Molecular approach to problems of high-altitude, high-speed flight

MAY, 1957

MCLECULAR APPROACH TO PRCBLEMS CF HIGH-ALTITUDE, HIGH-SPEED FLIGHT

BY G . N. PATTERSON

<

m :;00 VlO m() C :;o ~ :> m "UZ - i "U:>

0 ...

~m UTIA REVIEW NC. 10 - " rt"l

,....

.. ,, : < z ,...v;

ffiS:

!l~ c - Z

°0

O:I~ O ~ c V'I :E'-" _{A Z} c O z O O r -mo

5

·

,.

,

MAY, 1957

HIGH-ALTITUDE, HIGH-SPEED FLIGHT

BY

G. N. PATTERSON

FORWARD

This review was prepared for a meeting of the Wind Tunnel and Model Testing Panel, Advisory Group for Aeronautical Research and Development, scheduled for July 8-11, 1957 at Scheveningen, Holland.

This review was made possible by the U. S. Air Force Office of Scientific Research under Contract No. AF 18(600) -1185.

'

.

\,

.SUMMARY

In order to obtain a better understanding of the aerodynamics

.

of high performance aircraft it is necessary to investigate the effects of rarefaction and high temperatures on the flow of gases. Both effects show' the increasingly important role played by the molecular structure of a gas in its macroscopie motion.

In the introductory sections of this review it is pointed out that flight at very high speed can only occur at extreme altitudes. The zones of the upper atmosphere, the regimes of fluid mechanics based on considerations of rarefaction and temperature, and the modern experimental methods for investigations in some of these regimes are discussed. Rarefaction effects are then discussed in terms of gas-surface interactions , free molecule flow and slip flow, and Newtonian flow is considered as an extension of free molecule motion. The effects of high temperature are indicated for undissociated air and air in

dissociation equilibrium, and the important effects of relaxation are outlined.

" Page NOTATION ii 1. Il. INTRODUCTION 1. Flight Corridor

2. 'Physics of the Upper Atmosphere 3. Regimes of Fluid Mechanics 4. Exper'imental Methods

RAREFACTION EFFECTS 1 1 1 3 4 6

5. Interaction of a Gas Molecule with aSolid Surface 6 6. The Ve locity Distribution Function 9

7. Free Molecule Aerodynamics 12

8. Free Molecule Probes 17

9. Slip Flow 21

10. Free Molecule Properties of Flow at Very High 25 Mach Number

lIl. mGH TEMPERATURE EFFECTS 27

11. Distribution of the Internal Energy 0 f Air in 27 Equilibrium

12. Relaxation Effects 28

IJ-l .. Properties of Undissociated Air at High 29 ' ... '1'

Temperature

14. Properties of Air in Dissociation Equilibrium 31 15. Shock Wave Structure and Associated Relaxation 32

Phenomena

16. Relaxation Phenomena in Hypersonic Flows 33 17. Relaxation Phenomena in Hypersonic Wind Tunnels 35

\ .' A

A

.

1 A .. IJ c C. 1 C

C

n

C. "I d e

ER

f, f I f 0 fi 0 g h k

'

"

rn M ( ii ) NOTATION ,

area or coefficient (Eq. 6.9) according to context coefficient related to heat flux (Eq. 6.6)

coefficient related to viscous action (Eq. 6.6)

molecular speed

components of molecular velocity most probable molecular speed molecular speed referred to cm drag coefficient

components of molecular velocity referred to cm specific heat at constant pressure or pressure coefficient according to context

diameter

emissivity or expon,ential according to context

1 Ö Ui

0

ti

.'

,

1

0

u ' (.

the tensor-('lI!...₂ _

1-~)

- -3

-

ö

K._,C iJ'

(:JX' x· xw

J "",· 1 n

energy of molecule or 'energy flux according to _

context

rotationaJ energy <of a molecule

general velocity distribution function

:Maxwe.G.lJ.'!siv-ètlo&ä.t.yddiSlt:Hibuition:'tundion

Maxwell form of velocity distribution function s uitable for diatomic gas

acceleration due to .gravity height

Boltzmann constant length of body

mass of molecule Mach riumber (q/ a)

n N p Pr q r R Re S St t T t max. u. 1

x

₁

·

d..

R

number of molecules per llll.it volume

number of molecules striking unit area in unit time

pressu~e

Prancitl number macroscopie speed

heat conducted by molecules heat flow due to .radiation

recovery factor or radius according to eontext gas constant Reynolds number speed ratio (q/ cm) StàhtOh nUmbei' time temperature equilibrium temperature stagnation temperature maximUIn reaction time

components of ma<troscopic velocity position coordinates

thermal accommodation coefficient or angle of

incidence

accommodation coefficient associated with rotational energy of the molecule

the ratio of specifie heat at constant pressure to specific heat at constant volumè

,

..

~

_m

E

e

A

cr

_T

dw

( iv )

thiekness of the shock wave

the ratio ER/kT

inclination of normal of a surface to direction of

macroscopie motion or molecular vibration

temperature according to cont.".;;;'

coefficient of heat conduction

mean free path

ratio of internal coefficient of heat conduction to

the translational value (see Eq. 6.10) coefficient of viscosity

coefficient of bulk viscosity

number of excited internal degrees of freedom of

a molecule

components of molecular velocity

'".q.

normal momentum accommodatic~m c ffieient

tangential momentum accommodation coefficient

shearing stress or relaxation time according to

context

element of volume in physieal space

'

.

1. INTRODUCTION

1. Flight Corridor

The requirements of modern sustained flight involve certain limitations which must be carefully considered.. At a given altitude the flight speed must be sufficiently high to provide adequate lift but low enough to avoid excessive skin temperatures , For flight speeds above

1000 ft./ sec. we may take the dynamic pressure necessary for lift to be about 80 lb. /sq. ft. If a skin temperature of about 10000_{C is}

permissible, then a "corridor of flight" exists as shown in Fig. 1. On one side of this corridor the aircraft will be too slowand on the other side too hot. This corridor would be cut off for dynamic pressures

below 40 lb./sq. ft. and a permissible skin temperature of about 8000C. _ Figure 1 is a modification of a diagram prepared by Haber (Ref. 1) and improved by Williams (Ref. 2).

lt appears therefore that steady flight at high speed

automatically implies flight at high altitude. In this paper we will be concerned with the aerodynamic problems associated with rarefaction effects (high altitude) and high temperature effects (high speed). For convenience these effects will be considered separately but it should be noted that in flight they can occur simultaneously. Our object will be to present a survey of these effects and try to indicate where the emphasis in research should be placed in the future .

