The effect of non-equilibrium effects on the measurement of flow properties in the longshot hypersonic tunnel

FOR FLUID

DYNAMICS

1 ~ NOV. 197Z

NOTE

83

THE EFFECT OF NON-EQUILIBRIUM EFFECTS

ON THE MEASUREMENT OF FLOW PROPERTIES

IN THE LONGSHOT HYPERSONIC TUNNEL

by

Edgard BACKX

RHODE-SAINT-GENESE, BELGIUM

THE EFFECT OF NON-EQUILIBRIUM EFFECTS ON THE MEASUREMENT OF FLOW PROPERTIES

IN THE LONGSHOT HYPERSONIC TUNNEL

by

Edgard BACKX

"Navorsingsstagiair van het N.F.W.O."

ACKNOWLEDGMENT SUMMARY NOTATION LIST OF FIGURES 1 • INTRODUCTION TABLE OF CONTENTS.

2. VIBRATIONAL FREEZING IN NOZZLE EXPANSION 2.1 Theory

2.2 Calculation method 2.3 Results

3. NON EQUILIBRIUM EFFECTS IN A BLUNT BODY STAGNATION REGION 3. 1 Theory 3.2 Calculation method 3.3 Results

4.

CONCLUSIONS REFERENCES TABLE FIGURES.

ACKNOWLEDGMENTS.

I wish to thank the fOllowing persons who

contributed in one way or another to the report presented:

-Prof. T. Van Der Waeteren, Prof. B.E. Richards, Prof.

M. Lewis, Mme Toubeau.

The N.F.W.O. (.ational fonds voor

wetenschap-pelijk onderzoek) has supported my work during the year

This report gives an estimation of vibrational frozen temperatures in the testing reg10n for the range of

running conditions used in the von Karman Institue Longshot hypersonic wind tunnel. To assess the extent of the non-equilibrium effects caused by the vibra.tional freezing, the statie temperature and pressure variation along the nozzle are calculated.

An approximate method has been developed to find the influence of vibrational relaxation on the tempera-ture and pressure alon~ the stagnation line between a bow shock and a body. This has been done to determine the errors involved in calculating test section conditions 1n which the flow is assumed to be in equilibrium and to aid selection of an appropriate technique of measuring stagnation temperature required to study the nozzle wall turbulent bouridary layer in the Longshot tunnel.

a A

•

A c g h L m M P q p

*

r t

T

T

a

T

v u x x s y Ö ö·

e

e

p 0 0 T

frozen sound speed area

throat area

=

I

y R T a

specific heat at constant pressure defined by equation 10

enthalpy

relaxation length mass flow rate Mach number pressure = 00 - 0 t hroat radi us body radius gas const an t

entropy of a perfect gas at unit temperature and pressure

time

temperature

translational temperature = rotational temperature

vibrational temperature velocity

distance

shock stand-off distance ratio of specific heats boundary layer thickness

boundary layer displacement thickness characteristic vibrational temperature nozzle semi-angle

density

vibrational energy

equilibrium vibrational energy relaxation time

a d eq o op ov st 2

active

=

translational + rotational modes dimensionless

equilibrium total eondition

total equivalent perfect condition

total equivalent eondition for a vibrationally excited gas

statie eondition stagnation condition

1- First approximation of T _{C1rst case)} v

2. The expansion of _{N 2} l.n the nozzle

3. First approximation of T ( 2nd case)

v

4. Determination of T using the full set of equations

v

5. Vibrational relaxation times for N2

6. Influence of vibrational non-equilibrium on statie conditions

7.-12. Variation of non-equilibrium temperature and pressure along the nozzle

7-8

9-10 11-12 T_ov = 20000_K T_ov

=

25000 K T

=

30000_K

°v

13. Frozen vibrational temperature

14. T /T at a fixed station A/A-

=

a§qa function of p r·/tan

e

ov

1000 for different T ov

15. p /p at

eq a fixed station A/A*

=

1000 for different

Tov as a functl.on of p

.

r

*/

tan

e

ov

16. The temperature variation on the stagnation line in front

of a blunt body (Rb

=

9

cm)

