Second Sound Attenuation in a Liquid Helium Counterflow Jet Citation Laguna, Glenn Alan (1975) Second Sound Attenuation in a Liquid Helium Counterflow Jet. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7C0G-HK08. https://resolver.caltech.edu/CaltechTHESIS:09282010-140423372 Abstract The attenuation of a beam of high frequency second sound traversing a counterflow jet in liquid helium has been measured in the temperature range 1.6 to 2.06°K. Combined use of thin film superconducting thermometers with specially developed low noise amplifiers allowed a temperature resolution of better than one part in 10^8 °K. The additional attenuation due to the jet was found to be less than 10 percent of the predicted value using the theory of mutual friction in a supercritical counterflow, and consistent with the result of earlier temperature gradient and ion beam attenuation measurements. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Applied Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Liepmann, Hans Wolfgang Thesis Committee: Unknown, Unknown Defense Date: 10 March 1975 Record Number: CaltechTHESIS:09282010-140423372 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09282010-140423372 DOI: 10.7907/7C0G-HK08 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6069 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 28 Sep 2010 21:18 Last Modified: 05 Aug 2024 22:08 Thesis Files Preview PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

SECOND SOUND ATTENUATION IN A LIQUID HELIUM COUNTERFLOW JET Thesis by Glenn A. Laguna In Partial Fulfillment of the Requirements For the Deg ree of Doctor of Philos ophy California Institute of Technology Pasadena, California 1975 (Submitted March 10, (975) ii ACKNOWLEDGEMENTS It is a great pleasure to acknowledge all of the people who made this work possible. adviser and friend, In particular, I would like to thank my Professor Hans Wolfgang Liepmann for his advice and encou ragement. Drs. James E. Broadwell and Paul Dinlotakis have also contributed many helpful suggestions, help of Drs. Harris A. and the Notarys and David Palmer of Low Temper- ature Physics was indispensible for the thin film fabrication. The author also wishes to thank Mrs. help with the figures and Mrs. Betty Wood for her Jacquelyn Beard for typing the manuscript and contributing her good humor. Lastly, I would 1ike to thank the graduate students of GALCIT for leaving the experirnental apparatus alone long enough for rnc to take sorne measurenlents. The initial phase of this work was supported by the Alfred P. Sloan Foundation. The final pha se of the work was supported by the Air Force Office of Scientific Research. iii ABSTRACT The attenuation of a beam of high frequency second sound traversing a counterflow jet in liquid helium has been measured in the temperature range 1.6 to 2.06°K. Combined use of thin film superconducting thermometers with specially developed low noise amplifiers allowed a temperature resolution of better than one part In ] 0 80 K. The additional attenuation due to the jet was found to be less than 10 percent of the predicted value using the theory of rnutual friction in a supercritical counterflow, and consistent with the result of earlier temperature gradient and ion hearn attenuation measurements. iv TABLE OF CONTENTS PART TITLE PAGE Acknowledgements 11 Abstract iii Table of Contents iv I. Introduction 1 II. The Two-Fluid Model and Mutual Friction 6 II. A Review of the Two Fluid Model 6 II. B Experiments Motivating the Present Research 11 III. IV. V. Experimental Apparatus 21 III. A Counterflow Jet 21 III. B Second Sound Equipment 21 III. C Bath Temperature Regulation 30 III. D Electronics 32 Method of Measurement and Preliminary Experiments 37 IV. A Second Sound Resonance Technique 37 IV. B Initial Checkout Experiments 39 Attenuation Measurements 48 V. A Data Taking Procedure 48 V. B Discussion of the Experimental Resonance Curve 49 V. C Da ta Reduction 52 VI. Results and Discussion 60 VII. Future Experiments 66 VIII. Conclusion 69 v TABLE OF CONTENTS (cont.) PART TITLE PAGE Appendices A. B. Second Sound Detectors 70 A. 1 Types of Resistive Detectors 70 A. 2 Superconducting Thin Film Fabrication Details 72 A.3 Photoresist Processing 74 Deflection of Second Sound by a Turbulent Jet 79 B. 1 Some Facts about Turbulent Jets 79 B. 2 Mean Square Deflection Angle 81 B.3 Index of Refraction of Second Sound 82 B.4 Correlation Length 83 B.5 Completion of the Calculation 83 B.6 Conclusion 84 1 I. INTRODUCTION Liquid helium below the A. -point has been perhaps the most closely scrutinized of all fluids because of its unique position as a fluid whose behavior on a macroscopic scale is governed by quanturn mechanical effects. Despite this intensive study, the experi- mental investigation of the hydrodynamics of liquid helium has been extremely limited, and many of the interesting phenomena observed in classical fluids remain unstudied. Although it is now agreed that the linearized two-fluid equations are accurate in the proper limit, the nonlinear phenomena of shock waves and turbulence have just recently begun to be studied. For example, (Ref. 1) in 1951, and Dessler and Fairbank (Ref. although Osborne 2) in 1956, meas- ured finite amplitude effects in second sound (temperature or entropy waves), shock waves were not observed in liquid helium until the work of Cummings in 1973 (R ef. 3). Turbulence in both the superfluid and normal fluid has often been invoked as an explanation for various observed phenomena in liquid helium, yet no experiments have measured any random fluctuating quantities. Vinen I s (Ref. In 4) model of the Gorter-Mellink force the fact that the superfluid is supposedly turbulent is almost irrelevant. Vinen does not introduce any stochastic functions or averaging into his model. There can be no doubt about the need to look closely at these effects of nonlinea rity. The present investigation is third in a group of studies which deals with another hydrodynamic problem. With few excep- tions, the experiments on the fluid mechanics of liquid helium have 2 been on confined flows, i. e., capillaries or wider channels. While this type of geometry is advantageous in the investigation of critical velocities, it is not a clean situation hydrodynamically because there is no way to separate the flow properties fron1 the boundary conditions at the walls, and these boundary conditions are not yet known with certainty (see Ref. experiments (Ref. 5). The two previous 6,7), as well as the present one, have been con- cerned with nozzles and jets m flow geometry without walls, from the boundary conditions. liquid helium. The jet provide s a thus separating the flow properties As it is unlikely that any applica- tions of liquid helium as a coolant will use narrow capillaries, it is essential to know the heat transfer characteristics of the bulk fluid for technological purposes. This experiment is a study of the counterflow jet similar to the jet shown in Figure 1. 1. The jet is produced by thermal counterflow or internal convection and is supercritical in the sense that at the orifice the heat flux is high enough to cause a breakdown of the simple two-fluid model. For supercritical counterflow in a channel, the temperature difference between lhe hot and cold ends of the channel is proportional to the cube of the heat flux, with a constant of proportionality dependent only on the bath temperature and not on geometry. Gorter and Mellink (Ref. two-fluid equations. This enlpirical result prompted 8) to append mutual friction terms to the This accounts for the temperature gradient and produces an additional linear attenuation of second sound. However, the geometry independent tempe rature gradient is not 3 L He Bath Figure 1. 1. The counterflow jet 4 consistent with the investigation of Dimotakis and Broadwell (Ref. 7) who measured the temperature gradient a long the axis of the jet and found that the gradient disappeared within one jet diameter of If the jet remains collimated, the orifice. (Ref. as measured by Kapitza 9), and if the superfluid is not entrained by the normal fluid (which was shown by Dimotakis to be improbable), then the mutual friction idea rnust either be modified or abandoned. Additional details of the expe riments and mutual friction theory are provided in Chapter II. In the present work a beam of high frequency second sound was propagated transverse to the jet, and the additional attenuation as a function of heat flux at the orifice was measured. Zero addi- tional attenuation would be consistent with the Dimotakis and Broadwell experiment and would be :-1 further indication of the geometry dependence of mutual friction. The other extreme would be a measurement of the full attenuation predicted by the GorterMellink force. An additional goal of this research was to develop a simple system for the detection of high frequency second sound specifically for use In flow investigations of liquid helium. Even under the most ideal conditions and with maximum power input to the second sound emitte r. the emitter the amplitude of a 1 MHz wave one centimeter from would be 10 -S ~ to 10 -7 0 K, and this could be substan- tially reduced by the Kapitza boundary effect. in electronic spectral Recent developments analysis equipment have made a simple detection scheme feasible. This will pe rmit the extensive use of 5 high frequency second sound as a local probe for heat transfer studies in liquid helium. REFERENCES 1. D. V. Osborne. Proc. Phys. 2. A. J. Dessler and W. M. Fairbank, 3. J. C. Cummings, Thesis, California Institute of Ph. D. Soc. (London) 64, Phys. Rev. 114 (1951). 104, 6 (1956). Technology (1973). 4. W. F. Vi nen, 5. C. C. Lin, New York, 6. J. E. Proc. Roy. Soc. (London) 242, 493 (1957). Liquid Heliuln, editor Careri (Academic Press, 1963) page 93. Broadwell and H. W. Phys. Fluids g, Broadwell, Phys. Fluids ..!i., 1787 Liepmann, 1533 (1969). 7. Dimotakis and J. E. P. E. (1973). 8. C. J. Gorter and J. H. 9. P. L. Kapitza, J. Mellink, Phys. Physica USSR~, .!.2., 285 (1949). 181 (1941). 6 II. THE TWO-FLUID MODEL AND MUTUAL FRICTION II. A. Review of the Two- Fluid Model The hydrodynaITlics of liquid helium below the Ie -point can be described by a two-fluid ITlodel suggested by Landau (Ref. Tisza (Ref. 2). 