An analysis of turbulent boundary layer flow characteristics and the computational procedure used are discussed. The integrated mass and momentum flux profiles and differentials of the integral quantities are used in the computations so that local evaluation of the streamwise velocity gradient is not necessary. The computed results are compared with measured shear stress data obtained by using hot wire anemometer and laser velocimeter techniques. The flow measurements were made upstream and downstream of an adiabatic unseparated interaction of an oblique shock wave with the turbulent boundary layer on the flat wall of a two dimensional wind tunnel. A comparison of the numerical analysis and actual measurements is made and the effects of small differences in mean flow profiles on the computed shear stress distributions are discussed

Childs, M. E.

Sun, C. C.

English

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https://ntrs.nasa.gov/search.jsp?R=19740021536 2020-03-23T06:09:56+00:00Z
N74-29649"
O
Unclas
BOUNDARY LAYER FLOWS
by
Chen-Chih Sun
and-
Morris E. Childs+
University of Washington, Seattle, Washington
Index Categories: Boundary Layers and Convective Heat Transfer -
Turbulent; Supersonic and Hypersonic Flow;
Shock-Waves
(NASA-CR-138302) CLLCULATION OF TURBULENT
SHEA-d STRESS IN SUPERSONIC BOUNDARY LAYER
FLOWS (Washington Univ.) 15 p HC $4.00
CSCL 20D
-
	
G3/12
Research Assistant Professor, Department of Mechanical Engineering,
f.	 University of Washington, Seattle, Washington. Member ALAA.
S
+ Professor and Chairman, Department of Mechanical Engineering,
UniversiLy of Tashington, Seattle, Washington. Memos ALA&.
This work was supported by HASA Grant NGR-48-002-047 under administration
of th aerodynamics Branch, Ames Res.,arch Center.
(I
Pr wry	 .^^...........o....»_..,..^.:...:._.._.....a.._...r.
2
SYMBOLS
C - skin friction coefficient
M - Mach number
P - pressure
r - distance normal to centerline
R a radius of the dact
Tij = total stress tensor
u - time averaged velocity in primary flow direction
V - time averaged velocity normal to centerline
x = distance parallel to centerline
y - R-r
b = boundary layer thickness
u = molecular viscosity
= y/6
P = time averages density
T - shear stress
Subscripts
e	 = boundary layer edge condition
OD	
= free stream condition
Superscript
<( )'>	 = time av z:; •aged flUctu;".t.i.:,,n value
PRIDING PAUL BLANK NOT FILL
won
3INTRODMION
in the study of supersonic turbulent G-undary layer flow the turbulent
shear stress distribution has always been ®2 great importance and interest.
Thu direct seawranent of the turbulent shear stress ism however, quite difficult.
A natural alternative is to compute the shear from experimental swan flow data
by numerically integrating the momentum equation. Such computations have been
performed in recent studies by Bushnell and Morrisla Horatman and Owen2 , and
Sturek3 . This note describes results obtained by u computational procedure
which differs from those previously reported in that integrated mass and momen-
tum flux profiles and differentials of these integral quantitieu ozo used in the
computations so that local evaluation of the streamwise velocity gradient is not
necessary. The cr •aputed results are @&dared with measured shear stress data
obtained by using hot wire anemometer and laser velocimster techniques in recent
studies by Rose and Johnsond ' S. The moasurements of Rote and Johnson were made
upstream and downstream of an adiabatic unseparated interaction of an oblique
shock wave with the turbulent boundary layer on the flat wall of a too -dimen-
sional, M. - 2.9 wind tunnel. The shock wave was generated by a 7' wedge.
The turbulance data obtained from the two independent systems of measurement were
in reasonably good agreement, indicating that the data should be reliable. The
computational procedure developed here is easy to use and the computed results
show reasonably good overall agreement with those obtained by direct measurement.
As would be expected for any method of computing shear stress from mean flow
data, the computed values of shear stress are quite sensitive to small differences
In mean flow profiles and to simplifving assumptions Vo'ch may be made in de-
veloping the rel8tionzihip s zo be used in the computations. The effect of some
of these differences .,a  owputed shear. stress distribltti^ons is discussed.
r	 i
[
4
BASIC EQUATIONS AND BOUNDARY CONDITIONS
The time-averaged equations for the conservation of mass and momentum
for steady compressible turbulent boundary layer flow in an axisymmetric
chataiel are, respectively,
ax (Pu) + 2x ^_P ' u'> + r 3r (rpv) + r 2r (r <p'v'>) - 0	 (1)
and
8x (PU2) + r 3r (rpuv) _ - 2x + 2x Txx + r ar Trx	 (2)
waere
xx	 (TV)xx - (P<u' 2> + 2u <p 'u' > )	 (3)
TM
 _ (TV) rx - (P <u'v' > + u <P'v'> + v <plu'>) 	 (4)
with TV representing the viscous stress.
If we assume that b <Plul>l << IP <u'v'>[ and I
	
