The gas kinetics of very high flight speeds


Material Information

The gas kinetics of very high flight speeds = Gaskinetik sehr hoher fluggeschwindigkeiten
Portion of title:
Gaskinetik sehr hoher fluggeschwindigkeiten
Physical Description:
49 p. : ill. ; 28 cm.
Sanger, Eugen, 1905-
United States -- National Advisory Committee for Aeronautics
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aeronautics   ( lcsh )
Gas flow   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


The aerodynamic forces on bodies of arbitrary shape were investigated under conditions such that the mean free path of the air molecule is greater than the dimensions of the body. Air pressures and friction forces were calculated from gas kinetic theory for surfaces facing both toward and away from the air stream at any angle. Air forces for an atmosphere of definite composition (molecular hydrogen) were calculated as a function of the flight velocity. The results indicate that the friction stresses between the air and the body surface are of the same magnitude as the dynamic pressure and as the air pressures normal to the surface. The application of the general method to special cases such as thin airfoils and projectiles leads to high drag coefficients and poor glide ratios even for the theoretically best wing sections.
Statement of Responsibility:
by Eugen Sanger.
General Note:
"Technical memorandum 1270."
General Note:
"Report date May 1950."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
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sobekcm - AA00006220_00001
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Full Text





By Eugen Sanger


In ordinary gas dynamics we use assumptions which also agree with
kinetic theory of gases for small mean free paths of the air molecules.
The air forces thus calculated have to be re-examined, if the mean free
path is comparable with the dimensions of the moving body or even with
its boundary layer. This case is very difficult to calculate. The con-
ditions, however, are more simple, if the mean free path is large com-
pared to the body length, so that the collisions of molecules with each
other can be neglected compared to the collisions with the body surface.

In order to study the influence of the large mean free path, calcu-
lations are first carried out for the case of extreme rarefaction.
Furthermore, the calculations on this "completely ideal" gas will be
carried out under consideration of Maxwell's velocity distribution and
under the assumption of certain experimentally established reflection
laws for the translational and nontranslational molecular degrees of

The results thus obtained allow us to find, besides the air pressure
forces perpendicular to the surface, also the friction stresses parallel
to the surface.

Their general results are calculated out for two practically
important cases: for the thin smooth plate and for a projectile-shaped
body moving axially.

The mathematical part of the investigation was carried out primarily
by Dr. Irene Bredt.

*"Gaskinetik sehr hoher Fluggeschwindigkeiten." Forschungsbericht
Nr. 972, May 31, 1938.

NACA TM 1270









With the motion of bodies at very great atmospheric heights, the
air can no longer be considered a continuous medium, in the sense of
flow theory.

At over 50 kilometers altitude, the mean free path of the air
molecules will be of the magnitude of boundary layer thickness and,
at over 100 kilometers altitude, of the magnitude of the moving body

The mean free path at greater heights will be definitely greater
than the body dimensions of the moving body, and the especially simple
conditions of very rarefied "completely ideal" gases are valid where
the effect of the collisions of the molecules with each other disappears
compared to the effect of the collisions with the moving body.

The air molecules collide then against the moving body as individual
particles, Independent of each other, and are reflected with a mechanism
whi.h deviates more or less from the known Newtonian principle of air
drag, as shown by the results of the kinetics of very rarefied gases.

The velocity of the body will be denoted by v (meters per second)
in the following discussion.

If we Imagine as usual that for the consideration of the flow
process the body stands still and the medium moves, then v equals the
uniform undisturbed flow velocity of all the air molecules.

NACA TM 1270

The air molecules have also their ideal random thermal motion.

The individual molecular thermal velocities are distributed com-
pletely at random in all directions and are of completely arbitrary
magnitudes, where the various absolute velocity magnitudes group around
a most probable value, c (meters per second), according to Maxwell's
distribution, which has the ratio r?2: JT to the most applied gas
kinetic value of "average molecular speed V' (square root of the average
of the squares of all the velocities present).

According to the known Maxwell speed distribution law, letting
p(kg s2/M4) equal the total molecular mass per unit volume, the
mass dp of those molecules having velocities between c,
and c, + dc is:
dp 4 c 2 C2
X= e dc, (1)
P It c3 x

If we consider only this mass dp of an otherwise motion-
less (v = 0) gas, the molecules of which are moving with the particular
speed cx in random directions, then the quantity dp of molecules
striking per second on a unit surface of a motionless plane wall can
be calculated, imaging that all velocities cx are plotted from the
center of a sphere with radius ex. The conditions of figure 1 are the
result of an inclination angle $ between the wall normal and the
velocity direction under consideration, and of the molecular quan-
tity dp dF4 which passes through the striped area dF of the sphere
surface. Then:

dp = -d cx cos 0 dF = d Cx
4c,2 0 4

4 NACA TM 1270

If, with the aid of Maxwell's equation, we include in our calcu-
lation all molecules with the various possible speeds cx between 0
and o, the total mass p of the molecules colliding per second will

dp -.

cxJ C2
- dc

2\ n

as one can find in any textbook of gas kinetics. The pressure of the
motionless gas against the stationary wall can be calculated similarly.

The impulse dip perpendicular to the wall with which molecules of
particular speed cx are striking the wall at an angle 0 will be:

dp /2

2 o2 = dp x2
c2 cos2 0 F = cx2

and the impulse of all molecules striking the wall:


c 2 P
6 do 3
0 3 4-i

c e


dc c2
x 4

If we double this impulse value because of the elastic rebound
always assumed for motionless gases, then we obtain the gas rest pressure

p 2 P -2
p = c or -c
2 3

The value of the total impulse is also interesting, i.e., the sum
of all single molecular impulses which strike per second on the unit


P =

NACA TM 1270

The total impulse of the molecular mass dp corresponding to a
particular ex will be:

dp 2
di = dpcx = p C2

It is thus 1.5 times greater than the effective impulse dip
against the plate. The total impulse i is also greater in the same



If we consider again the mass dp of molecules with speeds
between cx and Cx + dcx (almost equal) and if we examine its action
on a flat plate in an air stream with an angle of attack a, then this
process can be illustrated by figure 2, if we further assume
that v sin a < cx.

