Development of a laminar boundary layer behind a suction point

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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1336


DEVELOPMENT OF A LAMINAR BOUNDARY LAYER

BEHIND A SUCTION POINT*

By W. Wuest


1. INTRODUCTION


Boundary-layer suction originally was applied to reduce the
boundary-layer thickness and therewith the inclination to flow sepa-
ration; however, since the properties of bodies with small drag have
been improved more and more, attention was drawn to an increased extent
to the reduction of surface friction. One now strived toward keeping
the boundary layer laminar as long as possible, thus to defer the tran-
sition point to turbulence as far as possible. Boundary-layer suction
was recognized to have a favorable effect in this sense, and therewith
the velocity distribution in a laminar boundary layer behind a suction
point acquired heightened interest. The stability of a laminar velocity
profile is very severely affected by the shape of this profile.

In a considerable number of theoretical reports (reference 1) the
case of continuous suction was treated for reasons of mathematical
simplicity; permeability of the wall surface was assumed. In further
reports, the stability of laminar boundary-layer profiles in case of
continuous suction was treated and a considerable rise in the stability
limit was determined; however, a technical realization of such perme-
able walls with sufficiently smooth surface and adequate material
strength characteristics is difficult. For structural reasons, it is
simpler to arrange single-suction slots. In addition to the suction
effect proper, there appears here the sink effect first discussed in
detail by L. Prandtl and 0. Schrenk (reference 2) and recently treated
by Pfenniger (reference 3) in an instructive experimental investigation.

Below, the pressure variation along the wall as well as, in partic-
ular, the sink effect are disregarded. Figure 1 shows the practical
realization of such a case. We assume that on a flat plate A, a laminar
boundary layer ("Blasius boundary layer") develops at constant pressure.
We assume a second plate B arranged beginning from a certain point xo
at the distance yo parallel to the first plate so that a suction slot

*"Entwicklung einer laminaren Grenzschicht hinter einer
Absaugestelle." Ingenieur Archiv, Vol. 17, 1949, pp. 199-20o.


rz7/ f







NACA TM 1336


is formed between the two plates. The magnitude of the power require-
ment for suction is assumed to be precisely such that merely the part
of the boundary layer situated between the two plates is removed. Thus,
there begins above the plate B a new laminar boundary layer which is
distinguished from the Blasius boundary layer by another initial condi-
tion. The new boundary layer forms at its start the outer part of a
Biasius boundary layer.


2. BOUNDARY-LAYER EQUATION AND ASYMPTOTIC BEHAVIOR


By introduction of the stream function and the total pressure, the
boundary-layer equation may be transformed by the well-known method
(reference 4) into


3g = Vu (i)g
ox (32


where g = p + u and u. We limit ourselves to the case that
oy
the flow takes place outside of the boundary layer at the velocity
a1 = const., thus to the flat plate and put furthermore


g = 2 u2(l q(x,*)) + Const. (2)

or, respectively

u = q (3)


This statement has been chosen so that for large i-values, q assumes
the value 1. Equation (1) is thereby transformed into

dq -_q
7 =VUl q (4)
Ox 6*2


From the definition of the stream function and from equation (3), one
further obtains with q = '/vu1x
I/v77






NACA TM 1336


ou _1 Uli uix oq
By 2 xV V TI


-2u
Cy2


1 ul ulx
2 2 V--)


2


y = xdq
Y dqr -
0^J rq


In order to investigate the asymptotic behavior of the differential
equation (3), we put for large values of *

q = 1 q. with q << 1


In first approximation, one then obtains

aq, a qw
vul 2
OX


This differential equation, however, is mathematically identical with
the differential equation of a nonsteady flow independent of x which
has been treated before (reference 5); the time t is now replaced by
the stipulated space coordinate x. It also corresponds to the well-
known heat-conduction equation. The general solution is therefore given
by


q( *,x) =


L J q''(
2


x
A0


_, O- ) -- dq
*1 ^ (xl x 0vul(x x


Cd'( x
4vul(x x')


(10)


qw(ojx')1o






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Therein


0 = 2 e-y dy



is the known error integral. W. Tollaien (reference 6) has investigated
this solution for two special cases where the first integral disappears.
For the boundary layer with suction, however, this will no longer be the
case.


