Turbulence In He Research Articles

A Mathematical Analysis of the Intermediate Behaviour of the Energy Cascades of Quantum Turbulence

We propose a mathematical interpolation between several regimes of energy cascade in quantum turbulence in He II. On the basis of a physical interpretation of such mathematical expression we discuss in which conditions it is expected to appear an intermediate k^{2} regime (equipartition regime) in the transition region between the hydrodynamic regime and the Kelvin wave regime (namely, between the k^{-5/3} and k^{-1} regions in coflow situations and between the k^{-3} and k^{-1} regions in counterflow situations). It is seen that if the energy rate transfer from the hydrodynamic region to the Kelvin wave region is sufficiently slow, such equipartition region will be present, but for higher values of such energy rate transfer it will disappear. For high rates of the energy rate transfer, the transition regime between the hydrodynamic and the Kelvin wave regimes will be monotonous, characterized by a negative exponent of k between -5/3 and −1 (or between −3 and −1), instead of the positive 2 exponent of the equipartition regime.

Statistical properties of homogeneous and isotropic turbulence in He II measured via particle tracking velocimetry

Despite being a quantum two-fluid system, superfluid helium-4 (He II) is observed to behave similarly to classical fluids when a flow is generated by mechanical forcing. This similarity has brought up the feasibility of utilizing He II for high Reynolds number classical turbulence research, considering the small kinematic viscosity of He II. However, it has been suggested that the non-classical dissipation mechanism in He II at small scales may alter its turbulent statistics and intermittency. In this work, we report our study of a nearly homogeneous and isotropic turbulence (HIT) generated by a towed grid in He II. We measure the velocity field using particle tracking velocimetry with solidified deuterium particles as the tracers. By correlating the velocities measured simultaneously on different particle trajectories or at different times along the same particle trajectory, we are able to conduct both Eulerian and Lagrangian flow analyses. Spatial velocity structure functions obtained through the Eulerian analysis show scaling behaviors in the inertial subrange similar to that for classical HIT but with enhanced intermittency. The Lagrangian analysis allows us to examine the flow statistics down to below the dissipation length scale. Interestingly, strong deviations from the classical scaling behaviors are observed in this regime. We discuss how these deviations may relate to the motion of quantized vortices in the superfluid component in He II.

The Use of Second Sound in Investigations of Quantum Turbulence in He II

Starting from a historic perspective, we discuss the use of second sound in experimental investigations of quantized vorticity and quantum turbulence in the two-fluid temperature regime of superfluid $$^4$$ He. Starting with the theoretical prediction of second sound by Tisza and Landau and its experimental discovery by Peshkov, we briefly review the pioneering experiments of Hall, Vinen and others that have contributed in an essential way to our current understanding of quantum turbulence in superfluid $$^4$$ He, with an emphasis on relevant research performed over the last two decades in our laboratory in Prague. We then propose further dedicated experiments where second sound can be used both as a generator and detector of quantum turbulence.

An apparatus for generation and quantitative measurement of homogeneous isotropic turbulence in He ii.

The superfluid phase of helium-4, known as He ii, exhibits extremely small kinematic viscosity and may be a useful tool for economically producing and studying high Reynolds number turbulent flow. Such applications are not currently possible because a comprehensive understanding of the complex two-fluid behavior of He ii is lacking. This situation could be remedied by a systematic investigation of simple, well controlled turbulence that can be directly compared with theoretical models. To this end, we have developed a new apparatus that combines flow visualization with second sound attenuation to study turbulence in the wake of a mesh grid towed through a He ii filled channel. One of three mesh grids (mesh number M = 3, 3.75, or 5 mm) can be pulled at speeds between 0.1 and 60 cm/s through a cast acrylic flow channel which has a 16 mm × 16 mm cross section and measures 330 mm long. The motion of solidified deuterium tracer particles, with diameter of the order 1 μm, in the resulting flow is captured by a high speed camera, and a particle tracking velocimetry algorithm resolves the Lagrangian particle trajectories through the turbulent flow field. A pair of oscillating superleak second sound transducers installed in the channel allows complementary measurement of vortex line density in the superfluid throughout the turbulent decay process. Success in early experiments demonstrates the effectiveness of both probes, and preliminary analysis of the data shows that both measurements strongly correlate with each other. Further investigations will provide comprehensive information that can be used to address open questions about turbulence in He ii and move toward the application of this fluid to high Reynolds number fluid research.

