**Anisotropy in the Free Stream Region of Turbulent Flows through Emergent Rigid Vegetation on Rough Beds**

#### **Nadia Penna \*, Francesco Coscarella, Antonino D'Ippolito and Roberto Gaudio**

Dipartimento di Ingegneria Civile, Università della Calabria, 87036 Rende (CS), Italy; francesco.coscarella@unical.it (F.C.); antonino.dippolito@unical.it (A.D.); gaudio@unical.it (R.G.) **\*** Correspondence: nadia.penna@unical.it; Tel.: +39-0984-496-553

Received: 10 August 2020; Accepted: 30 August 2020; Published: 2 September 2020

**Abstract:** Most of the existing works on vegetated flows are based on experimental tests in smooth channel beds with staggered-arranged rigid/flexible vegetation stems. Actually, a riverbed is characterized by other roughness elements, i.e., sediments, which have important implications on the development of the turbulence structures, especially in the near-bed flow zone. Thus, the aim of this experimental study was to explore for the first time the turbulence anisotropy of flows through emergent rigid vegetation on rough beds, using the so-called anisotropy invariant maps (AIMs). Toward this end, an experimental investigation, based on Acoustic Doppler Velocimeter (ADV) measures, was performed in a laboratory flume and consisted of three runs with different bed sediment size. In order to comprehend the mean flow conditions, the present study firstly analyzed and discussed the time-averaged velocity, the Reynolds shear stresses, the viscous stresses, and the vorticity fields in the free stream region. The analysis of the AIMs showed that the combined effect of vegetation and bed roughness causes the evolution of the turbulence from the quasi-three-dimensional isotropy to axisymmetric anisotropy approaching the bed surface. This confirms that, as the effects of the bed roughness diminish, the turbulence tends to an isotropic state. This behavior is more evident for the run with the lowest bed sediment diameter. Furthermore, it was revealed that also the topographical configuration of the bed surface has a strong impact on the turbulent characteristics of the flow.

**Keywords:** anisotropy; rigid vegetation; sediments; turbulent flow

#### **1. Introduction**

The flow through emergent rigid vegetation has been widely investigated by researchers, both experimentally and numerically, aiming at analyzing the effects of vegetation on the flow structure and its implications on hydraulic resistance, turbulent structures, mixing processes, and sediment transport [1–5]. This particular type of vegetation (rigid cylinders) can simulate rigid reeds or trees in riparian environments [6,7], when the flow does not hit the foliage, since the dynamic plant motions exhibited by real vegetation is neglected [8].

Most of the existing works were primarily conducted on smooth channel beds with staggered-arranged vegetation stems (e.g., [1,2,6,9–16]). However, special interest must be devoted to studies on vegetated flows with rough beds, since the interactions between flow, vegetation, and sediments permit achieving a better understanding of the turbulence structures in natural environments, which have a key role in the sediment transport mechanism. In fact, the flow and turbulence characteristics through emergent rigid vegetation on rough beds are still poorly understood. As reported by Maji et al. [17], flow conditions and the solid volume fraction of emergent vegetation affect the individual contributions of sweep and ejection coherent structures, which play a dominant

role in dislodging bed particles. Therefore, an in-depth description of turbulent structure is imperative for the correct understanding of the sediment transport process and for the development of new sediment transport theories in emergent vegetated flows.

Recently, Penna et al. [18] analyzed the flow field, the turbulent kinetic energy (TKE), and the energy spectra of velocity fluctuations around a rigid stem on three different rough beds. They showed that, in the region below the free surface region, the flow is strongly influenced by the vegetation. Moving toward the bed, the flow is affected by a combined effect of both vegetation and bed roughness. At the same time, Penna et al. [18] noted a strong lateral variation of TKE from the flume centerline to the cylinder in the intermediate region. Finally, the analysis of the energy spectra revealed that, in the near-bed flow region at low wave numbers, the macro-turbulence is governed by the bed roughness, regardless of the measurement point location with respect to the vegetation stem. In the region of wake vortexes (i.e., downstream of the vegetation stem), the macro-turbulence is extended at smaller scales, implying a strong influence of the vegetation.

Nevertheless, a comprehensive characterization of the turbulence structures in the different flow layers cannot disregard from the turbulence anisotropy investigation. In fact, one of the most frequently analyzed quantities in turbulence studies is the anisotropic behavior of turbulence, in terms of the degree of departure from the isotropic turbulence. This is a common feature of complex fluid flows [19], such as those that characterize vegetated channels. The 'isotropic turbulence' refers to an idealized condition, in which the velocity fluctuations do not vary regardless of the rotation of axes [20]. This means that the Reynolds normal stresses along the streamwise, spanwise, and vertical directions (σ*uu*, σ*vv*, and σ*ww*, respectively) are the same. Conversely, in the 'anisotropic turbulence' the Reynolds normal stresses cannot be considered as invariant, because the temporal velocity fluctuations along the three axes follow a preferential direction [20].

To characterize the type of turbulence, one of the most used methodologies is the definition of the Reynolds stress anisotropy tensor. In particular, the diagonalization of the tensor provides three eigenvalues (λ1, λ2, and λ3) and three eigenvectors (*e*1, *e*2, and *e*3) of the turbulence anisotropy [19]. The anisotropy invariant map (AIM) describes the domain of all potential turbulent flows considering the second and third invariants. In fact, it is a 2D domain with a triangular shape, whose boundaries are characteristic of turbulence state (1D, 2D, and 3D turbulence) and related processes (axisymmetric expansion, axisymmetric contraction, and two-component turbulence) [3].

Hence, the driving idea of the present study was the description of the turbulence anisotropy (with the AIMs) of flows through emergent rigid vegetation on rough beds. Indeed, exploring for the first time this crucial aspect in vegetated flows may advance the current understanding of the flow–vegetation–roughness interaction by describing the evolution of the stress ellipsoid formed by the Reynolds stresses. To this end, an experimental campaign was performed in a uniformly vegetated channel varying the bed sediment size (coarse sand, fine gravel, and coarse gravel), under the same hydraulic conditions. Additionally, in order to better comprehend the overall flow conditions, the time-averaged velocity, the Reynolds shear stresses, the viscous stresses, and the vorticity fields were analyzed and briefly discussed.

#### **2. Experimental Program**

#### *2.1. Flume Set-Up and Bed Sediments*

The experimental study was performed in a 9.6 m long, 0.485 m wide, and 0.5 m deep flow recirculating tilting flume at the *Laboratorio* "*Grandi Modelli Idraulici*" (GMI), *Università della Calabria*, Rende, Italy. Three experimental runs were performed under the same approach flow conditions and with the same vegetation arrangement, but with different bed sediments. In particular, the flume bed was covered with a 20 cm thick layer of uniform, very coarse sand (*d*<sup>50</sup> = 1.53 mm), fine gravel (*d*<sup>50</sup> = 6.49 mm), and coarse gravel (*d*<sup>50</sup> = 17.98 mm) for Runs 1, 2, and 3, respectively. Figure 1 shows

the grain-size distribution of the mixtures used to create the bed, which were obtained by analyzing three samples for each Run.

**Figure 1.** Grain-size distribution curve of the bed sediments for each experimental run.

The flow rate *Q* was controlled by a submerged pump. It was measured using a calibrated sharp-crested V notch weir installed in a downstream tank, where the outflow was collected. To reduce the disturbance at the flume entrance and to dampen the related turbulence level, honeycombs with a diameter of 10 mm were used. The flume slope was set equal to 1.5% by maneuvering a hydraulic jack. Furthermore, the flow depth *h* was regulated with a downstream tailgate and measured with a point gauge.

The experimental runs were carried out by using a uniformly distributed vegetated channel bed, where the emergent vegetation was simulated with vertical, rigid, and circular wooden cylinders. A total number of 68 cylinders, each 2 cm in diameter, were inserted into a 1.96 m long, 0.485 m wide, and 0.015 m thick Plexiglas panel, which in turn was fixed to the channel bottom. The test section was located 6 m downstream of the flume inlet. The cylinders were arranged in an aligned pattern where the axis-to-axis distance between the stems was equal to 12 cm in both the streamwise and spanwise directions. Therefore, the total number of stems per unit area was 71.6 m<sup>−</sup>2. The frontal area per canopy volume was *a* = *d*/Δ*S*<sup>2</sup> = 1.4 m−1, where *d* is the stem diameter and Δ*S* the axis-to-axis distance between the stems. The solid volume fraction occupied by the stems was φ = (π/4)*ad* = 0.02; thus, the vegetation can be considered as dense [21]. Further details of the experimental setup were recently reported by Penna et al. [18].

