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Quadrant analysis in the turbulent far-wake of a cylinder
Fluid Dynamics North-Holland
Research
2 (1987) 3-14
Quadrant analysis in the turbulent far-wake of a cylinder R.A. ANTONIA
and L.W.B. BROWNE
Department of Mechanical Engineering, Received
10 November
University of Newcastle, N.S. W., 2308, Australia
1986
Abstract. The quadrant analysis technique of Willmarth and Lu (1972) has been used to estimate contributions from the four quadrants to the average momentum and heat transfers in the self-preserving region of a turbulent wake. The importance of sweeps near the wake centreline and of ejections at the edge of the wake becomes clearer when sweeps and ejections are viewed in the context of the topology of the large scale motion. The application of the quadrant analysis to the products of velocity and temperature fluctuations conditioned on the large scale motion, clarified the picture inferred from the conventional quadrant analysis approach by clearly identifying the regions of space which are important for transferring momentum and heat.
1. Introduction The quadrant analysis technique, introduced by Willmarth and Lu (1972) for a turbulent boundary layer, and extended by Lu and Willmarth (1973), has subsequently been used in different flows (e.g. Brodkey et al., 1974 for turbulent channel flow; Sabot and Comte-Bellot, 1976 for turbulent pipe flow; Sreenivasan and Antonia, 1979 for a turbulent circular jet). Recently, Fabris and Nakayama (1985) applied the technique to the product uu (U and u are the longitudinal and lateral velocity fluctuations) measured in the self-preserving region of a cylinder wake. The main aim of the work was to investigate the Reynolds stress structures and features of the large eddies. The results indicated that from about the half velocity defect point to the edge of the wake, ejections (slow speed fluid directed away from the centreline) contribute the most to the Reynolds shear stress z, in much the same manner as in the outer region of a boundary layer. It was also inferred that the centreline region was characterized by sweeps (high speed fluid directed towards the centreline). A significant amount of information already exists on several aspects of the organised motion in the turbulent far wake of a cylinder. This information stems both from flow visualisations (Taneda, 1959; Papailiou and Lykoudis, 1974; Cimbala, 1985) and other investigations (Grant, 1958; Keffer, 1965; Townsend, 1979; Mumford, 1983; Browne et al., 1986). The emerging picture suggests that the predominant flow mode consists of three-dimensional large scale rotational bulges which are asymmetrical about the centreline. A simplified two-dimensional view of this arrangement is depicted in Fig. 1. The information shown on the sketch is based on the topology of the large scale motion, determined in the far wake of a circular cylinder by Browne et al. (1986) and Antonia et al. (1986a). The topology revealed local regions, cross-hatched in Fig. 1, in which shear stresses and heat fluxes are relatively large. One such region, identified as an ejection, is associated with the fluid ejected on the upstream side of a large scale structure. Another region, identified as a sweep, is associated with the fluid swept in on the downstream side of the following structure. The physical locations of these regions correspond approximately with the regions of the wake where the quadrant technique 0169-5983/87/$4.00
0 1987, The Japan
Society
of Fluid Mechanics
R.A. Antonia, L. W.3. Browne / Qrrndrant analysis in the turbulent far-wake ---wEngulfed
heated
of a cylinder
Flow Direction ambient
fluid
cylinder
COLD
Fig. 1. Simplified two-dimensional view of dominant organised motion in the turbulent far-wake as seen by observer moving in the x direction with the approximate velocity of the centre of the structures.
used by Fabris and Nakayama (1985) indicated that ejections and sweeps may be important in the context of contributions to shear stresses. This correspondence suggests that results obtained by the quadrant technique, which suffers from the obvious limitation of drawing information from only one point in space, would be best explained in the context of information related to the large scale motion. Fabris and Nakayama foreshadowed such a possibility by concluding that “future work which incorporates other types of conditioning together with the present quadrant type analysis will further clarify the Reynolds structure in the far wake”. In this paper we extend the results obtained by Fabris and Nakayama by showing that the quadrant analysis technique applied to signals u, u and 8 (the temperature fluctuation), are qua~tatively consistent with the topology delineated by Browne et al. (1986) and Antonia et al. (1986a) for a turbulent wake. Results obtained by applying the quadrant analysis technique to conditional averages of U, u and 8, conditioned on the large scale motion, show unambiguously the physical significance of ejections and sweeps in this particular flow.
2. Experimental
details
Full details of the experimental set-up are given in Browne and Antonia (1986). In summary, the wind tunnel used was of the open return type with a working section of 350 mm X 350 mm, 2.4 m long. A stainless steel tube of diameter d = 2.67 mm was mounted in the midplane of the working section, normal to the flow and 20 cm after the end of the contraction leading into the working section. This tube was heated at a rate of 100 W to provide temperature as a passive marker of the flow. The free stream velocity U, was 6.7 m/s so that the Reynolds number Re, based on the cylinder diameter was 1170. The floor of the working section was slightly tilted to achieve a zero pressure gradient. All measurements were made at x/d = 420. At this station, the centreline mean temperature T,, relative to ambient, was 0.82 K. An assembly of an X-wire and a cold wire was moved across the wake from about y * = - 2 to +2, where y* = y/L and L (= 12.3 mm) was the value of y, Fig. 2, where the velocity defect was half its centreline value U, (= 0.36 m/s). The cold wire associated with the X-wire was located 0.5 mm in front of the crossing point of the X-wires. This did not interfere with the X-wire since statistics of U, the velocity fluctuation in the x direction, U, the velocity fluctuation in the y direction, and uu were virtually the same with and without the cold wire in position. Also the spatial resolution of the three wire assembly was satisfactory since ~6 statistics were within 3.5% of those obtained independently using a parallel arrangement of a single hot wire and a single cold wire separated in the J’ direction by 0.2 mm.
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake
of a cylinder
5
x
(al
(bl
Fig. 2. Definition
sketch for the quadrant
(c) analysis
technique:
(a) uu; (b) u6’; (c) ue.
All probes were made in our laboratory. The cold wires were all 0.63 pm diameter Pt-10% Rh Wollaston wires with an active length of 1.0 mm. The hot wires were 5 pm diameter Pt-10% Rh Wollaston wires with an active length of 0.85 mm, separated by 0.8 mm. In-house anemometers provided 0.1 mA current for the cold wires and constant temperature anemometers with an overheat of 1.8 were used for the X-wires. Careful velocity, yaw and temperature calibrations were carried out using a personal computer/data logger arrangement. Data were low-pass filtered at 1 kHz and sampled at f, = 2 kHz per channel directed into a DEC 11/34 computer using an 11 bit plus sign sample and hold A/D unit. Before filtering, the signals were suitably offset and gained. Data sampling time at each station was typically of 50 s duration.
