Subsonic tests in free molecule flow on impact tube pressure probes having a wide range of tube lengths at angles of attack up to 90º

•

SUBSONIC TESTS IN FREE MOLECULE FLOW ON IMPACT TUBE PRESSURE PROBES HAVING A WillE RANGE OF

TUBE LENGTHS AT ANGLES OF ATTACK UP TO 90°

by

P. C. Hughes

ACKNOWLEDGEMENT

The author wishes to express his sincere appreciation to his supervisor, Dr. J. H. deLeeuw, for his capable direction and friendly encouragement, and for his auditing of this report.

The author is also indebted to Dr. G. N. Patterson for making this research possible. This work was supported by the USAF Cambridge Research Laboratories under Contract No. AFC 9(628)- 363.

..

SUMMARY

Subsonic tests were performed in the UTIAS low density wind tunnel to obtain data on the free molecule properties of impact tube pressure probes at subsonic speeds over a wide range of tube lengths .

The data substantiated recent theoretical ca1culations. For non-zero angles of attack it was found that the probe pressure ratio can be approximated by assuming that the probe is at zero angle of attack to the axial component of the speed ratio. This approximation will yield a value which is too high and will be more in error as either the speed ratio or the impact tube length/diameter ratio is increased ..

TABLE OF CONTENTS

Page No.

NOTATION v

1. INTRODUCTION 1

2. EXPERIMENTAL APPARATUS 1

2. 1 The Low Density Wind Tunnel 1

2. 2 Pressure Probes 2

2. 3 The Thermistor Gauge 2

3. CALIBRATION OF THE SUBSONIC NOZZLE 2

3.1 Preliminary Discussion 2

3. 2 Nozzle Calibration 4

3. 3 The Near-Orifice Probe 5

4. DESCRIPTION OF PROBE TESTS 5

5. THEORY 6

6. ANALYSIS OF EXPERIMENTAL ERRORS 9

7. DISCUSSION OF RESULTS 11

7.1 Results at Zero Angle of Attack 11

7.2 Results at Non-Zero Angles of Attack 11

8. CONCL USIONS 12 REFERENCES 13 APPENDIX A 14 APPENDIX B 17 APPENDIX C 18 TABLES I, II • FIGURES 1 to 30

NOTATION

Note: Symbols of limited use are explained in the text. Cm d D erfS Kn

L

M P 1 P₂ P t P stag P ts Rl' R 2 S Tl T 2 T 0 u ~

D

E

most probable random speed of a molecule diameter of impact tube

dil

symbol for the error function defined as

kfe~v.."'du..

Y1f

À

0

Knudsen number defined by Kn =

d

length of impact tube

Mach number

static pressure at position of probe

pressure registered in probe gauge volume symbol for P₂ at zero angle of attack

pressure in the stagnation chamber of wind tunnel static pressure surrounding the flow in the test section of wind tunnel

defined on page 15

Speed Ratio defined by

~

and therefore =

j

r

M

Static temperature at position of probe

temperature in probe gauge volume

stagnation temperature (uniform throughout flow) flow velocity

angle of attack, see Fig. 8a

specific heat ratio, equal to 1. 4 for air angle of flow inclinatibn, see Fig. 8a

relative errors, see page 18

final relative error in estimate of S (page 20)

molecular rnean free path

probe angle with respect to a reference direction (Fig. 8a)

function defined on page 16

1. INTRODUCTION

During the past few years, considerable interest has been taken by the low density group at UTIAS in the properties of free -molecule impact a~d orifice probes (Refs. 1 to 7). Under free molecular conditions, the mean free path in the flow around the probe, and in the flow in its

interior, is assumed to be large compared to probe dimensions so that molecule-molecule collisions are rare compared to molecule-surface c ol1is ions.

In Ref. 7 a numerical solution is presented for the free-molecule impact pressure probe relatiohs for tubes of arbitrary length and for speed ratios up to 20. The present report describes a series of subsonic tests performed on probes having a range of tube lengths with a view to experimental verification of the theory of Ref. 7. Also presented in th is report are data which indicate the manner in which the probe

pressure ratio, P2

f!j,

was found to vary with angle of attack; these

PlAt T2

results are then compared with the theoretical predictions of Ref. 8.

2. EXPERIMENT AL APP ARA TUS 2. 1 The Low Density Wind Tunnel

The tests reported here we re performed in the UTIAS low density wind tunnel, a complete description of which is given in Ref. 9. The tunnel is of the continuous, open circuit type with a vacuum pump drive and is designed to operate at Mach numbers up to 5 over astatic pressure range from 1 to 70 micron Hg. Atmospheric air passes through a dryer past a needle valve which is used to regulate the mass flow rate by throttling the air down to low pressure before it enters the stagnation chamber of the tunnel. In this chamber any desired value of stagnation temperature up to 1500 F can be set by means of a heated liner. A series of six booster diffusion pumps on the downstream side of the nozzle maintain a continuous flow of air through the test section.

For the present tests a subsonic nozzle 5.5" in diameter was used (Fig. 1). The calibration of this nozzle is described in detail in Sec. 3.

The stagnation chamber and test section pressures were measured throughout by means of two mercury MacLeod gauges having ranges of 0 - 2 mm Hg and 0 - 50 ;M-Hg respectively. Two thermocouple elements, one on the stagnation chamber wa11 and the other at the nozzle entrace, were used to measure and control the stagnation temperature at a constant value of 850 _{F. A traversing gear allowed positioning of the}

probes from outside the tunnel to within O. 001" and the probe (and the gauge volume to which it was clamped) could be rotated to any angle of

attack to within one-third of a degree. 2.2 Pressure Probes

The overall dimensions of the ten pressure probes used are shown in Fig. 2; the probes were identical except for the lengths and

diameters of the impact tubes which are tabulated in Table 1. Care was taken to ensure that the impact tube entrance was on the axis of rotation of the probe.

