TECHNISCHE HOGESCHOOL VLIEGTUIGBOUWKUNDÉ'
TECHNISCHE U^ ^ELFT
umm: ^^ \2 Juli 1950
Kluyverweg 1 - 2629 HS ^Sm^ No.3
August, 1947 T H E C O L L E G E O F A E H O W A U T I C S C R A N F I E L DNote on the velocity and temperature distributions attained with suction on a flat plate of infinite extent
in ^compressible flow
-by-A.D. Young, M.A.,
of the Department of Aerodynamics
-SUMMRY-The problem considered by Griffith and Meredithl for incompressible flow is here
considered for compressible flow, it boing assumed that there is. no heat transfer by
conductior. at the plate. Essentially, the method consists of estabiishing 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 fi;üactions of the ordinates and of Mach number.
The results of calculations covoring a range of Mach niAmboPs up to 5.0 aro shown in Figs. 1 and 2.
-2-1. Notation X
u
VT
k
c VJ
r
r
distance measured parallel to the plate in direction of main stream upstream of plate
distance measured normal to the plate from the surface of the plate
velocity component in x direction velocity component in y dj.-rection density
temperature
coefficient of visocity thermal conductivity
specific heat at constant volume specific heat at constant pressure
M,c.-/k (Prandtl number, assumeji constant) mechanical equivalent of heat
Cp/c^ (assumed constant) J Op T^. /^ -— (shear gtress) dy
i
s x j f f i xsuffix
r
M
&1 refers to quantities measured at lar-ge normal distances from the plate (y--^ ex- ), w refers to quantities measured at the plate d e f i n e d Isy y V 1 - ^
ar,
Un ( T - 1) J «p T,(I-1)
v
(a^L is the spee$i of sound in the main stream)
-3-2.
IntroductionThe classic solution due to Griffith and Meredith-^ of the velocity distribution attained v/ith
suction on a flat plate of infinite extent in incompiessible flow is of special interest, since it is a solution of the genoral equations of motion and does not depend on the usual assumptions of boundary layer theory. The
corresponding problem for compressible flow is )f)y no
means as simple in its most general form. ïiowey>j^, if the usual assumptions of bo^mdary layer thJË^ory afe made, it permits of an exact solution which is easily c^btained, This solution may have no practical importano'e at the moment, but it v/as felt to have sufficient intrinsic interest to be worth recording.
3. Analysis
The equation of motion in the boimdary layer of a flat plate at zero incidence in steady compressible flow is ^ u , U •- -4- V
ex
'ÓU^ y
^ •>) y (/^ ^ y^(1)
The equation of b o n t i n u i t y i s
ex ^ d y \
= 0 ( 2 )The energy equation i s
>> T ^ T
J Op u £_t, ^ J Cp V
1) X "^y
l . f k l£\ju
( 3 )We are interested in the problem of the final velocity and temperatijire profiles far downstream from the plate leadiiig edge when
A s 0.
"Sx
>4-Hence, the above equations become
Ov d u d / ,, du / ^ ' ^ ( 4 )
"" d y ay i^' dy
p V = c o n s t . = P 1 "^1 (5) ^ dy dy V f^ <iy/ M ^ y / • • • • • • v ^ O y where i = J Cp ï , a n d o ' = A^ Qp ( P r a n d t j L ' s number k assumed constant),The gas equation leads to
-^- - ïi - _ii. •••• (7)
^ 1 T i
It will be assumed thfiit the variation of ^ with T is given by
ü
J^^JjJ\
(8)
whereoo= c o n s t . For a i r a t normal t e m p e r a t u r e s LO i s al^out 0 , 7 6 , b u t i t i n c r e a s e s s l i g h t l y w i t h T .
The boundary c o n d i t i o n s a r e
u = u , p = p , V = V •, i = i , du = 0 a t y = Oo 1 ^ \ 1 1 1 dy
VI, = 0, ^ = 0 at y = 0, if no heat transfer by dy conduction is assumed to occur at
the plate.