3. Conventional
quadrant analysis
The quadrant analysis technique which is used here is similar to that introduced by Willmarth and Lu (1972) and Wallace et al. (1972). Contributions to the product a/3 are sorted into the four quadrants of the (a, /?) plane (see Fig. 2). We focus mainly on the ejection and sweep quadrants (identified in Fig. 2) although, for completeness, results for the other quadrants are also presented in this section. After digital sorting of the data, the occupancy time and the contribution that each quadrant makes to q were calculated. The occupancy time is defined as the time for which the pair of fluctuations ((Y, /?) can be found in a given quadrant i ( = I, II, III, IV) of the ( LY,fi) plane. The occupancy time is shown in Fig. 3 as a fraction of the total record duration. The contribution to G was then estimated from the joint probability density function p(cx, fi) of the fluctuating pair ((Y, fl). By definition,
aP ~(a, a= JjDo -+CC The contribution
to 3
P> da
from a particular
(1)
dfl.
quadrant
i of the ((w, p) plane
was calculated
by
R.A. Antonia, L W.B. &owe
/ Quadrant analysis in the t~r~~le~t~ur-~,ake
of a cylinder
Fig. 3. Occupancy
time in the four quadrants by the products UD, ~6’ and uB. d, i = I; b, II; p, III; 9, IV.
integrating c@p(a, /3) in that quadrant. For example, quadrant of the (u, u) plane for y z=-0 is given by
the contribution
Contributions
are shown in Fig. 4, n~rma~zed
from each quadrant
of the ((u, /3) plane
overall rms values fy’ (= VT) and /3’ determined the overall correlation coefficient pap = $/(~‘/3’, from each quadrant since
to G from the ejection
by the
from the total record. Also shown in Fig. 4 is which is equal to the sum of the contributions
Values of u’/U,, u’/UO and B’/T, are given in Table 1 for the y* values used in Figs. 3 and 4. Also shown in Table 1 are the values of the correlation coefficients pap (a, /I = U, u or 8). The results for (&)r/u’u shown in Fig. 4 are very similar to those already presented and discussed in some detail by Fabris and Nakayama (1985). Here only the main features will be highlighted. In the outer part of the wake, the ejection quadrant (III for y* < 0) is the major contributor to puu even though the occupancy time in this quadrant (Fig. 3) is much smaller than for the sweep (I) quadrant. Near the centreline, it is the contribution from the sweep quadrant which is slightly larger than that from the ejection quadrant, while the occupancy times for the two quadrants (Fig. 3) are nearly equal. Cont~butions from the sweep and ejection quadrants cross over at appro~mately y* = - 0.8, where the shear is close to its maximum value. For (?$)i/u’8’, distributions of the ejection (i = II) and sweep (i = IV) contributions follow very closely those of (uU),/u’u’ (Fig. 4). There is a correspondingly close similarity between the distributions for occupancy times in the ejection and sweep quadrants. It should also be mentioned that the results for (u, 6) [ejection quadrant II, sweep quadrant IV] differ from
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake
b b
of a cylinder
7
b”
lE~ectm.11)
qlsweep, IV I 9
0.6
P P d
d
Fig. 4. auadrants
Contributions from to the correlation
the four coefficient
u, .R= u; (b) u, 8;
those of (u, U) or (u, ~9) in only a minor way: there is relatively little difference between the importance of ejection and sweep quadrants near the centreline and the interaction quadrants (I and III) make a contribution of the same sign, albeit quite small, as that from the ejection and sweep quadrants.
Table 1 Variation
0.08 -0.16 - 0.41 - 0.65 - 0.89 - 1.14 - 1.38 - 1.63 - 1.87 -2.11
across
the wake of rms values of u, v, 19 and of mean values of products
0.29 0.31 0.32 0.34 0.34 0.32 0.28 0.21 0.14 0.08
0.24 0.24 0.25 0.24 0.23 0.22 0.19 0.15 0.12 0.09
0.17 0.18 0.19 0.22 0.24 0.25 0.24 0.21 0.16 0.10
0
0.27 0.44 0.52 0.56 0.57 0.55 0.53 0.48 0.36
MU, MO, vr3
-0.51 - 0.54 -0.56 - 0.61 - 0.65 - 0.68 - 0.70 - 0.70 - 0.64 - 0.47
0.02 - 0.24 - 0.42 - 0.53 - 0.56 -0.57 -0.55 - 0.52 - 0.48 - 0.41
8
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent jar-wake
/\ @ 1-17’ --L
/
/
/ I \
of a cylinder
1,
-,’
-,’ / // /
I
R.A. Antonia, L. WB. Browne / Quadrant analysis in the turbulent far-wake
of a cylinder
9
4. Joint probability density functions Equi-probability density contours of the pairs of fluctuations ((Y, /3) are shown in Fig. 5 for three locations in the wake: near the centreline (y * = +0.08), close to the position of maximum shear (y * = - 0.65), and in the outer flow region (y* 2: - 1.63). For the latter location, the flow is intermittent, the intermittency factor, or fraction of time the flow is turbulent, being equal to 0.43. This value was estimated by averaging the intermittency function (= 1 when the flow is turbulent and 0 when it is non-turbulent) obtained by comparing the temperature signal with an arbitrary threshold. Details for the selection of this threshold have been given in Antonia and Browne (1987). At y* = - 1.63, the joint probability density function was also calculated for fluctuations (a,, &) in only the turbulent part of the record (Fig. 6). The fluctuations CX~and /3, are centered so that ‘Y,= & = 0. Also shown in Figs. 5 and 6 are contours of constant values for the Gaussian joint probability density function
a+i= i2’
PG
-1
-1 exp
a2 -at2 - 2Pap
2(1_p;B)
2(1 - P$3)1’2
[
aP
PZ
m’p’z
)I (3) ’
(
with (Y~and & replacing cy and p respectively in Fig. 6. The Gaussian contours are drawn for to the measured values. values of pap corresponding The contours in Fig. 5 indicate that there are small departures from the Gaussian distributions at y* = 0.08 in the case of (u, u) and (u, 19). These departures appear to be associated mainly with the sweep quadrants (I for UU, IV for ve), reflecting the incursion of fluid from either side of the centreline. The contours for p(u, 0) at this location are closer to the Gaussian contours presumably because less emphasis is given to the lateral motion directed towards the centreline and more importance given to older and well mixed turbulence travelling along the centreline. the measured contours depart from the elliptical Gaussian contours. This is Aty * = - 163 ., especially pronounced in the sweep quadrant. As can be inferred from Fig. 1 and as will become clearer in the following Section, in the outer part of the wake, the sweep quadrant comprises mainly engulfed irrotational fluid or fluid which is about to be entrained and does not make a significant contribution to the Reynolds shear stress in this region of the wake.
