2.1 Vortex shedding frequency

To obtain the frequency of vortex shedding, a hot-wire probe was mounted $6h$ downstream of the body, $3h$ above the wake centreline. The hot-wire probe, with a wire that is $0.5~\text{mm}$ long and $2.5~\unicode[STIX]{x03BC}\text{m}$ in diameter, is built in-house. It was operated in combination with a Dantec Dynamics StreamLine 90N10 system, equipped with a Constant Temperature Anemometer (CTA) module. A total of 91 250 samples were recorded at a sampling frequency of $1\;\times \;10^{4}\;\text{Hz}$ . The time signal was converted to the frequency spectrum employing a Welch algorithm. A bin size of $2^{15}$ was used, which corresponds to a frequency resolution of $0.3~\text{Hz}$ .

Figure 1.

A schematic drawing of the experimental set-up, with: (1) the perforated surface of the rectangular forebody, (2) the hot-wire probe, and (3) and (4) the cameras and laser of the PIV system, respectively, aligned as to capture the field of view delineated with dash-dotted lines.

Figure 2.

Frequency-premultiplied power spectral energy estimated via the Welch method for two hot-wire signals measured at (a) $2.5~\text{m}~\text{s}^{-1}$ (case U2.5Dp0) and (b) $20.0~\text{m}~\text{s}^{-1}$ (case U20Dp0) without suction from the body wall.

Two examples of frequency-premultiplied power energy spectra obtained from hot-wire signals at the lowest (2.5  $\text{m}~\text{s}^{-1}$ , case U2.5Dp0) and highest (20.0  $\text{m}~\text{s}^{-1}$ , case U20Dp0) speed, without suction from the wall, are shown in figure 2. As can be seen from the spectra in figure 2, vortex shedding is strongly periodic, the power spectra are narrow-banded around the vortex shedding frequency, and the related energy peak in the spectra are by far dominant over all other frequency components. Note that in figure 2(b) there are other peaks distributed at $150~\text{Hz}$ , $250~\text{Hz}$ , $350~\text{Hz}$ and so on. These peaks are due to the PIV system which was activated at a frequency of $150~\text{Hz}$ while the hot-wire signal was measuring, and the higher frequencies are caused by the interaction between the PIV frequency at $150~\text{Hz}$ and the vortex shedding frequency, which is close to $100~\text{Hz}$ , and thus they are not related to flow structures. More details on the spectral characteristics of hot-wire signals for this flow case, even when suction is applied from the wall, can be found in Trip & Fransson (Reference Trip and Fransson2014).

2.2 Phase-averaged flow

The time-resolved hot-wire signal was recorded simultaneously to the PIV experiment, and can therefore be conveniently used as a phase indicator to sort the PIV images. The PIV snapshots of a complete ensemble are subdivided into 16 discrete phase bins, based on their phase angle in the shedding cycle. The mean of the snapshots in each bin is approximately equal to the phase average of the velocity field. The independence of the phase average from the number of phase bins was checked by increasing the number of phase bins up to 128. An example of the phase-averaged flow field, at a generic phase, obtained for the flow case U2.5Dp0 is given in figure 3, where the velocity vectors and the vorticity fields are reported, together with the identification of the wake vortices through the $\unicode[STIX]{x1D706}_{2}$ criterion (see Jeong & Hussain Reference Jeong and Hussain1995) plotted with a black line on the figures. As highlighted by figure 3, the phase-averaged flow field is smooth enough for the application of differential operators (i.e. so as to derive the vorticity field) and the vortices which dominates the wake dynamics are clearly visible.

Figure 3.

Flow case U2.5Dp0: phase-averaged velocity (a) and vorticity (b) fields at a generic phase angle; black line indicates the wake vortices identified by the $\unicode[STIX]{x1D706}_{2}$ criterion (see Jeong & Hussain Reference Jeong and Hussain1995).

As additional information on the unsteadiness of the flow field, the components of the Reynolds stresses resolved by PIV for case U2.5Dp0 have been reported in figure 4(a), together with the contribution given solely by the phase-averaged velocity field (b) and that given by the residual velocity fluctuations (c), in the spirit of the triple decomposition of the flow variables (see the following for more details).

