaite2005). For the additive uncertainty $\hat {\varDelta }^k$ , the Nyquist stability requirement that $-\hat {T}_{yu}^{k,\textit{real}}K$ does not encircle the critical point (−1,0) is then equivalent to satisfying the condition

(4.5) \begin{align} |1-\hat {T}^k_{yu}\hat {K}^k|&\gt |\hat {K}^k\hat {\varDelta }^k| \quad \forall \omega, \nonumber \\ &\Leftrightarrow \nonumber \\ \|\hat {K}^k\hat {S}^k\hat {\varDelta }^k\|_{\infty }&\lt 1. \end{align}

In summary, robust stability for boundary-layer flows can be addressed by shaping both $\hat {S}^k$ and $\hat {K}^k\hat {S}^k$ . While $\hat {S}^k$ captures only specific, well-defined uncertainties within the bandwidth – such as pure gain or phase shifts – $\hat {K}^k\hat {S}^k$ accounts for more general discrepancies, particularly outside the bandwidth where the nominal plant has low gain.

4.3. Robust performance

The concept of robust performance consists here in maintaining performance in terms of amplitude reduction despite differences between on-design and off-design operating conditions: the lower the ratio $\|\hat {T}^{k,c}_{z_{\kern-1pt j}w}\|_2$ / $\|\hat {T}^{k}_{z_{\kern-1pt j}w}\|_2$ despite for example free stream variations, new noise sources or nonlinearities, the better the performance robustness. Note that only model flaws on $\hat {T}_{yu}^k$ , which lies within the feedback loop, can lead to instability; uncertainties on other transfers outside the loop (e.g. $\hat {T}^k_{z_{\kern-1pt j}u}$ , $\hat {T}^k_{z_{\kern-1pt j}y_w}$ , $\hat {W}^k_{yw}$ ) merely degrade performance without compromising stability.

In case of maintaining performance when free stream condition variations are considered, using a feedback control structure (instead of a feedforward one) designed for the nominal operating point may be sufficient provided that the variations around this point are not too large (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024). In such cases of free stream uncertainties, it may not be necessary to constrain other transfer functions in the synthesis step and just the natural ability of the feedback to handle modelling errors is sufficient to ensure robust performance.

In case of new noise source $\hat {n}^k$ modelling a noise measurement on the estimation sensor (displayed in figure 4), a common practice is to constrain $\hat {K}^k\hat {S}^k$ to limit the actuator activity on certain bandwidth. Indeed, the quantity $\hat {K}^k\hat {S}^k$ may represent the transfer from noise $\hat {n}^k$ (corrupting $\hat {y}^k$ ) to $\hat {u}^k$ . To desensitize $\hat {u}^k$ to noisy measurements, one can seek to reduce $\|\hat {K}^k\hat {S}^k\|_2$ as this means reducing the root-mean-square (r.m.s.) value of $\hat {u}^k(t)$ (i.e. the power spent by the control) in case of white noise $\hat {n}^k$ on estimation sensors $\hat {y}^k$ . Minimizing $\|\hat {K}^k\hat {S}^k\|_2$ appears as the most general way to limit actuator activity when the noise $\hat {n}^k$ is unknown; if one seeks to desensitize the controller on a particular frequency range (see e.g. Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024) or if the properties of $\hat {n}^k$ (e.g. frequency spectrum) are known, it may be of interest to constrain $\|\hat {K}^k\hat {S}^k\|_\infty$ . It is worth noting that the same transfer, here $\hat {K}^k\hat {S}^k$ , can be associated with both robust stability and robust performance, depending on the source of uncertainty. In the specific case of a new noise source $\hat {n}^k$ , the perturbation cannot destabilize the closed loop, since it is uncorrelated with the input–output dynamics. Its effect is therefore purely related to performance. By contrast, modelling errors that directly affect the feedback loop via $\hat {T}_{yu}^k$ can compromise both performance and stability, as they alter the closed-loop dynamics themselves.

Nonlinearities can also be viewed through the lens of robust performance, as they may degrade control objectives. However, since the present study uses classical tools of control theory which are mathematically well-established in a linear framework, nonlinear effects cannot be explicitly incorporated in the synthesis step and can only be assessed a posteriori, by evaluating their impact on the achieved performance.

5.1. Uncontrolled flow

The evolution of the velocity wave packet at the generalized inflection point $x_{2,\textit{GIP}}$ and the pressure wave packet near the wall, both generated by an impulse of $w^0$ (the exogenous noise source at the centre spanwise position of the domain, see figure 1) is plotted in figure 5. Both wave packets are convected downstream with the same group velocity of approximately $0.92V_{\infty }$ and grow in amplitude. Additionally, both wave packets spread in the streamwise and spanwise directions and each wave packet is symmetrical with respect to the spanwise position $x_3=0$ (this is consistent with the symmetry (3.19) between transfer functions justified in Appendix B). The two wave packets have different 2-D structure. As the amplification rates of the first mode and second mode are, respectively, maximal for oblique and 2-D perturbations (Mack Reference Mack1984), the velocity wave packet at $x_{2,\textit{GIP}}$ and the pressure wave packet near the wall can be associated with the first and second instability modes, respectively. This assumption is justified in more detail in the following.

Figure 5.

(a,b,c) Velocity and (d,e,f) pressure wave packets at times $tV_{\infty }/\delta _{0}^{*}=$ (a,d) $455$ , (b,e) $834$ and (c,f) $1365$ , generated by an impulse of the exogenous input $w^0$ at time $t=0$ . The velocity and pressure wave packets are at $x_{2,\textit{GIP}}$ and $x_2=1.99\delta _0 ^*$ , respectively.

Figure 5. Long description

Panel A: A scatter plot shows velocity wave packets at a normalized time of approximately 455. The x-axis represents the normalized streamwise position, and the y-axis represents the normalized wall-normal position. The color scale indicates the normalized streamwise velocity fluctuation, with red representing positive values and blue representing negative values. The wave packet appears as a concentrated cluster of points around the streamwise position of 400 and wall-normal position of 0. Panel B: Another scatter plot shows velocity wave packets at a normalized time of approximately 834. The axes and color scale are the same as in Panel A. The wave packet has spread out more along the streamwise direction compared to Panel A. Panel C: A scatter plot shows velocity wave packets at a normalized time of approximately 1365. The axes and color scale are the same as in previous panels. The wave packet has further spread out and shows a more elongated shape along the streamwise direction. Panel D: A scatter plot shows pressure wave packets at a normalized time of approximately 455. The x-axis represents the normalized streamwise position, and the y-axis represents the normalized wall-normal position. The color scale indicates the normalized pressure fluctuation, with red representing positive values and blue representing negative values. The wave packet appears as a concentrated cluster of points around the streamwise position of 400 and wall-normal position of 0. Panel E: Another scatter plot shows pressure wave packets at a normalized time of approximately 834. The axes and color scale are the same as in Panel D. The wave packet has spread out more along the streamwise direction compared to Panel D. Panel F: A scatter plot shows pressure wave packets at a normalized time of approximately 1365. The axes and color scale are the same as in previous panels. The wave packet has further spread out and shows a more elongated shape along the streamwise direction.

