Turbulence strength in ultimate Taylor–Couette turbulence

Rodrigo Ezeta,Chao Sun,Detlef Lohse,Sander G Huisman

doi:10.1017/jfm.2017.795

Abstract

We provide experimental measurements for the effective scaling of the Taylor–Reynolds number within the bulk $\mathit{Re}_{\unicode[STIX]{x1D706},\mathit{bulk}}$, based on local flow quantities as a function of the driving strength (expressed as the Taylor number $\mathit{Ta}$), in the ultimate regime of Taylor–Couette flow. We define $Re_{\unicode[STIX]{x1D706},bulk}=(\unicode[STIX]{x1D70E}_{bulk}(u_{\unicode[STIX]{x1D703}}))^{2}(15/(\unicode[STIX]{x1D708}\unicode[STIX]{x1D716}_{bulk}))^{1/2}$, where $\unicode[STIX]{x1D70E}_{bulk}(u_{\unicode[STIX]{x1D703}})$ is the bulk-averaged standard deviation of the azimuthal velocity, $\unicode[STIX]{x1D716}_{bulk}$ is the bulk-averaged local dissipation rate and $\unicode[STIX]{x1D708}$ is the liquid kinematic viscosity. The data are obtained through flow velocity field measurements using particle image velocimetry. We estimate the value of the local dissipation rate $\unicode[STIX]{x1D716}(r)$ using the scaling of the second-order velocity structure functions in the longitudinal and transverse directions within the inertial range – without invoking Taylor’s hypothesis. We find an effective scaling of $\unicode[STIX]{x1D716}_{\mathit{bulk}}/(\unicode[STIX]{x1D708}^{3}d^{-4})\sim \mathit{Ta}^{1.40}$, (corresponding to $\mathit{Nu}_{\unicode[STIX]{x1D714},\mathit{bulk}}\sim \mathit{Ta}^{0.40}$ for the dimensionless local angular velocity transfer), which is nearly the same as for the global energy dissipation rate obtained from both torque measurements ($\mathit{Nu}_{\unicode[STIX]{x1D714}}\sim \mathit{Ta}^{0.40}$) and direct numerical simulations ($\mathit{Nu}_{\unicode[STIX]{x1D714}}\sim \mathit{Ta}^{0.38}$). The resulting Kolmogorov length scale is then found to scale as $\unicode[STIX]{x1D702}_{\mathit{bulk}}/d\sim \mathit{Ta}^{-0.35}$ and the turbulence intensity as $I_{\unicode[STIX]{x1D703},\mathit{bulk}}\sim \mathit{Ta}^{-0.061}$. With both the local dissipation rate and the local fluctuations available we finally find that the Taylor–Reynolds number effectively scales as $\mathit{Re}_{\unicode[STIX]{x1D706},\mathit{bulk}}\sim \mathit{Ta}^{0.18}$ in the present parameter regime of $4.0\times 10^{8}<\mathit{Ta}<9.0\times 10^{10}$.

Highlights

Taylor–Couette (TC) flow, the flow between two coaxial co- or counter-rotating cylinders, is one of the idealized systems in which turbulent flows can be paradigmatically studied due to its simple geometry and its resulting accessibility through experiments, numerics and theory
We have measured local velocity fields using particle image velocimetry (PIV) in the ultimate regime of turbulence. We showed that both structure functions yield similar energy dissipation rate profiles that intersect within the bulk, to what is observed in Rayleigh–Bénard convection
When averaging these profiles within the bulk, this leads to an effective scaling ofbulk ∼ Ta1.40±0.04, which is the same scaling as obtained for the global quantitymeasured from the torque scaling (Huisman et al 2014; Ostilla-Mónico et al 2014)

Summary

Introduction

Taylor–Couette (TC) flow, the flow between two coaxial co- or counter-rotating cylinders, is one of the idealized systems in which turbulent flows can be paradigmatically studied due to its simple geometry and its resulting accessibility through experiments, numerics and theory. In its rich and vast parameter space, various different flow structures can be observed

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Results

Conclusion