#### *2.2. Experimental Procedure*

Initially, the flume was filled in with sediments and was subsequently screeded flat to obtain a bed with a mean surface elevation having the same longitudinal slope of the channel. All the experimental runs initiated with a steady flow rate equal to 19.73 l s−<sup>1</sup> and a water depth of 14 cm (measured 50 cm upstream to the vegetation array), designed to prevent sediments motions and to satisfy the fixed bed condition. Thus, the average flow velocity *U* was 0.30 m s−<sup>1</sup> (=*Q*/(*Bh*), where *B* is the flume width) and the flow Froude number *Fr* was 0.26 (=*U*/(*gh*) 0.5, where *g* is the gravitational acceleration). For each run, Table 1 shows the details of the experimental conditions for the approaching flow (measured 50 cm upstream to the test section), where *u*\* is the shear velocity determined extending the linear trend of the Reynolds shear stress (RSS) distribution down to the maximum crest level (= −*uw*- 0.5, where *<sup>u</sup>*- and *w* are the temporal velocity fluctuations in the streamwise and vertical directions, respectively, and the symbol · indicates the time average), *T* is the water temperature (measured with an integrated thermometer with an accuracy of 0.1 ◦C) and ν is the water kinematic viscosity, determined as a function of the water temperature [22]. Furthermore, the Reynolds number of the sediments *Re*\* (=*u\**ε/ν, where

ε is the equivalent Nikuradse sand roughness height, equal to about 2*d*50) and the Reynolds number of the vegetation stems *Red* (=*Ud*/ν, where *d* is the stem diameter) were calculated.


**Table 1.** Details of the experimental conditions.

The instantaneous three-dimensional flow velocity components were measured with a down-looking Vectrino probe (Acoustic Doppler Velocimeter, ADV) manufactured by Nortek, Vangkroken, Norway. The measurements were performed along the flume centerline at various relative streamwise distances *x*/*Ls* = 0, 0.17, 0.33, 0.50, 0.67, 0.83, 1.00, where *x* is the streamwise direction and *Ls* is the length of the study area equal to 12 cm. Note that *x*/*Ls* = 0 is the origin of the study area, located 6.78 m downstream of the channel entrance, and *x*/*Ls* = 0.50 corresponds to the vegetation stem axis. The 3D velocity components (*u*, *v*, *w*) refer to (*x*, *y*, *z*), where *y* and *z* are the spanwise and vertical direction, respectively.

The ADV probe was installed on a motorized 3-axis traverse system (HR Wallingford Ltd., Wallingford, Oxfordshire, UK) to easily move the probe within the study area during the experimental run. The Vectrino was operated with a transmitting length of 0.3 mm and a sampling volume constituted of a cylinder of 6 mm in diameter and 1 mm high. The sampling duration was equal to 180 s (the sampling frequency was fixed to 100 Hz), assuring statistically time-independent time-averaged velocities and turbulence quantities. It was not possible to perform velocity measurements within the flow zone 5 cm below the free surface, because the ADV beams converge at 5 cm below the probe. Thus, the vertical resolutions were 3 mm for *z* ≤ 15 mm and 5 mm above, where *z* is the vertical axis starting from the maximum crest elevation in the study area.

The ADV data were pre-processed for detecting potential spikes with the phase-space thresholding method. Spikes were replaced with a third-order polynomial through 12 points on both sides of the spike itself, as suggested by Goring and Nikora [23].

#### **3. Results and Discussion**

#### *3.1. Time-Averaged Flow*

The dimensionless time-averaged velocity fields and 2D velocity vectors, having magnitude *<sup>u</sup>*<sup>ˆ</sup> = *u*<sup>2</sup> + *w*<sup>2</sup> 0.5/*<sup>u</sup>*<sup>∗</sup> (where *<sup>u</sup>* and *<sup>w</sup>* are the time-averaged velocity components) and direction tan−1(*w*/*u*), on the vertical central plane are illustrated in Figure 2 for each Run. Here, the horizontal axis is represented as *x*ˆ = *x*/*Ls*, and the vertical axis *z*ˆ was made dimensionless dividing *z* by the local flow depth *hl*.

**Figure 2.** Contours of dimensionless time-averaged velocity and 2D velocity vectors measured on the vertical central plane for (**a**) Run 1, (**b**) Run 2, and (**c**) Run 3. The black broken lines indicate the edge of the vegetation stem.

In Figure 2, a streamwise variation of the velocity field is detected in the three Runs. Specifically, in correspondence of the vegetation stem, *u*ˆ increases with respect to the areas upstream to and downstream of the cylinder. This denotes the presence of: (1) a convergent flow between two stems and toward the flume centerline [24]; (2) a retarded flow owing to a divergent flow downstream of the stems. The changes of magnitude and direction of velocity vectors suggest the presence of a near-bed flow heterogeneity, which are more pronounced looking from Run 3 to Run 1. The streamwise and vertical variations of the velocity field is in agreement with Maji et al. [17], since significant velocity gradients were found in all the experimental runs. This is due to the different bed roughness that characterizes the three experimental runs. In essence, the flow velocity increases with the vertical distance, reaching the maximum values in correspondence of the vegetation stem and at the elevation *z* ≈ 0.15*hl*, regardless of the bed roughness, implying that this region is mainly influenced by the presence of vegetation. Here, the production of turbulence by the canopy exceeds the production by the bed shear [21].

Figure 3 presents the contours of the dimensionless Reynolds shear stresses τˆ*uw* (= −*uw*-/*u*<sup>2</sup> <sup>∗</sup>) on the vertical central plane for the three experimental Runs. They exhibit small magnitudes in the flow area dominated by the vegetation (for *z* > 0.15*hl*) [6]. Moving toward the bed surface, the Reynolds shear stresses increase with a high gradient, owing to the bed roughness. As *d*<sup>50</sup> increases, this zone becomes more extended. However, close to the bed at the crest level, the Reynolds shear stresses become negligible. The contours of τˆ*uw* reveal that they are not influenced by the position of vegetation stems, since their spatial distribution is quite uniform. Indeed, this agrees with the findings of Ricardo et al. [6], who demonstrated that the Reynolds stresses are not sensitive to local spatial gradients of the stem distribution, because they depend on the local number of stems per unit area. Analogous patterns can be noticed in Figure 4, where the dimensionless viscous stresses τˆ<sup>ν</sup> (= ν(d*u*/d*z*)/*u*<sup>2</sup> <sup>∗</sup>) on the vertical central plane for the three experimental Runs are presented. In fact, the streamwise distribution of τˆ<sup>ν</sup> is almost uniform in each Run. As the roughness decreases, the viscous shear stress increases at the crest level. Then, it diminishes as the vertical distance *z* increases. This agrees with the findings of Nepf [21], who stated that the viscous stress is negligible with respect to the vegetative drag over most of the depth, excluding a thin layer near the bed of a scale comparable to the stem diameter.

**Figure 3.** Contours of dimensionless Reynolds shear stresses measured on the vertical central plane for (**a**) Run 1, (**b**) Run 2, and (**c**) Run 3. The black broken lines indicate the edge of the vegetation stem.

**Figure 4.** Contours of dimensionless viscous shear stresses measured on the vertical central plane for (**a**) Run 1, (**b**) Run 2, and (**c**) Run 3. The black broken lines indicate the edge of the vegetation stem.

The effects of the bed roughness structures were investigated through the analysis of the dimensionless vorticity of the time-averaged flow ω*yd*/*u*<sup>∗</sup> on the vertical central plane (Figure 5). Here, ω*<sup>y</sup>* is the vorticity of the time-averaged flow, given by ∂*u*/∂*z* − ∂*w*/∂*x*. Positive values of the vorticity indicate clockwise fluid motion; on the contrary, negative values refer to counterclockwise direction. Specifically, the rotational direction provides information about flow acceleration and deceleration in the near-bed flow: counterclockwise rotation induces flow acceleration, with a downward transport of momentum in the downstream direction; clockwise rotation causes flow deceleration, with upward transport of momentum in the upstream direction [25,26]. It is evident that the vorticity changes its signs alternatively in the flow layer affected by the presence of both vegetation and bed roughness. This implies the heterogeneity of the time-averaged near-bed flow: fluid streaks move alternatively in both clockwise and counterclockwise directions [25]. Furthermore, it is possible to note that the changes of the vorticity rotational direction are more frequent in Run 1 than in the other two runs along the streamwise direction. As observed by Ricardo et al. [27], the cylinders induce a regular structure of vortex patterns independently from the space between cylinders also in the horizontal plane.