Linearized DNS is also performed with white-noise from all $35$ uncorrelated sources $w^i$ . The profiles at $x_1=1766.7\delta _0^{*}$ of the r.m.s. fluctuations averaged in the spanwise direction are plotted in figure 6(a). The greatest fluctuations are for thermodynamic quantities (density and temperature fluctuations) which peak near the generalized inflection point (dashed line); the velocity profile also takes its highest value close to $x_{2,\textit{GIP}}$ , while the pressure profile has two peaks: the highest one at the wall and a second one at $x_{2,\textit{GIP}}$ . These results for the position of the profile peaks at $x_1=1766.7\delta _0^{*}$ are true for all streamwise positions (see figure 6 b). Note that upstream of the domain, the pressure maxima in the wall-normal direction are not at the wall; external noise sources generate an acoustic wave leaving the boundary layer. Although the velocity peaks near the generalized inflection point and can therefore be associated with a predominance of the first mode according to the input–output analysis of Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019), it is not possible to determine the dominant mechanism based solely on the r.m.s. profiles for density and temperature because both the first and the second modes lead to a peak around $x_{2,\textit{GIP}}$ (Bugeat et al. Reference Bugeat, Chassaing, Robinet and Sagaut2019). Hence, in order to classify more precisely the dominant linear mechanisms involved in our 3-D linearized DNS, the frequency spectra at $x_{2,\textit{GIP}}$ and near the wall are analysed in figure 7. The frequency spectra of the primitive variables are normalized by their respective maximal values for comparison purposes, averaged in the spanwise direction, and

(5.1) \begin{align} F=2\pi f \delta _0^*/V_{\infty } \end{align}

represents the dimensionless frequency. At the generalized inflection point (see figure 7 a), all the primitive variables are dominated by the low frequencies associated with the first mode (Bugeat et al. Reference Bugeat, Chassaing, Robinet and Sagaut2019; Poulain et al. Reference Poulain, Content, Rigas, Garnier and Sipp2024). Near the wall (see figure 7 b), all the primitive variables are dominated by the higher frequencies associated with the second Mack mode. Hence, the highest r.m.s. values of the velocity, density and temperature are mainly due to first instability mode while the highest r.m.s. value of the pressure is due to the second instability mode. The second peak of the r.m.s. profile of pressure near $x_{2,\textit{GIP}}$ is associated with the first mode. The chosen exogenous noise sources thus trigger the amplification mechanisms of both the first and the second instability modes that coexist in the boundary layer with the dominant mode depending on the wall-normal location. This is consistent with the findings of Guo et al. (Reference Guo, Hao and Wen2023) and Chen et al. (Reference Chen, Guo and Wen2023), who showed that when both first and second instability modes are initially excited with the same amplitude, oblique waves of the first mode are prominent in the outer boundary layer but remain comparatively weak near the wall, where the 2-D structures of the second Mack mode dominate.

Figure 6.

(a) Profiles at $x_1=1766.7\delta _0^{*}$ of r.m.s. variables averaged in the spanwise direction. (b) Evolution of ${\rm arg}_{{x}_2}\ \max $ of the r.m.s. variables averaged in the spanwise direction. Black solid, dashed and dotted lines represent the boundary layer thickness, the generalized inflection point and the compressible displacement thickness, respectively.

Figure 7.

Spanwise-averaged PSD (each curve normalized by its own maximum) of fluctuating variables at $x_1=1775\delta _0^{*}$ , (a) $x_{2,\textit{GIP}}$ and (b) $x_2=0.015\delta _0^*$ (below $x_{2,\textit{GIP}}$ and very close to the wall) when the 3-D boundary layer is excited simultaneously by the $35$ uncorrelated white-noise signals $w^i$ .

5.2. Selecting sensors

5.2.1. Performance sensors

As we wish to control both the first and the second modes, we select performance sensors capable of observing both phenomena. Therefore, sensors $\boldsymbol{z}_1,\boldsymbol{z}_2,\boldsymbol{z}_3$ which measure the wall-pressure, essentially observe the second Mack mode. Since the significance of oblique waves of the first mode may not be fully captured by wall-pressure measurements (see figure 7) (Chen et al. Reference Chen, Guo and Wen2023; Guo et al. Reference Guo, Hao and Wen2023), the additional row of performance sensors $\boldsymbol{z}_4$ is introduced to better capture the first instability mode by measuring the longitudinal velocity fluctuation at the generalized inflection point. These statements are confirmed by examining, in figure 8, the module of the transfer function $\hat {T}^k_{\phi (x_1) w}$ (obtained as the Fourier transforms of the impulse responses, see figure 5) from the noise sources to sensors $\phi (x_1)$ measuring (a) $v_{x_1}'$ at $x_{2,\textit{GIP}}$ and (b) $p'$ at the wall. Note that $|k|=17$ corresponds to $(|k|\beta /n)\delta _0^*\simeq 0.92$ . It is clear that the sensor $v_{x_1}'$ at $x_{2,\textit{GIP}}$ measures the first mode, since the dominant mode numbers correspond to oblique waves $k\neq 0$ . The wall-pressure sensors observe both the first ( $k\neq 0$ ) and second ( $k=0$ ) instability modes, with a predominant contribution from the latter. Figure 7 demonstrated that the frequency contents of the near-wall primitive variables are very similar (except for lower values of velocity fluctuations at low frequencies); therefore, it is expected that reducing the r.m.s. values of wall-pressure with destructive interference will also attenuate the near-wall velocity, density and temperature fluctuations. Conversely, reducing the r.m.s. values of longitudinal velocity perturbations should also attenuate pressure, density and temperature fluctuations at $x_{2,\textit{GIP}}$ as the frequency contents of primitive variables are similar. Indeed, the PSDs are also very similar, except near the peak at $F\sim 0.25$ which is more pronounced for pressure and less for velocity.

Figure 8.

Modules of $\hat {T}^k_{\phi (x_1) w}$ for (a) velocity fluctuation performance sensors at the generalized inflection point and (b) wall-pressure fluctuation performance sensors.

