Three-dimensional Flow Structures Research Articles

Experimental and Numerical Research on End-Wall Flow Mechanism of High-Loading Tandem Stators

Tandem blades have been recognized for their potential to enhance the loading capacity of compressors. However, tandem stators currently do not exhibit advantages due to insufficient understanding of the complex end-wall flow mechanisms. To address this, an extensive study was conducted on tandem stators using experimental and numerical methods. The analysis focused on loss development, three-dimensional flow structures, and interaction mechanisms between front and rear blades. The results indicated the following: (1) The rear blade’s influence on the front blade has contrasting effects in mid-span and end-wall regions. The stagnation action of the rear blade increases front blade load, leading to greater corner separation losses. (2) Mid-span flow losses account for over 60% of total losses near stalls, primarily due to increased mixing losses from the migration of front blade corner separations towards the mid-span region in the rear blade channel. (3) At higher mass flow rates, corner separations occur in the rear blade, driven by significant circumferential pressure differences at the front section of the rear blade, causing end-wall fluid migration towards suction surfaces. (4) Corner stalls predominantly occur in the front blade, with associated losses exceeding those in conventional blades.

Three-dimensional flow structures modelling based on a depth-integrated method in a sharply curved open channel over topography

In this study, we introduce the bottom velocity calculation (BVC) technique, a depth-integrated approach for modeling three-dimensional flow systems in the two-dimensional (2D) river management model framework. The method has been expanded to a general coordinate system and its applicability to flow in bends and meanders for the applications to rivers. The method was validated to a laboratory experiment conducted in a sharply curved channel over topography. The pattern of water surface elevation and vertical velocity distribution can be replicated by the BVC method’s models, which also show strong qualitative agreement with the experimental dataset and 3D model. The benefit of using the BVC technique instead of the 2D model is verified; the 2D model is unable to replicate the profile since it does not take into account three-dimensional flow structures. As seen above, the BVC method is helpful in evaluating the river environment because it can account for the complicated material transports caused by three-dimensional flows in the meandering sections of the river channel.

Experimental study of flow dynamics in a dual-stage annular swirling jet

In this study, three-dimensional flow structures at a Reynolds number of 16 000 are measured by stereoscopic particle image velocimetry to reveal the dynamic structures of a dual-stage co-rotating annular swirling jet with a blunt separating wall. The swirler number of the outer swirling jet is fixed at around 0.5 and that of the inner swirling jet is varied from 0 to 0.7. Spectral proper orthogonal decomposition is used for extracting organized structures and the topological evolution from a spatiotemporal flow field. The evolution of mean-flow topography is initially depicted as the flow transition from the center body wake to the presence of a central recirculation zone, with an increase in the inner swirl number varying from 0 to 0.7. Two precessing vortex cores (PVCs) with different frequencies are observed in the presence of a central recirculation zone, and the temporal independence of the two PVCs is identified. A transition region exists between the two PVCs because of the different dominant axial regions of the two PVCs, which is located between the central recirculation induced by inner swirling and that by the merged swirling flow. The conservation of circulation for both PVCs was confirmed in the transition region, and the two PVCs exhibited independent single-helical modes. Furthermore, the main frequencies of the two PVCs are proportional to the inner swirl number; however, they are higher than those of the corresponding single swirling jet. As predicted by the Landau equation, both PVCs had the same critical swirl number, suggesting that the two structures occurred simultaneously.

Considerations for Use of Three-Hole Probes for Turbomachinery Applications

Three-hole probes are frequently employed to characterize the flow field upstream, within, and downstream of turbomachines. From three pressure measurements, the static pressure, total pressure, Mach number, and a single axis flow angle can be resolved. Despite this utility, there are induced errors related to the use of these probes that are particularly pertinent to use in turbomachines. Small passages cause end-wall proximity effects, the rotor induces flow unsteadiness, and complex flow three-dimensional flow structures are developed. These effects are considered in this effort by contrasting experimental inlet distortion data from an auxiliary power unit inlet of a centrifugal compressor with computational inlet-only models. The aerodynamic interface plane was measured using a rotatable rake assembly with trapezoidal three-hole probes and fast response pressure transducers. Two computational models are presented, one with and one without the aerodynamic interface plane instrumentation used in the experiment. The measurements from the simulated traverse are compared to the inlet-only computational model and experimental results. End-wall effects were the dominant error driver for the static pressure measurements. Swirl was predominantly altered by perpendicular flow angle magnitudes above 10 deg. Indicated total pressure was impacted by radial flow angle and unsteadiness that were highly coupled.

