Cryogenic helium-4 has long been recognized as a useful material in fluids research. The unique properties of helium-4 in the gaseous phase and the normal liquid phase allow for the generation of turbulent flows with exceptionally high Reynolds and Rayleigh numbers. In the superfluid phase, helium-4 exhibits two-fluid hydrodynamics and possesses fascinating properties due to its quantum nature. However, studying the flows in helium-4 has been very challenging largely due to the lack of effective visualization and velocimetry techniques. In this article, we discuss the development of novel instrumentation for flow-visualization in helium based on the generation and imaging of thin lines of metastable tracer molecules. These molecular tracers are created via femtosecond-laser field-ionization of helium atoms and can be imaged using a laser-induced fluorescence technique. By observing the displacement and distortion of the tracer lines in helium, quantitative information about the flow field can be extracted. We present experimental results in the study of thermal counterflow in superfluid helium that validate the concept of this technique. We also discuss anticipated future developments of this powerful visualization technique.
Cryogenic helium-4 has three phases that are of particular interest in fluid mechanics research: a gas phase, a normal liquid phase (He-I), and a superfluid phase (He-II). The gaseous helium and He-I are classical Navier-Stokes viscous fluids.1 The critical point of helium-4 (Pc = 2.24 bars, Tc = 5.19 K) is much easier to attain than for any other test fluids. By adjusting the pressure and temperature around the critical point, the density of helium can be varied in a wide range, and so can its kinematic viscosity. This unique property allows the generation of flows in gaseous helium and He-I with exceptionally high Reynolds and Rayleigh numbers as well as wide tuning ranges of these flow parameters, creating new opportunities for turbulence research and hydrodynamic model testing.2–5 He-II, on the other hand, is a quantum liquid consisting of two intermiscible fluids: a viscous normal-fluid component whose fraction drops from unity to zero as the temperature decreases from the lambda point (about 2.17 K) to zero and an inviscid superfluid component whose rotational motion is possible only in the presence of quantized vortex filaments.6 Turbulence in He-II is an active research area that is important both in fundamental science and in practical applications of He-II as a coolant.7 Due to the restriction in rotation, superfluid turbulence takes the form of a tangle of quantized vortex lines (quantum turbulence).8 Turbulence in the normal fluid is more conventional, but its interaction with the quantized vortices can lead to the non-classical force of mutual friction between the two fluids.9 Turbulence in such a system can exhibit a behavior that is similar to that found in a classical fluid, when the two fluids are coupled and have a common velocity field,10,11 but it may take forms that are unknown in classical fluid mechanics when the two fluids have different velocity fields.12 Studying turbulence in He-II may therefore enrich our knowledge of turbulence in general.
Despite the great potential of cryogenic helium-4 as a working fluid, the lack of quantitative flow measuring tools has hampered the application of helium in fluid research. Typical measurement techniques for common test fluids include pitot tube pressure measurements and hot-wire anemometry. These single-point diagnostics normally have limited spatial resolution due to the size of the probes, and they rely on the Taylor frozen flow hypothesis13 which may not hold in highly turbulent flows. In He-II, the motion of both fluid components can contribute to the sensor response. When the two fluids have different velocity fields, data analysis can become very complicated.