If in eqiiation (6) we change the independent variat)le from y to u. writing'f (u) = U, ^u ^ j_ _ ^^^^^ ^,^^
I dy
eliminate pv by means of equation (4), we obtain
(i_.r) a < 3^+.T/i2'C+c)=o (9)
du du Vdu'" J
5 -From (4) and (5) d T ' ^ D T :/ rr. du
^
'
- ri ^1
f ]
and hencer = ^^ r^ u-w 0^
where C i s a c o n s t . I f v/e v ; r i t e Y = v a l u e of'T" a t t h e w a l l , w 1 ^^ F u r t h e r , , s i n c e i = O, when u = u , G - - p -^r u ^ 1 ^ 1 1 1 T h e r e f o r e , ~ ^ 3 V ^ ( u ^ - u ) ^v i l w
(10) E q u a t i o n (9) can t h e n be w r i t t e n d i du d 2 i(1 - c r ) ^ V f. - o V (u _ u) ^ - i - o ' = 0 ..(11)
T h i s e q u a t i o n i s r e a d i l y s o l v e d t o g i v e ^ 1 - ^ = - (T ^ 2 2 (2 - O " ) 1 - .vU Un / NOT ^ 2 h „ }Lcr
Un . ( 1 2 )satisfying the conditions i = 1, , when u --^ u, , and
di _ 0, v;hen u = 0. ou I t i s of i n t e r e s t t o n o t e t h a t a t t h e w a l l where u = 0 , 2
i^^.
i t h
1 2 w (13) / c i X l Q . a • n e ,* • •-6-and hence the total energy at the wail differs from
that in the main stream only by the quantity
/ 2 2 ' f V - V I w 1 du From (10) we h a v e , s i n c e ' T ' = M . — , / dv dy u Ü - = 1 - exp
1 '
»y p V, I Ê1 Let r? - - "^1 -^ P (14) and l e tI
1T
with J a 0, when 75= o.
Then, from (14)
u
u.
= 1 - exp
.(f)
1
and from (12)
(15)
(16)
i . i =
x r : i _
^ 2(2
-cr)
exp.
{2S).- ^ exv,((rS)
.(17)
Writing © = i/i , b = (-y - 1) ] \ ,
9-1 (T
then
b ^
(2 - (T)
F iS)
where F ( ^ ) =
''•t > • • • • > : oDxp.^rï) - exp. (2^)
(18)
(O 1F-T-om (16) and (18) we-.can express — & — as
"1 b/2
functions of ^ only, independent of Mach number,
To derive the actual velocity and temperature
distributions for any given Mac^ number we need to
evaluate the relation between ^ a n d t) (or y) given
by (15). \
/From.-7-From (15)
= r5
i t b r
2(2 -.r)
dr
(19)
In general, the integral on the right hand side of (19) must be evaluated either numerically or graplxLcally, giving 'j! a,s a function of S and M-, . Since Vi
is neg.ative, only negative values of S need bei^|
considered an.d it will be found that values of/j j greater than 10 may be ignored. Having determined Tp (or y) for a comprehensive range of values of ^ and M
we can then, for each Mach number, replot
u , Q- 1
—- and
"--\ b/2
as functions of 7/ , usin^ the
basic (or incompressible) profiles given by (16) and (18)'
For the special case CO = 1,0, (19) can be integrated outz'ight to give
-^1 =
^
f
2 (2 -cr-) 2 exp(crƒ)
-
exp, (2^1
2 -t 1"
5 ^ ^
2
2 (20).4. CalciiLati.^ns and results
The velocity and temperature disti'ibutions have been calculated for6a= 0,76 and M-^ = 0, 1.0, 2.0, 3.0, 4,0 and 5,0, CT' being taken as 0.72. For comparison, calculations have also been made for CA.Ï = 1.0 and Mj^ = 1.0, 3.0 and 5.0. The resulting velocity distributions
3 /as,,,.
K Tiiis process of establishing a transformation of the lateral ordinate y , which converts the temperature and v;el«city profiles for incompressible flow to those for compressible flow, was first used by Hantsche a.nd V/endt in Ref.2, They then applied it to the boundary layer on a flat plate in compressible flov/ without
suction for the special case where CO = 1.0. However, it seems capable of much v\?idür application, and in a
'• latür paper it is hopeu to use it for more general
problenTS of the boundary layer on a finite flat plate both v/ith and without suction in compressible flow,
-8-as functions of TJ are shown in Pig.l, ai-.d the corresponding temperature distributions are sho\;a in P i g . 2 , It will be noted that there is a
thickening of the velocity and temperatui'e ^o^onda^-j layer with increase of Mach number, and thJ.s process is enlianced by an increase of 'oO .
o O o —
REFEREIN CES
-No. A u t h o r T i t l e
Griffith and Meredith The possible improvement in
air ci'af t p e r f o rman c e due to the use of boundary layer
suction, .iy H.A.E.. Report iTo ^ E..35OI. (A.R.C..25]-?)' See .also ' M o d e m
Developments in Plviid Dynamics' Vol„II. p.534
(Claren don Pre s s ) .
Hantsche and VJendt ^..-um liompressj.b-litatseinflus" bei der laminaren Grenzschicht der ebenen Plantte» Jahrbuch der Deutschen L u f t f alir t f o r 3 oh"an,Q; 1940, P-517. oOo
COLLEGE REPORT No. 8. I;0 0-8 01-6
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