Pergamon
MeehardcsResearchCommunieatiotts,Vol.25, No. 1, pp. 3348, 1998 Copyright© 1998EhevierScleneeLtd Printedin the USA. All rightsreceived 00936413/98$19.00+ .00 PII S0093 6413(98)000056
LOW REYNOLDS NUMBER AXISYMMETRIC TURBULENT B O U N D A R Y L A Y E R ON A C Y L I N D E R
Fouad A. Youssef', Sadek Z. Kassab ~, and Sami F. AIFahed Center of Research for Experimental Thermal Sciences Mechanical & Industrial Engineering Dept. College of Engineering & Petroleum Kuwait University P.O. Box 5969 Safat 13060 Kuwait
(Received 19 November 1996; accepted for print 11 November 1997) Abstract
In the present study, an axisymmetric turbulent boundary layer growing on a cylinder is investigated experimentally using hot wire anemometry. The combined effects of transverse curvature as well as low Reynolds number on the mean and turbulent flow quantities are studied. The measurements include the mean velocity, turbulence intensity, skewness and flatness factors in addition to wall shear stress. The results are presented separately for the near wall region and the outer region using dimensionless parameters suitable for each case. They are also compared with the results available in the open literature. The present investigation revealed that the mean velocity in near wall region is similar to other simple turbulent flows (flat plate boundary layer, pipe and channel flows); but it differs in the logarithmic and outer regions. Further, for dimensionless moments of higher orders, such as skewness and flatness factors, the main effects of the low Reynolds number and the transverse curvature are present in the near wall region as well as the outer region. © 1998ElsevierScienceLtd 1 Introduction
The study of turbulent boundary layer that develops as the fluid flows over surfaces with curvature normal to the mean flow gains its importance from the many applications it covers in practice. Axial flow along slender bodies of revolution, ship models, missiles, aircrafts, towed submerged cables or cylindrical bodies are some examples. Although this type of flow is important, it received less attention than the flow over a flat plate. Therefore, the body of experimental data available for this type of flow is limited. This may be due to the fact that the analysis of boundary layers on a cylinder (a simple type of this category) is complicated by the additional length scale related to the transverse curvature of the wall. As pointed out by Neves et al. (1994) this extra length scale gives rise to several flow regimes that are characterized by the ratio of the cylinder radius, a, to the flow length scales: the boundary layer thickness, 8, and the viscous length scale, ~v, (where v is the kinematics u
• On Leave from Mech. Eng. Dept.. Faculty of Engineering. Mattaria, Helwan University, Cairo, Egypt. "On leave form Mech. Eng. Dept.. Faculty of Engineering, Alexandria University. Egypt. 33
34
F.A. YOUSSEF, S. Z. KASSAB and S. F. ALFAHED
viscosity of the fluid, and u' is the friction velocity). Consequently, this leads to several au" 6 aU ~ parameters namely, a" =   , ¥ =  , R, = , and ~ = (where, U= is the free Vu a 2 V a V stream velocity, and x is the distance from the leading edge of the cylinder where 8=0) Although Lueptow (1990) noted that the appropriate nondimensional parameter that incorporates the effect of transverse curvature is still debatable, he preferred using the 6 transverse curvature ratio, ? =  , in his review because of its clear interpretation. a
Considering both the boundary layer on a cylinder, and the flat plate boundary layer the results of Willmarth and Yang (1970) showed that the velocity profile is much "fuller" when transverse curvature is present. Patel (1973) observed a remarkable similarity between the influence of a favorable pressure gradient on the flow in the wall region of a two dimensional boundary layer and the influence of transverse surface curvature on the flow in the wall region on an axisymmetric turbulent flow In addition, the fundamental differences in the functional dependence of the turbulent quantities measured on a cylinder and a flat plate made Lueptow et al(1985) demanding further work in this area of research to understand the underlying mechanism of turbulence. Moreover, from the variable interval time averaging "VITA" and uvquadrant event detection, Lueptow and Hafitonidis (1987) reported that the burst mechanism for the generation of turbulence near the wall in the cylindrical boundary layer is similar to that for other wallbounded flow. Meanwhile, the flow visualization suggested that the turbulent transport of fluid in the boundary layer on the cylinder is very different from that in the planar boundary layer. This is because the flow is less constrained by the wall in boundary layer on a cylinder than in the boundary layer on a flat plate. Furthermore, Lueptow (1990) found that the distribution of turbulent quantities in the boundary layer on a cylinder is somewhat different from a planar boundary layer. He also realized that very little data are available based on parameter variation accomplished by measuring the boundary layer characteristics at different axial location, given the same transverse curvature in order to vary a*,7, or V, independently from R a. Meanwhile, Snarski and Lueptow (1995) found that transverse curvature has little effect on the fundamental turbulent structure of the boundary layer for the moderate transverse curvature ratio used in their study They further noticed that transverse curvature may enhance the large scale motion owing to the reduced constraint imposed on the flow by the smaller cylinderical wall. Low Reynolds number flows play an important role in many fluid flow problems including flow through turbomachinery, flow over wings, numerical modeling and model testing in wind tunnels. Coles (1962) found comparatively large change in turbulent boundary layers in zero pressure gradient at momentum thickness Reynolds number, Ro, less than 5000. Further, Purtell et al. (1981) noted that the Reynolds number effect penetrates the boundary layer much deeper in terms of the turbulence intensity than it does for the mean velocity. Moreover, Murlis et al. (1982) supported the suggestion of Huffman and Bradshaw (1972) that the main source of low Reynolds number effects is the viscous superlayer (the region at the instantaneous edge of the turbulent boundary layer in which vorticity is transferred to originally irrotational fluid by the action of viscosity).
