. * CP No 89 (14.215)
A R C Technml Report
AERB LfBRARy copy.
MINISTRY OF SUPPLY
AERONAUTICAL RESEARCH COUNCIL
CURRENT ‘PAPERS
Notes and Graphs for Boundary Layer
Calculations in Compressible Flow
W. F Cope, MA, A.F RAe.S.,
of the Aerodynamics Dlvlsmn, N.P L.
LONDON HER MAJESTY’S STATIONERY OFFICE
1952
PRICE 3s 6d NET
C.P. No. 89.
Notes and Graphsfor Boundory Layer Calculations m Cmpqessible Flow
I- By - W. F. Cope, MA, A.B.R.Ae.S., of the
~croQmmios Division, N.P.L.
24th Awust, 1951
Formulae WC gxvcn for calculating the skin friction oocfflcionts and the various pnrmeters nssociatcd mlth the bowdnry layor up to EIach nmbors of 4, The calculations have been made for a lminar boundnry layer nxth n sinusoidal veloolty dmtribution; for o turbulent boundary layer with a "power 1s~" velocity distribution, the distribution following "one fifth", “one seventh", "one ninth" rind "one elcvonth" power Inns; and for a turbulent boundary layer with a "log law" velocity distribution.
Introduction
In 7634' some calculations m-c node on the effect of cmprcsslbility on n turbulent boundary layer. At the tine (1942-43) this work was dmnc no ixnsuremnts of the velocity dlstrlbutisn oxistcd and the cmdysis hnd to be boscd on the assmption that it was not opprccinbly offccted by the rminstrccm being supersonic. This nsslnption cwld be rcndorcd plausible, but It was n pure spcculatim and nocordingly the onnlysrs was only cnrrled for enough to ostabllsh trends.
Since that ttiJo onalyticnl rcnsons supportmg this assmption have been grven2, end DCDS at the N.P.L.3, R.A.E.4 Y
mcnts hnve been xdo rind in Ammzcn covering botmeen thm
Xaoh nmbors up to 2s and. Reynolds mmbcrs up to 20 ililllon. ,Thcsc nensurcmnts hcve cstnbl1shed the genercll truth of tho asswJptlons rnnde in 7634 rind accordingly it sccw north while to collplcte the calculations pnd collate the infonlntion in one docuncnt in the hope it nny be of use to designers.
The general method
(1) It is ossmed that nrltten:-
of oolculntion is as follows:-
the velocity distribution con be
f(n) nxth TJ s y/6 .
(2) A quantity a is defined by:-
(1 - a) = 1 06 Mi" a
06 a ( 1. (3)/
Published by permission of the Director, National Physical Laboratory.
(3) It is assumed that throughout the'layer:-
ua + 2 Cp T 'I Constant.
In particular
= (1 - a) -I T2
and .
(since p is ocnstant through the lpiyer). as usual, fhe stream and wall values.
Vh2re I and R denote,
(4) It is assured that:-
m TW
0 is a fraction whose value ranges from $ at very high temperatures to e/s in wind tunnelnork with atmospherio stagnation temperature. It seems unlikely that aqy error cf practical importance to the ai~mf't designer will be introduced by putting:- w = 2.
@oLiLation of DispL3#xwmt and LIomntw lbickneSses
The stanlard definitions me:-
. ..(I)
*That is that the Prandtl number is unity. It is also assumed that there is no heat transfer between tho layer and the wall.
-3 -
On the e.ssmptmns ju2t given these becoma:-
H = f/4
The analytical and numerical work is subdivided ut&r the heads: -
Lnminar Boundary Layer
Turbulent Boundary Layer, Power La\?
Turbulent Boundaq Layer, Log Law.
Laminar pimdn2-y Layer
It is assumed th&
n: f = sin
0 -11 . 2
The values SE? S'/S, a/s and H are plotted a.a fvnot5c.n~ of
EII wer the mnge 0 6 M 6 5 in figures 2, 3 and 4 rcspeotivcly.
