An approximate method for calculation of the laminar boundary layer with suction for bodies of arbitrary shape

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Title:
An approximate method for calculation of the laminar boundary layer with suction for bodies of arbitrary shape
Series Title:
NACA TM
Physical Description:
85 p. : ill ; 27 cm.
Language:
English
Creator:
Schlichting, H
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
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Subjects / Keywords:
Aerodynamics   ( lcsh )
Laminar boundary layer   ( lcsh )
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federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: a method of approximation for calculation of the laminar boundary layer with suction for arbitrary body contour and arbitrary distribution of the suction quantity along the contour and arbitrary distribution of the suction quantity along the contour of the body in the flow is developed. The method is related to the well-known Pohlhausen method for calculation of the laminar boundary layer without suction. The calculation requires the integration of a differential equation of the first order according to the isocline method. The method is applied to several special cases for which there also exist, in part, exact solutions: Plate in longitudinal flow and plane stagnation point flow with homogeneous suction. Furthermore the circular cylinder and symmetrical Joukowsky profile with homogeneous suction were calculated as examples.
Bibliography:
Includes bibliographic references (p. 45-46).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by H. Schlichting.
General Note:
"Report date March 1949."
General Note:
"Translation of "Ein näherungsverfahren zur berechnung der laminaren grenzschicht mit absaugung beig beliebiger körperform" from Aerodynamisches Institut der Technischen Hochschule Braunschweig, Bericht 43/13, Jun 1943."

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A icA m (i\








NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM NO. 1216


AN APPROXIMATE METHOD FOR CALCULATION OF THE

LAMINAR BOUNDARY LAYER WITH SUCTION FOR

BODIES OF ARBITRARY SHAPE*

By H. Schlichting


Outline: I. Introduction: Statement of the Problem

II. Symbols

III. The Boundary Layer Equation with Suction

IV. The Generail Approximation Method for Arbitrary Pressure
Distribution and Arbitrary Distribution o:' he Velocitj
of Suction
(a) The expression for the velocity distribution
(b) The differential equation for the momentum thickness
(c) Stagnation point and separation point
(d) Execution of the calculation for the general case

V. Special Cases
A. Without suction
(a) The plane plate in longitudinal flow
(b) The two-dimensional stagnation.point flow
B. With suction
(a) The growth of the boundary layer for the plane plate
in longitudinal flow with homogeneous suction
(b) The two-dimensional stagnation point flow with
homogeneous suction

VI. Examples
(a) The circular cylinder with homogeneous suction for
various suction quantities
(b) Symmetrical Joukowsky profile for ca = 0 with
homogeneous suction

VII. Summary

VIII. Bibliography

IX. Appendix

*"Ein N'lherungovtrahren aw-^- Berechnunr der laminaren Grenzschich.
mit Absa',ung be! belieblper Kjrper'oriam. Aerodynamisches Institut dr.
Technischen Hochschule Brounse-nweip, Bericht 43/13, June 12, 1943.






NACA TM No. 1216


I. INTRODUCTION


Various ways were tried recently to decrease the friction drag of
a bndy in a flow; they all employ influencing the boundary layer
(reference 1). One of them consists in keeping the boundary layer
laminar by suction; promising tests have been carried out by Holstein
(re!'erpnces 2 and 3) and Ackeret (reference 4). Since for large Reynolds
n unbers the friction drag of the laminar boundary layer is much lower
than that of the turbulent boundary layer, a considerable saving in
drag results from keeping the boundary layer laminar, even with the
blower power required for suction taken into account. The boundary layer
is kept laminar by suction in two ways: first, by reduction of the
thickness of the boundary layer and second, by the fact that the
sjc-ion changes the form of the velocity distribution so that it becomes
more stable, in a manner similar to the change by a pressure drop
(reference 7). Thereby the critical Reynolds number of the boundary
layer (US*/V)crit becomes considerably higher than for the case without
suction. This latter circumstance takes full effect only if continuous
suction is applied which one might visualize realized through a porous
wall. Thus the suction quantities required for keeping the boundary
layer laminar become so small that the suction must be regarded as a
very promising auxiliary means for drag reduction.

Various partial solutions exist at present concerning the theoretical
investigation of this problem. Thus H. Schlichting (references 5 and 6)
investigated the plane plate in longitudinal flow with homogeneous
suction. At large distance from the leading edge of the plate a constant
boundary layer thickness and an asymptotic suction profile result. Later
H. Schlichting and K. Bussmann (reference 9) investigated the two-
dimensional stagnation point flow with homogeneous suction and the plate
in longitudinal flow with vo 1//x (x = distance from the leading edge
of the plate). In all cases a strong dependence upon the mass coefficient
of the suction resulted for the velocity distribution and the other
boundary layer quantities. K. Bussmann, H. Minz (reference 8), and
A Ulrich (reference 16) calculated the transition from laminar to turbulent
(stability) of the boundary layer with suction for several cases; in all
of them the stability limit was found to have been raised considerably by
the suction. As is known from earlier investigations (reference 7), the
same amount of influence on the transition from laminar to turbulent is
exerted by the pressure gradient along the contour in the flow for
Impermeable wall. Both influences (pressure gradient and suction) will
be present simultaneously for the intended maintenance of a laminar
boundary layer for a wing. Both influences have a stabilizing effect for
'th suction n nL:c- region of pressure drop; in the region of pressure rise,
h,'wever, pressure ri'ed'cn'. and suction have opposite influences. Whereas
w!.hut nu.ct 'on, "'r, rrr-rA.re rise, mostly transition in the boundary






NACA TM No. 1216


layer occurs; here the important problem arises whether this transition can
be suppressed by moderate suction.

The solutions for the laminar boundary layer with suction existing
so far are not sufficient for answering these questions. An exact
calculation of the boundary layer with suction encounters insuperable
numerical difficulties just as in the case of the impermeable wall. Thus
it is the more important to have an approximation method at disposal
which permits one to check the calculation of the boundary layer with
suction for an arbitrary body. Such a method will be developed in the
present treatise. The method given here is an analogon to the well-known
Pohlhausen method for impermeable wall. It permits the calculation of
the laminar boundary layer with suction for an arbitrarily prescribed
shape of the body and an arbitrarily prescribed distribution of the
suction velocity along the contour in the flow.


II. SYMBOLS

(a) Lengths


x,y coordinates parallel and perpendicular, reL,,ctively, to
the wall wetted by the flow (x = y = 0: stagnation
point and leading edge of the plate, respectively)

6* displacement thickness of the boundary layer ( (l u/U)dy)


0 momentum thickness of the boundary layer

f u/U(1 u/)ldy)

51 measure of the boundary layer thickness

I plate length or wing chord, respectively

b plate width


(b) Velocities

u,v velocity components in the friction layer, parallel and
perpendicular to the wall

U(x) potential velocity outside of the friction layer

Uo free stream velocity


given suction velocity at the wall vo< 0: suction






NACA TM No. 1216


(c) Other Quantities

T 0 wall shearing stress

T wall shearing stress for asymptotic solution of boundary
B> layer on plate in longitudinal flow with homogeneous
suction

total suction quantity b. o vodx

CQ dimensionless mass coefficient of suction; c > : suction

C*Q reduced mass coefficient of suction ( V)

7 dimensionless distance from wall (y/61)

Fl(&), F2(j) basic functions for velocity distribution in boundary layer,
equations (8), (9)

K form parameter of boundary layer profiles, equation (6)

X,IX dimensionless boundary layer thickness, equations (12), (13)

Ii1 dimensionless momentum thickness, equations (22), (23)

Sdimensionless length of boundary layer 5J 2

III. THE EQUATI O NS OF THE B UNDARY LAYER

WITH SUCTION

Following we shall consider the plane problem, thus the boundary
layer on a cylindrical body in a flow (fig. 1). x, y are assumed to be
the coordinates along the wall and perpendicular to the wall, respectively,
Uo the free stream velocity, U(x) the potential flow outside of the
friction layer, and u(x, y), v(x, y) the velocity distribution
in the friction layer. Suction and blowing is introduced into the
calculation by having along the wall a normal velocity Vo(x) prescribed
which is different from zero and generally variable with x:


vo(x) > 0: blowing; Vo(x) < 0: suction

vo/Uo may be assumed to be very small (0.01 to 0.0001). Only the case of
continuous suction will be considered, where, therefore, vo(x) is a
continuous function of x. One may visualize this case as realized by a porous
wall. The tangential velocity ar. the wall should, for every case, equal
zero. The boundary layer differential equations with boundary conditions
are for the steady flow case






NACA TM No. 1216 5

au au dU v 2u
u + dxv +V (1)


au &y
+ = (la)



y = 0: u = 0 v = vo(x)
( (2)
y =0 : u = U(x)



The system of equations (1), (2) differs from the ordinary boundary
layer theory merely by the fact that one of the boundary conditions for
y = 0 is changed from v = 0 to v = vo(x) J 0. Thereby the character
of the solutions changes decisively: the solutions differ greatly
according to whether it is a case of vo> 0 (blowing) or vo< 0 (suction).

A special solution of the equations (1), (2) which forms the basis
for the theory of the boundary layer with suction and is used again
below is the solution for the plane plate in longitudinal flow with
homogeneous suction, thus vo(x) = vo = const < 0 and U(x) = Uo. For
this case the boundary layer thickness becomes constant at same distance
from the leading edge of the plate; also, the velocity distribution
becomes independent of x (reference 5). From = 0 follows because
of the continuity y = 0 and hence



v(x, y) = vo = Constant


From equation (1) then follows for the velocity distribution



u(x, y) = u(y) = Uo e = Uo y/+ e (3)



with

*1 = -- (4)
-VO






NACA TM No. 1216


signifying the displacement thickness of the asymptotic solution. The
wall shearing stress for this solution is:


S c = -pUovo (4a)

It is independent of the viscosity. This asymptotic solution is one of
the very rare cases where the boundary layer differential equations can
be integrated in closed form.

For solution of the boundary layer differential equations (1), (2)
for the general case where the contour of the body and hence U(x) and
also vo(x) are prescribed arbitrarily one could consider developing
the velocity distribution from the stagnation point into a series in
terms of x in the same way as for impermeable wall (reference 11);
the coefficients of this series then are functions dependent on y for
which ordinary differential equations result. K. Bussmann (reference 17)
applied this method for the circular cylinder; with a very considerable
expenditure of time for calculations the aim was attained there. However,
for slenderer body shapes the difficulties of convergence increase so
much that this method which works directly with the differential equations
is useless for practical purposes.


IV. THE GENERAL APPROXIMATION METHOD FOR

ARBITRARY PRESSURE DISTRIBUTION

AND ARBITRARY DISTRIBUTION OF

THE SUCTION VELOCITY

(a) The Expression for the Velocity Distribution


For that reason one applies an approximation method which uses
instead of the differential equations the momentum theorem which represents
an integral of these differential equations. By integration of the
equations (1), (2) over y between the limits y = 0 and y = one
obtains in the known manner (reference 18):



U2d + (2 + *) Ud Uv -= (
dx dx o )






NACA TM No. 1216 7


0 sign!fies the momentum thickness, 5* the displacemeir: thickness,and

To= the wall shearing stress. The approximation method :'or

calculation of the boundary layer to be chosen here proceeds in such a
manner that a pleasible expression is given for the velocity distribution
in 'he boundary layer u(x, y) which is contained in equation (5)
in 6, 5* and To. Thus an ordinary differential equation for 3(x)
results from equation (5); after this differential equation has been
solved one obtains the remaining characteristics of .he boundary layer
6*(x), To(x), and the velocity distribution u(x, y) in the boundary
layer. The usefulness of this approximation method depends to a great
extent on whether one succeeds In finding for u(x, y) an expression
by appropriate functions.

