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Evidence for a low dimension strange attractor for temperature variations inside helium-II counter-flow jets
Physica B 165&166 North-Holland
(1990)
761-762
EVIDENCE FOR A LOW DIMENSION COUNTERFLOW JETS*
Paul J. DOLAN, Department
STRANGE ATTRACTOR
JR.+ and Charles
FOR TEMPERATURE
VARIATIONS
INSIDE
HELIUM-II
W. SMITH
of Physics and Astronomy,
University
of Maine,
@one,
ME
04469
U.S.A.
Time series measurements of the temperature of liquid helium inside the chamber of thermal counterflow jets driven by a constant heat source show four distinct regions of fluid flow as a Detailed analysis suggests that the temperature variations for the function of heater power. lowest power oscillatory region may be characterized by a strange attractor of fractal dimension, n,4.
1. INTRODUCTION Thermal counterflow of helium II has been the subject of numerous experimental and theoretical studies (l-4) beginning with Kapitza's pioneer work on heat transfer in 1941 (5). This famous paper reports several observations including the displacement of a vane to monitor the flow of the normal fluid component at the nozzle of a thermal counterflow jet and a transition to "erratic" flow above a fairly well defined critical heat flux. Our work was motivated in part by this historic paper, together with an interesting series of measurements in the mid-1970's on steady state thermal counterflow jets (6). The latter studies were able to conclude, among other things, that the jet plume into the isothermal bath, above a critical heat flux, was characterized by some type of turbulent velocity fluctuations. We report here an extension of this type of study to higher heater power while monitoring both the temperature inside the thermal counterflow jet chamber and the flow into the bath. 2. EXPERIMENTAL TECHNIQUES AND OBSERVATIONS We employed thermal counterflow jets of ncminal chamber size, 2.5 cm in cast acrylic: length by 1.0 cm in diameter and nozzle length 1.0 cm bv 1.0 mm diameter (see insert. Fiaure 2). Each contained a 52.3 ohm metal film resigtor, as a heater and a 3300 ohm carbon composite resistor as a thermometer. Time series measurements of the temperature of the liquid helium inside the jet chamber were digitally recorded by logging the current through the thermometer resistor at constant voltage as a function of time. Mass flow at the jet nozzle was indicated by the displacement of a sensitive microbalance vane located in the temperature controlled bath and viewed
directly through a window in the dewar. Four different regions of jet behavior were observed. In the lowest power range, a constant efflux from the jet was indicated as a constant deflection of the microbalance vane and a constant temperature inside the jet chamber. Kapitza's pioneer work on heat transfer in helium 4s carried out in this ranae. Above a critical Dower (nominallv Q25mW). the-efflux from the nozzle and the tempe;ature " inside the jet chamber become oscillatory. Each decrease in temperature is in phase with a strong efflux from the nozzle, corresponding to out-soino flow. This behavior is exoected based upon conventional phenomenology for thermal counterflow (7). For a range of input power (up to about 50 mW) oscillations persist with the temperature inside the jet chamber never exceeding the lambda point. We therefore term this power range the superfluid region, S. As the power is increased further, again oscillations are observed, however the temperature in the jet chamber is above the lambda point but below the boiling temperature. The oscillations of the microbalance vane and variation of the temperature are not well correlated and show little variation in period with power. We term this power range the normal fluid region, N. If power is increased still further, the liquid helium around the heater is driven into the vapor phase. 3. RESULTS AND DISCUSSION The remainder of this paper deals exclusively with the variation of the temperature inside the jet chamber in the superfluid region S, just above onset for oscillation. Figure 1 shows a typical plot of the temperature as a function of time. Each oscillation shows a slow warm-up interval, at the end of which flow from the nozzle appears restricted (the microbalance
--Supported by the National Science Foundation (USA), DMR-8312492, Low Temperature Physics Program. +Present Address: Department of Physics, Northeastern Illinois University, Chicago, IL 60625 U.S.A.
0921-4526/90/$03.50 @
1990 - Elsevier
Science
Publishers
B.V. (North-Holland)
P.J. Dolan Jr., C. W. Smith
762
1
:
:
:
20
30
40
.I
50
SECONDS
FIGURE 1 Temperature inside the jet chamber as a function Bath temperature 1.77 K. of time. vane shows little displacement) and the temperature steeply rises. Flow then restarts as observed by a sharp temperature decrease caused by the inflow of superfluid component from the bath and a microbalance vane deflection away from the nozzle indicating outflow of normal The cycle then repeats. As the fluid component. input power is increased, the warm-up interval decreases, the magnitude of the temperature excursion increases and we observe a power range that will support continuous oscillations. The warm-up interval for fixed power monatonically increases with chamber volume, as expected.
