Temperature response of a hot-wire anemometer to shock and rarefaction waves

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m Ä l O (,1)0 m() C Äl~ »om "tl Z _-t "tl»o O::! Äl m ' - t m Z

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5

TEMPERATURE RESPONSE OF A HOT-WlRE ANEMOMETER TO SHOCK AND RAREFACTION WAVES

BY '

S. Go DATAR

TEMPERATURE RESPONSE OF A HOT-WIRE ANEMOMETER TO SHOCK AND RAREFACTION WAVES

BY

S. G. DATAR

ACKNOWLEDGEMENT

The author wishes to express his sincere appreciation to Dr 0 G.N. Patterson for providing the opportunity to pursue this investigation.

The author is also greatly indebted to Dr. 1.1. Glass and Dr 0 1.J 0 Billington for their supervision of the project and

construct-ive criticism during the progress of the work 0

Thanks are also due to Dr 0 1.J 0 Billington for his assistance in conducting experiments 0

The financial assistance received from the Defence

Research Board of Canada and the Office of Naval Research (U 0 S oN.)

is gratefully acknowledged 0

.

'

·SUMMARY

The use of the hot-wire anemometer in the study of tem-peratures behind the incident and reflected shock waves and incident and reflected rarefaction waves is described . The measured tem-peratures in both cases show good agreement with theory at low diaphragm pressure ratios. The agreement is -better for incident. waves than for reflected waves. The dis agreement progressively increases with increasing diaphragm press ure ratios .

The results indicate that the recovery temperature behind shock waves is lower than that predicted by perfect flow theory. while in the case of rarefaction waves it is higher. In the latter case. if the actual measured piezo-pressur~ profile through·the rarefaction wave is used to determine the flow tempèrature. then the measured recovery temperature is in good agreement with the cornputed values .

,

( i ) TABLE OF CONTENTS NOTATION I. 1I. IN TRODUC TION I THEORETICAL RE LA TIONS

2. 1 Shock and Rarefaction Waves 2.2 Hot-Wire Anemometer Page ii 1 2 2 6

lIl. EXPERIMENTAL EQUIPMENT AND PROCEDURE: 12 3.1 3 inch x 3 inch Shock Tube

3.2 Rot-Wire Anemometer Set

IV ~ HOT-WIRE OSCILLOSCOPE RESULTS FOR SHOCK

12 12

AND RAREFACTION WAVES 14

V. DISCUSSION AND CONCLUSIONS 17

REFERENCES 19

( ii ) NOTATION For Shock Tube Flows

a speed of sound

cp specific heat at constant pressure Cv specific heat at constant volurn e e internal energy =

C,,"T

T statie ternperature

.

'

_{T o stagnation ternperature} K thermal conductivity p pressure L length of charnber t time u particle velocity x distance

~ ratio of specific heats cp/cv

e

de:nsity

Dimensionless _. Ratios

A··

=

O::~. E·· = ei:.

IJ (). ... d IJ

-e:-tt

.!~L. x. Mi = _{(1 . N} =

o...

....

t:

-

... p .. ₌

i;-

_T··

₌

TL

l.J IJ T,' ct Uij

=

-1~ _O-~ T _OlJ ., ::: Toe: _T'

6'

0(.

₌

0;.+1

~i

- èJ',( -I 1 ~ ~ l

ï'

ij = ~L

_e·

f\

_IJ

..

-

ei

_{e~ \A.~} Ll.~ _~

d-I

= OAt.

X

:::

:x:

'-.

-C-

L

( iii )

For Hot-Wire Anemometer

C_w wire specific heat

d wire diameter

e v0ltage across wire

I wire heating current

l

wire length

Ro wire resistance (unheated)

Rs impedance of heating current source Rw wire resistance

Te wire equilibrium temperature (unheated) T

_w

wire temperature (heated)

Uo. 10' n wire calibration constants

0( _{temperature coefficient of resistance}

6h rneasured oscHloscope trace jump

À finite circuit correction factor

Special Symbols

--

R

-

S

-

R

--

S

c

ê

represent forward facing rarefaëtion wave and shock wave

respectively in which particles enter from right to left represent ha.c~war(l:facing rarefaction wave. j afld Ijpegk wave: respectively in which the partieles enter fraro left

to right

represent contact regiQnl3 trij,vellillg

te

the left

a.nd rilht

respective ly ,

'

.

( 1 ) 1. INTRODUCTION

The hot-wire anemometer has found its widest application as a valuable tool in turbulence research both at low and high speeds (Refs. 1. 2 and 3). The hot wire has also been extensively used for the measurement of mean flow velocities in pipe flow s and flows from jets and orifices. The hot wire senses those changes in the dynamic and thermodynamic pruperties that affect changes in the rate of heat transfer between the flow and the wire. These in turn alter the

resistance of the wire which ean be recorded as a measurable electrical signal.

In recent years attempts have been made to apply the ho

t-wire technique to studies of one-dimensional unsteady flows and the transient wave phenomena in shock tubes (Rei". 4). Shock tubes pro -vide a very convenient means of producing transient flows of very

short duration with extreme variation in flow properties like press ure. temperature and density (Ref. 5). The s udden' removal of the diaphragm instantly sets up ÎloV{ fields of short duration

w

ith different

thermo-dynamic properties . It therefore becomes necessary to know the response of t11e hot-wire to a step function change in flow conditions if it is to be used as a flow probe. When a shock wave passes over the

wire mounted at some convenient distance from the origin of the wave. it experiences a sudden change in mass flow. pressure • density and

temperature and the flow remains uniform for a short interval until it

is disturbed by another front like a reflected shock or a rarefaction 'I

wave or a contact front. /

In practice the hot-wire can be operated as a constant

temperature or constant current instrument. In the former case the wire resistance is held constant at a known finite value and the heating current is altered to compensate for the variations in flow parame.,ters while in the latter the current is held constant at a known value and the resistance is allowed to vary.

The wire response to the applied step function is not ins

tan-taneous but ta,kes a finite time before coming to equilibrium with the surroundings because of the therrnal lag of the wire. The response is

usually exponential in nature and is a function of mass and thermal capacity of the wire., The effect of thermal lag is less pronounced if wires of small diameters are used but the high mortality of such wires sets some limitation on their use.