**Figure 5.** Contours of dimensionless vorticity of the time-averaged flow in the vertical central plane for (**a**) Run 1, (**b**) Run 2, and (**c**) Run 3. The black broken lines indicate the edge of the vegetation stem.

#### *3.2. Anisotropy Invariant Maps*

To investigate the anisotropic behavior of the flow through emergent rigid vegetation on rough beds, the AIM was examined along the flume centerline.

Originally introduced by Lumley and Newman [28], this map (also called the Lumley triangle) is a two-dimensional domain based on the invariant properties of the Reynolds stress anisotropy tensor *bij*, which can be defined as follows:

$$b\_{ij} = \frac{u\_i' u\_j'}{\overline{u\_i' u\_i'}} - \frac{1}{3} \delta\_{ij} \tag{1}$$

where δ*ij* is the Kronecker delta function (δ*ij*(*i* = *j*) = 1 and δ*ij*(*i j*) = 0) and, adopting the Einstein notation, *u*- *i u*- *<sup>i</sup>* is twice the TKE. The shape of the AIM is a triangle on a (*III*, −*II*) plane (Figure 6), where *II* is the second invariant of *bij* and represents the degree of anisotropy and *III* is the third invariant of *bij* and signifies the nature of anisotropy.

**Figure 6.** Conceptual diagram of the anisotropy invariant map.

The two invariants can be expressed, respectively, as follows:

$$II = -\frac{b\_{ij}b\_{ij}}{2} = -\left(\lambda\_1^2 + \lambda\_1\lambda\_2 + \lambda\_2^2\right) \tag{2}$$

$$III = \frac{b\_{ij}b\_{jk}b\_{ki}}{3} = -\lambda\_1\lambda\_2(\lambda\_1 + \lambda\_2) \tag{3}$$

where λ<sup>1</sup> and λ<sup>2</sup> are the anisotropy eigenvalues.

The AIM is delimited by two curves and an upper line. The left curve is characterized by negative values of the third invariant and can be described as follows: *III* <sup>=</sup> <sup>−</sup>2(−*II*/3)3/2. This curve refers to the pancake-shaped turbulence, since two diagonal components of the Reynolds stress tensor are greater than the third one. The right curve is defined as *III* = 2(−*II*/3)3/<sup>2</sup> and corresponds to the cigar-shaped turbulence, that is, one diagonal component of the Reynolds stress tensor is greater than the other two. Lastly, the function that expresses the upper line is *III* = −(9*II* + 1)/27; it describes a two-component turbulence. The bottom cusp of the AIM indicates, instead, the 3D isotropic turbulence.

Recently, Dey et al. [29] proposed an additional classification of the turbulence anisotropy, considering the shape of the ellipsoid formed by the Reynolds principal stresses σ*uu*, σ*vv*, and σ*ww* along the *x*-, *y*-, and *z*-axis, respectively. The Reynolds principal stresses are expressed respectively as ρ*uu*-, ρ*vv*-, and ρ*ww*-, where ρ is the mass density of water and *v*' is the temporal fluctuation of the velocity in the spanwise direction.

Specifically, in the case of σ*uu* = σ*vv* = σ*ww*, that is an isotropic turbulence, the stress ellipsoid is a sphere. If σ*uu* = σ*vv* > σ*ww* (on the left curve of the AIM, which is termed as the axisymmetric contraction limit), the stress ellipsoid takes the shape of an oblate spheroid. The two-component axisymmetric limit lies on the left vertex of the AIM, where the conditions σ*uu* = σ*vv* and σ*ww* = 0 prevail. In this case, the shape of the stress ellipsoid is a circular disk. On the right curve (the axisymmetric expansion limit), one component of the Reynolds stresses is larger than the other two (σ*uu* = σ*vv* < σ*ww*), thus the stress ellipsoid takes the form of a prolate spheroid. At the top boundary, which indicates the two-component limit, the stress ellipsoid is an elliptical disk, since σ*uu* > σ*vv* and σ*ww* = 0. Finally, the one-component limit lies on the right vertex, where only one component of Reynolds stress sustains (that is, σ*uu* > 0 and σ*vv* = σ*ww* = 0 or σ*ww* > 0 and σ*uu* = σ*vv* = 0). This means that the stress ellipsoid assumes the shape of a straight line.

Figure 7 shows the data plots of −*II* versus *III* and the AIMs for Run 1, at the following streamwise relative distances: *x*/*Ls* = 0, 0.17, 0.33, 0.50, 0.67, 0.83, 1.00. Note that the plots were zoomed in the area of the AIM in which the data were concentrated. In the same way, Figures 8 and 9 depict the AIMs for Runs 2 and 3, respectively. At a given streamwise distance, each subplot illustrates the evolution of the turbulence anisotropy along the dimensionless vertical distance *z*ˆ. In all the Runs, moving from the crest level upwards, the data points of the Reynolds stress tensor describe a particular path, with some differences as the bed roughness changes.

**Figure 7.** AIMs of Run 1 at (**a**) *x*/*Ls* = 0, (**b**) *x*/*Ls* = 0.17, (**c**) *x*/*Ls* = 0.33, (**d**) *x*/*Ls* = 0.50 (at the vegetation stem axis), (**e**) *x*/*Ls* = 0.67, (**f**) *x*/*Ls* = 0.83, (**g**) *x*/*Ls* = 1.00.

Looking at Figure 7, near the bed, the turbulence anisotropy is prevalent as the data lie close to the right side of the Lumley triangle. This means that the velocity fluctuation in the vertical direction predominates owing to the bed roughness height, which enhances σ*ww*. Then, the data plots move toward the line of plane-strain limit, which is characterized by the condition *III* = 0. As the vertical distance increases, the turbulence anisotropy shows a feeble tendency again toward the axisymmetric expansion limit. The described path occurs at each streamwise distance, implying a similar behavior of the Reynolds stress tensor, regardless of the location with respect to the vegetation stem. Therefore, considering the classification based on the ellipsoid shape [29], it is evident that, near the bed, a prolate spheroid axisymmetric turbulence is predominant. Subsequently, as the vertical distance increases, an axisymmetric contraction develops, tending to the 3D isotropic turbulence in the region mainly affected by the vegetation.

As regards Run 2, although near the crest level, the position of the data points in the AIMs does not vary from that of Run 1; moving toward the free surface, their path slightly changes. In fact, it is evident that the data plots tend to move toward the line of plane-strain limit, but they rapidly turn back to the right side of the Lumley triangle. This implies that one component of the Reynold stresses prevails on the others for almost the entire investigated flow depth. Thus, in Run 2, the ellipsoid is basically a prolate spheroid along *z*ˆ. The same trend is visible at the different streamwise distances.

**Figure 8.** AIMs of Run 2 at (**a**) *x*/*Ls* = 0, (**b**) *x*/*Ls* = 0.17, (**c**) *x*/*Ls* = 0.33, (**d**) *x*/*Ls* = 0.50 (at the vegetation stem axis), (**e**) *x*/*Ls* = 0.67, (**f**) *x*/*Ls* = 0.83, (**g**) *x*/*Ls* = 1.00.

Akin to both Runs 1 and 2, the data plots of Run 3 start from the axisymmetric expansion limit (that corresponds to the right-curved side of the triangle). Increasing the vertical distance, they rapidly move toward the line of plane-strain limit. Then, the turbulence anisotropy tends to the isotropic state (the data move toward the bottom cusp of the AIM). This is due to the bed roughness influence on turbulence anisotropy, which vanishes moving toward the free surface. Thus, initially the ellipsoid shape is a prolate spheroid. As the vertical distance increases, an axisymmetric contraction develops, tending to the 3D isotropic state and, as a result, the stress ellipsoid becomes a sphere.

