4.1. Phase-averaged one-point quantities

Figures 2(a) and 2(b) show the ratios of kinetic energy of coherent motion to the total fluctuating field (i.e. $\overline {\tilde {u}^{2}}/\overline {u^{2}}$ and $\overline {\tilde {v}^{2}}/\overline {v^{2}}$ ) at $x/d=10$ , 20 and 40. Also included in this figure are the data of Thiesset et al. (Reference Thiesset, Danaila and Antonia2014) at comparable $Re$ values. Figure 2(c) shows the ratio of the scalar variance of the coherent motion to the conventional scalar variance (i.e. $\overline {\tilde {\theta }^{2}}/\overline {\theta ^{2}}$ ). There is reasonable agreement between the present distributions of $\overline {\tilde {u}^{2}}/\overline {u^{2}}$ and $\overline {\tilde {v}^{2}}/\overline {v^{2}}$ and the data of Thiesset et al. (Reference Thiesset, Danaila and Antonia2014) (figure 2 a,b). The maximum of $\overline {\tilde {v}^{2}}/\overline {v^{2}}$ ( $\approx 67\,\%$ ) occurs at the wake centreline and gradually decreases as $y/d$ increases. In contrast, $\overline {\tilde {u}^{2}}/\overline {u^{2}}$ is very small at the wake centreline, even at $x/d=10$ where $\overline {\tilde {u}^{2}}/\overline {u^{2}}\approx 2\,\%$ . Its maximum occurs at $y/d\approx 1{-}2$ and shifts further from the centreline with increasing downstream distance. Similarly, the distributions of $\overline {\tilde {\theta }^{2}}/\overline {\theta ^{2}}$ in figure 2(c) resemble those of $\overline {\tilde {u}^{2}}/\overline {u^{2}}$ , with maxima occurring at transverse regions away from the centreline. However, these maxima are closer to the centreline. This is because the maximum $\tilde {u}$ occurs at the upper boundary of the positive vortex (Matsumura & Antonia Reference Matsumura and Antonia1993), while the maximum $\tilde {\theta }$ is located at the vortex centre (Chen et al. Reference Chen, Zhou, Zhou and Antonia2016). Notably, at $x/d=10$ , $\overline {\tilde {\theta }^{2}}$ accounts for approximately $ 15\,\%$ of $\overline {\theta ^{2}}$ at the centreline, far exceeding the corresponding 2 % observed in the case of $\overline {u^{2}}$ . This highlights the larger contribution of the coherent motion to $\overline {\theta ^{2}}$ .

Figure 2.

Contributions of the coherent motion to total fluctuating field: (a) $\overline {\tilde {u}^{2}}/\overline {u^{2}}$ , (b) $\overline {\tilde {v}^{2}}/\overline {v^{2}}$ , (c) $\overline {\tilde {\theta }^{2}}/\overline {\theta ^{2}}$ . Plus symbols indicate data of Thiesset et al. (Reference Thiesset, Danaila and Antonia2014) at $Re=2540$ .

To highlight the spatiotemporal dynamics of the coherent motion in the wake, we examine the coherent scalar $\tilde {\theta }$ contours in the $ (\phi ,y)$ plane at $x/d=10$ , 20 and 40. These contours are shown in figure 3(a–c). The contours of the phase-averaged strain $\langle S \rangle =1/2(\partial \langle U \rangle /{\partial y } +\partial \langle V \rangle /{\partial x })$ are shown in figure 3(d–f). The contours of the phase-averaged vorticity $\langle \omega _{z} \rangle =\partial \langle V \rangle /{\partial x } -\partial \langle U \rangle /{\partial y }$ are presented in figure 3(g–i), where the black arrow lines indicate phase-averaged sectional streamlines, and blue curves denote the $Q$ -criterion isocontours drawn at $40\,\%$ of the maximum vortex strength level. We can observe from figure 3(a,b) that the maximum $\tilde {\theta }$ occurs at $\phi =\pi /2\pm 2k\pi$ , with its transverse position shifting away from the centreline as $x/d$ increases. The minimum $\langle S \rangle$ (figure 3 d–f) also occurs at $\phi =\pi /2\pm 2k\pi$ . This suggests that the region of maximum $\tilde {\theta }$ corresponds to the region of minimum $\langle S \rangle$ . In figure 3(g–i), the maximum $\langle \omega \rangle$ occurs at the same phase as the maximum $\tilde {\theta }$ , and for $x/d=10$ and 20, both maxima appear at approximately the same $y/d$ . This indicates that heat is contained within the vortex core, which is approximately identified using the $Q$ -criterion and the $\lambda _2$ method (Jeong & Hussain Reference Jeong and Hussain1995). This agrees with the findings of Chen et al. (Reference Chen, Zhou, Zhou and Antonia2016), who reported that the positive $\tilde {\theta }$ contours form a concentric distribution within the spanwise roller, and the negative contours occur between adjacent rollers. Notably, this behaviour no longer holds at $x/d = 40$ (figure 3 c,i), where heat is instead transported along the vortex periphery towards the free-stream region. This transition is due to the considerable weakening of the spanwise vortex at $x/d = 40$ , reducing its ability to concentrate heat. In the $ ( \phi ,y )$ distributions of coherent velocity vectors, the maximum coherent longitudinal velocity $\tilde {u}$ (see figure 3(d–f) in Thiesset et al. (Reference Thiesset, Danaila and Antonia2014), or figure 10(a–c) in Chen et al. (Reference Chen, Zhou, Zhou and Antonia2016)) and the maximum coherent transverse velocity $\tilde {v}$ (see figure 3(g–i) in Thiesset et al. (Reference Thiesset, Danaila and Antonia2014) or figure 10(d–f) in Chen et al. (Reference Chen, Zhou, Zhou and Antonia2016)) exhibit phase difference $\phi =\unicode{x03C0} /2$ . This phenomenon is also evident in figure 4, which shows the distributions of $\tilde {u}/U_{\infty }$ , $\tilde {v}/U_{\infty }$ and $\tilde {\theta }/\varTheta _{0}$ along the centreline at $x/d=10$ . The maximum values of $\tilde {v}$ occur at $\phi =0\pm 2k\pi$ , whereas the maximum values of $\tilde {u}$ and $\tilde {\theta }$ occur at $\phi =\pi /2\pm k\pi$ . In what follows, we focus on the wake centreline to isolate the effects of the coherent motion on the scale-by-scale distribution of the scalar, where the mean shear is negligible (Praskovsky et al. Reference Praskovsky, Gledzer, Karyakin and Zhou1993; Sreenivasan & Dhruva Reference Sreenivasan and Dhruva1998; Thiesset et al. Reference Thiesset, Danaila and Antonia2014).

Figure 3.

(a–c) Contours of $\tilde {\theta }/\varTheta _{0}$ , (d–f) contours of $ \langle S \rangle d/U_{\infty }$ , (g–i) contours of $\langle \omega \rangle d / U_{\infty }$ , for (a,d,g) $x/d=10$ , (b,e,h) $x/d=20$ , (c, f,i) $x/d=40$ . In (g–i), black arrow lines indicate phase-averaged sectional streamlines. Blue curves denote the $Q$ -criterion isocontours drawn at $40\,\%$ of the maximum vortex strength level, inside which the normalised circulation $\varGamma /(U_{\infty }d)$ is computed, yielding values 0.37, 0.23 and 0.002, respectively. Plus symbols represent the locations of vortex centre identified using the $\lambda _2$ method (Jeong & Hussain Reference Jeong and Hussain1995), and cross symbols represents the locations of saddle points. The flow direction is from left to right.

Figure 4.