Visualization of grid-generated turbulence in He II using PTV

Due to its low viscosity, cryogenic He II has potential use for simulating large-scale, high Reynolds number turbulent flow in a compact and efficient apparatus. To realize this potential, the behavior of the fluid in the simplest cases, such as turbulence generated by flow past a mesh grid, must be well understood. We have designed, constructed, and commissioned an apparatus to visualize the evolution of turbulence in the wake of a mesh grid towed through He II. Visualization is accomplished using the particle tracking velocimetry (PTV) technique, where μm-sized tracer particles are introduced to the flow, illuminated with a planar laser sheet, and recorded by a scientific imaging camera; the particles move with the fluid, and tracking their motion with a computer algorithm results in a complete map of the turbulent velocity field in the imaging region. In our experiment, this region is inside a carefully designed He II filled cast acrylic channel measuring approximately 16 × 16 × 330 mm. One of three different grids, which have mesh numbers M = 3, 3.75, or 5 mm, can be attached to the pulling system which moves it through the channel with constant velocity up to 600 mm/s. The consequent motion of the solidified deuterium tracer particles is used to investigate the energy statistics, effective kinematic viscosity, and quantized vortex dynamics in turbulent He II.

Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium

We report observation of an inverse energy cascade in second sound acoustic turbulence in He II. Its onset occurs above a critical driving energy and it is accompanied by giant waves that constitute an acoustic analogue of the rogue waves that occasionally appear on the surface of the ocean. The theory of the phenomenon is developed and shown to be in good agreement with the experiments.

On Decaying Counterflow Turbulence in He II

We review available experimental and numerical data on decaying counterflow turbulence in He II and present our current understanding of the underlying physics. We take into account the temperature gradient present in the steady-state counterflow turbulence, the fact that in the early stage of the decay the turbulence is still thermally driven, and the fact that at the beginning of the decay the vortex tangle is strongly polarized. When the heater that generates the counterflow turbulence is switched off, the vortex tangle decays, the vortex lines randomize their spatial orientation and the tangle’s polarization decreases. The process of depolarization slows down the recovery of the transverse second sound signal which measures the vortex line density; at some values of parameters it even leads to a net decrease of the amplitude of the transverse second sound prior to reaching the universal −3/2 power temporal law decay typical of classical homogeneous isotropic turbulence in a finite-sized channel.

Depolarization of decaying counterflow turbulence in He II

We present experimental evidence backed up by numerical simulations that the steady-state vortex tangle created in He II by heat-transfer counterflow is strongly polarized. When the heater that generates the counterflow turbulence is switched off, the vortex tangle decays, the vortex lines randomize their spatial orientation and the tangle's polarization decreases. The process of depolarization slows down the recovery of the transverse second sound signal which measures the vortex line density; at some values of parameters it even leads to a net decrease of the amplitude of the transverse second sound prior to reaching the universal -32 power temporal law decay typical of classical homogeneous isotropic turbulence in a finite-sized channel.

Energy spectra of quantum turbulence in He II and 3He-B: A unified view

Quantum turbulence in superfluid He II and in 3He-B that can be regarded as nearly isothermal, isotropic, and homogeneous is discussed within the two-fluid model. A general form of the 3D energy spectrum is proposed: at large length scales, where normal and superfluid eddies are locked together by the mutual friction force, the energy spectrum is essentially classical and includes an inertial range of a Kolmogorov K62 form. With increasing wavenumber k, the normal fluid part of the spectrum terminates due to finite viscosity, while the superfluid part of the spectral energy density changes towards k −3 and then back into Kolmogorov-like k −5/3 again. Agreement with computer simulations and experiments is claimed if account is taken of the turbulent box size and of the energy decay rate.

Decaying Counterflow Turbulence in He II

Decaying Counterflow Turbulence in He II

Decay of counterflow He II turbulence in a finite channel: possibility of missing links between classical and quantum turbulence.

Decay of He II counterflow turbulence generated by applying a heat pulse to the closed end of a circular channel is investigated using second sound attenuation. Several orders of magnitude of decaying vortex line density display various regimes, starting with the inverse time decay predicted by Vinen towards distinct classical t(-3/2) power law occurring after saturation of the energy containing length scale by the size of the channel. The decaying counterflow turbulence displays a surprisingly close resemblance to recently reported decaying grid turbulence in He II, and both of them appear closely linked to classical grid generated turbulence.

