Measurement and Characteristic Analysis of Turbulent Flow near Permeable Spur Dike

As a waterway regulation structure, the spur dike has been widely used to divert water flow, protect river banks, and restore waterway ecology [1, 2]. The turbulence has a great impact on the erosion of river bottom and the life of aquatic organisms. To fully understand the influence of turbulence around a spur dike, it is necessary to thoroughly analyze the turbulence features.

The structure and flow mechanism of the turbulence around spur dike have been extensively studied through flume experiments. For example, Rajaratnam and Nwachukwu [3] conducted flume experiments to ascertain the structure of turbulence near the spur dike, using the model of the three‐dimensional (3D) turbulent boundary layer. Ohmoto et al. [4] experimentally clarified the momentum exchange between main flow and the spur dike regions, revealing the obvious correlation between the water surface oscillation around the spur dike and velocity fluctuation. Jeon et al. [5] investigated the flow structure and turbulence mechanism around a non-submerged spur dike. Koken and Gogus [6] examined the different shear stresses and pressures on river bottom at three different spur dike lengths. Mehraein et al. [7] analyzed the turbulent kinetic energies and Reynolds stresses at different positions near the bottom of straight spur dike. Gu et al. [8] carried out flume experiments to observe the properties of turbulence structures on permeable and impermeable dikes. Through lab experiments, Hasegawa et al. [9] found that permeable pile spur dike can reduce the longitudinal velocity and turbulence intensity in the near-bank area.

Some scholars measured the 3D flow fields and turbulence features around the spur dike through acoustic doppler velocimetry (ADV). For instance, Duan et al. [10-12] captured the turbulence intensity and Reynolds stress of spur dikes by the ADV, and tackled the changes of turbulent states before and after the formation of the scour hole, including outward interaction, ejection, inward interaction, and sweeping. With the aid of the ADV, Jeon and Kang [13] obtained the 3D instantaneous velocities of the 3D turbulences around a spur dike, and derived the turbulence stresses from the obtained data.

Some other scholars explored the turbulence features around spur dikes through numerical simulation and theoretical analysis. Soulsby and Dyer [14], Keshavarzy and Ball [15], and Marchioli and Soldati [16] determined the correlations between the scour and near-bed turbulence parameters by means of quadrant analysis and triple correlations. Nikora and Goring [17], and Nezu [18] formulated the relations of velocity, turbulence intensity, and Reynolds stress after measuring the turbulence intensity and kinetic energy. Cui and Zhang [19] combined the k-epsilon (ε) model and the wall function into a new model, simulated the flow field around spur dike with free surface, and calculated the turbulent kinetic energy and its dissipation rate. Zhang [20] developed a 3D non-linear k-ε model to simulate the turbulence in the local scour hole around a single non-submerged spur dike. Acharya and Duan [21] adopted the renormalization group (RNG) k-ε model to simulate the turbulence field around spur dikes. Kang et al. [22] simulated the turbulence fields around multiple-arm instream structures, using the large-eddy simulation (LES) model, which solves the 3D Navier-Stokes equations, and the curvilinear immersed boundary (CURVIB) method.

To sum up, many experiments and simulations have been conducted to reveal the turbulence characteristics around spur dikes. However, the experimental studies have two common defects: the measuring points are close to the scour hole or the river bottom; the turbulent data are often collected from one horizontal plane, failing to reflect the turbulence features along the water depth. As for the simulation analyses, the turbulence features (e.g. turbulence intensity and Reynolds stress) obtained by most simulation models lack support from measured data, which dampens the simulation accuracy and applicability. What is worse, the previous studies on the turbulence features around spur dikes mostly emphasize on solid spur dikes, paying little attention to permeable spur dikes.

This paper measures the flow field around a permeable spur dike, which is installed at an open channel, through Nortek Vectrino Plus ADV. The time history of 3D velocities at all measuring points around the dike was recorded. Based on the measured data, the variation rules of turbulence intensity at different parts of the dike were analyzed along the water depth. In addition, the turbulence structure was discussed through quadrant decomposition. In this way, the authors quantified the turbulent Reynolds stress contribution of different flow regimes, and disclosed the relationship between Reynolds stress and turbulent kinetic energy.

