5.1 Phase-averaged vorticity

Rib-like structures, which are located in the alleyway between successive Kármán vortices and oriented mostly along the diverging separatrix, have been identified in both experiments (e.g. Hayakawa & Hussain Reference Hayakawa and Hussain1989; Bays-muchmore & Ahmed Reference Bays-muchmore and Ahmed1993; Mi & Antonia Reference Mi and Antonia1996; Scarano & Poelma Reference Scarano and Poelma2009) and numerical simulations (Jeong, Grinstein & Hussain Reference Jeong, Grinstein and Hussain1994; Brede, Eckelmann & Rockwell Reference Brede, Eckelmann and Rockwell1996). However, the streamwise evolution of the ribs, their interaction with the rollers and their effect on the heat and momentum transport are, probably for the first time, captured in the present experiment. In this section, the capture of the ribs will be verified in terms of both the coherent vorticity topology and the spectral correlation of $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{y}$ . The interaction between ribs and rollers and their effect on the heat and momentum transport will be discussed in detail in § 5.7.

The isocontours of the three phase-averaged coherent vorticity components, i.e. $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ , $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ and $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ , are presented in figure 7. The phase $\unicode[STIX]{x1D719}$ , ranging from $-2\unicode[STIX]{x03C0}$ to $+2\unicode[STIX]{x03C0}$ , can be interpreted as a longitudinal distance based on Taylor’s hypothesis, $\unicode[STIX]{x1D719}=0$ to $2\unicode[STIX]{x03C0}$ corresponding to the averaged vortex wavelength. To avoid any distortion of the physical space, the same scales are used in the $\unicode[STIX]{x1D719}$ and $y^{\ast }$ directions in this and subsequent figures. The positions of the centres and saddle points (e.g. Zhou & Antonia Reference Zhou and Antonia1994), identified from the phase-averaged sectional streamlines (not shown), are marked by ‘ $+$ ’ and ‘ $\times$ ’ respectively. The thicker broken line denotes the outermost vorticity contour of $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ , which is approximately 25 % of the maximum magnitude $|\tilde{\unicode[STIX]{x1D714}}_{z,max}^{\ast }|$ . This contour provides a reference for the periphery of the Kármán vortex. The broken line passing through the saddle point is the diverging separatrix, i.e. the braid connecting the consecutive roller structures (Hussain & Hayakawa Reference Hussain and Hayakawa1987). The contours of vorticity are similar to those in Zhou et al. (Reference Zhou, Zhou, Yiu and Chua2003), in which a similar experimental arrangement was employed. Because of the improved phase-average method used in this work, the organized motion is more accurately extracted at all stations, providing a clearer picture of the streamwise evolution of the coherent structures, especially the rib structures, which could hardly be identified in Zhou et al. (Reference Zhou, Zhou, Yiu and Chua2003).

Figure 7.

Phase-averaged vorticity isocontours: (a–c) $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ , contour increment $=$ 0.042, 0.013 and 0.012 respectively; (d–f) $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ , 0.031, 0.020 and 0.016; (g–i) $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ , 0.102, 0.051 and 0.019. Here and in subsequent figures, upstanding crosses $+$ and $\times$ represent the locations of the vortex centre and saddle points respectively. The thick broken contour corresponds to the contour of 25 % of the maximum spanwise vorticity and the broken line through the saddle point indicates the diverging separatrix.

The $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ contours display the well-known Kármán vortex street from $x^{\ast }=10$ to 40 (figure 7 g–i). The vortices decay rapidly, and the maximum concentration of $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ at $x^{\ast }=40$ drops to only 26 % of that at $x^{\ast }=10$ . However, its size grows considerably, and the vortex centre shifts away from the centreline, increasing from $y^{\ast }=0.39$ at $x^{\ast }=10$ to $y^{\ast }=1.38$ at $x^{\ast }=40$ . This observation is consistent with the downstream growth in the wake width. It should be noted that the $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ concentrations at $x^{\ast }=40$ are hardly identifiable in Zhou et al. (Reference Zhou, Zhou, Yiu and Chua2003) but are easily discernible in figure 7(i) because of the improved phase averaging.

In figure 7(a–f), the contours of $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ and $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ display concentrations along the diverging separatrix and wrap around the consecutive opposite-signed rollers, as also observed in Williamson (Reference Williamson1996) and Scarano & Poelma (Reference Scarano and Poelma2009), thus highlighting the signature of the rib-like structures. The organized structures at $x^{\ast }=10$ are more complicated than and quite different from those at $x^{\ast }=20$ and 40. As shown in figure 7(g), rollers of opposite sign are closely arranged near the centreline, resulting in the vigorous vorticity transport between them (this is discussed in the next section) and subsequently causing the rollers to contort, as delineated by the opposite-signed $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ contours that lean against each other. The deformed rollers can make an extra contribution to $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{y}$ in addition to that resulting from the rib structures, which is why both the $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ and $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ contours within the Kármán vortex (figure 7 a,d) exhibit a similarity to those of $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ . The concentration of $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ within the vortex at $x^{\ast }=10$ is much larger than that of $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ (figure 7 a,d), indicating that the interactions between the opposite-signed vortices occur mostly in the streamwise direction, as observed from the instantaneous isocontours of $\unicode[STIX]{x1D714}_{z}$ from the DNS data of Djenidi & Antonia (Reference Djenidi and Antonia2009) (their figure 17c). At $x^{\ast }=10$ , the additional $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ from the deformed rollers smears the $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ contours linked to the ribs. As a result, the $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ contours are not concentrated along the diverging separatrix (figure 7 a). At $x^{\ast }=20$ and 40, the opposite-signed vortices drift away from the centreline, and the braid region between them expands. Consequently, both $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ and $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ resulting from the longitudinal rib structures show comparable maximum concentrations in the braid region. As such, although the roller distortion may be further enhanced by the spanwise instability (Williamson Reference Williamson1996), the presence of the ribs is clearly identifiable from the distributions of $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ and $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ along the diverging separatrix. The capture of the ribs is further confirmed by comparing the cospectra between $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{y}$ at the centre and saddle point locations at $x^{\ast }=20$ and 40 (figure 8). The cospectra corresponding to the saddle point display a pronounced peak at $St(\equiv \,fd/U_{1})=0.2$ , indicating a strong correlation between $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{y}$ at $f_{s}$ , which is expected given the presence of the rib-like structures. On the other hand, such a peak is absent in the cospectra corresponding to the vortex centre at $x^{\ast }=20$ and 40, as $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{y}$ are unlikely to be correlated within the spanwise vortices. The same sign of $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ and $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ as $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ within the spanwise vortex is connected to how the rollers are distorted in the ( $x,z$ ) and ( $y,z$ ) planes, which will be discussed in § 5.3.

