Transport properties of free molecule (Knudsen) flow

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TRANSPO~T PROPERTIES OF FREE MOLECULE (KNUDSEN) FLOW

BY

G. N. PATTERSON

TRANSPORT PROPERTIES OF FREE MOLECULE (KNUDSEN)FLOW

BY

G. No PATTERSON

.'

SUMMARY

The subject is introduced by a brief review of sorne physical properties of the upper ahnosphere and the three regimes of low density flow. The present state of knowledge of the interactioll of gas molecules with the surface of a solid body as expressed by various acc.ommodation coefficients is reviewed. The importal'lce of the function which describes the distribution of .m olecular velocities is indicated for both isentropic and nonisentropic flows. Consideration is then given to the determin-ation of the aerodynamic forces and heat transfer in free molecule flow. The application of the theory to free molecule probes and the use of the equilibrilLlAl te::nperature probe in observing the properties of the shock transition are discussed. A wind tunnel for low density studies is also described. Some extrapolation into the realm of hypersonic flow con -cludes this discussion.

( i )

TABLE OF CONTENTS

Page

NOTATION ii

1. INTRODUCTION 1

1I. REGIMES OF LOW DENSITY FLOW 1

lIl. MOLECULAR REFLECTION - MOMENTUM

ACCOMMODATION COEFFICIENTS 2

IV. THERMAL ACCOMMODATION COEFFICIENT 4

V. VELOCITY DISTRIBUTION FUNCTION 5

VI. TRANSPORT PROPERTIES OF FREE MOLECULE

FLOW AT A SURFACE ELEMENT 8

VII. FREE MOLECULE AERODYNAMICS 11

VIII. FREE MOLECULE PROBES 12

IX.

LOW DENSITY WIND TUNNELS 16

X. INVESTIGATION OF THE SHOCK TRANSITION 17 XI. THE HYPERSONIC LIMIT OF FREE MOLECULE

FLOW AND NEWTONIAN FLOW 18

A A· ₁ c c· ₁

c

e E

f'

_o k Kn 1 m M ( ii ) NOTATION

area or coefficient (Eq. 15) according to context

coefficient related to heat flux (Eq. 12)

coefficient related to viscous action (Eq. 12) molecular speed

components of molecular velocity most probable molecular speed

rnolecular speed referred to cm drag coefficient

components of molecular velocity referred to Cm lift c oeffic ient

pr ssure coefficient

diameter

emissivity or exponential accord:ing to context 1

(d

u.

d

U

"

j

l

dUk

the tensor _ .1

+

J. - - - -

b"

2

0

Xj () xi 3

d

Xk IJ

energy of molecules or energy flux according to context

rotational energy of a molecule

general velocity distribution functions

Maxwell 's velocity distribution function

Maxwell form of velocity distribution function

suitable for diatomic gas

Boltzmann constant Knudsen number length of body mass of mol~cule

n N p pr q R Re S St t T x' 1.

O<R

( iii )

number of molecules per unit volume

nu.rnber of molecules striking unit area in unit time

pressure

Prandtl number

macroscopie speed

hea.t conducted by molecules

heat flow due to radiation

gas constant or flux of radiant energy according to context Reynolds number speed ratio (q/ cm) Stanton nu.mber time temperatur0 stagnation templE'rature

compene4 ts of macroscopie velocity position coordinaies

thermal accommodation coefficient or angle of incidGilCe according te context

accommodation coefficient associated with

rotational en.ergy of ths molecule thiclmess of the shock wave

the ratio ER

I

kT

inclination of normal of a surface to direction

of macroscopie motien

coefficient of heat conduction mep.n free path

A

A

./All

"

_1/

si

~ <TT T

Je:

c/w

( iv )

ratio of internal coefficient of heat conduction to the translation value (see Eq. 14)

coefficient of viscosity

coefficient of bulk viscosity

number of excited internal degrees of freedom of a molecule

components of molecule velocity

normal momentum accommodation coefficient tangential momentum accommodation coefficient shearing stress or relaxation time according to context

element of volume in physical space element of volume in the velocity space

( 1 )

I. INTRODUC TION (

This paper is presented from the point of view of the aero-nautical engineer. It is hoped that the following discussion of the

trans-port properties of very low density flows and their relation to problems of high altitude flight will be of interest and assistance to the chemical engineer who may wish to use the theory of free molecule flow in other applications.

From a consideration of adequate lift and acceptable skin temperatures the aeronautical engineer is led to the conclusions that

sustained flight at high speed implies flight at high altitude. Ultimately

the aeronautical engineer ' must focus hls attention on free molecule flow at hypersonic speeds (M> 5) and associated real gas effects due to

mole-cular vibration. dissociation, and ionization. However, in the present state of the subject the transport properties of free molecule flow are

considered from the point of view of the kinetic theory of gases which has been found to be adequate for supersonic speeds (1 ë. M <. 5).

REGIMES OF LOW DENSITY FLOWS !

Il.

The problems of high altitude flight have turned the attention of the aeronautical engineer to the physical properties of the earth's

atmosphere at great heights. These properties can be outlined briefly in terms of the variation of temperature with height indicated in Fig. 1.

The temperature faUs almost adiabatically in the lowest region of the

atmosphere. At about 2 km this trend is reversed and the temperature rises until an altitude of 50 km is reached. This rise results fr om an

increase in the ozone content of the atmosphere and the associated

absorption of solar radiation. From 50 km to 80 km this effect drops off as the ozone content decreases. Finally the rise in temperature

above 80 km is due to dissociation and ionization of the air in the upper atmosphere. Figure 1 is illustrative only and should not be taken too literally. A more exact knowledge of the physical properties of the atmosphere is one of the objectives of the current International Geophysical Year.

The mean free path (or the average distance traveUed by

the molecules between collisions) is of greater interest from the point of view of rarefaction effects in aerodynamics. The variation of mean free path with height is indicated in Fig. 1. C learly this quantity can have eventually a magnitude larger than the characteristic dimension of an aircraft. The ratio of the mean free path (

À.

)

to a characteristic body dimension

(p.. )

is called the Knudsen number (Kn

=

.A

/.J...

).

When

À is no longer negligible compared with.( • the gas does not behave like a continuum. As a gas becomes more rarefied~ the Knudsen number

increases and the associated rarefaction effects give rise to three essentially different kinds of flow. The first deviation from continuum flow called slip flow occurs in the range O. Ol

f.

Kn

f

0.1. When the Knudsen number is very large (Kn

>

10), free molecule flow occurs.

("

( 2 )

Transition flow is found in the intermediate raage O. 10 (Kn 1..10. These:: '.

ranges of Kn are arbitrary but they appear to correspond wUh existi,ng experimental information (Refs. 1, 2).

