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RM No. L9C02
NACA
RESEARCH MEMORANDUM
AN APPROXIMATE METHOD FOR ESTIMATING THE INCOMPRESSIBLE
LAMINAR BOUNDARY-LAYER CHARACTERISTICS
ON A FLAT PLATE IN SLIPPING FLOW
By
Coleman duP. Donaldson
Langley Aeronautical Laboratory
Langley Air Force Base, Va.
UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
1 20 MARSTON SCIENCE LIBRARY
RO. BOX 117011
GAINESYtU.E. FL 32611-7011 USA
NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
WASHINGTON
May 2, 1949
/jr '^vf^ 3^/^
NACA EM No. L9C02
NATIONAL ADVISORY COMMITTEE FOE AERONAUTICS
RESEAECH MEMORANDUM
AN APPEOXIMATE METHOD FOR ESTIMATING THE INCOMPEESSIBLE
LAMINAE BOUNDARY-LAYER CHARACTERISTICS
ON A FLAT PLATE IN SLIPPING FLOW
By Coleman duP. DonaldBon
SUMMARY
An approximate method is presented for the estimation of the
properties of the incompressihle iBmlinav "boundary layer on a flat
plate in the slip— flow region iising Earman's momentum, method.
At eq^uivalent stations on the same "body at the same Eeynolds
numher, the total thicJoiess and the skin friction of a slipping
"boundary layer are less than that of the normal hoijndary layer at the same
Reynolds n-umber. However^ the difference "between the slip and normal
"boundary layer is amn 1 1 until slip velocities at the wall are encountered
which are large in comparison with the free— stream velocity; that
is, u-jf ^ 0.3Uq. An Important effect of slip is that on the displacement
thickness of the "boundary layer.
The following criterion is presented for determining the importance
MX
.ip phenomena: If — >
de3crl"blng viscous phenomena,
MX
of slip phenomena: If — > 0.0^4-, slip becomes an important factor in
INTRODUCTION
Recent interest in very high altitude flight has led to an interest
in and considerable speculation as to the nature of gas flows when the
mean free path of the gas molecules is of the order of magnitude of the
boundary— layer thickness on a body and also for which the mean free
path of the molecules is of the order of magnitude of the length of the
body itself. Tsien (reference l) has described these two types of flow,
the former being called the slip— flow regime and the latter the free-
molecule— flow regime .
MCA RM No. L9C02
There has "been considerahle work done on shear or drag forces
in the slip— flow region (see references 2, 3, and k) , hut most of this
work has heen done on hounded flows such as the flow hetween two
concentric rotating cylinders or the flow through long tubes. It is
the purpose of this paper to investigate the general nature of the
slip flow on a flat plate when the mean free path is of the order of ^
hut less than, the "boundary— layer thickness. The analysis is only an
approximation, hut the properties of a laminar boundary layer in the
slip— flow regime and the magnitude of the drag reduction due to slip
are evaluated. Insofar as a simple method of evaluation is useful,
such an approximate analysis may he Justified.
SYMBOLS
C-Q drag coefficient
k ratio (s/X)
I length of flat plate
L length of order of mean free path
m mass of molecule
u horizontal velocity
V vertical velocity
X horizontal coordinate
y vertical coordinate
5 boundary— layer thickness
\ mean free path
H viscosity
V kinematic viscosity
p density
T shear stress
NACA EM No. L9C02
Subscripts:
n normal flow
o free-stream values
s slip f low
w wall values
ANALYSIS
Karman's momeiitum theory for the "boundary layer on a flat plate
may be expressed (see reference 5) by the formula
d
\ = j^ I p^(^o - ^)^y (1)
This formula may be used if the normal boundary— layer assumptions are
valid; that is^ that the boundary— layer thickness is small compared with
the distance to the leading edge of the plate, that the flow in the
boundary layer is almost parallel to the surface, and that the major
viscous terms are of the same order of magnitude as the inertia terms.
It will be seen as the analysis progresses that these conditions are met,
and so a velocity profile for the laminar boundary layer consistent
with the boundary conditions in slip flow will be assumed and used to
solve eq^uation (l) for the rate of growth of the boundary layer and the
value of the surface friction at a given station.
The velocity at the wall in a slip flow from elementary kinetic
considerations (see reference k, pp. 291—299) ™a-y ^Q taken as
% = 4^) (2)
^ '^ w
so that if a boundary— layer profile of the form
Ji- = A +Bf + Cfff (3)
u„ 5 \S/
MCA EM No. L9C02
is assumed, the following "boundary conditions may "be taken
y = u =
y = 5 u =
- = <ni
Uo and ^
dy
>
=
W
The application of these "boundary conditions results in
Uq 2X, + & 2\ + 6 5(2X + 6)
jzi
(5)
The friction at the wall is
w
-.('^
2|au,
Vdy/w 2\ + 8
(6)
"Upon su"bstituting eq^uations (5) and (6) in equation (l) and assuming the
flow to "be incQmpressihle, the following result is obtained "by carrying
out the integration
2nu
o
^^ " 2X, + 5
dx
2 2 2 3
3 15
(2A. + &)2
(7)
Equation (7) is differentiated to o'btain
g^^^o
2\ + b
= PUo
^X5 + 2^2
3 5 3
^6^ , A^3
15
(2\ + 6)2 (2;^ + 5)3
d5
dx
(8)
NACA RM No. L9C02
This eq^uation is ncfw integrated and yields
X =
51 - Ax% 16
\-
1
s3
8.