2, Physics of the Upper Atmosphere

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 (Ref. 3), The term upper atmosphere may be loosely used to denote the regions of the earth's atmosphere above 30 km. (18.6 mi.). The lowest region of the atmosphere is called the troposphere (Fig. 2). As the altitude increases through this zone the temperature and pressure drop almost adiabatically. The tropopause is reached at temperatures of 200-2200K. lts height varies from 6.km. (3.7 mi.) above the poles to 18 km. (11.2 mi.) above the equator. Above the tropopause is the stratosphere, an isothermal region of about 10 km.

(6.2 mi.) height depending on latitude. The region above the stratosphere (30 km. or 18.6 mi.) extending outward to interplana..tary space is termed the upper atmosphere.

Above 80 km. (50 mL) the atmosphere is ionized and this region is called the ionosphere. The ionosphere is subdivided accord-ing to the intensity of ionization. The E - layer is moderately ionized and it is situated at about the 100 km. (62 mi.) level. The F - layer is more strongly ionized and occurs at about 200-300 km. (124-187 mi.),

A pictorial representation of the physical properties of the atmosphere is given in Fig. 2 (Ref. 4). This figure is illustrative only and should not be taken too literally.

( 2 )

Sounding rockets are used to obtain data regarding the upper

atmosphere. The 'ambient pressure is measured by a tap on the surface

of the rocket located at a point where the local surface pressure is

within a few percent of the ambient value and is essentially independent

of the Mach and Reynolds numbers. The ambient air density is

determined from the measured pressure according to the relation

p

=

- U/

g) (d p/dh). This formula does not involve the mass of the molecule

which is not known for great heights. However, above 100 km. (62 mi.)

the mean free path of the molecules is no longer small and at this and

greater altitudes the de11sity must be calculated from kinetic theory

considerations (Ref. 5). The temperature is determined from the

pressure and density.

Some physical properties of the upper atmosphere are shown

in Fig. 3 whlch is based on data given in Ref. 3. In obtaining this

information certain assumptions must be made regarding the molecules.

It is considered that no change in molecular weight occurs up to a height

of 80 km. (50 mi.). The dissociation of diatomic oxygen into atomie

oxygen takes place a altitudes between 80 km. (50 mi.) and 120 km.

(75 mi.) and the dissociation of diatomic nitrogen into atomie nitrogen

occurs at 130-250 km. (80 - 155 mi.).

Referring to Fig. 3 it wil! be seen that the temperature falls

in the troposphere. At about 20 km. (12.5 mi.) this trend is reversed

and the temperature !'lSes to about 2700K at 50 km. (31 mi.). This rise

resu.lts from an increase in the ozone content of the air and the

associated absorption of solar radiation. From 50 km. to 80 km. (31

-50 mi.) this effect drops off as the ozone content decreases. Finally

the rise in temperature above 80 km. (50 mi.) is due to the dissociation

and ionization of the llpper atmospheric air.

The composition of day air under standard sea level

conditions is (NACA standard atmosphere}

Constituent nitrogen (N 2) oxygen (02) argon (A) Mole Fraction 0.7812 0.2095 0.0093 Molecular Weight 28.016 32.000 39.944

The analysis of samples of the a.tmoRphere at various. altitude,s has

indicated that the composition of the air remains practically constant

up to 80 Km. (50 mi.). A consideration of the action of gaseous

diffusion in the atmosphere under equi.librium conditions and no mass

motion indicates that the lighter gases should be more prevalent at

higher altitudes. However. the macroscopic motion of the atmosphere

produces a considerable mixing effect which leaves the compoBition

effectively unchanged. Above 80 km. (50 mi.) the content of the

3. '.Regimes of Fluid Mechanics ,.

Classical hydrodynamics was the beginning of flu.id mechanics. At this stage of progressa fluid was assumed to be frictionless,

incompressible. continuous , and chemicaUy invariant, and considerable attention was given to the lift force on .a body. When the restriction of an inviscid fluid was dropped. the subject of aerodynamics emerged which provided .a better understanding of the drag of a body. The steady improvement in the performance of aircr-aft soon made it necessary to remove the limitation imposed by the assumption of incompressibility. and the modern subject of gas dynamics was born.. The emphasis in research was now placed on the study of major compressibility effects such as the critical Mach number and shock waves. The stage has now been reached when theassumption of continuity must be carefully

s,crutinized. It has become apparent th at flight through theearth's

atmosphere involves rarefaction and high temperature effects which can only be explained on the ba.sis of the molecular properties of gases. The advent of very high temperatures indicates that even chemical invariance may no longer be valid.

It is natural to consider the 'molecular flow of gases initially

from the point of view of the 'ktinetic theory (Reft; 6). In this theory the molecules are perfectly elastic spheres or point centers of force which possess only three translational degrees of freedom. Theaverage dista,nce travelled between collisions (or the mean free path) is a básic characteristic of the molecular motion. The ratio of the 'mean free· path ({\) to a characteristic body dimension (~) is called the Knudsen number

(Kn

=

'AI;' ).

When

l\

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 threeessentially different kinds of flow. The first

deviation from continuum flow called slip flow occurs in the range 0.01

<

Kn

<

0 . 1. When the Knudsen number is very lar ge (Kn> 10)>> free molecule flow occurs. Transition flow is found in the intermediate range O. 10

<

Kn

<

10. These ranges of Kn are arbitrary but they appear to correspond with existingexperimental information (Ref. 7).

The fundamental assumption of the kinetic theory of gases is that the internal energy of the gas is contained in the random

translational motion of the molecules. As the temperature of the gas .

. rises above 1000oK, it becomes increasingly apparent that considerable internal energy is also stored up in the internal motions of the molecule

(rotation, vibration) , and in the excited electroiüc states and ionization.