Various means were employed in the construction of the probes to cover a wide range of diameter /length (i. e. D). The four basic methods are illustrated in Fig. 3. For the longer tubes, a length of stainless steel hypodermic tubing was hard soldered to the probe. When these lengths became too small the probe wall was build up with hard solder, flattened, and a hole drilled through. For D in the neighbourhood of unity it was possible merely to drill a hole in the probe wall itself (made of hypodermic tubing as well). Finally for the shortest test probes, the probe wall was slightly flattened, a hole drilled, and then the wall was honed down to secure the desired tube length.

2.3 The Thermistor Gauge

A thermistor gauge which operated on the principle that at very low densities the heat conducted away from an element varies directly as the pressure, was found to be a suitable instrument for measuring the pressure in the gauge volume connected to the impact tube.

A diagram for the mechanical arrangement and a circuit diagram are given in Fig.s. 4 and 5. The dummy gauge compensated for zero drift once equilibrium testing conditions existed. The electrical circuit is the same as that recommended by the manufacturer (Western Electric). The output from the bridge was fed directly to a Minneapolis Honeywell millivolt recorder which had a total range of 0 - 70 mv. in 1 mv. steps. The smallest division on the scale was 0.002 mv.

Calibration showed that the recorder output was linear with pfressure.

3. CALIBRATION OF THE SUBSONIC NOZZLE 3. 1 Preliminary Discussion

A means had to be found for the measurement of the speed ratio of the flow past the impact tube. Some measurements by Harris (Ref. 10) in this nozzle seemed to indicate that if slight axial gradients in

ol

'

..

speed ratio (Fig. 6) were to be avoided, the optimum position for cali-bration would be in a plane one inch from the exit plane of the nozzle.

By varying the tunnel operating conditions, several radial traverses were made at this station using the near-orifice probe described below. A typical traverse is shown in Fig. 7. It can be seen that at these low pressures the boundary layer tends to fill the nozzle and that, as a

result, there is no region of uniform speed ratio. This observation suggests that the flow is not isentropic. Quantitatively, the test section static

pressure (as measured by an orifice probe) was found to be essentially constant across the jet; this agrees with similar measurements in Ref. 2, page 47. Furthermore, when the value of the speed ratio was calculated using the isentropic relation

V-I

-

_Y'

it was found that the result was a value of S in excess of the measured value. Note also that since the static pressure was found to be constant across the flow, the test section pressure, i. e., the pressure just out-side the jet, measured with a MacLeod gauge, P ts ' is identical with the static pressure in the flow at the nozzle centerline, Pl.

It was noted further that the flow, at points off the axis, was inclined to the axis and that by rotating the probe through

±

900,

this angle of flow inclination could be determined (details are given in Appendix A). The results, for the same flow conditions as in Fig. 7 are shown in Fig. 8.

From this preliminary investigation of the nozzle flow, the following conc1usions were drawn:

(1) The considerable flow inclination discussed above would be troublesome for points off the nozzle axis.

(2) The large speed ratio gradient off the nozzle axis would necessitate very accurate positioning of the probe, particularly in the angle of attack investigations.

(3) Although the flow was everywhere nonisentropic, isentropic conditions were most closely approached on the nozzle centerline. Isentropy is desirable in maintaining a Maxwellian velocity distribution.

(4) The optimum axial location for testing was a position one inch from the exit plane of the nozzle.

Under these circumstances, it was decided to restrict the test location to a single point in the flow, namely, the point on the axis one inch from the nozzle exit plane. The static pressure and speed ratio at this point could then be varied by alterations in tunnel operating

conditions. Accordingly a calibration relating statie pressure and speed ratio at this point to tunnel operating conditions was indicated.

3. 2 Nozzle Calibration

It was considered that if the tU:!1nel was operating at equilib-rium conditions, then the speed ratio at any point in the nozzle, on the centerline at one inch from the nozzle exit plane, in particular, would dep end only on

(1) Stagnation Temperature, T 0

(2) Pressure in the stagnation chamber, P stag (3) Pressure in the test section, Pts

Throughout the tests, the stagnation temperature was kept at a value of 850 _{F thus allowing only (2) and (3) to vary. By performing}

calibrations for many values of Pts and Pstag' while determining S with a near orifice probe (Sec. 3.3), a set of expê'rimentally determined nozzle characteristics was obtained. A calibration diagram containing the obtained results, much reduced in size Îrom the one actually used, is shown in Fig. 9.

The dashed curves in Fig. 9 show the variation of speed ratio with test section pressure for a given number of booster diffusion pumps, N. These curves had to be used with caution as they deper..d on the pump characteristics which may change over a period of time. The true nozzle characteristics are therefore based on the measured va.lue of Pstag, and the curves of constant Nare only used as a general guide.

The interesting feature of these curves is that they

demonstrate the manner in which the speed ratio increases to a maximum and then slowly decreases as the mass flow through the nozzle, for a fixed number of pumps, is increased from zero. The explanation of this behaviour lies in the fél.ct that for _{P ts} :>' 10 micron Hg the volume flow, Q, through the booster pumps is essentially constant. Now we may say, approximately,

QN S • Aeff

where the effective area of the nozzle, Aeff' will increase as Pts increases due to the reduction in boundary layer thickness. The inverse dependence of S and Pts is thus demonstrated. For Pts

<

10 micron Hg, the volume

flow pumped by the boosters begins to fall off rapidly due to backstreaming. The maximum in the curve occurs where these two effects tend to cancel

.'

one another.

Plots of Pstag vs. Pts for a fixed N and a varying mass flow showed a nearly linear relationship; it was thus readily possible to find a best-fit curve and. in so doing. most of the random error in P stag and Pts was removed. Two curves of constant mean free path in the test flow are sketched in Fig. 9 as well. from calculations given in Appendix B.