1) and The ITlotion of the liquid is considered to consist of the ITlotion of two interpenetrating but noninteracting fluids, described by its own density and velocity fields. has a density P s and velocity v The superfluid As the absolute zero of s temperature is approached, all the fluid becomes superfluid, p s -> rj as T 0 . -> Therefore, i. e. , the superfluid carries no entropy ann rnust flow reversibly without viscous dissipation. quence, each As a conse- Landau postulated that the velocity field of the supcrfluid must be irrotational. velocity v n The normal fluid with density is nornlal 10 P nand the sense that it has a viscosity and Yj carries all the entropy. With the above understanding the equations of motion for the two fluids can be derived (see Ref. The total density is the 3). sum of the sllperfluid plus normal fluid densities, and the mass flux is the sum of the superfluid and the normal fluid mass fluxes, i. e. p -:> -+ J = f.l v s -> Ps v s + P n v~ (1) Thus the continuity equation can be written 1n a way identical to an ordinary fluid, .z.e. at .... + 'V • P v = 0 (2) 7 The momentum equation is also a straightforward generalization. Neglecting viscous dissipation it can be written simply as rlj.1 ot (3) 8 + o(x TIik = 0 TT ik k = P n v ni v nk + P s v si v sk + P 6 ik . Since the norrnal fluid carries all the entropy, the entropy flux must be P sv where n s is the entropy per unit mass. Neglecting source s of entropy we have (4 ) Lastly one must consider how the superfluid moves. A s it is presumed irrotational, its convective derivative can be derived from a scalar potential, 0";s i. e. , ,v 2 (s &+9\2 + iJ-) = 0 Arguments of Galilean invariance identify potential per unit mass (see Ref. 4). (5) iJ- with the chemical Upon rearrangement and inclusion of the normal fluid viscosity the equations appear as in Table II. 1 where they have been summarized for easy reference. There are two consequences of the equations of motion which are of particular importance to the present work. First, through the chemical potential term in the equation for the superfluid, a temperature gradient acts as a driving force on the superfluid causing it to change its velocity. of walls, Thus, in the absence ordinary heat conduction is excluded as a mechanism of heat transport. Instead it is replaced by the highly efficient heat 8 Table II. 1 Two-Fluid Equations of Motion ££. at + \ j . p v = 0 Mass Conservation Monlenturn Conservation 8 -> ->-> at (p v) + V • (P vv + PsP n ->-> + VP w w) p r1 'I 17 2 (V Ps + P Entropy Conse rva tion Sps at + V • (psv ) n Supe rOuid -> av s c)t v r V 2 s \2 t 11. . ) 0 -+ , v wbc re ){ p n - the rrnal v n + ps v s C onducti vity w v n - vs 1\ - bulk viscosity. -> w) 9 transfer method of thermal counterflow or internal convection, which is an isothermal, convective heat transport (see Ref. 5). The heat flux vector is the product of the entropy flux and the a bsolute temperature, i. e. , q psTv (6) n If there are solid boundaries a temperature gradient proportional to the heat flux can be maintained owing to the viscous interaction of the normal component with the walls. For a channel of con- stant diameter this is found to be oT ( 7) ox where G is a geonletry-dependent constant. A second consequence of the two-fluid equations is the type of wave motion which the liquid helium will support (see Ref. 6). If the two-fluid equations are linearized and a wave solution is substituted for the superfluid and normal fluid velocities, and the perturbation parts of the pressure and ternperature, iw (t-~) P = P o + p' e u T - T o + T' e iw (t-~) u one gets a quartic characteristic polynonlial for the phase velocity. Upon neglecting the coefficient of thermal expansion, which is small for al1 liquids and anomalously small for liquid helium, the quartic separates into two quadratics giving the two velocities Ts2 p cp n s (8 ) 10 Thruugh an analysis due to Lifschitz (Ref. Th,-~ of these waves becomes clear. speed 7) the physical nature u is associated with l a pressure wave with the temperature remaining constant to the approxirnation that neglected. P, the coefficient of thermal expansion, can be The superfluid and normal fluid oscillate in phase so this mode is identical to sound propagation in ordinary fluids. Further consideration shows that this oscillation, first sound, which is called corresponds more to isentropic than isothermal con- ditions. The other wave speed, is associated with a non- dispersive ten1perature or entropy wave. The superfluid and normal fluid oscillate out of phase in a way that the mass flux e q ua] s ze r 0 , 1. e. , vs = - p n /p s v n The pressure and density remain constant to first order in 13 These temperature waves are unique to liquid helium and have no relation to the highly damped temperature waves In a thermally conducting merlium which die out within one thermal wavelength. This new wave motion, called second sound for historical reasons, was predicted first by Tisza's two-fluid theory and discovered experimentally by Peshkov in 1944 (Ref. coupling to pressure is of order 8). Because the 13 , second sound is most effi- ciently excited by a heater rather than a vibrating plate, and a sensitive therrnorneter is needed for detection. A s a unique property of liquid helium, second sound has 11 been extensively studied. sound, (Ref. The atten'-1ation coefficient of svcond which is usually denoted by 9) in his helium. (1 , \I,'as derived by I<.halatnikov consideration of the kinetic coefficients of liquid This was ITleasured by Hanson cud Pellam (Ref. Their frequencies extended into the 100 kHz range. (0). Perhaps the ITlost drarnatic deITlonstration of the wave nature of second sound was the work of Mercereau, Notarys, and Pellam (Ref. 11). Using frequencies up to 600 kHz they m,-,asured the diffraction (li second sound frorn a heater which consisted of a.n array of equd.lly spaced pa rallel elen1ents. In a late!" work, N(ltarys (Ref. 12) measured the speed and attenuation r)f second sound with frequenci.es up to 25 MHz. Using a resonance techni.que. Nota rys found that even at these high frequencies the spcl'd ;lnd sallIe as at: lower frequencies. A a1h~lll!ati[)n were th.· sirnilal" resunance technique hds been used in this inve stigation but: with g rva tl y si.Jn plificd electrunics. II. B. Experirnents Motivating the Present Research Despite the successes of the two-flui.d model, a s has been remarked by C. T. Lane (Ref the equations, 13), we re proposed on thE' basis of a specific set of experiments and il is nut unlikely that they win have to be modified in the CHlrse ur lillie. Sorne of the l'l'cent exp<'rinlents and the modifications to the two-fluid nl0del which they suggest ~re considered in this section. 12 Equation (7) for the teITlperature gradient in a channel can be integrated, providing the coefficient stant (sm~all arT) is rea"onably con- temperature difference), to give = - f, aq Li T (0 ) for the temperature difference between the hot and cold ends of a channel in pure counterflow with no net mass flux. This e xpre s - sion was verified for small heat fluxes by Keesom (Ref. 14), but as lhe heat flux is increased a critical h('at flux is reached beyond which the exp(·rimentally determined relation between the temperature diffe rence and the heat flux is To account for the cubic dependence of the temperature diff(~rence on the h(~at flux, Gorter and Mellink (Ref. 15) in 1949 proposed appending mutual friction terms to the two-fluid equations. The frictional force was proportional to the cube of the relative velocity of th" saperfluid and normal fluid, and given by the expression ... F sn 2-> =-Appww w s n -~ v n v (10) s The equations of motion become Bv s v 2 s at + \l \-2- + JJ.) ;'y 11 i-lt (v n V) v n -1 \.vp p 1'1n p n \l 2 A --t v n f} n ~; p s n 2-+ w w s v T - A,) (1 l) s W G . W ( 12 ) These equations give a temperature gradient in the supercritical region 'V T :.:.: - aq - b(T) q 3 Ap b(T) = s (p Note that b(T) s n s T) ( 13) 3 is independent of the size or shape of the channel. This law seems to apply to wide channels over a wide range of heat fluxes and channel dirTIensions. mea sured by Vinen (Ref. The coefficient A was 16) and found to be tempe rature dependent, but only very weakly dependent on the size of the channel. On the basis of his experimental observations Vinen proposed a mechanism for the mutual friction force (Ref. Acunding to Vinen, 16). the critical relative velocity was representative of a tran",iiion to a turbulent state of the superfluid. This turbu- lent stale is characterized by a tangled mass of quantized vortex lines and is r~)Ugh]y analogous to the ranclOlTI fluctuations in the vorticity of a turbulent classical fluid. but with circulation a round each vortex core quantized, as suggested by Onsager (Ref. Feynman (Ref. 19). 18) and The mutual friction resulted from the inter- action of the excitati ons which make up the normal fluid with the tangled mass of vortices. Using dimensional analysis and frequent appeals to classical fluid mechanics, Vinen was able to extract the essential dependence of the Gorter-Mellink force on the relative velocity. Although intuitively appealing, the idea of rnutual fril,tion does not appear applicable in all situations where there is a tive velocity. rela- Temperature difference measurements in a channel 14 where v s and v could be varied independently (Refs. n 20 and 21) show a more complicated dependence on the velocities than can be expressed as a simple function of their difference. To explain these experiments and others, more friction tenns were added to the equations of motion; however, the predictive value of Vinen ' s model is lost and its physical basis somewhat open to question since it relies heavily on geom,etry independence. One cannot tell a priori which mutual friction terms should be added to describe a flow situation. Still, the Gorter-Mellink law provides results in excellent agreement with pure counterflow in wide channels. To further investigate the validity of the equations of motion with the Gorter-Mellink force appended, (Ref. Dimotakis and Broadwell 22) measured the temperature gradient in a counterflow jet. In 1941 Kapitza (Ref. 23) observed that a well-defined, collimated jet emerged from the mouth of a counterflow channel and persisted for many channel dianleters downstream. That the jet did not spread appreciably was observed by measuring the deflection of a vane moved across the jet. The two-fluid model would identify the effluent as normal fluid while the orifice must represent a sink for the superfluid. Consideration of the superfluid and normal fluid streamlines shows that the jet is a region of high relative velocity extending far into the fluid away from solid bounda ries. If the mutual friction is truly a volume force independent of the walls, a temperature gradient should be obse rved extending all the way to the free surface of the liquid. Dimotakis and Broadwell measured the temperature gradient 15 by traversing a small carbon thermometer along the axis of the jet from inside the chamber to well outside the chamber into the free The sta rtling result was that the temperature gradient wa s fluid. confined to the vicinity of the orifice and disappeared completely Note, within one diameter of the orifice. however, that this result is not at all unusual if one thinks in te rm s of the analogy between temperature in a counterflow jet and pressure in an ordinary jet. For the classical jet there is no mechanism for the support of a downstream pres sure gradient. A simple explanation, that w =0 lU consistent with mutual friction, the jet; i. e., the superfluid is entrained by the normal fluid and so there is no relative velocity. was considered untenable by Dimotakis (Ref. reason. Let is T(x) This explanation 24-) for the following be the measured temperature at a distance x from the orifice along the axis of the jet. One can form a dimen- sionless variable Teo ature