Spatially Resolved Analysis of the Influence of Three-Dimensional Flow Field Structures on Loss Processes in PEMFCs

During operation of large-area polymer electrolyte membrane fuel cells (PEMFCs) under high loads, strong gradients occur along the gas channel and mass transport becomes the performance limiting factor. Towards the gas outlet the concentration of the reactants decreases, while the relative humidity increases and liquid water is formed. Despite these high gradients, the aim is to achieve a high performance over the entire channel length.The flow field geometry, which controls the distribution of the reactant, the removal of excess liquid water and the electrical contact, has a significant influence on the overall performance and the power density distribution along the channel. Conventionally, channel-rib structures are used, but these are limited in terms of the distribution and transportation of the reactants, especially under high loads.Due to their specific structure and open geometry, three-dimensional flow structures have an optimized transport of reactants to the active layer as well as removal of water and thus a strongly improved mass transport. At the same time, such structures have a highly reduced contact area, which leads to increased ohmic losses. In order to fully understand the impact on the performance of the cell under different operating conditions, a quantification of the local performance as well as resistances distribution and their dependencies is essential.For this purpose, we are examining both channel-rib and three-dimensional flow structures using a segmented cell. The performance distribution is determined by the local measurement of current and voltage. The measurement of electrochemical impedance spectra of each segment and impedance data analysis by the distribution of relaxation times (DRT) [1,2,3] enables the local deconvolution of the loss processes.In this contribution, an experimental analysis of the influence of three-dimensional meshes compared to a channel-rib flow field design on cell performance is presented. In particular, the effects of high loads, high humidities and low oxygen concentrations on the performance and resistances and their distribution along the gas channels are discussed. Based on the physical interpretation of the results, consequences and conclusions for the further development of three-dimensional flow field geometries are drawn.

3D-3C measurements of flow reversal in small sessile drops in shear flow

Sessile drops play an important role in nature and the operation of many technical and biological systems as for instance fuel cells, cleaning processes, lab-on-a-chip devices and single-cell analysis. Nonetheless, their dynamic behavior in a shear flow is still not fully understood (e. g. detachment mechanism). Challenges exist regarding the precise simulation of two-phase flows as well as difficulties in conducting flow measurements with sufficient temporal resolution of more than 1 kHz. In this article, we present the first three-dimensional flow measurements in strongly oscillating drops stemming from a shear flow by using a monocular 3D localization microscope based on a Double-Helix Point Spread Function combined with Particle Tracking Velocimetry. The high temporal resolution and the large measurement volume - in terms of microscopy - make it possible to measure the time- and phase-averaged flow in small drops with Bond numbers (Bo) smaller than 1. Water drops were placed in an air flow channel and measurements were conducted for Reynolds numbers (Red) from about 750 to 1700. Our measurements show that the results of previous investigations for drops with Bo>1 concerning vortex pattern and flow reversal inside the drop apply for smaller drops as well. In addition, we are able to reveal the three-dimensional flow structure and multiple vortex-pattern in time and space. We discover a periodic 3D vortical flow pattern that corresponds to the first and second eigenfrequency of the drop. Moreover, we demonstrate the potential of adaptive optics to correct measurement errors stemming from time-varying light refraction when conducting measurements through the fluctuating drop surface, in particular for opaque substrates. The results may help in understanding the coupling of inner and outer flow for sessile drops in shear flow which allows for an analysis of the onset motion of these drops, i.e. drop removal. Removal of sessile drops plays a crucial role in many applications, which includes among other things the water management of fuel cells where small drops with Bo<1 predominantly occur.

Combining Computational Fluid Dynamics and Experimental Data to Understand Fish Schooling Behavior.

Understanding the flow physics behind fish schooling poses significant challenges due to the difficulties in directly measuring hydrodynamic performance and the three-dimensional, chaotic, and complex flow structures generated by collective moving organisms. Numerous previous simulations and experiments have utilized computational, mechanical, or robotic models to represent live fish. And existing studies of live fish schools have contributed significantly to dissecting the complexities of fish schooling. But the scarcity of combined approaches that include both computational and experimental studies, ideally of the same fish schools, has limited our ability to understand the physical factors that are involved in fish collective behavior. This underscores the necessity of developing new approaches to working directly with live fish schools. An integrated method that combines experiments on live fish schools with computational fluid dynamics (CFD) simulations represents an innovative method of studying the hydrodynamics of fish schooling. CFD techniques can deliver accurate performance measurements and high-fidelity flow characteristics for comprehensive analysis. Concurrently, experimental approaches can capture the precise locomotor kinematics of fish and offer additional flow information through particle image velocimetry (PIV) measurements, potentially enhancing the accuracy and efficiency of CFD studies via advanced data assimilation techniques. The flow patterns observed in PIV experiments with fish schools and the complex hydrodynamic interactions revealed by integrated analyses highlight the complexity of fish schooling, prompting a reevaluation of the classic Weihs model of school dynamics. The synergy between CFD models and experimental data grants us comprehensive insights into the flow dynamics of fish schools, facilitating the evaluation of their functional significance and enabling comparative studies of schooling behavior. In addition, we consider the challenges in developing integrated analytical methods and suggest promising directions for future research.

