Transient heat transfer of superfluid ^4He in nonhomogeneous geometries -- Part I: Second sound, rarefaction, and thermal layer
aa r X i v : . [ c ond - m a t . o t h e r] F e b Transient heat transfer of superfluid He in nonhomogeneous geometries - Part I: Secondsound, rarefaction, and thermal layer
Shiran Bao a , Wei Guo a,b, ∗ a National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive. Tallahassee, FL 32310, USA b Mechanical Engineering Department, Florida State University, Tallahassee, FL 32310, USA
Abstract
Transient heat transfer in superfluid He (He II) is a complex process that involves the interplay of the unique counterflow heat-transfer mode, the emission of second-sound waves, and the creation of quantized vortices. Many past researches focused onhomogeneous heat transfer of He II in a uniform channel driven by a planar heater. In this paper, we report our systematic studyof He II transient heat transfer in nonhomogeneous geometries that are pertinent to emergent applications. By solving the HeII two-fluid equation of motion coupled with the Vinen’s equation for vortex-density evolution, we examine and compare thecharacteristics of transient heat transfer from planar, cylindrical, and spherical heaters in He II. Our results show that as the heaterturns on, an outgoing second-sound pulse emerges, in which the vortex density grows rapidly. These vortices attenuate the secondsound and result in a heated He II layer in front of the heater, i.e., the thermal layer. In the planar case where the vortices are createdthroughout the space, the second-sound pulse is continuously attenuated, leading to a strong thermal layer that di ff usively spreadsfollowing the heat pulse. On the contrary, in the cylindrical and the spherical heater cases, vortices are created mainly in a thinthermal layer near the heater surface. As the heat pulse ends, a rarefaction tail develops following the second-sound pulse, in whichthe temperature drops. This rarefaction tail can promptly suppress the thermal layer and take away all the thermal energy depositedin it. The e ff ects of the heater size, heat flux, pulse duration, and temperature on the thermal-layer dynamics are discussed. We alsoshow how the peak heat flux for the onset of boiling in He II can be studied in our model. Keywords:
Superfluid He, Transient heat transfer, Nonhomogeneous geometry, Second sound, Rarefaction, Quantized vortices
1. Introduction
Saturated liquid He transits to the superfluid phase (knownas He II) below about 2.17 K [1]. In He II, two miscible fluidcomponents co-exist: an inviscid and zero-entropy superfluidcomponent (i.e., the condensate) and a viscous normal-fluidcomponent (i.e., the collection of thermal excitations). Thistwo-fluid system possesses many fascinating thermal and me-chanical properties [2]. For instance, He II supports two distinctsound modes: an ordinary pressure-density wave (i.e., the firstsound) where the two fluids move in phase, and a temperature-entropy wave (i.e., the second sound) where the two fluids moveoppositely. Furthermore, heat transfer in He II is via an ex-tremely e ff ective counterflow mode instead of convection [3]:the normal fluid carries the heat away from a source at a veloc-ity v n = q /ρ sT , where q is the heat flux, and ρ and s are the HeII density and specific entropy, respectively; while the super-fluid moves in the opposite direction at a velocity v s = − v n ρ n /ρ s to balance the mass flow (here ρ n and ρ s are the densities ofthe respective fluid components). When the relative velocity ofthe two fluids exceeds a small critical value, a chaotic tangle ofquantized vortex lines can be spontaneously created in the su-perfluid, each carrying a quantized circulation κ ≃ − cm / s ∗ Corresponding author
Email address: [email protected] (Wei Guo) around its angstrom-sized core [4]. A mutual friction force be-tween the two fluids appears due to the scattering of the thermalexcitations o ff the quantized vortices [5]. This mutual frictioncan profoundly a ff ect the heat transfer and turbulence charac-teristics in both fluids [6–12].Due to its low temperature and extraordinary heat-transfercapability, He II has been widely utilized in scientific and engi-neering applications such as for cooling superconducting parti-cle accelerator cavities, superconducting magnets, and satellites[3]. Many of these applications involve transient heat transfer inHe II, a process that is complicated due to the interplay of coun-terflow, second-sound emission, and vortex nucleation. Therehave