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Producing and recording converging second-sound shock waves
Producing and recording converging second-sound shock waves G. Stamm, Th. Olszok, M. v. Schwerdtner and D.W. Schmidt Max-Planck-lnstitut Germany
fL~r StrSmungsforschung,
Bunsenstr.
10, W-3400 GSttingen,
Received 31 October 1991
Two different designs of convergent counterflow channels are presented. The imploding second-sound wave was produced by a vacuum-deposited chromium film on the inner curved surface of these channels, and the temperature was measured by a bolometer whose distance from the heated surface could be externally adjusted. If the average heat input is small enough to avoid the production of superfluid vorticity, which is the main dissipative process, acoustic cylindrical waves can be observed. The temporal evolution of the perturbation temperature follows the geometrical predictions.
Keywords: helium; converging shock waves; temperature measurement
The propagation of second-sound pulses in He II under different flow geometries has been the subject of numerous investigations I-3, but only a few experiments have been performed on converging second-sound shock waves. Torczynski4 measured the temperature jump as a function of the Mach number and pointed out the great difficulty in producing film heaters with an appropriate geometry and homogeneity and also in recording the converging temperature waves. We now present two different possibilities to produce stable converging temperature shock waves, to record the corresponding signals and to compare them with geometrical predictions and results from numerical calculations and analytical models.
moveable temperature probe
plate II
0011 - 2 2 7 5 1 9 2 / 0 6 0 5 9 8 -.Cr3 © 1992 Butterworth -',Heinernann Ltd
598
Cryogenics 1992 Vol 32, No 6
plate IV
l
plate III
/
Flow channel 1 Flow channel 1 (FC1) (Figure 1) was built on a Lucite bottom plate in which was milled a 60 ° segment of a circular cylinder with a radius of 45 mm. Two plane Lucite plates (I and II) with the shape of a 60 ° segment form the walls that terminate the channel in the axial direction while another pair of Lucite plates (III and!I~)~form the walls terminating it in the tangential direction. The heater consists of an (originally plane) Kapton foil of 70 ~m thickness, serving as a substrate, on which a chromium film of about 1000 A thickness was vacuum deposited at a pressure of 2 x 10-5 mbar. In a second .deposition stage the supplying leads, a layer of 100 ,~ chromium, 5000 A copper and 500 A gold, were evaporated on to the heating film. After gluing the four side walls (I-IV) of the channel together, they were mounted on the bottom plate.as indicated in Figure 1,
~"
'
)
sensor
,,
\
P
converging shock wave
!IIii '
/
/
plate I
/ ' kapton (foil (70/~m) with film heater
bottom plate Figure ;1 Schematic drawing of the experimental setup Of FCI (hOt,to,scale). Extension in the axial direction is 42 mm
Converging second-sound shock waves: G. Stamm et al.
bending the Kapton foil according to the curvature of the bottom plate; to provide a tight connection the downward borders of the walls are knife-edged. By this procedure we obtain a large-scale cylindrically formed and highly homogeneous heating film which perfectly keeps its form also by cooling down to liquid helium temperatures. Through a slit of 3 mm width we can introduce our temperature probe. The distance between the lower end of the probe carrying the sensor and the heated surface can be adjusted from outside the cryostat in the range from 1 mm up to 45 mm. The temperature waves were produced by passing a rectangular electric current through the heating film. Owing to its fast thermal response we achieve rise and fall times better than 100 ns. With the dynamically calibrated sensors which work as superconducting bolometers as described by Stamm 5 we can obtain the temporal evolution of the perturbation temperature T ' = T - T o (To: temperature of the helium bath) at the location of the sensor. As the front surface of the probe is small (only 0.1 mm wide), its influence on the stability of the shock waves can be neglected. Following Fiszdon et al. 6 the governing equation for the evolution of the temperature is in first order approximation aT at
1 a(rw) r Or
(1)
Assuming the counterflow velocity w at the heated surface to be constant we recognize that according to the above equation the temperature must rise after the shock front, leaving a warm tail at the end of the pulse. In fact, this is what we measure for weak heat fluxes when the influence of the vortex-line density (VLD) can be neglected (see Figure 2). If we increase the input heat flux and reduce the time tR between two consecutive
30
T
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.
.
.
I
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.
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.
.
I
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I
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T
20
10
0 t [ms]-I0
0
1
2
3
4
Figure 3
Temperature signals recorded in FCI at distances d = 2, 10, 20 and 30 mm from the heated surface. O = 3 W cm -2, tH = I ms, tR = 0 . 2 5 S and T o = 1.4 K. Close to the heater the signals show a large temperature overshoot
pulses, the pulse shapes become dominated by the influence of the VLD, and the effects due to geometry are then completely governed by the so-called 'overshoot' as described by Fiszdon et al. 6 (see Figure 3). Flow channel 2 Flow channel 2 (FC2), shown in Figure 4, consists of a ring 2 mm thick manufactured from a quartz glass tube of 36 mm inner diameter. Fixed on a specially designed device this ring was placed in the vacuum deposition
conducting strl )s
14
T
T' [mK]
.
