Self-Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding in the Cylinder Wake

The Bénard–von Kármán vortex shedding instability in the wake of a cylinder is perhaps the best known example of a supercritical Hopf bifurcation in fluid dynamics. However, a simplified physical description that accurately accounts for the saturation amplitude of the instability is still missing. Here, we present a simple self-consistent model that provides a clear description of the saturation mechanism and quantitatively predicts the saturated amplitude and flow fields. The model is formally constructed by a set of coupled equations governing the mean flow together with its most unstable eigenmode with finite size. The saturation amplitude is determined by requiring the mean flow to be neutrally stable. Without requiring any input from numerical or experimental data, the resolution of the model provides a good prediction of the amplitude and frequency of the vortex shedding as well as the spatial structure of the mean flow and the Reynolds stress.

Figure 1

Growth rate ${\sigma }_1$ for the converged coupled system of equations of the self-consistent model (4) for different Reynolds stress forcing amplitudes $A_f^2$ at $\mathrm{Re}=100$. The insets show the spatial distribution of the divergence of the Reynolds stress in the $x$ direction and the boundary of the recirculation region for different $A_f^2$, as indicated in the figure.

Figure 2

Comparison of the $x$ component of the mean flows (a) and the Reynolds stress divergence (b) for $\mathrm{Re}=100$ computed from DNS (top half) and predicted from Eq. (5) (bottom half, SC). Plot (c) shows horizontal and vertical cuts for $y=0$ (top), $x=1$ (bottom left), and $x=3$ (bottom right), for the $x$ component of the mean flows ($\mathit{U}$, ${\mathit{U}}^{*}$), the base flow (${\mathit{U}}_B$), and Reynolds stress divergence ($\mathit{F}=⟨({\mathit{u}}^{\prime }·\nabla ){\mathit{u}}^{\prime }⟩$), as detailed in the legend.

Figure 3

Comparison of the frequency predictions of the self-consistent model for different Reynolds numbers. Vortex shedding frequency from experiments by Williamson [12] (dashed black line), from the present result (blue squares), and from the most unstable mode of the base flow (solid red line) and the mean flow obtained from DNS (red circles).

Figure 4

Comparison of the vortex shedding amplitude predictions of the self-consistent model for different Reynolds numbers. Saturation amplitude obtained from DNS (red circles), predicted by the self-consistent model (blue triangles), and as given according to weakly nonlinear expansion around threshold (dash-dotted line).