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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{d}comprises the stoss-side length L

_{s}and the leeside length L

_{l}(L

_{d}= L

_{s}+ L

_{l}). The dune height H

_{d}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 ū(z) = 0, whereas the upper dashed line signifies the boundary layer (ū = ū

_{0}) in the wall-wake flow. Here, ū(z) is the time-averaged streamwise flow velocity in the wake flow and ū

_{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.

## 2. Experimental Design

^{–4}, was prepared by gluing uniform gravels of median size d

_{50}= 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 H

_{d}were 0.09 m and 0.03 m, whereas the dune lengths L

_{d}were 0.4 m (L

_{S}= 0.24 m and L

_{l}= 0.16 m) and 0.3 m (L

_{S}= 0.24 m and L

_{l}= 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 Ū

_{0}were maintained as h ≈ 0.3 m and Ū

_{0}≈ 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

_{*}[= (τ

_{0}/ρ)

^{0.5}], obtained from the streamwise bed slope, was 0.03 m s

^{−1}. Here, τ

_{0}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.

_{d}= −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 S

_{df}(k

_{w}) of streamwise velocity fluctuations u′. Here, k

_{w}is the wavenumber (= 2πf/ū) and f is the frequency. Figure 3a,b illustrates the data plots of S

_{df}(k

_{w}) 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/L

_{d}= 0.7 and a relative vertical distance z/L

_{d}= 0.13. The S

_{df}(k

_{w}) curves of decontaminated signals compare well with Kolmogorov ‘–5/3’ scaling law in the inertial subrange for k

_{w}≥ 30 rad s

^{−1}. In addition, it appears that the discrete spectral peaks are prominent for k

_{w}< 30 rad s

^{−1}. This indicates that the signals corresponding to k

_{w}< 30 rad s

^{−1}contained large-scale turbulent structures, while those for k

_{w}≥ 30 rad s

^{−1}confirmed a pure turbulence. Therefore, a high-pass filter with a cut-off wavenumber of 30 rad s

^{−1}was used to filter the data.

^{0.5}, ($\overline{{v}^{\prime}{v}^{\prime}}$)

^{0.5}, ($\overline{{w}^{\prime}{w}^{\prime}}$)

^{0.5}] in (x, y, z) and the Reynolds shear stress τ per unit mass density of fluid (= −$\overline{{u}^{\prime}{w}^{\prime}}$). 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 ū

^{+}(= ū/u

_{*}) at the upstream of isolated dunal bedforms for both Runs 1 and 2. The data plots compare well with the classical logarithmic law ū/u

_{*}= κ

^{−1}ln(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.

## 3. Time-Averaged Flow

#### 3.1. Streamwise Flow Velocity

^{+}at upstream and various downstream relative streamwise distances x/L

_{d}in Runs 1 and 2. Immediate downstream of the dune (x/L

_{d}= 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/L

_{d}≈ 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/L

_{d}= 2.1 to 2.5). At x/L

_{d}≈ 3.3, the velocity profile appears to follow the undisturbed upstream velocity profile. It is also evident that above the relative vertical distance z/H

_{d}= 1.5, the values of ū

^{+}remain almost the same irrespective of x/L

_{d}. 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].

#### 3.2. Reynolds Shear Stress

^{+}(= τ/${u}_{*}^{2}$) at the upstream and various downstream relative streamwise distances x/L

_{d}in Runs 1 and 2. Upstream of the dune (x/L

_{d}= −0.5), theτ

^{+}profile follows a linear law. The τ

^{+}is approximately unity at the relative vertical distance z/H

_{d}= 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/L

_{d}= 1), the τ

^{+}is negative in the near-bed flow zone. Thereafter, it increases with an increase in z/H

_{d}, attaining a positive peak at the dune crest (z/H

_{d}= 1). Above the crest, the τ

^{+}decreases with an increase in z/H

_{d}and attains almost similar pattern to the upstream profile for z/H

_{d}> 1.5. It appears that for a given z/H

_{d}, the τ

^{+}decreases with an increase in x/L

_{d}. In particular, at x/L

_{d}≈ 3.3, the τ

^{+}profile becomes almost similar to the upstream profile at x/L

_{d}= −0.5. It may be noted that for z/H

_{d}> 1.75, the values of τ

^{+}at various x/L

_{d}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.