LOW REYNOLDS NUMBER TURBULENCE
35
In an effort to extend both the experimental data base and the understanding of the low Reynolds number turbulent boundary layer flow on a cylinder in axial flow, the present experimental investigation was performed. It includes the study of two different effects on the turbulent boundary layer namely, the transverse curvature effect and the low Reynolds number effect. Each effect was studied separately by other investigators before. However in the present study they are combined and investigated experimentally using hot wire anomemetry. In addition, the obtained results are discussed and compared with some other published results.
2 Experimental Setup and Procedure The present study was performed using the 460mmx460mm blower tunnel facilities available at the Fluid Mechanics Lab, Mechanical Engineering Department, Kuwait University. A brief description of the present setup is considered, see Fig 1.a. The thyristor controlled variable speed fan delivers air to a short diffuser of the constant pressure type containing three screens. This leads to a settling length having a crosssection of 1.5m2 and fitted with a smoothing screen. The nozzle is connected directly to the end of the settling chamber. The outlet of the nozzle is of a square shape of internal dimensions 460mmx460mm and discharges directly to the atmosphere. Static pressure tappings at the entrance to and exit from the nozzle permits the free stream air velocity to be calculated. The maximum free stream velocity that can be obtained is 30 m/s. An investigation of the central plane of the working section showed that the root mean square value of the axial velocity fluctuations was not greater than 1%. of the mean velocity, indicating an acceptable level of turbulence. In order to reduce the model disturbance to the flow, the measurements were carried out using two pipes with open and sharp ( internally chamfered 30 ° ) ends, and having inner/outer nominal diameters of 15.6/21.2m, and 22.4/26.7mm. Further, to ensure a straight and smooth surface the length of each pipe was limited to 80 cm The pipe was tightly fastened by a hanging mechanism as shown in Fig. 1.b. It is fixed by means of a 0.2mm diameter piano wire, three at each end. Fine adjustments were carried out using tension screws attached to each hanging wire. An additional wire ( wire number seven) is placed at an angle to the free stream in order to resist any axial movement. Alignment of the pipe in the flow direction is carried out roughly prior to starting the experiment. It is then precisely alignment by comparing the mean velocity profiles obtained at a certain distance from the leading edge of the pipe at three different positions around its circumference separated by 90° . Measurements were carried out at three nominal free stream velocity of 5, 8, and 12 m/s. This range of velocities was chosen to obtain a low Reynolds number turbulent boundary layer. The flee stream velocity was maintained constant during each run.
