4.2.1. Centreline turbulence intensity

The centreline turbulence intensities for different tree models at the same Reynolds number as well as different Reynolds numbers for the same tree model were compared. The turbulence intensity was defined as follows:

(4.2) \begin{align} \begin{split} \frac { u_{avg}^{\prime} } { U_\infty } = \frac { \sqrt {\frac {1}{3} \overline {u_i^{2}} } } { U_\infty }, \end{split} \end{align}

where $u_{avg}^{\prime}$ is calculated based on $\overline {u_i^{2}}$ and $u_i$ denotes the velocity fluctuation in the $i$ direction.

In a study on fractal-grid-generated turbulence, Gomes-Fernandes, Ganapathisubramani & Vassilicos (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012) used a method to normalize the centreline turbulence intensity. This method has also been used in the present study, where normalization is performed by dividing the turbulence intensity by its peak value and then dividing $x$ by the $x$ -coordinate where the magnitude of turbulence intensity is maximum ( $x_{peak}$ ). Figure 11 illustrates the results using this normalization. As observed in the figure, the turbulence intensity collapses more effectively when the Reynolds number is high. At low Reynolds numbers (figure 11 a), it does not collapse if $x /x_{peak} \gt\,\rm 1.0$ . The role of viscosity was considered strong, and the flow field was not yet sufficiently developed for turbulence. Hence, this normalization method is unsuitable for cases with low Reynolds numbers. At intermediate Reynolds numbers (figure 11 b), the collapse improves. A more complete collapse appeared at higher Reynolds numbers (figure 11 c,d). As observed in the figure, except for the case of Basic $n\,=\,4$ , the turbulence intensities for $n\,=\,6$ and $n\,=\,8$ collapsed effectively. However, this collapse has only been confirmed for the centreline turbulence intensity of $n\,=\,6$ and 8. Further research must be conducted to determine whether it applies to $n\,=\,9$ , 10 or higher.

Figure 12 illustrates the dependence of turbulence intensity on the Reynolds number for the same tree model. In all three models, $x_{peak}$ moves upstream as the Reynolds number increases. This trend converges at $Re_H =10\,000$ when $n\,=\,\rm 4$ and 8, as there is virtually no difference in the turbulence intensity beyond $Re_H = 10\,000$ . For $n\,=\,\rm 6$ , it converges at $Re_H =60\,000$ . For $n\,=\,\rm 4$ and 8, the peak turbulence intensity increases with an increasing Reynolds number; however, it stagnates at $Re_H =10\,000$ . This suggests that the centreline turbulence intensity hardly changes after a critical Reynolds number (e.g. $Re_H =10\,000$ ). Similarly, for $n\,=\,\rm 6$ , the peak turbulence intensity also grows with increasing Reynolds number and stagnates at $Re_H =60\,000$ . At $Re_H\,=\,\rm 60\,000$ and 120 000, the peak turbulence intensity is significantly greater at $n\,=\,\rm 6$ than at $n\,=\,\rm 4$ and 8. This discrepancy can be attributed to the fact that the peak turbulence intensity does not always occur at $y /H\,=\,\rm 0$ and $z /H\,=\,\rm 0.5$ (the centreline) when viewed along the $y$ - and $z$ -axes. For instance, as shown in figure 13, the peak value of turbulence intensity does not always occur at $y/H\,=\,\rm 0$ , and similarly, the peak value of spatially averaged turbulence intensity does not always occur at $z/H\,=\,\rm 0.5$ , as shown in figure 14 (discussed later).

In the study by Gomes-Fernandes, Ganapathisubramani & Vassilicos (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012), they defined an equation that predicts the peak centreline turbulence intensity and $x_{peak}$ position in the case of grid turbulence. For fractal tree turbulence, an equation for predicting the peak turbulence intensity and location of $x_{peak}$ may exist, which needs to be further investigated.

Figure 12. Comparison of streamwise evolution of centreline turbulence intensity at various $Re_H$ in the case of $(a)$ Basic $n\,=\,4$ , $(b)$ Basic $n\,=\,6$ and $(c)$ Basic $n\,=\,8$ .

4.2.2. Height dependency of turbulence intensity

The height dependency of turbulence intensity for different tree models is presented. As discussed in § 4.2.1, in the case of the same tree model, the turbulence intensity exhibits almost the same behaviour beyond $Re_H = 60\,000$ . Hence, only the case of $Re_H = 120\,000$ is explored in this subsection.

