3.1. Overview of the Wiener–Hopf technique

The Wiener–Hopf technique (Noble Reference Noble1958; Martini et al. Reference Martini, Jung, Cavalieri, Jordan and Towne2022) is applied for the jet control problem, and its performance is compared to that of a wave-cancelling approach (Sasaki et al. Reference Sasaki, Morra, Fabbiane, Cavalieri, Hanifi and Henningson2018; Brito et al. Reference Brito, Morra, Cavalieri, Araújo, Henningson and Hanifi2021; Maia et al. Reference Maia, Jordan, Cavalieri, Martini, Sasaki and Silvestre2021). The same control parameters and transfer functions are used for both methods. In this subsection, the Wiener–Hopf technique is presented briefly; further details about this method and its application to control problems can be found in Martini et al. (Reference Martini, Jung, Cavalieri, Jordan and Towne2022). Wiener–Hopf equations are typically associated with problems where a half-domain condition needs to be satisfied, such as a half-convolution problem (Noble Reference Noble1958; Martini et al. Reference Martini, Jung, Cavalieri, Jordan and Towne2022)

(3.1)\begin{equation} \int_{0}^{\infty} {{S}}(t-\tau)\,{{W}}_+(\tau) \,{\rm d}\tau= {T}(t), \quad t > 0, \end{equation}

where ${S}$ and ${T}$ are known functions, and ${{W}}$ is a unknown function (kernel). Since we are dealing with only one reference sensor and one actuator, this is a single-input/single-output control system, thus the variables considered here become scalars. This equation can be extended to $t < 0$ by writing Zich & Daniele (Reference Zich and Daniele2014)

(3.2)\begin{equation} \int^{\infty}_ {-\infty} {S}(t-\tau)\,{W_+}(\tau)\,{k}(\tau)\,{\rm d}\tau = {T}(t)\,{k}(t) + {W_-}(t)\,{k}({-}t), \quad {-\infty}< t<\infty, \end{equation}

where ${k}(t)$ is the Heaviside step function, and ${W_-}(t)$ is an additional unknown function, with the property ${W_-}(t>0)=0$. Taking a Fourier transform leads to

(3.3)\begin{equation} {\hat{S}}(\omega)\,{\hat{W}}_+(\omega) = {\hat{W}}_-(\omega)+{\hat{T}(\omega)}, \end{equation}

where $+$ and $-$ subscripts indicate that the function is analytical in the upper and lower complex half-planes, respectively. This is ensured for ${{W}}_+$ and ${{W}}_-$ as these functions are defined to be zero for negative and positive times, respectively. We may also refer to ${{W}}_+$ and ${{W}}_-$ as causal and anti-causal functions, respectively; these definitions carry over to all $+$ and $-$ functions.

Solution of such a problem can be achieved with multiplicative and additive factorization:

(3.4)$$\begin{gather} {\hat{S}}(\omega)= {\hat{S}_-}(\omega)\,{\hat{S}_+}(\omega), \end{gather}$$

(3.5)$$\begin{gather}{\hat{S}}_-^{{-}1}(\omega)\,{\hat{T}}(\omega) = ({\hat{S}}_-^{{-}1}(\omega)\, {\hat{T}}(\omega))_+{+} ({\hat{S}}_-^{{-}1}(\omega)\,{\hat{T}}(\omega))_-. \end{gather}$$

Here, the multiplicative factorization of ${\hat {S}}(\omega )$ is solved numerically, following Daniele & Lombardi (Reference Daniele and Lombardi2007). With these two factorizations, the solution of (3.1) is given by

(3.6)$$\begin{gather} {\hat{W}}_+(\omega) = {\hat{S}}_+^{{-}1}(\omega)\,({\hat{S}}_-^{{-}1}(\omega)\, {\hat{T}}(\omega))_+, \end{gather}$$

(3.7)$$\begin{gather}{\hat{W}}_-(\omega) = {\hat{S}}_-(\omega)\,({\hat{S}}_-^{{-}1}(\omega)\,{\hat{T}}(\omega))_-, \end{gather}$$

where ${\hat {W}}_+$, which is regular in the upper half-complex plane, is the term related to causal control.