36
F.A. YOUSSEF, S. Z. KASSAB and S. F. ALFAHED
All mean and turbulent measuremems performed through the course of the present study were done using Dantec miniature single hot wire probe type 55 P11. The probe is ptplated tungsten wire with a nominal length of 1.25mm and a nominal diameter of 51.tm. A standard 4mm Dantec probe holder was used. Following the recommendation of Bruun(1995, p. 266), the probe was inclined at a small angle ( 25 ° ) relative to the wall to remove the stem from the near wall region. Measurements were carried out using complete Dantec hot wire data acquisition system and traversing mechanism. It includes 56C17 CTA Bridge ( with resistance measurement accuracy < 5%), 56N22 main value unit ( resolution 100 ~tv and measurement error or nonlinearity + 200 p.v maximum), 66N20 signal conditioner( accuracy of gains +1%) and 56HOO Traversing system ( backlash < 0.3 mm with repeatability and minimum movement _+ 0.04mm ). The system is connected with PC type IBM 486. Following the analysis of George (1978) and after conducting the preliminary measurements, a suitable sampling frequency and data rate were used for each run. The hot wire was calibrated in terms of the free stream velocity before and after each run by positioning it with a 54N60 Flow Master (used as a calibration device, with 0.01 rrds resolution and an accuracy of + 2 5 % of reading), outside the boundary layer but in the core region of the jet. The tunnel speed was adjusted to obtain various known values of free stream velocity. The tunnel speed was then fixed to produce the desired free stream velocity. The next step was to carefully position the hot wire as close to the surface of the cylinder as possible which is followed by slowly traversing the wire away from the cylinder surface, and perpendicular to it, using the automatically preprogrammed traversing mechanism. At each point the output of the hot wire was recorded for further analysis of mean and turbulent quantities Throughout the present investigation, following Azad and Kassab (1986), the experimental data within the first halfmillimeter were examined carefully with respect to the linearity of the mean velocity within the viscous sublayer. The data points which seem to be in error due to proximity of the wall, are ignored, figure 2 illustrates this procedure. The source of errors and the uncertainties may arise in any experimental investigation should be studied carefully and evaluated to reveal to what extent the data obtained throughout the work is reliable. For the present study the estimation of errors ( random and systematic ) may be broken down into many categories such as; Uncertainties due to finite number of samples taken at each point, digital resolution, errors due to signal processing and errors due to fluid properties. Considering the errors due to finite number of samples, these errors equal to the root mean square value (RMS) of the measuring parameters divided by the square root of the number of samples, N This error is called the standard error and its value can be evaluated from the relation ( Bendat and Piesel 1971 and Taylor 1981):
LOW REYNOLDS NUMBER TURBULENCE
37
RMS According to this formula, the maximum expected error in velocity for the present study does not exceed + 0.03 m/s ( i.e. _=_0.4% for U~o=Sin~s). Regarding the errors due to digital resolution and errors due to signal processing, it can be seen that they are dependent on the specifications of the equipment used. In the light of the DANTEC equipment specifications one may conclude that this error is expected not to exceed 1%. In addition, the errors due to fluid properties were minimized by carrying out the experiment under atmospheric pressure and temperature. In general, an estimation of the uncertainty of the other measured values showed that they are comparable to the values reported by Bruun ( 1995 ). 3 Results and Discussion
In order to present and analyze the velocity data measured for cylindrical boundary layer, one should first measure and determine accurately the wall shear stress, %, (or the friction velocity, u ° =
, where p is the density of the fluid). It is believed that a reasonable
estimate is obtained in the present work. This is accomplished by measuring the mean velocity at a sufficient number of positions within the viscous sublayer from which the slope f
N
for the mean velocity profile, [d_~_,] , is obtained. Consequently, the wall shear stress xw, w
(or the friction velocity, u*=
, where v is the fluid kinematic viscosity) w
together with the y origin (absolute position of the hot wire from the cylinder wall) are determined. This technique is presented in Fig. 2. In addition, the friction velocity, u °, obtained using this method is given in Table 1. Another important parameter, which one should consider, is the Reynolds number range. As emphasized before, the turbulent flow is said to be in the low Reynolds number range if the Reynolds number based on the momentum thickness, 0, satisfyRo = U®0 < 5000. v Following Moore (1952) the momentum thickness,0, was estimated in the present study using the expression
38
F.A. YOUSSEF, S. Z. KASSAB and S. F. ALFAHED
(O+a) ~ a " =
1 U , ) l:,
where a is the radius of the cylinder, 8 is the boundary layer thickness at which =U 0.99 , and r is the radius from the cylinder centerline. The values of the momentum thickness,0, along with the momentum thickness Reynolds number, R0, the nominal free stream velocity, U~, the boundary layer thickness,& and the cylinder radius, a, are given in Table 1 for each run. It is clear from this table that the range of R~ for the present study is less than 2050. Consequently, the flow under consideration in the present investigation is a low Reynolds number flow. The mean velocity data for the present study are illustrated in Figs. 3, 4 and 5, using familiar format y ( yu"/l? ) versus :!" ( ::/;1") Also included in these figures are the plots tbr two well known relations: U" = y '
and
l!"  A l n y '
+("
where A = l, k is the von Karman constant = 0.41, and C=5.0. These adopted numerical k values of the loglaw constants k and C are the values suggested by the 1968 Stanford Conference on the computation of turbulent boundary layer [see Coles and Hirst (1968)]. Figure 3a presents the results for the first cylinder (d1=21.2 mm) and for three different nominal free stream velocities namely 5, 8, and 12 m/s and at four different axial locations, x ,namely 20, 30, 40, and 50 cm from the leading edge of the cylinder. While Fig 3b presents the results for the second cylinder (d2=26.7 ram) for the same velocities and the same axial locations as the first cylinder. In order to have a clear presentation for the data, the profiles presented in figure 3 are plotted as different line types. It is clear from Fig. 3 that, in the viscous sublayer (y_<_5) and the beginning of the buffer layer, the experimental data for various runs are collapsed. In addition, they are in agreement with the relation Uy This behavior of the present experimental data agrees with most of the previous experiments (for example, Rao and Keshavan 1972, Willmarth et al. 1976 and Lueptow et al. 1985) where, as noted by Neves et al. (1994), they find no effect of curvature in the mean velocity profile for y'<20. Further away from this region there is no collapse of the data and they spread This spread of the data covers both the logarithmic region as well as the outer region. This may be related to the fact that, as indicated in Table 1, each run has its own values for dimensionless parameters, 7, R 0, and a , different from the other runs. Moreover, Fig. 3 demonstrated clearly that the mean velocity profiles in the outer region deviate from
LOW REYNOLDS NUMBER TURBULENCE
*. Vm,~++~ . t a . 2 Dilex+~
39
6.0 I ~ RUN IZ4 5.0
;
'
 

   
4.0
.