Figure 13 illustrates an example of the turbulence intensity distribution in the spanwise direction for Basic $n\,=\,8$ for $z /H = 0.5$ at various downstream distances. As observed in the figure, the peak value of turbulence intensity does not appear on the centreline ( $y=0$ ). The asymmetric profiles can be attributed to the asymmetric tree shape, as shown in figure 3. The tree branches are located closer to the inflow direction (smaller $x$ coordinate) on the $-y$ side than on the $+y$ side. Simultaneously, the centreline turbulence intensity first grows rapidly to a peak value and then decays gradually along the $x$ -direction, as shown in § 4.2.1. It is reasonable to assume that this is also applicable for the $y \neq 0$ cases. Thus, it can be surmised that the turbulence intensity on the $-y$ side reaches a peak value and starts to decay at a more forward position than that on the $+y$ side, resulting in asymmetric profiles depicted in figure 13. We also considered that the evaluation of turbulence intensity in the canopy part is insufficient only for the centreline, as the thin branches also spread in the spanwise direction. For a given height (e.g. $z/H\,=\,0.5$ , as shown in figure 13), we plotted the transverse profiles of turbulence intensity at various $x$ coordinates. At each $x$ coordinate, we identified the peak turbulence intensity on the line along the $y$ direction. We defined the half-width of turbulence intensity as the range where the turbulence intensity exceeds half of this peak on that line. At each $x$ coordinate, turbulence statistics, including turbulence intensity, the Taylor-microscale-based Reynolds number and isotropy parameters, were averaged over $y$ within the half-width of turbulence intensity. We repeated these steps for different heights to obtain the height dependency of various turbulence statistics.

Figure 13. Turbulence intensity distribution in the spanwise direction for Basic $n\,=\,8$ at $z /H=0.5, Re_H = 120\,000$ at various downstream distances.

Figure 14 illustrates the height dependency of turbulence intensity. Similar to the centreline’s results, it is shown to decrease gradually in the downstream direction in all three models. Near the wake locations behind the fractal tree ( $x /H = 0.6$ ), turbulence intensity is markedly more potent at $z /H\,=\,0.4-0.65$ for $n\,=\,4$ , $z /H\,=\,0.3-0.55$ for $n\,=\,6$ and $z /H\,=\,0.3-0.5$ for $n\,=\,8$ , compared with other heights. This suggests that the turbulence intensity is stronger near the tree trunk and second branch height than at other altitudes. We considered that the large-scale turbulence generated by the trunk is dominant in near-wake locations. This trend is maintained after $x /H = 0.6$ , although the overall turbulence intensity gradually decreases. Generally, at sufficiently distant downstream locations ( $x /H \geqslant 2.2$ ), the height at which turbulence intensity is the strongest is approximately $z /H\,=\,0.35{-}0.5$ . The difference in turbulence intensity between the central part of the trunk and the canopy part also becomes smaller. This may be due to the fractal nature of the tree geometry, which promoted vertical mixing within the wake, as reported in a previous study (Schröttle & Dörnbrack Reference Schröttle and Dörnbrack2013).

Figure 14. Turbulence intensity as a function of height ( $z /H$ ) at $Re_H = 120\,000$ : $(a)$ Basic $n\,=\,4$ ; $(b)$ Basic $n\,=\,6$ ; $(c)$ Basic $n\,=\,8$ . The angle bracket, $\langle \cdot \rangle _{\text {half-width}}$ , represents the spatial average over $y$ within the half-width of turbulence intensity, which defined in § 4.2.2.

4.2.3. Transverse plane average of turbulence intensity

Although we have analysed the centreline turbulence intensity and its height dependency, the complex geometry of the tree presents a highly 3-D flow field. Similar to the method employed by Laizet & Vassilicos (Reference Laizet and Vassilicos2015), we considered the average turbulence intensity within the $y$ – $z$ plane for further analysis. The $y$ – $z$ plane was selected as a rectangular area that closely follows the tree’s outer boundary, capturing its $x$ -direction projection while minimizing the rectangle’s area, as shown in figure 15 $(a)$ . This selection ensures that the plane captures the tree’s influence on the flow field with the smallest cross-sectional area. The turbulence intensity is averaged over $y$ and $z$ within this plane. Figure 15 $(b)$ shows the streamwise evolution of spatially averaged turbulence intensity.

Figure 15. $(a)$ The $y$ – $z$ plane used to obtain the spatial average. $(b)$ Streamwise evolution of turbulence intensity (with angle brackets $\langle \cdot \rangle _{yz}$ denoting averaging over $y$ and $z$ ) for Basic $n\,=\,4$ , 6 and 8 at $Re_H = 120\,000$ .