been extensive experimental and numerical studies of one-dimensional (1D) transient heat transfer of He II in a uniformchannel driven by a planar heater, due to the simplicity of thisgeometry [13–17]. These studies have revealed that the tran-sient heating from the heater generates a second-sound pulsethat propagates in He II. A counterflow establishes in the pulse,which produces tangled quantized vortices. These vortices thenattenuate the second-sound pulse, converting the energy carriedby the pulse to the internal energy of He II. This heated regionin front of the heater is termed as the thermal layer. When theheat flux is relatively high, the continuous attenuation eventu-ally curtails the second-sound pulse to a limiting profile [15],and the heat produced by the heater largely gets deposited in1he thermal layer which gradually di ff uses along the channelfollowing the second-sound pulse.It has been recognized that the heat transfer of He II in non-homogeneous geometries can exhibit new features. For in-stance, Fiszdon et al. conducted transient heat transfer experi-ments in He II using cylindrical heaters [14]. They found thata rarefaction tail of the second-sound pulse can develop, whichexhibits a drop in temperature. The thermal layer in this geom-etry can be significantly suppressed as compared to that in theplanar geometry. These observations were examined and repro-duced in numerical simulations by Kondaurova, et al. [18–20].Nevertheless, there lacks a systematic characterization of thethermal-layer dynamics and how the heat energy is divided be-tween the thermal layer and the propagating second sound. Pro-ducing this knowledge could benefit applications pertinent tocylinder shaped systems cooled by He II, such as superconduct-ing transmission lines and magnet coils [21, 22]. An emergente ff ort in developing hot-wire anemometry for studying quantumturbulence in He II [23] has further strengthened this need.Besides the cylindrical geometry, transient heat transfer ofHe II in spherical geometry is also relevant to practical appli-cations. In particular, it has been known that superconductingaccelerator cavities cooled by He II can quench due to transientheating from tiny surface defects [24]. Locating these surfacehot spots for subsequent defect removal is the key for improv-ing the cavity performance. Our team has recently developed aninnovative molecular tagging technique for locating surface hotspots via tracking thin lines of He ∗ molecular tracers [25, 26].These tracers move with the normal fluid [27–30], and there-fore the transient radial heat transfer from a hot spot can leadto line deformations that contain accurate information about thespot location. In order to extract this information, it is critical tounderstand how the heat energy is partitioned between the ther-mal layer and the second-sound pulse [25]. However, despitesome limited studies on steady-state vortex distribution near aspherical heater [31, 32], the transient behaviors of the thermallayer and its interaction with the second sound in this geometryhave remained largely unexplored [18].In this paper, we present a numerical study of transient heattransfer in all three (i.e., planar, cylindrical, and spherical) ge-ometries in He II. Our goal is to examine and compare theheat-transfer characteristics in these geometries so that a bet-ter understanding of the energy partition and thermal-layer dy-namics can be achieved. The paper is organized as follows. InSec. 2, we introduce our model, which is based on the govern-ing equations of the two-fluid system and the Vinen’s equationfor vortex-density evolution. In Sec. 3, we validate our modelby comparing the simulation result of transient heat transfer inplanar and cylindrical geometries with the experimental mea-surements by Fiszdon et al. [14]. The systematic study of theheat transfer in all geometries is discussed in Sec. 4. We firstpresent in Sec. 4.1 the calculated spatial profiles of the second-sound wave, the vortex-line density, and the thermal energy inthe three geometries under the same heating conditions. Thiscomparison clearly shows the unique features of He II heattransfer in nonhomogeneous geometries. We then discuss thethermal-layer dynamics in the cylindrical and the spherical ge- ometries and the e ff ects of various heat-pulse parameters inSec. 4.2. In Sec. 4.3, we illustrate how our model can also beused to determine the peak heat flux for the onset of boiling inHe II in di ff erent geometries. A summary is included in Sec. 5.
