4.1 Boundary-layer characteristics

Table 1 lists the main characteristics of the investigated boundary layers. The logarithmic-law parameter aerodynamic roughness length, $z_{0}$ , was determined by fitting the vertical streamwise velocity profile to the logarithmic law (Perret et al. Reference Perret, Basley, Mathis and Piquet2019). As described by Perret et al. (Reference Perret, Basley, Mathis and Piquet2019) the zero-plane displacement height, $d$ , is estimated directly from the calculation of the moment of pressure forces on the roughness elements while the friction velocity, $u_{\unicode[STIX]{x1D70F}}$ , is also estimated from the measured form drag. The independence of $u_{\unicode[STIX]{x1D70F}}/U_{e}$ and $z_{0}/h$ from the Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ indicates that the three flow configurations are in the fully rough regime. The boundary-layer thickness, $\unicode[STIX]{x1D6FF}$ , shown in table 1 defines the wall-normal location at which the mean velocity is equal to 99 % of the free-stream velocity $U_{e}$ . In the measurement cross-section, for all the configurations, the non-dimensional pressure gradient $K=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D70C}U_{e}^{3})\,\text{d}P/\text{d}x$ along the wind tunnel was found to be below $-2.9\times 10^{-8}$ . The aerodynamic parameters $d$ and $z_{0}$ can be used to classify the roughness flow regime with the model derived by Macdonald et al. (Reference Macdonald, Griffiths and Hall1998) or the data compiled by Grimmond & Oke (Reference Grimmond and Oke1999). The three canopies studied here represent the three near-wall flow regimes as defined by Grimmond & Oke (Reference Grimmond and Oke1999) where the $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ represents isolated wake flow, $\unicode[STIX]{x1D706}_{p}=25\,\%$ represents wake-interference flow and $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ represents skimming flow (see figure 3 of Perret et al. (Reference Perret, Basley, Mathis and Piquet2019)). For further details the reader is referred to Perret et al. (Reference Perret, Basley, Mathis and Piquet2019).

Table 1.

Scaling parameters. The coloured symbol chart will be used in all of the following figures.

Figure 3.

Wall-normal profiles of the (a) mean, (b) variance, (c) skewness and (d) kurtosis of the streamwise velocity component. Vertical solid lines show the wall-normal location of the canopy top $z=h$ for the three roughness configurations (being negligible when normalizing by $\unicode[STIX]{x1D6FF}$ , variation of $(h-d)/\unicode[STIX]{x1D6FF}$ with $Re_{\unicode[STIX]{x1D70F}}$ is not shown).

Figure 3 shows the wall-normal profiles of the main statistical characteristics of the streamwise velocity component including mean velocity, variance, skewness and kurtosis for the six cases shown in table 1. Scaling using the roughness length and displacement height results in a collapse of the mean streamwise velocity component in regard to both canopy geometry and Reynolds number. The remaining statistics show agreement within the outer-layer scaling using the displacement height and boundary-layer thickness. However, both the variance and skewness are influenced by the canopy geometry within the inner layer close to the roughness. Perret et al. (Reference Perret, Basley, Mathis and Piquet2019) conducted detailed scaling analysis for these six cases, but were unable to find a scaling that collapses the variance and skewness close to the wall. One salient feature of the present flow configurations put forward by these authors is the variation of the wall-normal extent of the roughness sublayer as a function of $\unicode[STIX]{x1D706}_{p}$ . While classically defined as the region where the flow statistics are non-homogeneous in the horizontal plane, Squire et al. (Reference Squire, Morill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) recently proposed defining its upper limit $z_{RSL}$ as the lower limit of the inertial region in which the velocity variance follows a logarithmic law. Following this approach and based on the data shown in figure 3(b), Perret et al. (Reference Perret, Basley, Mathis and Piquet2019) found that $z_{RSL}$ varies with the roughness configuration and Reynolds number (table 1). This suggests that the densest canopy configuration prevents the canopy-induced coherent structures from developing in the wall-normal direction. This matches well with the well-recognized picture of the skimming flow regime in which a thin shear layer develops at the canopy top with very limited penetration of the flow within the canopy and is consistent with the recent results of Basley et al. (Reference Basley, Perret and Mathis2019).

