4.1 Control approach

Based on insights gained from the baseline characterization of the wall-normal vortex, we develop an active flow control strategy to weaken the vortex core. We aim to widen the vortex core radius by inducing unsteadiness in the core region. This modification will reduce the azimuthal velocity and hence increase the vortex core pressure.

To trigger the emergence of vortical unsteadiness, we use unsteady actuation from the bottom wall at the vortex centre. We have observed from the above DMD analysis that modal structures appear at frequencies between 0.13 and 0.81. To introduce unsteady perturbation with these characteristics, we add rotary blowing and suction perturbation by prescribing a velocity boundary condition at the wall,

(4.1) $$\begin{eqnarray}\frac{u_{z}^{c}(r,\unicode[STIX]{x1D703},z=0,t)}{u_{\unicode[STIX]{x1D703},max}}=A_{c}\cos (2\unicode[STIX]{x03C0}f_{c}t+m_{c}\unicode[STIX]{x1D703})\exp \left(-\frac{r^{2}}{a^{2}}\right),\end{eqnarray}$$

where $A_{c}$ , $f_{c}$ and $m_{c}$ are the actuation amplitude, frequency and azimuthal wavenumber respectively. The control frequency is chosen in the range $0.08\leqslant f_{c}\leqslant 0.3$ to span across the dominant frequencies identified from the above DMD analysis of the baseline flow. The azimuthal wavenumber is set to $m_{c}=1$ (co-rotating) and $-1$ (counter-rotating), as DMD analysis reveals the largest fluctuations at these wavenumbers. Here, there equal and opposite amplitudes of blowing and suction are applied to the flow, with zero net mass addition. For the sake of completeness, we also consider actuation with azimuthal wavenumbers of $m=0$ and $|m|>1$ , as well as steady forcing with $f_{c}=0$ . This particular form of the function is chosen to implement suction and blowing in a spatially compact manner near the vortex centre. As we briefly discuss later, unsteady actuation with $f_{c}>0$ is indeed required to achieve effective control.

To quantify the control input, we introduce a circulation-based oscillatory momentum coefficient, defined as

(4.2) $$\begin{eqnarray}C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}\equiv \frac{J^{\prime }}{\unicode[STIX]{x1D6E4}_{\infty }^{2}},\quad \text{where}~J^{\prime }\equiv f_{c}\int _{0}^{1/f_{c}}\int _{0}^{R}\int _{0}^{2\unicode[STIX]{x03C0}}(u_{z}^{c})^{2}\,\text{d}\unicode[STIX]{x1D703}\,\text{d}r\,\text{d}t\end{eqnarray}$$

is the oscillatory momentum input. We choose a control amplitude in the range $0.1\leqslant A_{c}\leqslant 1$ , which corresponds to $0.02\,\%\leqslant C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}\leqslant 2.03\,\%$ , to evaluate the momentum coefficient effect.

4.2 Controlled flow

Let us examine the effects of forcing on the wall-normal vortex and its core-pressure distribution. We first set the actuation amplitude at $A_{c}=1$ ( $C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}=2.03\,\%$ ) and study the influence of unsteady forcing at frequencies of $f_{c}=0.11$ and 0.27. These control frequencies are selected to be close to those of baseline DMD modes A and I, with the aim of stimulating the emergence of instabilities around the vortex. We also consider co-rotating and counter-rotating actuation with $m=\pm 1$ . Figure 4 presents the controlled flows with the chosen control parameters. The vortex cores using the $Q$ -criterion isosurface (blue) and the overall vortical structures with the transparent grey vorticity norm isosurface are visualized. The corresponding time-averaged pressure profiles are also shown in figure 4 to assess the effectiveness of modifying the pressure distribution.

Figure 4.

Visualizations of the controlled flows with isosurfaces of $\Vert \unicode[STIX]{x1D74E}\Vert =2$ (grey) and a $Q$ -criterion of $Q=2$ (blue). Co-rotating (a,d) and counter-rotating (b,e) control cases using $C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}=2.03\,\%$ at $f_{c}=0.11$ (a,b) and 0.27 (d,e) are shown. The corresponding time-averaged pressure contours are also shown in (c) and (f).

By perturbing the flow at a frequency of $f_{c}=0.11$ , the vortex profile can be modified significantly with co-rotating and counter-rotating inputs. As shown in figure 4(a–c), the vortex core is spread radially in an unsteady manner and the vortex breakdown region is eliminated. In the case of co-rotating control, the vortical structure becomes more complex and radially spread out compared with the baseline flow. Hence, the core region of the flow has a lower azimuthal velocity and leads to an increase in the core pressure, which can be seen in figure 4(c). With counter-rotating control, the vortex core is modified, while its width is not significantly altered. Instead, the core deforms into a braid-like helical structure. The corresponding minimum pressure in the counter-rotating controlled flow increases by 33 % compared with the baseline flow, but not as much as what is achieved with the co-rotating forcing case, which accomplishes a 54 % increase.