The average turbulence intensity for $n\,=\,4$ , 6 and 8 exhibits both a region of rapid growth and a region of decay, with the peak marking the boundary between these regions, as shown in figure 15 $(b)$ . The peaks are located at approximately $x /H\,=\,0.4$ , similar to the results from the centreline analysis (figure 12). Regarding the peak values, $n\,=\,4$ exhibits a higher peak value of 10.2 $\%$ when compared with 7.0 $\%$ and 6.9 $\%$ for $n\,=\,6$ and $n\,=\,8$ , respectively. This difference might be attributed to the dominance of the large-scale turbulence in the wake region of $n\,=\,4$ , which tends to exhibit higher turbulence intensity than the small-scale turbulence. For $n\,=\,6$ and 8, the higher fractal iteration number create thinner and denser branches than $n\,=\,4$ , which might lead to the dominance of small-scale turbulence in their wake regions. Furthermore, figure 14 shows that the turbulence intensity in the canopy region of $n\,=\,4$ ( $z /H\,=\,0.75-0.95$ ) is significantly higher, approximately 10 $\%$ –13 $\%$ , when compared with 8 $\%$ –11 $\%$ for $n\,=\,6$ ( $z/ H\,=\,0.65-0.95$ ) and 6 $\%$ –10 $\%$ for $n\,=\,8$ ( $z /H\,=\,0.6-0.95$ ). Additionally, since all the tree heights were scaled to 1 m, the trunks of $n\,=\,6$ and $n\,=\,8$ are thinner than those of $n$ = 4. The turbulence intensity near the trunks of $n$ = 4 ( $z /H\,=\,0.35-0.65$ ) is approximately 18 $\%$ –20 $\%$ , while it is approximately 16 $\%$ –17 $\%$ for $n\,=\,6$ ( $z /H\,=\,0.3-0.55$ ) and 13 $\%$ –16 $\%$ for $n\,=\,8$ ( $z /H\,=\,0.3{-}0.5$ ). These factors might contribute to the higher peak value observed in $n\,=\,4$ .

In terms of the decay rate, $n\,=\,8$ exhibits a noticeably faster decay in the near wake region (approximately $x /H \leqslant 2.0$ ). For example, at $x /H\,=\,1.0$ , the turbulence intensity for $n\,=\,8$ drops to 3.8 $\%$ , which is 54.8 $\%$ of its peak value, while for $n\,=\,4$ and $n\,=\,6$ , the values are 7.3 $\%$ and 4.7 $\%$ , corresponding to 71.5 $\%$ and 66.4 $\%$ of their respective peak values. As mentioned earlier, in the wake region of $n\,=\,8$ , small-scale turbulence might be dominate. The small-scale turbulence tends to dissipate faster than the large-scale one, which might contribute to the rapid decay of the average turbulence intensity in the near-wake region (approximately $x /H \leqslant 2.0$ ) of $n\,=\,8$ .

Figure 16. Comparison of streamwise evolution of centreline Taylor-microscale-based Reynolds number as a function of $x /x_{peak}$ at $(a)$ $Re_H=2500$ , $(b)$ $Re_H=10\,000$ , $(c)$ $Re_H=60\,000$ and $(d)$ $Re_H=120\,000$ .

4.3.1. Centreline turbulent Reynolds number

Figure 11(c,d) illustrate an effective collapse in the turbulence intensity between Basic $n\,=\,6$ and $n\,=\,8$ . Therefore, we employed $x_{peak}$ to normalize streamwise distances in all subsequent centreline comparisons between tree models. Figure 16 depicts the centreline Taylor-microscale-based Reynolds number, $Re_\lambda = u^{\prime} \lambda / \nu$ . At low Reynolds number (figure 16 a), in all three models, the $Re_\lambda$ are all small, with peak values below 30 and cannot be considered as even turbulent flow. At medium Reynolds number (figure 16 b), $Re_\lambda$ becomes larger overall.

At high Reynolds number (figure 16 c,d), $Re_\lambda$ increases further, with $n$ = 6 showing the largest $Re_\lambda$ in most decaying regions. The data do not collapse properly, so it can be considered that $x_{peak}$ is not suitable for collapsing centreline $Re_\lambda$ . At $Re_H\,=\,60\,000$ and 120 000, the peak $Re_\lambda$ is significantly greater at $n\,=\,6$ than at $n\,=\,4$ and 8. This discrepancy likely arises because the peak $Re_\lambda$ does not always occur at $y /H\,=\,0$ and $z /H\,=\,0.5$ (the centreline) when viewed along the $y$ - and $z$ -axes.

As shown in figure 16, $Re_\lambda$ remains approximately constant following a certain downstream location, except for the case of $n\,=\,6$ at $Re_H\,=\,2500$ . Specifically, in figure 16(a,b), $Re_\lambda$ remains approximately constant at approximately $x \geqslant\,3$ $x_{peak}$ and 6 $x_{peak}$ for $n\,=\,4$ and 8 at $Re_H=2500$ and at approximately $x \geqslant\,6$ $x_{peak}$ for $n\,=\,4$ , 6 and 8 at $Re_H=10\,000$ . Similarly, in figure 16(c,d), $Re_\lambda$ remains approximately constant at approximately $x \geqslant$ 6 $x_{peak}$ , 12 $x_{peak}$ and 10 $x_{peak}$ for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,60\,000$ and at approximately $x \geqslant$ 7 $x_{peak}$ , 12 $x_{peak}$ and 12 $x_{peak}$ for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,120\,000$ . This phenomenon may be attributed to the gradual decrease in $u^{\prime}$ and the gradual increase in $\lambda$ in the downstream direction, causing $Re_\lambda$ to remain approximately constant.