The uniform velocity v of the individual molecule combines with
the ideal random velocity (which can have any space direction) to give
a resultant, the components of which are:

perpendicular to the plate: v sin a + cx cos
parallel to the plate and to v cos a: v cos a + ex sin cos
parallel to the plate and perpendicular
to v cos a: ex sin 0 sin
and for which the absolute value is therefore:
w = v2 + 2vcx (sin a cos 0 + cos a sin 0 cos + cx2)

From the sphere of all possible directions of qx, a spherical sector
with the half opening angle cosae = v sin a/cx is excluded, in which
the speed component v sin a + cx cos 0 is directed away from the
plate, i.e., the molecules of this 0 range do not strike the plate.
The integration over the velocity directions of all colliding molecules
is not from 0 = 0 to A as with the motionless gas, but from = 0
to n = C.

6 NACA TM 1270

The molecular mass colliding per second against the unit plate
area with the selected speed cx is therefore:

dp = dp (v sin a + ex cos 0 )2cx2 sin 0 do
x If=0

dp + 2v sin a
T- + 2v sin a +

c /

For v sin a > cx the integration extends over the whole sphere
from 0 = 0 to n and the molecular mass colliding against the plate
with selected speed cx is

dp = 2 2

S =0

(v sin a + ex cos 0)2Cx.2 sin 0 do = dpv sin a

NACA TM 1270

Both equations naturally give the same value for v sin a = cx.
With the aid of Maxwell's distribution equation the total molecular
mass colliding on the unit area per unit time within the speed
range cx = 0 to a will be:

S s
C=V sin a

1x + 2v sin a + sinC dp +

C=v sin a

s ina a
p 1
+ 4_
f o=0

( 3
\ 3


- sin a e

sin a + -v-



sin2 M)e2 dcx


S c

v2s In2a

v sin a
Sv sin + 2 v sin a
+ +
c c2

e c2 dc

Thus the number of the colliding molecules is known, and for cal-
culation of the forces acting an the plate, we must now determine what
impulse the molecular mass under consideration produces in the direc-
tions in question.

The Impulse perpendicular to the plate, ip and the impulse
parallel to it iT will be examined separately.

v sin a dp

'v sin a


NACA TM 1270

We find for the impulse per second perpendicular to the plate

dip = dp
cx L3-0

(v sin a + cx cos 0)2dF

= 2 + -- in a)3
6 Cx

2. If v sin a > Cx:

di dp=
p x2 j ^

(v sin a + cx cos 0)2dF

dP c 2 + 6 sin2~
6 cy2

This summation of the impulse components over all possible direc-
tions yields the total impulse, perpendicular to the plate, of the air
molecules striking the unit area per second with a speed c, deter-
mined by dp.

1. If v sin a < cx:

NACA TM 1270

The further summation of the impulses over all speeds cx with the
aid of Maxwell's distribution equation results in the total impulse ip
acting per second perpendicular to the plate:

ip =

o 6
T- cx2

d 2(1+

sin2) +

v sin a

1v sin a


1 Cx4 -
S-r e
V c3

+ x1 Cx e
cx=v sin A c3

x 2
c2. ( +
ex+ -
G ^

sin a)3dcx

v2sin2a -

1 cv sin a +
3 2

c2)e c2

Sv sin a -

2 2 2 e

If we set a = 0 then equation (5) gives the impulse
less gas against the motionless wall, equation (3).


dx (5)

of the motion-

Also a = 0 gives the impulse of the motionless gas, which will
not be influenced through uniform motion of the gas mass parallel to
the flat plate. We find the impulse parallel to the plate in the direc-
tion of v cos a in the following manner:

The molecular beam with a particular speed cx and with a particu-
lar direction (the latter determined by the inclination angle 0 between
the velocity cx and the perpendicular to the plate, and by the angle *

v- sin

"2 (2

+ 6




10 NACA TM 1270

between the projection cx cos 0 and the direction v cos a) gives the
molecular mass colliding per second according to figure 2.

S c2sin 0 do d*
dp = dp 4Cx2 rv sin a + cx cos

The velocity component of this beam parallel to the plate is:

v cos a + cx sin 0 cos *

and the impulse of the beam with a particular cx, 0, and will be

dp sin do d (v sin a + cx cos 0)(v cos a + cx sin cos )

If one integrates over all J and 0, one obtains the impulse of
the beam with a particular cx.

For v sin a <-cx:

diT= J ( sin a +c cos I cos a+Cx sin 0 cos *)sin 0 do d
=0 00

dp 2(3 v3 s i
dp_ c2 sin2a cos a + 3 sin a cos a + v cos a
S \2 x cX3 c x2 2 cx

and for v sin a >cx:

'21t n
di= sin a + Cx cos .Xv cos a +cx sin 0 cos sin 0 do d

= dp v2sin a cos a



NACA TM 1270

The integration over all cx with the aid of Maxwell's distribution
equation gives finally the total impulse in the required direction
parallel to the plate:

v sin a
IT =

dpv- sin a cos a +

dp C13 2v oa
dp sin2 co3
\C x

SC=v sin a

cx2 2 x /

v sin a

v sin
+ v2in a cos + -2 s -
2 cx=
\ ^1 ex=0

a v2sin2
e dcx

We can start out from the impulse of the gas stream, given by
equations (5) and (6), (perpendicular and parallel to the plate) in
order to calculate the forces produced by the air on the front side of
the plate oblique to the air stream, including the force perpendicular
to the plate (normal pressure p) and the force parallel (friction
stress r ).

