2 the status of research in turbulent...
TRANSCRIPT
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2
T H E S T A T U S OF R E S E A R C H I N T U R B U L E N T B O U N D A R Y L A Y E R S
W I T H F L U I D I N J E C T I O N
L. O. F. JEROMIN
Engineering Department, University of Cambridge*
Summary. The purpose of this paper is to discuss the state of research in turbulent boundary layers with fluid injection and to present a starting point for further research in this topic. All available theoretical as well as experimental investigations have been considered. The main assumptions of the theories and the similarities of the various theories to each other are discussed and experimental data are compared with the predictions of the theories.
L I S T OF S Y M B O L S
A Eq. (2.32) constant A Eq. (3.18) constant , A Eq. (2.41) integration constant B Eq. (2.12.1) injection paramete r B Eq. (2.41) integration constant B Eq. (3.18) integration constant B, Section 2.1.2 blowing rate parameter b Eq. (2.36) Rannie 's constant C Eq. (2.42) constant in law o f the wall c / = ~w/ O ~ u ~ skin friction coefficient
X l/ cF = - - c F d x total skin friction coefficient
X 0
c h = q. , /Ooouo.cpoo(T,--Tw)
c h = qw/oo.uooCroo(T,~--To.)
cF
D
t Now at Messer Griesheim GmbH,
mass fraction of injected gas Stanton number (compressible flow)
Stanton number (incompressible flow)
specific heat at constant pressure diffusion coefficient
Frankfurt am Main, C~rmany.
65
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66
D d Eq. (2.44) E Eq. (3.23) F = ~ . ,v , , /~ou~
F Eq. (3.23) F c Eq. (2.15) FRx Eq. (2.15) FRo Eq. (2.15) H = h + u2/2
H = 6"/0
h
M = u /a
n t Eq. (2.44.2)
P Pt Eq. (2.44.2) P r = #ce/~t
P r r = eMCe/eR
q = 2 ( a T / ~ y )
Q Eq. (3.22) R = u..~**/la**
R Eq. (3.1) R D = u**e**D/#. .
Rs = u s Q y J # ~
R x = u**~**x/#**
R o = uo.Qo.O/#..
r
S c = # / o D
S c r = eu /~eD
T
U
u, = /Lle. u, Eq. (2.51) V Eq. (3.20) V
W Eq. (3.21) X
Y Yi Eq. (2.44.2)
L. O. F. JEROM1N
diameter integration constant integration constant injection parameter integration constant similarity parameter similarity parameter similarity parameter stagnation enthalpy boundary layer shape parameter static enthalpy Mach number parameter of bilogarithmic law static pressure parameter of bilogarithmi¢ law Prandfl number turbulent Prandtl number heat flux total heat flux Reynolds number per unit length effectiveness parameter Reynolds number based on diameter Coles's substructure Reynolds number Reynolds number based on x Reynolds number based on 0 Reynolds number based on t~ s recovery factor Sehmidt number turbulent Schmidt number Reynolds analogy factor absolute temperature velocity in x-direction friction velocity McQuaid's universal velocity temperature distribution parameter velocity in y-direction temperature distribution parameter coordinate along the wall coordinate normal to the wall parameter of bilogarithmie law
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~ C H IN TURBULENT BOUNDARY LAYERS 67
~l Eq. (3.16) 7' Eq. (2.55) 7" = c~,/c v .F A* Eq. (2.35) A t Eq. (2.61)
0
6" = S (1 - Ou/o**u**) dy 0
6o
6,
= 8 o -
= - Ocpv'T' / ..~T 6 H I ~y
Eq. (2.27) Q
o= o- 1 u 0
Eq. (2.42) 2
Eq. (2.45) '~i Eq. (3.23) P 1'
Eq. (2.28) 0 a Eq. (2.26)
¢ Eq. (2.53)
dy
Greek symbols
constant intermitteney factor ratio of specific heats molecular weight effective displacement thickness pressure parameter boundary layer thickness
displacement thickness
value of "boundary layer thickness"
at which uo.-__u_ 1 U~
value of "boundary layer thickness" for twice the distance from the wall to the position at which 7' = 0.5
eddy diffusion coefficient
eddy conductivity of heat
eddy viscosity
transformation parameter
momentum thickness
mixing length constant thermal conductivity parameter of bilogarithmic law constants viscosity kinematic viscosity transformation parameter density transformation parameter shear stress function representing wall effects
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68
~p
02i
Eq. (3.22) Eq. (2.25) Eq. (2.53)
Eq. (3.18)
L. O. F. JEROMIN
temperature distribution parameter stream function function representing reduction of wall effects constants
,4 B F L P g
8
8
T w
0 0 o o
Subscripts
air (mean stream fluid) starting points of integration foreign gas (injected fluid) laminar flow potential flow recovery condition at the edge of the sublayer substructure turbulent flow wall condition zero injection stagnation condition free stream condition
J
Superscripts
transformed flow (incompressible) reference condition fluctuating quantity mean value
1. I N T R O D U C T I O N
The possible applications of the compressible turbulent boundary layer with fluid injection cover a wide engineering field; in addition fluid injection has considerable theoretical interest. Transpiration cooling, the fluid injection of a secondary gas through a porous surface into the main stream, results in a deformation of the boundary layer such that skin
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RESEARCH IN TURBULENT BOUNDARY LAYERS 6 9
friction and heat transfer are reduced, t Moreover, the high-speed re-entry of rockets and satellites into the atmosphere requires the development of techniques for protecting their payloads against the intense heat produced, There are three promising ways in which this cooling may take place:
(i) Ablation of heat shield materials ("ablation cooling", see ref. 2). (ii) Injection of secondary fluid through one or more slots into the
main stream ("film cooling", see refs. 3 to 7). (iii) InJection of secondary gas through a porous surface into the main
stream ("transpiration cooling", the subject of the present review).
Transpiration cooling is very similar to the first two processes and a detailed knowledge of the mechanism should therefore assist the under- standing of both ablation and film cooling. In the former case, the injection air is replaced by the gaseous decomposition products of the ablating materials (such as glass-reinforced Teflon) and in the latter case by the vapour from the liquid (for example, liquid helium) which is on the surface. Transpiration cooling is a suitable method of cooling any surface exposed to a high-temperature gas stream and typical processes occur in electric arcs (9), gas turbine blades (1°' 11) combustion chamber walls, hyper- sonic ram-jet intakes, and in rocket motor nozzles (12-14). In all that follows the consideration is restricted to fluid injection.
In the last few years many scientists have turned their attention to the transpired turbulent boundary layer, and a large numer of theoretical and experimental analyses have been published. Eckert and Livingood/a) Sellers (~) and Mager and Divoky (~8) have compared transpiration and film cooling with conventional cooling methods like convection cooling. Their investigations showed that cooling methods which influence the temperature and velocity profiles directly, like transpiration cooling, are much more effective than conventional wall cooling. Sometimes it is only necessary to protect certain areas exposed to a supersonic, or very hot air stream. One example is the stagnation point of rockets and satellites (re-
t Injection of a cool gas into the boundary layer not only cools the surface but reduces the skin friction coci~cient. This has led some investigators to conclude that blowing may be preferable to the normal suction method which will keep the boundary layer laminar for a longer period. Experimental evidence indicates that the turbulent skin friction coefficient can be reduced to a value which is smaller than the corresponding laminar one; but this approach must nevertheless be regarded with caution. In the presence of injection or suction, the drag of a turbulent boundary layer is higher than that caused by skin friction alone (Edwards(l)). Fluid injection therefore causes two opposing effects, a decrca~ in the heat transfer and skin friction ~tticient and an in- crease in the effective drag. The optimum conditions for any particular problem must form the subject for further investigations.
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7 L. O. F. JEROMIN
entering the atmosphere) where it is necessary to reduce the wall temperature considerably in those regions where the velocity decreases to zero. This can be effected by constructing the region around the stagnation point from a porous material and forcing air from inside into the main stream. This stagnation point injection would mainly result in a laminar and tran- sitional boundary layer problem and only to a minor extent (in the region further downstream) in a turbulent one. This region further downstream is not necessarily porous so that the turbulent boundary layer developing along this impermeable surface would be a very special case, namely a turbulent boundary layer along a solid wall with a "transpired history" in the laminar and transitional region. A brief survey of this topic is given by Eckert eta/ . (4) The downstream effects of transpiration cooling are the subject of papers by Bernicker (17) and Woodruff and Lorenz O°s).
Craven °s), Squire (tg) and Spalding et al. (2°) presented reviews of analyti- cal and experimental analyses of the turbulent boundary layer with air and foreign gas injection. Craven simply offers, without comment, a review of all the available theories for determining the skin friction coefficient for boundary layers with suction and injection, whereas Squire presents some calculations to show the advantages of transpiration cooling for aircraft and rockets in the Mach number range of 2.5 ~< M ~< 5. Spalding et al. (2°~ give a brief systematical survey of the theoretical approaches which predict the skin friction coefficient and the Stanton number variation of the com- pressible turbulent boundary layer with air and foreign gas injection. But all the information summarized in refs. 18 to 20 must be regarded as rather incomplete since recent investigations are not considered. All the available analytical and experimental information concerning turbulent boundary layers with fluid injection will be summarized in the next sections. The main aim is to establish a survey as up-to-date, and as systematic, as possible, ~o that further investigations in this topic can begin on the basis of this summary.