Figure 17 depicts the dependence of $Re_\lambda$ on the Reynolds number for the same tree model. In all three models, $Re_\lambda$ increases overall with increasing Reynolds number, similar with the turbulence intensity trend illustrated in figure 12. Notably, in figure 12, $u_{avg}^{\prime}$ is normalized by uniform flow velocity $U_\infty$ , so $u_{avg}^{\prime}$ or $u^{\prime}$ will increase with increasing Reynolds number. The position at which $Re_\lambda$ is maximum always appears at approximately $x /H\,=\,0.4{-}0.5$ , independent of the Reynolds number.

Figure 17. Comparison of streamwise evolution of centreline Taylor-microscale-based Reynolds number as a function of $x /H$ at various $Re_H$ in the case of $(a)$ Basic $n\,=\,4$ , $(b)$ Basic $n\,=\,6$ and $(c)$ Basic $n\,=\,8$ .

4.3.2. Height dependency of turbulent Reynolds number

Similar to the height dependency of turbulence intensity depicted in § 4.2.2, the height dependency of $Re_\lambda$ is also given in figure 18. The spatial average, $\langle Re_\lambda \rangle _{\text {half-width}}$ , was taken over $y$ within the half-width of turbulence intensity, which is defined in § 4.2.2.

Figure 18. Taylor-microscale-based Reynolds number as a function of height ( $z /H$ ) at $Re_H = 120\,000$ : $(a)$ Basic $n\,=\,4$ ; $(b)$ Basic $n\,=\,6$ ; $(c)$ Basic $n\,=\,8$ . The angle bracket, $\langle \cdot \rangle _{\text {half-width}}$ , represents the spatial average over $y$ within the half-width of turbulence intensity, which defined in § 4.2.2.

Here $Re_\lambda$ decreased gradually in the downstream direction in all three models. In near-wake locations behind the fractal tree ( $x /H =0.6$ ), $Re_\lambda$ is considerably more potent at $z /H\,=\,0.4{-}0.65$ for $n\,=\,4$ , $z /H\,=\,0.3{-}0.55$ for $n\,=\,6$ and $z /H\,=\,0.3{-}0.5$ for $n\,=\,8$ , compared with other altitudes. This is because the turbulence intensity is stronger near the tree trunk than at the tree crown, as mentioned before. Generally, at sufficiently distant downstream locations ( $z /H \geqslant 2.2$ ), similar with the turbulence intensity, the height at which $Re_\lambda$ is most vigorous is approximately $z /H\,=\,0.35{-}0.5$ in all three models.

4.4.1. Centreline global and local isotropy parameters

In this section, the isotropy of the wake region of fractal trees is assessed. First, the global isotropy associated with the large scales is assessed, followed by the local isotropy related to the small scales. For clarity, we only present Basic $n\,=\,8$ data. Notably, all the other models also exhibited very similar values and the same trend.

Figure 19 compares the centreline global isotropy parameters $u^{\prime} /v^{\prime}$ , $u^{\prime} /w^{\prime}$ and $v^{\prime} /w^{\prime}$ at various Reynolds numbers for Basic $n\,=\,8$ . At low Reynolds numbers (figure 19 $a$ ), except for $x /x_{peak} =\,4.0{-}6.0$ , $u^{\prime} /v^{\prime}$ , $u^{\prime} /w^{\prime}$ and $v^{\prime} /w^{\prime}$ are significantly away from 1.0, indicating that the wake region is not globally isotropic overall. At intermediate Reynolds numbers (figure 19 b), an improvement in global isotropy is observed, with most parameters lying between 0.9 and 1.2. At high Reynolds numbers (figure 19 c,d), except for the region $x /x_{peak} \lt 1.0$ , all three global isotropy parameters generally hover around 1.0 and are marginally less than 1.2. This result is close to that of grid turbulence (Gomes-Fernandes, Ganapathisubramani & Vassilicos Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012) with $u^{\prime} /v^{\prime}$ around 1.2 when $x /x_{peak} \gt 1.0$ . For Reynolds numbers that are medium and high, the flow is globally anisotropic if $x /x_{peak} \lt 1.0$ , while it is almost globally isotropic if $x/x_{peak} \gt 1.0$ . This concurs with the grid turbulence case.

Figure 19. Global isotropy parameters of centreline for Basic $n\,=\,8$ at $(a)$ $Re_H=2500$ , $(b)$ $Re_H=10\,000$ , $(c)$ $Re_H=60\,000$ and $(d)$ $Re_H=120\,000$ .

