Vortex nucleation, transition to turbulence, and cavitation: “system failure” experiments in liquid helium and extreme value statistics

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Abstract

It is suggested that recent experiments in liquid helium, like single vortex nucleation, transition to turbulent flow around a sphere at a critical velocity, and cavitation of the liquid in a sound wave belong to the type of “system failure” experiments which is well known in reliability testing and whose statistical properties are described by extreme value statistics. This leads to far reaching consequences for the interpretation of the critical velocities and of the voltage threshold for cavitation.

Introduction

The high-voltage breakdown of an insulator, the breaking strength of a piece of material, the lifetime of vacuum tubes or semiconductor devices, e.g., are typical “system failure” experiments in which a system (a device consisting of components, a block of material, etc.) is tested either for its lifetime under constant conditions of operation or by changing some external load (voltage, temperature, pressure, etc.) until it fails to function. In all these experiments the so-called “weakest link” principle is at work: the system fails when its weakest part fails. Similarly, when superfluid helium flows through an orifice or around a solid body, vortex lines will be generated first at that position on the solid surface where the barrier for nucleation (or the critical velocity) is minimal. And in experiments on cavitation, like in high-voltage breakdown, the threshold will be determined by the weakest nucleation site. Therefore, it is plausible to analyze these experiments in a similar way as the other failure experiments.

Section snippets

Extreme value statistics

The statistical analysis of an experiment in which the minimum (or maximum) of a sequence of independent random variables xi(i=1,…,n) having a common cumulative distribution function (CDF) F(x), is measured (all other values xi need not be observable) requires the mathematical tool of “extreme value theory” or “extreme value statistics” (EVS) [1]. It is a central result of EVS to derive the particular extreme value distribution (EVD) of the minimum in the limit of large sample size n.

Data analysis

Suppose the measured CDF is the Gompertz as in vortex nucleation [2], [3] and cavitation [5]. It is determined by the two parameters an and bn:H1(x)=1−exp(−exp(an(x−bn))),an is the shape parameter which is related to the “width” of the CDF and bn is the shift parameter which enters into the definition of a “critical” velocity or a “threshold” voltage, respectively, by the median of the CDF. Both parameters depend not only on the particular parent distribution but also on the sample size n.

Conclusions

Two things are clearly necessary for a more detailed interpretation of the measured EVDs:

  • 1.

    A physical model for the parent distribution F(x) of the nucleation sites, barriers, or critical velocities. In case of vortex nucleation this will include information on the structure of the surface, e.g., the statistical distribution of its roughness [1], [8]. How the roughness then affects the distribution of the nucleation barrier or the critical velocity is an open question. This problem appears to be

Acknowledgments

A helpful correspondence with Professor D. Pfeifer (Oldenburg University) on extreme value statistics is gratefully acknowledged.

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