Figure 8.

Cospectral density functions of $\unicode[STIX]{x1D714}_{x}$ and $\unicode[STIX]{x1D714}_{y}$ at (a) $x^{\ast }=20$ , $y^{\ast }=0.79$ and (c) 40, 1.38, i.e. the most likely positions of vortex centres, and (b) 20, 1.38 and (d) 40, 1.77, i.e. the most likely positions of saddle points.

Figure 9.

The vorticity flux density vector $\boldsymbol{J}$ ( $J_{x},J_{y}$ ) at (a,b) $x^{\ast }=10$ and (c,d) 20. The vectors in (a,c) are calculated from the 3-D data with the $w$ component included, and those in (b,d) from the 2-D data without $w$ .

5.2 Vorticity transport

As discussed in § 5.1, the vigorous interactions between the neighbouring opposite-signed vortices may be responsible for the distortion of the spanwise rollers at $x^{\ast }=10$ . One way to evaluate the interactions between the vortices is to examine the turbulent vorticity flux density vector, $\boldsymbol{J}=(J^{x},J^{y})$ , which provides a measure for the transport of vorticity, as defined by (Kolar, Lyn & Rodi Reference Kolar, Lyn and Rodi1997)

The magnitude of the vector reflects the strength of the vorticity flux density (Kolar et al. Reference Kolar, Lyn and Rodi1997; Zhou et al. Reference Zhou, Zhang and Yiu2002). The vector is generally directed from higher to lower vorticity concentration regions. Most previous studies (e.g. Kolar et al. Reference Kolar, Lyn and Rodi1997; Zhou & Yiu Reference Zhou and Yiu2006) of the vorticity flux were based on the 2-D data, i.e. without the information on the gradient of $\langle w_{r}^{2}\rangle$ in (5.1) and (5.2), since the spanwise velocity component was not measured. In fact, Kolar et al. (Reference Kolar, Lyn and Rodi1997) noted that the ‘total’ turbulent vorticity flux should include gradients in $\langle w_{r}^{2}\rangle$ . The present data, which include all of the terms in (5.1) and (5.2), should lead to a more accurate picture of the vorticity transport. The full flux density vector has been calculated and compared with the estimation from the 2-D data in figure 9. The correspondence between the magnitude and length of the vectors is given in the upper left corner of each plot. In the 2-D case (figure 9 b,d and also figure 18b in Kolar et al. (Reference Kolar, Lyn and Rodi1997)), the vorticity transport appears to be very weak especially at $x^{\ast }=20$ in the downstream half, away from the centreline, of the spanwise roller at $\unicode[STIX]{x1D719}=0$ . However, the full flux density vectors (figure 9 a,c) show an unequivocally strong activity. There is another important difference that emerges from the comparison. While the vectors calculated from the 2-D data appear to cross the border downstream of the vortex, implying a vorticity exchange or vorticity cancellation between neighbouring opposite-signed vortices, this is not the case if the vectors are calculated from the 3-D data. Instead, the vorticity flux vectors are now predominantly linked to the rotational motion of the vortex. That is, the 2-D data could be misleading. A close examination of figure 9(a,c) points to the vorticity transport between the neighbouring vortices on the downstream side of the vortex ( $\unicode[STIX]{x1D719}=0$ ) being ‘blocked’, at least partially, by the engulfed non-turbulent fluid from the free stream immediately downstream of the rollers. This inference is corroborated by the maximum negative concentrations of $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ in figure 10(j,k), i.e. the engulfed non-turbulent cold fluid from the free stream, which occurs downstream close to the vortex at $\unicode[STIX]{x1D719}=0$ . Consequently, the intense vorticity exchange or cancellation between neighbouring vortices occurs mostly on the upstream half of the vortex ( $\unicode[STIX]{x1D719}=0$ ) and below the diverging separatrix. One should note the small magnitude of the negative $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ contours below the diverging separatrix between $\unicode[STIX]{x1D719}=1$ and 0 in figure 10(j,k).

Figure 10.

Isocontours of phase-averaged velocity and temperature fluctuations at $x^{\ast }=10$ , 20 and 40: (a–c) contour intervals $=$ 0.020, 0.008, 0.004; (d–f) 0.041, 0.017, 0.007; (g–i) 0.0066, 0.0038, 0.0052; (j–l) 0.089, 0.018, 0.020.

Previous studies (Cantwell & Coles Reference Cantwell and Coles1983; Hussain & Hayakawa Reference Hussain and Hayakawa1987; Williamson Reference Williamson1996) have highlighted the importance, in the context of the flow dynamics, of the saddle region where turbulence production occurs and the contact region between the ribs and the spanwise rollers where turbulent mixing takes place. The present data shown in figure 9(a,c) point to the fact that the upstream half of the vortex is also of great importance, as it is where most of the vorticity cancellation occurs, thus contributing greatly to the decay of the vortex strength, at least in the near wake ( $x^{\ast }\leqslant 20$ ). At $x^{\ast }=40$ (not shown), this vorticity cancellation becomes quite weak, mainly due to the consecutive positive and negative vortices drifting away from each other. As a result, the vorticity decay downstream is mainly dominated by the vortex breakdown process (Hussain & Hayakawa Reference Hussain and Hayakawa1987).