Figure 4.

Flow case U2.5Dp0: experimentally resolved Reynolds stresses in column (a), contribution given by the phase-averaged velocity field in column (b) and by the residual velocity fluctuations in column (c).

2.3 Maps for passive control

To study the effect of a small body on the vortex shedding frequency, a control cylinder is introduced into the wake of the body. The control cylinder, with a diameter equal to $d^{\ast }=3$  mm, is rigid such that it can be clamped between the wind tunnel walls. The minimum diameter of the cylinder, which is the one adopted here, is dictated by technological realisability of the experiment. However, from an aerodynamic viewpoint this cylinder is rather intrusive. Indeed, the maximum Reynolds number, based on its diameter, ranges between $5\times 10^{2}$ and $4\times 10^{3}$ depending on the flow case. An experimental sensitivity map for the considered passive control is obtained by recording the vortex shedding frequency with the control cylinder placed on a $15\times 11$ rectangular grid extending $1h$ from the wake centreline in the wall-normal direction and $2h$ downstream of the trailing edge of the plate in the streamwise direction. The resulting map is reported later in figure 10(b) for case U2.5Dp0 and commented in § 5.4.

3.1 A model for the mean flow field

In the present work the mean flow field and its phase average (with respect to vortex shedding) is always available from the experiments through dedicated PIV measurements. Relying on the available experimental information we want to derive a mathematical model for the mean flow field. The model proposed here is based on the governing equations (3.7)–(3.8) and makes use of all the information available from experiments.

In particular the only unknown term in (3.7)–(3.8) is the eddy-viscosity field $\unicode[STIX]{x1D708}_{t}(\boldsymbol{x})$ , which cannot be obtained experimentally since the time resolution of the PIV measurements inevitably acts as a filter on the flow variables which prevents obtaining fully resolved Reynolds stresses. For this reason $\unicode[STIX]{x1D708}_{t}(\boldsymbol{x})$ is a free field on which it is possible to act so as to change the output of (3.7)–(3.8). Consequently, $\unicode[STIX]{x1D708}_{t}(\boldsymbol{x})$ is estimated so as to minimise the differences between the target mean flow field coming from the experiments, $\overline{\boldsymbol{U}}^{(\text{T})}$ , and the simulated flow field, $\overline{\boldsymbol{U}}$ . The target flow $\overline{\boldsymbol{U}}^{(\text{T})}$ can be the raw experimental mean field, $\overline{\boldsymbol{U}}^{(E)}$ , or a regularised field, $\overline{\boldsymbol{U}}^{(R)}$ , as explained later. The considered objective function to be minimised in the resulting optimisation problem is the following one:

The cost function $I$ must be minimised considering that $\overline{\boldsymbol{U}}$ is the solution of (3.7)–(3.8), and this constraint is taken into account by a classical Lagrangian approach, i.e. defining an augmented functional as follows:

where $\vec{V}$ and $q$ are Lagrangian multipliers.

By applying standard variational methods to (3.12) it is possible to derive the adjoint equations for the Lagrange multipliers $(\vec{V},q)$ and the associated boundary conditions. In particular, the equations for $(\vec{V},q)$ are written as follows:

As we will see in the following sections, two types of boundary conditions are used for the mean flow equations. One type is a Dirichlet type (boundary $\unicode[STIX]{x1D6E4}_{D}$ ), such that the mean velocity $\overline{\boldsymbol{U}}$ is forced to assume a prescribed value. The other type, on $\unicode[STIX]{x1D6E4}_{o}$ , is a stress-free condition, and it can be written as follows: $\overline{p}\boldsymbol{n}-\left((1/Re)+\unicode[STIX]{x1D708}_{t}\right)[\boldsymbol{n}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\overline{\boldsymbol{U}}+\unicode[STIX]{x1D735}^{\text{T}}\overline{\boldsymbol{U}})]=\mathbf{0}$ . On the boundary $\unicode[STIX]{x1D6E4}_{D}$ it can be shown that the natural boundary conditions for $\vec{V}$ are homogeneous Dirichlet conditions, i.e. $\vec{V}=\mathbf{0}$ . On $\unicode[STIX]{x1D6E4}_{o}$ we have the following conditions: $q\boldsymbol{n}-\left((1/Re)+\unicode[STIX]{x1D708}_{t}\right)[\boldsymbol{n}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\vec{V}+\unicode[STIX]{x1D735}^{\text{T}}\vec{V})]=-\!\left(\overline{\boldsymbol{U}}\boldsymbol{\cdot }\boldsymbol{n}\right)\vec{V}$ .