At high temperature part of the internal energy of the gas will arise \

from diss,ociation and the formation of new components. During a transition in the macroscopic properties ,of a gas in a flow, the gas requires.a finite "relaxation time" to reach equilibr~um in both

composition and distribution of energy between the various degrees of freedom of its molecules. Below a M.bEnumber of 6 ~md analtitude of

4ëf.1J.H

1

Lft

:

.L

(l

;,2;

km;}no':appreciable.rrelaxation effects occur in flight since

translational and rotational degrees of freedom. This is the regime of

supersonic flight. When T> 1000ÛJ{ relaxation effects are important

and we enter the hypersonic region of flight ior which 7

<

M

<

24 (2000oK

<

T

<

80000K) .

4. Experimental Methods

We shall consider briefly the major items of equipment used

in the experimental investigation of rarefaction and high temperature effects . It is fundamental to modern developments in fluid mechanics

that adequate test facilities be availa'ble for checking new theories. Two

sucn faci.lities are the low density wind tunnel and the shock tube.

Up to about 1947 no experimental work had been done on the hi.gh speed flow of rarefied gases. Work then began concurrently at the Ames Aeronautical Laboratory (NACA) and the University of California on the developmen of low density wind tunnels. Some time later (1953),

with helpful advice and' encouragement from the research groups in the

above two organizations. the Institute of Aerophysics, University of

Toronto. undertook to develop a wind tu.nnel capable of operation at veyy

low dens i tie s .

Low density tunnels now in operation are of the continuous flow, nonreturn, open~jet type. The test gas may be air taken from the

room through a dust filter and drier or bottled gas rendered dry by passing it through a refrigerated trap at high pressure . The mass flow of inlet gas is controlled and measured. It is heated to a desired

temperature and then it passes into a large settling chamber which contains 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

instrumenta ion may be placed. The gas then proceeds to a surge chamber connected to the vacuum pumps.

The low density tunnels developed by the Ames Aeronautical Laboratory and the University of California are described in Refs. 8 and 9, respectively. A report on the UTIA low density wind tunnel will be published soon (Ref. 10). In the subsonic range (0.

l(

M{ 1. 0) this wind tunnel was designed for operation at pressures between 1 and 70 microns Hg .• Reynolds numbers per inch from 0.08 to 70 and mean

free paths between 2.0 and 0.02 inches. For supersonic operation

(1.0~M~5.0) the pressure range was the same, and the Reynolds

number per inch and mean free path varied from 10 to 4000, and 0.2 to 0 .. 002 inches. ,respectively ..

Photographs of the UTIA low density tunnel are shown in Figs. 4a ;and 4b: : ~ The primary pumping system consists of six booster-type oil diffusion pumps having 'a combined pumping speed of about 7200 liters per second over the range of operating pressures indicated above. The booster pumps are conn,ected to two large mechanical pumps which form 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 but the core of uniform flow was limited to a relatively small reg,ion due to the large thickness of the boundary layer. In fact, measurement at,

pressures below about 10 micron,s 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. 8). Special instrumentation is required such as free molecule probes (described below in Section 8) and the

electron gun. Figure 5 shows how a beam of electrons may be used to

survey the density field near a flat plate placed perpendicular to the dire ction of flow.

The shock tube is one of the most useful facilities for the investigation of high temperature effects . It can reproduce both the

temperatures and velocities of hypersonic flight. The shock tube was originally used by Vieille in 1899 in detonation experiments. More general interest in the subject followed the work of Payman and Shepherd (Ref. 11) in 1937. Since that time 'considerable pioneering work has been done in the United States beginning with that of Bleakney at PrillCeton University. Surveys of the shock tube field are given in Refs. 12, 13, 14and 15., Abroad review of shock tube facilities and

their applications is presently being prepared by Glass and Hall at UTIA.

At the In.atitute of Aerophysics two shock tubes (1 in.

diameterand 2 in. square) are in operation covering the Mach number

range from 5 to 20 in~J·air, argon, and nitrogen (Figs. 6a, 6b). The square shock tube consists of four steel plates mounted ina 6 in.

diameter steel pipe with a wall 7/8 .in. thick. The intervening space is

filled with Woods metal. A total length of 24 ft. is con,structed in this

manner with a series of ten observation ports situaied ',at one foot intervals . The design pressure is 3000 p. s. i. A time-position

schlieren optical system is used in cónj1lD.ction with a compressed air

driven drum-camera which produces film speeds up to 700 f. p. s. Flows at high Mach number are produced by igniting a

stoichiometric mixture of hydrogen and oxygen diluted with either helium

or hydrogen at initial pressures of about 150 p. s. i. Two methods of ignition have been investigated; (a) a spark located at the clos~r end of ,the chamber containing the mixture, (b) a wire gtretched lengtl'lwis.e in

the chamber which dissipfltes about 50 joules of energy. The latter method has given the best résults with the least shock wave attenuation

(about 0.1 Mach number per ft.). The increasing pressure of the ignited

mixture finally ruptures a metal diaphragm and a strong shock wave is produced.

The measured physical parameters are shock velocities, particle velocities and pressures. The shock wave velocity is

determined by ionization probes andagrees with that measured from...;: .

wave speed schlieren photographs to within ,about 1%. A contact surface which moves at particle speed is produced by passing a shock wave

( 6 )

through a perforated disc. lts speed can be obtained from schlieren records. The pressures are measured by a diaphragm type gauge and a piezo-electric gauge whose time responses are about 1~ secs.) and

1~ sec., respectively.

The simple shock tube consisting of a diaphragm in a tube of constant cross section is suitable for the study of high temperature effects at low Mach number. However, it can produce the required hypersonic conditions for flow around blunt bodies . It is a valuab1e asset in the study of heat transfer and real gas effects (relaxation) , but it may not be possible to reproduce these effects on a model as they would be in flight, particularly if the relaxation distance is large for some degree of freedom (see Section 12).

In order to simulate the flight Reynolds numbers and relaxation effects , a large model is needed as wen as a much longer running time than that which can be achieved in the simple shock tube. This requirement has led to the development of the hypersonic shock tunnel which is basically a shock tube with a diverging nozzle. This facility has a relatively long running time. In one form it operates from a reservoir of uniform hot gas produced by a reflecting shock wave. Another design has an internal nozzle which scoops up the air from behind the shock front. However, many problems remain to be solved before this type of tunnel will reach its fun potential (Ref. 15).

