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GUGGENHEIM AERONAUTICAL LABORATORY
CALIFORNIA INSTITUTE OF TECHNOLOGY
HYPERS~IC RESEARCH PROJECT
Memorandum No. 63
September 15, 1961
HOT -WIRE MEASUREMENTS IN
LOW REYNOLDS NUMBER HYPERSONIC FLOWS
by
C. Forbes Dewey, Jr.
ARMY ORDNANCE CONTRACTS
DA-04-495-0rd-1960 and DA-04-495-0RD-3231
GUGGENHEIM AERONAUTICAL LABORATORY
CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena. California
HYPERSONIC RESEARCH PROJECT
Memorandwn No. 63
September 15. 1961
HOT-WIRE MEASUREMENTS IN
LOW REYNOLDS NUMBER HYPERSONIC FLOWS
by
C. Forbes Dewey, Jr.
Guggenheim Aeronautical Laboratory
ARMY ORDNANCE CONTRACTS
DA-04-495-0rd-1·960 and DA-04-495-0RD-3231
ACKNOWLEDGMENTS
The author would like to express his deep gratitude to Professor
Lester Lees for suggesting this investigation and supervising the
research program. Dr. Anthony Demetriades contributed greatly to
the investigation in many ways; his hot-wire experience and assistance
in obtaining the wake data were invaluable. Professor Toshi Kubota
and Professor Edward Zukoski contributed many important suggestions
and criticisms to the research program. The author wishes to thank
Paul Baloga and the hypersonic staff for their many hours of expert
assistance, Mrs. Geraldine VanGieson for her help in preparing the
manuscript, and Mrs. Truus van Harreveld for performing part of the
computations. The cooperation of Dr. John Laufer and Dr. Thomas
Vre balovich of the Jet Propulsion Laboratory is also gratefully
acknowledged.
A prelirr.inary report on this work was presented to the American
Rocket Society Annual Meeting, Washington, D. C., December 5-8, 1960.
ii
ABSTRACT
Measurements were made of the heat loss and recovery tem
perature of a fine hot-wire at a nominal Mach number of 5. 8. Data
were obtained over an eight-fold range of Reynolds numbers in the
transitional regime between continuum and free-molecule flow. At
high Reynolds nurn ber s, the heat transfer data agree well with the
results of Laufer and McClellan, which were obtained at lower Mach
numbers. At lower Reynolds numbers, the results indicate a monotonic
transition between continuum and free molecule heat transfer laws.
The slope of the heat transfer correlation also appears to vary mono
tonically, with Nu -..J )"Re at high Reynolds numbers and Nu ;v Re for
Re < < 1.
Data on the wire recovery temperature (corresponding to zero
net heat transfer) were obtained for free-stream Knudsen numbers
between 0. 4 and 3. 0. Comparison with previous data suggests that for
Mach numbers greater than about two the normalized variation of
recovery temperature in the transitional regime is a unique function
of the free-stream Knudsen number.
The steady-state hot-wire may be used to obtain two thermody
namic measurements: the rate of heat transfer from the wire and the
wire recovery temperature. An illustrative experiment was performed
in the wake of a transverse cylinder, using both hot-wire and pressure
instruments in a redundant system of measurements. It was shown that
good accuracy may be obtained with a hot-wire even when the Reynolds
number based on wire diameter is small.
111
TABLE OF CONTENTS
PART PAGE
Acknowledgments ii
Abstract iii
Table of Contents iv
List of Figures v
List of Symbols vi
I. Introduction 1
II. Experimental Method 4
II. 1. Equipment 4
II. 2. Measurements 6
II. 3. Wire Calibration and Annealing 7
Ill. Data Reduction 11
IV. Results and Discussion 21
IV. 1. Heat Transfer Measurements 21
IV. 2. Recovery Temperature Measurements 28
v.