In the present state of the subject, rarefaction effects are considered either from t~e point of view of highly rarefied .gases (free molecule flow) or as a roodification of continuum flow produced by

low-ering the pressure. The tra.nsition region between these two extremes has been the subject of empirical investigation mainly,

since

no

adequate theory has been developed for th!s regime of flow.

Deviations from contimmm flow becorne appa.rent in

flight

at

altitudes from 30 to 80 km. Figure 1 indicates that free molecule now might be expected above about 120-140 km.

lIl. MOLECULAR REFLECTION - MOMENTUM ACCOMMODATION COEFFICIENTS

We can conceive of two kinds of reflection of gas molecules from asolid boundary. If the wan were perfectly smooth. it is possible that "Mirror-like" or specular reflection might occur in which the com-ponent of momentum of the incident" molecule norm al to the surface is reversed in direction but unchanged in magnitude on contact ~Nith the wall. In practice, howeverJ ths surface is rough and contains ÏI!ter-stices in which a gas molecule may be temporarily trapped.

Further-/-.."- more, the ultimate direction of reflection may have no relation to the

incident direction. This type of reflection will be described as diff~u5e in character. In diffuse reflection all directions of emission about the

normal to t.."f.J.e surface are equàlly p~obable, regardless of the dirl!ction

of impingement. More specifica.Uy. the probability that a molecule wi11 leave the surface at a particu ar angle is proportional to the cosine of the angle wUh respect to the norma.l. ln general the speeds of diffusely

re-flecting molecules are grouped according to a MaxweUian distribution corresponding to a temperature which can be diff.erent from that of thc surface.

Gas-surface interaction.s have been studied mainly by the molecular beam technique. The .method is surveyed in Refs. 3. 4. In

this method a stream of molec'.ll:es is direc!ed: on aplane surface element

and measurements of the flux of scattered molecules are made at

various angles relative to the incident beam (Fig. 2). The beam is

pro-duced by an orifice or a tube. The molecules emitted by the source move along diverging rectilim~ar paths. On reaching the orifice the properly orientated moleculas pass through and constitute the molecular beaml and those stopped by the orifice are drawn off by a pump •. The

bea.m passes through a region of high vacuum

UO-

6 mm4 Hg.) and strikes the test surface at a selected angle. The scattered molecules which reflect within a s.mall soHd angle pass into a detector and produ.ce a smal! inc.re.ment in pressure. 'I':!::e detector (essentially an ionization gauge) can be moved to various positions to determine the complete flux distribution.

(3 )

Of special interest to designers are the molecular beam tests of air molecules on typical materials used in aircraft construction. Hurlbut (Ref. 4) finds that the cosine law of scattering is valid for the spatial flux distrihution of air and nitrogen molecules reflected from

polished low carbon steel, etched low carbon steel and polished aluminum, independent 'of outgassing or surface temperature. Furthermore, the reflected molecules possessed a mean energy closely consistent with the thermal conditions of the wall.

On the other hand when air and nitrogen molecules reflected fr om a glass surface, small deviations from the cosine scattering

dis-tribution were detected. These deviations could be explained py assum-ing that while most of the molecules are reflected diffusely, the remain-tiel' are reflected specularly. However, experiments using other surfaces and gases show that large deviations from diffuse reflection can occur, and the above method of explaining the difference is in doubt since the large deviation may not be due entirely to pure specular reflection, inter-mediate types of interaction being quite possible.

We can characterize the actual reflection process in a given case in terms of over-all average deviations from "completely diffuse

reflection" for which the scattered (reflected) flux distribution obeys the cosine law and the emitted molecules are in Maxwellian equilibrium with the surface. The transport of momentum to and from a surface in dir-ections normal and parallel to the wall can be characterized in the form of accommodation coefficients. For tangential momentum exchanges we de fine

'C. _I. (1)

where

r;

and

'tI'"

are the actual incident and reflected fluxes of tangential momentum, respectively. Note that the definttion of

ar

has essentially the same form as 0( since

cr;.

= ('G-T

r ) / (Ti - 'Lw) and, for

per-fectly diffuse reflection,

r'w

=

O. Similarly, for the normal momentum exchange we may write

0-₁₁ =

-Pi - "Pr

-Pi -

"P'III

( 2)

where Pi and Pr are the actual incident and reflected fluxes of ;normal momentum, respectively, and Pw is the emitted normal momentum flux for completely diffuse reflection. These momentum accommodation coefficients have the following limiting values:

For completelyspe.cul~r reflection, For completely diffuse reflection

0;.=0;=0

0;..

ON

=

I

If the interaction involves a combination of completely diffuse and spec-ular reflection, then

cr;,

is not independent of

0;.

and only one of these

co-('4 )

efficients is needed (Ref. 5). In an actual physical case, however, it is

expected that OIt

J ~) o.'f'\Q.. ~ will be independent.

The above accommodation coefficients will be useful for the calculation of aerodynamic forces and heat transfer. They have been defined in terms of macroscopie quantities that can be determined

ex-perimentally. The molecular bearn experiments of Hurlbut (Ref. 4)

in-dicate that for ordinary aircraft materiais,

cr

_T

=

I •

IV. THERMAL ACCOMMODATION COEFFICIENT

We define the thermal acco.mmodation coefficient as follows.

(3)

where Ei and Er are the actual incident and reflected energy fluxes,

respectively, and Ew is the emitted (reflected) energy flux for completely

dilfuse reflection. We have thc following lirniting values:

For completely specular reflection. Ei = Er'

For cornpletely diffuse reflection, Er = Ew,

b(. = 0,

~=1.

The characteristics of the molecular motion of a highly

r,e.refied gas can be illustrated by considering an experiment for the

measurement of tbe thermal accommodation coefficient (Ref. 6). We

investigate the properties of the motion b~tween two infinitely long,

con-centric cylinders. When no gas exists between the two cylinders, the

flow of heat wiU be in accordance with the Stefan-Boltzmann law. If the

reflection of radiation is diffuse and the cylinders .are made of the same

material. then the heat flow due to radiation is

:

~

AI

(~'t_ 1";.4) _

i(\+fu)_~

A~

Aa.

(4)

where e is thc gray body emissivity, A is the area of a cylinder, and k

is Boltzmann's constant (see Ref. 6). The subscripts 1, 2 refer to the inner and outer cylinders, respectively. Tests can be run to determine

e, and QR can t.~en be calculated in subsequent tests in which a gas is

present.

If now a gas is allowed to fiU the space between the two

cy1i:nders such that its .mean free path is many times tbe intervening distance, then heat will be conducted between the surfaces by the mole-cules. By extending...Knudsen' s theory of conduction through highly

rarefied gases (Ref. 7). we obtain

9~

=

F(o<)

l~~ (~-T.)

