An approximate method for estimating the incompressible laminar boundary-layer characteristics on a flat plate in slippi...

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Title:
An approximate method for estimating the incompressible laminar boundary-layer characteristics on a flat plate in slipping flow
Series Title:
NACA RM
Physical Description:
19 p. : ill. ; 28 cm.
Language:
English
Creator:
Donaldson, Coleman duP
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Laminar boundary layer   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: An approximate method is presented for the estimation of the properties of the incompressible laminar boundary layer on a flat plate in the slip-flow regime using Kármán's momentum method. These results are compared with results for a normal boundary layer at the same Reynolds number to bring out the characteristics of a slip boundary layer. A simple criterion for the importance of slip is presented.
Bibliography:
Includes bibliographic references (p. 13).
Statement of Responsibility:
by Coleman duP. Donaldson.
General Note:
"Report date May 2, 1949."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003810747
oclc - 135157485
sobekcm - AA00006223_00001
System ID:
AA00006223:00001

Full Text





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NACA RM No. L9C02

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

RESEARCH MEMORANDUM


AN APPROXIMATE METHOD FOR ESTIMATING THE INCOMPRESSIBLE

LAMINAR BOUNDARY-LAYER CHARACTERISTICS

ON A FLAT PLATE IN SLIPPING FLOW

By Coleman duP. Donaldson


SUMMARY


An approximate method is presented for the estimation of the
properties of the incompressible laminar boundary layer on a flat
plate in the slip-flow region using KarmAn's momentum method.

At equivalent stations on the same body at the same Reynolds
number, the total thickness and the skin friction of a slipping
boundary layer are less than that of the normal boundary layer at the same
Reynolds number. However, the difference between the slip and normal
boundary layer is small until slip velocities at the wall are encountered
which are large in comparison with the free-stream velocity; that
is, uw 0.3uo. An important effect of slip is that on the displacement
thickness of the boundary layer.

The following criterion is presented for determining the importance

of slip phenomena: If > 0.04, slip becomes an important factor in

describing viscous phenomena.


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 1) has described these two types of flow,
the former being called the slip-flow regime and the latter the free-
molecule-flow regime.








NACA RM No. L9C02


There has been considerable work done on shear or drag forces
in the slip-flow region (see references 2, 3, and 4), but most of this
work has been done on bounded flows such as the flow between 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,
but less than, the boundary-layer thickness. The analysis is only an
approximation, but 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 be Justified.


SYMBOLS


CD drag coefficient

k ratio (5/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

X mean free path

M viscosity

V kinematic viscosity

p density


T shear stress








NACA RM No. L9C02


Subscripts:

n normal flow

o free-stream values

s slip flow

w wall values


ANALYSIS


Karman's momentum theory for the boundary layer on a flat plate
may be expressed (see reference 5) by the formula




T = f pu(uo u)dy (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 equation (1) 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 (.ee reference 4, pp. 291-299) may be taken as


= X d) (2)


so that if a boundary-layer profile of the form


u= A +B +C (3)
uo o







NACA RM No. L9C02


is assumed, the following boundary conditions may be taken


y =0 U = =\ 1
((,YNI


y = 6 u = uo and d = 0
dy


The application of these boundary conditions results in


u 2X 2y2
Uo 2= + 5 2. + 6 (2X. + 7)


The friction at the wall is


(du\ _w 2+uo
TW yv 2X + B


Upon substituting equations (5) and (6) in equation (1) and assuming the
flow to be incompressible, the following result is obtained by carrying
out the integration


2p u
= o 2 d
w = puo d
2. +8 5x


S2 + 3
-A& + --
3 15
(2X + 5)2


Equation (7) is differentiated to obtain


4 + 2 2
21Vuo 2 3 5
= pu0 (
2 + 6 (2X + 5)2







NACA RM No. L9C02


This equation is now integrated and yields


S= u 2 2 1 X3 152b3 + 1l 2+
V 10 15 15 (2k + 5) 15 2X + 15 2


This equation relates the distance from the leading edge of a
and the boundary-layer thickness for a given mean free path.
may be simplified by the introduction of the plate length 1
ratio


whereby equation (9) becomes


flat plate
Equation (9)
and the


x u k2 8 16 1 1 k3 8 2 + k)
T V t 1 15 2 + 5 +k 15 2 + k 15 g 2





x= 2fk)
T n, k


(10)





(11)


Equation (11) 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


2uo
2 + k


(12)


and from equation (6) the friction stress at the wall divided by twice
the dynamic pressure is

S_= 1I 2 (13)
puo2 R 7 2 + k







NACA RM No. L9C02


The displacement thickness of the boundary layer is found to be



1 k
= d = k (14)



It is readily seen that if the mean free path X is placed equal
to zero in equation (5) the boundary-layer profile becomes


S= 2. -
uo E 4


and the boundary-layer thickness is found, by putting X = 0 in
equation (9), to be


= 5.48, =


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
number 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 number around 2.9 at an altitude of 250,000 feet,
since (see appendix A)

Rn 1
M (15)
1.37 2


A lower velocity might have been chosen for a i-foot body at 250,000 feet,
so that the flow would be incompressible, but the resulting lower Reynolds
number would have given a larger boundary 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. Indeed, this fact indicates the desirability of extending the
analysis to include the effects of compressibility.

