Experiments in Fluids 19 (1995) '73-'87 c Springer-Verlag 1995

The evolution of initially uniform shear flow through a nearly two-dimensional 90 0 curved duct

H. J. Lim, M. K. Chung, H. J. Sung

Abstract An experimental study has been made in a nearly two-dimensional 90' curved duct to investigate the effects of interaction between streamline curvature and mean strain on the evolution of turbulence. The initial uniform shear at the entrance to the curved duct was varied by an upstream shear generator to produce five different shear conditions; a uniform flow (UF), a positive weak shear (PW), a positive strong shear (PS), a negative weak shear (NW) and a negative strong shear (NS). The variations of surface pressure and the mean velocity profiles along the downstream direction under different initial shears are carefully measured. The responses of turbulent Reynolds stresses and triple velocity products to the curvature and the mean strain are also investigated. The evolution of turbulence under the curvature with the different shear conditions is described in terms of the turbulent kinetic energy and the various length scales vs the angular distance eor a curvature parameters Sc which is defined by Sc= (UlR)/(dUldy- UlR). The results show that the turbulent kinetic energy and the integral length scale are augmented when Sc < 0.054 whereas they are suppressed when Sc > 0.054. It is also observed that the micro-length scales of Taylor and Kolmogoroff are relatively insensitive to the curvature.

List of symbols

wall static pressure coefficient, Cpw=2(P-Prer)/ pU~

D duct width = 190 mm d turbulent diffusion H duct height = 600 mm k turbulent kinetic energy, k = q2/2 L streamwise integral length scale or duct length 1 characteristic length scale n normal distance from curved surface P wall static pressure

wall static pressure at x = 8D p pressure fluctuation

Received: 22 June 1994 IAccepted: 20 February 1995

H. J. Lim, M. K. Chung, H. J. Sung Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Yusong-ku, Taejon, 305-701, Korea

Correspondence to: M. K. Chung

S

t U,V,W

u,v,w

uy

u3 , y3, U2y, uv x,y,z

Greek symbols

i:

II J.

v ¢ o P T

Subscripts

o w

1 Introduction

U2+V+~

(q2)lfl

radius of curvature Reynolds number based on Uc and D, Ren = UcD/v ratio of curvature strain to straight mean strain, S=(UlR)/(dUldy) ratio of curvature strain to total mean strain, Sc= (UlR)/(dUldy- UlR) time mean velocity components in x,y, z coordinates, respectively centerline mean velocity at x = 8D velocity fluctuations in x,y, z coordinates, respectively characteristic velocity scale

Reynolds normal stresses in the x,y, z coordinates, respectively

Reynolds shear stress

triple velocity products streamwise, transverse and spanwise coordinates

boundary layer thickness dissipation Kolmogoroff micro-length scale Taylor micro-length scale kinematic viscosity pressure-velocity correlation term flow angle in curved duct density of air non-dimensional development time, T = (xlU) I dUidyl modified non-dimensional development time, Tc=(X/U) IdUldy- UlRI

initial value at 0 = 10°, y/D = -0.1 surface wall

The effects of streamline curvature on turbulence have been intensively investigated both experimentally and theoretically due to the dramatic change in the turbulence structure under

174

the stabilizing or destabilizing curvature. Most experimental studies have been carried out in turbulent boundary layers over a convex surface (So and Mellor 1973; Meroney and Bradshaw 1975; Ramaprian and Shivaprasad 1977, 1978; Muck et al. 1985), a concave surface (Meroney and Bradshaw 1975; So and Mellor 1975; Ramaprian and Shivaprasad 1977, 1978; Hoffmann et al. 1985; Barlow and Johnston 1988) or in a curved duct of high aspect ratio (Eskinazi and Yeh 1956; Ellis and Joubert 1974; Hunt and Joubert 1979).

A consensus among the experimental findings can be summarized as follows: The convex (or concave) curvature suppresses (augments) the diffusion of turbulent energy away from the wall, decreases (increases) drastically the intensity of turbulence, shear stress and skin friction, reduces (amplifies) the large scale motion normal to the wall and the length scale of turbulence and shifts the spectral distribution of turbulent energy toward high (low) wave number regions. Another characteristics of the curvature effects are that the velocity fluctuations parallel to the radius of curvature (v-component) are more strongly affected than those in the streamwise direction (u-component).