In order to highlight the effects induced by the presence of vegetation, Figure 10 shows the AIMs for the undisturbed flow conditions detected 50 cm upstream to the vegetation array in all the three Runs. It is revealed that, without the influence of vegetation, the turbulence anisotropy tends to the plane-strain limit in all the Runs, independently from the bed roughness, which, however, is the main cause of a prolate spheroid axisymmetric turbulence near the bed surface. This latter is more pronounced in Run 3, owing to a higher roughness height than in the other two Runs.

**Figure 9.** AIMs of Run 3 at (**a**) *x*/*Ls* = 0, (**b**) *x*/*Ls* = 0.17, (**c**) *x*/*Ls* = 0.33, (**d**) *x*/*Ls* = 0.50 (at the vegetation stem axis), (**e**) *x*/*Ls* = 0.67, (**f**) *x*/*Ls* = 0.83, (**g**) *x*/*Ls* = 1.00.

#### *3.3. Anisotropic Invariant Function*

Choi and Lumley [30] introduced a function, called the anisotropic invariant function *F*, with the aim of providing an insight into the turbulence anisotropy from the two-component limit to the isotropic limit [31]. The function can be calculated as follows:

$$F = 1 + 9II + 27III\tag{4}$$

The main peculiarity of the anisotropic invariant function is that it vanishes when the turbulence anisotropy prevails (*F* = 0), whereas it reaches unity (*F* = 1) when the turbulence reaches the three-dimensional isotropic state.

The contours of the anisotropic invariant function on the vertical central plane in the test section are shown in Figure 11 for all the Runs. The streamwise variation of the anisotropic invariant function is quite uniform, regardless of the location of the vegetation stem. However, it is possible to note that the topographical configuration of the bed surface has a strong impact on the turbulence characteristics of the flow. In fact, on the uphill stretches *F* is almost null, indicating a strong two-dimensional turbulence, since one velocity component is limited by the bed. Then, the anisotropic invariant function becomes greater than 0 on the downhill stretches, where the turbulence can develop in the three directions.

**Figure 11.** Contours of the anisotropic invariant function *F* in the test section for (**a**) Run 1, (**b**) Run 2, and (**c**) Run 3. The black broken lines indicate the edge of the vegetation stem.

Moving toward the free surface, the anisotropic invariant function gradually increases, reaching approximately *F* = 0.9. This confirms that the bed roughness influence on turbulence anisotropy vanishes moving toward the free surface.

#### *3.4. New Research Prospects*

The experimental results obtained in this study represent a new dataset that may be used for the calibration of advanced numerical models, which are usually based on isotropic turbulence hypothesis. In fact, as it was demonstrated, vegetation and bed roughness can hinder flow by acting as an obstruction, generating turbulence, and affecting the entire flow velocity distribution [32], modifying the turbulence behavior from isotropic to anisotropic moving toward the bed surface.

The modelling of such vegetated flows on rough beds clearly gets complicated if natural and complex channel cross-sections with different shapes are considered. Recent researches showed that different rectangular and trapezoidal shapes as well as the corner angles can exert a strong impact on the flow velocity distribution and its induced secondary flow [33–35]; hence, they may affect the sediment transport process [36]. Therefore, new analytical models of sidewall turbulence effect on streamwise velocity profile have been recently proposed (e.g., [35,37]) for uneven narrow and wide channels. These analytical models could be improved by considering extra turbulence zones represented by both vegetation and rough beds.

However, flow structures become even more complex when vegetation appears in channels with bedforms [38]. How the existence of vegetation changes flow pattern over gravel bedforms still remains poorly understood and could be considered as another potential development of the present research work.

#### **4. Conclusions**

The aim of the present work was the study of the turbulence anisotropy in the free stream region of turbulent flows through rigid emergent vegetation on rough beds using AIMs. Three experimental runs were performed in a uniformly distributed vegetated channel with three different bed sediments. The principal findings are summarized below.


**Author Contributions:** Conceptualization, N.P., F.C., A.D., R.G.; methodology, N.P., F.C., A.D., R.G.; formal analysis, N.P., F.C.; data curation, F.C.; writing—original draft preparation, N.P.; writing—review and editing, N.P., F.C., A.D., R.G.; supervision, R.G.; funding acquisition, R.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the "*PreFluSed—Prevenzione del rischio di alluvioni in un bacino Fluviale calabrese in presenza di trasporto di Sedimenti*" Project (*Ministero dell'Ambiente e della Tutela del Territorio e del Mare, Direzione Generale per la Salvaguardia del Territorio e delle Acque*, Italy).

**Acknowledgments:** The authors would like to thank Davide Garigliano for his valuable work during the performance of the experimental runs and the anonymous referees for their suggestions and comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Turbulence in Wall-Wake Flow Downstream of an Isolated Dunal Bedform**

#### **Sankar Sarkar 1, Sk Zeeshan Ali <sup>2</sup> and Subhasish Dey 2,\***


Received: 1 September 2019; Accepted: 19 September 2019; Published: 22 September 2019

**Abstract:** This study examines the turbulence in wall-wake flow downstream of an isolated dunal bedform. The streamwise flow velocity and Reynolds shear stress profiles at the upstream and various streamwise distances downstream of the dune were obtained. The results reveal that in the wall-wake flow, the third-order moments change their signs below the dune crest, whereas their signs remain unaltered above the crest. The near-wake flow is featured by sweep events, whereas the far-wake flow is controlled by the ejection events. Downstream of the dune, the turbulent kinetic energy production and dissipation rates, in the near-bed flow zone, are positive. However, they reduce as the vertical distance increases up to the lower-half of the dune height and beyond that, they increase with an increase in vertical distance, attaining their peaks at the crest. The turbulent kinetic energy diffusion and pressure energy diffusion rates, in the near-bed flow zone, are negative, whereas they attain their positive peaks at the crest. The anisotropy invariant maps indicate that the data plots in the wall-wake flow form a looping trend. Below the crest, the turbulence has an affinity to a two-dimensional isotropy, whereas above the crest, the anisotropy tends to reduce to a quasi-three-dimensional isotropy.

**Keywords:** hydraulics; turbulent flow; wall-wake flow; dunal bedform

#### **1. Introduction**

Turbulent flow over dunal bedforms fascinates researchers. The topic is important not only from the viewpoint of intrinsic scientific reasons, but also owing to its far-reaching applications in engineering. In addition to its practical applications, it allows a significant theoretical understanding of wake flows. Despite impressive advances over the past years, an inclusive picture of the flow and turbulence characteristics over a dunal bedform remains far from complete [1]. The dunes are created by an interaction between the flow and bed sediment particles. Dunes are kind of bedforms that are found when the flow variables, such as flow velocity and bed shear stress over a sediment bed surpass their threshold values.

Over the decades, a large corpus of experimental and numerical studies has been reported to grasp the flow features over dunal bedforms. Researchers studied the velocity field over dunes to acquire an insight into the physical features, including the reattachment point, wake region and internal boundary layer [2,3]. The experimental observations of flow over a series of twoand three-dimensional dunes revealed that the two-dimensional dunes induce stronger turbulence compared to their three-dimensional counterparts [4,5]. However, the flow characteristics over a natural dune were found to be quite different from those over an artificial dune [6]. Best [7] found that over the dune crests, the ejections dominate the instantaneous flow field.

*Water* **2019**, *11*, 1975

In a natural streamflow, an isolated dunal bedform acts as a bluff-body, producing wall-wake flow at its downstream. The wake flow downstream of an isolated dunal bedform persists up to a certain stretch until the local wake flow diffuses to and becomes the part of the undisturbed upstream flow. Figure 1 presents a conceptual representation of flow past an isolated dunal bedform in *xz* plane. Here, *x* is the streamwise distance measured from a convenient point *O* and *z* is the vertical distance from the bed. The dune length *Ld* comprises the stoss-side length *Ls* and the leeside length *Ll* (*Ld* = *Ls* + *Ll*). The dune height *Hd* is the vertical distance of the dune crest from the bed. Downstream of the dune, a flow reversal takes place, called the *near-wake flow*. Afterward, the flow is called the *far-wake flow*. In Figure 1, the lower dashed line denotes the locus of *u¯*(*z*) = 0, whereas the upper dashed line signifies the boundary layer (*u¯* = *u¯*0) in the wall-wake flow. Here, *u¯*(z) is the time-averaged streamwise flow velocity in the wake flow and *u¯* 0(*z*) is the time-averaged streamwise flow velocity in the undisturbed upstream flow. In the far downstream of the dunal bedform, the flow achieves the undisturbed upstream state, called the *fully recovered* open-channel flow.