This nate gives the results of a study of the temperature response of hot wires in the regions behind incident and reflected shock waves and incident and reflected rarefaction waves , The orig-inal purpose wá.s to investigate the feasibility of using hot wires in the measurement of temperatures behind spherical shock ·waves generated by explosion of pressurized glass spheres. In such an application the chances of a wire surviving two runs. needed for the calculation of

( 2 )

both mass flow and stagnation ternperature, would be very remote. Since the literature leaves some doubt about the validity of the stan-'dard hot-wire heát 10ss equation when applied'to qperation at very

low Reynolds number I some preliminary checks of w ire response in

regions of ~ero mass flow appeared desirable'. Äs a syart, it was decided to make a few meas rements on the behaviour of hot,-Wires in the 'regions behind shocks reflected from tlie clos~d énd of a shock tube, where there is no mass flow and where one run would be enough to evaluate the temperature. Since the res uits did tiot ~eem encour-aging, a further investigation was made both fn incident and reflected shock wave and rarefaction wave regioml, with particular emphasis on the temperature response of the wires.

Il. THEORETICAlj RESULTS

2.1 Shock and Rar~faction Waves

-, The U1:eory of one -:&îrnensional unsteady flow has been treated in detail in Ref. 5 and the applications of the theory to shock-tube floVis have been given in R ef. 6 .

.' Brief).y, the simple wave model and the associated flow regions af ter the rupture of diaphragm at x = ,0 are shown in Fig. 1. The important _, components of the flow are as foilows:

-

--a) A centred rarefaction wave ( R ) (principal rarefaction wave~ rnoving to the left in the compression chamber at the local speed of sound setting in motion the gas orig-inally at rest in the chamber.

-b) A shock wave ( S ) (principal or priI!lary shock) moving ; at constant speed to the right setting in motion the gas , originaJ.1y at rest in the channel.

c)

A

contact surface (which is the boundary between the masses of the gases which were originally separated by the diaphragm) moving te' the right ,behind the shock in the chamber. The velocity of the contact surface is theoret-'

ically equal to the local partiele speed.

-c;i) A, shock ( S ) reflected from the far end of the closed tube " travelling upstream to the left. If the end is open a,

reflected rarefaction wave travels upstream for the range of shock strengths and initial pressures used in t-h.is

investigation.

-e) A 'rarefactlon wave ( R ) reflected fràm the far end of the chamber travelling downstream to the right. '

( 3 )

Wave fronts arising from the refraction at the contact surface have been omitted since they' do not appear in the present investigátion. The quantitative value~ of various thermodynarnic parameters for various flow fields requiredfor the determination of the re,sponse of hot·wires is preserited here without detailed derivations .'

(Ref. '6).

Prirnary Shqck Wave

The region between the shock wave and the contact surface represents state 2. (Fig. 1). The state 1 in fr-ont of the prirnary shock is at rest. The shock strength P21 is given by the Rankine-Hugoniot relation

where o(-=. ~- I 0+1

( 1 )

A convenient form of the relation between ~hock strength P21 and diaphragm pressure ratio P 41 for the purposes of platting is

R

4

=

_1-

~

-

(Pl., - \)

J

..

(34

EI4

JY~4

(

2 )

PZ1

L

o<,P2..,+1

e

where E14

=--d-and charnber

4-is the internal energy ratio of the gas in the channel respectively.

(i

=

i, 2. 3 etc.)

-'

A plot of P 14 against P21 for lower shock strengths is shown in Fig. 2.

( 3 ) since u.f = 0 ( 4 ) Density Ratio ~ =

fa

2.1 f', ( 5 )

Speed of Sound and Ternperature Ratio

A

=

O-z.

=

[12

j

Y2..

-::"t

Y2. _

~H.,

(0<,

-t "Pz.1

~

Y2.

21 0-, T. Z.I - I -t 0( TI

I 1'.2..1

( 4 )

Particle Velocity or ,Contact Surface 'Velocity

U

= ~ -

--;=:B;;:-2.,'-:--_'_----,,=;-,

2.1 a.

I -

~I[f

(o<.,pz

_,

+'HV2..

( 7 ) As it will be assumed that the hot-wire responds to stagnation

temper-atures (Ref. 4), it is convenient to have a relation between stagnation

temperatures and static ternperatures

2.

102..

-= \ +

~

(~J

T Z 2.. 0..2.. 2.

Pz,(b+"'P2.,)

+

~

P2.I -I) 1+ b PZI 7 1+ b""P2..1 ( 8 ) ( 9 ) where 0 = 1.4

or this can be rearranged to give

( 10 )

This gives the theoretical rise in stagnation temperature across the

primary shock moving in still air. A plot of T02/T1 versus P21 is

shown in Fig. 12.

Primary Rarefaction Wave

At the instant (t

=

0) the diaphragm is punctured a centred

rarefaction wave is generated because of the ~udden expansion of the

high pressure gas in the chamber. The ideal shock tube theory

assumes the expansion as isentropic and the wave as centred . But

in reality the wave is not a centred wave) and is' weaker than the

theoretically expected wave (Ref. 8). However, it does appear to

retain its isentropic character throughout the expansion. The

theoret-ical analysis (Refs. 5 and 6) by the method of characteristics, shows

that such a centred wave can be represented in the x-t plane as a

pencil of rays emanating from a common origin (Fig. 1). At this

cornmon centre. u, f' and p which are functions of x and t are

dis-continuolls but this discontinuity is quickly smoothed out and a'

continuous flow is established. Along a characteristic line_J particle

velocity , the speed of sound" and hence the flow properties.l are

constant. With increasing pressure ratio the rarefaction wave

be-comes thicker until, when u3

>

a3) the tail of the wave is actually

carried downstream from the diaphragm station. For wa.ves of this

strength steady state region 3 does not exist upstream of the

dia-phragm and measurements taken through the initial rarefaction wave

are not possible.