Pohlhausen (reference 15) first carried out this method for the
boundary layer with impermeable wall. The velocity profiles in the
boundary layer were approximated by a one-parametric family and the approxi-
mation function for the velocity distribution expressed as a polynome of
the fourth degree. The coefficients of this polynome are determined by
fulfilling for the velocity profiles a few boundary conditions which
result from the differential equations of the boundary layer. This
method proved to be satisfactory for the boundary layer without suction.

Thus one proceeds in the same way for the boundary layer with
suction. For the velocity distribution in the boundary layer one chooses
the one-parametric expression



*= Fl() + KF2(h) = (6)
U 81(x)



Fl(7) and F2(6) are fixed prescribed functions which are immediately
expressed explicitly; K = K(x) is a form paramet.er of the boundary
layer profiles,the distribution of which along the length is dependent on
the body shape and the suction law; 51(x) is a measure for the local
boundary layer thickness. The connection between 61 and 5* and 4 is
given later. It proved useful to choose other expressions for the
functions F1() and F2(j) than Pohlhausen for the impermeable wall.
Fur the velocity profile according to equation (6) the following five
b- ndary conditions are prescribed; they all follow from the differential
equations of the boundary layer with suction, equation (1), (2):

JLA 6u 2
y = C: u = 0; v, Ut +V (7a,b)
:oy oX (y2

Ou 22u
y = m: u = Uo,; = 0 2 0; (7c,d,e)
OY ,-jv'2






NACA TM No. 1216


The selection of F1(n) and F2(1) is to be made from the view point
that a few typical special cases of velocity profiles of the boundary
layer with suction are represented by equation (6) as satisfactorily as
possible. In particular we shall require the asymptotic suction profile
according to equation (3) to be contained in the expression (6). This
condition is satisfied if one puts


Fl(n) = 1 e-' (8)


and correlates the values K = 0 and 81 = 8* to the asymptotic suction
profile. Furthermore, the expression (6) naturally should yield usable
results also for the limiting case of disappearing suction. To this
purpose a good presentation of a typical boundary layer profile without
suction is required. One chooses as this profile the plate flow for
impermeable wall according to Blasius (reference 11). Since no convenient
analytical formula exists for the exact solution of this case, a good
approximation formula for Blasius' plate profile is needed. It is found
u
that the function -o = sin(aq) gives a very good approximation to the
Blasius profile (a = Constant).1 Thus one puts



0 < r 3: F2() = Fl sinn)

(9)

n > 3: F2(h) = FI 1 = -e'



and then obtains with K = -1 a good approximation for the plate flow
without suction. The corresponding value of 51 is given later. The
functions Fl(j) and F2(n) are given in figure 2 and table 1. Thus
one has for the velocity distribution in the boundary layer the expression:




IThat the sine function is a good approximation for the velocity
distribution at the plane plate without suction, resulted from an
invest iptie n of Mr. Iglisch about the asymptotic behavior of the plane
stagnetion point flow for large blowing quantity (reference 20).






NACA TM No. 1216


0 < T 3: = 1 + K [ i

(10)
> 3: = 1 (K + l)e-


By selection of the functions F1 and F2 the boundary conditions (7a,c,d,e)
are per se satisfied. The last boundary condition,equation (Tb), results,
because of


I- V [ K 1- (13)



in the following qualifying equation for K:


v-U1 + K 1 =UU vU(1 + K)



and from it with

1 U (12)



Xl = v5



for K the equation


K + X1 -
K : -(1k)




X and X1 are two dimensionless boundary layer parameters. A quantity
analogous to X was already used by Pohlhausen for the boundary layer
without suction; X1 is newly added by the suction. For the asymptotic
suction profile with 51 = 6, X1 = 1 according to equation (4). Tne
form parameter K as a function of X and X1 is represented in
figure 3.






NACA TM No. 1216


(b) The Differential Equation for the Momentum Thickness


In order to obtain by means of the expressions (6), (8;
from equation (5) the differential equation for the momentum
one must first set up the relations between ,6*, and 51.
displacement thickness there results:


=
1 "=0


(1 Fl)dq K


SF2d.
0


The calculation of the integral gives:


= 1- K = g*(K)



For the momentum thickness one obtains:


1 (FI + KF2)(1 F1 KF2)dT
1 ,0



=1 Co + ClK + C2K2 = g(K)



The calculation of the integrals gives:


Co = Fl(1 Fl)di =
10


2!

C1= (F2 ,FIF2)dT, = -1 + = .6656
^CT


and (9)
thickness
For the


(15)


(16)


(17)


(18a)





(18b)






NACA TM No. 1216


F22d = -3* 12 -


Cl C2 = 2 -


I
1 -
6e = -0.02358
1 +\


6
=- 0.09014


Thus there Is


i=
61- =


For the form parameter
one obtains therefore


t + o.06656K 0.0235K2 = g(K)


of the boundary layer profiles 5*/.


(17a)


used later


0,- =
I. ^ -


S+ C1K + C2K


1 0.09014K

S+ o.o0656 o.02358K2
2


Furthermore there results according to equations (ll) and (17):



1 = g[ K (1- =f(K) (20)



The functions g(K), 8*/3 and To01/,U according to equation (11) are
represented in figure 4 and table 2.

In order to derive from equation (5) a differential equation
for 8(x) one writes equation (5) in the form


US d0 U is 2 T 0
d + 2 +-
9 dx v 0 U


(21)


Furthermore one introduces according to Holstein and Bohlen (reference 1?)


C2 = -
0l"


Hence


(18c)


(19)






NACA TM No. 1216


2-
U1- = K = Xg2



-Vo3
= El = %lg


(22)




(23)






(24)


S2
Z =
v


K = ZU'; KI = -Vo Z


(25)


is valid. With equations (22) to (25) as well as equation (20) the
differential equation (21) is transformed into


1 K(2 6
U + L = f(K)
dx g(K) -


If one finally puts for abbreviation:



G(K iK) = 2f 2K 2 -- -- 2K1


(K)the differential equation for () become:

the differential equation for Z(x) becomes:


(26)


(27)


(28)


With


then






NACA TM No. 1216


If the function G(K, Il) is known, the integral curve Z(x)
can be calculated from this equation by means of the isocline methAd.
For carrying out the calculation in practice it is useful t,.
introduce dimensionless quantities. One forms them with the aid ,of
the free stream velocity Uo and a length of reference 1 (for instance,
chord of the wing). Thus one puts

ZUo x -vo(*) JJ
z* = --; x* =; Uo V-) -= (x*). (29)


Then equation (28) becomes:


Z C(K, K1) z z ( dU
G( = u/uol) = Uo 6 K = fl(x*) f7 (*0)
dx' U/U0 ZUdx'

The function G(K, cl) is calculated as follows: First, one obtains K
and Ki as functions of X and X1 from equations (22) and (23), if
one takes the connection between K and X, XI according to equation (1I)
into consideration:

K = g(K)X = g2(X, X1)X
(1)
-K = g(K)xl = g(x, xl)j1


From equations (27) and (31) follows:


21 =g[1+K( -] 2 K( )21 2Kj- Xg


-G = g 1 + Kl ) 2 [ 2 ] x


G = 2gF(X, il)


(32)






NACA TM No. 1216


with


F(X, Xj) 1 + K( 2g X [1 K(2 J 1 (33)


Hence G can be calculated first as function of ), X, and then,
because of equation (31), also as function of K, Kl.

The functions K(X, X1) and Kl(X, Xi) are represented in figure 5
and table 3. The function thus determined G(K, Kl) is given in
figure 6 and table 3.


(c) Stagnation Point and Separation Point

The behavior of the differential equation (28) at the stagnation
point where U = 0 requires special considerations. In order that the
initial inclination of the integral curve (dZ/dx)o at this point be
of finite value, G( i, KI) must equal zero. This gives the corresponding
initial values Ko, %lo. Since the function g(K) does not have a zero
for the values of K considered (compare fig. 4) the determination
of the initial values no, 'lo amounts to the zeros of


F(ko, Xlo) = 0 (34)


The resulting initial values at the stagnation point \o, Xlo are given
in table 4, together with the initial values Ko, Klo calculated
additionally according to equation (31). To each pair of values Ko, Ilo
corresponds a mass coefficient of suction which results from

,o2 o(o0)o
V-y = *o v *10



as
-vo(o) K10
-Co



In figure 7 the initial values Ko and lo are plotted against the
local mass coefficient at the stagnation point. The initial value Zo
corresponding to Ko is obtained by






NACA TM No. 1216


Zo = _)
Uo '


The following connection exists between the distribution function of

the suction fl(x) = -vx) and Co: Uo' = KlUo/, K, being
Uo
a profile constant. Thus there is






and
S-o(o) -vo(o) / 1
UV Uo0



fl(o)
Co (36)



The determination of the initial values of the integral curve proceeds,
therefore, as follows: With the given initial value of the suction
velocity at the stagnation point fl(o) one first determines Co
according to equation (36). One obtains the corresponding initial
values Ko and Klo from figure 7, and according to equation (35) the
initial value Zo of the integral curve. If the suction does not
begin at the stagnation point but further downstream, Co = 0;
Klo = 0 and according to figure 7

0.0709
Ko = 0.0709; Zo = -
Uo '


Separation point.- The separation point is defined by the fact that
the wall shearing stress there equals 7pro. This gives for K, according
-6
to equation (11), the value K = = -2.099. For the asymptotic suction
profile K = 0; this is simultaneously the greatest possible value of K.
-11
To K = -2.099 corresponds the value = = -1.099 for all X1, and
6 -






NACA TM No. 1216


K = -0.0721 for all Ml. However, if one would want to carry the
boundary layer calculation up to this point, certain difficulties result
in the last part shortly ahead of this point, since the correlation
between K and X is not unequivocal there (compare fig. 4). The
function G(K, 1i) against K also is not unequivocal shortly ahead
of this point. Thus it is useful to select a point situated somewhat
further upstream as separation point where the boundary layer calculation
has to stop. Such a point results if one chooses the K- value of an
exact separation profile according to Hartree (reference 13). For this
latter there is:



separation: KA = ( A = -0.0682 (37)



One defines this point as separation point of the present boundary
layer calculation for all mass coefficients of suction. The following
table gives a survey of the values of 4 and 5* at the separation
point for four different calculation methods:


The selection of the separation point thus made is somewhat arbitrary;
however, it may be accepted unhesitatingly since, as is well known, the
approximation methods for the boundary layer calculation in the region
of the pressure rise are always somewhat uncertain and only a rough
estimate but no exact calculation of the boundary layer parameters is
possible here. For the same reason one may also accept the fact that
for the present case the velocity distribution u/Uo partly assumes,
shortly ahead of the separation point, values which are slightly
larger than 1.