**Figure 1.** Conceptual sketch of flow over an isolated dunal bedform.

In this context, it is pertinent to mention that for a shear-free flow, Schlichting [8] pioneered the similarity theory of the velocity defect profile in the free-wake flow downstream of a circular cylinder. The wall-wake flow downstream of an isolated dunal bedform in an approach wall-shear flow, being different from a free-wake flow, is rather intricate. The turbulence characteristics and the vortex shedding downstream of bed-mounted bluff-bodies in both near- and far-wake flows were studied by various researchers. Some of these bluff-bodies include plate [9,10], hemisphere [11], sphere [12,13], circular cylinder [14–21] and pebble cluster [22].

It is worth noting that most of the former studies were dedicated to understanding the flow features over a continuous train of dunes. In fact, little is known about the flow and turbulence characteristics over an isolated dunal bedform. This study specifically puts into focus the flow and turbulence characteristics downstream of an isolated two-dimensional dunal bedform over a rough bed in order to advance the present state-of-the-art. In addition to time-averaged streamwise flow velocity, the salient features of turbulence, including the Reynolds shear stress, turbulent bursting, turbulent kinetic energy budget and Reynolds stress anisotropy, are greatly discussed. It may be noted that the preliminary studies of flow and turbulence characteristics downstream of an isolated dunal bedform have been recently presented elsewhere [23,24].

#### **2. Experimental Design**

Experiments were performed in a re-circulatory flume, having a rectangular cross-section, at the Fluvial Mechanics Laboratory in the Indian Statistical Institute, Kolkata, India. The length, width and height of the flume were 20 m, 0.5 m and 0.5 m, respectively. The inflow discharge, supplied by a centrifugal pump, was measured by an electromagnetic gadget. The transparent sidewalls of the flume provided visual access to the flow. The flume bed, having a streamwise bed slope of 3 <sup>×</sup> 10–4, was prepared by gluing uniform gravels of median size *d*<sup>50</sup> = 2.49 mm. In the experiments, two types of isolated two-dimensional dunal bedforms, classified as Runs 1 and 2, respectively (Figure 2), were mounted on the flume bed at a distance of 7 m from the inlet. In Runs 1 and 2, the dune heights *Hd* were 0.09 m and 0.03 m, whereas the dune lengths *Ld* were 0.4 m (*LS* = 0.24 m and *Ll* = 0.16 m) and 0.3 m (*LS* = 0.24 m and *Ll* = 0.06 m), respectively. In both the runs, the same approach uniform flow condition was maintained. The approach flow depth *h* and depth-averaged approach flow velocity *U¯* <sup>0</sup> were maintained as *<sup>h</sup>* <sup>≈</sup> 0.3 m and *U¯* <sup>0</sup> <sup>≈</sup> 0.44 m s−1. The flow depth and the free surface profile were measured by a Vernier point gauge, having a precision of ±0.1 mm. The approach shear velocity *u*\* [= (τ0/ρ) 0.5], obtained from the streamwise bed slope, was 0.03 m s−1. Here, τ<sup>0</sup> is the bed shear stress and ρ is the mass density of fluid. However, the values of *u*\* in both Runs 1 and 2, determined from the Reynolds shear stress profiles, were 0.027 m s–1 and 0.025 m s−1, respectively. It is worth noting that to find the *u*\* from the Reynolds shear stress profiles, the profiles were extrapolated up to the bed. In both the runs, the flow Reynolds number was 528,000, whereas the flow Froude number was 0.256 (subcritical). The shear Reynolds number *R*\* (= *d*50*u*\*/ν, where ν is the coefficient of kinematic viscosity of fluid) was preserved to be 74.7 (> 70), setting a hydraulically rough flow regime.

**Figure 2.** Photographs of isolated dunal bedforms in (**a**) Run 1 and (**b**) Run 2. Flow direction is from left to right.

A 5 cm down-looking *Vecrtino* probe (acoustic Doppler velocimetry), also called *Vectrino plus*, was used to capture the instantaneous three-dimensional flow velocity components along the flume centreline at various relative streamwise distances *x*/*Ld* = −0.5, −0.25, 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.3, 1.7, 2.1, 2.5 and 3.3. The Vecrtino system, having a flexible sampling volume of 6 mm diameter and 1 to 4 mm height, was operated with 10 MHz acoustic frequency and 100 Hz sampling rate. The velocity components (*u*, *v*, *w*) correspond to (*x*, *y*, *z*), where *y* is the spanwise direction. It may be noted that up to the dune crest, the lowest sampling height was set as 1 mm, whereas beyond the crest, it was 2.5 mm. The closest measuring location of the data points was 2 mm. A sampling duration of 300 s was found to be adequate to obtain the time-independent flow velocity and turbulence quantities. The minimum signal-to-noise ratio was maintained as 18, whereas the minimum threshold of signal correlation was maintained as 70%. The measured data were filtered whenever required applying the *acceleration thresholding method* [25]. This method could separate and substitute the unwanted data spikes in two phases. The threshold values of 1 to 1.5 for decontaminating the measured data were ascertained by satisfying Kolmogorov '–5/3' scaling law in the inertial subrange for the spectral density function *Sdf*(*kw*) of streamwise velocity fluctuations *u*- . Here, *kw* is the wavenumber (= 2π*f*/*u¯*) and *f* is the frequency. Figure 3a,b illustrates the data plots of *Sdf*(*kw*) for velocity fluctuations (*u*- , *v*- , *w*- ) in (*x*, *y*, *z*) before and after decontaminating the data in Run 1, respectively, at a relative streamwise distance *x*/*Ld* = 0.7 and a relative vertical distance *z*/*Ld* = 0.13. The *Sdf*(*kw*) curves of decontaminated signals compare well with Kolmogorov '–5/3' scaling law in the inertial subrange for *kw* <sup>≥</sup> 30 rad s<sup>−</sup>1. In addition, it appears that the discrete spectral peaks are prominent for *kw* < 30 rad s<sup>−</sup>1. This indicates that the signals corresponding to *kw* < 30 rad s−<sup>1</sup> contained large-scale turbulent structures, while those for *kw* <sup>≥</sup> 30 rad s−<sup>1</sup> confirmed a pure turbulence. Therefore, a high-pass filter with a cut-off wavenumber of 30 rad s−<sup>1</sup> was used to filter the data.

**Figure 3.** Spectral density function *Sdf*(*kw*) versus wavenumber *kw* (**a**) before and (**b**) after decontaminating the data in Run 1 at a relative streamwise distance *x*/*Ld* = 0.7 and a relative vertical distance *z*/*Ld* = 0.13.

In order to find the uncertainty of Vectrino data, 15 samples were collected at a sampling rate of 100 Hz for a duration of 300 s at a vertical distance *z* = 5 mm. Table 1 summarizes the results of uncertainty estimations of the time-averaged velocity components (*u¯*, *v*, *w*) and the turbulence intensities [(*uu*-) 0.5, (*vv*-) 0.5, (*ww*-) 0.5] in (*x*, *y*, *z*) and the Reynolds shear stress τ per unit mass density of fluid (= −*uw*-). It is pertinent to mention that to avoid bias and random errors, the samplings were done every time after resuming the experiments. The errors for the time-averaged velocity components, turbulence intensities and Reynolds shear stress were within ±4%, ±7% and ±8%, respectively. This confirmed the appropriateness of the data sampling with 100 Hz sampling rate. Further, it was necessary to ascertain the fully-developed undisturbed approach velocity profiles for both the Runs. Figure 4 shows the vertical profiles of nondimensional streamwise flow velocity *u¯*<sup>+</sup> (= *u¯*/*u*\*) at the upstream of isolated dunal bedforms for both Runs 1 and 2. The data plots compare well with the classical logarithmic law *u¯*/*u*\* = κ−1ln(*z*/*d*50) + 8.5 for a hydraulically rough flow regime. Here, κ is the von Kármán constant (= 0.41). This confirmed the acceptability of the fully-developed undisturbed approach flow velocity profiles for a hydraulically rough flow regime. τ

**Figure 4.** Vertical profiles of nondimensional streamwise flow velocity *u¯*<sup>+</sup> at the upstream of isolated dunal bedforms for Runs 1 and 2.