and 6T temperatures. (T(x) - Too) /6 T where is the bath tempe r- is the difference between the chamber and bath If this dimensionless temperature is plotted against x , the resulting profile is self-similar for all heat fluxes and bath temperatures (see Ref. 22, The normal fluid Figure 15). would have to entrain the superfluid while v s varies over three orders of magnitude in order to produce this simila rity with In summary, the jet represents a region w -= or high relative velucity with no temperature gradient and is therefore inconsistent with a geometry independent mutual friction. This appears to be a sig- nificant failure of the Vinen lTIodel for supercritical counterflow. o. 16 In 1968 another experiment involving counterflow through an orifice was done by Careri, Cerdonio, and Dupr~ (Ref. 25). This time the observation was by means of a negative ion trapping. Negatively charged ions in liquid helium had been observed to be trapped on the superfluid vortex lines (for explanation see Donnelly, Ref. 26), so that the attenuation of a negative ion beam is a direct measure of the amount of vorticity in the superfluid. menters mounted diodes, detector, The experi- each consisting of an ion source and on both sides of an orifice. The experi ment consisted of measuring the ion current as a function of the heat flux through the orifice. At a critical heat current they observed a sharp increase in the attenuation of the ion beam in the diode on the normal fluid downstream side (hereafter called simply downstream side) of the orifice with no change in attenuation in the upstream diode. The tem.perature difference across the orifice was also monitored at the same tirne. The change in attenuation of the ion beam oc- e1.l ned while the temperature difference was still linear in the heat flux. The interpretation of this result was that vortex lines created at the orifice interact with the normal fluid and are observed on the downstrearn side. Normal fluid turbulence could not play an impor- tant role since one would then observe a temperature independent Reynolds number for the transition using the normal fluid velocity at the critical heat flux. Although this experiment again suggests entrainment of the superfluid, the interpretation is not wholly convincing. The first objection is that the diodes are mounted sym- metrically and close to the orifice, but because of the small size 17 of the orifice these are still 50 diameters away from the exit plane. Thus the diode on the downstream side is in a region of high while the upstream diode is in a region where v n v n Since the ::::: 0 vortices are part of the superfluid it is unclear why they should all move with the normal fluid. One would expect only a fraction of the superfluid vortices to be entrained, depending on v n Thus it is still unclear why the upstream diode saw no attenuation of the beam. In view of the Vinen model of mutual friction and the observation by Careri of superfluid vorticity downstream of the orifice even at small heat fluxes, one cannot simply explain the temper- ature gradient in the Dimotakis and Broadwell experiment. this reason the present research was undertaken. For It is found from the equations of 111otion with the Gorter-Mellink terms that mutual friction causes an additional linear attenuation of second sound given by e -(6a )x Ap 2 - 2u w 2 ( )4-) This has been verified by Vinen for pure counterflow in wide channels and interpreted as the scattering of the normal fluid quanta in second sound by the quantized vortices in the turbulent supe rfluid. In the present expe riment an emitte r and detector for second sound were placed in a Fabry-Perot resonator configuration on the normal fluid downstream side of the jet. A second sound beam propagated transverse to the jet in a way similar to the ion beam of Ref. 25. The basic experiment consisted of rneasuring 18 the attenuation of the second sound beaITl as a function of the heat flux in the jet. If, as observed by Kapitza and DiITlotakis, the jet does not spread, then zero additional attenuation would indicate again that the Gorter-Mellink force ITlust turn itself off outside the jet. Thus zero additional attenuation would be consistent with the results of DiITlotakis and Broadwell. This was the expected result. uation of the second sound was observed, If a tten- this would confirrn Careri's ITleasureITlent and could be compa red to Vinen' s data for wide channels as a test of the geoITletry independence of the GorterMellink force. REFERENCES 1. L.D. 2. L. 3. L. D. Landau, J. Phys. Tisza, Nature 141, Landau and E. M. Press, 4. Ibid. 5. F. London, USSR 5, 71 (1941). 913 (1938). Lifschitz, Fluid Mechanics, (pergamon 1959), pages 510-517. page 513. London, Superfluids vol. 2, (Dover, New York, 1954), pages 155-164. 6. L. D. Landau and E. M. Press, 7. Ibid. 8. V. 9. I. M. London, Lifschitz, Fluid Mechanics, (Pergamon 1959), pages 517-522. page 519. Peshkov, J. Phys. USSR !..Q., 389 (194()) Khalatnikov, Introduction to the Theory of Superfluidity. (Benjamin, New York, 1965), pages 78-80. 19 10. W. B. and J. R. Hanson Pellarn, Phys. Rev. 95, 321 (1954) . 11. J. Notarys, and J. R. H. Mercereau, Pellarn, Proceedings of the Eighth International Conference on Low Temperature Physics. 12. (Toronto, 1960), page 552. H. A. Notarys, Ph. D. Thesis, California Institute of Technology (1964). ] 3. C. T. Lane, Superfluid Physics (McG raw- Hill, 14. W. A. Kees om IS. C, J. Gorter and J. H. 16. W. F. Vinen, Proc. Roy. Soc. A 240, 114 (1957). 17. W. F. Vinen, Proc. Roy. Soc. A ';;42, 493 (1C)S7). ]8. L. Onsager, NUDVO Cirnento ~, 19. R. P. and A. P. Feynman, Kees am, Mellink, 20. II. C. 249 (]949). Physics, vol. Supedluid Helium, editor J. F. Krarnet'S, ll)(j()) Allen page 199. P. E. 22" 473 (1972). Dimotakis and J. E. Broadwell, Phys. Fluids li" 1787 (1973). i, 181 (1941). 23. P. L. Kapitza, J. Phys. USSR 24. P. E. Dirnotakis, Ph. D. Thesis, California Institute of Technology (1972) page 24. 2.5. 1, r i . Van Dcr Heijden, W. J. P De Voogt, and H. C. Kramers, Physica 22. L85 (l94l)). Physica..!.2, 19'15). (Academic Press, New York, 21. Physica .i,) 59 (1l) ~(»). Prog ress in Low T clnp. (North l!ulland, Amsterdam., New York, 1962). G. Careri, 233 (1968). M. Cerdonio, and F. Dupr~, Phys. Rev. 167, 20 26. R. J. Donnelly, Experimental Superfluidity, (Univ. Press, Chicago, 1967), pages 172-207. of Chicago 21 III EXPERIMENTAL APPARATUS III. A. Counterflow Jet The counterflow jet is shown in Figure III. 1. structed of lucite with a wall thickness of 1/8 inch. It is con- The heat flux th rough the walls is then un d e r one percellt of the total heat flux. ThE" heate r at the base of the wide cham be r is 25 ohms and is constructed of .OOS inch Evanohrn wire. Uniform heat distribution over the bottom of the channel was insured by potting the heater in Cerelow, a low melting point alloy silnila r to W Dod I s metal. The orifice is a slit l/16 inch wide and 1/4 inch long. is the only region of high heat flux. It This shape was chosen for easier alignment of the second sound beam with the jet, while the total area was dictated by the desire to produce heat fluxes comparable to previous experiments and still be able to maintain the bath tempe rature within the requi red 1 i mils. III. B. Second Sound Eguipment III. B. 1 Emitte r and Detector The second sound emitte r and detector a re shown in Figure o III. 2. The emitter was a gold film les s than 100 A thick and 1 2 with 1500A • thick copper leads evaporated under high vacuum zcm onto a glass microscope slide. Electrical contact to the copper was made with indium solder and a drop of silver print paint. The o resistance of the emitter was kept very close to 50 ohms at 4.2 K to match the impedance of the subminiature coax, second harmonic generation. thereby reducing Second sound was generated by Joule 22 --I r-- 1/16 in. 25.n Heater Fi.gure III. 1. Design details of the counterflow jet used in the present expe riments. 23 !1-_..---1.27cm --~~.-..I o 100 A Au Glass Substrate .11-..----+--1500 'l-LAUP Acu Indium Solder Emitter Glass Substrate Au- Sn Film 1500 A Sn Indium Solder Detector Figure III. 2. Second sound emitter and detector. 24 w heating of the resistive gold film at a frequency 2 A supe rconducting thin film biased on its trans ition tempe ratu re se rved a s the second sound detecto r. Typical detector transition curves are shown in Figure III. 3. When the film is fed with a constant current the second sound temperature fluctuations cause a fluctuating resistance of the film and therefore an oscillating voltage drop across it. The films were approximately lOOOA of tin on 250A of gold. Leads are l500A of pure tin which o has a transition ten1perature of 3. 72°K and therefore is not sensitive to the temperature fluctuations of secund sound. The sensit ive area is the one millirneter square flag-shaped region in the cent.('T. Fabrication detd.i1s arc supplied in Appendi", A. The basic 1ransitiol1 teIllperatun' of the IIJm was set by the ratio of tin to gold, but this transition cOllld be lowered by means of a nlagnctic field as seen in Figure 111. :'. In this way one film served as a dl'1ector over a wide tenlperature range. The deviation from linearity 'If the resistance versus tenlperature curve in the center of the transition is less than nne pl'rcent. tht, bias cu rrent the detector efficiency, dR It) dT ' By optimizing could be made as 0 V nlts 1 K. high as 0.6 III. B. 2 Second Sound Cell The se< ond sound cell served the dual purposes of holding the emitter and detector parallel and supporting the magnet which biased the detecto r. The assembled cell is shown in Figure III. 4. It was machined entirely out of Type 304 stainless steel and has Figure III. 3. o 5 10 15 20 mV 25 30 35 20 TORR 30 40 Typical family of superconducting detector transition curves. Magnetic field ipcreases to L1'2 left, Bias current = 1 mAo 10 U1 N 26 Figure III. 4. Second sound cell 27 provision for varying the emitter detector spacing from four rnillimeters to one centimeter. Alignment of the emitter and detector is by three set screws for each side arranged similar to a laser mirror mount. Tension is provided by stainless steel leaf springs which hold the emitter and detector against the alignment screws. The magnet was fashioned by winding 40 A. W. G. magnet wire on a s oft iron core. copper This is capable of providing fields at the detector of up to 200 gauss. The magnet yoke fits onto the detector mount in close proxinlity to the detector, but is easily demounted for alignment of the emitter and detector. Alignment was accomplished at room temperature by meanS of a helium-neon laser and an optical setup. Because the enti rc structure is of the same material, there can be no differential contraction so alignnlent at room telnperature is mailJtained at helium temperatures. The optical path length from the laser to the cell was two meters. By bringing the reflected light from the surfaces of the emitter and detector back along the optical axis and into the dumping mirror of the laser (0.5 cm diameter) the maximum misalignment was better than o. 15 degrees which corresponds to an alignment of O. I A for second sound frequencies up to one mega- hertz. The entire assembly of the jet, second sound cell, and supporting structure is shown in Figure III. 5. a top view schematic, drawn to scale, detector. Figure 111.6 shows of the orifice, emitter, and 28 Figure III. 5. Support structure with jet 29 r-:--- ---- --------, L___ _ __ -1 . Emitter ( tern 1 ) Orifice ~Detector r- -------, L-. _ _ _ _ ._.