2. Numerical model
To study the flow field and the heat transfer in He II, we adoptthe Hall-Vinen-Bekarevich-Khalatnikov (HVBK) model [33],which is based on the conservation equations for the fluid mass,momentum, and entropy, as listed below: ∂ρ∂ t + ∇ · ( ρ v ) = ∂ ( ρ s ) ∂ t + ∇ · ( ρ s v n ) = F ns · v ns T (2) ∂ v s ∂ t + v s · ∇ v s + ∇ µ = F ns ρ s (3) ∂ ( ρ v ) ∂ t + ∇ ( ρ s v s + ρ n v n ) + ∇ p = η n ∇ v n (4)The definitions of the relevant parameters are provided in theNomenclature table. In the above equations, ρ v = ρ s v s + ρ n v n represents the total momentum density. The Gorter-Mellinkmutual friction F ns per unit fluid volume depends on the vortex-line density L and the relative velocity v ns = v n − v s betweenthe two fluids as [5, 34]: F ns = κ ρ s ρ n ρ B L L v ns (5)where B L is a known temperature-dependent mutual friction co-e ffi cient [35]. The chemical potential µ ( P , T , v ns ) of He II in-cludes a correction due to the counterflow velocity v ns , as pro-posed by Landau [2]: µ ( P , T , v ns ) = µ ( P , T ) − ρ n ρ v ns (6)This HVBK model represents a coarse-grained description ofthe two-fluid hydrodynamics, since the action of individual vor-tices on the normal fluid [36, 37] is smoothed out. When thevortex-line density is relatively high, this model has been shownto describe non-isothermal flows in He II very well even in non-homogeneous geometries [38, 39].To provide a closure to the above HVBK model, we adopt amodified version of the Vinen’s phenomenological equation todetermine the temporal and spatial variations of the vortex-linedensity L ( r , t ) [5, 34]: ∂ L ∂ t + ∇ · ( v L L ) = α V | v ns | L / − β V L + γ V | v ns | / (7)where α V , β V and γ V are temperature-dependent phenomeno-logical coe ffi cients introduced by Vinen [5]. The term ∇ · ( v L L )accounts for the drifting of the vortices [40, 41], where the vor-tex mean velocity v L is taken to be the local superfluid velocity v s , as originally proposed by Vinen [5, 34] and later utilized by2 eatingsurfaceQuantizedvorticesThermallayer a) b) c) He II 2 nd soundwave F l o w c h a n n e l P r o p a g a ti o n d i r ec ti o n He II H ea ti ng s u rf ace He IIHeatingsurface2 nd soundwaveThermallayerQuantizedvorticesRarefactionwave Figure 1: Schematic diagrams showing the transient heat transfer in He II from (a) a planar heater, (b) a cylindrical heater, and (c) a spherical heater. many others [42, 43]. The first two terms on the right-hand sideof Eq. 7 respectively account for the generation and the decayof the vortices, and the third source term serves to trigger theinitial growth of the line density [5].If one ignores the vortices and linearizes Eqs. (1)-(4) as-suming small-amplitude wave-form variations of the entropyand the counterflow velocity, it is straightforward to derive atemperature-entropy wave mode (i.e., the second sound) [2]. Atransient heating from a heater surface then generates a second-sound pulse in He II whose amplitude ∆ T is determined by theheat flux. When this amplitude is relatively high, the second-sound speed c can be written as c = c [1 + ε ( T ) ∆ T ], where c is the speed in the zero-amplitude limit and the nonlinearcoe ffi cient ε ( T ) takes the form [2]: ε ( T ) = ∂∂ T ln c C p T (8)At T < .
88 K where ε ( T ) is positive, the second-sound wavewith a higher amplitude travels faster. Therefore, a front shockcan appear at the leading edge of the second-sound pulse at suf-ficiently large ∆ T . At T > .
88 K where ε ( T ) is negative, a rearshock can form at the tail of a second-sound pulse. This phys-ical picture gets complicated when vortices are present, whichcan attenuate and distort the second-sound pulse profile.Here we consider the transient heat transfer from planar,cylindrical, and spherical heaters in He II based on all the cou-pled governing equations (i.e., Eqs. 1-7), as shown schemati-cally in Fig. 1. For simplicity, we ignore small-scale turbulentfluctuations and assume 1D flow in all three geometries, i.e., 1Dflow perpendicular to the heater in the planar case and along theradial direction in the cylindrical and the spherical cases. Fora rectangular heat pulse with a surface heat flux q h and a du-ration ∆ t , we set the boundary conditions at the heater surfaceto be v n = q h /ρ sT for the normal fluid and v s = − v n ρ n /ρ s for the superfluid during 0 < t < ∆ t and v n = v s = t > ∆ t . All the thermodynamic properties of He II are calcu-lated using the Hepak dynamic library [44]. The values of thecoe ffi cients α V and β V as recommended by Kondaurova et al. are used in Eq. 7, which appear to produce simulation resultsin good agreement with experimental observations [19]. Wethen evolve the governing equations using the MacCormack’spredictor-corrector scheme, which is accurate to the second or- der in time and space [45]. A flux-corrected transport approachis also adopted to suppress the numerical oscillations due to thediscontinuity at the shock front [45]. We have tested variousspatial steps ∆ r and time steps ∆ t s and found that the calculatedresults converged well when ∆ r < × − m and ∆ t s < × − s. In order to balance the result fidelity and the computationalcost, ∆ r = − m and ∆ t s = − s are used in all the reportedsimulations.
