Open-loop control of a global instability in a swirling jet by harmonic forcing: A weakly nonlinear analysis

Highly swirling flows are often prone to precessing instabilities, with an azimuthal wave number of $m=-1$. We carry out a weakly nonlinear analysis to determine the response behavior of this instability to harmonic forcing. An incompressible flow is considered, where an annular inlet provides a swirling flow into a cylindrical region. For high swirl a vortex breakdown is induced, which is found to support an $m=-1$ instability. By expanding about the Reynolds number where this instability first occurs, a Stuart-Landau equation for the critical mode amplitude can be found and the effect of forcing can be assessed. Two types of forcing are considered. First, a Gaussian forcing confined to the inlet nozzle is used to study $m=0$ and $m=-1$ forcings. Second, optimal forcings (measured by the two-norm) with azimuthal wave numbers in the range $-3\le m\le 3$ are considered. It is found that modal stabilization is highly dependent on the azimuthal wave number $m$, which governs whether the forcing is counter- or corotating with the direction of swirl. Counterrotating forcings are able to stabilize the mode for a wide range of forcing frequencies, while corotating forcings fail to yield a stable flow. In all cases, it is the base-flow modification induced by the forced response that is the dominant underlying feature responsible for the observed stabilization. This base-flow modification seeks to reduce axial momentum near the recirculation region for corotating forcings and increase it for counterrotating forcings, thus changing the size of the recirculation bubble and producing the two distinct response behaviors.

Figure 1

Sketch of the computational domain for the simulations. The domain size is given by $x_{\text{inlet}}, x_{\text{outlet}}$, and $r_{\text{outlet}}$. We show the reflection of the domain across the line $r=0$, but the flow is not solved there; instead, a symmetry condition is applied.

Figure 2

Computational grid consists of three regions of different mesh densities, with $x_1=-0.5, r_1=1.5$, and $x_2=3$, and mesh densities $n_1=12n_r, n_2=20n_r$, and $n_3=8n_r$, where $n_r=1.5$ is a mesh-refinement parameter.

Figure 3

(a) Axial velocity component of the steady axisymmetric base flow ${\mathit{q}}_0$ for $S=0.87$ at the threshold of instability ${\text{Re}}_c=295.95$ and (b) close-up of the recirculation bubble with streamlines superimposed.

Figure 4

Axial velocity component of the unstable mode for the base flow in Fig. 3: (a) $\text{Re}\left(u_A\right)$ and (b) close-up of $\text{Re}\left(u_A\right)$ with streamlines superimposed.

Figure 5

Axial velocity component of the unstable adjoint mode for the base flow in Fig. 3: (a) $\text{Re}\left(u_A^{†}\right)$ and (b) close-up of $\text{Re}\left(u_A^{†}\right)$ with streamlines superimposed.

Figure 6

Axial velocity component of the harmonics and base-flow modifications caused by the unstable mode in Fig. 4: (a) $\text{Re}\left(u_{AA}\right)$, (b) $\text{Re}\left(u_{A\overline{A}}\right)$, and $\text{Re}\left(u_{\delta }\right)$.

Figure 7

Gain, real part of $\mu$ and absolute value of $\gamma$ shown for a range of frequencies: gain of the linear frequency response for (a) $m_f=0$ and (b) $m_f=-1$, weakly nonlinear coefficient $\mu$ for (c) $m_f=0$ and (d) $m_f=-1$, and frequency shift term $\gamma$ for (e) $m_f=0$ and (f) $m_f=-1$. Calculated frequencies are shown as pluses in the gain plots and as crosses and circles in the $\mu$ and $\gamma$ plots. Circles denote a positive sign for $\mu$ (stabilizing) or $\gamma$ and crosses denote a negative sign. The vertical solid black lines shows the resonant frequency ${\omega }_c$ for which, in the case of $m_f=-1$, the amplitude equation is not valid.

