Forced wakes far from threshold: Stuart-Landau equation applied to experimental data

In this Rapid Communication, we study with the Stuart-Landau (SL) amplitude equation, a wake flow control scenario using experimental data from a cylinder wake forced by plasma actuators. Given the formal framework recently discussed by Gallaire et al. [Fluid Dyn. Res. 48, 061401 (2016)] on pushing amplitude equations far from threshold, we analyze experimental data of a forced wake in order to test the SL reduced order model. Linear stability theory and global mode concepts are used to determine the SL parameters. The extension to forced wakes of the SL model had been proposed by Thira and Wesfreid [J. Fluid Mech. 579, 137 (2007)] in the context of their study on stability properties, but its employment still remained an open question. Here, we show that a forced wake at a Reynolds number far from the first threshold can also attain the critical behavior described by the SL model.

Figure 1

Schematic diagrams of (a) a forced wake where the forcing is represented by a perturbation $\delta u$ on the upper and lower parts of the cylinder that modifies the mean flow and so $L_R$. (b) Actual experimental setup, when forcing is obtained through plasma actuation.

Figure 2

Isocontours of the instantaneous 2D velocity modulus of the wake behind the cylinder for different duty cycles. (a) $\mathrm{dc}=0$ (no forcing), (b) $\mathrm{dc}=10$, (c) $\mathrm{dc}=20$. While increasing the forcing amplitude, an extension of the recirculation region length $L_R$, and an increase of the wavelength are observed.

Figure 3

(a) Evolution of the mean velocity measured along the $x$ axis as a function of $\mathrm{dc}$, where the different labels are displayed on (b). The definition of $\mathrm{\Delta }{u}_{min}$ is plotted in the case of $\mathrm{dc}=20$. The inset plot shows the values of $\mathrm{\Delta }{u}_{min}$ as a function of the forcing parameter $\mathrm{dc}$. (b) Evolution of the spatial shapes of the global modes with the forcing $\mathrm{dc}$. For each curve, its maximum defines a pair of values $(x_{max},\rho )$. (c) Relationship between the squared amplitude ${\rho }^2$ and the characteristic modification of the mean flow given by $\mathrm{\Delta }{u}_{min}$. As $\mathrm{\Delta }{u}_{min}$ increases, ${\rho }^2$ tends to 0, corresponding to the wake stabilization. Linearity ceases to hold when the forcing weakens, which is displayed in the graph through a diminished color intensity of the plot markers. (d) Behavior of the selected frequency $f$ of the global mode ($◯$ and $\square$) and the growth rate $\sigma$ ($▴$) as a function of the squared amplitude ${\rho }^2$; $◯$ and $▴$ are obtained through a linear stability analysis, and $\square$ are determined from a Fourier decomposition of the experimental values. As ${\rho }^2\to 0$, the stable mean flow frequency $f_0\sim 0.115$. In the same sense, while increasing the forcing, the temporal growth rate decreases.

Figure 4

Schematic representation of the necessary steps involved in the linear stability analysis: (a) Mean flow streamwise velocity contours, with a slight loss of symmetry is due to forcing, where a cut a $x=x_c$ gives (b) a mean velocity profile (square symbols) $\langle v_x(y,x=x_c)\rangle$, which is fitted by a function (dashed line) $u\left(y\right)=1-a_0+a_{0}tanh\left[a_1\left({y/d)}^2-a_2\right)\right]$, where $a_0$, $a_1$, and $a_2$ are fitting parameters. Solving the corresponding dispersion relation, the Rayleigh equation for this inviscid instability, for a range of wave numbers $k=k_r+i{k}_i$, we obtain (c) a map for $k_i=\text{const}$ in the $\omega$ plane. At a critical point, a cusp on the $k_i=-0.7$ curve, the point $({\omega }_{0r},{\omega }_{0i})$ is determined for $x_c$. In (d) the results for each $x$ position($\circ$ marks) are plotted in a $\omega$ plane. Criteria for the prediction of a global frequency in a spatially developing flow is given by the condition ${\partial \omega /\partial x|}_{x=x_s}=0$ [34]. The saddle point ${\omega }_s=\omega (x=x_s)$ is obtained through an analytic continuation to complex values of $x=x_r+i{x}_i$ by inspecting the when a cusp point takes place (point-dashed curve) [36]. Considering all the cases, the points ${\omega }_{\mathbf{s}}=(\omega ,\sigma )$ provide the data for Fig. 3.