1

,

J-rr

J

( 5 )

where Pi. Ti are the gas pressure and temperature and the function F

( Cl( ) contains a constant whichdepends on molecular weight. the ratio

of specific heats. and the geo:metry of the apparatus. This equation is

determined from a consideration of the translational energy transported

to and from each surface by the molecules. the effect of intermolecular

collisions being neglected.

Heat can be introduced into the inner cylinder by an

electri-cal method and drawn off the outer cylinder by a water jacket (Q). Then Qc = Q - QR' and we can find the thermal accommodation coefficient

from Eq. 5. By changing the outer cylinder. the value of

c:<

for a

number of materials can be found.

Values of c:>( obtained by this method are as follows {Ref. 6): Type of Surface

Machined Aluminum Etched Bronze Polished Bronze Etched Aluminum Etched Cast Iron Machined Bronze

Flat Laquer on Bronze Polished Aluminum Polished Cast Iron Machined Cast" Iron

Value of 0( O. 95 - O. 97 O. 93 O. 95 . 0.91 - 0.94 0.89 - O. 97 0.89 - 0.96 0.89 - O. 93 0.88 - 0.89 0.87 - O. 95 0.87 - O. 93 0.87-0.88

At present both ~ and 0( are assumed to be constant for

a given gas and surface. independent of stream velocity or the

temper-atures of gas and surface. More comprehensive data are needed at

high stream (macroscopie) velocities and high temperatures. At present it appears that we may take

cr;.

=

1 and 0( = O. 9 for ordinary aircraft

materiais.

The above considerations hold strictly for a monatomic gasr In a diatomic gas other forms of the thermal accommodation coefficient must be considered. For example. if the energy of a gas arised from

the translation and rotation of its molecules, then the collision of such

molecules with a wa11 will involve accommodation of the rotationa!

energy and we must define a new coefficient

CE

~

_')

i -

(E

R

)r-(E

RL: -

(E"R

\v

V. VELOCITY DISTRIBUTION FUNCTION

(6)

In the kinetic theory approach to fluid mechanics the

---_._-~---( 6 )

scopic properties of a flowing gas can be determined from the collision information for assumed molecules and the distribution of molecular velocities (Ref. 1). Simple spherical or point-center molecules are sufficient to establish the coefficients of the basic velocity distribution functions from which may be calculated the characteristics. of isentropic and nonisentropic flows.

Let us consider an element of volume d

-r

containing a very large number of molecules nd-r . The fundamental question of kinetic theory is: How many of the molecules in d-C have velocities in a pres-cribed range

t

J t.,-I~

f,

(i

=

1. 2, 3) at a specific time t? Let us repres-ent the above rkngJ as ~n element of volume in a velocity space, d "<.)

=

d

f

~

t

dt

r.

.

If we plot the velocities of the nd"t molecules in the veloc1ity

s~ac:.

they will be scattered over all possible values. The number of points which are plotted in d ~ is the answer to our question above. If f is the density of points in dW. then the number of molecules in d"( which have velocities between

r:.

and

f,.

+

ol

l.,

(th at is. in dl.() ) is ndT. f. de..>. The symbol f is called the velocity distribution function. In general f depends on ti. xi. t.

The significance. of the velocity distribution function is seen in the following relation for the components of the macros~opic (stream. or mass) velocity,

"'lA,' l">ti

lt)

t

~

Ï ..

of

~"-l

(7)

where the integration is made over all pos~ible velocities of the molecule

( - ..0

<:.

~.

<..

o ( ) . The velocity distribution function is a "bridge" between th~ microscopic and macroscopic motion of the molecules. The velocity of a molecule can be referred to the macroscopic velocity in d ~

by the relation

(8)

The translational energy of the molecules with velocity components in dtO is 1/2 m

f..

f..

and he\ce the mean energy in d-r due to translational m otion is t (

~

r..

=;

+;:

~

+

~

l. ),

-krr

..

f.+dw

"l~f+

-:t""''ë'

(9) where 1/3 c 2 = RT. This result shows that the energy in d'r is partly macroscopic (visible) and partly microscopic (invisible). The latter is the internal (therrnal) energy.

The basic characteristic of isentropic flow is that the num-ber of molecules having velocities in the range

r;. .

ft

+"1:.

(that is.

{ 7 )

the number in d~) is unaltered by molecular collisions as we "follow

the molecules". The number lost by some collisions is regained by others and a condition of molecular equilibrium exists. This is called Maxwellian motion and is characterized by the velocity distribution function.

(è~ c/~)

(10)

Cl

Hol

Here cm is the most probable speed or the maximum point on the dis-tribution curve (Ref. 1, p. 36).

Nonisentropic flow involves viscosity effects and heat

con-duction and can be regarded as a ".slight deviation" from equilibrium or Maxwellian motion. The velocity distribution function for slightly non~ isentropic flow is (11) where

A.~

I

,

(12)

and eij is given in the list of symbols. Note that Ai and Aij result from heat conduction and viscous action, respectively.

The above velocity distributions apply strictly to a monatomic gas. The distribution funetion in Maxwellian form for diatomic mol

e-cules may be taken to be (Ref. 8)

r

l

-+0

_e.-f.

₍₁₃₎

i

lËJ

-=-

-c

o ~e t

ct.

where

ê

= ER

J

kT and (ER>' is the rotational energy of a molecule

in the

'l..

th rotational state.

~s

relation is based on the assumption that f may be factored into a translational term and a rotational term.

Note that we now have a translational temperature Tt and a rotational

temperature T R •

A distribution funetion which applies to both monatomic gases and a class of diatomic gases for which the exchange of energy between the translational and internal degrees of freedom is rapid may be expressed as follows (Ref. 9):

+:c. "

t I (EL) [\

+

A

(~~

-

1:'-)

+A ..

s{ (\ -

~

t')+1I.(I-

:<:ry

(14)

( 8 )

This function gives the num ber of molecules in a 6

+

Y dimensional

phase space having positions between xi, xi

+

cbq, velocities in the range

s:,.,

r;

+clJ,

and internal enrgy between EI~ EI

+

dEI' Note

that l

A

=

b

d'"\.\~

(15)

-p

J

-:>l'l<;

where / b is the bulk viscosity (Ref. 10). AlsoL\. is the ratio of the heat conduction coefficient for the internal energy to that for the random translational energy.

VI. TRANSPORT PROPERTJES OF FREE MOLECULE FLOW AT A SURFACE ELEMENT

. We shall now consider the transfer of mass, momentum, and energy to and from an element of surface of a body in a free molecule flow. The fundamental characteristic of free molecule flow is that on the average molecular collisions are very remote from the body and the transport of mass, momentum, and energy to a surface by the inçident·

molecules is independent of the transfer of these quantities away from the surface by the reflected molecules. In other words we can treat the incoming and emergent streams of molecules separately (Refs. 11,

12). The absence of molecular collisions irnplies that no macroscopic

changes wiU be produce'd in the gas motion by the body - there wiU be no boundary layer or shock .waves. In fact the gas cannot macroscop-ically sense the presence of the body. Basically, all that is involved is a reflection process.