EMBEDDING
We have applied the techniques of nonlinear dynamics to attempt to characterize the tnperature variations in the S region. Using the Grassberger-Procaccia algorithm (8), we have measured the fractal dimension of many time series records. The slopes for embeddinq plots (log of the correlation integral versuslog of the correlation length) for embedding dimensions 1 thru 12 are shown in Figure 2. While the results for the N region exhibit substantial variation, the dimensionality for the S region consistently suggest behavior characterized by a strange attractor of low dimension.
DIMENSION
FIGURE 2 Slope from embedding plots versus embedding Insert shows thermal counterflow gr2ension.
4. CONCLUSION We have investigated the oscillatory behavior of thermal counterflow jets operating under a range of conditions but primarily just above the constant flow region. Analysis of time series records of the temperature of the liquid helium inside the jet chamber suggests that the complicated oscillations observed can be characterized by a strange attractor of low fractal dimension, ~4. REFERENCES
(1)
W.F. Vinen, Proc. Roy. Sot. A240 (1957) 114, 128; A242 (1957) 493; A243 (1957) 400.
(2)
R.J. Donnelly, Proc. Roy. Sot. A281 (1964) 130; with P. H. Roberts, A312 (1969) 519; with C.E. Swanson, J. Low Temp. Phys. 61 (1985) 363; J. Fluid Mech. 173 (1986) 387.
(3)
K.W. Schwarz, Phys. Rev. 818 (1978) 245; Phy. Rev. Lett. 49 (1982) 283; Phys. Rev. 1331 (1985) 5782.
(4)
J.T. Tough, Prog. in Low Temp. Phys., ed. D.F. Brewer (North-Holland, Amsterdam) Vol. VIII (1982) 133.
(5)
pi:.
(6)
P.E. Dimotakis and J.E. Broadwell, Phys. Fluids 16 (1973) 1787; G.A. Laguna, Phys. Rev. 812 (1975) 4874; P.E. Dimotakis and G.A. Laguna, Phys. Rev. 1315 (1977) 5240.
(7)
R.J. Donnelly and C.E. Swanson, Mech. 173 (1986) 387.
J. Fluid
(8)
P. Grassberger and I. rrocaccia, Rev. Lett. 50 (1983) 346.
Phys.
Kapitza,
J.
OT
dys.
USSR 4 (1941)
(1990)
761-762
EVIDENCE FOR A LOW DIMENSION COUNTERFLOW JETS*
Paul J. DOLAN, Department
STRANGE ATTRACTOR
JR.+ and Charles
FOR TEMPERATURE
VARIATIONS
INSIDE
HELIUM-II
W. SMITH
of Physics and Astronomy,
University
of Maine,
@one,
ME
04469
U.S.A.
Time series measurements of the temperature of liquid helium inside the chamber of thermal counterflow jets driven by a constant heat source show four distinct regions of fluid flow as a Detailed analysis suggests that the temperature variations for the function of heater power. lowest power oscillatory region may be characterized by a strange attractor of fractal dimension, n,4.
1. INTRODUCTION Thermal counterflow of helium II has been the subject of numerous experimental and theoretical studies (l-4) beginning with Kapitza's pioneer work on heat transfer in 1941 (5). This famous paper reports several observations including the displacement of a vane to monitor the flow of the normal fluid component at the nozzle of a thermal counterflow jet and a transition to "erratic" flow above a fairly well defined critical heat flux. Our work was motivated in part by this historic paper, together with an interesting series of measurements in the mid-1970's on steady state thermal counterflow jets (6). The latter studies were able to conclude, among other things, that the jet plume into the isothermal bath, above a critical heat flux, was characterized by some type of turbulent velocity fluctuations. We report here an extension of this type of study to higher heater power while monitoring both the temperature inside the thermal counterflow jet chamber and the flow into the bath. 2. EXPERIMENTAL TECHNIQUES AND OBSERVATIONS We employed thermal counterflow jets of ncminal chamber size, 2.5 cm in cast acrylic: length by 1.0 cm in diameter and nozzle length 1.0 cm bv 1.0 mm diameter (see insert. Fiaure 2). Each contained a 52.3 ohm metal film resigtor, as a heater and a 3300 ohm carbon composite resistor as a thermometer. Time series measurements of the temperature of the liquid helium inside the jet chamber were digitally recorded by logging the current through the thermometer resistor at constant voltage as a function of time. Mass flow at the jet nozzle was indicated by the displacement of a sensitive microbalance vane located in the temperature controlled bath and viewed