Hunt and Joubert (1979) discussed the effects of curvature on turbulence by investigating the governing equations in curvilinear s-n coordinates for the mean momentum and the Reynolds shear and normal stresses. The extra curvature strain UIR does not contribute any to the total turbulent kinetic

energy production. Since an extra production term 4uvU/R

appears in both the equations for Du21Dt and Dv2/Dt with different sign, the extra strain is thought to produce an 'energy pump' effect by which some of the existing energy is transferred without any loss between the components in the plane of streamline curvature.

Another important contribution of curvature strain to turbulence is closely associated with the third-order terms in the Reynolds stress equations. The order of magnitude analysis reveals that the curvature terms in the third-order moments are scaled by q31R while other terms are scaled with q31l, where

q = (u 2 +-;;: + w2 ) \/2 and I is the length scale. If the curvature is small or IIR ~ 1, then the curvature terms are negligible. However, in the separated recirculating region of a backward-facing step or in the corner regions of a duct or cavity, for example, the magnitude of R has the same order as I and thus curvature term is comparable to other terms. Even in the curved boundary layer flows (MR ~ 0.01), Ramaprian and Shivaprasad (1978) observed remarkable influence of the streamline curvature on the third-order moments.

In uniformly sheared straight flows, the sign of shear makes no difference in producing the turbulence kinetic energy (Ahmad et al. 1976). However, when the shear flow is associated with streamline curvature, depending on the relative directions between the shear and curvature, the turbulence is either enhanced or suppressed. Therefore a quantitative understanding about the interaction between mean shear and curvature is crucial to the analysis of the complex turbulent flow field with streamline curvature.

Recently, an attempt has been made by Holloway and Tavoularis (1992, hereinafter referred to as HT) in which the effect of curvature on uniformly sheared turbulence was experimentally investigated in isolation from any

other interacting mechanism such as the entrainment of irrotational flow, the proximity of viscous wall layers and the non-uniformity of the mean shear. A homogeneous shear flow for studying the interaction of pure shear and curvature was realized in a curved duct of which the cross section is 250 x 457 mm and the centerline radius R of 2.0 m and 5.0 m. The shear rate dUldy, was varied with a shear generator in a range, - 64 ~ dUIdy ~ 65, and the curvature parameter 5 between - 0.5 and 0.64, where 5 is defined as the ratio of the curvature strain to the straight mean strain,S = (UlR)1 (dUldy). Before discussing the experimental results, they analyzed the nondimensional equations describing the temporal evolution of the Reynolds normal and shear stress under the 'homogeneous shear' condition. They asserted that all dimensionless turbulence statistics in a curved homogeneous shear flow would depend on two parameters: namely, the dimensionless time T = (xl U) I dUIdy I and S. When 5 < 0, the curvature would enhance the production of turbulent kinetic energy and the Reynolds shear stress by the mean shear whereas the curvature opposes it when 5> o. The critical value of 5 for the onset of such opposite effects turned out to be 0.05 in their experimental result.

Although there have appeared a vast number of investigations in the past, quantitative information about the extent of curvature effects on turbulence structure is still lacking. The present experiment has been carried out to acquire quantitative information about the effects of interaction between the shear and the streamline curvature on the evolution of turbulence by investigating the turbulent flow field in a nearly two-dimensional 90 curved duct with a uniform shear inlet.

The present paper consists of four parts. The description of the experiment is given in Sect. 2. The Sect. 3 presents the wall static pressure and mean velocity profile. Turbulence structures in the curved duct under different shear conditions are discussed in Sect. 4. Parametric studies have been made to scrutinize the downstream evolution of turbulence under the curvature influence, which is described in the final part of Sect. 4.

2 Experimental apparatus and procedure

2.1 Wind tunnel and shear generator Figure 1 shows a schematic layout of the present experimental apparatus. Air from a 7.5 hp centrifugal blower passed through a diffuser, a settling chamber and a two-dimensional contraction of area ratio 3.2: 1. An adjustable shear generator was equipped at the exit of the contraction to provide a uniform shear flow at any desired shear rate. The basic design concept is similar to that of Chung and Kyong (1989). It consisted of nine equally-spaced channels, of which width was 19 mm. The flow passing through each channel was individually controlled by adjusting the opening of a damper which was composed of two parallel perforated acrylic bands with equally-spaced rectangular holes. The opening area ratio of the damper can be adjusted by sliding the one perforated band over the other in a range of 31.7% to 68.8%. The maximum mean shear rate, IdUldyl, obtained by the shear

Fig. 1. Schematic diagram of wind tunnel 1 concentration nozzle; 2 shear generator; 3 developing straight duct; 4 curved duct; 5 redeveloping straight duct. Duct height = 600 mm. All dimensions are in mm