3.2. Control law design based on the Wiener–Hopf approach

The Wiener–Hopf based control is considered for a linear dynamical system. An optimal control for such system can be obtained by means of a quadratic cost functional that expresses the control cost and the cost of state deviation from the original condition, with this functional minimized with respect to the control kernel. For simplicity, we first consider a case without feedback contamination i.e. the effect of $u$ on $y$ is neglected. Causality can be enforced by including Lagrange multipliers $\varLambda _-$ to a quadratic cost functional as

(3.8)\begin{equation} J = \int^{\infty}_ {-\infty}(\langle{{{Q}{z^{*}}}{{z}}} + {Ru_+^{*}}{u_+} \rangle )\, {\rm d}t + \int_{-\infty}^{\infty}[{{\varGamma_+}}(t)\,{\varLambda_-}(t)]\,{\rm d}t + \int_{-\infty}^{\infty}[{{\varGamma_+^*}}(t)\,{\varLambda_-^*}(t)]\,{\rm d}t, \end{equation}

where the $*$ superscript denotes the complex conjugate, ${z}$ is the target for the control problem, $Q$ is a weighting of the state deviation from the original condition, $R$ is a control penalization, ${\varLambda _-}$ is a Lagrange multiplier, $\varGamma$ is the control kernel, and ${u}$ is the actuation signal. As $\varLambda$ is anti-causal, the last two integrals in (3.8) force $\varGamma$ to be causal. A control law $\varGamma$ relates flow measurements $y$ to actuations $u$. In the frequency domain, this relation reads

(3.9)\begin{equation} {\hat{u}}(\omega) = {\hat{\varGamma}}(\omega)\,{\hat{y}}(\omega). \end{equation}

We also have that ${z} = {z_{un}} + {G_{uz}}*{u} = {z_{un}} + {G_{uz}}*{\varGamma *y}$, i.e. the output is a linear combination of the uncontrolled signal and the actuation response, where $z_{un}$ is the uncontrolled signal at the target location, ${G_{uz}}$ is the transfer function from the control signal ${u}$ to the output ${z}$, and $*$ is the convolution operator. For the optimal solutions, (3.8) is minimized with respect to $\varGamma ^*$, yielding the following Wiener–Hopf equation (we refer the reader to Martini et al. Reference Martini, Jung, Cavalieri, Jordan and Towne2022 for details):

(3.10)\begin{equation} {\hat{H}_l}{\hat{\varGamma}}_+{\hat{S}_{yy}} + {\hat{\varLambda}}_-{=} {\hat{H}_r}{\hat{S}_{yz}}, \end{equation}

where

(3.11)$$\begin{gather} {\hat{H}_l}(\omega) = \hat{G}_{uz}^{*}(\omega)\,{Q}\,{\hat{G}_{uz}}(\omega)+{R}, \end{gather}$$

(3.12)$$\begin{gather}{\hat{H}_r}(\omega) ={-}\hat{G}_{uz}^{*}(\omega)\,{Q}. \end{gather}$$

Here, ${\hat {S}_{yy}}(\omega )$ is the power spectral density (PSD) of the input signal ${y}$, and ${\hat {S}_{yz}}(\omega )$ is the cross-spectral density (CSD) between the observation signal ${y}$ and the target signal ${z}$ without actuation. Applying the factorization presented earlier, the following expression is obtained for ${\hat {\varGamma }}_+$:

(3.13)\begin{equation} {\hat{\varGamma}}_+{=} {\hat{S}_{yy\,+}^{{-}1}}({\hat{S}_{yy \,-}^{{-}1}} {\hat{S}_{yz}} {\hat{H}_{r}} {\hat{H}_{l\,-}^{{-}1}})_+ {\hat{H}_{l\,+}^{{-}1}}. \end{equation}

The expression above is thus used for obtaining the optimal causal control law under the Wiener–Hopf formalism (Noble Reference Noble1958; Martinelli Reference Martinelli2009; Martini et al. Reference Martini, Jung, Cavalieri, Jordan and Towne2022).

In the present system, there is a feedback contamination of the sensors $y$ by the actuators $u$ due to their proximity in the experimental set-up. To account for this, and avoid appearance of a Larson effect (Sanfilippo & Valle Reference Sanfilippo and Valle2013), the Wiener–Hopf kernel is modified as (Martini et al. Reference Martini, Jung, Cavalieri, Jordan and Towne2022)

(3.14)\begin{equation} {\hat{\varGamma'}}_+{=} ({{I}}+{\hat{\varGamma}}{\hat{G}_{uy}})^{{-}1}\hat{\varGamma}, \end{equation}

where ${\hat {G}_{uy}}$ is the transfer function between the actuation signal and the microphone readings.

In the time domain, the actuation shown in (3.9) is given by

(3.15)\begin{equation} {u}(t) = \int^{\infty}_{-\infty} {\varGamma}(\tau)\,{y}(t-\tau)\,{\rm d}\tau, \end{equation}

such that with $\varGamma$ constructed with the Wiener–Hopf approach detailed above, we have $\varGamma (\tau <0)=0$, by construction, i.e. actuations are a function only of previous measurements, thus the control law is causal. This is, however, not the case if causality is not enforced, which requires the integral to be truncated for $\tau >0$ when used for real-time applications. In the following subsections, causal and non-causal kernels will be shown.