Distributions of $\tilde {u}/U_{\infty }$ , $\tilde {v}/U_{\infty }$ and $\tilde {\theta }/\varTheta _{0}$ along the centreline at $x/d=10$ .

4.2. Phase-averaged second-order structure functions

Following (3.3), we show the phase-averaged second-order structure functions for the total scalar field (figure 5 a–c), the coherent scalar (figure 5 d–f), and the stochastic scalar (figure 5 g–i) in the $ ( \phi ,r )$ plane at $x/d=10$ , 20 and 40, respectively. Note that the separation $r$ is inferred from Taylor’s hypothesis and normalised by $L_{v}$ , the streamwise distance between two consecutive vortices of the same sign: $L_v =4.15d$ , $4.13d$ and $4.24d$ for $x/d=10$ , 20 and 40, respectively. The scale-by-scale distributions of the total scalar variance exhibit significant phase characteristics (figure 5 a–c). Specifically, at $x/d=10$ , $\langle (\Delta \theta )^2\rangle$ is significantly enhanced for the smallest scales at $\phi =\pi /4\pm k\pi /2$ . At $x/d=20$ , a clear phase dependence appears only for $r/L_{v}\gtrsim 0.2$ . At $x/d= 40$ , the phase dependence is nearly imperceptible at all scales. The distributions of stochastic scalar variance $ \langle ( \Delta \theta ^{\prime } ) ^{2} \rangle$ in figure 5(g–i) are very similar to those of $\langle (\Delta \theta )^2\rangle$ in figure 5(a–c). In figure 5(d,e), the coherent distributions of $(\Delta \tilde {\theta })^2$ are significantly enhanced at $\phi =\pi /4\pm k\pi /2$ , even for the smallest scales, while $(\Delta \tilde {\theta })^2$ is close to zero at $\phi = 0 \pm k\pi /2$ and $r/L_{v}=0.5 \pm k/2$ . This suggests that the periodicity of $\tilde {\theta }$ is half the distance between two consecutive vortices of the same sign, such that $\tilde {\theta } ( x ) = \tilde {\theta } ( x +k ( L_{v}/2 ) )$ , leading to $ \langle ( \Delta \theta ) ^{2} \rangle ( r=L_{v}/2+k ( L_{v}/2 ) ) = \langle ( \Delta \theta ^{\prime } ) ^{2} \rangle ( r=L_{v}/2+k ( L_{v}/2 ) )$ . At $x/d=40$ , due to the very weak coherent motion, phase averaging fails to capture the periodicity of $(\Delta \tilde {\theta })^2$ . Notably, the contours of $(\Delta \tilde {\theta })^2 / \overline {\tilde {\theta }^2}$ in figure 5(d,e) exhibit high similarity and regularity at $x/d=10$ and 20. This pattern suggests that an expression may be used to describe and predict the distributions of $(\Delta \tilde {\theta })^2 / \overline {\tilde {\theta }^2}$ across different scales. By integrating the phase-averaged structure functions over the period $\phi \in [ -\pi , \pi ]$ , the resulting time-averaged structure functions depend only on the scale $r$ and can be expressed as a sine function parametrised by $L_{v}$ :

(4.1) \begin{equation} \overline {\left ( \Delta \tilde {\theta } \right )^{2} } \Big/ \overline { \tilde {\theta }^{2} }=4\sin^{2}\left ( 2\pi \frac{r}{L_{v}} \right )\!. \end{equation}

Here, the use of $2\pi$ reflects the fact that the $\overline { ( \Delta \tilde {\theta } )^{2} } / \overline { \tilde {\theta }^{2} }$ distributions exhibit two oscillations within $r/L_{v}\lt 1$ , so that phase $2\pi$ naturally corresponds to the characteristic scale $L_{v}$ . The prefactor 4 was chosen to match the observed amplitude (approximately 4) of these distributions. The only difference between (4.1) and (3.3) in Thiesset et al. (Reference Thiesset, Danaila and Antonia2014) for $\overline { ( \Delta \tilde {v} )^{2} } / \overline { \tilde {v}^{2} }$ lies in the use of $2\pi$ in (4.1), compared to $\pi$ in (3.3) of Thiesset et al. (Reference Thiesset, Danaila and Antonia2014). Figure 6 compares the time-averaged structure functions at $x/d=10$ and 20 with those predicted by (4.1). At $x/d=10$ , the sine-type model is in almost perfect agreement with the time-averaged structure function at all scales. At $x/d=20$ , this model generally captures the time-averaged structure function, though minor discrepancies persist. For instance, in the range $ 0.18\lesssim r/L_{v} \lesssim 0.27$ , the predicted values slightly exceed the experimental values, whereas in the range $ 0.43\lesssim r/L_{v} \lesssim 0.53$ , they are slightly lower. Overall, this model adequately captures the distributions of the time-averaged second-order structure functions of $\tilde {\theta }$ , demonstrating its reliability in predicting $\overline { ( \Delta \tilde {\theta } )^{2} } / \overline { \tilde {\theta }^{2} }$ in the scalar field of a cylinder wake where coherent motion is discernible. Notably, we compared datasets for $Re_d=2500, 5000, 10\,000, 15\,000$ at $x/d=10$ and $y/d=0$ with (3.3) of Thiesset et al. (Reference Thiesset, Danaila and Antonia2014); the resulting distributions for all Reynolds numbers collapsed onto a single curve (not shown), indicating that (3.3) is Reynolds number independent. Based on this evidence, we reasonably infer that (4.1) is likewise insensitive to the Reynolds number.

Figure 5.

Phase-averaged second-order scalar structure functions (in logarithmic scales $\log_{10}$ ) at the centreline, where $L_{v}$ is the streamwise distance between two consecutive vortices of the same sign: (a–c) $\langle (\Delta \theta )^2\rangle / \overline {\theta ^2}$ , (d–f) $(\Delta \tilde {\theta })^2 / \overline {\tilde {\theta }^2}$ , (g–i) $ \langle (\Delta {\theta }' )^2 \rangle / \overline {{\theta }'^2}$ , for (a,d,g) $x/d=10$ , (b,e,h) $x/d=20$ , (c, f,i) $x/d=40$ .

Figure 6.

Time-averaged second-order structure functions of the coherent scalar $\tilde {\theta }$ . Solid line indicates (4.1); squares indicate $x/d=10$ ; circles indicate $x/d=20$ .

