BOUNDARY
LAYER
ON A FLAT
WITH
SUCTION
PLATE
Br P. L. BHATNAGAR, F.A.Sc., R. SANKAR AND A. C. JAI~
(Department of Applied Mathematics, Indian lnstitute of Science, Bangalore)
Received Oetober 1, 1959
INTRODUCTION
IN this paper we discuss the problem of evaluating the thickness of boundary
layer and skin friction on a thin plate with suction, placed parallel to an
otherwise uniform flow. A number of workers have studied this problem.
For example, Blasius' classical work discusses the boundary layer on a plate
at some distance from the leading edge, while Schlichting has given the
solution of the problem taking into account uniform suction in which the
velocity distribution is independent of the distance from the leading edge.
Recently Carrier and Lin x have obtained the solution in the neighbourhood
of the leading edge, while Torda ~ has deduced the necessary equations giving
the distribution of suction at the plate which will ensure constant thickness
of boundary layer. He has represented this suction velocity distribution
in a graphical form for an aerofoil for which experimental data are available. We have obtained the stream function by the method of successive
approximations assuming a linear law of suction which holds strictly in the
neighbourhood of the leading edge. In passing, we may mention that in
the absence of suction the present solution reduces to a forro similar to that
given by Carrier and Lin. We have obtained the thickness of the boundary
layer and the skin friction explicitly in terms of suction velocity and distante
from the leading edge without putting any restriction on the distribution
of suction speed on the plate. To ensure symmetry we have assumed the
suction to take place with the same speed at the corresponding points on
the two sides of the plate.
It is evident that our airo is in a sense different from that of Torda.
While Torda has attempted to specify the distribution of suction speed over
the plate to ensure a constant thickness of boundary layer, we have attempted
to evaluate the boundary layer thickness in terms of a specified distribution
of suction speed. The present point of view may be found more useful in
pracfice besides being more general than that of Torda.
Al
1
2
P . L . BHATNAGARAND OTI-IERS
The explicit expressions for velocity boundary layer thickness $ (x)
and the skin friction ~o (x) for the downstream flow as obtained in this
paper are:
+,o~(~~),+
....
(~~ ~91
(~x
and
9 o (xi =
~
+
= ~
1/-0
-
v0' (x
>)
,
where
a = 0.332.
1.
DESCRIPTION OF THE FLOW NEAR THE LEADING EDGE
The equations governing ah incompressible, viscous, fluid flow in twodimensional cartesian co-ordinates xi, yl are:
Equation of continuity :
aUx d- ~-~~
~91 = O,
~--~1
(1.1)
Equations of momentum:
~ul
~ux
1 ~ P l _ vXT~ul'
u , ~ + v ~ ~ + o ~xl
(1.2)
ul ~~vl + vi ~~Vl + o1 ~Pl
by~
(1.3)
vV~vl.
_
The introduction of the following dimensionless variables:
Pi
XxUo
p=~-Pª191 ~,
x-
V ~
-- ~
"01
~~b
Vo
~X '
~
ylUo
,
y-
~
ul
,
u . . Uo. .
~~b
~y'
(1.4)
where u0 is the free stream velocity, reduces the above equations to:
_ _ _ (~~~),
0.5)
Boundary Layer on a Flat Plate with Suction
3
where
V~~ =~ ~
a'ff + ayr
~'~
(1.6)
It is found convenient to work through polar co-ordinates defined by
x=rcosO,
y=rsinO.
With these, (1.5) and (1.6) reduce to:
~r--~-t- r ar a
r ~ ar a + r a ar
1 aaff
2 ~2~' t
1 a'ff
r arCO~ + -~ ~0~1 + r* ~6~
2
= r t~-0 ~ (V~~)
3r
(U2ff) ,
(1.7)
where
V2r = ~-~ + 7 ~ + r" a0~"
(1.8)
Following Carrier and Lin we write this equation as
L (~b) = L* (~b),
(1.9)
where L denotes the biharmonic operator and L* denotes the non-linear
operator on the right. For a linear law of suction,
v (x, 0) = - Vx,
(1.10)
the boundary conditions are
(~)e-,
= Vr,
a~
(~)e-,,
= - vr,
(1.11)
(r~0),.~ = O,
(~),.,
=o
We assume a solution of (1.9) in the forro:
= r176+ r + ~~ + ....
where
~o, ~1, #,, ....
(1.12)
4
P.L.
BRATNAGAR
A N D OTHERS
are defined by
L (r
= 0,
(1.13 a)
L (~b~)= L* (~bo),
(1.13 b)
L (~,) = L* (~o + ~1) -- L* (~o),
(1.13 c)
L (~~) = L* (~o + ~1 + . . . + ~~-0 -- L* (~o + ~1 + . . . + ~~-~),
0.13 n)
We shall ehoose ~o sucia that it satisfies the boundary eonditions (1.11)
eompletely, i.e..