The ratio of particle velocity in region 3 to the speed of

sound in region 4 is given by (Ref. 6 )

-

~

II-RJ3 ]

( 5 )

Similarly the other flow parameters can be expressed in t~rms of the

press ure ratio across the wave

2~,,-T

34

=

"P

34 ( 12 )

( 13 ) The stagnation temperature behind the rarefaction wave in region 3

can be obtained by using

-r: "t -\ 2-103 -- \

+

~- M"3

'3 -

--To3 -l~~ _.. 1S"A ':'.L ~'\;: - ,_ - 2. lTi;..; 3

The Eq. 14 when combined with Eqs. 12 and 13 yields

l-034-__!;.;:L -

-=r:

-

-

4-( 14 )

( 15 )

( 16 ) Therefore it is possible to calculate density. partiele velocity and Ma.ch number in region 3 if the rarefaction wave strellgth P34 is known. In theory (Ref 0 6) P34 is related to diaphragm and shock

pressure ratios by

( 17 ) So knowing the diaphragm pressure ratio P41 and shock strength P21' the rarefaction wave strength is given by Eq. 1'7. In practice (Ref. 7). measurernents taken upstrea.rn of the diaphragm show P34 to be

higher than the theoretical value . Normal Reîlection of a Shock Wave

The primary shock wave îormedwhen the diaphragrn is

ruptured advances into still air and undergoes normal reflection from the closed end of the channel. In order to satisfy the boundary

condition that the partiele velocity behind the reflected shock wave is zero. the partiele velocity induced behind the reflected shock must be equal to that behind the incident shock·wave. Hence

U_2. =U5

( 18 ) and

u -

2.,- LLz. -a.₁ - U",2..'" · AZI ( 19 )

Using the expression for U21 (Eq. '1) this can.he written as

( 6 )

Using Eqs. 7 and 20 the pressure ratio across

wave can be obtained in the form

the reflected shock

p. -

o{,+2.-"A2.

52. - _{1-+0(\ P,2. ( 21 )}

Similarly the temperature ratio across the reflected shock can be

expressed as .

ï ' - _ P52.

(0(,+1='52.)

'52. -

-\ +0(. \ "P52.

(22 )

Norma~ Reflection of a Rarefaction Wave

The primary rarefaction wave formed at the instant when

the diaphragm is ruptured advances into still air and undergoes

norm al reflection from the closed end of the chamber. The

tempera-ture ratio in terms of P34 across the reflected rarefaction wave can

be obtained in the form

y.

(3

(Tb 4

î

2.

=

2

"P

₃₄ - \ _{( 23 )}

or in terms of the Mach number

~4=

r1:

~:3lZ

[04-1

3

J

( 24 )

In practice, steady state region 6 behind the reflected

rarefaction wave can only be achieved for low diaphragm pressure

rat ios. As the pressure ratio increases, so does the time required

for complete reflection,. A long channel is then required to delay

rèflection of the primary shock, and the available shoc1t tube length

becomes a limiting factor. 2.2 Hot-Wire Anemàmeters

The application of the hot-wire anemometer to ·shock tube'

flows is of recent origin (Refs. 4 and 9). If a Vv4.re heated by an

electric current is placed in a flowing gas the wire voltage reflects

the variations in flow conditions (heating or cooling). T he response e'quations have been developed (Refs. 1, 2 and 10) to describe the

béhaviour of hot wires. The equations are based on the princip:lé. of

éonservation of energy, the difference in heat input and heat 10ss be ing.

equated to the rate of change of s.tored thermal energy in the wire.

where W

H

E

d.E

W-H= dt (25)

heat input to the wire per second

heat 10ss of the wire per second

( 7 )

o

If Rw is the wire l.~esistance at a tem_{perature T w in} K and I the heating current$ then

Following the derivation of King (Ref. 10) the heat loss can be

expressed as

where

H:::

l

(-\"'''112)

(~

+

ft

TI

l.cpf'd. )

e.

length of the wire diameter of the wire

density of surrounding medium

specific heat

thermal cO!lductivity

( 26 )

( 27 )

This expression was derived for small diameter wires placed in in

-compressible potential flow and takes into account the heat loss due to

forced and free convection • radiation and conduction but does not account for the losses due to the conduction to the wire supports.

Calling B

=

~b

..

---A =

f..~211

~c/:Jed-the equatioll 27 takes the form

H:= (i4Jll4-

13)

(T

w -

Te)

( 28 )

For steady state condition

_d

·

J:

. :::.

C d.l'''''

=

.

0

dt

-·w

dt

( 29 )

where C_w is the the!r'mal capacity of the wire. and Eq. 25 takes the

f'')rro

(30 )

Te represents the equiJibrium temperature attained by an unheated ~ire in a flow of velocity u and in general is very close to the stag

-nation _{temperature T o for sm} all diameter wires (Ref. 12). Dis-crimination between T 0 and Te will not be made in the remainder of

this report. The resistance of the wire Ro corresponding to this temperature _{T o} (I --- 0) is called the "cold resistance" of the wire.

The rnethod of deterrnining the cold resistance of the hot -wire is

explained in ReL 11.

Use of the Hot Wire _, in a Shock Tube

'

The hot wire in a shock tube needs to be brought to the

-( 8 )

is fired the ·resfstance of the wire. and consequently the voltage across the wire. changes because of changes in flow quantities . These changes

can be obtaihed as measurable voltage signals. The magnitude of the voltage change depends on the region in which the hot··\vire is situated. From the voltage jump the flow quantities in any region can be obtained in terms of the initial conditions in the shock tube. Since only the constant current operation was used throughout this pre$ent investi-gation. the ·pertinent results are developed here for that mode of operation.

Consider a wire initially in region ï:that is heated by a current Ii (e. go regions 1 and 4) and, s ubsequently is found in region j (e. g 0 regions 2 and 3) where it cornes into equilibrium with the flow in

that region . If the hot wire is operated twice with tw 0 different heating currents for waves of the same strength (i. e. same stagnation temper-ature T oj and mass flow ~jUj ) we can write King's law for the steady state

r{"R~J

=::

1

(T

_Wd

-TOd)

(A

+BJfjLLJ)

Tt

B~d

=

l

(T~J

-T

_{oJ )}

(R

-+-

Bj

E'j

Ll})

( 31 ) ( 32 )

where primes denote two different overheating ratios. Eqs. 31 and 32 represent two equations in two unknowns Toj andj~juj ;:md can be

solved. Deviding one .by the other. we have I

/R"

TI

--L:î

=

WJ

-Tod-I"

R"

Til - T. '

L W}

wJ

o~

Using the resistance temperature relation

"Rw

T_w

=

Kj-

~-+

0(

(Tw-TtTI

and solving for Roj

RI ,

Ril,

(I'2._

I/I2..)

"R . -

~~W:..:..lJL-.l.._

---;;;----;-;-°d -

T'2. I _

I"

2.

R " .