17. The temperature variation on the stagnation line in

a high temperature reservoir gas is used such that af ter expans10n to the flow testing region, condensation of the test gas is

prevented. These temperatures are usually so high that not only translational and rotational energ1 modes are excited but also vibrational and higher energy excited modes. During the expansion in the nozzle, the equilibrium state changes continuously. The adjustment of translational and rotational energy modes occurs almost instantaneously but excitation of the vibrational energy mode normally requires a time, several

orders of magnitude greater for relaxation towards equilibrium. The magnitude of this relaxation time is dependent on

temperature and pressure. When the relaxation rate becomes much smaller than the expansion rate, the energy in the vibrational mode and the vibrational temperature cannot change significantly during the expansion. This non-equilibrium effect is called

vibrational temperature freezing. Similarly, the higher excited modes of internal energy may be ~rozen in the flow, since these will have even lower relaxation rates.

Another non-equilibrium phenomenon occurs when a blunt body is placed in a hypersonic test flow. A normal shock wave will be formed at a finite distance from the stagnation point of the body. The kinetic energy of the stream will be converted into internal energy in the shock layer and the region downstream, hence causing an increase in enthalpy of the gas. The translational and rotational energy modes wil 1 be excited within the shock thickness, but vibrational equilibrium viii not be fully excited until the gas has undergone enough molecular collisions whether the initial vibrational excitation be frozen or in equilibrium. Downstream of the shock, the

vibrational temperature will increase with a rate dependent on the relaxation time. The excitation of the vibrational mode lS supplied by the translational and rotational energies and by

the kinetic energy of the gas.

The first problern of vibrational freezing in a hypersonic nozzle is treated quite extensively in the

literature. There exist a number of sophisticated treatrnents (Refs. 1,2,3,4), but the simple method of Stollery and Smith (Ref.5) predicts the frozen vibrational ternperature fairly

accurately. The use of a more sophisticated method than that of Ref.5 does not necessarily provide improved accuracy because of the dependence of all calculations on uncertain knowledge

of the relaxation time. Also uncertainties arise in calculations of the flow in hypersonic nozzles from the large wall boundary layer growth which must be predicted to determine the change

~n stream tube area. This uncertainty can be reduced however

if the nozzle wall pressure can be measured.

The second problem of vibrational relaxation

behind a shock is treated by Marrone (Ref.6) in a very detailed way. Sedney (Ref.7) and Johanessen (Ref.8) have also treated some aspects of the problem.

The VKI longshot free-piston hypersonic wind tunnel (Refs.9,10) has such a high reservoir temperature that considerable vibration excitation is present. However the tunnel is unusual in that the reservoir pressure is so

high that dense gas effects are important. It is desired in this study to determine the extent to which the flow parameters

deviate from an assumption of equilibrium flow which has been

assurned ~n presently available flow calculation. Knowledge

of this is desired particularly in order to assess which are the most accurate methods of determining the stagnation

temperature.

This report presents a study of non-equilibrium effects 1n the longshot nozzle carried out to determine their

importance on measurements and estimates of the properties of the test gas flow. In particular it is desired to aid selection of a method of measuring the total temperature ~n the

2. VIBRATIONAL FREEZING IN NOZZLE EXPANSION

2. 1 Theory

The fOllowing set of equations describes the

expansion of a quasi-one-dimensional inviscid flow in a nozzle.

Continuity equation

Momentum equation

Energy equat ion

E qua t ion 0 f state

Rate equation Nozzle geometry Vibrational energy puA = const = Ut -pu( du)

=

~ dx dx c T + aCT ) pa a

v

p

=

pRT a u2 + - -

=

const

=

ho 2 ( 1 ) (2 )

( 4 )

( 6)

The internal energy of the gas is divided into the active (translation and rotation) and the inert modes.