The wall-normal vortex has also been modified by forcing with a higher frequency of $f_{c}=0.27$ , as illustrated in figure 4(d–f). With counter-rotating control, the vortex breakdown region is eliminated and the vortical flow becomes unsteady throughout the entire vortical region. For co-rotating control, the vortex breakdown is essentially removed but shows slight bulging near $z/a\approx 6$ . For this reason, the pressure increase at the base of the vortex is not achieved as in the lower-frequency forcing case. This suggests that instigation of fluctuations at low frequency leads to a high level of unsteadiness and mixing. On the other hand, counter-rotating actuation at this higher frequency does not exhibit noteworthy difference from the low-frequency actuation.

We further analyse the influence of control frequency and amplitude on the wall-normal vortex. As discussed above, pressure increase along the vortex core is achieved by spreading the vortex core radially and reducing the azimuthal velocity  $u_{\unicode[STIX]{x1D703}}$ . Hence, we can use the time-averaged axial circulation $\unicode[STIX]{x1D6E4}_{z}(r,z)=\int _{0}^{2\unicode[STIX]{x03C0}}\bar{u}_{\unicode[STIX]{x1D703}}(r,z,\unicode[STIX]{x1D703})r\,\text{d}\unicode[STIX]{x1D703}$ , normalized by $\unicode[STIX]{x1D6E4}_{\infty }$ , as a metric to assess the effectiveness of the current flow control approach. In what follows, we focus on the co-rotating control set-up, as it is more effective than the counter-rotating control set-up in alleviating the low-pressure core.

Figure 5.

(a) Mean circulation of baseline and co-rotating control cases at $A=1$ . (b) Time-averaged pressure distribution along the vortex centre for different control frequencies.

While keeping the actuation amplitude constant at $A_{c}=1$ ( $C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}=2.03\,\%$ ), let us vary the actuation frequency $f_{c}$ from 0.08 to 0.3. We present the time-averaged axial circulation $\unicode[STIX]{x1D6E4}_{z}(r,z)/\unicode[STIX]{x1D6E4}_{\infty }$ at representative axial locations in figure 5(a). With actuation, we observe that the core of the wall-normal vortex is spread over the radial direction. The actuation introduced from the bottom wall is able to generate a significant level of unsteadiness around the vortex core to eliminate the vortex breakdown region and decrease the azimuthal velocity, which in turn raises the core-pressure profile, as can be seen in figure 5(b). Among the actuation frequencies considered, those near $f_{c}\approx 0.1$ are found to be beneficial for pressure increase. In fact, the controlled flow with $f_{c}=0.08$ achieves the highest increase in pressure for $z/a<5$ . We will further elaborate on the reason for low-frequency actuation being effective in modifying the vortical flow.

We further examine the impact of the control amplitude at the low frequency of $f_{c}=0.08$ using co-rotating actuation, which achieves significant core-pressure increase. The forcing amplitude is varied from $A_{c}=0.1$ to 1, as shown in figure 6. The time-averaged axial circulation $\unicode[STIX]{x1D6E4}_{z}(r,z)/\unicode[STIX]{x1D6E4}_{\infty }$ at low-pressure locations of $z/a=2$ and 8 is shown in figure 6(a). For $A_{c}\leqslant 0.3$ ( $C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}\leqslant 0.18\,\%$ ), the actuation does not substantially alter the circulation profile. For $A_{c}\geqslant 0.5$ ( $C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}\geqslant 0.51\,\%$ ), the circulation is significantly reduced and the core distribution becomes wider in the radial direction. The control becomes highly effective past $A_{c}=0.5$ ( $C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}=0.51\,\%$ ). On further increasing the control amplitude, the circulation continues to decrease but the amount of reduction saturates.

The corresponding time-averaged pressure distribution along the vortex core is presented in figure 6(b). For $A_{c}=0.1$ and 0.3, the time-averaged pressure profile does not deviate much from that of the baseline. As the control amplitude increases to $A_{c}=0.5$ ( $C_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6E4}}=0.51\,\%$ ), the vortex breakdown region is removed and the time-averaged pressure substantially increases. This indicates the existence of a critical threshold amplitude $A_{c}\approx 0.5$ to achieve effective flow control for modifying the pressure distribution. Once the control amplitude surpasses $A_{c}>0.5$ , there is no significant gain from using the additional control input. This tells us that the pressure profile of this type of flow can be altered effectively at the threshold amplitude using low-frequency actuation.