2. T U R B U L E N T B . O U N D A R Y L A Y E R S W I T H A I R I N J E C T I O N
2.1. Compressible flow
2.1.1 Experimental investigations
The compressible turbulent boundary layer with fluid injection was the subject of quite, ~ ~number of experimental investigations in the past few years. The mCst ~interesting facts about these experimental studies, like 'geometry of ~he:experimental set-up, injected gas, injection parameter
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RESEARCH IN TURBULENT BOUNDARY LAYERS 71
F = O,vw/O~uoo, temperature, Mach number, Reynolds number and meas- ured quantities are summarized in Table 1. For the sake of completeness Table 1 includes some information about investigations of minor interest in the present context, like preliminary investigations, and those investigat- ing the influence of injection on heat transfer in supersonic nozzles and rocket motors.
The boundary layer along porous cones or hemispheres were the subject of most of the experimental investigations. The reason for this choice is the possible application of fluid injection in the region of the stagnation point of rockets and satellites, and the relatively easy and fairly accurate determination of overall skin friction coefficients and Stanton numbers from momentum and energy balances for the whole cone. Such a relatively simple arrangement has the disadvantage that valuable information about the local behaviour of the boundary layer cannot be obtained. How- ever, only detailed information about local skin friction and heat transfer coefficients, boundary layer profiles and recovery factors can be con- sidered as reasonable checks for the reliability of the predictions of theories. Moreover, most of the theories are derived for two-dimensional flows along flat plates so that this experimental set-up is more important than the axisymmetrical flow along cones.
Only a few measurements by Rubesin ~21), Danberg ~22~ and Jeromin has meas- ured local skin friction coefficients, local Stanton numbers and the influ- ence of injection on the recovery factor for the Mach numbers 0, t 2"0 and 2"7. At a much higher Mach number of 6.2 an experimental investigation was undertaken by Danberg ~2~ covering a large range of heat transfer rates at the wall. He evaluated local boundary layer characteristics like skin friction and heat transfer coefficients, momentum and displacement thickness, boundary layer shape parameter H and the Reynolds analogy factor from measured velocity and temperature profiles. The skin friction and heat transfer coefficients were determined from the slope at the wall of the boundary layer profiles so that c I and c h show scatter in the order of -/-20 per cent. Jeromin's ~23~ investigation at Mach numbers of 2.5 and 3"5 concentrates more on the ease of zero and low heat transfer rates at the wall. From measured boundary layer profiles the usual boundary layer parameters c/, 0, 6, 6" and H have been determined. The skin friction coefficient was evaluated here from the law of the wall for tran- spired boundary layers in connection with a boundary layer transformation
t A M a t h number of 0 refers to incompressible flow.
-
TA
BL
E
1.
-,.}
EX
PER
IME
NT
AL
ST
UD
IES
OF
T
HE
C
OM
PRE
SSIB
LE
T
UR
BU
LE
NT
B
OU
ND
AR
Y
LA
YE
R
WIT
H
FL
UID
IN
JEC
TIO
N
to
Inve
stig
ator
(s)
Sul
liva
n,
Cha
uvin
and
R
umse
y(a~
) 19
53
Rub
esin
, Pappasand
OkunotU)
1955
Cha
uvin
and
C
arte
r(SS
) 19
55
Rub
esin
uo
19
56
Lea
don an
d So
ott(
~, 2
5)
1956
Geo
met
ry
8 ° c
one
flat
pl
ate
8 °
con
e
fiat
pl
ate
co
ne
flat
pl
ate
Inje
cted
ga
s
air
air
nitr
ogen
he
lium
air
air
air
heli
um
F.
10 a
0 to
2 0 to 3
0 to
14
0 to
0.5
0
to
0.5
0 to 3 0 to
3
0to
4
0to
15
Tem
p.
[°K
]
480
T~
535
300
~<
To~
34
5
477
~ T,
~ 53
3 T
,,/T
~
=
1.5
300 T
,~
345
300 3
45
-{
= 28
0
Mea
sure
d q
uan
tity
redu
ctio
n of
wal
l te
mpe
ratu
re d
ue t
o in
ject
ion
Sta
nto
n n
umbe
r,
skin
fri
ctio
n co
effi
cien
t, r
ecov
ery
fact
or
Sta
nto
n n
umbe
r,
reco
very
fa
ctor
Sta
nto
n n
umbe
r,
loca
l sk
in
fric
tion
co
effi
cien
t,
reco
very
fac
tor,
av
erag
e sk
in
fric
tion
coe
ffic
ient
, S
tan
ton
nu
mb
er
Sta
nto
n n
um
ber
, re
cove
ry f
acto
r
Mac
h
Rey
nold
s N
o.
nu
mb
er
R =
8 )<
106
2.
05
[per
met
re]
1"5
~
2-7
R~)
< 10
-6
~7
R=
2"
05
8 X
106
[p
er m
etre
]
0 2-0
2.7 ?o
2"7
R=
3
4)<
10 6
[p
er m
etre
]
Rem
ark
s
prem
ilin
ary
inve
stig
atio
n,
not
too
accu
rate
. E g
ood
agre
e-
men
t w
ith
Ru
bes
in's
th
eory
prel
imin
ary
inve
stig
atio
n,
not
too
ac-
cura
te
r-'
©
©
z
-
Ten
dela
nd a
nd
Oku
no(e
O~
1956
Sta
ider
an
d
Inou
yo (a
e)
1956
Lea
don,
Sco
tt a
nd
An
der
son
(a¢)
195
7
Ras
his (
as)
1957
War
d a
nd
H
arm
on
(m 1
959
Sco
tt,
An
der
son
an
d E
lgin
(~°)
19
59
10 °
cone
hem
i=
sphe
re
20 °
cone
40
°
doub
le
wed
ge
cone
fiat
pl
ate,
co
ne
air
air
air
heli
um
nitr
ogen
he
lium
Tef
lon
air
heli
um
0 to
2.6
7.5
0 to
32
0 to
20
297
Too
. 34
5
Too
. =
975
T0*
* =
1895
T./T
O*
= 3.
3
aver
age
skin
fr
icti
on
coef
fici
ent
Sta
nto
n n
um
ber
Sta
nto
n n
um
ber
, re
cove
ry
fact
or
heat
tra
nsfe
r,
wal
l te
mpe
ratu
re
redu
ctio
n
loca
l sk
in
fric
tion
co
effi
cien
t
heat
tra
nsfe
r,
Sta
nto
n n
umbe
r,
velo
city
an
d
con
cen
trat
ion
pr
ofil
es,
reco
very
fa
ctor
2-7
I
4-8
2×
10
~
~R
~
2.8
×1
0 ~
[p
er m
etre
]
2"7
× l0
s R
~
6-5
× 10
5
e
4X
lO e
[p
er m
etre
]
good
ag
ree.
m
cnt
wit
h R
ub
esin
's
theo
ry
stag
nati
on
poin
t in
jec-
ti
on,
D
= di
amet
er o
f he
mis
pher
e
lam
inar
flo
w
and
tra
nsi-
ti
on r
egio
n st
udie
d
lam
inar
, tr
ansi
tion
al
and
tu
rbu
len
t fl
ow,
blo
w-
off
stud
ies
m
o z o~
-
Tab
le I
con
t.
Tem
p.
Mea
sure
d
Mac
h
Rey
no
lds
Inv
esti
gat
or(
s)
Geo
met
ry
Inje
cted
F
. 10
3 R
emar
ks
gas
[°K
] q
uan
tity
N
o.
nu
mb
er
air
0 to
12
-....i
Gre
en a
nd
N
alF
m
Pap
pas
an
d
Ok
un
o m
} 19
60
Bar
tle
and
L
eado
n~l~
) 19
60
War
ner
an
d
Em
mo
ns C
tx)
1964
sup
erso
nic
n
ozz
le
air
hel
ium
F
reo
n- 1
2
282
to
805
nit
rog
en
T,,
=
335
To~T
**
=
1.9
=
3.3
Sta
nto
n n
um
ber
, co
nd
itio
n f
or
sep
arat
ion
, sk
in f
rict
ion
co
effi
cien
t
1X
l0 e
8X
106
15 °
con
e
flat
p
late
cyli
nd
rica
l ro
cket
m
oto
r ch
amb
er
wat
er,
anh
yd
rou
s am
mon
ia,
eth
yl
alco
ho
l,
Fre
on
-I 1
3
0to
8
0to
4
0to
5
0 to
1"9
2600
~
To*
* ~
4100
aver
age
skin
fr
icti
on
co
effi
cien
t
Sta
nto
n n
um
ber
, re
cov
ery
fac
tor
hea
t tr
ansf
er
0-3
0"7
3-5
4.5
2"0
3.2
0.9
X1
0 e
R
x~
6
X 1
0 e
R
8"6
X 1
0 8
12
X1
0 e
[p
er m
etre
]
=
6"8
X 1
0 *
inje
ctio
n
mas
s fl
ow
var
ied
alo
ng
su
rfac
e, g
oo
d
reco
ver
y
fact
or
mea
s-
ure
men
ts
D
=
dia
- m
eter
of
cham
ber
r-
9 O
-
Librizzi a
nd
Crc
sci t
m
1963
Pap
pas
and
Okuno ~m
1964
McRee,
Pete
rson
and
Bra
slaw
(w
1964
Dan
berg
tt~
1960
, 19
64
supe
rson
ic
nozz
le
(2<
lim
en-
sion
al)
15 ° cone
flat
pl
ate
flat
pl
ate
flat
pl
ate
heli
um
nitr
ogen
air
heli
um
Fre
on-1
2
air
air
air
0 to
8 0 to
1"5 0 to
1"8 0 to
2"
5
895
T0
oo
~
1485
290
to 4
17
300
T0oo
30
0 4-
2~<
T
./T
.o ~
5
To.o
=
550
4.1
~<
T,
IT~,
, 7.
