rus
DOI: https://doi.org/10.22184/1993-8578.2023.16.6.384.392
A several methods are used for formation of quantum vortices in superfluid helium, including oscillation bodies placed in a liquid helium. In the article we analyze the features of these methods, presents the possibilities of their application for the study of liquids, the transition of the flows into a turbulent state, and the use of such oscillators in technology.
A several methods are used for formation of quantum vortices in superfluid helium, including oscillation bodies placed in a liquid helium. In the article we analyze the features of these methods, presents the possibilities of their application for the study of liquids, the transition of the flows into a turbulent state, and the use of such oscillators in technology.
Теги: quantized vortices quartz tuning fork second sound superfluid helium второй звук квантованные вихри кварцевый камертон сверхтекучий гелий
Original paper
FORMATION OF QUANTUM VORTICES IN SUPERFLUID HELIUM
V.B.Efimov1, Doct. of Sci. (Physics and Mathematics), Leading Researcher, ORCID: 0000-0002-9195-2458 / victor_efimov@yahoo.co.uk
A.A.Esina1, Junior Researcher, ORCID: 0000-0001-5700-3729
Abstract. A several methods are used for formation of quantum vortices in superfluid helium, including oscillation bodies placed in a liquid helium. In the article we analyze the features of these methods, presents the possibilities of their application for the study of liquids, the transition of the flows into a turbulent state, and the use of such oscillators in technology.
Keywords: superfluid helium, second sound, quantized vortices, quartz tuning fork
For citation: V.B. Efimov, A.A. Esina. Formation of quantum vortices in superfluid helium. NANOINDUSTRY. 2023. V. 16, no. 6. PP. 384–392. https://doi.org/10.22184/1993-8578.2023.16.6.384.392.
INTRODUCTION
The study of turbulent phenomena, peculiarities of propagation of nonlinear waves, formation of vortex currents at their interaction, especially in relation to applied problems, is one of the most investigated problems of modern physics. Turbulent phenomena can be of both vortex character and vortex-free. Vortex phenomena lead to formation of such structures as cyclones and anticyclones in the atmosphere of the Earth and other planets, tornadoes and vortexes, and whirlpools. The vortex-free manifestation of turbulent phenomena is usually determined by interaction of nonlinear waves, due to which, in particular, waves of abnormally high height, called "freak waves", can be formed on the water surface.
One of the questions arising in the study of turbulent phenomena is the question of energy transfer along the energy spectrum from the pumping region to the dissipative region, as well as the possibilities and ways of influencing such energy flows for different types of turbulence. However, turbulent phenomena, unlike laminar flows, are very difficult to describe theoretically. Real turbulent flows are subject to a large number of random influences, although for experimental verification of theoretical conclusions or for phenomenological conclusions one tries to make measurements under controlled and reproducible conditions. Superfluid helium is one of the media that allows experiments to be performed quite easily under the same conditions. 4Не can be purified to incredibly low concentrations of impurities, including the 3Не isotope. All other impurities are insoluble in liquid helium and freeze out at higher temperatures. In addition, superfluid helium has a number of unique properties. The viscosity of liquid and superfluid helium is two orders of magnitude lower than water viscosity, which reduces the experimental system parameter L·v, where L is the characteristic size of the system, v is the liquid flow rate. Rotational motion of the fluid flow forms vortices, for which in helium at temperatures below Tλ the flow of the superfluid component of fluid along any trajectory around the vortex cortex obeys the laws of quantization, which significantly simplifies description of the ensemble of quantum vortices:
, (1)
where h – Planck’s constant, m is mass of the helium atom, and n is an integer.
Another unique property of superfluid helium is the possibility of propagation of strongly nonlinear weakly damped thermal waves – second sound waves. The nonlinearity coefficient of these waves α2 varies from –∞ at Tλ to a large positive value at lower temperatures, passing through 0 at T=1.88 K. Thus, any harmonic wave of the second sound propagating in superfluid helium is distorted according to the relation:
, (2)
where δT is wave amplitude, and forms a whole cascade of multiple harmonics, which corresponds to energy transfer to a higher frequency range – energy transfer corresponding to the Kolmogorov inertial mechanism. Such energy transfer (called acoustic turbulence) in the resonator of second sound waves, as well as the processes of formation, existence, and decay of such cascades, have been observed earlier [1].
