NOTE ON THE VELOCITY AND TEMPERATURE
DISTRIBUTIONS ATTAINED WITH SUCTION ON
A FLAT PLATE OF INFINITE EXTENT IN
COMPRESSIBLE FLOWf
By A. D. YOUNG (Department of Aerodynamics,
the College of Aeronautics, Cranfield)
[Received 4 November 1947]
SUMMARY
The problem considered by Griffith and Meredith (1) for incompressibleflowis here
considered for compressible flow of a perfect gas with constant specific heats, it
being assumed that there is no heat transfer by conduction at the plate. Essentially,
the method consists in establishing a correspondence between the velocity and
temperature profiles for incompressible flow and those for compressible flow, the
lateral ordinates being scaled by factors which are functions of the ordinates and
of Mach number.
The results of calculations covering a range of Mach numbers up to 5-0 are shown
in Figs. 1 and 2.
1. Notation
x distance measured parallel to the plate in direction of main stream
upstream of plate
y distance measured normal to the plate from the surface of the
plate
u velocity component in a;-direction
v
velocity component in ^/-direction
p density
T temperature
/x coefficient of viscosity
k thermal condm tivity
cv specific heat ui, constant volume (assumed constant)
c p specific heat at constant pressure (assumed constant)
a H£p/k (Prandtl number, assumed constant)
J mechanical equivalent of heat
y cp/cv (assumed constant)
» Jcp T (enthalpy)
T
ft— (shear stress)
dy
suffix 1 refers to quantities measured at large normal distances from the
plate (y -*• oo)
t This paper is substantially Report No. 8 of the College of Aeronautics.
NOTE ON VELOCITY AND TEMPERATURE DISTRIBUTIONS
71
suffix w refers to quantities measured at the plate
w
defined byf—) = (—)
V
—
1
u1=
a,
*
'" x iis the speed of sound in the main
/(
u\2
\ (a
stream)
«'\(y-l)Jcv,p-i
pTf;
0
»/»!
b
(y-l)M\.
2. Introduction
The solution due to Griffith and Meredith (1) of the velocity distribution
attained with suction on a flat plate of infinite extent in incompressible
flow is of special interest, since it is a solution of the general equations of
motion and does not depend on the usual assumptions of boundary layer
theory. The corresponding problem for compressible flow is by no means
as simple in its most general form. However, if the usual assumptions
of boundary layer theory are made, it permits of an exact solution which
is easily obtained. In the following it is assumed that the gas is perfect
and that the specific heats and Prandtl number are constant.
3. Analysis
The equation of motion in the boundary layer of a flat plate at zero
incidence in steady compressible flow is
8
The equation of continuity is
|(PW) + | (
H
=
O.
(2)
The energy equation is
,. 8T t T dT J 8 LdT\ fi/duY
Jcnu
\-Jcnv— =
Ik— + - [ — .
..,
(3)
We are interested in the problem of the final velocity and temperature
profiles far downstream from the leading edge of the plate when all
quantities are independent of .T.
72
A. D. YOUNG
Hence, the above equations become
du
d
pv = const. = pxvv
di
(5)
d
where i = Jcp T and a = ^2- (Prandtl's number, assumed constant).
The gas equation leads to
£- = % = *!
(7)
Pi T
iI t will be assumed that the variation of /x with T is given by
KV
£-©"-{3"where w = const. For air at normal temperatures w is about 0-76, but it
increases slightly with Tx.
The boundary conditions are
u = Ui, p = P!, v = vlt
i = ilt
u = 0, ^ - = 0 at
dy
-£- = 0 at
y = co,
y = Q,
if no heat transfer by conduction is assumed to occur at the plate.
If in equation (6) we change the independent variable from y to u,
ntJ
writing r{u) = /x—, i = i(u) and eliminate pv by means of equation (4),
dy
we obtain
From (4) and (5),
and hence
j
j-
— = pv = pxvlt
du
T =
where C^ is a constant.