Since the control strategy works by destructive interference, we need to make sure the signature of actuation on performance sensors is not too dissimilar to that from noise sources. Therefore, we compare in figure 9 the module of the transfer functions from the noise sources to those from the actuators, for sensors $\boldsymbol{z}_3$ (wall-pressure) and $\boldsymbol{z}_4$ (velocity at $x_2=x_{2,\textit{GIP}}$ ) at the end of the numerical domain. Figure 9(a,b) compares $\hat {T}^k_{z_4 u}$ with $\hat {T}^k_{z_4 w}$ : the main difference is that the signature of the second Mack mode is visible in the transfer from the actuators, while it is not in the transfer from the noise sources. Conversely in figure 9(c,d) comparing $\hat {T}^k_{z_3 u}$ with $\hat {T}^k_{z_3 w}$ , the signature of the first mode is not visible at the wall in the pressure signal from the actuators, while it is for the signal from the noise sources. These differences in the responses can be attributed to the fact that the disturbances generated by $\boldsymbol{u}$ do not take advantage of the full length of the instability domain contrary to the disturbances generated by $\boldsymbol{w}$ (see figure 3), leading to reduced amplification of the low frequency of the first mode and a relatively stronger contribution of the second mode. In addition, the spatial structure of the actuator forcing $b_u(x_3)$ favours low spanwise wavenumbers (in contrast to $b_w(x_3)$ , which corresponds to a Dirac distribution in the spanwise direction), further modifying the observed spectral response. Note that the differences in the responses from $\boldsymbol{w}$ and $\boldsymbol{u}$ have a detrimental effect for noise-cancellation by destructive interference. In future work, one may seek to optimize actuators in order to minimize these differences.

Figure 9.

Comparison of the modules of transfer functions from $\hat {u}^k$ and $\hat {w}^k$ with the sensors at the end of the domain $\hat {z}_3^k$ (wall-pressure) and $\hat {z}_4^k$ (velocity at $x_{2,\textit{GIP}}$ ): (a) $\hat {T}^k_{z_4 u}$ , (b) $\hat {T}^k_{z_4 w}$ , (c) $\hat {T}^k_{z_3 u}$ , (d) $\hat {T}^k_{z_3 w}$ . Note that $|k|=17$ corresponds to $(|k|\beta /n)\delta _0^*\simeq 0.92$ .

Figure 9. Long description

Panel A: A heat map showing the comparison of transfer function modules for wall-pressure sensors. The x-axis represents the variable F on a logarithmic scale ranging from 10^−2 to 10^0. The y-axis represents the variable (k beta/n) delta 0* ranging from −0.4 to 0.4. The color scale ranges from 0 to 340, with lighter colors indicating higher values. The heat map shows two distinct regions of high intensity. Panel B: A heat map showing the comparison of transfer function modules for wall-pressure sensors. The x-axis represents the variable F on a logarithmic scale ranging from 10^−2 to 10^0. The y-axis represents the variable (k beta/n) delta 0* ranging from −0.4 to 0.4. The color scale ranges from 0 to 91, with lighter colors indicating higher values. The heat map shows two distinct regions of high intensity. Panel C: A heat map showing the comparison of transfer function modules for velocity sensors. The x-axis represents the variable F on a logarithmic scale ranging from 10^−2 to 10^0. The y-axis represents the variable (k beta/n) delta 0* ranging from -0.4 to 0.4. The color scale ranges from 0 to 5500, with lighter colors indicating higher values. The heat map shows a single region of high intensity. Panel D: A heat map showing the comparison of transfer function modules for velocity sensors. The x-axis represents the variable F on a logarithmic scale ranging from 10^−2 to 10^0. The y-axis represents the variable (k beta/n) delta 0* ranging from −0.4 to 0.4. The color scale ranges from 0 to 38, with lighter colors indicating higher values. The heat map shows two distinct regions of high intensity.

For identification and control, we only consider a subset of Fourier modes $|k|\leqslant 5$ , capturing 90 % of the perturbation energy, as we shall see below. Figure 10 shows the evolution of $\|\hat {T}^{k}_{\phi (x_1)w}\|_{\infty }=\underset {F\in \mathbb{R}}{\sup }\ |\hat {T}^{k}_{\phi (x_1)w}(iF)|$ for $|k|\leqslant 8$ and $\phi (x_1)$ measuring either (a) velocity at $x_2=x_{2,\textit{GIP}}$ or (b) wall-pressure. After reaching a peak, the magnitude associated with a particular mode $k$ decreases for the highest modes. For the velocity sensors at $x_{2,\textit{GIP}}$ , spanwise modes that have a negligible energy contribution upstream of the domain are dominant downstream and the most energetic spanwise mode varies as a function of $x_1$ , contrary to the wall-pressure performance sensors, which are continuously dominated by $k=0$ . The number of modes $|k|\leqslant k_{90\,\%}$ necessary to capture $90\,\%$ of $\|T_{\phi (x_1)w}\|_2$ is depicted in figure 10(c,d). Given the selected position $\boldsymbol{u}$ of the actuators (see dashed vertical lines), we only consider modes $|k|\leqslant 5$ (see horizontal dotted line) for identification and synthesis because trying to control higher spanwise modes would have a negligible effect on the r.m.s. values of the performance sensors, which are located farther downstream. Note that even if $k_{90\,\%}=5$ , a greater number of sensors need to be chosen to avoid aliasing (see § 3.5). The choice $n=35$ is here more than sufficient. Of course, zero-padding in Fourier space is used to work with 35 actuators/sensors while controlling only 11 transverse modes.

Figure 10.

Evolution of (a,b) $\|\hat {T}^{k}_{\phi (x_1)w}\|_{\infty }$ for $|k|\leqslant 8$ and (c,d) number of spanwise modes $k_{90\,\%}$ necessary to capture at least $90\,\%$ of $\|T_{\phi (x_1)w}\|_{2}$ for $\phi (x_1)$ measuring (a,c) velocity fluctuations at $x_{2,\textit{GIP}}$ and (b,d) wall-pressure fluctuations. The vertical dotted line represents the streamwise position $x_{1,\boldsymbol{u}}$ of the actuators. The spanwise modes $|k|\leq 5$ (below the horizontal dotted line in c and d) are those used in the identification and synthesis step.

In this paragraph, we justify how the number of performance sensor rows was chosen, and how their longitudinal positions were selected. Of course, the set of performance sensors $\boldsymbol{z}_{\!j}$ must be representative of the dynamics to be controlled from the actuators to the end of the numerical domain. For the velocity sensors at $x_{2,\textit{GIP}}$ , as the frequencies $F$ contributing the most to the energy of a sensor close to the actuators are still non-negligible close to the end of the numerical domain used (see figures 3 and 8), minimizing the r.m.s. values of a single row $\boldsymbol{z}_4$ of velocity sensors downstream of the domain at $x_{1,\boldsymbol{z}_4}=1775\delta _0^*$ should also have a positive effect on the rows of sensors farther upstream. Choosing a row of sensors sufficiently far downstream also makes sense because the energy of the disturbances increases as they are convected, so the control must reduce consequently the energy downstream to avoid triggering nonlinearities which appear from a certain energy threshold: the farther downstream the disturbances are from the actuators, the more their energy must be reduced (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024). For the wall-pressure sensors, the frequencies contributing to the energy of the second Mack mode near the actuators have a negligible impact downstream of the domain (see figures 3 and 8). To cover the entire amplified bandwidth of the second Mack mode from the actuators to the end of the domain, it is necessary to use several rows of sensors. Indeed, if all the dominant frequencies from the actuators to the end of the domain are not taken into account in the synthesis step, a harmful waterbed effect may occur: some frequencies can be amplified by the controller whereas they could be predominant in some abscissa of the plate (see Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023). In practice, three rows of wall-pressure sensors $\boldsymbol{z}_1,\boldsymbol{z}_2,\boldsymbol{z}_3$ are used for the synthesis, respectively, at $(x_{1,\boldsymbol{z}_1},\, x_{1,\boldsymbol{z}_2},\, x_{1,\boldsymbol{z}_3})=(1022.6,1312.2,1775.0)\delta _0^*$ . These positions are close to the ones used in the 2-D studies of Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024) that allowed to obtain an efficient control of the second Mack mode in terms of disturbance amplitude reduction.