Equilibrium is assumed to exist within each group but not between the two groups. Writing Eqs. (1)-(4) in differential

form and eliminating p, u and cr g~ves :

T a 1 p dT a dx ~

₌

dx where rM2

(1-1 _ M2 A g

=

rRT a M2 dA ( - - -- + (1 - YM 2 ) g) A dx dA (r - 1 ) g)

-dx da dx

( 8 )

( 9) ( 10)

g 1S calculated from the rate equation. From these derivatives of Tand p, a step by step calculation can be programmed on a computer. This is the method used by Stollery and Park (Ref.4). Very small step sizes have to be used in order to obtain a sufficient accuracy.

An approximate method, proposed by Stollery and Smith also has been used in the present study. It is based on the fa ct thRt the vibrational energy constitutes only a small part of the total internal energy at relatively low temperatures (4% at 1000oK, 10% at 20000 _{K). The energy} equation may be rewritten as :

c

T

pa a c pa T

o

+ q

where q

=

a(To) - aCT )

v

( 1 1 )

( 12)

Eqs. (1), (2),

(4)

-(7) and (11) describe the expansion of a perfect gas with heat addition q. The problem of a perfect gas flow with heat addition and area change is treated in the literature (e.g. Ref.ll). Lewis anq Arney (Ref.12) have used this method in a slightly different form. The calculation procedure is as complicated as the solution of the full equations, but two limiting cases can be considered.

1.

a(T

_{v )}

=

aCT).

The gas flow is in thermal equilibrium. This is the case of an expanding real gas.

2. a(Tv )

=

ö(To). Here, the total vibrational energy is conserved. The remaining system is an expansion of a perfect gas.

For each case, one can find

a(x)

and T(X) from the local values of pand T thus found. The

a(x)

distribution is

a

calculated from the rate equation, which has to be written in a differen~ form. The distributions obtained cannot be

similar because of the energy conservation equation. A second approximation can be calculated using the

a(x)

distribution obtained in the second case and Eqs. (1), (2), (3), (4). But, if the difference between both distributions is small, this

second approximation is unnecessary, because of the uncertainties in relaxation constants.

It 15 not possible to calculate the influence of vibrational freezing on flow statie conditions using the above mentioned methods. In this case, the full Eqs.(l)- (7) have to be solved.

2.2 Calculation method

A

computer program for the equilibrium expansion of real nitrogen, including dense gas effects, has been written by Culotta and Richards (Ref.13). One can avoid dealing with the real gas equation of state and s t i l l obtain the same result by using equivalent perfect conditions. P a n d T

op op

are fictitious pe~fect ~as values of reservoir pressure and temperature from which the flow properties can be determined using the perfect gas formulas. The flow properties in the test section will be the same i f the entropy and enthalpy are equal in both cases. P a n d T are therefore calculated from So

op op

'and ho using the perfect gas relations

T

=

ho/c ( 13) op p I P

₌

exp (s~ IR

-

sO/R + c IR In T ) ( 14) op p op o I 5 0

=

3.1379 P 1n atm. op T 1n oK op

Using the expansion tables for a perfect gas, the same static conditions are obtained for the same free stream Mach number

(Fig.1,2). Sin ce static conditions are already close to each other in the region where freezing starts (see further), we can use the simpler calculation instead of the Culotta-Richards program.

To find the vibrational energy from caleulated statie eonditions, a linear rate equation is used :

da dx

=

o - 0

L

written ~n a difference form, this yields

a - 0 n+1 n

=

la + a )-n + L ) n + a ) n

We assume 0

=

0 ahead of the throat. The initial conditions

-for Eq.(16) being the throat conditions, 0n+1' L

n+1 and ~x

are known from the statie conditions so that 0 1 is the

n+

only unknown in Eq.(16). This step by step process can be

carried out until L becomes so great that a 1 ~ a

=

frozen

n+ n

( 1 6 )

vibrational energy. The vibrational temperature is related to the vibrational energy by :

Re

a

=

_alT

e v

-where

a

is assumed to be independent of pressure.

