Experimental investigation of turbulent flows through a boulder array placed on a permeable bed

In mountain rivers, immobile boulders are regarded as large bed obstacles, largely influencing the spatial and temporal variability in flow and topographic dynamics (Euler et al. 2017; Afzalimehr et al. 2019). The investigation of the effects of boulders on flow structures is difficult to perform due to the complexity of the bed surface geometry configuration. Many laboratory studies focused on the mean and turbulent flow characteristics in their vicinity and the complex eddy structures around boulders (e.g. Shamloo et al. 2001; Sadeque et al. 2009; Dey et al. 2011). Regarding an isolated boulder, it is found that three different regions can be identified along the centre line of the boulder, including downstream of the boulder, an intermediate region and upstream of the boulder. For the downstream region of a boulder, the velocity profile appears to be complex over the entire flow depth and a near-wake region forms when flow separates at the edge of the boulder, resulting in downstream flow recirculation. Meanwhile, a far-wake region beyond the near-wake region enables flow to return to a normal state near the bed as upstream. In real situations, boulders in mountain streams commonly occur in arrays. Particularly, multiple arrays of isolated boulders in a regular staggered pattern can be observed in mountain streams (e.g. Lawless & Robert 2001; Papanicolaou et al. 2012).

In terms of turbulent flow characteristics solely past a boulder array, researchers point out that flows around an individual boulder in an array, different from those around an isolated one, are impacted by flow retardation and backwater effects upstream of the array. Therefore, flow structure around an individual boulder would be largely modified in comparison with the isolated-boulder configuration. Baki et al. (2014, 2015) investigated the flow velocity and turbulence characteristics in a staggered-boulder array in a typical fish pass, finding that the boulder array tended to increase the water depth and reduce the flow velocity. Compared with the flow around an isolated boulder, the wake region behind a boulder decayed slowly in the longitudinal direction and had the higher turbulent intensity and turbulent kinetic energy affected by neighbouring boulders. To further understand the effects of a boulder array, Baki et al. (2016) using a three-dimensional numerical model investigated the impacts of variations in boulder size, spacing, and pattern on flow characteristics. Sarkar et al. (2016) used double-averaging methodology (DAM) to successfully resolve the spatial heterogeneity in flows over a boulder array on a gravel bed. Sarkar (2016) found that turbulence characteristics vary insignificantly above the crest level of the array in comparison with those below the crest level. Also, macro-bed-roughness can result in the variability of the curve shape of the velocity profiles (Wang et al. 2015). Yager et al. (2007) found that boulders in a staggered pattern magnified the reach-averaged total bed shear-stress. Papanicolaou et al. (2011) found that the additional form drag created by the array produced a greater increase in reach-averaged total bed shear-stress compared with the isolated roughness conditions. Fang et al. (2017) investigated flows over submerged boulder arrays and found that boulder arrays promoted the local bed shear-stress distribution.

Boulders can promote the formation of discrete clusters, which serve as standalone anchor particles atop the finer particles and on the bed. The fine particles around boulders create a rough, permeable bed. Conventionally, turbulent flows over permeable beds are considered similar to those over impermeable beds since the effect of bed permeability on flows is usually considered negligible. However, when the flow is over a permeable bed, the surface and subsurface flow regions for a permeable bed exist above and within the bed, respectively. The flow over a permeable bed penetrates the bed, driving the exchange of mass, momentum, and energy between the bed surface and subsurface (Zippe & Graf 1983; Choi & Waller 1997; Liu et al. 2017). Bed permeability also controls flow mixing, encouraging a more organized near-wall flow structure and simultaneously governing flow physical processes within porous media (e.g. Cheng et al. 2011; Manes et al. 2011). It was observed that the von Karman constant over permeable beds is much lower than its standard value (=0.41) for terms of permeable smooth and rough beds (Breugem et al. 2006). Manes et al. (2009, 2011) found that bed permeability could enhance the wall-normal fluctuating velocity. This enhancement led to an increase in turbulent shear-stresses and thus further induced higher friction factors over permeable beds.

