Behavioral patterns of turbulent kinetic energy budget and quadrant analysis for flows over bedform under the influence of downward seepage

C_d

coefficient of discharge.

D₅₀

median particle diameter.

f_ku and f_kw

turbulent kinetic energy flux in streamwise and vertical directions, respectively.

g

acceleration due to gravity.

h

total depth of the flow over the bedform.

H_b

height of the flow above the notch.

H

hole size

I_i,H

indicator function

L_n

length of the notch.

n

number of samples taken.

S_i,H

invariant factor (F)

TKE

turbulent kinetic energy

t_P

turbulence production

t_D

turbulence diffusion

U, V, and W

time averaged velocity in the streamwise, spanwise, and vertical directions, respectively.

U_i, Vi, and W_i

the instantaneous velocities in the streamwise, spanwise, and vertical directions, respectively.

U_*

bed shear velocity

u′, v′, and w′

fluctuating components associated with the flow in streamwise, span-wise, and depth-wise directions, respectively.

$u′u′̄0.5,v′v′̄0.5,w′w′̄0.5$

root mean square values of u′, v′, and w′, respectively.

ɛ

turbulence dissipation

ν

kinematic viscosity of the fluid

σ_g

geometric standard deviation in the size of the sand particle.

ϕ

dry angle of repose for sand.

Flow in the alluvial channels is always associated with turbulent characteristics. These turbulent characteristics govern the rate and amount of sediment transport in the near riverbed region, which determines the morphological changes in the river. Therefore, studying the higher-order turbulent characteristics becomes vital to predict sediment transport and morphological changes in the river. Higher order turbulent statistics study involves analysis of turbulent kinetic energy, its production, dissipation, diffusion, and so forth. In addition, studies on quadrant analysis near the bed zone can also be performed, which reveal information on the occurrence of bursting events and the formation of coherent structures in turbulent flow. The most significant events in the sequence of quasi-cyclic events that constitute the bursting phenomena are ejections and sweeps. Because of local and temporal pressure differences, fluid parcels having low speed progress into the outer layer from the channel bed. This phenomenon is known as the ejection event. As these low-speed fluid parcels eventually lose their coherence, the fluid parcel with high-speed sweeps the remaining ejected fluid and rushes toward the channel bed comprising the sweep events (Kline , 1967; Chatwin and Allen, 1985; Gad-el-Hak and Bandyopadhyay, 1994; and Wu and Shih, 2012). Studies have been conducted on turbulent kinetic energy and quadrant analysis by many researchers on the plane bed as well as over bedforms.

Experimental studies over the plain bed revealed that the formation of hairpin vortices at the interface of flow and bed governs the formation of multiple sweep events, which are responsible for the commencement of sediment motion in the plain bed and the development of bedforms (Williams and Kemp, 1971; Raudkivi and Witte, 1990; Best, 1992; and Raudkivi, 1997). Nelson (1995) confirmed that sweep has a greater capacity to carry sediment than other bursting events. Investigation of flow behaviors over the flatbed surface with uniform sand transitioning to ripples and dunes suggests a gradual rise in the turbulent kinetic energy throughout the transition phase (Robert and Uhlman, 2001). Arfaie (2018) examined the impact of bed roughness on the sediment-carrying capacity of the flow. The results demonstrated that as the aspect ratio of roughness elements increases, the focal point of the recirculation zone shifts further downstream, strengthening the flow separation and turbulent kinetic energy in that zone. Numerical modeling using large eddy simulation over rough beds suggests that ejection events are more violent over the rough bed than over smooth beds. Visualization of turbulent structures in the rough bed also reveals the presence of a hair-pin vortex, which governs the generation of sweep and ejection events over the entire flow lengths (Bomminayuni and Stoesser, 2011; Taye , 2023). Mignot (2009) stated that in a narrow channel, large-scale coherent motions mainly contribute to the turbulent kinetic energy flux.

Previous studies conducted on the plain bed under seepage conditions reveal that injection enhances turbulent kinetic energy, accelerating the rate of diffusion, while suction shows opposite characteristics (Antonia , 1995; Krogstad and Kourakine, 2000; Yoda and Westerweel, 2001; Chen and Chiew, 2004; and Kim and Sung, 2003). However, Rao and Sitaram (1999) have argued about a reverse pattern in turbulent characteristics. Lu (2008), Chen and Chiew (2004), and Faruque and Al (2009) investigated the effect of seepage on the turbulent kinetic energy (TKE) budget, TKE flux, and quadrant analysis over a rigid rough bed. The results highlight an increase in turbulent kinetic energy with the domination of ejection events throughout the flow depth. According to numerous research studies (Richardson , 1985), injection tends to promote bed erosion while suction generally prevents sediment movement. In contrast to that, other researchers claimed that injection prevented bed particle mobility while suction increased the rate at which the sediment was transported (Watters and Rao, 1971; Willetts and Drossos, 1975).