Another key feature is that the distributions of $(\Delta \tilde {\theta })^2$ in figure 5(d–f) closely resemble those of $(\Delta \tilde {u})^2$ (see figure 4(d–f) in Thiesset et al. Reference Thiesset, Danaila and Antonia2014). For instance, $(\Delta \tilde {u})^2$ is also significantly enhanced at $\phi =\pi /4\pm k\pi /2$ , even for the smallest scales, and it is very small at $\phi = 0 \pm k\pi /2$ and $r/L_{v}=0.5 \pm k/2$ . This confirms that $\tilde {\theta }$ and $\tilde {u}$ exhibit similar characteristics in the context of the phase-averaged second-order structure functions. These characteristics are also evident in figure 7, which shows the distributions of $E_{uu}(f)$ , $E_{vv}(f)$ and $E_{\theta \theta }(f)$ at $x/d=10$ . A distinct peak appears in $E_{\theta \theta } (f)$ and $E_{uu} (f)$ at $fd/U_{\infty } \approx 0.4$ , corresponding to twice the vortex shedding frequency. This indicates that the periodicity of $\tilde {\theta }$ and $\tilde {u}$ corresponds to half the distance between two vortices. To better highlight this similarity, we present the co-spectra $E_{u\theta }(f)$ and $E_{v\theta }(f)$ at $x/d=10$ in figure 8(a), together with the corresponding spectral coherence in figure 8(b). The coherence (Wang et al. Reference Wang, Zhou, Wang and Jin2004) is defined as $Coh_{\alpha \beta }= ({( E_{\alpha \beta })^{2} + ( Q_{\alpha \beta } )^{2} })/({ E_{\alpha \alpha } E_{\beta \beta } })$ , where $\alpha$ and $\beta$ denote the fluctuation components (here, the velocity $u$ or $v$ , and the scalar $\theta$ ), and $Q_{\alpha \beta }$ is the quadrature spectrum. In figure 8(a), $E_{u\theta }(f)$ is generally negative at low frequencies, whereas $E_{v\theta }(f)$ remains close to zero. At $fd/U_{\infty } \approx 0.4$ , both co-spectra exhibit extrema: $E_{v\theta }(f)$ attains a positive peak, whereas $E_{u\theta }(f)$ reaches a deeper negative minimum. These observations suggest that $u$ and $\theta$ are strongly correlated but are predominantly out of phase, with a phase difference close to $\pi$ (not shown). In contrast, the correlation between $v$ and $\theta$ is noticeably weaker. Consistently, in figure 8(b), the coherence between $u$ and $\theta$ is markedly higher than that between $v$ and $\theta$ , with a distinct peak at $fd/U_{\infty } \approx 0.4$ . These results provide clear evidence that $\theta$ is more strongly coupled dynamically with $u$ than with $v$ , albeit with an opposite phase relationship. Similarly, Chen et al. (Reference Chen, Zhou, Antonia and Zhou2017) noted that the spectra of $\theta$ and $u$ exhibit similarity in the near field of a cylinder wake. In contrast, the distributions of $\langle (\Delta \theta )^2\rangle$ in figure 5(a–c) differ significantly from those of $\langle (\Delta u)^2\rangle$ (see figure 4(a–c) in Thiesset et al. Reference Thiesset, Danaila and Antonia2014). For instance, at $x/d=10$ , the energy enhancement for $\langle (\Delta u)^2\rangle$ occurs at $\phi =\pi /2\pm k\pi$ , whereas for $\langle (\Delta \theta )^2\rangle$ , it is located at $\phi =\pi /4\pm k\pi /2$ . This discrepancy arises primarily from the substantial contribution of $\overline {\tilde {\theta }^{2}}$ to $\overline {\theta ^{2}}$ , in contrast to the relatively minor contribution of $\overline {\tilde {u}^{2}}$ to $\overline {u^{2}}$ along the wake centreline (see figure 2).

Figure 7.

Distributions of $E_{uu}(f)$ , $E_{vv}(f)$ and $E_{\theta \theta }(f)$ at $x/d=10$ . The vertical dashed lines represent $fd/U_{\infty } \approx 0.2$ and 0.4, respectively.

Figure 8.

(a) Co-spectra $E_{u\theta }(f)$ and $E_{v\theta }(f)$ at $x/d=10$ . (b) Spectral coherence between $u$ and $\theta$ , and between $v$ and $\theta$ , at $x/d=10$ . The vertical dashed lines represent $fd/U_{\infty } \approx 0.4$ .

Figure 9(a) shows the contribution of the coherent component to total scalar field (i.e. $\overline {(\Delta \tilde {\theta } )^{2}} /\overline {(\Delta \theta )^{2}}$ ) in the time-averaged second-order structure functions at $x/d=10$ , 20 and 40. For reference, the distributions of $\overline {(\Delta \tilde {u} )^{2}} /\overline {(\Delta u )^{2}}$ and $\overline {(\Delta \tilde {v} )^{2}} /\overline {(\Delta v )^{2}}$ at $x/d=10$ are also included. Figure 9(b) shows the enlarged distributions of $\overline {(\Delta \tilde {\theta } )^{2}} / \overline {(\Delta \theta )^{2}}$ at $x/d = 20$ and 40. At $x/d=10$ , $\overline {(\Delta \tilde {\theta } )^{2}}$ accounts for $\approx 29\,\%$ of $\overline {(\Delta \theta )^{2}}$ at $r/L_{v}\approx 0.25$ , and is still perceptible at the smallest scales (i.e. $r/L_{v}\approx 0.02$ ), where $\overline {(\Delta \tilde {\theta } )^{2}} /\overline {(\Delta \theta )^{2}} \approx 11\,\%$ . At $x/d= 20$ , $\overline {(\Delta \tilde {\theta } )^{2}}$ is no longer discernible at the smallest scales but still accounts for approximately $ 2\,\%$ of $\overline {(\Delta \theta )^{2}}$ at $r/L_{v}\approx 0.25$ (figure 9 b). At $x/d=40$ , $\overline {(\Delta \tilde {\theta } )^{2}}$ is not discernible at all scales. The distribution of $\overline {(\Delta \tilde {u} )^{2}} /\overline {(\Delta u )^{2}}$ is similar to that of $\overline {(\Delta \tilde {\theta } )^{2}} /\overline {(\Delta \theta )^{2}}$ at $x/d=10$ , albeit with smaller corresponding values. In contrast, at $x/d=10$ , $\overline {(\Delta \tilde {v} )^{2}} /\overline {(\Delta v )^{2}}$ is significantly larger than $\overline {(\Delta \tilde {\theta } )^{2}} /\overline {(\Delta \theta )^{2}}$ across all scales. Specifically, the maximum value of $\overline {(\Delta \tilde {v} )^{2}} /\overline {(\Delta v )^{2}}$ occurs at $r/L_{v}\approx 0.54$ , reaching approximately $ 78\,\%$ . Even at the smallest scale, $\overline {(\Delta \tilde {v} )^{2}}$ still accounts for approximately $ 18\,\%$ of $\overline {(\Delta v )^{2}}$ . Additionally, $\overline {(\Delta \tilde {\theta } )^{2}}$ and $\overline {(\Delta \tilde {u} )^{2}}$ are nearly zero at $r/L_{v}\approx 0.54$ , which is expected since the periodicity of $\tilde {\theta }$ and $\tilde {u}$ corresponds to half the distance between two consecutive vortices. These observations suggest that the contribution of the coherent component to the total fluctuating field depends on the specific quantity and scale.

Figure 9.

(a) Distributions of $\overline {(\Delta \tilde {\theta } )^{2}} /\overline {(\Delta \theta )^{2}}$ at $x/d=10$ , 20 and 40. For reference, also shown are the distributions of $\overline {(\Delta \tilde {u} )^{2}} /\overline {(\Delta u )^{2}}$ and $\overline {(\Delta \tilde {v} )^{2}} /\overline {(\Delta v )^{2}}$ at $x/d=10$ . The dashed line represents $r/L_{v} = 0.54$ . (b) Enlarged distributions of $\overline {(\Delta \tilde {\theta } )^{2}} / \overline {(\Delta \theta )^{2}}$ at $x/d = 20$ and 40, corresponding to the zoomed-in version of (a). The dashed line represents $r/L_{v} = 0.54$ .

4.3. Phase-averaged third-order structure functions

We now focus on the behaviour of phase-averaged third-order mixed structure functions, particularly the term $ \langle \Delta u\,(\Delta \theta )^{2} \rangle$ , which appears explicitly in (3.21). Substituting (3.1) into $ \langle \Delta u\,(\Delta \theta )^{2} \rangle$ yields

(4.2) \begin{align} \langle \Delta u\,(\Delta \theta )^2 \rangle =\Delta \tilde {u}\,(\Delta \tilde {\theta })^2+ \langle \Delta u^{\prime } (\Delta \theta ^{\prime })^2 \rangle +\Delta \tilde {u} \langle (\Delta \theta ^{\prime })^2 \rangle +2\, \Delta \tilde {\theta }\left \langle \Delta u^{\prime }\, \Delta \theta ^{\prime }\right \rangle \!. \\[-25pt] \nonumber \end{align}

Figure 10.