For this calculation, we have to make some assumptions on the trans-
fer of this impulse to the plate and on the change of the kinetic trans-
lational energy of the molecules into other energy forms for which pre-
sent gas kinetics furnish only a partial basis.

If we first assume monatomic gases, so that inner degrees of free-
dom for energy absorbance do not exist, and further assume that the
struck molecules of the wall are in such a temperature condition that
they also cannot take over any energy from the colliding molecules,

NACA TM 1270

then the molecules have to leave the wall again with the same speed with
which they arrived. The collision is thus completely elastic and we have
only to derive the direction of reflection.

Gas kinetics distinguishes two different possibilities for this:

Mirror Reflection where the assumption is made (following Newton)
that the angle of incidence and the angle of reflection are equal and
both beams are in the same perpendicular plane.

Diffuse Reflection where it is assumed (following Knudsenl) that
the reflection direction is not at all dependent on the direction of
the impinging beam and is completely diffuse, i.e., that the colliding
molecules first submerge in the wall surfaces, then after a finite time
of "adherence" leave again, in a completely arbitrary direction.

This last hypothesis is generally accepted in flow theory, where
the adherence of the frictional boundary layer on the surface is
explained by diffuse recoil of the molecules.

In the case of a motionless gas (v = 0) both assumptions lead to
the same distribution of rebound molecules and thus to the same forces
on the wall, as the striking molecules are in completely random direc-
tions and this then is also true for the rebound molecules under both

In the case of a gas in motion, the two assumptions lead to very
different air forces.

With elastic "mirror" reflections, the impulse iT of the gas
flow parallel to the wall stays unchanged. Shear stresses on the front
side of the plate Tv are not transferred to the wall. The friction
forces are zero.

T = 0 (7)

The impulse ip of the impinging molecules normal to the wall is
destroyed completely and an equal but opposite impulse is produced by

1Knudsen M.: Annalen der Physik, Vol. 28, p. 75, (1909); Vol. 28,
p. 114, (1909); Vol. 28, p. 999, (1909); Vol. 31. pp. 205, 633, (1910);
Vol. 35, P. 389, (1911); Vol. 34, p. 593, (1911); Vol. 48, p. 1113,
(1915); Vol. 50, p. 472, (1916); and Vol. 83, p. 797, (1927).

NACA TM 1270

the completely elastic recoil. The pressure on the wall caused by this
process thus equals twice the impulse ip:

pv = 2ip (8)

With diffuse-elastic reflection, the impulse iL of the gas stream
parallel to the wall will be given up entirely to the wall and the fric-
tion stress equals iT:

T = i-r (9)

The wall normal impulse ip of the arriving molecules is destroyed
again, whereby is created a partial pressure pi = ip.

The second part of the pressure, due to the diffuse-elastic recoil,
has to be investigated more closely. To imitate the real process, an
impulse value of the magnitude of the complete impulse i of the beam
striking per unit plate area is distributed evenly on a hemisphere as
if all gas particles started from the center of this hemisphere, and
finally the resultant of this impulse distribution perpendicular to the
plate is ascertained.

The total impulse 1 of the arriving beam is derived, according
to the proceeding impulse calculations, in the following manner:

The molecular mass striking per unit time on the known area
section df = cx2 sin d d de selected for a certain Cx, 0, and is:

dp = dp(cx2 sin 0 do di/4c 2 (v sin a + Cx cos

The effective speed of this ray is

w = I2 + 2 cx(sin a cos 0 + cos a, sin 0 cos +) + c2

NACA TM 1270

and the impulse per second thus:

dw dp in 0 d* (v sin a

+ cxcos v \2 + 2vcx(sin a cos 0 + cos a sin 0 cos +) + c

This impulse integrated over all cx, 0, and I will give finally the
total impulse of the beam. The actual carrying out of this integration
is so difficult that the impulse will evaluate by successive approxi-
mation. It is started with the vector sum of the impulse resultants i
and i-, perpendicular and parallel to the plate.

Sip2 + 42

This impulse resultant is smaller than the total impulse. It is
found in connection with equation (3) that the total impulse of a gas
at rest (v = 0) is 1.5 times the impulse resultant. Total impulse and
resultant are equal for uniformity flowing gas without heat
motion (cx = 0). For conditions lying in between, we assume a constant
relationship for the factor with which the impulse resultant has to be
multiplied to obtain the total impulse i. For instance:2

1.5 + c- Bin a + 1.5(-)2
i = i
1 + l sin a + 1.5(v)2

This total impulse, according to our assumption, is now considered as
the completely uniform impulse radiation per area unit in all directions
outward from the surface.

2Interpolation by Prof. Busemann, Braunschweig.

NACA M 1270

The beam pressure p2 perpendicular to the surface is then equal
to 1/2, as shown by a simple integration over all normal components.

Thus, the pressure vertical to the wall in the case of diffuse-
elastic reflection is:

Pv = i + i/2 (10)

In order to get a first view on the numerous conditions of the air
forces just found, figure 3 shows the relationship between the pres-
sure p or the shear force T and the dynamic pressure q = e v2 for
a hypothetical atmosphere of monatomic hydrogen with t = 00 C tempera-
ture (c = 2124 m/s) and for flight speeds up to v = 8000 m/s, for
either mirrorlike or diffuse recoil.

It is seen how different the air forces can be according to the
assumptions made: mirrorlike or diffuse.