Turbulent bursting events show distinct behavior under seepage conditions, not just near the bed region but also beyond the bed region (Faruque and Balachandar, 2011). Sharma and Kumar (2017) conducted experiments to study the structure of turbulence over a non-uniform sand bed channel in the presence of downward seepage and observed an increase in turbulent kinetic energy production, while the turbulent kinetic energy dissipation and diffusion decreased in the presence of downward seepage. The turbulent kinetic energy fluxes also increased near the bed zone with the introduction of downward seepage. Analysis of bursting events suggests the dominance of sweep events near the bed zone, which is mainly responsible for the motion of sediments. Deshpande and Kumar (2016) in their experimental studies on alluvial channels with curved cross sections also concluded that the increase in the contribution of sweep events and turbulent kinetic energy near the bed zone, in the presence of seepage, plays a major role in the increase in bed material transportation and change in the cross-sectional shape of the curved channel. Singha (2012) in their experiment over a rough plain bed concluded that suction impinged energy into smaller-scale turbulent structures, weakening the large-scale turbulent structures. It was also observed that both suction and injection weaken the large-scale structure but the extent to which the structure weakens is different.

Dunes are one of the major features of alluvial rivers. The development of dune features from the plain bed is due to the initiation of sediment motion. Turbulent kinetic energy production, diffusion and dissipation, and bursting events play a major role in the initiation, rate, and amount of sediment movement. Researchers have suggested that the majority of turbulent kinetic energy is produced in the recirculation zone downstream of the dune crest level (Mendoza and Wen Shen, 1990; Johns , 1993; Nelson , 1993; Yoon and Patel, 1996; Bardina , 1997; Wilcox, 1998; Tan Brinke , 1999; Lyn and Altinakar, 2002; Yue , 2005; Xie , 2014; and Kadia , 2022). Fourniotis (2009) also conducted studies on the effect of dunes on the flow characteristics utilizing the k–ε turbulence model and concluded the same. Hanmaiahgari (2017) conducted quadrant analysis over the bedforms and concluded the dominance of sweep and outward events in mobile bedforms, which governs the bedform motion. Bustamante-Penagos and Niño (2020) in their study of flow over a rough bed surface revealed that ejection events increase the magnitude of turbulent kinetic energy in the production zone and inertial subrange zone, leading to changes in the turbulent structure pattern and turbulent interaction with the bed region.

To analyze the influence of surface roughness of the dune bedform on the flow near the bed region, Kirca (2020) conducted experimental investigations on fully developed flow over higher angle dunes with smooth and rough beds. The investigation reveals that the re-attachment point shifts further downstream as compared to the rough wall case, affecting the near-bed flow and turbulence characteristics qualitatively and quantitatively, altering the sediment entrainment characteristics.

Elgamal (2022) compared two depth average k–ɛ turbulence models employing numerical analysis techniques for predicting the turbulent kinetic energy over bedforms. The investigation concluded that in the conventional 1D k-Rastogi–Rodi [standard depth averaged k-e turbulence model (SDAKE)] model, the bed shear velocity determines the rate of turbulent kinetic energy production and dissipation. However, as the bed topographical characteristics changes due to sediment movement, the shear flow zone shifts away from the bed region, producing the majority of turbulent kinetic energy away from the bed region. Thus, the turbulent kinetic energy near the bed region is overestimated by the conventional model. The authors propose a newly designed depth-averaged TKE model [moment based depth averaged k-e turbulence model (MDAKE)] based on the moment concept, which substitutes the integral moment velocity scale over a train of bedforms for bed shear velocity.

Hardy (2021) in their study of flow patterns over the three-dimensional dune mentioned that three-dimensional bedforms generate twice as many vortices as two-dimensional bedforms, which are larger and induce more turbulence close to the bed. Significant amounts of the sediment can be transported by these larger flow superstructures, which further increases the three-dimensionality of the bedform structure.