4
,
I
I
-4
-2
1
(al 0.1
/---\
/+----L\ /
7 \ ‘\
’
7 I
//
a=ut
0
2
a-ut
.-4
-2
‘\
0
2
-4
-2
0
2
4
a/a’ Fig. 6. Joint probability density functions of the turbulent part only of the flow, of u and o, u and 0.1, 0.05, 0.01. -, y* = -1.63. (a) n, = ut, 8, = ol; (b) u,, et; (c) u,, e,. Inner to outer contours: - -, Gaussian, eq. (3).
9, v and B at experiment;
R.A. Antonia, L. W.B. Browne / Quadrant analysis m the turbulent far-wake of a cylinder
10
There is a smaller departure from the Gaussian contours in the ejection quadrant. This departure is noticeably reduced when only the turbulent fluid is considered (Fig. 6) although the departure in the sweep quadrant remains important. The main implication of Fig. 6 is that in the outer wake region, the ejection quadrant (III for WJ, II for UB and II for ~6) focuses on fluid with a relatively long history in the fully turbulent part of the flow. By contrast, sweep regions near the centreline are associated with newly entrained fluid apparently causing the departures from the Gaussian contours, observed even at the centreline. The behaviour of the present joint probability density functions is similar to that reported by Sreenivasan and Antonia (1979) in a slightly heated round jet with a co-flowing external stream. Departures from Gaussian distributions were found to be less pronounced for the outward radial motion than for the inward radial motion which is likely to be associated with the newly entrained fluid.
5. Conditional
quadrant analysis
In studies of coherent structures in turbulent shear flows, it is usual to write instantaneous variables using either the double or triple decompositions (e.g. Hussain, 1983). In order to better highlight the contribution of the organised motion to the Reynolds stresses and heat fluxes, Browne et al. (1986) and Antonia et al. (1986b) used the triple decomposition, e.g. u=U+(u)+us,
(4)
where U is the instantaneous velocity, G its mean value, the angular brackets denote the coherent large scale component, and u, is assumed to be identified with the small scale motion. Implicit in eq. (4) is the assumed independence between u, and (u). Since the fluctuation (Yhas been defined here so that Cu= 0, eq. (4) can be re-written as (Y= (a)
+ (Y,.
Contours of (a) and of the product (a)(p) were presented by Antonia et al. (1986b), where quantities within angular brackets were conditioned with respect to the occurrence of temperature fronts, as depicted in Fig. 1, detected with an array of temperature sensors ‘. The contours were displayed in the (Ax *, y *) plane, where Ax * = - U,r/L, r being the time measured from the instant of detection and U, (2: 0.97Ut) being a convection velocity of the large structure. The results obtained by applying the quadrant analysis technique to quantities (u), (u), (0) are shown in Fig. 7 in the (Ax *, y * ) plane. Also included in this figure are the locations of the temperature fronts used for detection. These fronts are roughly aligned with the diverging separatrices (Fig. 1). Contours of the product (a* )( fi * ) in Fig. 7 correspond, as shown, to the four quadrants of the ((a* ), (/3 *)) plane. From this figure, the contribution from each quadrant can be identified. For example, at large values of ( y * I, the contours of (u * ) (u * ) are associated only with the ejection quadrant (II for positive y * and III for negative y *, cf. Fig. 2). By contrast, for values of 1y* 1 near the centreline, the contours coincide only with the sweep quadrant (I and IV for negative and positive y * respectively). A similar result holds for u0 and ~0. Thus, for the three pairs of variables considered, the ejection quadrant contributes nearly totally (although only the large magnitude contours are plotted for clarity) to the product (a)(p) at the edge of the wake whereas the sweep quadrant is the major contributor close to the centreline. There is a range of 1y* ) over which contributions from ejections and sweeps overlap, although the domains of contribution retain their spatial identity. ’ Detection details can be found in Browne et al. (1986)
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake
----c
of a cylinder
11
Flow Direction
Y”
-6
-4
-2
0
2
4
0
2
4
0
2
4
Ax*
-4
-2 Ax"
-6
-4
-2 Ax"
Fig. 7. Contributions to the shear stress and heat fluxes obtained by applying the quadrant analysis technique to conditional averages of velocity and temperature. (a) (( u)( o ) contours are shown for 0.01 and 0.02. Contour values are negative for y * > 0 and positive for y * i 0. (b) ( u)( f7) contours are for - 0.02 and - 0.04. (c) ( u)( 0) contours are for 0.02 and 0.04. Contour values are positive for y * 1 0 and negative for y * < 0.
The previous observations can be quantified by integrating distance corresponding to one period of the large scale motion,
the products viz.