The vibrational temperature found from this calculation

~s too high since static conditions were calculated assuming

equilibrium. To caleulate the statie conditions assuming an infinite relaxation time, we have to convert the gas in the

reservo1r into a real gas with the same amount of energy but

without dense gas effects. This conversion can be done, using the equations of enthalpy and entropy for a gas with an excited

vibrational mode. h

=

c T + pa c

Ra

alT

e v u2 + 2 - 1

alT

s

=

~ lnT +

R

R

v + ln

alT

e v

-alT

e v

alT

e v -- ln p + 5

o '

o

( 1 8 )

The enthalpy and entropy are calculated from the conditions in the test section where no dense gas effects exist. A reservoir temperature and pressure is then calculated from Eqs. (18) and

(19) which is valid for a vibrationally excited gas (Fig.2). From these conditions, the second case proposed by Stollery and

Smith can be calculated (Fig.3). Instead of calculating a

second approximation, based upon the

a(x)

distribution, to find the changes in statie conditions, the method outlined by

Stollery and Park (Ref.4) was used. The calculation was started a distance x downstream of the throat, up to this point,

equilibrium is assumed. This simplifies the calculation and can be justified by the fact that when dense gas effects are

present, the vibrational relaxation time is sa small compared to transit times that the vibrational temperature will mostly follow the translational temperature ( as illustrated in Fig.1). This justifies the use of the total equivalent perfect conditions used in calculating the frozen temperature. The result is shown in Fig.4. The small difference between T predicted in Fig.1

v

compared to Fig.4 is due to the low value of T since the frozen v

vibrational energy constitutes only 1% of the total energy.

Some comments on Eqs. (5) and (15) will be g1ven before discussing the results.

Vibrational time constants.

The relaxation length which has to be known from

gas properties, is the product of the relaxation time and the velocity of the gas.

sure and temperature.

The relaxation time is a function of pres-In all cases,

inversely proportional to p, so that

T for N

2 is assumed to be T p is only a function of

temperature. The formulas T p

=

f(T) in the literature are

mostly obtained from relaxation studies behind a shock wave.

For an expanding gas, the relaxation time is much smaller. The

formula used 1n here, is the only equation given specifically for relaxing N

2 (Ref. 14)

Tp

=

3 x 10-12 exp (181/T 1/3 ) sec.atm. . (20)

This 1S compared with other values of relaxation rate in Fig. 5 Another uncertainty arises from the fa ct that impurities 1n the gas will increase the relaxation rate.

The effective nozzle shape differs from the actual one ( a cone with

6

0 half-angle) because of the boundary layer on the wall. To find the function A

=

f(x) we made the assumption that the displacement thickness var1es linearly

with distance and equals zero 1n the throat. Thus, if we know

the displacement thickness at one point in the nozzle, we

can define an effective nozzle with a smaller cone angle. The

displacement thickness at the exit is found from the Reynolds number based on nozzle length, using a correlation found in

Ref. 15. Experiments have shown that this calculated ö* is

2.3. Results.

Fig. 1 and 3 indicate that the vibrational tem-perature is suitably predicted by the approximate method

pro-posed by Stollery and Smith. The difference in frozen

tempe-rature for the two limiting cases is only

60

o K. The frozen

temperature calculated from the full equations is

873°K,

the

corresponding energy is less that 1

%

of the total energy.

Therefore, Fig. 1 prediets almost the same temperature. The

influence of freezing on statie conditions is shown in Fig.

6.

The dotted lines indicate the kind of error in data reduction

made, when vibrational freezing is not considered. The

pres-sure (A) meapres-sured is assumed to be the result of an equilibri-um expansion (B) so that the Mach nequilibri-umber is 14.81 and the

tem-perature 59.1.

are 14.97 and

57.3

The corrected Mach number and ternperature

oK. The accuracy of the assessment of

static temperature frorn the measured pitot pressure and total

temperature is not better than

5

%.

The difference of 3

%

predicted by our calculations ~s therfore less than the

mea-sur~ng accuracy. However, i t ~s not negligible and has to be

taken into account, for instance, when condensation is consi-dered.

To generalize the calculation, the effect of vibra-tional freezing on statie conditions is calculated for a whole range of conditions (equivalent conditions for vibrationally

excited gas) and plotted in Figs.

7 -

12. The solution is

also applicable to other conical nozzles through the

sirnila-rity parameter p r x / tan

e

Fig. 13 gives the vibrational

o

ternperature as a function of the similarity parameter. The

results frorn Ref.