directly through a window in the dewar. Four different regions of jet behavior were observed. In the lowest power range, a constant efflux from the jet was indicated as a constant deflection of the microbalance vane and a constant temperature inside the jet chamber. Kapitza's pioneer work on heat transfer in helium 4s carried out in this ranae. Above a critical Dower (nominallv Q25mW). the-efflux from the nozzle and the tempe;ature " inside the jet chamber become oscillatory. Each decrease in temperature is in phase with a strong efflux from the nozzle, corresponding to out-soino flow. This behavior is exoected based upon conventional phenomenology for thermal counterflow (7). For a range of input power (up to about 50 mW) oscillations persist with the temperature inside the jet chamber never exceeding the lambda point. We therefore term this power range the superfluid region, S. As the power is increased further, again oscillations are observed, however the temperature in the jet chamber is above the lambda point but below the boiling temperature. The oscillations of the microbalance vane and variation of the temperature are not well correlated and show little variation in period with power. We term this power range the normal fluid region, N. If power is increased still further, the liquid helium around the heater is driven into the vapor phase. 3. RESULTS AND DISCUSSION The remainder of this paper deals exclusively with the variation of the temperature inside the jet chamber in the superfluid region S, just above onset for oscillation. Figure 1 shows a typical plot of the temperature as a function of time. Each oscillation shows a slow warm-up interval, at the end of which flow from the nozzle appears restricted (the microbalance
--Supported by the National Science Foundation (USA), DMR-8312492, Low Temperature Physics Program. +Present Address: Department of Physics, Northeastern Illinois University, Chicago, IL 60625 U.S.A.
0921-4526/90/$03.50 @
1990 - Elsevier
Science
Publishers
B.V. (North-Holland)
P.J. Dolan Jr., C. W. Smith
762
1
:
:
:
20
30
40
.I
50
SECONDS
FIGURE 1 Temperature inside the jet chamber as a function Bath temperature 1.77 K. of time. vane shows little displacement) and the temperature steeply rises. Flow then restarts as observed by a sharp temperature decrease caused by the inflow of superfluid component from the bath and a microbalance vane deflection away from the nozzle indicating outflow of normal The cycle then repeats. As the fluid component. input power is increased, the warm-up interval decreases, the magnitude of the temperature excursion increases and we observe a power range that will support continuous oscillations. The warm-up interval for fixed power monatonically increases with chamber volume, as expected.
EMBEDDING
We have applied the techniques of nonlinear dynamics to attempt to characterize the tnperature variations in the S region. Using the Grassberger-Procaccia algorithm (8), we have measured the fractal dimension of many time series records. The slopes for embeddinq plots (log of the correlation integral versuslog of the correlation length) for embedding dimensions 1 thru 12 are shown in Figure 2. While the results for the N region exhibit substantial variation, the dimensionality for the S region consistently suggest behavior characterized by a strange attractor of low dimension.
DIMENSION
FIGURE 2 Slope from embedding plots versus embedding Insert shows thermal counterflow gr2ension.
4. CONCLUSION We have investigated the oscillatory behavior of thermal counterflow jets operating under a range of conditions but primarily just above the constant flow region. Analysis of time series records of the temperature of the liquid helium inside the jet chamber suggests that the complicated oscillations observed can be characterized by a strange attractor of low fractal dimension, ~4. REFERENCES
(1)
W.F. Vinen, Proc. Roy. Sot. A240 (1957) 114, 128; A242 (1957) 493; A243 (1957) 400.
(2)
R.J. Donnelly, Proc. Roy. Sot. A281 (1964) 130; with P. H. Roberts, A312 (1969) 519; with C.E. Swanson, J. Low Temp. Phys. 61 (1985) 363; J. Fluid Mech. 173 (1986) 387.
(3)
K.W. Schwarz, Phys. Rev. 818 (1978) 245; Phy. Rev. Lett. 49 (1982) 283; Phys. Rev. 1331 (1985) 5782.
(4)
J.T. Tough, Prog. in Low Temp. Phys., ed. D.F. Brewer (North-Holland, Amsterdam) Vol. VIII (1982) 133.
(5)
pi:.
(6)
P.E. Dimotakis and J.E. Broadwell, Phys. Fluids 16 (1973) 1787; G.A. Laguna, Phys. Rev. 812 (1975) 4874; P.E. Dimotakis and G.A. Laguna, Phys. Rev. 1315 (1977) 5240.
(7)
R.J. Donnelly and C.E. Swanson, Mech. 173 (1986) 387.
J. Fluid
(8)
P. Grassberger and I. rrocaccia, Rev. Lett. 50 (1983) 346.
Phys.
Kapitza,
J.
OT
dys.
USSR 4 (1941)