The fact is that numerous studies have reported hydrodynamics of boulder flows, and turbulent flow structure over permeable beds has been extensively investigated. However, the impact of the bed permeability on flows deflected by boulders has rarely been documented. This may lead to a knowledge gap between existing open channel flow theory and realistic flow configurations. Therefore, the present study designs a novel complex porous bed configuration, which describes flow propagating through a boulder array placed on a permeable bed. This study aims to investigate the impacts of the coexistence of boulders and bed permeability on the hydrodynamics. The investigated parameters are mean flow velocity distribution, Reynolds stresses and turbulent kinetic energy (TKE). To deepen the understanding of the interaction mechanism between flow, boulder and bed permeability, we further conduct a quadrant analysis of the turbulence structure, indicating the features of coherent structure over the permeable boulder-array bed. Finally, the budget of TKE is shown to describe the transfer of TKE by different mechanisms.

Experiments were carried out in a rectangular flume equipped with a recirculation system. The flume was 0.5 m wide, 0.6 m deep, and 16 m long with a streamwise bed slope of 0.1%. The flume commenced with a head tank, capably filtering large-scale eddies and providing a smooth flow. The incoming flow was adjusted by an inlet valve to produce various flow configurations and the flow discharge was measured by a sharp-crested weir. The smooth bed (SB) was configured with a layer of glass beads with a constant diameter of d_b = 14 mm to mimic the gravel bed configuration, which is commonly observed in mountain river systems. Using glass beads packed in multiple layers to function as a permeable bed can refer to the setup of Manes et al. (2009), in which the bead diameter is 12 mm. Because of the one-layer set up, the bed with one-layer of beads is referred to as the impermeable bed (IMPB). To simply construct the permeable configuration, the second layer of beads was well packed on the IMPB, which was able to enhance the vertical motion of flows. Thereafter, it is referred to as the permeable bed (PB) in the current study.

Representing a natural boulder array, spherical plastic balls of a diameter of D = 40 mm were initially glued atop the PB in a staggered pattern (Figure 1(a)). The ball shape and dimension were designed similarly to other studies of flow around boulders (Papanicolaou et al. 2012; Liu et al. 2017). In total 55 boulders were arranged in 22 rows and each row includes either two or three boulders. The transverse spacing between two neighbouring boulders was set equal to the streamwise spacing between the centres of two neighbouring boulders (Figure 1(b)). This arrangement resulted in an aerial density of the boulder array of ∼4.7%. Unlike other studies (Papanicolaou et al. 2012; Tsakiris et al. 2014), where boulders were directly placed atop the glass bead layer, boulders in this study were buried in glass-bead beds (Figure 1(c) and 1(d)). This treatment can provide a better description of natural scenarios of boulders and the neighbouring small-size particles (e.g. pebbles and gravels). In this sense, the boulders were exposed with different heights from the top of the bead layer (Figure 1(c)). The exposure height of the boulder can be simply defined as the difference (D₁) between the boulder and bead layers, as described in Figure 1(c). The state of boulders, therefore, can be completely exposed, slightly buried and largely buried. The exposure height, correspondingly, is D₁ = D, D−d_b and D−2d_b.

The flow velocities and turbulence quantities were obtained from the measurement of 3D instantaneous flow velocities using a SonTek^®10 MHz type Acoustic Doppler Velocimeter (ADV). The sampling frequency was set at 30 Hz. Within a sampling duration of 70 seconds, 2,100 points can be recorded so that the turbulent characteristics around a boulder can be well measured. The ADV used was able to measure flow velocities in a range of 0.001 m/s–5 m/s, with an accuracy of ±1%. To improve the data quality, those data with a ratio of signal to noise less than 70% were removed. In addition, high instantaneous velocity spikes were filtered to further refine the quality of the measured velocity data, using the phase-space thresholding technique proposed by Goring & Nikora (2002). Due to the employment of the one-point-measurement ADV, the measurement efficiency was not as high as with other velocity profilers such as ADV profiler and particle image velocimetry (PIV). For each run, measurements were carried out at every 1 cm across the entire water depth. This measurement resolution is similar to the setup of Nepf & Vinoni (2000), who used a one-point-measurement ADV to measure flow velocities controlled by near-canopy-top coherent structure. Under this resolution, a good analysis of the TKE budget depending on the spatial derivatives was made.

