Detached eddy simulations with a deformable free-surface are used to predict flow and vortical structure in a pump station and to determine the severity of adverse hydraulic conditions inside the pump bays and pump columns. The model includes the approach channel to the five pump bays. Two of the bays contain vertical pump columns with a suction bell. The other bays contain a draft-tube-like structure (formed suction inlet) leading to a vertical pump column. The simulations are conducted at lab scale using 1:10 geometrical scaling and Froude number similarity. To mimic different operating scenarios, simulations are conducted with all five pumps operating and with only three pumps operating. Additionally, one simulation is conducted with several flow training devices present. Simulations accurately predict the type of vortices forming inside the pump bays, the level of flow nonuniformity inside the pump columns and the swirl angle at the level of the pump impeller. This study demonstrates the feasibility of using eddy-resolving simulations as a tool to evaluate and improve design of new and existing pump stations.

Barrier lakes typically form in mountain valleys after landslides block rivers. They develop abruptly and are prone to structural failure, posing threats to nearby areas. Here we propose a plain-region barrier lake type, termed river-capturing barrier lakes. Lakes near the Yellow River, including Hongze and Nansi Lakes, whose beds are elevated above the surrounding landscape, provide representative examples. Historical records, sedimentary analyses and numerical modeling indicate that these lakes formed after Yellow River avulsion and capture, and that a substantial difference in sediment load between the two rivers promoted upstream barrier lake formation on centennial timescales. Human interventions expanded their extent and accelerated their evolution into suspended lakes. While these lakes pose flooding risks like most barrier lakes, they can enhance water storage and allocation capacity for plain regions when properly managed. This dual functionality highlights the need for effective management to balance risks with the socio-economic advantages they provide. Plain-region barrier lakes form when sediment-rich rivers avulse and capture weaker rivers, creating long-lived, flood-prone yet beneficial suspended lakes, shown through historical records, geomorphic, sedimentary and hydrological analyses.

Eddy-resolving numerical simulations are used to investigate the fully developed open-channel flow over an array of large-scale bed roughness elements placed on a rough or on a smooth surface that mimics freshwater mussels partially buried in a gravel or in a sand bed, respectively. The rough surface corresponds to the scanned surface of a gravel bed with uniformly distributed bed roughness. In this paper we analyse how the surface mussel coverage density, BC, the roughness of the bed surface on which the mussels are placed and the filtering activity of the mussels (i.e. the local mass exchange occurring through the mussels' syphons) affect the double-averaged profiles of the streamwise velocity, turbulent kinetic energy, Reynolds and dispersive shear stresses, and the equivalent bed roughness height, $ K_{S} $ . Results show that similar to rough-bed boundary layers forming over sparse roughness elements in a deep environment (uniform free-stream velocity) and to those developing in a depth-limited environment (e.g. open channel), a multilayer analytical model can be used to approximate the double-averaged profile of the streamwise velocity over the flow depth, D, in the case of fully developed flow over a mussel bed. For similar values of BC and height of the protruding mussels, h, the scaling coefficient of the law-of-the-wake component supplementing the law-of-the-wall inside the inertial layer ( $h \lt z \lt D$ ) is found to be lower than values estimated for developing boundary layers over mussel beds. Results show that the equivalent roughness height increases monotonically with the surface mussel coverage density until BC approximate to 0.8, when the average distance between the mussels becomes sufficiently low for a skimming flow regime to develop over the top of the mussels. Bed roughness effects on the double-averaged variables are significant only for cases with $ extit{BC} \lt 0.3$ . Results also show that mussel-induced velocity streaks are generated over the top of the mussels and the average transverse spacing of the streaks, lambda, decays with increasing BC for constant h. The variation of $ \lambda / K_{S} $ with the non-dimensional distance from the bed surface is similar to that observed for fully developed flow over a rough bed with distributed roughness except for very low surface mussel coverage densities (i.e. $ extit{BC} \lt 0.02 $ ) when lambda remains constant (i.e. $ \lambda / K_{S} $ approximate to 10).