3.3. Control law design based on the IFFC method

In the wave-cancelling approach, here also referred to as the IFFC method, the strategy for control involves a direct superposition of the predicted uncontrolled disturbance at the objective position with the actuation. The IFFC control law can be obtained by minimizing a quadratic functional cost, similar to what is shown in (3.8), but without the constraints imposed to obtain a causal kernel, i.e. without the Lagrange multiplier terms, which yields the equation (Sasaki et al. Reference Sasaki, Morra, Fabbiane, Cavalieri, Hanifi and Henningson2018; Brito et al. Reference Brito, Morra, Cavalieri, Araújo, Henningson and Hanifi2021)

(3.16)\begin{equation} {\hat{\varGamma}} = \frac{({\hat{G}_{uz}})^{*}{Q}{\hat{G}_{yz}}}{({{\hat{G}_{uz}})^{*}}{Q}({\hat{G}_{uz}})+{R}}, \end{equation}

where $\hat {G}_{yz}$ is the transfer function between the microphones $y$ and the control target $z$, which is simply ${\hat {S}_{yz}}/{\hat {S}_{yy}}$. Once the inverse Fourier transform of the IFFC kernel is taken, the non-causal part is neglected, i.e. we consider $\varGamma (\tau <0) = 0$ in order to be able to apply the IFFC kernel in experiments.

In order to account for the contamination of the sensor readings caused by the actuators, it is also necessary to take into consideration the transfer function between the actuators and the microphones ($G_{uy}$). In this case, the IFFC kernel can be obtained as

(3.17)\begin{equation} {\hat{\varGamma}} = \frac{({\hat{G}_{uz}} -{\hat{G}_{uy}}{\hat{G}_{yz}})^{*}{Q}{\hat{G}_{yz}}}{({{\hat{G}_{uz}} -{\hat{G}_{uy}}{\hat{G}_{yz}})^{*}}{Q}({\hat{G}_{uz}} -{\hat{G}_{uy}}{\hat{G}_{yz}})+{R}}. \end{equation}

3.4. Further considerations about the control laws

For the present work the CSD and PSD were obtained using the Welch method, using a Hanning window with 218 blocks and 50 % overlap. Since we have used sampling rate $10$ kHz ($St = 29.9$) and measurements of $30$ s to identify the transfer functions, we obtained a frequency resolution $11$ Hz ($St = 0.03$), with a total of $2730$ data points per block.

Some of the energy content captured by the sensors is not related to the axisymmetric mode, including random fluctuations in the flow field, ambient and electrical noise, and also some undesired noise introduced by the actuators and forcing system. In order to prevent this undesired energy content from affecting the kernel, a zero-phase band-pass filter was applied to the raw data before the calculation of the transfer functions. The application of a filter was important because even very-small-amplitude noise in the kernel could result in a significant increase of the actuation amplitudes due to the acoustic feedback contamination. A frequency-based control penalization was used for the control kernels, wherein large penalizations were applied outside the desired frequency band, defined previously. This helps to avoid undesired frequencies that can lead to feedback contamination, and also regularizes the problem, avoiding divisions by zero (or by values close to zero) in (3.13) and (3.16). To further regularize the problem, a noise term of constant value $1\times 10^{-4}$ was added to the PSD of $u$ and $y$. For the case where a forcing is applied up to Strouhal number $0.85$, better results where obtained using a constant value $1\times 10^{-3}$.

The expressions used to obtain the control penalization are

(3.18)$$\begin{gather} R_1 = R_{ctrl} + a(R_{noise}-R_{ctrl})(\tanh(f + f_{c1})-\tanh(f - f_{c1})), \end{gather}$$

(3.19)$$\begin{gather}R_2 = R_{ctrl} + a(R_{ctrl} - R_{noise})(\tanh(f + f_{c2})-\tanh(f - f_{c2})), \end{gather}$$

(3.20)$$\begin{gather}R=\begin{cases} R_1, & \text{if } f_c \leq (f_{c1} + f_{c2})/2, \\ R_2, & \mathrm{otherwise}, \end{cases} \end{gather}$$

illustrated in figure 3. In the expressions above, $f$ is the frequency, $f_{c1}$ and $f_{c2}$ are cut-off frequencies, $R_{ctrl}$ defines the penalisation within the region of interest, and $R_{noise}$ is the penalisation out of the region of interest. We have used $a = 30$, a parameter that defines how fast and smooth is the transition from $R_{ctrl}$ to $R_{noise}$. The parameters used in each case are summarized in table 1.