In the study of grid turbulence, local isotropy was assessed using two relations ( $K_1, K_3$ ) derived by Taylor (Reference Taylor1935). Namely, small-scale turbulence is locally isotropic if the Reynolds number is sufficiently high and $K_1, K_3$ should equal to 1. We also evaluated $K_1$ and $K_3$ , and additionally, we defined $K_2$ and $K_4$ , as follows, to assess the local isotropy of the fractal tree wake region:

(4.3) \begin{align} \begin{split} K_1 = 2 \, \overline {\left (\frac {\partial u}{\partial x}\right )^2} / \overline {\left (\frac {\partial v}{\partial x}\right )^2},\,\,\,\, K_3 = 2 \, \overline {\left (\frac {\partial u}{\partial x}\right )^2} / \overline {\left (\frac {\partial u}{\partial y}\right )^2}, \end{split} \end{align}

(4.4) \begin{align} \begin{split} K_2 = 2 \, \overline {\left (\frac {\partial u}{\partial x}\right )^2} / \overline {\left (\frac {\partial w}{\partial x}\right )^2},\,\,\,\, K_4 = 2 \, \overline {\left (\frac {\partial u}{\partial x}\right )^2} / \overline {\left (\frac {\partial u}{\partial z}\right )^2}. \end{split} \end{align}

Figures 20 and 21 compare the centreline local isotropy parameters $K_1, K_3$ and $K_2, K_4$ at various Reynolds numbers for Basic $n\,=\,8$ .

Similar with global isotropy parameters, at low Reynolds numbers (figures 20 a and 21 $a$ ), $K_1, K_2, K_3$ and $K_4$ are significantly away from 1.0, indicating that the wake region is not locally isotropic. At medium Reynolds numbers (figures 20 band 21 $b$ ), an improvement in the local isotropy is observed, but $K_1$ , $K_2$ and $K_4$ are still considerably away from 1.0, which does not fit the definition of a locally isotropic fluid. At high Reynolds numbers (figures 20 c,d and 21 $c, d$ ), all four parameters generally hover around 1.0, which is close to the value in a previous grid turbulence study (Gomes-Fernandes, Ganapathisubramani & Vassilicos Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012). Generally, the wake region for $x/x_{peak} \gt 1.0$ can be regarded as essentially isotropic at high Reynolds numbers ( $Re_H \geqslant 60\,000$ ).

Figure 20. Local isotropy parameters of centreline for Basic $n\,=\,8$ at $(a)$ $Re_H=2500$ , $(b)$ $Re_H=10\,000$ , $(c)$ $Re_H=60\,000$ and $(d)$ $Re_H=120\,000$ .

Figure 21. Local isotropy parameters of centreline for Basic $n\,=\,8$ at $(a)$ $Re_H=2500$ , $(b)$ $Re_H=10\,000$ , $(c)$ $Re_H=60\,000$ and $(d)$ $Re_H=120\,000$ .

Figure 22. Global isotropy parameters as a function of height ( $z /H$ ) at $Re_H = 120\,000$ for Basic $n\,=\,8$ : $(a)$ $u' /v'$ ; $(b)$ $u' /w'$ ; $(c)$ $v' /w'$ . The angle bracket, $\langle \cdot \rangle _{\text {half-width}}$ , represents the spatial average over $y$ within the half-width of turbulence intensity, which defined in § 4.2.2.

4.4.2. Height dependency of isotropy

The global isotropy parameters at $Re_H = 120\,000$ for Basic $n\,=\,8$ as a function of height ( $z /H$ ) are illustrated in figure 22. The spatial average was taken over $y$ within the half-width of turbulence intensity, which defined in § 4.2.2. As observed in the figure, the distributions of $u^{\prime} /v^{\prime}$ and $v^{\prime} /w^{\prime}$ ranged from approximately 0.8–1.2 and 1.0–1.2, respectively. After $x /H = 2.2$ , no obvious change with altitude could be observed. Here $u^{\prime} /w^{\prime}$ does not exhibit a clear tendency to change with height in the near wake region. After $x /H = 2.2$ , it showed a tendency to decrease from large to small from the crown to the trunk. We confirmed that this trend is also broadly consistent, but not as pronounced, for $n\,=\,6$ and almost absent for $n\,=\,4$ . Overall, the distribution of all three parameters ranged from approximately 0.8–1.2, suggesting that the wake region was essentially global isotropic at all different heights. We also confirmed this trend for Basic $n\,=\,4$ and 6.

The local isotropy parameters at $Re_H = 120\,000$ for Basic $n\,=\,8$ as a function of height ( $z /H$ ) are shown in figures 23 and 24. In these figures, $K_1$ and $K_2$ exhibit approximately the same values at different heights. However, $K_3$ and $K_4$ tended to increase from the crown to the trunk. This trend is also confirmed for $n\,=\,6$ but almost absent for $n\,=\,4$ . This suggests that the shear deformation of streamwise velocity fluctuations ( $\partial u / \partial y$ , $\partial u / \partial z$ ) is stronger near the canopy than the spatial gradient of streamwise velocity fluctuations ( $\partial u / \partial x$ ). Overall, the four parameters ranged from around 0.8–1.2, suggesting that the wake region was essentially local isotropic at all different heights. We also confirmed this for Basic $n\,=\,4$ and 6.

Figure 23. Local isotropy parameters as a function of height ( $z /H$ ) at $Re_H = 120\,000$ for Basic $n\,=\,8$ : $(a)$ $K_1$ , $(b)$ $K_3$ . The angle bracket, $\langle \cdot \rangle _{\text {half-width}}$ , represents the spatial average over $y$ within the half-width of turbulence intensity, which defined in § 4.2.2.