5.2.2. Estimation sensors

The estimation sensors $\boldsymbol{y}$ are wall-pressure fluctuation sensors. This choice is supported by the fact that both instability modes are observable by this type of sensor (see figure 8 b) and not only the oblique first mode as with the velocity sensors at $x_{2,\textit{GIP}}$ (see figure 8 a). In addition, unlike performance sensors $\boldsymbol{z}_{\!j}$ which are only used during synthesis and whose associated transfer functions could be modelled (e.g. by numerical calculations), estimation sensors $\boldsymbol{y}$ need to be physically implementable in realistic set-ups to compute the actuator signal $\boldsymbol{u}(t)$ from $\boldsymbol{K}$ . Wall-pressure sensors are indeed commonly used in hypersonic experimental set-ups (Lugrin et al. Reference Lugrin, Nicolas, Severac, Tobeli, Beneddine, Garnier, Esquieu and Bur2022). Only a feedback set-up where the estimation sensors are downstream of the actuators is employed in this 3-D study as a feedforward set-up where the estimation sensors are upstream of the actuators is unable to ensure performance robustness when free stream velocity variations are considered (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023) even with multimodel or gain scheduling methods (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024).

To control the high frequencies of the second Mack mode, the convection delay $\tau _{\boldsymbol{yu}}$ of the disturbances from $\boldsymbol{u}$ to $\boldsymbol{y}$ has to be small enough (Barbagallo et al. Reference Barbagallo, Dergham, Sipp, Schmid and Robinet2012; Juillet et al. Reference Juillet, Schmid and Huerre2013; Freire et al. Reference Freire, Cavalieri, Silvestre, Hanifi and Henningson2020). Considering that frequencies up to $F\sim 0.3$ must be controlled (see figure 3), the actuator/estimation sensor distance $V_{\infty }\tau _{\boldsymbol{yu}}$ must be less than $ V_{\infty }/f_c=2\pi \delta _0^*/F\sim 21\delta _0^*$ with $f_c$ the controllable bandwidth (Glad & Ljung Reference Glad and Ljung2000). For frequencies outside the bandwidth, the controller cannot attenuate their amplitude. Therefore, the greater the distance between the sensor and the actuator, the less effective the controller becomes in terms of disturbance amplitude reduction (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024). The estimation sensors are placed at a distance of $20.4\delta _0^*$ from the streamwise position of the actuators, i.e. $x_{1,\boldsymbol{y}}=887.6\delta _0^*$ . By examining $|\hat {T}^k_{yu}|$ in figure 11, we see that there is energy up to $(|k|\beta /n)\delta _0^*=0.486$ (i.e. $|k|=9$ ), so at least 18 sensors should be used to avoid aliasing (see § 3.5). Therefore, our choice $n=35$ is more than enough to avoid aliasing. Zero-padding in the Fourier domain is used since only modes up to $|k|=5$ will be controlled.

Figure 11.

Module of $\hat {T}^k_{yu}$ .

6.1. System identification

Our synthesis methods require low-dimensional state-space realizations for all SISO transfer functions, for $k=0,\ldots, 5$ . These are obtained in a data-driven fashion. More specifically, we apply the eigensystem realization algorithm (Juang & Pappa Reference Juang and Pappa1985) on impulse response data as in Belson et al. (Reference Belson, Semeraro, Rowley and Henningson2013). Impulse responses are obtained from inverse Fourier transform of frequency responses as in Sasaki et al. (Reference Sasaki, Morra, Fabbiane, Cavalieri, Hanifi and Henningson2018, Reference Sasaki, Morra, Cavalieri, Hanifi and Henningson2020) and Morra et al. (Reference Morra, Sasaki, Hanifi, Cavalieri and Henningson2020).

The frequency response of the optimal filter $\hat {T}^k_{z_{\kern-1pt j} y_w}$ is obtained using estimates of the (cross-)PSDs from stochastic input/output signals, using Welch’s approach (Welch Reference Welch1967). A Savitzky–Golay of order two (Savitzky & Golay Reference Savitzky and Golay1964) and low/high pass filters (Butterworth Reference Butterworth1930) are also applied on $\hat {T}^k_{z_{\kern-1pt j} y_w}$ to smooth the frequency spectrum and to focus only on the dominant frequencies of the instability modes, respectively. The spectral factor $\hat {W}^k_{yw}$ is build from the magnitude data $\sqrt {\hat {\varPhi }^k_y}$ (estimated using the Welch method) and the MATLAB function fitmagfrd. For the frequency responses of $\hat {T}^{k}_{z_{\kern-1pt j}u}$ and $\hat {T}^{k}_{yu}$ , they are calculated by Fourier transform of the responses obtained from an impulse of one of the actuator in the absence of noise sources. As discussed in § 3.4, we remove a common deadtime $\tau _{\boldsymbol{z}_{\!j}\boldsymbol{y_w}}$ (convection delay from $\boldsymbol{y_w}$ to $\boldsymbol{z}_{\!j}$ ) for the impulses from $\hat {y}_w^k$ and $\hat {u}^k$ to $\hat {z}_j^k$ prior to identification. This allows for a significant reduction in the order of the corresponding state-space realization.

The comparison of the ROMs (circles) and frequency/impulse data (solid lines) for $\hat {T}^k_{z_{\kern-1pt j} y_w}$ and $\hat {W}^k_{yw}$ used in the synthesis step is plotted in figure 12. To avoid overburdening the article, figures for $\hat {T}^{k}_{z_{\kern-1pt j}u}$ and $\hat {T}^{k}_{yu}$ are omitted. A good agreement is obtained and the ROMs capture most of the important dynamics. For a particular spanwise mode $k$ , the sum of the orders of each SISO ROM is $\sim 160$ . As we have to control modes up to $k=5$ , this would amount to work on a MIMO system potentially of order $\sim 960$ . Even though balanced truncation may be used to reduce the number of states, this illustrates the need to work on multiple decentralized SISO problems in the wavenumber space.

Figure 12.

Comparison of the ROMs used in the synthesis (circles) and frequency/impulse data (solid lines) for (a,b) $\hat {W}^{k}_{yw}$ and (c,d) $\hat {T}^{k}_{z_3y_w}$ . For the ROMs of $\hat {T}^{k}_{z_3y_w}$ , the time axis of the impulse responses are shifted by $\tau _{\boldsymbol{z}_{\!j} \boldsymbol{y_w}}V_{\infty }/\delta _0^*\simeq 892$ which corresponds to the suppression of unnecessary dead times.