6
heat
tra
nsfe
r co
effi
cien
t
loca
l S
tant
on
num
ber,
rec
over
y fa
ctor
loca
l sk
in
fric
tion
coe
ffic
ient
an
d op
tim
um
slot
con
figu
rati
on
velo
city
and
te
mpe
ratu
re
prof
iles
, lo
cal
skin
fr
icti
on
coef
fici
ent
loca
l sk
in
fric
tion
coe
ffic
ient
, S
tant
on n
umbe
r,
boun
dary
lay
er
para
met
ers
R
1-3
X l
0 T
2-
3X10
7 1-
2) ~s
-
Tab
le I
con
t.
Tem
p.
Mea
sure
d
Mac
h i
R
eyno
lds
Inve
stig
ator
(s)
Geo
met
ry
Inje
cted
F
. 10
3 R
emar
ks
gas
[°K
] qu
anti
ty
No.
I i
nu
mb
er
air
Jero
min
ha)
1966
Fog
arol
i an
d
Say
dah
~)
1966
fiat
pl
ate
7.5
° c
on
e
air
0 to
1.7 0 to
2"2 0 to
20
T0
oo
=
295
1"9
~ T
¢/T
~
~2
"3
2.9
~ T
,/T
o,
~3
"5
T0
• =
45
0
to
'10
50
velo
city
an
d
tem
pera
ture
pro
file
s,
loca
l sk
in f
rict
ion
coef
fici
ent,
b
ou
nd
ary
lay
er
para
met
ers
aver
age
skin
fr
icti
on a
nd
hea
t tr
ansf
er c
oeff
icie
nt
2"5
3-5
5"3
and
8"
1
1"6
X 1
07
R~
~ 2"
5X
107
1.4>
(107
R
~ ~
2"4
X 1
0 7
R
0.9
7X
10
~
0.72
x 1
06
[per
met
re]
r~
©
©
z
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 77
(see Section 2.1.5) and checked by the momentum equation so that these values must be considered at present as the most realistic and accurate information about cf; however, there is still some scatter in the results.
The local reduction of heat transfer due to injection through a porous surface is the subject of quite a number of experimental investigations using the flat plate configuration. These are the experimental studies by Leadon and Scott (24' ~5), Scott, Anderson and Elgin C~°) and Bartle and Leadon t27) who measured Stanton numbers and recovery factors and the already mentioned investigation by Danberg (~2). All the other investigators considered in Table 1 measured average skin friction coefficients or average Stanton numbers for porous cones. It is rather doubtful whether the measurements of the average skin friction coefficient or Stanton number for a porous cone are a reasonable check for theories predicting the corresponding local quantities for a flat plate even when plots of cy/cfo against 2F/cj~, or C~/CFo against 2F/cro, or corresponding plots for c h are used which might reduce this uncertainty by assuming that the reduction of the overall skin friction coefficient and Stanton number are the same as for the local quantities under equal conditions.
All available data for the skin friction coefficient and the Stanton num- ber are plotted in Figs. 1 and 2. In Fig. 1 skin friction coefficients for boundary layers with air injection through a porous surface are compared at R x = constant with those obtained along solid surfaces by plotting cf/Cfo against 2F/clo. Such a simple plot shows best the reduction of the skin friction coefficient due to injection compared with the case of zero injection which is characterized in Fig. 1 by the point 2F/cfo = 0 and cf/Cfo = 1. The data for the incompressible flow considered in Fig. 1 will be discussed in Section 2.2.1. Despite the scatter of the points the tendency is clear: the reduction of the skin friction coefficient due to air injection for 2F/Cfo = constant becomes smaller with increasing Mach number.*
The four regions marked in Fig. 1 represent the reduction of the skin friction coefficient for the Mach numbers M = 0, M = 2-5, M = 3.5 and M = 6.2. Only data for skin friction coefficients measured along flat plates have been considered when the borders of the regions were fixed, namely the investigations by McQuaid (2s) (M = 0), Rubesin t21) and
t It should be noticed that clo and C~o decrease with increasing Math number as well so that it cannot be directly deduced from plots of cj/clo against 2F/cso, CA/CAo against F/cho, respectively, that fluid injection becomes less effective with increasing Mach number. Plotting, for example, cl/clo against F gives very roughly about the same reduction of the skin friction coefficient due to injection independent of the Mach number.
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78 L. O. F. .IEROMIN
AL [-]
I
0.9
OS
0,7
0.6
0.5
0.4
0.3
0.2.
0.1
, - r , | ........ , . - - ,
o I " 1'.o 2'.0 3.0 2F r.] Cf 0 k
FIe. 1. The reduction of the skin friction coefficient due to air injection (experimental data).
Jeromin (2s) (M = 2.5, M = 3.5) and Danberg (12) (M = 6.2). Regions rather than lines have been chosen to represent the experimental data because o f their scatter which becomes considerable for Danberg ' s data at M = 6.2. Only Rubes in '# ~z) and Tendeland and Okuno ' s (a°) data for porous cones fit with some good will in these four regions. The measure- ments of Pappas and Okuno ~z) at a porous cone show a reasonable agreement with other experimental investigations only for M = 0.3 and M = 0.7, whereas their data for M = 3.5 and M = 4.5 give a much smaller reduction o f the skin friction coefficient than found by other investigators.
I t must be stressed in this connection that two parameters cannot be considered in all details in Fig. I : the Reynolds number R x and the temper-
-
RESEARCH IN TURBULENT BOUNDARY LAYERS
EXPLANATION OF THE SYMBOLS USED IN FIG. 1
79
syrup1
+
×
*
II
O
A
V [>
4
k,
Investigator Mach number Geometry
Mickley and Davies (~) 1957
RomanenkoandKharcbenko
-
80
ch[_] Cho l &
L. O. F. JEROM1N
0.9
0.8
0.7
5'105,~ Rx~ 107
0.6
0.5
0.4
0.3
0-2
0.1 M=0
o 1 2 3 c~o [d
FIG. 2. The reduction of the Stanton number due to air injection (experi- mental data).
to represent the experimental data in the next sections where they are needed for comparisons with available theories.
In Fig. 2 the available experimental data about the reduction of the Stanton number due to air injection have been summarized. Despite the scatter the data show corresponding to Fig. 1 a clear tendency: the reduc- tion of the Stanton number due to air injection for F/e~ = constant be- comes smaller with increasing Mach number.* Four regions for M = 0, M = 2, M = 3 and M = 6 have been outlined in Fig. 2. They are based on experimental data for the flat plate configuration taking again into consideration the scatter of the published data. These four regions will
~" See footnote on page 77.
-
RESEARCH IN TURBULENT BOUNDARY LAYERS
EXPLANATION OF THE SYMBOLS USED IN FIG. 2
Symbol Investigator Mach number Geometry
X incompressible flat plate
+
r~
O
A
-
82 L. O. F. JEROM1N
factor due to injection whereby the reduction decreases with increasing Mach number for F = constant. Despite the scatter of the data this tendency is clear. This result is confirmed by the experimental data published by Rubesin, Pappas and Okuno ~3s}, Leadon and Scott {24' 25~ and Pappas and Okuno ~31~. Moreover, Rubesin, Pappas and Okuno ~z3~ found an additional influence of the Reynolds number on the reduction of the recovery factor such that the recovery factor drops slightly with increasing Reynolds number for F = constant. This result can be con-
[-Ij_~______.Q~_L ~Y . . . . . f° ..... No. G., 10 . . . . Math Injected .~ w - - ^zz'~..~,a.,,._ ~ x m r t e a n d 2'0
v n t rogen
~ .o~ J ~n ~ ~ ~ , - Leadon a, Scott .
J • " ~ ~ ~ - - Rubesin, Pappas, .
. 0 L~aC[on, Scott . [24], [ ' )5], 1956 3'0 hehum
~ . 9 0 ' o
0.2 0.4 o.6 o.e 1.o ~.:, 1.4 1.6 1.e z.o
F.10' [,,.]
FIo. 3. The reduction of the recovery factor due to air injection (experi- mental dam).
firmed to a certain extent by measurements of the recovery factor along a solid fiat plate (see for example Stalder, Rubesin and Tendeland [136] or Schubauer and Tchen [137]). These experimental results have shown that the recovery factor for a Mach number of M.. = 2.4 drops slightly from r = 0 . 9 0 at R x = 1 × 1 0 6 to r = 0 . 8 8 5 at R x = 6 × 1 0 6 . As the Reynolds number range for the experiments considered is about the same for the solid fiat plate case and the porous plate one (see Fig. 3 and Table I) it can be deduced that the Reynolds number effect on the re- covery factor is less pronounced but still existing for boundary layers along solid surfaces. It must be stressed considering the present experi- ence that it is mainly speculation rather than sound knowledge to pro-
-
RI~EARCH IN TURBULENT BOUNDARY LAYERS 83
vide a proper interpretation for the increased reduction of the recovery factor for transpired boundary layers; hence an acceptable physical ex- planation has not been formulated yet. Apart from unavoidable experi- mental difficulties and possibly errors or at least uncertainties in deter- mining the recovery factor for transpired boundary layers--the scatter of the data for the recovery factor is much wider for the case of boundary layers along permeable surfaces compared with those along solid ones-- the most likely reason might be the surface roughness of the porous plates which is several orders higher than for ground surfaces normally used for boundary layer experiments in supersonic wind tunnels and which might produce a different energy spectrum close to the wall.
2.1.2. The reference state concept The simplest approach for predicting the variation of the boundary
layer parameters with Reynolds number, for example, is the reference state concept. Two examples of the compressible turbulent boundary layer with fluid injection are the methods proposed by Nash (4° and Knuth and Dershin (4.).