Once the Lagrangian multipliers $(\vec{V},q)$ are found, the variation $\unicode[STIX]{x1D6FF}I$ of the cost function $I$ in (3.11) with respect to a generic variation in the spatial distribution of $\unicode[STIX]{x1D708}_{t}(\boldsymbol{x})$ , $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}_{t}(\boldsymbol{x})$ , is given as follows:

The optimal distribution of $\unicode[STIX]{x1D708}_{t}$ can thus be obtained using a gradient-based method based on (3.15). At each step of the gradient method, $\unicode[STIX]{x1D708}_{t}$ is varied moving towards the steepest descent for $I$ and the length of the step in the direction of the gradient is found by searching numerically for the optimality condition in the direction specified by the gradient, which is computed by (3.15).

It is important to underline that the distribution of $\unicode[STIX]{x1D708}_{t}$ obtained by the procedure described above is aimed at modelling the unresolved small-scales Reynolds stresses, which is the most important contribution needed in (3.8) so as to reproduce accurately the reference mean flow. However, since the field $\unicode[STIX]{x1D708}_{t}$ is the result of an optimisation process, it also compensates for all the elements, which prevents equations (3.7)–(3.8) from reproducing exactly the experimental mean flow even if the exact unresolved stresses were introduced directly in the equations. These elements can be, for instance, the experimental uncertainty or the errors due to the numerical discretisation of the equations, including domain truncation.

Since the mean experimental flow fields are affected by noise, which may amplify as they are differentiated in space, they have been usually regularised before being employed as target flows in the calibration of the mean flow model. The regularisation applied here is a variational form of the Tikhonov regularisation smoothing first-order spatial derivatives, as proposed for instance in Wang (Reference Wang, Kirby and Johnson2011) for inverse electrocardiographic applications. If not explicitly stated, regularised fields are always used here instead of raw experimental data. Moreover, again if not explicitly commented, we have also imposed that the regularised mean velocity fields are divergence-free. In principle, since we are measuring the flow in the symmetry plane and, moreover, the considered flow should approximate a nominally plane wake, it is expected that the experimental mean flow will be two-dimensional and, thus, divergence-free in the measurement plane. This may not be the case due to measurement and differentiation errors. For this reason, even if tests reported later prove that this is not an important aspect, we decided to add the divergence-free constraint to the regularised fields. Lastly, even if the target flow was not divergence-free, the tuned model flow is so by definition (see equation (3.7)) and, thus, the projection of the reference flow on a divergence-free subspace has no significant influence on the resulting tuned flow model. This aspect is also investigated later. Indicating with $\overline{\boldsymbol{U}}^{(E)}$ the PIV measurements and with $\overline{\boldsymbol{U}}^{(R)}$ the regularised field, the regularisation problem consists in searching for $\overline{\boldsymbol{U}}^{(R)}$ which minimises $I_{G}$ given as follows:

where $\unicode[STIX]{x1D706}_{r}$ is an arbitrary penalisation constant which regulates the amount of regularisation of the procedure and $q$ is a Lagrangian multiplier. Here the value of $\unicode[STIX]{x1D706}_{r}$ has been selected heuristically.

3.2 Stability analysis of the mean flow field

The stability problem, which is related to the existence of a periodic flow component, is governed by the linearised equations (3.9) and (3.10).