175

2.3 Test of present curved duct In the vicinity of the curved-sidewalls, the secondary flows are generated due to the imbalance between the centrifugal force and the pressure force. In order to reduce these secondary flows, So and Mellor (l973) installed eight small high pressure wall jets on both endwalls at the entrance to the curved section. The amount of momentum added in this way was adjusted to compensate the momentum defect of the end-wall boundary layers. By this method, the yaw angle could be reduced from 1O~20c to less than 4°. On the other hand, Gillis et al. (l980) introduced a boundary layer fence method to reduce the pre-existing secondary flow. The present study adopted the latter method due to its simplicity and more effectiveness. The boundary layer fences were made of acrylic plates of 2 mm thick and 25 mm wide. Two boundary layer fences were mounted horizontally in the streamwise direction along the convex surface at the distance 50 mm from the respective end-walls. The effectiveness of this method was ascertained by measuring the yaw angle variation with a 3-hole Pitot tube. The result shown in Fig. 2 reveals that the secondary flow is slightly asymmetric about the centerplane. Since the yaw angle due to the secondary flows is less than lOin the core region far from the wall (n > 20 mm), present measurements in this region are virtually free from the three-dimensionality by the secondary

probe for u and v components and TSI-1260AT1~miniature probe fOr w component. The spanwise intensity tV was measured by rotating the probe as in Fujita and Kovasznay (1968). The sensing elements of the hot wires were made of tungsten wire of diameter 5 ~m and its length 1.2 mm. The wires of X-probe were separated by 1.0 mm with nominal inclinations of ± 45° with respect to the axis of the probe body. The wire overheat ratio was adjusted to 1.7 and the frequency response was set at 40 kHz.

The hot-wires were calibrated in a specially-designed jet

which has relatively low turbulence level (fllU ~ 0.006) in the potential core region. Instantaneous velocities were directly derived from the bridge voltage by using a modified King's law (Bruun and Tropea 1985). The coefficients were calculated by a least-square fitting from the calibrated results. The bridge voltage was compensated for the temperature variation between the calibration and the experiment. The analog bridge voltage was amplified with gain 2, low-pass filtered with the cut-off frequency of 5 kHz and subsequently digitized by a 12-bit analog to digital converter (TSI-IFA200). The sampling rate was 10 kHz and the digitized data were stored in a computer.

In the curved duct flow, the probe alignment with the main stream direction is one of the difficult problems. In the present study, the maximum error of yaw angle was estimated to be about 1°. HT found that a~o ~ror of yaw angle would lead to

1%, 2% and 5% errors in u2 , v2 and uv, respectively, and that the accuracy in the measurement of U would not be significantly affected by slight misalignment but the error in V could exceed nearly 100%. The relative uncertainties of the present data are estimated by the standard method (Yavuzkurt

1984; Moffat 1988). The results are: U, ± 1.5%; u2 , ±5%; v2 , uv, ± 12%; w 2 , ±20%; and triple products, ±30%.

Negative shear

dU 0 dy

=>

generator was about 35 sec -1. In order to reduce the length scale of turbulence and to remove quickly the non-uniformity in the mean shear, a coarse screen with the solidity of 36% was installed acrOSS the flow at the exit of the shear generator. It

was observed that the streamwise turbulent intensity u2 at xlD = 8 were nearly homogeneous in the transverse direction with about 3 ~ 5% of the centerline mean velocity.

The test section consisted of a straight duct, a curved duct and an exit straight redeveloping duct in series. In order to describe the flow field, a general rectilinear coordinate system is adopted in the present study. These are denoted by x, y and z, as shown in Fig. 1. The transverse y-coordinate is normal to the streamwise x-coordinate even in the curved section and the spanwise z-coordinate is normal to both x- and y-coordinates with a direction following the right-hand rule. The straight developing duct has a rectangular crOSS section; the transverse width D = 190 mm, spanwise height H =600 mm and the length in the streamwise direction L = 1890 mm which corresponds to 9.9D. A 90° curved duct has a centerline radius R of 1000 mm and has the same CrOss section as the straight duct. Thus, the inner wall radius is Rw = 905 mm and the outer one Rw = 1095 mm. In order to prevent ambient air currents from affecting the flow in the curved section and to investigate the relaxation process after the curved duct, an exit redeveloping duct is attached which has the same cross section and the streamwise length of 800 mm.

