3.1 Mean flow velocity during boulder sinking
Figure 2 shows the mean flow velocity contour maps in an x-z plane situated along the symmetry of the boulder. Both longitudinal and vertical components (U and W) of the mean velocity are used to assess how the mean flow fields evolve as a boulder vertically sink during the local bed erosion process. At a very early moment (t = 5min), the boulder nearly remained the initial position as the local bed erosion was weak (Fig. 2a, d). The contour lines of the longitudinal velocity beneath the boulder top plane transformed more vertically when approaching the boulder. The vertical distribution of contour lines indicates a deceleration of primary flow (or momentum) due to the boulder blocking. Over the boulder top plane, a concentrated high velocity zone appears, suggesting the dominant mass and momentum corridor in the local region around a boulder. Despite incomplete visualization, we infer that this concentration zone can extend downstream to a position more than one diameter of the boulder. In the wake of (behind) the boulder, a stagnate zone of the same order of boulder diameter indicated by extremely small or even negative longitudinal velocity is generated, suggesting deposition or long retention of passive scatters such as sediments and contaminants. The small velocity zone gradually decays when leaving the boulder (in the far wake).
The contour map of the vertical velocity well corresponds to its primary counterpart. In the lower part, a negative zone is formed, suggesting downflow parallel to the front face of the boulder. This downflow structure well accounts for the bed scour in front of the boulder due to the evolution of the mean flow fields. Similar hydrodynamics and associated bed scour have been reported for scour around bridge pilers (Dey and Raikar, 2007). A positive vertical velocity zone is formed above the downflow zone, fitting the surface of the boulder. This upflow zone squeezes the upper flow, well accounting for the acceleration of the primary flow over the boulder. This is similar to the edge flow dynamics when flow approaches a vegetation canopy (Moltchanov et al, 2015). At the same level, a downflow (negative velocity) zone exists over the downstream part of the boulder, which essentially suggests the flow separation off a boulder body. The near-wake region is also stagnant relatively, while the far wake is dominated by the negative vertical velocity, together with the near-boulder-surface negative velocity suggesting the boundary of the vertical separation around the boulder.
As the bed scouring continues, the mean flow fields largely adjust with the gradual sinking of the boulder(Fig. 2b,e). After t = 4hours, the boulder seems to sink below the original bed level by half a diameter due to the local bed scour, where the mean flow fields (U and W) behave quite similar regarding the spatial distribution pattern but still different regarding values. Also, it can be observed that the boulder is slightly displaced downstream by nearly half a diameter. The contour lines of the longitudinal velocity tend to be less distorted around the boulder, particularly for the upper open layer, suggesting less influential on the entire mean flow field. However, it can be observed that the quasi-horizontal shear layer above the boulder top level in the near-wake region becomes skewer (contour lines becoming more inclined) and slightly shift upper relative to the bounder position than that at the early moment, which will be well indicated by the turbulence characteristics later on.
In correspondence, the downflow in front of the boulder tends to expand vertically and horizontally relative to the boulder scale (note that the flow field below the original bed level cannot be measured). However, the concentrated upflow zone in the upstream part of the boulder tends to shrunk. In the wake of the boulder, a positive zone emerges due to the sink of the boulder indicating upflow, which substantially differs from the downflow at the very early momentum. This hydrodynamic shift might be attributed to the strength development of the horse-shoe vortices over a distorted bed (Koken and Constantinescu, 2008). The above change regarding the hydrodynamic features are further enhanced after t = 16hours despite the boulder being relatively stabilized spatially (Fig. 2c,f).
With the continuous subsidence of boulders, the exposure height of boulders decreases. As shown in the vertical velocity distribution around boulders at different times in Fig. 3, the influence range of boulders on the vertical velocity distribution is roughly the exposure height of boulders. The deformation amplitude of the vertical velocity distribution in the boulder near the wake decreases with decreasing exposure height and then gradually recovers to a near logarithmic distribution. However, due to the exposure height of boulders and the topography of riverbed erosion and deposition, the vertical velocity distributions at x/D = 2.5 are slightly deformed near the bed, which is close to the traditional logarithmic distribution. This is because the enhanced horse-shoe vortices over the deformed bed (Koken and Constantinescu, 2008) enhances the mixing extent in the wake and shortens the separation length (dicated by the comparison of velocity profiles between t = 10min and 24hours at x/D = 1, seen in Fig. 3-a,c).