-----Vr,
( ~~b~
~r/o=o
\ ~r Io=~,~
(~~
(~~o'~
~0 1o=o
=0,
: -- Vr,
(1.14)
\ ~O)e=~" = 0
Besides the velocity distribution (as determined by ~bo) is to be symmetric
about the plate. The boundary conditions to be satisfied by ~t for i > 0,
ate
~--~ = ~-0 = 0,
at
0 = 0,
2*r.
(1.15)
ALIso the velocity distribution (as determined from each one of the $i's)
is to be symmetrical about the plate.
The first two conditions in (1.14) suggest a solution for (1.13 a) in the
form
~o -= r~f (O).
(1.16)
Using this form wr find that f(O) has to satisfy
f l v + 4 f " = O,
(1.17)
with the conditions
V
f(O) = ~ ,
V
f(21r) = -- ~ ,
(1.18)
f' (0)=0,
f ' (2~) = 0 .
Boundary Layer on a Flat Plate with Suction
5
With these we easily see that the solution of (1.13 a) of the assumed type
is
Vr I
Vr I ..
~bo = --2- + ~ " tsm 20 -- 20) -- 2Cr t sinl0,
(1.19)
where the constant C is yet undetermined.
The symmetry of velocity distribution demands
(~~o~
(a~o~
r50,]r.p = ~r~O,]r.p
'
(1.20)
ar ]r.-p
\ ~r/r-o
0-a
for all p a n d
'
0-2f-4
a.
These conditions yir
li .20
C=0.
'l~us
Vr2
Vr2 (sin 20 -- 20).
O . = T +~7
(1.22)
If we put V ---- 0 in (1.22), we get,
~bo ---- 0.
(1.23)
Hence we find that the above stream function is unabl• to account for the
flow without suction.
The boundary eonditions for the flow without suction are
(~)e.o =~
(~710.,. =~
(~),-o = o,
(~)0-~. = 0.
0.24)
Besidr the velocity distribution is to be symmctric about the plato, It is
found by trial that the lcast value of n for which a function of the forro
rnf(O) satisfies thcse conditions is 3/2 and the corresponding function f(O)
is
given b y
6
P.L.
O
BHATNAGARAND OTHER8
30
f(O) = cos E -- cos-~ .
(1.25)
Thus to get a complete expression for ~k0 which will also account for the
flow in the absence of suction without affecting the boundary conditions
we must add to (1.22).
(o
~)
2Ar 3/2 cos ~ -- cos - -
,
(1.26)
where A is yet an undetermined constant which is independent of suction.
Hence
~bo---- ~-Vr~+-~-Vr2 (sin 20 -- 20) + 2Ar a/~ cos ~. -- cos - -
.
(1.27)
The presence of the constant A allows us a choice to fix up the flow without
suction.
With this value of ~bo, (1.13 b) becomes
2A ~.
L (~bl) = - 7 - (2 sin 0 -- 3 sin 20)
+ ~A V ( 2 Ÿ
-
. ~0 + 7 c o s ~0 sin
2 0 sin 0 -
6cos~
cos~)
§ ~~ (~ _ ~~§ ~ s~~~0)
~~~~~
It appears that a solution of (1.28) is of the forro
~b1 = raA (0) + rV/ZA (0) + r4fa (0);
(1.29)
but sucia a solution is found not to satisfy boundary conditions (I. 15) and
the symmetry conditions unless V-----0. Thus we cannot proceed with the
above forro of solution for ~1 if we ate interested in a solution for the problem with suction. Ir is found that a function of the form
~b1 = A2r s {f~ (0) + ~~ (0) log r}
+ AV r,/~ {f~ (0) + ~91 (0) log r}
77"
Boundary Layer 011 a Flat Plate with Suction
7
q- 2V---2r ' {)Ÿ(0) + 43 (0) log r},
(1.30)
~rt
satisfies (1.28) and the eonditions (1.15)~along with the symmetry of the
velocity distribution. Using this we find that fx, f2, A, 4,1, 4,2, ~b.~have to
satisfy
~b~TM + 10~1" + 94a = O,
(1.31)
fx ~v + lOf~" + 9f~ = -- 24~1 -- 8~~" + 4 sin 0 -- 6 sin 20,
(1.32)
~,xv + ~ ~91 + ~161 ~~ := 0,
(1.33)
29.,
441
105
.f2TM + ~- J~ + ~ A = -- -~- ff~ -- 10r
. 0
+ 2rr sin
. ]0 -- 6 cos ~30 -- cos ~50- ,
+ 7 cos ~0 -- 20 sin
~aTM + 20r
+
64 r
(1.34)
= O,
(1.35)
0
fa TM + 20jŸ + 64fa = -- 96 ~91 -- 12~a" + 1 -- 7r1
+ ~-~ sin 20,
(1.36)
with the conditions
)q (0) = l i (2~r) -----fi' (0) = fi' (2rr) = O,
r
(o) = r ( 2 . ) = ~~' (0) = r
(2,0 = 0,
i=1,
(1.37)
2, 3.