II. RAREF ACTION EFFECTS

5. Interaction of a Gas Molecule with aSolid Surface

We can conceive of two kinds of reflection of gas molecules from asolid boundary. lf the wan were perfectly smooth it is possible that "mirror-like" or specular reflection might occur in which the

component of the incident molecule normal to the surface is reversed in direction but unchanged in. magnitude on contact with the wan. In

practi..ce, however, the surface is rough and contains interstices in which a gas molecule may be temporarily trapped. Furthermore ~ the ultimate direction of reflection may have no relation to the incident direction. This type of reflection will be described as diffuse in character. In diffuse reflection all directions of emission about the normal to the surface a.re equally probabl~ J regardless of the direction

of impinge·ment. More specifically, the probability that a molecule will leave the surface at a particular angle is proportional to the eosine of the angle with respect to t he normal. In general the speeds of diffusely reflecting molecules are grouped according to a Maxwellian distribution

corresponding to a temperature which can be different from that of the

surface.

molecular be·am technique. The method is surveyed in Refs. 16, 17. In this method a stream of molecules is directed on a plane surface element and measurements of the flux of scattered molecules is made of various .angles .relative to the incident beam (Fig. 7). The beam is produced by the thermal effusion of molecules from a small souree chamber through an orifice or a tube. The molecules emitted 'by the souree move along diverging rectilinear paths. On reaching tl\e orifice the properly orientated molecules pass through and constitute the

molecular beam and those stopped by the orifice are drawn off by a

pump. The beam passes through a region of high vacuum 00- 6 mmo Hg.)

and strikes the test surface at a selected .angle. The scattered molecules which reflect within a small solid angle pass into a detector and produce a small increment in press ure. The detector (essentially an ionization gauge) can be moved to various positions to determine the complete flux

distribution .

Of special interest to designers are the molecular beam tests of air molecu.1es on typical materials used in .aircraft construction.

Hurlbut (Ref. 17) finds that the cos.ine law of scattering is valid for the spatial.flux distribution of air and n,itrogen molecules reflected from polished low carbon steel, etched low carbon steel and polished

aluminum independent of outgassing or surface temperature. Further

-more J the reflected molecules posse~sed a mean energy closely consistent with the thermal condition of the wall. On the other hand when .air and Jiitrogen molecules reflected from a glass surface J small deviations from the eosine scattering distribution were detected. These deviations could be explained by assuming that while most of the

molecules are reflected diffusely, the remainder 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, intermediate types of interaction being quite possible.

We can çharacterize the actual reflection process in a given case in terms of ov~-all average deviations from "completely diffuse reflection" for which. 'the scattered (reflected) flux distribution obeys the eosine law and the emitted molecules are in Maxwellian equilibrium with the surface . We define the thermal accommodation coefficient as follows .

(1)

where E. and E are the actual incident and reflected energy fluxes,

1 r

respectively J and Ew_ is the emitted (reflected) energy flux for completely

diffus·e reflection . We have the following limiting values: For completely specular reflection, Ei

=

Er' ct

=

0, For completely diffuse reflection, Er

=

Ew' (Á

=

1.

( 8 )

The transport of momentum to and from a surface in directions

normal and parallel to the wall can be characterized in a similar way.

For tangential momentum exchanges we define

C' - L

L r

(2)

where 7;

.

and

-c

are the actual incident and reflected fluxes of

tangential momerltum, respectively. Note that the definition of

cr

T has

the same form ase/.... since 0 T

=

(C; i -

T

r)/(

7:

i -

-C

w) and for

perfectly diffuse reflection,

7:

w = O. Similarly for the normal momentum

exchange we may write

r

(3)

where p. and pare the actual incident and reflected fluxes of normal

momentfrm, rl~pectively. and Pw is the emitted normal momentum flux

for completely diffuse reflection. These momentum accommodation

coefficients have the following limiting values:

For completely specular reflection, () T

=

()

N

=

0

For completely diffuse reflection,

cr

T =

cT

N = 1

If the in eraction involves a combination of completely diffuse and

specular reflection, then () N is not independent of () Tand only one of

these coefficients is needed (Ref. 18). In an actual physical case,

however , it is expected that

eX.

,

()

T' () N wiU 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 macroscopic quantities that can be determined

experimentally. We shall consider experimental methods for

determin-ing the accommodation coefficient in Sections 7 and 9 below. The

follow-lng table lists the results of tests described in Refs. 19 and 20.

Table 1. Accommodation Coefficients for Air on Various Surfaces (a) Accommodation Coefficient for Tangential Momentum (Ref. 19)

Type of Surface Machined Brass Old Shellac Mercury Oil Glass Fresh Shellac Value of

0'

T 1. 00 1. 00 1. 00 0.90 0.89 0.79

(b) Thermal Accommodation Coefficient (Ref. 20) Type of Surf~e Machined .Aluminum Etched Bronze Polished Bronze Etched Aluminum ·:j~lc:hè(CCa'St Iron Machined Bronze Flat Laquer on Bronze Polished Aluminum Polished Cast Iron Machlned Casl" Iron

·.Value of 0<. 0.95 - 0_.97 0.93 - 0.95 0.91 - 0.94 0.89 - 0.97 0.89 - 0.96 0.89 - 0.93 0.88 - 0.89 0.87 - 0.95 0.87 - 0.93 0.87 - 0.88

A t present both ( j Tand

\I...

are assumed to be constant for a given gas and surface , independent of stream velocity or the temperatures 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 () T :: 1 and cl... = 0.9 for ordinary aircraft materials.

The 'above considerations hold strictly for a monatomic gas. In a diatomic gas other forms of the thermal .accommodation coefficient must be considered. For example, if the energy of a gas arises from the tr'anslation and rotation of its molecules, then the collision of such molecules with a wall wil! involve accommodation of the rotational energy and we must define a new coefficient.f

.t.

(4)

6. The Velocity Distribution Function

In the kinetic theory approach to fluid mechanics the macroscopic properties of a flowing .gas can be determined from the collision information for assumed molecules and the distribution of molecular velocities (Ref. 6) . . 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 "G containing a very large number of molecules nd""C. The fundamental question of kinetic

theo'ry is: How many of the molecules in d 1:" have velocities in a

prescribed range

cS

i.