4

are also included. Note however that these

A

=

1 + { X tan

e

~

r

and a different relaxation time formula

Tp

=

.01402 exp ( -T /401.1)

a sec. atm.

( 2 1 )

(22)

dense gas effects ~n the reservo~r we re absent due to the low total pressures. Therefore To

=

Tov

Figs. 14. and 15 give the effect of freezing on temperature and pressure at a fixed station (A/A·

=

1000) as a function of the similarity parameter and include also the results of Ref.

4.

The unexpected variation of the pressure ratio (Fig.

15) can be explained. In Ref.

4,

freezing started always ahead of the throat, in the subsonic region, while in the Longshot, freezing starts weIl downstream of the throat. Going back to the analogy with a perfect gas flow with heat transfer, one can see that the influence of heat transfer gives different effect in subsonic and supersonic flows (Ref. 11).

3. NON-EQUILIBRIUM EFFECTS IN A BLUNT BODY STAGNATION REGION.

3.1 Theory.

In front of a stagnation probe, a normal shock wave 1S formed at a stand-off distance, dependent on probe

dimensions and Mach number. The translational and rotational temperature will increase suddenly in the shock layer (Ref. 16) The vibrational temperature, which 1S the frozen value ahead

of the shock wave, will not have changed significantly in passing the shock layer, since the thickness of this layer 1S

small compared to the relaxation length. Between shock and body, the vibrational temperature will increase, the rate of change depending on relaxation time.

The non-equilibrium relaxation region behind a normal bow shock is described fully by the following set of

equations. - equation of state - continuity equation - energy equation - momentum equations - rat e equat i on

The form of the equations differ from author to author, depen-ding on the choice of coördinate system. For an ideal gas,

Ref. 17 gives a solution for the equilibrium flow which is valid for the stagnation region and Refs. 18 - 20 solve the full

shape of the bow shock is an input, while the shape of the body is a result of the calculation. All methods (except Ref. 17.) reqmre a careful step by step process because of inherent insta-bility.

The appearance of vibrational relaxation compli-cates the problem. A very sophisticated method has been de-velopped by Marrone (Ref.

6.).

Dissociation and recombination of elements as weIl as electronic and vibrational excitation phenomena are included. Sedney (Ref.

7.)

treats some aspects of the problem of vibrational relaxation as an example for

more complicated non-equilibrium effects. Johanessen (Ref.

8.)

solves the problem of vibrational relaxation behind a normal shock wave (not caused by a body) by means of the Rayleigh line method .

The purpose is to develop a method that does not ne-cessary yield exact values of the parameters behind the shock but rather indicate the differences between the equilibrium and non-equilibrium cases.

3.2. Calcuàtion method.

None of the methods referenced in section 3.1 could be adapted for the V.K.I; I.B.M. 1130 computer, due to the complexity of the programs involved. The interest was 1n the values on the stagnation line, because only these

influence the stagnation point values that are measured by a pitot, a total temperature probe or a stagnation point heat transfer gauge of relevance in this study.

The following method, only applicable on the stag-nation line because of the use of quasi one dimensional equa-tions was developed. The equations used are :

•

Continuity equation puA

=

m (23)

Equation of state p

=

p RT a (24) du _~

-

pu

=

dx dx Momentum equation 2 C T + 0 + ~

=

const. pa a 2 Energy equation

(26)

do f ( u ,p,T ,0 )

=

-0 dx a Rate equation

The unknown'5 are p, u,p;

The independent variabIe is X

T , A, o. a

There is one equation less than unknowns. In the exact method,

the systern of equations can be closed by using a second momen-tum equation and a given shock shape.

sult of the calculation.

The body shape is a

re-The velociy béhind a bow shock decreases to zero

nearly linearly at least for an ideal gas. Therefore, we will

solve the equations in the following approximate way. Given

the conditions behind the shock from ideal gas equations, the velocity at a distance x downstream is calculated assum1ng a

linear decrease to zero at the stagnation point. Afterwards,

the other properties except 0 are calculated from the equations

with 0 n+1

=

on From the rate equation, a new 0n+1 is

cal-culated. Then new conditions at this point are found assuming

a perfect flow with cooling (the heat 1055 being 0n+1 - On).