## 4. Third-Order Moments

_{jk}= $\overline{{\tilde{u}}^{j}{\tilde{w}}^{k}}$, where $\tilde{u}$ = u′/($\overline{{u}^{\prime}{u}^{\prime}}$)

^{0.5}, $\tilde{w}$ = w′/($\overline{{w}^{\prime}{w}^{\prime}}$)

^{0.5}and j + k = 3. Therefore, depending on the values of j and k, the third-order moments are given as, m

_{30}= $\overline{{u}^{\prime}{u}^{\prime}{u}^{\prime}}$/($\overline{{u}^{\prime}{u}^{\prime}}$)

^{1.5}, m

_{03}= $\overline{{w}^{\prime}{w}^{\prime}{w}^{\prime}}$/($\overline{{w}^{\prime}{w}^{\prime}}$)

^{1.5}, m

_{21}= $\overline{{u}^{\prime}{u}^{\prime}{w}^{\prime}}$/[($\overline{{u}^{\prime}{u}^{\prime}}$)×($\overline{{w}^{\prime}{w}^{\prime}}$)

^{0.5}] and m

_{12}= $\overline{{u}^{\prime}{w}^{\prime}{w}^{\prime}}$/[($\overline{{u}^{\prime}{u}^{\prime}}$)

^{0.5}×($\overline{{w}^{\prime}{w}^{\prime}}$)]. Here, the m

_{30}signifies the skewness of u′, indicating the streamwise flux of the streamwise Reynolds normal stress $\overline{{u}^{\prime}{u}^{\prime}}$. The m

_{03}defines the skewness of w′, suggesting the vertical flux of the vertical Reynolds normal stress $\overline{{w}^{\prime}{w}^{\prime}}$. In addition, the m

_{21}represents the advection of $\overline{{u}^{\prime}{u}^{\prime}}$ in the vertical direction, whereas the m

_{12}demonstrates the advection of $\overline{{w}^{\prime}{w}^{\prime}}$ in the streamwise direction.

_{30}and m

_{03}at the upstream and various downstream relative streamwise distances x/L

_{d}in Runs 1 and 2. Upstream of the dune (x/L

_{d}= −0.5), the m

_{30}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/H

_{d}without changing their signs. Downstream of the dune (x/L

_{d}= 1 to 2.1), for a given x/L

_{d}, the m

_{30}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/H

_{d}until they attain their respective positive and negative peaks at z/H

_{d}≈ 0.75 and 0.5. As the z/H

_{d}increases further, the m

_{30}and m

_{03}reduce quickly, changing their signs at z/H

_{d}= 1, and for z/H

_{d}> 1, they become independent of z/H

_{d}. However, these features disappear gradually with an increase in x/L

_{d}. It may be noted that the m

_{30}and m

_{03}profiles at x/L

_{d}= 3.3 remain almost similar to those in the upstream.

_{21}and m

_{12}at the upstream and various downstream relative streamwise distances x/L

_{d}in Runs 1 and 2. It appears that upstream of the dune (x/L

_{d}= −0.5), the m

_{21}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/H

_{d}up to a certain height. Subsequently, they reduce with an increase in z/H

_{d}, becoming independent of z/H

_{d}for z/H

_{d}> 1.1. Downstream of the dune (x/L

_{d}= 1 to 2.1), for a given x/L

_{d}, the m

_{21}and m

_{12}, in the near-bed flow zone, are negative and positive, respectively. Then, they increase with an increase in z/H

_{d}attaining their respective peaks. Afterward, they reduce quickly, changing their signs at the dune crest (z/H

_{d}= 1). Thereafter, the m

_{21}and m

_{12}profiles recover their upstream profiles. Downstream of the dune, an advection of $\overline{{u}^{\prime}{u}^{\prime}}$ in the upward direction and that of $\overline{{w}^{\prime}{w}^{\prime}}$ 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 $\overline{{u}^{\prime}{u}^{\prime}}$ in the downward direction. By contrast, above the crest, the streamwise deceleration is associated with an upward flux producing ejections with an advection of $\overline{{u}^{\prime}{u}^{\prime}}$ in the upward direction.