2.o

1.0
 ~
'=i = 
 
.

.
    ~
:
 
.
I I
....
1 
Fig l,a Layout of the blower tmmel .
.
.
.
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Y (mm) Fig. 2 M e a n velocity m e a s u r e m e n t v e r y close to t h e wall.
1 Nozzle 2+ 5uppo~ng s,/sc~ 3 Cyllnde~
Fig I.h C y l i . d e ~ m,spensio~[ systcr,t
25
(a)
L
/
25
(n)
20 20'

15 m.~

__
u•
y
15

U*
IO
10 /
×~ I
o
~.3
+
,uN,x
I
o
~t.rl~ zla
~
ittr~ xJi
I
~
RtrN taz
×
nUN t33
I
~
R W ~z
m
Jmur
,~
o
/ 5

0 25 25 2o
(b)
20
15 U÷
U+ lO
/~* / .,i
10
o =?,;!: :7: °
@.,~1
o ~
" ,o,:, RUN214 i
+ /
RUN Rur~ Z~
RU~221
L
0
o 10
100
1000
I0
10000
velocity profiles in wall coordinates
a) Cylinder
#1
........
i ....... I000 I0000
y+
y+ Fig. 3 Mean
I00
b) Cylinder
for different #2
runs
Fig. 4 M e a n velocity profiles f o r t h r e e d i f f e r e n t free s t r e a m velocities. a) C y l i n d e r #1, Position #2
b) C y l i n d e r #2, P o s i t i o n #2
40
F.A. YOUSSEF, S. Z. KASSAB and S. F. ALFAHED
the logarithmic velocity profile by falling below it. This behavior has been termed a "negative wake" (Lueptow 1990). It is important to note that for a boundary layer growing on a flat plate the profiles rise above the logarithmic portion of the law of the wall in the outer region. Figure 4a presents the mean velocity data for different free stream velocities at certain location for the first cylinder, whereas, Fig. 4b represents it for the second cylinder. It is clear from these figures that in the logarithmic region, as the free stream velocity, U~ (or the momentum thickness Reynolds number, R0) increases the velocity profile rises up with nearly the same slope. In addition, one should also note that, in agreement with Fig. 3, the data collapsed in viscous sublayer and the beginning of the buffer layer and show a negative wake in the outer region. Moreover, in order to know which parameter leads to the shift in the mean velocity profile, figure 4a suggests that it is not the transverse curvature ration, y, since in runs 122 and 132 ~, is not greatly differed and yet the mean velocity profiles show a great difference. On the other hand, in order to isolate the effects of each parameter and show which one among them have a major eft?ct on the mean velocity profile, figure 5 is plotted. In figure 5a the results of run 131 are compared with run 121 where R o is similar ( see Table 1 ) but the other parameters are not. It is clear from this figure that the results of the two runs agreed with each other. Therefore variations of the values o f ' / a n d a* have no effect on the mean velocity profile Meanwhile, figure 5b compare the results of run 121 with 223 where a is almost the same but the other parameters are not. This figure shows clearly that varying R 8 and "r' have a marked effect on the mean velocity. However, as mentioned earlier( see figure 4a, runs 122 and 132 ) within the range of the present study, ~, has no effect. Conseqently, the variations of the mean velocity shown in figure 5b are mainly due to variations in R o . In addition, figure 5c compare the results of run 124 with run 134 where 7 is almost the same but other parameters are not. Similar to figure 5b, figure 5c shows that the variation of R o has the dominant effect on the mean velocity ( a ' i s also varied but figure 5a demonstrated that it has negligible effect ). Finally, the results presented in figure 5 show that within the limits of the present study, R o is the parameter which leads to the shift in the mean velocity profile. The results of the root mean square of the longitudinal fluctuating velocity component, /
u ' / = ~/1~7") , nondimensionalized by the friction velocity, u*, are presented in Figs. 6 and 7. In order to have a clear presentation for the data, the profiles presented in figure 6 are ploted 14 t
as different line types.