__ ._-1 I...-..----1.27cm -----....-II Figure III. 6. Top view schematic of the emitter, detector, and 0 dfice geometry 30 III. C. Bath Temperature Regulation Temperature regulation was accomplished by means of regu- lation of the bath pres sure using a condom regulator. The pres sure above the bath was monitored with a lOO mm full scale Barocell pressure transducer and converted to temperature using the 1958 He 4 temperature scale. To maintain constant temperature for the duration of the experiment, it was necessary to maintain a constant heat input into the Dewar. This necessitated multiplexing between the jet heater and a ballast heater of approximately the same resistance wound around the base of the supporting structure. The efficiency of this system is shown in the graph of bath tempe rature vs. an experimental run (Figure III. 7). time for This graph represents a worst case of low temperature and high heat flux. It is worthwhile to note that at 1.6°K and comparable heat flux it would take a bath temperature change of approximately thirty millidegrees to produce a change in bulk attenuation at 100 kHz equal to 10% of the Go rte rMellink attenuation. Therefore, the limitation imposed by bath temperature regulation is O. 1% of the Gortcr-Mcllink attenuation. The current for the two heaters wa s supplied by a HewlettPackard Model 6299A Harrison DC power supply, and was monitored by measuring the voltage drop across a 100 ohm resistor In serie s with the heaters. This voltage was fed into one channel of an analog channel selector (see Ref. The DC l, Appendix 1 for details). output from the Barocell formed a second channel. The analog channel selector multiplexed the two voltages into the guarded Q.. r- a '" A, A ('f') co °1Wo CD • ('f') If) A Figure III. 7. 50 E-J!,. 100 Time (Seconds) A 6. "''' "," 150 J!i "'"AAN'"'"" Bath temperature during an experiment for a heat flux of 2. 88 Watts / cm 2 . d.~6. A ~4:. ~ ",~Mo;""":"6.A ~ ",," A ":!. A A A '""",A "'" A .,..,. - "'~ '"~ ",""" '" ,,= 0 0:: 0:: - CD ('f') c• 0 CD ~I g v ,...... I !- l~ I! ~J 2JJ I - ·1- I 'Il I II I I I I j ,- -:';'J ! f("\ IN () . \.D -- j ,~ ~~ !~ w 32 input of a Hewlett-Packard 3450A multifunction meter with 6 digit resolution. This digitized the two voltages and printed thenl at one second intervals on a Hewlett-Packard 5050 B printer, providing a record of tIl(' b,d:h telnperature and heater current during the run. Altf' rnatively, the digital output {rorn the voltmeter could be for- matted into four 8-bit words and written onto magnetic tape using a Kennedy incremental tape recorder. This was used also to record the second sound detector voltage-ternperature characteristics. A block diagram of the pressure sensor and heater' electronics is shDwn in Figu rc III. H. III. D. Electronics A block diagram of the electronics for the generation and detection of second sound is shown in Figure III. 9. It is easiest to discuss the ernitter and detector electronics separately. two systems are, in fact, The integrally couplt>d through the wave analyzer's frequency output. IJL D. \ l;:mitte I' Electronics The hear! of the emitter system is a General Radio 1310 B sine wave generator with a frequency range of 2 Hz to 2 MH:/, and a short term drift of 0.3 p. p. m. over 10 minutes. This oscillator is capable of being synchronized to an external reference signal which was derived from the wave analyzer's beat frequency oscillator (B. F. O. ) output. The B. F. O. signal was converted to a squa re wave a tld its frequency digitally divided in half in a device Figure III. 8. Ballast Heater ACS Channel Heater • DVM .... I Printer Block diagram of the heater and bath ternperature regulation electronics. ACS is the analog channel selector which selectively s,\"itches input to the digital voltmeter. Power Supply Barocell J ..tC '.;J ' I ~ Detector 10K F L.O.Output Wave ~--J X-y ~ I Analyzer I IPlotter w Second sound electronic instrun1entation. DDT is a digital frequency di"vider. SSCA stands for small signal cur rent amplifier. counter ~ w ~ Figure III. 9. DDT ' - - - - - - ! ! Frequency 2 W Osci i!ator I ---1111 . ..--. 1-1 Emitter Low Pass ...----11 Fi It er 4- '-' ~ 3S labeled II Frequency DDT". This synchronized the oscillator to exactly half the center frequency of the wave analyz~r's passband and also enabled the scanning local oscillator of the wave analyzer to sweep the second sound frequency. This method of frequency division is digitally precise as compared to the usual frequency doubling by a nonlinear element which has a typical accuracy of about one percent. The other parts of the emitter circuit are a counter and filter. The counter was a General H.aelio Model 1192 B which gave the second sound frequency accurate to O. 001 kHz, was an Ithaco Model 4213, and the fille r operated as a low pass filter whose corner frequency was set just above the maxin1um oscillator frequency. circuit, This reduced second harmonic generation 111 the emiUe r and its 50 ohm output impedance matched to the cables and emitter impedance. Pa rticula r ca re had t.o be t.aken to a void any second harmonic generation because elect.ron,agnetic radiation of the second harmonic of the oscillator fl·(~guency would appear to the det.ector the same as second sound. lII. D. 2 Detector Electronic s The detector electronics consisted of a small signal current amplifier (SSCA) and wave analyzer. The amplifier had a band- width ofiO MHz with a noise figure of 3. 3 nanovolts rJ Hz, served as a preamplifier. and It was originally built as a current amplifie I." for a photomultiplier tube but since its 2200 ohm input impedance was large compared to the detector resistance it served 36 equally well as a voltage anlplifier. With detector efficiencies as a high as 0.6 volts! K this current amplifier gives a si.gnal to noise ratio of unity for 50 nanodegree second sound amplitudes. Minor changes in the electronics would improve this by a factor DC [0 but amplifier noise did not represent a limitation in this rnent. nH~aSurt.'- The SSCA and the detector formed the low level part of the signal detection systern and so were provided with a separate ground. The twu wave analyzers used were a Hewlett-Pack;tnl I\1od"l 3590A with V:,Q4A sweeping 10ca1 oscillator for frequencies below 600 kHz and a Hewlett-Packard Model for frequencies up to 1.2 MIT::--" )10A with !')71\ sw('(~p drive These provided narrow band filters and also the reference signal used to sweep the second sound cavity through its resonance. Both analyzers provided a DC rarnp pro- portional to the center frequency of the filter and a DC voltage proportional to the RMS value of the signal conlponent within the passband. The two outputs from the wave analyze r were fed into an X- Y Plotter (Hewlett Packard 7004 B). Thus the wave analyze r generated a pl()L of second sound arnplitude versus frequency; the second sound spectrum of the resonant cavity. REFERENCES 1. Paul E. Dimotakis, Technology (1972). Ph. D. Thesis, California Institute of 1. C .• 37 IV. METHOD OF MEASUREMENT AND PRELIMINAR Y EXPERIMENTS IV.A. Second Sound Resonance Technique A traveling x second sound wave in the direction can be represented as e -a.x e iw(t - x/u ) 2 The second sound resonator is ideally formed by an infinite plane emitter with the detector being an infinite perfect reflector having a The geometrical optics small sensillg elelnent in its center. approxinlation is valid here provided the second sound wave length Notarys (Ref. is much smaller than the emitter dimensions. 1) found that his detectors exhibited negligible reflective loss even at very high frequencies. emitte r is If the power input per unit a rea to the Q / A cos eDt, then the amplitude at the detector will be Q 1 1 A P cU 2 ( cos {JJt - ~} 1 2 2 WD) 2 \ cosh a.D - cos - - , u z WD) Q -- tan -1 (, cotha.D tan \ u In this formula c 2 is the heat capacity and D is the spacing between the emitter and detector. A resonance condition is satisfied whenever = nl1 38 At a fixed temperature this can be accornplished by varying J) however it can also be accomplished by holding the frequency fixed and allowing the bath temperature to drift, u ' Z thus changing The speed of second sound can therefore be Ineasured at a fixed temperature by measuring the frequencies of two succes sive resonances, i. e. , (2) An expression for the attenuation can be derived from Eq. 2 cosh aD by the expansion of slon of 61JJ , cos Z IuD /u for small argUIT1ent and the expan- near a resonance. Z 1 1 ;'fz of the res onance curve at By measuring the width, of the maximum height one finds for a frequency resonance a ::: b.w 2u (T constant) (3) Z For a temperature resonance the expansions give w b. T a =2--2 u (IJJ then a t:.a, when the jet is in the above formulas is replaced by There is an easier way to get mea su red. (4 ) 2 If there is an additional attenuation, turned on, constant) One can find b.a b.a provided a a t t:.a has already been by a compa ris on of the height of the resonance peaks for the jet on and jet off, i. e, , 1 2 ( b. T ss)on ] cosh aD - 1 b.T 2 cosh (a + 6.a) D SS) off ( ( 1, 39 Expansion of the cosh terms gives t::.a. IV. B. (5 ) Initial Checkout Experiments Several experiments were performed in order to test the limitations and reliability of the second sound apparatus and to allay some of the fears of the experimenter about competing effects. These experiments will be described in this section. IV. B. 1 Second Sound Experiments Three experiments were performed to test the second sound apparatut->. The first was a simple experiment designed to demon- strate that second sound was, in fact, being measured and not electromagnetic or inductive heating of the detector which was an anticipated problem. Provision was made for the insertion of a small shutter between the emitter and detector. Because the wavelength of light at 1 MHz is approxima tely 300 meters the shutter blocks second sound but does not affect the electromagnetic radiation. The results at a frequency of 1 MHz are shown in Figure IV. 1. This clearly shows the resonance pattern and also the limitations imposed by electromagnetic pickup. were also measured at 100, Similar spectra 300 and 500 kHz. Another quicke r method of determining the relative amplitude of second sound and radiation is to measure the amplitude of the received signal ve rsus the input signal. Second sound increases as J 4 I 1.522 n/ 1.740 torr 1.833 1.909 OK ~~M'MI~~~.