5.3 Phase-averaged velocity and temperature fluctuations

All three phase-averaged components, $\tilde{u} ^{\ast }$ , $\tilde{v}^{\ast }$ and $\tilde{w}^{\ast }$ , of the velocity fluctuation and their evolution from $x^{\ast }=10$ to 40 are presented in figure 10, along with the phase-averaged temperature fluctuation $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ . In contrast, MA showed only $\tilde{u} ^{\ast }$ , $\tilde{v}^{\ast }$ and $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ at $x^{\ast }=10$ . The contours of $\tilde{u} ^{\ast }$ , $\tilde{v}^{\ast }$ and $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ at $x^{\ast }=10$ resemble those in MA. For example, the $\tilde{u} ^{\ast }$ contours display approximate up–down antisymmetry about the vortex centre, while the $\tilde{v}^{\ast }$ contours are antisymmetric about $\unicode[STIX]{x1D719}=0$ , the maximum occurring in the alleyways between opposite-signed vortices. The positive contours of $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ show a concentric distribution within the spanwise roller and the negative contours occur between adjacent rollers, indicating that heat is contained within the Kármán vortex and cold fluid is entrained from free stream into the wake. The resemblance between the two sets of data provides a validation for the present phase-averaged results.

It is $\tilde{w}^{\ast }$ , missing in MA, that is most interesting in the evolving coherent velocity fluctuations. At $x^{\ast }=10$ , the contours of $\tilde{w}^{\ast }$ (figure 10 g) exhibit markedly different features from those of either $\tilde{u} ^{\ast }$ or $\tilde{v}^{\ast }$ , with the opposite-signed contours almost evenly distributed along the two sides of the diverging separatrix. At $x^{\ast }=20$ and 40, however, the contours of $\tilde{w}^{\ast }$ evolve into a distribution similar to those of $\tilde{v}^{\ast }$ . The distinct behaviours of $\tilde{w}^{\ast }$ at $x^{\ast }=10$ from those at $x^{\ast }=20$ and 40 imply that the mechanism of inducing $\tilde{w}^{\ast }$ changes from $x^{\ast }=10$ to $x^{\ast }=20$ and 40. Djenidi & Antonia (Reference Djenidi and Antonia2009) studied the momentum and heat transport in a transitional wake ( $Re=200$ ) via DNS. They found that, while the structures of $u$ and $v$ are both related with the spanwise rollers, the structures of $w$ result from the induction of ribs as well as the contorted rollers. The present distribution of $\tilde{w}^{\ast }$ along the diverging separatrix at $x^{\ast }=10$ , also observed by Djenidi & Antonia (Reference Djenidi and Antonia2009), is clearly a signature of ribs, with a caveat that the small $\tilde{w}^{\ast }$ concentrations near the vortex centre should result from the distorted rollers. With increasing $x^{\ast }$ , the ribs decay rapidly, as is evident in figure 7 and also in previous measurements, e.g. Brede et al. (Reference Brede, Eckelmann and Rockwell1996) and Huang, Zhou & Zhou (Reference Huang, Zhou and Zhou2006); also, the distortion of the spanwise rollers is enhanced (Williamson Reference Williamson1996; Zdravkovich Reference Zdravkovich1997). A possible scenario of the distorted spanwise rollers is sketched in figure 11. Without distortion (figure 11 a), the measurement plane ( $x$ , $y$ ) is perpendicular to the axis of the rollers, and the coherent velocities $\tilde{v}_{1}$ and $\tilde{v}_{2}$ induced by rollers will stay in the measurement plane, making no contribution to the spanwise velocity component. However, in reality, the spanwise rollers are in general distorted. One possible scenario for the distortion is illustrated in figure 11(b): the axis of the roller is non-perpendicular to the measurement plane, $\tilde{v}_{1}$ and $\tilde{v}_{2}$ producing velocity components in both the $y$ and $z$ directions. Specifically, $\tilde{v}_{1}$ can be decomposed into a positive lateral component $\tilde{v}_{+}^{m}$ and a negative spanwise component $\tilde{w}_{-}^{m}$ , where the superscript $m$ denotes measured quantities and the subscript ‘ $+$ ’ or ‘ $-$ ’ indicates the sign of the measured quantities. The scenario is consistent with the positive $\tilde{v}^{\ast }$ and negative $\tilde{w}^{\ast }$ contours between $\unicode[STIX]{x1D719}=1$ and 0 at $x^{\ast }=20$ and 40 (figure 10 e,f,h,i). Further, $\tilde{v}_{2}$ can be decomposed into a negative lateral component $\tilde{v}_{-}^{m}$ and a positive spanwise component $\tilde{w}_{+}^{m}$ , which is in agreement with the negative $\tilde{v}^{\ast }$ and positive $\tilde{w}^{\ast }$ contours between $\unicode[STIX]{x1D719}=0$ and $-1$ in figure 10(e,f,h,i). All of the observations point to the strong likelihood that $\tilde{w}^{\ast }$ results mainly from the distorted spanwise rollers at $x^{\ast }=20$ and 40. It should be noted that the roller in figure 11(b) can also result in a negative lateral vorticity component ( $\tilde{\unicode[STIX]{x1D714}}_{y}<0$ ) within the roller, as evidenced by the negative $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ contours within the spanwise vortex in figure 7(e,f).

Figure 11.