Figure 24. Local isotropy parameters as a function of height ( $z /H$ ) at $Re_H = 120\,000$ for Basic $n\,=\,8$ : $(a)$ $K_2$ , $(b)$ $K_4$ . The angle bracket, $\langle \cdot \rangle _{\text {half\hbox{-}width}}$ , represents the spatial average over $y$ within the half-width of turbulence intensity, which defined in § 4.2.2.

4.5.1. Integral length scale

The integral length scale in the $x$ -direction, $L_u$ , was calculated by integrating the longitudinal correlation function $f(r, x)$ from $r$ = 0 to the first zero-crossing point of $r$ . Here, $f(r,x) = \overline {u(x)u(x+r)} / \overline {u(x)^2}$ , where $u(x)$ represents the velocity fluctuation in streamwise direction at $x$ .

As mentioned in § 4.4, when the tree-height-based Reynolds number $Re_H \leqslant 10\,000$ , the wake region was not isotropic, and $L / \lambda \propto Re_\lambda$ was not applicable. Hence, this discussion does not include cases with low Reynolds numbers. On the other hand, the wake region can be considered essentially isotropic if $Re_H \geqslant 60\,000$ . In this case, $L / \lambda \propto Re_\lambda$ is applicable and can be verified by plotting the dependency of $L_u /\lambda$ on $Re_\lambda$ . If $L_u / \lambda \propto Re_\lambda$ holds true, then $\epsilon = C_\epsilon u^{\prime3}/{L}$ also holds true with a constant $C_\epsilon$ . Conversely, $C_\epsilon$ is not a constant value but varies with position in the wake region.

Figure 25 shows the variation of $L_u /\lambda$ in the streamwise distance as a function of $x /x_{peak}$ . As observed in the figure, in the decaying region of $x \geqslant x_{peak}$ where $Re_\lambda$ decreases, $L_u /\lambda$ does not significantly decrease for any of the three models. At $Re_H=60\,000$ , it is approximately 3.5 for all three models. This trend is sustained at $Re_H = 120\,000$ . For $n\,=\,4$ , $L_u / \lambda$ remains approximately 4.0 until $x / x_{peak} = 5.0$ . Then, for an unknown reason, it marginally increases, and $L_u / \lambda$ is approximately 6.0. For $n\,=\,6$ , it increases gradually within the 2.5–4.5 range and then becomes approximately 4.5. For $n\,=\,8$ , it approximately fluctuates in the 3.0–4.5 range.

Figure 25. Integral length scale to Taylor microscale ratio $L_u /\lambda$ on the centreline for different tree models in relation to $x /x_{peak}$ at $(a)$ $Re_H=60\,000$ and $(b)$ $Re_H=120\,000$ .

If $Re_\lambda$ and $C_\epsilon$ are constant, it would not be important that $L_u /\lambda$ remains approximately the same value. However, as illustrated in figure 16, at high Reynolds numbers ( $Re_H \geqslant 60\,000$ ), $Re_\lambda$ continues to decrease over a long region beyond $x_{peak}$ . In particular, $Re_\lambda$ keeps decreasing approximately until $x \approx$ 6 $x_{peak}$ , 12 $x_{peak}$ and 10 $x_{peak}$ for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,60\,000$ and until $x \approx$ 7 $x_{peak}$ , 12 $x_{peak}$ , 12 $x_{peak}$ for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,120\,000$ . In the above decay regions of $Re_\lambda$ , we plot the variation of $L_u /\lambda$ as a function of $Re_\lambda$ , as shown in figure 26. Here $L_u /\lambda$ remains approximately constant and does not increase as $Re_\lambda$ increases. This result qualitatively concurs with the findings of fractal-grid-generated turbulence (Seoud & Vassilicos Reference Seoud and Vassilicos2007; Mazellier & Vassilicos Reference Mazellier and Vassilicos2010; Valente & Vassilicos Reference Valente and Vassilicos2011b ; Gomes-Fernandes, Ganapathisubramani & Vassilicos Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012). Therefore, $\epsilon = C_\epsilon u^{\prime3}/{L}$ does not hold, primarily because $C_\epsilon$ is not a constant in the decaying region of the fractal-tree-generated turbulent flows studied here.

Figure 26. Integral length scale to Taylor microscale ratio $L_u /\lambda$ on the centreline for different tree models in relation to $Re_\lambda$ in the decay region of $Re_\lambda$ at $(a)$ $Re_H=60\,000$ and $(b)$ $Re_H=120\,000$ .