T' [rnK]
glass plates
/
2mm \
12 I0
8 2
6
2O
I0
4
\
2 0
film i n t e r r u p t i o n f o r s u p p l y i n g leads
t[ms]-2
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0
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I
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•
•
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2
Figure 2
Temperature signals recorded in FCI at distances d = 2, 10 and 20 mm from the heated surface. The heat flux of the applied rectangular heat pulse was Q = 2 W cm -2, the pulse duration tH = 0 . 8 ms, the repetition time tR = 0 . 4 S and the bath temperature To = 1.4 K. The signals s h o w a nearly linear increase after the shock f r o n t and a warm tail at their end
i nner d i a m e t e r 36 mm inner surface covered with thin-film heater
Figure 4
Schematic exploded view of the experimental setup of FC2 (not to scale). The glass ring can be moved up and down
Cryogenics 1992 Vol 32, No 6
599
Converging second-sound shock waves: G. Stamm et al.
~
X
neither completely rigid nor completely soft, and detected at the time t2. This singularity is then reflected at the heater and reaches the probe at the time t3. After a further reflection at the centre the pulse is re-formed to its original rectangular shape but reverse in sign, and recorded at the time t4 Elongating the wave path by another identical sequence of reflections, the pulse will again look like the incoming one recorded at t]. These results could be reproduced theoretically by an analytical model using the Fourier integral method, and by numerical calculations with a VLD close to zero, as described in detail by M0hring and Fiszdon 7.
t
/
4 1 T'[mK]
/ t [m~]- 6
.
0
.
.
.
i
1
.
.
.
.
.
2
.
.
i
,
,
,
3
Figure 5 Temperature curve, recorded by the sensor after releaseof a short rectangular heat pulse, due to direct reception of the propagating pulse at time tl and reflection at the centre (t2), the wall (t3) and at the centre again (t4). O = 1 W c m - 2 , tH = 0.2 ms and To = 1.85 K
apparatus and, while rotating at a frequency of 3 rps, the inner surface was covered with a chromium film of about 300 A . The ends of this tube are closed by two quartz glass plates, and on to one of them a secondsound receiver with supplying leads is vapour deposited. As we have to supply the electric current to the heating film on the inner surface, we get a small irregularity at the junction of the heating film to the supplying leads. However, this interruption of the film does not have a recognizable influence on the form and stability of the converging waves, as has been verified by numerous experiments. While the two glass plates are fixed, the position of the ring heater relative to the position of the sensor can be varied from outside the cryostat. If the average heat input is small enough to avoid significant production of superfluid vorticity we observe acoustic cylindrical converging waves as shown in Figure 5 together with the wave path. After release of a rectangular heat pulse at the heater, the sensor first detects, at the time tl corresponding to the distance from the heater, a rectangular pulse travelling with a velocity of about 20 m s-] towards the centre of the tube. As the pulse duration is only 200 #s, the geometrical effects shown in Figure 2 can be neglected. At the centre this incoming jump is reflected as a logarithmic singularity, which means that the centre is
600
Cryogenics 1992 Vol 32, No 6
Conclusions We have shown two methods of producing and recording converging temperature shock waves. Using only weak heat pulses the recorded temporal evolution of the perturbation temperature can be described by theoretical models which neglect the influence of the superfluid vorticity.
Acknowledgements The authors would like to express their thanks to Professor Dr. E.-A. Miiller for his constant and helpful support of this investigation and to Professor Dr W. Fiszdon and Dr W. Poppe for the valuable discussions and suggestions during all stages of this work. Special thanks are due also to Mr. Gaads for his vacuum deposition work. This research was supported in part by the Deutsche Forschungsgemeinschaft.
References 1 Schwerdtner,M. v. Mitt MPlfar StrOmungsforsch G6ttingen(1988) Nr. 90 2 Stature, G. MPI Far StrOmungsforsch G6ttingen(1988) Ber. 16 3 Gulyaev,A.I.Z.hEksp Teor Fiz (1969) 57 59 [Soy Phys JEPT(1970) 30 37 ]
4 Torczynsld,J.R. Phys Fluids (1984) 27 1138 5 Stature, G. Mitt MPlfiir Strt~mungsforsch G6ttingen(1991) Nr. 103 6 Fiszdon, W., Schwerdtner, M. v., S l a m , G. and Poppe, W. J Fluid Mech (1990) 212 663 7 Miihring, W. and Piszdon, W. Converging axi- and spherically-
symmetrictop-hatpulses (in He II), in preparation
fL~r StrSmungsforschung,
Bunsenstr.