Sketch of (a) non-distorted and (b) distorted spanwise vortex rollers in the $y$ – $z$ plane. Superscript ‘ $m$ ’ denotes the measured velocity and subscripts ‘ $+$ ’ and ‘ $-$ ’ indicate the positive and negative signs respectively of the measured quantities; ‘A–A’ and ‘B–B’ represent the sectional views of the vortex roller.

The evolution of the $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ contours exhibits a close correlation with the variation in the strength of the spanwise vortex. At $x^{\ast }=10$ and 20, the positive contours of $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ due to heat separated along with the vortex from the electrically heated cylinder coincide well with the $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ concentrations, indicating that the Kármán vortex retains heat effectively with its strong rotational motion. This changes dramatically at $x^{\ast }=40$ . Heat is transferred out of the periphery of the vortex, the main trend being towards the free stream. This is directly connected to the fact that the spanwise vortex becomes quite weak at $x^{\ast }=40$ , with its maximum magnitude of $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ dropping to only approximately 26 % of that at $x^{\ast }=10$ (figure 7 g,i), and hence cannot retain heat as effectively as at $x^{\ast }=10$ and 20.

5.4 Coherent momentum and heat fluxes and remainders

The topologies of the heat and momentum transport are examined in this section, including both large-scale coherent quantities and relatively small-scale remainders. In contrast to only two components ( $u\unicode[STIX]{x1D703}$ and $v\unicode[STIX]{x1D703}$ ) available in MA, all three components ( $u\unicode[STIX]{x1D703}$ , $v\unicode[STIX]{x1D703}$ , $w\unicode[STIX]{x1D703}$ ) of the heat flux vector are examined here, thus providing a more complete picture of 3-D momentum and heat transport.

Figure 12.

Isocontours of phase-averaged coherent momentum and heat fluxes. Contour intervals: (a–c) 0.0026, 0.0005, 0.00016; (d–f) 0.0076, 0.0006, 0.00047; (g–i) 0.015, 0.0016, 0.00088; (j–l) 0.0015, 0.0005, 0.00034.

Figure 12 presents the coherent Reynolds shear stress $\tilde{u} ^{\ast }\tilde{v}^{\ast }$ and the three components, i.e. $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ , $\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ and $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ , of the heat flux vector at the three stations. The structures of $\tilde{u} ^{\ast }\tilde{v}^{\ast }$ , $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ and $\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ are similar to those reported by MA, and need not be discussed individually. Similarly to the behaviours of $\tilde{w}^{\ast }$ , the $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ contours exhibit quite different features between $x^{\ast }=10$ and the other two $x^{\ast }$ positions. At $x^{\ast }=10$ , the oppositely signed $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ concentrations occur on the two sides of the diverging separatrix, opposite to each other, apparently resulting from the motion of ribs. It should be noted that, due to the high concentration of $\tilde{\unicode[STIX]{x1D703}}^{\ast }$ within the roller, a high concentration of $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ occurs near the vortex centre, though with a weak $\tilde{w}^{\ast }$ (figure 10 g), at $x^{\ast }=10$ . At $x^{\ast }=20$ and 40, the high-level contours of $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ do not seem to show any correlation with the diverging separatrix and, instead, exhibit a great resemblance to those of $\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ at the same $x^{\ast }$ . This difference in $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ is consistent with that in $\tilde{w}^{\ast }$ (figure 10 g–i) between $x^{\ast }=10$ and the other two $x^{\ast }$ positions. The similarity between the contours of $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ and $\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ at $x^{\ast }=20$ and 40 is connected to that between the contours of $\tilde{w}^{\ast }$ and $\tilde{v}^{\ast }$ (figure 10 e,h,f,i). The $\tilde{v}\tilde{\unicode[STIX]{x1D703}}$ and $\tilde{w}\tilde{\unicode[STIX]{x1D703}}$ contours are of opposite sign, as are the $\tilde{v}$ and $\tilde{\unicode[STIX]{x1D714}}$ contours, due to the spatial contortion of the spanwise rollers as seen in figure 11. The evolution of $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ reflects the fact that both rollers and ribs influence the coherent part of the heat transport at $x^{\ast }=10$ . However, the ribs appear to be weakened at $x^{\ast }=20$ and 40; so does their ability to induce the spanwise motion and heat flux. Consequently, the coherent heat transport is overwhelmingly dominated by the distorted spanwise rollers at $x^{\ast }=20$ and 40. The result is in agreement with the observation of Huang et al. (Reference Huang, Zhou and Zhou2006), from PIV data ( $Re=2000$ ), that the ribs become weak and less organized from $x^{\ast }=4$ to 12.