When two streams of equal densities but of unequal velocities and different flow directions come into contact, large-scale coherent structures are generated downstream of the confluence apex. They include vertical vortices inside the region of high mean shear between the fluids originating in the two tributaries [i.e., the mixing interface (MI)] and streamwise-oriented-vortical (SOV) cells inside and in the vicinity of the MI. The flow physics becomes even more complex when the bed elevations of the tributary channels are not the same as that of the downstream channel. Eddy resolving simulations are used to investigate how the gradual increase in the level of bed discordance between the minor tributary channel and the main downstream channel affect flow, turbulence structure, sediment entrainment mechanisms and mixing at an asymmetrical confluence with a confluence angle of 70 degrees. To better isolate the effects of bed discordance, simulations are performed with idealized conditions in which the bottom surfaces of the open channels are horizontal except near the region where the minor tributary connects with the main channel. The other main flow parameter that is varied is the discharge ratio between the two streams, QR. Results show that for constant QR, the coherence of the vertical MI vortices increases with increasing bed discordance, and the wake mode becomes gradually dominant even though the velocity and momentum ratios are not close to unity. Increasing the bed discordance reduces the coherence and number of SOV cells forming on the minor tributary side of the MI and promotes the formation of a near-bed intrusion of mixed fluid into the minor tributary side of the main channel near the confluence apex. This is the main mechanism that is responsible for the increase in the rates of mixing between the two streams with increasing bed discordance for constant QR. While for small bed discordance levels, sediment entrainment inside the confluence hydrodynamic zone is driven primarily by the SOV cells, for sufficiently large bed discordance levels and QR, sediment entrainment on the minor tributary side is mainly driven by the plunging core of high free surface velocities as it enters the main channel.

Numerical simulations are conducted to evaluate the three-dimensional flow field and the bed shear stress in the vicinity of a multiple-pier bridge located in the Po river (Italy), considering the naturally-deformed bathymetry. The use of Detached Eddy Simulations (DES) allows to explicitly resolve the unsteady motion of the energetically important turbulent eddies, and the Volume of Fluid (VoF) method is used to consider the deformations of the free-surface. Simulations are conducted in different hydrodynamic regimes, including free-surface flow and pressure flow that generates in case of deck overtopping. The objective is to investigate the applicability of the DES approach and the VoF technique for simulating the flow dynamics in a full-scale river reach with irregular geometry and a man-made structure on the riverbed. The complex interplays among the river flow, the deformed bathymetry, and the bridge structure are explicitly accounted for, with a precision that far exceeds the typical level of detail achieved through standard methods used for the simulation of river flows (e.g., two-dimensional depth averaged models). In the case of free-surface flow, the deformed bathymetry, typical of natural rivers, as well as the non-zero angle of attack and the complex shape of the bridge piers, influence the flow field at the bridge site and the distributions of bed shear stresses. This aspect highlights some limitations that arise when canonical cases (i.e., piers of regular shape and angle of attack of 0° over a flat bed) are considered in place of real complex geometries. The impact of the lateral flow contraction on the flow fields and on the potential of sediment erosion is limited in the present case due to the reduced width of the piers and the large distance between them, resulting in a low blocking ratio. Transitioning to the pressure-flow regime increases the free surface elevation upstream of the bridge and induces the formation of a high-velocity orifice flow beneath the deck, with regions of high velocity extending far downstream. Recirculation regions are observed below and downstream of the deck. Compared to an equivalent free-surface case with the same discharge and stage, pressure-flow induces much higher bed shear stresses at the bridge site, entailing an increased erosion potential. In these conditions, the flow acceleration around the piers and the lateral flow contraction have a lower impact on the erosive capacity, as confirmed by a pressure-flow simulation conducted by removing the piers.

