**5. Energy Spectra Analysis**

To provide a further insight into the interaction between the turbulent characteristics of the vortex system and developing scour hole, this section presents an energy spectra analysis of the time series of velocity fluctuations at four representative points. Graftieaux et al. [20] suggested that the velocity fluctuations are their maximum near the mean vortex center within the region of a well-established vortex. For this reason, the vortex centers (marked as VC1, VC2, and VC3 in the right column of Figure 5) were selected to examine the turbulent energy cascade associated with the vortex system. Additionally, a characteristic point located in the shear layer in the vicinity of the propeller (marked as TV in Figure 5e1) was also selected for the sake of comparison. The TV position is located within the shedding path of the tip vortex originating from the propeller blades, as observed by Felli et al. [24] and Wei and Chiew [11], thus serving as a reference for the flow nature that is exclusively associated with the free jet diffusion.

Figures 7 and 8, respectively, illustrate the energy spectra distributions of the velocity fluctuations at point TV and the three vortex centers. The left column (*S*u) and right column (*S*w) in the figures denote the spectral energy density of the horizontal and vertical velocity components, respectively. Since the shear layer structure close to the propeller possesses a steady attribute throughout the entire scouring process (see Figure 5), Figure 7 only presents the power spectra of point TV at the asymptotic state (*t* = 24 h), whereas Figure 8 includes the power spectra at the vortex centers associated with the evolving vortex system at different time intervals of *t* = 0, 0.5, 2, 12, 24 h. Figure 7 reveals that in the near wake region of the propeller, both *S*u and *S*w exhibit two obvious spectra peaks superimposed on a broadband spectrum. The former is conjectured to be associated with the dominant frequencies of the tip vortex shedding originating from the rotating propeller blade. The latter reveals a less steep slope than the well-known Kolmogorov energy cascade slope of −5/3 as the turbulent energy is enhanced at small scale eddies (larger frequencies). One may, therefore, expect a local unbalance between turbulence production and dissipation close to the propeller, which is not surprising since the turbulence there is neither isotropic nor fully developed. Moreover, a side-by-side comparison between Figure 7a,b show that the turbulent energy associated with the horizontal velocity fluctuation appears to be greater than its vertical counterpart for the entire frequency band, confirming the dominant role of the axial (horizontal) velocity within a propeller jet.

**Figure 7.** Power spectra of horizontal and vertical velocity fluctuations at a typical point of tip vortex shedding (*x* = 0.5*D*p, *z* =0.7*<sup>D</sup>*p) at *X*w = <sup>2</sup>*D*p.

**Figure 8.** Power spectra of horizontal and vertical velocity fluctuations at vortex core centers of VC1, VC2, and VC3 at *X*w = <sup>2</sup>*D*p: (**a1**,**a2**) *t* = 0 h; (**b1**,**b2**) *t* = 0.5 h; (**c1**,**c2**) *t* = 2 h; (**d1**,**d2**) *t* = 12 h; (**e1**,**e2**) *t* = 24 h.

On the other hand, away from the near wake region of the propeller, Figure 8 shows that no perceptible peak frequency can be observed from the spectra distributions associated with the vortex centers, indicating that the periodicity emanated from the propeller rotation has been attenuated. In the meantime, for all three vortices, Figure 8 reveals that both *S*u and *S*w exhibit a comparable power spectrum distribution which obeys the −5/3 law within the inertial range defined by Kolmogorov [25], implying a fully developed turbulence and an asymptotic between production and dissipation. This means that unlike the near wake region (e.g., point TV), where the small-scale eddies hold a significant amount of turbulent energy, it is the large-scale turbulent eddies that matter in governing the formation of the coherent structures, i.e., the well-established vortices (V1, V2, and V3). Furthermore, it can be observed that, compared to V1 (in red line) and V2 (in blue line), the energy spectra of V3 (in black line) gradually decreases with the development of the scour hole. Given that V3 is directly responsible for enlarging the scour hole as already pointed out, this observation implies a direct coupling between the vortex evolution and the developing scour hole, that is, an energy transfer from the turbulent energy dissipation to the scouring action. On the other hand, V1, which initiated the onset of scour process, reveals an opposite scenario as its energy spectra in small-scales appears to be enhanced and exceed its two counterparts during *t* = 12–24 h. One may surmise that under the confinement e ffect exerted by the two larger vortices (V2 and V3) on both sides, V1 is subject to a more intense velocity fluctuation and thus has higher turbulent kinetic energy, albeit in a smaller size.