Turbulent Boundary L&ycr. Pswer Law
It is assurredthat:-
f I .,ll/m
where m is an integer and often m = 7.
A Accurate solutions exid6
< Ths!q , but arc lnborious to obtain. The abwe
asslrmptiom is suffioicnkly accurate for the purpose in vim and has the double ndvcmtago o* being annlyticn~)ly simple and eosdy computabl..e.
. ..(2)
-4-
Thentho'cqmtions (2) yield:-
Fix/6 = I - z; (1 -CL) [, + Y-2 a + , . , + “‘,-” an + . . . m+3 mt2n+l 3
1 a" = -^- + 2m Z ----------------
lW1 n=l (mt2n-l)(nt2ntl)
m c
-(2utl+n+l) a" ----------
= (D+i)(mt2) ' - 2 z --------_----_------__________ ,
n=l (iz+2n-~)(nt2n)(mt2ntl)(r.w2ntZ) 3
Those series becom mm and more slmily coma-gent as a increases; this difficulty can be overoo~~ when it 1s soen that equations (2) can also be mitten:-
bX/6 = 1 - (1 - a) m J
1 P -------- a-f
0 1 -UP
9/s = L.a? - !L:-a!: (1 a a s 0 1 -aP
both of nhich cm bo integrated by +mentazy mthods in tho fo& of a polynonrLal plus n log tom. This polynor.~d, however, nill contain sevokl. terms ard so the mthcd is only of Ipractical u3c when tho series expansion converges slowly.
!ib? VS&EG Of 6'16, D/6 and H for n = 5, 7, 9 and 11 are plotted as functions of 11 over the range 0 < 114 4 in figurei 5, 6 and 7 respectlvoly.
Turbulent Boundary Lwer. Log Law
It is assuicd that:-
f = 1 t A In-?-
-5-
The integrals in equations (2) can be expanded in pomrs~of A nith coefficients functions of a to give:- I , _.J
ita 6*/t? 3 -..- A f
2434 6a~1+6wp12) 2@a(5t10ata2) 1 .a ----s-m A + w-m-------- Aa
I-2 (I-2,)(1-a) -,~;:aji;:a~;-- A3 t . . .
Z.(lta) m--w-- A + _6:!?::!, Aa '- .":!%"! A3 + ': 1.) 1-a (l-a)” (l-al3
are quite useless for %oh numbers much &eater than
l-u c
,
w =A f- c
but these series unity.
NOV oquations (2) can also be aritton:-
and thus dopend on:-
1 1
f 7% i
ch7 I we.,--z----m ,
a olda 4
J xt1nv"- '
that is to sq on the exponential-integrd f~tion.
Tn‘fdot:-
l-a f/6 =' 1 + ---
2cCA [ lq;Ei (eg)s+.exp (-,;iT.> Ei (.I$!)]
l-a O/6 = --- I t
a L
-I,.
14 14 --- Ei ( > - ----
7 ( VG
.* .
I+& Ida -we- &i a”’ x II * The/
The values of (14/alk& and (I+.&)& nre given below in the table and as figure 0. '. "
M ,a
0 0 0.5 0.0476 1.0 0.1667 I.5 0.3103
i:; 3.0 ~:~
;t:Z 0:7101 0.7619 4.5 0.8020 5.0 0.8333
A typical value for A
So (I+&)/& varies
an?. (Ii&)/&i varies
2.500
EC; . ;?Ija-/ 2.146
0.0954 2.1;7 2.095
is 0.1.
from1 to IO 1 omitting tho law
&on 21 up?ards J Uach nuder region.
The exponentid-integral funotion is tab&&ad up to values of 10 for the ind.ependent variable nrd has the aQynptotic expansion
exp (4
c
12 Ei (x)~
‘6 24 ------- I + - + - + -- + -- + . . .
X OX 2 2 x4 3
;k$;;e~drapidly oonvergent for x > 10 if only 3- or44igUra accuracy .