.
For all runs presented in figure 6, ; increases first up to a U
LOW REYNOLDS NUMBER TURBULENCE
41
maximum value and then it starts to decrease as y+ increase towards the edge of the boundary Ut
layer. A close look of the results indicated that the maximum position for ; occurs in the U
range ll_
U~ (or Ro) increases the magnitude of ; increases. //
U t
Regarding the position of maximum 7, Lueptow et al. ( 1985 ) reported that U
occurred at y+_16. While the results of Lueptow and Haritonidis ( 1987 )
u~/ //
J
max
demonstrated that u5) u
that u1 l/*
J
has occurred at y+_=13. In addition, Neves et al. ( 1994 ) indicated max
has occurred at y+_=13. Moreover, the position of u) max
Z/*
boundary layer, is nearly the same as u~ U *
for a cylindrical max
for the fiat plate, which occurs at y+=13 as max
U'
estimated from Klebanoffs (1955) data. Thus, the present results of 7 are in close U
agreement with these results. Consequently, one should recall here the statement of Lueptow U'
and Haritonidis ( 1987 ) regarding 7, that is " The similarity in the magnitude of the U
maximum turbulence intensity and its distance from the wall between the cylindrical boundary layer and other wall bounded flows suggest that the mechanism for the generation of turbulence is the same at least in the wall region ". Away from the wall and in the range U'
100_
The results of skewness of u,
S(u) =  
are presented in Figs. 8 and 9. The
experimental results in proximity of the wall are shown in Fig. 8 compared together with the simulation results of Neves et al. (1994). Before discussing these results, it is important to emphasize that the instantaneous value of odd moments can have either a positive or a negative sign and very slight offsets can adversely affect the accuracy of their measurements. Another difficulty with the measurements of skewness (and also flatness) is I
that the nondimensionalizing parameter in the case of u is(u 2)2, which amplifies the error if
42
F.A. YOUSSEF, S. Z. KASSAB and S. F. ALFAHED
it is small. Taking these effects into consideration the scatter of S(u) is understandable. Figure 8 shows that for y+<10 S(u) is positive. Further, away from the wall (y~>40) there are strong negative streamwise velocity fluctuations, as denoted by the negative skewness. In addition, Fig. 8 includes the results ofNeves et al (1994) for ~, = 5 and 11. It is clear from this figure that the present experimental data for ?"= 0.67 to 1.35 followed the same trend as Neves et al. simulation results. Moreover, the present results supported the results of Neves et al. in which an increase in the curvature does not significantly affect the skewness of u close to the wall ( f
< 20).
While away from the wall (y+>40) skewness of u decreases
with increasing curvature. The distributions of the skewness of u versus y are presented in 5 Y Fig. 9. The scatter in the results for  ?_0.5 indicates the effect of intermittent region of the 6 boundary layer Finally, the results of the flatness factor, F(u) = u4
(u2)2 

, are shown in Figs. 10 to 12.
Figure 10 presents a sample of the data in the region close to the wall. It shows that the data for various runs collapses in this region. Further, F(u) decreases in the near wall region, y~ _<10, before it starts to increase slowly for y" > 15. This trend is in close agreement with the simulation results ofNeves et al (1994) as shown in Fig. 11. In addition, it is clear from this figure that away from the wall ( y ' > 30 ) the results of the present study along with the results ofNeves et al. indicate that F(u) increases with increasing curvature. The results of the flatness factor, F(u), in the outer region of the boundary layer are shown in Fig. 12. This figure clearly defines the onset of the intermittent nature of the outer layer of the boundary layer at about Y_O.5, where the collapse of F(u) data starts to 5 disappear (the intermittency is the fraction of the total time for which the flow is turbulent). One should note that, for the case of a flat plate turbulent boundary layer, Frenkiel and Klebanoff (1975) and Sabot and ComteBellot (1976) have shown that the intermittency of the outer region begins around )' = 0.4 In addition, Huffman and Bradshaw (1972), judging S by the intermittancy measurements and smoke photographs of Fiedler and Head (1966), indicated that the irregularity of the interface between the turbulent and irrotational flow increases at low Reynolds numbers. While, for the case of turbulent flow growing on a cylinder, the result of Lueptow and Haritonidis (1987) showed that the intermittency level is Y nearly one until _y_ ~0.7 compared to _0.4 in the planar case. Although, they did not 8 8 make any conclusive statement about this difference, Leuptow (1990) pointed out that the unconstrained motion of eddies, in the case of turbulent boundary layer growing in a cylinder, allows them to "fill out" the boundary layer resulting in the interface being further
LOW REYNOLDS N U M B E R TURBULENCE
2.5 20
'i
I
43
i
(a)
2,0
÷ 15""
u
i 1.5
l0 / S • o
/

\
   _ U'IZ.441nY y*+~
I
t.0
ii
in.ill
®
ilu~m
+
llutml
,
ilumz, l
x
ilum~
o
tunll3
t
RUNI~
w~
'
0.5 (b) 20
~
÷ 15
0.0
~
U
2.5
~
(b)
1o 1.0 5
U*12,44In y*+ 5 ,
G
lllz~
i:x
RU~2~
[
0
1.5
0.5
0.0
:: ::,o .+, 0
........