Figure 8

Axial velocity components of the forced response, mean-flow modification, and harmonics for forcing with $m_f=0$ and ${\omega }_f=1.3$: (a) $\text{Re}\left(u_F\right)$, (b) $\text{Re}\left(u_{F\overline{F}}\right)$, (c) $\text{Re}\left(u_{A\overline{F}}\right)$, and (d) $\text{Re}\left(u_{AF}\right)$.

Figure 9

Axial velocity components of the forced response, mean-flow modification, and harmonics for forcing with $m_f=-1$ and ${\omega }_f=1.7$: (a) $\text{Re}\left(u_F\right)$, (b) $\text{Re}\left(u_{F\overline{F}}\right)$, (c) $\text{Re}\left(u_{A\overline{F}}\right)$, and (d) $\text{Re}\left(u_{AF}\right)$.

Figure 10

Weakly nonlinear interaction terms due to base-flow modifications arising from the Gaussian forcing with $m_f=0$ at ${\omega }_f=1.3$ (axial velocity components shown): (a) $\text{Re}\left({\mu }_{u,F\overline{F}}\right)$ and (b) close-up of $\text{Re}\left({\mu }_{u,F\overline{F}}^I\right)$ with streamlines superimposed.

Figure 11

Weakly nonlinear interaction terms due to base-flow modifications arising from the Gaussian forcing with $m_f=-1$ at ${\omega }_f=1.7$ (axial velocity components shown): (a) $\text{Re}\left({\mu }_{u,F\overline{F}}\right)$ and (b) close-up of $\text{Re}\left({\mu }_{u,F\overline{F}}^I\right)$ with streamlines superimposed.

Figure 12

Optimal gains of the forced linear response for (a) positive and (b) negative wave numbers, along with the values of the optimal weakly nonlinear coefficient $\mu$ for (c) positive and (d) negative wave numbers and the optimal frequency shift parameter $\gamma$ for (e) positive and (f) negative wave numbers. The vertical lines show the resonant frequencies ${\omega }_c$ and $2{\omega }_c$ where the amplitude equation may not be valid, depending on the wave number of the forcing. Circles are used to show that $\mu ,\gamma >0$, while crosses indicate that $\mu ,\gamma <0$.

Figure 13

Axial velocity components of the optimal forcing and induced base-flow modification for most stabilizing frequencies for negative wave numbers: (a) $f_u, m_f=-1$, and ${\omega }_f=1.7$; (b) $\text{Re}\left(u_{F\overline{F}}\right), m_f=-1$, and ${\omega }_f=1.7$; (c) $f_u, m_f=-2$, and ${\omega }_f=3.9$; (d) $\text{Re}\left(u_{F\overline{F}}\right), m_f=-2$, and ${\omega }_f=3.9$; (e) $f_u, m_f=-3$, and ${\omega }_f=5.1$; and (f) $\text{Re}\left(u_{F\overline{F}}\right), m_f=-3$, and ${\omega }_f=5.1$.

Figure 14

Base-flow modifications for various azimuthal wave numbers, showing the longitudinal velocity components in each case: (a) $\text{Re}\left(u_{F\overline{F}}\right), m_f=1$, and ${\omega }_f=0.8$; (b) $\text{Re}\left(u_{F\overline{F}}\right), m_f=-1$, and ${\omega }_f=1.6$; (c) $\text{Re}\left(u_{F\overline{F}}\right), m_f=2$, and ${\omega }_f=0.4$; (d) $\text{Re}\left(u_{F\overline{F}}\right), m_f=-2$, and ${\omega }_f=3.9$; (e) $\text{Re}\left(u_{F\overline{F}}\right), m_f=3$, and ${\omega }_f=0.4$; (f) $\text{Re}\left(u_{F\overline{F}}\right), m_f=-3$, and ${\omega }_f=5.1$; and (g) $\text{Re}\left(u_{F\overline{F}}\right), m_f=0$, and ${\omega }_f=1.3$.