The associated boundary conditions are of the same type as those imposed on the base flow equations, but they are homogeneous. Considering the boundary conditions used in the present work, which are either boundaries where the velocity field is specified, $\unicode[STIX]{x1D6E4}_{D}$ (i.e. inflow boundaries or solid walls) or stress-free boundaries, $\unicode[STIX]{x1D6E4}_{O}$ (i.e. outflow boundaries), we have the following conditions:

A modal form for the periodic flow component is assumed: When the modal form for the flow perturbation is introduced in (3.9)–(3.10), the following stability equations are found: with the same homogeneous boundary conditions specified for ( $\tilde{\boldsymbol{u}}$ , $\tilde{p}$ ) in (3.17). Equations (3.19), together with the boundary conditions, form an eigenvalue problem. Any solution $(\hat{\boldsymbol{u}},\hat{p})$ , associated to the eigenvalue $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D706}+\text{i}\unicode[STIX]{x1D714}$ , is a global mode with time growth rate equal to $\unicode[STIX]{x1D706}$ ( $\unicode[STIX]{x1D706}>0$ indicates an unstable mode) and angular frequency $\unicode[STIX]{x1D714}$ . Note that the associated frequency is given by $f=\unicode[STIX]{x1D714}/(2\unicode[STIX]{x03C0})$ , and is given directly in non-dimensional form as a Strouhal number based on the same velocity and length reference quantities used to normalise the Navier–Stokes (NS) equations.

3.3 Passive control of vortex shedding instability

The objective of this section is to estimate the effect of placing a small control rod in the flow on the eigenvalue identified by the stability analysis of the mean flow that is related to the vortex shedding instability. This is carried out using an adjoint-based sensitivity analysis analogous to that proposed in Marquet et al. (Reference Marquet, Sipp and Jacquin2008).

As a device for passive control we consider a cylinder of diameter $d^{\ast }$ positioned on a generic point $(x_{0},y_{0})$ . The cylinder diameter is assumed to be small enough so that its action on the flow can be described by a linearised approach. Moreover, it is not directly simulated, but its effect is modelled by introducing the force that it exerts on the flow explicitly in the NS equations. This force is assumed to be a pure drag force, which depends on the local flow conditions. Consequently the control cylinder, when invested by a flow velocity $\boldsymbol{U}$ , is assumed to exert a localised drag force of the following form:

where $\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})$ is the Dirac function centred on the position of the control cylinder, $\boldsymbol{x}_{0}$ . The notation $\unicode[STIX]{x1D6FF}\boldsymbol{F}$ is used in (3.20) so as to emphasise that this force is a small perturbation of the uncontrolled configuration. As concerns the drag coefficient, this can be generally assumed to be a function of the flow Reynolds number, i.e. $C_{D}(Re_{d\ast })$ . In this paper we have assumed a constant value equal to $C_{D}\simeq 1.2$ , which is representative of the drag of the control cylinder in the considered flow, as will be illustrated in the following. For quantitative indications on the drag coefficient of circular cylinders, we refer to the ESDU database (ESDU 1986). Note that (3.20) is a quasi-static model, in the sense that the unsteady load $\unicode[STIX]{x1D6FF}\boldsymbol{F}$ is proportional to the local time-varying velocity field $\boldsymbol{U}$ through a constant drag coefficient $C_{D}$ , which in this case is an averaged drag coefficient. The use of a quasi-static model for similar cases is common in the literature (see e.g. the review in Camarri Reference Camarri2015, and its bibliography) and is loosely justified by the scale separation existing between the flow to be controlled and the control cylinder.

In the framework of the triple decomposition of the flow variables, assuming further that turbulent fluctuations can be neglected in the estimation of the force experienced by the control cylinder, we consider the drag force given by (3.20) linearised with respect to $\tilde{\boldsymbol{u}}$ (see the decomposition in (3.1)), which is consistent with the assumptions made for the stability equations (3.19):

in which $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}=(1/2)d^{\ast }C_{D}(Re_{d^{\ast }})$ .