-

RWJ

w~ ( 33 ) "' 34 ) ( 35 )

If ROi is the cold resistance of the wire before the tube is fired.

therefore

Roj

=

Ro i..

~ -+o(tTö~

-

TOLTI

(

36 )

0<.

where ~

=

\ I

+

0( (T_o L - ï

_f)

and T oi is the temperature in reg,ion i before the firing of the tube

~

_T

_

_{- To.l. -ToL =}

(ROl

_{~ - I}

)_1

₀₍

o ROL I

( 37:)

For constant current operation Ij = Ii and Rwj is the quantity which must be obtained from the voltage jump

'

.

( 9 )

( 38 )

The quantities I' and lil of Eq. 35 are directly measurable while

q~antities R/wj and R'<Vj can be obtained from oscilloscope records of the hot-wire response. The lheight of the jump due to the wave rep-resents a voltage change t .6. e across the wire. If a calibrating signal (sinusoidal wave) of known amplitude is photographed along with the hot-wire trace. the magnitude cf the voltage jump across the

wire can be obta.ined by meas uring the height of the jUI!1p in the

oscilloscope record.

In the Eq. 35 D it is noticed that when r/I is much larger

than I I • the equation reduces to

R

_0J~

.. "-)/

_hl""J

This means that wllen heating currents are small the wire is fairly

insensitivE: to mass floV! but predorn5.nantly sensitive to the temper

-ature of the flow since T_V1

=

_{T o . In} this case the wire is actually oper2..ting as a resist3.llce thermometer. When the operating temper -ature of the wire is very close to the equilibrium temperature (small current the voltage change across the wire for constant current operation can be written as

L\e

=-

·t

I <-~R

=:t IRoo(óT

This mode of operation allows an approximate measurement of

temperature with only one shock--tube run if mass flow information is

not required. Figure 3 shows the shape of the temperature curve for

an uncompensated w:re. Ir electronic compensation is applied (Ref 0 1)

it merely controls the lae of the wire but has no effect on the final equilibrium value. The method of calculating the necessary comp

en-sation and the precautions to be observed regarding the overloading of amplifiers are discussed in Ref. 4. In the present study. uncompen-=

sated wires we re used.

With t11.e assumption Roj :: Rwj the Eq. 37 takes the

simple farm

Finite Circuit Correction

( 39 )

The assumption made in the constant heating current system that the heating current source impedance is infinite is not

strictly true since most practical systerns have souree impedances

small enough to introduce appreciable current fluctuations . F'or shock tube work this effect must oe taken into account since the transition

across the wave system c auses significant changes in resistance 0 A

finite circuit eorrection factor can be developed (Ref. 4) for the

( 10 )

for a ho~ wireand current s upp1y. Erepresents the open circuit

voltage and Rs the interna1 source resistance. The heating current is

given by hence or

1=

E

Rw+Rs

6

I

-=

_--=E _ _ _ Rs+Rw+~Rw

and the output voltage

e

="RwI

The measured voltage jump, therefore, is

.

,

6.e

_m

-=

Kw

AT

+

1b.

"Rw

E

N eg1ecting the second order terms, we obtain from Eqs.

A

e

=. I 6."Rw 'Rs m

Rs+Rw

(, 40 ) ( 41 ) 40 and 41 ( 42 )

We have assumed constant current to calcu1ate the resistance Roj

(Eq. 38) so tha:t

~e

=

ID.Rw _{( 43 )}

( 44 )

This gives the correction factor by which measured voltages are to be multiplied to obtain constant current voltage jump • .6.e

Deviations from King~s Law

Thè King's heat 10ss equation

(Eb.

27) can be rearranged

in a more usefu1 form (Ref. 12)

'

NlA.=-J~

(Be)(Pt\.)+-i

(45)

where Nu is Nusselt number defined by

H

Nu. -

_{- ut}

(Tw

-T_o) ~

Re is Reyno1ds number based on wire diameter and Pr is Prandtl number.

For diatomic gases Pr = 0.72 over a wide range of temp-eratures (Ref. 12). So Eq. 45 can be written as

Nll

=

0·'3

\8

+0' b77lRe .

There is a linear relation between Nusselt number and the square root of Reynp1ds number. The first term represents 10sses due to

(11)

radiation and free convection while the second term represents the conduction to the wire supports. The measure of free convection is expressed by Grashof's lll1mber (Ref. 14)

G

-

~ f 2.cf3 (Tw -T,,)

where

1\.-

r-2..

_T

o

g acceleration due to gravity

fA

coefficient of viscosity

Since the diameter of the wire appears in the third power . the Grashof nurnber is vei7 small because of smallness of the diameter of the wire and hence the normal free convection effect (Ref. 14) is also very small.

Another quantity that influences free convection is the Knudsen nUI!"lber which is a measure of the degree of gas rarefaction and gives the eifect of molecular motions on the flow field

r\::: -

~

\. molecular free path d diameter of the wire

The Knudsen number ean also be expressed in terms of Mach number and Reynolds number (Ref. 15).

"

I t . - \V;

~\ -=.- ~ .. ::, ~~e

Fully developed free moleeular flow approximately occurs for K

=

2

or more (Ref. 15). When K

<

0.01 the effect of rnolecular motions can be neglected and the now ean be treated as continuum. Since ~ is inversely proportional to the density of gas. the Knudsen number is greater for the same wire ai: lower density.

The literature contains very little data concerning the hot-wire characteristics as the mass flow tends to zero. Refs. 16.

17 and 18 give an account of the effect of low Reynolds nurnQer on heat 108S. At very low velocities the free convection currenf from

the wire and the radiation losses become appreciable and the form of heat 10S8 departs from the King's equation. References 16 and 18

also suggest that the aspect ratio of the wire has a significant effect on heat loss at low Reynolds numbers. Wires of shorter length were

found to lose more heat by conduction to the supports. However", for

most practical purposes aspect ratio effects are accounted for in the wire calibration procedure.

It is important to Hote that the deviations from King's law

are of no importance when the hot wire is calibrated at conditiol1s similar to the operating condition~. In this case the constants A and

B in Eq. 30 are determined experimentally and the physical signifi-cance of these terms can be ignored. It is further seen in E qs. 35

and 37 that the form of these constants does not influence ternperature

( 12 )

IIl. EXPERIME NTAL EQUIPMENT AND PROCEDURE

3. 1 3 inch x 3 inch Shock Tube

The experimental investigations were carried out i:n the 3 inch x 3 inch shock tube which is shown in Fig. 5. The tube

con-sists of several sections of one and two-foot lengths to facilitate

adjustment of the required lengths for various purposes . Reference

7 gives some main features of the shock tube. A solenoid operated

plunger was used to puncture the diaphragm separating high pressure and low pressure gases.