**Table 1.** Uncertainty estimation for Vectrino.

\* Standard deviation. † Average of maximum (negative and positive) percentage error.

#### **3. Time-Averaged Flow**

#### *3.1. Streamwise Flow Velocity*

Figure 5 shows the vertical profiles of nondimensional streamwise flow velocity *u¯*<sup>+</sup> at upstream and various downstream relative streamwise distances *x*/*Ld* in Runs 1 and 2. Immediate downstream of the dune (*x*/*Ld* = 1), the wall-shear flow separates from the dune crest, giving rise to a flow reversal owing to negative streamwise flow velocity. The near-wake flow zone extends up to *x*/*Ld* ≈ 1.7. As the flow reaches further downstream, the flow reversal disappears. In addition, the streamwise flow velocity, having a velocity defect, starts to recover the undisturbed upstream velocity profile in the far-wake zone (*x*/*Ld* = 2.1 to 2.5). At *x*/*Ld* ≈ 3.3, the velocity profile appears to follow the undisturbed upstream velocity profile. It is also evident that above the relative vertical distance *z*/*Hd* = 1.5, the values of *u¯*<sup>+</sup> remain almost the same irrespective of *x*/*Ld*. However, the extents of the near- and far-wake flow zones in Runs 1 and 2 are different because of the effects of dune dimensions. It is worth mentioning that in wall-wake flows downstream of a sphere and a horizontal cylinder, the velocity profiles appear to follow their corresponding undisturbed upstream velocity profile at streamwise distances equaling roughly 8.5 and 7 times the diameter of sphere and cylinder, respectively [12,21].

**Figure 5.** Vertical profiles of nondimensional streamwise flow velocity *u¯*<sup>+</sup> at the upstream and various downstream relative streamwise distances *x*/*Ld* of isolated dunal bedforms for (**a**) Run 1 and (**b**) Run 2.

#### *3.2. Reynolds Shear Stress*

Figure 6 presents the vertical profiles of nondimensional Reynolds shear stress τ<sup>+</sup> (= τ/*u*<sup>2</sup> <sup>∗</sup>) at the upstream and various downstream relative streamwise distances *x*/*Ld* in Runs 1 and 2. Upstream of the dune (*x*/*Ld* = <sup>−</sup>0.5), theτ<sup>+</sup> profile follows a linear law. The <sup>τ</sup><sup>+</sup> is approximately unity at the relative vertical distance *z*/*Hd* = 0 and then, it reduces with an increase in relative vertical distance to become zero at the free surface (if the profiles would be extended up to the free surface). Immediate downstream of the dune (*x*/*Ld* = 1), the τ<sup>+</sup> is negative in the near-bed flow zone. Thereafter, it increases with an increase in *z*/*Hd*, attaining a positive peak at the dune crest (*z*/*Hd* = 1). Above the crest, the τ<sup>+</sup> decreases with an increase in *z*/*Hd* and attains almost similar pattern to the upstream profile for *z*/*Hd* > 1.5. It appears that for a given *z*/*Hd*, the τ<sup>+</sup> decreases with an increase in *x*/*Ld*. In particular, at *x*/*Ld* <sup>≈</sup> 3.3, the <sup>τ</sup><sup>+</sup> profile becomes almost similar to the upstream profile at *x*/*Ld* = <sup>−</sup>0.5. It may be noted that for *z*/*Hd* > 1.75, the values of τ<sup>+</sup> at various *x*/*Ld* are nearly similar. Therefore, it may be concluded that the Reynolds shear stress in the wall-wake flow is influenced by the dune up to a vertical distance of approximately 1.75 times the dune height and a streamwise distance of approximately 2.5 times the dune length.

**Figure 6.** Vertical profiles of nondimensional Reynolds shear stress τ<sup>+</sup> at the upstream and various downstream relative streamwise distances *x*/*Ld* of isolated dunal bedforms for (**a**) Run 1 and (**b**) Run 2.

#### **4. Third-Order Moments**

The third-order moments of velocity fluctuations offer relevant probabilistic information about the flux and the advection of Reynolds normal stresses. In addition, they give an indication of the predominance of turbulent bursting events [26]. The third-order moments, in the generalized form in *xz* plane, is expressed as *mjk* <sup>=</sup> &*u<sup>j</sup> <sup>w</sup>*&*k*, where &*<sup>u</sup>* <sup>=</sup> *<sup>u</sup>*- /(*uu*-) 0.5, *<sup>w</sup>*& <sup>=</sup> *<sup>w</sup>*- /(*ww*-) 0.5 and *j* + *k* = 3. Therefore, depending on the values of *j* and *k*, the third-order moments are given as, *m*<sup>30</sup> = *uuu*-/(*uu*-) 1.5, *m*<sup>03</sup> = *www*-/(*ww*-) 1.5, *m*<sup>21</sup> = *uuw*-/[(*uu*-)×(*ww*-) 0.5] and *m*<sup>12</sup> = *uww*-/[(*uu*-) 0.5×(*ww*-)]. Here, the *m*<sup>30</sup> signifies the skewness of *u*- , indicating the streamwise flux of the streamwise Reynolds normal stress *uu*-. The *m*<sup>03</sup> defines the skewness of *w*- , suggesting the vertical flux of the vertical Reynolds normal stress *ww*-. In addition, the *m*<sup>21</sup> represents the advection of *uu* in the vertical direction, whereas the *m*<sup>12</sup> demonstrates the advection of *ww*in the streamwise direction.

Figure 7 shows the vertical profiles of *m*<sup>30</sup> and *m*<sup>03</sup> at the upstream and various downstream relative streamwise distances *x*/*Ld* in Runs 1 and 2. Upstream of the dune (*x*/*Ld* = −0.5), the *m*<sup>30</sup> and *m*03, in the near-bed flow zone, are negative and positive, respectively. Then, they increase with an increase in relative vertical distance *z*/*Hd* without changing their signs. Downstream of the dune (*x*/*Ld* = 1 to 2.1), for a given *x*/*Ld*, the *m*<sup>30</sup> and *m*03, in the near-bed flow zone, start with positive and negative values, respectively. Thereafter, they increase slowly with an increase in *z*/*Hd* until they attain their

respective positive and negative peaks at *z*/*Hd* ≈ 0.75 and 0.5. As the *z*/*Hd* increases further, the *m*<sup>30</sup> and *m*<sup>03</sup> reduce quickly, changing their signs at *z*/*Hd* = 1, and for *z*/*Hd* > 1, they become independent of *z*/*Hd*. However, these features disappear gradually with an increase in *x*/*Ld*. It may be noted that the *m*<sup>30</sup> and *m*<sup>03</sup> profiles at *x*/*Ld* = 3.3 remain almost similar to those in the upstream.

**Figure 7.** Vertical profiles of third-order moments *m*<sup>30</sup> and *m*<sup>03</sup> at various relative streamwise distances *x*/*Ld* in Runs 1 and 2.

Figure 8 depicts the vertical profiles of *m*<sup>21</sup> and *m*<sup>12</sup> at the upstream and various downstream relative streamwise distances *x*/*Ld* in Runs 1 and 2. It appears that upstream of the dune (*x*/*Ld* = −0.5), the *m*<sup>21</sup> and *m*12, in the near-bed flow zone, attain positive and negative values, respectively. Then, they increase with an increase in relative vertical distance *z*/*Hd* up to a certain height. Subsequently, they reduce with an increase in *z*/*Hd*, becoming independent of *z*/*Hd* for *z*/*Hd* > 1.1. Downstream of the dune (*x*/*Ld* = 1 to 2.1), for a given *x*/*Ld*, the *m*<sup>21</sup> and *m*12, in the near-bed flow zone, are negative and positive, respectively. Then, they increase with an increase in *z*/*Hd* attaining their respective peaks. Afterward, they reduce quickly, changing their signs at the dune crest (*z*/*Hd* = 1). Thereafter, the *m*<sup>21</sup> and *m*<sup>12</sup> profiles recover their upstream profiles. Downstream of the dune, an advection of *uu* in the upward direction and that of *ww* in the upstream direction prevail below the crest. In fact, below the crest, there appears a streamwise acceleration, which is linked with the downward flux causing sweeps

with an advection of *uu* in the downward direction. By contrast, above the crest, the streamwise deceleration is associated with an upward flux producing ejections with an advection of *uu* in the upward direction.