Since the flow of the superfluid component of vortices dissipates the second sound waves quite effectively, attenuation of these waves can be used as a means to register system transition into a turbulent state. On the other hand, introduction of vortices into propagating nonlinear waves of the second sound can be used as a dissipative element affecting the acoustic turbulence process.
The aim of this work was to find a way to influence the energy transfer processes in acoustic turbulence through controlled introduction of quantum vortices into the resonator volume by oscillators, and to estimate the number of vortices generated around oscillators.
RESEARCH METHODS
As an oscillator of vortices, we used quartz tuners. Such or similar oscillators are used as frequency standards for handheld electronic clocks. The calibration of quartz chamberpieces is performed by electrical response corresponding to their mechanical oscillations (current through the oscillator when an alternating voltage is applied in resonance). The electromechanical coefficient a is determined from measured half-width of the oscillator resonance Δω under ideal conditions (in vacuum):
, (3)
where M is mass of the tuning fork support and R is electrical resistance of the system in resonance. Knowing a, it is possible to determine the movement rate of the supports of the quartz tuning fork from charging current I of the electrodes of the tuning fork, and from the alternating current voltage on the oscillator U0 – bending force of the tuning fork supports, corresponding to resistance force of the medium [2]:
, (4)
. (5)
We have previously verified correspondence of the calculated rate value of the tuning fork supports to their real value using a Michelson interferometer. The results of this measurement showed a good agreement between the calculated and measured values of the maximum rate of the tuning fork support [3].
At increasing the oscillations amplitude the excitations transfer to environment has two sharply different modes. At small amplitudes of oscillations, the flow of liquid or gas around the oscillating supports of the tuning box is laminar, and resistance force is proportional to the motion speed and density of the medium entrained by the oscillator. The electromechanical relations for the motion of such a tuning fork are given in Fig.1a. The resonant frequency in this case is defined as:
, (6)
or for small additions of mass compared to the oscillating body mass [4]:
. (7)
The measurements results performed in different environments on one of the quartz tunings are shown in Fig.1b. The peculiarity of this tuning box was that the direction of the crystal axes was chosen during its fabrication, allowing the supports to oscillate along with bending modes (fB ≈ 76 kHz) and torsional oscillations (fT ≈ 363 kHz). The graph shows a good correlation between medium density and the decrease in the resonance frequency of the tuning box in laminar flow. Two-mode tuning devices allow one mode to perturb the environment and register the response (including formation of a turbulent state) by the other mode.
The increase of the oscillation amplitude above the critical value leads to the birth of quantized vortices. At that, as well as for any turbulent motion, resisting force of the medium is proportional to the motion rate squared:
, (8)
where CD is the resisting coefficient, AS is the cross section of the moving body. The measurements results have shown that transition to turbulent vortex generation occurs for quartz chamberpieces at rates of several tens of cm/s, depending on the size of the chamberpieces and their resonance frequencies. Fig. 2 shows measured results for a tuning element with resonance frequency of f = 8 kHz in vacuum at room temperature and liquid helium temperature, as well as in liquid helium at different temperatures.
The graphs clearly show that a decrease in the helium temperature and the subsequent transition of helium to superfluid state leads to decreasing in the resisting force in the laminar flow (solid red line on the graph), which correlates with a decrease in excitations concentration in helium (exponential freezing of rotons) and, consequently, in dynamic rate. Thus, in these experiments we have determined the characteristic rates of motion transition of quartz chambers into the turbulent mode, which allows us to set the modes of introduction of the dissipative component into the superfluid helium volume.
DISCUSSION
In the experiments, a principal possibility for introducing vortices by oscillating quartz oscillators into the space of frequency energy transfer at acoustic turbulence of second sound waves was determined. Let us try to determine the growth rate of the number of vortices generated in the helium volume at oscillation of such oscillators. The total energy transferred to the vortex system by an oscillating body will be determined by the mechanical or electrical power of the loss of forces causing oscillations. It is not difficult to see that multiplication of charging current of the plates on tuning box by the voltage of the alternating signal that excites these oscillations is equal to multiplication of the calculated force by the calculated rate of the tuning box supports. Then it is possible to determine the maximum amplitude of oscillation of the tuning fork and to determine the total energy transferred to the medium at one period of oscillation:
. (9)
It should be taken into account that at small vibration amplitudes (and small velocities) the flow is laminar and only exceeding the rate limit values vcr leads to vortices. Subtraction of laminar motion from the general dependence F(v) is shown in Fig.3. The resistance force increase to the oscillator motion at the transition to turbulent vortex formation is close to the quadratic dependence , with vcr being a value of about 10–15 cm/s. For the results shown in Fig.3 at a 30 cm/s rate of 30 cm/s, the oscillation amplitude of 4 kHz tuning fork is 12 μm and the maximum force is 4.2 · 10–6 N; for a 32 kHz tuning fork it is 1.5 μm and 3 · 10z–7 N, respectively. The energy transferred to the vortex system by a 4 kHz chamberpot during one period of oscillation will be determined by the integral under the curve labeled as a dashed line on the graph, and it will be ~ 10–10 J. For 32 kHz energy will be ~ 10–12 J.