These equations admit of direet solution and we get
~bl = A r 8 [E_~_fl (cos 30 - - c o s 0) -t- 2 (sin 20 - - 2 sin 0)
+~1 (3
sin 0 - - sin 30) log Kr)]
+AVrwz[ 1 (
0
50
7V)
rr
i - ~ 49 cos ~ + 6 cos ~- -- 55 cos - 0
~)
+ ~2-~00 (20 sin ~ + 89 sin ~ -- 41 sin -
4, (oo~~o oo~~)~o~~]
8
P.L.
BHATNAGAR AND OTHER$
V~
+ ~
r ' [(~r -- 0) (3 -- 8 cos 20 q- 5 cos 40)
+ 5 (2 sin 2 0 - - sin 40) log t~r],
(1.38)
where K, ~, t~ are constants yet undetermined.
It is difficult to find the domain of convergenee of the series
r
"Jf- r
-]- r
"-~ "'"
But the formation of the terms show that there exists a eirele of eonvergence
for the series.
DETERMINATION OF THE CONSTANT A
2.
In the case of no suction, i.e., V = 0, the stream funetion is given by
r
2Ar3/2 ( c o s 0 -- cos ~ )
§ A~r 8 [ ~
(cos 30 -- cos 0) § 25(sin 28 -- 2 sin 0)
+ 1(3 sin 0 -- sin 30) log ~r]
(2.1)
So for small values of 0 the velocity u is given by
u = 4Ar 1/2 0 q- 8~ ArX/~ 0a + A2r~, [ -- ~r0 h- 3 (1 q- 5 log r) 0 a
2
14
The velocity profile obtained by Blasius for flow downstream is given by
u = ~n
2.4 ! + "'"
Y
~=x-~
and
(2.3)
where
a = 0-332.
For small values of r and O,
u --- ~r 1/" 0
~2r~0~
48
-I- . . . ,
(2.4)
Boundary Layer on a Flat Plate with Suction
9
For sufficiently small values of r and 0 the leading term of the expansion
describes the flow. Hence comparing the coei¡
of rX/tO in the above
expansions (2.2) and (2.4) we get
4A = ~ = 0" 332.
Therefore
A = 0.083.
(2.5)
Now that A is found we can calculate the vorticity ~ and skin friction ~0.
We get
2V
(~-
0) -
4A
cos
0
+ A'r { (,r -- O)cos 0 + ~ sin 0 + 2 sin 20
+ 3 (sin 0) log ~r}
30) + 4 cos 0
__ AV,
r rs/2 {( ,r -- 0 ( sin ~0 8+9 ~-4 sin ~-
+ i~ cos ~- +71
V2r2 {
27r~ (~r -- 0) (1 -- 2 cos 20) + sin 20
+ 25 (sin 20) log t~r}
(2.6)
The skin friction in the case of continuously varying suction is given by,
bO
ro : t ~ | ( ~ + , ~ ) I t . ~
(2.7)
Hence we get
[
c~
r o -----q xi3a
3.
~~Trx
(
aV xa/Z 13 +
16 + 12-~~
4~ log Ax) ~wx'l
V
2 ~r J"
(2.8)
DESCRIPTION OF THE FLOW DOWNSTREAM
In the downstream region the Navier Stokes equations in the dimensionless forro are represented approximately by the boundary layer equation
10
P. L. BHATNAGAR AND OTHERS
bu
bu
~tu
u T~ + ~' ~ = ~y---~,
(3.1)
along with the equation of eontinuity (1.1).
be satisfied ate
u=0,
v=--vo(x),
at
The boundary conditions to
y=0,
and
(3.2)
U'---~ l,
as
y---r o<3,
where yo (x) is the suction speed.
To solve this equation we assume a stream function
~b = "~ ah (x) yn,
(3.3)
The first two of the boundary conditions (3.2) give
ao' (x) = Yo (x),
al (x) = O.