Si

+

d~ i (i

=

1, 2, 3) at a specific time t? Let us represènt the above range as an element of volume in a velocity space, d w

=

d ~

l'

d ~ 2: d

S

3' If we plot the velocities of the nd"'C molecules in the velocity space, they wiU be scattered over all possible values. The number of points which are plotted ill.j:lw is the answer to our

( 10 )

question .above. If f is the density of points in d UJ, then the number of molecules in d t; which have velocities between

S

i and ~ i

+

d

<f

i

(that is. in dIJ..) is nd

-C

.

.

f. dUJ. The symbol f is called the velocity

distribution function. In gener'al f depends on

$

i, xi' t.

The significance of the velocity distribution function is seen

in the following relation for the components of the macroscopic (stream

or mass) velocity ,

( 1)

where the integration is made over all possible molecule velocities

(-cO

<::

Si <00). The velocity distribution function is a "bridge 11 between

the microscopic and macroscopic motion of the molecules. The velocity

of a molecule can be referred to the macroscopie velocity in d"C by the

relation

(2)

The translational energy of the molecules with velocity components in

d

iJ.>

is } m S i S i and hence the mean energy in d

7:

due to translational motion is

(c5

i

~

i

=

S

1 2

+

~

2 2

+

~

3 2),

.

/

f~'

?:.

P

d

w

=

_I

m

q

2.

-I-

L

In

"2

'), (.

Z

{)

.z

(3)

where -} 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 (thermal) ener gy .

. The basic characteristic of isentropic flow is that the

number of molecules having ve locities in the range

~

i'

~

.

+

d

~

i

(that is. the number in d w ) is unaltered by molecular co111sions as we

"follow the molecules ". The number lost by some collisions is regained by óthers anda condition of molecular equilibrium exists. This is

called Maxwellian motion and is characterized by the velocity

dis tribution function

., '2.

-c

e

Here cm is the most probable speed or the maximum. point on the

distribution curve (Ref. 6, p. 36).

Noni7entropic flow involves viscosity effects and heat

conduction and can be regarded as a "slight deviation" from

equilibrium or Maxwellian motion. The velocity distribution function for slightly nonisentropic flow is

(5)

where

"

A

1.\

,

·::_2..u

~

e··

L.J

cl '

f

(6)

and eij is given in the list of symbols. Note that Ai and A _{ij result from}

heat conductionand viscous action, respectively.

The above velocity distributions apply strictly to a monatomic gas. The distribution function in Maxwellian form for diatomic molecules may be taken to be (Ref. 21)

-é:

Fo'

(E)::-

Po

e

(7)

~ _E~

~

e

,

where ~

=

ER/kT and (ER) is the rotational energy of a molecule in

the

ct

th rotational state,

~is

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 tran,slational temperature Tt and a rotational temperature TR .

A distribution function 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. 22)

~I

=

F:

(4)[1

+

A(~J;

-

C')+A,C,f(l-t

C')

+ A

(1-

~)

J

+

A'j ti C

j ]

(8)

This function gives the number of molecules in a 6 +

V

d.f.~en.s,i9nal phasespace having Eositions between xi, xi

+

dXi' velocHtles ~ the range

f

i.

-5

i

+

d

<;

i and internal energy between EI' EI

+

dEI" N ote that

( 12 )

where~ b is the bulk viscosity. We shall have more to say about this

quantity in Section 12. Also

K

{internal}

1<

(translational)

Thus there will be heat conduction coefficients for the random

translational and internal energies.

7 . Free Molecule Aerodynamics

(10)

We have seen in Section 3 that free molecule flow will exist

if the characteristic dimension of a body is less than about one -tenth of

the mean free path. Referenee to Fig. 3 showà that very large mean free paths occur in the upper atmosphere and that free molecule flow might

be expected above 120-140 km. (or 75 - 100 mi. say). It should be noted,

however, that although the mean free path may be considerable, the

number of molecules in the element of volume d 1; is still large. For

example, when the mean free path in the upper atmosphere is 10 ft. , the number of molecules in a cubic inch is still 1013 . The definition of

the macroscopie properties of a gas (p, ~ , T,

'b )

through the velocity

distri.bution function is still valid.

The basic 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 incident

molecules is independent of the transfer of these tquantities. away from

the surface by the reflected molecules. In other words we can treat the

incoming and emergent streams of molecules separately (Refs . . 213., 24);! The

absence of molecular collisions implies that no macroscopic changes

will be produced in the gas motion by the body - there will be no boundary

layer or shock waves. In fact the gas cannot macroscopically sense the presence of the body. Basically, all that is involved is a reflection process.

The characteristics of the molecular motion of a highly

rarefied gas can be illustrated by considering an experiment for the

measurement of the thermal accommodation coefficient (Ref. 20). We J

investigate the properties of the motion between two infinitely long,

concentric cylinders. When no gas exists between the two cylinders, the

flow of heat will 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

Q :::

J..

A,(T.

4 -

T ..

4

) (1)

R

..L(/+..&) _

.&

e

A2

A.,.

where e is the gray body emissivity,A is the area of a cylinder, and k is Boltzmann's constant (,see 'R.ef. ::'20}) · The subscripts 1, 2 refer to the

inner and outer cylinders. respectively. Tests can be run to determine e. and

Qa

can then be calculated in subsequent tests in which a Ras is present.

If now a gas is allowed to fiU the space between the two cylinders such that its mean free path is many times the intervening distance. then heat wiU be conducted between the surfaces by the molecules. By extending Knudsen's theory of conduction through

highly rarefied gases (Ref. 25). we obtain

Q

_e

=

(2)

where Pi' Ti are the gas pressure and temperature and the function F(ot ) contains a constant which depends on molecular weight. the ratio of specific heats. and the geometry 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 i.ntroduced into the center cylinder by an

electrical method and drawn off the outer cylinder by a water jacket (Q). Then Qc :::: Q - Q and we can find the thermal accommodation

coefficient from

~q

.

2. By changing the outer cylinder. the value of ç:J.. for a number of materials can be found.

dA

Let us calculate the exchange of mass • . mbmentum .and energy in a free molecule flow at the surface of a body assuming that the molecular velocities are distributed according to Maxwell's law (Eq. 6.4). The number of incident molecules striking unit are'a in unit time having velocity components in the range

~

i.

