Self-similar decay of high Reynolds number Taylor-Couette turbulence

Abstract

We study the decay of high-Reynolds number Taylor-Couette turbulence, i.e. the turbulent flow between two coaxial rotating cylinders. To do so, the rotation of the inner cylinder (Rei = 2×10 6 , the outer cylinder is at rest) is stopped within 12 s, thus fully removing the energy input to the system. Using a combination of laser Doppler anemometry and particle image velocimetry measurements, six decay decades of the kinetic energy could be captured. First, in the absence of cylinder rotation, the flow-velocity during the decay does not develop any height dependence in contrast to the well-known Taylor vortex state. Second, the radial profile of the azimuthal velocity is found to be self-similar. Nonetheless, the decay of this wall-bounded inhomogeneous turbulent flow does not follow a strict power law as for decaying turbulent homogeneous isotropic flows, but it is faster, due to the strong viscous drag applied by the bounding walls. We theoretically describe the decay in a quantitative way by taking the effects of additional friction at the walls into account.Turbulence is a phenomenon far from equilibrium: Turbulent flow is driven in one or the other way by some energy input and at the same time energy is dissipated, predominantly (but not exclusively) at the smaller scales. For statistically stationary turbulence, this balance is reflected in the famous picture of the Richardson-Kolmogorov energy cascade [1,2]. While the driving on large scales clearly is non-universal, depending on the flow geometry and stirring mechanism, the energy dissipation mechanism has been hypothesized to be self-similar [3][4][5][6][7][8].How exactly is the energy taken out of the system? A good way to find out is to turn off the driving and follow the then decaying turbulence, as then all scales are probed during the decay process. This has been done in various studies over the last decades for homogeneous isotropic turbulence (HIT). Experimentally, the focus of attention was on grid-induced turbulence [8][9][10][11][12][13][14][15], whereas in numerical simulations periodic boundary conditions were used [16][17][18][19]. To what degree the decay of the turbulence depends on the initial conditions [20][21][22] and whether or not it is selfsimilar has controversially been debated [5,11,16,[23][24][25][26][27]. We note that for HIT, already from dimensional analysis one obtains power laws for the temporal evolution of the vorticity and kinetic energy in decaying turbulence, namely ω(t) ∝ t −3/2 and k(t) ∝ t −2 , respectively, in good agreement with many measurements [10,12,28]. These scaling laws are also obtained [29] when employing the 'variable range mean field theory' of Ref. [30], developed for HIT. In that way, the late-time behavior, when the flow is already viscosity dominated, can also be calculated, allowing for the calculation of the lifetime of the decaying turbulence [29].However, real turbulence is neither homogeneous nor isotropic, but it has anisotropies and is wall-bounded, with a considerable fraction of the dissipa...

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

“…We find that the decay regime observed and discussed in Ref. [22] corresponds to an intermediate stage of decay. We do not find the proposed self-similar regime until the last life-stage which can be described as a quenching problem.…”

“…The experiment also has secondary flows due to residual cooling effects, which become important at later times and cause deviations from the models in Ref. [22].…”

“…The experiment also has secondary flows due to residual cooling effects, which become important at later times and cause deviations from the models in Ref. [22].In this manuscript, we numerically simulate Taylor-Couette turbulence, analogous to the experiments of Ref. [22],…”

“…[22].In this manuscript, we numerically simulate Taylor-Couette turbulence, analogous to the experiments of Ref. [22],…”

“…We simulate three cases: for the first case, which we denote as sudden stop as in the experiment (SS-EXP), we instantaneously stop the inner cylinder at t = 0, and allow the turbulence to decay. The main difference between the simulation and the experiment is that we stop the cylinders instantaneously, while the stopping time of the experiment by Verschoof et al [22] is around 12 s. Even if this is still orders of magnitude smaller than the viscous time (d 2 /ν = 6 · 10 3 s) it corresponds to over three-thousand large eddy turn-over times as mentioned previously (d/U = 3.2 · 10 −3 s).…”