Nash introduced a simple empirical similarity parameter with which he succeeded in correlating compressible experimental data with Turcotte's sublayer hypothesis for incompressible flow. t He modified Turcotte's
injection parameter v~[u~o into Owv~ = uo..F assuming that the new
parameter suppresses the effect of compressibility. Plotting *,/z~o against uo.[U,o" F he found that the experimental data of Tendeland and Okuno {s°~ and Pappas and Okuno (~) for compressible flow and the data of Mickley and Davies (d° for incompressible flow lie within a scatter on one single curve, namely
I:, _exp[_6 .94 (u~o .F~( l+v /~ -~) ] (2.1) • ,o \ U,o /
But it must be stressed that this correlation could be fortuitous, moreover Turcotte's approach is suspect for reasons which will be discussed in the next section. In Nash's concept there are too many questionable simpli- fications without any theoretical basis: for example
(i) In Turcotte's theory, the skin friction velocity must be calculated from the local skin friction coefficient, whereas Nash used for his
T Turcotte's sublayor hypothesis will be discussed in more detail in Section 2.2.
-
84 L. O. F. JEROMIN
correlation the overall skin friction coefficient as measured for a porous cone in a supersonic flow.
(ii) Experimental data measured on a cone are compared with a theory derived for the case of a flat plate.
(iii) The compressibility effects are only considered by a slight modifi- cation of one coordinate, namely by changing the incompressible
variable * * uo. v,,/u,o into .F. u~i
Knuth and Dershin t45) tried to suppress the compressibility effects in a more sophisticated way by extending Eekert's (4" and Sommer and Short's (~s) reference state concept to compressible transpired boundary layers. The usual simplifications of the boundary layer model in the pres- ence of fluid injection (see Section 2.1.3) led Knuth and 13~rshin to the conclusion that it might be possible to correlate compressible and in- compressible skin friction coefficients by plotting c~/c~o against
--2V'c 0! B* = l l ' 5F c?" (2.2) where the superscript * refers to reference condition. If incompressible and compressible data collapse in such a plot on one single curve--the so-called constant property curve--the reference concept might be justi- tied.
Knuth and Dershin determined their constant property curve from incompressible data (Goodwin (51) and Smith(52)). The compressible flow is represented only by the heat transfer data of Bartle and Leaden (27) since it was believed that heat transfer measurements in supersonic flow are much more reliable than skin friction measurements. This advantage of greater accuracy of the heat transfer data is at least lost by the fact that they had to introduce a modified Reynolds analogy with all its uncer- tainties in order to relate heat transfer with skin friction coefficients. The reference state expression for turbulent flow with air injection necessary to reduce compressible into incompressible flow is the same as that de- veloped by Knuth (4') for laminar flow, namely
T* = 0"5(T,,+T.o)+O.2(T,o-Too)+O.10wvw 1 (T , -To . ) (2.3) O'u** C~o
This assumption is based on experience obtained for compressible flow without injection where the reference temperature formulae for laminar flow correlate more or less successfully the quantifies for turbulent flow as well.
-
RESEARCH IN TURBULENT BOUNDARY LAYERS
c; ~t-J 1"0 symbol Dat| " I I~lach No. I Reynolds No. I Ref.
skin friction 0 1.3.1O s ~ R x ~ 1"!.101 S1
0 skin friction 0 2 S. lOS ~/~x ~ I '0.106 $2
) • heat trlnsfer 2"0 R x - 7"0.106 27
0. . hh::t**t;:::,:* *
I ( 1 ~ . - ~ . " he| t t r ln i fe r 3"2 .... Rx - 7"7.10 7
0.6 ~ x
o-41- ~
0 ~ '~ 0 2 4 6 8 10 B;[-]
FIG. 4. Correlation between compressible and incompressible flow using the reference state concept after Knuth and Dershin. (4s)
85
Introducing their Reynolds analogy and applying the reference temper- ature formulae Knuth and Dershin succeeded in correlating the compres- sible and the incompressible data on one single line as shown in Fig. 4 where c~./C~o is plotted against B~*. This result seems to justify the reference state concept. A similar good agreement was obtained for the Stanton number whereas the prediction of the reference state concept for the recovery factor is not convincing since its predicted reduction is much bigger than found by experimental investigations.
Finally one can say that much more extensive studies in this field are necessary before final conclusions can be drawn about the reference tem- perature concept in its present form. The two key questions are
(i) is the reference temperature formula for laminar flow generally valid for turbulent boundary layers ? and
(ii) is there a Reynolds analogy between heat and momentum transfer?
These questions can be solved at present only by appropriate experiments. Anyway the reference concept must be always considered as a pseudo-
theoretical approach which certainly becomes obsolete when a general theory for turbulent boundary layers with air injection has been formu- lated.
-
86 L . O . F. JEROMIN
2. 1.3. Theories based on mixing length concept
A number of theories have been proposed based on the assumptions that the turbulent boundary layer can be divided into a laminar sublayer and an adjoining fully turbulent region. The first approaches in this direction were the boundary layer analyses presented by Rannie c5~ and Crocco ~54). They assumed a Couette flow model (all changes in x-direction are negli- gible compared with changes in y-direction) and a simplified Reynolds analogy between momentum and energy transfer. Both theories are based on a simple film model, replacing the boundary layer by a film of constant thickness, and hence neglecting any variations in flow direction as com- pared with those across the film. The aim of both theories is to establish relations with which the heat transfer coet~cient (and hence the necessary amount of cooling medium to protect the surface) can be calculated from the skin friction coet~cient which is assumed as known. Despite the fact that the skin friction coefficient was evaluated from isothermal, incom- pressible turbulent boundary layer formulae, the theoretical predictions agree reasonably well with experimental results.
More sophisticated contributions to a theoretical analysis of the com- pressible turbulent boundary layer with air injection are the papers by Rubesin ~55~ and Dorrance and Dore c~6~. Considering the present state of knowledge of transpired turbulent boundary layers their approach of solving the boundary layer equations with Prandtl's mixing length theory must be considered as the best possible theoretical solution in integrating directly the boundary layer differential equations. Both theories have much in common in that they are practically all extensions of earlier compressible boundary layer theories for turbulent flow without injection where the latter ones are extended incompressible mixing-length theories taking into account compressibility effects. The starting point for both theories are the boundary layer equations, namely the continuity equation
~ u ~- O~v ~x ~y = 0 (2.4)
momentum equation
with
Ou ~u dp ~ e u -~ -+ Ov ~--~-= -d---x- + Oy
-=- (1~ + eM) ~u ~u'v' - ~ ' 8M --- OU
~y
(2.5)
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 87"
energy equation
with
OH OH O ~u vx-xzY- +~v Ox = Oy (q+uT) (2.6>
2 +eH Oh ~ce'v'T' q - : 8H - -
cp Oy ' OT Oy
These three equations have to be solved simultaneously. The two theories proposed by Rubesin and by Dorrance and Dore are almost identical. Their main assumptions are:
(i) the boundary layer is divided in the usual way into a laminar sublayer close to the wall where molecular transport processes pre- dominate, and a fully turbulent region which is governed by turbu- lent transport processes only. No buffer layer is considered;
(ii) the turbulent Prandtl number Pr T - eM" Ce is equal to unity; £H
(iii) a Couette flow model is assumed, so that the partial differential equations describing the boundary layer behaviour reduce to ordi- nary ones;
(iv) the gases are ideal; (v) the shear distribution through the boundary layer can be approx-
imated by Prandtl's mixing length theory, namely
eM = 0~y 2
(vi) only the case of constant pressure dp/dx = 0 is considered; (vii) the injection mass flow F is constant along the surface.
With the help of assumption (iii) the partial differential equations (2.4) and (2.5) can be simplified to an ordinary one, namely
du d [ du ] (2.7)
Setting the Prandtl numbers for the sublayer and the outer region equal to unity leads to a direct relationship between velocity and temperature: so that only (2.6) and (2.5) have to be solved simultaneously and eq. (2.6) is not required.
Neglecting the eddy transport term eat in the sublayer and the viscous term # in the outer region leads to expressions defining the velocity
-
88 L . O . F . JEROMIN
distribution throughout the boundary layer. It follows for the sublayer (0 ~ y ~ Ys, subscript s refers to conditions at the edge of the sublayer)
du / ~ - ~ - = ~wvwu+constant = Qwv~u+~,~ (2.8)
and for the outer part
_ / du ~ @wvwu = 0~ Y~ ~--d'y-y) +constant (2.9)
after introducing Prandtl's mixing length theory. The velocity and shear stress distribution must be continuous at the edge of the sublayer, hence the constants in (2.8) and (2.9) must be identical. Therefore one obtains
[ du~ ~ @wv.u+ lrw = O~2y 2 [--d-y) (2.10)
or on integrating
f u@ tl' du Y = Ys ~v/Owv,,u+~: w for y ~ y,. (2.11)
u,
A corresponding expression can be derived for the sublayer region. With the velocity distribution throughout the boundary layer known, the skin friction coet~cient can be determined from the momentum equation
dO c f + F = 2 d---~- (2.12)
with the momentum thickness
8
O= o--f~ l U
0
Up to this point gubesin's and Dorrancc and Dore's theories are almost identical. The only differences between the theories are the extra- polation of the constants u, y, and us and the process of the numerical integration of eq. (2.11) itself. Rubesin made three empirical assumptions based on measurements in incompressible flow:
(i) the sublayer "Reynolds number" usy~ = 13.12 = constant; the factor 13.1 s is evaluated from experimental results in incompres- sible flow without injection;
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 89
(ii) u, = 13.1 for T w --- T.~ and u, = 13"1~/~/T~o for Tw ~ To. and y, according to eq. (2.11);
(iii) the mixing length constant is not influenced by compressibility effects and fluid injection at the wall and was set ~ -- 0.392.