.2 5*2
Case 2-U' = A --U' =


New method: (sine
(New method: in -0.0721) (-1.55)
approximation)

Pohlhausen P4
(reference 15) -1-92

Exact Hartree 682
(reference 13)

Exact Howarth -1
(reference 14)







NACA TM No. 1216


(d) Performance of the Calculation for the General Case

By means of the system of formulas given above one may perform the
calculation of the boundary layer for an arbitrarily prescribed body
shape and an arbitrary distribution of the suction velocity along the
wall in the flow. It takes the following course:


To the distribution of the suction velocity
the total suction quantity


Q1=
x=0


-v,(x)


corresponds


Vo(x) dx = -cqUobl


and the reduced mass coefficient


C* = cQ^ Rv


and the reduced suction distribution function according to equation (29):


-v (x*) U
fi(x*) =-


Thus there is


cQ* =
*=o


fl(x*) dx*


(39)


If the suction begins at the stagnation point, one determines

with KI = ( the mass coefficient C, at the s-aar.na;on pint

according to equation (36). Then one obtains Ko and K from
figure 7 and Zo according to equation (3'.) Witn these initial value!
the differential equation (30) can now be graphically inLegrated by means
of the diagrem in figure 6. The calculation is carried o-t up to the
point where i reaches the value KA = -0.0682. This intehre-.:n
immediately yields Z*, i, ~i as function of x*, wi h


(3')






NACA TM No. 1216


r= -. (40)



The remaining boundary layer parameters then result as follows:
By means of figure 5 one obtains after i and Kl the parameters X
and X1 and additionally from figure 3 the form parameter K. After K
one obtains from figure 4 the form parameter 6*/0 and thus






From equation (20) one then also obtains the wall shearing stress To


To Vi f(K)
I -= (40a)
U Uo U


Finally, the parameter 61 is required for the velocity distribution in
the boundary layer. According to equations (6), (7), and (40):


Y- Y (K)gl g (41)


Examples of such boundary layer calculations are given in chapter VI.


V. SPECIAL CASES


A. WITHOUT SUCTION


Our general system of formulas is to be specialized in this section
for a few typical special cases for which one can partly give solutions
in closed form. First, the case without suction in particular shall be
treated for which, naturally, our equations also must give satisfactory
results. This case one obtains for vo(x) 0; then






NACA TM No. 1216


X1 O l1 5

and equation (14) is transformed

K

Therewith, according to equation

S1
s = g(x) = +


0 (without suction)

into

= X 1


(17):

C1(x 1) + C2(X 1)2


(42)


= + 0.06656 (x 1) 0.02358 (x 1) (43)


The differential equation (28) for the momentum thickness becomes


dZ G9I) zu
dx u


G(K) is, according to equation (32) and (33):

c = 2gF(X)


F() = 1 + (X 1)(- ) 2x + Cl( 1) + C2( 1)2]



1( + (i )2 ,6)]


F(X) = -2C23 + (2C2 -2 + )2 + )2 +


(44)





(46)









(46)


Furthermore, according

i = g2A


to equation (31):

= X + Cl(x 1) + C2(X 1) 2


(47)






NACA TM No. 1216


The values of G and K calculated according to equations (43), (45),
(46), (47) are given in table 3.


(a) The Plane Plate in Longitudinal Flow


The boundary
(Blasius) which U
to equations (42)


layer at the plate in longitudinal flow without suction
= Uo is obtained for X = K = 0. Then, according
to (46):


K = -1

1 6=
g(O, 0) =2 C C2 2


F(0, O) =
6


G(O, 0) = 2 -(- = 2 = 0.429
6 n ; 2


(1486)


With the Initial value Zo = 0 the integration of equation (44) then

gives: Z = 2 -) or


4 /
= ,- = 0o.655 Cv



For the form parameter 8*/a follows from equation (19):


S --2
= 2- = 2.66
4 -


(49)


(o0)






NACA TM No. 1216


and thus for the displacement thickness


w = ( 2) = 1.740 (l)


wH
For the coefficient of the total friction drag cf = WUo of the
QUo2b
plate of the width b and the length 1, wetted on one side, one
26
obtains because of cf = -



cf = = 1.308 (52)


Finally, the velocity distribution is, according to equation (10),


0 < <_ 3: u = Uo sin (


) 3: u = U

y
Therein Is = g and


1 = = 1. 60i (54)


In figure 8 the velocity distribution according to equations (53)
and (CA) is compared with the exact solution of Blasius; the agreement is
very good. Furthermore the characteristics of the boundary layer according
to the present approximation calculation are compared with the values
of the exact solution of Blasius in the following table. For further
comparison the values according to the approximation method of Pohlhausen
(reference li) also have been given. The agreement of our new epjroxl-
ma:-,in nmehod with the exact solution is excellent for all boundary layer
paerametLers; the drag coefficient, in particular, shows an error of
onl: 2 percent.






NACA TM No. 1216


Coefficients of the Boundary Layer at the Flat Plate

in Longitudinal Flow Without Suction


Calculation method V U -
-* r b / uo





Pohlhausen Ph
(reference P ) 1.750 0.685 2.55 1-370 0.234
(reference 15)

Exact (Blasius) 1.721 0.664 2.59 1.328 0.220


The deviations of our sine approximation from the exact solution are,
for most characteristics, even somewhat smaller than in the Pohlhausen
method.


(b) The Plane Stagnation Point Flow

For the plane stagnation point flow the velocity of the potential
flow U(x) = ulX. All boundary layer characteristics are in this case
independent of the length x. The initial value of the momentum
thickness Zo is obtained from equation (44) for G(Ko) = 0. Since g(K)
does not vanish in the range of the values of K considered, there
must be F(Xo) = 0. From equation (46) one finds as zero of F(X) the
value


Xo = 0.3547


(stagnation point without suction)


(54)


The corresponding values of K and K according to equations (42)
and (47) are Ko = -0.6453 and Ko = 0.0709; furthermore there is,
according to equation (43): g(Xo) = C.447. Therewith the momentum
thickness for the plane stagnation point flow becomes:


= = orii = 0.66/


(55)


The form parameter 6*/l results from equation (19) as 5*/8 = 2.37;
therewith one has







NACA TM No. 1216


6* = 2.37 = 0.630


Furthermore, according to equation (43): 61 = 0.595 r .
results from equation (11) for the wall shearing stress:


T v
-- = 1.163



The velocity distribution results from equation (10) as:


o0 T

Thus there


(-7)


- = 0.3574(1 e-') + 0.6453 sinCfi



1 21 3: *= 1 0.3574e-n
Uo j


(58)


(58a)


TI = = 1.68y
b 1


Figure 8 gives a comparison between velocity distribution according to
equation (58) and the exact solution by Hiemenz (reference 12); here also
the agreement is satisfactory. Furthermore the characteristics of the
boundary layer according to the present calculation are again compared
with the exact solution by Hiemenz and with the approximate calculation
by Pohlhausen in the following table.


(56)


with







NACA TM No. 1216


Coefficients of the Boundary Layer of the Plane

Stagnation Point Flow without Suction


Calculation method 13 [ 8* U 1 i -0'

New method (sine
approximation) 0.266 0.630 2.37 1.163 0.310

Pohlhausen P4
(reference 15) 0.278 0.661 2.31 1.19 0.331

Exact (Hiemenz) 0.292 0.648 2.21 1.234 0.360


The agreement of the new method with the exact solution is for this
case somewhat less satisfactory than for the plane plate; neither is it
quite as good as the approximation of Pohlhausen. But even here the new
method yields still very useful values.


B. WITH SUCTION


In this section a few cases with suction will be treated far which
the solutions can be given in closed form. First we shall treat the
boundary layer at the plate in longitudinal flow with homogeneous suction,
already investigated formerly (reference 6). The following results are
considerably more accurate than those former ones.


(a) Growth of the Boundary Layer for the Plate

in Longitudinal Flow with Homogeneous Suction

For this case the boundary layer is at large distance from the
leading edge of the plate independent of x; hence all boundary layer
parameters are constant. The corresponding asymptotic solution has
been given already in equations (3), (4), (4a). The applying values
are:


S-V
g5 *


5-'
rg~


81 = 5*


E Tom -PUovo


1 ; = i l = i
1


K = 0


(59)






NACA TM No. 1216


One now calculates the growth of the boundary layer from the value
zero at the leading edge of the plate to the given asymptotic value.
In our system of formulas one has to put for it:



Therewith bees 0;to equation () with the abbreviation

Therewith becomes according to equation (1i) with the abbreviation


S1- = c = 0.764
6


K = 1 1
1 c01


(6c)



(61)


and according to equation (17)


= g(O, X1)
1


Po + Pl1 + P2)kl2
(1 cXl)2


Po = = 0.30986


6= 1 = 0.06656

p2 = -2 + + 2 + C = 0.05819
7C 2 \ V


Furthermore there is according to equation (16)



8* = 6- +
51 -


with


(62)


(62a)


(63)






NACA TM No. 1216


The wall shearing stress becomes according to equations (11) and (13):


Uo 6
To =
S 1] 1 c11


ii
-pvoUo
j(1 cX1)


The differenLial equation (28) assumes for the present case the form:


dZ G( (, C1).
dx Uo


i = "VoV


vo = Constant < 0


(65)


The integration of this differential equation requires the explicit
expression for G(o, "l). According to (32) and (33):


G(o, AI) = 2g( -


ScX
= 2g(l X1)
1 cX1


Thus G(o, Ki) = O for X1 = 1. Therefore X1 = 1 is a solution of
the momentum equation; it corresponds to the asymptotic solution. The
initial value at the leading edge of the plate is X1 = 0. For the
length of growing boundary layer X1 varies from 0 to 1.

If one introduces as dimensionless distance along the plate

2


(67)


the differential equation (65) can be written in the form:


d(=i2)
= CG(K1)
dS


Initial value: 9 = O0 K1 = 0


(64)


(66)


(68)


=-T-
V3 V7)


(68a)






NACA TM No. 1216


The connection between l1 and. I is given by


1r = klg(Xl)


(69)


with g(Xk) according to equation (62).

The differential equation (68) can be solved according to the
isocline method. For the present case, however, an analytical solution,
too, is possible which is preferable. From equation (69) first follows:


2 !K1 Gd 1 dKl
dXl d


Here all quantities can be best expressed by
equation for xl() results. With dal/d1l
one obtains from equation (70) after division
does not disappear in the range 0 I X, 1:


= (1 x1)


XI so that a differential
according to equation (69)
by 2g since the latter


- ckI

1 cX1


Initial value: I = 0: X1 = O


Because of
of t = 0,


(71)


(71a)


6 ?
g(o, o) Po = "- one obtains from it in the neighbourhood
X1 = 0:


dS 6 66 0 1
"= n- p = 2)- l


t = 2 12 = .391Xi2
it O


(70)


and


(72)


(7?)


X1 8(


dg dX
dXl dC1I






28 NACA TM No. 1216


Hence follows for the neighborhood of the leading edge of the plate
(P = 0)


Because of


A1 = 1.60o or


6
- follows hence:
11


8*
81


-v 8*
--- = ( 2)


As the ccmIparison with equation (51) shows, Lhe boundary
starts, therefore, at the leading edge of the plate with
the plate without suction.


layer Lhickness
the value for


In order to integrate the equation (71), one has to insert the
explicit values of g(X1) and dg/dXi according to equation (62).
After some intermediate calculating (compare appendix I) one obtains


1pP0 + Pll P212 + PI3)
(1 cX)2(l Xi) cX


Po = Po = 0.409e6

Pl = 2p + Poc = -0.4667

P2 = 3p2 = 0.17157

P, = -cp2 = -0.02772


(74)


(75)


Y 1= 1.60V
V


14-I'
=2
_/ It-


with


8 (t 2) T_
U


d
@71 '





NACA TM No. 1216


The breaking up into partial fractions yields:


d_ P;
S- +
dX c'3


Kl K2 K.
>1 1 C11 (cl 1)2


The integration with the initial value X1 = 0


= c3


K4
+ = f'(Xl)
cA1 1


for t = 0 yields


+ KI In (1 X1) + 2- In -



- K X + -- In (1 ca1) = f(k1)
3 C lA 1


The Kl, ..., K4 result from the breaking up into partial fractions as


K1 = -6.9560;


K2 = 3.4704;


K3 = -0.2284;


K4 = -0.1569


Thus the solution finally reads


I = -0.2564i 6.956 In (1 x1) + 7.2846 In (1 0.9099X1)


4 0.2284 l
+ 0.22846 x 0.3293 In (1 0.476411)
0.4764x1 1


(79)


2
For development of this solution in the neighborhood of t = 0,
I1 = 0, the coefficient of X1 must, because of equation (73), equal zero;


the coefficient of 2
J.


must equal ( ) = 0.391.
n d


The result is:


3 K1 + X K =
3 3 K4 0
C

K1 -c
2 K2 + CKIKC 4it -\V/


(77a)


(77b)


WiTh the numerical values of equation (78) one may verify that these
equations are satisfied.