4.5.2. Non-dimensional energy dissipation parameter

Figure 27 shows the variation of the non-dimensional energy dissipation parameter $C_\epsilon$ as a function of $Re_\lambda$ in the $Re_\lambda$ decay region. It is observed that $C_\epsilon$ decreases with the increase in $Re_\lambda$ . For $n\,=\,4$ , 6 and 8 at $Re_H=60\,000$ , part of the decaying region occurs at $Re_\lambda \leqslant 100$ , where the low-Reynolds number effect (viscosity effect) may also influence $C_\epsilon$ (Sreenivasan Reference Sreenivasan1984, Reference Sreenivasan1998; Bos, Shao & Bertoglio Reference Bos, Shao and Bertoglio2007; Kitamura et al. Reference Kitamura, Nagata, Sakai, Sasoh, Terashima, Saito and Harasaki2014; Vassilicos Reference Vassilicos2015; Kitamura et al. Reference Kitamura, Nagata, Sakai, Sasoh, Terashima, Saito and Harasaki2014). For $Re_\lambda \gt 100$ , we applied the least-squares fit method to evaluate the relationship, $C_\epsilon \propto Re_\lambda ^{-1}$ . We confirmed that for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,60\,000$ , the exponents are approximately $-1.28$ , $-1.33$ and $-1.01$ , respectively. Similarly, for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,120\,000$ , the exponents are approximately $-0.96$ , $-1.17$ and $-0.96$ , respectively. Notably, for $Re_H\,=\,60\,000$ , limited data points used for the fit may affect the representatives of the results. For $n\,=\,4$ at $Re_H\,=\,120\,000$ , we excluded four data points where $C_\epsilon \gt 0.4$ . Overall, for $Re_H\,=\,120\,000$ , $Re_\lambda \gt 100$ , $C_\epsilon$ is approximately proportional to $Re_\lambda ^{-1}$ , which qualitatively agrees with the previous studies conducted on fractal-grid-generated turbulence (Seoud & Vassilicos Reference Seoud and Vassilicos2007; Mazellier & Vassilicos Reference Mazellier and Vassilicos2010; Valente & Vassilicos Reference Valente and Vassilicos2011b ; Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012).

Figure 27. Non-dimensional energy dissipation parameter $C_\epsilon$ on the centreline in relation to $Re_\lambda$ in the decay region of $Re_\lambda$ for the fractal trees studied at $(a) Re_H=60\,000$ and $(b) Re_H=120\,000$ .

Figure 28. Non-dimensional energy dissipation parameter $C_\epsilon$ on the centreline in relation to $x /H$ for the fractal trees studied at $(a) Re_H=60\,000$ and $(b) Re_H=120\,000$ .

Figure 28 shows the variation of $C_\epsilon$ as a function of $x /H$ . Each fractal tree exhibits a near wake region where $C_\epsilon$ increases rapidly, followed by a downstream region where $C_\epsilon$ remains approximately constant. The boundary between these two regions occurs at approximately $x\,=\,4.0$ $H$ for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,60\,000$ and approximately $x\,=\,4.0-5.0$ $H$ for $n\,=\,4$ , 6 and 8 at $Re_H\,=\,120\,000$ .

4.5.3. The TKE production and transverse transport

This section presents additional analysis of the transverse scans behind all three fractal trees to obtain a full picture of the evolution of wake turbulence. It is worth investigating whether the non-equilibrium dissipation scaling observed here is accompanied by non-zero turbulence production and/or triple correlation transport as in Valente & Vassilicos (Reference Valente and Vassilicos2011b ), Nagata et al. (Reference Nagata, Sakai, Inaba, Suzuki, Terashima and Suzuki2013) and Hearst & Lavoie (Reference Hearst and Lavoie2016). These two parameters are evaluated in the context of the following TKE equation:

(4.5) \begin{align} \begin{split} \frac {U_k}{2} \frac {\partial \overline {q^2}}{\partial x_k} = - \overline {u_i u_j} \frac {\partial U_i}{\partial x_j} - \frac {\partial }{\partial x_k} \left ( \frac {\overline {u_k q^2} }{2} \right ) - \frac {\partial }{\partial x_k} \left ( \frac {\overline {u_k p} }{\rho } \right ) + \frac {\nu }{2} \frac {\partial ^2 \overline {q^2} }{\partial x_k \partial x_k} - \epsilon. \end{split} \end{align}

The left-hand side term represents the advection term (TKE decay). The right-hand side terms represent production, triple-correlation transport, pressure transport, viscous diffusion and dissipation. Here, $U_1$ = $U$ , $U_2$ = $V$ and $U_3$ = $W$ denote the mean velocities, while $u_1$ = $u$ , $u_2$ = $v$ and $u_3$ = $w$ and $p$ represent the fluctuating velocities and pressure.