Phase-averaged third-order mixed structure functions at the centreline: (a–c) $- \langle \Delta u\,(\Delta \theta )^{2} \rangle /\bar {\varepsilon }_{\theta }r$ , (d–f) $-\Delta \tilde {u}\, (\Delta \tilde {\theta } )^{2} /\bar {\varepsilon }_{\theta }r$ , (g–i) $- \langle \Delta {u}'\, (\Delta {\theta }')^{2} \rangle /\bar {\varepsilon }_{\theta }r$ , for (a,d,g) $x/d=10$ , (b,e,h) $x/d=20$ , (c, f,i) $x/d=40$ .

Equation (4.2) shows that phase-averaged third-order mixed structure functions consist of coherent and stochastic terms, specifically $\Delta \tilde {u}\,(\Delta \tilde {\theta })^2$ and $\langle \Delta u^{\prime }\,(\Delta \theta ^{\prime })^2\rangle$ , along with interaction terms between the coherent and stochastic components (i.e. $\Delta \tilde {u}\,\langle (\Delta \theta ^{\prime })^2\rangle$ and $2\, \Delta \tilde {\theta }\,\langle \Delta u^{\prime }\, \Delta \theta ^{\prime }\rangle$ ). This fundamentally differs from the phase-averaged second-order structure functions. We show the phase-averaged third-order mixed structure functions for the total scalar (figure 10 a–c), the coherent scalar (figure 10 d–f) and the stochastic scalar (figure 10 g–i) in the $ ( \phi ,r )$ plane at $x/d= 10$ , 20 and 40, respectively. Note that (4.2) is normalised by $\overline {\varepsilon }_\theta r$ . Figure 10(a–c) clearly show that the distributions of $\langle \Delta u\,(\Delta \theta )^2\rangle$ exhibit significant phase correlation, even at the smallest scales and at even $x/d=40$ . A similar feature is also observed in the distributions of $\langle \Delta u(\Delta v)^2\rangle$ (see figure 8(a–c) in Thiesset et al. Reference Thiesset, Danaila and Antonia2014). This strongly supports the finding of Thiesset et al. (Reference Thiesset, Danaila and Antonia2014) that phase correlation is significantly stronger in third-order structure functions than in second-order structure functions.

Another key feature of the distributions in figure 10(a–c) is that the value of $\langle \Delta u\,(\Delta \theta )^2\rangle$ gradually decreases as $x/d$ increases. In a locally isotropic scalar field at moderate scales, the third-order mixed structure functions should not exceed the isotropic value $4/3$ at finite $Re$ (Antonia et al. Reference Antonia, Zhu, Anselmet and Ould-Rouis1996, Reference Antonia, Tang, Djenidi and Zhou2019; Danaila & Mydlarski Reference Danaila and Mydlarski2001; Danaila et al. Reference Danaila, Anselmet and Antonia2002), i.e.

(4.3) \begin{equation} -\overline {\Delta u\,(\Delta \theta )^2} \leqslant \dfrac {4}{3}\overline {\varepsilon }_\theta r\approx 1.33\overline {\varepsilon }_\theta r. \end{equation}

However, figure 10(a–c) show that the maximum value of $- \langle \Delta u\,(\Delta \theta )^{2} \rangle /\bar {\varepsilon }_{\theta }r$ significantly exceeds the isotropic value $4/3$ . Specifically, it exceeds $4/3$ by approximately factors of 10 and 2 at $x/d=10$ and 20, respectively. The maximum value occurs at $\phi \approx \pi /2\pm k\pi$ , corresponding to the phase of maximum coherent strain (see figure 3), suggesting that coherent strain is a key factor in inducing anisotropy in the scalar field. This is fully consistent with the conclusions of Thiesset, Danaila & Antonia (Reference Thiesset, Danaila and Antonia2013) in the velocity field. At $x/d=40$ , the maximum value of $- \langle \Delta u\,(\Delta \theta )^{2} \rangle /\bar {\varepsilon }_{\theta }r$ is approximately 1.56, which is only 1.1 times the isotropic value $4/3$ .

In the phase-averaged third-order mixed structure functions of the coherent component (figure 10 d–f), a phase correlation is observed at $\phi =\pi /4\pm k\pi /2$ . The magnitude of $-\Delta \tilde {u}\, (\Delta \tilde {\theta } )^{2} /\bar {\varepsilon }_{\theta }r$ decreases as $x/d$ increases. At $x/d=40$ , $-\Delta \tilde {u}\, (\Delta \tilde {\theta } )^{2} /\bar {\varepsilon }_{\theta }r\sim 10^{-4}$ , which is nearly three orders of magnitude smaller than $- \langle \Delta u\,(\Delta \theta )^{2} \rangle /\bar {\varepsilon }_{\theta }r$ , and thus is negligible. Similarly, figure 10(g–i) show a clear phase correlation from $x/d=10$ to 40, even at the smallest scales. A detailed examination of figures 10(a,b) and 10(g,h) reveals a very clear difference between the distributions of $ \langle \Delta {u}'\, (\Delta {\theta }')^{2} \rangle$ and $ \langle \Delta u\,(\Delta \theta )^{2} \rangle$ at $x/d=10$ to 20. This difference is attributed to the non-negligible interaction terms $\Delta \tilde {u}\,\langle (\Delta \theta ^{\prime })^2\rangle$ and $2\, \Delta \tilde {\theta }\,\langle \Delta u^{\prime }\, \Delta \theta ^{\prime }\rangle$ in (4.2), particularly $2\, \Delta \tilde {\theta }\,\langle \Delta u^{\prime }\, \Delta \theta ^{\prime }\rangle$ . This behaviour is more clearly observed in figure 11, which illustrates the contributions of each component in the time-averaged third-order structure functions from $x/d=10$ to 40. In figure 11(a), the distributions of $\overline {2\, \Delta \tilde {\theta }\,\langle \Delta u^{\prime }\, \Delta \theta ^{\prime }\rangle }$ contribute significantly to $\overline {\langle \Delta u\,(\Delta \theta )^2\rangle }$ across all scales at $x/d=10$ . Although its contribution is smaller than that of $\overline { \langle \Delta {u}^{\prime }\, (\Delta {\theta }^{\prime })^{2} \rangle }$ , it is much greater than that of $\overline {\Delta \tilde {u}\, (\Delta \tilde {\theta } )^{2}}$ and $\overline {\Delta {\tilde {u} }\, \langle (\Delta {\theta }^{\prime })^{2} \rangle }$ . In figure 11(b), the contribution of $\overline {2\, \Delta \tilde {\theta }\,\langle \Delta u^{\prime }\, \Delta \theta ^{\prime }\rangle }$ decreases significantly at $x/d=20$ . In figure 11(c), the contributions of $\overline {2\, \Delta \tilde {\theta }\,\langle \Delta u^{\prime }\, \Delta \theta ^{\prime }\rangle }$ , $\overline {\Delta \tilde {u}\, (\Delta \tilde {\theta } )^{2}}$ and $\overline {\Delta {\tilde {u} }\, \langle (\Delta {\theta }^{\prime })^{2} \rangle }$ are negligible across all scales, leading to $\overline { \langle \Delta u\,(\Delta \theta )^{2} \rangle }\approx \overline { \langle \Delta {u}^{\prime }\, (\Delta {\theta }^{\prime })^{2} \rangle }$ at $x/d=40$ .