3. 3 The Near-Orifice Probe

Since the behaviour of the orifice probe is well understood. such a probe was considered suited as a calibration instrument. The general relations for impact tubes (Ref. 3) can be solved analytically in the limit as liD --"0. The expressions are

(1) True orifice probe:

.

p~r

--

X(S~o~)-

_is

(\+er.f.

q

(2) Near-orifice probe:

-p.

11.

1-

t

This latter expression has been shown experimentally to be valid in Ref. 10. and was used in the calibration procedure.

The near-orifice probe as developed by Rothe was made in the following manner. A hole 0.052" in diameter was drilled in the side of a length of #15 stainless steel hypodermic tubing (0.072" O. D. x 0.054" 1. D.). This h9le was then filled with Wood's metal and honed smooth. This surface was subsequently covered electrolytically with a thin nickle film (0.00025"). A small hole (0.004" diameter) was then drilled through this nickle plate backed by the Wood's metal. It was then only necessary to remove the Wood's metal by heating (not easily done) to produce a near-orifice probe.

The calculated value of D for this probe was O. 004/0. 00025

=

16. However. microscopic examination of the orifice revealed that a more -realistic value would be in the range 8-.10 due to slight burring of the hole.

4. DESCRIPTION OF PROBE TESTS

Each probe to be tested was c1amped in a vacuum seal onto the gauge volume which could be rotated to allow the angle of attack to be

changed. The tunnel was pumped down and held at high vacuum with an

oil diffusion pump for 48 hours. Since the time involved in changing

probes (which were themselves kept in the surge tank of the tunnel) was

only a minute or so, only a small amount of gas was absorbed by the

tunnel interior during this process. For this reason the tunnel was

outgassed in tWQ days. During this two-day period a solenoid valve was actuated which opened several holes leading to the gauge volume

aiding in more rapid outgassing of the probe. Furthermore, the heat·

from this solenoid raised the probe to a high temperature thus further

increasing the outgassing rate.

Af ter this outgassing period, a check calibration was made of the thermistor gauge, and the (small) outgassing pressure of the probe was recorded for correction purposes. (Typical values are in the range

O. 05 - O. 1 micron Hg.) A mass flow was then pumped through the nozzle

at a desirable set of operating conditions. To allow temperature equilib-rium of the probe, nozzle, and stagnation chamber, this condition was maintained for a period of between four and six hours.

When the two thermocouple elements (Sec. 2.1) in the stagnation chamber indicated the same temperature, the tunnel flow was

considered ready for testing. Gauge volume readings were taken at angles

of attack from 900 _{to 0}0 _{in 10-degree increments, care being taken at}

each position to ensure pressure equilibrium in the probe. The alignment of tb.e probe for zero angle of attack was checked by comparing the

readings at

!

900 _{• During this period MacLeod gauge values of Pstag}

and Pts were frequently recorded, to check on pressure equilibrium and to help eliminate random errors. Finally, a post-check calibration was made.

Each probe was tested in this fashion at several values of

speed ratio, ·subject to the condition that the test section pressure

should be greater thé?ll 6 micron Hg for accuracy and less than 10 micron

Hg to avoid as much as possible deviation from free molecular flow conditions·.

5. THEORY

At zero ar..gle of attack the theoretical calculations of

Ref. 7 were compared. The assumptions made in the theory were stated

in Ref. 7 as follows:

"(1) Free-molecule flow exists throughout the length of the tube (i. e .• the Knudsen number based on the tube

dimensions is approximately five or greater). This

assures that the molecular flows from the free stream

considered independently of each other. Only collisions between molecules and the walls and not between molecules themselves are considered.

(2) The re-emission of the molecules from the wall follows the eosine law of diffuse reflection. such that. af ter the first collision with the wall of the tube. a free -stream molecule has lost its preferred (macroscopie) direction of motion.

(3) On either side of the impact-tube. in the free stream and in the gauge volume. the microscopie random motion of the molecules is Maxwellian.

(4) A state of equilibrium has been reached with respect to the mass flows into and out of the gauge volume. "

For the experimental situation being considered assumptions (3) and (4) are easily satisfied and it is expected that (2) will be valid as well for the low speed ratios involved. The Knudsen number effects referred to in (1) are discussed in the next section.

When the probe is not aligned with the flow. recourse may be had to the theory of Ref. 8. The same assump~ions were made there as in Ref. 7, but the theory was extended to include situations where the probe is at an arbitrary angle of attack to the flow. The solution to the problem is form ulated in Ref. 8 as follows.

Let (3 ) Then

1C

R

IS

D

ei\

=

2!K(4)l

b )

P(4='l~lri)J~

l \

, J

wtD')

o

(4) where (5 )

K(,p,D)

=

a.(\))+:.7r

(1-<..c:L(I))J )

tI>l)

\( Up,

\»)

=

I

+

~

[\ -

2"-

t {))

J

1

1-

(1-

T '")

~1.}

(IaJ

- i:'

L1-

a..({)~[\/I-T1. ~

5'''-'

Tl

(T<:I)

1t

+

~s,,,+U)o;+ 5GO$Q.e-5"~'"':O.é+S~D.XI'"

erf

Sc.nsQ),U

where

a(D)

=

a function derived by Clausing (Ref. 11)

T

=

tan0

D

cosU

=

cos OCcos0

+

sin~

sin0 cose

.

(8)

For the present purposes, the most significant result of the above analysis is that the probe gauge volume registers a pressure which is less than the pressure it would register if the probe were sensitive only to the axial component of the speed ratio. That is

R(S, D,

ct)

<

R(Scosot, D, 0), ( ~>o) (9)

In particular, when the probe is at right angles to the flow,

R (S, D,

~)

<

1 (10)

since R (0, D, 0)

=

1. For low speed ratios and short tubes the approxi-mation

R (S, D,

<X )

R (Scos OG, D, 0) (11)

is not seriously in error.