4.2 Scale decomposition

In the case of the atmospheric surface layer developing over large roughness elements, the outer and inner peaks in the energy spectrum are rarely separated. The cubical obstacles induce energetic structures with typical frequencies smaller than that of the near-smooth-wall turbulence in a range closer to those attributed to the large-scale structures developing in the logarithmic and outer region. It should also be pointed out that although the outer peak is not clearly visible this does not mean that large-scale influence does not exist, but rather that scale separation is not clear and significant overlap exists between the different coherent structures interacting with each other. This has been shown by Perret et al. (Reference Perret, Basley, Mathis and Piquet2019) and is the reason why the scale-separation method based on a two-point measurement approach is favoured here (Baars, Hutchins and Marusic Reference Baars, Hutchins and Marusic2016b ; Pathikonda & Christensen Reference Pathikonda and Christensen2017). Using the method described in § 2 the large-scale signal, $u_{L}^{+}$ , is extracted from the raw near-wall velocity signal, $u_{NW}^{+}$ , at each of the moving HWA probe wall-normal locations in each of the six cases using a transfer function. The modulus and phase of the transfer function for the moving probe location of $z/h=1.25$ in each of the six cases are shown in figure 4 where it is clear that the modulus and phase of the transfer function depend on the canopy geometry, but not on the Reynolds number. In this section, the focus is on the main statistical characteristics of $u_{L}$ and $u_{S}$ and their contribution to the skewness, which is an indicator of the existence of amplitude modulation (Duvvuri & McKeon Reference Duvvuri and McKeon2015). A thorough analysis of the spectral content of the flow and of its large- and small-scale components has been performed by Perret et al. (Reference Perret, Basley, Mathis and Piquet2019) and Basley et al. (Reference Basley, Perret and Mathis2019) in the same flow configurations as here. These authors demonstrated the co-existence of very large-scale motions (VLSMs), large-scale motions (LSMs) and canopy-generated coherent structures whose characteristics obey different scaling laws. For the sake of conciseness, these results are not recalled here, the reader being referred to these studies.

Figure 4.

(a) Modulus and (b) phase of the transfer kernel $|H_{L}|$ at $z/h=1.25$ for configurations with $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ , 25 % and 44.4 % at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and 49 900. The colour chart is as per table 1 for canopy configurations; solid and dashed lines correspond respectively to the low and the high Reynolds numbers.

Figure 5.

Spectra of $u^{\ast }$ (solid line) and $u_{L}^{+}$ (dashed line) for configurations with $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ , 25 % and 44.4 % at (a) $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and (b) 49 900 at $z/h=2.1$ . Vertical solid lines show the streamwise wavelength corresponding to the obstacle height $\unicode[STIX]{x1D706}=h$ .