In gas kinetics an attempt is made to approach the real conditions
by assuming that the reflection for a fraction f of all striking
molecules is diffuse, while the remainder (1 f) will be repelled
mirrorlike. The fraction of diffuse reflections depends on the kind
of striking molecules and particularly on the material, surface conditions,
and temperature of the struck wall.

According to numerous measurements3, the plate can be considered as
completely rough under conditions usually prevailing in flight technique,
i.e., the mirror reflected part (1 f) is negligibly small, so the
reflection will be almost completely diffuse.

An experimentally obtained dependence of f on the angle of attack,
such that the reflection will be more mirrorlike with smaller angle of
attack, is according to previous measurements of flight relations, too
insignificant to be considered.

Knauer and Stern4 assume from optical analogies that the angle of
attack at which mirror reflection begins is such that the surface roughness

3For example, Karl Jellinek, Lehrbuch der physikalischen Chemie,
Vol. 1, p. 270, 1928.

Xnauer, F., and Stern, 0.: Zs. f. Phys. Vol. 50, pp. 766, 799

NACA TM 1270

height, projected on the beam, must be smaller than De Broglie's
wavelength X of the molecular beam.

With a wave length of 10-8 cm and a roughness height of 10-5 to 10-6,
one obtains for the angle of attack which is of interest sin a = X/h = 10-3,
i.e., an angle range of a few minutes, which is insignificant in flight

With regard to the reflection direction, we assume in the following
analysis that f = 1, i.e., completely diffuse recoil.

So far, for the reflection speed, perfect elasticity of the recoil
was assumed, which means individual recoil speed is equal to the
colliding speed.

The struck wall will, in fact, be much colder than the gas molecule
temperature after its submergence in the wall surface (and so after its
complete braking on the plate to the velocity v), so that we have to
assume heat transmission to the wall molecules from the colliding
molecule which remains a finite time in the plate surface.

Figures 4 and 5 show the internal energies U for molecular
nitrogen or hydrogen and their combination from the individual degrees
of freedom of the molecular motions as a function of the temperature,
starting from an internal energy Uo of the gas at rest corresponding
to a temperature of 0C, with the other internal energy values equal
to the kinetic energy corresponding to v.

For this, the relation between U and v is stated
as U = Uo + Av2/2g. The graph goes up to a U = 8000 kcal/kg, corres-
ponding to a flight speed range up to v = 8000 m/s. The specific heat
at constant volume cv was calculated under the usual assumptions on
energy absorption by translation, rotation, and oscillation of the
molecules (the latter according to Planck's formula) after the collision.

We see from both figures that, especially for the N2, very high
temperatures correspond to the high flight speeds. A complete tempera-
ture equalization of the colliding molecule to the wall temperature
would be equal to a total annihilation of the recoil speed, or an
almost completely inelastic collision. It should be observed that the
wall accommodates itself in a short time to the temperature of the
colliding molecules, because of the very small heat capacity of the
thin metal walls of the moving body.

The molecule mass, colliding on the oblique unit surface per second
at very high flight speeds is, for example, about equal to pv sin a,

NACA TM 1270

and thus the arriving energy E = Uppv sin a. With pp = 7 = 10-6 kg/m3,
a = 7, and v = 8000 m/s the value of the energy brought in is
E = 7.8 kcal/m2 sec, almost independent of the composition of the atmos-
phere. If the accommodation coefficient of the arriving gas molecules
is one, then the wall would obtain this energy in the form of heat and
this heat quantity should be given away by radiation, where a temperature
increase AT of approximately 5800 is necessary for black body radia-
tion, i.e., the plate stays in fact pretty cold compared to the colliding
molecules, and a very intensive, lasting energy delivery by the colliding
molecules is out of the question. According to existing measurements5,
this temperature equalization is not 100 percent, however, an accommodation
coefficient of 30 percent was found under certain conditions, i.e., the
reflected gas mass contains still 70 percent of its internal energy U
which it possessed at the moment of collision.

The reflection velocity for a monatcmic atmosphere is established
this way.

The remaining internal energy of a molecular atmosphere will dis-
tribute itself quite differently over the existing degrees of freedom of
the reflected molecule than assumed for the colliding energy, which con-
sisted primarily of kinetic energy 1/2 Av2/g, and only in small measure
of the internal energy U, of the gas at rest, which latter was distri-
buted evenly over all degrees of freedom.

For a diatomic molecule with three translational and two rotational
degrees of freedom, the individual shares of Uo, for an "average"
velocity E, are for each kilogram of gas 3/6 Ac2/g for the three
translational degrees and 2/6 Ac2/g for the two rotational degrees of
freedom of the molecules.

On collision, all degrees of freedom will take part in the energy
distribution change, and it can be assumed for further estimation of
the diatomic molecule between the perfectly elastic and the perfectly
inelastic collision conditions, for instance, that the total energy
A/g v2 + 3 C2 + 2 62) = A/g( v2 + 22) distributes itself on the

average evenly over all these five degrees of freedom.

It can be taken from figures 4 and 5 that very high temperatures
are associated with the high colliding molecular speeds at which another
motional degree of freedom is excited, that of mutual molecular oscillation.

5Wien-Harms, Handbuch der Experimentalphysik, Vol. VIII/2, p. 638,

NACA TM 1270

The known Boltzmann's equilization rule on the energies is not of
value for these oscillational degrees of freedom.