3' i

+

,d

3'

i is

n.

i

~

1X

f d

S

1 d ~ 2 d

$'

3. Then the total number of molediles incident on unit ~area in unit time is

cO ~ oe

N,:=:n,

r

d

S,

(dS,

~

Çd

fg

_OC -00

( 14 )

These integrals can be evaluated with the help of Eq. 6.2. The result

is

where

Su.

-=

~

=

S~e

CI'Y"I

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

reflection 0 00 0()

Nw:::n"

)dc.

)dC.

(c,~dC~=hwV~;"

_~ _00 _00 (4) (5) (6)

Assuming that all incident molecules are reflected. then Ni

=

_{N w ·} The molecules which'have velocities between

~

. and

~

i

+

d

3

i transport momentum in m

5

l

normal to the surface.

~hen

the

contribu.tion to the normal pressure due to the momentum transported

by the incident molecules is

ol) oQ e>D

p,

=

rvo

n, )

d {,

î

d

S, (.;,'

~

d

g'~

(7)

o _0/) _00

Similarly the normal pressure due to the perfectly diffusely reflecting

molecules is .t

pw

==

rn

N,V

lt:

T",

(8)

where we make use of Eq. 6. Now the total normal pressure acting on

the surface is (9) (see Eq. 5.3). Therefore 00 00 00

p=~n

,

(d~, ~g. (P[(2-~N)f+

0-;.

V

JT:T

w

s]

J~

_00 (10) IJ _00

which permits us to ca1culate the normal force per unit area due to the

incident and reflected molecules.

The incident molecules having velocities in the range

S

i;

3

i

+

d

~

i transport to the surface a y - component of tangential

momentum of amount of m $2. The tangential force in the y - direction

resulting from all incident molecules is

oe

0.0 C>O

rL,==mn,'

[d5;

[J~

(e

f.

P

d~

o - o U __oO

(11)

From the characteristics of completely diffuse reflection.

-C

w

=

0 and the total tangential force in the y - direction is (see Eq. 5.2)

(12)

These relations. applied to an element of area on a body and integrated over the whole surface • win provide the resultant forces on various aerodynamic shapes in free molecule flow.

Let us now consider the balance of energy for theelement of surface dA. Referred to unit area of 1lA. energy balance requires that

(13)

where Ei' Er are the incident and reflected fluxes of molecular

energy. Ri. R are the incident and emitted fluxes of radiant energy. and Q is the hlat removed from unit area of <iA from inside the body (a known quantity).

If the gas is diatomic and if we assume that only the rotational component of the internal energy is significant compared with the translational energy J then the incident energy takes the form

Et:

=

(ba)~

+

N

L [~

.

₍₁₄₎

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

00 00 00

(r')l=fmnLIdr.(d~. (~(SS,JPd~

(15)

o

_cc _00

For completely diffus.e reflection the corresponding expression ior Ew is

(16)

( 16 )

lf we assume gray body. r.:adiation and a body small compared with its surroundings, then

RL -

R

r ::

e

B

(T

_s

4- -

T '" ... )

(18)

where _{T s} is the temperature of the surroundings and TB is the body

temperature.

All the quantities in the energy balance equation applied to

unit area of dA are now known or can be ca1culated. If we multiply

Eq. 13 by dA and integrate over the surface of the body, we obtain the over-all energy balance equation.

The above analy.sis has been applied to the circular cylinder in Refs. 8, .26 (see also Ref .. 27). For a monatomic gas the coefficient

Of~:t:~1::~5'tI.

+

(S'+t)(r.

+1,)} + ;:

{it ]

( 19)

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

respectPvely. Thus in free molecule flow the drag coefficient depends on the speed ratio (S) and the _{temperature ratio (Tr/Ti). The}

temperature ratio can be obtained from the energy balance equation for

the cylinder

2

%(2,+2.)_[(S\2)t,

+

(S'+f)

?<J

+~

(.n:

)3/

z

[e

8

(T

_w

If_

T

_s

4) -

Q]

=

0

oLp~ RT~

.

(20) where (21)

and the reflection process is such that only the thermal accommodation

coefficient is retained (see Section 5).

The insulated cylinder is of special interest since it lends

itself to _{a simpler experimental approach. In this case T r}

=

_{T w ' and}

only the transfer of translational energy by the molecule is involved. Then

(22) and substitution in Eq. 19 gives

The drag coefficient and temperature ratio are plotted against speed ratio in Figs. 8 and 9, respectively. The experiments

described in Ref. 8 gave drag coefficients and temperature ratios

which agreed reasonably weU with theory. In particular the

experiments verified a prediction of the above theory that CD is

independent of the Reynolds and Knudsen numbers. A small correction

to the Cn relation to allow for a diatomic molecule with both

translational and rotational energy in accordance with Eq. 14 was found

to be in the right direction (Fig. 8).

The interesting fact that arises fr.om the energy balance

equation is that the insulated cylinder attains a temperature higher

than the stagnation temperature of the stream (Ref. 26) _ For an

insulated flat plate aligned in the direction of flow of a monatomic gas,

the energy balance equation becomes

Tw -

1+

J...

S~

-

-

2

T~

The stagnation temperature for á monatomic gas is

1.

To -

I+LS

Tt.: -

5'

Therefore T w

?

T o'

8. Free Molecule Probes

(24)

(25)

Orre 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 low 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 from 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 macroscopie motion of the gas.

The work of developing free molecule probes was initiated

( 18 )

(Section 7) was carried a stage further to include the case of the more general velocity distribution function for nonisentropic flows (Refs. 28.

29). Thus the properties of a cylindrical wire are known when the probe

is placed in a boundary layer or shock wave. Free molecule pressure probes were subsequently deve loped at the Institute of Aerophysics (Refs. 30. 31).

Let us consider the pressure probe in the form of an orifice

or/hu!

7

_ps

A:)

Tt: "- /, -'--~

"

,,-Po

"-ol. / /

in the side of a tube of small diameter (Ref. 30). The speed ratio. pressure and temperature of the external gas are S. Pi' Ti' respectively. Inside the tube the gas is at rest and the pressure and temperature become Pr and T r' Equation 7 . 4 gives the number of molecules which pass through unit area of the orifice in unit time if the molecular velocities of the incident molecules are distributed according to Maxwell's law. The number of molecules emerging from

the tube through unit area of the orifice in unit time is given by Eq. 7.6 with the subscript r replacing w. Equilibrium corresponds to the

condition of no resultant flow of mass through the orifice. Therefore

m Ni

=

E:

J~r.O:

e-

s·c....·'\

Vrr

SÜ><. ...