(76)


(77)


(78)






NACA TM No. 1216


The solution Xl(S) calculated accordingly is given in table 5.
From X1(t) all remaining boundary layer parameters can then be calculated
immediately according to equations (61), (62), (63), and (64). They also are
"-vo8* 705*
given in table 5. In figure 9 5- and. are plotted against (.

The displacement thickness of the boundary layer reaches 0.95 of
its asymptotic value after an extent of the growing boundary layer of

tA -)2 UV2 4.5. The velocity profiles in the growing boundary
.voy
layer in the plotting u/Uo against --- = ~11 are represented in
figure 10. For the wall shearing stress one obtains from equations (64)
and (0a)

TO 6
= 6 (80)
o. X1(1 chl)


The wall shearing stress is plotted in figure 11 as a function
of /

Drag.- In view of the reduction of the drag by maintenance of a
laminar boundary layer the friction drag in the extent of growing
boundary layer is of particular interest for this solution. For the
asymptotic solution the local friction drag along the wall is constant
with Tom = -PVoUo; thus the coefficient of the total friction drag
also equals this value

W TOG v
S- -- (81)
Uo2 b o20


For small suction quantities -vo/Uo the extent of growing boundary
layer is sometimes so large that the growth is not finished by far at the
end of the plate. According to former investigations (references 8 and 10)
it is to be expected that for homogeneous suction at the plate the
maintenance of a laminar boundary laier is possible even for Reynolds
U.-
numbersof the order of magnitude -j- = 107 to 10 with a very small
suction quantity of the order of magnitude Q = = 10- For 10
Uo0 7 U Uo
and = 107 or 108 one has at the end of the plate
I = 2 U! = 0.1 or 1, that is, the growth of the boundary layer
is not finished by far.






NACA TM No. 1216


Since the friction layer over the extent of growing boundary layer
is thinner, the friction drag there is considerably larger than for the
asymptotic solution. For this reason the calculation of the drag over
the extent of growing boundary layer will be given completely.

The total friction drag for the plate wetted on one side is:


W = b To dx (82)
o(82)

and with the value of o7 according to equation (80) and with To,
according to equation (4a):


W = -oUobl l)
'Oo6 L0 Xl(0 cxl)


With dx =


dt according to equation (67) this equation becomes:


2 v
W = pbUo2
V -


S (1 c)dt
6 X1(1 ci)
t=0


with E, signifying the value of t at the end of the plate, thus:


(L2
St lD


U0 = f(lo)
V= r(X0)


Therein Xlo signifies the value of X1 at the end of the plate which
is obtained from equation (77) for t = ~1i therefore


f(klo) 3 lo + Kl n (1 lo) + In 1-
c33 3xio 4 i (- (%.itl0)


-K3 lo + In (1 cXlo) (85)
cXlo 1


(83)


(84)


)g2 2L






NACA TI No. 1216


Introducing in equation (83) for = f'(x)
dX1
to equation (76) one obtains:


the expression according


W = pbUo2 --F(klo)
-To


and


W
cf = -
Uo2 bl


(86)


= 2-- IF(olo)
-vO1


with F()io)


signifying


=lo


1Xl=0


F(0lo) = "I
6


fl'(X) d x
Xl(1 cxi)


Finally introducing -V -. r2 f(Xlo) according to equation (84) into

equation (86) one obtains


=-v F(Xlo) -To
c = 2 = 2- G(Xlo)
S Uo F(Xlo) Uo





cf = cfW G(Alo)


(88)


Because of the connection between Xlo and EZ according to
equation (84), the total drag coefficient for the extent of growing
boundary layer is thereby given as a function of the dimensionless

distance along the plate =- ) On the other hand, equations (84)


(87)






NACA TM No. 1216


and (88) give for prescribed mass coefficient of the suction -vo/Uo
the drag law cf also against Uol/V in the form of a parameter
representation. The parameter Xlo is the dimensionless boundary

layer thickness at the end of the plate: lo =1-v The values
Sv x=
of Xlo lie between 0 and 1, the first value being valid at the
leading edge of the plate, the latter for the asymptotic solution, after
the growth of the boundary layer has ended.

The calculation of the integral F(Xlo) according to equation (87)
gives (appendix II)


F(Xio) =


KlC
1 c


K2n( clo)
-c4 K + In(l cklo)


+ 1 Zn(1 lo) + AK2 In (- lo)


cXlo
- (-K3 + K _)
1 cXlo


K2
2+


cX1o(2 c l)
(1 cXlo)


(89)


c is, according to equation (60), c = 1 -, and Kl) ..., K4 are
given by equations (75) and (78). After insertion of the numerical values
according to equations (60) and (78) follows:



F(Xlo) = -0.3288 In (1 0.4764X1o) 6,956 In (1 10o)


+ 7,2846 In (1 0.9099X1o)

0. 764Xlo 0.4764X1o (2 0.476411o)
0.0374 0.05980
1 0.4764Xlo (1 o.4764xlo)2


The values of F(Xlo) and G(Xlo) are given in table 6. For Xlo --1
that is S--->o (growth of boundary layer ended) one has, as can
immediately be seen from equations (89) and (77):






NACA TM No. 1216


F(Xlo)
lo --->:1 o(klo) = 1
f(klo)


(89a)


and thus cf-- cr for s --)4 On the other hand one has in the
neighborhood of the leading edge of the plate, that is, for Xlo--> 0
according to equation (73)


\lo ->0: f(klo) =


( C lo2
It 2] 0


and thus according to equation (87)


lo -->0:


F(o10) = i- )l


and therefore


o--- 0: c(Xlo) = 1
= lo


(89b)


If one substitutes this value into equation (88) and takes into consideration
that



i tI 2)


is valid for small Xlo according to equation (73), one obtains




Cf = 2 /IF
Uo 4 /Ir"
rU0 2"/






NACA TM No. 1216


or


c = 2 U y-1/2



thus the drag law of the plate without suction according to equation (52).
The drag law of the length of growing boundary layer is therefore for
very small lengths of growing boundary tj asymptotically transformed
into the drag law of the plate without suction.

The drag law according to equation (88) is represented in figure 12,
where cf/cf, is plotted against Et. Furthermore figure 13 gives the
drag law in the form cf against Uol/V for various values of the
mass coefficient -vo/Uo. The larger the suction quantity the smaller
the Reynolds number at which the respective cf curve separates from
the drag curve of the plate without suction and is transformed, after a
-2vo
certain transition region, into the asymptotic curve cf = Uo The
Reynolds number at which the latter is reached is the larger, the
smaller the suction quantity.

The drag coefficients given here represent the total drag of the
plate with suction. No special sink drag is added (compare reference 10)
since for continuous suction, as in the present case, the sucked particles
of fluid have already given up their entire x- momentum in the boundary
layer so that this momentum is contained in the friction drag.

In order to obtain the total drag power of the plate with suction,
however, one must, aside from the drag given here, take into account the
blower power of the suction.


(b) The Plate Stagnation Point Flow with Homogeneous Suction

Another special case which can be solved in closed form is the plane
stagnation point flow with homogeneous suction. Since for this case the
exact solution from the differential equations of the boundary layer has
been given elsewhere (reference 9) it shall also briefly be treated here.
The potential flow is U(x) = ulx and the suction velocity
Vo(x) = vo(o) = voo <0. If the integral curve of equation (28) is to
have a finite value at the stagnation point x = 0, there has to be
G(K i) = 0; this in turn requires, as was discussed in detail in
chapter IV c, F(X, Xl) = 0. F(x, 1i) Is given by equation (33). The
values of Xo, Xlo which belong together follow from it; they give for
the general case the initial values of the boundary layer calculation at
the stagnation point; for the present case of stagnation point flow they






NACA TM No. 1216


immediately give the complete solution since the boundary layer thickness
and all other parameters are independent of the length of growing boundary
layer x. Besides Mi, X one further obtains K according to
equation (14), g according to equation (17), 8*/C according to
equation (19), and so and Klo according to equation (31). The mass
coefficient Co = _-oo is obtained according to equation (34). From Ko

finally follows o and therewith 8*. The results are compiled in
table 4. Naturally, momentum and displacement thickness decrease with
increasing suction quantity. With Co--> the form parameter 8*/C
approaches the value 2 of the asymptotic suction profile. In figure 14

9 fuj/V and 8* ul/v are plotted against Co and compared with the
exact solution. The agreement is quite satisfactory.

As conclusion of these considerations of the special cases the
characteristic boundary layer parameters for these special cases are
compiled in the following table.








NACA TM No. 1216


I--- i- OJ 0 -
o0 t- 0
-O CJ 0 r
o lo o












00 00 Q0


r-4
0 0 0
oo 0t
r-I 03r OP






H 0 0 c0 U H1 |
C U















o 0 O I
I* -- C\
























S0 0 0 0
o I













S-I V C .)
0 ,C 0 :: pq 0 0
m 0 0 P4 4) a --
0 M +)
C 0 4 -
S 43 4, 0
0 1 0) 40
0 0l 0
pq I I3 <






tBI 0v ; +J C~


I4t
0


a,








3 ,
C:



0 +
Sa







fo
m*r







o 0
a,,



,C
0 a
OH e








0 u
CO T-'


-Q






5









a
oC a



0
. *rI
04-'
0 0



0
*r4





NACA TM No. 1216


are given in figure 15. For the case without suction P = 101.70 results
as separation point; this is slightly further to the front than for the
customary Pohlhausen method (cp = 108.9) for which the calculation was
performed elsewhere (reference 7). With increasing suction quantity
results a reduction of the boundary layer thickness and a shifting of the
separation point toward the rear.

In order to completely avoid the separation for the circular cylinder,
it is probably useful to select not a homogeneous suction along the
contour, as in the present case, but a distribution of vo(x) which has
considerably larger values on the rear than on the foreside. Euch
calculations may also be carried out according to the present method
without additional expenditure of time.

A comparison of the present approximate calculation with an exact
calculation by K. Bussmann (reference 17) for the displacement and
momentum thickness is given in figure 16. The latter calculation is a
development in power series starting from the stagnation point, as first
indicated by Blasius (reference 11). Except for the neighborhood of
the separation point the agreement is quite satisfactory.


(b) Symmetrical Joukowsky Profiles for ca = 0

As second example a symmetrical Joukowsky profile of 15 percent
thickness has been calculated for ca = 0, also with homogeneous
suction. The suction extends over the entire contour. The same profile
without suction has been calculated elsewhere (reference 7), also
according to the Pohlhausen method. Here, too, a reduction of the
boundary layer thickness and a shifting of the separation point toward
the rear results with increasing suction quantity. For the suction
quantity Co = 0.417, that is fl(0) = 3, a separation does no longer
occur.