To estimate density of the created vortices, it is necessary to take into account the birth energy of the unit length of the vortex. The basic energy of a quantum vortex is determined from the kinetic energy of a moving superfluid, determined by the quantization laws:
, (10)
where integration is limited either by the working size for a single vortex or by the distance to the nearest neighboring vortices. Estimating n = 1 (it is energetically favorable for vortices to take the minimum possible n, in 4Не a vortex with one circulation quantum n = ±1, ln(l/rc) ~ 10 (which is quite reasonable at sufficiently dense packing of vortices), ρs = 0.147 g/cm3, we obtain energy per unit length of the vortex filament .
For the above energies it turns out that for one oscillation in the vortex system increases by ~108 μm at oscillation amplitude of 12 μm for 4 kHz of the tuning fork and ~106 μm at oscillation amplitude of 1.5 μm for 32 kHz. Thus, we see that the vortex system length at a single oscillation of the oscillator increases by many orders of magnitude more than the simple elongation of a single vortex, i.e., the process of the turbulent vortex flow generation is a collective one involving many spinning vortices simultaneously. Let us try to estimate density of the formed vortex space around the oscillator.
The motion speed of a vortex ring of radius R0 at zero temperature is given by the expression [5]:
, (11)
where κ=h/m4He =9.998 · 10–8 m2/s – circulation quant. It is not difficult to see that ring rate is smaller the its larger radius. For a vortex ring of radius 1 µm, the vortex velocity is of the order v0 = 8 cm/s, for 10 µm – v0 = 1 cm/s, for a vortex with a radius of 100 µm – v0 = 1.2 mm/s.
At non-zero temperature of superfluid helium, the moving vortex ring decreases its size due to mutual friction of the normal and superfluid components. The rate of change of the vortex ring radius is defined as [5]:
. (12)
The existence time of the vortex can be estimated as:
, (13)
where α is mutual friction coefficient between the normal and superfluid components of superfluid helium. The flight distance of the ring before its disappearance is defined as [6]:
.
Taking into account the smallness of the vortex crust radius in comparison with the vortex radius, we finally obtain. Therefore, at sufficiently strong dissipative processes in superfluid helium, detection of vortices at a distance D is possible only for vortices with sizes larger than R0 determined by the relation 2R0 > 2αD. It can be assumed that the vortex ring radius is the order of the oscillation amplitude of the tuning forks or ≈ 10–20 μm for a tuning fork with f = 4 kHz tuning fork and of the order of a few μm for f = 32 kHz. From the estimates of [7], at T = 1.2 K, vortices with a diameter of 18 µm travel a distance of 0.5 mm for a time of the order of τ ≈ 40 ms. At T = 1.6 K, the same vortex will travel a distance of ≈ 0.3 mm for the same time.
Taking into account the rate of filling of the volume near the oscillator for τ/2 ≈ 20 ms, the total length of vortices will be 106 cm. This allows us to estimate the average number of vortices in the space near such an oscillator with an area of oscillating planes ~1 cm2 in n ~ 107 cm–2, at a distance of ≈ 1 mm from the oscillator, which should create a sufficient dissipative contribution to acoustic turbulence. For a small quartz tuning fork f = 32 kHz with an area of ~ 1 mm2 for an oscillation amplitude of a few µm, the vortex ring rate will be an order of magnitude larger and its lifetime of the order of 1 ms. The total length of vortices at such a time will be 3 · 103 cm and average density of vortices near the oscillator at a depth of 0.1 mm (distance where the vortex rings with a radius of 1 µm disappear) will be the same n ~ 107 см–2.