**Figure 8.** Vertical profiles of third-order moments *m*<sup>21</sup> and *m*<sup>12</sup> at various relative streamwise distances *x*/*Ld* in Runs 1 and 2.

#### **5. Quadrant Analysis**

Lu and Willmarth [27] suggested that the bursting events can be quantified by performing the quadrant analysis of velocity fluctuations *u* and *w* on a *u*- *w* plane. The turbulent bursting includes four events in four distinct quadrants *i* = 1 to 4, such as (i) *Q*1 events or *outward interactions* (*i* = 1 and *u*- , *w*- > 0), (ii) *Q*2 events or *ejections* (*i* = 2 and *u*- < 0, *w*- > 0), (iii) *Q*3 events or *inward interactions* (*i* = 3 and *u*- , *w*- < 0) and (iv) *Q*4 events or *sweeps* (*i* = 4 and *u*- > 0, *w*- < 0). Outside the *hole size H*, the contribution of *uw*- *<sup>i</sup>*,*<sup>H</sup>* from the quadrant *i* to *uw* is ascertained by averaging the quantity *u*- (*t*)*w*- (*t*)*Fi*,*<sup>H</sup>* over the sampling duration. Here, *Fi*,*<sup>H</sup>* is the *detection function*, defined as *Fi*,*<sup>H</sup>* = 1 if the pair (*u*- , *w*- ) in the quadrant *i* satisfies the condition |*u*- *w*- |≥ *H*(*uu*-) 0.5(*ww*-) 0.5 and *Fi*,*<sup>H</sup>* = 0 otherwise. The *relative fractional contributions Si*,*<sup>H</sup>* toward the Reynolds shear stress production is expressed as *Si*,*<sup>H</sup>* = *uw*- *<sup>i</sup>*,*<sup>H</sup>* /*uw*-. It turns out that for *H* = 0, the sum of *S*1,0, *S*2,0, *S*3,0 and *S*4,0 becomes unity.

Figures 9 and 10 show the vertical profiles of *Si*,0 at the upstream and various downstream relative streamwise distances *x*/*Ld* in Runs 1 and 2, respectively. Upstream of the dune (*x*/*Ld* = –0.5), the *Q*2 and *Q*4 events remain the most and the second-most contributing events, respectively, to the production of Reynolds shear stress. However, the *Q*1 and *Q*3 events are trivial across the flow depth. Downstream of the dune (*x*/*Ld* = 1 to 2.1), all the four events contribute largely below the dune crest with prevailing *Q*4 events in the form of arrival of high-speed fluid streaks. At *x*/*Ld* = 2.5, contributions from the *Q*2 and *Q*4 events appear to be nearly equal below the crest. Further downstream (*x*/*Ld* = 3.3), the *Q*2 events dominate over *Q*4 events in the form of arrival of low-speed fluid streaks. It may be noted that above the crest (*z*/*Hd* > 1), the *Q*2 events are the most contributing events regardless of *x*/*Ld*.

**Figure 9.** Vertical profiles of relative fractional contributions *Si*,0 at various relative streamwise distances *x*/*Ld* in Run 1.

**Figure 10.** Vertical profiles of relative fractional contributions *Si*,0 at various relative streamwise distances *x*/*Ld* in Run 2.

Figure 11a,b shows the variations of relative fractional contributions |*Si*,*H*| with hole size *H* in Run 1 for different relative vertical distances *z*/*Hd* (=0.05, 0.25 and 0.5) at relative streamwise distances *x*/*Ld* = –0.5 (uninterrupted upstream flow) and 1 (near-wake flow), whereas Figure 12a,b shows those at *x*/*Ld* = 1.7 (far-wake flow) and 3.3 (near to fully recovered flow). It appears that upstream of the dune (*x*/*Ld* = –0.5), the *Q*1 and *Q*3 events for *z*/*Hd* = 0.05 contribute minimally to the Reynolds shear stress production as compared to the *Q*2 and *Q*4 events. However, for *z*/*Hd* = 0.05, the pairs (*Q*1, *Q*3) and (*Q*2, *Q*4) are equal, indicating that they mutually cancel the dominance of each other. At *x*/*Ld* = –0.5, the *Q*2 events remain dominant for *z*/*D* = 0.25 and 0.5. Immediate downstream of the dune (*x*/*Ld* = 1), the *Q*1 and *Q*3 events, for a given *z*/*Hd*, are smaller than *Q*2 and *Q*4 events. However, at the downstream, the *Q*4 remain the most dominant events for *z*/*Hd* = 0.05, 0.25 and 0.5. At *x*/*Ld* = 1.7, these features remain similar to those at *x*/*Ld* = 1, but with relatively smaller *Q*4 events. Far downstream of the dune (*x*/*Ld* = 3.3), the events, for a given *z*/*Hd*, follow the upstream trend. The contributions from the events are considerable for lower values of *H*. In essence, for *H* ≥ 12, all the events are trivial at different streamwise and vertical distances.

**Figure 11.** Relative fractional contributions |*Si*,*H*| as a function of hole size *H* in Run 1 at relative streamwise distances (a) *x*/*Ld* = –0.5 and (b) *x*/*Ld* = 1 for relative vertical distances *z*/*Hd* = 0.05, 0.25 and 0.5.

**Figure 12.** Relative fractional contributions |*Si*,*H*| as a function of hole size *H* in Run 1 at relative streamwise distances (a) *x*/*Ld* = 1.7 and (b) *x*/*Ld* = 3.3 for relative vertical distances *z*/*Hd* = 0.05, 0.25 and 0.5.

#### **6. Turbulent Kinetic Energy Budget**

The turbulent kinetic energy budget reads *tP* = ε + *tD* + *pD* − *vD*, where *tP* is the turbulent kinetic energy production rate (= –*uw*-∂*u¯*/∂*z*), ε is the turbulent kinetic energy dissipation rate, *tD* is the turbulent kinetic energy diffusion rate (= ∂*fkw*/∂*z)*, *fkw* is the vertical flux of turbulent kinetic energy, *pD* is the pressure energy diffusion rate [= ρ−1∂(*pw*-)/∂*z*], *p* is the pressure fluctuations, *vD* is the viscous diffusion rate (=ν∂2*k*/∂*z*2) and *k* is the turbulent kinetic energy. In an open channel flow, the *vD*

is insignificant compared to other components of the turbulent kinetic energy budget. In this study, Kolmogorov second hypothesis was applied to determine the ε from the velocity power spectra [28]. The *tP* and *tD* were determined from the experimental data, whereas the *pD* was obtained from the relationship *pD* = *tP* − ε − *tD.* In nondimensional form, the set of variables (*tP,* ε*, tD, pD*) is expressed as (*TP*, *ED*, *TD*, *PD*) <sup>=</sup> (*tP*, <sup>ε</sup>*, tD*, *pD*) <sup>×</sup> (*Hd*/*u*<sup>3</sup> ∗).

Figure 13 illustrates the vertical profiles of nondimensional components of the turbulent kinetic energy budget at various relative streamwise distances *x*/*Ld* in Run 1. Upstream of the dune (*x*/*Ld* = –0.5), all the components of the turbulent kinetic energy budget, in the near-bed flow zone, are positive with a sequence of magnitude *TP* > *ED* > *PD* > *TD* and then, they reduce with an increase in relative vertical distance *z*/*Hd*. Above the dune crest (*z*/*Hd* > 1), they are quite small. Downstream of the dune (*x*/*Ld* = 1 to 2.1), the peaks of *TP*, *ED*, *PD* and *TD* are found to appear at the crest. In the near-bed flow zone, the *TP* and *ED* are positive, whereas the *PD* and *TD* are negative for *x*/*Ld* = 1 to 2.1. Downstream of the dune, the absolute values of *TP*, *ED*, *PD* and *TD* decrease with an increase in *x*/*Ld*. In particular, at *x*/*Ld* = 3.3, the *TP*, *ED*, *PD* and *TD* profiles are almost similar to those of the undisturbed upstream flow at *x*/*Ld* = −0.5.

**Figure 13.** *Cont*.

**Figure 13.** Vertical profiles of the nondimensional components of turbulent kinetic energy budget *TP*, *ED*, *TD* and *PD* at various relative streamwise distances *x*/*Ld* in Run 1.