3. BLASrUS BOUNDARY LAYER


Although we presupposed that the velocity ul at the edge of the
boundary layer is constant, the problem of the suction boundary layer
to be treated here nevertheless differs from the flow on a simple flat
plate ("Blasius boundary layer") by the fact that other initial condi-
tions exist; rather, the Blasius boundary layer is contained as special
solution among the suction boundary layers since there x0 = 0, thus
suction point and beginning of the plate A (fig. 1) coincide. Since we
shall make use of this special solution for the later calculation, we
shall first consider the Blasius boundary layer. It is distinguished
by the fact that q may be regarded as dependent merely on a quan-
tity 11 = /f-vuix. One then obtains from equation (4) the following
differential equation of the Blasius boundary layer




TI + 0 (11)




The solution may be written in the following form





qB = 3r(l + a+ j) + 3 )2 + a3 (- + (12)






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The constants ai therein have the following values


a_ 2


a2
S"90


-7
a3 (990 x 15)

a4 = 1.60333 x 10-0


a, = 0.57627 x 10-


a, = 3.890( x 10-9


a- = -1.398o .< 10-9


a8 = -3.9135 x 10-11


a9 = 3.7282 x 10-12


a10 = 2.2383 x 10-13


all = -0.3104 ;., 0-14


a12


= -1.081 x 10-15


Due to the boundary condition at th wall, one integration constant is
zero. The second integration constant is determined from the asymptotic
behavior for large values of r. 'Because of q.w(,0) = 0 the first
integral in equation (10) is eliminated. The second integral, however,
yields by partial integration, with consideration of the asymptotic
behavior of the error integral, just as in the case treated before by
W. Wuest the solution


(13)


The constants 3 in equation (12) and 7 in equation (13) are deter-
mined by the fact that for large I values q and oq/f.r according
to equation (12) and equation (1-l) agree with each other. The recalcu-
lation of the two constants yielded the following values


0 = .0.66.42


7 ='0.828


For comparison, L. Prandtl (reference 7) givesc\thy f6flowing values
calculated by Blasius and ToA-er which read, converted to the above
designations

0 = 2 / 0.332 = o.664


S= 2 F 0.231 = 0.819


qwB'.~) 1 'Y-1
[ -_ V4=Vulx]






NACA TM 1336


F. Riegels and J. A. Zaat give in a new report (reference 8) for 7
the following value


y = 0.342 24 = 0.857

The function q with first and second derivative has been tabulated
and plotted in numerical table 1 and figure 21.


1. ASYMPTOTIC BEHAVIOR OF THE SUCTION BOUNDARY LAYER


For calculation of the asymptotic behavior of the suction boundary
layer, we divide the function qw defined by equation (8) into two
parts


w = v1 + w2

The first part is to be selected so that it satisfies the initial condi-
tion at the suction point x = xO; this is done by extending the asymp-
totic solution of the Blasius boundary layer to x > xO as well. From
equation (13) one then obtains


q wl = 7 1 0
4vulx]


The numerical table has been calculated with the values o = 0.664
and 7 = 0.819.







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Numerical Table 1.


Blasius Boundary Layer


q'(i)


I "(T)


-4 4 ______--


0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.5
3.0
3.5
4.0
4.5
5.0


.06606
.13106
.1939
.2546
.3135
.3695
.4228
.4681
.5211
.5659
.6081
.6469
.6831
.7167
.7474
.7758
.8017
.8252
.8465
.8657
.9352
.9715
.9881
.9962
.9988
.9997


o.6640o
.6555
.6427
.6206
.5981
.5734
.5471
.5195
.4912
.4625
.4337
.4062
.3766
.3487
.3217
.2955
.2705
.2466
.2238
.2022
.1820
.1013
.0509
.0217
.0085
.0029
.0009


.12751
.17750
.2114
.2369
.2561
.2700
.2796
.2856
.2883
.2883
.2857
.2809
.2743
.2660
.2563
.2457
.2340
.2248
.2088
.1955
.1303
.0774
.0381
.0169
.0066
.0022


0
.2570
.3620
.4404
.5046
.5599
.6079
.6502
.6842
.7212
.7523
.7798
.8o4 3
.8265
.8466
.8645
.8807
.8954
.9084
.9200
.9304
.9671
.9857
.9940
..9981
.9994
.9998


Therein 4 = 0
quantity. The
ary condition


forms the new wall streamline and r0 the suction
second part qw2 then must be chosen so that the bound-
qw = qw(O,x) is satisfied. If the asymptotic rela-


tion q ~ 1 q would rigorously apply in the entire domain of the
boundary layer, there would have to be at the wall qw,(0,x) = 1, because
of q = 0; however, the asymptotic solution deviates from the rigorous
solution if it is continued up to the wall. Therefore qw(O,x) is an
unknown function regarding which we merely make the assumption that it
does not become infinite. As initial condition for the part qw2 one


1 qq)






