3.1 Phase averaging

A phase-averaging technique is used to extract the coherent structures. The technique is an improvement over that used by Zhou, Zhang & Yiu (Reference Zhou, Zhang and Yiu2002b ) and Zhou & Yiu (Reference Zhou and Yiu2006). Briefly, the transverse velocity fluctuation signal $v$ , which is used as the reference phase, from the probe at each lateral measurement location is filtered using a fourth-order Butterworth filter with the centre frequency set at $f_{shedding}$ . A cross-correlation between the measured signal $v$ and the filtered signal $v_{f}$ is carried out to determine the average phase shift between $v_{f}$ and the measured $v$ . This shift is then applied to $v_{f}$ and a new reference signal $v_{f}^{\prime }$ is formed. Figure 3 illustrates the $v$ signals at three typical lateral locations at $x/d=10$ and the corresponding shifted $v_{f}^{\prime }$ signals.

Figure 3.

Comparison of the measured signal $v$ (the thin line) and the shifted filtered signal $v_{f}^{\prime }$ (the thicker line) at $x/d=10$ .

Two phases of particular interest are identified on $v_{f}^{\prime }$ , namely

and

The two phases correspond to instants $t_{A,i}$ and $t_{B,i}$ respectively. The origin of time is arbitrary. The phase of the measured fluctuation signals is then determined as

and

The interval between phases A and B is made equal to $0.5T_{s}=0.5/f_{s}$ by compressing or stretching and is further divided into 20 equal intervals. The total number of detections is approximately 1980, 1700 and 1050 for $x/d=10,20$ and 40 respectively.

The instantaneous quantity $\unicode[STIX]{x1D6E4}$ may be viewed as the sum of the time-averaged component $\overline{\unicode[STIX]{x1D6E4}}$ and the fluctuating component $\unicode[STIX]{x1D6FD}$ , which can be further decomposed into a coherent fluctuation $\widetilde{\unicode[STIX]{x1D6FD}}$ and a remainder $\unicode[STIX]{x1D6FD}_{r}$ , namely

The coherent part $\widetilde{\unicode[STIX]{x1D6FD}}\equiv \langle \unicode[STIX]{x1D6FD}\rangle$ reflects the effect from the large-scale coherent structures, while the remainder $\unicode[STIX]{x1D6FD}_{r}$ reflects largely the random incoherent motions in the flow. After multiplying (3.5) by $\unicode[STIX]{x1D6FE}=\widetilde{\unicode[STIX]{x1D6FE}}+\unicode[STIX]{x1D6FE}_{r}$ , we obtain

By phase averaging (3.6) and assuming a zero correlation between coherent and remainder fluctuations, i.e. $\langle \widetilde{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FE}_{r}\rangle =0$ and $\langle \widetilde{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6FD}_{r}\rangle =0$ , we obtain

where $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6FE}$ represent either $u_{i}$ , $\unicode[STIX]{x1D703}$ , $\unicode[STIX]{x1D714}_{i}$ or $\unicode[STIX]{x1D703}_{,i}$ .

Figure 4.

Distribution of mean streamwise velocity and temperature at $x^{+}=10$ , 20 and 40. Dashed lines are drawn in the mean velocity plot to mark out the positions $y^{+}=0$ and $1-\overline{U}^{+}=0$ respectively.

3.2 Structural averaging

Once the coherent components are extracted from the examined quantities, the averaged coherent contributions to the products of fluctuating components within one period can be given in terms of the structural average. The conditionally averaged structure begins at $n_{1}$ samples (corresponding to $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}$ ) before $\unicode[STIX]{x1D719}=0$ and ends at $n_{2}$ samples (corresponding to $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}$ ) after $\unicode[STIX]{x1D719}=0$ . The structural average, denoted by a double overbar, is defined by

5.1 Spectral similarity between enstrophy and scalar dissipation rate

Corrsin (Reference Corrsin1953) was the first to point out the similarity between the transport equations for the mean enstrophy $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}$ and the mean scalar dissipation rate $\overline{\unicode[STIX]{x1D703}_{,i}\,\unicode[STIX]{x1D703}_{,i}}$ , which underpins a possible similarity between $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}$ and $\overline{\unicode[STIX]{x1D703}_{,i}\,\unicode[STIX]{x1D703}_{,i}}$ in either spectral or physical space. The similarity in spectral space is examined in this section, while the similarity in physical space will be considered in the next section.