This step by step method can be continued until the stagnation point is reached.

There are many formulas for the vibrational relaxa-tion times in the literature. S 0 me 0 f th e se ( Re f.

4,

1

4,

21,

22,-23) are plotted in Fig. 5. The following much used for-mula was ·used in this calculation

T P = 1. 1

1

T 2 e xp (1

54/

T 1/3) sec . at m.

(Ref. 21) (28)

The formula proposed by Gaydon and Hurle (Ref. 23, Fig. 5) glves almost the same results. It is difficult to say which formula lS the most appropriate to use because of the unknown effects of impurities in the gas.

2.4 Results.

The temperature distribution behind a bow shock 1n front of a sphere mounted in typical flows of Longshot is plotted in

ág.16.

and

17.

for two different body radii. For the large st body, the te~perature will be in equilibrium at the stagnation point. Due to the small stand-off distance, the vi-brational temperature remains lower than the equilibrium tempe-rature for the smaller radius. It is interesting to note that also the stagnation pressure is affected, although in a very small way (TabIe 1). From these results, we can conclude that for the Longshot test conditions

- the temperature sensed by a shielded thermo-couple will be the equilibrium stagnation temperature, since

thermal equilibrium will be attained a few millimeter downstream of tube inlet.

- the temperature measured with a Sodium line reversal method will be too low since this measuring

tech-nique glves an average value of the vibrational temperature (which changes significantly from the equilibrium temperature) between shock and body.

- a hot wire will have too small a dimension to

measure the equilibrium stagnation temperature. Shielding should increase the relaxation distance so that equilibrium is attained at the wire.

The influence on stagnation point heat transfer has not been investigated. It is obvious from the Figures that

for a large body, the influence will be very small. For a small stagnation probe, there might be an influence. The eva-luation of this difference requires the temperature gradient at the wall to be known. Therefore, the boundary layer has to be taken into account and coupled with the non-equilibrium effects. The approximate method used here lS no longer applicable. The experimental difficulties to use stagnation point heat transfer gauges in the Longshot also makes this method less attractive. The th in film gauges in previous tests have been damaged even at less severe conditions than envisaged due to abrasion. The calorimeter gauges which are less accurate because of dif-ficulties in calibration, must also have a spherical surface to fit flush in the stagnation probe. The small dimensions

(necessary for a boundary layer probe) and the shaping make this method problematic.

4. CON C LUS ION S.

The effects of freezing during the expansion 1n the Longshot nozzle have been examined. The most important consequence is that the statie temperature in the test sec-tion decreases a few percent as compared to the actual pre-dicted temperature. This effect has to be taken into account when one considers optimising the test conditions in order to obtain a Reynolds number as high as possible. Condensation of the test gas together with material problems limit this opti-misation.

Due to the lag of vibrational temperature, behind

a normal shock caused by a flow stagnation probe, the sodium line reversal method is not an appropriate method for measuring

stagnation temperatures in the Longshot test section. Stag-nation point heat transfer gauges on probes of large diameter

may be appropriate but experimentally are difficult to achieve with sufficient accuracy. The most reliable method to measure this important value seems to be a shielded probe.

R E F ERE N CES •

1. Wilson J.L., Schofield D., Lapworth K.C. "A computer program for non-equilibrium convergent-divergent nozzle flow". N.P.L. Aero report 1250, 1967

2. Erickson W.D.: "Vibrational non-equilibrium flow of nitrogen in hypersonic nozzles." NASA TN D-1810, 1963

3. Reddy, N.M., Daum F.L.: "Similar solutions in vibrational non-equilibrium nozzle flows." ARL-0144, 1970

4.