Two flow discharges, i.e. Q = 28 L/s and 48 L/s, were specified and a constant flow depth was maintained by an adjustable tailgate located at the exit of the flume, thus resulting in two Reynolds numbers (Re = 51,638 and 88,523). Therefore, the experiments are referred to as the low-rate and high-rate flows, respectively (Table 1). In experiments, the effective depth h=h_o + z_o, the distance from the water surface to the zero-velocity plane, where z_o = 0.25d is the displacement position and h_o is the distance from the water surface to the top level of the glass beads. As shown in Figure 1(a), the test section has a length of 4 m located at 5 m downstream of the flume inlet to ensure a fully developed flow before entering the boulder array, and this can be confirmed by pre-experiments (examining the water depth). This is also similar to the setup of Papanicolaou et al. (2012) using similar geometry. The velocity measurement was conducted in the region where the flow became developed after moving through the boulder array. In the longitudinal direction, the test section started at a distance of 2 m from the most upstream test section and extended up to 2.205 cm (the ending vertical is located 5.125D downstream of the centre of the boulder). Along the distance of 5.125D, five verticals (0.625D, 1.75D, 2.8D, 4D and 5.125D) were measured.

Figure 2 shows time-averaged streamwise and vertical velocities normalized by the shear velocity (u/u_*) for an IMPB (one-layer beads) and PB (two-layer beads) without a boulder array for the high-rate flow. The presentation firstly provides a basic understanding of the flow behaviour over an IMPB and PB. It is observed that the difference in the streamwise velocity was relatively small (Figure 2). For a PB of the high-rate case, the time-averaged velocity decreases by ∼0.8u_* and bulk flow resistance increases by ∼4% (Table 1). The performance of streamwise velocity and bulk flow resistance for IMPB and PB is similar to the observation of Zippe & Graf (1983) and Manes et al. (2009).

Figure 3 shows the vertical distribution of streamwise velocity in regards to different bed configurations. Firstly, the time-averaged velocity at different locations downstream of a boulder is shown in Figure 3(a)–3(d). Generally compared with the IMPB, the averaged flow velocity above the PB downstream of the boulder (x/D = 0.625–5.125) is greater by roughly 0.4 ∼ 1u_∗. The increase of the friction factor, however, was 5% and 7% for Q = 48 L/s and Q = 28 L/s, respectively (Table 1). Due to the different layers of glass beads laid on the bed, the magnitude of the streamwise velocities can be observed shifted upward as the layer number increases for both low-rate and high-rate flows (Figure 3(e) and 3(f)). However, the maximum velocities for the three bed configurations hold no significant difference. This indicates that the exposure height of boulders might have only a small impact on the mean streamwise velocities. It should be noted that as the bead layer number decreases, the variation rate in velocity near the boulders becomes larger. This is attributed to the boulders deflecting the near-bed velocity more as the exposure height increases. Interestingly, the deflection in velocity is more significant for the low-rate case. The possible cause is that for the high-rate case, vortices generated around the boulders might facilitate the exchange of flow momentum between the lower and upper flows. For the vertical velocity, the distributions for the SB and IMPB have similar profiles with no shift of distance observed even if a layer of beads is present for the IMPB. However, for the PB a clear upward shift of the profile can be observed. It can be noted that as the number of bead layers increases, the magnitude of the vertical velocity tends to decrease in the near-bed region. This might be because the presence of bed permeability may weaken the vertical flow motion by providing more pores for mass flux (Manes et al. 2009).