6.2. Constrained minimization problem

The structure of the six independent controllers $\hat {K}^{k}$ is imposed in the following way: each controller $\hat {K}^{k}$ is searched in state-space form with a strictly proper structure (inducing a roll-off of $-20$ dB per decade at high frequencies allowing to be robust to high-frequency noisy estimation sensors) and a tridiagonal state matrix; each controller $\hat {K}^{k}$ is stable and stabilizing with an order equal to $6$ . These specifications are similar to those imposed in the structured mixed $H_2/H_{\infty }$ syntheses of Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024) for the 2-D control of the second Mack mode (so they were searching the ‘spanwise invariant’ controller $\hat {K}^{0}$ ).

For each controller $\hat {K}^{k}$ with the structure imposed above, the MATLAB function systune (Apkarian et al. Reference Apkarian, Noll and Rondepierre2010, Reference Apkarian, Gahinet and Buhr2014) solves the following mixed $H_2$ / $H_{\infty }$ problem:

(6.1) \begin{align} \begin{split} \text{Step 1.}\ &\text{minimize}\ \mathrm{max}(\|\hat {T}^{k,c}_{z_4w}\|_{2},\ \|\hat {T}^{k,c}_{\boldsymbol{z}_{1:3}w}\|_{2})\\ &\text{subject to}\ \|\hat {W}_{\hat {S}^{k}}\hat {S}^{k}\|_{\infty }\lt 1 \text{ and } \|\hat {W}_{\hat {K}^k\hat {S}^{k}}\hat {K}^k\hat {S}^{k}\|_{\infty }\lt 1,\\ \\ \text{Step 2.}\ &\text{minimize}\ \|\hat {K}^k\hat {S}^{k}\|_2\\ &\text{subject to}\ \|\hat {W}_{\hat {S}^{k}}\hat {S}^{k}\|_{\infty }\lt 1,\ \|\hat {T}^{k,c}_{z_4w}\|_{2}\lt 1.01\|\hat {T}^{k,c}_{z_4w}\|^{\text{Step1}}_{2}\\ &\text{and}\ \|\hat {T}^{k,c}_{\boldsymbol{z}_{1:3}w}\|_{2}\lt 1.01\|\hat {T}^{k,c}_{\boldsymbol{z}_{1:3}w}\|^{\text{Step1}}_{2}\ \end{split} \end{align}

with $\boldsymbol{z}_{1:3}=[z_1,z_2,z_3]^T$ . Hence, for a spanwise mode $k$ , $\hat {T}^{k,c}_{\boldsymbol{z}_{1:3}w}$ is a single-input–multiple-output (one input and three outputs) transfer function. This two step CMP is the transcription of the fluidic specifications established in § 4 and is explained in the following.

In step 1 of the CMP, the minimization of $\|\hat {T}^{k,c}_{z_4w}\|_{2}$ and $\|\hat {T}^{k,c}_{\boldsymbol{z}_{1:3}w}\|_{2}$ allows the reduction of r.m.s. values of both velocity sensors at the generalized inflection point and wall-pressure fluctuation sensors. To be able to compare these two types of sensors in the ‘ $\text{minimize}\ \mathrm{max}$ ’ formulation, their $H_2$ norms are normalized by the uncontrolled values (approximated by the six spanwise modes $k$ used in the synthesis) $\|\boldsymbol{T_{z_4w}}\|_{2}$ and $\|\boldsymbol{T_{z_{1:3}w}}\|_{2}$ , respectively. The results are therefore weighted by the importance that a spanwise mode has in the total energy. The $H_{\infty }$ constraint on $\hat {W}_{\hat {S}^{k}}\hat {S}^{k}$ maintains adequate stability margins (see § 4.2). The constant frequency template $\hat {W}_{\hat {S}^{k}}=0.5$ , implying $\|\hat {S}^{k}\|_{\infty }\lt 2$ , is used for all spanwise modes; this template provided substantial stability margins despite variations of $\pm 5\,\%$ of the free stream velocity in the 2-D studies of Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023, Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024) (i.e. for the case $k=0$ ). The frequency template $|1/\hat {W}_{\hat {K}^k\hat {S}^{k}}|=80\ \mathrm{dB}$ linked to the $H_{\infty }$ constraint on $\hat {W}_{\hat {K}^k\hat {S}^{k}}\hat {K}^k\hat {S}^{k}$ is just chosen as a very large constant so as not to constrain too much the actuator activity while rejecting ill-conditioned controller. In summary, this first algorithmic step returns controller $\hat {K}^{k}$ cancelling as much as possible velocity and pressure disturbances while ensuring important stability margins in the control bandwidth.

In step 2 of the CMP, a minimization of $\|\hat {K}^k\hat {S}^{k}\|_2$ is realized under the constraint of recovering similar results to step 1 in terms of pressure/velocity disturbance rejection. The same minimal stability margins than in step 1 are required through the frequency template $W_{\hat {S}^{k}}=0.5$ . The minimization of $\|\hat {K}^k\hat {S}^{k}\|_2$ serves a dual purpose: first, ensuring robust performance by desensitizing the controller to white noise corrupting the estimations sensors; second, by reducing the $H_2$ norm, one can also expect a reduction in the $H_\infty$ norm thereby contributing to robust stability objectives for additive uncertainties, particularly outside the bandwidth. This minimization complements the strictly proper structure of $\hat {K}^{k}$ which brings natural performance and stability robustness in high frequencies. This two-step resolution is explained by the fact that it is difficult to find weighting functions to compare $\|\hat {T}^{k,c}_{z_{\kern-1pt j}w}\|_{2}$ with $\|\hat {K}^k\hat {S}^{k}\|_2$ in a single minimization problem. The optimal results of the step 1 in terms of disturbance rejection (i.e. $\|\hat {T}^{k,c}_{z_4w}\|^{{Step1}}_{2}$ and $\|\hat {T}^{k,c}_{\boldsymbol{z}_{1:3}w}\|^{{Step1}}_{2}$ ) are raised by a factor 1.01 in step 2 in order to obtain several controllers that respect these constraints before the algorithm starts the minimization procedure of $\|\hat {K}^k\hat {S}^{k}\|_2$ . In summary, the second algorithmic step is a trivial way to obtain controllers $\hat {K}^{k}$ reducing as much as possible amplitude of velocity/pressure disturbance, while ensuring desensitization of the loop to white noise corrupting the estimation sensors and robust stability in and outside the bandwidth. In other words, each controller $\hat {K}^k$ is designed to produce a control action in phase opposition to perturbations at its associated spanwise wavenumber $k$ (Sasaki et al. Reference Sasaki, Morra, Fabbiane, Cavalieri, Hanifi and Henningson2018, Reference Sasaki, Morra, Cavalieri, Hanifi and Henningson2020; Morra et al. Reference Morra, Sasaki, Hanifi, Cavalieri and Henningson2020). The $H_2/H_\infty$ structured feedback and decentralized design do not alter this principle but rather enforce deviations from ideal wave-cancellation to ensure feasibility, stability and robustness.