Stollery J .L., Park ~.: "Computer solutions to the problem

of vibrational relaxation in hypersonic nozzle flows" Imperial College of Science and Technology Rep. 115, 1963

5. Stollery J.L., Smith J.E.: "A note on the variation of

V1-brational temperature along a nozzle." Journ. of

Fl. Mech., vol. 13, p. 225, 1962

6.Marrone P.V. :"Inviscid, non-equilir'bium flow behind bow and normal shock we.ves".part I, Cal rep. n° QM-16-26 A - 1 2 (I), M ay 1 963

7. Sedney R.: "Some aspects of non-equilibrium flows" Journ. of th e A er. Sc. vo 1. 28, p. 189, 1961

8. Johanessen, "Analysis of vibrational relaxation regions by means of the Rayleigh line method." Journ. of Fl. Mech., vol. 10, p. 25, 1961

9. Richard~ B.E.; Enkenhus K.R.: "HYpersonic testing in the

VKI Longshot free-piston tunnel." AIAA Journ. vol. 8, n° 6, p. 1020-1025, July 1970

10. RichardsB.E., Enkenhus K.R. " The Longshot free-piston hypersonic tunnel." VKI, TN ~9, 1968

11. Shapiro A.M.: " The dynamics and thermodynamics of compres-sible fluid flow", Vol. 1, 1953, p 219-260, 199 The Ronali press company, New York.

12. Lewis A.D., Arney G.D. "Vibrational non-equilibrium with nitrogen in low density flow." AEDC-TDR-63-31, 1963

13. Culotta S., Richards B.E.: " Methods for determining condi-tions in real nitrogen expanding flows" VKI TN 58, 1970

14. Sebacher D.I.: " A correlation on N

2 vibrational to

trans-lational relaxation times." AIAA Journ. vol. 5, p. 819, 1967.

15. Perry J.H.: "An experimental study of the turbulent hyper-sonic boundary layer at high rates of wall heat trans-fer". Thesis submitted for the degree of Doctor of Philosophy, June 1968, Univ. of Southampton.

16. Borrel P.: "Relaxation processes in gases". Advances in

molecular relaxation processes, vol. ,no 1, p. 69

nov. 1967

17. Li T.L, Geiger R.E.: "Stagnation point of a blunt body

~n hypersonic flow." Journ. of Aer. Sc. vol. 24 n° 1, p. 25, 1957.

18. Van Dyke M.D., Gordon R.D.: "Supersonic flow past a farnily of blunt axisyrnmetric bodies." NASA TR R-l, 1959

19. Fuller, F.B.: "Numerical solutions for superson~c flow

of an ideal gas around blunt two-dimensional bodies" NASA TN D-791

20. Maslen S.H., Moeckel W.E.: "Inviscid hypersonic flow past bI un t bod i es." Jou r n. 0 f t he A er. Sc. vol. 24, n° 9

p. 683, 1957

21. Huber P. W., Kantrowitz A.: "Heat capacity lag measurements ~n various gases." Journ. of Chem. Phys. vol. 15, p. 275, 1947; Recent advances in Aero-thermochem-istry, Agard Oslo 1966, p. 507

22. Millikan R.C., White D.R. ;"Journ. of Chem; Phys. vol. 39, 1963, p. 98-101; Recent advances in Aerothermochemis-try, Agard Oslo 1966, p. 507

23. Gaydon A.G., Hurle I.R.: 8° sympos1um on combustion, Baltimore, 1960, p. 309; Recent advances in Aero-thermochemistry, Agard Oslo 1966, p. 507.

Pressure and Temperature at the Stagnation Point.

,

rb T 2

/

Tst P 2

/

Pst cm 9

40. 1

286.2

.6

26.2

284.81

x see fig. 17.

1500 1000 500

•

_•

•

P.=42600 pI; 8=60 Top =2650

P

o,r49/J00 pI;

R.al N2 .xpan

1/0"

program

p,.rl.,f N2

,

~

66J

~~---174

5

10 H

a. CL 0 Cl.

Q

a.

,

CL ~

·

... 0 0 • _•

~

·

•

_·

·

.... 0 \

·

\

,

\ \ \ \ \ \ \ \ \ <:) \ I <:) \ <:) \

.--\ \ \ \ \ \ \ \ \ \ ~ \

...

\

....

\

....

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