## 5. Quadrant Analysis

_{i}

_{,H}over the sampling duration. Here, F

_{i}

_{,H}is the detection function, defined as F

_{i}

_{,H}= 1 if the pair (u′, w′) in the quadrant i satisfies the condition |u′w′|≥ H($\overline{{u}^{\prime}{u}^{\prime}}$)

^{0.5}($\overline{{w}^{\prime}{w}^{\prime}}$)

^{0.5}and F

_{i}

_{,H}= 0 otherwise. The relative fractional contributions S

_{i}

_{,H}toward the Reynolds shear stress production is expressed as S

_{i}

_{,H}= ${\overline{{u}^{\prime}{w}^{\prime}}|}_{{}^{i,H}}$/$\overline{{u}^{\prime}{w}^{\prime}}$. 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.

_{i}

_{,0}at the upstream and various downstream relative streamwise distances x/L

_{d}in Runs 1 and 2, respectively. Upstream of the dune (x/L

_{d}= –0.5), the Q2 and Q4 events remain the most and the second-most contributing events, respectively, to the production of Reynolds shear stress. However, the Q1 and Q3 events are trivial across the flow depth. Downstream of the dune (x/L

_{d}= 1 to 2.1), all the four events contribute largely below the dune crest with prevailing Q4 events in the form of arrival of high-speed fluid streaks. At x/L

_{d}= 2.5, contributions from the Q2 and Q4 events appear to be nearly equal below the crest. Further downstream (x/L

_{d}= 3.3), the Q2 events dominate over Q4 events in the form of arrival of low-speed fluid streaks. It may be noted that above the crest (z/H

_{d}> 1), the Q2 events are the most contributing events regardless of x/L

_{d}.

_{i}

_{,H}| with hole size H in Run 1 for different relative vertical distances z/H

_{d}(=0.05, 0.25 and 0.5) at relative streamwise distances x/L

_{d}= –0.5 (uninterrupted upstream flow) and 1 (near-wake flow), whereas Figure 12a,b shows those at x/L

_{d}= 1.7 (far-wake flow) and 3.3 (near to fully recovered flow). It appears that upstream of the dune (x/L

_{d}= –0.5), the Q1 and Q3 events for z/H

_{d}= 0.05 contribute minimally to the Reynolds shear stress production as compared to the Q2 and Q4 events. However, for z/H

_{d}= 0.05, the pairs (Q1, Q3) and (Q2, Q4) are equal, indicating that they mutually cancel the dominance of each other. At x/L

_{d}= –0.5, the Q2 events remain dominant for z/D = 0.25 and 0.5. Immediate downstream of the dune (x/L

_{d}= 1), the Q1 and Q3 events, for a given z/H

_{d}, are smaller than Q2 and Q4 events. However, at the downstream, the Q4 remain the most dominant events for z/H

_{d}= 0.05, 0.25 and 0.5. At x/L

_{d}= 1.7, these features remain similar to those at x/L

_{d}= 1, but with relatively smaller Q4 events. Far downstream of the dune (x/L

_{d}= 3.3), the events, for a given z/H

_{d}, 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.

## 6. Turbulent Kinetic Energy Budget

_{P}= ε + t

_{D}+ p

_{D}− v

_{D}, where t

_{P}is the turbulent kinetic energy production rate (= –$\overline{{u}^{\prime}{w}^{\prime}}$∂ū/∂z), ε is the turbulent kinetic energy dissipation rate, t

_{D}is the turbulent kinetic energy diffusion rate (= ∂f

_{kw}/∂z), f

_{kw}is the vertical flux of turbulent kinetic energy, p

_{D}is the pressure energy diffusion rate [= ρ

^{−1}∂($\overline{{p}^{\prime}{w}^{\prime}}$)/∂z], p′ is the pressure fluctuations, v

_{D}is the viscous diffusion rate (=ν∂

^{2}k/∂z

^{2}) and k is the turbulent kinetic energy. In an open channel flow, the v

_{D}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 t

_{P}and t

_{D}were determined from the experimental data, whereas the p

_{D}was obtained from the relationship p

_{D}= t

_{P}− ε − t

_{D}. In nondimensional form, the set of variables (t

_{P}, ε, t

_{D}, p

_{D}) is expressed as (T

_{P}, E

_{D}, T

_{D}, P

_{D}) = (t

_{P}, ε, t

_{D}, p

_{D}) × (H

_{d}/${u}_{*}^{3}$).