While the three translational and the two rotational degrees of
freedom of a diatomic molecule have each the same energy admission

Ux = 1 APT = 1 Ac2/g (kcal/kg)
2 6

that is together

U = A5 T = Ai2/g
(trans. + rot) 2 6

the energy admission of the oscillational degree of freedom Us at low
temperatures is practically zero and increases at higher temperatures
according to Planck's equation:

U A _RQ_ 2 A2 _
Se/T 6 T(e/T 1)

approaching the limiting value, valid for high temperatures, of full
exitation of the oscillational degree of freedom,

=2 APT= A 52
2 o g

In the last equation:

A = the mechanical equivalent of heat, 1/427 kcal/kg

R = the individual gas constant m/

T = the absolute temperature OK

0 = a temperature characteristic for each material, which is, for

example, for nitrogen N2 = 3350K, for hydrogen H2 = 6100K.

The temperatures of the colliding molecules at high flight speeds
are so high (according to figs. 4 and 5) that the molecular gas here

NACA TM 1270

already dissociates strongly into its atoms under normal equilibrium

The transformation of the gases hydrogen and nitrogen, which are of
importance in the higher atmospheric layers, into their monatomic, active
modification belongs to the most energetic endothermic chemical processes
which are known (H2 = 2H 51300 kcal/kg; N, = 2N 6050 kcal/kg), and
the dissociation (if it actually occurs) would absorb extraordinary amounts
of energy and would make the collision almost completely inelastic.

So far it has not yet been proven by experiments that these
molecules really dissociate on collision against a fixed wall at the
speeds here considered.

However the results of known tests with electrons colliding
against the molecules of very rarefied H2 or N2 gases let us
guess that the energy of the collision with a molecular speed up
to v = 8000 m/s is not sufficient to disturb the molecular bond.

With the electrons colliding against N2 or H2 molecules dis-
sociations are observed6 only when the energy of the colliding electron
was several times the dissociation energy of the struck molecule.
This transferred on our case would yield colliding speeds of
over v = 10,000 m/s for a nitrogen atmosphere or over v = 35,000 m/s
for a hydrogen atmosphere, which lies outside of the range of our

For the calculation of the forces on a plate oblique to the air-
stream, we shall therefore not assume the dissociation of the colliding

The degree of elasticity of the recoil will only be derived from
the energy distribution of the wall molecules and of the proper rotational
and oscillational degrees of freedom of the colliding molecule.

This degree of elasticity, i.e., the ratio of the molecular
reflection velocity, when energy division occurs, to the reflection
velocity when io energy division occurs, is estimated as follows:

Wien-Harms; Handbuch der Experimentalphysik, Vol. VIII/1,
pp. 704, 706 (1929).

NACA TM 1270

First, corresponding to the measured accommodation coefficient,
'0 percent of the collision energy corresponding to v be communicated
ie wall molecules. The remaining collision energy,

A/g( 0.7 2 + 3 2 + A/ 7 2 + ( 52)
2 6 6 V +6

.d be distributed over the three translational degrees, the two
ional degrees and the oscillational degree of freedom evenly and
'ding to the exitation degree so that each translational degree of
om has the following energy:

A/(7 2 +5
Te' 1)

condition of motion of the diffusely reflected molecules is the same
sumed with equation (3), only there the energy content of a trans-
nal degree of freedom was A/g 2. Here the internal energy
goingg to the three translational degrees of freedom is given
= 3Er and the translational speed is now:

6 (0.7 2 5 -2
o = _=
F+A 20

T(e 1)

ad of: cr =v2 when no energy distribution takes place
en the wall and the translational degrees of freedom. (The latter
comes from the energy balance:

S 1 v2 +3
2 2 61

NACA TM 1270

The recoil impulses should fall off approximately like speeds
obtained from energetic considerations, so that:


This degree of elasticity is drawn in
hydrogen as a function of the flight speed
the relation between v and T.

figure 6 for nitrogen and
v, using figure 4 or 5 for

The pressure vertical to the wall on the plate oblique to the
air stream, in the case of diffuse-inelastic reflection, can be obtained
from equation 11:

Pv = i + E

Corresponding to figure 3, figure 7 shows the relation between the
pressure p or the shear T and the dynamic pressure q for hydrogen
(c = 1508 m/s) and nitrogen (c = 406 m/s) for air angle of attack a = 40
and for flight speeds to v = 8000 m/s.

In figure 7 the two impulse contributions due to collision (ip)
and recoil (ei/2) are separated.

NACA TM 1270

The collision impulse could be given without objection from purely
mechanical relationships.

The reflection impulse is estimated on the basis of a series of
rather arbitrary assumptions on direction and speed of the reflection.

Further information, proceeding out of the conjectures presented
here, on the retention of single air molecules after collision with
the wall with high speed should be found by experiment. Such tests may
be joined with the well-known molecular beam investigations, where the
usual beam speeds are to be increased greatly by correspondingly greater
energy delivery to the molecules under investigation. One can thus
obtain a type of wind tunnel investigation in which single molecules
fly with extremely high velocities, and the effects of their collision
with a solid body can be observed.

With the help of very rapid molecular beams, a number of questions
should be cleared up experimentally: how often a reflection of the
molecules from the struck wall actually occurs, what factors change the
completely elastic collision into a more or less inelastic collision
through the transformation of translational energy into other forms of
energy (i.e., rotational, oscillational or dissociational energy of
the gas molecules or the wall molecules), and what direction law the
final reflection follows, whether mirrorlike reflection, or predominantly
diffuse reflection, or following a different law.

These investigations can use to advantage De Broglie's analogy
between molecular beams and x-rays.



Vertical flow against the flat plate (a = ) represents the limiting

of oblique flow.

The relations obtained in section 2 are preferably discussed here
for this special case.

NACA TM 1270 2

The total molecule mass, colliding per unit area per unit time is
given by equation 4:

P= .x + 2 + )dp +

cf=v cx=0



2 '
v2cx 2
+ -3 e c2 dc + 4
c3 /tx

2 2
P C 2 v v
= I e (c + Cr-+2--

1 c2
2 v2

1x3 c4
+ -V-+ .