More straightforward velocity measurements can be made via direct flow visualization. However, visualization of flows in cryogenic helium is very challenging, in part due to the extremely low temperature and low density of helium.14 In the past, particle image velocimetry (PIV) and particle tracking velocimetry (PTV), using micron-sized solidified hydrogen particles as tracers, have been developed and applied to the study of flows in He-II.15–19 These methods, however, are not applicable to helium gas owing to the large variations and small values of the gas density. Even in He-II, the micron-sized particles can interact with both the normal fluid and the quantized vortices, rendering their motion hard to interpret.20 Recently, the feasibility of using metastable triplet molecules as tracers via fluorescence imaging has been validated through a series of experiments.12,14,21,22 These molecules have exceptionally long radiative lifetime (about 13 s).23 They form tiny bubbles (∼6 Å in radius24) in liquid helium or dense helium gas and behave as part of the gas in dilute helium gas.25 Due to their small size and effective mass, molecular tracers always follow the fluid motion in gaseous helium or He-I. In He-II, the tracers can bind to quantized vortex lines with a binding energy of about 30 K.26 However, above 1 K, thermally assisted escape process becomes exponentially frequent. In this situation, the tracers are essentially entrained by the normal-fluid component, since Stokes drag dominates other forces.27 We have found that the binding lifetime of the tracers on quantized vortices can become appreciable only below 0.6 K when the normal fluid is absent.28 Considering all these superior tracer properties, developing a high precision flow visualization technique using molecules, applicable to both gaseous and liquid helium, will foreseeably break new ground for substantial quantitative studies of classical flows as well as quantum hydrodynamics.
In this paper, we discuss the development of novel instrumentation for flow-visualization in helium. Thin lines of tracers are created in helium via femtosecond-laser field ionization of the helium atoms. The density of the tracers so created is sufficient such that high quality single-shot fluorescence images of the tracer lines can be obtained. These tracer lines are followed in time so that quantitative information about the flow field can be extracted.29 In Section II, we describe the experimental apparatus used to create the thin tracer lines and discuss the concept of laser-field ionization. The measured threshold laser intensity for field-ionization in helium as a function of the helium density is reported. Typical images of the tracer lines obtained by laser-induced fluorescence (LIF) are presented. In Section III, we show that the tracer-line imaging (TLI) technique has been successfully applied in the study of thermal counterflow in He-II. This application has allowed the observation of some novel normal-fluid behaviors that were not possible to study in the past. In Section IV, we discuss anticipated future developments of this visualization technique.
II. EXPERIMENTAL METHOD
A. Experimental setup
Metastable molecules can be created in helium as a consequence of ionization or excitation of ground-state helium atoms30,31
In order to produce molecules via laser-field ionization, laser intensity as high as 1013 W/cm2 is needed.32 This high instantaneous laser intensity can be achieved by focusing a strong laser pulse with short pulse duration. In our experiment, a commercial femtosecond regenerative amplifier laser system has been used. This system comprises a Spitfire Ace amplifier seeded by a Mai Tai pulsed laser and pumped by an Empower-45 laser.34 The output of the amplifier is 35-fs pulses with adjustable repetition rate up to 5 kHz. The output pulse energy is adjustable with a maximum of about 4 mJ. To test laser-field ionization in helium, we have adopted an experimental setup shown schematically in Fig. 1(a). A stainless steel square channel (inner side width 9.5 mm and total length 300 mm) is mounted at the bottom of a pumped helium bath whose temperature can be controlled. This channel can be filled with either gaseous helium or liquid helium at an adjustable temperature down to about 1.4 K. The channel runs through a stainless steel cube where the flow visualization is performed. The cube has two cylindrical side flanges with indium-sealed sapphire windows. A pair of vertical slots (height ∼4 mm and width ∼1.5 mm) is cut through the cube and the channel wall so that the laser beams can pass through. This design helps to avoid laser light scattering in the channel and laser-induced damage of the windows. A sapphire view-window is installed on the front side of the visualization cube, which allows an intensified charged-coupled device (ICCD) camera (PI-MAX4 camera system from Princeton Instruments35) to view the molecules perpendicular to both the laser beams and the channel axis. The femtosecond laser beam is first expanded to have an appropriate entrance beam diameter and is then focused to the center of the channel using a convex lens. The spatial profile (1/e2 intensity contour) of the femtosecond laser beam inside the channel is schematically shown in Fig. 1(b).
