Local origin of mode-B secondary instability in the flow past a circular cylinder

We perform a three-dimensional, short-wavelength stability analysis on the numerically simulated two-dimensional, unsteady wake resulting from a uniform flow past a circular cylinder at Reynolds numbers in the range $50\le \text{Re}\le 300$; here, $\text{Re}=U_{\infty }D/\nu$ with $U_{\infty }$, $D$, and $\nu$ being the free-stream velocity, the diameter of the cylinder, and the kinematic viscosity of the fluid, respectively. For a given Re, inviscid local stability equations from the geometric optics approach are solved on all the closed fluid particle trajectories with the same period as the time period of the flow. For each of the six distinct closed trajectories, denoted orbits 1–6, the growth rate is plotted as a function of Re for purely transverse perturbations. The inviscid instability on orbits 1 and 2, which are symmetric counterparts of one another, is shown to undergo bifurcations at $\text{Re}\approx 50$ and $\text{Re}\approx 250$. The bifurcation at $\text{Re}\approx 250$ from an asynchronous to a synchronous instability is argued to be related to the emergence of mode-B secondary instability, which experimental, numerical, and theoretical studies have shown to occur at $\text{Re}\approx 260$. Incorporating finite-wave-number, finite-Reynolds-number effects in the local stability framework, and comparing the corrected local stability growth rates with the global stability growth rates further support our main conclusion: The bifurcation to three-dimensional short-wavelength synchronous instability on orbits 1 and 2 is a possible local mechanism for the emergence of mode-B secondary instability. The relevance (or lack thereof) of the local instabilities on the other orbits is also discussed.

Figure 1

A schematic of the computational domain and its boundaries in the numerical simulations of the base flow.

Figure 2

The six closed trajectories with the same period as the time period of the unsteady base flow at $\text{Re}=300$ (left panel). The frame within the dashed line is magnified on the right to show the shape of the trajectories. Trajectories that are topologically similar to orbits 1, 2, and 3 are found for all $\text{Re}\in \left[50,300\right]$, whereas those similar to orbits 4, 5, and 6 are found for all $\text{Re}\in \left[180,300\right]$. The uniform free-stream veloctiy $U_{\infty }$ is along the positive $x$ axis.

Figure 3

$\left(\text{a}\right)$ Inviscid growth rate ${\sigma }_0^{re}$ and $\left(\text{b}\right)$ the imaginary part ${\sigma }_0^{im}$ of the complex growth rate [Eq. (6)], for orbits 1 and 2 (solid line) and orbit 3 (dashed line) plotted as a function of Re for purely transverse perturbations. Corresponding values from Giannetti [27] for $\text{Re}=190$ and 260 are indicated by $◯$ (orbits 1 and 2) and $▵$ (orbit 3). The markers on the curves indicate the actual values of Re at which the growth rate calculations were performed.

Figure 4

Path traced by the $\left(\text{a}\right)$ Floquet multiplier $E_1$, and $\left(\text{b}\right)$ the corresponding Floquet exponent ${\sigma }_0$ [Eq. (6)] on the complex plane as Re is varied in the vicinity of $\text{Re}=250$. Indicated right next to every datapoint is the corresponding value of Re, and the arrows on the curves indicate the direction of increasing Re. The dashed curve in panel $\left(\text{a}\right)$ is the unit circle, outside of which lie the unstable Floquet multipliers. In panel $\left(\text{b}\right)$, ${\sigma }_0^{re}>0$ corresponds to instability. Both $E_1$ and ${\sigma }_0$ in this figure correspond to orbits 1 and 2.

Figure 5

$\left(\text{a}\right)$ Inviscid growth rate ${\sigma }_0^{re}$ and $\left(\text{b}\right)$ the imaginary part ${\sigma }_0^{im}$ of the complex growth rate [Eq. (6)], for orbits 4 and 5 (solid line with square markers), and orbit 6 (dashed line with star markers) plotted as a function of Re for purely transverse perturbations. The markers on the curves indicate the actual values of Re at which the growth rate calculations were performed.

Figure 6

[(a)–(d)] The growth rate with only the viscous correction: ${\sigma }_{\nu }^{*}={\sigma }_0^{re}-{\beta }^{*2}/\text{Re}$ and the maximum corrected growth rate: ${\sigma }_c^{*}={\sigma }_0^{re}-3{\beta }^{*2}/\text{Re}$ (with ${\beta }^{*}=7.64$ corresponding to critical wavelength of $0.822D$ for mode B) as a function of Re (solid lines with markers) for orbits 1, 3, 4, and 6, respectively. The inviscid growth rate ${\sigma }_0^{re}$ is reproduced here for comparison (solid line with no markers). The range of Re for which the complex growth rate [Eq. (6)] has nonzero imaginary part (asynchronous instability) has been highlighted by the green background color in panels (a) and (c).

Figure 7

The corrected growth rate ${\sigma }_c$ [Eq. (8)] as a function of $\beta$ using two methodologies M1 and M2 for orbits 1 and 3 at $\text{Re}=280$. For comparison, the global stability growth rates from Barkley and Henderson [12] are plotted using open circles, and referred to as B&H in the legends.