- e c dcx

dc x
e C cX

1 2
. e c 2+2 *


The complete integration of equation 4(a) (in contrast to equation 4)
is possible by series development, because one can put v/c>>l for the
high flight speeds under consideration, which was not possible for
v sin a/c in equation 4.



P c

NACA TM 1270

The total impulse of this gas mass striking vertically against the
plate per unit time and unit area is given by equation 5:

dp- CX2
d c x 2

+6-2 +
ex 2 I -

ip = f

= pv2 +

pc2 p
pc + P
f .7 1


vc v2
2 3

lx3 c4
+ +
22 ~4

)1 -2
. e c2

If, as assumed at the start of the integration v/c 1, the terms
of the equation multiplied by e c2 can be neglected compared with
the first two terms of the equation.

For instance, the influence of the variety of absolute molecular
velocities (following Maxwell's distribution), which is presented by
these higher terms, is less than 0.3 percent of the value of the first
two terms, when v/c is 2 or more.

Therefore, for v/c > 2, the impulse can be calculated as if all
the molecules had the same velocity c; thus, considering only the
first two terms of equation 5(a),

p = pv2 + 1 pc2
P 2


Under the assumption of a completely inelastic collision, this
impulse is equal to the required pressure on the plate:

P = = p2 + 2 (lOa)
p- 2

dp 2C
6 x


NACA TM 1270

If it is assumed that the air molecules have not lost their velocity
after the collision with the plate, but that they rebound perfectly
elastically and "mirror-like," then the impulse given to the plate is
simply doubled, and equation 10(a) for elastic mirrorlike collision
is written:

p = 21 = 2(P2 + c2)


The influence of the random molecular motion, represented by the
second term, is at = 2 approximately 12.5 percent of the pure
Newtonian air force, which is represented by the first term. However,
the influence decreases, from Y.= 5 on, to under 2 percent of the
Newtonian value, contrary to the oblique conditions with small angles
of attack, where the influence of the molecular velocity is still great
even with high flight speeds.

If it is assumed that the molecule reflection is completely elastic
but diffuse, then the pressure on the plate decreases to a value:

2 v (v\2\
3c \/

p = ip + = 1 1 + 0.75

= p(v2 + c2) 1 + 0.T7

3 sn 3(
1 + sin +
c 2 )

1 +v sin a + V
3 0



1 + / sin a +
c 2(c


NACA TM 1270

If finally the molecule reflection becomes not only diffuse but also
partly inelastic in the sense of equation 12, then the pressure on the
plate reduces again to:

p = iP + e I= 2
p 2

+ 1c2) 1 + 0.75
2 /

1 + vsin m +()2
v V
3 c \c

1 + sin a + -
c 2 c

The graphs in figures 8 and
to equations 10(a) to 10(d)

9 show the pressure conditions according
versus the dynamic pressure of nitrogen or

For the calculation of the pressure of a motionless gas on the walls
of the gas container which has the same temperature as the gas kinetics
assumes completely elastic collisions for the normal range of molecular
speeds, i.e., the molecule keeps the same translational energy, on an
average, as it had before the shock. It is not important to know if
the reflection is mirrorlike or diffuse, because both assumptions lead to
the same molecular picture for the gas at rest.

If the molecules are polyatomic then
energy to each possible degree of freedom
shock; and this uniform distribution need

the distribution of the total
is already uniform before the
not change after the shock.

In equation 10(a) (for completely inelastic collisions) for the air
pressure against a plate vertical to the air stream p = pv2 + c2
the first term pv2 corresponds to the dynamic pressure of molecules having
no random motion against the plate (Newton), while the second term I pc2
corresponds exactly to the pressure of the static atmospheric air.



20 2 2
--+ -(v

NACA TM 1270

However, this explanation of the individual terms is only formally
correct, since the resting air pressure is calculated assuming elastic
molecular collisions. With the inelastic collision, the decrease in
the "stopping" pressure due to loss of the recoil impulse will be off-
set by the mixed term in the square of the sum of the two speeds (v and c),
in the velocity range under consideration.

This condition can be recognized more clearly from equation 10(b)
for the completely elastic collision p = 2pv2 + po2 where after sub-
traction of the static air pressure I pc2 there remains a pressure

of 2pv2 + pc2, which contains besides the Newtonian term, 2pv2,
1 2
also an additional term of 2 pc which reflects the effect of the
mixed term.

No resulting impulse is given parallel to the vertical plate
because of the symmetry of the total system, i.e., friction forces are
transferred in the plate plane, but the sum of these forces outward
is zero.


If the mean free path of a molecule is small compared to the
dimensions of an empty space into which the gas is flowing, then the
flow-in speed can be greater than the most probable molecule speed. In
the flow of diatomic gases into a complete vacuum, the directed
velocity, max, of the total flowing mass can surpass the probable
molecule speed c by a factor of about 1.87, according to the laws of
gas dynamics.

If, however, the molecular mean free path is comparable to the
empty space dimensions or even greater than these as assumed here, then
the number of molecule collisions behind the rapidly moving plate during
the flow-in is not sufficient to produce the mentioned acceleration,
and the molecules move with their usual speed c into the empty space
behind the plate.

According to figure 10, collisions between the air molecules and
the back side of the plate cannot take place, i.e., the pressure against
the "suction side" of the plate must have become zero, as soon as

> c
sin a

NACA TM 1270

This border line is, however, strongly blurred because of Maxwell's

The effective forces against the back side of the plate can be
derived by the same process which led to equations 4 to 12, where now
the uniform velocity v is directed away from the plate at the angle a,
while before it was directed toward the plate (fig. 11).