#### **7. Reynolds Stress Anisotropy**

An *isotropic turbulence* refers to an idealized condition, where the velocity fluctuations at a specific point remain invariant to the rotation of axes. In a lucid way, this condition indicates that the Reynolds normal stresses are identical (σ*<sup>x</sup>* = σ*<sup>y</sup>* = σ*z*), where (σ*x*, σ*y*, σ*z*) = (*uu*-, *vv*-, *ww*-). By contrast, in an *anisotropic turbulence*, the Reynolds normal stresses are dissimilar, because the velocity fluctuations *u*- *i* [= (*u*- , *v*- , *w*- ) for *i* = (1, 2, 3)] are directionally preferred.

The *Reynolds stress anisotropy tensor bij* is expressed as *bij* = *u*- *iu*- *<sup>j</sup>*/(2*k*) − δ*ij*/3, where δ*ij* is the Kronecker delta function [δ*ij*(*i* = *j*) = 1 and δ*ij*(*i j*) = 0]. To ascertain the degree and the nature of anisotropy, the second and third principal invariants, *I*<sup>2</sup> (= –*bijbij*/2) and *I*<sup>3</sup> (= *bijbjkbki*/3), respectively, are introduced. The Reynolds stress anisotropy is determined by plotting –*I*<sup>2</sup> as a function of *I*3, called the *anisotropy invariant map* (AIM). In an AIM, the possible turbulence states are confined to a triangle, called the *Lumley triangle* (Figure 14). The left-curved and the right-curved boundaries of the Lumley triangle, given by *<sup>I</sup>*<sup>3</sup> <sup>=</sup> <sup>±</sup>2(−*I*2/3)3/2, are symmetric about the *plane-strain limit* (*I*<sup>3</sup> <sup>=</sup> 0). In addition, the top-linear boundary of the Lumley triangle obeys *I*<sup>3</sup> = −(9*I*<sup>2</sup> + 1)/27. Dey et al. [29] envisioned the Reynolds stress anisotropy from the perspective of the shape of ellipsoid formed by the Reynolds normal stresses (σ*x*, σ*y*, σ*z*) in (*x*, *y*, *z*). In an isotropic turbulence (σ*<sup>x</sup>* = σ*<sup>y</sup>* = σ*z*), the stress ellipsoid becomes a *sphere* (Figure 14). On the left-curved boundary, called the *axisymmetric contraction limit*, one component of Reynolds normal stress is smaller than the other two equal components (σ*<sup>x</sup>* = σ*<sup>y</sup>* > σ*z*), forming the stress ellipsoid an *oblate spheroid*. On the left vertex, called the *two-component axisymmetric limit*, one component of Reynolds normal stress disappears (σ*<sup>x</sup>* = σ*<sup>y</sup>* and σ*<sup>z</sup>* = 0) to make the stress ellipsoid a *circular disc* (Figure 14). On the right-curved boundary, called the *axisymmetric expansion limit*, one component of Reynolds normal stress is larger than the other two equal components (σ*<sup>x</sup>* = σ*<sup>y</sup>* < σ*z*), making the stress ellipsoid a *prolate spheroid* (Figure 14). Further, on the top-linear boundary, called the *two-component limit*, one component of Reynolds normal stress is larger than the other component together with a third vanishing component (σ*<sup>x</sup>* > σ*<sup>y</sup>* and σ*<sup>z</sup>* = 0), producing the stress ellipsoid an *elliptical disk*. The point of intersecting of the plain-strain limit and the two-component limit is called the *two-component plain-strain limit*. Moreover, on the right vertex of the Lumley triangle, called the *one-component limit* [(σ*<sup>x</sup>* > 0, σ*<sup>y</sup>* = σ*<sup>z</sup>* = 0) or (σ*<sup>x</sup>* = σ*<sup>y</sup>* = 0, σ*<sup>z</sup>* > 0)], only one component of Reynolds normal stress sustains to make the stress ellipsoid a *straight line* (Figure 14).

**Figure 14.** Conceptual representation of Reynolds stress anisotropy.

Figure 15 shows the data plots of −*I*<sup>2</sup> versus *I*3, confined to the AIM boundaries, at various relative streamwise distances *x*/*Ld* in Runs 1 and 2. Upstream of the dune (*x*/*Ld* = −0.5), the data plots initiate from the near left vertex, moving toward the bottom cusp, and then, with an increase in vertical distance, they cross the plain-strain limit to shift toward the right-curved boundary. The trends of the data plots for both Runs 1 and 2 are almost monotonic. The AIM of the upstream indicates that as the vertical distance increases, the turbulence anisotropy tends to reduce to a quasi-three-dimensional isotropy. Immediate downstream of the dune, the data plots tend to create a stretched loop inclined to the left-curved boundary. However, below the dune crest (*z*/*Hd* < 1), the data plots in the near-bed flow zone initiate from the plain-strain limit and with an increase in vertical distance up to the crest, they shift toward the left vertex following the left-curved boundary. This suggests that the turbulence anisotropy has an affinity to a two-dimensional isotropy. Above the crest, the data plots turn toward the right and as the vertical distance increases further, they move toward the bottom cusp following the left-curved boundary. This demonstrates that the turbulence anisotropy tends to reduce to a quasi-three-dimensional isotropy. Further downstream (*x*/*Ld* = 1.7), the size of the loop created by the data plots reduces forming a tail, and the loop disappears at *x*/*Ld* = 3.3, signifying a recovery of the undisturbed upstream trend. It therefore appears that that below the crest, the turbulence has an affinity to a two-dimensional isotropy, whereas above the crest, a quasi-three-dimensional isotropy prevails.

From the perspective of the shape of stress ellipsoid, Figure 15 shows that below the dune crest, an oblate spheroid axisymmetric turbulence is predominant in the wall-wake flow. The line of plain-strain limit (*I*<sup>3</sup> = 0) is touched by the curve through the data plots in the near-bed flow zone. This reveals that the axisymmetric contraction to the oblate spheroid enhances as the vertical distance increases up to the crest. However, the axisymmetric contraction to oblate spheroid lessens with a further increase in vertical distance above the crest.

**Figure 15.** AIMs at various relative streamwise distances *x*/*Ld* in Runs 1 and 2.

#### **8. Conclusions**

This study puts into focus the turbulence in wall-wake flow downstream of an isolated dunal bedform. The vertical profiles of streamwise flow velocity reveal that the near-wake flow zone extends up to 1.7 times the dune length, whereas the streamwise flow velocity profile follows the undisturbed upstream velocity profile beyond 3.3 times the dune length. The Reynolds shear stress in the wall-wake flow is affected by the dune up to a vertical distance of 1.75 times the dune height and a streamwise distance of 2.5 times the dune length. The third-order moment of velocity fluctuations reveal that downstream of the dune, a streamwise acceleration having a downward flux prevails below the dune crest, whereas a streamwise deceleration having an upward flux persists above the crest. Below the crest, the sweeps are found to be the predominant events, whereas above the crest, the ejections are the major events. The components of the turbulent kinetic energy budget reveal an amplification of the magnitudes of the turbulent parameters, which attain their maximum peaks at the crest. The anisotropy invariant maps show that the data plots in the wall-wake flow start from the plain-strain limit in the near-bed flow zone, shifting toward the left vertex of the Lumley triangle up to the crest to show an affinity to a two-dimensional isotropy. Above the crest, the data plots show an affinity to a quasi-three-dimensional isotropy.

In essence, this study advances the current understanding of flow and turbulence characteristics in wall-wake flow downstream of an isolated dunal bedform. The experimental results provide guidance to numerical simulations of wall-wake flow. In addition, this study may be helpful, at least qualitatively, to simulate the mobile-bed flow downstream of a dunal bedform.

**Author Contributions:** Conceptualization, S.S. and S.D; data curation, S.S.; formal analysis, S.S., S.Z.A. and S.D; funding acquisition, S.S.; investigation, S.S., S.Z.A. and S.D; methodology, S.S. and S.D; resources, S.S., S.Z.A. and S.D; writing—original draft, S.S., S.Z.A. and S.D; writing—review and editing, S.S., S.Z.A. and S.D; supervision, S.D.

**Funding:** This research was funded by Indian Statistical Institute, Kolkata.

**Acknowledgments:** The third author (S.D) acknowledges the JC Bose fellowship project (project code: JBD) to coordinate this research program.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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