2.2 Instrumentation and data acquisition The distributions of wall static pressure on both curved walls along the midspan (z = 0 mm) were measured by using the tapped holes of 0.7 mm diameter. The measured pressure was converted to the voltage by a pressure transducer (Furness Controls, FCOI2) and a 20-channel scanner (Furness Controls, FC09l). The analog voltage signals were digitized by a system voltmeter (HP-3437A) and recorded on a computer (IBM/486) through an IEEE-488/GPIB board. A standard Pitot-tube with the sensing part diameter of 2.35 mm was used for the meaSUrement of the mean velocity distribution along the cross-section.

Standard hot-wire technique was used with TSI-IFAI00 constant temperature anemometer, TSI-1246T1.5 cross-wire

--

60 ,

\\ 50 Z= a mmI\\

=±30 mmI I:

I\ =±75 mm40 I :i

S 30 S

......... I, !\ ~ J!! I20 ' ,

1/ \I 176

10 // :\ j,/ t\

'I I" I0 -80 _40 00 40 80

(X-

Fig. 2. The yaw angle variation at 8= 800 over the convex wall for uniform flow, where IX = tan -I ( W/U)

motion. However, since the near wall region (n < 10 mm) is contaminated by the secondary flows, only the mean velocity data which are less sensitive to the secondary motions are presented in this report.

In the preliminary experimental stage, the problem of longitudinal vortices over the concave surface was investigated. Meroney and Bradshaw (1975) and Hoffmann et al. (1985) have reported large spanwise variations in boundary layers on concave surfaces, while the variations were quite persistent in the streamwise direction. The spanwise variations are believed to be caused by the positionally stable longitudinal vortices (Taylor-Gorder vortices) whose wavelength is of the same order as the boundary layer thickness and the streamwise extent of about 4 ~ 5

177

UF0.2 •

~,m ijJ ~ ~ ~ ~'$

u Q.

0

-0.2 '\,6 6

PW0.2 .G1 lEI 6 66 6 6 • 6 6 •'$ Q. 0u

-0.2 'b6 6 6 6 9 0°

PS.m lEI 6 6 60.2 6 6 ~"

'$

u Q.

0 • III '00 6 6-0.2 6 6 000

0.2 '$ ~ ~ lEI 13Q. "" U 0

-0.2 6 6 6 6 ~ 0°

8=90°

NW

NS

8= 0°0.2 66 6'$

Q. 0 iI " u -0.2

'lD 66 6 6

2 4 6 8 10 12 14 16 18 20 22 x/D

Fig. 3. Static pressure distribution along the two surface walls. 0: convex, D: concave

analysis that the curvature effect is isolated from the streamwise pressure gradient in this region. It is noted that the pressure gradient is still favorable in the curved section for the cases UF, PW and PS, but that the cases NW and NS have slightly adverse pressure gradients within the curved duct. The adverse pressure gradients at the concave inlet and convex outlet regions did not cause the flow separation for all flow cases.

As displayed in the figure, the overall trends of Cpw for shear flows are similar to that for the uniform flow. However, since negative shear flows have faster speed than the positive shear flows over the convex wall, the magnitude of Cpw becomes smaller due to the stronger centrifugal force. The opposite is applicable to the concave wall. As a result, the values of Cpw at both the convex and concave walls increase with increasing the shear rate for positive shear cases in comparison with those of UF, whereas the values of Cpw decrease for negative shear cases. Note that the static wall pressure difference between two walls is quite insensitive to the inlet flow conditions.

Figure 4 represents the streamwise mean velocity profiles at midspan (z = 0) along the duct. As shown in the figure, the evolutions of mean velocity profiles exhibit many different features depending on the inlet mean shear rate. For the case of the uniform inlet (UF), the mean velocity near the convex wall is increased after the inlet station of the curved duct whereas

the mean velocity in the neighborhood of the concave wall is decreased. As noted by Townsend (1976), the core region becomes close to being irrotational, i.e., UR = constant. Accordingly, the overall shear rate (dU/dy- U/R) appears to be negative in the central region. However, this shear flow is just being recovered to a uniform flow in the redeveloping duct. These observations are in general conformity with the findings of Ellis and Joubert (1974) and Hunt and Joubert (1979).