~~ i tJ'{fA"vi\~·~I"""'~1- --itO- -/ t5 20- 25 mV -30-- - A temperature resonance curve at 1 MHz. The signal is a convolution with the slope of the detector cha racte ristic also shown. Right vertical scale is for the detector; left scale is for the signal. - ~~--- - _m_ ---- -----~- Figure IV. 1. -~ --;i5~ - _~ut mV _~UI --;35- \ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -__ i 45-- ,j:., o 4] the input voltage squared while radiation increases linearly with voltage input. After the installation of the emitter circuit filter and crossed current geometry, which significantly reduced radiative effects, the noise due to radiation was ins ignificant compared to bath temperature drift which was the m.ajor lim itation. The second check on the second sound apparatus concerned diffraction of the second sound beam. spread due to diffraction, li the s~cond sound beam this would look like an additional attenua- tion although it should not change the attenuation due to the jet. The volunle attenuation of second sound was measured versus temperature at several frequencies. plotted in Figure IV. 2. (Ref. The results at 250 kHz are For comparison Hanson and Pellarn's 2) data are als 0 plotted. diffraction is a negligible loss. One can conclude that at 250 kHz However, it was found that at 100 kHz there is a spreading of the beam due to diffraction. A simple optics argument establishes that at 100 kHz the spreading angle is 2.3 degrees while at 250 kHz it is only 0.8 degrees. An additional check is the measurement of the speed of second sound shown in Figure IV. 3. The error bars represent the unavoidable uncertainty in the emitter-detector spacing. It can be seen that a uniform translation of all the points by a small distance would bring this data into essentially perfect agreement with the results of earlier experimente 1'S (data taken from A hle rs, IV. B. 2 Ref. 3). Temperature Gradient Measurements The counterflow jet used in the investigation of Dimotakis 42 3.5 3.0 _2.5 E o ""'- C\I o ~ 2.0 rt) '0 1.0 0.5 o 1.40 1.50 Figure IV. 2. 1.60 1.90 2.00 2.10 Bulk attenuation of second sound vs. T. Frequency was 250 kHz. Solid line is an interpolation of Hanson and Pellam's data (Ref. 2). 43 N N - - - - -. ---.,-----.------.-r-----------. O~~~~~--~-----4L - N o Q) en ....... ~ N ::> CO .-1 CD ::::I' -~----------------~-------------~-------------~~ 1.500 1.700 Figure IV. 3. T (OK) 1.900 2.100 Second sound speed vs. T. Solid line is an interpolation of G reywall and Ahlers data (Ref. 3). 44 and Broadwell (Ref. 4) had an axisymmetri.c shape while the jet used in the present experirnent was s0111ewhat rnore two-dirnensional. Therefore, it was necessary to establish the behavior of the tent·· perature gradient along the center line of the two-dimensional jet. To do this a slTIall carbon thern10meter (a cube approximately O. 01 inches on each side) was traversed along !h' method of measurement was essentially thl' jet center line. sanh~ The as that used by Dimotakis and Broadwell. A typical traverse 1S shown in Figu re IV. 4. The distinctive feature once again is that the tenlperature gradient takes place almost entirely within the straight section of the jet close to the orifice and disappears within one jet width of the orifice. For a further comparison the measu red tenlperature gradient In the straight section was fitted to the equation VT c b(T) q 3 which describes the gradient due to mutual friction in a channel. The measured values of with S C'.J(' ral b(T) are plotted in Figure IV. 5 along points from the measurements of Broadwell and Liepmann (Ref. 5) in the convergent-divergent nozzle geometry and Model I data of Dimotakis and Broadwell (Ref. 4). The solid line is computed [rOln the theoretical equation A(T)p b(T) - A(T) IS n the coefficient of mutual friction in Equation II. 11 which is given by the equation log10 A(T) = 1.10 + 3.1210g 10 T +- [0.0076/(1 - T/TA)l 45 1.660 r-------rT----,....-I----r\-------- ++++++ + + + 1.650r- - + - + ~ o + - 1.6401- + + ++ 1.63 90 .4 r I -0.2 0.0 +++,++++++ 0.2 0.4 Position (em) Figure IV.4. Temperature gradient measured in the jet. The exit plane is at O. 0 and positive positions 2 Heat flux = 3. 46 Watts I em a re in the free jet. 46 o a o · ~ r------------------,------------------~-------------------- o a o -- x + • A X I- . .0 o o o ..J '8 o N • o o o ....• 1.1I00 1.600 Figure IV. 5. 1.800 Gradient coefficient from. the measurements in the straight nozzle section. XiS are measurements of Ref. 4 and Ref. 5. 2.000 47 based on the Model II data of Dimotakis and Broadwell. Their Model II had a long straight section for more a ccurate gradient measurernents. The data shown in Figure IV. 5. the refore, dernon- strate s the consistency of the gradient mea surements on the twodimensional jet with those of the axisyrnmetric jet, and also the accuracy of the Gorter-Mellink law in a channel. REFERENCES 1. H. A. Notarys, Ph. D. Thesis, California Institute of Technology (1964). 2. W. B. Hanson and J. R. 3. D.S. Greywall and G. 4. P. F. Dirnotakis and J. E. Pellam, Ahlers, Phys. Phys. Rev. H.ev. 2-~, 2, 321 (1954). 2145 (1973). Broadwell, Phys. Fluids ~, Liepmann, Phys. 1787 (1973). 5. J. E. Broadwell and H. W. 1533 (1969). Fluids 12, 48 V. ATTENUA TION MEASUREMENTS v. A. Data- Taking Procedure This section is an outline of the basic experir:nental run procedure for measuring the added attenuahon of second sound due to the counterflow jet. The emitter detector spacing and distance from the orifice were fixed while the apparatus WclS at rOO1l1 tem- pe rature before a set of expe rimenta 1 run s. At liquid helium temperatures the run was started by setting the bath temperature using the condom regulator. The detector was then biased in the middle of its transition at that temperature by means of a ten-turn potentiometer on the magnet current supply. A resonance peak was found by scanning the wave analyz ~r close to the desired operating frequency. Then the arnpl itudt" of the Gene raJ Radio oscillator was adjusted so that the noise level was less than one percent of the second sound amplitude on reSI!l1ance. This could always be accomplished with a puwv r input less than 0.01 Watts Icm 2 into the emitter and it was experimentally verified that this caused no finite amplitude effects or power broadening of the resonance. Finally the heat flux of the jet was set by adjusting the current into the ballas t heate r. The bath was allowed to reach its new equilibrium pressure, usucl1ly within a few millitorr of the initial pressure and not enough to cause a change in the detector bias pOint. A frequency sweep was then made with the jet off. At the end of this sweep the sweeping local oscillator of the wave 49 analyzer was returned to its starting point and the jet turned on. Although it took less than two seconds for the jet to turn un, thirty seconds usually elapsed between the off and on f-3weeps allowing the bath to come fully to equilibrium once again. A seri.es of such runs was made at each ten1perature with heat fluxes increasing from 0.3 Watts/cm 2 until it was no longer possible to maintain constant bath tem.perature. 2.5 Watts/cm 2 This occurred at heat fluxes between at lower tenlperatures and -+ Watts/em 2 at the highest ternperatures 111easured. V. LL Di scus sion of the Expe rinlental Res onance Curve A typical experimental run at Figure V.I. T 2. 05() OK is shown in The higher trace was with the jet off and the lower trace with the jet on. For this run the resonant frequency was approximately 100.74 kHz and the heat flux at the orifice was 2 1. 23 Watts/em. The additional attenuation is quite noticeable. There is a slight shift in frequency of about ] 0 Hz; between the peaks with the jet on and the jet off. This was almost always present but not consistent in either size or direction, shift was never more than 30 Hz. jet on was shifted to a high(~r although the Sometilllc'S the peak with the frequency. Severa] e[[cels could change the index of refraction for second sound, but none of thes e would give a change in resonant frequency of more than one Hertz. It was also noticed that if a measurement was repeated twice with the jet off both times there was still a small frequency shift. One can only conclude that the major portion of this frequency shift is -_" lJ I Figure V. 1. t mV! 1M lJl- I I : # 100.6 -1-____ -; 100.8 .\\: ') 10[0 frequency(K Hz) i -- I -l .--~-+--~-- -I ',: 1< ;;~nd heel t flux of l. 23 Watts/cm~. Frequency -esonance at T Uppe r trace is with the jet ::,ff c~nd ;'::-.ver trace is with the jet on. 100.4 i - -I I ,-----1 - j_. -- - -- --t \.Tl o 51 due to the short term drift of the wave analyzer sweep oscillator. Note that one is asking for un reasonable stability from an analog instrument. With a better frequency source such as a digital syn- thesizer this effect could be exarnined further, however, the cost of such instrunlentation precludes furthe r inve stiga tion at this tinlE'. Some justification is needed for measurements at frequencies as low as 100 kHz since non-negligible diffracti:m was obse rved at this frequency. Diffraction should not affect the relative attenuation measurements since the same los s of beanl would be present both with the jet on and with the jet off. check was made just to be sure. Howcvpr, The resonance curves we re measured with the jet on and jet off and 250 kHL:. an experimental ~lt two frequencies, 100 kIIz Frorn these measurer:nents the relative attenuation was computed in two ways. First, the total attenuation was com- puted for each peak using the width at .707 of the maximum in Eq. IV.3. jet on. Subtraction then gave the addi ti ona1 attenuation with the In addition, the change in attenuat lOn was cornputed from the ratio of the resonant amplitudes and bulk attenuation using Eq. IV.5. The bulk attenuation at 250 kHy, is the same as the attenuation measured with the jet off (see Figure IV. 2). These four values of the additional attenuation we re the same to within experilnental accuracy, 100 kHz measurements. and this provides the justification for the It also shows that the additional attenua- hon due to the jet is independent of frequency. When the second sound beam was close to the orifice, multiple peaks were observed where only one had been previously. 52 These coale sced into one peak at higher frequencies so the effect is probably due to reflections of the diffracted wave from the lucite structu re. Therefore measurements at a distance downstreanl of 1/2 jet width were made only at frequencies above 250 kHz where diffraction was negligible. V. C. Data Reduction It was found that for a given temperature and position down- stream of the orifice, the additional attenuation heat flux (q) (Lla) versus the fell on a straight line when plotted on logarithmic graph paper; i. e., log (fla) = log I + n log q An exampl e is shown in Figure V. 2. Therefore, one can consider all of the data in the form Lla where I = Iq n represents the additional attenuation due to the jH at a 2 heat flux of 1 Watt/cm . The measured values of Figures V.3 to V.b. I for four positions are shown In In addition, a line which is computed from the best polynomial fit to the data is als 0 plotted. the measured value of the exponent, Table V. 1 gives n . For the 20 diameter measurements one straight line would not fit all the data on the log-log pInt; huwever, the data at high heat fluxes and low heat fluxes cDuld be fit separately by two straight lines of different slope andinterc('pt. This pos sibly 53 0.05 0.03 I E o tS <l 0.01 0.003 0.001 0 .1 0.3 0.5 Heat Flux Figure V. 2. 