The power spectrum density function of a quantity $\unicode[STIX]{x1D6FC}$ is defined such that $\int _{0}^{\infty }E_{\unicode[STIX]{x1D6FC}}(k)\,\text{d}k=\overline{\unicode[STIX]{x1D6FC}^{2}}$ , i.e. the area under the curve $E_{\unicode[STIX]{x1D6FC}}(k)$ represents the variance $\overline{\unicode[STIX]{x1D6FC}^{2}}$ . It should be noted that $kE_{\unicode[STIX]{x1D6FC}}(k)$ is plotted using a linear scale along the ordinate and log $k$ along the abscissa so that the area under the curve is still proportional to $\overline{\unicode[STIX]{x1D6FC}^{2}}$ , and thus the wavenumber corresponding to the peak of the spectrum is related to the most energetic scale. Following Fulachier & Dumas (Reference Fulachier and Dumas1976), Fulachier & Antonia (Reference Fulachier and Antonia1984) and Antonia et al. (Reference Antonia, Abe and Kawamura2009), we use the normalized spectra such that $\int _{0}^{\infty }F_{\unicode[STIX]{x1D6FC}}(k)\,\text{d}k=\int _{0}^{\infty }E_{\unicode[STIX]{x1D6FC}}(k)/\overline{\unicode[STIX]{x1D6FC}^{2}}\,\text{d}k=1$ .

The spectra of $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}$ and $\overline{\unicode[STIX]{x1D703}_{,i}\,\unicode[STIX]{x1D703}_{,i}}$ are defined as

and

where $F_{\unicode[STIX]{x1D714}i}$ and $F_{\unicode[STIX]{x1D703},i}$ are the power spectral density functions of the $i$ components of $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}$ and $\overline{\unicode[STIX]{x1D703}_{,i}\,\unicode[STIX]{x1D703}_{,i}}$ respectively, and are normalized to a unity integral, i.e. $\int _{0}^{\infty }F_{\unicode[STIX]{x1D714}i}\,\text{d}k_{1}^{\ast }=1$ and $\int _{0}^{\infty }F_{\unicode[STIX]{x1D703},i}\,\text{d}k_{1}^{\ast }=1$ .

The distributions of $k_{1}^{\ast }F_{\unicode[STIX]{x1D714}}$ and $k_{1}^{\ast }F_{\unicode[STIX]{x1D712}}$ (figure 5) follow each other closely at all three typical transverse positions, i.e. $y/L=0$ , 1 and 2, especially at $x^{+}=20$ and 40. The departures at small wavenumbers, particularly evident at $x^{+}=10$ and 20, are related to the organized Kármán vortices that occur at the Strouhal number $St=0.21$ . At $x^{+}=10$ , $F_{\unicode[STIX]{x1D714}}$ displays a pronounced peak at $f^{+}~(\equiv fU_{1}/d)=St$ . However, the most pronounced peak in $F_{\unicode[STIX]{x1D712}}$ occurs at a wavenumber corresponding to $f^{+}=2St$ . It will be shown and discussed later that the $\widetilde{\unicode[STIX]{x1D703}}_{,i}^{+}\widetilde{\unicode[STIX]{x1D703}}_{,i}^{+}$ contours are characterized by twin peaks caused by the entrainment of cold fluid into the heated wake from both sides of the free stream. As a result, the predominant frequency of $F_{\unicode[STIX]{x1D712}}$ is twice that of $F_{\unicode[STIX]{x1D714}}$ . At $x^{+}=20$ , a peak is still evident at $f^{+}=St$ for $F_{\unicode[STIX]{x1D714}}$ when $y/L\leqslant 1$ , although it is much less pronounced than at $x^{+}=10$ . However, the peak at $f^{+}=2St$ is hardly discernible in $F_{\unicode[STIX]{x1D712}}$ at $f^{+}=2St$ . By $x^{+}=40$ , the peaks $f^{+}=0.21$ or 0.42 have virtually disappeared in both $F_{\unicode[STIX]{x1D712}}$ and $F_{\unicode[STIX]{x1D714}}$ , and the two spectra follow each other more closely than at $x^{+}=10$ or 20. The streamwise evolution of the spectra at low wavenumbers suggests that the contribution from the coherent structures to $\overline{\unicode[STIX]{x1D703}_{,i}\,\unicode[STIX]{x1D703}_{,i}}$ and $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}$ plays a crucial role in degrading the spectral similarity. This is, in fact, substantiated by $F_{\unicode[STIX]{x1D712}}$ and $F_{\unicode[STIX]{x1D714}}$ at $y/L=2$ , where the coherent motion is rather weak and there is no evident peak at either $f^{+}=0.21$ or 0.42 even for $x^{+}=10$ and 20. As such, $F_{\unicode[STIX]{x1D712}}$ and $F_{\unicode[STIX]{x1D714}}$ follow each other well at low wavenumbers.