Assuming an error of less than 1 per cent Rubesin used only the velocity distribution in the outer part of the boundary layer, i.e. eq. (2.11), to determine the momentum thickness and neglected the contribution of the sublayer. Thus u~ is made zero and y, remains finite in such an integration process. The momentum thickness known, the skin friction coet~cient was calculated from the momentum equation (2.12). The result is shown and compared with experimental data in Fig. 5 by plotting cs/c/o against 2F/c/o. The theory predicts a reduction of cf due to injection but unfor-
¢._~f [ . ] cto
1;
O ' S "
0~-
0 ' 7 "
0'6
0"5
0'4"
0:3"
0.2-
0"1-
O~ i i I 0 I 2 3 2F "
~,o E ]
FIe. 5. The reduction of the skin friction cocfllcient due to air injection (Rubesin's (~) theory in comparison with experimental data).
-
90 L.O. F. JEROMIN
tuna*ely not in the right order of magnitude. The agreement is poor, for both incompressible flow (M = 0) and compressible flow (M = 4, T w ~ Tr) especially for high injection rates. The discrepancy is mainly due to the fact that the constants were evaluated from available information about incompressible flow without injection, since there was barely any informa- tion about transpired turbulent boundary layers accessible at the time when the theory was derived. But as more recent experiments have shown, these constants are in fact functions of the injection rate and heat transfer. Danberg (2z) for example, modified these theories by changing the constants according to his measurements and succeeded in transforming* his meas- ured velocity profile into a form similar to the semilogarithmic profile of incompressible flow, the so-called law of the wall. His coordinates are essentially derived from Rubesin's and Dorrance and Dore's theories, or are at least based on the same boundary layer model. He obtained reason- able agreement between his experimental and theoretical profiles for the fully turbulent region by adjusting his 'constants' of integration of the mixing length velocity profile so as to make them increase with heat transfer and decrease with increasing mass transfer rate at the wall.~
It must be pointed out that two parameters influence the representation of the theory in Fig. 5, namely the Reynolds number R~, and the tempera- ture ratio Tw/T~. Their influence on the skin friction coefficient is shown on Figs. 6 and 7. With increasing Reynolds number for Tw/T~o = const. and Moo = const. Rubesin's theory predicts an increasing reduction of the skin friction coefficient due to injection (see Fig. 6). The influence of heat transfer at the wall is such that with increasing cooling (T w --- T~) the reduc- tion of the skin friction coefficient increases slightly (see Fig. 7). These influences cannot be checked at present by experimental data because the influence of the different parameters on the boundary layer growth have not been investigated systematically enough.
Dorrance and Dore determined the momentum thickness numerically and included the contribution of the sublayer for the velocity distribution in the integration process. The integration constants were evaluated by
t Danherg called his theoretical analysis a transformation. His approach is certainly not a transformation in the same sense as used by Coles ¢e2) and Jeromin t:8~ where the compressible boundary layer equations are transformed mathematically into their corresponding incompressible forms.
Another theoretical approach, where the constants are the same as for incompressible flow without injection, has been proposed by Stevenson cert. His approach is also based on Prandtl's mixing length theory, but the calculation follows a different path so that the final result is different. This theory will he discussed in Section 2.2.3.2.
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 9 1
c_Jt [ - ] CfO
1 0 -L
0 . 9
0 . 8
0 . 7 ,
0 . 6 '
0 , 5 ,
0 . 4 .
O,3 ¸
0 , 2 ¸
0 . 1 ¸
~ ~ : Rut,.~l. [ss] (M - 2, T . , r ~ 1. R; - 10')
: R u t ~ l n [5 R ( M . 2, TwIT" = I, R, = 10 7) ~ . ~ : Rub. in [55] (M - 2. T . / T = I, Rx = lOl)
. . . . I . . . . I . . . . I
1 2 3 ~_~o(. j
FlO. 6. The influence of the Reynolds number on the reduction of the skin friction coefficient (air injection: Rubesin's (m theory, M -- const.,
T w / T . . = const.).
assuming that the resulting skin friction law eventually derived from eq. (2.12) must reduce to Karman's skin friction law for incompressible flow when T~ = T~ and M.. = 0 c5". The mixing length constant x was assumed again to be unchanged by compressiblity effects and fluid injection and was set to 0.393. The main result of Dorrance and Dore's theory for the skin friction coefficient is shown on Fig. 8 and compared with available experimental data. The reduction of the skin friction coemcient due to in- jection is again predicted by the theory but unfortunately only in the right order of magnitude for the incompressible flow with M., - 0 whereas a far bigger reduction of the skin friction coefficient is predicted for compres-
-
92 L. O. F. JEROMIN
1.0
0.9
0.8
0.7
0.6
0 .5
0 . 4
0.3
0.2
0'1
Cfo
, " " ~
. . . . R0b,,,. tssl IM = , . o 'QI T. o T.>
- - : R ~ , ~ , [SSl ( M - 2, R. = ~ o , r . = r , . ) " ~
. . . . I . . . . I ' ' ' ' I
1 2 3 2F ' :--E [ ' ]
FIo. 7. The influence of the temperature ratio TWIT** on the reduction of the skin friction coet]lcient (air injection: Rubesin's (~) theory M = const,
R~ = const.).
sible flow, represented in Fig. 8 by a Math number of 5, and heat transfer at the wall (T w = T**) than found by experimental investigations.
Rubesin, and Dorrance and Dore extended their theories to include the case of heat transfer at the wall by assuming a Reynolds analogy with the heat flux proportional to the transport of momentum. In the second part of his analysis Rubesin investigated the influence of fluid injection on the Stanton number now allowing the Prandtl numbers for the sublayer and the outer region to vary throughout the boundary layer. So he determined the relation between the local Stanton number and the local skin friction eoemcient when Pr and Pr r are not unity by taking instead the ratio of corresponding terms of the energy and momentum equation, The influence
-
93
c-~H 1
of the Prandtl number on the Stanton number is adopted from investiga- tions without injection (see Rubesin (sS)) assuming that the skin friction coefficient is independent of the Prandtl number. This process leads to rather complicated ordinary differential expressions for the Stanton num- ber which he solved numerically assuming again that the contribution of the sublayer is negligible compared with the contribution of the outer region. The result for the Stanton number is presented by Rubesin in the form of graphs c h = f(Rx) with the parameters Moo, Tw/Too and F. Only a typical example for the reduction of the Stanton number due to air injec- tion will be shown here on Fig. 9 where Ch/Cho is plotted against F/cho for R x = 10 7, M = 0 and M = 4 and compared with experimental data. A slightly smaller reduction of the Stanton number is predicted by
0"9
0'8
0'7
0"6
0'5
0'4
RESEARCH IN TURBULENT BOUNDARY LAYERS
0.3
0.2
0.1
0 o 1 2 3 z~ [o 3
Cfo
Fxo. 8. The reduction of the skin friction coefficient due to air injection (comparison of Dorrance and Dore's (6e) theory with experimental data).
-
94
c_~_h [.] c h 0
~-o-k
0"9
0 '8
0 '7
0-6
0"5
0 '4
0 '3
0"2
0'1
L. O. F. JEROMIN
M=O
} experimen,:at dat~ (see Fig. 2)
: Rubesln [55] (hi = ~, T ~ T w ~ T r , 10 6 ~:: R. - 10 a) ~ mean
~ , ~ : Rubesln [S5] (M ~ O, T ~ T w, 10 t ~: R x ~ 10e) / c u r v e s
. . . . I . . . . I ' ' ' '
1 2 I
3 ~ H Cho
FIG. 9. The reduction of the Stanton number due to air injection (compari- son o f Rubes in ' s cu) theory with experimental data).
Rubesin's theory for incompressible flow than shown by experimental data. The agreement between theory and experiment is excellent for a Math number of 4. The curve representing Rubesin's theory for M = 4 must be considered as a mean eurve for 10 6 ~ Rx.~ 10 8 and T~. ~ Tw~ T,. The prediction of Rubesin's theory for the recovery factor does not agree with experimental data.
Dorrance and Dore extended their theory to include the ease of heat transfer at the wall by assuming a complete Reynolds analogy
Ch = c f l 2 (2.13)
a n d r = 1 = c o n s t a n t .
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RESEARCH IN TURBULENT BOUNDARY LAYERS 95
Their result is shown for M = 0 and M = 5 with T w ~ T.. and 9× l0 s R x ~; 3 .3×10 6 in Fig. 10. The agreement is excellent for incompressible flow whereas the prediction for compressible flow is wrong probably due to the fact that the constants were evaluated for incompressible flow without injection. Moreover the choice of a complete Reynolds analogy as expressed by eq. (2.13) becomes rather suspect for compressible flow as several investigations have shown. Instead a Reynolds analogy factor
el~2 s = (2.14) Ch
should be used which is in most cases smaller than one. Finally one can say that Dorrance and Dore's theory can be recommended only for in-
1,0 "[
0.9
0'8
0.7
O ' S '
0"5
0.4
0'3"
0.2
M=O
0"1
~ }+,~p ........ ~,J+. (+o+ F,e 2) ' ~ .
: Dorrance and Dote [ 5 6 ] ( M = S. T . = T . 10 s ( R x ~ 3 . 3 . 1 0 +)
, i : D o r r a n c e a n d D o r e [ 5 6 ] ( M = 0 . T , = T . 10 s ~ R ~ ~ 3 ] . 10 ~)
. . . . I . . . . I . . . . I '
0 1 2 3 ._~._F i'." I ChO
FIG. 10. The reduction of the Stanton number due to air injection (com- parison of Dorrance and Dore's c66) theory with experimental data).