10, W-3400 GSttingen,
Received 31 October 1991
Two different designs of convergent counterflow channels are presented. The imploding second-sound wave was produced by a vacuum-deposited chromium film on the inner curved surface of these channels, and the temperature was measured by a bolometer whose distance from the heated surface could be externally adjusted. If the average heat input is small enough to avoid the production of superfluid vorticity, which is the main dissipative process, acoustic cylindrical waves can be observed. The temporal evolution of the perturbation temperature follows the geometrical predictions.
Keywords: helium; converging shock waves; temperature measurement
The propagation of second-sound pulses in He II under different flow geometries has been the subject of numerous investigations I-3, but only a few experiments have been performed on converging second-sound shock waves. Torczynski4 measured the temperature jump as a function of the Mach number and pointed out the great difficulty in producing film heaters with an appropriate geometry and homogeneity and also in recording the converging temperature waves. We now present two different possibilities to produce stable converging temperature shock waves, to record the corresponding signals and to compare them with geometrical predictions and results from numerical calculations and analytical models.
moveable temperature probe
plate II
0011 - 2 2 7 5 1 9 2 / 0 6 0 5 9 8 -.Cr3 © 1992 Butterworth -',Heinernann Ltd
598
Cryogenics 1992 Vol 32, No 6
plate IV
l
plate III
/
Flow channel 1 Flow channel 1 (FC1) (Figure 1) was built on a Lucite bottom plate in which was milled a 60 ° segment of a circular cylinder with a radius of 45 mm. Two plane Lucite plates (I and II) with the shape of a 60 ° segment form the walls that terminate the channel in the axial direction while another pair of Lucite plates (III and!I~)~form the walls terminating it in the tangential direction. The heater consists of an (originally plane) Kapton foil of 70 ~m thickness, serving as a substrate, on which a chromium film of about 1000 A thickness was vacuum deposited at a pressure of 2 x 10-5 mbar. In a second .deposition stage the supplying leads, a layer of 100 ,~ chromium, 5000 A copper and 500 A gold, were evaporated on to the heating film. After gluing the four side walls (I-IV) of the channel together, they were mounted on the bottom plate.as indicated in Figure 1,
~"
'
)
sensor
,,
\
P
converging shock wave
!IIii '
/
/
plate I
/ ' kapton (foil (70/~m) with film heater
bottom plate Figure ;1 Schematic drawing of the experimental setup Of FCI (hOt,to,scale). Extension in the axial direction is 42 mm
Converging second-sound shock waves: G. Stamm et al.
bending the Kapton foil according to the curvature of the bottom plate; to provide a tight connection the downward borders of the walls are knife-edged. By this procedure we obtain a large-scale cylindrically formed and highly homogeneous heating film which perfectly keeps its form also by cooling down to liquid helium temperatures. Through a slit of 3 mm width we can introduce our temperature probe. The distance between the lower end of the probe carrying the sensor and the heated surface can be adjusted from outside the cryostat in the range from 1 mm up to 45 mm. The temperature waves were produced by passing a rectangular electric current through the heating film. Owing to its fast thermal response we achieve rise and fall times better than 100 ns. With the dynamically calibrated sensors which work as superconducting bolometers as described by Stamm 5 we can obtain the temporal evolution of the perturbation temperature T ' = T - T o (To: temperature of the helium bath) at the location of the sensor. As the front surface of the probe is small (only 0.1 mm wide), its influence on the stability of the shock waves can be neglected. Following Fiszdon et al. 6 the governing equation for the evolution of the temperature is in first order approximation aT at
1 a(rw) r Or
(1)
Assuming the counterflow velocity w at the heated surface to be constant we recognize that according to the above equation the temperature must rise after the shock front, leaving a warm tail at the end of the pulse. In fact, this is what we measure for weak heat fluxes when the influence of the vortex-line density (VLD) can be neglected (see Figure 2). If we increase the input heat flux and reduce the time tR between two consecutive
30
T
.
.
.
.
I
.
.
.
.
.
I
.
.
.
.
.
.
.
.
.
I
.
.
.
.
.
.
.
.
T
20
10
0 t [ms]-I0
0
1
2
3
4
Figure 3
Temperature signals recorded in FCI at distances d = 2, 10, 20 and 30 mm from the heated surface. O = 3 W cm -2, tH = I ms, tR = 0 . 2 5 S and T o = 1.4 K. Close to the heater the signals show a large temperature overshoot
pulses, the pulse shapes become dominated by the influence of the VLD, and the effects due to geometry are then completely governed by the so-called 'overshoot' as described by Fiszdon et al. 6 (see Figure 3). Flow channel 2 Flow channel 2 (FC2), shown in Figure 4, consists of a ring 2 mm thick manufactured from a quartz glass tube of 36 mm inner diameter. Fixed on a specially designed device this ring was placed in the vacuum deposition
conducting strl )s
14
T
T' [mK]
.