6. ANALYSIS OF EXPERIMENT AL ERROR

Although every opportunity was taken to eliminate random errors, this type of error persisted to some degree together with the more subtle systematic errors. The important sources of these errors are listed here:

detail:

(1) Errors in estimation of D for the near- orifice probe used in the nozzle calibration.

(2) Uncertainty in D of probe under investigation. (3) Errors in MacLeod readings of Pstag and Pts. (4) Outgassing errors in test probes.

(5) Errors in thermistor readings of gauge volume pressure. (6) Knudsen number effects.

These sources of error are now discussed in some

(1) Since the hole in the calibration probe was slightly burred, some doubt exists as to its effective D. Fortunately, as shown in Appendix C, the effects of relatively large uncertainty in D are small for the speed ratios under consideration. This error is a systematic one.

(2) For each of the probes under investigation the methods used by the precision machinist in their manufacture assured that only small errors existed in their values of D.

· (3) Random errors in P stag and Pts are considered in Appendix C where it is found that, at S

=

1, a 1/2% error is

intro-duced into the value of the speed ratio for each 1% error in these quantities. (4) As pointed out in Sec. ·4, the outgassing error was determined at the beginning of each test and taken into account. However a tendency was observed for the probe to absorb gas during the higher gauge volume pressure levels (say at 50 microns Hg). For this reason, the angle of attack on a particular run was set at 900 _{and then readings}

taken as the probe was rotated toward the 00 position.

(5) Check calibrations before and af ter runs agreed so c10sely with each other and with the calibrations of other runs that they will be considered accurate in this discus sion.

(6) To achieve free molecular conditions it is neces-sary to have the local mean free path several times larger than a signifi-cant local probe dimension. This is equivalent to insisting on Knudsen numbers, Kn >5 say. It is clear from Table 1 that the Knudsen numbers based on the free stream mean free path and impact tube diameter are more than adequate. The case which deserves closest scrutiny occurs when the longest tube (D = 0.01) faces the highest speed flow (S = 1) at zero angle of attack. Since the pressure in the gauge volume under these circumstances is about five times the free stream pressure, the Knudsen

number(based again on tube diameter) at the high pressure end of the impact tube is approximately Kn

=

6.

Furthermore the pressure ratio in this case is quite sensitive to the number of free stream molecules which experience their first collision with the tube wall deep inside the tube. One should there-fore bear in mind the possibility of intermolecular collisions for these molecules. Since the local mean free path varies by a factor of five as these molecules travel down the tube, it is difficult to assign a single Knudsen number.

From the discus sion above it is evident that free mole-cular conditions exist in the probe, except perhaps for the longest tube at zero angle of attack and S

=

1. In this case some slight transition effects might be present.

None of the errors discussed above are large. As a summary, the errors due to causes (1) to (5) are tabulated in Table II fromthe relations derived in Appendix C.

...

7. DISCUSSION OF RESULTS

The results obtained as described in Sec. 4 are now compared with the theory outlined in Sec. 5, bearing in mind the limitations

on experimental accuracy.

7. 1 Results at Zero Angle of Attack

FtfTi

In Figs. 10 to 19, the measured values of

p

1T2

for the various probe geometries are plotted as a function of S

~nd

2 compared with the theory of Ref. 7. It is clear that within the experi-mental accuracy good agreement between the two is found and it can be said that there is no experimental evidence here at all to make the theory suspect.

Figure 20 contains a plot of the ratio of impact tube

pressure te> the pressure measured by aJl orifice probe at the same con-ditions. This is done by dividing

~'qï

by

-x.

(S).

This plot eliminates

1112

the basic dependence on speed ratio and effectively shows the variation

with D. A plot of this nature has a tendency to emphasize the experimental errors since it does not directly involve the quantities actually measured,

but it still shows general tendencies in agreement with theory.

7.2 Results at Non-Zero Angle of Attack

Analysis of the results at non-zero angles of attack show that the probe pressure ratio is consistently lower than the valu.e expected based on the normal component of the speed ratio (see Sec. 5).

The size of this decrement tended to lllcrease with speed ratio and decrease

with D; not only was this effect consistently in the same direction, but was also of such a magnitude, .particularly for the longer tubes, that it would be difficult indeed to explain it away as being experimental error.

In Figs. 21 to 30 the experimental results are com-pared with the theory of Ref. 8. Although there is a tendency for the experimental points to be somewhat high for 400

_<

_Q!,

_<

_{700 , the data shows}

general agreement with theory. There is no doubt that the probe pressure ratio at

ct

=

900 is below unity.

A qualitative explanation of this result is as follows. Suppose a long tube is placed at right angles to a high speed free molecule flow. It is clear that by far the majority of the molecules entering the

tube arrive at the wall very near the entrance where they have but a small chance of eventually passing down the tube. On the other hand, the gas in

the gauge volume is at rest; a large number of these molecules therefore penetrate weU down the tube to make their first collision with the wall where there is a relatively higher probability of escape from the probe.

In this manner the gauge pressure reaehes an equilibrium value whieh is lower than the free stream statie pressure.

8. CONCL USIONS

For subsonie flows the theoretieal calculations of Refs. 7 and 8 have been sub stantiated in the laboratory for a wide range of impact tube lengths. However data at higher speed ratios are needed to study eompletely the speed range of interest.

It appears from the discussion of the results obtained when the probe was at an angle of attaek,

ol. ,

that the probe pressure ratio can be approximated by assuming that the probe is at zero angle of attaek and in a flow having the speed ratio S cos

ot.