Once $u_{L}^{+}$ is extracted using triple decomposition the small-scale signal, $u_{S}^{+}$ , can be computed. (Triple decomposition was first introduced by Hussain (Reference Hussain1983) to decompose the instantaneous velocity field into mean, large-scale and small-scale components.) Finally, $u^{\ast }$ is computed using the method described in § 2. The spectra of the universal and large-scale signal (figure 5) of the six cases show the differences in energy content of the two signals. No significant change occurs in the energy distribution between the different canopies and different Reynolds numbers. Finally, an increase in Reynolds number does not affect the magnitude of energy contained in the universal and large-scale signals. This last point may be tempered by the narrow range of Reynolds number used here, as it has been shown previously that the large-scale content increases as the Reynolds number increases (see Mathis et al. (Reference Mathis, Hutchins and Marusic2009, Reference Mathis, Hutchins and Marusic2011a ) among others). The statistics of the $u^{\ast }$ signal including variance, skewness and kurtosis are compared in figure 6 with the statistics of the raw near-wall velocity signal $u_{NW}^{+}$ , $u_{L}^{+}$ and $u_{S}^{+}$ showing only the $\unicode[STIX]{x1D706}_{p}=25\,\%$ case as an example. In all six cases (not shown here) $u_{S}^{+}$ captures the majority of the variance in the inner layer while the large-scale contribution becomes important only in the outer layer. The skewness is shown to be almost completely captured by $u_{S}^{+}$ with the contribution from $u_{L}^{+}$ close to zero. The kurtosis of the raw signal is shown to be a result of both $u_{L}^{+}$ and $u_{S}^{+}$ with the contribution of $u_{S}^{+}$ increasing with wall-normal distance in the outer layer. Mathis et al. (Reference Mathis, Hutchins and Marusic2011a ) noted that the universal signal is the signal that exists in the absence of the influence of large-scale structures while $u_{S}^{+}$ is the signal that exists in the absence of any superposition. Therefore a comparison between the $u_{S}^{+}$ and $u^{\ast }$ signals provides insight into the influence of the amplitude modulation on the $u_{S}^{+}$ structures. The presence of amplitude modulation causes no influence in the variance or kurtosis as $u_{S}^{+}$ and $u^{\ast }$ have similar profiles. In the absence of amplitude modulation the magnitude of skewness of $u^{\ast }$ is significantly lower throughout the boundary layer. These trends are true for each of the six configurations except in the case of the skewness of $u^{\ast }$ . The wall-normal location at which the profile of the skewness of $u^{\ast }$ crosses the profile of the skewness of $u_{L}^{+}$ changes depending on the roughness configuration. In roughness configurations with $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ or 25 % the $u^{\ast }$ profile crosses the $u_{L}^{+}$ profile at a wall-normal distance of approximately $(z-d)/\unicode[STIX]{x1D6FF}=0.09$ while in roughness configurations with $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ this crossing occurs at $(z-d)/\unicode[STIX]{x1D6FF}=0.05$ . As $u^{\ast }$ is the signal that exists in the absence of influence of the large scales it should correspond to a signal from a low Reynolds number flow where large-scale influence is weak. The decrease of contribution of $u^{\ast }$ to the skewness in the configuration with $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ is a result of increased large-scale activity. No significant differences are found between cases when varying Reynolds number as both Reynolds numbers are sufficient to generate significant large-scale activity and differ by less than a factor of two.

Figure 6.

Comparison of $u^{\ast }$ , $u_{L}^{+}$ , $u_{S}^{+}$ and $u_{NW}^{+}$ statistics (a) variance, (b) skewness and (c) kurtosis for configuration with $\unicode[STIX]{x1D706}_{p}=25\,\%$ at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ .

4.3 Influence of canopy geometry and Reynolds number

Figure 7.

Triple decomposition of the skewness of the streamwise velocity component including (a)  $\overline{u_{S}^{+3}}$ , (b)  $\overline{u_{L}^{+3}}$ , (c)  $3\overline{u_{L}^{+}u_{S}^{+2}}$ and (d)  $3\overline{u_{L}^{+2}u_{S}^{+}}$ .

Skewness decomposition as shown in (4.1) has been used to investigate the nonlinear interactions between large- and small-scale structures in turbulent flows (Blackman & Perret Reference Blackman and Perret2016).