When the inlet flow is sheared, the mean velocity variation is substantial along the duct depending on the shear rate. For positive shear flows (PW, PS), the mean velocity near the convex wall increases whereas it decreases near the concave wall along the downstream. Thus the magnitude of shear rate becomes smaller with the downstream distance. In the case of PW, the uniformity of shear rate is deteriorated from the station e= 20 0 as the boundary layer grows very rapidly on the concave wall. But the initially imposed shear rate remains almost unchanged in the region -0.3

UF

o ). 0

178 PW

o ). 0

PS

o ). 0

NW

o ). 0

NS

o " "" 0

Fig. 4. Variations of the streamwise mean

to to velocity profiles at midspan along the duct u/uc

Dv2 -U pay-=4uv-+2--+d 2-2e 2Dt R pay Y ,Y

Dw2 paw --= 0 +2 - -a +dw2 - 2ew 2 ,Dt P z

Dq2 _(dU U)m= -2uv dy -Ii. +0+d-8,

p (au av)+- -a +-a +du,-eu,'p y ,x

In Eqs. (2)-(6), the right hand side represents the production, (3) redistribution, diffusion and dissipation terms in turn.

In the discussion which follows, main concern is about the interaction between the uniform mean strain and the curvature in isolation from any effect of wall presence. From this point of(4) view, only the profiles within -0.3

0.5

-0.5

-0.5 0.5

-0.5 0.5

D ~ a

o;.. a

D ~ a

~ '?J" I Cil Cil

~

I

IP , I ~

~~ ~

I \ \ (

I ~e

~ ~ ~

x=8D 8=20· 8=40· 9=60· 8=80" x=20D 0.5

Ii ,~ 1I UF D ..... a>.

. ~ ~ 4 Id-0.5

0.5 , rI' e) PWl 179eSI' JD ..... a>.

~ d d I-0.5 ,, PS

I/) ICil

,

) NW\\

J \Cil

iP~l~ ~ ~ '"

NS\III \CilCilCili ee

e) ,\ Cil

e c;lil

lillil

III Fig. 5a---{}. Distributions of Reynolds normal and ~I!l etD ee Gle dJ~ shear stresses. a uv; b u1 ; C ;72; d w1

-0.5 a 0.5 a a a a a 2(uv/U~) x 10

Fig.5a

smaller turbulence intensity than those in the region y < O. The positive, it leads to positive u-fluctuations due to the negacontrary is true for the case of negative shear (Rohr et al. 1988; tive velocity gradient. The converse is true for negative Tavoularis and Karnik 1989). Suc~nhomogeneity is evident in v-fluctuations. Moreover, the production term in Eq. (6)

is always positive for negative shear cases. Therefore, the Figs. 5b-d for the profiles of u1 , V and wcomponents at xlD = 8. Therefore, when the profiles in Figs. 5b-d and 6 are correlation uv remains most probably positive, which can be discussed, the difference in the non-dimensional development seen in the figure. When the mean shear is positive, the time during which the fluid particle at different y-positions is centrifugal motion contributes to the positive correlation exposed must be taken into account. between u- and v-fluctuations. However, the positiveness of

Now consider the Reynolds shear stress distribution in the mean velocity gradient counteracts the centrifugal Fig. Sa first. In a curved duct flow with negative shear, the contributions to u. Thus, if the shear rate is relatively weakly positive u-fluctuations give rise to positive v-fluctuations due positive, the centrifugal force is dominant in maintaining the to the centrifugal force. And when the v-fluctuations are correlation to be positive, which is the cases of PW and PS. As

••

180

)(=8 D 9=20· 9=40· 9=60· 9=800 x=20D 0.5

a ~ 0

-0.5 0.5

a ~ 0

-o.s O.S

a ~ 0

-0.5 O.S

a ~ 0

-o.S O.S

a ~ 0

-o.S.o o.S 0 o o

;/DID #lID' ,,/ UF#'" I

t

..,,~IDe \a

pEl1iI 1D IiIEl.IDe ID~

," r

El I PWI.e rl .,,~ ~ "e"

e/Dr#J rJI el

( r~ (e PS\00

t~,\ ~ , ,ga t

r5J(J

NW.f,. ,5 r, eIe _~IDe ~e ICII .. • iii III .. l ( ( NS(I

• e ID..ID ~ S ~,

) \(0 ~

IDe IDIDe

1!I(eeIDIDj!) ~efee

Fig.5b

soon as the curvature strain is removed at the end of the 90°

duct, the positive mean shear enforces the uv correlation to become negative as can be seen at xlD = 20 inJ:ig. Sa.