I
........
i
10
........
100
~
........
1000
y+
b) a+
i
........
i
l0
........
100
i
. . . . . . . .
1000
10000
y+
i
10000
Fig. 5 C o m p a r i s o n between different mean velocity profiles having similar a) R e
........
Fig. 6 Distribution of dimensionless t u r b u l e n t fluctuating velocity, a) C y l i n d e r #1 b) C y l i n a e r #2
u~ u*
c) y
2.5
RUN III
(a)
RUN 121 2.0
1.0
RUN t31
i
1.5
(a)
0.5
2
0.0
N
1.0
0.5
!
o.s 1.0
[] RUN,.
b
o
RUN 113
/
O
RUN 114
I
~ ~
°
0.0 1.5 2.5
(b)
[
2.0
L
~
a RUN 112 ~ RUN
2,0
~ RUN 122 132
l.O 0.5
1.5
I.o
~o
0.0
/
~ ®
i
o
(
b
)
0.~ 1.0 0.o
........
i
I0
........
i
I00
........
i
I000
~ RUN231
. . . . . . .
I0000
y+
2.0
.
0.0 Fig. 7 D i s t r i b u t i o n of dimensionless t u r b u l e n t flactuating velocity for t h r e e different free stream velocities. a) C y l i n d e r #1, Position #1 b) C y l i n d e r #1, Position #2
L I
o R ~ 233 o RUN234 .
.
.
/ .
.
.
.
I
. . . .
i
.
.
.
.
0.5 Y/~ Fig. 9 Distribution of skewness factor, S(u).
1.0
44
F.A. YOUSSEF, S. Z. KASSAB and S. F. ALFAHED
from the wall. Consequently, taking into consideration the two opposite effects on the position of the interface between turbulent and nonturbulent motion (namely, the cylindrical wall effect and the low Reynolds number effect), the results shown in Fig. 12 can be easily understood. 4 Concluding Remarks From the preceding analysis and associated discussion of the experimental results of low Reynolds number turbulent boundary layer growing along a cylinder in axial flow, the following concluding remarks can be deduced:  The mean velocities for different runs are collapsed in the viscous sublayer and the beginning of the buffer layer However, they spreaded in the logarithmic and outer regions due to the variation of the momentum thickness Reynolds number and other dimensionless parameters  The mean velocity profiles in the outer region deviate from the logarithmic velocity profile by falling below it The present experimental results are in agreement with results of other previous investigators regarding the position of maximum dimensionless root mean square velocity ll' II
 For a certain position, within the logarithmic and outer region, as Reynolds number, Ro, increases the value of ~ increases II
 Increasing the curvature has nearly no effect on the skewness of u very close to the wall, while away from the wall skewness ofu decreases with increasing curvature.  Within the near wall region, the flatness factor increases as the curvature increases  In the outer region, the results of both the skewness and flatness factors indicate clearly the beginning of the intermittent region of the boundary layer Nomenclature:
a a A
cylinder radius dimensionless parameter [= au °/o] loglaw constant [=l/~c]
C
loglaw intercept constant
C,
skin friction coefficient [=2(u"
IU~):
F(u) flatness factor of u I= ~ / (~')z l
[=U/u'] f
1
U U,
dimensionless velocity free stream velocity
X y
distance from the leading edge dimensionless distance r[=
6
boundary layer thickness
(5'
displacement thickness
yu"/u]
H
shape factor [= d" / O]
7
transverse curvature ratio [=6 / a]
r
radius from the cylinder center line
K
yon Karman constant
LOW REYNOLDS NUMBER TURBULENCE
45
Table 1: Summary of data for all runs
Run
a
U~xx
x
8
0
u"
C~
a+ [a u'/u]
?