Figure 11.

Contributions of coherent and stochastic components in time-averaged third-order structure functions. Red indicates $-\overline {\Delta \tilde {u}\, (\Delta \tilde {\theta } )^{2} } /\bar {\varepsilon }_{\theta }r$ , magenta indicates $-\overline { \langle \Delta {u}^{\prime } (\Delta {\theta }^{\prime })^{2} \rangle } /\bar {\varepsilon }_{\theta }r$ , blue indicates $-\overline {\Delta {\tilde {u} }\, \langle (\Delta {\theta }^{\prime })^{2} \rangle } /\bar {\varepsilon }_{\theta }r$ , green indicates $-\overline {2\,\Delta {\tilde {\theta } }\, \langle \Delta {u}^{\prime }\, \Delta {\theta }^{\prime } \rangle } /\bar {\varepsilon }_{\theta }r$ , black indicates $-\overline { \langle \Delta u\, (\Delta \theta )^{2} \rangle } /\bar {\varepsilon }_{\theta }r$ , for (a) $x/d=10$ , (b) $x/d=20$ , (c) $x/d=40$ .

For a comprehensive comparison of scalar and velocity third-order structure functions, we present the phase-averaged third-order streamwise velocity structure function $ \langle (\Delta u)^{3} \rangle$ . Substituting (3.1) into its definition yields

(4.4) \begin{equation} \langle (\Delta u)^{3} \rangle =(\Delta \tilde {u})^3+3\,\Delta \tilde {u} \langle (\Delta u^{\prime })^2 \rangle + \langle (\Delta u^{\prime })^3 \rangle . \end{equation}

Compared with (4.2), aside from the common coherent and stochastic terms – $(\Delta \tilde {u})^3$ and $\langle (\Delta u^{\prime })^3\rangle$ – (4.4) contains only one interaction term between the coherent and stochastic components, namely $3\,\Delta \tilde {u}\,\langle (\Delta u^{\prime })^2\rangle$ . We show the phase-averaged third-order streamwise velocity structure functions in the $ ( \phi ,r )$ plane at $x/d=10$ , 20 and 40 for the total fluctuating field (figure 12 a–c), the coherent component (figure 12 d–f) and the stochastic component (figure 12 g–i). We normalise (4.4) by $\overline {\varepsilon }r$ , where $\overline {\varepsilon }$ is the mean energy dissipation rate, with values 4.1, 1.6 and 0.39 $\text{m}^{2}\ \text{s}^{-3}$ at $x/d=10$ , 20 and 40, respectively (Zhou et al. Reference Zhou, Zhou, Yiu and Chua2003).

Figure 12.

Phase-averaged third-order streamwise velocity structure functions at the centreline: (a–c) $- \langle (\Delta u)^{3} \rangle /\bar {\varepsilon }r$ , (d–f) $-(\Delta \tilde {u} )^{3} /\bar {\varepsilon }r$ , (g–i) $- \langle (\Delta {u}')^{3} \rangle /\bar {\varepsilon }r$ , for (a,d,g) $x/d=10$ , (b,e,h) $x/d=20$ , (c, f,i) $x/d=40$ .

In figures 12(a–c) and 12(g–i), the distributions of $ \langle (\Delta u)^{3} \rangle$ and $\langle (\Delta u^{\prime })^3\rangle$ exhibit nearly identical phase correlations, similar to those of $\langle \Delta u\,(\Delta \theta )^2\rangle$ and $ \langle \Delta {u}'\, (\Delta {\theta }')^{2} \rangle$ (see figure 10 a–c,g–i). However, for the $(\Delta \tilde {u})^3$ distributions in figure 12(d–f), the phase correlation is markedly stronger at $\phi =-(3/4)\pi \pm 2k\pi$ and $-\pi /4\pm 2k\pi$ than at $\phi =\pi /4\pm 2k\pi$ and $(3/4)\pi \pm 2k\pi$ . In contrast, for the $\Delta \tilde {u}\, (\Delta \tilde {\theta } )^{2}$ distributions in figure 10(d–f), the difference between the two phases is much weaker. This discrepancy arises because $\Delta \tilde {u}$ is enhanced unequally at both phases (see figure 4(d–f) in Thiesset et al. Reference Thiesset, Danaila and Antonia2014), whereas $\Delta \tilde {\theta }$ is enhanced nearly equally at both phases (see figure 5 d–f). Furthermore, in a locally isotropic velocity field at moderate scales and finite Reynolds numbers, the third-order streamwise velocity structure function should not exceed the isotropic value $4/5$ , i.e.

(4.5) \begin{equation} -\overline {\left \langle (\Delta u)^{3} \right \rangle } \leqslant \dfrac {4}{5}\overline {\varepsilon } r. \end{equation}

Figure 12(a–c) show that the maximum value of $-\overline { \langle (\Delta u)^{3} \rangle }/\bar {\varepsilon }r$ exceeds the isotropic value to varying degrees: by factors of 6, 2 and 1.1 at $x/d=10$ , 20 and 40, respectively. Interestingly, the factors at $x/d=20$ and 40 match those obtained for $- \langle \Delta u\,(\Delta \theta )^{2} \rangle /\bar {\varepsilon }_{\theta }r$ in figure 10(b,c), further underscoring the similarity between the two quantities. Analogously, figure 13 displays the contributions of each component to the time-averaged third-order streamwise velocity structure function from $x/d=10$ to 40. As expected, the distributions of each term across scales are broadly similar to those of the time-averaged third-order mixed structure functions (see figure 11).

Figure 13.

Contributions of coherent and stochastic components in time-averaged third-order streamwise velocity structure functions: red indicates $-\overline {(\Delta \tilde {u})^{3} } /\bar {\varepsilon }r$ , magenta indicates $-\overline { \langle (\Delta {u}^{\prime })^{3} \rangle } /\bar {\varepsilon }r$ , blue indicates $-3\,\overline {\Delta {\tilde {u} }\, \langle (\Delta {u}^{\prime })^{2} \rangle } /\bar {\varepsilon }r$ , black indicates $-\overline { \langle (\Delta u)^{3} \rangle } /\bar {\varepsilon }r$ , for (a) $x/d=10$ , (b) $x/d=20$ , (c) $x/d=40$ .

4.4. Phase-averaged second-order mixed structure functions

Scalar transport plays a crucial role in turbulent mixing processes. Matsumura & Antonia (Reference Matsumura and Antonia1993) and Chen et al. (Reference Chen, Zhou, Zhou and Antonia2016) investigated scalar transport in the intermediate wake on the $ (\phi ,y )$ plane, focusing on two heat flux components: $u\theta$ and $v\theta$ . Their findings indicate that large-scale coherent structures contribute more to passive scalar transport than to momentum transport. Next, we analyse the scale-by-scale distributions of the heat flux ( $\langle \Delta u\,\Delta \theta \rangle$ and $\langle \Delta v\,\Delta \theta \rangle$ ) to provide a more comprehensive characterisation of the scalar transport.

Applying the same derivation approach as in (3.3), we obtain

(4.6) \begin{align} \left \langle \Delta u\Delta \theta \right \rangle =\Delta \tilde {u} \Delta \tilde {\theta }+ \langle \Delta {u}'\Delta {\theta }' \rangle , \end{align}

(4.7) \begin{align} \left \langle \Delta v\Delta \theta \right \rangle =\Delta \tilde {v} \Delta \tilde {\theta }+ \langle \Delta {v}'\Delta {\theta }' \rangle . \end{align}

Figure 14.