This approximation win yield a value whieh is too high however, and win be more in error as either the speed ratio or the tube length/diameter ratio is inereased.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Harris, E. L. Enkenhus, K. R. Harris, E. L. Patterson, G. N. Patterson, G. N. Enkenhus, K. R. Harris, E. L. Patterson, G. N. Enkenhus, K. R. Harris, E. L. Patterson, G. N. deLeeuw, J. H. Rothe, D. E. Hughes, P. C. Enkenhus, K. R. Harris, E. L. Clausing, P. Sreekanth, A. K. REFERENCES

Investigation of the Time Response and Out-gassing Effects of Pressure Probes in Free-Molecule Flow. UTIA Tech. Note No. 6, October 1955.

Pressure Probes at Very Low Density. UTIA ·

Report No. 43, January 1957.

Properties of Impact Pressure Probes in Free-Molecule Flow. UTIA Report No. 52, April 1958. Theory of Free-Molecule, Orifice-Type Pressure Probes in Isentropic and Non-Isentropic Flows. UTIA Report No. 41 (Revised), April 1959. Pressure Probes in Free-Molecule Flow. UTIA Report No. 62, June 1959.

Pressure Probes in Free-Molecule Flow. UTIA Review No. 19, July 1961.

A Numerical Solution for the Free- Molecule Impact Pressure Probe Relations for Tubes of Arbitrary Length. UTIA Report No. 88,

December 1962.

Theory for the Free Molecule Impact Probe at an Arhitrary Angle of Attack. UTIAS Report No. 103, June 1965.

The Design, Op er ation, and Instrumentation of the UTIA Low Density Wind Tunnel.

UTIA Report No. 44, June 1957.

Investigation of Free- Molecule and Transition Flows Near the Leading Edge of a Flat Plate. UTIA Report No. 53, November 1958.

Ueber die Stroemung oehr verduennter Gase durch Roehren von beliebiger Laenge AP..n. der Physik, Vol. 12, 1932, p. 961.

Drag Measurements on Circular Cylinders and Spheres in the Transition Regime at a Mach Number 0f 2. UTIA Report No. 74, April 1961.

APPENDIX A

In. this appendix it is shown how small flow angles of incli-nation to a reference direction can be measured with the use of a

near-orifice pressure probe.

Suppose the angles 9,

€ , ei

be as defined in Figure 8a

(longtube probe shown). Then it is c1ear that

(A. 1)

, We shall also require the following relations which can be verified in several ways:

l6

c::c::

I)

erf

~

=-

~

S

(A-2)

~

( 7-

s"-)

'l.

lUl

[sc \

-+

<0)

j

=

~r

f

S

-+

r;

5

e -

6

_(A.~)

'X..(è)

~

\

+

{tr

6

+

'blo

'X. tese \

+

8)1

-=

t (

s)

-t

{i

S (

I

+

.(Lr

f

s)

~,.

(a) Calculations for a True Orifice Probe

We have, for a true orifice probe, (Ref. 4)

P"L(I1)E)jl;

=

X

(S~(e+6))

Ft

Tl.

(A.6)

Now suppose readings are taken at values of 00, +90°,

and -90°. Then

(A,,)

The quantities of interest are Rl and R2, defined as

•

)

(A,

11)

Then, from (A. 7), (A. 8), and (A. 9)

D

=

X(SC.OSE)

(~

)

~.

\"3) 2

Realizing that to second order sin

~

= E and cos é = 1 -

~

we may make use of Eqs. (A. 12) and (A. 13) to obtain the relations

Z

f\ ,

V -(

,

\~.

Il..

~

~ -SSII"l~)

)

R\

=

'X,(S)ll+f1r

Se

+(1ï-i)S~~J-tfS(I+e.r-f5)é2.

R'l...

= yeS)

[I-..JT Sf. ...

(Ii-I)

s"'E}4

s(

I H r

f-'S

}€'

from which

R \

~ R

1.

=

2.

V;

s

t(

s)

€

Similarly it may be shown,that

Note that to first order,

0·'4-)

(1\.\Sj

~.I'J

from which

5

can be found very simply if a curve of

X-(ç)

is available. The value of

é

can then be found to first order from (A. 16):

é

=

R~-R~

(b) Calculations for a Near-Orifice Probe

Instead of Eq. (A. 6), we must write (Ref. 6)

R(e

l

E)R

1

=

...h

'CS

(

)~_ x.[SCc~{~+G)J-tLH-~r{5lOS{~+~~

D

T

'-t"L,.

l

~ö"5 &+~~

-

J)(A

2-0\

\ \ 1.. \ -

J..

·

')

D

In a manner exactly paral.leling the treatment for a true orifice probe we must culculate

When this is done it is found that

and

- pLolS) ( \-

S>'."LJ -

~

S

(I+.er

t

S) E ....

-t

4t(~) (-~)(I+i

+

2'{t-)

lf..11,.)

Note that to first order

from which

5

Cfu"1 readily be found if a curve of

~1.l.S)

is available.

value of , can then be found from

The

If G turns out to be between 50 and 150 _{greater accuracy}

can be obtained by applying an iterative procedure to Eqs. (A. 16a) and

APPENDIX B

Details are given in this appendix concerning the calcu-lation of the molecular mean free path as a function of static pressure

and speed ratio.

In Ref. 12 it is shown that, for air,

where a

=

5.328 X 10- 3, where b

=

210. 6 for air.

Á

is in inches P in microns Hg T is in oR

We may write for adiabatic conditions

[

y-\

T=

Tc

\+

r

A=-Substituting (B. 2) in (B. 1) we obtain

À

=

I

P

To+

b

a·:t'1

)

(B.l)

(6.2.)

(8."3)

c=

b('YY-a.. Tc'l.