3.2 Mean flow velocity under variable submergence scenarios at the final moment
Figure 4 visualizes the mean flow velocity contour maps in an x-z plane situated along the symmetry of the boulder for different submergence scenarios. In this section, for the experimental set-ups, three flow stages, corresponding to three submergence ratios, were operated to implement the boulder-bed scouring experiments. However, the local bed scour around the boulder seems inconsistent as the submergence ratio consistently changed, which led to boulder sinking of different degrees. For example, the height that is blow the bed level for the boulder for Run 2 is smaller than that for other runs. This is mainly because when a scour hole was formed upstream, the boulder was likely to fall into the hole due to gravity. However, the boulder for Run 2 was nearly stabilized at this original position by accidence, and its sink is only caused by sediment erosion right below the boulder. However, this accidence would not limit the assessment of the mean flow fields under different submergence ratio. At equilibrium (t = 24hours), the boulders were still under different submergence conditions. Therefore, this section aims to assess the effect of boulder submergence on the mean flow fields. For all runs of different submergence ratio, h/D = 1 has the greatest influence on boulder exposure (Fig. 4b,e). The extreme value of the flow velocity at different submergence degrees is located near the bottom of the downstream of the boulder, which is consistent with the above discussion.
3.3 Turbulent kinetic energy
Figure 5a-c shows the contour of the turbulence kinetic energy (TKE) at the centerline of the boulder at different time moments during bed scouring. Again, because scouring only occurs near the boulder and the bed incision is negligible in the far-region of the boulder, the region above the original bed is illuminated. Similar to the evolution of the mean flow velocity, the TKE contour adjusted as the bed around the boulder eroded locally and the boulder sank. At the earlier moment (t = 10min) of the scouring when the boulder was slightly submerged by the bed topography, an elongated high-magnitude core of TKE was formed in the near wake and near the boulder top(Fig. 5a). Below the elongated core, the TKE was diffused towards the bed. The TKE pattern suggested that the centerline-plane turbulence is mainly dominated by the vertical dimension. As the scouring continued (t = 3hours), the boulder sank into the bed and leaned to the downstream (Fig. 5b). The elongated high-magnitude core of TKE can be identified to shift slightly upward compared with the earlier moment. Meanwhile, the elongated core seemed to grow in width along the downstream direction, which allows the significant TKE region in the wake like jet diffusion differing largely from that for the earlier moment. For a later moment (t = 4hours), the TKE contour remained nearly constant despite that the deposited sediment downstream the boulder was eroded (Fig. 5c). The change in the TKE contour suggested that the turbulence affecting the near-bed region became more significant when the scouring reached the final state that the downstream bed was scoured with an apparent sinking of the boulder. This pattern shift of the boulder wake TKE might be explainable corresponding to the upwelling flow when the boulder was embedded into the bed topography.
The influence of the boulder on turbulence is concentrated downstream of the boulder, and the TKE decreases with increasing water depth. This may be due to the water flow forming a horseshoe vortex at the bottom of the boulder, which is active and extends into the wake area downstream of the boulder and gradually reduces its strength. When the boulders are submerged, trailing vortices are formed downstream of the boulders, and the effect range is consistent with the exposure height of boulders. Therefore, in the downstream area of the boulder, the water depth below the exposed height of the boulder increases the TKE of the water flow, thereby increasing the erosion and transport of sediments (Yager et al., 2007; Baki et al., 2015). Furthermore, the TKE increased downstream of the boulder, which may be attributed to the fact that the near-bed counterflow downstream of the boulder met with the sand-carrying water formed by the stagnant pressure in this area, resulting in a surge of turbulent kinetic energy. As the scouring time proceeds and the exposed height of the boulder decreases, the magnitude and location of the TKE change, with the magnitude of the turbulent kinetic energy first increasing and then decreasing and gradually moving up from the downstream of the boulder to near the surface of the boulders. This may be due to the change in riverbed scour around the boulder, especially the range of scour pits and the boulder itself moving upstream while sinking, which changes the distribution of the TKE around the boulder.