Various turbulent models are also used to investigate the much-needed information on coherent structures (vortices), their interactions with walls or beds, and their downstream propagation (Bardina , 1997; Wilcox, 1998; Yue , 2005; and Xie , 2014). Babakaiff and Hickin (1996) demonstrated that turbulence-generated vortices reached the water’s surface and formed boils. As dunes occupy a greater proportion of the flow depth, more boils emerge. Yue (2005) conducted a large-eddy simulation of turbulent flow over a fixed two-dimensional dune and observed that for the complex flow environment used in the experiment, turbulence kinetic energy has larger peaks in the shear layer in the separation zone than the profile obtained from experimental data. Other aspects of flow, such as turbulent production, diffusion, dissipation, and realistic simulations of coherent flow patterns in flow passing over the dunes, can also be generated using advanced numerical modeling. Thus, utilizing turbulence modeling approaches, it is possible to gain a comprehensive understanding of the dynamics of dunes, including processes such as dune formation, turbulent features, sediment transport, and deposition over dunes (Yalin, 1977; 1992; Richards, 1980; Nelson and Smith, 1989; and Gyr and Kinzelbach, 2004).

In addition, field investigations have also demonstrated that due to the generation of eddies in the wake zone at the lee side of the dune, energy production is nearly two to four times higher than it is over the crest of the dune (Venditti and Bauer, 2005; Keylock , 2014). Lefebvre (2019) used high-resolution numerical modeling to examine the flow over bedforms in the Rio Paraná (Argentina), and the results showed that the maximum angle of the dune’s slip face rather than the height of the bedform determines the magnitude and extent of the turbulent wake zone. According to Motamedi (2014), for a low-angle dune (lee angle less than 10°), topographical factors such as crest shape and dune height do not affect the separation zone; however, for a higher lee angle, these factors have an impact on the extent of the separation zone on the lee side of the dune. The investigation also concluded the dune with a higher lee angle generates more turbulent energy than lower-angle dunes. As a result, larger-angle dunes experience more intense bursting events than lower-angle dunes, which enhances the rate of sediment flow.

Although numerous studies have been performed on the turbulent kinetic energy budget and quadrant analysis in flow over plain beds and bedforms, studies on these parameters in the presence of seepage over bed forms are yet to be explored. The current work focuses on the investigation of the dynamics of turbulent kinetic energy, its production, diffusion, and dissipation, and quadrant analysis to predict the bursting events. This information will reveal how the previously mentioned flow behaviors changed in the presence and absence of seepage.

The experimental setup incorporates a tilting flume made up of glass. The flume has dimensions of a length of 20 m, width of 1 m, and depth of 0.72 m. The slope of the flume is maintained at 0.002 49. A collection tank measuring 2.8 × 1.5 × 1.5 m³ (length × breadth × depth) was installed at the flume’s upstream end. To prevent highly turbulent flow from entering the channel, two wooden baffles were implanted in the collecting tank. In addition, it allows the flow to enter the experimental zone in the flume very slowly to minimize the impact of entering turbulence. Except for a 2 m stretch at the upstream limit, the entire length of the flume bed was made porous by covering it with a fine mesh (0.1 mm) reinforced by a steel tube system. Steel tubes were stacked in a basal pressure chamber having dimensions of 15.20 × 1 × 0.22 m³ (length × width × height) to provide separation between the channel’s bottom and the fine mesh. By adjusting the tailgate placed downstream of the flume, the required depth of flow is maintained in the flume. Figure 1(A) shows schematic depiction of the top and side perspectives of the experimental flume.

Wooden boards spaced 1 m apart are installed on either side of the flume to create bedforms. The size of these wooden planks is identical to the dune considered in this experiment. Sand was compacted together between them to develop a bedform [see Fig. 1(B)]. To create a fixed bedform and prevent sediment from being carried by the flow, the upper portion of the bedform structure was covered with galvanized iron (GI) grids with 0.1 mm apertures across the whole width of the flume. Five dunes are positioned between 5 and 10 m downstream of the flume. An individual dune is 85 cm long, which has a stoss side of 77 cm and a lee side of 8 cm. The height of the bedform gradually rises from 0 cm at the beginning to 5 cm at 77 cm length. Thus, the crest of the dune is formed at 77 cm from the initial point of the flume. Following the crest, the geometry rapidly decreases in height from 5 cm at 77 to 0 cm at 85 cm. Figure 2 illustrates a schematic of a solitary bedform along with the grid diagram of the measurement locations above the dune. Table I provides an explanation of the terminologies used to designate various parts along the dune’s length.