(/+4

5""'''-)

(1)

~t.'

Tt..

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

.

..fr

f

T

~

=

e -

S·~

'cl.

~

Vrr

S

(.04~'

(

1-

_

,~

ç

(,od,cI.')

(2)

p~

1;-Now it has been found experimentally that various orientations of the orifice have no appreciable effect on the temperature inside the tube / (T r := T r '). Furthermore if we evaluate Eq~. 1 and 2 above for cJ...

=

01...=0

':( and

d..

= ei... '

=

Tf

/2. we obtain the useful result (see diagram)

1

5

=

~o-

po

₍₃₎

2VrT

Ps

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

Equation 3 holds strictly when the external molecular motion is Maxwellian (see Eq. 6.4). The usefulness of the probe can be considerably increased by calculating its properties in a free

molecule. nonisentropic flow (see Eq. 6.5). The calculation for a two dimensional boundary layer yields the same result as Eq. 3 which can

.

•

be deduced from the following relations which hold for the

two-dimensional boundary layer if T r ~ Tr ' and 0/..

=

0( "

r,

)

S"'~2ot

J

po\+~=2t=>SL(I+AI2..~d.~ol

e -

-+Vtf

5C1>4~~s~c:;..

(4)

(5)

For weak shock waves, Eq. 3 will also hold, but when Al' All become appreciable compared with 1, the more general relations must be used.

The above results suggest that the pressure probe may be used to

determine the individual values of the coefficients of the velocity

distribution function (Ai, Aij)· In general this will be difficult since

these coefficientsare small for "slightly nonisentropic flow" (see Section 6).

Let us now consider the properties of a cylinder in nonisentropic, free molecule flow. The calculations outlined in Section 7 are repeated for the more general velocity distribution

function (Eq. 6.5), with the following results (Ref. 28)

Co:

C

Do - (

A,/z

CS)

C

_D1

+

All

C

_DII

+

A2.2.

C

_D22 (8)

(9)

where the partial drag and lift coefficients (CD ' . . , CL2) are functions of the speed ratio (S) and arise from fue deviation from

Maxwellian (isentropic) flow. The relative importance of the drag terms is indicated in Fig. 10. Note that CD is the drag of the cylinder

corresponding to Maxwellian motion. °We conclude that the aerodynamic

I

forces on a cylinder in nonuniform, free molecular flow are affected iby

the non-Maxwellian components of the distribution function. In general

the nonisentropic effe·cts are small except in region,s of low speed ratio

( 20 )

The energy balance equation for the cylinder must also be

reconsidered using the velocity distribution function for nonisentropic

flow (Eq. 6.5). The Stanton number is defined as follows, .

(10)

where Q is the net heat 10ss per unit time per. .. unit;le~gith~i.i.-d)isdth.& ~~l~

diameter, _{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

St::

r;1;~+(A,j2S)<St,+

AI/Sc

_lJ -tA.22

<5

t

2.'2. (11)

The partial Stanton numbers corresponding to the nonisentropic terms

in the velocity distribution function are small compared with Sto (Fig. 11).

The energy balance is only slightly influenced by the viscous stress

terms but, as in the 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 (cX..). The basic fact

indicated by the above analysis is that if we know the speed ratio, the

accommodation coefficient, and the velocity distribution function for a given flow (Ai, _{A ij}), 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

velocity distribution function (Eq. 6.7). 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

(cl...

,0{ R' Section 5). For molecular eqJtilibrium, t~e

accommodation _{coefficients and temperatures (T, TR ) are essenhally}

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. 21)

( 12)

where the original definition of speed ratio is retained and a, bare

functions of S (Ref. 21) .

Let us consider two special cases of the energy balance equation for a diatomic gas. If there is no accommodation of rotational energy, then

d..

RI

d...

=

0, and

T

w

_..L

a.('5)

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 rotational degrees of freedom attains the equilibrium state. On the other hand, if the rotational accommodation is as food as the translational, then

ci...

R =

0<.

and the energy balance equation becomes

T

w _

...L.a(S)

+...L(TR)

Tt: -

6

beS)

3

T~ (14)

The actual value of 0( R is a subject for further investigation. This question and the form of the energy balance equation for a cylinder in the nonisentropic, free molecule flow of a diatomic gas is discussed in Ref. 21.

9. Slip Flow

In the present sta.te of the subject, rarefaction effects are

considered either from the point 'of view of highly rarefied gases (free molecule flow) or as amodification of continuum flow produced by lowering the pressure . The transition region between these two

extremes has been the subject of empirical investigation mainly since no adequate theory has been developed for this regime of flow.

Deviations from continuum flow become apparent in flight at altitudes from 20 to 50 mi. The boundary layer is affected in two ways; (a) the bounda~y conditions allow for a small but finite slip

velocity along the wall plus a corresponding temperature jump; (b) some of the terms in the equations of motion which were previously considered to be of negligible order must now be retained.

Details of the method for determining the conditions of flow at a wall (boundary condiÜQ...~) from the kinetic theory are given in Ref. 6. Let us consider an element

of

volume of -gas with one face on the wall

(interface). Reflection of molecules froIll1he wall is assumed to be of

,--=-1 {;a. 5 Flo IN I dL I

///

mm:t-~r

:///T/7

Wa 1/ _{. _.---}L.d~~ _{- - "}

of the modified type discussed in Section 5 for which we know the tangential

momentum and thermal accommodation c oe fficients

«J

T,

ei..

)

.

The analytical method consists in writing down a relation which makes the transfer of mass, momentum and energy by the

reflecting molecules compatible with what the transfer would be if it were produced by molecules whîch crossed the interface from an adjacent element of volume similar to d -C in all respects. The compatibility relation for the transfer of tangential momentum yields the equation for slip velocity on the wall,

( 22 )

US~~(;TÓT), À(~~J.r,.,

(1)

where k is a numerical constant very close to 1, and the subscript s

refers to the condition of the gas at the wall (taken to be at rest). The

application of the compatibility condition to the transfer of translational

energy provides a relation for the temperature jump at the wall,

T. -

I:

=

~ [~] [~

A

(ö

T)

'S w

~

ei...