Figure 1 shows that very large mean free paths occur in the upper atmosphere. It should be noted, however, that although the mean free path may be considerable~ the nurnber of molecules in the element of volume d is still large. For example, when the mean free path in .the upper atmosphere is 10 ft., the numbe r of molecules iri' a cubic inch

is still 1013 • The definitions of the macroscopic properties of a gas (p,

f

,~T. q) through the velocity distribution function are still valide

Let us calculate the exchange of mass, momentum, and energy in a free molecule flow at the surface of a. body assurning that

the molecular velocities are distributed according to Maxwell's law (Eq. 10). The number of incident molecules striking unit area in unit time ha ving velocity components in the range

f, ,

.

r

.

+d'F.

is

Î' l. I J , '

n( ~1"d.f.

d;f":j..df

3 · Then t~e total number of molecules incident on

unlt .tJiI'M I

.i

O,\-"I2.CL \ 0 IA..,.

~t

-t-IMe

\s

.

Ni

~

n,.

~

r,

ç~

Î>-

(~,+

cA

S.

(16)

( 9 )

(17)

where

(18)

The quantity S is called the speed ratio. For completely diffuse reflectron

(19)

Assu.ming that all incident molecules are reflected. then Ni

=

_Nw .

. The molecules which have veloeities between ~, and

f,

A ' t

+

dit,

transport momentum m (.1 normal to the surface. Then the con-tribuhon. to the normal pressure due to the momentum transported by the incident,molecules is

~ ~

1,> -;.

",",ni

~

A

f,

~~

;

ç

f,'-.f'

~

f'3

Cl _ <;;IoC - gQ r/'

(20)

Similarly the normal pressure due to the diffusely reflecting molecules is

(21)

where we make use of Eq. 19. Now the tot al normal pressure acting on the surface is

(22)

(see Eq. 2). Therefore ...0 oCI

-1'~Mni ~~Sl ~s~ ~~L-~)r'\~~~\)3(23)

which permits us to calculate the norm al force per unit area due to the incident and reflected molecules.

The incident molecules. having velocities in the range ~J •

{. +d-r.

transport to the surface a y-component of

tangentialmom-ent~m of arnount m

5: .

The tangential force in the y-direction result-ing from all incident

:n

olecules is

( 10 )

0 ( ) - 0 C><)

-z;;

~~""fs;

y(

~{-PJ ~

o -cP -00

(24)

From the characteristics of completely diffuse reflectionl

r

loV = 0 and

the total tangential force in the y-direction is (see Eq. 1)

T':.

T, -..,- - ,.,..- - r

t L r - 11 ï C /' (25) These relationsl applied to an element of area on a body and integrated over the whole surfacel wiU provide the resultant forces on various aerodynamic shapes in free molec"ule flow.

Let us now consider the balaIice of energy .for the element of surface ciA. RC2ferred to. unit area of dA, energy balance requires that

(26)

wher"e Eil Er are the incident and reflected fluxes of molecular energYI Ril Rr are the incident and emitted fluxes of radiant energyl and Q is the heat removed from unit area of dA from inside the body (a known quantity).

If the gas is diatornic and if we assume that only the rotation-al compDnent of the internrotation-al energy is significant 'compared with the translational energYI then the incident energy takes" the form

(27)

where according to the kinetic theory ER' the rotational energy of the molecules l has the value.kT. In accordance with preceding analysis

Qó 000 d>

(Er),

=

~

Mn,

)~Sl l~5L

G

cr"

t'.,)~

d!3

(28)

For completely diffuse reflection the corresponding expression for Ew is

(29)

Then

( 11 )

vn.

FREE-·MOLECULE-AElWDYNAMICS

The above analysis has been applied 0 the circular cylinder

in ttefs. 13, 14, 15. For a monatomic gas the coefficient of total drag

is.

(3I)

where 10 (S) and 11 (S) are modified Bessel functions of order 0 and 1,

respectively. Thus in free molecule flow the drag coefficient depends

on the speed ratio {S) and the temperature ratio CT r

I

Ti). The

ternper-ature ratio can be obtained from the energy balance equation for the

cylinder. 2

~

(Z,H.,)

_~St+2)Z",

-+

(S>+V"ZtJ

+

_~~l'

E- (

_RTt

3:..

fh.[

_J

_e.

e

_(T

_~

«t

_-

_{-ç ) -}

~

_{R -}

]

0 {32) where _{_ .L}

_st..

'\...

'Z1::1ïe.

IoCtS

2

)

Z

~ ~

Ir

S

2.

e -

t

$

2.[

16 (

t

~ ~)

-+

T \ (

ts ,) ]

{33)

and the reflection process is such that o:nly the thermal accommodation coefficient is retained ~see Section

4).

I .

The insulated cylinder is of special interest sllce it lends itself to a simpier experimental approach. In this case _{T r} "" T_W1 and

only the transfer of transla ional energy by the molecule is involved.

Then

Iw

--

I [

( :;; 2.-1-2

)-:t

I

+

(S'

~-J. ~

)

Z

2-2-

CZ,+""Z).)

and substitution in Eq. 31 gives

(34)

(35)

The drag coefficient and tem perature ratio are plo' ted

against speed ratio in Figs. 3 and 4, respectively. The experiments

( 12 )

agreed reasonably well with theory. In particular the experiments ver-ified a prediction of the above theory that Cn is independent of the Rey-nolds and Knudsen numbers. A small correction to the CD relation to allow for a diatomic molecule with both translational and rotational energy in accordance wUh Eq. 27 was found to be in the right direction (Fig. 4).

The interesting fact th at arises from the energy balance equation is that the insulated cylinder attains a temperature higher than the stagnation temperature of the stream (Ref. 14). For an insulated. flat plate aligned in the direction of flow of a monatomic gas, the energy balance equation becom es ..

-I+{S~

(36)

The stagnation temperature for a monatomic gas is

-l+~S4.

S

(37)

Therefore T w

>

T o.

Vill. FREE MOLECULE PROBES

One of the most important results of modern research in the field of free molecule flow has been the development of the free molecule probe. The construction of lQW density wind tunnels has made it possible to obtain flows in which the mean free path is appreciably larger than the probe. At the same time the nozzle and test section are large enough to ensure continuum or slip flow around models. Thus the performance of the probe can be calculated fr om free molecule theory, and then the probe can be applied to the investigation of more complex flows such as slip flow and shock transition. Such probes have the fundamental advantage that they possess no boundary layer or wave system and they do not disturb the macroscopic motion of the gas.