As the bed material, medium-sized river sand with an average size of 0.61 mm was chosen. To keep them from accessing the basal pressure chamber, these particles are laid on the fine mesh. The geometric standard deviation (σ_g) of sand particles was found to be 1.3, which is less than 1.4, describing the bed material to be uniform (Marsh , 2004). The dry angle of repose (ϕ) of sand particles was found to be 32.2°.

The flume’s discharge and flow velocity are calculated as 0.0508 m³/s and 0.423 m/s, respectively. A valve that controls the discharge rate in the case of downward seepage is installed into the base pressure chamber’s downstream end. 10% and 15% of the flume discharge were allowed to seep down through the pores. The kinematic viscosity (ν) of water is 1.0034 mm²/s at 20 °C ambient temperature. The Reynolds number (Re = Uh/ν) and Froude number (⁠$Fr=U/gh$⁠) of the flow are found to be 50.588 × 10³ and 0.348, respectively.

A Nortek four-beam down-looking Vectrino + Acoustic Doppler Velocimetry (ADV) sensor was used to assess the flow velocity at 11 locations on the central dune in both the absence and presence of downward seepage. The instrument has a sampling rate of up to 200 Hz. The sapling rate of the instrument for the present studies is maintained at 100 Hz. Near the bed region, the sampling volume’s height was decreased to 1 mm in order to prevent it from interacting with the bed surface. The height of the sampling volume was adjusted to 4 mm away from the bed region. Measurements near the bed were taken with extreme care. At each vertical section, the depth gap between two measurement locations was kept as close as possible in the region near the bed surface so as to increase the number of readings and obtain better information on the flow behavior in the region close to the bedform surface.

Over 11 portions of the central dune (nine on the stoss side and two on the lee side sections), flow rates were measured under conditions of no seepage, 10% seepage, and 15% seepage. The seepage amounts (10% and 15%) considered in the present study correspond to the percent component of the total flow, which passes through the bedform made of sand particles in the form of downward seepage.

The central dune was located 8 m from the flume’s downstream end. 30 000 readings at each height were noted at all the measured locations. Throughout the measuring process, a minimum correlation of 70% and a signal-to-noise ratio of 15 were maintained. The correlation can be reduced by 5% while considering a measuring location very close to the bed.

By gathering ten sets of 15 pulses of 30 000 samples at 100 Hz at a height of 3 mm above the bed surface, the standard deviation and uncertainty associated with ADV data are evaluated first before carrying out the actual measurements at different sections over the dune.

The results are listed in Table II. U, V, and W in Table II are the time-averaged velocities in the streamwise, lateral, and vertical directions, respectively. Furthermore, u′, v′, and w′ are the fluctuating components of velocities in the streamwise, lateral, and vertical directions, respectively.

Results obtained from the studies of patterns of turbulent kinetic energy, its production, dissipation diffusion, flux components, and quadrant analysis are described and discussed below.

To facilitate observations and comparative studies in flow patterns under no seepage and seepage conditions, all the data are normalized with a constant value of U_* (=3.28 cm/s), which is estimated at the crest portion of the dune under the no seepage condition.

Patterns of non-dimensional turbulent kinetic energy distributions in the near bed surface region, at some critical sections, are presented in Fig. 3.

Results illustrate that at the initial and lee side sections of the dune, the average turbulent kinetic energy along the whole depth increased with increasing seepage discharge. At the initial sections on the stoss side of the dune, a rising trend of turbulent kinetic energy is observed for both 10% and 15% seepage. Turbulent kinetic energy increased by ∼48%–62% for 10% seepage and ∼50%–72% for 15% seepage higher than the no seepage condition, at initial sections of the dune, with a maximum increase at the 0 cm section of the stoss side of the dune. This rise in turbulent kinetic energy is due to the influence of the wake zone generated at the preceding dune’s lee side.

Similar trends can also be observed at the lee side sections of the considered dune. At lee side sections of the dune, the turbulent kinetic energy increases from ∼16% to 72% under the 10% seepage and from ∼48% to 82% under the 15% seepage conditions. A maximum increase in turbulent kinetic energy is observed at the 83 cm section on the lee side section under the influence of seepage discharge. This provides evidence that at the sections of the dune where the wake zone prevails, turbulent kinetic energy increases with the increase in downward seepage discharge, which generates a higher amount of turbulence than under no seepage conditions. At the lee sections of the dune, maximum variation in the turbulent kinetic energy is observed at the zones ∼0.2 < z/h < 0.3 under the 10% seepage condition and ∼0.1 < z/h < 0.2 under the 15% seepage condition, suggesting that with an increase in the seepage amount, the peak of the turbulent kinetic energy pattern shifts closer to the bed region. Here, z/h is the non-dimensional flow depth, with z being the vertical distance from the bedform surface and h being the flow depth over the bedform.