The proposed permeable spur dike can regulate the waterway and restore the ecology. On the one hand, the dike slows down the erosion of sediment around the dike; on the other hand, the dike provides aquatic organisms, especially fish, with small habitats and migration channels. It is important to analyze the turbulence intensity and structure near the dike, because the two factors directly bear on the movement and transportation of sediment [23], as well as the behaviors of fish, e.g. movement, spawning, and migration [24].

3.1 Turbulence intensity

Turbulence intensity (σ) indicates the intensity of the velocity pulsation in the water flow:

${{\sigma }_{u}}={{\left( {{n}^{-1}}\sum\limits_{i=1}^{n}{u_{i}^{\prime 2}} \right)}^{1/2}}$     (1)

${{\sigma }_{V}}={{\left( {{n}^{-1}}\sum\limits_{i=1}^{n}{V_{i}^{\prime 2}} \right)}^{1/2}}$     (2)

${{\sigma }_{W}}={{\left( {{n}^{-1}}\sum\limits_{i=1}^{n}{w_{i}^{\prime 2}} \right)}^{1/2}}$     (3)

where, σ_u, σ_v, and σ_w are turbulence intensities in X, Y, and Z directions, respectively; $u_{i}^{\prime}, v_{i}^{\prime}$, and $w_{i}^{\prime}$ are the fluctuating velocities in X, Y, and Z directions, respectively; n is the sample size.

To disclose the rule of turbulence intensity along the water depth, the dimensionless turbulence intensity (σ/U₀) distribution at each point of profiles V2 and V4 were analyzed under different submersion degrees (z/h) (Figure 6):

$U_{0}=Q / H B$      (4)

Under the submergence degree of 1, the height of the measuring point equals that of the dike; under the submergence degree of 0.8, the measuring point lies right on the top of the dike hole. In addition, the positions of z/h>1, 0.8<z/h<1, and z/h<0.8 were defined as the free layer, dike crest layer, and dike body layer, respectively. 

As shown in Figure 6, on the same vertical line, the variation law of 3D turbulence intensity was basically the same along water depth, and $\sigma_{v}$ was smaller than $\sigma_{u} \text { and } \sigma_{w}$. Section V2 is located at the upstream of the dike, and the water flow is relatively stable.

In the vertical direction, with the growing submersion degree, the turbulence intensity first decreased and then increased. With the decline in submersion degree, the turbulence intensity decreased in the free layer, but changed suddenly in the dike crest layer. Below the dike crest layer, the turbulence intensity increased with the falling submergence degree. The minimum turbulence intensity appeared in the dike crest layer, while the maximum turbulence intensity was observed on the water surface or river bottom. There was not much difference between the two values.

On profile V4 lies in the downstream of the dike. The variation in turbulence intensity behind the dike was complicated by the dike permeability. At v4-(1) and v4-(2), turbulence intensity peaked at z/h=0.8. With the decline in submersion degree, the turbulence intensity decreased in the free layer, but increased continuously across the dike crest layer. This rise of turbulence intensity is attributable to the fact that: the water flow in the dike hole spreads at the outlet, forming a hydraulic jump, and interacts with the falling water flow over the dike top to produce severe turbulence. Within 0.5<z/h<0.8, the turbulence intensity decreased with submersion degree. From z/h=0.5 to the bottom, the turbulence became increasingly intense. Overall, the variation of turbulence intensity at other positions of profile V4 is similar to that of profile V2.

To further clarify the variation in turbulence intensity at different positions of the dike, the authors compared the turbulence intensities of profiles V2 and V4. Considering the similarity in the variation of turbulence intensity and the fact that σ_u and σ_w are obviously greater than σ_v, only the σ_u at different positions of the dike was analyzed in details (Figure 7).