( CX)(/3) over a
where A* = 3.4 (Browne et al., 1986). Estimates were also obtained for the contribution from each quadrant to (w). The results for the ejection and sweep quadrants are shown in Table 2, where, since there is close symmetry about the centreline, only the results for y* < 0 are shown. The sum of the two contributions exceeds (Ly)(p) slightly. This excess is offset by small negative contributions, not shown here, from the interaction quadrants. It is evident from Table 2 that, as concluded from Fig. 7, the sweep quadrant makes a negligible contribution near the edge of the wake ( 1y * 1 = 2) and the ejection quadrant provides only a small contribution close
12
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake of a cylinder
Table 2 Percentage L’*
- 2.03 - 1.83 - 1.62 ~ 1.42 -1.22 - 1.01 - 0.81 - 0.61 - 0.40 -0.20
contributions a=u,
to m
from ejection
and sweep quadrants,
a=u,p=e
a=u,p=o
p=u
using eq. (5)
Ejection
Sweep
Ejection
Sweep
Ejection
Sweep
(III)
(I)
(II)
(IV
(II)
(Iv)
115 112 90 68 55 50 31 40 11 15
0
105 103 87 68 51 48 32 34 10 14
0
109 105 87 14 60 53 38 41 26 16
2 5 21 31 41 48 62 59 74 113
0
15 35 46 50 64 61 91 96
0
13 32 43 52 68 66 90 87
to the wake centreline. The contributions from the two quadrants shown in Table 2 cross over at _v*= + 1, reflecting the behaviour exhibited by the contributions in Fig. 4 from the unconditioned quantities. Note however that, quantitatively, the difference between the two quadrants is much more marked for conditional than conventional quantities. For example, Fig. 4 indicates that near y* = 0, the contribution from the sweep quadrant is only slightly larger than that from the ejection quadrant for uu and ~0. The difference between the sweep and ejection quadrants is almost negligible in the case of conventional ~0 values. Table 2 indicates that, for the conditional values, the contribution from the sweep quadrant is much larger than from the ejection quadrant in all three cases. These large and important differences between the conventional and conditional results may be explained as follows. Near the centreline, one would expect a reduction in the flow organisation due partly to the possible interaction between large structures on opposite sides of the centreline and perhaps to the smaller scale “old turbulence” being convected along the central region of the wake. This may then be why the small scale contribution, in so far as it masks the contribution from the large scale organised motion, significantly reduces the effectiveness of the conventional quadrant analysis technique in distinguishing between different flow features. It is possible that the use of low-pass filtering may allow the conventional quadrant technique, when applied to one point in space, to focus better on the large scale organised motion. We have not attempted this here since we believe that the use of information obtained over a significant region of space circumvents, to a large extent, the need for low-pass filtering. The spatially localised contributions from ejections and sweeps in Fig. 7 has an important implication with regard to how the quadrant analysis technique may be used to detect the large scale structures, depicted in Fig. 1, unambiguously. For example, an array of X-wires could be deployed so that there is a sufficient number of wires in the range 1 < 1y* 1 ,< 2. One would then expect the passage of the large structures to be detected, more or less simultaneously, by the ejection quadrant uu signals from these wires. This multipoint approach should, in principle. permit the mean period between structures to be determined less ambiguously than with the one point approach where the mean period various monotonically with the magnitude of the threshold which is applied to the single uu signal (Fabris and Nakayama, 1985). We should however note that there already exist several techniques (e.g. Townsend, 1979; Hussain. 1983; Antonia et al., 1986b; Browne et al., 1986) which can adequately detect large scale structures on the basis of multipoint information. In this context, the use of the quadrant technique, for the purpose of detection, would seem superfluous. The technique remains nevertheless important in that it can identify large contributions to the average momentum and
R.A. Antonia, L. WB. Browne / Quadrant analysis in the turbulent far-wake of a cylinder
heat transfers; it can certainly be used to quantify been made by another method.
these contributions
after the detection
13
has
6. Concluding remarks The results obtained with the quadrant analysis technique, as applied to conventional fluctuations at one point in the flow, can be interpreted with less ambiguity when information on the topology of the large scale motion is available. When the quadrant analysis technique is applied to fluctuations which are conditionally averaged with respect to the large scale motion, the results identify regions of space which correspond uniquely to ejection and sweep quadrants. In connection with the simplified physical representation in Fig. 1, ejections are associated with the outer upstream part of a spanwise vortex while sweeps can be identified with the inner, close to the centreline, downstream part of the same vortex. The relative contributions to the momentum and heat transports are much larger when the quadrant analysis technique is applied to the conditional data than when applied to the conventional data. The measured joint probability density functions suggest that the outward motion associated with ejections is closer to Gaussian than the inward motion associated with sweeps, perhaps because of the newly entrained fluid in the sweeps. It should be emphasised that Figs. 1 and 7 are two-dimensional cuts of what are likely to be three-dimensional vortices which vary in size, strength, age, etc. Although the details of the three-dimensional structures are beyond the scope of this paper, it can be speculated that the pictures in Figs. 1 and 7 are consistent with the possibility of three-dimensional vortex loops (e.g. Roshko, 1976; Antonia et al., 1986a). The legs of these loops would be aligned with the direction of the principal rate of strain, which is also the direction of the temperature fronts. The tips of the loops would be connected in the spanwise direction, consistent with the regions of spanwise vorticity that are suggested by the two-dimensional views of Figs. 1 and 7. The topology of the large scale motion in a plane jet (Antonia et al., 1986b) is similar to that sketched in Fig. 1, except for the obvious change of sign in vorticity. In this case one would expect to identify ejections with the outer downstream part of a spanwise vortex and sweeps with the inner upstream part of the same vortex. The notion that the large scale motion is more coherent in a plane wake or a plane jet than in a boundary layer can be supported qualitatively if not quantitatively. Consistent with this notion is Fabris and Nakayama’s (1985) suggestion that the large structure in a wake is more efficient in transferring momentum than that in a turbulent boundary layer.
Acknowledgments The support of the Australian Research Grants Scheme is gratefully contributions by D. Shah and D. Bisset are much appreciated.
acknowledged.
References Antonia, Antonia, Antonia, Brodkey, Browne, Browne,
R.A. and L.W.B. Browne (1987). Internat. J. Heat Mass Transfer, to appear. R.A., L.W.B. Browne, D.K. Bisset and L. Fulachier (1986a). J. Fluid Mech., to appear. R.A., A.J. Chambers, D.H. Britz and L.W.B. Browne (1986b). J. Fluid Mech. 172, 211. R.S., J.M. Wallace and H. Eckelmann (1974). J. Fluid Mech. 63, 209. L.W.B. and R.A. Antonia (1986). Phys. Flui& 29, 709. L.W.B., R.A. Antonia and D.K. Bisset (1986). Phys. Fluids 29, 3612.
The
14
R.A. Anton&, L. W. B. Browne / Quadruni analysis in the turbulent far-wake
Cimbala, J.M. (1985). Proc. Fifth Symposium on Turbulent Shear Flows, Cornell Fabris, G. and A. Nakayama (1985). Turbulent Shear Flows 4, 192. Grant, H.L. (1958). 1. Fluid Mech. 4, 149. Hussain, A.K.M.F. (1983). Phys. Fluids 26, 2816. Keffer, J.F. (1965). J. Fluid Mech. 22, 135. Lu, S.S. and W.W. Willmarth (1973). J. Fluid Mech. 60, 481. Mumford, J.C. (1983). J. Fluid Mech. 137, 447. Papailiou, D.D. and P.S. Lykoudis (1974). J. Fluid Mech. 62, 11. Rosbko, AS. (1976). AIAA Paper 76-78. Sabot, J. and G. Comte-bellot (1976). J. Fluid Mech. 74, 767. Sreenivasan, K.R. and R.A. Antonia (1979). AIAA Jnf. 16, 867. Taneda, S. (1959). J. Phys. Sot. Japan 14, 843. Townsend, A.A. (1979). J. Fluid Mech. 95, 515. Wallace, J.M., H. Eckelmann and R.S. Brodkey (1972). J. Fluid Mech. 54, 39. Willm~tb, W.W. and S.S. Lu (1972). J. Fluid Mech. 55, 65.
of a cylinder
University,
4.1.