Thus ne oannrite
The values of fY 6 for Reynolds ntiers of ? $,hf~610yk3f zti3?3zyot:2 :," 4 figures 9, 10 a@ II.
Skin Friotion Gpeffioients
Beoauee of the tenpratwa difference betixxn the nallandthe mainstream, the several ooeffioients require careful definition. The standard defitition is in terms, of mainstream values as follows:-
but/
-7-
but it is somatincs convenient to define thorn in terms of %all vjlu~~, thus:-
Once a velocity distribution has been fimA, thon the skin friction ooeffioients can be doter-mined from the relationships:-
2ae/ax = of A"$? : cg
28/x = cp 2; i Cf.
the disposable constants being adjusted SO that the formulae roduoe to the incompressible form on putting a = 0. Figures 12, 13 ana 14 sh01-1 curves of Cf and
2 caldated in this my. In this case the
constants have been a just& so that the formulae reduce for a = 0 to the von l-C.&c&-KempfSchoenhehen. type.
The saro curves enable the thickness of thd leyor to bc calculated since the relationship just given shows that figures 13 and 14 arc a plot of 28/x against R. Thus 9 can be obtainednhon x is known and, since H is also kno~m, 6 ma sx also by using tho appropriate ourve.
It seams likely that oxprossions in terms of free streamvalues are most likely to be of use to designers since tho free stroem values of the wind tunnel correspond to tho atmospheric conditions of real life. But it is north pointing out that Cfh and C, attain their incompressible values for a Reynolds number equal to
and that therefore the values of CY, and CmJ oan be read off the LI small curve of figures 12 to 14 by interpreting R in this sense,
No. Author(s)
1 AT. F. Cope
2
3
4
A. D. Young
-a-
BIBLIWWfiY .
Title, etc.
The Turbulent Boundary Layer in Ccanpressible Flow. pork Perfomed for Ordnance Board. Canmtiated by Superintendent, Engineering Dap+xmnt, N.P.L. (To be publish&)
A.R.C. 7634.
The Equations of Notion and l?mergy and the Velocity Profile of a Turbulent Boundary Layer in a Impressible Fluid. College of Aeronnutics Report No. 42. (January, 1951). A.R.C. 13,921.
J. A. Cole and Boundary Layer Measurements at a Xach iv. F. Cbpe Nmber of 2$ in the Presence of
Pressure Gradients. N.P.L. Engineering Divismn Report Eng. Div. No. QO/l+Y. by, 191c9). LFLC. 12,618.
R. J. Monaghan and The Ueasumnent of Heat Tronsfor and Skin J. E. Johnson Friction at Supersonic Speeds. Pert II.
Boundary Layer Measurements on a Flat Plate at M = 2.5 and Zero Heat Transfer. (December, 1949). Current Paper No. 64.
Robert E. Wilson Turbulent Boundary Layer Charectcristics at Supersonic Speeds. Journal Ae. SC. 1950, 17, 585-594.
W. F. Cope and The Lsninnr Boundary Layer in Canprcssible D. R. Hmtreo Flow. Phil. Trans. R.S.(A) 19&3, 241, 1.
AA.
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0 40 Turbulent Boundory Layer Log low Velocky Distribution.
f
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Turbulent; Boundary Loyer Log Low Veloc;Ly Dkkrfbution
0. IO
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SKIN FRICTION OF FLAT PLATES, COMPRESSIBLE FLOW
FLAT PLATE (LAMINAR)-BLASIUS FLAT PLATE&JRBULEN$ kh POWER LAW. CjJR = KM) Cf XT = Gu)
CM WI A f w SMALL I.320 SMALL 0.073
; 1.308 1265 1.5 I .060 066
316 1209 20 .054 25 ‘048 3.0 042
SKIN FRICTION OF FLAT PLATES, COMPRESSIBLE FLOW
FLAT PLATE (LAMINAR), BLASIUS.
CrJff - f(o((> CM f(d)
SMALL I*328
FLAT PLATE (TURBULENT), LOG LAW. -l F
I I.308 I.269 r203