10 15
15 (2X- + S)
+ -^X log 22l_L^
15 2X. + 6 15 2X
(9)
This equation relates the distance from the leading edge of a flat plate
and the "boundary— layer thickness for a given mean free path. Equation (9)
may he simplified by the introduction of the plate length I and the
ratio
I-
wherehy equation (9) "becomes
X
T
ao
_8_ 16 1
15 "*■ 15 2 + k
1 k3 8 , 2 + kA ,,-^
or
T - «n(T)'f "='
(11)
Equation (ll) gives the position on a flat plate at which the
"boundary layer is k times thicker than the mean free path. The value
of f (k) is plotted against k in figure 1.
From equation (2) the velocity at the wall at any point is given "by
2u.
"w =
2 + k
(12)
and from equation (6) the friction stress at the wall divided "by twice
the dynamic pressure is
'w
piio'^
EnVV2 + k
(13)
6 NACA RM No. L9C02
The displacement thickness of the "bouadary layer is found to "be
(li^)
It is readily seen that if the mean free path X is placed eq^ual
to zero in equation (5) the ho-undary— layer profile 'beGomea
and the boundary— layer thickness is foiuid, "by putting X = in
equation (9)j "to "be
6 = 5-^8
/?
which is in general agreement with the Blasius solution.
EXAMPLE
A specific example is now worked out to illustrate for a particular
case the difference "between a slip and a normal flow. Thus a Reynolds
num"ber of 100 was assumed and the ratio of plate length to mean free
path was chosen as 25- This might correspond to a 1— foot— chord plate
traveling at a Mach num'ber around 2.9 a-"t an altitude of 250,000 feet,
since (see appendix A)
M = ^ ^ (15)
A lower velocity might have been chosen for a 1— foot body at 250,000 feet,
so that the flow would be incompressible, but the resulting lower Reynolds
number would have given a larger boimdary layer and the slip— flow region would
have been confined to a somewhat smaller region near the leading edge so
that it would have been more difficult to demonstrate the results of the
NACA RM No. L9C02
slip. Indeedj this fact indicates the desiralDility of extending the
analysis to include the effects of compressihillty.
Figure 2 shows the solution for the thickness of the "boundary layer
along the plate in terms of the dimensionless ratios b/X and x/Z .
The velocity profiles are also plotted at their proper positions. It
is seen that there is a slip velocity at the wall over the entire plate.
Figure 3 shows this solution compared with the normal boundary— layer
solution at the same Reynolds number. It is seen that the slipping
houndary layer is thinner at eq^ulvalent stations than the normal "boundary
layer (taken in this example to "be the solution when X, = O) .
Figure k shows a comparison of the displacement thiclmesses in termB
of the ratio B*/l for the two cases. It can "be seen that there is a
large effect on the displacement thickness due to slip, as might be expected.
Finally, figure 5 shows a comparison of the local skin frictions in
the two cases. It may "be seen that as the thickness of the "boundary layer
"becomes large with respect to the mean free path, the slip skin friction
approaches the normal value. But at the leading edge where, since there
is no "boundary layer, the flow must "be a free-molecule flow, the result
of this analysis is compared with the free-molecule stress coefficient
given "by (see appendix B):
M 2.91
It is seen that the present method yields a stress at the leading
edge that is just twice the value derived from free molecule considerations.
If these skin frictions are integrated over the surface of the plate, the
drag coefficient for the slip flow on one surface is found to be
Cd3 = 0.1312
while for the normal flow at the same Reynolds number it is found to be
Cfl = 0.1460
°-n
The figure shows that the effect of slip has little effect on the skin
friction over most of the plate.
8 MCA EM No. L9C02
DISCUSSION
Strictly speaking, the eq^uation for the velocity at the vail
M
^" ^ Hit
I
w
holds only for the houndary layer when the mean free path is conalderahly
less than the boimdary— layer thickness. The error is principally that,
in deriving the equation for the velocity at the vail, the momentuni "brought
in to the vail hy molecules at an average distance L from the wall ia
(see reference h, p. 1^4-0)
m
-^K^.
It may be seen from the shape of the velocity profile in figure 6 that
this assumption becomes feasible when the boundary— layer thickness is
approximately twice the length L or approximately twice the mean free
path. It iSj therefore, obvious that this type of analysis is only
applicable to boundary— layer regions when the mean free path is about
one— half or less than one— half the boundary— layer thickness.
From the analysis it is seen that the thicknesses and rate of growth
of the slip boundary layer are of the same order of magnitude as those
of a normal boundary layer at the same Reynolds number, and hence the
boundary— layer assumptions necessary for eq^uation (l) must be eq\;ia.lly
valid for the slip boundary layer.