Research
2 (1987) 3-14
Quadrant analysis in the turbulent far-wake of a cylinder R.A. ANTONIA
and L.W.B. BROWNE
Department of Mechanical Engineering, Received
10 November
University of Newcastle, N.S. W., 2308, Australia
1986
Abstract. The quadrant analysis technique of Willmarth and Lu (1972) has been used to estimate contributions from the four quadrants to the average momentum and heat transfers in the self-preserving region of a turbulent wake. The importance of sweeps near the wake centreline and of ejections at the edge of the wake becomes clearer when sweeps and ejections are viewed in the context of the topology of the large scale motion. The application of the quadrant analysis to the products of velocity and temperature fluctuations conditioned on the large scale motion, clarified the picture inferred from the conventional quadrant analysis approach by clearly identifying the regions of space which are important for transferring momentum and heat.
1. Introduction The quadrant analysis technique, introduced by Willmarth and Lu (1972) for a turbulent boundary layer, and extended by Lu and Willmarth (1973), has subsequently been used in different flows (e.g. Brodkey et al., 1974 for turbulent channel flow; Sabot and Comte-Bellot, 1976 for turbulent pipe flow; Sreenivasan and Antonia, 1979 for a turbulent circular jet). Recently, Fabris and Nakayama (1985) applied the technique to the product uu (U and u are the longitudinal and lateral velocity fluctuations) measured in the self-preserving region of a cylinder wake. The main aim of the work was to investigate the Reynolds stress structures and features of the large eddies. The results indicated that from about the half velocity defect point to the edge of the wake, ejections (slow speed fluid directed away from the centreline) contribute the most to the Reynolds shear stress z, in much the same manner as in the outer region of a boundary layer. It was also inferred that the centreline region was characterized by sweeps (high speed fluid directed towards the centreline). A significant amount of information already exists on several aspects of the organised motion in the turbulent far wake of a cylinder. This information stems both from flow visualisations (Taneda, 1959; Papailiou and Lykoudis, 1974; Cimbala, 1985) and other investigations (Grant, 1958; Keffer, 1965; Townsend, 1979; Mumford, 1983; Browne et al., 1986). The emerging picture suggests that the predominant flow mode consists of three-dimensional large scale rotational bulges which are asymmetrical about the centreline. A simplified two-dimensional view of this arrangement is depicted in Fig. 1. The information shown on the sketch is based on the topology of the large scale motion, determined in the far wake of a circular cylinder by Browne et al. (1986) and Antonia et al. (1986a). The topology revealed local regions, cross-hatched in Fig. 1, in which shear stresses and heat fluxes are relatively large. One such region, identified as an ejection, is associated with the fluid ejected on the upstream side of a large scale structure. Another region, identified as a sweep, is associated with the fluid swept in on the downstream side of the following structure. The physical locations of these regions correspond approximately with the regions of the wake where the quadrant technique 0169-5983/87/$4.00
0 1987, The Japan
Society
of Fluid Mechanics
R.A. Antonia, L. W.3. Browne / Qrrndrant analysis in the turbulent far-wake ---wEngulfed
heated
of a cylinder
Flow Direction ambient
fluid
cylinder
COLD
Fig. 1. Simplified two-dimensional view of dominant organised motion in the turbulent far-wake as seen by observer moving in the x direction with the approximate velocity of the centre of the structures.
used by Fabris and Nakayama (1985) indicated that ejections and sweeps may be important in the context of contributions to shear stresses. This correspondence suggests that results obtained by the quadrant technique, which suffers from the obvious limitation of drawing information from only one point in space, would be best explained in the context of information related to the large scale motion. Fabris and Nakayama foreshadowed such a possibility by concluding that “future work which incorporates other types of conditioning together with the present quadrant type analysis will further clarify the Reynolds structure in the far wake”. In this paper we extend the results obtained by Fabris and Nakayama by showing that the quadrant analysis technique applied to signals u, u and 8 (the temperature fluctuation), are qua~tatively consistent with the topology delineated by Browne et al. (1986) and Antonia et al. (1986a) for a turbulent wake. Results obtained by applying the quadrant analysis technique to conditional averages of U, u and 8, conditioned on the large scale motion, show unambiguously the physical significance of ejections and sweeps in this particular flow.
2. Experimental
details
Full details of the experimental set-up are given in Browne and Antonia (1986). In summary, the wind tunnel used was of the open return type with a working section of 350 mm X 350 mm, 2.4 m long. A stainless steel tube of diameter d = 2.67 mm was mounted in the midplane of the working section, normal to the flow and 20 cm after the end of the contraction leading into the working section. This tube was heated at a rate of 100 W to provide temperature as a passive marker of the flow. The free stream velocity U, was 6.7 m/s so that the Reynolds number Re, based on the cylinder diameter was 1170. The floor of the working section was slightly tilted to achieve a zero pressure gradient. All measurements were made at x/d = 420. At this station, the centreline mean temperature T,, relative to ambient, was 0.82 K. An assembly of an X-wire and a cold wire was moved across the wake from about y * = - 2 to +2, where y* = y/L and L (= 12.3 mm) was the value of y, Fig. 2, where the velocity defect was half its centreline value U, (= 0.36 m/s). The cold wire associated with the X-wire was located 0.5 mm in front of the crossing point of the X-wires. This did not interfere with the X-wire since statistics of U, the velocity fluctuation in the x direction, U, the velocity fluctuation in the y direction, and uu were virtually the same with and without the cold wire in position. Also the spatial resolution of the three wire assembly was satisfactory since ~6 statistics were within 3.5% of those obtained independently using a parallel arrangement of a single hot wire and a single cold wire separated in the J’ direction by 0.2 mm.
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake
of a cylinder
5
x
(al
(bl
Fig. 2. Definition
sketch for the quadrant
(c) analysis
technique:
(a) uu; (b) u6’; (c) ue.
All probes were made in our laboratory. The cold wires were all 0.63 pm diameter Pt-10% Rh Wollaston wires with an active length of 1.0 mm. The hot wires were 5 pm diameter Pt-10% Rh Wollaston wires with an active length of 0.85 mm, separated by 0.8 mm. In-house anemometers provided 0.1 mA current for the cold wires and constant temperature anemometers with an overheat of 1.8 were used for the X-wires. Careful velocity, yaw and temperature calibrations were carried out using a personal computer/data logger arrangement. Data were low-pass filtered at 1 kHz and sampled at f, = 2 kHz per channel directed into a DEC 11/34 computer using an 11 bit plus sign sample and hold A/D unit. Before filtering, the signals were suitably offset and gained. Data sampling time at each station was typically of 50 s duration.