CONCLUSIONS
In this work, we experimentally determined the energy transfer rate from quartz tuning forks of different sizes and different resonance frequencies into a vortex system, estimated possible vortex density for such processes, and showed applicability of such oscillators to create sufficient dissipative processes for further study of energy transfer processes in acoustic turbulence. The work was supported by RSF grant No. 22-22-00718.
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
FORMATION OF QUANTUM VORTICES IN SUPERFLUID HELIUM
V.B.Efimov1, Doct. of Sci. (Physics and Mathematics), Leading Researcher, ORCID: 0000-0002-9195-2458 / victor_efimov@yahoo.co.uk
A.A.Esina1, Junior Researcher, ORCID: 0000-0001-5700-3729
Abstract. A several methods are used for formation of quantum vortices in superfluid helium, including oscillation bodies placed in a liquid helium. In the article we analyze the features of these methods, presents the possibilities of their application for the study of liquids, the transition of the flows into a turbulent state, and the use of such oscillators in technology.
Keywords: superfluid helium, second sound, quantized vortices, quartz tuning fork
For citation: V.B. Efimov, A.A. Esina. Formation of quantum vortices in superfluid helium. NANOINDUSTRY. 2023. V. 16, no. 6. PP. 384–392. https://doi.org/10.22184/1993-8578.2023.16.6.384.392.
INTRODUCTION
The study of turbulent phenomena, peculiarities of propagation of nonlinear waves, formation of vortex currents at their interaction, especially in relation to applied problems, is one of the most investigated problems of modern physics. Turbulent phenomena can be of both vortex character and vortex-free. Vortex phenomena lead to formation of such structures as cyclones and anticyclones in the atmosphere of the Earth and other planets, tornadoes and vortexes, and whirlpools. The vortex-free manifestation of turbulent phenomena is usually determined by interaction of nonlinear waves, due to which, in particular, waves of abnormally high height, called "freak waves", can be formed on the water surface.
One of the questions arising in the study of turbulent phenomena is the question of energy transfer along the energy spectrum from the pumping region to the dissipative region, as well as the possibilities and ways of influencing such energy flows for different types of turbulence. However, turbulent phenomena, unlike laminar flows, are very difficult to describe theoretically. Real turbulent flows are subject to a large number of random influences, although for experimental verification of theoretical conclusions or for phenomenological conclusions one tries to make measurements under controlled and reproducible conditions. Superfluid helium is one of the media that allows experiments to be performed quite easily under the same conditions. 4Не can be purified to incredibly low concentrations of impurities, including the 3Не isotope. All other impurities are insoluble in liquid helium and freeze out at higher temperatures. In addition, superfluid helium has a number of unique properties. The viscosity of liquid and superfluid helium is two orders of magnitude lower than water viscosity, which reduces the experimental system parameter L·v, where L is the characteristic size of the system, v is the liquid flow rate. Rotational motion of the fluid flow forms vortices, for which in helium at temperatures below Tλ the flow of the superfluid component of fluid along any trajectory around the vortex cortex obeys the laws of quantization, which significantly simplifies description of the ensemble of quantum vortices:
, (1)
where h – Planck’s constant, m is mass of the helium atom, and n is an integer.
Another unique property of superfluid helium is the possibility of propagation of strongly nonlinear weakly damped thermal waves – second sound waves. The nonlinearity coefficient of these waves α2 varies from –∞ at Tλ to a large positive value at lower temperatures, passing through 0 at T=1.88 K. Thus, any harmonic wave of the second sound propagating in superfluid helium is distorted according to the relation:
, (2)
where δT is wave amplitude, and forms a whole cascade of multiple harmonics, which corresponds to energy transfer to a higher frequency range – energy transfer corresponding to the Kolmogorov inertial mechanism. Such energy transfer (called acoustic turbulence) in the resonator of second sound waves, as well as the processes of formation, existence, and decay of such cascades, have been observed earlier [1].
Since the flow of the superfluid component of vortices dissipates the second sound waves quite effectively, attenuation of these waves can be used as a means to register system transition into a turbulent state. On the other hand, introduction of vortices into propagating nonlinear waves of the second sound can be used as a dissipative element affecting the acoustic turbulence process.
The aim of this work was to find a way to influence the energy transfer processes in acoustic turbulence through controlled introduction of quantum vortices into the resonator volume by oscillators, and to estimate the number of vortices generated around oscillators.