VII. SUMMARY


A method of approximation for calculation of the laminar boundary
layer with suction for arbitrary body contour and arbitrary distribution
of the suction quantity along the contour of the body in the flow is
developed. The method is related to the well-known Pohlhausen method for
calculation of the laminar boundary layer without suction. The calculation
requires the integration of a differential equation of the first order
according to the Isocline method. The method is applied to several special
cases for which there also exist, in part, exact solutions: Plate in longi-
tudinal flow and plane stagnation point flow with homogeneous suction.
Furthermore the circular cylinder and symmetrical Joukowsky profile with
homogeneous suction were calculated as examples.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics






NACA TM No. 1216


V I I. A P P E ND I X ES

APPENDIX I

Concerning the Length of Growing Boundary Layer for the Plane Plate

with Homogeneous Suction

According to equation (71) is



~ 6 dXE d1 -1 cX1
l + (1 (1 c-l-) (71)
a(d) dt 1 cx1


From equation (62) one finrs:


dg (1 cX)(pl + 2p2X1) + 2c(1 cxl)(po + pi*X p2X2)
dX1 (1 cx 1)


S+ 2cpo + (Plc + 2p2)X1
= (I,1)
(1 c )3


.Substitution of equations(I,l) and (62) into equation (71) gives:







40
40 RACA Th No. 1216




*
H







whl rA ^i
cu

Hm H














O 0
H pH
u so
,e4
-I i




II H


_r-_ +___ _
H + r 4



m



+o++ 1 o R
Hu H P a







U H
11 H4 C -H







+ A-- 2









i 0 A
H H + r
S)' + + -









I-I .






.0-


+ H H
.4 1

-) 0
.4 0 c














35;






NACA TM No. 1216


APPENDIX II

Concerning the Calculation of the Drag of the Plane Plate

with Homogeneous Suction


The calculation of


f'(Xl) dX1
X1(1 cxl)


(87)


F llo
F(Xlo) = |
x1=0


gives with


f'(xl) according to equation (76), if 1I


is replaced by


03(1 !) c l dz-
c3 z(1 cz)
1z=0


+ K1 l
z=0


dz
(z l)z(l cz)


+ K2 J





- K Z


The integrals are
finds:


03


dz
cz 4)z(l cz)


+ 1 1
+ K3 I-z0
Vz=O


dz
(cz 1) 2z( cz)


diz
z = = I + II + III + IV + V
(cz l) z


(II,1)


solved by breaking up into partial fractions. One



in z In z ~~1
z=0


II = -K In z + n (z 1) +
c 1


----- 2In
z=0






NACA TM No. 1216


II3 -K2 1 n z + I n z -) n ( -.
z=0



IV = K3 [n z In(cz 1) z- 2]
cz -1 (cz -1)2




V = Kg -In z + --- + In(cz 1
I cz -1


When summed up, all terms with [n z lo cancel each other, because
0
of equation (77a). After insertion of the limits the remaining terms give:


n(cz 1) Il = In(1 cxlo); [n(z 1 lo
-0 0


= In(l Xlo)


[n z o


= In x


cr 11lo
L" ^


F 1 XAlo
(cz 1) 2O


cXlo(2 c1lo)

(1 cXlo)2


S- clo
1 'Xlo






NACA TM No. 1216


Thus there results by simplification from equation (II,1):


(Xloc) = 1 -
^c-3 1 c


- -K3 K
c 3


K4) In(l clo)


+ 6 In(l Alo) + k2 In(l -ilo




ScXlo K cklo(2 cklo)
- (-K3 + K) + 2
1 c0o (1 cAXo)


(89)






NACA TM No. 1216


APPENDIX III


To page 30.- Table 10 gives the numerical table concerning the
velocity distribution of figure 10.

To page 35.- For the boundary on the plate in longitudinal flow
with homogeneous suction the exact solution from the differential
equations also was given in an unpublished report by Iglisch. A
comparison of the approximate solution above with that exact solution
is given in figures 18 and 19. Figure 18 gives the comparison of the
displacement and momentum thickness; particularly for the displacement
thickness the agreement is good. Figure 19 gives the comparison for the
wall shearing stress; here also the agreement is satisfactory. (So far,
these comparisons can be carried out only for the front part of the
length of growing boundary layer, up to 6 = 0.5, since the exact
solution does not yet completely exist.)






NACA TM No. 1216


REFERENCES


1. Betz, A.: Beeinflussung der Grenzschicht und ihre praktische
Verwertung. Jahrb. Dtsch. Akad. Luftfahrtforschung 1939/40, p. 246
and Schriften d. Dtsch. Akad. Luftfahrtforschung Heft 49 (1942).

2. Holstein, H.: Messungen zur Lamlnarhaltung der Grenzschicht durch
Absaugung an einem Tragflrgel. Bericht S. 10 der Lilienthal-
Gesellschaft fur Luftfahrtforschung 1942.

3. Holstein, H.: Messungen zur Laminarhaltung der Reibungsschicht durch
Absaugung an einem Tragfliigel mit Prof l NACA 0012-64, FB 1654, 1942.

4. Ackeret, J., Ras, M., and Pfenninger, W.: Verhlnderung des Turbulent-
werdens einer Reibungeachicht durch Absaugung. Die Naturwissen-
schaften 1941, p. 622.

5. Schlichting, H.: Die Grenzschicht mit Absaugung und Ausblasen.
Luftfahrtforechung Bd. 19, p. 179 (1942).

6. Schlichting, H.: Die Grenzschicht an der ebenen Platte mit Absaugung
und Ausblasen. Luftfahrtforschung Bd. 19,p. 293 (1942).

7. Schlichting, H., and Ulrich, A.: Zur Berechnung des Umschlages
laminar/turbulent. Bericht E. 10 der Lilienthal-Gesellschaft fir
Luftfahrtforschung, p. 75, 1942 and Jahrb. 1942 der Dtsch. Luftfahrt-
forschung, p. I 8.

8. Busamann, K., and Minz, H.: Uber die Stabilitft der-laminaren
Reibungsschicht. Jahrb. 1942 der dtsch. Luftfahrtforschung, p. I 36.

9. Schlichting, H., and Bussmann, K.: Exakte Lisungen fur die laminare
Grenzschicht mit Absaugen und Ausblasen. Schriften der Dtsch.
Akad. d. Luftfahrtforschung, 1943.

10. Schlichting, H.: Die Beeinflussung der Grenzschicht durch Absaugen
und Ausblasen. Lecture to the Deutschen Akademie der Luftfahrt-
forschung,May 7, 1943; to be published soon.

11. Blasius, H.: Grenzschichten in FlUssigkeiten mit kleiner Relbung.
Zschr. Math. u. Phys., Bd. '6, p. 1 (1908).

12. Hiemenz, K.: Die Grenzschicht an einem in den gleichmissigen
Flissigkeitestram eingetauchten Kreiszyllnder. Dingl. Polytechn.
Journal. Bd. 326, p. 321 (1911).






NACA TM No. 1216


13. Hartree, D. R.: On an Equation Occurring in Falkner and Skan's
Appr";imate Treatment of the Equations of the Boundary Layer.
Cambridge Phil. Soc. Vol. 33, p. 223 (1937).

14. Howarth, L.: On the Solution of the Laminar Boundary Layer Equations.
Proc. Roy. Soc. London A No. 919, Vol. 164 (1938), p. 547.

15. Pohlhausen, K.: Zur nKherungsweisen Integration der Differential-
gleichung der laminaren Grenzschicht. Zechr. angew. Math. u.
Mech. Bd. 1, p. 252 (1921).

16. Ulrich, A.: Die Stabilitit der laminaren Reibungsschicht an der
langsangestramten Platte mit Absaugung und Ausblasen. Bericht 43/9
des Aerodynamischen Institute der T. H. Braunschweig; to be
published soon.

17. Bussmann, K.: Exakte Lbsungen fUr die Grenzschicht am Kreiszylinder
mit Absaugen und Ausblasen. Not published.

18. Prandtl, L.: The Mechanics of Viscous Fluids. Durand, Aerodynamic
Theory vol. III, Berlin 1935.

19. Holstein, H., and Bohlen, T.: Ein vereinfachtes Verfahren zur
Berechnung laminarer Relbungsschichten, die dam Ansatz von
K. Pohlhausen geniigen. Bericht S. 10 der Lilienthal-Gesellschaft
fir Luftfahrtforschung 1942.

20. Igllsch, R.: Uber das asymptotische Verhalten der Ldsungen einer
nichtlinearen gewbhnlichen Differentialgleichung 3. Ordnung.
Bericht 43/14 des Aerodynamischen Institute der T. H. Braunschweig;
to be published soon.






NACA TM No. 1216


TABLE 1

THE BASIC FUNCTIONS FI


AND F2


FOR THE VELOCITY DISTRIBUTION

IN THE BOUNDARY LAYER

WITH SUCTION


SFI F2


0 0 0
.2 .1813 .0768
.4 .3297 .1221
.6 .4512 .1423
.8 .5507 .1452
1.0 .6321 .1322
1.2 .6988 .1112
1.4 .7534 .0843
1.6 .7981 .0551
1.8 .8347 .0263
2.0 .8647 -.0017
2.2 .8892 -.0240
2.4 .9093 -.0416
2.6 .9257 -.0524
2.8 .9392 -.0552
3.0 .9502 -.0498
3.5 .9698 -.0302
4.0 .9817 -.0183
4.5 .9889 -.0111
5.0 .9933 -.0067
6.0 .9975 -.0025
7.0 .9991 -.0009
m 1 0






NACA TM No. 1216


TABLE 2

PARAMETER OF BOUNDARY LAYER WITH SUCTION

K 6 6* ToS0 Tro
-K g -_ g s E T rU u


a0 0.5 1 2 1 0.5
.1 .4931 1.009 2.05 .9524 .4696
.2 .4856 1.018 2.10 .9047 .4393
.3 .4779 1.027 2.15 .8571 .4096
.4 .4696 1.036 2.21 .8094 .3801

.5 .4608 1.045 2.27 .7618 .3510
.6 .4516 1.054 2.33 .7142 .3225
S.6453 .4472 1.058 2.37 .6926 .3097
-7 .4419 1.063 2.41 .6665 .2945
.8 .4317 1.072 2.48 .6189 .2672

.9 .4210 1.081 2.57 .5712 .2405
C 10. .4099 1.090 2.66 .5236 .2146
1.1 .3982 1.099 2.76 .4760 .1895
1.2 .3862 1.108 2.87 .4283 .1654
1-3 .3736 1.117 2.99 .3807 .1461
1.4 .3606 1.126 3.12 -3330 .1201
1.5 .3471 1.135 3.27 .2854 .0991
1.6 .3331 1.144 3.43 .2378 .0792
1.7 .3186 1.153 3.62 .1901 .0606
1.8 .3037 1.162 3.82 .1425 .0433

1.9 .2883 1.171 4.09 .0948 .0273
2.0 .2724 1.180 4.33 .0472 .0129
d2.099 .2562 1.189 4. 6 0 0

Asymptotic suction profile.
bStagnation point without suction.
cPlane plate without suction.
dSeparation point.