1.0 3.0 (Watts / em 2) Additional attenuation vs. heat flux 0 at T = 1. 623 K and 9.6 diameters downstream. 5.0 54 represents some type of transition between two flow regimes, possibility that will be explored further in Section VI. Figure v.6 and the values in Table V. 1 are for the line which fit the higher heat flux data. a 55 g - .r-----~----------.------------------.-----------------, In (:) o • _0 eo rtl' '0 t:t <l o o • l/) o • O~ ______________L -______________L-____________--J 1.500 Figure V. 3. 1.700 1.900 2.100 .A dditional attenuation vs. T for a heat flux of 1 Watt/em at 3.2 diameters downstream. 56 g o • CX) ....-----..,.---..--..-----T--- 80 • CD - E CJ etta I 0 0 0 • ::r tS <J 9o • o o• 1.600 1.700 Figure V.4. 1.900 2.000 Additional attenuation vs. T for a heat flux of 1 Watt/crn 2 at 1/2 diameter downstream. 2.100 57 o o o • ~~r-------~--------~---------r---------T----------~ 8o • CD E 0 ...... •0 ", 0 0 0 • ::I' ~ <l 80 • C\I o • o 1.600 1.700 1.800 1.900 2.000 T(OK) Figu re V. 5. A dditional attenuation vs. T for a heat flux of 1 Watt/em at 9.6 diameters downstream. 2.100 58 oo er----------r----------r----------r----------r----------, lJ) 8 e - t:J 0 e E co 0 (f)....... I 0 (j <l 0 0 e N oo e o o• ~ ________ 1.600 ~ __________ 1.700 Figure V.6. ~ ________ ~ __________ 1.800 1.900 T(OK) ~ _________ 2.000 Additional attenuation vs. T for a heat flux of 1 Watt / cm 2 at 20 diameters downstream. 2.100 2.017 2.056 2.12 2. 05 1.983 2. 061 ':< Too few points to be considered a reliable measurement. 1.946 2. 35 ':< 1. 66 1. 62 1. 57 2.45 2. 066 1. 914 1. 858 2. 21 1. 911 2.41 2.006 ~:< 1. 813 1. 76 1.854 1. 98 1. 858 2.20 1.946 1. 893 1. 30 1. 753 1. 67 1.798 1. 90 1. 811 2.01 1. 911 2.21 1. 30 1. 714 1. 61 1. 712 1. 64 1. 713 2. 59 1.854 1. 32 1. S3 2.068 1. 60 ':' 1.89 ':' 1. 49 1. 17 1. 47 2. 006 1. 951 1. 910 1. 651 1. 25 1. 56 1. 627 1. 653 1. 652 1. 69 n 1. 676 T 2.23 = 20 1. 755 x/d 1. 47 T 1.625 n 1. 43 = 9. 6 1. 593 x/d 2. 39 = 3. 2 T n n x/d OBSERVED POWER DEPENDENCE T x/d ::: O. 5 Table V. 1. U1 -D 60 VI. RESULTS AND DISCUSSION The experimental results are sununarized by the plot of fja/a gm vs. T ila shown in Figure VI. 1. 2 tion for a heat flux of 1 Watt/cm , i. e., values shown in Figures V. 3 to V. 6. is the change in attenuathe q =: I The quantity inte rcept a. is the grn attenuation due to mutual friction that would be present for the same heat flux in a channel. Recall that this heat flux must be If the supe rfluid is not carried by the normal fluid in the jet. entrained by the normal fluid, and If the Gorter-Mellink force is a true volUlue force which acts in the absence of walls, a. then gm should also be the attenuation of second sound in the free jet. There Cl rc several observations that can be made on the basis of Figure VI. 1. The observed attenuation is always less than 8 percent of the Gorter-Mellink attenuation, and a/a gm is clearly temperature dependent with the fraction depending on the distance In addition, from the orifice. the observed power law dependence on the heat flux than Mellink attenuation is proportional to uation goes as gn with g 2 t.a follows a different a. gm The Gorter- while the observed atten- n = 1. 3 to 1. 6. The geometry dependence can be accounted for, extent, by spreading of the jet. to some The spreading would have to be considerably more than has been observed by Kapitza (Ref. Dinwtal,is (l\d. not large, l). lkcaus(' the length to width the ordinary spreading !'elations for jet are not entirely applicable, jet. d J) and ratill of t.he j(~t 18 plane turbulent nor a re those for an axisymmetric Although all turbulent jets spread linearly with distance, the 61 0 I -- ~ • - T ...~---- (!) [!] ~ 0 CD 0 (!) I!I • I!l • ~ . (!)o g ...... tJ ts I!I (!) ~ <l A AA- A 0 N 0 I!\:!) A ~ A- • I!I [!]~ (!) (!) ~ I!I A A 0 I!I A- A • 1.500 1.700 Figure VI. 1. T ( OK) 1.900 2.100 Ratio of the measured attenuation at 1 Watt/cm 2 to attenuation predicted from the mutual friction theory. 62 angle of spreading depends on the geometry. No geometric scaling can bring the observed attenuation into agreement with the attenuation in a channel because there is already a substantial difference at x/d = 1/2 where the jet cannot have spread appreciably. Thus the obse rved attenuation cannot be due to mutual friction in the sense that this describes the temperature gradient in a channel. However, this conclusion does not shed any new light on how the mutual friction hypothesis is violated. The question of whether superfluid is entrained by the normal fluid, th(~ or wheth(:r the Gorter- Mellink force acts only in the presence of walls is still unanswe red. The attenuation re sults are qua.litatively con s i stent with both the axial gradient measurements of Dimotakis and Broadwell (Ref. 3) and Careri's ion beam attenuation measurements (Ref. :1:). Comparatively little attenuation of second sound was observed which would be expected from the lack of temperature gradient m the jet. The fact that some attenuation was observed indicates that the flow in the jet is inhomogenous, thus confirming Ca reri' s observation of ion beam attenuation on the downstrearn side of hi s orifice. A further comparison with Careri's data docs not !,ccm pos sible. All of Careri's counterflow measu rernents we re below 1. 4 oK, and because of the small size of the orifice and low heat flux his Reynolds numbers were always below 1000. the Reynolds number based on normal fluid density, R n = P dv n n T] In particular, 63 is only 20-30 at these low temperatures, compared with several thousand for the second sound attenuation measurements. It is unclear how to find another dimensionless group to scale the data, or even if one can be found. It is possible to make a quantitative comparison between the second sound attenuation and temperature gradient measurements. According to the mutual friction theory, the temperature gradient can be related to the attenuation by using the equations \7T 3 agm = c(T) q b(T) is given in Eq. = Ps sTw. One finds that where q = b(T) q II. 13 and \7 T = Y (T) q a For comparison, o K/cm 2 Watts/ern, and 1n Eq. II. 14 with gm the gradient is computed from the observed atten- Aa • instead of uation, c(T) 2 a grn A s a typical example, \7 T = 6. 66 0 for a bath temperature at 1.6 , heat flux of 2.66 t::.a measured at 9.6 diameters downstream. is within the noise level of the ca rbon the rmomete r, gradient would be undetectable. i. e., This such a The calculation illustrates the increased sensitivity of the second sound attenuation compared to the gradient measurements. The second sound attenuation in the jet was observed to be frequency independent. If the attenuation is viewed as a scattering process due to inhomogeneities in the flow field, this would indicate 64 that the second sound wavelength is smaller than the length scale of the inhomogeneities responsible for the scattering. (Ref. 5) and Schmidt (Ref. Both Kovasznay 6) have obse rved frequency independent scattering of ultrasonic waves in turbulent flows when the vvavelength was less than the microscale of the turbulence. Thi s con- dition is just barely fulfilled in the second sound attenuation experiments, the wavelength being com para ble to or s J ightl Y less than the D1icroscale. independence, However, in view of the obse rved frequency a calculation of the attenuation based on geometrical optics is given in Appendix B. This closely follows the calculation for the diffusion of a light (or sound) beam passing through a turbulent bounda ry layer given by Lieprnann (EeL 7). REFERENCES i, 1. P.L. Kapitza, J. Phys. USSR 2. P. E. Dimotakis, Ph. D. Thes is, 181 (1941). California Institute of Technology (I972),page 24. 3. P. E. Dimotakis and J. E. Broadwell, Phys. Fluids l§.. 1787 (1973). 4. ;' G. Careri, M. Ce rdonio and F. Dupre, Phys. Rev. 167, 233 (1968). 5. Chih-Ming Ho and L. S. G. Kovasznay, Acoustic wave propaga- tion through a two-dimensional turbulent jet, unpublished report (1972). 6. Dieter W. Schmidt, Recent experimental investigations on the scattering of sound by turbulence, AGARD Report 4-61 (1963). 65 7. H. W. Liepmann, Deflection and diffusion of a light ray passing through a turbulent boundary layer, (1952). Douglas Report SM-l4397 66 VII. FUTURE EXPERIMENTS High frequency second sound has enormous potential as a It seems almost tool for investigating the flow of liquid helium. as if the range of applications is limited only by the imagination of the experimenters. There are three experiments which would be of particular value for understanding the counterflow jet and these will be discussed briefly in this section. The first set of proposed experiments deals with some refinements of the fluid mechanics of the jet itself. £]uid, tu rbulet1ce quantities and meal1 veloci.ties do In an ordina ry not exhibit self-presCl'vatioll until nlany channel dian)cters dm'.nstrearn. Unl il the flow is self-preserving the velocity profile ()J the jet he) s a memory of the initial velocity distribution which, in turn, IS erned by the shape of the pipe or channel producing the jet. the jet become s self -prese rving, gov-· After the veloc ity dist ribut ions are In- dependent of the distance from the orifice when nondimensionalized with a suitable local length and velocity scale. It is not practical within the confines of a cryostat to investigate self-preservation. However, conside rable improvement in the design of the ori fice could be made. There have been good wind tunnel rneasurements on the jet s produced from long pipes and true orifices. While nothing can sub- stitute for a velocity profile measurement in the liquid helium itself, the ability to draw on classical fluid results would be at least helpful in interpreting attenuation measurements. Some of the uncer- tainties and assumptions about the flow field could be eliminated. 