For an ideal Gaussian beam, if we denote ω0 to be the radius of the 1/e2 irradiance intensity contour at the focal plane where the wavefront is flat, the 1/e2 intensity contour radius ω(z) at a distance z from the focal plane is then given by36
where λ is the wavelength of the beam (about 800 nm). One can define a distance, called the Rayleigh range ZR, over which the beam radius spreads by a factor of as . The laser intensity only drops by a factor of 2 from the focal plane to the Rayleigh range boundary, and the molecules are thus expected to be produced essentially within the Rayleigh range from the focal plane. In order to characterize the profile of our femtosecond laser beam near the focal plane, we have measured the beam cross section intensity profile using an infrared camera. The beam cross section intensity profile measured at the focal plane is nearly a Gaussian with a typical M2 factor of about 1.04 (M2 values close to 1 indicate that the beam profile is very close to a Gaussian profile37). The beam radius ω0 can be determined from a fit of the Gaussian profile. We repeat the measurement at a different location z along the beam prorogation direction. The variation of the beam radius ω with z is shown in Fig. 2, which agrees very well with the profile given by Eq. (2) (the red solid line in Fig. 2).
The thickness of the molecular tracer line is controlled by the width of the femtosecond laser beam ω0. One can easily derive that ω0 is related to the focal length f of the lens and the incident beam radius on the lens ωf via ω0 = λ f/πωf. By using a lens with an appropriate focal length f or by adjusting the incident beam radius ωf, a desired thickness of the trace-line can thus be achieved. In principle, the line thickness, which sets the resolution limit in our flow visualization experiment, is only limited by optical diffraction which occurs when the femtosecond laser beam diameter 2ω0 becomes comparable to its wavelength (about 800 nm).
B. Laser-field ionization in helium
Laser-induced ionization and dielectric breakdown in both gaseous and liquid helium have been the subject of extensive studies.38–41 For individual ground state helium atoms, the transition from the perturbative multiphoton ionization process to tunneling ionization process in a laser field is determined by the Keldysh adiabaticity parameter42
where I.P. = 24.6 eV is the ionization threshold energy of a helium atom, and Up is the ponderomotive potential of an electron in the laser electric field E = E0cos(ωt). The switch-over from the multiphoton ionization regime (γ ≫ 1) to the tunneling ionization regime (γ ≪ 1) occurs at field intensities of about 2 × 1014 W/cm2, which is higher than the maximum laser intensity we used. Thus, the ionization of ground-state helium atoms in our experiment should be in the multiphoton regime (about 17 photons at 800 nm, hν = 1.53 eV). The electrons generated in the multiphoton ionization process are subsequently accelerated by the radiation field and can undergo ionizing collisions with surrounding helium atoms. When the density of the helium is low (i.e., in warm helium gas at low pressures), the number of ionizing collisions is limited during the laser pulse. Breakdown in this case is caused essentially by multiphoton ionization of individual helium atoms, and the threshold laser intensity for breakdown is expected to be independent of the helium density and insensitive to the laser pulse duration.38 When helium density is high (i.e., in dense helium gas or liquid helium), ionized electrons undergoing ionizing collisions with nearby He atoms may produce more electrons, and an electron cascade ionization could occur. Furthermore, following a laser pulse, the ionized electrons and helium ions can recombine, which leads to the generation of molecules in both the singlet and triplet states. The triplet molecules can survive during the time interval between laser pulses. As pointed out by Benderskii et al.,32 the triplet molecules can be easily ionized (I.P. = 4.26 eV) since the Coulombic potential that binds the Rydberg electron can be completely suppressed with a laser intensity as low as 1.5 × 1012 W/cm2. The triplet molecules created by an initial femtosecond laser pulse thus can serve as seeds for the initiation of the cascade ionization during subsequent pulses. As a consequence, the breakdown threshold laser intensity in dense helium gas or liquid helium should be much lower.