In steps 1 and 2, respectively, a dozen and a hundred random controller initializations are, respectively, performed to deal with the non-convex nature of the optimization problems. With the non-smooth algorithm employed and a parallel computation on 12 central processing unit cores (Intel(R) Xeon(R) CPU E5–2650 v3 @ 2.30 GHz), the six controllers $\hat {K}^{k}$ at the end of step 2 are obtained in approximately 2 hr (so around 20 min per controller $\hat {K}^{k}$ ), which seems very reasonable for the control of a 3-D configuration.

7.1. Linear regime

The results of the controlled system in a linear regime are analysed in this section, first on the nominal full-system (i.e. after implementation of the total 66th-order controller $\boldsymbol{K}$ in the CFD solver FastS), then in off-design operating conditions. The analysis of the results on the ROMs is provided in Appendix C.

7.1.1. On-design operating conditions

The frequency spectra of the spanwise modes for the performance sensors $\boldsymbol{z}_4$ and $\boldsymbol{z}_3$ after an impulse of the exogenous noise $w^0$ (at the centre spanwise position) are represented in figure 13. The control action reduces the amplitude of the dominant frequencies and spanwise modes of both the first and second instability modes. The amplitude of the dominant spanwise modes are reduced at each abscissa of the plate and control does not modify modes higher than $k=5$ (see figure 14) because we focused only on smaller modes in our synthesis ( $\hat {K}^k=0$ for $k\geq 6$ ). The quantity $\underset {x_1\gt x_{1,\boldsymbol{u}},\ k}{\max } \ln {\|\hat {T}^{k}_{\phi (x_1)w}\|_{\infty }}-\underset {x\gt x_{1,\boldsymbol{u}},\ k}{\max } \ln {\|\hat {T}^{k,c}_{\phi (x_1)w}\|_{\infty }}$ from the streamwise position of the actuators $x_{1,\boldsymbol{u}}$ until the end of the domain is, respectively, equal to $\sim 0.9$ and $\sim 0.85$ for the velocity and pressure performance sensors. From a local linear stability point of view, this quantity is akin to $\underset {x_1\gt x_{1,\boldsymbol{u}}}{\max }\ \tilde {N} - \underset {x_1\gt x_{1,\boldsymbol{u}}}{\max }\ \tilde {N}^c$ with $\tilde {N}$ and $\tilde {N}^c$ the $N$ -factor envelope of the uncontrolled and controlled cases, respectively. Performance improvements would mainly require modifying the actuator spatial support as well as the forced components to better match the instability structure induced by exogenous noise, thereby enhancing wave cancellation (Sasaki et al. Reference Sasaki, Morra, Cavalieri, Hanifi and Henningson2020), which is beyond the scope of the present study. By contrast, in the present configuration, increasing the number of actuators or reducing the streamwise distance between $\boldsymbol{u}$ and $\boldsymbol{y}$ is not expected to yield further gains, as these aspects are already near optimal (see § 3.5–5.2.2).

Figure 13.

Comparison of the frequency spectra of the spanwise modes of the performance sensors (a,b) $\boldsymbol{z}_4$ and (c,d) $\boldsymbol{z}_3$ for the (a,c) uncontrolled and (b,d) controlled case.

Figure 13. Long description

Panel A: A heat map showing the frequency spectra of spanwise modes for performance sensors in the uncontrolled case. The x-axis represents the frequency (F) on a logarithmic scale, and the y-axis represents the normalized wavenumber (kβ/nδ0*). The color scale ranges from black to yellow, indicating increasing values of the magnitude of the spanwise mode (|T̂zw|δ0*). Two distinct regions of high intensity are visible, one above and one below the center line. Panel B: A heat map showing the frequency spectra of spanwise modes for performance sensors in the controlled case. The axes and color scale are the same as in Panel A. The regions of high intensity are less pronounced compared to Panel A. Panel C: A heat map showing the frequency spectra of spanwise modes for another set of performance sensors in the uncontrolled case. The x-axis represents the frequency (F) on a logarithmic scale, and the y-axis represents the normalized wavenumber (kβ/nδ0*). The color scale ranges from black to yellow, indicating increasing values of the magnitude of the spanwise mode (|T̂zw|δ0*). A central region of high intensity is visible, with additional smaller regions above and below. Panel D: A heat map showing the frequency spectra of spanwise modes for another set of performance sensors in the controlled case. The axes and color scale are the same as in Panel C. The central region of high intensity is less pronounced compared to Panel C.

Figure 14.

Evolution of $\|\hat {T}^{k}_{\phi (x_1)w}\|_{\infty }$ for $|k|\leqslant 8$ for $\phi (x_1)$ measuring (a) velocity fluctuation at $x_2=x_{2,\textit{GIP}}$ , (b) wall-pressure fluctuation. The vertical dotted line represents the streamwise position of the actuators. The norms $\|\hat {T}^{k}_{\phi (x_1)w}\|_{\infty }$ in (a) and (b) are normalized by $\|\hat {T}^{4}_{v'_{x_{1}}(x_{1,\boldsymbol{u}})w}\|_{\infty }$ and $\|\hat {T}^{0}_{p'(x_{1,\boldsymbol{u}})w}\|_{\infty }$ , respectively, to express the potential gain in terms of the $N$ factor.

Beyond these results for the performance sensors, the performance is also evaluated by considering the r.m.s. values averaged in the spanwise direction when the 3-D boundary layer is excited simultaneously by the 35 uncorrelated white-noise signals $w^i$ .

For the r.m.s. velocity field (see figure 15 a), whose high values are located around the generalized inflection point due to the predominance of the first mode, the control reduces the amplitude of the perturbations. For the r.m.s. pressure field (see figure 15 b), the high values are located both close to the generalized inflection point and close to the wall due to the first and second mode, respectively. The reactive law controls simultaneously these two modes but an acoustic wave is generated at the location of the actuators. We will see in § 7.2 that this acoustic wave does not prevent delaying transition to turbulence so the resulting controller is deemed sufficiently satisfactory and performance robustness is addressed next.

Figure 15.

Contours of (a) $v'_{x_1,\textit{rms}}$ and (b) $p'_{\textit{rms}}$ averaged in the spanwise direction for the (i) uncontrolled and (ii) controlled cases. The white solid and dashed lines, respectively, represent the boundary layer thickness $\delta$ and the generalized inflection point position $x_{2,\textit{GIP}}$ .