Mass and Heat Exchange in Rotating Compressor Cavities With Variable Cob Separation

Abstract Next generation aeroengines will operate at ever-increasing pressure ratios with smaller cores, where the control of blade-tip clearances across the flight cycle is an emerging design challenge. Such clearances are affected by the thermal expansion of the compressor disks that hold the blades, where acute thermal stresses govern operating life. The cavities formed by corotating disks feature a heated shroud at high radius and cooler cobs at low radius. A three-dimensional, unsteady and unstable flow structure is induced by destabilizing buoyancy forces. The radial distribution of disk temperature is driven by a conjugate heat transfer at Grashof numbers of order 1013. Such flows are further influenced by the heat and mass exchange with an axial throughflow of cooling air at low radius, where the interaction depends on the Rossby number and separation of the disk cobs. This paper is the first to study the effect of cob separation ratio on mass and heat exchange for compressor cavities. A model is developed to predict the cavity-throughflow interaction, and disk and fluid-core temperatures. The judicious use of a physics-based methodology provides reliable, reduced-order solutions to the complex conjugate problem, thereby making it appropriate for practical engine thermo-mechanical design. The model is validated by detailed experimental measurements using the Bath Compressor Cavity Rig, where variable disk cob spacings were investigated over a range of engine-representative conditions. The unsteady pressure measurements collected in the frame of reference of the rotating disks reveal new insight into the fundamentally aperiodic nature of the flow structure. This new understanding of heat transfer informs an expedient reduced-order model and enables more efficient design of future high pressure-ratio aeroengines.

Direct Numerical Simulation of Tandem-Wing Aerodynamic Interactions at Low Reynolds Number

A three-dimensional direct numerical simulation was conducted to investigate the vortex–wing interactions through two NACA 0012 stationary wings placed in tandem at a low Reynolds number Re=10,000. The aerodynamic characteristics and three-dimensional flow structures were analyzed for the tandem wings. The back wing disturbed by the upstream vortices gained an evident increase in aerodynamic performance, where the advantage is related to the suppression of the large-scale vortex formation near the trailing edge. The Liutex method was applied to visualize the vortical structures for investigating the three-dimensional evolution and instability when interacting with the back wing. The upstream wake triggered dual-secondary vortices and intensified the secondary instability on the back wing. The induced vortices contributed to the lift enhancement because they provided an extra low-pressure region when propagating downstream on the suction side of the back wing. Because of the three-dimensional destabilization, the vortex interaction in the evolution process accelerated the transition and injection of the high-momentum flow into the boundary layer attached to the back wing, energizing the turbulent boundary layer and eliminating the large-scale separation near the trailing edge. This study provided a new perspective on the enhanced aerodynamic performance of tandem layout.

Numerical investigation of a three-dimensional flow structure influenced by the lateral expansion of a partially distributed submerged canopy

This study employed numerical simulations to investigate the three-dimensional hydrodynamic structure of a channel obstructed by submerged rigid canopies. After validating the model using existing experimental data, a series of numerical experiments varying the canopy density and the blocking ratio were conducted. The results indicate that submerged canopy, different from emergent canopies, triggers the generation of three-dimensional coherent vortices. When canopies expand laterally, the scale, location, and evolution characteristics of cross-sectional secondary flows vary. The direction of secondary circulation at the canopy top of the fully covered vegetation channel is affected by the river width/depth ratio. Three-dimensional coherent vortices are jointly controlled by vertical and transverse Reynolds stresses. When the channel is seriously blocked, the degree of momentum exchange in the mixing layer does not change significantly. The boundary friction effect dominates the momentum exchange process near the wall. An equation coupling the drag length scale and blocking ratio with the vertical coherent vortex penetration length is proposed, indicating that the penetration length increases exponentially with both variables. Higher canopy density exhibits a stronger resistance to the vertical coherent vortex penetration length. In the process of canopy lateral expansion, the evolution time of the outer vortex to scale stability is longer than that of the inner vortex. The significance of research on the three-dimensional vortex structure of rigid submerged vegetation is established based on previous studies and existing conclusions.