3. Conventional
quadrant analysis
The quadrant analysis technique which is used here is similar to that introduced by Willmarth and Lu (1972) and Wallace et al. (1972). Contributions to the product a/3 are sorted into the four quadrants of the (a, /?) plane (see Fig. 2). We focus mainly on the ejection and sweep quadrants (identified in Fig. 2) although, for completeness, results for the other quadrants are also presented in this section. After digital sorting of the data, the occupancy time and the contribution that each quadrant makes to q were calculated. The occupancy time is defined as the time for which the pair of fluctuations ((Y, /?) can be found in a given quadrant i ( = I, II, III, IV) of the ( LY,fi) plane. The occupancy time is shown in Fig. 3 as a fraction of the total record duration. The contribution to G was then estimated from the joint probability density function p(cx, fi) of the fluctuating pair ((Y, fl). By definition,
aP ~(a, a= JjDo -+CC The contribution
to 3
P> da
from a particular
(1)
dfl.
quadrant
i of the ((w, p) plane
was calculated
by
R.A. Antonia, L W.B. &owe
/ Quadrant analysis in the t~r~~le~t~ur-~,ake
of a cylinder
Fig. 3. Occupancy
time in the four quadrants by the products UD, ~6’ and uB. d, i = I; b, II; p, III; 9, IV.
integrating c@p(a, /3) in that quadrant. For example, quadrant of the (u, u) plane for y z=-0 is given by
the contribution
Contributions
are shown in Fig. 4, n~rma~zed
from each quadrant
of the ((u, /3) plane
overall rms values fy’ (= VT) and /3’ determined the overall correlation coefficient pap = $/(~‘/3’, from each quadrant since
to G from the ejection
by the
from the total record. Also shown in Fig. 4 is which is equal to the sum of the contributions
Values of u’/U,, u’/UO and B’/T, are given in Table 1 for the y* values used in Figs. 3 and 4. Also shown in Table 1 are the values of the correlation coefficients pap (a, /I = U, u or 8). The results for (&)r/u’u shown in Fig. 4 are very similar to those already presented and discussed in some detail by Fabris and Nakayama (1985). Here only the main features will be highlighted. In the outer part of the wake, the ejection quadrant (III for y* < 0) is the major contributor to puu even though the occupancy time in this quadrant (Fig. 3) is much smaller than for the sweep (I) quadrant. Near the centreline, it is the contribution from the sweep quadrant which is slightly larger than that from the ejection quadrant, while the occupancy times for the two quadrants (Fig. 3) are nearly equal. Cont~butions from the sweep and ejection quadrants cross over at appro~mately y* = - 0.8, where the shear is close to its maximum value. For (?$)i/u’8’, distributions of the ejection (i = II) and sweep (i = IV) contributions follow very closely those of (uU),/u’u’ (Fig. 4). There is a correspondingly close similarity between the distributions for occupancy times in the ejection and sweep quadrants. It should also be mentioned that the results for (u, 6) [ejection quadrant II, sweep quadrant IV] differ from
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake
b b
of a cylinder
7
b”
lE~ectm.11)
qlsweep, IV I 9
0.6
P P d
d
Fig. 4. auadrants
Contributions from to the correlation
the four coefficient
u, .R= u; (b) u, 8;
those of (u, U) or (u, ~9) in only a minor way: there is relatively little difference between the importance of ejection and sweep quadrants near the centreline and the interaction quadrants (I and III) make a contribution of the same sign, albeit quite small, as that from the ejection and sweep quadrants.
Table 1 Variation
0.08 -0.16 - 0.41 - 0.65 - 0.89 - 1.14 - 1.38 - 1.63 - 1.87 -2.11
across
the wake of rms values of u, v, 19 and of mean values of products
0.29 0.31 0.32 0.34 0.34 0.32 0.28 0.21 0.14 0.08
0.24 0.24 0.25 0.24 0.23 0.22 0.19 0.15 0.12 0.09
0.17 0.18 0.19 0.22 0.24 0.25 0.24 0.21 0.16 0.10
0
0.27 0.44 0.52 0.56 0.57 0.55 0.53 0.48 0.36
MU, MO, vr3
-0.51 - 0.54 -0.56 - 0.61 - 0.65 - 0.68 - 0.70 - 0.70 - 0.64 - 0.47
0.02 - 0.24 - 0.42 - 0.53 - 0.56 -0.57 -0.55 - 0.52 - 0.48 - 0.41
8
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent jar-wake
/\ @ 1-17’ --L
/
/
/ I \
of a cylinder
1,
-,’
-,’ / // /
I
R.A. Antonia, L. WB. Browne / Quadrant analysis in the turbulent far-wake
of a cylinder
9
4. Joint probability density functions Equi-probability density contours of the pairs of fluctuations ((Y, /3) are shown in Fig. 5 for three locations in the wake: near the centreline (y * = +0.08), close to the position of maximum shear (y * = - 0.65), and in the outer flow region (y* 2: - 1.63). For the latter location, the flow is intermittent, the intermittency factor, or fraction of time the flow is turbulent, being equal to 0.43. This value was estimated by averaging the intermittency function (= 1 when the flow is turbulent and 0 when it is non-turbulent) obtained by comparing the temperature signal with an arbitrary threshold. Details for the selection of this threshold have been given in Antonia and Browne (1987). At y* = - 1.63, the joint probability density function was also calculated for fluctuations (a,, &) in only the turbulent part of the record (Fig. 6). The fluctuations CX~and /3, are centered so that ‘Y,= & = 0. Also shown in Figs. 5 and 6 are contours of constant values for the Gaussian joint probability density function
a+i= i2’
PG
-1
-1 exp
a2 -at2 - 2Pap
2(1_p;B)
2(1 - P$3)1’2
[
aP
PZ
m’p’z
)I (3) ’
(
with (Y~and & replacing cy and p respectively in Fig. 6. The Gaussian contours are drawn for to the measured values. values of pap corresponding The contours in Fig. 5 indicate that there are small departures from the Gaussian distributions at y* = 0.08 in the case of (u, u) and (u, 19). These departures appear to be associated mainly with the sweep quadrants (I for UU, IV for ve), reflecting the incursion of fluid from either side of the centreline. The contours for p(u, 0) at this location are closer to the Gaussian contours presumably because less emphasis is given to the lateral motion directed towards the centreline and more importance given to older and well mixed turbulence travelling along the centreline. the measured contours depart from the elliptical Gaussian contours. This is Aty * = - 163 ., especially pronounced in the sweep quadrant. As can be inferred from Fig. 1 and as will become clearer in the following Section, in the outer part of the wake, the sweep quadrant comprises mainly engulfed irrotational fluid or fluid which is about to be entrained and does not make a significant contribution to the Reynolds shear stress in this region of the wake.