As shown in Figure 7, the turbulence intensity in the downstream of the dike was generally twice that in the upstream, except in V2-(6) and V4-(6), which are 40cm from the right of the dike head. In the main channel of the river, the dike has less influence on the water flow. That is why V2-(6) and V4-(6) have little difference in turbulence intensity.

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Figure 6. The distribution of 3D turbulence intensity

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Figure 7. The σ_u distribution at different positions

3.2 Turbulence structure

3.2.1 Quadrant analysis

Quadrant analysis is a classic, stable, and popular way to analyze turbulence structure [25, 26]. The quadrant analysis of Reynolds stress was conceptualized by Lu and Willmarth [27] to examine the shear stress features on the outer boundary of the viscous bottom of turbulence. Since then, this approach has been widely applied to study the turbulence around spur dikes [28, 29].

Quadrant analysis aims to distribute instantaneous pulsation velocity (u', w') in the four quadrants of the plane coordinate system (Q1, Q2, Q3 and Q4), and obtain the corresponding Reynolds stress contribution S_{i, M} in each quadrant (Figure 8). In the first quadrant Q1 (u'>0, w'>0), the high-speed fluid moves away from the bed surface, belonging to the outward interaction state; In the second quadrant Q2 (u'<0, w'>0), the low-speed fluid moves away from the bed surface, belonging to the ejection state; In the third quadrant Q3 (u< 0, w'<0), the low-speed fluid moves towards the bed surface, belonging to the inward interaction state; In the fourth quadrant Q4 (u'>0, w'<0), the high-seed fluid moves towards the bed surface, belonging to the sweeping state.

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Figure 8. The four quadrants and the hole region

From the time distribution of pulsation velocities u' and w', it is clear that various flow states alternated with the elapse of time. The time distribution of pulsation velocity at vertical line V4-(5) in dike crest layer (z/h=1) was selected for detailed analysis. Under the above conditions of state identification, the four flow states were observed clearly in Figure 9.

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Figure 9. The time distribution of pulsation velocity

The hole region [27], that is, the shaded part in Figure 8, refers to the region bounded by four hyperbolas $\left|u^{\prime} w^{\prime}\right|=M\left|\overline{u^{\prime} w^{\prime}}\right|$, where M is the size of the hole region. Under different M values, the contribution of each flow state to Reynolds stress S_i,M (i=1,2,3,4) can be calculated by:

${{S}_{i,M}}=\frac{1}{T}\int_{0}^{T}{{{C}_{i,M}}}(t){{u}^{\prime }}(t){{w}^{\prime }}(t)dt$      (5)

If $\left|\mathrm{u}^{\prime}(\mathrm{t}) \mathrm{w}^{\prime}(\mathrm{t})\right|>\mathrm{M}\left[\overline{\mathrm{u}^{\prime} \mathrm{w}^{\prime}}\right] \text { and }\left[u^{\prime}(t) w^{\prime}(t)\right]$ fall in the Qi quadrant, C_i,M=1; otherwise, C_i,M=0.

All instantaneous pulsation velocities are counted when M=0. Next, the turbulence distribution of each vertical line of profiles V2 and V4 was analyzed to clearly understand the distribution of pulsation velocities (u' and w') in each quadrant (Figure 10).

As shown in Figure 10, the transverse pulsation velocity (u') and vertical fluctuating velocity (w') of the flow in the upstream of the dike were both smaller than those in the downstream. This echoes with the finding in subsection 3.1 that the turbulence intensities in the downstream were greater than those in the upstream.

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Figure 10. The distribution of pulsation velocities $\left(u^{\prime}, w^{\prime}\right)$

In the scoure hole, the sediment partciles are lifted by sweeping and ejection flows (u'>0, w'<0; u'<0, w'>0), and rotated by the transverse pulsation velocity [28]. Therefore, ejection (Q2) and sweeping (Q4) dominate the total sample, and require further analysis.