Interestingly, the ribs seem to play a role that is increasingly important downstream in both momentum and heat flux of the remainders (figure 13). At $x^{\ast }=10$ , the high-level $\langle u_{r}^{\ast }v_{r}^{\ast }\rangle$ contours tend to run along the diverging separatrix. The concentration of positive $\langle u_{r}^{\ast }v_{r}^{\ast }\rangle$ near the vortex centre is the footprint of the shear stress that is associated with the diverging separatrix below the centreline ( $y^{\ast }<0$ ). Because the positive and negative vortices are close to each other near the centreline at $x^{\ast }=10$ , the contours of the shear stress generated in the saddle region have actually protruded into the Kármán vortex on the other side of the centreline, as also observed by Hussain & Hayakawa (Reference Hussain and Hayakawa1987) and Matsumura & Antonia (Reference Matsumura and Antonia1993). As $x^{\ast }$ increases, the maximum of $\langle u_{r}^{\ast }v_{r}^{\ast }\rangle$ is gradually shifted to the saddle point. This change is correlated with the observation that the vortices drift away from the centreline. It should be noted that the saddle region at $y^{\ast }>0$ corresponds laterally to the positive vortex roller at $y^{\ast }<0$ . As a result, the influence of the rollers at $y^{\ast }<0$ on the saddle region at $y^{\ast }>0$ diminishes as both positive and negative vortices move away from the centreline and the contours of $\langle u_{r}^{\ast }v_{r}^{\ast }\rangle$ are predominantly concentrated along the diverging separatrix, with a maximum at the saddle point at $x^{\ast }=40$ (figure 13 c). Hussain & Hayakawa (Reference Hussain and Hayakawa1987) observed that the turbulent production occurs mostly at approximately the same location as the peak of $\langle u_{r}^{\ast }v_{r}^{\ast }\rangle$ . The shift of the location in the maximum $\langle u_{r}^{\ast }v_{r}^{\ast }\rangle$ implies the improved importance of the saddle region in the turbulence production. On the other hand, while the maximum $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ is also near the vortex centre at $x^{\ast }=10$ , the $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ concentrations occur between spanwise vortices. However, at $x^{\ast }=20$ and 40 the concentrations of both $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ and $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ occur about the diverging separatrix. Hussain & Hayakawa (Reference Hussain and Hayakawa1987) asserted, without providing the experimental evidence, that the most effective turbulence mixing occurs where ribs and spanwise vortices are in contact with each other as a result of the small-scale fluctuations produced by the direct interaction between the streamwise vorticity and the spanwise vorticity. The present observation from the $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ and $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ contours at $x^{\ast }=20$ and 40 (figure 13 h,i,k,l) seems consistent with their assertion, given that the remainder is composed mainly of small scales responsible for turbulent diffusion and mixing (Tennekes & Lumley Reference Tennekes and Lumley1972). Generally, $\langle u_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ and $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ are of opposite sign above the centreline, reflecting the negative correlation between $u$ and $v$ in the region where $\text{d}\bar{U}/\text{d}y>0$ (e.g. Li et al. Reference Li, Balaras and Wallace2010, also figure 13 c). The interesting exception is that the $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ concentrations are of opposite sign on either side of the diverging separatrix at $x^{\ast }=20$ and 40 (figure 13 k,l), again confirming the claim that the ribs are dynamically important in these $x^{\ast }$ positions.

Figure 13.

Isocontours of momentum and heat fluxes associated with the remainder motion. Contour intervals: (a–c) 0.001, 0.0007, 0.00034; (d–f) 0.0048, 0.0021, 0.0029; (g–i) 0.0037, 0.0015, 0.0015; (j–l) 0.0014, 0.0011, 0.0013.

5.5 Productions of turbulent kinetic energy and temperature variance

The turbulence energy production $\langle P_{e}\rangle ^{\ast }$ plays the role of an energy sink for the mean flow and an energy source for the fluctuating velocities, defined by (e.g. Cantwell & Coles Reference Cantwell and Coles1983; Pope Reference Pope2001)

Previous calculations (e.g. Cantwell & Coles Reference Cantwell and Coles1983; Antonia et al. Reference Antonia, Browne, Bisset and Fulachier1987; Hussain & Hayakawa Reference Hussain and Hayakawa1987) of $\langle P_{e}\rangle ^{\ast }$ were based on only $u$ and $v$ , so that only part I in (5.3) was estimated. However, even in the intermediate wake, the three-dimensionality of the flow (e.g. Williamson Reference Williamson1996) cannot be ignored. Here, both parts I and II in (5.3) are measured directly with the vorticity probe, thus providing a more complete picture of the turbulence energy production and its streamwise evolution.

Figure 14 shows the isocontours of $\langle P_{e}\rangle ^{\ast }$ at $x^{\ast }=10$ and 40. For comparison, both part I and parts I $+$ II in (5.3) are presented. The topological features of part I (figure 14 a,b) resemble, in general, those reported in Hussain & Hayakawa (Reference Hussain and Hayakawa1987), who studied $\langle P_{e}\rangle ^{\ast }$ in a turbulent intermediate wake ( $x^{\ast }=10$ and 40) at $Re=1.3\times 10^{4}$ . For example, the concentrations of $\langle P_{e}\rangle ^{\ast }$ at both $x^{\ast }=10$ and 40 occur largely in the saddle region and along the divergent separatrix; there are insignificant negative concentrations mainly within the vortex. These similarities provide a validation of the present data. Not surprisingly, the isocontours of parts I $+$ II of $\langle P_{e}\rangle ^{\ast }$ (figure 14 c,d) exhibit a similar pattern to part I (figure 14 a,b). However, the maximum concentrations of parts I $+$ II at $x^{\ast }=10$ and 40 are 16 % and 22 % larger than their counterparts of part I respectively; that is, the contribution from part II cannot be neglected especially at $x^{\ast }=40$ . The larger contribution from part II to the full $\langle P_{e}\rangle ^{\ast }$ at $x^{\ast }=40$ is associated with the more pronounced three-dimensionality of the flow than at $x^{\ast }=10$ . One can expect that the contribution from part II to the full quantity of $\langle P_{e}\rangle ^{\ast }$ will increase further downstream where the three-dimensionality of the flow will be enhanced with increasing $x^{\ast }$ (e.g. Williamson Reference Williamson1996).

Figure 14.

Isocontours of the production $\langle P_{e}\rangle ^{\ast }$ of turbulent kinetic energies: (a,b) part I in (5.3), i.e. without $w$ -related terms; (c,d) parts I $+$ II. Contour intervals: (a) 0.00083, (b) 0.000033, (c) 0.00107, (d) 0.000054.