Now = 1. 4 for air and to was maintained throughout at 5450 _{R; hence the numerical values of A, B, and Care}

A = 0.478

B = 0.174 C

=

0.011

APPENDIX C

In the following it is shown how estimates of the relative errors in speed ratio and

~

J~

can be deduced by tracing through the calculations to find the effects on these quantities of inaccuracies in the basic measured quantitie s.

As discussed in Sec. 6, the errors are largely due to the following:

(1)

~

=

relative random error in test section pressure.

(2)

€~

= relative random error in stagnation chamber pressure. (3) ~ ...

=

relative systematic error in the estimate of the

effective value of D for the near- orifice probe used in nozzle calibration.

(1) Relative Errors in Determination of Nozz1e Characteristic s

Let us first calculate the relative error in the values of S (6os,c)

used to obtain the characteristic curves (Fig. 9). The quantity S was measured by taking the ratio of two gauge volume pressure readings, one at zero angle of attack and the other at 900 .

so that

Again from reference 6 we have for a near- orifice probe

X

(Scos

fi) -

i

(I

+

~rf Sc.at:l~)

l -

-L

D

X(s) -

is (

1-+

~rf5)

1_

-L

o

(C.I)

We may assume that the two gauge volume pressure readings are (relatively) accurate so that the assumed -D error ~ results directly in the error in S 6~\c. as follows:

tlS) -

MI+e.rf:L

X:[S(,+G.jl-

15

(1+

EL)[\+.e..rt {

"'{H.€ ..

~~

(C.~)

To first order this reduces to (see Appendix A, Eqs. A.2, A.3, A.4, A.5)

(e.f)

For the values of S and D used, (S ~ 1, D

=

10) this can be approximated by

.

I.e.

_(C.s)

As discussed in Sec. 3.2 almost all the errors in the

characteristics are due to

6:.

_{IC since the errors in P ts and Pstag have been} largely removed. Now to a good degree of approximation the curves of Fig. 9 can be represented as a family of straight lines given by

where a = 0.59

and b

=

1. 70 microns Hg, and where the pressures are in microns Hg. The errors in the characteristics may now be dealt with in terms of relative errors in a and b (

Eo..

and

Gb

respectively). We ean

therefore write that

from whieh

,

a.

€~

R.+o.)

-t

b

~.b

a.'f;-\-.".,

+

b

(2) Final Relative Error in S

We arrive at a value of S by using values of P ts and P stag

(having errors

EI

and G~ respectively) in conjunction with a set of

curves which themselves have errors(

ea.

and ~b)' It is clear then that the resulting ·error in S must be found from

~

(l~ê

... )

-

R-~ (I~()

~(1+4

")

~+co CI+~

... )

-t

~C,+G,)

Upon _{employing Eq. (C. 6) to eliminate P stag and Eq. (C. 8) to eliminate}

e:~ and GA:. ' this reduces to

where

iEl

If we now assume that D

=

10, a

=

O. 59, b

=

1. 70 and P_ts

=

lOt- '

Eq. (C. 11) becomes

=

-1,~2.

(1-

O.~9S)

El

S

~.IOJ

(c.\\ )

(c.

12.) EL =

O.O~b4

P 2

/Ti

(3) Relative Error in

Pï

V

T2

P

ff2

To e stimate the relative error in P

~

f

~

we as sume that P2 is relatively accurate and that T2 is very nearly T 0 (as

dis-cussed in Sec. 4). Then employing the adiabatic law for total temperature

-:::. e

-t (

2.S4. \

tE

I

1+

_2s

'a.)

S

P2

ffi

(4) Using the above relations, the maximum errors in S and P1

1

1'2

Probe No. 1 2 3 4 5 6 7 8 9 10 Note:

.

.

TABLE I Probe Data

t,

Impact Tube d, Impact Tube

D d Knd

Length, inches I. D., inches

=l

0.600 0.006 0.01 33.3 0.060 0.006 O. 10 33. 3 0.013 0.004 0.31 50 0.008 0.004 0.50 50 0.006 0.004 0.67 50 0.006 0.006 1.0 33. 3 0.005 0.006 1.2 33. 3 0.004 0.006 1.5 33. 3 0.003 0.006 2.0 33.3 0.002 0.006 3.0 33.3

The Knudsen number is based on the impact tube diameter and a nomina! statie pressure of 10 microns, i. e., a mean free path of O. 2 inches .

TABLE II

P2

[T;.

Relative Errors in S and

Pi

1/

T2

Assume that the _{relative errors in P ts' Pstag' and D are}

E'

1 =

+

1%. €:

2 = :: 2'}'.. and

G

I. =

+

20% T7eCtiVelY. Then the

-

P

T

relative areas in thefinal values of

S

and

Pi

T~

are given by Eqs.

(C. 10) and (C. 13).

S Relativë Error in_ S

(%)

Relative Er-ror in P 2 _P1

%

0. 2 23.44 1. 26 0.3 14.96 1. 38 0.4 10.70 1. 47 0.5 8. 14 1. 54 0.6 6.42 1. 60 0. 7 5. 18 1. 64 0.8 4.25 1. 66 0. 9 3.52 1. 66 1.0 2.93 1. 65

I

-CD

-.... "-\ \ \ \ t - - - t - - + _ / ' . /

--/

I

I / / /

9/32" DRILL. HOLES SPACED 45° APART ON 13.25" P.C.D. = _IC) I~ _ _ _ _

o_~

L

= in -- +

-FIG. I

MACHINE NOZZLE FROM BRASS CASTING.

SMALL VOIDS VISIBLE AT THE MACHINED SURFACE ARE PERMISSIBLE EXCEPT IF THEY OCCUR ALONG THE INTERN AL CONTOUR A-B-C.

Nen

-v

2. TOLERANCES: AIDECIMAL FRACTIONS' ±.OO3" BI COMMON FRACTIONS .! .010·

FIG. 1 THE SUBSONIC NOZZLE

/7777777777A ..-7

o.