Application of the Steady-State Hot-Wire to Wake
Measurements 30
References 33
Appendix A End Loss Corrections for the
Transitional Regime 36
Appendix B -- Measurement of the Wire Support
Temperature 45
Appendix C -- The Effect of Overheat on Nusselt
Number 48
Appendix D -- Measurements at Reynolds Numbers
Approaching Free-Molecule Flow 52
Figures 53
iv
LIST OF FIGURES
NUMBER PAGE
1 Variation of Free Stream Mach Number with
Tunnel Pressure 53
2 Variation of Stagnation Temperature with
Tunnel Pressure 54
3 Hot-Wire Probe and Tunnel Installation 55
4 Typical Hot-Wire After Tunnel Exposure 56
5 Diagram of Calibration Oven 57
6 Electrical Circuit Diagram 58
7 Empirical Correlation of Hot-Wire Heat Transfer
at Low Reynolds Numbers 59
8 Nusselt Number-Reynolds Number Relation for
Supersonic and Hypersonic Flow 60
9 Slope of Nusselt Number-Reynolds Number
Relation as a Function of Reynolds Number 61
10 Normalized Variation of Recovery Temperature
with Knudsen Number 62
11 Data Obtained from Cylinder Wake Traverse 63
12 Variation of Flow Quantities Across the Cylinder
Wake 64. 65
13 End Loss Relations for Nusselt Number Correction 66
14 End Loss Relations for Recovery Temperature
Correction 67
15 Wire Support Temperature as a Function of
Tunnel Pressure 68
16 Variation of Nusselt Number with Overheat 69
17 Nusselt Number-Reynolds Number Relation
Approaching Free-Molecule Flow 70
v
LIST OF SYMBOLS
cp specific heat
d hot-wire diameter
D cylinder diameter
f denotes a functional relation between two quantities
f( s ), g( s ) functions of the molecular speed ratio s
1 1 1
q
h convective heat transfer coefficient,
Tw- T
aw
i wire current
k air thermal conductivity
k wire thermal conductivity at temperature T w
w
Kn Knudsen nwnber, (M/Re )
00
wire length
M Mach nwnber
n exponent in Nu - Re relation: Nu r-.; Re n
0 0 0 0
Nu Nusselt nwnber measured with finite length wire, h d/k
m m o
Nu Nusselt nwnber for infinite length wire, hd/k
0 0
p static pressure
pitot pressure
Pro Prandtl nwnber,
q heat transfer rate per unit area
R wire resistance
Re Reynolds number,
vi
s resistance parameter.
molecular speed ratio ~M
T-T
awm
t non-dimensional temperature,
T temperature
u free-stream velocity
00
v support temperature parameter. (Eq. 12)
X distance from cylinder axis in streamwise direction
y distance from cylinder axis in transverse direction
0. accommodation coefficient
=
0. temperature-resistivity coefficient. R~Rr 1 + o.r (Tw - Tr)
r
ratio of specific heats
boundary layer displacerr,ent thickness
recovery ratio parameter,
end loss correction parameter, (Eq. 11)
experimental recovery ratio for continuum flow
theoretical recovery ratio for free molecule flow
measured recovery ratio, (T /T )
awm o
non-dimensional support temperature, (T s/T )
0
recovery ratio for infinite length wire, (T */T )
0
n. n
rt - Yl I
normalized recovery ratio. < c> < f - c>
'fiT )
viscosity; also ( 2/ (1/Kn) log ( 1/Kn) in near free
molecule flow analysis
dimensionless quantity, (Eq. A-17)
vii
measure of Nusselt number change with overheat, (Eq. A-19)
p density
7:' overheat, (T - T ) I T
w aw aw
dimensionless quantity in near free molecule flow analysis,
(Eq. 19)
dimensionless quantity in near free molecule flow analysis,
(Eq. 20)
end loss correction factor, (Eqs. 2 and 3)
abbreviation for (tanh l) /l.J)
Subscripts
( zero current
>aw
{ )m measured
evaluated at stagnation temperature
>o
( )r evaluated at reference temperature T
r
( )s evaluated at support temperature T s
)w evaluated at wire temperature T w
)* refers to infinitely long wire
free stream
)00
( )2 behind a normal shock
evaluated at wake centerline
>t
Superscripts
~
( ) evaluated at zero overheat
local value at edge of wake mixing region
( ) I
( ) (n) the nth approxirrtation of an iterative solution
n
length-average or norxnalized quantity
1
I. INTRODUCTION
Hot-wire probes have been used for a number of years to
measure the mean properties of a fluid stream. The hot-wire is
capable of providing two thermodynamic measurements: the rate at
which heat is transferred from the wire to the stream and the wire
recovery temperature for zero heat transfer. If sufficiently accurate
relations are known for the heat loss and recovery temperature as a
function of the properties of the flow, only one additional thermodynamic
measurement is necessary to specify the flow field uniquely.
The present investigation was undertaken with two purposes in
mind. First, in using the hot-wire to measure mean quantities in low
density compressible flow, it is necessary to have an accurate calibration
of the wire heat loss and recovery temperature as a function of Mach
number and Reynolds number. For high Reynolds number flows, these
1 *
relations are well represented by the data of Laufer and McClellan .
Often, however, it is necessary to use the hot-wire in regions where
the Reynolds number based on wire diameter is small. Previous
3 4 6 7
investigations • • • have illustrated the general qualitative features
of the Nu - Re relation at small Reynolds numbers, but the experimental
uncertainties are too large for calibration purposes. The first objective,
therefore, was to obtain an accurate hot-wire calibration at high Mach
number in the transition regime between continuum and free molecule
flow.
*
Superscripts denote references at the end of the text.