The variations of streamwise normal stress u2 are represented in Fig. sb. As was mentioned previously, the initial intensity distribution at xlD = 8 is inhomogeneous due to the different development time for different y. In the curved duct,

it is seen that u2 is continuously suppressed in the positive shear flows (PW, PS), whereas it is augmented in the negative shear flows (NW, NS). In the UF case, although

not apparently discernible, u 2 is slightly enhanced. These suppression or enhancement can be understood by considering the right hand side in Eq. (2). For the cases PS and

PW, uv> 0 and (dUldy+ UlR) > O. Therefore the production

1.0

term is negative and consequently u2 is suppressed. In a similar

manner, the enhancement of u2 in the cases NW and NS can be explained by the facts that uv> 0 and (dUldy+ UIR)

l( .80 e·20· e·40· e·&o· e·80· X= 20 0 0.5

a ~ 0

-0.5 O.S

0)

a ~ 0

-O.S O.S

fJ

" " ,I }, ~ ~ "

.., ~ ,

~ J rafil

0',0 \!I Il'

(;)

" " ~ , eP" fill )0

(;) G)

(;) ",e 0 f

8 5 5

I ) )IJ 0

l {

5 It!e,

(

, " t 5

\ '1 ~, 0 '" ~8 (;)!!l

l 0• 0 E '" i

e , 6fJ8 «

)8

-0. O.

"a >. '" )

.e

-0. O. .,

"

a ~ 0 1

,e \!II -05 . 0 0.5 0 o o

) UF

, } PW , ) PS Itt'

~e , " NW ,

l(,BO 9,20· 9,{.0· 9 ,60· 9=BO· l(,200 0.5 ;a

f~ ~e' a) eJ ) e ~e ~

~ , -., , , ~.

~. 13~ rsfJ' ele I J,e ,13 ~

~ & ., " l!\;) 0,

,{1J#'l cfJl J

\ ,0 rf I

~ 0 I IC!l

I I,

ill 1 I'0. J I }J \, , f I t 1

- - -

pressure-velocity correlations and dissipation terms which are denoted symbolically. Note that the diffusion terms are dropped due to spatial homogeneity.

As has been pointed out by Gibson et al. (1984), the initial response of the production terms to the sudden changes in curvature is evidently not matched by the corresponding change in the dissipation terms which persist at levels set by the turbulence generated upstream. This fact is supported by the present experimental results that the large scale motion which is most responsible for the production mechanism responses rapidly to the curvature, whereas the small scale motion for the dissipation mechanism remains more or less unaffected by the curvature (see subsection 4.4). Since the pressure-velocity correlations could not be experimentally measured, the behaviors of the triple velocity products between curvature and mean shear are discussed below by considering the production terms only.

When the turbulent characteristic velocity and length scales are represented as u and I, respectively, the production terms can be categorized into three groups which are scaled with u4 /l, u4/R and u3UIR. In the present situations, l!R < 1 and thus the terms scaled with u4/R can be neglected for simplicity. Furthermore, in the cases PS and PW, (u/U) (RIl) ~ 0(10- 1 ) and the terms associated with dU/dy and Reynolds stress gradients are much smaller than the terms with U/R. Therefore, in the cases PS and PW, the production of the triple products is dominated by the curvature strain. However, for the cases NW and NS, (u/U) (RIl) - 0(1), which implies that all terms are of comparable magnitudes to each others. Therefore, the triple

products u3 depends directly on the straight shear dU/dy and

only indirectly on the curvature through the terms u1v and

uv, which is evidenced in Fig. 6a by noting that u1v < O. On the contrary, the triple products v3 depends directly on the curvature and is only indirectly dependent on the straight shear, which is also revealed in Fig. 6b. In a turbulent boundary layer over a complex surface, Gibson et al. (1984) found that

the most ~iking changes occur in v3 among the triple products

and that v3 was still changing with the distance downstream

when the -u2v and

-uv profiles were adjusted to the external

curvature conditions. It was also observed on a concave wall by

Barlow and Johnston (1988) that v was augmented much more than u3 through the destabilizing curvature. In our experiment,

the integral of the triple product v changed about 30% more than that of u3 within the range -O.3

x=8D 8 =20° 8=40°

PS

o ~ 0

- O. 5 '--_'------J'------'_----'_---'_---'-_---'-_---'-_---'-_---'-_---'-_---'-_---J

0.5 r--r------,,.------,------r------r---~------,

NW

o ~ 0

#1 -0.5 '--_'------J"-::-----'_----'_---'_---'-_----":_---'-_---'-_---'-_----',_---l_---J - 0.25 0 0.25 0 0 0 0

a (u3/U~lx 103

x= 8 0 8=20° 8=40° 8=60° 0.5

~

l

I

( (;)

0~ 0

0 0

J 0

0 (;)

(;)o 0 ~ 0

-0.5

x =20 08=80°, ~ PS 0 0

~ I(;) (;) e

(;) 0~

~

0.51r---.----....------y----.-----....------y-------,

NW

o ~ 0

- 0.5 '--_'------J'------J'------'_----'_---'_---'_---'-_--'-_--'-_--'-_---'-_---J -0.25 0 0.25 0 0 0 0

b (v3/U~ I x 103 o

Fig. 6a---0.054.