A series of simulations with a circular cylinder of diameter D placed in an open channel flow of depth H are conducted in both purely oscillatory flow (one-mode forcing with maximum orbital velocity, U-m) and in combined oscillatory and current flow regime (mean velocity, U-s). The paper discusses how the phase-averaged flow, the coherent structures generated around the cylinder, and the forces induced on the cylinder change with the ratio between the steady current velocity and the oscillatory velocity (0 <= U-s/U-m <= 1.4) and with the Keulegan-Carpenter number, KC = UmT/D (1.5 <= KC <= 30.8), where T is the period of the oscillatory flow. Results show that for constant U-s/U-m, the capacity of the horseshoe vortex to erode the bed increases with the KC number. For constant KC number, the overall coherence and lifespan of the main horseshoe vortex forming during each oscillatory cycle increase with U-s/U-m. The same is true for the number and initial circulation of the wake vortices shed on the downstream side of the cylinder in the simulations conducted with KC = 15.4 and 30.8. An instability in the form of large-scale hairpin-like vortices develops along the vertical cores of the wake vortices in the KC = 1.5 simulations with U-s/U-m >= 1.0. For combined wave-current flows with U-s/U-m >= 0.4, the regions of relatively high bed shear stress magnitude and the peak values inside these regions increase monotonically with the KC number for constant U-s/U-m. Results show that even for cases with a steady current, the oscillatory component of the phase-averaged in-line force is reasonably well described by Morisson's equation. A drag coefficient can be defined for the steady component of the in-line force with U-s as the velocity scale. For high KC numbers and U-s/U-m > 1 this drag coefficient is comparable to the standard drag coefficient in steady flow. The phase lag between the in-line force and the oscillatory velocity component decreases with increasing KC and is independent of U-s/U-m. For cases where the number of vortices shed in the wake is the same during all cycles, the phase-averaged spanwise force remains small. However, at times, the cylinder is subject to instantaneous forces of comparable magnitude along the streamwise and spanwise directions for KC >= 15.4, which has important consequences for the design of piles in combined wave-current flow.

Detached Eddy Simulations (DES) with deformable free surface are used to analyze the three-dimensional flow fields and the bed shear stresses at a multiple-pier bridge in a river reach with natural bathymetry. The bed shear stresses and the dynamics of coherent structures are analyzed for different flow regimes (free surface and pressure flow with deck overtopping) and different scenarios, which include the removal of some of the piers and/or the bridge deck. Using data from this full-scale case study, the paper discusses how the presence of pressurized flow beneath the bridge deck and the proximity of piers affect the scour potential. In the present case study, the contraction caused by adjacent piers has a minor effect, since piers are relatively thin and far from each other. The bed shear stresses and the erosion potential are found to be more than twice larger when the flow is pressurized compared with the corresponding simulation in which the bridge deck is removed. When the flow is pressurized, the piers have small effects on the erosive capacity of flow; hence, for relatively high flow conditions, pressure scour is dominant compared to local and contraction scour. The results from DES are also compared with those obtained using the Reynolds-averaged Navier Stokes (RANS) approach. The mean flow field and mean bed shear stresses are fairly close to each other, particularly for pressurized-flow conditions at the bridge deck. Simulation results show that in regions with large-scale coherent structures, RANS computations underestimate the turbulent kinetic energy and, in turn, the fluctuations of bed shear stress that play a role in local scouring.

Three-dimensional eddy-resolving simulations are used to study the structure of turbulent boundary layers developing in an open channel where an array of sparse roughness elements in the form of partially burrowed mussels is placed on the smooth, flat channel bed starting at a certain streamwise location. The identical mussels are oriented with their major axis parallel to the mean flow. Their positions are randomised while making sure that the mussels are close to uniformly distributed inside the array. The turbulent flow approaching the leading edge of the array is fully developed. The protruding parts of the mussels play the role of sparse roughness elements and generate a rough-bed, internal boundary that is characterised by zero net flow exchange but non-zero local flow exchange due to active filtering. A double-averaging technique is used to obtain an equivalent, width- and time-averaged, boundary layer over a ‘flat’ rough surface containing no mussels. The paper discusses the effects of varying the mussel array density, protruding height of the mussels and filtering discharge on the spatial growth of the two-dimensional boundary layer. With proper scaling, the profiles of the (double-averaged) streamwise velocity are close to self-similar inside the inertial layer (e.g. for h $\lt$ z $\lt$ $\delta$ , where h is the height of the protruding part of the mussels and $\delta$ is the boundary layer thickness) starting some distance from the leading edge of the array. The scaled turbulent kinetic energy and concentration profiles associated with the scalar advected through the excurrent syphons are also found to be self-similar above the vertical location where the maximum value is reached. An analytical model containing three subzones is proposed for the streamwise velocity in between the bed ( z $=$ 0) and z $=$ $\delta$ . The velocity profile inside the inertial region contains a law-of-the-wall component supplemented by a law-of-the-wake component. The scaling coefficient of the law-of-the-wake component is found to be larger than typical values used to describe velocity variation in turbulent boundary layers developing in a surrounding flow with close-to-uniform free stream velocity. The equivalent roughness height for this particular type of boundary layers developing over sparse roughness elements increases monotonically with h and the mussel array density, $\rho$ N . The paper also discusses the effect of the mussel bed density on the average refiltration fraction and the phytoplankton removal efficiency of the mussel bed.