-
96 L . O . F. JEROMIN
compressible flow whereas Rubesin's theory gives reasonable results only for compressible flow. The experimental data suggest a far bigger influence of the Mach number on the reduction of the skin friction coefficient due to injection than predicted by either theory.
A theory very similar to these of Rubesin and Dorrance and Dore was proposed by Ness tS~). This theory is derived for foreign gas injection, but Walker and Schumann t co) have shown that it can be used for air injection as well. Ness's analysis is an improvement over Rubesin's since it employs no mathematical approximations in solving the Couette type boundary layer equations. The usual starting condition that the skin friction coeffi- cient and hence the velocity at the interface approach infinity at the leading edge of a plate for example, has been replaced by a more realistic starting condition assuming that the velocity at the interface between the laminar sublayer and the fully turbulent region is equal to the velocity at the edge of the boundary layer at the start of the turbulent region. Despite these improvements, the agreement between theory and experiment, as shown by Walker and Schumann t6°~, is still not very convincing (but better than for Rubesin's theory, especially for incompressible flow) perhaps mainly due to the choice of the constants which are assumed again to be independ- ent of the injection rate. The discrepancy between theory and experiment becomes more and more significant as the injection rate and the Math number increase.
Finally one can draw the conclusion that Rubesin's, Dorrance and Dore and Ness's theories can be improved by considering a buffer layer between the laminar sublayer and the fully turbulent region and adjusting the con- stants according to the information now available. The 'constants' are most probably functions of the wall temperature and the injection rate, so that this influence must be considered during the integration process. Moreover the Prandtl number could be allowed to vary through the laminar sublayer and the turbulent outer region. But this approach would in- crease the mathematical problems enormously so that the most interesting application of these possible extended theories would be rather difficult.
2.1.4. Analysis based on similari ty parameters
The skin friction coefficient for compressible turbulent boundary layers with zero injection and zero pressure gradient is in general a function of the Reynolds number R 0 based on the momentum thickness, the temper- ature ratio TwIT ~ and the free stream Maeh number; hence one can write
C ro = C.ro(Ro, T, , /T~, M ~ ) .
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 97
Moreover most theoretical expressions for the skin friction coefficient can be reduced to the form (see for example Spalding caa))
OF
cm Fc = ~Oo(Ro.FRo) 2
c/o Fc = ~ox(Rx.rR~) 2
(2.15)
(2.15.1)
where the functions ~P0 and ~Px are independent of Mach number and tem- perature ratio. The influence of these two parameters is only considered by the functions F c and FRo (or FRx respectively) which were eventually found from experimental data and set
and
Fc= d (2.16)
FRo = #~ (2.17) #w
The main advantage of treating the boundary layer in such a way is the resulting relatively simple calculation process for skin friction and heat
Ro FRO 101 10 2 10 3 10 4 10 5 10 6
10-2
10"3. - - I
10_2 I
10"°103 104 105 10 s 107 108 Rx FRx
FIG. 11. The universal tu rbu lent boundary ]ayer law after Spalding. (e3)
transfer coefficients. For example, only two diagrams are necessary to determine cs: one containing c/.F c as a function of Ro'FRo or Rx" FRx which is universal and hence independent of M~ and T.,/T~ (see Fig. 11); the other containing F c FRo and FRx as functions of the parameter M~ and Tw/T~. Another advantage is the fact that the FRo- and FR~-functions were chosen so as to fit best available experimental data and are not based on more or less arbitrary assumptions.
-
98 L. O. F. JEROMIN
Spalding et al. ~2°> extended this skin friction calculation method to transpired turbulent boundary layers, postulating that F c, FRo and FR~ are
~wVwU~o additionally a function of an injection parameter defined by B -
so that Fc = Fc(M**, Tw/T~, B)
FRo = FRo(M~, Tw/T~, B)
F ~ = FR~(M., T~/T., B).
~w
(2.18)
(2.19)
(2.20)
The last two equations are linked by the momentum equation (2.12) which can be written in the form
dR0 dRx = (1 +B). (2.12.1)
Fo F~o and FR~ can be regarded as constants when M**, T~/T~, and Fare independent of x, so that (2.12.1) can be written as
d(Ro.FR0) = cf Fc (1 + B) FRo (2. 12.2) dfgx.FRx) 2 FR~.Fc
which is the wanted connection between FRo and FRx. The problem is now the determination of the functions F c and Fse or FRx in terms of M.., Tw/T ~ and B. In principle systematical analyses of experimental data would lead to these functions plotting c I against R x or R 0 for fixed values M**, Tw/T~ and B. But unfortunately there is not enough information available to permit this procedure. Instead Spalding et al.
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 99
Equation (2.21) has been solved in a rather complicated calculation process, t The result is presented by Spalding et al. in the form of tables and graphs where F c is given in terms of (1 + B) for the parameter M~ and Tw/Too. The ranges of variables covered by their calculation are
0 ~ M o o ~ 12
0"05 ~ Tw/Too ~ 20
0 ~ B ~ 19.
It must be pointed out here that the extension of the present calculation process to hypersonic flow with M.~ ,,~ > 5 is rather questionable since real gas effects are not considered. Analysing the available theories to obtain an expression for FRo was not successful since the resulting relationships
M.
1'0
0.1
0"01
•. .:.: ' - .
~./!;_\ v
;/
ee • •
Folnt I ,,~h No. I ^ ' ~ I R , ~ . ~ • ~= 0 Mickley and DIvles 46
"F == 0 Tewflk el; al. ~10
A 2"0 Bartle and Le~lon 27
X 2'7 Tendehnd a.d Okuno 30
0 2.7 P.ubesln et I I . 33 t
O 3"0 Ludon and Scott 24
V 3"2 h r t l e and L¢~ion 27
13 3.2 h r t l e and Lea(Ion 27
• 3,& Leadon and Scott 24
I " Note: These data for helium to air injection ;
=11 others for air to air.
# . - =1= FRO = --~w (l,B) .. ° : . °
10 100 I + B
FIO. 12. The determination of the Fj 0 function after Spalding et al. (t°)
loo0
t See also Section 3.4.
-
lO0 L.O.F . JEROMIN
10-2 r- . " =,:.=, ~ ~ . ! j D r a g l a w from I
~ . " , . . .~ . . " • ~ o ~ r b . ~ , . . ~ ~ ~ , I
o ":':.:" . . . . . .. - . . . _ ~ - ~ . j _ . . ~ ~ _ ~ I =. 10 "3 I#oi°,l.=o,.o-I "A°'~°r , I " . ! ' ~ ' 0 ~ 1 O • ~.0 Mlckley and Davis 1 4 . . _ ~ 7
M U ~ 2'0 ,Barcle and Leadon 27
~" I & I 3 " 0 i Barcleand Le=don
10 z 10 3 10 ~ Re FRe
10 ~
1,3
._u
10"~ ..u G"
x
Point Math No. ...... I I Author O ~0 Tewflk et. af.
O 2.7 Tendel=nd and Okuno & 2.7 Rubesln el, al 9 30 Lezdon Ind Scott x 3.0" Lezdon and Seot~
I Reference 90 30 33
24
* Note: These data for helium to air injection; air others for air to air,
10 ~ 10 ~ 101 10 7 Rx FRx
Fzo. 13. Comparison of experimental data with the universal turbulent boundary layer law after Spalding e t al. cs°~
-
c~o H
1-0 - ~
0.9 ¸
0.8 ¸
0.7
0.6
0.5
0.4
0.3 ¸
0.2
0 .1
RESEARCH I N T U R B U L E N T B O U N D A R Y LAYERS
M = '
~ } data (~ee Fig. 1) experimenta~
~ , ~ : Spalding et al. [20] (M = O. T w = T . , R x = 106)
: Spalding et al. [20] (M = 4, T w = T r, R~ = ~06)
M = 2 .5 " ~ E E T _ L . - -
0 1 2 3 2! [.] cfO
FIG. 14. The reduction of the skin friction coefficient due to air injection (comparison of Spalding et a/.'s (2°) theory with experimental data.)
101
differ considerably from theory to theory so that only the analysis of experimental data remains. Assuming the equations (2.18) to (2.20) hold, the universal plot on Fig. 11 can be used to analyse the available experi- mental data and to determine FRo as a function of B. The result is shown on Fig. 12. There is obviously a Math number effect in Fig. 12 with M.. increasing in the direction of the arrow when one only considers data measured along a fiat plate. But nevertheless Spalding et al. neglected the parameter M~. in Fig. 12 and put a mean curve through the data, namely
FRo = #w (1 + B) =/' (2.22) # -
-
102 L.O.F. JEROMIN
resulting in a correlation demonstrated in Fig. 13 where the drawn line represents the universal skin friction law from Fig. 11. As one must expect there is again a well-defined Mach number effect in Fig. 13 so that eq. (2.22) needs an improvement taking into account the parameter M~. This fact becomes even more obvious in Fig. 14 where Spalding et al.'s calculation method is compared with experimental data by plotting es/Clo against 2F/czo. Their theory predicts an increase of the skin friction reduc- tion with increasing Math number for the same injection rate and Reynolds number. All measurements and theories indicate an opposite tendency. Introducing a Reynolds analogy as defined by eq. (2.14) Spalding et al. extended their calculation method based on the similarity parameters
~ . - L
0"9
0"8
0"7
0 '6
0.5
0 . 3 '
M = = q
0 . 2 .
0 . 1 " : sp=4,~i,,g ~ ,~ [zo] (,~ = o. R~ = lo', r . / r . = 0.4)
• ~ o ~ ; S p a l d i n g o r aL [ 2 0 ] ( ~ . 4, R x = I06 , T . = T )
' 0 ' I ' ~ ' ' I " '
o 1 Z 3
l~o. 15. The reduotion of tlw Stamton number due to air injection (com- parison of Spalding e t al.'s cs°~ theory with experimental data).