(4.1) $$\begin{eqnarray}\displaystyle \overline{u^{+3}}=\overline{u_{S}^{+3}}+\overline{u_{L}^{+3}}+3\overline{u_{L}^{+}u_{S}^{+2}}+3\overline{u_{L}^{+2}u_{S}^{+}}. & & \displaystyle\end{eqnarray}$$

Here it is used to determine the influence of the canopy geometry and Reynolds number on these nonlinear interactions. Figure 7 shows the small-scale skewness, large-scale skewness and two scale-interaction terms. The influence of the canopy geometry is particularly apparent in the contribution of the small scales close to the canopy where there is a clear separation between the cases (figure 7 a). This separation is a result of the distinct canopy flow regimes in each of the cases. As mentioned, within the skimming flow regime ( $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ ) there is a thinner shear layer (or roughness sublayer) whereas in the isolated wake ( $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ ) and wake-interference ( $\unicode[STIX]{x1D706}_{p}=25\,\%$ ) flow regimes the shear layer wall-normal extent is larger, increasing the importance of the small scales. Away from the canopy, in the outer layer, the influence of the canopy geometry or flow regime is not significant. Moreover, throughout the boundary layer the canopy geometry does not significantly influence the large-scale contribution or the contribution of the nonlinear term $\overline{u_{L}^{+2}u_{S}^{+}}$ , which represents the influence of the small scales on the large scales. However, an increase in Reynolds number increases the contribution of this nonlinear term within the outer layer (figure 7 d). Finally, the nonlinear term $\overline{u_{L}^{+}u_{S}^{+2}}$ has been shown to represent the amplitude modulation (Mathis et al. Reference Mathis, Marusic, Hutchins and Sreenivasan2011c ; Duvvuri & McKeon Reference Duvvuri and McKeon2015). Here, it is clear that although the canopies with $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ and 25 % display similar amplitude modulation, the amplitude modulation of the canopy with $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ is significantly modified at both Reynolds numbers (figure 7 c). Throughout the boundary layer, except close to the canopy, the amplitude modulation is weaker in the $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ canopy. As mentioned in § 4.1, this flow configuration has the finest roughness sublayer. This is confirmed if one considers the wall-normal location of the zero crossing of the skewness of the streamwise velocity component as the upper limit of the roughness sublayer (figure 7 a). It is also where the small-scale component $u_{S}$ is the least energetic relative to the large scales (Perret et al. Reference Perret, Basley, Mathis and Piquet2019). In this flow configuration, the small scales are less energetic and more confined to near the canopy top, the amplitude modulation imprint is therefore weaker than the two other cases.

The coefficients $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ of the predictive model computed for each of the cases listed in table 1 using the method in § 2 are shown in figure 8 along with the coefficient $\unicode[STIX]{x1D6E4}$ . The roughness configuration affects the superposition coefficient, $\unicode[STIX]{x1D6FC}$ , close to the roughness in the inner layer where differences in the flow regimes are important. However, in the outer layer the superposition is consistent in all roughness configurations. In the outer layer the influence of the roughness flow regime disappears and the large-scale structures become similar, thereby resulting in similar superposition. The amplitude modulation coefficient, $\unicode[STIX]{x1D6FD}$ , depends on roughness configuration in the inner layer, but in the case of the roughness configuration with $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ the amplitude modulation is decreased both in the inner layer and the outer layer. This is consistent with the nonlinear term $\overline{u_{L}^{+}u_{S}^{+2}}$ , which shows lower magnitudes of amplitude modulation in the $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ configuration. As discussed, the characteristics of the shear layer in the skimming flow regime change the characteristics of the small-scale structures and their interactions with the large-scale structures in the outer layer above. The dependence of the superposition and amplitude modulation on the roughness configuration close to the roughness is a result of changes to the dynamics of the shear layers that develop at the top of the roughness elements in the different flow regimes. Within the skimming flow regime the shear layer does not penetrate the roughness elements resulting in a thin, but strong shear layer, whereas the spacing between roughness elements in the isolated and wake-interference regimes result in a shear layer that penetrates the canopy layer increasing the vertical transfer of momentum of small-scale structures in this region (Basley et al. Reference Basley, Perret and Mathis2019). The shear layer in the wake-interference flow regime also experiences a strong flapping phenomenon that promotes the transfer of momentum between the canopy layer (small scales) and outer layer (large scales). The results show that an increase in Reynolds number does not increase the superposition or the amplitude modulation, in contradiction to Mathis et al. (Reference Mathis, Hutchins and Marusic2011a ), who found that increased Reynolds number increases the large-scale activity in the outer layer thereby increasing the amplitude modulation. These results should be tempered by the fact that the Reynolds numbers used here are not sufficiently separated to significantly affect the large scales and therefore the scale interactions.