Figure 15. Long description

Panel A: Two heat maps showing the distribution of a variable related to flow dynamics. The x-axis is labeled x1/delta0* and ranges from 0 to 1750. The y-axis is labeled x2/delta0* and ranges from 0 to 20. The color scale on the right indicates the magnitude of the variable, with colors ranging from blue (low values) to red (high values). The white solid and dashed lines represent the boundary layer thickness and the generalized inflection point position, respectively. The left heat map shows the uncontrolled case, while the right heat map shows the controlled case. Panel B: Two heat maps showing the distribution of another variable related to flow dynamics. The x-axis is labeled x1/delta0* and ranges from 0 to 1750. The y-axis is labeled x2/delta0* and ranges from 0 to 20. The color scale on the right indicates the magnitude of the variable, with colors ranging from blue (low values) to red (high values). The white solid and dashed lines represent the boundary layer thickness and the generalized inflection point position, respectively. The left heat map shows the uncontrolled case, while the right heat map shows the controlled case.

7.1.2. Off-design operating conditions

In this section, performance robustness to off-design operating conditions is assessed by considering free stream velocity variations. The free stream velocity variations are the most problematic ones for noise-amplifier flows because they change the convective delays involving phase variations in the transfer functions, which may convert destructive interference into constructive ones (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2023). Moreover, Nibourel et al. (Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024) showed that with free stream velocity variations of only $5\,\%$ , the magnitudes of the dominant frequencies of the 2-D second Mack mode can vary by up to $80\,\%$ compared with the nominal case. Therefore, we will consider in this 3-D study free stream velocity variations of $2.5\,\%$ , which already represent a significant challenge for the system’s stability and performance.

In addition to modifying the aerodynamic delays, the variations of $V_{\infty }$ imply variations of $\pm 2.5\,\%$ of $M_{\infty }$ and $Re_{x_1}$ , since $\rho _{\infty }$ and $T_{\infty }$ are kept constant. New boundary layer profiles are used as inlet conditions and the previous controller $\boldsymbol{K}$ is directly implemented on the full 3-D system which is still excited by the 35 uncorrelated white noise sources $w^i$ with the same amplitudes as in the $M_{\infty }=4.5$ linearized DNS. It appears that the feedback control law is still stable, thanks to the important stability margins imposed in the synthesis (see § 6.2). The maxima along the wall-normal direction of $\lt v'_{x_1,\textit{rms}}\gt _{x_3}$ and $\lt p'_{\textit{rms}}\gt _{x_3}$ are plotted in figure 16. This figure shows that free stream velocity variations result in more significant amplitude changes for pressure instabilities compared with velocity instabilities, even in the absence of control. Concerning the controlled cases, the disturbance amplitudes are still reduced in off-design conditions thanks to the feedback set-up and its natural ability to maintain performance robustness over wide phase variations (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024). Note that the higher the free stream velocity, the lower the intensity of the acoustic wave generated by the control. For this range of free stream variations, our simple feedback law is considered to be sufficiently robust in performance so that it is not necessary to further make robust some $\hat {K}^{k}$ (e.g. by multimodel synthesis or gain scheduling (Nibourel et al. Reference Nibourel, Leclercq, Demourant, Garnier and Sipp2024)).

Figure 16.

Evolution of (a) $\max _{x_2} \lt v'_{x_1,\textit{rms}}\gt _{x_3}$ and (b) $\max _{x_2} \lt p'_{\textit{rms}}\gt _{x_3}$ after a variation of $V_{\infty }$ of $\pm 2.5\,\%$ . The nominal cases are in solid lines; the $+ 2.5\,\%$ and $- 2.5\,\%$ cases are, respectively, in dotted and dashed lines. The vertical magenta and black dotted lines represent, respectively, the streamwise positions of the actuators (with the sensors $\boldsymbol{y}$ nearby) and the performance sensors $\boldsymbol{z}_{\!j}$ used for synthesis.

Performance robustness with respect to noise on the estimation sensors and with respect to the amplitude level of nonlinearities has also been evaluated but are reported in Appendix D and Appendix E, respectively. As a result, the control law still achieves high level of performance in terms of amplitude reduction despite a signal-to-noise ratio (SNR) of 0.53 (respectively, 2) in the case of high-frequency coloured noise (respectively, white noise) corrupting the estimation sensors $\boldsymbol{y}$ (see Appendix D) and despite exogenous noises leading to nonlinearities and velocity fluctuations up to $2\,\%$ of $V_\infty$ at the streamwise position of the estimation sensors (see Appendix E). Robustness to measurement noise is attributable to the CMPs (see § 6.2), whereas robustness to nonlinearities is just an observed outcome.

One could also assess the robustness of the controller to spanwise perturbations of the base flow but this is beyond the scope of our study. Note that the presence of a spanwise velocity component $v_{x_3,\infty }$ would not break the spanwise invariance of the present infinite-spanwise configuration. It would therefore still be possible to design a controller for this specific case by decomposing the MIMO system into independent SISO systems with the following specifications: firstly, the symmetry relation $\hat {T}^{-k}=\hat {T}^{k}$ would no longer hold, requiring independent synthesis of the positive and negative wavenumbers; secondly, the impulse responses between complex inputs and outputs would become also complex (contrary to Appendix B) leading to design each $\hat {K}$ as a $2\times 2$ MIMO controller (to treat the real and imaginary parts).

7.2. Transition to turbulence

Finally, the ability of the feedback law to delay transition to turbulence of a supersonic boundary layer is assessed. The length of the previous 3-D computational domain is doubled in the streamwise direction in order to obtain transition in the new domain while remaining weakly nonlinear from the streamwise position of the actuators to the end of the previous domain used in linearized DNS. By maintaining weak nonlinearity over this length, we mitigate both pressure and velocity disturbances up to at least $x_1=1910.2 \delta _0^*$ ; otherwise, the reduction of pressure disturbances risks dropping if the nonlinear effects are too significant from the actuator location (see Appendix E). For this new computational domain used for transition to turbulence, the boundary conditions and the wall-normal and spanwise lengths remain unchanged. The new streamwise length is $x_1 \in [0., 3820.4 \delta _0^*]$ ; a sponge zone is added downstream of this domain, with the stretching of 30 cells in the streamwise direction and $L_{\textit{sponge}}=91.9\delta _0^*$ . A resolution of $6400\times 220\times 176$ is adopted (not including the sponge zone) and the spacings in the $x_1$ , $x_2$ , $x_3$ directions are the same as in the previous linearized DNS. This resolution ensures that $\varDelta x_1^+\lt 4$ , $\varDelta x_3^+\lt 3.5$ and $\varDelta x^+_{2,w\textit{all}}\lt 1$ in the entire domain (dimensionless grid size is defined as $\varDelta x^+_i=\sqrt {\tau _{w\textit{all}}\rho _{w\textit{all}}}\varDelta x_i/\mu _{w\textit{all}}$ with $\varDelta x_i$ corresponding to the grid spacing in direction $i$ of the Cartesian coordinate system).