The results illustrated in Fig. 3 also reveal that from the central sections up to the crest zone of the dune, the average turbulent kinetic energy along the whole depth decreases under the influence of downward seepage as compared to the no seepage condition. The decrease in turbulent kinetic energy at these sections ranges from 1% to 6% under the 10% seepage condition and from 5% to 48% under the 15% seepage conditions. Thus, the intensity of turbulent kinetic energy decreases with the increase in seepage discharge from the central portion to the crest portion of the dune as compared to the no seepage condition.

Figures 4 and 5 represent the vertical distribution of non-dimensional streamwise (F_ku) and vertical (F_kw) turbulent kinetic energy flux, respectively.

Results illustrate that the value of average F_ku along the flow depth increases at the initial and lee side sections of the dune with the increase in seepage discharge. At the initial section of the considered dune, the average F_ku along the flow depth increases by ∼1%–3% under the 10% seepage condition and by ∼8%–9% under the 15% seepage condition as compared to the no seepage condition. Similarly, at the lee side sections of the dune, the average F_ku along the flow depth value increases by ∼13%–16% and 14%–36% with the introduction of 10% and 15% seepage discharge, respectively. The values of F_ku near the bed zone at the initial and lee side sections tend to be on the negative side under the no seepage condition; however, the values shift toward the positive side under downward seepage conditions. A decrease of about ∼2%–4% under the 10% seepage condition and ∼16%–39% in the average value of F_ku is observed from the middle sections to the crest portion of the dune.

In contrast to the pattern of F_ku, the average value of F_kw along the flow depth decreases at the initial and lee side section of the dune and decreases along the flow depth with an increase in seepage discharge. The average value of F_kw decreases by ∼22%–44% under the 10% seepage and by ∼44%–62% under the 15% seepage condition to no seepage condition. At the initial section, the average value of F_kw reduces by ∼11%–20% under both 10% and 15% seepage conditions. The average value of F_kw increases by ∼6%–7% and 28%–34% under 10% and 15% seepage conditions, respectively, at the middle sections and crest portion of the dune.

As the trends of F_ku and F_kw determine the mobility of sediment particles, shifting of values of F_ku toward the positive side and F_kw toward the negative side in the near-bed region at initial and lee side sections under the influence of downward seepage conclude the possibility of a higher rate and volume of sediment transport from these regions.

Turbulence production defines the magnitude of the interchange of mean flow to the fluctuations. Patterns of the vertical distribution of turbulent production shown in Fig. 5 reveal that with downward seepage, the average increase in turbulent kinetic energy production along the whole depth ranges from ∼27% to 37% and from ∼26% to 49% under 10% and 15% seepage conditions, respectively, at the initial sections of the dune, with maximum turbulent production occurring at the zone between z/h ∼ 0.1–0.3 under all the conditions. Similarly, the increase in average turbulence production ranges from ∼3% to 59% and 9%–68% for a seepage of 10% and 15%, respectively, at the lee side sections of the dune. However, from the middle sections to the crest portion of the dune, the average turbulence production along the depth of the flow decreases by ∼2%–15% under the 10% seepage and by ∼16%–27% under the 15% seepage condition. At the lee side section, the maximum value of turbulent production is observed at the region z/h ∼ 0.3–0.4. These results justify the fact that with the introduction of seepage, the momentum exchange near the bed region increases, leading to a higher degree of turbulence and surge in turbulent kinetic energy production.

A reduction in the dissipation rate of turbulent kinetic energy can be observed from the patterns, showing the vertical distribution of turbulence dissipation at some critical sections, as shown in Fig. 7.

The average value of the turbulence dissipation rate along the flow depth reduces by ∼7%–23% and ∼5%–39% under 10% and 15% seepage conditions as compared to no seepage conditions at the initial section of the dune. The average turbulence dissipation rate decreased by ∼24%–35% with the introduction of a downward seepage of 10%, while for 15% seepage, it reduces by ∼33%–44% with respect to the no seepage condition. The maximum value of the turbulence dissipation rate can be observed at zone z/h < 0.2 both in the absence and presence of seepage. At the middle sections and crest portion of the dune, the average turbulence dissipation rate increases by ∼18%–27% under the 10% seepage condition and by ∼29%–31% under the 15% seepage condition.