Phase-averaged second-order mixed structure functions: (a–c) $\langle \Delta u\Delta \theta \rangle / \overline {u\theta }$ , (d–f) $\Delta \tilde {u} \Delta \tilde {\theta } / \overline {\tilde {u}\tilde {\theta }}$ , (g–i) $ \langle \Delta {u}'\Delta {\theta }' \rangle / \overline {{u}'{\theta }'}$ , for (a,d,g) $x/d=10$ , (b,e,h) $x/d=20$ , (c,f,i) $x/d=40$ .

These equations describe the scale-by-scale dynamics of interactions between coherent velocity components ( $\tilde {u}$ or $\tilde {v}$ ) and the coherent scalar $\tilde {\theta }$ from a phase perspective. We first examine the behaviour of $\langle \Delta u\Delta \theta \rangle$ . Figure 14 shows the phase-averaged second-order mixed structure functions in the $ ( \phi ,r )$ plane for the total turbulent field, coherent field and stochastic field at $x/d=10$ , 20 and 40. We can observe a clear phase correlation in $\langle \Delta u\Delta \theta \rangle$ at $\phi =-\pi /2\pm k\pi$ (figure 14 a–c). For instance, at $x/d=40$ , phase correlation is still perceptible at $r/L_{v}\gtrsim 0.03$ . The distributions of $\langle \Delta u\Delta \theta \rangle$ exhibit staggered enhancements at $x/d=10$ , which are likely due to differences in the phase dependence of $\langle (\Delta u)^2\rangle$ (see figure 4(a–c) of Thiesset et al. Reference Thiesset, Danaila and Antonia2014) and $\langle (\Delta \theta )^2\rangle$ (see figure 5 a–c). In figure 14(g–i), the stochastic distributions of $ \langle \Delta {u}'\Delta {\theta }' \rangle / \overline {{u}'{\theta }'}$ closely resemble those of the total fluctuating field. In contrast, the coherent distributions of $\Delta \tilde {u} \Delta \tilde {\theta }$ in figure 14(d–f) exhibit features similar to those of $(\Delta \tilde {\theta })^2$ . This is expected given the similarity between $(\Delta \tilde {\theta })^2$ (see figure 5 d–f) and $(\Delta \tilde {u})^2$ (see figure 4(d–f) of Thiesset et al. Reference Thiesset, Danaila and Antonia2014). Thus, given the high similarity and regularity of the $\Delta \tilde {u} \Delta \tilde {\theta }$ distributions at $x/d=10$ and 20, a sine-type model – the same as (4.1) – is adopted to describe the time-averaged mixed structure functions:

(4.8) \begin{equation} \overline {\Delta \tilde {u} \Delta \tilde {\theta }} / \overline {\tilde {u}\tilde {\theta }} =4\sin^{2}\left ( 2\pi \frac{r}{L_{v}} \right )\!. \end{equation}

Figure 15(a) compares (4.8) with the experimental phase-averaged structure functions at $x/d=10$ and 20. The data and the model agree closely at all scales. Notably, (4.1) and (4.8) are identical, suggesting that $\overline { ( \Delta \tilde {\theta } )^{2} } / \overline { \tilde {\theta }^{2} }$ and $\overline {\Delta \tilde {u} \Delta \tilde {\theta }} / \overline {\tilde {u}\tilde {\theta }}$ follow the same distribution.

Figure 15.

Time-averaged second-order mixed structure functions of the coherent components: (a) $\overline {\Delta \tilde {u} \Delta \tilde {\theta }} /\overline { \tilde {u} \tilde {\theta }}$ , (b) $\overline {\Delta \tilde {v} \Delta \tilde {\theta }} /\overline { \tilde {v} \tilde {\theta }}$ . Solid line indicates (4.6); squares indicate $x/d=10$ ; circles indicate $x/d=20$ .

Figure 16.

Distributions of $\overline {\Delta \tilde {u} \Delta \tilde {\theta }} /\overline {\Delta u \Delta \theta }$ at $x/d=10$ , 20 and 40. Vertical dashed line indicates $r/L_{v} = 0.54$ .

Figure 16 shows the contribution of the coherent component to the total scalar field (i.e. $\overline {\Delta \tilde {u} \Delta \tilde {\theta }} /\overline {\Delta u \Delta \theta }$ ) in the time-averaged mixed structure functions at $x/d=10$ , 20 and 40. The contribution of $\overline {\Delta \tilde {u}\Delta \tilde {\theta }}$ to $\overline {\Delta u \Delta \theta }$ is more pronounced than that of $\overline {(\Delta \tilde {\theta } )^{2}}$ . Specifically, at $x/d=10$ , $\overline {\Delta \tilde {u}\Delta \tilde {\theta }}$ accounts for approximately $ 66\,\%$ of $\overline { \langle \Delta u \Delta \theta \rangle }$ at $r/L_{v}\approx 0.25$ , and remains approximately $ 34\,\%$ at the smallest scales. Even at $x/d=20$ , the contribution of $\overline {\Delta \tilde {u}\Delta \tilde {\theta }}$ remains significant, with $\overline {\Delta \tilde {u}\Delta \tilde {\theta }} /\overline { \langle \Delta u \Delta \theta \rangle }$ reaching approximately 17 $\,\%$ at $r/L_{v}\approx 0.25$ , and 7 $\,\%$ at $r/L_{v}\approx 0.02$ .

Figure 17.

Phase-averaged second-order mixed structure functions: (a–c) $\langle \Delta v\Delta \theta \rangle / \overline {v\theta }$ , (d–f) $\Delta \tilde {v} \Delta \tilde {\theta } / \overline {\tilde {v}\tilde {\theta }}$ , (g–i) $ \langle \Delta {v}'\Delta {\theta }' \rangle / \overline {{v}'{\theta }'}$ , for (a,d,g) $x/d=10$ , (b,e,h) $x/d=20$ , (c, f,i) $x/d=40$ .

Next, we examine the behaviour of $\langle \Delta v\Delta \theta \rangle$ . The phase-averaged structure functions for the total turbulent field (figure 17 a–c), the coherent field (figure 17 d–f) and the stochastic field (figure 17 g–i) in the $ ( \phi ,r )$ plane are shown for three values of $x/d$ . The phase correlation of $\langle \Delta v\Delta \theta \rangle$ in figure 17(a–c) differs significantly from that of $\langle \Delta u\Delta \theta \rangle$ in figure 14(a–c). For example, the $\langle \Delta v\Delta \theta \rangle$ distributions exhibit $2\pi$ -periodicity in phase. Within a single phase period, $\phi \in [ 0,2\pi ]$ , the $\langle \Delta v\Delta \theta \rangle$ distributions are approximately symmetric about $\phi =\pi$ . Similarly, the coherent distributions of $\Delta \tilde {v} \Delta \tilde {\theta }$ in figure 17(d–f) exhibit features similar to those of $\langle \Delta v\Delta \theta \rangle$ . However, the magnitude of $\Delta \tilde {v} \Delta \tilde {\theta } / \overline {\tilde {v}\tilde {\theta }}$ does not remain of the same order from $x/d=10$ to 40. This is due to the irregularity of $\overline {\tilde {v}\tilde {\theta }}$ . Specifically, the coherent velocity $\tilde {v}$ and the coherent scalar $\tilde {\theta }$ are out of phase (figure 4), leading to an irregular time-averaged product. This is more evident in figure 15(b), which shows the experimental distributions of the time-averaged structure functions $\overline {\Delta \tilde {v} \Delta \tilde {\theta }} /\overline { \tilde {v} \tilde {\theta }}$ at $x/d=10$ and 20. Since $\overline {\Delta \tilde {v} \Delta \tilde {\theta }} /\overline { \tilde {v} \tilde {\theta }}$ follows different distributions at these two positions, no single expression can accurately fit the data.