RESEARCH METHODS
As an oscillator of vortices, we used quartz tuners. Such or similar oscillators are used as frequency standards for handheld electronic clocks. The calibration of quartz chamberpieces is performed by electrical response corresponding to their mechanical oscillations (current through the oscillator when an alternating voltage is applied in resonance). The electromechanical coefficient a is determined from measured half-width of the oscillator resonance Δω under ideal conditions (in vacuum):
, (3)
where M is mass of the tuning fork support and R is electrical resistance of the system in resonance. Knowing a, it is possible to determine the movement rate of the supports of the quartz tuning fork from charging current I of the electrodes of the tuning fork, and from the alternating current voltage on the oscillator U0 – bending force of the tuning fork supports, corresponding to resistance force of the medium [2]:
, (4)
. (5)
We have previously verified correspondence of the calculated rate value of the tuning fork supports to their real value using a Michelson interferometer. The results of this measurement showed a good agreement between the calculated and measured values of the maximum rate of the tuning fork support [3].
At increasing the oscillations amplitude the excitations transfer to environment has two sharply different modes. At small amplitudes of oscillations, the flow of liquid or gas around the oscillating supports of the tuning box is laminar, and resistance force is proportional to the motion speed and density of the medium entrained by the oscillator. The electromechanical relations for the motion of such a tuning fork are given in Fig.1a. The resonant frequency in this case is defined as:
, (6)
or for small additions of mass compared to the oscillating body mass [4]:
. (7)
The measurements results performed in different environments on one of the quartz tunings are shown in Fig.1b. The peculiarity of this tuning box was that the direction of the crystal axes was chosen during its fabrication, allowing the supports to oscillate along with bending modes (fB ≈ 76 kHz) and torsional oscillations (fT ≈ 363 kHz). The graph shows a good correlation between medium density and the decrease in the resonance frequency of the tuning box in laminar flow. Two-mode tuning devices allow one mode to perturb the environment and register the response (including formation of a turbulent state) by the other mode.
The increase of the oscillation amplitude above the critical value leads to the birth of quantized vortices. At that, as well as for any turbulent motion, resisting force of the medium is proportional to the motion rate squared:
, (8)
where CD is the resisting coefficient, AS is the cross section of the moving body. The measurements results have shown that transition to turbulent vortex generation occurs for quartz chamberpieces at rates of several tens of cm/s, depending on the size of the chamberpieces and their resonance frequencies. Fig. 2 shows measured results for a tuning element with resonance frequency of f = 8 kHz in vacuum at room temperature and liquid helium temperature, as well as in liquid helium at different temperatures.
The graphs clearly show that a decrease in the helium temperature and the subsequent transition of helium to superfluid state leads to decreasing in the resisting force in the laminar flow (solid red line on the graph), which correlates with a decrease in excitations concentration in helium (exponential freezing of rotons) and, consequently, in dynamic rate. Thus, in these experiments we have determined the characteristic rates of motion transition of quartz chambers into the turbulent mode, which allows us to set the modes of introduction of the dissipative component into the superfluid helium volume.
DISCUSSION
In the experiments, a principal possibility for introducing vortices by oscillating quartz oscillators into the space of frequency energy transfer at acoustic turbulence of second sound waves was determined. Let us try to determine the growth rate of the number of vortices generated in the helium volume at oscillation of such oscillators. The total energy transferred to the vortex system by an oscillating body will be determined by the mechanical or electrical power of the loss of forces causing oscillations. It is not difficult to see that multiplication of charging current of the plates on tuning box by the voltage of the alternating signal that excites these oscillations is equal to multiplication of the calculated force by the calculated rate of the tuning box supports. Then it is possible to determine the maximum amplitude of oscillation of the tuning fork and to determine the total energy transferred to the medium at one period of oscillation:
. (9)
It should be taken into account that at small vibration amplitudes (and small velocities) the flow is laminar and only exceeding the rate limit values vcr leads to vortices. Subtraction of laminar motion from the general dependence F(v) is shown in Fig.3. The resistance force increase to the oscillator motion at the transition to turbulent vortex formation is close to the quadratic dependence , with vcr being a value of about 10–15 cm/s. For the results shown in Fig.3 at a 30 cm/s rate of 30 cm/s, the oscillation amplitude of 4 kHz tuning fork is 12 μm and the maximum force is 4.2 · 10–6 N; for a 32 kHz tuning fork it is 1.5 μm and 3 · 10z–7 N, respectively. The energy transferred to the vortex system by a 4 kHz chamberpot during one period of oscillation will be determined by the integral under the curve labeled as a dashed line on the graph, and it will be ~ 10–10 J. For 32 kHz energy will be ~ 10–12 J.