NACA TM No. 1216


TABLE 3


THE FUNCTION G(i, 1) FOR THE INTEGRATION

OF THE DIFFERENTIAL EQUATION

OF THE MOMENTUM THICKNESS


mI = 0 without suction 0. = 0.1

IX 1 ,( K,K ) x x 11I G(K,rl)


0.1062
.0937
.0816
.0709
.0586
.0477
.0373
.0273
.0177
.0086
o
-.0159
-.0298
-.0419
-.0528
-.0602
-.0666
-.0711
-.0738
-.0748
-.0742
-.0721
-.0682


1 1 ,--


-0.2042
-.1323
-.0621
0
.0729
.1375
.2001
.2607
.3191
-3753
.4292
.5301
.6213
.7023
.7730
-8330
.8820
.9198
.9470
.9617
.9656
.9584


0.5
.4
.3
0
.2
.1
0
-.2
-.4
-.6
-.8
-1.099


0.206
.210
.215

.221
.227
.235
.250
.270
.290
.320


0.1132
.0875
.0627
.0520
.0401
.0190
0
-.0330
-.0550
-.0710
-.0775
-.0721


-0.333
-.192
-.057
0
.072
.190
.300
.490
.635
.742
.792
.758


S 0.2
x IX I .% G


0.5
.4
.3
.2


0.50
.45
.40
.3547
.30
.25
.20
.15
.10
.05
0
-.1
-.2
-.3
-.4
-.5
-.6
-.7
-.8
-.9
-1.0
-1.099


0.405
.415
.424
.432


.443
.454
.480
.515
.557
.620


0.1213
.0936
.0674
.0431
.0350
.0205
0
o
-.0355
-.0590
-.0770
-.0835
-.0721


-0.450
-.310
-.172
-.095
o
.077
.188
.370
.516
.620
.657
.558


.1
0
-.2
-.4
-.6
-.8
-1.099


I I







NACA TM No. 1216


TABLE 3 Concluded


THE FUNCTION G( K, i) FOR THE INTEGRATION

OF THE DIFFERENTIAL EQUATION OF THE

MOMENTUM THICKNESS Concluded


'K = 0.3 Kg = 0.5

X 1 K G(t,c1) x X1 1 G( C, IL)


0.5
.4
.3
.2


0
-.2
-.4
-.6
-.8
-1.099


0.4
.3
.2


0
-.2
-.4
-1.099


0.590
S.611

.622
.637

.652
.688
.738
.808
.892
1.172


0.770
.782
.800
.815

.835
.875
.935


0.1292
.100
.0722
.0463
.0222
.0192
10
-.0380
-.0660
-.0830
-.0910
-.0721


0.1o68
.0777
.0500
.0202
.oo8
0
-.0415
-.0730
-.0721


-0.542
-.400
-.265
-.136
-.016
0
.099
.294
.438
.520
.555
.358


-0.460
-.324
-.200
-.080
o
0
.035
.232
*375
.158


' '


0.4
-3
.2
.1


-.4

-1.099


0.3
.2
.1
0
-.2
-.4


0.935
.950
.965
.982
1.0
1.045
1.115


0.113
.0825
.0538
.0220
0
-.0460
-.0810

-.0721


-0.475
-.350
-.225
-.100
0
.195
.320

-.042


K = 0.6

x Xl I G( K, )


1.116
1.122
1.135
1.154
1.200
1.254


0.088
.056
.020
0
-.048
-.086


KI = 0.7

X I X1 K G(l,1)


0.3
.2
.1
0
-.2
-.4


1.294
1.292
1.296
1.304
1.338
1.401


0.089
.059
.030
o
-.054
-.096


-o.327
-.206
-.072
0
.190
.340


-0.125
-.082
-.005
.089
.262
.402


-_ _I I


?1 = 0.4
X X1 1 G(K, l)








NACA TM No. 1216


on
I t- 0 r H




o o.
a Ol H Cu 4





m 0

coT 00 O
S0 0 0
v a m t- ru
E* *


o( 0
1n %-0 cu fn fn CC) a%

S Do (n C O CU O


Co o Cu r l 0













0. 0
t- rr) 00 tt CU N O Cu
| n c Cu cu cu H H









0o m 0 0
S .\ -. 0 ;- 0








o0 .-Q \ C O O
0o H Ch cn om







T0 01 C 0\ 0\ f\ C0
"o i
0o O- i n H C O
S0 0 0 0 0 0










SCU o U n (Y )- ot


k C cu H- H H t 0 0
0 0


0 1H1






NACA 4M No. 1216


TABLE 5

THE BOUNDARY-LAYER PARAMETERS AT THE PLANE PLATE

IN LONGITUDINAL FLOW WITH HOMOGENEOUS SUCTION;

LENGTH OF GROWING BOUNDARY LAYER


5* 5* -ro5* TO5* To
l 1 v gUo -pvoUo

0 0 1 1.090 2.66 0 0.572 o 0
0.000171 .02 .989 1.089 2.65 .0218 .576 26.43 -0131
.ooo662 .04 .979 1.088 2.64 .0435 .581 13.34 .0257
.00154 .06 .968 1.087 2.63 .0652 .586 8.98 .0392
.00291 .08 .956 1.086 2.62 .0869 .591 6.80 .0540

.00456 .10 .945 1.085 2.61 .1085 .597 5.50 .0676
.01139 .15 .916 1.083 2.58 .1624 .610 3.76 .1068
.02037 .20 .884 .o080 2.55 .2159 .625 2.89 .1426
.0341 .25 .851 1.077 2.53 .2692 .640 2.38 .1845
.0517 .30 .817 1.074 2.50 .3221 .656 2.04 .227

.0783 .35 .780 1.070 2.47 -3746 .673 1.80 .280
.1124 .40 .741 1.067 2.44 .4267 .690 1.617 -335
.1551 .45 .700 1.063 2.41 .4784 .707 1.481 .394
.2127 .50 .656 1.059 2.37 .5296 .728 1.375 .461
.2879 .55 .610 1.055 2.34 .5802 .749 1.290 .536

.3883 .60 .560 1.050 2.31 .6303 -770 1.222 .624
.5209 .65 .507 1.046 2.27 .6797 .793 1.167 .722
.7091 .70 .450 1.041 2.24 .7284 .817 1.122 .842
.9756 .75 .389 1.035 2.20 .7763 .843 1.086 .988
1.373 .80 .323 1.029 2.16 .8233 .871 1.058 1.172
2.007 .85 .252 1.023 2.12 .8693 .900 1.035 1.416
3.163 .90 .175 1.016 2.08 .9142 .931 1.018 1.780
5.840 .95 .091 1.oo8 2.o4 .9578 .964 1.007 2.415
10.556 .98 .037 1.003 2.02 .9833 .985 1.002 3.250
14.733 .99 .019 1.002 2.01 .9917 -993 1.001 3-835
1.00 0 1 2 1 1 1 "






NACA TM No. 1216


TABLE 6


DRAG LAW OF THE PLANE PLATE IN LONGITUDINAL FLOW

WITH HOMOGENEOUS SUCTION


x1 I = f(xL1) F(X5) G(X1)


0
.01
.02
.03
.04
.06
.08
.10
.15
.20
.25
-30
.35
.4o
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
.98
.99
.992
.994
.996
.997
.998
.999
.9995


.0000406
.oooowo6
.0001706
.0003682
.0006612
.001535
.002909
.00456
.01139
.02037
.03405
.05172
.07833
.1124
.1551
.2127
.2879
.3883
.5209
.7091
.9756
1-3731
2.0075
3.1630
5.8403
10.556
14.73
16.15
18.01
20.69
22.62
25.37
30.11
34.90
m


I. J


.004665
.oo8883
.012732
.016623
.02646
.03608
.04719
.07427
.10506
.13949
.18066
.23006
.2872
*3539
.4357
.5357
.6614
.8199
1.0347
1.3284
1.7538
2.4165
3.6018
6.3098
11.043
15.23
16.64
18.51
21.19
23.12
25.87
30.61
35.40
Go


Cf
C f


S Uo
cfeo = -


11".02
52.08
34.56
25.14
17.29
12.40
10.35
6.528
5.159
4.096
3.493
2.937
2.556
2.281
2.048
1.861
1.704
1.574
1.459
1.362-
1.277
1.204
1.139
1.080
1.046
1.034
1.031
1.0275
1.0241
1.0220
1.0196
1.0166
1.0143
1


= G(X1)






NACA TM No. 1216


TABLE 7

PARAMETERS FOR THE VELOCITY DISTRIBUTION OVER THE LENGTH OF GROWING

BOUNDARY LAYER FOR THE PLANE PLATE IN LONGITUDINAL FLOW

WITH HOMOGENEOUS SUCTION (TO FIG. 12)


S-K T -K

0 0 1 1.0 0.754 0.384

.1 .143 .919 1.4 .848 .255

.2 .267 .840 1.8 .902 .171

.4 .453 .696 3-0 .973 .053

.6 .590 .573 1 o

.8 .685 .467






NACA TM No. 1216


TABLE 8

RESULTS OF THE BOUNDARY LAYER CALCULATIONS

FOR THE CIRCULAR CYLINDER WITH SUCTION

(a) Co = 0


Po 2 Ic ). K 1j -K

0 0 0.1883 0.442 0.0709 0 0.355 o 0.653
4 .0698 .1884 .443 .0708 .355 .653
8 .1396 .1888 .446 .0705 .353 .657
12 .2094 .1892 .447 .0700 .350 .660
16 .2793 .1897 .448 .0696 .348 .662
20 .349 .1913 .452 .0668 .345 .664
25 .436 .1944 .459 .0685 .343 .665
30 .524 .1985 .469 .0682 .342 .667
35 .611 .2030 .479 .0675 .340 .669
40 .698 .2071 .488 .0657 .333 .673
45 .785 .2128 .502 .0641 .325 .680
50 .873 .2129 .522 .0612 .312 .690
55 .959 .2259 .540 .0585 .300 .703
60 1.047 .2326 .563 .0541 .280 .720
65 1.134 .2438 .591 .0502 .260 .740
70 1.222 .2552 .623 .0446 .232 .770
75 1.309 .2698 .669 .0377 .202 .805
80 1.396 .2881 .726 .0288 .156 .850
85 1.484 .3087 .790 .0166 .094 .905
90 1.571 .3332 .886 o 0 1.000
95 1.658 -3583 1.010 -.0223 -.141 1.15
100 1.746 .3937 1.240 -.0538 -.420 1.34
S 101.7 1.776 .4062 1.290 -.0682 -.682 1.44






NACA TM No. 1216


TABLE 8 Continued


RESULTS OF THE BOUNDARY-LAYER CALCULATIONS Continued


(b) Co = 0.5


UoO R */Uo 9 1 -K


.0698
.1396
.2094
.2793
.349
.436
.524
.611
.698
.785
.873
.959
1.047
1.134
1.222
1.309
1.396
1.484
1.571
1.658
1.746
1.833
1.854


0.1573
.1575
.1587
.1598
.1605
.1619
.1634
.1658
.1695
.1726
.1766
.1821
.1871
.1937
.2012
.2097
.2181
.2289
.2470
.2683
.2881
.3123
.3421
.3522


0.3681
.3686
*3714
.3739
-3756
-3788
-3824
.3880
.3966
.404
.413
.426
.440
.455
.473
.495
.515
.543
.593
.645
.709
.835
.978
1.004


0.0495
.0495
.0494
.0493
.0492
.0490
.0487
.0482
.0471
.0456
.0441
.0430
.0403
.0375
.0331
.0287
.0230
.0170
.0109
0
-.0144
-.0338
-.0606
-.0682


0.1112
.1114
.1122
.1130
.1137
.1145
.1155
.1172
.1199
.1220
.1249
.1288
-1323
-1370
.1422
.1481
.1542
.1619
.1747
.1897
.2037
.2208
.2419
.2490


0.240
.240
.240
.240
.240
.240
.238
.234
.225
.220
.215
.208
.195
.185
.164
.141
.115
.089
.057
0
-.075
-.185
-.415
-.452


0.250
.250
.250
.250
.250
.250
.252
.255
.260
.267
.275
.290
.297
.305
.315
-329
.350
.360
-375
.410
.462
.525
.610
.625


0.580
.580
.580
.581
.582
.584
.585
.587
.589
.592
.595
.601
.608
.615
.622
.630
.640
.665
.695
.735
.781
.884
1.06
1.18