67 An unambiguously two-dimensional orifice would also be better for interpretation than the small aspect ratio orifice used in this investigation. The shape used here was an attempt to com- promise on several conflicting requiren'1ents. The temperature gradient measurements require a wide opening; second sound bearn alignment is easier with an elongated opening; and a small area is needed for high heat flux with low total power input. A very narrow slit would make the shape of the odfice unambiguous, would also increase However, x/d for a given distance, since the attenuation goes as considerable loss in sensitivity if d jet. x , downstream. exp[ -UdJ there would be a is made too small. It may be possible, using second sound, tion of whether the relative velocity and W to settle the ques- is zero in the counterflow One can show from the linearized equations that second sound propagating in a cavity at an angle to a flow will have a shifted resonant frequency. = W = 0 is V W, and the resonant frequency of the cavity with W o then the resonant frequency change will be lIw = ± W U f(T) f(T) \! o where . Th e f unchon if the mass flow velocity is ..... relative velocity v Specifically, n. V .t f(T) n.li2W) 2u 2 2 z ps s jJ 8 aT ! f.' n' \7)s goes to y;ero a't a I)(Hlt close to this temperature, l.C).'lov. ) [" By working one should be able to separate the con- tributions of the mass flow and relative velocities to this frequency 68 shift. A lthough V /u Z and W /u Z are small, at 100 kHz the change in frequency should be easily within the resolution of the wave analyzer. For the development of a nlodel of the counterflow jet it is essential to have accurate information about the spreading rate of the jet. The roost promising way to obtain thi.s inforrnation seerns to be schlie ren photography. Gulyaev has oemonstraied the fcasi- biJity of this technique for observing counterflows in liquio heliurn. l His heat fluxes were very high (18-21) Watts/cm ) but his Dptical dewar was also homebuilt and not nearly as well adapted to the measurenlent as the commercially nlade opti cal deVvCl r s nnw a vai1able in this country. It is hoped to have a working schlieren system at Galcit sometime within the next year fur this study. 69 VIII. CONCLUSION A technique for the use of high frequency second sound to study the fluid mechanics of liquid helium has been developed and applied to study the additional attenuation of a second sound beam traversing a supercritical counterflow jet. of the beam was clearly measurable, A lthough the attenuation it was less than lO percent of that predicted by the mutual friction theory and measured in a counterflow channel. The size of the attenuation is quantitatively consistent with measurements of the temperature gradient in the same jet and with earlier temperature gradient measurements. This once again suggests that either the Gorter-Mellink force acts only in the presence of walls, or the relative velocity is nearly zero in the free jet. The additional attenuation of second sound by the jet indicates that the flow is inhomogeneous, thus qualitatively confirming earlier investigators' measurements of the ion beam attenuation due to counte rflow thr~)ugh an orifice. Frequency independence of the second sound attenuation suggests that the inhomogeneities are larger than the second sound wavelength. The attenuation vs. ternperature at various distances downstream frorn the orifice follow similar, wcll-dt~fined curves indicating that future experimental and theoretical study is warranted. 70 APPENDIX A. SECOND SOUND DETECTORS Several methods have been used for the detection of second sound. Recently, Brillouin scattering (Ref. Brillouin scattering (Ref. 1) and stimulated 2) of laser light have been used. Another novel approach was to make a Helmholtz type resonator using metallized millipore filters (Ref. 3). However, the bolometer still remains the simplest method of second sound observation, and the only one (with the exception of laser scattering) that has the capability of detecting high frequency second sound. The purposes of this appendix are to review some bolometer materials, and to explain the fabrication technique used to produce a bolometer which satisfied the experimental requirements of high frequency second sound detection. The fabrication technique should al s 0 be appl i- cable to the hot film velocimeter used for bounda ry layer studies in ordinary fluids. A.. I. Types of Resistive Detectors A.ny material whose electrical resistance is a strong function of temperature in the liquid helium range is a likely candidate for a second sound balometer. two clas ses, These materials fall naturally into the semiconductors and supe rconductors. conductor the intrinsic carrier concentration, In a senl i- hence the conductivity, is proportional to (kT)3/2 where E tance, R g , exp [-E /kT] g is the gap energy (Ref. 4). Consequently, increases with decreasing temperature, the re sis- and at liquid 71 helium temperatures R is so high that doping becomes necessary to make a useful detector. carbon and germanium. The most often used materials are Semiconductor bolometers have the rnajor advantage of excellent sensitivity ove r the entire liquid helium temperature range. The figure of merit is the slope of the voltage drop versus temperature curve, i. e., dV :: I dR dT 0 dT where I o is the bias current. Sensitivities up to 1 Volt/oK can be achieved fairly easily with semiconductors. However, the semi- conductor materials also have several disadvantages n'laking them inapplicable to high frequency second sound work. The high resis- tance makes it difficult to impedance match these detccto rs to transmission lines. Consequently, they are susceptible to the same noise problems which plagued the old style piez.oelectric pressure transducers as well as the 1 If noise characteristic of semiconductors. Electron1agnetic crosstalk is also an impedance related problem. The large thermal mass seve rely limits the frequency response, although with special precautions thin carbon films have been used to detect second sound at frequencies up to 500 kHz (Ref. 5). was anticipated, Since the need to detect second sound at 1 MHz semiconductor detectors were ruled out. The other commonly used class of bolometer materials is the superconductors. from a high value, R tion temperature, T For a superconductor the resistance drops n c ,to zero in a small range near the transiThe principal drawback of the 72 superconductor is that it has a high sensitivity only over a limited temperature range. This can be overcome by taking advanLtge of the thermodynamics of superconductors; the transition temperature is a function of the applied magnetic field. The advantages of superconductors are that they have a low resistance reducing both impedance mismatch and crosstalk, and in the form of thin films they have a low thermal mass which gives superconducting thin films the capability of megahertz. frequency response. fabrication, With careful sensitivities of 0.6 Volts/OK can be reached; however, an orde r of magn itude below this is mo re typical. Superconducting thin films have been rather extensively studied. Specific application to second sound detection is discussed by Notarys (Ref. Rudnick at UCLA has used an anomalous jump 6). in the transition of granular aluminum films to investigate third sound in liquid helium (Ref. 7). The dV /dT at such a jump is essentially infinite; however, we have been unable to reproduce Rudnick's results. A.2. Superconducting Thin Film Fabrication Details The films are made by evaporating a known volume of pure material to completion in a high vacuum. The most convenient way to do this is with a measured length of pure wire. The actual evaporation is done by Joule heating of a molybdenum boat containing the metal sample. The substrate is mounted approximately five inches above the boat. less than 5 x 10 -5 torr. During the evaporation, the pressure was Film thickness calibrations were 73 originally done on an interferom~eter. In order to make fillll with reproducible transition temperatures it was found necessary to keep the heater current and tirnc of evaporation the sallle. The leads, being thicker, entailed a separate evaporation which had to be done quickly after the first evaporation. Otherwise an oxide layer developE'd which prevented the leads from making electrical contact. There are no elemental supcrconclU\·tllrs with transition temperatures in the working range between and the Ie -point. 1.:' ()K This necessitates the use of either a compound superconductor or a supe rposition of superconductor on top of a nonsupc rconducting llletal. The lllost satisfactory cOlllbination for ease l)f fabrication and good sensitivity seelllS to be tin un gold in a A film o o consisting of 1000 A +: 1 ratio. of sn on 2S0A of has a norlllal resistance of 0.5 U/square al Au characteristically 1. ZOK ilnd a /,er() fit,tel 0 transition telllperature of about 2. 1 K. almost the entire telllperature range can be covt'rcd \\ith on .. [drn. A slight change in the ratio of the two metals also changes the transition telllperature. Increasing the ratio moves the transition temperature closer to that of the bulk superconductor. This ratio change can best be accolllplished by va rying the tin thickness because there is a minilllum thickness of gold which lllUst be o present to be effective. With 250 A Q reproducible. However, of gold the results were very 0 with 125A of gold and 5aOA of tin the results were not at all consistent and the slope of the transition was conside ra bly deg raded. 74 As mentioned above, approximately per square. 1/2U wants a resistance at resistance is 1/2 R of 200 is necessary. the normal resistance of the film is N T c ) of For impedance matching one (defined as the temperature where the 50~6 . Therefore a length-to-width ,'atio This, plus the requirement of a local meas- urement, necessitates miniaturization on a scale comparable to thick film hybrid electronic technology. Sonie comprornise s in specifications still had to be made, even with a photolithographic technique similar to that used in the Inanufacture of integrated circuits. This process involves the application of photoresist, a photoscnsitivl' plastic coating, over the entire film. The phot()- resist is then exposed through a con1;td.mask under a mercury vapor ultraviolet light source. Developing removes the exposed portions of the photoresist but leaves a coating of plastic over the unexposed portions which protects that part of the film during the etching process. More detail on the photoresist technique is pro- vided in the next section for future reference. Figure A. 1 is a photug raph of one of the detectors used in this investigation. A.3. Photoresist Processing Th(, photo res ist used was Shipley AZ- 1350 (manufactured by the Shipley Company Inc., Newton, Mass. 02162). This is a positive working photoresist which mea!1S that the exposed portions are removed by the developer. Positive photoresist has the advan- tage of greater line resolution using a thick photoresist coating than could be achieved with negative working photoresist. The 75 s ...... ..... ~ 76 AZ 1350 can be safely handled in gold fluores cent light which is A uniform coating was another advantage of positive photoresists. applied by spinning on a rotating platform at 3000 RPM. The coating was air dried since baking can change the film prope rties, although baking also makes the photoresist coating more resistant to chemical action. Exposure of the photoresist is done through a contact mask. This mask is a positive photographic reduction of a pattern either drawn on a white background or laid out on a plastic backing with printed circuit layout tape. Reductions of up to 30: 1 are The emulsion side practical at the Caltech graphic arts facility. of the mask must be towards the thin film, and the mask is held flush against the photoresist by a piece of suprasil fused quartz which is transparent to ultraviolet. The exposure time under the mercury vapor lamp is determined by the proximjty of the lamp and the size of the pattern. In this application, 15 to 30 seconds After exposure, the