#### **6. Comparison of Near-Bed Flow Characteristics**

The near-bed flow characteristics are always of grea<sup>t</sup> importance in understanding the interaction between the flow and scouring bed. According to the adopted resolution of the PIV measurement (as shown in Table 1), this study was able to extract the near-bed data at a distance of 2.7 mm (i.e., the location of the nearest data point close to the bed boundary) above the scoured bed boundary. The so-obtained results are plotted in Figure 9, in which AKE is the averaged kinetic energy (= 1 2 (*u*<sup>2</sup> + *w*<sup>2</sup>)/*U*<sup>2</sup> o ); TKE is the turbulent kinetic energy (= 1 2 (*u*<sup>2</sup> + *w*<sup>2</sup>)/*U*<sup>2</sup> o ); and RSS is the Reynolds shear stress (= −*uw*/*U*<sup>2</sup> o), in which *u* , *w* are the fluctuating velocity components in the horizontal and vertical directions, respectively. It should be stated that these three parameters only consider the in-plane components of the three-dimensional jet flow due to the nature of the planer PIV measurement. For the sake of clarity, the scour bed profile is also superimposed in the figure whose scale follows the secondary ordinate axis on the right.

At the onset of the scouring action, Figure 9a reveals a dominant TKE (blue filled circle) prevailing along the initial flatbed with a local maximum value between *<sup>x</sup>*/*D*p = 1–1.5. This location is reasonably correlated with the core of the near-bed vortex (V3) as shown in Figure 5a2. As the scour hole develops, the overall TKE profile undergoes a significant reduction (see Figure 9b–d), which reflects a weakening of the scouring capacity associated with the scour-driving vortex. This inference is in conformity with the inference that one may deduce from the turbulent energy spectra plots. The local increase in TKE (around *<sup>x</sup>*/*D*p = 2 at the quay wall), on the other hand, is likely attributed to the presence of the corner vortex that formed at the junction between the scour bed and the quay wall. At the asymptotic state (*t* = 24 h), when the scour process almost ceases altogether, the near-bed TKE distribution drops to approximately zero (see Figure 9e).

Conversely, the development of the near-bed AKE distribution (black filled square) reveals an opposite trend to that of its turbulent counterpart. Initially, the flatbed acts as a blockage and restrict jet expansion; this constraint facilitates an energy transfer from the mean flow to the turbulent flow field, thus resulting in a relatively small AKE value along the initial bed (see Figure 9a). Thereafter, during *t* = 0.5–2 h, Figure 9b,c show that the AKE reveals a local increase a round *<sup>x</sup>*/*D*p = 1–1.5, which is probably caused by the growing near-bed vortex (V3). Entering the stabilizing phase (*t* = 12–24 h), the larger scour hole seems to be able to stabilize the well-established near-bed vortex, which, in turn, conserves the mean flow energy as the AKE is found to exceed its TKE counterpart (see Figure 9d,e). In other words, being di fferent from the initial state of the scouring process, less energy is extracted

from the mean flow to the turbulence during the later scouring phases, which eventually retards the scouring process.

**Figure 9.** Comparison of AKE, TKE, RSS distributions along the developing scour bed at *X*w = <sup>2</sup>*D*p: (**a**) *t* = 0 h; (**b**) *t* = 0.5 h; (**c**) *t* = 2 h; (**d**) *t* = 12 h; (**e**) *t* = 24 h.

As for the RSS profile, Figure 9a,b show that during the initial scouring stage the near-bed RSS exhibits a negative value due to the reverse flow associated with the clockwise vortex (V3). As the scour evolves, Figure 9c–e reveal that the RSS distribution gradually decreases to a near-zero value. On the whole, in contrast to the distributions of AKE and TKE, the RSS exhibits a relatively insignificant change during the scouring process, which may indicate its less important role in driving the scour development.