_{d}in Run 1. Upstream of the dune (x/L

_{d}= –0.5), all the components of the turbulent kinetic energy budget, in the near-bed flow zone, are positive with a sequence of magnitude T

_{P}> E

_{D}> P

_{D}> T

_{D}and then, they reduce with an increase in relative vertical distance z/H

_{d}. Above the dune crest (z/H

_{d}> 1), they are quite small. Downstream of the dune (x/L

_{d}= 1 to 2.1), the peaks of T

_{P}, E

_{D}, P

_{D}and T

_{D}are found to appear at the crest. In the near-bed flow zone, the T

_{P}and E

_{D}are positive, whereas the P

_{D}and T

_{D}are negative for x/L

_{d}= 1 to 2.1. Downstream of the dune, the absolute values of T

_{P}, E

_{D}, P

_{D}and T

_{D}decrease with an increase in x/L

_{d}. In particular, at x/L

_{d}= 3.3, the T

_{P}, E

_{D}, P

_{D}and T

_{D}profiles are almost similar to those of the undisturbed upstream flow at x/L

_{d}= −0.5.

## 7. Reynolds Stress Anisotropy

_{x}= σ

_{y}= σ

_{z}), where (σ

_{x}, σ

_{y}, σ

_{z}) = ($\overline{{u}^{\prime}{u}^{\prime}}$, $\overline{{v}^{\prime}{v}^{\prime}}$, $\overline{{w}^{\prime}{w}^{\prime}}$). By contrast, in an anisotropic turbulence, the Reynolds normal stresses are dissimilar, because the velocity fluctuations ${{u}^{\prime}}_{i}$ [= (u′, v′, w′) for i = (1, 2, 3)] are directionally preferred.

_{ij}is expressed as b

_{ij}= $\overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{j}}$/(2k) − 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

_{2}(= –b

_{ij}b

_{ij}/2) and I

_{3}(= b

_{ij}b

_{jk}b

_{ki}/3), respectively, are introduced. The Reynolds stress anisotropy is determined by plotting –I

_{2}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 I

_{3}= ±2(−I

_{2}/3)

^{3/2}, are symmetric about the plane-strain limit (I

_{3}= 0). In addition, the top-linear boundary of the Lumley triangle obeys I

_{3}= −(9I

_{2}+ 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

_{x}= σ

_{y}= 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

_{x}= σ

_{y}> 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

_{x}= σ

_{y}and σ

_{z}= 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

_{x}= σ

_{y}< 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

_{x}> σ

_{y}and σ

_{z}= 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

_{x}> 0, σ

_{y}= σ

_{z}= 0) or (σ

_{x}= σ

_{y}= 0, σ

_{z}> 0)], only one component of Reynolds normal stress sustains to make the stress ellipsoid a straight line (Figure 14).

_{2}versus I

_{3}, confined to the AIM boundaries, at various relative streamwise distances x/L

_{d}in Runs 1 and 2. Upstream of the dune (x/L

_{d}= −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/H

_{d}< 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/L

_{d}= 1.7), the size of the loop created by the data plots reduces forming a tail, and the loop disappears at x/L

_{d}= 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.

_{3}= 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.

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Photographs of isolated dunal bedforms in (

**a**) Run 1 and (

**b**) Run 2. Flow direction is from left to right.

**Figure 3.**Spectral density function S

_{df}(k

_{w}) versus wavenumber k

_{w}(

**a**) before and (

**b**) after decontaminating the data in Run 1 at a relative streamwise distance x/L

_{d}= 0.7 and a relative vertical distance z/L

_{d}= 0.13.