Turbulent diffusion can be defined as the spread of turbulence. A reduction in turbulent diffusion can be observed from the patterns, showing the vertical distribution of turbulence diffusion at some critical sections, as shown in Fig. 8.

Similar to the patterns of the dissipation rate, at initial sections of the dune, the average turbulence diffusion along the depth of the flow also reduces by ∼12% and ∼21% under 10% and 15% seepage conditions, respectively, as compared to the no seepage condition. At the lee side sections of the dune, the reduction in average turbulence diffusion ranges from ∼12% to 21% under the 10% seepage condition and ∼16%–26% under the 15% seepage condition. However, at the middle sections and crest portion of the dune, there is a surge in average turbulence diffusion by ∼13% and ∼23% under 10% seepage and 15% seepage conditions, respectively.

The analysis of different components of the turbulent kinetic energy budget reveals that application of downward seepage leads to higher momentum exchange near the bed zone of the initial and lee side sections of the dune, which causes a rise in turbulence production as more flow energy gets converted to turbulent fluctuation. This surge in turbulence production reduces the turbulence diffusion and dissipation rate as most of the flow energy gets utilized in turbulence production near the bed zone of the initial and lee side sections.

Conditional statistics of Reynolds shear stress provide information about the coherent structures formed due to turbulence, which may cause more rapid distribution of Reynolds stress than the small-scale turbulent structures. The importance of ejections and sweep events was described by Cellino and Lemmin (2004), which defines the mixing and transport processes in the flow. These coherent structures cannot be identified using time-averaged analysis due to the short life span. Thus, both time and space correlation measurements are required for investigating these structures. Conditional statistics of the velocity fluctuations u′ and w′ provided insight into bursting events, using four quadrants, namely,

• outward interactions (i = 1; u′ > 0, w′ > 0),

• ejection (i = 2; u′ < 0, w′ > 0),

• inward interactions (i = 3; u′ < 0, w′ < 0), and

• sweep (i = 4; u′ > 0, w′ < 0).

Patterns of vertical distribution showing the fractional contribution of bursting events toward Reynolds shear stresses (S_i,H) for hole size (H) = 0 are shown in Figs. 9–12.

Figures 9 and 11 present the vertical distributions of outward interaction (S_1,0) and inward interactions (S_3,0) under no seepage and seepage conditions at some important locations over the dune. The distribution patterns reveal that at the initial and lee side sections of the dune, the contributions from outward interactions and inward interactions decrease with increase in downward seepage discharge. At the lee side sections of the dune, contribution from outward interaction decreases nearly by 9% under the 10% seepage condition, while it decreases by ∼3%–12% under 15% seepage conditions with respect to the no seepage condition. Similarly, at the initial sections of the dune, contribution from inward interaction decreases by ∼5%–7% under the 10% seepage condition, while it decreases by ∼12%–14% under the 15% seepage condition. A reverse trend is observed from the central sections to the crest portion of the dune. Contribution from inward interaction increases by ∼19% under the 10% seepage condition and by ∼15%–23% under the 15% seepage condition from central sections to the crest portion of the dune. It was also observed that at the lee side sections of the dune, the trend of outward interaction events shifted from the positive side under the no seepage condition to the negative side under the influence of seepage.

Similar to the patterns of contributions from inward interaction, at lee side sections of the dune, the contributions from outward interaction also decreased by ∼7%–11% under the 10% seepage condition and ∼10%–11% under the 15% seepage condition as compared to the no seepage condition. At the initial sections of the dune, the decrease in the inward interaction’s contribution ranges from ∼7% to 12% under the 10% seepage condition, while under the 15% seepage condition, the decrease ranges from ∼13% to 14%. However, an increase of about ∼10%–16% under the 10% seepage condition and ∼18%–20% under the 15% seepage condition was observed from the central portions to the crest section of the dune.

Trends showing contradictory behavior are observed for ejection (S_2,0) and sweep (S_4,0) bursting events (Figs. 10 and 12). Opposite to the behavior of inward and outward interactions, the distribution patterns reveal that at the initial and lee side sections of the dune, the contributions from ejection and sweep events increase with the increase in downward seepage discharge. The contribution of the ejection event increases by ∼23%–28% under the 10% seepage condition and by ∼20%–47% under the 15% seepage condition at the lee side sections of the dune. At the initial sections of the dune, the contribution of the ejection event increases by ∼9%–13% under the 10% seepage condition and by ∼15%–18% under the 15% seepage condition. In the near bed zone z/h < 0.2, the values showing the contribution of the ejection event shift from the negative side to the positive side under the influence of seepage at the lee side sections of the dune. From the central portion to the crest section of the dune, the contribution from the ejection event decreases by ∼5%–9% and ∼3%–26% under 10% and 15% seepage conditions, respectively, as compared to the no seepage condition.