NACA TM 1336


further has qw2(*,x0) = 0 since qwl(*,xo) already satisfies the
initial condition

q w = 7 1 0


which insures the connection wtth the Blasius solution. The contri-
bution qw2 to the solution al'b must obey the differential equa-
tion (9). In the solution (equation (10)) the first integral is elimi-
nated, because of qw2(,x0o) = 0, whereas in the second integral one
has to put


qw%2(0,x) = q%(0,x) ql(O,x) = q(0,x) 7 1 -

so that the asymptotic solution reads

{ + \- /[ c -/ 2O))
, = 7 1 +f q (O,x') 7 1 x'



By partial integration one obtains with consideration of the asymptotic
behavior of the error integral (by W. Wuest, elsewhere)


7[ (D**o + (oxO) 7 K i- 4 1 ( \
vulx ul Iul(x x0)^I

Because of the connection with the Blasius solution, however,
q(O, xo) = 7, if the asymptotic solution is continued up to the wall,
so that one finally obtains as the asymptotic solution for the suction
boundary layer


qv Y + y( ( o)[ (14)







NACA TM 1336


Instead of the error integrals 4 one may for large values of 4
again go back to the Blasius solution if one takes the asymptotic
behavior of the latter according to equation (8) and equation (13) into
consideration


q + *0\ 0 q
q ~ qB 1 qB X (1)
Vul /ulx0 Vul x x0


In this formula qB represents the Blasius solution. The last form of
the solution proves to be particularly expedient for the further con-
siderations.


5. APPROXIMATE SOLUTION FOR THE SUCTION BOUNDARY LAYER


It suggests itself to generalize the asymptotic solution which is
valid for large values of 4 in the following manner



q = qB* F(,x) 1 --B (16)
Vu /SVul x x0)

Due to q = 0 for 4 = 0 and because of equation (15) the func-
tion F(i,x) must satisfy the following conditions


F(O,x) = qB \ F(),x) = u 0 (17)



It was hoped at first that one could choose for F, as in the nonsteady
analogue by W. Wuest, elsewhere correspondingly an exponential func-
tion as the simplest formulation; besides equation (17) the disappear-
ance of the second derivative of q at the wall would be added as a
further condition; however, it was shown that such a formulation does
not meet with success and even, in a certain domain, does not yield any
solution at all.






10 NACA TM 1336


For the further calculation we introduce the following simplified
notation

+ t0 + 0 x Xo
= = = + (18)
_ulx VulX- x) vu x

so that the solution (equation (16)) reads


q = q(nl) F 1 qB') (19)

According to a suggestion by A. Betz, we equate as first approxi-
mation q1 the function F to the value dependent only on x


FO(X) = F(O,x) = QB o) (20


at the wall. -Thus the first approximation reads


l= qB() Fo 1 qB(')) (21)

This formulation does not fulfill the boundary-layer equation (4) exactly.
In particular, the second derivative of ql at the wall does not disap-
pear; however, the dependency on the second derivative of the stability
of the velocity profile is of a very sensitive nature so that one has to
look for a more accurate solution. By substitution of the approximate
solution .(equation (21)) into the boundary-layer equation (4), one
obtains

qB' )x x 2 {[o } 2 (x xO) 2I




Hence there results 2 61 in2 as the error of this first approxi-
mation. By subtraction of the exact solution in which F stands
for FO and q for q1, while el disappears, one then obtains






NACA TM 1336


r2 =) 2-1 22E.
2 F FFO) [1 ( + (22)


where

S2 =( t 2(x xo)~2L



is an unknown function. The quantity e2'' disappears for q' = 0
and ij' = m. By integration of equation (22) one obtains

(F Fo)E 1- qB(T')] = i + '2 (23)


Therein eI is to be determined graphically or numerically by repeated
quadrature

El j fr2 2 l2 (24)
I = o 61 d11 dT2


One may determine the asymptotic behavior of e2 by substituting in the
above definition of e2'' for and q the asymptotic
values q 1 qwl and yq 1 qw. Thereby one obtains


2"x x0) -(qv- qwl)
ox

Hence follows with use of equations (9), (19), (21), and repeated inte-
gration with respect to q' = i/vux x0)


E2 F(F Fo) [ V (I-o ) (25)






NACA TM 1336


As before, 0 denotes the error integral. Generally we visualize g2
as represented in the following manner
o I-
2 -Z a(x) Q ) (26)
X.1


By way of approximation we limit ourselves to the first two terms,
with aI = 7(F. FO) and a2 determined by the fact that q must
disappear at the wall. We determine accordingly the function F
approximately to be


F = FO + q ') 1(',x) + Y(F. FO) + a2 -

(27)

a2 = -1(0,x) 7(F% FO) (28)