The channel was evacuated by a Kinney pump and

com-mercial air cylinders and helium cylinders were used for pressurizing

the chamber. A bank of Wallace and Tiernan gauges were employed to measure channel and chamber pressures.

3.2 Hot- Wire Anemometer

The hot-wire anemometer used in the present experiments

has been dealt with in great detail in Refs. 8 and 11. It essentially

cons ists of four units, the control unit, the compensating amplifier, the calibration unit and the power supply.

The control unit incorporates precise bridges and a

current control circuit which enable accurate measurement of the

resistances of the hot 'wire under cold and hot operating conditions . Provision has been made to apply square wave heating current for

the purpose of calibrating the wire dynarnica11y if compensation is to

be used.

The compensating amplifier is a five stage push - pull

amplifier with variable amplification up to 10. 000 and variable

compensation up to 0.995 milliseconds in steps of O. 005 milliseconds

Five low pass filters with different cut-off frequencies are incor--porated to limit high frequency noise.

The calibration unït contains a direct current

potentio-meter employing a standard ce 11 and is used for accurate measure-ment of the current flowing through a precision resistor in series

with the hot 'wire. A test signal circuit allows a signalof given amplitude to be fed from an external generator through the control

unit to the compensating amplifier for amplifier calibration. Hot-Wire Probes

The hot-wire probes were the same as those described

in Ref. 12. A sectional view of the hot-wire probe and the plug mounted ~n the shock tube is shown in Fig. 6. A composite

( 13 )

7. The tips of the sewing needles, the unetched portion of the silver sleeve and the bare etched wire are all seen in the figure. The description of mounting the wire can be found in Ref. 12.

Signal Generator

A Hewlet-Packard audio oscillator was used to provide the calibration signal for determining the amplitude of the hot-wire oscilloscope records. The frequency range of thi.s sine wave

generator was 6 cps to 10 KC. Recording Equipment

The recording of the hot-wire traces was done OIT a

Dumont model 322 double beam è,athode ray oscilloscope and record camera model 299 with a lens off: 2.8. 2 1/2 x 3 1/4 inch Super XXX and Super x...X panchromatic films were used to make the

photo-graphic records.

Experimental Procedure

For e~periIrjental work the probes were a1ways mounted in such a way that the wire axis was horizontal. Measurements wer~

made at various positions along the top wall of the shock tube. The length of the tube was so adjusted to give maximum steady flow at the probe station. Because of shock attenuation the diaphragm pressure ratio required to give a certaill shock strength is higher than that pred cted by the theory (Ref. 6). At low shock strength (P21

<

2. 5),

the agreement between theory and experim ent is reasonably good. The values of diaphragm press Llre ratios requ.ired to give shock

strengths P₂₁

>

2.5 were taken frorn experirnentaJ. results -previously obtained with the same shock tube.

The temperature measurements made cover a range of shock strengHls P21 from 1.3 to 4 and rarefaction wave strengths P 34 from 1.:3 to 2. The high mortality of the hot wires set some / limitation on the range of diaphragm press ure ratios used. The

breakage of wires can be considerably reduced if the low pressure end of the shock tube is kept open. However this could not be done when the measurement on the reflected shocks were made.

It can be seen from Fig. 1 that a hot wire initially in the region 1 will be traversed by regions 2 and 5 on the low pressure side)

and a wire initially in region 4 will be traversed by regions 3 and 6 on the high pressure side. A channel length of 8 feet was used to e liminate the possibility of the reflected shock wave interfering with the reflected rarefaction wave. Similarly a cl1.amber length of 3 feet was used to prevent the rarefaction wave from interfering with the desired shock results .

( 14 )

Since the interest centred on the final equilibrium values

of the temperature, the wir es were not calibrated before each run was

made except for a determination of coid resistance. The tube was

pressurized to the required ~lalue and a ruI'.. was made w ith a low heat-ing current (1 to 2 ma). Ir the wi!'e did not break, a second rün was

made with higher heating current (about 8 to 12 m. a.). Even if the wire broke in the first run, thc temperature jump could be estimated using Eq. 39. Thi.s approximate result obtained for low heating cur-rents is fairly good~ the error eing not r.:lOre than 4 to 510 from the values obtained from t\70 runs on the same wire.

WUh each hot-wire trace, a calibrating signal (sine wave)

of known r. m. s. amplitude Vlas also photographed as s lawn in Figs . 8, 9 and 10. The representative voltage jumps 6 h (see Fig. 4) of the

hot-wire traces were measured on Hilger T-500 Universal measuring projector. Frorn t~le measured amplitude of the sine wave, the jumps Ah were converted into the rneasured voltage jumps .6.e_rn across the

ho~ vlire. The finite circuit correction (Eq. 44 was applied to each trace and the temperature jllmp Ä T vas calculated as outlLned in section 2 above.

IV. HOT-WIRE OSCILLOSCOPE RESULTS FOR SHOCK AND

RAREFACTION WAVES

Figure B is typical of the response of a hot-wire to- incident and re"1ected shock waves and Figs. 9, 10 and 11 are those of incident and reflected rarefaction waves. The position of the probe and the

configuration of incident and reflected waves in each case are shown on the x-t diagrams.

Figure 8 sI IJ 'IS tt e oscilloscope record of a hot-wire

response where tlle initial jump (A) is posHive when the primary shock wave passes o~.rer the w ire, but tL ere are combinations of wire

tem-perature and shock stren,Jtl T here the ir:i.tial jump could te negative.

The d ·rection of the initial signal (increaae or decrease of resista.nce)

for a given shock strength is governed by the overheating ratio and the cooling effect of mass flow. In t' e region 5 behlnd tt e reflected shock

wave there is no maSG flow and therefore the resistance of the wire in this region wiU always be higher than tLe steady state resistance

following the incident shock wave.