'6

+

I

J

Pr

è

~

5 (2)

where Pr is the Prandtl number.

The equations for the flow in the boundary layer are obtained

by establishing the order of each term in the exact equations relative to

two dimensionless numbers,

1\

/

~ and ~ /) , where ~ is the boundary

layer thickness (see Ref. 6). According to this analysis both the slip

velocit~ and temperature jump referred to free stream conditions have

order

?\

/

~

.

In continuum flow slip velocity and temperature jump are

neglected along with all terms in the equations of motion of order ~ /

b

or

b

/

.J...

or higher . and the wel! known boundary layer equations are obtai.ned for which the boundary conditions are Us

=

_{0, T s}

=

Tw' lf

terms of the order of the dimensionless slip velocity and temperature jump are retained in the boundary conditi.ons, then terms of order

A /

&

and

b

/

J..

should be retained in the equations of motion. It will be seen

that rarefaction not only produces slip and temperature jump at the wall,

but it is also responsible for "interaction effects " due to changes in the

boundary layer equations .

Before continuing further with the effect of rarefaction on

skin friction ànd heat transfer . let us consider the simple case of the

incompressible. two-dimensional flow between two concentric cylinders.

the inner one rotating and the outer one stationary (Ref. 32). This is a

particularly interesting flow since it can provide us with an experimental

method for determining the accommodation coefficient for tangential

momentum (Ó T)' For such a flow the equation of motion is

-.L.~

(r

'dLt) -

~

~

àr

or

r

(3)

and the s 0 lu tion is

(4)

This solution must satisfy the following boundary conditions:

(5)

(b) On the surface of the outer cylinder

( 6)

Applying these boundary conditions we find that the drag of

a cylinder with slip compared to its value without slip is as follows

00 :::

I

+

~I

Kn .

(2-

ó

T)

D

cJT

(7)

where k depends only on the geometry of the apparatus, and Kn

=

A

/h

wher~

h is the distance separating the two cylinders. Since

'A

=

1.

26V'fA

}

Eq.

'!l

cä.n:.,éils.o 1àe'writlen 'r : , : " ' . . C>

a.

,~) '

,.1r \

D

_{o -}

}

_-t--

..A'(2-6""T)

0-

p

cr,

(

8

)

where k' depends on,.,A, R, T.

't

.

and the geometry of,the apparatus .

It wil! be seen therefore that the inverse drag is a linear function of the

inverse pressure (Ref. 32).

Th,e concentric. rotating cylinder method for studying

rarefaction effects has been used by Stacy (Ref. 33), Van Dyke (Ref. 34)

and others. In general the method consists in supporting the stationary

cylinder on a calibrated torsion fibre. The amount of twist of the fibre

produced by a given rotational speed of the inner cylinder is measured

directly on a .scale. Low pressures are held for sufficient time to

minimize outgassing effects . Merlic (Ref. 32) was interested in the

case of air on aluminum because of the importance of this combination

in tlight problems. But he has also made a study of the important

effect or-''''pressure history" which was suggested by Ref. 33. In general

Merlic found that for 'air held at 0.01 microns on a clean aluminum

surface 'a value for 0 T of 0.9 gave good agreement between theory and

experiment. On the other hand when the system was held at 250 microns,

0;-

was found to decrease with time until a value of

0"

T

=

O. 6 was

reached after six days. Figure 12 summerizes Merlic's results . The

reason for the decrease in 0-T with time at the higher pressure is still

obscure and more tests are needed. It has been conjectured that the

decrease in drag may be due to a thin surface film which increases the

amount of specular reflection from the surface . Lower vacuums remove

this film. It should be noted that Hurlbut (Ref. 17) obtained <:J T

=

1 by

( 24 )

The more general problem of the effect of rarefaction on the

skin friction on a flat plate at zero angle of incidence has been reviewed

by Schaaf and Sherman (Ref. 35). One of the difficulties in analysing

the boundary layer in slip flow is the existence of two basic numbers

1\

I.

~

,

S

I

~

or

A

I$;,

A

I ) .

These parameters are

fre quently expressed in terms of the Mach and Reynolds numbers. Thus

since

b

l

1.~ ~e-1/2,

then

ti

l

è

rvM/{Rë.

Alsof..

I~/V M/Re.

In presenting research results M/{Re is used for Re

>-

1 and

M/Re

is

used for Re

<

1. Some experimental results obtained in the slip flow

regime are summarized in Fig. 13.

A comparison between theory and experiment is difficult at present because of the underdeveloped state of the theory. No

satisfactory theory is available for the test range shown in Fig. 13. The Reynolds number is too small for the Prandtl theory of the boundary

layer to be applicable and too large for Oseen's theory of viscous flow. Furthermore M

I

Re

(or ()

IJ..)

is not high enough to permit the

application of free molecule flow theory. The direct application of the

slip boundary conditions (Eqs. 1, 2) to the ordinary boundary layer

equations for incompressible flow has yielded some results . The

general effect of the modified boundary conditions is to reduce the

displacement thickness, skin friction and heat transfer. The decrease

in the boundary layer displacement is given by (Ref. 36)

(9)

but the change in the skin friction coefficient is of lower order.

On the other hand some results show a tendency to depend on Mand Re separately. In fact Refs. 37. 38, 39 indicate that an

increase in the local skin friction coefficient can be expected due to the

interaction between the boundary layer and the free stream pressure

disturbance induced by the boundary layer. Although Fig. 13 suggests

that rarefaction effects can be analysed in terms of the single parameter

MI

Re, a closer inspection shows additional interrelations and we must

conclude that no such single correlation of CD M vs

MI

VRe actually

exists.

It is apparent, therefore, that the effects of rarefaction on

the skin friction coefficient have at least two aspects. Interaction

effects increase the skin friction, the effects becoming larger as the Mach number goes up. and they are dominant for Re in the vicinity of

1000. However. when Re is reduced to something of the order of 50,

slip effects become predominant and the skin friction decreases with

increasing Mach number. The latter is certainly true for M

I

~ ~ 1.

As regards the more general problem it may be possible to

find an asymptotic series type of solution which will correct the ordinary boundary layer theory for the neglected terms in the flow