Figure 5.

Spectral density functions, multiplied by $k_{1}^{\ast }$ , of $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D712}$ measured at $y/L=0$ , 1 and 2, where $L$ is the wake half-width: (a) $x^{+}=10$ , (b) 20, (c) 40. It should be noted that the origins for the spectra at $y/L=1$ and 2 are shifted by 0.4 and 0.8 for the ordinate respectively.

The similarity between $F_{\unicode[STIX]{x1D712}}$ and $F_{\unicode[STIX]{x1D714}}$ is more tenable at $y/L=1$ than at $y/L=0$ or 2 over the high-wavenumber range. This is especially evident at $x^{+}=20$ and 40. Antonia et al. (Reference Antonia, Abe and Kawamura2009) found that in the outer region of a channel flow the spectral similarity between $\overline{\unicode[STIX]{x1D703}_{,i}\,\unicode[STIX]{x1D703}_{,i}}$ and $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}$ is not as good as near the wall due to the higher mean velocity and temperature gradients in this region. This improved similarity at $y/L=1$ (figure 5 b–c) suggests that the higher mean shear at $y/L=1$ may act to promote the similarity between $F_{\unicode[STIX]{x1D712}}$ and $F_{\unicode[STIX]{x1D714}}$ . This effect is more evident at high wavenumbers since the influence of the coherent motion may overwhelm that due to $\unicode[STIX]{x2202}\overline{U}/\unicode[STIX]{x2202}y$ and $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D6E9}}/\unicode[STIX]{x2202}y$ at low wavenumbers. One way of quantifying the similarity between spectra is to calculate the ratio of the two spectra. For wavenumbers that contribute most to $\overline{\unicode[STIX]{x1D703}_{,i}\,\unicode[STIX]{x1D703}_{,i}}$ and $\overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}}$ , $F_{\unicode[STIX]{x1D712}}/F_{\unicode[STIX]{x1D714}}$ is approximately 1 within an error of approximately $\pm 10\,\%$ at $y/L=1$ at $x^{+}=20$ and 40.

Figure 6.

Spectral density functions, multiplied by $k_{1}^{\ast }$ , of $\unicode[STIX]{x1D712},\unicode[STIX]{x1D714}$ , and their components at $y/L=1$ : (a,d) $x^{+}=10$ , (b,e) $x^{+}=20$ , (c,f) $x^{+}=40$ .

Figure 7.

Probability density function of $\cos (\unicode[STIX]{x1D719})$ , where $\unicode[STIX]{x1D719}$ is the angle between the vorticity vector $\unicode[STIX]{x1D74E}$ and the scalar gradient vector $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ , and $\cos (\unicode[STIX]{x1D719})=\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}/(|\unicode[STIX]{x1D74E}||\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}|)$ : (a) $x^{+}=10$ , (b) $x^{+}=20$ , (c) $x^{+}=40$ .