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 103
F c, FRo and FRx to include the influence of heat transfer. Analysing experi- mental data they found with a considerable scatter that the Reynolds analogy factor s can be set approximately to 0.825 independent of Mach number, injection rate and heat transfer at the wall. Consequently Fig. 11 and Fig. 13 can be used to predict Stanton numbers c h when the ordinate Fcc / is replaced by schF c.
The prediction of Spalding's theory for the Stanton number c h is shown on Fig. 15 and compared with experimental data plotting Ch/Cho against F/cho for Rx = 10L As for the skin friction coefficient the Mach number effect is predicted wrongly, especially for incompressible flow (M = 0) where the discrepancy between theory and experiment is considerable. Even for a Mach number of 4 the agreement between theory and experi- ment is not very good.
Spalding et al.'s method should not be used for incompressible flow and flows at Mach numbers smaller than two. For these cases the predic- tion of the skin friction and Stanton number and especially their variation with the Mach number is questionable and disagrees with every other theory and, what is more significant, with all measured data for incompres- sible flow. The advantage of Spalding et al.'s theory is the relatively quick calculation of skin friction and heat transfer coefficients so that it might be interesting for quick engineering calculations. The last point might justify an improvement of the present calculation concept to include in more detail the Mach number effects, especially the influence of M_ o n FRo.
2.1.5. Boundary layer transformations
2.1.5.1. The present status of boundary layer transformations. A com- pletely different approach to solve the problem of compressible turbulent boundary layers with air injection is the application of a boundary layer transformation. All the transformations between compressible and in- compressible boundary layers are inspired by the work of Dorodnitsyn cv°) and Howarth c~1) who showed that a simple change of the coordinate y normal to the wall into
Y
Y* = f -~* dy (2.23)
could reduc the compressible laminar boundary layer equation to ad quasi-incomepressible form. This work was extended by IUingworth (n) an Stewartson ¢v3) who showed that a transformation of both coordinate
-
104 L . O . F . JEROMIN
could produce an exact mathematical correspondence between two-dimen- sional laminar incompressible and compressible boundary layers for the case of zero heat transfer, unit Prandtl number and assuming that the viscosity can be taken proportional to temperature. In this work it was assumed that the stream functions in the two flows are invariant under the transformation.
In the case of a turbulent layer it is, of course, impossible to get a complete mathematical transformation since the mechanism determining the shear stress distribution through the boundary layer cannot be ex- pressed in mathematical terms at present so that only a theory-experi- ment compromise remains. A number of authors (Spence ~74~, Mager t75~, Burggraf t78~) have extended the transformation concept for laminar flows to turbulent flows. These authors assumed that the stream functions remain invariant. Despite these restrictive assumptions all these authors did succeed in getting good agreement with experiment for some of the boundary layer properties.
A notable advance in the transformation technique was made by Coles ~62~ who assumed that the stream functions in the two corresponding flows were not equal. This work was further extended and rationalized by Croc- CO (77). Obviously the fact that the stream functions are not equal in the two flows increases the mathematical complexity of the transformation but reduces the number of arbitrary assumptions. It is true that their substructure hypothesis does represent a compromise with the unknown nature of turbulence, but this hypothesis may be readily adopted as the theory of turbulence is developed.
Coles t62) succeeded in reducing all available experimental data for the skin friction coefficient of compressible boundary layers to corresponding incompressible values. Recently Baronti and Libby tTs) investigated the correctness of a point-to-point mapping of compressible turbulent bound- ary layer flow into its corresponding incompressible form using Coles's transformation. They did not use Coles's substructure hypothesis but preferred to employ the assumption that the Reynolds number associated with the laminar sublayer is the same for compressible as well as incom- pressible flow. Baronti and Libby succeeded in correlating the compres- sible boundary layer profile (up to a Maeh number of M = 6) with the law of the wall for the region where it normally holds for incompressible flow. Discrepancies arose, however, when they attempted to transform the outside part of the compressible boundary layer profile into the veloc- ity defect law. They attributed these discrepancies to favourable pressure gradients, but it is quite possible that the reasons are deeper and might
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 105
be related to the sublayer concept itself or a failure of the boundary layer transformation in this region.
Rosenbaum t79~ tried to extend Coles's transformation concept to com- pressible boundary layers with heat transfer and mass diffusion. He assumed that the incompressible flow field is completely known when the boundary layer is divided in the usual manner into a laminar sublayer, a region where the law of the wall holds and another where the velocity defect law is valid. The corresponding compressible flow field can be easily determined from the incompressible one with the help of Coles's transformation. However, contrary to Croceo tt~n Rosenbaum used the now known compressible flow field to solve numerically the equations of conservation of energy and species by an implicit finite difference solution technique. This approach is questionable because his calculations are based on the same incompressible boundary layer with zero heat transfer and zero mass diffusion whether, or not, heat transfer and mass diffusion are present in the corresponding compressible flow. This approach postulates that the compressible boundary layer with heat transfer and mass diffusion can be reduced completely to a corresponding incompressible flow with zero heat transfer and zero mass diffusion. This has not been proved. In this connection one must remember that Rosenbaum used Coles's transforma- tion which is strictly valid only for the case of zero heat transfer. Despite these doubtful assumptions Rosenbaum obtained reasonable agreement between experimental data and his theoretical values for some boundary layer characteristics.
2.1.5.2. Colds transformation. It will increase the understanding of the transformation for transpired turbulent boundary layers if Coles's trans-~ formation for the solid flat plate case is discussed first. Coles's transforma- tion is valid for the case of zero pressure gradient and zero heat transfer, The idea of this transformation is to establish a correspondence between the compressible boundary layer described by the equations (2.4) to (2.6) and a simpler form of flow--in general an incompressible flow described by the equations (2.38) to (2.40)mwhich has been more thoroughly inves~ tigated experimentally as well as analysed theoretically. In order to reduce the compressible boundary layer equations (2.4) to (2.6) to the corre- sponding equations (2.38) to (2.40) for incompressible flow Coles introduced three transformation parameters ~, ~/ and ~ which are derived from the stream functions ~ and ~0* where starred symbols refer in general to the
t Crocco introduced transformation rules for the enthalpy distribution as well in ordet~ to transform the energy equation.
-
106 L.O.F. JEROMIN
transformed stage which is identical here with incompressible flow. The stream functions are defined in usual way by
Ou = ~ y , O v - ~x (2.24)
0*u* = B~p* ~*v* -- ~v* (2.25) By* ' Bx*"
The connection between the stream functions ~0 and ~p* are the transforma- tion parameters ~, ~7 and ~ which are in general functions of the coordinates x and y. They are defined by
~0*(x*, y*) tT(x, y) - - a(x) (2.26) ~o(x, y)
~* By* ~(x, y) - Q By - ~(x) (2.27)
dx* ~(x, y) -- dx - ~(x). (2.28) and
Coles showed that the three functions a, ~ and ~ were functions of x only, or rather that a, x* and By*/~ By are independent of y when second order derivatives are neglected. It can be proved that an exact correspondence for the inertia and pressure gradient term of the compressible and incom- pressible flow can be established mathematically when the transformation parameters are introduced into eq. (2.4) and (2.5). The weak point of the transformation is the deduction that the shear stress terms can be trans- formed as well, which cannot be proved in general because of the unknown shear stress distribution in physical as well as mathematical terms. Assum- ing that the shear stress distribution of the compressible boundary layer can be reduced to a corresponding one in incompressible flow and postulat- ing that a boundary layer with constant pressure transforms into an incompressible one with zero pressure gradient, the transformation leads to two expressions connecting the transformation parameters o, ~ and with the usual boundary layer variables, namely
and
. . . . = constant (2.29) U ~ U
(2.30)
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 107
with ~w -- 0 for the case of zero injection. Moreover, transformation rules can be derived for the interesting boundary layer characteristics like for example cf, O, R x connecting them with corresponding quantities in incom- pressible flow.
In order to determine the three unknown quantities ~, ~ and ~ one needs a third relationship, which Coles derived from his substructure hypothesis. He assumed that there exists a certain turbulent substructure correspond- ing to a well-defined Reynolds number R s within the boundary layer which is independent of compressibility effects when the density and viscosity are evaluated at a suitably chosen mean temperature 7",. This is a rather nebulous approach, and the physical interpretation of this concept rather doubtful, even when the substructure hypothesis is considered in its more rationalized form as presented by Crocco ~ . The substructure concept is mainly justified by its application since it succeeds in reducing all available data for the skin friction coefficients measured along solid flat plates to an equivalent incompressible value. This correlation must be regarded as excellent considering the published raw data obtained in different wind tunnels with their unavoidable irregularities of tunnel con- ditions and the effect of tripping devices of the experimental set-up, for example.
2.1.5.3. Jeromin's transformation. Jeromin ca' ~a) extended Coles's trans- formation concept for compressible turbulent boundary layers on solid surfaces to layers with air injection. The present transformation concept is practically identical to that of Coles since it is based on the same definitions for the transformation parameters ~, ~7 and ~ so that the equa- tions (2.29) and (2.30) hold here as well. The difference occurs in the deter- mination of the third relation connecting the boundary layer parameters with the usual boundary layer variables.