Figure 6.

Time-averaged (a) axial circulation at representative locations and (b) pressure distribution along the vortex core $r=0$ for varied control amplitudes (co-rotating cases) at $f_{c}=0.08$ .

We have observed that low-frequency forcing is effective in modifying the wall-normal vortex profile and increasing its core pressure. To shed light on the influence of frequency on altering the base flow, let us revisit the Burgers vortex model. In the three-dimensional stability analysis of the Burgers vortex performed by Schmid & Rossi (Reference Schmid and Rossi2004), it was reported that significant transient growth of perturbation can take place (while being neutrally stable). They performed the stability analysis by considering the Lundgren transformation (Lundgren Reference Lundgren1982) to reduce the three-dimensional flow to be described by a mapped two-dimensional diffusion equation. In this reduced state, they quantified the gain of the transient growth ${\hat{G}}$ as a function of the axial wavenumber $\hat{k}_{z}$ , which is non-dimensionalized as $\hat{k}_{z}=\overline{k}_{z}\unicode[STIX]{x1D705}(t_{max})$ , where $\overline{k}_{z}$ is the wavenumber in the mapped state and $\unicode[STIX]{x1D705}$ is a scaling parameter that is dependent on the time of maximum energy amplification (for details, see Schmid & Rossi Reference Schmid and Rossi2004).

Here, we consider their stability analysis as a simplified model for the present study, as we expect some commonality in behaviour especially away from the wall. The spatial minimum pressure values achieved with $A_{c}=1$ for $0.02\leqslant f_{c}\leqslant 0.17$ (co-rotating) and $0.08\leqslant f_{c}\leqslant 0.3$ (counter-rotating) over the non-dimensional axial wavenumber $\hat{k}_{z}$ are shown in figure 7. Also plotted is the transient growth gain for the Burgers vortex at $Re_{\unicode[STIX]{x1D6E4}}=5000$ reported by Schmid & Rossi (Reference Schmid and Rossi2004) to show a qualitative comparison. We can note that sizable transient growth can be expected for $\hat{k}_{z}\leqslant 1$ , which corresponds to $f_{c}\leqslant 0.17$ (co-rotating) and $f_{c}\leqslant 0.23$ (counter-rotating) in the present study. The pressure changes attained by co-rotating and counter-rotating control cases are provided, and exhibit qualitative agreement with the insights on transient growth. Hence, we infer that the transient growth mechanism (secondary hump around $\hat{k}_{z}=0.7$ ) plays an important role in ensuring that small forcing inputs are greatly amplified to effectively modify the mean pressure profile of the wall-normal vortex. Because co-rotating flow control creates large unsteadiness in a nonlinear manner, it is expected that there will be some difference in the wavenumber that achieves the highest mean flow modification and transient growth gain distribution. The counter-rotating case, on the other hand, agrees well with the transient gain distribution, while the perturbation does not grow as much as in the co-rotating control cases, as we can observe from figure 4. Since the forcing is harmonic in time, resolvent analysis (Jovanović & Bamieh Reference Jovanović and Bamieh2005; McKeon & Sharma Reference McKeon and Sharma2010; Yeh & Taira Reference Yeh and Taira2018) would also be required to gain a deeper understanding of the control mechanism. The insights from non-modal analysis can offer guidance on finding an effective control set-up to modify the mean flow and reducing the control parameter space to search over.

Finally, we note that cases with $m=0$ and $|m|>1$ , as well as steady control with $f_{c}=0$ , were also considered. We have found that cases with $m=0$ and $|m|>1$ do not modify the vortical flow as effectively as the cases with $m=\pm 1$ . The observation of the control set-up with $m=1$ working most effectively agrees qualitatively with the non-modal analysis of the Burgers vortex conducted by Schmid & Rossi (Reference Schmid and Rossi2004). With the steady actuation $f_{c}=0$ , the vortex core is displaced away from $r=0$ and still exhibits axial variation (i.e. $\hat{k}_{z}\neq 0$ ). The results imply that unsteady actuation plays an important role in effectively alleviating the low vortex core pressure.

Figure 7.

Minimal spatial pressure shown as a function of the non-dimensional axial wavenumber  $\hat{k}_{z}$ . Also shown is the energy amplification (transient growth gain) from the stability analysis of Schmid & Rossi (Reference Schmid and Rossi2004). All results are shown for $Re_{\unicode[STIX]{x1D6E4}}=5000$ . The dashed line shows the minimal core pressure for the baseline flow.