SummaryDam‐Break waves arise from the sudden release of the water stored behind a wall, and they cause destruction and losses on nearby areas. Characterisation of the impact force on obstacles is very important for risk assessment. In this work, we performed a series of lock‐release experiments to evaluate the wave‐induced impact force on a vertical end wall situated some distance from the lock. Laboratory data were used to evaluate the performance of several types of numerical models to predict the wave‐induced force on the vertical end wall. The selected models were as follows: the hydrostatic 1D De Saint‐Venant model, the multi‐layer hydrostatic model and the Navier–Stokes two‐phase flow. For the latter two models, both 2D (in the ‐ plane) and 3D simulations were performed. The results show that, independently of the hydrostatic assumption, the 2‐D simulations generate nonphysical oscillations which are not present in the measured temporal histories of the pressure and force values. By contrast, the predictions of both 3D numerical models better reproduce the temporal variation of the pressure forces on the end wall. The present study also shows that from a practical point of view, the 1D De Saint‐Venant model predicts with sufficient accuracy the impact time and the magnitude of the peak value of the impact force. Therefore, if the use of a fully 3D models is not a feasible option, in terms of a trade‐off between computation cost and accuracy, the use of a simple hydrostatic 1D De Saint‐Venant model is preferable to a 2D model.

The study investigates the flow structure, dynamics of the large-scale coherent structures, drag forces and sediment entrainment mechanisms generated by a circular array of diameter D containing rigid emerged circular cylinders of diameter d placed in a smooth-bed channel of depth h under strong shallow flow conditions (D/h = 20, d/D = 0.0125 and 0.025). Eddy resolving simulations are conducted with different values of the solid volume fraction 0.025 <= SVF <= 0.1 and non-dimensional frontal area per unit volume (2.56 <= aD <= 10.2, where for the present configuration, a = SVF(1/d)4 pi) and a fixed channel Reynolds number (Reh = 10 000). The flow conditions are such that a vortex street (VS)-type of shallow wake is expected to form for a solid cylinder of diameter D. Findings are compared with previous results obtained for cases with moderately shallow conditions 2.5 <= D/h <= 3.5 and with the limiting case of flow past a solid cylinder with D/h = 20. For moderately shallow conditions, the core of the main horseshoe vortex (HV) forming around the array occupies a small fraction of the water column and its coherence is the largest in front of the array. By contrast, for very shallow conditions, multiple HVs form around the array and their cores occupy a large fraction of the water column. Moreover, the coherence of most of these HVs peaks close to the sides of the array. Only for relatively large aD values, the main HV extends over the whole upstream face of the array, similar to the limiting case of a solid cylinder. For sufficiently high aD, a secondary instability is present inside the near wake that leads to the formation of parallel horizontal vortices in the vicinity of the wake roller vortices. The changes in the wake structure with decreasing SVF and aD are qualitatively similar to those observed for cases with moderately shallow flow conditions, with the antisymmetric wake shedding mode being suppressed for aD <= 2.5. However, the Strouhal number associated with the shedding of wake rollers, which is still close to 0.2 for a solid cylinder with D/h = 20, can be as high as 0.4 for aD = 2.5. The paper also discusses how the steady wake and total wake lengths and the strength of the bleeding flow vary with the SVF. Simulation results show that the capacity of the flow to entrain sediment inside and around the array peaks for SVF = 0.05 and aD = 5.2, with sediment entrainment monotonically increasing with the SVF outside the array and monotonically decreasing inside the array. Despite the differences in the flow structure next to and inside the array, the variation of the mean, time-averaged streamwise drag coefficient of the solid cylinders C-d with aD is close to that observed for arrays with moderately shallow flow conditions. The combined drag coefficient for the array decreases with increasing flow shallowness, with the decay being stronger for relatively large values of aD.