Evolution of length scales Figure 8a displays the evolution of the streamwise integral length scale 1. The integral scale was computed by integrating

the auto-correlation curve of u-ftuctuations to the first zero crossing point. The figure shows that L increases in UF, NW and NS and decreases in PW and PS, nearly linearly in the initial curved region. Therefore, the integral length sca~ depends strongly on the curvature. In order to investigate the effect of curvature on the length scale variation, changes in the length scale during Te = 3 are plotted vs the curvature parameter, 50 in Fig. 8b. Although there are some scatters in the data of HT, a simple correlation curve can be drawn which also crosses the line LILo = 1 near 5e = 0.054, as in Fig. 7b. Adopting the same functional form as for klko, the curve may be represented by LlLo = exp {0.035 TeO -19.55e + 18.55~)}.

The Taylor micro-length scale ), was obtained by differentiating the auto-correlation curve for u-component at t = 0, as has been done by Rose (966). The results are shown in Fig. 9a, which reveals that the Taylor micro-length scale

4.4

NW

NW

PS

o

x =20 08=80°8=60°

o 0 (~/U~) x103

8=40°8= 20°x =80

0.5 L-_"--_L-----l_----'_----I._---'-_---'-_---L._-'--_--'-_--'---_-'---_ -0.25 0 0.25 0

0.5 ,----,--------,.------:r---::-r-----=-,------:-,--------,

0.5 ,-----,-,,--------,.-----:r------:-,-----:--.-----,-------,

- 0.5 I....-_L-----l_----'_----I._---'-_---L._-'--_---'-_-'---_...L.-_-'---_"-----'

-0.5 L-_L-_L-----l_----'_----I._---'-_---L._-'--_---'-_---'-_-'---_-'------J

0.5,.--.------,------,.------r----,------,-------,

o ~ 0

o ~ 0

o ~ 0

d

-0.5 -0.25 0 0.25 0 0 0 0 0

-2 3 3 C (u vi Uc ) X10

X= 80 8= 20° 8=40° 8=60° 8=80° x=20 0 0.5

PS

0

" 0,.,

0.6

800 900

800 900

0.4

400 500

8

400 500 600

8

o 0.2 Sc

-0.2

a

3

b

b

OL------L_---'-_..l...-_L----l._---'-_...l..-_L-_

4,-------------------

3,------------------

oL------'-_---'-_.L-_~_ ___'___ __'____L___'__ ___'__ -0.4

2 A

o -' ...... -'

Fig. 8a, b. Evolution of streamwise integral length scale along ylD= -0.1. a with downstream distance (}; b with curvature parameter 5,. -: LlLo=exp{0.035r,(1-19.55,+ 18.55~)}. Symbols as in Fig. 7

900

0.6

10

0 UF GJ PW 0 PS 3 & NW 0/ NS

0 0

-" -' ...... ...... 2-" -'

Case ko Lo Ao '10 (m2/s2 ) (mm) (mm) (mm)

UF 0.012 37.3 4.2 0.43 PW 0.064 30.2 5.2 0.31 PS 0.167 44.7 7.4 0.29 NW 0.056 31.3 5.5 0.33 NS 0.109 34.5 5.4 0.28

Table. 2. Some reference turbulence quantities (}= 10\ ylD = -0.1

Fig. 7a, b. Evolution of turbulent kinetic energy along ylD= -0.1. a with downstream distance (}; b with curvature parameter 5,. EEl: data of Holloway and Tavoularis (1992). -klko= exp {0.078r,(1-19.55, + 18.55~) }

Sc

discussed with their transport equations. The variation of

u3 depends directly on the straight_~hear dU/dy and only

indirectly on a curvature, whereas y3 changes depending

directly on the curvature. The measurements of u2v and uv revealed that the curvature offers an inertia to maintain the radially inward transport of both the kinetic energy and the Reynolds stress in the present configuration.

The turbulent kinetic energy and the integral scale are enhanced when a curvature parameter So defined by

Fig. 9a, b. Evolution of streamwise small scales along ylD= -0.1. a Taylor micro-length scale },; b Kolmogoroff length scale '1. Symbols as in Fig. 7

186 a

00

8

6

0

-" 4...... -"

2

b 0-0.4

(UlR)/(dUldy - UlR), is less than 0.054 and they are suppressed for Sc> 0.054. Although the integral length scale changes significantly depending on the curvature, the Taylor micro-length scale and the Kolmogoroff scale change relatively small through the curved duct. The latter implies that the small scale eddies are not strongly affected by the curvature.