It is assumed first that v sin a
The uniform velocity v of the individual molecule combines with
the ideal random velocity cx of the molecule (which can have any space
direction) to a resultant whose components are:

perpendicular to the plate: c cos v sin a

parallel to the plate and to v cos a: v cos a + cx sin sin

parallel to the plate and perpendicular to v cos a,: cx sin 0 sin

From the sphere of all possible directions of cx, a spherical sector
with the half angle cos X = v sin is taken, inside of which the
velocity component v sin a cx cos 0 is directed towards the plate, so
that the molecules of this 0-range do in fact collide against the plate.
For all 0 > X the resulting molecular velocity is directed away from the
plate. Therefore no collision with the plate takes place.

The integration over all colliding directions is extended from = 0
to = x.

The molecular mass with the chosen speed cx colliding per second
against the unit area of the plate is therefore:

dp = --~ (ex cos v sin a )dF = -cyx 2v sin a + v2sin2a/cx)


For v sin a >cx there are no collisions with the plate, so that
this case need not be treated.

NACA TM 1270

The total molecule mass colliding on the unit area per unit time
is obtained with the aid of tMaxwell's distribution equation for the
velocity range cx = v sin a to :

P = x s 2v sin a + v2sin2a/cx)dp
rX=v sin a

-2 2Cx
- 2- sin a + --- a
c3 c3


- p


in2a) e

sin a

v2s in2Ca
p c c
= a2

- v sin a/c

e dc,

=v sin a

i ;=V 8in e

v2sin2c / sin a

S+ 2v s L /cx 2


02 .
e dcx -- -


P c
- ff 2


NACA TM 1270

Similarly for the impulse vertical to the plate:

dip d (cx cos v sin a)2dF
4cz r f (C=0

= d c 2 -3 sin a +- 3 -2 sin2 -
6 c cx2

v so i

Ucx- v sin a

- p

sill a i 1 1
i-n a 1 v2sin a + cv sin a -
c 3 3 2

x3 /

ain 3a dp


c2) e


v sin --a -
--7 dcxf

cx2 3 -v- sin a + 3 2 sina -
cx C7 cx3


NACA TM 1270

Finally, for the impulse parallel to the plate:


x os C vc sin a)v cos a dF

v cos a 2V2sin a cos a + sin2a cos
Cx. )

4 cx v cos a 2v2sin a coB a +

v2s iln2a
= cos a e
V2 3


v ein a
-- in
c sin a

Vw sin2a cos a dp

e dc

= ^

V cos c: e

The impulse resultant vertical
elastic-diffuse molecular recoil is
colliding molecules:

sin a 2
e dcx
- c



to the wall as a consequence of the
from the total impulse i of the

1 + sin + 2
1 = 1.5 lp2 + 1T2
1 + Y sina + a. )2

iT =

LX =v Sin a

NACA TM 1270

Equation (10) again holds for the total pressure against the
suction side of the plate with elastic-diffuse recoil. (Translator's
note: Formula missing in original German report.) While for the
total shear stress on the suction side of the plate, equation 9 is used:

Tr iT

The influence of a certain inelasticity of recoil can be estimated,
particularly for small angles of attack by the same procedure which led
to equation (ll), according to which the degree of inelasticity can also
be specified.

For the total pressure against the suction side, equation (12) is
pr = ip + E~

Corresponding to figure 7, the graphs in figure 12 show the relation
between the pressure p or the shear T and the dynamic pressure q
for hydrogen and nitrogen at an angle of attack a = 40 and flight
speeds up to v = 8000 m/s.


With the help of the previously mentioned relations, it is possible
to estimate all the air forces acting on the surfaces of a flying body
of any shape, which is moving at flight altitudes of over 100 km with
speeds between about 2000 m/s and 8000 m/s, if definite assumptions are
made on the composition of the air at this height.

The air forces were differentiated into those which act perpendicular
to the surface under observation (pressures), and those which act parallel
to the surface (friction).

The pressure stresses as well as the shear stresses were found to
be a function only of the angle of attack and the flight speed, for a
particular gas.

In figures 13 and 14 is shown this dependence of the air forces on
all possible angles of attack and on flight speeds between v = 2000 m/s
and v = 8000 m/s for an atmosphere of molecular hydrogen.

NACA TM 1270 33

It is to be noted in figure 13 that the air pressure vertical to
the plate strongly increases with increasing velocity, even with an
angle of attack a = 0, if the molecular recoil is diffuse.

This representation can be used as a basis for the calculation of
air force coefficients for certain flight bodies in hydrogen, treating
each flat surface section separately, with its own angle of attack, or,
if the body surface is 2urved, analyzing it into a great number of
small areas with individual angles of attack (flat areas or symmetrical
cone areas), and then investigating theee.

As the simplest example, the flat, infinitely
treated first. The Usual symbols are

thin plate will be



wing surface

and the air force coefficients are:

ca = =

cw =


- -1 cos

Pv Pr'
!1- t sin

+ -)sin a

- cos a

and the glide ratio:


lP Pr.
- -cos
\ q 1

a. + cos a

a- + )r sin a



a+ -- +

NACA TM 1270

In figure 15 are drawn the lift coefficients and in figure 16 the
glide ratios of the flat, thin plate according to the above relations.

On account of the extraordinarily great shear forces very bad
glide ratios result, which are approximately e = = 1.9 at 2000 m/s
with the most favorable angle of attack and which got worse at higher
speeds, for example, at 8000 m/s, e = about 2.7.