The work of developing free molecule probes was initiated at the University of California. The aerodynamics of the cylinder

(Section 7) was carried a stage further to include the case of the more general velocity distribution function for nonisentropic flows (Refs. 5, 16). Thus the properties of a cylindrical wire are known when the probe is placed in a boun~ary layer ~r shock wave. Free molecule pressure probes were subsequently developed at the Institute of Aerophysics (R,efs. 17, 18).

Let us consider the pressure probe in the form of an ori-fice in the side of a tube of small diameter (Ref. 17). The speed ratio, pressure, and temperature of the external gas are S, Pi, Ti, respect-ively. Inside the tube the gas is at rest and the pressure and temper-ature become pr and T r • Equation 17 gives the number of molecules

( 13 )

which pass through unit area of the orifiee in unit time if the random velocities of the incident molecules are distributed accordh'1g to Max-well's law. The nu.mber of molecules emerging from the tube through unit area of the orifice in unit time is given by Eq. 19 with the subscript

r replacingw. Equilibrium corresponds to the condition of no resultant

flow of mass through the orifice. Therefore m Ni = m Nr. or

(38)

The above result applies to the case in which the macroscopie motion of the external gas is directed toward the orifice. If the stream flows away from the opening. then

,

_

SL~~'

e.

Where

0<..

and o-t are the angles between the normal to the orifice and

the x-axis (~, r:>( f ~ ~). Now it has been found experimentally that

various orientations ,of the orifiee have no appreciabie effect on the

temperature inside the tube _{(T r =} T'r)' Furthermore, if we evaluate E qs. 38 and 39 above for: c< = C>( I = 0 and r;t( =0< I = 7T /2. we obtain the useful result

(40)

2-

GriJs

It appears therefore that a free molecule pressure probe can give a direct measurement of speed ratio in terms of the forward

(1

₀ ),

sideways (~), and rearward

(fo'>

facing pressures only. Equations 38 and 39 have been checkea experimentally (Fig. 5) in the UTIA low density wind tunnel (Ref. 18).

Equati'on 40 holds strictly when the external molecular motion is Maxwellian (see E.q. 10). The usefulness of the prohe can be

con-siderably increased by calculating its properties in a free molecule, nonisentropic flow (see Eq. 11). The calcul.a.tion for a two-dimensional

boundary layer) i. ld. !l 1 1 ! i ! l i F ; hl h can be deduced

from the following relations which hddfor the two-dimensional boundary layer-!fTr ='T'r and.~ =;.,,(1,

(41)

! .

( 14 )

(43)

(44)

For weak shock waves Eq. 40 will a1so hold. but when Al' AU become significant compared with 1, the more general relations must be used. The above results suggest that the pressure probe may be used to de-termine the individual values of the coefficients of the velocity distri b-ution function (Ai. Aij). In general this wiU be difficult since these cö-efficients are small for "Slightly nonisentropic flow" (see Section 5).

Let us now consider the properties of a cylinder in nonisen-tropie free molecule flow. The calculations outlined in Section 7 are repeated for the more general velocity distribution function (Eq. 11) with the following results (Ref. 5).

Ct>

~

C

Co -

(AI /2 S)

CD. -\-

AnC

bn -t-

A2-1.

CP

_Ll.. (45)

(46)

where the partial drag and Uft coefficients (CD! • . . . • . . . CL2) are functions of the speed ratio (S) and arise from the deviation from

Maxwellian (isentropie) flow. The relative importance of the drag terms is indicated in Fig. 6. Note that CD is the drag of the cylinder corresponding to Maxwellian motion. We c8nclude that the aerodynamic forces on a cylinder in nonuniform. free molecular flow are affected by the non-Maxwellian components of the distribution function. In general the nonisentropic effects are small except in regions of low speed ratio where forces due to the heat flux terms (AD become significant.

The energy balance equation for the cylinder must also be reconsidered using the velocity distribution function for nonisentropic flow (Eq. 11). The Stanton number is defined as follows,

l~

5

TT"

~

UI

R (

~w

-

-ç)d

(47)

where Q is the net heat 10ss per unit time per unit length, d is the dia-meter, T w is the cylinder temperature. and T aw is the equilibrium cylinder temperature. Then the heat transfer in nonisentropic flow is governed by the relation.

( 15 )

(48)

The partial Stanton numbers corresponding to the nonisentropic terms in the velocity distribution functiQn are small compared wUh Sto (Fig. 7). The energy balance is only slightly influenced by the viscous stress terms but. as in thè' case of aerodynamic forces, the heat flux contribution can

be appreciable when the speed ratio is low. It should be noted that the

Stanton numbers. are all functions of the speed ratio (S) and the thermal

accommodation coefficient ( C<). The basic fact indicated by the above analysis is th~ if we know the speed ratio, the accommodation

coeffi-cient, and the velocity distribution function for a given flow (Ai. Aij), then the temperature of the cylinder (or wire) can be calculated.

, The above results apply to a monatomic gas. For a diatomic

gas flowing in Maxwellian equilibrium we must use the appropriate

vel-ocity distribution function (Eq. 13). The method proceeds as outlined

above except that we now consider that the temperatures associated with translation and rotation are different and we must use two accommodation

coefficients (O(,o<'R' . Section 4). For molecular equilibrium the accom-modation coefficients and temperatures (T. TR) are essentially constant

in the integration since variations of these quantities have no effect on the integrations over the velocity space and around the cylinder (wire).

The energy balance equation now becomes (Ref. 8).

(49)

where the original definition of speed ratio is retained and a. bare functions of S (Ref. 8).

Let us consider 'two special cases of the energy balance

equation for a diatomic gas. If there is no accommodation of rotational

e.t:lergy, then c( R/!>( = 0, and

Iw

_-

(50)

which is the result for a monatomic' gas. This might occur if a very large nUmber of collisions must take place before the partition of

energy between the translational and rotaüpnal degrees of freedom

attains the equilibrium state; On the other hand, if the rotational

ac-commodation is as good as the translational, then o(~

=

0( and the

energy balance equation becomes

-

6t

(S)

J,(S)

(51 )

( 16 )

This question and the form of the energy balance equation for a cylinder

in the nonisentropic, free molecule flow of a dfatomic gas is discussed

in Ref.· 8,

IX. LOW DENSITY WIND TUNNELS

We shall consider briefly the m·ajor item of equipment used

in the laboratory investigations of rarefaction effects. It is fundamental

to modern developments in fluid mechanics that adequate test facilities

be available for checking new theories. The low density wind tunnel is

particularly important from this point of view.