<P'u'>1 << 1 
ax 
(Pu) 1
ana transform to an x-y coordinate system, the continuity and momentum equations
may be combined and integrated in a direction normal to the surface to yield
2
T	 6	 Cf	 d6	 6 dpeue 6 dR (n put
P n2 
(1-n R) = 2 + (dx + P u2 dx + R dx) o P u2 do
e e	 e e	 e e
26d6	 62 dpeue rp. Pct( R dx + u2R ax ) ^o	 P u2 dop
e e	 e e
- (d6 + 6 dR + 6 dp eue u (n
 Pu do
dx R dx peue dx ue o peue
26 V	 62dpeue u rn	ou
+ ( R Tx + peueR dx ) ue o n pace do
P	 [^ Y^f
A^J
5+ d f -(R-nd) 
e- 
da T..) do
peu^II a
'o
n	 2	 d2 	 s1	 2
+ a ^ P u do - -Ijo n ^ do	
-
Pan*
n	 2	 n
" 3 a j 
ppu 
do + u 
aII a j n p do	 (s)
so n e	 a	 o e a
tqustiaa (5) becomes applicable to two-dimensional flop as II ^.
The normal stress, Tom , which appears in equation (5), is not known from
-am profile data. However, computations which have been made in this study
show that its effect is small. In the results shown, the streamwise gradient
of Txx has been neglected.
Tswwledge is also required of the static pressure distribution in the
boundary layer. In many studies of supersonic boundary layer flow no attempt
is made to measure the static pressure variation normal to the wall, even though,
for some adverse pressure gradient flows, the variation may, in fact, be rather
large. In most instances the static pressure at the boundary layer adge may be
determined with confidence. If this is done, the normal pressure variation may
than be represented in approximate fashion by assuming a linear distribution
between the wall static pressure and the pressure at the boundary layer edge.
Results are shown here for'both a linear static pressure distribution and a
constant static pressure.
Examination of equation (5) shows that an accurate value of boundary layer
growth rate is very importa•it for the calculation of the shear stress. However,
precise determination of tha boundary layer thickness from experimental mean data
is difficult. It is even iwre difficult to evaluate the ,ioandsry layer grmrb
rate accurately. 1"nis problem may be avoided by uwi.ng the condition that the
6 1
shear stress diminishes to zero at the boundary layer edge and solving
equation (5) for d8/dx:
2
dd _	 ff	 d	 dpene
dx	 dx
1
d dR
+ R dx) jo
Pu22 do -
d2 2	 1dpa a
dx	 jon
Put
—^ do2 + (Peue2 Pau2R
P
d dRd	 dPeue
- (R	 +dx
1(	 Pu	 d2) pane do +
dPeue
dx
1
Pujon do
Pee	
dx
J.
peueR paue
1	 2
+ d	 j ^ do -ax 2	 1aR 2x j n 2__T do - d 1 dojo	 e eue o Pee
±
R
	1
+	 'Fic j n PPu doJ0, dP+ PP a2 (1	 2R)e e ea
d	 d(Pe pw)	 1
+ 2
	 (2
_	 6	 2d
3.)l/[ aJ
1
(	 Pun Zdo-
1j PutZdodxPeue o	 Paua 0 Psue
1
do -+ j Pu	 28
1	 (p -p )
Pu	 e wdo + 1(2 -
d
3R)l (6)
R
o Pene on Peue P u2e e
In equation (6) a linear pressure variation normal to the wall has been assumed.
If the streamwise gradient ofrthe normal stress is ignored and if the
flow is assumed to be locally similar, equation (5) reduces to the expressi=n
used by Bushnell and Morrisl in their computation of shear stress.
Before solving for the shear stress from equation (5), it is also necessary
to know the coefficient of skin friction. This may bi- obtained by us4.ng the wall-
wake velocity profile proposed by Sun and Childs 6. The method of least-squares
may be used to fit the wall-waive profile to the experimental mean velocity pro-
files to provide -alnes .) C  and to provide a si:oothed representation of the
mean velocity diuLribat+LL.
RrSULTS
In ca-y.71+:;•, :-- 7`le ccn;,:r;riras 'or tha t	 downstream of the ..:a r ; w_­:._
boundary layer interactions, the departure from local similarity has been
i	 a
taken into account. For purposes of comparison, however, computations based
on local similarity have also been made. The effect of static pressure variation
normal to the wall has also been considered for the downstream stations by assum-
ing a linear variation in pressure. In these :omputations the static pressure
at the boundary lay edge has been computed from the free stream total pressure
and pitot pressure, with appropriate allowance made for the loss in total pressure
across the shock system. The velocity and density profiles needed for the compu-
tations were obtained from mean flow pitot profiles with appropriate allowance
made for static pressure variation across the boundary layer, and under the
assumption of constant to total temperature across the boundary layer. As was
mentioned in the previous section, the mean velocity profiles may be smoothed
by using a least squares fit of the wall-wake velocity profile to the experimental