It is worth noting that the negative concentration of parts I $+$ II (figure 14 c,d) is much larger than that of part I (figure 14 a,b). An examination of the contours of the individual terms of part I and part II (not shown) indicates that this negative production is mainly from the contribution of the normal stress terms, i.e. the terms related to $\langle u_{r}^{2}\rangle ^{\ast }$ , $\langle v_{r}^{2}\rangle ^{\ast }$ and $\langle w_{r}^{2}\rangle ^{\ast }$ . Both Antonia et al. (Reference Antonia, Browne, Bisset and Fulachier1987) and Hussain & Hayakawa (Reference Hussain and Hayakawa1987) also noted, based on 2-D data, the small negative concentrations of $\langle P_{e}\rangle ^{\ast }$ within the vortex, as figure 14(a,b) indicates; Hussain & Hayakawa (Reference Hussain and Hayakawa1987) confirmed that the negative contribution to the production is from the normal stress terms by studying the normal stress and shear stress terms separately. The present data further reveal that this contribution to the full $\langle P_{e}\rangle ^{\ast }$ comes mainly from the spanwise normal stress.

The production of the temperature variance associated with small-scale motions, $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ , is given by (Tennekes & Lumley Reference Tennekes and Lumley1972; Antonia et al. Reference Antonia, Browne, Bisset and Fulachier1987)

To the best of our knowledge, no attempt has yet been made to determine the topological features of $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ in the intermediate wake. The isocontours of $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ at all three $x^{\ast }$ stations are shown in figure 15. The contours of the three individual terms on the right-hand side of (5.4) are also studied, though not shown. At $x^{\ast }=10$ , the first two terms occur mainly within the Kármán vortex and make a dominant contribution to $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ ; on the other hand, the third term occurs mostly between neighbouring vortices with a maximum concentration of approximately 7 % of the total $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ , i.e. the sum of all three terms. That is, $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ at $x^{\ast }=10$ is mainly due to the contribution from the first two terms. This is also reflected in the relatively low magnitudes of the $\langle w_{r}\unicode[STIX]{x1D703}\rangle ^{\ast }$ contours compared with those of $\langle u_{r}\unicode[STIX]{x1D703}\rangle ^{\ast }$ and $\langle v_{r}\unicode[STIX]{x1D703}\rangle ^{\ast }$ at $x^{\ast }=10$ (figure 13 d,g,j). However, as $x^{\ast }$ increases, the third term gradually increases in magnitude. At $x^{\ast }=40$ , the maximum contour levels of the three terms are quite comparable, along with the topological similarity in the contours of $\langle u_{r}\unicode[STIX]{x1D703}_{r}\rangle ^{\ast }$ , $\langle v_{r}\unicode[STIX]{x1D703}_{r}\rangle ^{\ast }$ and $\langle w_{r}\unicode[STIX]{x1D703}\rangle ^{\ast }$ (figure 13 f,i,l). The maximum concentration of the $w$ -related term is 13 % of that of the total $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ . The $w$ -related term is expected to contribute more to $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ beyond $x^{\ast }=40$ as the flow and hence heat transport become more 3-D.

Figure 15.

Isocontours of the production of temperature variance associated with the small-scale motions: (a) $x^{\ast }=10$ , (b) 20, (c) 40. Contour intervals: (a) 0.0039, (b) 0.00051, (c) 0.00049.

Figure 16.

Distributions of the r.m.s. values of the three components of the heat flux vector across the wake: (a) $x^{\ast }=10$ , (b) 20, (c) 40. The symbol $\unicode[STIX]{x1D6FC}$ denotes $u,v$ or $w$ .

5.6 Evolution of the heat flux components

Since the instantaneous heat transport is highly 3-D and $w\unicode[STIX]{x1D703}$ plays a role in transporting heat, it is of fundamental interest to understand the streamwise evolution characteristics of the three components of the heat flux vector. Figure 16 compares the distributions of $(u^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ , $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ and $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ across the wake for the three stations. At $x^{\ast }=10$ , $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ is the largest of the three and $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ is the smallest. The difference is especially evident for $y^{\ast }<1$ . Nonetheless, $(u^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ , $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ and $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ are certainly of comparable magnitude. The magnitude of $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ drops rapidly with increasing $x^{\ast }$ , as does $(u^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ . These decreases are correlated with the rapidly decaying spanwise vortices (figure 7). On the other hand, $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ declines slowly even though the rib structures also decay downstream. As a result, the difference between $(u^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ , $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ and $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ shrinks. As a matter of fact, by $x^{\ast }=40$ ,  $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ even exceeds $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ , and $(u^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ is now slightly larger than the other two components. Further, the downstream evolution of $(u^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ , $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ and $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ suggests an approach to axisymmetry of the flow. Mi & Antonia (Reference Mi and Antonia2010) observed that both global and local axisymmetry, i.e. the axisymmetry of large and small scales respectively, is approximately satisfied at $x^{\ast }=40$ . It should be noted that axisymmetry implies invariance of properties with respect to rotation about a preferred axis; it is therefore much less constraining than isotropy, which implies spherical symmetry. George & Hussein (Reference George and Hussein1991) examined previously discussed data and found that local axisymmetry was more adequate than local isotropy in many turbulent shear flows, including the turbulent jet, the far wake and also regions of the boundary layer. A possible explanation for the onset of axisymmetry at $x^{\ast }\geqslant 40$ is that the coherent structures are quite weak in this region and the strong directional characteristics of the mean flow dominate. As such, the turbulent kinetic energy extracted from the mean flow is preferentially allocated to the streamwise velocity fluctuation. The similar behaviour of the r.m.s. values of the fluctuating velocity and heat flux indicates a close association between $(u^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ , $(v^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ , $(w^{\ast }\unicode[STIX]{x1D703}^{\ast })_{rms}$ and the corresponding relative behaviour of the fluctuating velocity components.

5.7 Discussion: topologies of heat transport and production

The characteristics of heat transport can be examined by means of the coherent heat flux vectors $\tilde{\boldsymbol{q}}^{\ast }\equiv (\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast })$ and the remainder vectors $\langle \boldsymbol{q}_{r}^{\ast }\rangle$   $\equiv$  ( $\langle u_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ , $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ , $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ ), along with the phase-averaged vorticity vector $\tilde{\unicode[STIX]{x1D74E}}^{\ast }\equiv (\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast },\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast },\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast })$ , as shown in figure 17. The 3-D vectors are presented in terms of 2-D vectors (streamwise and lateral components) and the isocontours of the spanwise component, which are represented by colour for clarity.