Figure 8.

Predictive model coefficients (a)  $\unicode[STIX]{x1D6FC}$ , (b)  $\unicode[STIX]{x1D6FD}$ and (c)  $\unicode[STIX]{x1D6E4}$ for configurations with $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ , 25 % and 44.4 % at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and 49 900.

Figure 9.

Comparison of $u^{\ast }$ statistics (a) variance, (b) skewness and (c) kurtosis, and $u_{L}^{+}$ statistics (d) variance, (e) skewness and (f) kurtosis for configurations with $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ , 25 % and 44.4 % at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and 49 900.

The variance, skewness and kurtosis of the universal signal, $u^{\ast }$ , and $u_{L}^{+}$ in each of the six cases are presented in figure 9. The influence of the roughness configuration can be seen in the profiles of variance and skewness in the inner layer close to the roughness, whereas this influence becomes negligible in the profile of kurtosis. The changes in variance and skewness are a result of changes to the small-scale structures produced by the roughness. Small scales in the wake-interference flow regime have larger magnitudes of skewness and smaller magnitudes of turbulence intensity compared to the skimming flow regime. Although there is an increase in magnitude of variance of the large-scale structures in the 44.4 % configuration these changes are not limited to the region close to the roughness as in the $u^{\ast }$ profile (figure 9 d). Excluding this slight increase in the variance the similarity of the other $u_{L}^{+}$ profiles suggests that the very-large-scale structures in each of the cases have similar characteristics. Using the outer-layer scaling a change in Reynolds number does not affect the statistics of the universal or large-scale signals. These results have shown that the model coefficients and universal signal are significantly influenced by the canopy geometry or canopy flow regime while the large-scale structures have been shown to be similar in each of the cases. Therefore, the universal signal is not universal for all rough-wall boundary layers and the predictive model must be calibrated for each of the roughness flow regimes.

4.4 Prediction and validation

Model coefficients provided by the calibration allow for the prediction of a statistically representative signal, $u_{p}^{+}$ , that hypothetically can be reconstructed at any Reynolds number, where the only required input is the large-scale reference signal, $u_{L}^{+}$ . In this section, a series of tests are performed in order to assess whether the above assumption, which works well in smooth-wall boundary layer, still holds in an atmospheric boundary layer over an urban canopy. To do so, a series of tests is performed to validate and assess the capabilities of the model, in which canopy configuration and Reynolds numbers are mixed, as seen in table 2.

Figure 10.

Comparison of $u_{NW}^{+}$ and $u_{p}^{+}$ statistics (a) variance, (b) skewness and (c) kurtosis for $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=49\,900$ where $u_{p}^{+}$ is determined using model coefficients calibrated at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and $u_{L}^{+}$ at $Re_{\unicode[STIX]{x1D70F}}=49\,900$ (Test 1).

Table 2.

Characteristics of the input (calibration parameters and large-scale signal) and output of the predictive model used for testing and validating the prediction capabilities of the model.

The capabilities of the predictive model, which has been calibrated for $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=32\,400$ , is first tested by predicting the near-canopy signal for the same plan density at the higher Reynolds number $Re_{\unicode[STIX]{x1D70F}}=49\,900$ (Test 1). To do this a large-scale reference signal measured at $Re_{\unicode[STIX]{x1D70F}}=49\,900$ is used to run the predictive model where the universal signal and model coefficients were determined from a calibration at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ . Additionally, the large-scale reference signal must be interpolated onto the non-dimensional time scale $t^{+}$ of the universal signal so that the time sampling of both signals is consistent. In addition, the two signals must have the same length, by clipping the longest of the two. Figure 10 shows the characteristics of the predicted signal (blue stars) compared with the characteristics of the measured near-canopy signal (black circles) up to the fourth order. Although there is some slight discrepancy between the prediction and the near-canopy signal, it is clear that the predictive model calibrated at a lower Reynolds number is able to reproduce the characteristics of the near-canopy signal at a higher Reynolds number. Finally, the spectra of the predicted signal are similar to the spectra of the measured signal as shown in figure 11. There is a slight shift in the wavelength of the spectra of the predicted signal that becomes more significant closer to the roughness. This might be due to the application of Taylor’s hypothesis which has questionable suitability close to the roughness. However, the similarity of the spectra further validates the model and suggests that the model can be calibrated at any arbitrary Reynolds number.