profiles. Computations have been made for both smoothed and unsmoothed profiles.
Figure 1 shows shear stress distributions computed for an upstream (x = 5.375 cm)
station in Johnson's and Rose's investigation, along with their shear stress data
from the hot-wire anemometer and the laser velocimeter measurements. The computed
shear values which are shown have been obtained under th assumption of local
similarity. As is shown ; the calculated results agree quite well over much of
the boundary layer, with the measured results obtained with the laser velocimeter.
The differences be :weer. the calculated results and the hot-wire results are
greater. For both the hot-wire and laser velocimeter measurements the peak values
of shear stress are seen to occur substantially farther from the wall than is
observed for the calculated distributions. Also shown in the figure are values of
the wall shear stress as determined by a least squares fit of the wall-wake profile
to the mean data and as measured by a Preston tube. The agreement between the two
shear stress values is good.
II
s
For purposes of comparison, a wall-wake profile propused earlier by Maise
and McDonald 7 h-s been used to represent the velocity profiles obtained by Rose
and Johnson. As is shown, the shear values computed from the Maise-McDonald
profile are substantially lower than those obtained with the unsmoothed data or
the wall-wake representation described in Reference 6.
Results of the computations for the data of Rose and Johnson for a station
downstream ofthe shock wave-boundary layer interaction (x = 9.375 cm) are shown
in Figure 2, along with their measured shear stress distributions. Several sets
of computations have been made from the mean data, including one, for purposes of
comparison, in which local similarity has been assumed. Results have been obtained
for both smoothed and unsmoothed profiles and for both linear and constant static
pressure distribution across the boundary layer. The velocity and density profiles
used in computing the shear stress are shown in Figure 3, along with the wall-wake
representations of the velocity profiles.
As is shown in Figure 2, the measured shear stress values obtained by the
two experimental techniques are in quite good agreement. These in turn agree
reasonably well with the computed values obtained by using smoothed nonsimilar
profiles and the assumption of a linear pressure variation across the boundary
layer. There is considerable d if f erence between the results for the smoothed
and unsmoothed profiles, with the peak shear stress value for the unsmoothed
profile occuring lower in the boundary layer. Near the wall the difference
apparently was due primarily to inaccuracy in the numerical integration proc",s
for the unsmoothed data. Much smaller step sizes could be used with the smoo_hed
data. The results obtained from the wall-wake profiles are valid, of course, oale
if the profiles provide accurate representations of the actual velocity dis-
tributions. In the outer part of the boundary layer the wall-wake profile 1 -ro, d
an excellent representation of the data at one station but did ruse fit c-,t <i'
19
_ A
r	
1
9
the other. This would account for soma; of the difference in computed shear stress
for smoothed and unsmoothed profiles in that region.
The computed shesx valuea are quite sensitive to the assumptions regarding
static pressure. The computed values of static pressure at the boundary layer
edge at the first and second measuring stations differed frou those at the wall
by 3.0 and 1.2 percent, respectively. As is shown, the computed values of shear
stress with no consideration given to the pressure difference across the boundary
layer were substantially lower than those obtained when the pressure difference
was considered.
As is also sho•m in Figure 2, the results obtained under the assumption of
local similarity are markedly different from those determined when similarity is
not assumed. The peak shear stress levels computed assuming local similarity are
less than half the values computed when similarity is not assumed. Furthermore,
Vie shapes of the shear stress distribution curves are quite different. These
results occur even though the differences between profiles at two closely spaced
stations are small. The density and velocity profiles in Figure 3 were obtained
at stresmwise stations located approximately one l-nundar y
 layer thickness apart.
Although the profiles at successive station:, appes at firs*_ glanc to be quite