The DNS of transition to turbulence is performed with exogenous noise sources with $\mathrm{Var}[w^i]=4$ . For comparison, the variance used in the linearized DNS was $\mathrm{Var}[w_{\textit{linear}}^i]=0.0016$ . With this noise intensity, the nonlinear effects are no longer negligible from $x_{1,{Nl}}=1284.6 \delta _0 ^*$ (see figure 17 a, first vertical black dotted line); this position is defined in this study when the relative difference between $\max _{x_2} \lt v'_{x_1,\textit{rms}}\gt _{x_3}$ of the linearized and nonlinear DNS is greater than $5\,\%$ . Hence, the nonlinear effects are low at the streamwise position of the actuators $x_{1,\boldsymbol{u}}$ . At $x_1=1910.2\delta _0^*$ (i.e. the end of the previous domain used for linearized DNS), the results differ by approximately $30\,\%$ . In the uncontrolled case, the start of transition to turbulence, defined by $\arg _{x_1} \min C_{\!f}$ (with $C_{\!f}= {\lt \tau _{w\textit{all}}\gt _{x_3,t}}/{0.5\rho _{\infty }V_{\infty }^2}$ ), is at $x_{1,{Tr}}=2736.0\delta _0^*$ (see figure 17 b, second vertical black dotted line). The dotted curves that represent the skin friction of a laminar and a fully turbulent flat-plate boundary layer are also presented in this figure: the laminar curve comes from the DNS without exogenous noise sources whereas the turbulent one comes from the van Driest II correlation (Franko & Lele Reference Franko and Lele2013). In contrast to studies where the boundary layer is excited by a single primary mode, leading to a single common breakdown scenario, transition here occurs in the presence of multiple primary instability waves. One possible mechanism that could be considered is nonlinear interactions between the first and second modes, as described by Guo et al. (Reference Guo, Hao and Wen2023). Yet, this remains an open question in the present work because no detailed analysis of the transition scenario has been performed.

Figure 17.

For exogenous noise sources with $\mathrm{Var}[w^i]=4$ , evolution of (a) $\max _{x_2} \lt v'_{x_1,\textit{rms}}\gt _{x_3}$ and (b) $C_{\!f}$ . The linear case (dashed line) in (a) is rescaled by multiplying its results by $\sqrt {\mathrm{Var}[w^i]/\mathrm{Var}[w_{\textit{linear}}^i]}$ for comparison. The magenta vertical line represents the streamwise position of the actuators. The first and second black vertical dotted lines represent the streamwise position from which the nonlinear effects are no longer negligible and the beginning of transition to turbulence, respectively. The black dashed curves in (b) represent the evolution of the skin friction coefficient for the fully laminar case (lower curve) and the turbulent case (upper curve).

The feedback control drastically reduces the maximum amplitude of disturbances all along the new numerical domain (see figure 18). At the end of the domain for $x_1=3820.4\delta _0^*$ , the quantity $\max _{x_2}\ \lt ._{\textit{rms}}\gt _{x_3}$ is reduced by approximately $61\,\%$ and $27\,\%$ for $v_{x_1}'$ and $p'$ , respectively. Note that the intensity of the acoustic wave generated by the control near $x_{\boldsymbol{u}}$ is negligible compared with the values of the pressure fluctuations farther downstream of the domain and this acoustic wave does not prevent mitigation of the disturbance growth farther downstream (see figure 18 b). The profiles of $\lt v'_{x_1,\textit{rms}}\gt _{x_3}$ at different streamwise locations are plotted in figure 19. In the weakly nonlinear region (see figure 19 a), the controlled r.m.s. profile takes lower values than the uncontrolled r.m.s. profile on the whole wall-normal direction thanks to the control of both the first and the second instability modes. Even at a more downstream position where the nonlinear effects are much more important (see figure 19 b), the maxima of the profiles are still located near the generalized inflection point as long as the boundary layer remains laminar. At the beginning of transition to turbulence (see figure 19 c), the uncontrolled profile $\lt v'_{x_1,\textit{rms}}\gt _{x_3}$ starts to drop towards the wall while the maximum of the controlled profile is still near $x_{2,\textit{GIP}}$ . Finally, at $x_1=3300.5\delta _0^*$ (see figure 19 d), the maximum of the uncontrolled profile is located close to the wall while it remains farther above in the controlled profile, with much lower r.m.s. values. The fact that the collapse of the velocity profile is delayed in the controlled case is due to the delay of the breakdown stage thanks to the amplitude reduction of the modes up to $k=5$ and so weaker nonlinear interactions.

Figure 18.

Evolution of $\max _{x_2} \lt ._{\textit{rms}}\gt _{x_3}$ for (a) $v_{x_1}'$ and (b) $p'$ for the transition case. For vertical lines, same caption as in figure 17.

Figure 19.

Profiles of $\lt v'_{x_1,\textit{rms}}\gt _{x_3}$ at (a) $x_1=x_{1,{Nl}}$ , (b) $x_1=1910.2\delta _0^*$ , (c) $x_1=x_{1,{Tr}}$ and (d) $x_1=3300.5\delta _0^*$ .

The instantaneous contours of streamwise velocity and numerical schlieren visualization are shown in figure 20. The uncontrolled flow presents chaotic structures while the controlled flow appears smoother. The emergence of turbulent structures is clearly delayed thanks to the control action. The laminar region is extended in the controlled case and transition to turbulence is delayed by $\varDelta x_{1,{Tr}}=564.4\delta _0 ^*$ (see figure 21); this distance represents $\sim 20 \,\%$ of the distance from the leading edge of the flat plate to abscissa of transition in the uncontrolled case (i.e. $19\delta _0^*+2736\delta _0^*$ ). The delay in terms of $Re_{x_1}$ is $\varDelta Re_{x_1}=1.2\times 10^6$ .

Figure 20.

Instantaneous contours of streamwise velocity ( $x_1x_3$ -plane at $x_2=1.99\delta _0^*$ ) and numerical schlieren visualization ( $x_1x_2$ -plane at $x_3=0$ ) between $2939.7\delta _0^*\lt x_1\lt 3820.4\delta _0^*$ for the (a) uncontrolled and (b) controlled cases.

Figure 20. Long description

Panel A: A heat map representing the uncontrolled case. The x-axis and y-axis are labeled with coordinates x1 and x3, respectively. The color scale ranges from 0 to 0.10 for the magnitude of the gradient of density and from 0.10 to 0.40 for the normalized streamwise velocity. The heat map shows a gradient of colors from blue to red, indicating varying velocities and density gradients. The highest values are concentrated in the center, with a noticeable transition from blue to red. Panel B: A heat map representing the controlled case. The x-axis and y-axis are labeled with coordinates x1 and x3, respectively. The color scale ranges from 0 to 0.10 for the magnitude of the gradient of density and from 0.10 to 0.40 for the normalized streamwise velocity. The heat map shows a more uniform distribution of colors, predominantly blue, indicating lower velocities and density gradients compared to the uncontrolled case.

Figure 21.

Evolution of the friction coefficient $C_{\!f}$ for the transition to turbulence case. The third black vertical dotted line represents the beginning of transition to turbulence for the controlled case. For other vertical lines, same caption as in figure 17.