photoresist is was found to be optimum. developed for 50 seconds In an aqueous develope r (provided by the manufacturer) and rinsed lh distilled water. photoresist exposed to the This washes away any uv and leaves a protective coating over the region corresponding to the dark part of the mask. The thin film is now ready for etching. Of the various etching procedures, most straightforwa rd. can be used. chemical etching is the For lllore accurate work, ion beam etching Plasma etching and anodization are also accurate 77 etching techniques but require special materials considerations and are not applicable to gold-tin films. The superimposed tin on gold forms a layer of compounds such as Au Sn 2 and Au Sn 3 These are resistant to commonly used chemical etchants, however, a double strength tin etching solution worked reasonably well. Some touching up with a glass needle under a microscope is necessary to complete the process. This was not irnpractical because of the comparatively large size of the pattern. of 30 drops Hel + 10 drops HN0 3 The tin etch consists + 20 drops H 2 0 2 diluted in 30 ml. of ethylene glycol to prevent undercutting. The solution must be cooled for at least 30 minutes before use. Large quantities of the etch cannot be stored because the acid breaks down the solvent in about I day. For tin-gold filrns the amounts of acid and pt'roxide should be doubled. Etching time is approximately 30 seconds. but this again depends on the size of the patte rn, and visual inspection is the only sure method to determine that the etching is completed. Etching for slightly too long undercuts the photoresist and destroy!:> the pattern as does not rinsing promptly and thoroughly in distilled water. It is best to work by successive approximations in short steps. The photoresist can be removed after etching with reagent grade acetone. This does not leave a residue on the film, completes the process. and With appropriate care and a different exposure and etching technique, half micron lines can be resolved. 78 REFERENCES 1. R.L. St. Peters, Communications 2. Dusan Petrac, 3. R. A. T.J. GreytakandG.B. Benedek, Optics L, 412 (1970). Ph. D. Thesis, Sherlock and D. O. UCLA (1971). Ed-wards, Rev. Sci. Instr . .il, 1603 (1970) . ..t. Charles Kittel, Introduction to Solid State Physics, (John Wiley & Sons, New York, 5. H. Snyder, Phys. 6. Harris Notarys, 1971), 4th ed., Fluids.§" Ph. D. Chap. ll, p. 270. 755 (1963) Thesis, California Institute of Technology (1964). 7. K. R. Atkins and 1. 37 (1970). Rudnick, Progr. in Low Temp. Phys. 6 79 APPENDIX B This appendix presents the results of a calculation of the Many ideas from deflection of second sound by a turbulent jet. classical turbulence and their justification in ternlS of the present expe riment will be dis cus sed. While this proba bi y is not a final thl'oretical answer to.vhat caused the attenuation, the ag reernent of the calculation with experirnent suggests that the basic physical idee1 n1ay be correll. B. I Some Facts A bout Turbulent Jets In a classical fluid, the transition from lan1inar flow to turbulent flow occurs at a critical value of the Reynolds number R = where J. some length scale in the flow. 1S ratio of inertial to viscous ftH'ces. This repre sents the For a classical fluid in a pipe, the critical Reynolds number is 2300, ;\nd the jet emerging from the end of such a pipe will always be turbulent. A s a fi rst step, it should be dctern1ineQ whether the counter- flow jet is laminar or turbulent. velocity, w It is assumed that the relative is zero in the jet but that there is no net rnass transfer so v is zero within the confines of the pipe leading to the orifice. Thus in the channel all momentum is associated with the relative motion while in the jet all momentum is carried by These momenta must be equal and so conservation of momentum defines the exit velocity. The stagnation point of the superfluid v. 80 occurs at or very near to the orifice as indicated by the temperatu re gradient measurements. Using the equation for the channel =~ P sT s w and the momentum equation, - v- one find s {f' ~~ P s psT Thl' Reyn old s num be r at the exit is the refo re is the normal fluid viscosity and th(' pipe. Thus the Reynolds nurnber is pt· ratu reo For;) heat flux of 1 Watt/cll1'~, i:1 d is the diameter of strong /\ltlclic)f} ul tcm- ) H, g oe s f r urn 'J! () () at o 0 2. 1 K to 1+, (Jon at 1. r:; K ba sed on the width ()f t.he orifice. Th,: jet should be turbulent at all temperatures wHh a heat flux of 2 .5 Watts jcn," , however, it may not be turbulent fur heat fluxes lower than this at the highest observed tenlperatures. factors affect the transition, the numbe r R .~ Sin c e Ilta 11 y 2')00 should not be taken too lite rally. The attenuation at a given ternpcrature, tion was obse rved to be frequency independent. heat flux, and POS1- In an ordinary fluid, frequency indqwndent attenuation of sound has been ubse I'ved whet] th(· wavelength is shorter' than t.he rnicroscale of the turbulence. The extent to which this is satisfied will be discussed in a later section, but the observed frequency independence suggests a geometrical optics calculation. 81 For a plane turbulent jet, the mean velocity profile becomes self-preserving at about 5 diameters downstream. Howeve r, the turbulence quantities do not appear to be self-preserving until x/d = 40 - 50 (Ref. Until the turbulence is self-prese rving, I). flow has a memory of the nozzle geometry. calculation is concerned with x/d < 20 the Even though this it will be assumed that the small s cal (' of turbulence has reached its asymptotic s ta te of local isotropy. Mean Square Deflection Angle B.2. The most important part of the calculation is based on an argument due to Liepmann (Ref. 2) for the mean square deflection of a light beam passing through a turbulent boundary layer, under the condition that the wavelength is much smaller than the corLet relation length. x be the streamwise direction, y the cross stream direction and let the thickness of the boundary layer be with () The index of refracti on is in the form n'« 1 n:= n (y) (I o In'), representing the random fluctuating part of the index of refraction due to the turbulence, y;T = 0 (ove rba r sind i- cate tilTle averaging). Light rays are assumed to enter the boundary layer in the norrnal direction. Therefore, slons of the beam is ignored. spreading of the jet over the dimenIn the expe riment, the part of the beam of interest is only I mm so this should be a good approxiInation. 82 Liepmann! s result for the mean square angle of deflection is b ((n (y) n 22 :;Z(b) n (0) o JJ 0 .z R ( I y - Eo, I) dy dS onl -) \ox (S) 0 0 provided that the fluctuations are homogeneous over regions larger R than the correlation length. onl 'onl' 1- :-) \ 3x \ 0x / Y B.3. - f" - is a correlation function, 1. c. , shown that for second sound in -;n 12 E(IY-.~.)I \0 x / " Index of Refraction of Second Sound Khalatnikov (Ref. 3) has the presence of a flow, v where v y w a r e the components of the mass flow and and y + Yw y y relative velocities in the direction of the second sound propagation. Y is a monotonically decreasing function of temperature. jet, w with v is taken to be zero. y = 0 y results. w :/: 0 'y v - Y The calculation has been tried but a different temperature dependence The index of refraction is n Let For the v I Y v I Y v = where the turbulence cOlnponent, =l---Y. v Y t. e. is the nlcan velocity and v I y tion can now be written In the 1'orn1 0 v IS Y The index of refrac- 83 v n=(J.. )\(1 \ u ZO B.4. Correlation Length It will become necessary to choose a Dimotakis (Ref. complete the calculation. correlation length to 4) has suggested a dimen- sionless group to scale the onset of mutual friction in a channel. critical heat flux q , c For a the relation Aq £ c ~ see nl S t 0 hoI d w her e 2, ilw 1S coefficient in the force of 1l1\11Ud! t' han n e I did 1)1 etc ran d T h (, fricti.()l1. « ( ) t' I' (' is I it t i () n the ! (' n g 1h prup()sed he rc is 1 A.q l\ :: where B,5. Completion of the Calculation can Rdy - ; \ ) b (\y 6 reel sonably tClke falls to ~',er() and s'T is a nun1ber of order unity. c For the j ct, Ollt' c - is outside thc regi()n of high velocity so (6): If the correlation function n o 1 rilpidly for increasing where \y with /\6(\y - :::\) :; \) is the Di rae i) - function. b 2~ (6 ) = Z/\ fn o 2 0 1\ then one can replace is the scale of fluctuations ,ani 2 dy (y) \ox, 84 We now use the mean value theorem and approximate the averages separately, 1. e. , ( no 2) ave = 1 'an' 2 1_ ,ax) ave' " where A is the The microscale microscale. vvill be approxi- mately the correlation length so that € 2 (6) (~ y For the experimentally dete rrnined relations for an ordind ry plane turbulent jet will be used. ~ where U s .078 U is the mean center tine velocity (Ref. formula J I) given by the 1 U U 2 s s = UJ d 2 \x) is the exit speed of the jet (Ref. 5). 2 B.6. Conclusion The attenuation is related to the rnec1 n squa rc deflection by the formula 85 Q whE' re d~ tUtiOll S fo r is the size of the beam. and t) , = f(T,O) :t q 3/2- the width of the jet, dO"\\d1strea.m fcon) the orifice. :F'j)~llre Finally, we get after substi- U. J Q Note that = is a function of the distance The functi on f(T,6) £. is plotted in I). 1 for a position 20 diameters from the jet exit. The has been chosen to be 1/3. C (, n ~ tall tin It is nece ssa ry Lo examine the assumption that thd 11 the \vavelength. J\ is greater With the above choice of the constant, l\. rang('s from O. IB to 0.41 mm while the second sound wavelength goes frorn 0.2 to 0.15 mm for a 100 kHz frequency in the temper0 ature range 1,50 to 2.05 K. Thus the condition on the coherence length is at least approximately satisfied. The result of this calculation is extremely suggestive and the [ol1ov\'inf~ 1I a things may be inferred: '1'11(' slllwrfluid must move with the normal fluid. l'C'lativc vc-lol'ity instead of v Using gives the wrong ternperature dc- pendencC', ,,:) The jet is turbulent. A laxnina r profile cannot give attenuation of the magnitude observed. Us ing the spreading rela- tions for a turbulent jet removes most of the geometry dependence of the attenuation, 86 or:; ..0 - - - 4.0 ~ -.. E u "" 3.0 r<> I 0 ....., ~ ~2.0 1.0 I _. ,,_,______ 1.900 2.100 .-J 0.01.500 Figure B. 1. 1.700 Attenuation calculated Iron, the mea n squa re deflection angle. 87 3) tht~ The coherence length in the jet is length scale. Gorter-Mdlink This sugge sts that the mutual friction acts a wa y from solid boundaries but in a previously unsuspected way. friction, as a source of entropy, assumes a Mutual role sOITlething like viscous dissipation in an ordinary fluid. REFERENCES 1. H. Tenl1ckes and J. L. (MIT Pres s, 2. H. VI. Lurnley, A First Cuursc 1972), Cambridge, Liepmann, HI page 1 H1. Deflection and diffusion of a light ray passing through a boundary layer, Douglas Report SM-14')97 (1')52). 3. 1. M. Khalatnikov, Sov. Phys. -JETPi, 649 (1956). 4. PoE. Dimotakis, Phys. Rev. A10, 5. H. Schlichting, Co., New Yo r k , Turbulence, 1721 (1974). Boundary-Layer Theory, 1 9 () 8 ), pa g c () 9 7 . (McGraw-Ifill Boo},