Figure 6 presents the spectra of the scalar dissipation rate, the enstrophy and their components. The peak at $f^{+}=St$ in $F_{\unicode[STIX]{x1D714}}$ results mainly from the contribution of the spanwise vorticity component, i.e. from $F_{\unicode[STIX]{x1D714}3}$ , while that at $f^{+}=0.42$ in $F_{\unicode[STIX]{x1D712}}$ is mostly contributed by $F_{\unicode[STIX]{x1D703},1}$ . Since none of the components of $F_{\unicode[STIX]{x1D712}}$ or $F_{\unicode[STIX]{x1D714}}$ can be neglected, the spectral similarity is a reflection of the summation of all three components. There is a difference between the contribution of $F_{\unicode[STIX]{x1D703},i}$ to $F_{\unicode[STIX]{x1D712}}$ and that of $F_{\unicode[STIX]{x1D714}_{i}}$ to $F_{\unicode[STIX]{x1D714}}$ . The most energetic wavenumbers, excluding the most pronounced peak, of $F_{\unicode[STIX]{x1D703},2}$ and $F_{\unicode[STIX]{x1D703},3}$ are comparable but lower than those of $F_{\unicode[STIX]{x1D703},1}$ ; however, the most energetic wavenumbers of $F_{\unicode[STIX]{x1D714}2}$ and $F_{\unicode[STIX]{x1D714}3}$ are higher than those of $F_{\unicode[STIX]{x1D714}1}$ . The difference is especially evident at $x^{+}=40$ . Antonia et al. (Reference Antonia, Abe and Kawamura2009) made a similar observation in the outer region of a turbulent channel flow and suggested that this difference in the occurrence of the most energetic wavenumbers between $F_{\unicode[STIX]{x1D714}_{i}}$ and $F_{\unicode[STIX]{x1D703}_{,i}}$ is linked to the different structures between the scalar dissipation rate and the enstrophy, i.e. the tubular structures in the enstrophy field and the sheet-like structures in the scalar dissipation rate field. The same explanation applies here. The presence of sheet-like structures in this flow is corroborated by the probability density function (p.d.f.) of $\cos (\unicode[STIX]{x1D719})~(=\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}/(|\unicode[STIX]{x1D74E}||\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}|))$ (figure 7). In the presence of sheet-like structures in $\unicode[STIX]{x1D712}$ , the vorticity and scalar gradient vectors tend to be aligned with the directions of the intermediate and compressive principal rates of strain respectively (Abe & Antonia Reference Abe and Antonia2009; Antonia et al. Reference Antonia, Abe and Kawamura2009). Then, one may expect that $\unicode[STIX]{x1D74E}$ and $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ should be statistically orthogonal to each other, as is shown in figure 7(b,c), where the peaks in the p.d.f.s of $\cos (\unicode[STIX]{x1D719})$ occur almost at $\cos (\unicode[STIX]{x1D719})=0$ for $x^{+}=20$ and 40. The p.d.f. of $\cos (\unicode[STIX]{x1D719})$ displays significant scatter at $x^{+}=10$ across the wake, suggesting that the scalar at this location does not yet feel the strain associated with the velocity field. This is not surprising since heat is mostly contained within the Kármán vortex rollers at $x^{+}=10$ , as observed by Matsumura & Antonia (Reference Matsumura and Antonia1993) and confirmed by the present phase-averaged data (figure 8 a), while the maximum strain rate occurs near the saddle points along the divergent separatrix (Hussain & Hayakawa Reference Hussain and Hayakawa1987; Thiesset, Danaila & Antonia Reference Thiesset, Danaila and Antonia2013). This is also internally consistent with the earlier observations from the spectra; that is, the mean shear has little effect in promoting the similarity between $F_{\unicode[STIX]{x1D714}}$ and $F_{\unicode[STIX]{x1D712}}$ at $x^{+}=10$ , and the departure between $F_{\unicode[STIX]{x1D712}}$ and $F_{\unicode[STIX]{x1D714}}$ is larger at $x^{+}=10$ than $x^{+}=20$ or 40. Evidently, when the passive scalar is more stretched due to the rate of strain at $x^{+}=40$ , $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ are statistically orthogonal to each other (figure 7 c) and the similarity between $F_{\unicode[STIX]{x1D714}}$ and $F_{\unicode[STIX]{x1D712}}$ is also best (figure 5 c). Observation indicates that the more the passive scalar is stretched, or more effectively mixed by the rate of strain, the better the similarity between $F_{\unicode[STIX]{x1D714}}$ and $F_{\unicode[STIX]{x1D712}}$ will be.

Figure 8.

Isocontours of phase-averaged (a–c) coherent temperature (contour intervals $=0.089$ , 0.018, 0.020), (d–f) scalar dissipation rate (0.18, 0.010, 0.004) and (g–i) enstrophy (0.11, 0.043, 0.005). Hereafter, $+$ indicates centres of vortices and $\times$ indicates saddle points. The thicker dashed lines denote the periphery of the coherent spanwise vorticity $\widetilde{\unicode[STIX]{x1D714}}_{z}^{+}$ . The dash-dotted line through the saddle point represents the diverging separatrix. The flow is from left to right.