Figure 17.

Phase-averaged vorticity, coherent and incoherent heat flux vectors. In (a–c), $\tilde{\unicode[STIX]{x1D74E}}^{\ast }:(\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }:\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast })$ is given by vectors and $\tilde{\unicode[STIX]{x1D714}}_{z}^{\ast }$ is given by the isocontours; in (d–f), $\tilde{\boldsymbol{q}}^{\ast }$ : ( $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ ) is given by vectors and $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ is given by the isocontours; in (g–i), $\langle \boldsymbol{q}_{r}^{\ast }\rangle$ : ( $\langle u_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ , $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ ) is given by vectors and $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ is given by the isocontours.

At $x^{\ast }=10$ , the vorticity vectors $(\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast },\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast })$ of relatively large magnitude occur largely within the spanwise vortices and appear to be nearly parallel to the centreline (figure 17 a). Presumably, the opposite-signed spanwise rollers are closely separated with respect to each other such that the deformation of the rollers in the ( $x$ , $z$ ) plane can lead to additionally produced $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ in addition to $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ originated from the predominantly longitudinal rib-like structures (cf. § 5.1 and also Hayakawa & Hussain Reference Hayakawa and Hussain1989; Hammache & Gharib Reference Hammache and Gharib1991). The ensuing $\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast }$ is naturally substantially larger than the lateral component $\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast }$ , resulting in almost horizontally oriented vectors $(\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast },\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast })$ within the Kármán vortex. With increasing $x^{\ast }$ , the opposite-signed rollers drift away from the centreline and hence from each other, and the braid region between two neighbouring rollers grows so that the ribs are more readily identifiable with the vectors $(\tilde{\unicode[STIX]{x1D714}}_{x}^{\ast },\tilde{\unicode[STIX]{x1D714}}_{y}^{\ast })$ , which are oriented along the diverging separatrix (figure 17 b,c).

The coherent heat flux vectors $\tilde{\boldsymbol{q}}^{\ast }$ exhibit a marked change from $x^{\ast }=10$ to the other two $x^{\ast }$ positions. At $x^{\ast }=10$ , the vectors $(\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast })$ within the spanwise rollers indicate unambiguously the rotational motion about the vortex centre. The vectors immediately downstream of the negative rollers point upwards, almost perpendicularly to the centreline, reflecting the arrival of cooler fluid engulfed from the free stream. The concentration of the $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ contours on one side of the diverging separatrix is opposite in sign to that on the other side of the separatrix. The observation that the maximum $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ occurs at a location different from that of the maximum magnitude of ( $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ ) corroborates the conclusion in § 5.4, i.e. the coherent flow structures that induce the heat fluxes ( $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ ) are different from those responsible for $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ . While ( $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ ) result mainly from the spanwise vortices, $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ is mostly attributed to the predominantly longitudinal ribs. In contrast, at $x^{\ast }=20$ and 40, the maximum concentration of $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ occurs at almost the same location as that of $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ and $\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ , an indication that the same coherent structures, i.e. the contorted spanwise rollers (cf. § 5.4), account for ( $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ ) and $\tilde{w}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ . It should be noted that at $x^{\ast }=40$ , the vectors ( $\tilde{u} ^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast },\tilde{v}^{\ast }\tilde{\unicode[STIX]{x1D703}}^{\ast }$ ) no longer show any rotational motion within the spanwise vortex; rather, they appear to drift out of the periphery of the vortex, moving towards the free stream. The result reflects the greatly weakened strength, due to decaying and breaking up, of the spanwise vortices. The weakened vortices are unable to retain heat as well as at $x^{\ast }=10$ and 20.

The behaviour of $\langle q_{r}^{\ast }\rangle$ differs markedly from that of $\tilde{q}^{\ast }$ . The vectors ( $\langle u_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ , $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ ) point, in general, upstream and are characterized by a large magnitude at the upstream half of the rollers near the centreline. Concentrations of $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ occur mostly between neighbouring rollers, where the ribs are connected to the spanwise rollers (figure 13 j–l). They result from direct interactions between the ribs and the rollers, which act to produce large random velocity fluctuations (Hussain & Hayakawa Reference Hussain and Hayakawa1987).

Knowing all three components of the vorticity and heat flux vectors, we can now propose a 3-D picture of the heat and momentum transport. The conceptual sketch in figure 18(a) is based on the present experimental data at $x^{\ast }=10$ . Compared with the 2-D model (see the inset of figure 18) in MA, the present model highlights the roles played by the rib structures in transporting heat and momentum in the braid region and also the vorticity flux density between neighbouring opposite-signed rollers. This model also underlines the importance of the upstream half of the spanwise rollers, rather than only one quadrant of the roller, as in the MA model, in diffusing heat out of the vortex. For simplicity, the spanwise rollers are drawn as 2-D structures, i.e. the spanwise contortion of the rollers is not shown. This simplification is reasonable since the roller at the beginning of the intermediate wake is less distorted compared with that downstream (Williamson Reference Williamson1996; Zdravkovich Reference Zdravkovich1997). The ribs are drawn conceptually as counter-rotating longitudinal vortices, which wrap around the spanwise vortices on the basis of the present coherent vorticity data (figures 7 a–f and 17 a–c) and previous reports (e.g. Brede et al. Reference Brede, Eckelmann and Rockwell1996; Huang et al. Reference Huang, Zhou and Zhou2006).

Figure 18.