The moat favorable angles of attack are comparatively great at
small speeds, i.e., at v = 2000 m/s, a = 250 approximately, and decrease
with increasing speed to about 7 at v = 8000 m/s.

Similarly to the infinitely thin plate, high speed profiles of
finite thickness can also be calculated, i.e., wedge-shaped and
lenticular airfoils. Their air force coefficients hardly deviate from
those of the smooth plate, if they are of moderate thickness.

In general, the wings investigated here in the gas kinetics flow
range behave worse than in the gas dynamics range, where already the
glide ratios are worse than in the usual aerodynamical flow region.

The full effect of this unfortunate behavior will be corrected to
some extent by a flight technique such that at the high flight velocities
under consideration, inertial forces are developed by the concave down-
ward flight path, which support the wing.

Figure 17 treats the question of how great the air drag is in the
gas kinetics flow range for a body of rotation (projectile form) moving
axially, with an ogival nose of three calibers radius and cylindrical
body, and how far the air drag can be improved by a truncated cone bevel
at the end of the missile.

These questions can be easily answered with the aid of figures 13
and 14 if the ogive is divided into a large number of truncated cones,
each of which represents a surface with a definite angle of attack.

The extraordinary value of the drag coefficient is again striking;
it can be traced to the very great friction forces in the extremely
rarefied air.

A noticeable improvement of the drag coefficient could be obtained
by beveling the end of the projectile; the improvement is about 7 per-
cent of the original value.

Somewhat more tediously but in basically the same manner, the air
forces on a projectile, airship, etc., at an oblique angle of attack
can be determined, using figures 13 and 14.

NACA TM 1270 35


The air forces on bodies of arbitrary shape are investigated when
the bodies move with speeds of 2000 to 3000 m/s in such thin air that
the mean free path of the air molecules is greater than the dimensions
of the moving body.

The air pressure acting perpendicular to the body surface, as well
as the friction forces acting parallel to the surface, are derived with
the aid of the calculation procedure of gas kinetics for surfaces facing
both toward and away from the air stream at any angle.

The air forces for an atmosphere of definite composition (molecular
hydrogen) are calculated as a function of the flight velocity at all
possible angles of attack of a surface and shown in graphs.

Thereby the friction stresses between air and body surface prove
to be of the same magnitude as the dynamic pressure and as the air
pressures vertical to the body surface, i.e., 300 times greater than in
the aerodynamic flow range.

The application of the general calculation results to particular
technically important cases, like thin airfoils and projectile shapes,
results in extraordinarily high air drag coefficients and poor glide
ratios even for the theoretically best wing sections

Translated by Bureau of Aeronautics
Technical Information
Navy Department

NACA TM 1270

dF= 2 cr sin o do


Figure 1.- Velocity vectors of the thermal motion of molecules of a motionless
gas and their position relative to a fixed boundary wall.

NACA TM 1270 37


,I I





MO-U A'. V Ni .V-ld 3H1l JO N011V1N31O0
\ x/ \u

K^----------^ N

NACA TM 1270

2000 4000 6000 8000

Figure 3.- Air pressures p and shear stresses T on the front side of a flat
plate at 40 angle of attack in an atmosphere of atomic hydrogen under the
assumption of elastic diffuse or mirror-like recoil of the atoms from the

NACA TM 1270

Figure 4.-

Colliding speed and associated internal energy of molecular nitrogen
in relation to the colliding temperature of the gas.





NACA TM 1270


DO 3000

Figure ,5.- Colliding speed and associated internal energy of molecular
hydrogen in relation to the colliding temperature of the gas.


U --

.4-II 4 -







Degree of elasticity of recoil for nitrogen
molecules from the struck wall.

or hydrogen

09- p-q .-- -----



06 ,


""- P 4--------
04 --__ _

0.3 \ ____________

N2r q

'N p.q
vC rn/s




Figure 7.- Air pressure p and shear stress r on the front side of a flat.
plate at 40 angle of attack in an atmosphere of molecular hydrogen or
nitrogen under the assumption of diffuse and semielastic molecular
recoil from the wall.

NACA TM 1270

1.0 i-


Figure 6.-




NACA TM 1270


0. _-_ ____-----------------











Figure 8.- Air pressure p on the wall vertical to a stream of molecular
hydrogen, under various assumptions on the collision process.



NACA TM 1270





vL m/sl




Figure 9.- Air

pressure p on the wall vertical to a stream of molecular
under various assumptions on the collision process.

Figure 10.- Velocity vectors of the molecular motion for collision on the
back side of the flat plate.



NACA TM 1270


-I 4





I 0-

I >

I 0

0 0
I a,

I .

.> --
0 -4



NACA TM 1270

Figure 12.- Air pressures p and shear stress T on the back side of a flat
plate at 40 angle of attack in an atmosphere of molecular hydrogen or
nitrogen, under the assumption of diffuse and semielastic molecular recoil
from the wall.

NACA TM 1270

Figure 13.- Coefficient p/q of the air pressure vertical to the plate for
all angles of attack and for flight speeds between 2000 m/s and
8000 m/s in atmosphere of molecular hydrogen.

NACA TM 1270

Figure 14.- Coefficients T/q of the shear stress between air and plate
for all angles of attack and for flight speeds between 2000 and
8000 m/s in an atmosphere of molecular hydrogen.

Lift coefficients for the flat infinitely thin plate.

Figure 15.-

NACA 'IM 1270

Figure 16.- Reciprocals of the glide ratios for the flat infinitely thin plate,
and best values of the glide ratio (dotted line) with corresponding angles
of attack.

NACA TM 1270

Figure 17.- Coefficients of the pressure drag, friction drag, and total drag
for a projectile-shaped body of rotation, with different missile b-ottoms.

NACA-Langley 5-10-50 .900




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