similar, striking differences are found for th_ cti-ainited :-hear stress distributions.
CJE?fDTS' n'O-'
An alternative t F	 a£	 .-.ut Inc; sa-..c i
mental mean profile: ,,',re
developed. Tne me, ?,t-	 t:	 -._rut from t?
integrs:ted r.,as sr,_	 a	 ai
quantities arm use:?
	 a^c :ions so t^,;r
aL_t;ibution from _:, eri-
bu rl y:'.-, laye_- f?r^-r ite:: beeo
^ ,	 : eported in tl,^r
le of t'^ '	r t.-.<<ul
e...!^uation nt the =t eat
wise
`'
11
a
i
i
j
10
turbulence data for two-dimensional et'iabatic boundary layer flow in the
i regions upstream and downstream of an oblique shuck wave interaction. Th.:i
j
computed results are quite sensitive to the accuracy of the numerical inte-
grations required in the computational procedure and to the mean property
I
distributions in the boundary layer. The assumption of bcal similarity may
I
cause large errors in computed shear stress values for flows subjected to
j;	 pressure gradients, even though adjacent profiles of the mean properties may
appear, on first examin&tion, to be quite similar. The computed shear stress
levels are quite sensitive to the static pressure distribution normal to the
I.
wall. Thus, if reliable shear stress distributions are to be obtained from
^i
mean profile data, the static pressure distribution must be known rather
j
accurately. The effect of the streamwise gradient of the normal stress on the
computed results is small and apD rently may be neglected.
r
L
11
REFERENCES
1. Bushnell, D.M. and Morris, D.J., "Shear-Stress, Eddy-Viscosity and
Mixing Length Distributions in Hypersonic Turbulen, Boundary
Layers," NASA TMX-2310, 1971.
2. Horstman, C.C. and Owen, F.K., "Turbulent Properties of a Compressible
Boundary Layer," AIAA Journal, vol. 10, no. 11, November 1972, pp.
1418-1424.
3. Sturek, W.B., "Calculations of Turbulent Shear Stress in Supersonic
Turbulent Boundary Layer Zero and Adverse Pressure Gradient Flow,"
AIM Paper No. 73-166, January 1973.
4. Johnson,D.A. and Rose, W.C., "Measurement of Turbulent Transport
Properties in a Supersonic Be: ,ndary-Layer Flow Using Laser Veloci-
meter and Hot Wire Anemometer Techniques," AIAA Paper 73-1045,
Seattle, Washington, 1973.
5. Rose, W.C. and Johnson, D.A., "A Study of Shcok Wave Turbulent
Boundary Layer Interaction Using Laser Velocimeter and Hot Wire
Anemometer Techniques," AIAA Paper 74-95, Washington, D.C., 1974.
6. Sun, C.C. and Childs, M.E., "A Modified Wall-Wake Velocity Profile for
Turbulent Compressible Boundary Layers," Journal of Aircraft, vol. 10,
no. 6, June 1973, pp. 381-383.
7. Maise, G. and McDonald, H., "Mixing Length and Kinematic Eddy Viscosity
in a Compressible Boundary Layer," AIAA Journal, vol. 6, no. 1, January
1968, pp. 73-80.
Lk
t 4
Figure 1.
Figure 2.
Turbulent shear stress distribution upstream
of a shock wave-boundary layer interaction,
two-dimensional tunnel [4]
Turbulent shear stress distribution
downstream of a shock wave-boundary layer
interaction, two-dimensional tunnel [5]
It
Figure 3.	 Density and velocity profiles downstream
of a shock wave-boundary layer interaction [5]
IV
1 I
V
2.5 G
M
2.0 Measured— (- ^<u'v'> /Peue] x 103
1
laser velocimeter [4]'
1 A	 hot -wire anemometer [4]
3
1. ` ® Cowputed — [(T - uau/ ay)/Peu.j x 103
^-
1 q 	 mean data (local similarity assumed)1
1 wall-wake presentation of mean a
i.0  *A data (local similarity assumed) 1
----- Maise-McDonald profile repre-
sentation of mean data [7]
1Q-Tw computed from wall-wake profile
` 0.5 (D -T 	 measured with Preston Tubei
♦ 1
I,
N
i 2
i
0.0 0.5 1.0
I- P <u'v' >/Peue] x 103 ,	 I(T - uau/ay) /peLa) x 103
Mea:iured - [- p<u'v'=/peue] x 103
Laser velocimater [5]
Hot-wire anemometer [5]
Computed - [( T - uau/ay ) /Peu_ x 103
q mean data, linear static pressure distribution,
Eq. (5)
— wall-wake representation of mean data, linear
static pressure distribution, Eq. (5)
mean data, aP/3y = 0, Eq. (5)
--•— wall-wake representation of mean data, aP /}y = 0,
Eq. (5)
O mean data, locally similar, linear static
pressure distribution
LM -^- - wall
-wake representation, locally similar, linear
^	 static pressure distribution
`J — T computed from wall
-wake profile
^' • w
\ A
O\ ` ^\ q
O\ o	 q
0 o \q Q
o^	 o ^ q
q	 ^
'.0
;u/ay) /a eu e ] x 103
O
2.,5
2.0
8
v
1.0
0.5
^a
P
1.0	 x a 8.105 cm
p	 velocity
—
	 wall- wake representation of
velocity
Adensity
x 9.375 cm
velocity
-- -	 wall-wake repre.Fentation of p
.0	 0.5
	
velocity	 ^1
^	 p
>+	 q 	 density	 LLJ
n 	 A


Calculation of turbulent shear stress in supersonic boundary layer flows

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