3. Model Validation
For model validation purpose, we first performed numeri-cal simulation on transient heat transfer of He II under thesame conditions as in the experiments conducted by Fizdon et al. [14]. These authors examined the transient heat trans-fer from both a planar heater and a cylindrical heater (radius r h = . ∆ t = q h = / cm . They measured the time variations of the He IItemperature at distances r = L seen by agiven heat pulse was relatively high, which is often treated asa tuning parameter in past numerical works [13, 14, 18]. Inour calculation, we set L = × cm − to achieve the bestmatch with the experimental observations. Fig. 2(a) shows themeasured temperature profiles together with our simulation re-sults. Since ε ( T ) > r − r h = ∆ t = q h = / cm were applied at 2 Hz repetition rate.Due to the nonhomogeneous geometry, a radial dependance of3 W P V ǻ 7 P . 6 L P X O D W L R Q ǻ 7 P . ( [ S H U L P H Q W D O P P P P P P W P V ǻ 7 P . 6 L P X O D W L R Q ǻ 7 P . ( [ S H U L P H Q W D O P P P P P P (a) Planar heaterPlanar heaterCylindrical heaterCylindrical heater (b)
Figure 2: Experimental and simulated temporal profiles of the temperature in-crement ∆ T = T − T ∞ at (a) 1, 2, and 5.4 mm from the surface of a planar heaterwith q h = / cm , ∆ t = q h = / cm , ∆ t = r h = T ∞ = . the initial line density L ( r ) = L h ( r h / r ) as recommended byKondaurova et al. [18] was adopted in our calculation, wherethe line density at the heater surface L h was set to 8 × cm − due to the increased repetition rate. Again, all the key featuresof the observed temperature curves are reproduced. This ex-cellent agreement between the experimental measurements andour simulation results has thereby validated the fidelity of ourmodel calculation.
4. Simulation results and discussion
In this section, we first present the simulation results to com-pare the key features associated with the transient heat transferin di ff erent heater geometries. We then examine the time evo-lution of the thermal layer in the cylindrical and the sphericalheater cases under various heating conditions. Since our focusis the heat transfer following a single heat pulse, a small initialvortex-line density L = cm − is assumed in the calcula-tions. This L is comparable to the typical density of remnantvortices pinned to He II container walls [46]. Indeed, it has been shown that in relatively high flux counterflow, the simu-lated temperature profile in He II is nearly independent of L when L is smaller than about 10 cm − due to the source termin Eq. 7 [20]. To avoid the complication of possible boiling inHe II near the heater surface, we have also assumed that theheater is placed at a 1-meter depth below the He II free surfacein all the cases. We will discuss in the last subsection how thishydrostatic head pressure ensures the helium to be always inthe He II state during the transient heat transfer. This discus-sion also provides a foundation for our future study of the peakheat flux for the onset of boiling in He II. ff erent heatergeometries To compare the heat transfer characteristics in di ff erentheater geometries, we show the simulated spatial profiles of thetemperature increment ∆ T = T − T ∞ , the vortex-line density L , and the thermal energy density W = ρ C p ∆ T at various time t in Fig. 3. In this calculation, we set the He II bath tempera-ture to T ∞ = .
78 K. A heat pulse with a surface flux q h = / cm and a fixed duration ∆ t = . t = r h =
1) Second-sound pulse:
As shown in Fig. 3(1a-1c), asecond-sound pulse with positive ∆ T emerges when the heaterturns on, which carries the heat energy and propagates awayfrom the heater surface at the known second-sound speed (i.e., c = . / s at 1.78 K [35]) in all three cases. Inside the pulseprofile, a counterflow establishes where the normal-fluid veloc-ity is determined by the thermal energy flux as v n = c W /ρ sT = ( c C p / s ) · ∆ T / T . This counterflow leads to a rapid generationof the quantized vortices. In the planar heater case, the second-sound pulse gradually evolves from a rectangular profile nearthe heater to a front-shock profile due to the combined e ff ectsof the vortex attenuation and the positive ε ( T ). In the cylindri-cal and the spherical heater cases, as the second-sound pulsepropagates outward, the cross-section area of the pulse A ( r ) in-creases as r and r , respectively. In regions where the vortexdensity is low and hence the mutual friction is negligible, thekinetic energy of each fluid component is nearly conserved. Forthe normal fluid, this means that v n in the second-sound pulsemust drop as 1 / √ r in the cylindrical geometry and as 1 / r in thespherical geometry. Since ∆ T is proportional to v n , it also dropsin a similar fashion as the pulse propagates, which is clearlyseen in Fig. 3(1b-1c).