References Ahmad QA; Luxton RE; Antonia RA (1976) Characteristics of

a turbulent boundary layer with an external turbulent uniform shear flow. J Fluid Mech 37: 369--396

Barlow RS; Johnston JP (1988) Structure of a turbulent boundary layer on a concave surface. J Fluid Mech 191: 137-176

Bruun HH; Tropea C (1985) The calibration of inclined hot-wire probes. J Phys E Sci Instrum 18: 405-413

Chung MK; Kyong NH (1989) Measurement of turbulent dispersion behind a fine cylindrical heat source in a weakly sheared flow. J Fluid Mech 205: 171-193

Crane RI; Sabzvari J (1984) Laser-Doppler measurements of Gorder vortices in laminar and low-Reynolds-number turbulent boundary layers. In Laser Anemometry in Fluid Mechanics (Ed. R.J. Adrian). Lisbon: Ladoan

Ellis LB; Joubert PN (1974) Turbulent shear flow in a curved duct. J Fluid Mech 62: 65-84

Eskinazi S; Yeh H (1956) An investigation on fully developed turbulent flows in a curved channel. J Aeronaut Sci 23: 23-35

Fujita H; Kovasznay LSG (1968) Measurement of Reynolds stress by a single rotated hot wire anemometer. Rev Sci lnstrum 39: 1351-1355

Gillis JC; Johnston JP; Moffat RJ; Kays WM (1980) Turbulent boundary layer on a convex curved surface. Stanford University Dept. Mech. Engng Rep. HMT-31

Gibson MM; Verriopoulis CA; Vlachos NS (1984) Turbulent boundary layer on a mildly curved convex surface: Part 1: Mean flow and turbulence measurements. Exp Fluids 2: 73-80

Hoffmann PH; Muck KC; Bradshaw P (1985) The effect of concave on turbulent boundary layers. J Fluid Mech 161: 371-403

Holloway AGL; Tavoularis S (1992) The effects of curvature on sheared turbulence. J Fluid Mech 237: 569-603

Hunt IA; Joubert PN (1979) Effects of small streamline curvature on turbulent duct flow. J Fluid Mech 91: 633-659

Jeans AH; Johnston JP (1982) The effects of concave curvature on turbulent boundary layer structure. Stanford University Dept. Mech. Engng Rep. MD-40

Meroney RN; Bradshaw P (1975) Turbulent boundary layer growth over a longitudinally curved surfaces. AIAA J 13: 1448-1453

Moffat RJ (1988) Describing the uncertainties in experimental results. Exp Thermal and Fluid Science 1: 3-17

Muck KC; Hoffmann PH; Bradshaw P (1985) The effect of convex surface curvature on turbulent boundary layers. J Fluid Mech 161: 347-369

Nagano Y; Tagawa M (1990) A structural turbulence model for triple products of velocity and scalar. J Fluid Mech 215: 639-657

Ramaprian BR; Shivaprasad BG (1977) Mean flow measurements in turbulent boundary layers along mildly curved surfaces. AlAA J 15: 189-196

Ramaprian BR; Shivaprasad BG (1978) The structure of turbulent boundary layers along mildly curved surfaces. J Fluid Mech 85: 273-303

Rohr JJ; Itsweire EC; Helland KN; van Atta CW (1988) An inves

tigation of the growth of turbulence in a uniform mean shear flow. J Fluid Mech 187: 1-33

Rose WG (1966) Results of an attempt to generate a homogeneous turbulent shear flow. J Fluid Mech 25: 97-120

So RMC (1975) A turbulence velocity scale for curved shear flows. J Fluid Mech 70: 37-57

So RMC; Mellor GL (1973) Experiment on convex curvature effects in turbulent boundary layers. J Fluid Mech 60: 43-62

So RMC; Mellor GL (1975) Experiment on turbulent boundary layers on a concave wall. Aero Q 26: 25-40

Tavoularis S; Karnik U (1989) Further experiments on the evolution of turbulent stresses and scales in uniformly sheared turbulence. J Fluid Mech 204: 457-478

Townsend AA (1976) The Structure of Turbulent Shear Flow. Cambridge: Cambridge University press

Yavuzkurt S (1984) A guide to uncertainty analysis of hot-wire data. J Fluids Eng 106: 181-186