Referring to the analysis methods of Valente & Vassilicos (Reference Valente and Vassilicos2011b ) and Hearst & Lavoie (Reference Hearst and Lavoie2016), figure 29 shows the centreline TKE production $\mathcal {P}$ and triple-correlation transport $\mathcal {T}$ , both of which are normalized by the energy dissipation $\epsilon$ . As seen in figure 29 $(a)$ , near the upstream region (around $x /H = 1.0$ ), the TKE production is strong, reaching approximately $-150\%$ of the dissipation for $n\,=\,4$ , 6 and 8, before gradually decreasing. In the range of approximately $x/H\,=\,2.0-4.5$ , the production term is at most approximately $-70\%$ , $-50\%$ and $-15\%$ of the dissipation term for $n\,=\,4$ , 6 and 8, respectively. For $x /H \gt 4.5$ , the production-to-dissipation ratio gradually increases again in absolute value. As shown in figure 29 $(b)$ , the TKE transverse transport remains strong in both the upstream and downstream regions.

Figure 29. $(a)$ Centreline TKE production $\mathcal {P}$ and $(b)$ centreline triple-correlation transport $\mathcal {T}$ in relation to $x /H$ for Basic $n\,=\,4$ , 6 and 8 at $Re_H\,=\,120\,000$ .

Figure 30 $(a, b)$ shows the transverse profiles of $\mathcal {P}$ and $\mathcal {T}$ for $n\,=\,6$ at $Re_H\,=\,120\,000$ , normalized by the spanwise mean dissipation $\langle \epsilon \rangle _{y}$ . For clarity, in figure 30, we only show the results for $n\,=\,6$ . Note that we also confirmed similar behaviour for $n\,=\,4$ and 8. From $x /H\,=\,3.0-7.0$ , the profiles of $\mathcal {P}$ along the $y$ -direction remain inhomogeneous, indicating that the flow field is also inhomogeneous, with the transverse profiles of $\mathcal {P} / \langle \epsilon \rangle _{y}$ showing an overall increase in absolute value. The $\mathcal {T}$ -to-dissipation ratio remains at a non-negligible level. The absolute values of these two ratios can reach approximately 150 $\%$ and 200 $\%$ of $\langle \epsilon \rangle _{y}$ , respectively, which is significantly higher than those observed in the near wake region of the fractal grids (Hearst & Lavoie Reference Hearst and Lavoie2016). Figure 30 $(c, d)$ show the transverse profiles of $\overline {uv} (\partial U / \partial y)$ and $\overline {uw} (\partial U / \partial z)$ , which contribute significantly to $\mathcal {P}$ , also normalized by $\langle \epsilon \rangle _{y}$ . As the turbulence evolves downstream, the profiles of $\overline {uv} (\partial U / \partial y)$ gradually approach zero and become approximately homogeneous, while the increase in $\overline {uw} (\partial U / \partial z)$ exceeds the decrease in $\overline {uv} (\partial U / \partial y)$ . This indicates that the flow field becomes approximately homogeneous in the $y$ -direction farther downstream but remains inhomogeneous in the $z$ -direction, which explains the significant increase in the TKE production-to-dissipation ratio downstream in figure 30 $(a)$ .

Figure 30. Transverse profiles of $(a)$ TKE production $\mathcal {P}$ , $(b)$ triple-correlation transport $\mathcal {T}$ , $(c)$ $\overline {uv} (\partial U / \partial y)$ and $(d)$ $\overline {uw} (\partial U / \partial z)$ at various downstream positions for Basic $n\,=\,6$ at $z /H=0.5, Re_H = 120\,000$ .

Based on the above analysis, the non-equilibrium near wake region (approximately within $x /H \lt 5.0$ at $Re_H = 120\,000$ ) is highly inhomogeneous for the fractal trees, with significant TKE production and transverse transport. This aligns qualitatively with previous studies on fractal-grid-generated turbulence (Hearst & Lavoie Reference Hearst and Lavoie2016), which compared fractal square element grids with regular grids. They found that the fractal square element grids’ non-equilibrium near wake region was significantly more inhomogeneous, with higher TKE production and transverse transport when compared with the regular grids. In this study, the fractal tree exhibits even more significant 3-D inhomogeneities (in both the $y$ - and $z$ -directions) than the fractal grids. Thus, it can be considered that the inhomogeneous shape of the tree leads to greater inhomogeneity and TKE production and transport in the non-equilibrium near wake region.

However, it should be noted that non-equilibrium dissipation is not necessarily due to one-point inhomogeneities (Nagata et al. Reference Nagata, Saiki, Sakai, Ito and Iwano2017). While these one-point inhomogeneities cause TKE production and transport, as observed in this study and previous research (Nagata et al. Reference Nagata, Sakai, Inaba, Suzuki, Terashima and Suzuki2013; Valente & Vassilicos Reference Valente and Vassilicos2014; Hearst & Lavoie Reference Hearst and Lavoie2016), they do not affect the two-point energy equation at the scales where the inertial energy cascade occurs (Valente & Vassilicos Reference Valente and Vassilicos2015). As Nagata et al. (Reference Nagata, Saiki, Sakai, Ito and Iwano2017) pointed out, the cascade mechanism of the turbulence dissipation is not affected by one-point inhomogeneities. The inhomogeneity of a two-point energy balance at the correct length scales is more important than the inhomogeneity of the one-point energy balance. However, analysing two-point statistics is beyond this paper’s scope and requires further investigation.