**5.1 Mode energy contribution**

**Figure 11** shows the energy contributions of the dominant POD modes and cumulative energy of the first three modes. Under CWP, the kinetic energy contribution of the first two POD modes was the most prominent, and both exceeded 15%, while that of the third-order mode sharply decreased to below 10%. The largest cumulative energy proportion of the first three modes (46.71%) appeared at the middle plane z = 1/2H, where the coherent structure was less affected by the free end and the ground effect. This observation is consistent with the finding reported in [20]. Affected by the twisted flow, the energy proportion taken by main POD modes was relatively uniform, especially at the mid-height of the building, and for increasing height, the cumulative energy percentages of the first three main modes gradually decreased from 34.95% to 28.60%.

#### **5.2 POD flow pattern**

The first three POD mode patterns of wake velocity field into horizontal plane 1/2 H under CWP and TWP30 are depicted in **Figures 12** and **13** respectively. For CWP,

**Figure 11.** *Energy contributions of dominant POD modes and cumulative energy of the first three modes. (a) z1=1/6 H. (b) z2=1/2 H. (c) z3=5/6 H.*

*Mode Interpretation of Aerodynamic Characteristics of Tall Buildings Subject to Twisted… DOI: http://dx.doi.org/10.5772/intechopen.103757*

**Figure 12.**

*The first three POD mode patterns of the wake velocity field at horizontal plane of 1/2 H under CWP (a) contour of u component; (b) contour of v component; (c) streamline pattern.*

the contour of the u component was asymmetrically distributed while that of the v component was a symmetry to the centerline. The first two POD modes were similar (i.e., streamline pattern) but had an opposite sign, and as a result, can be recognized as a pair of conjugate modes. The third mode was found to have a series of vortices arrayed along the streamwise direction with alternating rotation direction. These observations confirm the findings reported in the previous POD investigation on the wake flow of a cylinder [27, 28].

To further disclose the temporal and spectral characteristics of the main underlying modes in the velocity fields, the PSD of the first three POD mode coefficients at the mid-height plane of the building were analyzed and compared for CWP and TWP30 as shown in **Figure 14**. As can be observed, there were several peaks for the first two POD modes while the most pronounced one appeared at f11 = 0.04 and f12 = 0.11(see blue arrow), these manifests that although coherent structures were composed of different sizes of vortices, the most dominant types were the LEVS and TEVS. The third mode had the main frequency of 0.11, and more importantly, its modal shape was regularly distributed and closely resembled the Karman vortex. Subsequently, it is reasonable to speculate that the third mode is highly related to the TEVS phenomenon.

**Figure 13.**

*The first three POD mode patterns of the wake velocity field at horizontal plane of 1/2 H under TWP30 (a) contour of u component; (b) contour of v component; (c) streamline pattern.*

Under twisted flow, as shown in **Figure 13**, the mode topology and configuration were both largely modified, moreover, the vortex shedding was no longer aligned with the centerline but deviated towards the wind twisted direction. The vortex structures of the first two modes were very similar along and nearly had the same frequency of f3 = 0.048 (marked by green arrow in **Figure 13(c)**). This finding indicates that a pair of conjugate modes were well captured, with a high possibility to be controlled by the AEVS coherent structure. The twisted flow made the third mode shape less regular than that of CWP, and no distinctive peak was noted in this case.

To fully reveal the POD mode from a three-dimensional perspective, the mode results extracted from the velocity field at another horizontal plane z = 1/6 H under CWP and TWP30 are provided in **Figures 15** and **16**. It was found that the mode results at the plane near to the ground level z = 1/6 H basically shared a similar symmetrical distribution pattern to that of plane z = 1/6 H for the case of CWP and TWP30, despite some discrepancies (e.g., mode shape, magnitude and streamline) still being observed between these two horizontal planes due to the existence of the ground effect. An interesting observation is that because the twisted wind had a maximum twisted angle near the ground level, the twisted degree of the wake

*Mode Interpretation of Aerodynamic Characteristics of Tall Buildings Subject to Twisted… DOI: http://dx.doi.org/10.5772/intechopen.103757*

streamline pattern was much larger in the horizontal plane of z = 1/6 H than that of plane of z = 1/2 H, as indicated by the blue arrow in **Figure 13**(a3) and 16(a3).

As can be seen in **Figures 15** and **16**, the distribution of mode pattern at the vertical plane x = 0.12 m exhibited symmetrical properties along the centerline y = 0 for the case of CWP, possibly related to the "quadrupole" wake pattern. In the case of TWP30, these POD modes deviated towards the approaching wind twisting direction, namely, the distribution was concentrated in the positive direction of the y-axis. It should be noted that in comparison with the CWP case, the distribution range and amplitude of the POD mode of w velocity were significantly amplified for the case of TWP30. This indicated that the interaction between the vertical velocity components at different heights in the flow field was more intense, which could explain the increase of the aerodynamic correlation of the local wind loads in the vertical direction.

#### **5.3 Correlation of modal coefficients**

**Figure 17** shows the correlation of the first three POD mode coefficients at two horizontal planes (z1 = 1/6H and z2 = 1/2H) for CWP and TWP30.The x- and y-axis represent the modal coefficients of ith mode (i = 1,2 and 3) at the bottom height plane and mid-height plane respectively. For CWP, the phase-plane trajectory of the first three modes all exhibit a circular distribution (see the red dash line), thus, the

**Figure 15.**

*The first three POD mode patterns of the wake velocity field at horizontal plane of 1/6 H under CWP (a) contour of u component; (b) contour of v component; (c) streamline pattern.*

correlation of the same mode at different heights of the building is very close to zero, indicating that the coherent structure at these two locations is relatively independent. Conversely, affected by twisted flow, the distribution of the modal coefficients exhibits an inclined elliptic shape with a largely amplified correlation coefficient (see the black dash line). For example, the first predominant mode at these two different horizontal planes has the largest correlation coefficient *Cor* = 0.50, indicating that the predominant coherent structure simultaneously controls the flow motion at the bottom and middle planes. Notably, the correlation coefficients of the first mode (*Cor* = 0.50) and the second mode (*Cor* = 0.45) are very close to each other, demonstrating the existence of conjunction modes at the middle height of the building z2 = 1/ 2H, as demonstrated in **Figure 17**(a1) and (b1).