4.5. Experimental assessment of (3.21)

In this subsection, we first examine local isotropy along the centreline at $x/d=40$ . Following Monin & Yaglom (Reference Monin and Yaglom2007), the well-known isotropic relation between the second-order structure functions of the longitudinal and transverse velocity components is given by

(4.9) \begin{equation} \overline {(\Delta v)^{2}}_{iso} = \overline {(\Delta u)^{2}}+ \frac{r}{2} \frac{\partial }{\partial r} \overline {(\Delta u)^{2}} . \end{equation}

Figure 18 shows the ratios $\overline {(\Delta v)^2}/\overline {(\Delta v)^2}_{iso}$ and $\overline {(\Delta v^{\prime })^2}/\overline {(\Delta v^{\prime })^2}_{iso}$ at $x/d=40$ and $y/d=0$ . Values near unity indicate scales for which local isotropy holds approximately. At small scales ( $r/L_{v}\lesssim0.16$ ), the ratio $\overline {(\Delta v)^2}/\overline {(\Delta v)^2}_{iso}$ deviates from unity by less than 10 %; the discrepancy then increases with $r/L_{v}$ and is approximately 30 % at $r/L_{v}\approx 0.53$ . When the coherent contribution is removed, however, the ratio $\overline {(\Delta v^{\prime })^2}/\overline {(\Delta v^{\prime })^2}_{iso}$ is close to 1 within 10 % across all scales. This suggests that departures from local isotropy can be attributed to the coherent strain associated with the organised motion (Thiesset et al. Reference Thiesset, Danaila and Antonia2013). It is worth noting that at upstream locations ( $x/d=10$ and 20), the deviations from isotropy are larger across all scales (not shown), indicating that the assumption of local isotropy is not tenable. Based on the above analysis, the assumption of local isotropy is reasonable at $x/d=40$ for scales in the range $r/L_{v}\lesssim 0.16$ .

Figure 18.

Distributions of $\overline {(\Delta v)^2}/\overline {(\Delta v)^2}_{iso}$ and $\overline {(\Delta v^{\prime })^2}/\overline {(\Delta v^{\prime })^2}_{iso}$ at $x/d=40$ and $y/d=0$ . The horizontal solid line denotes the isotropic value 1, while the horizontal dashed lines represent a departure of 10 % from isotropy.

In figure 11(c), the coherent transport $-\overline {\Delta \tilde {u} (\Delta \tilde {\theta } )^{2} } /\bar {\varepsilon }_{\theta }r$ is negligibly small for all separations, indicating that the coherent motion is weak at $x/d=40$ . The total nonlinear transport $-\overline { \langle \Delta u(\Delta \theta )^{2} \rangle }/\bar {\varepsilon }_{\theta } r$ is entirely contributed by the stochastic transport $-\overline { \langle \Delta {u}^{\prime } (\Delta {\theta }^{\prime })^{2} \rangle } /\bar {\varepsilon }_{\theta }r$ . In particular, the magnitude of $-\overline { \langle \Delta u(\Delta \theta )^{2} \rangle }/\bar {\varepsilon }_{\theta } r$ increases at small scales, peaks at $r/L_{v} \approx 0.1$ , and then decreases rapidly as the scale increases. According to (4.3) and the analysis in the context of figure 10(a–c), the total nonlinear transport $-\overline { \langle \Delta u(\Delta \theta )^{2} \rangle }/\bar {\varepsilon }_{\theta } r$ should be smaller than $4/3$ . Here, the maximum value of $-\overline { \langle \Delta u(\Delta \theta )^{2} \rangle }/\bar {\varepsilon }_{\theta } r$ is approximately 1.1.

Figure 19.

The scale-by-scale distributions of each term in (3.21), divided by $-({4}/{3}) \overline {\varepsilon _{\theta }'} r$ , on the wake centreline at $x/d=40$ . The symbols are as follows: crosses for $({1}/{r^2}) \int _0^r s^2 A_{rm}\,{\textrm{d}}s$ (advection); triangles for $({1}/{r^2}) \int _0^r s^2 (D_{rr} + D_{rc})\,{\textrm{d}}s$ (diffusion); asterisks for $({1}/{r^2}) \int _0^r s^2 (P_{rm} + P_{rc})\,{\textrm{d}}s$ (production); squares for $T$ (total transport); circles for $-T_c - F_c$ (coherent transport and forcing); plusses for $K_r$ (viscous); diamonds for left-hand side (LHS) of (3.21); dash-dotted line for right-hand side (RHS) of (3.21).

Figure 18 shows that isotropy is satisfied approximately at small scales $r/L_{v}\lesssim 0.16$ . Therefore, the scale-by-scale distributions of (3.21) at this location can be evaluated without considering (3.20). Figure 19 shows the scale-by-scale distributions of each term in (3.21) on the axis at $x/d=40$ , normalised by $-({4}/{3}) \overline {\varepsilon _{\theta }'} r$ . At small scales ( $r/L_{v}\lesssim 0.049$ ), the viscous term $K_{r}$ dominates, indicating that viscosity plays a major role. Within this range, the total transport term $T$ exhibits a clear upward trend. Its contribution increases with scale, peaking at $r/L_{v}\approx 0.1$ before decreasing to 0 at larger scales. In the range $0.049\lesssim r/L_{v}\lesssim 0.28$ , the total transport term $T$ dominates, while $K_{r}$ decreases sharply and becomes negligible at larger scales. Meanwhile, the diffusion terms $({1}/{r^2}) \int _0^r s^2(D_{rr}+D_{r c})\, {\textrm{d}}s$ gradually increase. For $r/L_{v}\gtrsim0.28$ , the diffusion terms surpass the total transport term and become dominant. The contribution of the advection term $({1}/{r^2}) \int _0^r s^2 A_{rm}\,{\textrm{d}}s$ increases with $r/L_{v}$ and exceeds the total transport term $T$ at $r/L_{v}\approx 0.78$ . Since the mean temperature gradient is zero on the centreline and the coherent motion is negligible, the production terms $({1}/{r^2}) \int _0^r s^2(P_{rm}+P_{rc})\, {\textrm{d}}s$ , as well as the coherent transport and forcing terms $T_c + F_c$ , remain nearly zero for all separations. It is worth mentioning that (3.21) is satisfied adequately across all scales, with a maximum discrepancy of approximately 9 % between its left- and right-hand sides.