2) Quantized vortices:
The vortices are created as a con-sequence of the counterflow in the second-sound pulse. In theplanar heater case, the thermal energy flux W in the pulse re-mains high as the pulse propagates. Therefore, a dense tan-gle of vortices are created in the entire space traversed by thesecond-sound pulse (see Fig. 3(1a)), which continuously atten-uate the pulse. In the cylindrical and the spherical heater cases,since the thermal energy flux drops with r due to the diverg-ing geometries, the line density L is high (i.e., greater than 10 cm − ) only in a thin layer of He II near the heater surface (see4 igure 3: Profiles of (1) temperature increment ∆ T , (2) vortex-line density L , and (3) thermal energy density W compensated by the ratio of the cross section area A ( r ) to the heater surface area A h in (a) planar, (b) cylindrical, and (c) spherical geometries at 1.78 K. In all cases, q h =
23 W / cm , ∆ t = r h = Fig. 3(1b-1c)). Outside this region, the second-sound pulse ex-periences negligible attenuation.
3) Rarefaction tail:
A peculiar feature of the temperatureprofile in the cylindrical and the spherical geometries, as com-pared to the planar case, is the appearance of a tail region withnegative ∆ T following the positive second-sound pulse (seeFig. 3(1b-1c)). This negative ∆ T tail, which emerges after theheater is switched o ff , is known as the rarefaction wave [47–49]. The underlying physics can be understood as follows. Thetotal thermal energy carried by the second-sound pulse can beevaluated as Q s = R ∆ R W ( r ) A ( r ) dr , where ∆ R ≃ c ∆ t is thethickness of the pulse. Since W ( r ) A ( r ) is expected to increaseas √ r in the cylindrical geometry and as r in spherical geom-etry (confirmed in our simulation, i.e., see Fig. 3(3b-3c)), Q s increases as the pulse propagates. To supply this ever-growingthermal energy carried by the second-sound pulse, there mustbe a flow of the internal energy from the tail region towards thepulse front, which thereby leads to the formation of the negative ∆ T rarefaction tail. If we integrate Q s over both the positivepulse and the rarefaction tail, the total thermal energy carriedby them always equals the input heat energy, which fulfills theenergy-conservation law.
4) Thermal layer:
Near the heater surface where the vortex-line density L is high, the interaction between the vortices andthe second-sound pulse e ff ectively converts the thermal en- ergy carried by the pulse to locally deposited heat, resultingin a heated layer of He II, i.e., the thermal layer. To see thislayer clearly, we plot the ∆ T profile near the heater in all threecases in Fig. 4. As the heat pulse ends, ∆ T on the heatersurface reaches the highest value. In the planar heater case, ∆ T ( r h ) =
170 mK on the heater surface, which is about 7 timesthe ∆ T in the second-sound pulse. The heat content in thisthermal layer di ff usively spreads out [3]. On the contrary, inthe cylindrical and the spherical heater cases, the temperaturebuildup in the thermal layer is much weaker. Indeed, both thelayer thickness and the maximum ∆ T in the spherical geome-try are insignificant. Another important feature of the thermal-layer dynamics in the two nonhomogeneous geometries is thatthis layer dies out rapidly before it has time to undergo di ff usivespreading. This prompt suppression is due to the same mecha-nism for the formation of the rarefaction tail: the internal energyin these nonhomogeneous geometries is actively transferred to-wards the second-sound pulse front in order to supply the ever-growing thermal energy carried by the pulse. The depletion ofthe deposited heat in the thermal layer occurs simultaneouslywith the formation of the rarefaction tail.
5) Heat energy partition:
The partition of the heat energybetween the thermal layer and the second-sound pulse as well ashow this partition varies with time is of practical significance.To examine this partition, we calculate the heat energy in the5 U P P ǻ 7 P . D 3 O D Q D U K H D W H U P V P V P V P V P V U P P ǻ 7 P . E &