In thc case of rarefaction ·waves ~ tl e effect of the incident wave ön the wire mOlluted in the chamber will always e to reduce the

resistance of the 'vi:ce because oi decrease of temperature across the

incident rarefactlOrl wave and the cooling effect of maas flow. Figures 9 and 11 show th is behaviour for both k 'I arld high currenta. In the

case of reflected rarefaction waves, the wire might show a resis;tance

in region 6 increased from that of the steady state resi.stance following the incident wave. The oscilloscope record .in Fig. 1 dernonstrates this effect very clearly. The id.tial jump (A) on the figure 5-a due to

( 15 )

-.tho incident wave and is more negati'Ve than the jump (B) due to the reflected wave. The small fluctuations in the traces (Fig. 9) in the

region of reflected waves are due to noise in the electronic system.

Ths irregularity in the trace in the region of the teflected rarefactioll ,wave as aeen in Fig. 9 ""as observed in some of the runs at low h~at - ' ;.ng current. The irregularity occurs at nearly the same time irre~­

pective of diaphragm pressure ratios used. It has riO influence on tRe

final steady yalue~ as checked hy the rosults where it was not Sl8e&l. Since it occurs at dUferent virQ~ it is not due to defect in one

particular wire or support. Tno cru.\se seemli to have, its origin in

t1'1" electrcnic equ.ipment and could not be traeed. Figl!1.~'e' 10 shows a

trace without the irregularity, jump (A) being that of incident wave

and jump (B) that of reflected wave.

Fiiures II and 13 snow the curves tor expel"imental total

temperature ratioa T021 and T034 behind the incident shock wave anti incident rarefactior.. wave plotted against the shock strerlgth P21 and

rarefaction wave strength P:H' Figures 14 and 15 show experimental temperatura ratios T51 and T61 behind the reflected. shock wave and ,

}lE~h.ind the reflected rarefaction wave against P21 and P34 reapectivcly.

Tna points shown on tae Clll'V6S represent an averaf5e of man,. rune. The theoretical curves have alao been plotted fol' cornparison. The

agreement of experimental temperatures vlrUh theory in thc case of in

-cideat and reflected rarcfacUon wave ia fairly good at luw pressure ratloSi but the dis agreement increases with increasing diaphragm

preSBure ratio. In thiQ case of shock waves the agreement is re.e.swn-ably good at low shock strength~ but the disagreer.mmt becomes quite conSJid8rable at higher shock strengths for the case of ~eflGcted shock waves. The relative error la thc case of reflected shock waves for shock strer.gth P~H '" .4: is as high as 26 percent. From tha curves it

ap?oars that the expel'imeutal temperature ratio read~es 11 limiUng value. The results indicate that the recovery temperatilres behind the

incident and reflected shock -\ifavea are lower than those predicted by

thaory while ;.n the case of incident and reflected rarefaction waves they

seem to be !ügher than those predicted by theory. \

.~~---It should be

pOintád

out that unfortunately a discl"epancy

exists between the reEiUlts obtained during tI e present W ork and those.

obtained hy BillingtoE (Ref. ~ I~n the case of incident rarefaction' waves.

In Ref. 8 it t!:l stated taat the recovery temperature in the flow behind

the _ncident ra.refaction wave is lower than that predicted Iby theory. _~but the present investigation shows recovery temperaturea higher than

th..ose predicted by theory. Howeverg it should be noticed that the .

value oi tho temperature coefflcient of resistance.oc ~ lised in the calculations by Billington Is 0.0028 while a valile of oÇ ::0 0.0030 !las been lWed in the present r.eport. The magnitude of thc di~ference is

reduced by about 15 percent. if the values obtained by Billington are

modified using the value of 0.0030 for the ,f;oefficiant of resistance. But a di8crepancy still exists. The causa for this is not understood

despite the fact that many checks were made to trace the origin of Llis

16 )

In the case of rarefaction waves piezo pressure measure-ments made by Bi.lli.ngton behind the incident rarefaction waves (P34

were used to calcul'ate the total temperature ratio T034 using the

shock tube relations . From P34 the speed of sound in region 3 was

obtained using

From this the Mach number in region 3 was obtained employing (Ref.

6) and U ₃ _~ , , _

Q-ol

0..₄ - y- \

L

'

Q4J

M

-~ 3 - 0-3

The total temperature ratio as then calculated from

Tc y-\ t\.A2.

~=

1+

- -

,v'

3 T

3 2.

The temperature"ratios be ind the reflected rarefaction waves were

evaluated frbrn the relation

The "alues so obtain d are also shown in Figs. 13 and 15. In the case of incident rárefaction waves the _{value of T 034 computed}

from piezo pressure measurernents are in very good agreement with those from hot wire traces . The agreement in the case of reflected

waves is poorer but the variation is not large.

-ln Fig. 16 the time 6. t taken by the sho k wave to travel from the wire to the end of the tube and back to the w ire are plotted against incident shock strength. Figure 17 shows the time .6 t

required by the head of t erarefaction wave to travel from the wire to

the end of the tube and back to the wire against pressure ratio P34 across the incident wave.

The Knudsen nurnbers in the regions investigated were

very m uch lower than 0.01 and the flow under the conditions of oper-ation of the tube was in the continuum region. The variation öf the temperature res ults. therefore. CanJlot be at~r:ibuted to free -moIecule floweffects .

Reynolds numbers were calculated for various runs but cannot be plotted graphically because of the different s tarting press ures used in the tube at different diaphragm pressure ratios . The Reynolds nU,mbers based on the wire diameter were in the range 20 to 90.

( 17 )

V. DISCUSSION AND CONCLUSION

The present investigation indicates that the hot-wire

temperature response is as yet not sufficiently well understood to permit its use as a temperature rneasuring device in shock tube flows.

The measured response of a hot-wire to flow temperatures behind the incident and reflected shock waves and incident and reflec-ted rarefaction waves closely approximates that predicted by theory at very low wave strengths . The response of the wire in the case of incident shock and rarefaction waves is much better than for the corresponding reflected wäves where the overall temperature change is much larger. For waves of higher shock strengths the disagree-ment with theory in both incident and reflected waves becornes pro gressively worse. The assumption that a wire responds to total temperatures does not seem to hold at higher shock strengths .

The measured recovery temperature in the case of flows behind the inciqent and reflected shock waves are lower than those prediCted by perfect flow theory, while in the case of incident and reflected rarefaction waves they are higher. Although, in the case of rarefaction waves the temperatures evaluated from actua.l measured pressure ratios across incident waves with piezo gauges and the

Riemann relations seem to agree fairly well with those computed frorn hot-wire traces . Nevertheless, it does not explain the large differ:..ences found for the shock wave measurements in which case the properties across the wave are well known.