The vertical pulsation velocities (w') of the profile in the upstream of the dike were greater than the transverse pulsation velocity (u'). The ratios of (K2+K4)/(K1+K3) in V2-(1),V2-(2),V2-(3),V2-(4), and V2-(5) were 1.06, 1.24, 1.41, 1.39, and 1.07, respectively, where K_i is the percentage of S_i,0 in the total sample. Overall, the turbulence structure in the upstream of dike do not result in severe sediment scour.

In addition, transverse pulsation velocities (u') of the profile in the downstream of the dike were greater than the vertical pulsation velocity (w'). The ratios of (K2+K4)/(K1+K3) in V4-(1), V4-(2), V4-(3), V4-(4), and V4-(5) were 0.9, 0.94, 1.01, 1.06, and 1.08, respectively. Thus, it is difficult for the sediment to accmulate at the dike root. This explains the relatively slight scour at that position. Moreover, the sediment at other positions are susceptible to accumulation and transporting to the downstream. As a result, the downstream of the dike suffers serious scour.

To understand the distribution of different turbulence structures along water depth, the relationship between Reynolds stress contribution and submergence degree was quantified carefully. Under M=0, the Reynolds stress contribution was nondimensionalized by the square of frictional velocity $S_{i} / u *^{2}$ :

${{u}_{*}}=\sqrt{gHJ}$     (6)

where, H=19cm is water depth; J=0.0006 is the hydraulic gradient.

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Figure 11. The relationship between the Reynolds stress contribution and submergence degree

Figure 11 shows the relationship between the Reynolds stress contribution and the submergence degree.

As shown in Figure 11, on every vertical line in the upstream and downstream of the dike, the four turbulence structures (Qi) changed by basically the same law along the water depth. The Reynolds stress contribution $\mathrm{S}_{\mathrm{i}}$ of the upstream profile V2 of the spur dike was smaller than that of the downstream profile V4.

From water surface to river bottom, the turbulence structure on each vertical line of profile V2 was dominated by ejection (Q2) and sweeping (Q4). Besides, S₂ was the larget contribution. The $\mathrm{S}_{\mathrm{i}}$ on V2-(1) (dike root) increased continuously from the water surface to the river bottom.

At other positions, S_i decreased first and then increased with the growing submergence degree. The largest contribution appeared on the water surface or the river bottom, while the smallest emerged in the dike crest layer.

The downstream profile V4 exhibited more complex turbulence structures. Ejection (Q2) and sweeping (Q4) dominated the turbulence structure bear the free layer and the bottom, while outward interaction (Q1) and inward interaction (Q3) dominated near the dike crest layer. The greatest contribution appeared in the dike crest layer. However, there was no obvious law in the variation of S_i with water depth. A possible reason is the mixing and interaction of the backflow of water behind the dike, the flow around the dike head, the overflow over the dike crest, and the jet flow from dike body.

3.3 Reynolds stress and turbulent kinetic energy

Turbulent kinetic energy mirrors the turbulent state of water flow, and manifests the energy loss of water flow caused by turbulence. As a strong mixed water current, turbulence signifcantly promotes the growth and spawning of fishes [30]. In open channels, ejection (S₂) and sweeping (S₄) are the main sources of turbulent energy. They exist as the most important largescale structure in open channel turbulence, and boast a gigantic amount of energy from wall to free water surface [31]. Here, the Reynolds stress contribution is nondimensionalized by the frictional velocity $E / u *^{2}$ :

$E=\frac{1}{2}\left( u_{i}^{\prime 2}+w_{i}^{\prime 2} \right)$     (7)

Figure 12 illustrates the relationship between S₂, S₄ and turbulent kinetic energy.

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Figure 12. The relationship between Reynolds stress and turbulent kinetic energy

As shown in Figure 12, the data points formed an increasing curve, and the R² of the fitting formula stood at 0.91, suggesting the good correlation of the formula. Although the distribution of the points was relatively scattered after >1.0, the change trend remained consistent. Therefore, ejection (S₂) and sweep (S₄) states have consistent effects on turbulent kinetic energy.