2.34




2.342



2.343




2.35


2-37

2.40

2.67


0

8
12
16
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
Slo 6
s lo6. 4






NACA TM No. 1216


TABLE 8 Continued

RESULTS OF THE BOUNDARY LAYER CALCULATIONS Continued

(c) Co = 1

UoR
qp R v R* k I -K

0 0 0.1350 0.3071 0.0365 0.1909 0.170 0.415 0.525
4 .0698 .1350 .3071 .0364 .1909 .170 .415 .525
8 .1396 .1351 .3074 .0362 .1911 .170 .415 .525
12 .2094 .1360 .3094 .0362 .1923 .170 .415 .525
16 .2793 -1367 -3109 .0361 -1933 .169 .417 .525
20 .349 .1373 .3124 .0354 .1942 .166 .420 .525
25 -436 .1383 .3146 .0347 .1956 .162 .425 .524
30 .524 .1400 .3178 .0339 .1966 .156 .430 .523
35 .611 .1419 .3221 .0330 .2007 .153 .440 .522
40 .698 .1442 .3273 .0319 .2039 .150 .445 .521
45 .785 .1466 .3328 .0304 .2073 .142 .455 .520
50 .873 .1491 -3385 .0289 .2106 .135 .470 .519
55 .959 .1532 .3478 .0269 .2167 .127 .485 .517
60 1.047 .1595 .3613 .0254 .2256 .122 .495 .515
65 1.134 .1665 .3762 .0234 .2355 .107 .512 .509
70 1.222 .1729 .380 .0205 .2445 .098 -531 .500
75 1.309 .1795 .394 .0167 .2538 .077 .554 .500
80 1.396 .1869 .410 .0121 .2643 .055 .580 .508
85 1.484 .1942 .435 .0066 .2746 .031 .600 .515
90 1.571 .2017 .450 0 .2852 0 .630 .520
95 1.658 .2090 .476 -.0076 .2956 -.040 .668 .535
100 1.746 .2207 .507 -.0169 .3121 -.088 .710 .560
105 1.833 .2291 .539 -.0272 .3240 -.140 .775 .592
110 1.920 .2500 .598 -.0438 .3436 -.225 .825 .680
115 2.008 .2775 .700 -.0651 .3642 -.357 .850 .835
s 115.5 2.016 .2814 .715 -.0682 .3697 -.375 .870 .870






NACA TM No. 1216


TABLE 8 Continued


RESULTS OF THE BOUNDARY LAMER CALCULATIONS Continued

(d) Co = 2


PP IR "O1 X I -K


0
4
8
12
16
20
25
30
35
40
45
50
55
60
6.5
70
75
80
85
90
95
100
105
110
115
120
125
S 127.5
-


.0698
.1396
.2094
.2793
.349
.436
.524
.6n1
.590
.785
.873
-959
1.047
1.134
1.222
1.309
1.396
1.484
1.571
1.658
1.746
1.833
1.920
2.008
2.095
2.183
2.227


0.1030
.1030
.1031
.1034
.1039
.1044
.1050
.1055
.1068
.1078
.1091
.1106
.1123
.1153
.1187
.1217
.1252
.1289
.1334
.1393
.1460
.1543
.1631
.1721
.1860
.2015
.2205
.2371


0.2281
.2281
.2284
.2290
.2301
.2312
.2326
.2340
.2360
.2382
.2400
.2437
.2472
.2524
.2580
.2647
.2712
.2795
.2881
.3002
*3132
.3290
.3464
.3660
*3891
.415
.452
.472


0.0212
.0211
.0210
.0208
.0205
.0198
.0194
.0190
.0185
.0178
.0168
.0157
.0145
.0133
.0119
.0101
.0081
.0058
.0031
0
-.0040
-.0083
-.0138
-.0202
-.0306
-.0406
-.0557
-.0682


0.2913
.2913
.2918
.2925
.2939
.2953
.2970
.2984
.3021
.3049
.3086
.3128
.3176
-3261
.3357
.3442
.3541
.3646
-3773
.3940
.4129
.4364
.4613
.4868
.526
.570
.635
.671


0.090.
.090
.090
.087
.083
.079
.075
.073
.071
.069
.068
.066
.064
.060
.052
.045
.035
.025
.012
0
-.020
-.040
-.068
-.100
-.140
-.185
-.220
-.305


I I


0.620
.620
.620
.621
.623
.625
.628
.632
.638
.645
.654
.665
.677
.690
.710
.735
.755
.775
.795
.815
.845
.890
.925
.980
1.055
1.190
1.260
1.305


0.420
.420
.420
.419
.418
.416
.414
.412
.410
.405
-399
.394
.388
.380
.370
.360
-350
.340
.325
-310
.295
.278
.255
.225
.190
.120
.038
-.020


I I


---


i






NACA TM No. 1216



TABLE 8 Concluded


RESULTS OF THE BOUNDARY-LAYER CALCULATIONS Concluded

(e) Velocity Distribution






NACA TM No. 1216


TABLE 9


RESULTS OF THE BOUNDARY-LAYER CALCULATION FOR THE SYMMETRICAL

JOUKOWSKY PROFILE J 015 FOR ca = 0


(a) fi(o) = 0;


o = 0


ao L K 1 X X1 -K


180 0 0.0370 0.0878 0.0708 0 0.355 0 0.650
177.5 .0493 .0377 .0895 .0697 .349 .653
175 .00986 .0405 .0964 .0674 .335 .661
172.5 .0148 .0438 .1048 .0630 -319 .682
170 .0197 .0485 .1167 .0581 .298 .703
165 .0308 .0605 .1480 .0464 .247 .753
160 .o444 .0755 .1885 .0351 .197 .802
150 .0764 .1109 .284 .0193 .io8 .892
140 .1233 .1572 .411 .0053 -030 .964
136 .1445 .1772 .472 0 0 1.000
130 .1775 .2062 .559 -.0089 -.052 1.062
120 .2416 .2608 .742 -.0260 -.172 1.182
110 .3132 .3217 .960 -.0484 -362 1.367
S 101.9 -377 .3748 1.310 -.o68o 0 -.628 1.637






NACA TM No. 1216


TABLE 9 Continued

RESULTS OF THE BOUNDARY-LAYER CALCULATION Continued


(b) fi(o) = 0.5;


Co = 0.0695


B 0t 0T C X1 -K
t t rv t- v j1

180 o 0.0359 0.0846 0.0667 0.0180 0.335 0.044 0.640
177.5 .00493 .0366 .0867 .0658 .0183 -331 .045 .645
175 .00986 .0391 .0927 .0634 .0196 .322 .046 .654
172.5 .0148 .0430 .1025 .0603 .0215 -307 .047 .674
170 .0197 .0474 .1140 .0556 .0237 .283 .051 .690
165 .0308 .0580 .1416 .0430 .0299 .227 .060 .734
160 .0444 .0737 .1830 .0334 .0368 .180 .074 .773
150 .0764 .1072 .268 .0169 .0540 .100 .110 .833
140 .1233 .1446 .367 .0045 .0720 .024 .154 .867
136 .1445 .1613 .415 0 .0790 0 .173 .880
130 .1775 .1850 .483 -.0071 .0913 -.035 .203 .906
120 .2416 .2274 .620 -.0200 .1137 -.112 .255 .976
110 .3132 .2793 .764 -.0356 .1397 -.225 .316 1.075
100 .391 .3339 .961 -.0530 .1669 -.360 .384 1.200
s 92.4 .4575 .3742 1.125 -.0682 .1871 -.505 .440 1.310






NACA TM No. 1216


TABLE 9 Continued

RESULTS OF THE BOUNDARY-LAYER CALCULATION Continued


(c) f1(o) 1.0;


Co = 0.139


u 0i tx *t KKX -K

180 o 0.0348 0.0819 0.0626 0.0348 0.314 0.077 0.631
177.5 .00493 -0358 .0842 .0626 .0358 .313 .079 .631
175 .00986 .0382 .0900 .0605 .0388 .308 .085 .640
172.5 .0148 .0416 .0985 .0564 .0403 .293 .095 .653
170 .0197 .0460 .1095 .0516 .0450 .265 .104 .667
165 .0308 .0560 .1345 .0376 .0560 .205 .128 .703
160 .0444 .0686 .1678 .0292 .0678 .154 .156 .748
150 .0764 .0985 .243 .0152 .0985 .080 .223 .776
140 .1233 .1342 -332 .0038 .1340 .020 .300 .788
136 .1445 .1484 .368 0 .1478 0 -335 .795
130 .1775 .1682 .421 -.0059 .1682 -.032 .380 .8oo
120 .2416 .2030 .505 -.0162 .2025 -.084 .465 .806
110 .3132 .2385 .595 -.0270 .2400 -.142 .556 .809
100 .391 .2790 .696 -.0380 .28oo -.200 .650 .812
90 .475 .3200 .800 -.0499 .3203 -.267 .750 .818
80 .561 .362 .916 -.0636 .3622 -.343 .858 .830
S 76.4 .593 .376 .952 -.0682 .3766 -.380 .892 .860






NACA TM No. 1216


TABLE 9 Continued


RESULTS OF THE BOUNDARY-LAYER CALCULATION Continued


(d) fl(0) = 1.5;


Co = 0.2085


SPt 'B 1 x -K


180 0 0.0336 0.0792 0.0584 0.0504 0.295 0.110 0.635
177.5 .00493 -0351 .0824 .0592 .0526 .293 .116 .635
175 .00986 .0372 .0878 .0580 .0557 .281 .125 .639
172.5 .0148 .0404 .0958 .0531 .o6o6 .265 .136 .648
170 .0197 .0442 .1052 .0482 .0662 .202 .150 .658
165 .0308 .0544 .1301 .0376 .0816 .192 .181 .690
16o .0444 .0675 .1596 .0280 .0980 .145 .220 .705
150 .0764 .0927 .224 .0135 .1391 .070 .310 .722
140 .1233 .1249 .297 .0027 .1852 .016 .415 .718
136 .1445 .1368 .325 0 .205 0 .454 .702
130 .1775 .1543 -365 -.0046 .230 -.01 .512 .669
120 .2416 .1825 .430 -.0127 .272 -.060 .605 .630
110 .3132 .2107 .490 -.0202 .315 -.097 .698 .596
100 .391 .2388 .546 -.0276 .359 -.132 .785 .566
90 .475 .2683 .610 -.0356 .401 -.165 .870 .529
80 .561 .2981 .668 -.0432 .443 -.198 .950 .486
70 .645 .3255 .726 -.0o04 .487 -.230 1.034 .430
60 .734 -3555 .770 -.0578 .531 -.260 1.114 .350
50 .807 .3789 .808 -.0637 .568 -.286 1.190 .250
s 4o .885 .4027 .835 -.0682 .604 -.305 1.265 .150






NACA TM No. 1216


TABLE 9 Continued


RESULTS OF THE BOUNDAEY-LAYER CALCULATION Continued


(e) f1(0) = 3.0;


Co = 0.417


P a tl X X -K

180 0 0.0307 0.0720 0.0504 0.0921 0.242 0.210 0.602
177.5 .00493 .0323 .0755 .0511 .0940 .241 .213 .603
175 .00986 .0342 .0795 .0494 .1026 .229 .225 .604
172.5 .0148 .0365 .0852 .0433 .1095 .212 .240 .609
170 .0197 .0398 .0900 .0391 .1194 .195 .260 .612
165 .0308 .0486 .1150 .0300 .1398 .157 .310 .620
160 .0444 .0587 .136 .0216 .1705 .107 .376 .610
150 .0764 .0794 .184 .0099 .238 .051 .530 .570
140 .1233 .1015 .231 .0022 .309 .011 .666 .502
136 .1445 .1118 .248 0 -335 0 .714 .471
130 .1775 .1240 .270 -.0032 .370 -.016 .781 .422
120 .2416 .1435 .303 -.0080 .425 -.033 .880 .306
110 .3132 .1594 -333 -.0119 .476 -.049 .970 .153
100 .391 .1745 .358 -.0148 .522 -.060 1.045 .022
90 .475 .1881 .374 -.0172 .564 -.068 1.110 -.090
80 .561 .2006 .396 -.0197 .600 -.072 1.172 -.180
70 .645 .2109 .410 -.0212 .633 -.075 1.230 -.230
60 .734 .2218 .424 -.0226 .663 -.078 1.280 -.270
50 .807 .2302 .439 -.0234 .691 -.079 1.326 -.300
40 ..885 .2383 .452 -.0238 .715 -.080 1.336 -.330
30 .945 .2440 .464 -.0248 ..732 -.080 1.340 -.350
20 .990 .2486 .470 -.0256 .746 -.080 1.344 -.360
10 1.018 .2512 .476 -.0259 .754 -.080 1.348 -.360
0 1.028 .2522 .479 -.0261 .757 -.080 1.350 -.360
no separation