The present res ults cast some doubt on the validity of the standard hot-wire technique for shock tube temperature work especially at higher shock strengths . Many experimen ters have reported that hot-wire equilibrium temperatures are approxirnately equal to the flow stagnation temperature even at supersonic Mach numbers. The present results do not bear this out. Considering the case of the incident shock in Fig. 12, it is seen that the equilibrium temperature Te ,falls increasingly below _{T o as} the pressure ratio goes up. Eventually Te falls even be low the statie ternperature. The results behind the reflected shock suggest that the wire. on reaching equilibrium with its surroundings, is cooler than the air by a -very significant amount at higher shock strengths .

It is worthy of no'te that the rneasured temperature jurnp

ê:. T is always less than the theoretical 6 T, regardless of the sign of the temperature change. The deficiency in the meas urement increases with the size of the temperature jump. The shock ternperature jumps are greater than the rarefaction jumps and also exhibit a greater measurement deficiency. These results suggest that the solution lies in the wire rather than in the shock tube. One possible solution

appears to lie in the wire temperature distribution. The measured resistance of the wire is actually a summation of the resistances of the lengthwise elements of the wire_j

( 18 )

R

meo.-1lJ..,\€cL =

~A

K

before firing the tube the initial temperature distribution along the

wire will deterrn ine the initial res istance which is meas ured. Af ter

firing. the wire quickly reaches equilibrium with the flow. However,

the supports, because of their large mass, have a therrnal time con-stant which is greater probably by orders of magnitude than the wire time constant. Thus in a fe." milliseconds of the rneasurernent the

. s,upports will be sEll almost at the initial ternperature and there will

be considerable axial heat conduction througl the ends of the wire.

The wire temperature distributior. will be different in shape from the initial profile and the measured final temperature will not have

changeà as much as expected. This suggestion would explain the observed facts but requires further consideration. Experim ents

with Vlires of vario s aspect ratios might th:cow same light on this matter and wiU be considered at a latei~ date.

In conclü.sion it may be stateä tha.t the po or response

coupled with the ~igh mortality of the wire at higt er diaphragm

pres-sure ratios sets a very definlte limitation on the use of the hot Wires

as a device to measure total temperatures and mass flow a.t higher

wave strengths in a shock tube. The present rneasurements indicate

that the response of a hot vvire requires further investigation before

it 'can be accepted as a reliable instrument for rneasuring physical quantities in shock tube flows.

{ 19 } REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. Kovasznáy, L.S.G. Kovasznay, L.S.G. Dryden, H. L. , Keuthe, A.M. Dosanjh, D.S., Kovasznay, L.S.G., Clarken, P.C. Courant, R., Friedrics, K.O. Glas s, I. I. , Glas s, 1. 1. , Martin, W. , Patterson, G. N. Billington, 1. J . 9. Dosanjh, D.S. 10. Corrisin, S. 11. King, L.V.

Development of Turbulence Measuring Equipment, NACA TN 2839, Jan. 1953 The Hot-Wire Anemometer in Super-sonic Flow, J. Aero. Sci., Vol. 17, pp 565-572

The Measurement of Fluqtuations of Air Speed by the Hot-Wire Anemometer, NACA Report 320, 1929

Study of Transient Hot-Wire Response in a Shock Tube, Johns Hopkins

Univ-ersity, Dept. of Aeronautics, Report

No. CM 725, 1952

Supersonic Flow and Shock Waves. Interscience Publishers, 1948 Theory and Performance of Simple Shock Tubes, UTIA Review No. 12. Part I, 1958

A Theoretical and Experimental Study of the Shock Tube, UTIA Report No.

2, Nov. 1953

An Experimental Study of One

-Dimensional Refraction of á. Rare -faction Wave at a Contact Surface •

UTIA Report No. 32, June 1955

Use of Hot-Wire Anemometer in Shock Tube Investigations, NACA"TN 3163,

Dec. 1954

Extended Application of Hot-Wire Anemometer, NACA TN 1864, 1949

On the Convection of Heat from Small Cylinders in a Stream of Fluid.

Det-ermination of the Convection -Constants,,-_

'-of Small Platinum Wires with Appli-cations to Hot-Wire Anemometry. Phil. Trans. Roy. Soc. {London} ,

12. Billington, 1. J . 13. Hilsenrath, J. . 14. McAdams. W.H. 15. Stalder ,_J . R. , Goodwin. G. , Creager, M.O. 16. Cole. J .• Roshko, A. . 1 7 . Collis, D. C .• Williams, M. J . 18 . Co Uis, D. Co, Williams. M.J. ( 20 )

The Hot-Wire Anemometer and lts Use in Non-Steady Flow, UTIA""TN No. 5, Sept. 1955

Dry Air Prandtl Number

NBS-NACA, T able of Therrnal Pro-perties of Gases. July 1950

Heat Transmission. Second Edition, McGraw-Hill Book Co. lnc .• 1942 A Comparison of Theory and

Experi-ment for High-Speed, Free-Molecule Flow, NACA Rep. 1032. 1951

Heat Transfer of Wire at Reynolds

Numbers in

Osee

11 Range.

Heat Transfer and F luid Mechanics

Institute. University of Califorrria,

July 1954

Two-Dirnensional Forced Convection from Cylinders at Low Reynolds

Numbers, Aeronautical Research Labs .• Report A 105, Nov. 1957

Free Convection of Heat from Fine Wires. Aeronautical Research Labs ~ , Aerodynamic Note 140. Sept. 1954

6

t

/

/

,

-/

---

-,

/

5

5

,

/~.S

,

/

,

/

3

R

2

,

/

5

,

_._!!.

-I- -

_,

-

--R

_/

x

PRIMARY RAREFACTION WAV-yCONTACT SURFA~PRIMARY SHOCK

1=1,

[ 4

1

I1

3

1

7

2

1;7"

I

I

t=

12

1

6

1

/ 1

, )

k

5

REFLECTED RAREFACTION WAVE? CONTACT

SURFACE~FLECTED

SHOCK

S\:

6

:s

4

~I

3

2

I

o

~

---~

~

/

V'

/

/

/ '

V-I

10

20

30

40

so

60

70

80

P41

FIG 2 THEORETICAL VARIATION OF SHOCK STRENGTHS P21 WITH

RW

__ HOTWIRE I

..