(a) Conceptual model of momentum and 3-D heat transport at $x^{\ast }=10$ ; the 2-D model of Matsumura & Antonia (Reference Matsumura and Antonia1993) (MA) is shown in the inset to the figure. Thick and thin red arrows denote the coherent and remainder heat fluxes respectively. (b) Evolution of the topological features of turbulent energy and temperature variance productions from $x^{\ast }=10$ to 40 (structures are not to scale).

For clarity, we consider only one vortex shedding period. The cooler fluid from the free stream engulfed by the Kármán vortex can be largely divided into three parts. The first is entrained immediately downstream of vortex A towards the centreline, resulting in a strong heat transport from the centreline to the free stream, as indicated by the large arrows between $\unicode[STIX]{x1D719}=-1$ and 0 in figure 17(d). The second is drawn into the saddle region between vortices B and C and further bifurcated into two branches, due to the combined induced motion of the adjacent opposite-signed rolls, which lead to the opposite-signed spanwise velocities (figure 10 g) and heat fluxes (figure 12 j) on the two sides of the diverging separatrix. Although only one branch of the bifurcated flow is detected, the existence of the bifurcated flow near the saddle region has been demonstrated by the observation of Wu et al. (Reference Wu, Sheridan, Welsh and Hourigan1996) from the PIV data. The third (and last) part of the entrained flow is engulfed into the wake by vortex C. While the heat flux associated with the first part of the engulfed flow, as in the MA model, occurs mainly in the ( $x$ – $y$ ) plane and makes a predominant contribution to heat transport from the wake region to the free stream, the heat flux due to the second part should generally account for the heat transport in the spanwise direction.

The coherent motions play an important role in large-scale heat and momentum transport by entraining the cooler potential fluid from the free stream to the heated turbulent wake, while the small-scale random motions are crucial for heat diffusion and mixing between the warm wake and entrained cooler fluid. At $x^{\ast }=10$ , heat is mostly contained within the Kármán vortex (figure 10 j). Both the present (figure 17 g) and the MA results indicate that the remainder heat flux occurs mostly in quadrant II of vortex B and quadrant III of vortex C. The vectors pointing in the upstream centrifugal direction suggest that the remainder heat flux $(\langle u_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle ,\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle )$ serves to diffuse heat out of the vortex. However, what is overlooked by MA is the importance of quadrant II of vortex C (also quadrant III of vortex B), which contains the contact region between the rib and the roller, where effective turbulent mixing occurs (Hussain & Hayakawa Reference Hussain and Hayakawa1987). The heat flux $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ in quadrant II of vortex C is opposite in sign to $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ in quadrant III and points in the centrifugal direction (figure 17 g), implying a heat transport out of the vortex. Although relatively small at $x^{\ast }=10$ compared with that in quadrant III, the heat flux $(\langle u_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle ,\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle )$ in quadrant II of vortex C gains importance and in fact overwhelms that of quadrant III at $x^{\ast }=20{-}40$ where the maximum $\langle v_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ (and also $\langle w_{r}^{\ast }\unicode[STIX]{x1D703}_{r}^{\ast }\rangle$ at $x^{\ast }=40$ ) is shifted towards quadrant II (cf. figure 13 h,i,l).

Some comments regarding the downstream evolution of the proposed model are appropriate. As the rib structures decay and the distortion of the spanwise rollers intensifies with increasing $x^{\ast }$ , the large-scale heat and momentum transport is caused predominantly by the deformed rollers. This is why the coherent momentum flux and three heat fluxes are characterized by almost the same topologies at $x^{\ast }=40$ (figure 12 c,f,i,l). On the other hand, the most important heat diffusion region is gradually shifted from quadrant III to quadrant II of vortex C. This is consistent with the observations by Cantwell & Coles (Reference Cantwell and Coles1983) and Hussain (Reference Hussain1986) that the saddle region becomes more important as $x^{\ast }$ increases. Although based on the data at $x^{\ast }=10$ , the model is certainly expected to apply upstream of this location where the ribs are greater in strength and more organized (Huang et al. Reference Huang, Zhou and Zhou2006).

Figure 18(b) compares the streamwise evolution of the topologies associated with $\langle P_{e}\rangle ^{\ast }$ and $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ . While Hussain & Hayakawa (Reference Hussain and Hayakawa1987) have shown the typical topology of the turbulence production for $x^{\ast }=10{-}40$ based on their 2-D data, the topology of the temperature variance and a comparison with that of the turbulent energy production have not been reported hitherto. As indicated in figure 14(c), $\langle P_{e}\rangle ^{\ast }$ at $x^{\ast }=10$ is largely concentrated along the diverging separatrix below the saddle point; while exhibiting a similar distribution to $\langle P_{e}\rangle ^{\ast }$ , $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ occurs nearer to the centreline (figure 15 a). At $x^{\ast }=40$ , $\langle P_{e}\rangle ^{\ast }$ spreads along the diverging separatrix and has a maximum overlap with the saddle point (figure 14 d), but the concentration of $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ is located further away from the centreline than $\langle P_{e}\rangle ^{\ast }$ with a maximum above the saddle point. The more rapid lateral shift exhibited by $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ than by $\langle P_{e}\rangle ^{\ast }$ as $x^{\ast }$ increases is quite similar to the difference between $\langle v_{r}\unicode[STIX]{x1D703}_{r}\rangle ^{\ast }$ (figure 13 g–i) and $\langle u_{r}v_{r}\rangle ^{\ast }$ (figure 12 a–c). This is reasonable since $\langle v_{r}\unicode[STIX]{x1D703}_{r}\rangle ^{\ast }$ and $\langle u_{r}v_{r}\rangle ^{\ast }$ together with their interactions with the temperature and velocity gradients are the dominant terms of $\langle P_{\unicode[STIX]{x1D703}}\rangle ^{\ast }$ and $\langle P_{e}\rangle ^{\ast }$ respectively.