It is worth noting that the scale-by-scale distributions of the large-scale terms in (3.21) differ significantly from those in the transport equation for $\overline {(\Delta {q}^{\prime })^{2}}$ (see figure 9(c) in Thiesset et al. Reference Thiesset, Danaila and Antonia2014), where $\overline {(\Delta {q}^{\prime })^{2}}=\overline {\Delta {u}^{\prime }\Delta {u}^{\prime }}+\overline {\Delta {v}^{\prime }\Delta {v}^{\prime }}+\overline {\Delta {w}^{\prime }\Delta {w}^{\prime }}$ . In the transport equation for $\overline {(\Delta {\theta }^{\prime })^{2}}$ (figure 19), the dominant large-scale contribution arises from the diffusion term $({1}/{r^2}) \int _0^r s^2(D_{rr}+D_{r c})\, {\textrm{d}}s$ , rather than the advection term $({1}/{r^2}) \int _0^r s^2 A_{rm}\,{\textrm{d}}s$ . Specifically, at $r/L_{v}\approx 1.3$ , the ratio of the diffusion term to the advection term is approximately 3. A similar phenomenon is observed in the far-wake centreline ( $x/d=240$ ) of a circular cylinder at $Re=1400$ , where the diffusion term again dominates in the transport equation for $\overline {{\theta }^{2}}$ , with ratio (to the advection term) approximately 2.4 (see figure 5 in Lefeuvre et al. Reference Lefeuvre, Djenidi, Antonia and Zhou2014). In contrast, in the transport equation for $\overline {(\Delta {q}^{\prime })^{2}}$ (figure 9(c) in Thiesset et al. Reference Thiesset, Danaila and Antonia2014), the advection term $({1}/{r^2}) \int _0^r s^2 A_{rm}\,{\textrm{d}}s$ becomes the sole dominant contributor at large scales for $r/L_{v}\gtrsim 0.16$ . Another notable difference is that the magnitude of the coherent transport and forcing terms, $T_c + F_c$ , in the transport equation for $\overline {(\Delta {q}^{\prime })^{2}}$ is greater than that in the corresponding equation for $\overline {(\Delta {\theta }^{\prime })^{2}}$ . These observations suggest that at $x/d=40$ , the scale-by-scale distributions of the large-scale terms in the transport equations for $\overline {(\Delta {\theta }^{\prime })^{2}}$ and $\overline {(\Delta {q}^{\prime })^{2}}$ in the circular cylinder wake differ significantly, apart from the presence of the pressure term in the latter. Interestingly, a similar difference is also observed along the axis of a round jet. In the transport equation for $\overline {\theta ^{2}}$ , the diffusion and advection terms contribute comparably at $x/D=30$ , where $D$ is the diameter of the jet nozzle (see figure 7 in Antonia & Mi Reference Antonia and Mi1993). In contrast, for the transport equation of $\overline {(\Delta {q})^{2}}$ at $x/D=40$ , the advection term is the sole dominant contributor at large scales along the jet axis (see figure 2 in Burattini, Antonia & Danaila Reference Burattini, Antonia and Danaila2005).

Figure 20.

(a) Distributions of $\overline {vq^{2}}/U_{\infty }^3$ and $\overline {v\theta ^{2}}/(U_{\infty } \varTheta _0^2)$ at $x/d=40$ . (b) Distributions of the diffusion terms (i.e. $({1}/{2}) ({\partial \overline {vq^{2}}}/{\partial y})$ and $({1}/{2}) ({\partial \overline {v\theta ^{2}}}/{\partial y})$ ) and production terms (i.e. $\overline {uv}\,({\partial \overline {U}}/{\partial y})$ and $\overline {v\theta }\,({\partial \overline {\varTheta }}/{\partial y})$ ) in the transport equations for $\overline {q^{2}}$ and $\overline {\theta ^{2}}$ at $x/d=40$ , normalised by the centreline values of $\overline {\varepsilon }$ and $\overline {\varepsilon }_{\theta }$ , respectively. The horizontal dashed line represents value 0.

To clarify the diffusion term’s significant contribution in the scalar transport equation, we examine the third-order velocity and scalar statistics and the diffusion and production terms in the transport equations for $\overline {q^{2}}$ and $\overline {\theta ^{2}}$ (Antonia & Browne Reference Antonia and Browne1986; Lefeuvre et al. Reference Lefeuvre, Djenidi, Antonia and Zhou2014) in the circular cylinder wake. Figure 20(a) shows the distributions of $\overline {vq^{2}}/U_{\infty }^3$ and $\overline {v\theta ^{2}}/(U_{\infty } \varTheta _0^2)$ at $x/d=40$ . The diffusion terms (i.e. $({1}/{2})({\partial \overline {vq^{2}}}/{\partial y})$ and $({1}/{2}) ({\partial \overline {v\theta ^{2}}}/{\partial y})$ ) and the production terms (i.e. $\overline {uv}\,({\partial \overline {U}}/{\partial y})$ and $\overline {v\theta }\,({\partial \overline {\varTheta }}/{\partial y})$ ) are shown in figure 20(b). In figure 20(a), $\overline {vq^{2}}/U_{\infty }^3$ is approximately zero near the centreline, indicating that the associated diffusion term $({1}/{2}) ({\partial \overline {vq^{2}}}/{\partial y})$ is negligible around $y/d=0$ , as confirmed in figure 20(b). Since the production term $\overline {uv}\,({\partial \overline {U}}/{\partial y})$ is also negligible on the centreline, the advection term becomes the only large-scale contributor in the transport equation for $\overline {q^{2}}$ (Thiesset et al. Reference Thiesset, Danaila and Antonia2014). In contrast, $\overline {v\theta ^{2}}/(U_{\infty } \varTheta _0^2)$ in figure 20(a) varies substantially near the centreline, suggesting that the corresponding diffusion term $({1}/{2}) ({\partial \overline {v\theta ^{2}}}/{\partial y})$ contributes significantly to the transport equation for $\overline {\theta ^{2}}$ on the axis. The ratio $({1}/{2}) ({\partial \overline {v\theta ^{2}}}/{\partial y})/\overline {\varepsilon }_{\theta }$ in figure 20(b) is approximately 0.63, close to the corresponding value ( $\approx 0.67$ ) in the transport equation for $\overline {(\Delta {\theta }^{\prime })^{2}}$ at $r/L_{v}\approx 1.3$ (figure 19). Another noteworthy feature in figure 20(b) is that the maximum of $\overline {uv}\,({\partial \overline {U}}/{\partial y})$ occurs at $y/d\approx 1.4$ , approximately coinciding with the saddle points (figure 3 i). This is consistent with the peak of $ \langle u^{\prime } v^{\prime } \rangle$ occurring at the same location (see figure 13(c) of Chen et al. Reference Chen, Zhou, Zhou and Antonia2016), since both the Reynolds shear stress and the mean velocity gradient directly govern $\overline {uv}\,({\partial \overline {U}}/{\partial y})$ . Hussain & Hayakawa (Reference Hussain and Hayakawa1987) also observed that turbulent production occurs predominantly near the location of the maximum $ \langle u^{\prime } v^{\prime } \rangle$ . These findings highlight the crucial role of the saddle region in turbulence production. Furthermore, $({1}/{2}) ({\partial \overline {vq^{2}}}/{\partial y})$ is negative for $0.2\lesssim y/d\lesssim 2.6$ , with a minimum at $y/d\approx 1.2$ , and becomes positive for $y/d\gtrsim 2.6$ , suggesting that diffusion primarily transports turbulent kinetic energy towards regions farther from the wake centreline. In contrast, the maximum of $\overline {v\theta }\,({\partial \overline {\varTheta }}/{\partial y})$ occurs at $y/d \approx 3$ , farther from the centreline than the maximum of $\overline {uv}\,({\partial \overline {U}}/{\partial y})$ . A similar phenomenon is also observed by Chen et al. (Reference Chen, Zhou, Zhou and Antonia2016). This is reasonable because the peak of $\overline {v\theta }$ lies farther from the centreline than that of $\overline {uv}$ (see figure 13(c,i) of Chen et al. Reference Chen, Zhou, Zhou and Antonia2016), indicating that the production of scalar variance is displaced outwards relative to that of the kinetic energy. Moreover, $({1}/{2}) ({\partial \overline {v\theta ^{2}}}/{\partial y})$ is negative for $y/d\lesssim 1.3$ and becomes positive for $y/d\gtrsim 3.9$ , suggesting that the scalar variance is transported by diffusion both towards regions near the centreline ( $y/d\lesssim 1.3$ ) and towards the outer wake boundary ( $y/d\gtrsim 3.9$ ). These observations highlight the difference in spatial behaviour between the scalar and velocity fields in the wake.