NACA TM No. 1216



TABLE 9 Concluded


RESULTS OF THE BOUNDARY-LAYER CALCULATION Continued

(f) Velocity Distribution







NACA TM No. 1216


TABLE 10

VELOCITY DISTRIBUTION OVER THE REGION OF GROWING BOUNDARY LAYER

FOR THE PLATE WITH HOMOGENEOUS SUCTION


i-t = 0.1 (T o02 Ft = 0.' / = 0.6 f = o.8
-vo? u "-vo u "voy u -v3y -.vY .
v Uo v Uo v Uo v Uo v Uo

0 0 0 0 0 0 0 0 0 0
.0286 .1108 .0534 .1168 .0910 .1279 .1180 .1373 -137 -1455
.0572 .2176 .1068 .2271 .182o .2448 .236 .2598 .274 .2727
.0858 .3205 .1602 .3317 .273 .?522 .354 .3697 .411 .3848
.1144 .4173 .2136 .4287 .'64 .4497 .472 .4676 .548 .4830
.143 .5107 .267 .5211 .455 .5302 .590 .5506 .685 .5704
.1716 .5967 .2204 .6054 .546 .6215 .708 .6351 .822 .6469
.2002 .6760 .3738 .6826 .637 .6948 .826 .7051 -959 .7141
.2288 .7475 .-272 .7518 .728 .7598 .944 .7666 1.096 .7724
.2574 .8105 .4806 .8126 .819 .8164 1.062 .8196 1.233 .8224
.286 .8663 .5340 .8661 .910 .8659 1.180 .8657 1.370 .8655
.3146 .9112 .5874 .9094 1.001 .9059 1.298 .9029 1.507 .900o
.3432 .9475 .64o8 .9442 1.092 -9382 1.416 .9331 1.644 .9287
.3718 .9738 .6942 .9697 1.183 .9621 1.534 .9557 1.781 .9501
.4004 .9899 .7476 .9856 1.274 .9776 1.652 .9708 1.918 .9650
.4290 .9959 .8010 .9920 1.365 .9848 1.770 .9787 2.055 .9734
.5005 .9975 .934. .9952 1.5925 .9908 2.065 .9871 2.3975 -9839
.5720 .9985 1.0680 .9971 1.820 .9944 2.360 .9922 2.740 .9902
.6435 .9991 1.2015 .9982 2.0475 .9966 2.655 -9953 3.0825 .9941
.715 .9995 1.3350 .9989 2.275 .9980 2.950 .9971 3-425 .9964
.858 .9998 1.6020 .9996 2.730 .9992 3.540 .9989 4.11 .9987
1.001 .9999 1.8690 .9999 3.185 .9997 4.130 .9996 4.795 .9995
m 1.0000 1.0000 1.0000 m 1.0000 1.0000

/1t = 1.0 =1. = 1.8 3.0 =3 =0

0 0 0 0 0 0 0 0 0 0
.1508 .1518 .1696 .1617 .1804 .1682 .1946 .1772 .2 .1813
.3016 .2829 -3392 .2986 .3608 .-088 .1892 .3232 .4 .3297
.4524 .3966 .5088 .4149 .5412 .4269 .5838 -.437 .6 .4512
.6032 .4950 .6784 .5137 .7216 .5259 .7784 .5430 .8 .5507
.75401 .5814 .848 .5984 .902 .6095 .973 -6251 1.0 .6321
.9048 .6562 1.0176 .6704 1.0824 .6798 1.1676 .6929 1.2 .6988
1.0556 .7211 1.1872 .7119 1.2628 .7390 1.3622 .7489 1.4 .7534
1.2064 .7770 1.3568 .7840 1.4432 .7887 1.5568 .7952 1.6 .7981
1.3572 .8246 1.5264 .8280 1.6236 .8302 1.7514 .8333 1.8 .8347
1.5080 .8654 1.696 .8651 1.804 .8650 1.946 .8648 2.0 .8647
1.6588 .8984 1.8656 .8953 1.9e44 .8933 2.1406 .8905 2.2 .8892
1.8o96 .9253 2.0352 .9199 2.1648 .9164 2.3352 .9115 2.4 .9093
1.9604 .9458 2.2048 .9391 2.3452 .9347 2.5292 .9285 2.6 .9257
2.1112 .9614 2.3744 .9533 2.5256 .9486 2.7244 .9421 2.8 .9392
2.262 .9693 2.5440 .9629 2.7060 .9587 2.9190 .9528 3.0 .9502
2.639 .9814 2.968 .9775 3.1570 .9750 .34055 .9714 3-5 .9698
3.016 .9887 3.392 .9864 3.608 .9848 '.8920 .9827 4.0 .9817
3-393 .9932 3.816 .9917 4.059 .9908 4.785 .9895 4.5 .9889
3.770 .9959 4.240 .9950 4-510 .9944 4.865 .9931 5.0 .9933
4.524 .9985 5.088 .9981 5.412 .9979 5.838 .9976 6.0 .9975
5.278 .9994 5.936 .9993 6-314 .9993 6.811 .9992 7.0 .9991
1.0000 1.0000 cc 1.0000 1.0000 a 1.0000

The parameter K and l, (See table 7.)






NACA TM No. 1216




















Figure 1.- Explanatory sketch for the boundary layer with suction for
arbitrary body shape.


BI Figure 2.- The functions F1 and F2 for the velocity distribution in
boundary layer, see equation (9).






NACA TM No. 1216


-20- -- ---







ITI
S^ A -K -




-I------ _^ \^ ^ -


-I--
_,\N X4$-,

r: _-ll I\s^^ $ ^S -
*;s-l\,, ,,',,',,', \,,\ \\\\-\
,, _S -S .5 S S S ^


-0.5


a '. 70 \ .A
A,= 1 08 06 04 02 0 -02


Figure 3.- The form parameter K of the velocity profile as a function of
S, A 1, according to equation (14).






NACA TM No. 1216


/ 4.5

rts
ft/U


0.5












0


Figure 4.


Itj2L I I I _J I I--L
Oas 10 s. 2.0

6* To 1
The auxiliary functions G(K), and as functions

of K, according to equations (17a), (19), and (20).







70 NACA TM No. 1216



Ij *
__ -






,II 4
-- a:3 : '.j



ca d







S- t % -


\ o

-- r r 8
1 4C




_- I I- -J- t_ I cn
c-0 -- I, IA zt







c1



Ss '



p 0 __

1 r a y- "*
'? ~ ~~ -^ i- -- -- -- I








--- rfS^W ^ ^ --- \- --- --- ---, r ^~
l------l -<-- % Ji ~ ~ i 1 -


QsE






NACA TM No. 1216


Figure 6.- Diagram for solution of the differential equation for the momentum
thickness: G(s, x1).






NACA TM No. 1216


10 -

-0.0682, For all K! ,/
-i- -
Separation point" _
SHortree 71r



I




r.v







S---^-- -_- -_
-- -- ------ O-









0aoog' Stagnation point t (D
without-j suction _D--

mZ __
0~~t






NACA TM No. 1216 73





I"o ^"f 0
000 r08-
0.0709


006 -




004 04--- --




002 02----




0 / 2 3
-Vo K= o


Figure 7.- The initial values of the boundary layer at the stagnation point for
various suction quantities.







NACA TM No. 1216


LI
U0o .


Approxliate


-- i
/ i


----I--- --- --yf~
1 1 i i i- _


Figure 8.- Comparison of the velocity distributions according to the
approximate with the exact calculation.

(a) Plane plate in longitudinal flow, exact calculation according to Blasius,
approximate calculation according to equations (53) and (54).

(b) Exact calculation according to Hiemenz, approximate calculation
according to equations (58) and (58a).







NACA TM No. 1216


Figure 9.- The extent of growing boundary layer for the plane plate with
-V1T* 50 T05*
homogeneous suction: against j
v Uc i






NACA TM No. 1216


Figure 10.- Plane plate with homogeneous suction; region of growing
boundary layer; velocity distribution.






NACA TM No. 1216


16

1,2 ___







--^ .O -I
d00



04 -- -- ---- -- -- -



04 0 B 1.2 1.6 20 ?.4



Figure 11.- Plane plate with homogeneous suction; region of growing boundary
T -
layer; the local friction coefficient -- against t.
Oo







78 NACA TM No. 1216





41


a,
0
U a









II k o
*-- ---- u











8 h0o
.-I

ILrbO 6;:
S.--4










a,



0



0o 4
Cd



o h





0
CO
___g Cd
cc U4





Z V










a,
~ iv, V hCd

c'J .







NACA TM No. 1216 79





0
4'




-- -- "00



Sio/



I /


: -- I- O -
.:



---- --- ---- f W -/ )f |- ---- 1.4 0 0








S f i -'
z \
-4 Ul 0














^ f-- -^ ,.::I
/ 0
f Cd 0 0 0



S.D-I.l
Eo 0




A E:











-,-
V 0


Cd




14 4








Q)
.,- "- "



rri ~a~cl) ~ ~~ 8 ci c







NACA TM No. 1216


03 o.0.8
\





02 0,6






104


---- Exac t
SApproximate


--- -- ----0.400



Figure 14.- Plane stagnation point flow: comparison of the boundary-layer
thickness and momentum thickness of approximate and exact calculation
for various suction quantities, exact calculation according to reference 9.






NACA TM No. 1216


Figure 15.- The boundary layer on the circular cylinder with homogeneous

suction for various suction quantities Co = o U- Form

parameter K = 2 U With increasing suction quantity the
v ds
separation point shifts rearward.







NACA TM No. 1216





Ei



4-1



o

bf vl
--- / el'1, XO dddX S o .,C-.


sO











cco
00
^ \ \ -- ----4




0
I I _
















1.0
c\ \ h0








co
i1 a
,--0
C (D





d C L







0 0
,~1 o









Z-
l^ |---i ------ UiL!----4-4
*-saf q3 h ci *30'
*ok ^ ^ 2






NACA TM No. 1216


-008


Figure 17.- The boundary layer on a symmetrical Joukowsky profile J 015
with ca = 0 for homogeneous suction with various suction quantities

Co =fl according to equation (36): Co x fl(o)

a dU
(K1 = 51.7). = ds. With increasing suction quantity the
separation point shifts rearward.







84 NACA TM No. 1216




O
"-4

o








no
I b
la




\g s
\ ---

\ I o |


v \ I '




cdu


$I c

0
\I C
__ \ __ __a _._


0 0)
%\ \ / /

S~ca"









cd
,,-- --1--










\CV / -I' J


I- --"4
r- a







NACA TM No. 1216 85





I

-- -S--- n
9 0
o)



-F
o -.


Q 0


ci
SI C -


0 S h -'
S00 c I O
-o I




x L2 r =
a ~cua ob~

if C, C
Cd I





CC 00.-









a o
.5 ci
S-4 I

















O)





b-i
Scd





oa
_____ __ ____ ___ ___ ____ __ a
N~~O + ,- -

~r





























































































































































































siia~















2*







UNIVERSITY OF FLORIDA


3 1262 08106 319 9




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