If we write rw for the value of T at the wall, C^ = T U . Further, since
—p^v^v^. Therefore
T = 0, when u = v^,Cx=
NOTE ON VELOCITY AND TEMPERATURE DISTRIBUTIONS
73
Equation (9) can then be written
with the solution
satisfying the conditions i = iv when u = ux, and
di__
— = 0, when u = 0.
<2u
du
At the wall, where u = 0,
^,
(13)
and hence the total energy at the wall differs from that in the main stream
only by the quantity (vft,—vf).
From (10) we have, since T = u—,
dy
Let
"1
and let
—dt, = d-n—
with ^ = 0, when 17 = 0.
Then, from (14),
— = 1—exp(£),
(16)
«i
and from (12),
^-i
= _f!i-/exp(20--exp(a0).
(17)
Writing fi = •/*!, 6 = (y—l)Jff, then
2-<T
where
2
F(l) = -exp(<r£)—exp(2£)
(18)
a
From (16) and (18) we can express u/u^ and (6— l)/$b as functions of £
only, independent of Mach number. To derive the actual velocity and
temperature distributions for any given Mach number we need to evaluate
the relation between £ and i\ (or y) given by (15).
A. U.
From (15),
—1\ =
iUUJNli
(1) it, =
6(7
(19)
2(2-a)
o
o
In general, the integral on the right-hand side of (19) must be evaluated
either numerically or graphically, giving 77 as a function of £ and Mx.
Since vx is negative, only negative values of £ need be considered and it
will be found that values of |£| greater than 10 may be ignored. Having
determined 77 (or y) for a comprehensive range of values of £ and Mx we
can then, for each Mach number, replot u/ux and (0—1)/£6 as functions
of 77, using the basic (or incompressible) profiles given by (16) and (18).f
For the special case « = TO, (19) can be integrated outright to give
rs
«™<9,n
2
i)
4. Calculations and results
The velocity and temperature distributions have been calculated for
w = 0-76 and Mx = 0, 1-0, 2-0, 3-0, 4-0, and 5-0, a being taken as 0-72.
1-.0
^-~
•
•
^——
•k
08
t)/
- u-0761 . _
. J<r=O-72
-- tin 1-0 J
0/6
h
C?.
A
8
C
D
E
F
k V,-
0
10
20
30
40
50
7-yv.
60
70
80
M=1
M=3
M-S
90
100
V,
Fio. 1.
For comparison, calculations have also been made for w = 1-0 and
Mx = TO, 3-0, and 5-0. The resulting velocity distributions as functions
of 77 are shown in Fig. 1, and the corresponding temperature distributions
t This process of establishing a transformation of the lateral ordinate y, -which converts
the temperature and velocity profiles for incompressible flow to those for compressible flow,
was used by Hantsche and Wendt in ref. 2. They then applied it to the boundary layer on
a flat plate in compressible flow without suction for the special case where w = 1-0. However, it seems capable of much wider application, and it is hoped to use it for more general
problems of the boundary layer on a finite flat plate both with and without suction in
compressible flow.
NOTE ON VELOCITY AND TEMPERATURE DISTRIBUTIONS
75
are shown in Fig. 2. I t will be noted that there is a thickening of the
velocity and temperature boundary layer with increase of Mach number,
and this process is enhanced by an increase of <o.
1
10
A
B
0-8
0
E
f
-^^
0-6
1
04
A
u=V0
M=
M=
M=
M=<
M=S
•
^ \
\
02
^
_
0
10
20
S'O
40
50
FIG.
60
70
8-0
90
100
2.
REFERENCES
1. GRIFFITH and MEREDITH, The Possible Improvement in Aircraft Performance due
to the Use of Boundary Layer Suction. R.A.E. Report No. E. 3501 (A.R.C. 2315).
See also Modern Developments in Fluid Dynamics, 2, p. 534 (Clarendon Press).
2. HANTSCHE and WENDT, 'Zum Kompressibilitatseinfluss
bei der laminaren.
Grenzschicht der ebenen Platte': Jahrbuch der Deutachen Luftfahrtforschung, 1
(1940), 517.