FIG 4 EQUIVALENT CIRCUIT OF HOT-WIRE CURRENT SUPPLY

I

---I /,' /

,"

,

/ ' ....

-....

----... __________ Wz Tt T₀₂

Ah

t

FIG 3 TEMPERATURE JUMP THROUGH SHOCK WAVE AND UNCOMPENSATED

FIGURE 5

~LEADS

STANDARD PLUG ARALDITE RESIN

SHOCK TUBE WALL

HOTWIRE~

FIG 6 SECTIONAL VIEW OF A HOT-WIRE PROBE AND PLUG MOUNTED

FIGURE

7

COMPOSITE MICROSCOPE PHOTOGRAPH

OF 0.0001 INCH DIAMETER PLATINUM

WOLLASTON WIRE MOUNTED AND ETCHED.

THE TIPS OF THE TWO SEWING NEEDLE

MOUNTS, THE UNETCHED PORTION OF

THE SILVER SLEEVE AND THE BARE

PLATINUM WIRE ARE ALL SHOWN.

t

R-~...:'

CLOSED END

X

HOT WIRE

FIG 8 HOT-WIRE SHOWING UNIFORM STATE REGIONS 1, 2 AND 5. CASE

AIR/AIR P41 = 1.71; HOT-WIRE HEATING CURRENT: 2MA

t

/ '

~L-cs

,

CLOSED

END

x

FIG 9 HOT-WIRE SHOWING UNIFORM STATE REGIONS 4, 3 AND 6. CASE AIR/AIR P41 = 2.96; HOT-WIRE CURRENT: 1.42 MA CALJBRATING SIGNAL: 500 CPS; 4.51 MILLIVOLTS (R MS)

t

6

...L-cs

.-./

'"

/ ' CLOSED

END

~L---

----~---~~~~~~~~

______

__

x

FIG 10 HOT-WIRE SHOWING UNIFORM STATE REGIONS 4, 3 AND 6. CASE

AIR/AIR P41 = 2.96 HOT-WIRE CURRENT: 1.42 MA. CALIBRATING

t

6

. /

. /

,L..cs

CLOSED END

IL

__

~~_--=~E:::~~::2::::'_--,""",:X~

FIG 11 HOT-WIRE TRACE SHOWING UNIFORM STATES 4, 3 AND 6. CASE AIR/AIR. P 41 = 1. 71 HOT WIRE CURRENT: 10 MA. CALIBRATING SIGNAL: 500 CPS. 61.8 MILLIVOLTS

T

_OZI

T

₂₁ 2'°1 ~ 1·8

o

THEORETICAl T

₀₂₁

..

T

₂₁

EXPERIMENTAL T

OZI "","

"

1

-

61

I

'

~."",,,,,,

7" "

""","

"'"

1.41 7" '

I

;ti

,- I

&.2 1

1

7"'~"""

1

2

3

4

Pz.

FIG 12 THEORETICAL AND EXPERIMENTAL VARIATION OF mCIDEN T SHOCK

WAVE STAGNATION TEMPERATURE RATIO (T021) WITH DIAPHRAGM

PRESSURE RATIO (P41) THEORETICAL STATIC TEMPERATURE RATlOS T 21

'.0

O.g

T 034

O·

T₃₄

0·7

o·

~

~1

~

...

-.----:

--~

~

....

....

V

~

....

"

, / , / - '

v,....

, , /

,...."'"

,,~

"

THEORETICAL STAGNATION TEMPERATURE

,

RATIO T 034

0 T034 FROM HOT-WIRE TRACES

.

T034 FROM EXPERIMENTAL P 34 X (BILLINGTON)

r~

or

0.3 _{0·4 0·5 0·6 0·7 0.8 0·9}

~4

FIG 13 THEORETICAL AND EXPERIMENTAL VARIATION OF INCIDENT

RAREFACTION WAVE STAGNATION TEMPERATURE RATIO (T ₀₃₄)

WITH PRESSURE RATIO (P 34). THEORETICAL STATIC TEMP-ERATURE RATlOS (T34) ARE ALSO SHOWN.

THE VALUES CALCULATED FROM MEASURED (P34) BY BILLINGTON ARE COMPARED WITH IDEAL SHOCK TUBE THEORY

I I I I I , , I ; ,

'·0

T

₅₁ ~'O ~

I

I

THEORY 2· 6 0 EXPERIMENT

~

~

,? 2 I·

V

V

/

8

V

V

) 0 4

/ .

_..

V

a.o

1·0

V

O 2 3 4

F21

\'

FIG 14 THEORETICAL AND EXPERIMENTAL V ARIATION OF REFLECTED

SHOCK WAVE TEMPERATURE RATIO (T 51) WITH INCIDENT SHOCK STRENGTH (P21)

T

₆₄ "00

v

~

v~/

~

/

Q

) n GD

~

J[

V

V

THEORETICAL TEMPERATURE RATIO T64 )

/ '

0 T64 FROM HOT-WIRE TRACES

0·85 0·70 0·55 X T64 FROM EXPERIMENTAL P34 (BILLINGTON) ,. 0·4' ~~ ",,"-i

o

y 0·3

0'4

0'5

O·S

~4

FIG 15 THEORETICAL AND EXPERIMENTAL VARIATION OF REFLEC TED RAREFACTION WA VE TEMPERATURE RATIO (T64) WITH PRE SSURE RATIO (P3 4)

THE VALUES CALCULATED FROM MEASURED (P₃₄) BY BILLINGTON

ARE COMPARED WITH IDEAL SHOCK TUBE THEORY

àtXI0

3

(seconds)

,

2.0

1·8

1·6 I 1·4 I·

2

I·

('1

-

..

,

THEORY

~O

0

EXPERIMENT

o

~

~

I---...

~

_I

-'I

/<~CLOS<

••••

It~

.. s l .

1

I

I

x

2

3

4

~t

lltXI03

(SECONOS)

4

3

2

, 0

W.

o

'0·3

0 0

-I

.

_t

I

,

I

~

I

~

-CLOSED

/,C.S

THEORY

_END

_{- -}

~llt

~S

R-~~

EXPERIMENT

X

04

0·5

0·6

P34