Figure 4(a)–4(d) shows the streamwise turbulence intensity normalized by the shear velocity for the PB with a boulder array, compared with an exponential decay function in regards to open channel flows reported by Nezu & Nakagawa (1993). Downstream of the boulders, u_rms/u_* is greater in the near-bed region and then decreases with increasing height. Different from the predictions of Nezu & Nakagawa (1993), the presence of the boulder array leads to a shift of the maximum of u_rms/u_* away from the bed. The position of the maximum value for both discharges is above the top of the boulders (z/h = 0.2–0.3). This is attributed to the generation of vortices triggered by the boulders deflecting the flow, which are consistent with the observations of Papanicolaou et al. (2012) and Baki et al. (2015). Originating from the boulder (x/D = 0.625), the magnitude of the turbulence intensity does not differ much as the measurement location is moved downstream far away from the boulder. This might suggest that the vortices are maintained in the current boulder spacing. For both the IMPB and PB of the low-rate case, the vertical profile of u_rms/u_* has a similar shape to the exponential decay function, but the magnitudes are slightly smaller, consistent with Papanicolaou et al. (2012). For the high-rate case, after the turbulence intensity reaches its maximum above the boulder, the magnitude decays more rapidly with the vertical distance than that for the PB. This results in a greater deviation between the measured profiles and the original exponential decay function.

The impacts of different bed configurations on both streamwise (Figure 4(e) and 4(f)) and vertical (Figure 4(g) and 4(h)) turbulence intensities at x/D = 0.625 are illustrated. Herein, z/D₁ is taken as the y-axis to suggest the impact of the boulders on flows. Therefore, z/D₁ = 1 indicates the top of the boulders for the IMPB and PB. The distribution of u_rms/u_* has a negligible difference for the low-rate flow. However, for the high-rate flow, the near-bed peak of u_rms/u_* exhibits an apparent upward shifting off the boulder level as the bead layer increases. However, the similar magnitude and decay ratio for different bed configurations suggest a similar level of generated turbulence. The upward shifting is also identified in regards to w_rms/u_*. The effect is apparent for the low-rate flow, which is different from u_rms/u_*. This upward shifting of turbulence might be attributed to different patterns of vortices shedding from the boulders with different bed permeabilities, which needs to be further studied.

Figure 5 depicts S(f) in the near-bed region (z/h = 0.2) and compares the boulder wake at this point for the high-rate flow downstream of the boulder (x/D = 0.625 and 4). With plenty of fluctuations in the spectra within a boulder array for the SB, IMPB and PB, the range of the frequency could be fitted with a −5/3 slope line, reducing the confidence level of the line. However, the overall energy of the flow downstream of the boulder was very similar to the IMPB and PB. As with the flow over the PB, the turbulence continuously dissipates by penetrating into the permeable bed, like a low-pass filter (Manes et al. 2009). Thus, the gap between the IMPB and PB becomes small in the inertial sub-range. The data fitted the least-squares in the range shown to be parallel to a −5/3 slope. Therefore, the dissipation rate (e) was estimated using Equation (2).

Figure 6 shows the TKE budget immediately downstream of the boulder (x/D= 0.625) for the IMPB and PB. Similar to the distribution of the turbulence intensity (Figure 4), the TKE just above the boulder top prevails over other locations far away from the boulder top. It is clearly noted that the production term (T_p) plays a significant role in the generation of the TKE for different bed configurations and discharges. Interestingly, the pressure diffusion term (P_D) as the residual of all other terms behaves comparably to T_p. The turbulent diffusion (T_D) and dissipation (e) hold the function of the decay of the TKE, thus being negative. It should be noted that the dissipation (e) is greater when approaching the glass bead layer. This suggests that the generated vortices diminish in the near-bed region. The turbulent diffusion is small, indicating that the effect of vortices is insufficient to redistribute the TKE. For the high-rate case, every component to determine the TKE is greater than that for the low-rate case despite the presence of bed permeability. The bed permeability seems to have negative impacts on all terms of the TKE budget. Specifically, the magnitude of all terms is smaller for the PB than for the IMPB. However, the cause might be uncertain since the bed permeability (increasing bead layers) leading to a decrease in exposure height of the boulders might also decrease turbulent activity near the boulders. Another interesting observation is that the presence of the bed permeability tends to shift the maximum values of some terms a small distance away from the boulder top. This is consistent with the distribution of turbulence intensities (Figure 4).

Four events can be described by the above equations, which are the events in the quadrants that were outward interactions, Q1 (i = 1, ⁠); ejections, Q2 (i = 2, ⁠); inward interactions, Q3 (i = 3, ⁠); and sweeps, Q4 (i = 4, ⁠). Quadrant analysis was made for H = 0 and 2, respectively. This aims to investigate the contribution of high-frequency small eddies (for H = 0) and low-frequency large vortices (for H = 2) (Balachandar & Bhuiyan 2007).