For the case of fluid injection this relation follows from the definition equations for the stream functions (2.24) and (2.25). The stream functions ~p and v;* must hold all through the boundary layer and hence at the wall. Thus
X"
0
X
and ~w = - ~ Ov~ dx 0
when the virtual origin of the flow is set to zero for both the incompressible and compressible flow, which is no restriction to the problem. One obtains
8*
-
108 L. O. F. JEROMIN
assuming that o'v* and o~v w are constant along the wall,
o" . . . . (2.31) ~w OwVwX
making use of eq. (2.26). Equation (2.31) is the required third relationship defining, together with (2.29) and (2.30), all three transformation para- meters. The disadvantage of defining o, by eq. (2.31) is the fact that it does not include the case of zero injection. For the case of zero injection eq. (2.31) becomes indeterminate even when one tries to approach this limiting case by assuming very small injection rates Owv,, and O*v*~.
Eliminating a from (2.29) to (2.31) leads to
with the constant
do, 0 "2- Ao, dx
crx* + 2AF ~f - 2 A O*c~
A __ uL e * # * ewvw u~ ~ w Q*v~"
(2.32)
Once a is known the other parameters ~ and ~ can be determined from (2.29) and (2.30).
It is the principle of the present transformation that the properties for incompressible flow like u~, 0", #* and F*, which must be known to integrate eq. (2.32), can be chosen arbitrarily without affecting the trans- formation. Moreover, as Crocco pointed out one obtains the same trans- formation if the incompressible fluid chosen is a liquid or a gas. After establishing a starting condition a s --- f(Rxn, F, F*) and empirical relations defining c~ =f(R*,,, F*) and O * = f(R*, F*) equation (2.32) has been integrated leading to graphs a = f(Rx) with the parameters F and F*. These graphs are the key of the transformation and have been used to apply the transformation to measured compressible boundary layer pro- files with air injection.
The main advantage of a boundary layer transformation is the extension of the range of application of well-established theories for incompressible flow to compressible flow. As Stevenson (61) and McQuaid (zs) have shown,* Stevenson's law of the wall
2u* {~/1 . v'u* } 1 y*u*T -~-~- _ + - ~ - ~ --1 = --u In - - v , + C (2.33)
t For details see Section 2.2.
-
RESEARCH IN TURBULENT BOUNDARY LAYERS 109
with ~ = 0.410 and C = 5.00 t for transpired incompressible boundary layers is highly successful for determining the skin friction coefficient for measured boundary layer profiles. Equation (2.33) holds for the fully tur- bulent region of the boundary layer so that it can be considered as a for- mula defining u~* and hence c~ for a given boundary layer profile, for example by solving (2.33) for measured u* and y*. This concept can be used to determine the skin friction coefficient for the compressible bound- ary layer by transforming the quantities u and y into their corresponding incompressible form and evaluating u~* from eq. (2.33). The friction veloc- ity u~* is constant in the region where the law of the wall holds. Its value will be chosen to determine the skin friction coefficient c~ since it gives the best agreement with Stevenson's law. The compressible skin friction coeflicient follows from the incompressible one, again using the transfor- mation. Skin friction coefficients determined in this manner agreed ex- tremely well with those obtained from the momentum equation as Jero- mincs) has shown for compressible boundary layers at M = 2.5 and M = 3.5 and three different injection rates.
The success of the present transformation can be illustrated best by plots in Stevenson's coordinates. It was postulated that an incompressible flow with an injection rate F* can be chosen arbitrarily. This postulate has been checked on Fig. 16 for two representative compressible boundary layers measured by Jeromin (~3'89) at a Mach number of 3.5 for two different injection rates. These two profiles were chosen at random and can be considered as being representative for all other profiles measured by Jeromin. The fully turbulent part of the compressible transpired boundary layer profiles has been reduced to Stevenson's law of the wall for incompressible flow independent which injection rate F* was chosen. Only injection rates F* up to 3× 10 -a can be checked for the time being since the transformation parameter cannot be evaluated accurately enough for higher injection rates (for details see Jerominta)). In Fig. 16 the collapse for the fully turbulent region is excellent for all injection rates F* chosen. The skin friction coefficient evaluated for these compressible profiles from Stevenson's law of the wall is independent of the choice of F*. A collapse for the outer part of the compressible profile cannot be expected from such a plot since the transformed flows correspond to different Reynolds num- bers R~* and R~. Moreover, one would expect such a reduction only by
t The values for u and Care mean values since there is a considerable amount of disagree- ment in the literature regarding their actual values. Coles's values u = ff410 and C = 5.0 were mainly chosen because they are good mean values (see Black and Sarnecki(S°)).
-
1 10 L . O . F . JEROMIN
28
26
24
22-
20-
18-
1
ExFerirncntal data: Jeromin [~.¢.]
~ ~ 3"5, ~X ~ 2.10 7. T W > Tro Symbol RUN I F'-IO) I F. IO ) R;.10 "~
+ 3'5 -0-6 - 1-90 1 0,664 6'70 "~ 3"5 - 1'2 -1"90 1 1.176 13-05
O 3"5 - 0"6 - 1 "90 2 0"664 5-35 • 3"S - 1"2-1'90 2 1'176 7'67
A 3"$ - 0.6-- 1.90 2'8 0' ~ 6"41 A 3"5 -- 1'2 - 1"90 2"8 1"176 9"89
[ / ] ~ • i c
1680O
,'~ oo ° AAA AAAA • AA
. " e ° ~ o g ° A6 Aa
$ 1 r / l n f , o, ' l)$ | ¢ w o f t h ( W O I I
( E q . ( 2 . 3 3 ) ) with ~:o-410. e=s.oo (co~., [e2])
Assumed corrlll~Oedln| I incompressible f low
iJQ = 16.0 [m/s} 1 ~ F ' . 10 ) -" 2'8
p'ff i 1 [at] I
16-
14 i I n y~ 'U I " y l
4 5 6 7 8 9
FIG. 16. Comparison between transformed compressible turbulent boundary layer prof~es with air injection and Stevenson's (et) law of the wall
using Jeromin's (8) transformation. Mach number 3"5.
plotting the outer region of the profile in appropriate coordinates such as
._-i- -- f (2.34) U~
of the velocity defect law for example. This must remain the subject of a separate investigation and is mentioned here only for reasons of complete- ness.
Figure 17 is representative for all compressible boundary layers measured at Mach numbers o f 2.5 and 3.5 with three injection rates F. The fully turbulent part o f the compressible boundary layer profile is again com- pletely reduced to a corresponding incompressible one. This collapse is not as good for the highest injection rate of M = 3.5. But this profile is nearly separating (blown-off) so that it is rather surprising that it could be collapsed at all. The inj~tion rate for the incompressible flow was chosen to be F* = 2X l0 -3. The best possible skin friction coefficient was fitted again to these profiles by solving Stevenson's law of the wall for the trans- formed compressible flow.
-
- f -
RESEARCH IN TURBULENT BOUNDARY LAYERS ] 1 l
There seems to be a big discrepancy between the profiles measured at a Mach number of 2.5 and those obtained for M -- 3.5, especially for the outer part of the profile. This applies for all the diagrams concerning Stevenson's law of the wall. The reason for this discrepancy is the different Reynolds number for the transformed compressible flow. This number is about twice as high for the profiles obtained for a Mach number of 2.5 than for
Experlmentat data : ]eromln [69] M.35 R .2107 T >T - A t --xXXXXXaoD
t ' " ' " ' • &&a "Ax ~ _ng13a l Sy.bol I RUN I ~.10' I R'-IO-' ~ ,.m . t x
zet [-Jr [-J I H I H [-J ,,- ,~" ./o0,~ /
| t . ~ x q k ~
• t
.. I ~:: 2 ° - ~ J [ • , ",,T,M ~ (~. (233))
0 Experimencal d a ~ a " Jef'omin [69] M = 2"5, ~, = 2"101, T w > 7" r - - - -
10 t / ~ ~EMxRuP[ ~ 1 ~ { - - ~ e ] S y m b o l RUN F, 10 ] R~-IO-* R e
1G- t ~ . ;~s-1~z-i~o i.17o I 13~8 )sooo
1/., o x In ~mU?
1 I l l l
4 5 6 7 8 9
FIG. 1"/. C o m p a r i s o n be tween t ~ n s f o ~ e ~ l compress ib le t u r b u l e n t b o u n d -
ary layer profiles with air injection and Stevenson's (") law of the wall using Jeromin's ~s) transformation. Mach numbers 3"5, 2"5.
those at M = 3.5. McQuaid (~s) showed that the profiles which he measured in incompressible flow shift to the right (the level of y*u*/~* increases) for increasing Reynolds number R~ when the profile is plotted in Steven- son's coordinates. At the same time the overall length of the fully turbulent region increases with increasing R~. These are exactly the same features as found for the transformed flow as one can see for example on Fig. 17.
Experimental data for the skin friction coefficient and values for cj. evaluated from Stevenson's law in connection with the present boundary layer transformation are compared on Fig. 18 with each other. The agree- ment between theory and experiment is excellent for both Mach numbers
-
1 1 2 L. O. F. JEROMIN
c_~, [.] Cfo
1"0-
0'9 '
0 ' 7 -
0"6"
0'5"
0 * 4 "
0.3"
0 ' 2 "
0'1-
exper imenta l data (see Fig. 1) i i i 1 1 1 1 1 ~
: Jeromln 's [8][23] t ransformat . ion÷ Eq. (2.33)(M = 3.5, R x -" 10 7.
~ . ~ : Jeromin's [8] [23] t rans fo rmat ion+Eq. (2.33)(M = 2.5, xR .~ 10 l, T~. ~ Tr)
m ~ m : Eq. (2.33)(M = O)
0 . . . . ! , , , , I . . . . I
o 1 2 3 _z'F t-] CfO
FIG. 18. The reduction of the skin friction coetficient due to air injection (comparison of Jeromin's ~s,:3) transformation in connection with
St