T' [rnK]
glass plates
/
2mm \
12 I0
8 2
6
2O
I0
4
\
2 0
film i n t e r r u p t i o n f o r s u p p l y i n g leads
t[ms]-2
.
0
.
.
.
.
.
.
.
.
I
1
.
.
.
.
.
i
,
•
•
I
,
,
,
.
.
.
.
.
,
2
Figure 2
Temperature signals recorded in FCI at distances d = 2, 10 and 20 mm from the heated surface. The heat flux of the applied rectangular heat pulse was Q = 2 W cm -2, the pulse duration tH = 0 . 8 ms, the repetition time tR = 0 . 4 S and the bath temperature To = 1.4 K. The signals s h o w a nearly linear increase after the shock f r o n t and a warm tail at their end
i nner d i a m e t e r 36 mm inner surface covered with thin-film heater
Figure 4
Schematic exploded view of the experimental setup of FC2 (not to scale). The glass ring can be moved up and down
Cryogenics 1992 Vol 32, No 6
599
Converging second-sound shock waves: G. Stamm et al.
~
X
neither completely rigid nor completely soft, and detected at the time t2. This singularity is then reflected at the heater and reaches the probe at the time t3. After a further reflection at the centre the pulse is re-formed to its original rectangular shape but reverse in sign, and recorded at the time t4 Elongating the wave path by another identical sequence of reflections, the pulse will again look like the incoming one recorded at t]. These results could be reproduced theoretically by an analytical model using the Fourier integral method, and by numerical calculations with a VLD close to zero, as described in detail by M0hring and Fiszdon 7.
t
/
4 1 T'[mK]
/ t [m~]- 6
.
0
.
.
.
i
1
.
.
.
.
.
2
.
.
i
,
,
,
3
Figure 5 Temperature curve, recorded by the sensor after releaseof a short rectangular heat pulse, due to direct reception of the propagating pulse at time tl and reflection at the centre (t2), the wall (t3) and at the centre again (t4). O = 1 W c m - 2 , tH = 0.2 ms and To = 1.85 K
apparatus and, while rotating at a frequency of 3 rps, the inner surface was covered with a chromium film of about 300 A . The ends of this tube are closed by two quartz glass plates, and on to one of them a secondsound receiver with supplying leads is vapour deposited. As we have to supply the electric current to the heating film on the inner surface, we get a small irregularity at the junction of the heating film to the supplying leads. However, this interruption of the film does not have a recognizable influence on the form and stability of the converging waves, as has been verified by numerous experiments. While the two glass plates are fixed, the position of the ring heater relative to the position of the sensor can be varied from outside the cryostat. If the average heat input is small enough to avoid significant production of superfluid vorticity we observe acoustic cylindrical converging waves as shown in Figure 5 together with the wave path. After release of a rectangular heat pulse at the heater, the sensor first detects, at the time tl corresponding to the distance from the heater, a rectangular pulse travelling with a velocity of about 20 m s-] towards the centre of the tube. As the pulse duration is only 200 #s, the geometrical effects shown in Figure 2 can be neglected. At the centre this incoming jump is reflected as a logarithmic singularity, which means that the centre is
600
Cryogenics 1992 Vol 32, No 6
Conclusions We have shown two methods of producing and recording converging temperature shock waves. Using only weak heat pulses the recorded temporal evolution of the perturbation temperature can be described by theoretical models which neglect the influence of the superfluid vorticity.
Acknowledgements The authors would like to express their thanks to Professor Dr. E.-A. Miiller for his constant and helpful support of this investigation and to Professor Dr W. Fiszdon and Dr W. Poppe for the valuable discussions and suggestions during all stages of this work. Special thanks are due also to Mr. Gaads for his vacuum deposition work. This research was supported in part by the Deutsche Forschungsgemeinschaft.
References 1 Schwerdtner,M. v. Mitt MPlfar StrOmungsforsch G6ttingen(1988) Nr. 90 2 Stature, G. MPI Far StrOmungsforsch G6ttingen(1988) Ber. 16 3 Gulyaev,A.I.Z.hEksp Teor Fiz (1969) 57 59 [Soy Phys JEPT(1970) 30 37 ]
4 Torczynsld,J.R. Phys Fluids (1984) 27 1138 5 Stature, G. Mitt MPlfiir Strt~mungsforsch G6ttingen(1991) Nr. 103 6 Fiszdon, W., Schwerdtner, M. v., S l a m , G. and Poppe, W. J Fluid Mech (1990) 212 663 7 Miihring, W. and Piszdon, W. Converging axi- and spherically-
symmetrictop-hatpulses (in He II), in preparation