Since the Q2 and Q4 represent the dominant events of turbulence and carry the major mass, momentum and energy during the velocity fluctuations, hereby we only present the results of the Q2 and Q4 as shown in Figure 7(a)–7(d) for H = 0 and Figure 7(e)–7(h) for H = 2. For comparison, the results for bed configurations without the boulder array are firstly presented before that for boulder bed. For H = 0, without a boulder array, the near-bed Q4 dominates the entire turbulent activity compared with the Q2 (Figure 7(a) and 7(b)). This indicates that the downward sweeps are the major process in transporting the turbulence momentum and energy from the outer flow to the inner flow. However, the presence of the bed permeability seems to have negligible impact on the magnitude of Q2 and Q4, except for a short downward shifting of the maximum of Q2 and Q4. This identity is somewhat consistent with the distribution of turbulence intensity (see Figure 4). The short shifting might be from the bed porosity providing the extra space so that the Q4 event of turbulence can penetrate the deeper layer.

As the boulder array occurs, the near-bed Q2 behaves comparably to the Q4, suggesting that the turbulence momentum and energy exchange are driven equally by the ejections and sweeps. Even for the SB, the Q2 prevails over the Q4. Clearly, the presence of the glass beads appears to inhibit the generation of the Q2 and Q4 events. The magnitude of both turbulent events for the SB approximates 5–7 times those for the IMPB and PB. The increase in the glass bead layer also holds a similar but slight inhibiting effect. This suggests that the bed porosity may absorb the turbulent energy of the near-bed energy. However, the decrease in exposure height of the boulders may be the second cause of the weak turbulent activity when the glass bead layers exist. The presence of the bead layer somehow decreases the roughness of the bed configuration, thus weakening the turbulence.

As some small random fluctuating flow motions are filtered out (by setting H = 2), the Q2 for cases without the boulder array becomes significant, even exceeding the effect of Q4, and the Q4 is highly reduced compared with the values with respect to H = 0. This may be because the ejections (Q2) are dominated by low-frequency large eddies while the sweeps (Q4) are mainly assembled by high-frequency small eddies. With the filtering of small eddies, the large eddies take the major role. However, this change is not applicable to the boulder bed configuration. By setting H = 2, the distribution and magnitude are still similar to those in the case of no boulder array. This suggests that the presence of the boulders is the main cause of the near-bed large eddies.

To address the impacts of the coexistence of boulders and bed permeability on flow characteristics, this study designed an array of boulders (simulated by plastic spherical balls) placed within well-packed glass-bead-made permeable beds. The instantaneous flow velocities were measured to achieve mean flow and turbulence characteristics to investigate those impacts. For background reference, hydrodynamics investigation was made over smooth beds (SB) with the boulder array. The major flow characteristics such as mean flow velocity, turbulence intensity, turbulent kinetic energy (TKE) and instantaneous Reynolds stresses (through quadrant analysis) were presented in this study. The results show that the increase in bed permeability through decreasing the exposure height of boulders has little impact on the magnitude of streamwise velocity, but the near-bed velocity gradient seems to decrease, indicating the reduction of the bed shear-stress. Similarly to the previous studies, it is identified that bed permeability enables the downward shifting of the peak of turbulence intensity probably due to the absorption of the turbulence momentum and energy. The TKE budget analysis shows that bed permeability tends to weaken production, transport and diffusion intensities of the TKE. With bed permeability increasing (or the exposure height of the boulders decreasing), the peaks of all TKE terms shift upward off the boulder top. However, the TKE budget through the entire depth cannot attain equilibrium. Finally, the quadrant analysis of turbulence structure, by setting different hole sizes, clearly reveals that the ejections (Q2) and sweeps (Q4) with and without the boulder array are dominated by turbulence structure of different scales.

This research was supported by the National Key R&D Program of China (2017YFC1502504 and 2018YFC150540203) and the National Natural Science Foundation of China (51639007, 51909178, and 41771543).