The present study uses results of eddy-resolving numerical simulations to investigate the open channel flow over large clusters of freshwater mussels (Unio elongatulus) partially buried in a rough, gravel bed. The density of the mussels forming the array varies from 26 to 500 mussels/m2. The flow structure is analyzed at large distances from the leading edge of the mussel bed, where the flow can be considered fully developed. The effects of changing the mussel bed density, the filtering discharge, the burial level and the roughness of the bed surface in which mussels are burrowed, are investigated in terms of flow field, turbulent structures, drag forces, and bed shear stresses. It is found that strong interactions occur between energetic eddies generated by the larger gravels on the exposed bed surface and by the mussel shells. Simulations results show that for a burial depth close to 50% and a ratio between the average gravel size and the mussel protruding height of 0.13, the shell induced turbulence becomes dominant for mussel bed densities around 50 mussels/m2. The influence of the bed roughness becomes less relevant with increasing mussel density, as the generation of energetic eddies is mostly controlled by mussel-to-mussel interactions. For fixed bed roughness, burial level and filtering velocity, the mean streamwise drag force and the associated drag coefficient for the exposed part of each mussel decrease with increasing mussel density, even if strong variations are observed for individual mussels. For constant mussel bed density and burial level, the mean streamwise drag force and the mean drag coefficient decrease slightly with increasing bed roughness. Increasing the burial level decreases not only the drag forces but also the drag coefficients because of the more streamlined shape of the top of the mussels. Strong active filtering acts toward decreasing the mean streamwise force and the mean drag coefficient. The spanwise drag forces contribute significantly to the total drag force, especially for high mussel bed densities. Based on smooth bed calculations, bed-averaged shear stresses are reduced in highly dense clusters. Mussel-to-mussel interactions are important for dense arrays and influence flow structure and turbulence Eddy resolving simulations showed that the effect of bed roughness become less significant with increasing mussel bed density In dense clusters of mussels, forces on the shells and bed shear stresses are reduced thus favoring mussel stability

Failure of river bridges is often due to hydraulic reasons, such as the development of severe scouring around piers and abutments. In many cases, the presence of bridge structures produces complex flow fields, which are difficult to predict using standard hydraulic models. This entails difficulties in estimating the hydrodynamic actions on the structure and the potential scour on the riverbed. A three-dimensional Computational Fluid Dynamics (3D CFD) model is used to analyze the flow field and the bed shear stresses at an existing bridge. The Volume of Fluid (VoF) method allows resolving the deformable free-surface, and the Detached Eddy Simulation (DES) approach allows resolving the dynamics of the energetically important turbulent eddies in the flow. The analysis deals with modeling the flow around a realistic, full-scale, multi-pier bridge in the Po River. Simulations considered different hydrodynamic regimes, i.e., the free-surface flow regime and the hypothetical pressure-flow regime which would occur when water levels are higher than the low chord of the bridge deck. The numerical application highlights the potential of the DES technique for analyzing the hydrodynamics of flow at bridges in natural river channels. Besides the turbulent flow fields and the description of the vortical structures, relevant results include the time-varying spatial distribution of the hydrodynamic forces acting on the different parts of the bridge and the time-varying distribution of the shear stress on the bed. This aspect is of particular interest because i) the estimation of drag forces on the structure components may be oversimplified using canonical, averaged formulas, and ii) the fluctuations in time of the bed shear stresses play a key role in determining the removal of sediments from the bed and the scour hole close to the bridge piers.

River confluences contribute to the outflux of saturated dissolved gases in the water resulting from high dam discharges. This process is related to gas transfer across the water–air interface, which is primarily controlled by turbulent dissipation near the water surface. However, the near-surface turbulence dissipation is rarely reported in confluence hydrodynamics studies. This study conducted experiments with different discharge ratios to investigate near-surface turbulent motions at a laboratory-scale confluence. The higher dissipation rate ε H/U_m^3 of near-surface turbulence was mainly located inside the interfacial shear layer between the two incoming streams ( 10 –4 ) and the bank separation zone (10 –4 –10 –3 ) where high shear was found in the mean flow. By contrast, the dissipation rates were much lower inside the incoming flows and outside the two regions of high shear ( 10 –5 ). The magnitudes of the dissipation rate inside the shear layer were comparable in experiments where the mixing interface was in the Kelvin–Helmholtz mode or in the wake mode. The dissipation rate was found to increase away from the free surface outside the shear layer, while it was more uniformly distributed over the depth inside the layer possibly due to the presence of strongly-coherent, vertically-orientated vortices. In the far field, the mean shear within the shear layer was largely weakened. Nonetheless, the effects of flow separation persisted and laterally expanded to occupy the entire cross section. The dissipation rate ε H/U_m^3 of the confluent flow was more than 10 –4 even at a distance of 10 times the channel width in the post-confluence channel.

