Life stages of wall-bounded decay of Taylor-Couette turbulence

The decay of Taylor-Couette turbulence, i.e., the flow between two coaxial and independently rotating cylinders, is numerically studied by instantaneously stopping the forcing from an initially statistically stationary flow field at a Reynolds number of $\text{Re}=3.5\times {10}^4$. The effect of wall friction is analyzed by comparing three separate cases, in which the cylinders are either suddenly made no-slip or stress-free. Different life stages are observed during the decay. In the first stage, the decay is dominated by large-scale rolls. Counterintuitively, when these rolls fade away, if the flow inertia is small a redistribution of energy occurs and the energy of the azimuthal velocity behaves nonmonotonically, first decreasing by almost two orders of magnitude and then increasing during the redistribution. The second stage is dominated by non-normal transient growth of perturbations in the axial (spanwise) direction. Once this mechanism is exhausted, the flow enters the final life stage, viscous decay, which is dominated by wall friction. We show that this stage can be modeled by a one-dimensional heat equation, and that self-similar velocity profiles collapse onto the theoretical solution.

Figure 1

Top left: Temporal evolution of the kinetic energy in the azimuthal velocity. Bottom left: Temporal evolution of the wind kinetic energy. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red), and SS-V0 (blue).

Figure 2

Temporal evolution of the energy dissipation. The figures show the same data in semilogarithmic scale (right) and double logarithmic scale (left). Symbols: SS-EXP (green), SD (red), SS-V0 (blue), experimental data from Ref. [22] (right, data matches top temporal axis only).

Figure 3

Pseudocolor visualization of the azimuthal velocity in a constant-azimuth cut for the SS-V0 case at four different times (left to right): $\tilde{t}=0$, 10, 75, and 5000.

Figure 4

Pseudocolor visualization of the azimuthal velocity in a constant-azimuth cut for the SS-EXP case at five different times (left to right): $\tilde{t}=0$, 10, 100, 1000, and 17000.

Figure 5

Left: Azimuthally and axially averaged azimuthal kinetic energy dissipation for the SS-V0 case, and $\tilde{t}=0$ (blue), $\tilde{t}=10$ (orange), $\tilde{t}=30$ (green), and $\tilde{t}=50$ (red). Right: The same for SS-EXP case and $\tilde{t}=0$ (blue), $\tilde{t}=10$ (orange), and $\tilde{t}=50$ (green).

Figure 6

Pseudocolor visualization of the azimuthal velocity in a constant-radius cut at the mid-gap ($r=r_a$) for the SS-V0 case at two different times $\tilde{t}=75$ (top) and 1000 (bottom).

Figure 7

Left: Axially and azimuthally averaged normalized azimuthal velocity profiles for the SS-V0 case at $\tilde{t}=500$ (blue), $\tilde{t}=1000$ (green), and $\tilde{t}=5000$ (orange), as well as the theoretical solution based on Bessel functions (black dashed). Right: The same for the SS-EXP case at times $\tilde{t}=10$ (blue), $\tilde{t}=100$ (orange), $\tilde{t}=1000$ (green), $\tilde{t}=5000$ (red), and $\tilde{t}=15000$ (purple).