Figure 3.

Control penalization as function of the frequency.

Table 1.

Parameters considered for the control penalization.

The functions related to the actuation (${\hat {G}_{uy}}$ and ${\hat {G}_{uz}}$) were obtained with the forcing off, while $S_{yy}$ and $S_{yz}$ were obtained with the forcing on (for the forced jet cases) and the actuation off. The response of the flow to different amplitudes of actuation and forcing was tested in order to certify that the chosen actuation and forcing amplitudes used to obtain the transfer functions are within the linear range of the flow response. This was verified evaluating the integral of the PSD of the reference sensor $y$ and the target $z$, for various forcing inputs. Figure 4 shows the tests performed for the forced jet case with the control target for $0.3 \leq St \leq 0.45$, where we indicate the amplitudes selected to obtain transfer functions, clearly in a linear range. Figure 4 also indicates that such a linear range is wide for both forcing and actuation. Such linear behaviour is also observed for a broader frequency range, as observed in Maia et al. (Reference Maia, Jordan and Cavalieri2022), with the same experimental set-up considered here.

Figure 4.

Response of turbulent jet as functions of actuation and forcing amplitude for the bandwidth $0.3 \leq St \leq 0.45$, with the indication of the amplitude considered to obtain the transfer functions. The PSD integrals were normalized with respect to the maximal value in the tests.

The magnitude-squared coherence between two signals is a statistical measurement of their degree of linear dependence (Bendat & Piersol Reference Bendat and Piersol2010). The normalized coherence between two signals $i$ and $j$ is given as

(3.21)\begin{equation} \gamma_{ij}^2 = \frac{|\langle S_{ij}\rangle|^2}{|\langle S_{ii}\rangle|\,|\langle S_{jj}\rangle|}, \end{equation}

where $\langle \cdot \rangle$ is the expectation operator, $S_{ij}$ is the CSD between the pair of signals $i$ and $j$, and $S_{ii}$ and $S_{jj}$ are the PSD functions of signals $i$ and $j$, respectively. The coherences obtained between the sensors and actuators are shown in figure 5. Coherence levels must be sufficiently high for the suitability of the linear control approaches employed here. High coherences indicate a linear relationship between the input and output signals, with low extraneous noise at the input or output signal. Thus higher coherences allow accurate estimates of flow responses.

Figure 5.

Coherence between: (a) forcing $d$ and reference signal $y$; (b) reference signal $y$ and control target $z$; and (c) actuation signal $u$ and control target $z$.

As can be observed in the forced jet case, with the increase of the bandwidth, lower coherence levels between the reference sensors and the target were observed for Strouhal numbers below $0.45$. For $St = 0.37$, for example, $\gamma _{yz}$ dropped by approximately $50\,\%$ with the larger forcing bandwidth. For the forced jet with $0.3 \leq St \leq 0.45$, $\gamma _{yz}$ can also be considered relatively low for $St = 0.34$ and lower, where $\gamma _{yz}$ was below $0.6$. This lower coherence can be explained either by the microphones not being able to properly identify the $m = 0$ structures, e.g. due to azimuthal aliasing, or due to decoherence of the wavepacket along the jet. Unlike transitional flows, characterized by small disturbances, we deal here with a turbulent jet, with coherent structures that nonetheless display an intrinsic coherence decay, imposing additional challenges for flow control. Nevertheless, for the forced jet cases, the coherences between sensors were mostly above $0.8$ for the frequencies of interest. Regarding the natural jet case, low coherence levels were obtained for the entire frequency range considered for control, approximately $0.5$, which makes the control of natural turbulent jets a much more challenging task. The coherence between the actuation signal and the hot wire was higher than $0.85$ for the region of interest, mostly above $0.9$, when considering the forced jet case with $0.3 \leq St \leq 0.45$. However, as the actuation bandwidth increases, lower coherence levels are obtained, which was found to be a limitation of the actuation system. Differences in coherence levels observed between the forced and unforced cases ($0.3 \leq St \leq 0.85$) can be explained partially by differences in the amplitudes of the actuation signals when identifying the transfer functions, but it was observed that this does not have a significant impact in the obtained kernel. Finally, a very low coherence level can be observed between signals $d$ and $y$ for the unforced jet case. In this case, only electrical noise is captured by the acquisition system, which is not sufficient to activate the forcing system. Thus this low coherence occurs due to imperfect statistical convergence of the coherence, which is expected to go to zero as the number of blocks is increased. Hence the coherences observed between $d$ and $y$ serve as a metric for statistical errors, which are nonetheless low in the present application.