Up to about 1947 no experimental work had been done on the

high speed flow of rarefied gases. Work then began concurrently at the

Ames Aeronautical Laboratory (NACA) and the University of California

on the development of low density wind tunnels. Some time later U953),

wit)1 heipful advice and encouragement fr om the research groups in the

above two organizations, the lnstitute of Aerophysics. University of

Toronto, undertook to develop a wind tun..'1el capable of operation at very

low densities.

Low density tunnels now in operatian. áre of !he continuous

flow, nonreturn, open-jet type. The test gas may be air taken from the

room through a dust filter and drier or boUled gas rendered dry -by

pas-sing it through a refrigerated trap at high pressure. The mass flow of

iniet gas is controlled and rneasured. It is heated to a desired

temper-ature and then it passes into a large settling chamber which contalns a

liner_heated to the same temperature. The gas passes next through the

nozzle forming a jet in a large test chamber in which the instrumentation

may be placed. The gas then proceeds to a surge chamber c,onnected to

the vacuum pumps.

The facility descrfued in n,ef. 13 is typical of the low density

tunnels developed by the Ames Aeronautical Laboratory and the

Univer-sity of California. A report containb_g a description of the UTIA low

. density wind tunnel is also available (Ref. 18). In tÏl..e subsonic range

(0..1 ( M

<.1.

0.) this wind tunnel was designed for operation at pressures

petween 1 and 70. mocrongs Hg., Reynolds numbers per inch from 0..0.8

to 70., and mean free paths between 2.0. and 0..0.2 inches. For supersonic

operation (1.0. l.. M

<.

5.0) the pressure range was,the same. and the

Rey-nolds number per inch ahd .mean free path varied from 10. to 40.0.0., and

0..2 to 0..0.0.2 inches, respectiV'ely.

The primary pumping system consists of six booster-type

oil diffusion pumps having a combined pumping speed of about 720.0. liters

per second over the range of operating pressures indicated above. The

booster pumps are connected~ to two large mechanical pumps which forms

a second stage. Conventional axisymmetric nozzles for Mach~ numbers

of 2 and 4 have been used. The design Mach number was achieved in the

center of the jet bu the core of uniform flow was limited to a relatively

( 17 )

measurement at pressures below about 10 microns was not practicable

because the boundary layer covered the whole jet. Very large slip

velocities were observed on the walls of the subsonic nozzle. Provision

has been made for boundary layer suction (Ref. 13).

x.

INVESTIGATION OF THE SHOCK TRANSITION

Investigations of normal shock waves are desirable for

var-ious reasons. The one-dimensionality of the flow simplifies the mathe-matical treatment. No solid boundary is involved as a condition on the

transition equations. The deviation fr om Maxwellian mblecular motion

depends on a single shock-strength parameter. This deviation can be .

considerable and the properties of nonisentropic flow made readily

apparent. This deviation is contained in the Ai. Aij terms in the n on-Maxwellian velocity distribution functio!! (Eq. 11) which involve the

viscous stress and heat flux. In diatomic gases the deviation also arises

from the internal degrees of freedom of the molecule.

Sherrnan recognized the desirability of investigating the structure of the shock wave in a low densi1:y win~ tunnel in which

effect-ive use could be made of free molecule probes~ (Ref. 8). The study of the internal structure of the shock wave is facilitated by increasing the

me an free path so that the shock thickness is about 5000 times larger and free molecule probes are practicable. Devices called shock holders

were developed to produce a normal shock wave. This device is a

thin-walled circular cylinder or cone. frustrum located in the region of uniform

flow in the test section. A moveable section is used to vary the area of the e'xit opening. As the exit area is opened up. the curved detached

sl'Aock front changes to a stable plane wave of the tupe assumed by theory. The profile was me~;sured in terms of the equilibrium

temp-erature of a fine wire embedded in the shock zone and aligned parallel to

the plane of the front. According to free molecule theory (Section 8) the

ternperature of the wire can be calculated ti the velocity distribution

funcHon is kn.<?wn. In general. when the mean free path is several times

the diameter. the stream heats the wire to a temperature which depends on the local speed ration (Mach number)~ static ternperature. and the

number of excited degrees of freedom of the gas molecules. This is

true if the cylindrical 'Wire is a perfect heat'conductor internally and is 'ree of radiation and end losses. Free molecule temperature probes . can take the form of either a temperature - sensitive resistance wire or

a butt-welded thermocouple. The resistance wire responds to an inte.

-grated àverage temperature over the exposed length and is subjecfto

thermal end·losses. Thermocouple probes are sensitive only at the -junction. On the other hand the resistance wire probe can be made in

smaller diameters and can be subjected to tension for alignment

pur-poses.

.rhe transition in te.rnperature through the normal shock

wave was measured by Sherman who used an equilibrium temperature

( 18 )

with the variation of temperature through the shock front calculated on the basis of the Navier-Stokes equations. The comparison was satis-factory if the bulk viscc;>sity ( ) was inc1uded in the theory.

The shock wave thickness was determined from profiles of

this type, and a similar comparison between theory and experiment was made (Fig. 8). The thickness is basecl on the maximum slope of the transition curve and it ie plotted in reciprocal form against the Mach nurnber upstream from the shock front. Shock thicknesses obtained by Greene. Cowan. and Hornig (Refs. 10. 20. 21) are also shown in Fig. 8. These investigators used the measurement of the optical reflectivity of shock fronts in a shock tube to determine the shock thickness.

It is shown in Fig. 8 th at the Navier-Stokes equations are sufficient for the description of the transition through the shock front in air for upstream Mach numbers up to 2. The equations must include a

bulk viscosity. = 2/3 • to account for the effects of rotational relaxation phenomena which tend to increase the thickness of the wave. XI. THE HYPERSONIC LIMIT OF FREE MOLECULE FLOW AND

NEWTONIAN FLOW

The limit of tree molecule flow as the speed ratio or Mach

number tends to a very large value is now of considerable interest to

the aeronautical engineer. In order to overcome the pro lem of aero ... dynamic heating at hypersonic speeds (M>5). it is necessary to fly at very great altituQ.es. A ~owledge of free molecule flow at a large speed

ratio is alsa important in th.e problem of the re-entry of satéUites and

space eraft into thc eartn's atmosphere.

On the assumpUolrA fuat the transport processes of !he inci .. dent and reflected molecules are independent (free molecule now)~ the

coefficient of lUormal pressure acting Olll a flat plate inclined

at

an angle

G

to the direcUon of t~ maas flow is given by {Ref.

U

C

-=

~'l.e

[..L

(.1-

+

J-

J

1r-

)e.-Srv"L-f

SN

Ir

'2. .SN

r;.

(52)

The co:rresponding relat!on for the coefficient of skin friction iS

[ J

-Sr.")....

J

Cf

-=~é ~e Gr~e.

+1

+~SI'4

(53)

In theile relations S f'I :: S sin ~ where S is the speed ratio, and it bas been asswned that <:ï~

=

(J-T ; 1.