To study laser-field ionization of helium in our experiment, we have adopted an imaging method. Instead of detecting the ions or electrons, we synchronize the ICCD camera with the femtosecond laser and turn the camera on for 8 μs to catch the prompt visible scintillation photons produced in helium following every femtosecond laser pulse. These photons are generated as the molecules, the products of laser-field ionization in helium, quench from highly excited electronic states to their singlet and triplet ground states.32 The number of scintillation photons collected in an image thus scales with the total number of ionization events produced by a femtosecond laser pulse. For a given laser intensity and helium density, we normally sum up the scintillation counts over all pixels in an image and average this total count over 500 shots. Fig. 3(a) shows a representative sample of the averaged scintillation counts obtained in helium gas as a function of the laser intensity. When plotted on a log-log scale, the scintillation-count data always exhibit two regimes: a gently sloped background region at low laser intensity and a region with sharply increasing scintillation counts at higher laser intensity. Each region can be fitted by a power law, the lines of which can be extended until they intersect. We take the laser intensity at the intersection point as the threshold intensity of laser-field ionization in helium. This threshold laser intensity is measured at different helium densities. The result is shown in Fig. 3(b) as a function of the helium density ρ in units of density ρ0, where ρ0 = 0.145 g/cm3 is the density of liquid helium at T = 1.830 K. The essentially constant ionization threshold values for dilute helium gas with densities ≤10−3ρ0 agree with results reported by Ireland and Morgan for multiphoton ionization in gaseous helium.38 Further, we find that the ionization threshold drops roughly by a factor of 6.5 as the density approaches ρ0 (blue triangle), which agrees well with the threshold intensity previously reported in He-II by Benderskii et al.32
C. Imaging molecular tracer lines
To image the molecules created by the femtosecond laser pulses, a cycling-transition LIF technique that was first developed by McKinsey et al. has been used.27,43 A schematic diagram of the optical transitions of the molecules is shown in Fig. 4. These molecules can be excited by two infrared photons at 905 nm from their triplet ground state to the excited electronic state . Over 90% of the molecules in the d state decay to the b3Πg state in about 10 ns, emitting red photons at 640 nm,43 which can be detected by the ICCD camera. A 640 nm bandpass filter with a full-width half maximum (FWHM) bandwidth of 20 nm is installed on the camera to block unwanted laser light and to minimize background. From the b3Πg state, molecules quench back to the state. The time scale of the b → a quenching can be strongly affected by the vibrational and rotational levels that lie between these two energy states. For instance, some molecules may fall to excited vibrational levels of the state and are trapped in off-resonant a(1) and a(2) vibrational levels due to the long vibrational-relaxation time (about 1 s).44 Re-pumping lasers at 1073 nm and 1099 nm can be used to improve the cycling-transition efficiency by exciting these molecules to the c states where they essentially fall back to the a(0) state in a few nanoseconds and can be reused. This re-pumping method can reduce the b → a quenching time down to about 1 ms. Additional lasers for re-pumping the molecules in the excited rotational levels of state are likely needed if one wants to further reduce the quenching time.44
The imaging laser used in our experiment is an EKSPLA Nd:YAG pulsed laser with a fixed wavelength of 905 nm and is operated at 500 Hz.45 Its pulse duration is about 5 ns and its pulse energy is adjusted such that the laser intensity is about 2.5 mJ/cm2 per pulse in the flow channel, sufficient to saturate the optical transitions of molecules.44 Two fiber diode continuous wave (cw) lasers with wavelengths 1073 nm and 1099 nm are aligned with the imaging laser for enhancing the cycling transition efficiency. The output power of these cw lasers is about 1 W. An electronic shutter is used to control the illumination time of the imaging laser and the fiber lasers. We then overlap the imaging laser beams with the femtosecond laser using a thin-film polarizer-based beam-combiner (see Fig. 1(a)). The intensifier of the camera in this case is synchronized with the imaging laser and is turned on for 8 μs following every imaging laser pulse to collect the induced fluorescent light. Note that the quality of the tracer-line images depends on the density of the molecules and the number of the imaging laser pulses applied. The saturation molecule density is controlled by the balance between the generation due to femtosecond-laser illumination and the decay due to bi-molecular Penning ionization reaction.46 We find that typically 10-15 fs pulses are sufficient to achieve the saturation molecule density. When the femtosecond laser is operated at 5 kHz, this corresponds to an illumination time of only a few milliseconds. Subsequently, we determine that 6 pulses from the 905 nm imaging laser can provide a satisfying fluorescence signal. Due to the short illumination time of the laser beams and their high transmission through the coated windows in the flow channel, we did not observe any noticeable laser heating in our experiments.