Figure 11.

Spectra of $u_{NW}^{+}$ and $u_{p}^{+}$ at (a)  $z/h=1.25$ , (b)  $z/h=2.1$ and (c)  $z/h=3.2$ for $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=49\,900$ where $u_{p}^{+}$ is determined using model coefficients calibrated at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and $u_{L}^{+}$ at $Re_{\unicode[STIX]{x1D70F}}=49\,900$ (Test 1).

Another crucial question in making a predictive model for urban canopy flow is, to what extent is the calibration dependent on the plan area packing density at which the calibration is performed? Indeed, the previous section clearly evidenced that the universal signal and model coefficients are canopy dependent. In an attempt to shed light on this, the near-canopy signal is predicted for the $\unicode[STIX]{x1D706}_{p}=25\,\%$ at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ configuration using large-scale reference signals from the datasets of the $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ and 44.4 % configurations at the same Reynolds number (Test 2). To perform these predictions the calibrated model for the $\unicode[STIX]{x1D706}_{p}=25\,\%$ configuration is used along with a large-scale reference signal from a configuration with a different $\unicode[STIX]{x1D706}_{p}$ . As above, the large-scale reference signal from either the $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ or 44.4 % configuration is interpolated onto the non-dimensional time scale of the universal signal calibrated for the $\unicode[STIX]{x1D706}_{p}=25\,\%$ configuration. Figure 12 shows the characteristics of the predicted signal using a large-scale reference from the $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ configuration (blue triangles), $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ configuration (red squares) and the measured near-canopy signal of the $\unicode[STIX]{x1D706}_{p}=25\,\%$ configuration (black circles). The spectra of the predicted signals and measured near-canopy signal are shown at several wall-normal locations in figure 13. There is excellent agreement between the predicted signals and the near-canopy signal for the statistics up to the fourth order and the spectra in both prediction cases.

Figure 12.

Comparison of $u_{NW}^{+}$ and $u_{p}^{+}$ statistics (a) variance, (b) skewness and (c) kurtosis for $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=32\,400$ where $u_{p}^{+}$ is determined using model coefficients calibrated for $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $u_{L}^{+}$ from $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ or 44.4 % (Test 2).

Figure 13.

Spectra of $u_{NW}^{+}$ and $u_{p}^{+}$ at (a) $z/h=1.25$ , (b)  $z/h=2.1$ and (c)  $z/h=3.2$ for $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=32\,400$ where $u_{p}^{+}$ is determined using model coefficients calibrated for $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $u_{L}^{+}$ from $\unicode[STIX]{x1D706}_{p}=6.25\,\%$ or 44.4 % (Test 2).

To determine the error associated with these predictions the near-canopy signal was predicted within each canopy using a large-scale reference signal from each of the other canopy configurations for the lowest wall-normal location of $z/h=1.25$ as a test. The error for the statistics up to the fourth order was computed for each prediction using (4.2) where $\unicode[STIX]{x1D719}_{m}$ and $\unicode[STIX]{x1D719}_{p}$ are any statistics of the original measured and predicted signals, respectively.