4
,
I
I
-4
-2
1
(al 0.1
/---\
/+----L\ /
7 \ ‘\
’
7 I
//
a=ut
0
2
a-ut
.-4
-2
‘\
0
2
-4
-2
0
2
4
a/a’ Fig. 6. Joint probability density functions of the turbulent part only of the flow, of u and o, u and 0.1, 0.05, 0.01. -, y* = -1.63. (a) n, = ut, 8, = ol; (b) u,, et; (c) u,, e,. Inner to outer contours: - -, Gaussian, eq. (3).
9, v and B at experiment;
R.A. Antonia, L. W.B. Browne / Quadrant analysis m the turbulent far-wake of a cylinder
10
There is a smaller departure from the Gaussian contours in the ejection quadrant. This departure is noticeably reduced when only the turbulent fluid is considered (Fig. 6) although the departure in the sweep quadrant remains important. The main implication of Fig. 6 is that in the outer wake region, the ejection quadrant (III for WJ, II for UB and II for ~6) focuses on fluid with a relatively long history in the fully turbulent part of the flow. By contrast, sweep regions near the centreline are associated with newly entrained fluid apparently causing the departures from the Gaussian contours, observed even at the centreline. The behaviour of the present joint probability density functions is similar to that reported by Sreenivasan and Antonia (1979) in a slightly heated round jet with a co-flowing external stream. Departures from Gaussian distributions were found to be less pronounced for the outward radial motion than for the inward radial motion which is likely to be associated with the newly entrained fluid.
5. Conditional
quadrant analysis
In studies of coherent structures in turbulent shear flows, it is usual to write instantaneous variables using either the double or triple decompositions (e.g. Hussain, 1983). In order to better highlight the contribution of the organised motion to the Reynolds stresses and heat fluxes, Browne et al. (1986) and Antonia et al. (1986b) used the triple decomposition, e.g. u=U+(u)+us,
(4)
where U is the instantaneous velocity, G its mean value, the angular brackets denote the coherent large scale component, and u, is assumed to be identified with the small scale motion. Implicit in eq. (4) is the assumed independence between u, and (u). Since the fluctuation (Yhas been defined here so that Cu= 0, eq. (4) can be re-written as (Y= (a)
+ (Y,.
Contours of (a) and of the product (a)(p) were presented by Antonia et al. (1986b), where quantities within angular brackets were conditioned with respect to the occurrence of temperature fronts, as depicted in Fig. 1, detected with an array of temperature sensors ‘. The contours were displayed in the (Ax *, y *) plane, where Ax * = - U,r/L, r being the time measured from the instant of detection and U, (2: 0.97Ut) being a convection velocity of the large structure. The results obtained by applying the quadrant analysis technique to quantities (u), (u), (0) are shown in Fig. 7 in the (Ax *, y * ) plane. Also included in this figure are the locations of the temperature fronts used for detection. These fronts are roughly aligned with the diverging separatrices (Fig. 1). Contours of the product (a* )( fi * ) in Fig. 7 correspond, as shown, to the four quadrants of the ((a* ), (/3 *)) plane. From this figure, the contribution from each quadrant can be identified. For example, at large values of ( y * I, the contours of (u * ) (u * ) are associated only with the ejection quadrant (II for positive y * and III for negative y *, cf. Fig. 2). By contrast, for values of 1y* 1 near the centreline, the contours coincide only with the sweep quadrant (I and IV for negative and positive y * respectively). A similar result holds for u0 and ~0. Thus, for the three pairs of variables considered, the ejection quadrant contributes nearly totally (although only the large magnitude contours are plotted for clarity) to the product (a)(p) at the edge of the wake whereas the sweep quadrant is the major contributor close to the centreline. There is a range of 1y* ) over which contributions from ejections and sweeps overlap, although the domains of contribution retain their spatial identity. ’ Detection details can be found in Browne et al. (1986)
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake
----c
of a cylinder
11
Flow Direction
Y”
-6
-4
-2
0
2
4
0
2
4
0
2
4
Ax*
-4
-2 Ax"
-6
-4
-2 Ax"
Fig. 7. Contributions to the shear stress and heat fluxes obtained by applying the quadrant analysis technique to conditional averages of velocity and temperature. (a) (( u)( o ) contours are shown for 0.01 and 0.02. Contour values are negative for y * > 0 and positive for y * i 0. (b) ( u)( f7) contours are for - 0.02 and - 0.04. (c) ( u)( 0) contours are for 0.02 and 0.04. Contour values are positive for y * 1 0 and negative for y * < 0.
The previous observations can be quantified by integrating distance corresponding to one period of the large scale motion,
the products viz.
