A model for limit-cycle switching in open cavity flow

Abstract

A reduced mathematical model for the flow in an open cavity is presented. The reduction is based on the center manifold theory applied to a perturbation of the original system which allows for a codimension two bifurcation point. The model exhibits many of the key characteristics observed in the flow dynamics including unstable quasi-periodic edge states as well as switching of limit cycles with parameter variations. An explanation for the exchange of stabilities of the limit-cycles is presented based on the cross-coupling terms of the two amplitude equations.

Figures (4)

• Figure 1: (Top) streamwise velocity of the stationary base flow at $\mathrm{Re}_{c}=4131.33$ and (bottom) the spectrum (blue dots) obtained at the bifurcation point.
• Figure 2: Real part of the time varying response of the reduced system at (left) $\mathrm{Re}=4200$ and (center) $\mathrm{Re}=4500$. The labels $z_{1}$ (blue), and $z_{3}$ (orange) correspond to the amplitudes of the modes $\lambda_{1}$ and $\lambda_{3}$ respectively. The figure on the right shows the time evolution of peak of the oscillation amplitudes for the two modes $z_{1},z_{3}$, and for the two different Reynolds numbers, $Re=4200$ (solid lines) and $Re=4500$ (dashed lines).
• Figure 3: Evolution of the LCO null-clines as the Reynolds number is increased past the first bifurcation point. The solid blue line indicates the null-cline of the $\lambda_{1,2}$-LCO emerging from the first bifurcation. The solid orange line indicates the null-clines of the $\lambda_{3,4}$-LCO emerging from the second bifurcation point. The intersection of these LCO null-clines produces the quasi-periodic edge-state.
• Figure 4: (Left) Comparison of the angular frequencies of the full system and the reduced model. The blue line indicates the $\lambda_{1,2}$ LCO angular frequencies while the orange line indicates the $\lambda_{3,4}$ LCO angular frequencies. The solid black circles indicates the frequencies obtained from non-linear simulations. And (right), the bifurcation diagram obtained in terms of the equilibrium LCO amplitudes $|z|$. The blue lines indicate $|z_{1}|$ while the orange line indicates $|z_{3}|$. The solid lines indicate stable equilibria amplitudes while the dotted part of the lines indicate unstable equilibria amplitudes. The dashed lines indicate amplitudes for the (unstable) QP state. The $-\cdot$ gray vertical lines mark the successive bifurcations points identified in the work ($\widetilde{\mathrm{Re}}_{2}=4349.6$, $\widetilde{\mathrm{Re}}_{3}=4415.6$ and $\widetilde{\mathrm{Re}}_{4}=4853.8$).