0 o -IN """. ~ CD v

J

/ ---L -

----+---1

SECTION

''Ä'

J r ,

~

~.

11

I

_..

_I

~

IJ

I

11 :a

~

2

~

•

I

Note: All Test Probes are identical

in construction except for

Section

1'Ä

1

(see next figure)

FIG. 2

TEST

PROBE

g

co

(0

I :1 'Hord Sol der

I I

- - ' -

---~--,----

---I

: :I-Stoinless Steel Hypodermie Tubing (0.006" I.DJ

I I

j

'~-~tt-

----:1

t--- -- -- --

-j

L-_ ... _ - - ' _ 'Hord Solder Build-up

W!7u~- --

-,J - -- ---

=

=

=

=

~

,- ---

~

-- -

.1

'-Simple Hole

tlffiL" -

. ..."

- - -

- -

~=-=-- ~ ~ @~~u..

J

\

\ {--

-

-Tube Wall Honed Tube Woll Flottened

FIG. 3

CONSTRUCTION DETAILS

OF

SECTION

"Ä

1

I Ol

bra ss tube

("2

l'O')

dummy gauge plexiglass endpiece to bring out electrical leads plexiglass spacer brass fitting to connect to vacuum system • 0" ring

FIG. 4

DETAILS OF THERMISTOR GAUGE

FIG. 4 DETAILS OF THERMISTOR GAUGE

FIG. 5

dummy

thermistor

to millivolt t-""\:~----4 recorder

I---+-___

~~~I-x ~

I

0·8

x ...

x

""

~

~-~--+---+---______4

~5

boosters

~--+---+ ~

=

4·

21

J-L

Hg

- - - 4 0·7~----+---r----~---+---r---rx-~

4 boosters

_ _ _ ~O._~ ~ I r

0

'6~-~----+---=--=+==i--+--

p.

=

2 ·65

LL

Hg

Y--

I

e - -

-.-+ \

~

I

ro-'~---.I

1 - - - + - - - 1

en

0.5

~----+---+'\3

boosters

+~~

' "

o

_{J - - - + - - - + .J}

p.

=

4·

41

J-L

Hg

_{- "'---i} ~

1'0

<t

~0·4r---+---r----~---~---r---;-~+~--~

o

w

w

CL

en

0.3r---+---+----~~----+---~----~---4 ~

<t

....J

::>

~0·2r---+---+----~r---+---r---;---4

....J

o

~ 0·1r---+---+----~f---+---~----~---4

o

_1·0

_2·0

_3·0

INCHES DOWNSTREAM FROM NOZZLE EXIT PLANE

FIG. 6

0.7

o

:g

0:: -0 ~

0.6

a. (f)

..

(f)

0.5

0.4

0.3

0.2

0.1

o

r....fl)....

y

_.

7

I

J

,-\

.1

_I

~

c·

7-

.

'f

..

-4

-3

-2

-I

_~

1

2

3

₄

_r

~

I

~

.

,

I

Nozzle Inches Wall I

FIG. 7

Probe

Reference

Direction

a=8+E

FIG. 8a

-. Flow

Direction

Sketch Showing Angle Definitions

FIG. Ba SKETCH SHOWING ANGLE DEFINITIONS

~

_I

E,deg

1

t

~

10

1--1 (. i--

.

/

~

-

5

j / V

.

---

-

0

/ / 1

-~

.V

, -

-5

~

,~

1 -

-10

1

1

-3

-2

-I

o

₂

₃

_r,

_inches

FIG. Sb

:1..

30

•

cu

~p. en t. en

cu

~ 0..

2 5

~- ~---+- - - - + - 0 4 -U

-

en

cu

t-15

Ior---+ --~

5

I o . : : - - - - I ' _ _ + _ - +

-0.4

0.6

0.8

FIG. 9

1.0

S, Speed Ratio

1.2

lifT;

~1j1;

5.0

4.5

4.0

:3.5

:3.0

2.5

2.0

1.5

1.0

0.5

0

0

A

Y

.) Theory

Measured Values of

Probe Pressure Ratio

vs.

S

(a

=

0°)

0=0.01

0.2

0.4

0.6

0.8

1.0

L2

S,

Speed Ratio

v

4.5

4.0

0

3.5

3.0

Theory

2.5

2.0

1.5

Meosured Volues of

Probe Pressure Rot

i

0

vs.

S

(a =0°)

1.0

0=0.10

0.5

o

0

₀₂

_0.4

_0.6

_0.8

_1.0

_1.2

S,

Speed Ratio

FIG. 11

0

~

5

4.

0

3.5

3.0

Theory

2.

5

2.

0 1.5

V

Meosured Volues of

Probe Pressure Ratio

vs.

S

(a

=

0°)

t.O

0=0.308

O.

5

o

o

. Q2

_0.4

_0.6

_0.8

_1.0

_1.2

S,

Speed Ratio

FIG.

12

J

4.5

0

4.0

:')

3.5

3.0

Theory

2.5

2.0

1.5

Measured Values of

Probe Pressure Ratio

1.0

_vs.

_S

_(a

₌

_0°)

0=0.50

0.5

00

_0.2

_0.4

_0.6

0.8

1.0

1.2

S,

Speed Ratio

FIG.

13

IJ

4·0

0 3.5 3.0

2.5

v

7

2.0

1.5 1.0

1/

Meosured Volues of

Probe Pressure Ratio

vs.

S

<a =

_0°)

0=0.67

0.5

o

o

0.2

0.4

0.6

0.8

1.0

12

S,

Speed Ratio

FIG.

14

0 0

4.0

3.5

10

3.0

Theory

2.5

2.0

/l