**Figure 4.**Vertical profiles of nondimensional streamwise flow velocity ū

^{+}at the upstream of isolated dunal bedforms for Runs 1 and 2.

**Figure 5.**Vertical profiles of nondimensional streamwise flow velocity ū

^{+}at the upstream and various downstream relative streamwise distances x/L

_{d}of isolated dunal bedforms for (

**a**) Run 1 and (

**b**) Run 2.

**Figure 6.**Vertical profiles of nondimensional Reynolds shear stress τ

^{+}at the upstream and various downstream relative streamwise distances x/L

_{d}of isolated dunal bedforms for (

**a**) Run 1 and (

**b**) Run 2.

**Figure 7.**Vertical profiles of third-order moments m

_{30}and m

_{03}at various relative streamwise distances x/L

_{d}in Runs 1 and 2.

**Figure 8.**Vertical profiles of third-order moments m

_{21}and m

_{12}at various relative streamwise distances x/L

_{d}in Runs 1 and 2.

**Figure 9.**Vertical profiles of relative fractional contributions S

_{i}

_{,0}at various relative streamwise distances x/L

_{d}in Run 1.

**Figure 10.**Vertical profiles of relative fractional contributions S

_{i}

_{,0}at various relative streamwise distances x/L

_{d}in Run 2.

**Figure 11.**Relative fractional contributions |S

_{i}

_{,H}| as a function of hole size H in Run 1 at relative streamwise distances (a) x/L

_{d}= –0.5 and (b) x/L

_{d}= 1 for relative vertical distances z/H

_{d}= 0.05, 0.25 and 0.5.

**Figure 12.**Relative fractional contributions |S

_{i}

_{,H}| as a function of hole size H in Run 1 at relative streamwise distances (a) x/L

_{d}= 1.7 and (b) x/L

_{d}= 3.3 for relative vertical distances z/H

_{d}= 0.05, 0.25 and 0.5.

**Figure 13.**Vertical profiles of the nondimensional components of turbulent kinetic energy budget T

_{P}, E

_{D}, T

_{D}and P

_{D}at various relative streamwise distances x/L

_{d}in Run 1.

Ū (m s^{−1}) | $\overline{\mathit{v}}$ (m s^{−1}) | $\overline{\mathit{w}}$ (m s^{−1}) | ($\overline{{\mathit{u}}^{\prime}{\mathit{u}}^{\prime}}$)^{0.5} (m s ^{−1}) | ($\overline{{\mathit{v}}^{\prime}{\mathit{v}}^{\prime}}$)^{0.5} (m s ^{−1}) | ($\overline{{\mathit{w}}^{\prime}{\mathit{w}}^{\prime}}$)^{0.5} (m s ^{−1}) | τ (m^{2} s^{−2}) |
---|---|---|---|---|---|---|

2.94 × 10^{−3} *(±2.93 × 10 ^{−2} †) | 2.33 × 10^{−3}(±3.02 × 10 ^{−2}) | 1.75 × 10^{−3}(±3.95 × 10 ^{−2}) | 2.18 × 10^{−3}(±5.87 × 10 ^{−2}) | 1.34 × 10^{−3}(±6.72 × 10 ^{−2}) | 1.07 × 10^{−3}(±6.89 × 10 ^{−2}) | 4.37 × 10^{−5}(±7.48 × 10 ^{−4}) |

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## Share and Cite

**MDPI and ACS Style**

Sarkar, S.; Ali, S.Z.; Dey, S. Turbulence in Wall-Wake Flow Downstream of an Isolated Dunal Bedform. *Water* **2019**, *11*, 1975.
https://doi.org/10.3390/w11101975

**AMA Style**

Sarkar S, Ali SZ, Dey S. Turbulence in Wall-Wake Flow Downstream of an Isolated Dunal Bedform. *Water*. 2019; 11(10):1975.
https://doi.org/10.3390/w11101975

**Chicago/Turabian Style**

Sarkar, Sankar, Sk Zeeshan Ali, and Subhasish Dey. 2019. "Turbulence in Wall-Wake Flow Downstream of an Isolated Dunal Bedform" *Water* 11, no. 10: 1975.
https://doi.org/10.3390/w11101975