(4.2) $$\begin{eqnarray}\displaystyle \text{Error}=(\unicode[STIX]{x1D719}_{m}-\unicode[STIX]{x1D719}_{p})/\unicode[STIX]{x1D719}_{m}. & & \displaystyle\end{eqnarray}$$

Figure 14 shows the error averaged over the two predictions for each canopy configuration. The error is less than 3 % for all statistics and in all canopies with the largest error of 3 % for the kurtosis of the $\unicode[STIX]{x1D706}_{p}=25\,\%$ configuration. This confirms that a calibrated predictive model can be used to predict the near-canopy signal using a large-scale reference measured in any other canopy configuration.

Figure 14.

Error of $u_{p}^{+}$ statistics variance, skewness and kurtosis where $u_{p}^{+}$ is determined using model coefficients calibrated at a certain $\unicode[STIX]{x1D706}_{p}$ and $u_{L}^{+}$ at a different $\unicode[STIX]{x1D706}_{p}$ both at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ .

The final validation of the model combines both the Reynolds number and $\unicode[STIX]{x1D706}_{p}$ validation by predicting a near-canopy signal within the $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ configuration at $Re_{\unicode[STIX]{x1D70F}}=49\,900$ using the calibrated model at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and a large-scale reference signal from the $\unicode[STIX]{x1D706}_{p}=25\,\%$ configuration at $Re_{\unicode[STIX]{x1D70F}}=49\,900$ (Test 3). As in the previous validation the model is able to accurately reproduce the spectra of the near-canopy signal as well as the statistics up to the fourth order (figure 15). The model is able to accurately reproduce these statistics because, as has been shown here, the characteristics of the large scales in each of the canopies are similar. However, the differences in the characteristics of the universal signal and the predictive model coefficients prevent the application of a calibrated model at one $\unicode[STIX]{x1D706}_{p}$ to a prediction at another $\unicode[STIX]{x1D706}_{p}$ . The model must be calibrated using measurements from a canopy with the same configuration as the targeted one.

Figure 15.

Comparison of $u_{NW}^{+}$ and $u_{p}^{+}$ statistics (a) variance, (b) skewness and (c) kurtosis and (d) spectra at $z/h=1.5$ for $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=49\,900$ where $u_{p}^{+}$ is determined using model coefficients calibrated for $\unicode[STIX]{x1D706}_{p}=44.4\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=32\,400$ and $u_{L}^{+}$ from $\unicode[STIX]{x1D706}_{p}=25\,\%$ and $Re_{\unicode[STIX]{x1D70F}}=49\,900$ (Test 3).

In the smooth wall special attention has been paid to conserving the phase between the universal signal and large-scale signal used to run the predictive model (Mathis et al. Reference Mathis, Hutchins and Marusic2011a ). In these cases the large-scale reference signal used to run the predictive model was adjusted to retain the Fourier phase information of the large-scale signal used to build the universal signal. The phase information of the original large-scale signal is extracted using a Fourier transform and applied to the new large-scale reference signal. This process essentially re-synchronizes the new large-scale reference with the universal signal, $u^{\ast }$ (Mathis et al. Reference Mathis, Marusic, Hutchins and Sreenivasan2011c ). Here, this process was applied before performing the predictions detailed above. To determine influence of the phase shift on a prediction a test is performed using the large-scale reference signal used to build the predictive model. This signal is shifted out of phase with the universal signal and a prediction of the statistics made at each time shift (figure 16). As the phase shift increases the estimation of the variance, skewness and kurtosis worsen until they reach a plateau. The effect of the phase shift increases with increasing order of the statistic with the kurtosis showing the largest discrepancy. This suggests that conserving the phase information of the large-scale signal used to calibrate the model is important to the prediction.

Figure 16.

(a) Variance, (b) skewness and (c) kurtosis of $u_{p}^{+}$ and $u_{NW}^{+}$ for configuration with $\unicode[STIX]{x1D706}_{p}=25\,\%$ at $Re_{\unicode[STIX]{x1D70F}}=32\,400$ using phase shifted large-scale reference signal at $z/h=2.1$ ( $(z-d)/\unicode[STIX]{x1D6FF}=0.066$ ).