Von Kármán flows

The von Kármán swirling flow (generally mistaken for the vortex street named after the same Theodore von Kármán) is produced by the rotation of two face-to-face discs in a cylinder filled with fluid. For bladed discs and a low-viscosity fluid, the inertial stirring produces a high-intensity, large Reynolds number flow.

von Karman flow, seeded with bubbles. The two impellers (in black) are visible.
von Kármán flow seeded with air bubbles. The two impellers (in black) are visible.

This flow has been studied to answer a few fundamental open questions on turbulence. One of them is the symmetry restoration in turbulent flows. Despite their highly random instantaneous aspect, turbulent flows recover – statistically, and at small scales – the symmetries of the (forcing) boundary conditions. This flow proposes to answer the following question :

Is the small-scalle statistical symmetry restoration mechanism in turbulent flows valid at larger scales ?

One important result on this topic has been evidenced by Florent Ravelet, Louis Marié, Arnaud Chiffaudel et François Daviaud. They showed that the von Kármán flow kept a memory allowing to statistically remain in an asymmetric statistical state despite imposing back a symmetrical forcing. The flow exhibits a statistical hysteresis.

Hence, when the two impellers rotating at imposed frequencies f_1 (bottom) and f_2 (top), we can impose f_1 = f_2 and have a symmetrical forcing with respect to impeller exchange, but to obtain a flow that will indefinitely remain oriented towards one of the impellers. An analogy with ferromagnetic systems provides insights to plot the aspect the the hysteresis cycle when f_1 et f_2 are varied.

For large Reynolds number, the relevant dimensionless variables are :

\theta = \frac{f_1-f_2}{f_1 + f_2}

and the reduced torque difference :

\gamma = \frac{T_1-T_2}{T_1 + T_2}

The cycle has therefore this aspect :

VK_gth
Hysteresis cycle of the mean statistical states of the von Kármán flow. Black squares, states at imposed impeller frequency. Circles, states reached at imposed torques.

The frequency imposed cycle shows indeed how at well-chosen values of \theta, especially 0, more than one statistically steady state may coexist. The full cycle is obtained scanning \theta from -1 to 1 and 1 to -1 and shows three branches, the central one being metastable. Between the central and the external branches, a \gamma range is ``forbidden''.

During my PhD thesis, I focused my studies on torque imposed experiments, in which C_1 and C_2 are imposed. For well-chosen values of C_1 and C_2, we can impose \gamma in the forbidden region. We expect no statistical steady state to survive in this region. This result is confirmed by the experiments.

Evolution of the impeller frequency for various gamma values
Temporal series of the impeller rotation frequencies f_1 (dark) and f_2 (light) for several normalized torque differences \gamma. The color scale refers to the colors of the circles in the hysteresis cycle figure.

The study of the large-scale statistical properties of the multi-stable flow for (moderately) long times has yielded several scientific articles. Other statistical properties induced when changing the rotation sense of the impellers have also been published (see Pierre-Philippe Cortet et al., New Journal of Physics, 2011).

One PhD thesis (by Denis Kuzzay) has followed such work to investigate the dissipative structures down to the Kolmogorov (or below) scales. The SHREK collaboration investigates whether quantum turbulence dissipates energy in a similar fashion.

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