2(u'/U®f [5/a]
#
[mm] [m/s] [cm]
[mm] [mm] [m/s]
111
10.6
9.0
0.92
0.341 0.00870
0.85
3670
318
239
112
....
30
9.1
0.97
0.334 0.00883
0.86
3530
323
235
113
....
40
9.3
1.02
0.287 0.00931
0.88
2950
284
201
114
....
50
9.6
1.04
0.280 0.00899
0.91
2930
288
197
5
20
121
"
20
7.4
0.79
0.478 0.00746
0.70
122
....
8
30
7.8
0.84
0.476 0.00715
0.74
5500 5590
409 442
336 334
123
....
40
8.2
0.88
0.468 0.00702
0.77
5550
459
329
124
....
131
"
50
8.5
0.91
0.464 0.00691
0.80
5540
477
326
20
7.1
0.71
0.646 0.00616
0.67
8180
550
454
132
....
30
7.6
0.82
0.623 0.00582
0.72
8100
627
437
133
....
40
8.0
0.89
0.610 0.00568
0.76
8040
673
428
134
....
50
8.4
0.93
0.658 0.00519
0.79
9060
793
461
211
13.25 5
20
14.5
1.81
0.279 0.00621
1.09
4420
599
247
212
....
30
17.2
2.33
0.270 0.00611
1.29
4320
754
239
213
....
40
19.0
2.71
0.266 0.00591
1.42
4330
880
235
214
....
221
"
222
12
50
24.5
3.59
0.260 0.00554
1.84
4370
1175
23O
20
14.0
1.82
0.395 0.00529
1.05
6790
927
349
....
30
170
2.13
0.389 0.00478
1.27
7030
1123
223
....
40
18.5
2.55
0.381 0.0.0458
1.39
7050
1344
344 337
224
....
50
22.5
3.08
0.376 0.00446
1.69
7040
1625
332
231
"
20
13.0
1.51
0.55~ 0.00428
0.97
10670
1207
493
232
....
30
16.0
1.95
0.522 0.00389
1.20
10470
1527
462
233
....
40
18.0
2.26
0.505 0.00369
1.35
10470
1774
450
234
....
50
20.0
2.64
0.47~ 0.00333
1.50
10350
2049
423
8
12
xx Nominal Value x Based on actual values for U~ Is.
• urn,,, "
u
_ L ............
• n,.
 L
:: . . . . . . . . . . .
%
i ...........
::
o N,,,
~.~ '
o
(;,)
I.$. Oil
(¢)
.... o ......... .y
(d)
•
m~
ll,~
•
~a~, ~ % 0 o , % . .
%
" a '"a
.
,~_..~...........
zl
i
Ul y°
ii
im
~'J
:I
al
r,l y'
ii
+r,l
I
. ° ' o °
%
.....
11~ I
os
I',~ +I
.ii
61 y,
m
im
l
+I
am
all
II
y,
Fig. 8 Near wall d i s l r i l m t i o n of slcewne~ (actor, SOd ='~'T compared w i l h the reslllls of Neves el al. 0994).
Im
46
F . A . YOUSSEF, S. Z. KASSAB and S. F. ALFAHED 20
t
4
;Z 2
.......
i i
I0
i 5
0
. . . . . . . . .
(a) .
1
I
15
.
.
u l n zz"
.
.....
,
$
RUy 223
ta I
20
r  l ~ ~!4
.......
0
i
I
15
s
I0
5
0
(b) 5
rLr% 231
ZO
= runz2~ RU~222 _ _
15
ru~z~
10 20
lo
3O
40
5O
5' 0.
Fig.
10
Near
wall dist ribution
o f f l a t n e s s f a c t o r. FI u
:~
    i
5 0.0
0.5
Fig. 12 Distribution
" ~('~) 3
i
!
i
in ~
I
T
iiE!!i:!::.... i: (r)
I .......... tl.,.,,,: i .....
i
J
(llj
i
Y ~3
;
' 20
' 40
I
I
{111
8ll
I I10
N t ~ ~ s L ' h l t ( t~}941
,
]1
]
I
I
t
I
211
411
611
~1)
I Oil
l i B . I I ('4,1111~;11is(ill I)t'l~s ~u'll Ihe III t,selll r'c~ulfn td I1;lll~¢ss I;Icl(~l ;Ilnl the I t'slljls of N ~ ( : ~ t'l ;11. (1994) ill Ih¢ I)¢II1" s~all legh3l).
of flatness factor, F(u).