Measured Va lues

of

Probe Pressure Ratio

vs.

S

(a

=

OO)

1.5 1.0

0=1.0

0.5

0.2

0.4

0.6

0.8

1.0

1.2.

S,

Speed Ratio

FIG.

15

.0

!5

4.

0

3.

5

"

,-3.

2.5 2

_"

1.5

1.0

V

.5

o

o

o

0.2

/

'/

!

V

/

17

Measured Values of

Probe Pressure. Ratio

vs.

S

(0

=

(0)

0= 1.20

0.4

0.6

0.8

1.0

1.2

S,

Speed Ratio

5.0

4.5

F(

/f

~1T;4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

o

o

0.2

0 0 The~

Measured Values of

Probe Pressure Ratio

vs.

S

(0=

0°)

0=

1.50

0.4

0.6

0.8

1.0

1.2

5,

Speed Ratio

5.0

Firr;

Fi,T;

_4.5

4D

/i

.

3.5

0

30

Theory

2.5

2.0

1.5

Measured Values of

Probe Pressure Ratio

ID

vs.

S

(a

=

0°)

0= 2.0

0.5

Q2

0.4

0.6

0.8

1.0

1.2

S,

Speed Ratio

FIG.

Ie

4.0

3.5

3.0

2.5

2.0

1.5

1.0

V

0.5

o

o

Q2

-0 0 0 Theory .

Meosured Va lues of

Probe Pressure Ratio

vs.

S

(a

= 0°)

0=3.0

0.4

0.6

0.8

1.0

1.2

5,

Speed Ratio

f1,OI'ifice FIG. 20 1.40r---...---~---,.---.,..---..,.---""""

lid

~

1.35 .... - -... - - - -...

- - - - w - - - t - - - j

1.30

1.25

1.20

Comporison of Experimentol

Results ot Zero Angle of

Aftoek (nondimensionalized

With Respect

fo the Orifice

Probe Volues) with Theory.

lId

lId

100

®

<I>

I.

0

10

₀

11

0.83

3.3 CD

~

0.67

2D~

00.50

1.5

<lI>

00.33

0

1.15 t - - - f - - - - + - - - I - I ./-/r-/~

1.051---0.2

0.4

0.6

FIG. 20

cv

0

0.8

1.0

S,

Speed Ratio

1.2

5.0

4.5

~!+.

Probe Pressure Ratio vs.

a

(0= 0.01)

lf

T_{z Gl}

_s=

_1.048 4.0

à

0.961

0

0.795 EI 0.585 0 0.434 3.5 3.0

2.5

2.0 1.5 1.0 0.5

o

10 20 30 40 50 60 70 80 90

a

degrees FIG.21

5.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5

o

10 20 30 40

Probe Pressure Ratio

vs.

a (

0= O. 10) Q S= 1.036

à

0.802

o

0.573 El 0.4 17

o

0.226 50 60 70

a

degrees FI G.22 80 90

5.0

4.5

~K

Probe Pressure Ratio vs. 0 ( 0=0.308)

~ T2 Ó 5= 1.024 4.0 (;) 0.955 8 0.813

0

0.639 EI 0.456 3.5 0 0.213 3.0 2.5 2.0 1.5 1.0 0.5

o

10 20 30 40 50 60

70

80 90 Odegrees FIG.23

5.0

4.5

Probe Pressure Ratio

!ik

vs.

a

(0=0.50) ~ T₂

<:>

S = 1.02 I 4.0 G) 0.91 I 8 0.798

0

0.626 El 0.441 3.5 0 0.242 3.0 2.5 2.0 1.5 1.0 0.5

o

10 20 30 40 50 60 70 80 90

a

degrees FIG.24

5.0

3.5

3.0 2.5 2.0 1.5 1.0

0.5

o

10 20 30 40

50

FIG.25 Probe

vs.

a

(;) 8

o

I!J

o

Pressure Ratio (D= 0.67) S= 1.026 0.941

0.800

0

.

594

0.441 60 70

80

a

degrees

90

5.0

4.5

~Ji

Probe Pressure Ratio vs.

a

(0= 1.00)

P,

T₂ 8 S = 1.030 4.0

0

0.932 El 0.801 0 . 0.464 3.5 3.0 2.5 2.0 1.5 1.0 05

o

10 20 30 40 50 60 70 80 90

a

degrees FIG.26

5.0

4.5

Probe Pressure Ratio

Po[1.

vs.

a

(D = I. 20) ~ T₂ 8 S = 1.035 4.0 <i) 0.923 ~ 0.812 El 0.604 0 0.423 3.5 3.0 2.5 2.0 1.5 1.0

05

o

10 20 30 40 50 60 70 80 90

a

degrees FIG.27

5.0

4.5

Po/+.

Probe Pressure Ratio

vs.

a

(0= 1.50) ~ T₂ 8 s= 1.058 4.0 c:l 0.952 <:;> _OB20 El 0.652 0 0.454

3.5

3.0 2.5 2.0 1.5 1.0 . ("I . 0.5 ", ...

o

la

20 30 40 50 60 70 80 90

a

degrees FIG.28

5.0

4.5

Prebe Pressure Ratio

Por

vs.

a

(0=2.00)

Ft

T2

<:>

S = 1.025 4.0 t!l 0.917 <::> 0.783 Q _0.638 El 0.437 3.5 0 0.244 3.0 2.5 2.0 1.5 1.0 0.5

o

10 20 30 40 50 60 70 80 90

a

degrees FIG.29

5.0

4.5

Probe Pressure Ratio

Por

vs.

a

(0=3.00)

f\

T2

<:>

5= 1.01 2 4.0 c:l 0.931

A

0.792

0

0.638 0 0.449 3.5 0 0.236 3.0 2.5 2.0 1.5 1.0 0.5

o

10 20 30 40 50 60 70 80 90

a

degrees FIG.30