5.2 Phase-averaged scalar dissipation rate and enstrophy

6.1 Spectral similarity

The similarity between turbulent kinetic energy and temperature variance is explored in both the spectral and phase-averaged physical domains. The spectrum of $q$ is defined following Fulachier & Dumas (Reference Fulachier and Dumas1976), Fulachier & Antonia (Reference Fulachier and Antonia1984) and Antonia et al. (Reference Antonia, Abe and Kawamura2009), namely

where $q^{2}=u^{2}+v^{2}+w^{2}$ is the turbulent kinetic energy, and $F_{u},F_{v}$ and $F_{w}$ are the power spectral density functions of $u$ , $v$ and $w$ respectively, which are normalized so that $\int _{0}^{\infty }F_{u_{i}}\,\text{d}k_{1}^{\ast }=1$ . Fulachier & Antonia (Reference Fulachier and Antonia1984) studied the similarity between $F_{q}$ and $F_{\unicode[STIX]{x1D703}}$ in a turbulent far wake, where a comparison between $F_{u}$ and $F_{\unicode[STIX]{x1D703}}$ was also included as a reference. The similarity between $F_{q}$ and $F_{\unicode[STIX]{x1D703}}$ in the far wake is better than that between $F_{u}$ and $F_{\unicode[STIX]{x1D703}}$ . We also include $F_{u}$ in order to allow a comparison between the present results in the intermediate wake and the results in the far wake.

Figure 10 shows distributions of $k_{1}^{\ast }F_{q}$ , $k_{1}^{\ast }F_{u}$ and $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ at $y/L=0,1$ and 2 for $x^{+}=10$ , 20 and 40. A pronounced peak occurs at a wavenumber close to $St=0.21$ in all of the spectra for $x^{+}\leqslant 20$ . At $x^{+}=40$ , as the Kármán vortices break up, the peaks corresponding to $St=0.21$ almost disappear in $k_{1}^{\ast }F_{u}$ and $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ but are discernible in $k_{1}^{\ast }F_{q}$ , which is largely due to the contribution from $k_{1}^{\ast }F_{v}$ . It should be noted that a rather pronounced peak occurs at a wavenumber corresponding to $f^{+}=2St=0.42$ in $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ at $x^{+}=10$ . This is because heat is associated with both positive and negative vortices, which occur alternately near the centreline (figure 8 a–c). When advected downstream, the vortices move away from the centreline; the peak at $f^{+}=0.42$ is weak at $x^{+}=20$ and cannot be detected at $x^{+}=40$ .

Figure 10.

Spectral density functions, multiplied by $k_{1}^{\ast }$ , of $\boldsymbol{q}$ , $u$ and $\unicode[STIX]{x1D703}$ measured at $y/L=0$ , 1, and 2: (a) $x^{+}=10$ , (b) $x^{+}=20$ , (c) $x^{+}=40$ .

It should be noted that the comparison between $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ and $k_{1}^{\ast }F_{q}$ is not particularly good, with $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ following $k_{1}^{\ast }F_{u}$ more closely than $k_{1}^{\ast }F_{q}$ almost everywhere, except near $y/L=2$ at $x^{+}=10$ and 20. Further, $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ exhibits a closer similarity to $k_{1}^{\ast }F_{u}$ at $y/L=1$ than at $y/L=0$ , independently of $x^{+}$ , due to the larger mean gradients of both velocity and temperature near $y/L=1$ . Observation suggests that the mean shear acts to enhance the spectral similarity between $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ and $k_{1}^{\ast }F_{u}$ . The mean shear, however, does not affect to a significant extent the similarity between $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ and $k_{1}^{\ast }F_{q}$ . As $x^{+}$ increases from 10 to 40, the difference between $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ and $k_{1}^{\ast }F_{u}$ diminishes over almost the entire range of wavenumbers, especially near $k^{+}$ corresponding to $f^{+}=0.42$ ; on the other hand, the difference between $k_{1}^{\ast }F_{\unicode[STIX]{x1D703}}$ and $k_{1}^{\ast }F_{q}$ grows at high wavenumbers. In figure 11, we plot the spectra of $q$ at the three $x^{+}$ positions together with the weighted spectra of $u$ , $v$ and $w$ . At $x^{+}=10$ and 20, both $F_{u}$ and $F_{v}$ show a pronounced peak at or near $St=0.21$ . As $x^{+}$ reaches 40, the peak in $F_{u}$ retreats, as observed in the far wake (Fulachier & Antonia Reference Fulachier and Antonia1984). The peak of $F_{v}$ (also $F_{w}$ ) remains at $St=0.21$ . As a result, the spectra of $q$ show a peak at $St=0.21$ (figure 10 a–c) (It should be noted that the value of $k_{1}^{\ast }$ corresponding to $St=0.21$ for all spectra increases from $x^{+}=10$ to 40, due to the normalization of the wavenumber ( $k_{1}^{\ast }=2\unicode[STIX]{x03C0}f/U_{1}\unicode[STIX]{x1D702}$ ), where $\unicode[STIX]{x1D702}$ increases for larger $x^{+}$ .) The similarity between $F_{\unicode[STIX]{x1D703}}$ and $F_{u}$ indicates that the scalar variance in this flow is dictated by $u$ , which is different from the far wake, where the velocity vector dictates the behaviour of the scalar fluctuation (Fulachier & Antonia Reference Fulachier and Antonia1984).