Quantum turbulence in He II induced by second sound shock pulses

Direct measurements are presented of quantum turbulence in He II induced by second sound shock (SSS) pulses in a wide channel. Through the use of a leaky capacitor fit, a growth and decay characterization of the quantum turbulence is extracted from the measurements. As an example of this, a plot is shown of peak excess attenuation, which is proportional to the quantum turbulence, versus initial pulse energy. Also taken is an explicit energy account of the quantum turbulence as well as of the SSS pulses that induced the turbulence.

Techniques for the detection of second sound shock pulses and induced quantum turbulence in He II

This paper discusses the design and operation of thin-film thermometers and heaters for the detection and generation of second sound shock (SSS) pulses and second sound resonance (SSR) waves. The detection of quantum turbulence at multiple points along a channel is also discussed. An instrumentation package to simultaneously detect SSS pulses and induced quantum turbulence in He II has been developed. This system is able to generate and detect SSS pulses in a channel of He II while monitoring for induced quantum turbulence at multiple points along the channel. The system monitors for quantum turbulence by the attenuation of SSR waves. The experiments using this instrumentation package offer the first opportunity to directly measure the quantum turbulence time and distance evolution of SSS pulses. These experiments have applications in transient heat transfer problems in He II, such as the cooling and stability of superconductors.

Simplified hydrodynamic analysis of superfluid turbulence in He II: Internal dynamics of inhomogeneous vortex tangle.

The hydrodynamic theory of superfluid turbulence is presented in a new simplified form. It applies to flow situations frequently encountered in practice, in which the thermohydrodynamic environment of a superfluid turbulent tangle of quantized vortices may be considered, in a first order of approximation, as given. Flow quantities like the mass density, the entropy density, and the drift velocities of mass and elementary excitations act, accordingly, as external parameters with respect to the internal dynamics of the vortex tangle. The internal dynamics is completely specified by a kinematic equation governing the time evolution of the line-length density of the quantized vortices and a dynamic equation involving the impulse density of the vortex tangle. The derivation of these equations starts from a variational principle that is reminiscent of Hamilton's principle in classical mechanics and proceeds, in order to include dissipative effects, by using methods of the thermodynamics of irreversible processes. A new quantity called superfluid turbulent pressure is introduced which shows many properties that are familiar from the ordinary pressure in a classical fluid. Two important particular cases are considered in more detail, viz., homogeneous superfluid turbulent flow and flow situations in which the vortex tangle is in permanent internal equilibrium. When diffusion of the vortex-tangle impulse is taken into account and dispersive effects are disregarded, the dynamic equation of the vortex tangle assumes, in the case of internal equilibrium, the form of Burgers' equation with a nonlinear source term. This equation, which is new, may be considered as a natural generalization of Vinen's equation to inhomogeneous superfluid turbulence. Some exact solutions which represent uniformly propagating superfluid turbulence fronts are listed in the Appendix.

Free decay of homogeneous superfluid turbulence in Helium II

The equations governing the one-dimensional transient behaviour of homogeneous superfluid turbulence in He II that were originally considered by Vinen and Schwarz and recently extended by one of the authors are applied to the free decay of a homogeneous vortex tangle in which the gradients of the temperature and the thermodynamic potential vanish. It turns out that the equations admit a solution in closed form which may be expressed in terms of incomplete beta-functions. The general solution exhibits three distinct branches. On one of the branches the decay falls apart in an initial stage of fast decay and a final stage of slow decay separated by a short transition region. This mode of decay is qualitatively similar to the experimentally observed decay of a vortex tangle that has recently been reported by Schwarz and Rozen (1991) and by Olszok et al. (1994). An alternative mode of decay that is physically meaningful is represented by the other two branches when joined together. The behaviour of that mode resembles, in a qualitative way, the experimental decay originally observed by Vinen (1957).