Confluences are common components of all riverine systems and are characterized by converging streamlines and potential mixing of separate flows. The fluid dynamics of confluences is characterized by a highly complex structure with several common types of flow features observed. Many theoretical, field, laboratory and numerical studies tried to identify and clarify the main hydrodynamics, morphodynamics and ecological features of river confluences. Those studies spanned a wide range of flow and geometrical conditions ranging from laboratory-scale studies conducted in simplified geometries to field-scale studies of large-scale confluences. In particular, the identification of the main types of coherent structures generated within the post-confluence channel and of their dynamics has led to significant progress in understanding hydrodynamics and mixing processes at natural river confluences. Still, further research is needed to fully clarify how such structures interact among themselves and with the erodible bed and channel banks, how they redistribute momentum and affect transport of heat, suspended sediment, nutrients and what is their role in creating mixing patterns of dissolved and solid matter. In addition, the role of density differences between the incoming waters has garnered increasing attention in confluence studies, while the literature about the relationship between frequently observed species richness and confluence hydrodynamics is still limited. The present paper tries to review the literature on field and numerical studies of natural river confluences with a particular focus on the case in which density contrast affects confluence hydrodynamics and mixing and on large-scale river confluences. The paper discusses some main findings based on the field studies and eddy-resolving numerical simulations conducted by the authors. The application of bioenergetics metric at two large-size concordant bed confluences is also reported. Finally, the paper identifies some directions of future research that should lead to a better understanding of flow structure, transport processes, morphodynamics as well as of the ecological role of river confluences under different conditions.

The study investigates the evolution of a dam break wave of a mixture of clay and water in turbulent flow conditions and its interaction with a vertical rigid wall. The phenomenon reproduction, important for planning risk mitigation strategies and designing protective structures, is performed by means of three-dimensional large eddy simulations. The dynamical Smagorinsky model is employed and the volume-of-fluid technique is used to calculate the free surface evolution with time. A power-law model is considered to represent the mixture rheology. With reference to a small-scale setup, the paper analyzes the influence of the clay concentration and of initial fluid depth upstream of the lock gate on the temporal evolution of the front, the bed shear stresses, and the force acting on the vertical endwall. Moreover, the presence of three-dimensional effects and the turbulence role are also investigated. The occurrence of scale effects for the non-Newtonian dam break wave is finally examined by performing an additional test with a larger-scale setup. Results indicate that lobelike structures develop near the front of the current and near-bed streaks of low and high velocity are present behind the front. The variation of the wave front position with time is described by a power-law function, independently of both clay concentration and initial fluid depth. Over the body of the current, the nondimensional bed shear stress values in the simulations performed with non-Newtonian fluids are about two times larger than the ones in the clear water case. Moreover, the peak value of the nondimensional impact force, slightly lower than the one in clear water, decreases when clay concentration is increased. Comparing large- and small-scale tests, higher nondimensional propagation speeds and bed shear stress values in the front region are observed in the former one. Conversely, the nondimensional peak force is not affected by scale effects, suggesting that small-scale tests can be used to provide reliable predictions of this critical variable for engineering design.

Detached eddy simulations are used to characterize sediment entrainment mechanisms around a rectangular vegetation patch placed next to one of the sidewalls of a straight open channel. The plant stems are modeled as rigid circular cylinders and the incoming channel flow is fully turbulent. Simulations are performed with constant patch solid volume fraction (.=0.08) for the case the plant stems reach the free surface (emerged case) and then for a case with 50% patch submergence. Results show that the amplification of the streamwise velocity away from the lateral face of the patch decreases with increasing patch submergence. This is the main reason for the overall decay of the sediment entrainment capacity of the flow in the free stream region. Moreover, the large-scale patches of high and low instantaneous bed friction velocity present in the emerged case are becoming very weak in the 50% submergence case due to the decrease in the coherence of the shear layer vortices and of the instability that generates large-scale oscillations of the streamwise velocity inside the patch for emerged cases. The sediment entrainment capacity of the flow is also decaying inside of the patch, near its front face, with increasing patch submergence.