Similar to the behavior of ejection events, the contribution of sweep events also increases with the introduction of downward seepage. The contribution of the sweep event increases by ∼13%–36% under the 10% seepage condition and by ∼47%–65% under the 15% seepage condition at the lee side sections of the dune. At the initial sections of the dune, the contribution of the sweep event increases by ∼23%–29% under the 10% seepage condition and by ∼36%–38% under the 15% seepage condition. Near the bed surface zone at the lee side sections of the dune, the trend of the sweep event shifts from the negative side to the positive side. From the central portion to the crest section of the dune, the contribution from the sweep event decreases by ∼1%–2% and ∼1%–15% under 10% and 15% seepage conditions, respectively, as compared to the no seepage condition.

Thus, analysis of bursting events in the flow over dunes reveals that with the introduction of seepage, the contribution from inward and outward interaction events decreases while the contribution from sweep and ejection events increases at the initial and lee side sections of the dune. However, clear dominance of sweep events near the bed-surface region and at the initial and lee side sections can be observed from the analysis, which may lead to a surge in the rate and amount of sediment motion from these regions.

Variations in different stress fractions (S_i,H) near the bed region z/h < 0.1 under no seepage and seepage conditions are presented in Figs. 13–15. These plots define the strength of different bursting events in the presence and absence of seepage, which helps in determining the contribution of larger events to the turbulence intensity.

Figure 13 shows the variation in fractional contributions of different bursting events with hole sizes at the initial section on the stoss side of the dune under no seepage and seepage conditions. The analysis reveals that at the initial portion of the dune, the contribution of inward and outward interactions vanishes at H ≥ 4 both in the absence and presence of seepage. The contribution of the ejection event dominates up to H ≥ 4 under the no seepage condition while under the influence of downward seepage, the contribution vanishes at H ≥ 6 near the bed zone under both 10% and 15% seepage conditions. The sweep event prevails at the near-bed region of the initial sections of the dune under seepage conditions. In the absence of seepage, the sweep event vanishes at H ≥ 6, while under the influence of seepage, the contribution from the sweep event vanishes at H ≥ 8 and H ≥ 12 under 10% and 15% seepage conditions, respectively.

Variation in fractional contributions of different bursting events with hole sizes at the crest section of the dune in the presence and absence of seepage is shown in Fig. 14. At the crest portion of the dune, the hole size where the contribution of outward and inward interactions vanishes rises from H ≥ 2 under the no seepage condition to H ≥ 6 under seepage conditions and H ≥ 6 under the no seepage condition to H ≥ 8 under seepage conditions, respectively. The contribution of ejection and sweep vanishes at H ≥ 8 and H ≥ 10 for no seepage and seepage conditions, respectively.

Figure 15 shows the variation in fractional contributions of different bursting events with hole sizes at the initial section on the stoss side of the dune under no seepage and seepage conditions. Near the bed zone, z/h < 0.1, at the lee side sections, the contribution of both outward and inward interaction vanishes at H ≥ 10 under the no seepage condition while it vanishes at H ≥ 6 under both the seepage conditions. The contribution of the ejection event vanishes at H ≥ 4 under the no seepage condition, while for 10% and 15% seepage discharge conditions, it vanishes at H ≥ 6, and H ≥ 10, respectively. Sweep event contribution vanishes at H ≥ 4 under the no seepage condition, while it vanishes at H ≥ 6 and H ≥ 10 under 10% seepage and 15% seepage conditions, respectively.

Findings of the study highlight that downward seepage increases the rate of turbulent kinetic energy production at the initial and lee side sections in the near-bed region (primarily at 0.1 < z/h < 0.2 zone) as compared to the no seepage condition, confirming the increase in turbulent energy under both the seepage conditions. Contradictory results of the turbulent dissipation and diffusion rate at these sections show a decreasing trend in the presence of seepage compared with the no seepage condition. Opposite to the trend at the initial sections and lee side sections, turbulent kinetic energy production decreases, while the turbulent dissipation and diffusion rate increases in the presence of downward seepage. These results confirm that at initial sections and lee side sections, flow energy contributes more toward turbulent production than dissipation and diffusion processes, while at the middle sections and crest portion, most of the flow energy is dedicated toward dissipation and diffusion compared with the turbulent production process in the presence of seepage as compared to the seepage condition. In addition, the results of quadrant analysis also reveal that at the initial and lee side sections of the dune, the contribution from ejection and sweep events increases in the near bed region with downward seepage. However, the sweep event dominates the ejection event. Near the bed region at the crest portion, the contribution of outward interaction increases with the increase in downward seepage discharge.