Calculation example.- 0/F vu,x0 = 0.125 was selected as numerical
example;. F was calculated for the values x/xO = 1.234, 1.562, 4.34,
and 9.78 and plotted in figure 3. For x/xO = 1.562 the error was
determined by substitution of the approximated solution into the
boundary-layer equation, and compared with the first approximation
according to equation (21). Compared to the first approximation, a con-
siderable improvement results particularly in the region near the wall
(fig. 4). In figure 5 the results are converted to the velocity pro-
file, in figure 6 the second derivative is represented. As a supplement,
the connection between the degree of suction and the suction quantity of
the magnitude T0* = 01F/Vulx will be supplemented. By the degree of
suction e we here understand

8 *
S= 1 2 (29)
81"

81* being the displacement thickness immediately ahead of the suction
point and 62* immediately behind it. Therewith E is given by






NACA TM 1336


r of 1) d
e = Jo (30)

f 1 d*


The resulting values are tabulated in table 2 and plotted in figure 7.


Table 2. Degree of Suction


0q* O 0.1 0.2 0.4 0.6 0.8 1.0 2.0

8 0 0.392 0.520 0.671 0.762 0.824 0.870 0.974


The suction quantity 40 is, furthermore, given by the following
relation


:0 = tlxo0o* (31)


6. SUMMARY


The development of a laminar boundary layer behind a suction point
is investigated if by the suction merely the part of the boundary layer
near the wall is "cut off", without the slot exerting a sink effect.
As basis of the calculation, we used the boundary-layer equation in the
form indicated by Prandtl-Mises which is closely related to the heat
conduction equation or, respectively, to the differential equation of
the nonsteady flow which is independent of the coordinate x along the
wall. With consideration of the asymptotic behavior of the solution,
an approximate solution is developed which is similar in structure to
the solution of the nonsteady analogue which has been treated in an
earlier report by W. Wuest, elsewhere.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics






NACA TM 1336


REFERENCES


1. Tolimien, W., and Mangler, W.: StationKre laminare Grenzschichten.
Monogr. Fortschr. Luftfahrtforsch. Aerodyn. Vers.-Anst. GOttingen
(AVA-Monogr.), Vol. 1, 1946, CF. also H. Schlichting, Ing.-Arch.,
Vol. 16, 1948, p. 201.

2. Schrenk, 0.: Z. angew. Math. Mech., Vol. 13, 1933, p. 180.

3. Pfenniger, W.: Untersuchungen Uber Reibungsverminderungen an
Tragfl'Ugeln, insbesondere mit Hilfe von Grenzschichtabsaugung.
Mitt. Inst. Aerodyn. Tech. Hochschule Zurich Nr. 13, 1946.
(Available as NACA TM 1181.)

4. Prandtl, L.: Note on the Calculation of Boundary Layers. Z. angew.
Math. Mech., Vol. 18, 1938, p. 77-82. (Available as NACA TM 959.)

5. Wuest, W.: Beitrag zur instationaren laminaren Grenzschicht an
ebenen Wanden. Ing.-Arch., Vol. 17, 1949, pp. 193-198.

6. Tollmien, W.: Uber das Verhalten einer Str5mung langs einer Wand am
lusseren Rand ihrer Reibungsschicht. Betz-Festschrift AVA-G5ttingen,
1945, p. 218.

7. Prandtl, L.: F. W. Durand Aerodynamic Theory, Vol. 3, 1935, p. 88.

8. Riegels, F., and Zaat, J. A.: Zum Ubergang von grenzschichten in die
ungestb'rte str5mung. Nachr. Akad. Wiss. OGttingen. Math.-Phys.
Kl. 1947, pp. 42-45.






NACA TM 1336


I I I Suction

Figure 1.- Boundary-layer suction at the flat plate without sink effect.


i q,q,2q"


. JB -- -- --
^_ (_r__


N


2_q11?)


0 0.5 1.0 1.5 2.0 2.5 3.0 3.5



Figure 2.- The function q(71) of the Blasius boundary layer with first and
second derivative.


,4' (11)
4


--/ 1-f--f-I-----t-----I---*^*-1-----






NACA TM 1336


Figure .- Auxiliary function F(q',x) for calculation of the suction
boundary layer for ,0/\VUlXo = 0.125.


Figure 4.- Error of the first and second approximation for x/x0 = 1.562.







NACA TM 1336


Figure 5.- Velocity profiles of the suction boundary layer for
,0/J /vUlX = 0.125 and various distances from the suction
point.


0 0.050 0.100 0.150


Figure 6.- Second derivative of the velocity profiles of the suction
boundary layer for / vux- = 0.125.
13 i' 1 u






NACA 9I 1336


Degree of suction G


Figure 7.- Degree of suction (ratio of the cross-hatched and the total
shaded area in fig. 1).


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