Typical fluorescence images of the tracer lines created in helium vapor are shown in Figs. 5(a) and 5(b). As expected, the thickness of the tracer line matches the diameter of the femtosecond laser beam in the focal region, and the length of the line is comparable with twice the Rayleigh range of the beam. By using lenses with different focal lengths f, we can easily vary the thickness and the length of the tracer lines. Figs. 5(c) and 5(d) show typical fluorescence images obtained in liquid helium. We find that above the threshold laser intensity when dielectric breakdown occurs in liquid helium, diffuse clouds of molecules are formed, compromising the quality of the tracer line. This threshold laser intensity corresponds to a femtosecond-laser pulse energy of about 60 μJ. Slightly below the breakdown laser intensity, a controlled electron avalanche ionization regime exists as reported by Benderskii et al.32 and a thin line of tracers can still be produced. The tracer line shown in Fig. 5(c) is created using a lens with f = 75 cm. In this case, the tracer line has a length of about 2.2 cm and it extends across the whole channel and into the side flanges. The imaged portion of the tracer line inside the flow channel exhibits fairly uniform thickness. This optical setting is normally used in our subsequent flow visualization experiments.
III. APPLICATION OF THE TRACER-LINE IMAGING TECHNIQUE
Generating tracer lines in a fluid by exciting the fluid molecules to a metastable state or creating new molecules using powerful laser beams is not new in classical fluid research.47,48 For instance, Miles et al. have developed a molecular tagging velocimetry (MTV) technique for air flow research by tagging oxygen molecules via Raman excitation.49 Tracer-line patterns can be produced in air and followed in time. Their displacement provides information about the velocity field, while the dispersion of the tracer molecules quantifies mixing processes in the flow. Recognizing the great value of MTV, researchers have developed various flow tagging concepts for air and other gas flows.50–53 Compared to these MTV techniques, our TLI technique has some unique advantages. For instance, the usable tagging lifetimes of many existing MTV techniques are in the range of 1-100 μs. As a consequence, their applications require a sufficiently large mean flow velocity so that the displacement of the tracer lines within the short molecular lifetime may be resolved. On the other hand, the exceptionally long lifetime (13 s) of the molecules allows us to examine the velocity profile in cryogenic helium with a wide range of mean flow velocities from mild laminar flows to violent turbulent flows. Furthermore, molecular diffusion of tracers in cryogenic helium at above 1 K is very small27 and hence is much smaller an issue compared to typical room temperature MTV experiments. The unique power of our TLI technique combining the advantage of using helium as a test fluid has created new opportunities in fluid research. For instance, we have applied this TLI technique in the study of thermal counterflow in He-II and have revealed interesting normal-fluid behaviors that were previously unknown.54
Thermal counterflow is a flow mode that exists in He-II when a heat current is present. The normal-fluid component carries the heat and moves away from the heat source at a mean velocity , where q is the heat flux, T denotes the temperature, and s represents the specific entropy of helium.6 The superfluid component moves towards the heat source, serving to eliminate any net mass flow. This efficient heat transfer mode leads to an effective thermal conductivity of He-II much greater than any other materials. However, above a critical heat flux, the superfluid is known to become turbulent, taking the form of tangled quantized vortex filaments (quantum turbulence).8 A mutual friction between the two fluids arises through the interaction between quantized vortices and the normal fluid.9 Despite many studies on the quantized turbulence,55–62 a systematic characterization of thermal counterflow can hardly be possible without faithful flow visualization in the normal fluid.