Chaotic quantized vortices and hydrodynamic processes in superfluid helium

Superfluid turbulence in He II flows appears as a stochastic tangle of quantized vortex lines. Interest in this system extends beyond the field of superfluid helium to include the study of statistical physics of extended objects and classical turbulence. A wealth of information concerning the vortex tangle has been supplied by experiments, by phenomenological models, and by numerical simulation. In the restricted case of stationary homogeneous flows, a good correlation between theory and experiment has already been demonstrated. This review therefore concentrates on nonstationary and nonhomogeneous processes with the goal of understanding the dynamical properties of the vortex tangle. The first part of this paper reviews the concept of the stochastic vortex tangle with emphasis on the work of Vinen and Schwarz. Special attention is given to justifying Vinen's equation as a basis for the study of nonstationary processes. This analysis reveals that the interpretation of experimental results requires a careful account of the mutual influence of the vortex tangle and the flow of the supefluid. The second part of this paper reviews the development of the required hyrodynamic equations of superfluid turbulence. These equations are applied to a number of examples such as linear and nonlinear sound, heat pulses, and fluctuations. The role of stochastic vortices in the phase-transition problem is discussed. Attention is focused on the relation between hydrodynamic effects and the vortex tangle properties. Finally, the main results concerning the stochastic vortex tangle are resumed and some of the important questions are pointed out.

Hydrodynamics of superfluid turbulence in He II: three-dimensional theory

The one-dimensional hydrodynamic theory for superfluid turbulent flow of He II developed recently by one of the authors is extended to the three-dimensional case. The impulse density of the entangled mass of quantised vortices continues to play an important part in the analysis. The non-dissipative equations of motion are derived from an appropriate form of Hamilton's variational principle, while the dissipative effects are taken into account systematically according to the thermodynamics of irreversible processes. A number of helicity conservation theorems appears to be valid in the case of non-dissipative flow. Particular attention is paid to homogeneous superfluid turbulence. It turns out that the two states allowed in one-dimensional homogeneous flow, viz. the state in which the direction of the vortex tangle impulse is parallel and the one in which it is anti-parallel to the counterflow velocity, are, respectively, stable and metastable. The analysis might assist in clarifying the one-dimensional free decay of homogeneous superfluid turbulence.

Hydrodynamic theory of superfluid turbulence in He II and Schwarz's vortex modelling

A recent hydrodynamic theory of superfluid turbulence in He II developed by the author is applied to the dynamics of a homogeneous vortex tangle. The theory contains five unknown macroscopic coefficients. These coefficients are determined by relating equations of the hydrodynamic theory to similar equations obtained by K.W. Schwarz (1988) by means of an averaging procedure applied to a microscopically modelled homogeneous vortex tangle. A close correspondence is established between the two approaches. The validity of the hydrodynamic theory, however, is more general because it also applies to inhomogeneous flow situations. Some speculations about a possible numerical simulation of the voltex tangle impulse conclude the paper.

Hydrodynamics of quantum turbulence in He II: Vinen's equation derived from energy and impulse of vortex tangle

A macroscopic theory of superfluid turbulence is developed in which the vortex tangle is characterised by both its line-length density and its drift velocity. Three velocity fields are accordingly distinguished, viz. the mass velocity υ, the normal-fluid velocity υ n and the drift velocity υ e of the vortex tangle. The introduction of the drift velocity of the tangle as an additional variable seems to be new. The non-dissipative equations of motion are derived from a generalised form of Lin's variational principle. Both the energy and the impulse of the vortex tangle into account. It is shown that the effective mass density of the vortex tangle vanishes. In equilibrium the relative velocity of the vortex tangle is given by the derivative of the energy of the tangle with respect to the tangle impulse. A similar relation holds for a ring vortex in a fluid of infinite extent. When dissipative terms are added to the equations according to the thermodynamics of irreversible processes the Vinen equation follows immediately. The derivation suggests a new interpretation of the right-hand member of the Vinen equation in terms of the derivative of a potential energy. A similar potential energy has been used in investigations on vortex nucleation and critical velocities. The theory is extended by including the effects of large gradients of the line-length density. The corresponding generalisation of the Vinen equation is presented.

Noise and superfluid turbulence in He II: Theory

Noise may be used as a probe of the dynamics of a nonlinear system. Weak noise provides information on the local dynamics near a transition. This information can be helpful for identifying the underlying bifurcation. The normal form of the bifurcation can then be used as a starting point for a phenomenological model of the system. Such an approach furnishes a unified explanation of the experimental data near the transition between superfluid turbulent states TI and TII in liquid helium counterflow. Strong external noise probes global features of the nonlinearities of a system. It may also dramatically affect the dynamical behavior of the system and give rise to noise-induced transitions. The problems of modeling this phenomenon in the liquid helium counterflow system are discussed and a first model, providing a unified description of the experimental observations for weak and strong noise, is developed.