To estimate density of the created vortices, it is necessary to take into account the birth energy of the unit length of the vortex. The basic energy of a quantum vortex is determined from the kinetic energy of a moving superfluid, determined by the quantization laws:
, (10)
where integration is limited either by the working size for a single vortex or by the distance to the nearest neighboring vortices. Estimating n = 1 (it is energetically favorable for vortices to take the minimum possible n, in 4Не a vortex with one circulation quantum n = ±1, ln(l/rc) ~ 10 (which is quite reasonable at sufficiently dense packing of vortices), ρs = 0.147 g/cm3, we obtain energy per unit length of the vortex filament .
For the above energies it turns out that for one oscillation in the vortex system increases by ~108 μm at oscillation amplitude of 12 μm for 4 kHz of the tuning fork and ~106 μm at oscillation amplitude of 1.5 μm for 32 kHz. Thus, we see that the vortex system length at a single oscillation of the oscillator increases by many orders of magnitude more than the simple elongation of a single vortex, i.e., the process of the turbulent vortex flow generation is a collective one involving many spinning vortices simultaneously. Let us try to estimate density of the formed vortex space around the oscillator.
The motion speed of a vortex ring of radius R0 at zero temperature is given by the expression [5]:
, (11)
where κ=h/m4He =9.998 · 10–8 m2/s – circulation quant. It is not difficult to see that ring rate is smaller the its larger radius. For a vortex ring of radius 1 µm, the vortex velocity is of the order v0 = 8 cm/s, for 10 µm – v0 = 1 cm/s, for a vortex with a radius of 100 µm – v0 = 1.2 mm/s.
At non-zero temperature of superfluid helium, the moving vortex ring decreases its size due to mutual friction of the normal and superfluid components. The rate of change of the vortex ring radius is defined as [5]:
. (12)
The existence time of the vortex can be estimated as:
, (13)
where α is mutual friction coefficient between the normal and superfluid components of superfluid helium. The flight distance of the ring before its disappearance is defined as [6]:
.
Taking into account the smallness of the vortex crust radius in comparison with the vortex radius, we finally obtain. Therefore, at sufficiently strong dissipative processes in superfluid helium, detection of vortices at a distance D is possible only for vortices with sizes larger than R0 determined by the relation 2R0 > 2αD. It can be assumed that the vortex ring radius is the order of the oscillation amplitude of the tuning forks or ≈ 10–20 μm for a tuning fork with f = 4 kHz tuning fork and of the order of a few μm for f = 32 kHz. From the estimates of [7], at T = 1.2 K, vortices with a diameter of 18 µm travel a distance of 0.5 mm for a time of the order of τ ≈ 40 ms. At T = 1.6 K, the same vortex will travel a distance of ≈ 0.3 mm for the same time.
Taking into account the rate of filling of the volume near the oscillator for τ/2 ≈ 20 ms, the total length of vortices will be 106 cm. This allows us to estimate the average number of vortices in the space near such an oscillator with an area of oscillating planes ~1 cm2 in n ~ 107 cm–2, at a distance of ≈ 1 mm from the oscillator, which should create a sufficient dissipative contribution to acoustic turbulence. For a small quartz tuning fork f = 32 kHz with an area of ~ 1 mm2 for an oscillation amplitude of a few µm, the vortex ring rate will be an order of magnitude larger and its lifetime of the order of 1 ms. The total length of vortices at such a time will be 3 · 103 cm and average density of vortices near the oscillator at a depth of 0.1 mm (distance where the vortex rings with a radius of 1 µm disappear) will be the same n ~ 107 см–2.
CONCLUSIONS
In this work, we experimentally determined the energy transfer rate from quartz tuning forks of different sizes and different resonance frequencies into a vortex system, estimated possible vortex density for such processes, and showed applicability of such oscillators to create sufficient dissipative processes for further study of energy transfer processes in acoustic turbulence. The work was supported by RSF grant No. 22-22-00718.
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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