,-( 19 )

The resultant force on the flat plate depends on the ratio of the temperatures associated with the reflected and incident molElcules

CT; /-,;. )

as well as the speed ratio and angle of incidence. This

temper-ature ratio can be calculated from the energy balance equation usually

involving radiation effects, ànd the accommodation coefficient (Ref. 1).

When cthe speed ratio becomes very large (hypersonic

condit-ion in free molecule flow),

(54)

The first limit (ç.) will hold only if

-r;.

frt;

remains finite while S

becomes large.. The physical significance of these limits can be readily

seen. As the speed ratio becomes very large, the macroscopic (mass)

velocity of the molecules becomes much greater than the most probable

speed of the random motion, and the molecular motion assumes

effect-ively a simple form in which all molecules are moving in parallel paths

at the same speed. Then the number of molecules st:riking unit area in

unit time is V\'\1"si,n& • The total normal and tangential momentums

carried by the incident molecules to unit area of the surface aref""'~""e

(p

= mn) andt>v'a.~6 ~

e,

respectively where '\Tis the component of

-velocity normal to the surface. The limiting values given in Eqs. 54

above imply that the reflection process makes no appreciable contrib

-ution to the exchange of norm al momentum, that is, the norm al

momen-turn of the incident molecules is "destroyed". Of course. the random

nature of the reflection is such that the emergent molecules make no

resultant contr:ibution to .tangential momentum for any value of the speed

ratio (Ref. 1).

The above limits are determined on the assumption that the

kinetic theory of gases can provide an adequate description of the reflection

process. However. when the incident molecules have very high energies,

the collisions with the wan may deviate appreciably from the elastic

type, i. e. they may become "plastic". Wh en high energy molecules are

temporarily· trapped by the wan, they may be capable of exciting rotations

and vibrations of the crystal atoms. The vibrational amplitudê of the

atoms of an iron crystal may be increased to such an extent that local

mf1lting occurs on the wall. In this connection it is interesting to note ,.

that for melting of iron an energy of 0.6 eV per ato:m is required. On

the other hand the kinetic energies of a nitrogen molecule at speeds of

100 and10, 000 m/sec. are about 0.001 and 12.2 eV, respectively (Ref.

22). It appears to be possible that the temperature associated witH the

translational motion of the molecules emerging from the wall win be high

cornpared with the temperature of the incident molecules as the speed

ratio increases (Ir-

/T;.

becomes large as S ~oO ). There~ore, the

-limiting value for Cf' given in Eqs. (54) above may be more accurate if

the ratio Tr

I-rz.

is retained in some form .

It is also possible th at the reflection process will involve

( 20 )

of the reflection of high energy molecules from a wall must be left for further theoretical and experimental investigation.

It is interesting to compare the limit of free molecule flow as the speed ratio increases to a large value with a flow described by Newton. Newton assurned an inelastic collision between a gas molecule and the surface of a body in which the normal component of molecular velocity was destroyed and the tangential component remained unchanged. The nuniber of molecules striking unit area of a flat plate in unit time is n sin and each molecule experiences a change in normal momentum

of

1'l?

9..

sin

e.

Therefore the normal pressure on the surface is (Ref. 23)

(55) which agrees with Eq. 54.

According to Newton's theory, all thereflected molecules move along the surface with an unchanged tangential velocity q cos

e .

In this respect Newton's theory differs from the hypersonic limit of free molecule flow (see Eq. 53).

In Newton's theory no assumption is made about the density of the particles striking a body. If we consider a continuum flow in which the density is kept constant but the Mach number is allowed to become indefinitely large, then the random motion of the incident mol-ecules becomes insignificant compared with the macroscopic motion and the latter tends to become that described by Newton. Thus the transport process involves collisions of gas molecules wit~ the body and encounters between gas molecules themselves are not important by comparison. It appears, therefore, th at both rarefaction and high ~peed. tend to produce a free molecule transport process. .

. An interesting verification of the limiting expressioll lor C is given in Ref. 24. The normal pressure around a hemisphere-cylinder model was investigated in a shock tube which simulated condit10ns of .

hypersonic flight (Fig. 9). Since measurements. were made on Mach waves. the results apply only to the supersonic region of the flow. In spite of a complex flow pattern involving a detached bow wave, expansion waves, and a boundary layer, the val~e of Cf given by Eqs. 54, 55 was vel"ified.

1. 2. 3.

4.

5. 6. 7. 8. 9. 10. 11. 12. 13. Patterson, G. N. Schaaf~ S. A. Smith~ K. F. Hurlbut~ F. C. Bell~ S I' ~ and Schaaf~ S. A. Wiedmann~ M. L. and Trumpler~ P. R. Knudsen~ M. Sherman, F. S. Wang, Chang~ C. S. and Uhlenbeck, G. E. Rosenhead, L. and others Stalder ~ J. R., and Jukoff~ D:. Loeb, L. B. Stal der , J. R., Goodwin, G., and Creager, M. ( 21 ) REFERENCES

"Molecular Flow of Gases", John Wiley and Sons, New York. 1956

"Rarefied Gas Dynamics" , Applied Mech-anics Reviews, Vol. 9, No. 10, p. 413, 1956 "Molecular Beams", Methuen & Co., 1955 "An Experimental Molecular Beam Investi-gation of the Scattering of Molecules from Surfaces" , Report No. HE 150-118, Institute of Engineering Research, University of

California, 1953

"Aerodynamic Forces on a Cylinder for the Free Molecule Flow of a Nonuniform Gas," J. Am. Rocket Soc., Vol. 23, p. 314, 1953

"Thermal Accommodation Coefficients", Trans. A. S. M. E., Voi. 68, 'p. 57, 1956 "Die moleculare WM.rmeleitung der Gase und der Akkommodationskoeffizient", Annalen

-der Physik, Vol. 34, p. 593, 1911

"A Low-Density Wind-Tunnel Study of Shock Wave Structure andRelaxation Phenomena in Gases", NACA Tech. Note 3298, 1951 "Transport Phenomena in Polyatmic Gases" , Engineering Research Institute. University of Michigan, Report No. CM-681. 1951

'IA Discussion of the First and Second Vis' -cosities of Fluids, " Proc. Roy. Soc .• No. 1164, Vol. 226, Oct. 1954

"Heat Transfer to Bodies Travelling at High Speed in the Upper Atmosphere". NAG!\.

Report No. 944, 1949 .