In general, the effect of slip is to decrease the drag and the
boundary— layer thickness from what would be calculated for a nonnal
boundary layer at the same Reynolds number. The greatest effect of slip
is upon the displacement thickness. The growt-h of the boundary layer
and the skin friction at the wall may be very close to the normal values
even in the presence of a considerable slip velocity at the wall; that
is, uv = O.3U0.
It should be noted that further boundary conditions may be imposed
by assuming higher powers of y/& in the eq^uation of the velocity profile.
The most obvious condition neglected by this analysis is
NACA RM Wo. L9C02
This condition leads to reBults in somewhat "better agreement with the
Blasius results for the case of X, = where u^ = so that
"but leads to difficulties in the analysis when the mean free path is
apprecia'ble and there exists a slip velocity at the wall.
From the example it may "be seen that if the "boundary layer at the
end of the plate has a thicloaess less than a'bout 20 times the mean free
path, it might "be expected that the effects of slip would "be important .
From this fact it is possi"ble to construct, with the aid of eq^uations (11)
and (15), an approximate criterion for the Importance of slip
phenomena, "Upon su"bstituting equation (I5) into equation (11) and
putting — = 1.0 for the trailing edge, there results
I
1~ l.i*^f(k)iE
If k is to "be less than 20 at the trailing edge, then f (k)i]Tg must
"be leas than l6.5j or roughly
^>-^ (16)
1 25
From this approximate criterion for the importance of slip— flow
effects, it may "be seen that slip phenomena will "be more Important at
high Mach numbers and the present analysis should be extended to include
the effects of compressihility . This criterion agrees well with the
slip— flow regime as defined "by Tsien. Further, it may "be seen that the
two fundamental varia"bles most useful to descri'be gas flows at low
densities are Mach number and the ratio of mean free path to body length.
10 NACA EM No. L9C02
CONCLUSIONS
1. An approximate method of estimating the slip— flow "boundary layer
on a flat plate has heen presented.
2. At eq^uivalent stations the total thickness and the skin friction
of a slipping hoimdary layer are less than that of the norm.«?l houndary
layer at the same Reynolds number.
3. The difference "between the slip and normal "boundary layer is small
until slip velocities at the wall are encoimtered which are large in
comparison with the free— stream velocity, that is, Uy = O.3U0.
k. An important effect of slip is that on the displacement thickness of
the "boundary layer.
I
5. The following criterion is presented for determining the importance
ip phencBEena: If — >
descrihing viscous phenomena,
of slip phenomena: If — > G.Qii, slip "becomes an important factor In
Langley Aeronautical La"boratory
National Advisory Committee for Aeronautics
Langley Air Force Base, Va.
NACA RM No. L9C02 11
APPENDIX A
DERIVATION OF EQUATION (I5)
Reynolds number is defined as
R = P^2 _ u2 pk
and J from the kinetic theory of gases,
|i = O.U99pc\
so that practically
R^ = 2-i 1
^ c X,
Since the mean molecular velocity c for air is 1.462 times the
velocity of sound a, there results
Rn = 1.37Mf
A*
or
M = —
1.37
12 NACA EM No, L9C02
APPENDIX B
DERIVATION OF EREE-MOLECULE FRICTION STRESS COEFFICIENT
The momentum which strikes a unit area in a unit time of a flat
plate in a free-molecule flov is l/i^-pcuQ. If all the molecules are
reflected with zero velocity from the surface of the plate, the shear
stress at the surface is
■"w = j;^°^o
The stress coefficient is therefore
\ _ i J_ _ 1 ^-^^^ _ 0.366
puo2 ~ i| Uq " 1+ M ' M
41
NACA RM No. L9C02 I3
EHPERENCES
1. Tslen, Hsue-Shen: SuperaerodynamlcSj Mechanics of Rarefied Gases.
Jour. Aero. Sci., vol. 13^ no. 12, Dec. 1946, pp. 653-66^^.
2. Maxwell, James Clerk: Scientific Papers, vol. II, Cambridge Univ. Press
1890, p. 705.
3. Milllkan, R. A.: Coefficients of Slip in Gases and the Law of
Reflection of Molecules from the Surfaces of Solids and Lic[uids.
Phys. Rev., vol. 21, no. 3, 2d ser., March 1923, pp. 217-238.
k. Eennard, Earle H.: Kinetic Theory of Gases. McGraw-Hill Book Co.,
Inc., 1938, pp. lijO and 291-299-
5. Prandtl, L.: The Mechanics of Yiscous Fluids. Theorem of Momentum
and Karman's Approximate Theory. Vol. Ill of Aerodynamic Theory,
div. G, sec. I7, W. F. Durand, ed., Julius Springer (Berlin),
1935. pp. 103-105.
ll^
NACA EM No. L9C02
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1000
800
600
400
200
100
80
60
40
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NACA EM No. L9C02
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UNIVERSITY OF FLORIDA
3 1262 08106 576 4
GAINESVILLE FL ??«ii ^n
^c.»-L 32611-7011 USA