By inspecting equation (3.21) it is possible to notice that the force $\unicode[STIX]{x1D6FF}\boldsymbol{F}$ is composed of two contributions: $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{M}}$ , which depends on $\overline{\boldsymbol{U}}$ and is independent of $\tilde{\boldsymbol{u}}$ , and $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{F}}$ , which depends linearly on $\tilde{\boldsymbol{u}}$ . Consequently, the term $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{M}}$ acts as a forcing on the mean flow equations (3.7)–(3.8) and the term $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{F}}$ acts directly as a structural perturbation of the stability equations (3.19). Thus, the control cylinder affects the stability characteristics of the mean flow in two ways: via a modification of the mean flow itself caused by the term $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{M}}$ and via a modification of the stability equations caused by the term $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{F}}$ . The resulting global effect on a particular eigenvalue $\unicode[STIX]{x1D70E}$ can be estimated in a linear framework by an adjoint-based sensitivity analysis.

Let us focus on a particular global mode $\unicode[STIX]{x1D70E},(\hat{\boldsymbol{u}},\hat{p})$ of the uncontrolled mean flow field ( $\overline{\boldsymbol{U}}$ , $\overline{p}$ ). As the control is applied, both the mode and the mean flow field are perturbed, resulting in $\unicode[STIX]{x1D70E}^{\prime \prime },(\hat{\boldsymbol{u}}^{\prime \prime },\hat{p}^{\prime \prime })$ and ( $\overline{\boldsymbol{U}}^{\prime \prime }$ , $\overline{p}^{\prime \prime }$ ), respectively. The coupled governing equations for the perturbed mean flow and stability problem can be written as follows:

Here we are interested in estimating the perturbation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}$ of the eigenvalue $\unicode[STIX]{x1D70E}$ induced by control, i.e. $\unicode[STIX]{x1D70E}^{\prime \prime }=\unicode[STIX]{x1D70E}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}$ . Using an adjoint-based sensitivity analysis it is possible to show that (see Marquet et al. Reference Marquet, Sipp and Jacquin2008, for details):

where $\overline{\boldsymbol{U}}^{+}$ and $\hat{\boldsymbol{u}}^{+}$ are the adjoint mean flow and eigenmode, respectively, $\unicode[STIX]{x1D6FA}$ is the flow domain and $(~)^{\ast }$ stands for complex-conjugate quantities. Equation (3.23) is very convenient from a computational viewpoint because it is linear with respect to the parameter $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ and provides an answer for a generic position of the control cylinder ( $\boldsymbol{x}_{0}$ ) without the need to perform integration, since both $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{M}}$ and $\unicode[STIX]{x1D6FF}\boldsymbol{H}_{\boldsymbol{F}}$ , defined in (3.21), are proportional to $\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})$ . Thus, once the diameter of the control cylinder is fixed, equation (3.23) provides directly a control map $S(\boldsymbol{x})$ which gives the shift $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}$ of the identified eigenvalue versus a generic placement of the control cylinder in the flow domain. The use of equation (3.23) requires the estimation of the vector fields $\overline{\boldsymbol{U}}^{+}$ and $\hat{\boldsymbol{u}}^{+}$ . The field $\overline{\boldsymbol{U}}^{+}$ is solution of the following system of equations:

where $\unicode[STIX]{x1D735}_{\overline{\boldsymbol{U}}}\unicode[STIX]{x1D70E}$ is given by:

equations (3.24) are supplemented with homogeneous boundary conditions at the inlet and on the solid wall and the following condition on the outflow boundary:

The adjoint eigenmode $\{\unicode[STIX]{x1D70E}^{\ast },(\hat{\boldsymbol{u}}^{+},\hat{p}^{+})\}$ is the solution of the following eigenvalue problem:

As concerns the boundary conditions, $\hat{\boldsymbol{u}}^{+}=\mathbf{0}$ on the same boundaries on which $\hat{\boldsymbol{u}}=\mathbf{0}$ . At the outflow we have: $\hat{p}^{+}\boldsymbol{n}-Re_{eq}^{-1}[\boldsymbol{n}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\hat{\boldsymbol{u}}^{+}+\unicode[STIX]{x1D735}^{\text{T}}\hat{\boldsymbol{u}}^{+})]=(\overline{\boldsymbol{U}}\boldsymbol{\cdot }\boldsymbol{n})\hat{\boldsymbol{u}}^{+}$ . Moreover, the velocity field $\hat{\boldsymbol{u}}^{+}$ is normalised such that: