Rayleigh–Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II

Anisotropy in turbulent flows and in turbulent transport

We discuss, using simple analytical models and magnetohydrodynamic (MHD) simulations, the origin and parameters of turbulence and magnetic fields in galaxy clusters. Any pre-existing tangled magnetic field must decay in a few hundred million years by generating gas motions even if the electric conductivity of the intracluster gas is high. We argue that turbulent motions can be maintained in the intracluster gas and its dynamo action can prevent such a decay and amplify a random seed magnetic field by a net factor of typically 104 in 5 Gyr. Three physically distinct regimes can be identified in the evolution of turbulence and magnetic field in galaxy clusters. First, the fluctuation dynamo will produce microgauss (μG)-strong, random magnetic fields during the epoch of cluster formation and major mergers. At this stage pervasive turbulent flows with rms velocity of about 300 km s -1 can be maintained at scales of 100-200 kpc. The magnetic field is intermittent, has a smaller scale of 20-30 kpc and average strength of 2 μG. Secondly, turbulence will decay after the end of the major merger epoch; we discuss the dynamics of the decaying turbulence and the behaviour of magnetic field in it. Magnetic field and turbulent speed undergo a power-law decay, decreasing by a factor of 2 during this stage, whereas their scales increase by about the same factor. Thirdly, smaller-mass subclusters and cluster galaxies will produce turbulent wakes where magnetic fields will be generated as well. Although the wakes plausibly occupy only a small fraction of the cluster volume, we show that their area-covering factor can be close to unity, and thus they can produce some of the signatures of turbulence along virtually all lines of sight. The latter could potentially allow one to reconcile the possibility of turbulence with ordered filamentary gas structures, as in the Perseus cluster. The turbulent speeds and magnetic fields in the wakes are estimated to be of the order of 300 km s -1 and 2 μG, respectively, whereas the turbulent scales are of the order of 200 kpc for wakes behind subclusters of a mass 3 x 10 13 M ⊙ and about 10 kpc in the galactic wakes. Magnetic field in the wakes is intermittent and has the scale of about 30 and 1 kpc in the subcluster and galactic wakes, respectively. Random Faraday rotation measure is estimated to be typically 100-200 rad m -2 , in agreement with observations. We predict detectable polarization of synchrotron emission from cluster radio haloes at wavelengths 3-6 cm, if observed at sufficiently high resolution.

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Evolving turbulence and magnetic fields in galaxy clusters

Decaying homogeneous, isotropic turbulence is investigated using a phenomenological model based on the three-dimensional turbulent energy spectra. We generalize the approach first used by Comte-Bellot and Corrsin [J. Fluid Mech. 25, 657 (1966)] and revised by Saffman [J. Fluid Mech. 27, 581 (1967); Phys. Fluids 10, 1349 (1967)]. At small wave numbers we assume the spectral energy is proportional to the wave number to an arbitrary power. The specific case of power 2, which follows from the Saffman invariant, is discussed in detail and is later shown to best describe experimental data. For the spectral energy density in the inertial range we apply both the Kolmogorov −5/3 law, E(k)=Ce2/3k−5/3, and the refined Kolmogorov law by taking into account intermittency. We show that intermittency affects the energy decay mainly by shifting the position of the virtual origin rather than altering the power law of the energy decay. Additionally, the spectrum is naturally truncated due to the size of the wind tunnel tes...

On the decay of homogeneous isotropic turbulence

The term quantum turbulence denotes the turbulent motion of quantum fluids, systems such as superfluid helium and atomic Bose–Einstein condensates, which are characterized by quantized vorticity, superfluidity, and, at finite temperatures, two-fluid behavior. This article introduces their basic properties, describes types and regimes of turbulence that have been observed, and highlights similarities and differences between quantum turbulence and classical turbulence in ordinary fluids. Our aim is also to link together the articles of this special issue and to provide a perspective of the future development of a subject that contains aspects of fluid mechanics, atomic physics, condensed matter, and low-temperature physics.

https://www.pnas.org/content/pnas/111/Supplement_1/4647.full.pdf

Introduction to quantum turbulence

We develop a classical model for the decay of homogeneous and isotropic turbulence that takes into account the growth of the energy containing length scale and its saturation when it reaches the size of the containing vessel. The model describes our data on the decay of grid turbulence in helium II through 4 orders of magnitude in vorticity, as well as classical experiments on decaying turbulence.

Decay of grid turbulence in a finite channel

The results of a number of recent experiments on high Reynolds number grid turbulence in helium II suggest that its flow on large length scales resembles that of a classical fluid. It has been known for some time that the effective kinematic viscosity of this turbulent fluid, describing energy flow in the inertial range of wave numbers, is of the order of η/ρ where η is the normal fluid viscosity and ρ is the total density of the liquid. However, dissipation must be strongly influenced by quantum processes, and it cannot be associated simply with the normal-fluid viscosity. The importance of quantum processes arises because the dissipation occurs at small length scales, comparable with the spacing of the quantized vortex lines that allow turbulent motion in the superfluid component. We report an analysis of experimental data that allows us to deduce experimental values of the effective kinematic viscosity, which we call ν′(≠η/ρ), to which theories of the quantum dissipative processes can be compared.

Dissipation of grid turbulence in helium II

We have studied oscillatory flow through a 180° curved tube with the ratio of tube radius to radius of curvature equal to 1/7. The flow rate is constrained to vary sinusoidally about a non-zero mean at a specified period T, and mean flow rate Q. At a certain parameter range of interest Hamakiotes & Berger (1990) predict that fully developed flow undergoes a period-tripling bifurcation. Our measurements using laser-Doppler velocimetry found no bifurcation. An additional experiment was done to ensure that the flow was fully developed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211209300271X

Experimental investigation of periodic flow in curved pipes

Using superfluid helium II for the study of grid turbulence has many advantages. He II has the lowest kinematic viscosity of any known substance, enabling large Reynolds numbers to be achieved in a relatively small apparatus. In addition, the resolution of second sound attenuation measurements provides insights into the decay of turbulence behind a grid. From measurements of this decay, we report preliminary experimental observations pertaining to the effective kinematic viscosity of superfluid turbulence. We also provide a better interpretation of the relation between second sound attenuation measurements and vorticity.

Steven R. Stalp

Papers

Decay of grid turbulence in a finite channel

On the decay of homogeneous isotropic turbulence

Dissipation of grid turbulence in helium II

Experimental investigation of periodic flow in curved pipes

Effective kinematic viscosity of turbulence in superfluid 4He