Figure 5d-f shows the TKE contour map for different submergence ratios (h/D) at the final scouring time moment (t = 24hours). For shallowly submerged scenario (Fig. 5d), an elongated high-magnitude TKE core grew far below the boulder top despite an intermediate water depth but shallowly embedment. However, when h/D increased, the elongated TKE core shifted upward (Fig. 5e-f), growing closed to the boulder top. The difference in the movement of the location of high-magnitude TKE core might be because the shallow submergence of the boulder cannot promote the growth of the vertical (detached) shear layer, in contrast the horizontal (detached) shear layer became the major source for turbulence. However, when the boulder was more submerged, the vertical shear layer cannot grow infinitely in the vertical direction, which suggested that the main driving force is the shear near the boulder top.
3.4 Bed shear stress
As seen from the bed shear stress distribution in the near-tailflow zone in Fig. 6, with the center of boulder as the starting point x/D = 0, the bed Reynolds stress in the near-tailflow zone tends to change oscillatively and then surge with increasing x/D (Yager et al., 2007; Papanicolaou et al., 2012). At different moments, the differences are obvious (Fig. 6a). At the beginning of scouring (t = 10 min), the riverbed is rapidly deformed, the boulder position is shifted, and the bed shear stress undergoes continuous adjustment with undulating waves. In the later stage of scouring, t = 3 hours and 4 hours, both have the same bed surface orientation and the same size, and both have a surge at x/D = 1.3. Subsequently, at t = 24 hours, the camera was not able to take pictures due to the limitation of the camera's shooting range. However, within x/D ≤ 1.3, the bed shear stress direction is smooth without surge, indicating that the bed Reynolds stress starts to stabilize after t > 3 hours. After t > 3 hours, the boulder is obviously sinking and moving upstream, and the range and depth of the washout pits in front and on both sides of the boulder still have obvious changes, but the intensity of the changes is obviously smaller than the starting stage of the washout. This indicates that after t > 3 hours, the changes in bed siltation deformation and drift stone displacement have little effect on bed shear stress, and the bed shear stress in the near tail flow area of the drift stone changes drastically at the beginning of the scouring stage, where the exposure height of the drift stone is large and the bed deformation in the near tail flow area is small, so the effect on bed shear stress is also small. In turn, the influence of drift stones on the surrounding sediment transport is the greatest at the beginning of scouring, and the bed shear stress tends to be stable with time.
After 24 hours of scouring and stabilization, the bed shear stress under nonsubmerged conditions (h/D = 0.75) is significantly larger than the value under submerged conditions (h/D ≥ 1) (Fig. 6b). This is due to the more intense water–vapor mixing under the wake flow in the noninundated condition, and the wake vortex is more easily broken, so the scouring of the near-bed is more intense. As shown in Fig. 6c, for different particle sizes of boulder, except for the D = 10 cm experiment due to the limitation of the camera's shooting range, a surge occurs between x/D = 1.5 ~ 2 for D = 5 cm and 7.5 cm. This is because the bypass flow on both sides of the drifting rock meets the wake flow near the reattachment point at the tail position of the near wake flow area, producing a near-bed current flowing upstream.
Therefore, the directional reversal and surge of bed shear stress between x/D = 1.5 ~ 2 also laterally indicate that the near tail flow area is a sediment deposition area. This is consistent with the near wake region in the range of x/D < 2 downstream of the boulder. In Fig. 6d, the trends of shear stress at the three flow intensities are consistent, with roughly the same values, and the different flow intensities do not have a significant effect on the distribution of bed shear stress.