6.2 Coherent turbulent kinetic energy and temperature variance, and remainders

Figure 12 displays the coherent turbulent kinetic energy $\widetilde{q}^{+2}$ , streamwise velocity fluctuation variance $\widetilde{u}^{+2}$ and temperature variance $\widetilde{\unicode[STIX]{x1D703}}^{+2}$ at $x^{+}=10$ , 20 and 40. At each $x^{+},\widetilde{v}^{+2}$ overwhelms the other two components of $\widetilde{q}^{+2}$ . As a consequence, the contours of $\widetilde{q}^{+2}$ largely reflect those of $\widetilde{v}^{+2}$ (not shown), and the maximum concentration of $\widetilde{q}^{+2}$ is almost the same as that of $\widetilde{v}^{+2}$ , as also suggested by the most pronounced peak in $F_{q}$ and $F_{v}$ at a wavenumber corresponding to $St=0.21$ (figure 11). For both $\widetilde{\unicode[STIX]{x1D703}}^{+2}$ and $\widetilde{u}^{+2}$ , the maximum concentrations are at $\unicode[STIX]{x1D719}=0$ or $\pm 2\unicode[STIX]{x03C0}$ (e.g. figures 12 d,g and 12 e,h), where the minimum concentrations occur for $\widetilde{q}^{+2}$ (figure 12 a,b). As $x^{+}$ increases, the concentrations of both $\widetilde{\unicode[STIX]{x1D703}}^{+2}$ (figure 12 g–i) and $\widetilde{u}^{+2}$ (figure 12 d–f) drift away, following the Kármán vortices, from the centreline. On the other hand, the contours of $\widetilde{q}^{+2}$ (figure 12 a–c) remain symmetrical about the vortex centre for all $x^{+}$ . The above observations imply a closer connection between $\unicode[STIX]{x1D703}$ and $u$ than between $\unicode[STIX]{x1D703}$ and  $q$ .

Figure 11.

Spectral density functions, multiplied by $k_{1}^{\ast }$ , of $q$ and its three components at $y/L=1$ : (a) $x^{+}=10$ , (b) $x^{+}=20$ , (c) $x^{+}=40$ .

Table 4.

Averaged contributions from the coherent structures to the mean turbulent kinetic energy, variances of velocity fluctuations and scalar variance. The percentages are all calculated based the values of the corresponding $S_{c}$ and $S_{g}$ before they are rounded.

Figure 12.

(a–c) Isocontours of the phase-averaged coherent turbulent kinetic energy $\widetilde{q}^{+}\widetilde{q}^{+}$ (contour intervals: 0.0076, 0.0016, 0.0003), (d–f) variance of the velocity fluctuations $\widetilde{u}^{+}\widetilde{u}^{+}$ (0.0020, 0.0003, 0.00003) and (g–i) scalar variance $\widetilde{\unicode[STIX]{x1D703}}^{+}\widetilde{\unicode[STIX]{x1D703}}^{+}$ (0.028, 0.002, 0.001).

The coherent contributions to the turbulent kinetic energy and scalar variance are compared in table 4. A number of observations can be made. First, the coherent contribution $\unicode[STIX]{x1D6FC}$ to $\overline{u^{+2}}$ or $\overline{\unicode[STIX]{x1D703}^{+2}}$ is small, compared with that to $\overline{v^{+2}}$ . Second, the coherent contribution to $\overline{q^{+2}}$ greatly exceeds that to $\overline{\unicode[STIX]{x1D703}^{+2}}$ because of the highly organized $v$ component. Finally, $\unicode[STIX]{x1D6FC}$ across the wake of $\overline{u^{+2}}$ is comparable to that of $\overline{\unicode[STIX]{x1D703}^{+2}}$ , which explains at least partially why $F_{\unicode[STIX]{x1D703}}$ follows $F_{u}$ closely, rather than $F_{q}$ .