"The Kinetic Theory of Gases" , McGraw-HilI Book Co., New York, 1934

I'.A

Comparison of Theory and Experiment

for High-Speed, Free-Molecule Flow". NACA Report No. 1032, 1952

14. Stalder. J. R., . Goodwin. G .• and Creager, M. O. 15. Oppenheim. A. K. 16. ' BeU. S •• and Schaaf. S. A. 17. Patterson. G. N. 18. Enkenhus. K. R. 19. Co:wan, G. R. and Hornig. D. F. 20. Greene. E. F., Cowan._ G. R •• and Hornig. D. F. 21. 22. 23. 24. Greene. E. F. and Hornig. D. F. Sanger. E. Zahm. A. F. Rose. p. H. ( 22 )

"Heat Transfer to Bodies in a High-Speed Rarefied-Gas Stream", NACA Report No.

-1093. 1952

-"Generalized Theory of Convective Heat Transfer in a Free":Molecule Flowrr•

Insti-tute of Engineering Research. University of California. Report No. HE-150-93. 1952 "Heat Transfer to a Cylinder for the Free Molecule Flow of a Nonuniform Gas", Jet Propulsion, April, 1955

"Theory of Free-Molecule) Orifice-Type.

Pressure Probes in Isentropic and

Nonisen-tropic Flows~' Institute of Aerophysics.

University of Toronto. Report No. 41, 1956 "Pressure Probes at Very Low Density;! Institute of Aerophysics, University of Toronto. Report No. 43. 1957

"The Thickness of a Shock Front in a Gas". Metcalf Research Laboratory. Brown Univer-sity, Tech. Report No. I. April 6. 1949 "The Thickness of a Shock Front in Argon

and Nitrogen and Rotational Heat Capacity

Lags" , J. Chem. Phys •• Vol. 19. No. 4, p. 427. 1951

"The Shape and Thickness of Shock Fronts

in Argon, Hydrogen. Nitrogen and Oxygen", Metcalf Research Laboratory, Brown

University, Tech Report No. 4. August I.

1952

"Gaskinetik sehr grosser FlughBhen", Schweizer Archiv fur angewandte Wissen-schaft und Technik, Vol. 16. p. 43, 1950

"Superaerodynamics". J. Franklin Inst. ,

Vol. 217, No. 2, p. 153, 1934

"Physical Gas Dynamics Research at the AVCO Laboratory", Research Note No. 37. AVCO Research Laboratory. May 1957

MEAN FREE PATH, CM

o

500 1000 1500 2000 160 ~ ::::c::

-w o =>

r-IOO~---+----~~----~r---~ ~ 50r---+---+-~--~---~ O~

____

~

______

~

____

~

______

~ 150 200 250 300 350 TEMPERATURE, oK

FIG. 1 APPROXIMA TE PHYSICAL PROPERTIES OF THE UPPER ATMOSPHERE

FIG. 2 OCh ::u 0 -c: :!!::u 0 0 IT!IT! 0 0 ::UIT! -"'11 "'11-_z

MEASUREMENT OF THE FLUX OF MOLECULES SCATTERED FROM ASOLID SURFACE (REF. 4)

FIG. 3

T

_w Ti 4 o EXPERIMENT FREE MOLECULE 3 THEORY 2~---~---+~~---+----~ o / / ~ 0 / ~ CONTINUUM o /----THEORY ;;.-/

---I ~~~ ____ ~ __________ - L _ _ _ _ ~

o

2 SPEED RATIO

CONVECTIVE HEAT TRANSFER TO A CYLINDER IN FREE MOLECULE FLOW (REF. 13)

FIG. 4

15 0

10

\

0

5

- - MONATOMIC GAS THEORY - - - - DIATOMIC GAS THEORY

0 0 _HELIUM

•

NITROGEN

"'

o "" -- __ o 0 - _ o ~9:D • • • 0 • 0 2 SPEED RATIO

-3

DRAG OF AN INSULA TED CYLINDER IN FREE MOLECULE FLOW (REF. 13)

FIG. 5 3 ~~-.----.----.---. M = 0.86 P, = 8.4 microns Hg Knudsen No. = 6.8 ~ 2 r----t--~t---~E~X~PEQR!!!IM~E~N~T' !;;( 0 Righl side

a:: 6 Lef! side

~ - -Free molecule ::> flow Iheory en en LLJ a:: Cl. o o L -_ _ - L _ _ _ _ L -_ _ - L _ __ _ L -_ _ - L _ _ ~ o 30 60 90 120 150 180

ANGLE OF ROTATION 8 (DEGREESl

PRESSURES RECORDED BY AN ORIFICE PROBE IN FREE MOLECULE FLOW (REF. 18)

FIG. 6

4.---.-~----~---~---_.--~

3 r--+---+---f"'.C

Do/SENT ROPIC (MAXWELLlAN)

2r-~~--+---~---+---~~~

2 3 4

MOLECULAR SPEED RATIO, S =

V

r

'

M

COMPONENTS OF DRAG ON A CYLINDER IN NONISENTROPIC FLOW (REF. 5)

FIG. 7 STANTON NUMBER/ ex

g

0 0

o

0 0 o~---.---r---'

s:

0 I fTl (") C I l> :::0 Cf) "'U fTl fTl N 0 :::0 l> -i 0 Cf) VJ 11

~

s:

..t>

PARTlAL STANTON NUMBERS FOR A CYLINDER IN NON-ISENTROPIC, FREE MOLECULE FLOW (REF. 16)

FIG. 8

INITIAL MACH NUMBER

2

3

4

5

NAVIER-STOKES, BULK VISCOSITY = 0

.15f----"

A----/-/"'I-+/~

H

-

---l

;(' / / ~ -,~==~~====~--~ I NAVIER- STOKES, I BULK VISCOSITY = ~ fL I

6

·°'t-i;f----;:::::::C:::=:::::::::::::::::::±=======:::±::=~_____i

• WIRE TEMPERATURE THICKNESS - AIR o DENSITY THICKNESS, GREEN ET AL. - NITROGEN

o

'---i

t:. DENSITY THICKNESS, GREEN ET AL. - HYDROGEN fL* is a reference viscosity

VARIATION OF SHOCK TIllCKNESS WITH MACH NUMBER (REF. 8)

FIG. 9

t

1.0

p p.-p ... 2 X NEWTONIAN PRESSURE DISTRIBUTION P, • I-"""'P.'" Sin (Rl

SHOCK TUBE Ms =.8.8, PI = 10 cm, ('=1.19 o TOP SURFACE , }

D BOTTOM SURFACE MACH LlNE

MEASURE-MENTS SONIC PRESSURE

Ë

/

11---========~~~=~R~A~T1~O~F~OR~~~I.~191---_ • 0.5 --~. _--x _ D C IS 0 7'!f>' 90' 00 _{2 3} x

--

R

PRESSURE DISTRIBUTION ON A HEMESPHERE-CYLINDER MODEL CALCULATED FROM MACH LINES AND COMPARE.D WITH THE NEWTONIAN PRESSURE DISTRIBUTION (REF. 24)