In our experiment, we used a planar heater to generate counterflow in the channel as shown in Fig. 1. This heater is made of four thin film chip resistors that are connected in series with a total resistance of 400 Ω. A tracer line produced in the channel moves with the normal fluid for a certain “drift time” and is then imaged. Fig. 6(a1) shows a typical image of the tracer lines at small heat fluxes. Molecular diffusion in this case has a clear effect due to the long drift time. To enhance the image quality, five single-shot images are averaged. An initially straight tracer line (represented as the white dashed line) deforms into a nearly parabolic shape, indicating a laminar Poiseuille normal-fluid velocity profile in the channel.54 To obtain quantitative information of streamwise velocity u(x), we divide the line image into small segments (typically 40-60 segments) and determine the center location of each segment via a Gaussian fit of its fluorescence intensity profile. A local flow velocity is obtained by dividing the displacement of a segment by the drift time. The calculated velocity profile pertinent to Fig. 6(a1) is shown in 6(a2). One may also adopt a global line-fitting method to more accurately determine the velocity field.63
When the heat flux is large, a straight tracer line deforms randomly, indicating turbulent flow in the normal fluid. A typical single-shot image is shown in Fig. 6(b1). A probability density function (PDF) for the velocity can be generated based on the analysis of 200 such single-shot images (shown in (b2)) which exhibits a Gaussian distribution. Furthermore, by correlating the streamwise velocities of two line-segments (see schematic in Fig. 6(c1)), we can compute the second-order transverse structure function defined as , where the overline denotes ensemble averaging. The calculated for turbulent normal-fluid flow (see Fig. 6(c2)) always exhibits linear dependence on r when r is small and is independent of R when the reference location R is not too close to the channel wall. This behavior is different from what one would expect for turbulent flows in a viscous classical fluid with a Kolmogorov energy spectrum.64 This non-classical behavior of the normal-fluid turbulence in counterflow is believed to result from mutual friction between the two fluids. More details of this study on counterflow can be found in our recent publication.54
IV. FUTURE DEVELOPMENT
In turbulence research, it is often desirable to have the capability of measuring not only the transverse structure functions but also the longitudinal structure functions since they may obey different scalings.65,66 The longitudinal structure function of order n is defined as
In homogeneous isotropic turbulence, these structure functions are expected to behave as with universal scaling exponents.64 These scaling exponents are predicted to be ζn = n/3 for fully developed turbulence with a Kolmogorov energy spectrum.67,68 Measuring these exponents experimentally will allow one to obtain information about the turbulent energy spectrum as well as to examine how the turbulence may differ from the Kolmogorov form, the so-called intermittency phenomenon.69,70
For flows in a channel, our TLI technique allows convenient measurement of the streamwise velocity. To determine the longitudinal structure functions, we then need to make simultaneous velocity measurements at two locations aligned along the steamwise direction. This can be done with a setup shown schematically in Fig. 7. The output of the femtosecond laser can be divided into two beams using a beam splitter. The two beams are then focused into the flow channel to create two parallel lines of tracers. One of the two beams is reflected on a mirror mounted on a translation stage such that the separation r between the two beams can be continuously adjusted. The two tracer lines produced in the channel can be followed and imaged simultaneously. Based on the displacements of the two tracer lines, one can then correlate the velocities along the streamwise direction and calculate the longitudinal structure functions as (see Fig. 7). Indeed, this optical setup allows the determination of both longitudinal and transverse structure functions for displacements in the direction of the mean flow.
It is worthwhile to note that since the maximum output pulsed energy of our femtosecond laser system is 4 mJ, much greater than the energy needed to ionize helium atoms (slightly below 60 μJ), we may divide the laser beam into multiple beams to create more complicated tracer-line patterns such as multiple parallel lines, crosses, or grid structures. Quantitative measurements of the vorticity field and other complex flow field parameters can be made possible.
We acknowledge the startup support provided to W.G. by Florida State University and the National High Magnetic Field Lab (NHMFL), as well as support from the US Department of Energy under Grant No. DE-FG02 96ER40952 and the National Science Foundation under Grant No. DMR-1507386.