( CX)(/3) over a
where A* = 3.4 (Browne et al., 1986). Estimates were also obtained for the contribution from each quadrant to (w). The results for the ejection and sweep quadrants are shown in Table 2, where, since there is close symmetry about the centreline, only the results for y* < 0 are shown. The sum of the two contributions exceeds (Ly)(p) slightly. This excess is offset by small negative contributions, not shown here, from the interaction quadrants. It is evident from Table 2 that, as concluded from Fig. 7, the sweep quadrant makes a negligible contribution near the edge of the wake ( 1y * 1 = 2) and the ejection quadrant provides only a small contribution close
12
R.A. Antonia, L. W.B. Browne / Quadrant analysis in the turbulent far-wake of a cylinder
Table 2 Percentage L’*
- 2.03 - 1.83 - 1.62 ~ 1.42 -1.22 - 1.01 - 0.81 - 0.61 - 0.40 -0.20
contributions a=u,
to m
from ejection
and sweep quadrants,
a=u,p=e
a=u,p=o
p=u
using eq. (5)
Ejection
Sweep
Ejection
Sweep
Ejection
Sweep
(III)
(I)
(II)
(IV
(II)
(Iv)
115 112 90 68 55 50 31 40 11 15
0
105 103 87 68 51 48 32 34 10 14
0
109 105 87 14 60 53 38 41 26 16
2 5 21 31 41 48 62 59 74 113
0
15 35 46 50 64 61 91 96
0
13 32 43 52 68 66 90 87
to the wake centreline. The contributions from the two quadrants shown in Table 2 cross over at _v*= + 1, reflecting the behaviour exhibited by the contributions in Fig. 4 from the unconditioned quantities. Note however that, quantitatively, the difference between the two quadrants is much more marked for conditional than conventional quantities. For example, Fig. 4 indicates that near y* = 0, the contribution from the sweep quadrant is only slightly larger than that from the ejection quadrant for uu and ~0. The difference between the sweep and ejection quadrants is almost negligible in the case of conventional ~0 values. Table 2 indicates that, for the conditional values, the contribution from the sweep quadrant is much larger than from the ejection quadrant in all three cases. These large and important differences between the conventional and conditional results may be explained as follows. Near the centreline, one would expect a reduction in the flow organisation due partly to the possible interaction between large structures on opposite sides of the centreline and perhaps to the smaller scale “old turbulence” being convected along the central region of the wake. This may then be why the small scale contribution, in so far as it masks the contribution from the large scale organised motion, significantly reduces the effectiveness of the conventional quadrant analysis technique in distinguishing between different flow features. It is possible that the use of low-pass filtering may allow the conventional quadrant technique, when applied to one point in space, to focus better on the large scale organised motion. We have not attempted this here since we believe that the use of information obtained over a significant region of space circumvents, to a large extent, the need for low-pass filtering. The spatially localised contributions from ejections and sweeps in Fig. 7 has an important implication with regard to how the quadrant analysis technique may be used to detect the large scale structures, depicted in Fig. 1, unambiguously. For example, an array of X-wires could be deployed so that there is a sufficient number of wires in the range 1 < 1y* 1 ,< 2. One would then expect the passage of the large structures to be detected, more or less simultaneously, by the ejection quadrant uu signals from these wires. This multipoint approach should, in principle. permit the mean period between structures to be determined less ambiguously than with the one point approach where the mean period various monotonically with the magnitude of the threshold which is applied to the single uu signal (Fabris and Nakayama, 1985). We should however note that there already exist several techniques (e.g. Townsend, 1979; Hussain. 1983; Antonia et al., 1986b; Browne et al., 1986) which can adequately detect large scale structures on the basis of multipoint information. In this context, the use of the quadrant technique, for the purpose of detection, would seem superfluous. The technique remains nevertheless important in that it can identify large contributions to the average momentum and
R.A. Antonia, L. WB. Browne / Quadrant analysis in the turbulent far-wake of a cylinder
heat transfers; it can certainly be used to quantify been made by another method.
these contributions
after the detection
13
has
6. Concluding remarks The results obtained with the quadrant analysis technique, as applied to conventional fluctuations at one point in the flow, can be interpreted with less ambiguity when information on the topology of the large scale motion is available. When the quadrant analysis technique is applied to fluctuations which are conditionally averaged with respect to the large scale motion, the results identify regions of space which correspond uniquely to ejection and sweep quadrants. In connection with the simplified physical representation in Fig. 1, ejections are associated with the outer upstream part of a spanwise vortex while sweeps can be identified with the inner, close to the centreline, downstream part of the same vortex. The relative contributions to the momentum and heat transports are much larger when the quadrant analysis technique is applied to the conditional data than when applied to the conventional data. The measured joint probability density functions suggest that the outward motion associated with ejections is closer to Gaussian than the inward motion associated with sweeps, perhaps because of the newly entrained fluid in the sweeps. It should be emphasised that Figs. 1 and 7 are two-dimensional cuts of what are likely to be three-dimensional vortices which vary in size, strength, age, etc. Although the details of the three-dimensional structures are beyond the scope of this paper, it can be speculated that the pictures in Figs. 1 and 7 are consistent with the possibility of three-dimensional vortex loops (e.g. Roshko, 1976; Antonia et al., 1986a). The legs of these loops would be aligned with the direction of the principal rate of strain, which is also the direction of the temperature fronts. The tips of the loops would be connected in the spanwise direction, consistent with the regions of spanwise vorticity that are suggested by the two-dimensional views of Figs. 1 and 7. The topology of the large scale motion in a plane jet (Antonia et al., 1986b) is similar to that sketched in Fig. 1, except for the obvious change of sign in vorticity. In this case one would expect to identify ejections with the outer downstream part of a spanwise vortex and sweeps with the inner upstream part of the same vortex. The notion that the large scale motion is more coherent in a plane wake or a plane jet than in a boundary layer can be supported qualitatively if not quantitatively. Consistent with this notion is Fabris and Nakayama’s (1985) suggestion that the large structure in a wake is more efficient in transferring momentum than that in a turbulent boundary layer.
Acknowledgments The support of the Australian Research Grants Scheme is gratefully contributions by D. Shah and D. Bisset are much appreciated.
acknowledged.
References Antonia, Antonia, Antonia, Brodkey, Browne, Browne,
R.A. and L.W.B. Browne (1987). Internat. J. Heat Mass Transfer, to appear. R.A., L.W.B. Browne, D.K. Bisset and L. Fulachier (1986a). J. Fluid Mech., to appear. R.A., A.J. Chambers, D.H. Britz and L.W.B. Browne (1986b). J. Fluid Mech. 172, 211. R.S., J.M. Wallace and H. Eckelmann (1974). J. Fluid Mech. 63, 209. L.W.B. and R.A. Antonia (1986). Phys. Flui& 29, 709. L.W.B., R.A. Antonia and D.K. Bisset (1986). Phys. Fluids 29, 3612.
The
14
R.A. Anton&, L. W. B. Browne / Quadruni analysis in the turbulent far-wake
Cimbala, J.M. (1985). Proc. Fifth Symposium on Turbulent Shear Flows, Cornell Fabris, G. and A. Nakayama (1985). Turbulent Shear Flows 4, 192. Grant, H.L. (1958). 1. Fluid Mech. 4, 149. Hussain, A.K.M.F. (1983). Phys. Fluids 26, 2816. Keffer, J.F. (1965). J. Fluid Mech. 22, 135. Lu, S.S. and W.W. Willmarth (1973). J. Fluid Mech. 60, 481. Mumford, J.C. (1983). J. Fluid Mech. 137, 447. Papailiou, D.D. and P.S. Lykoudis (1974). J. Fluid Mech. 62, 11. Rosbko, AS. (1976). AIAA Paper 76-78. Sabot, J. and G. Comte-bellot (1976). J. Fluid Mech. 74, 767. Sreenivasan, K.R. and R.A. Antonia (1979). AIAA Jnf. 16, 867. Taneda, S. (1959). J. Phys. Sot. Japan 14, 843. Townsend, A.A. (1979). J. Fluid Mech. 95, 515. Wallace, J.M., H. Eckelmann and R.S. Brodkey (1972). J. Fluid Mech. 54, 39. Willm~tb, W.W. and S.S. Lu (1972). J. Fluid Mech. 55, 65.
of a cylinder
University,
4.1.