l.O
LOW REYNOLDS NUMBER TURBULENCE Ra
Reynolds number based on cylinder
Ro
radius [= Reynolds number based on the momentum thickness [= U~ 0 / o]
v
kinematic viscosity of the fluid
0
momentum thickness
p
density of the fluid
47
aU~/o]
u. u'
,o,,onvo,oo,ty root mean square of the longitudinal fluctuating
xw wall shear stress dimensionless parameter
velocity component [= xf~] U
longitudinal mean velocity
References
Azad, ILS. and Kassab, S.Z. 1986 Measurements of small distance in proximity of wall with a hot wire FED, Vol. 34, pp. 2325. Bendat, J. S. and Piersol, A. G., 1971 " Random data: Analysis and measurement procedure", John Wiley & Sons Inc, New York Bruun, H. H 1995 Hot wire Anemometry: Principles and Signal Analysis. Oxford University Press. Coles, D.E. 1962 The turbulent boundary layer in a compressible flow. Rand Report R403PR. Coles, D.E. and Hirst, E.A. Editors 1968 Proceedings of AFOSRIFP. Stanford Conference on Computation of Turbulent Boundary Layers (Stanford University, Stanford, Calif.). Erm, L.P. and Joubert, P.N 1991 LowReynolds number turbulent boundary layer. J Fluid Mech., Vol. 230, pp 144. Fiedler, H. and Head, M.R. 1966 Intermittency measurements in the turbulent boundary layer J. Fluid Mech. Vol 25, pp. 719734 Frenkiel, F.N. and Klebanoff, P.S 1975 On the lognormality of the smallscale structure of turbulence Boundary Layer Meteorology, Vol. 8, pp. 173200. George, W . K 1978 Processing of Random Signals. Proc. The Dynamic Flow Conference, pp. 757808. Huffmen, D. and Bradshaw, P. 1972 A note on von Karman's constant in low Reynolds number turbulent flow. J. Fluid Mech. Vol 53, pp. 4560 Klebanoff, P.S. 1955 Characteristics of Turbulence in a boundary layer with zero pressure gradient NACA Tech. Rep. No 1247.
48
F.A. YOUSSEF, S. Z. KASSAB and S. F. ALFAHED
Lueptow, 1LM. 1990 Turbulent boundary layer on a cylinder in axial flow. AIAA J., Vol. 28, pp. 17051706 Lueptow, ILM. and Haritonidis, J . H 1987 The structure of the turbulent boundary layer on a cylinder in axial flow. Phys. Fluids, Vol. 30, pp. 29933005. Lueptow, 1LM., Leehey, P. and Stellinger, T. 1985 The thick turbulent boundary layer on a cylinder: Mean and fluctuating velocities. Phys. Fluids, Vol. 28, pp. 34953505. Moore, F . K 1952 Displacement effect of a threedimensional boundary layer. NACA. Tech. Note No 2722 Murlis, J., Tsai, H.M. and Bradshaw, P 1982 The structure of turbulent boundary layer at low Reynolds numbers J. Fluid Mech., Vol. 122, pp. 1356. Neves, J.C., Moin, P. and Moser, R.D 1994 Effects of convex transverse curvature on wallbounded turbulence. Part 1 The velocity and vorticity. J. Fluid Mech. Vol. 272, pp. 349381. Patel, V.C 1973 A unified view of the law of the wall using mixinglength theory. Aero Quarterly, Vol. 24, pp 5570. Purtell, L.P., Klebanoff, P.S. and Bueldey, F.T. 1981 Turbulent boundary layer at low Reynolds number. Phys., Fluids, Vol. 24, pp. 802811 Rao, G.N.V. and Kesavan, N.R 1972 Axisymmetric turbulent boundary layer in zero pressure gradient flows. J. Appl. Mech., Vol. 39, pp. 2532 Sabot, J. and ComteBeilot, G 1976 Intermittency of coherent structures in the core region of fully developed turbulent pipe flow J Fluid Mech., Vol. 74, pp. 767796. Snarski, S. 1L and Lueptow,R. M 1995 Wall pressure and coherent structures in a turbulent boundary layer on a cylinder in axial flow J. Fluid Mech., Vol. 286, pp. 137171. Taylor, J. R., 1982," An Introduction to Error Analysis", University Science Books, Mill Valley, California. Willmarth, W.W. and Yang, C.S 1970 Wallpressure fluctuations beneath turbulent boundary layers on a flat plate and a cylinder. J. Fluid Mech., Vol. 41, pp. 4780. Wiilmarth, W.W., Winkel, R.E., Sharma, L.K. and Bogar, T.J. 1976 Axially symmetric turbulent boundary layers on cylinders: Mean velocity profiles and wall pressure fluctuations. J. Fluid Mech., Vol. 76, pp 3564