Figure 2 shows the solution for the thickness of the boundary layer
along the plate in terms of the dimensionless ratios 56/ and x/2.
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
boundary layer is thinner at equivalent stations than the normal boundary
layer (taken in this example to be the solution when X = 0).

Figure 4 shows a comparison of the displacement thicknesses in terms
of the ratio 6*/i 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):


T 0.366 0.366 1
T M 2.91 0.125


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


Cds = 0.1312


while for the normal flow at the same Reynolds number it is found to be


Cdn = 0.1460


The figure shows that the effect of slip has little effect on the skin
friction over most of the plate.








NACA RM No. L9C02


DISCUSSION


Strictly speaking, the equation for the velocity at the wall






holds only for the boundary layer when the mean free path is considerably
less than the boundary-layer thickness. The error is principally that,
in deriving the equation for the velocity at the wall, the momentum brought
in to the wall by molecules at an average distance L from the wall is
(see reference 4, p. 140)


m uw + L-



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 is, 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 equation (1) must be equally
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 normal
boundary layer at the same Reynolds number. The greatest effect of slip
is upon the displacement thickness. The growth 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, uw = 0.3uo.

It should be noted that further boundary conditions may be imposed
by assuming higher powers of y/5 in the equation of the velocity profile.
The most obvious condition neglected by this analysis is








NACA RM No. L9C02 9








This condition leads to results in somewhat better agreement with the
Blasius results for the case of X = 0 where u = 0 so that


E0



but leads to difficulties in the analysis when the mean free path is
appreciable 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 thickness less than about 20 times the mean free
path, it might be expected that the effects of slip would be important.
From this fact it is possible to construct, with the aid of equations (11)
and (15), an approximate criterion for the importance of slip
phenomena. Upon substituting equation (15) into equation (11) and
I
putting = 1.0 for the trailing edge, there results


1s 1. if(k)TE



If k is to be less than 20 at the trailing edge, then f(k)TE must
be less than 16.5, or roughly


MX > J (16)
2 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 compressibility. This criterion agrees well with the
slip-flow regime as defined by Tsien. Further, it may be seen that the
two fundamental variables most useful to describe gas flows at low
densities are Mach number and the ratio of mean free path to body length.








NACA RM No. L9C02


CONCLUSIONS


1. An approximate method of estimating the slip-flow boundary layer
on a flat plate has been presented.

2. At equivalent stations the total thickness and the skin friction
of a slipping boundary layer are less than that of the normal boundary
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 encountered which are large in
comparison with the free-stream velocity, that is, uw 0.3uo.

4. An important effect of slip is that on the displacement thickness of
the boundary layer.

5. The following criterion is presented for determining the importance
of slip phenomena: If 2-> 0.04, slip becomes an important factor in
describing viscous phenomena.


Langley Aeronautical Laboratory
National Advisory Committee for Aeronautics
Langley Air Force Base, Va.








NACA RM No. L9C02


APPENDIX A


DERIVATION OF EQUATION (15)


Reynolds number is defined as


Rn = pu = u2Z P%
I: II


and, from the kinetic theory of gases,


S= 0.499pkx


so that practically


Rn = 2 a


Since the mean molecular velocity
velocity of sound a, there results


Sfor air is 1.462 times the


Rn = 1.37M
X


M = n
1.37








NACA RM No. L9C02


APPENDIX B


DERIVATION OF FREE-MOLECULE FRICTION STRESS COEFFICIENT


The momentum which strikes a
plate in a free-molecule flow is
reflected with zero velocity from
stress at the surface is


unit area in a unit time of a flat
l/4pcuo. If all the molecules are
the surface of the plate, the shear


T =C1 -o



The stress coefficient is therefore


T 1 1 1.462 0.366
pU2 4 uo 4 M M
pu0 0







NACA RM No. L9C02


REFERENCES


1. Tsien, Hsue-Shen: Superaerodynamics, Mechanics of Rarefied Gases.
Jour. Aero. Sci., vol. 13, no. 12, Dec. 1946, pp. 653-664.

2. Maxwell, James Clerk: Scientific Papers, vol. II, Cambridge Univ. Press,
1890, p. 705.

3. Millikan, R. A.: Coefficients of Slip in Gases and the Law of
Reflection of Molecules from the Surfaces of Solids and Liquids.
Phys. Rev., vol. 21, no. 3, 2d ser., March 1923, pp. 217-238.

4. Kennard, Earle H.: Kinetic Theory of Gases. McGraw-Hill Book Co.,
Inc., 1938, pp. 140 and 291-299.

5. Prandtl, L.: The Mechanics of Viscous Fluids. Theorem of Momentum
and Karman's Approximate Theory. Vol. III of Aerodynamic Theory,
div. G, sec. 17, W. F. Durand, ed., Julius Springer (Berlin),
1935, PP. 103-105.








14 NACA RM No. L9C02












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NACA RM No. L9C02 15



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