All the findings obtained from the study physically signifies that near the bed region of the initial and lee side, higher momentum exchange occurs in the presence of downward seepage, resulting in a higher degree of turbulence. This further enhances turbulent production and gives rise to the contribution of the sweep event as compared to the no seepage condition. The increase in turbulent energy and specific sweep events may lead to a higher amount and rate of sediment motion from the initial and lee side sections of the dune than the no seepage condition. Therefore, the contribution of seepage in predicting the patterns of turbulent characteristics, quadrant analysis events, and sediment transport behavior in the formation of dune shaped bedforms cannot be neglected. The change in the amount and rate of sediment transport owing to the change in turbulent characteristics in the presence of downward seepage can significantly impact the morphological change in channel cross sections.

Although numerous studies were conducted to study the patterns of turbulent kinetic energy production, dissipation, and diffusion over bed forms, most of the studies did not consider the influence of downward seepage, which is present under natural conditions. Seepage acts as one of the governing factors in determining the patterns of turbulent characteristics and morphological changes associated with a channel. Researchers have carried out studies highlighting the influence of downward seepage on turbulent flow field over a plane bed, revealing the behavioral changes in turbulent characteristics under the influence of seepage. Thus, the main focus of this study is to determine the turbulent patterns of the flow field over fixed sand bed dunes under the influence of downward seepage and compare the findings under no seepage conditions, which can contribute to further improvement in knowledge on fluvial dynamics research. The results of turbulent patterns in the flow field over the dune in the absence of the seepage condition are in agreement with previous studies. The novelty of the work lies in studying turbulence flow patterns in the presence of downward seepage and comparing the behavior with patterns developed under no seepage conditions in a flow field over fixed two-dimensional dunes, which was not studied previously.

The present experimental studies address the behavioral patterns of turbulent kinetic energy, turbulent kinetic energy fluxes, components of the turbulent kinetic energy budget, and quadrant analysis. Experimental data measured over different sections of the dune are analyzed to reveal the anisotropic behavior associated with the flow. The results conclude the following:

1. At the dune’s initial and lee side sections, the average turbulent kinetic energy, and the average turbulence production, along the depth of the flow increases under the influence of downward seepage. However, a contradictory trend is observed in the middle portion and the crest portion of the dune.

2. The behavioral pattern at the initial section of the considered dune is similar to that on the lee side of the considered dune. This behavioral similarity is due to the influence of the preceding dune present just before the considered dune.

3. This surge in turbulence production in the presence of seepage confirms higher momentum exchange in the near bed-surface regions, 0.1 < z/h < 0.2, at the initial and lee side sections of the dune, increasing the strength of the turbulent intensity. This leads to a reduction in the dissipation and diffusion rate at these sections of the dune as most of the flow energy is dedicated to turbulence production rather than dissipation and diffusion processes. Opposite trends are observed from the middle sections to the central portion of the dune.

4. The results also confirmed that turbulent kinetic energy flux in streamwise direction increases while turbulent kinetic energy flux in the vertical direction decreases under the influence of seepage at the initial and lee side sections of the dune. The higher positive value of Fku and the higher positive value of Fkw indicate a greater change in sediment mobility from these regions than the no seepage condition.

5. The findings from quadrant analysis of bursting events suggest an increase in the occurrence of ejection and sweep events near the bedform zone of the initial and lee section, with an increase in seepage percentage. However, the maximum contribution to momentum transfer is from the sweep, which can enable higher volume and a higher rate of sediment transport from these regions.

6. All the results indicate that the introduction of downward seepage increases momentum exchange near the bed zone, causing a rise in turbulent kinetic energy intensity at the initial sections and the lee side of the considered dune. This leads to the increase in turbulent production with significant contribution from flow energy as the flow energy mainly contributes to turbulence production, turbulence dissipation, and diffusion rate decrease.

7. A higher rate of momentum exchange and dominance of the sweep event at the initial and lee side sections of the dune can lead to a higher rate of sediment mobility under seepage conditions than under the no seepage condition.

The authors have no conflicts to disclose.

Pradyumna Kumar Behera: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Vishal Deshpande: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Bimlesh Kumar: Conceptualization (equal); Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.