(3.4)
Substituting $ from (3.3) in (3.1) we get,
as (x) = -- ~ Yo (X) a2 (x),
(3.5)
a4 (x) = ~2 vos (x) a2 (x),
(3.6)
with the general recurrente relation
art-~ (x) =
1
1
n + 3 yo (x) an+~ (x) + (n + 1) (n + 2) (n + 3)
• ~ (n + 3 -- 2r) ra, (x) a ' n ~ - r (x), n > 1, (3.7)
ttt2
From the recurrence relation, we get
a6 (x) = -- ~ yo 3 (x) a2 (x) +
az' (x) a, (x),
(3.8)
1
4
1
ac (x) = ~-0 v~ (x) a~ (x) -- 6-0 Yo (x) aa (x) aa' (x)
1
90 Yo' (x) a~~ (x),
(3.9)
Boundary Layer on a Flat Plate with Suction
--1
a7 (x) = 2 ~
v~ (x) aa (x) + ~
1
B
yo (x) aa (x) aa' (x)
1
(3.10)
+ 12-6v0 (x) v0' (x) a2 z (x),
a~ (x) is left undetermined.
condition, viz.,
u---, 1 as
11
This is as to be expected since one boundary
y.-., co.
has not been used so lar.
u, as determined from (3.3), is
u = 2a~ (x) y + 3as (x) y~ + . . . ,
(3.11)
We determine a~ (x) by comparing the leading terms in the present velocity
profile with yo ( x ) = 0 and the Blasius' solution (2.3).
From (2.3) we get
ay
u = xX/Z
a~ y4
2.4! x z + "" "'
Equating the coei¡
(3.12)
of y from (3.12) and (3.13) we get
a~ (x) = 2 ~ '
where a = 0-332.
(3.13)
With this value of a~. (x), if we put yo (x) = 0 throughout the series, we get
the Blasius solution.
Now that a~ (x) is known we get
a8 (x) =
(3.14)
6xt/Z Yo (x),
el.
a4 (x) = 2 ~
(3. ~5)
vos (x),
O~
a5 (x) =
O~z
120xX/zyo 3 (x)
Og
a, (x) = 7 2 ~
(3.16)
240xZ,
C912
%` (x) + 4 ~
az
v~ (x) -- 360x %' (x),
(3.17)
12
P . L . Bm~,a'NAO~, Ah'D ormms
r191
(z I
a, (x) =" -- ~
v05 (X) -- ~
+ ~
V,' (X)
(3.18)
,,0 (x) ~o' (x),
a8 (x), a, (x), etc., can be similarly calculated. With the use of these values
we get
u=~-i7zy
2xx/~ ~ +
~
Y
y'
g6x)
[,~v? (x)
o,'Vo~ (x)
- - k72---6--~-~ +
240x ~
,..,o~176176176
+4~1
~~o' (x)~ y5
( ,~vd (x)
~~o (x)
+ k ~
+ 80x ~
"~
k ~
~'Vo (x) ~'o' (x)~ ,,
7-~---1"
(3.19)
9
Using Lagrange's reversion formula we get
Y=
Vx u +
a
2
us
ka/
+
u'
+
+48ka])
"'"
Since at the exige of the boundary layer the veloeity is approximately the
free stream velocity we have
u = 1, at y-----8.
(3.21)
Hence
~o (x) ( V x V
1
+ t ~ ~ ( V ) ' + ~(V)'} + ,
(3.22)
Thus ah explicit analytie expressŸ for 91(x) has been derived valid for
any suction velocity dist¡
The authors are not aware of any explieit expression having been given earlier by anyone.
The skin friction % is given by
C~~o1
(o
1"o(x) = t, ~ + ~;c v-o = 1. ~-x - v ; (x)
).
(3.23)
Boundary Layer on a Flat Plate with Suction
13
The solution near the leading edge has been given only for a linear
law of suction, but from the method it is clear that any other power law
can be used. Or in general any suction law can be dealt with by using the
power series expansion for the function. The detailed solution for the
leading edge with any suction is being worked out.*
S ~ Y
In this paper we have discussed the boundary layer on a plate with
suetion. The problem is solved near the leading edge as well as lar downstream. A linear suction law is assumed near the leading edge for simplicity, whereas no restriction is placed on the suction law in the region downstream. An explict expression for boundary layer thickness in terms of
suction speed and distance from leading edge is derived. It is found that
the thickness of the boundary layer depends on the derivative of the suction
speed. The skin friction also has been evaluated. Though near the leading
edge a linear law of suction is assumed, the method used in the paper can
be easily generalised for any other power law, for example, we may use a
power series expansion for the function de¡
the suction velocity.
~FEKENCF~
1. Carrier, G. F. and Lin, C . C . . . .
Quart. AppL Math., 1948, 6, 63.
2. Torda, T.P.
Jour. Math. and Physics, 1952, 31~ 206.
..
* One of the authors (A. C. J.) has found out the solution at the leading edge with constant
suction and bis solution will be publishr in a separate paper along with other results.