A genuinely three-dimensional Riemann solver based on self-similar variables in curvilinear coordinates

A genuinely three-dimensional Riemann solver named MuSIC-3DC (Multidimensional, Self-similar, strongly-Interacting, Consistent, Three-dimensional, Curvilinear coordinates) in curvilinear coordinates is proposed to simulate three-dimensional complex engineering problems. Following Balsara's idea, a three-dimensional Riemann problem at each grid vertex is studied. In order to accurately capture contact discontinuities, the linear variations are assumed in the strongly-interacting zone. Also, the self-similar variables are adopted to simplify the derivation according to the self-similar evolution of the Riemann problem, and the strongly-interacting state could be determined by combining the contributions of all the two-dimensional Riemann problems at the surfaces bounded the strongly-interacting state. Besides, the one-dimensional Riemann problem at the center of each cell interface is solved by the one-dimensional HLLE scheme. After these, the numerical flux at each cell interface is obtained by assembling the three-dimensional fluxes at the vertices and the one-dimensional flux at the center of the cell interface. Several numerical test cases are simulated to validate the proposed solver. The spherical blast wave and three-dimensional Riemann problems indicate that the MuSIC-3DC scheme is capable of capturing three-dimensional self-similar flow structures accurately and suppressing the mesh imprinting phenomenon effectively. The transonic case demonstrates the MuSIC-3DC scheme's high resolution in capturing shock waves. In the hypersonic cases, the MuSIC-3DC scheme is more accurate in predicting stagnation and reattachment heating loads. Moreover, the MuSIC-3DC scheme is capable of obtaining more physical surface heating distribution by considering the information traveling both normal and transverse to the cell interfaces.

Multi-Scale Flow Estimation Within Stereoscopic PIV Processing Based On Deep Learning

Classical methods of particle-image velocimetry (PIV) based on state-of-the-art cross-correlation algorithms constitute a spatial filtering operation that leads to a loss of spatial resolution of the estimated velocity fields. This is associated with numerous limitations, e.g., when it comes to aligning high-fidelity numerical simulations, since the detection of small turbulent scales is often precluded. An optical flow neural network adapted for PIV processing, called RAFT-PIV, can overcome the drawback of reduced spatial resolution while maintaining the reliability and robustness of the classical approach. The present work aims to extend the application of the aforementioned deep learning based method by integrating it within a stereoscopic PIV processing workflow. Experimental data from an optically accessible single-cylinder combustion engine the flow field of which is characterized by complex, three-dimensional flow structures and a broad range turbulent scales are processed using RAFT-PIV. The 2D-3C vector fields obtained from stereoscopic reconstruction are compared against a classical cross-correlation based PIV evaluation, providing insight into instantaneous quantities and turbulent structures. A qualitative comparison of the instantaneous in-plane and out-of-plane velocity components reveals that the reconstructed flow obtained from the deep learning based estimation includes a wide range of resolved turbulent length scales. It outperforms the classical method. Furthermore, the findings of the present study demonstrate that the pixel accuracy has an effect on both physical quantities and turbulence characteristics. In particular, an increase in turbulent kinetic energy is observed across the engine cycles of the investigated engine flow states. Moreover, a higher fraction of three-dimensional isotropic turbulent structures is derived from RAFT-PIV compared to the results from the cross-correlation based technique.

Flow dynamics and turbulent coherent structures around sediment reduction plates of a sewer system

In the management of urban drainage networks, great interest has been generated in the removal of sediments from sewer systems. The unsteady three-dimensional (3D) flow and turbulent coherent structures surrounding sediment reduction plates in a sewer system are investigated by means of the detached-eddy simulation (DES). Particular emphasis is given to detailing the instantaneous velocity and vorticity fields within the grooves, along with an examination of the three-dimensional, long-term, average flow structure at a Reynolds number of approximately 105. Velocity vectors demonstrate continuous flapping of the flow on the groove wall, periodically interacting with ejections of positive and negative vorticity originating from the grooves. The interaction between the three-dimensional groove flow and the shear flow leads to the downstream transport of patches of positive and negative vorticity, which significantly influence sediment transport. The high-velocity shear flows and strong vortices generated in undulating topography, as identified by the Q-criteria, are the key factors contributing to the efficient sediment reduction capabilities of the sediment reduction plates. The sediment reduction plates with partially enclosed structures exhibit low sedimentation rates in grooves on the plate, a broader acceleration region, and a lesser impact on the flow capacity. The results improve the understanding of the hydrodynamics and turbulent coherent structures surrounding the sediment reduction plates while elucidating the driving factors behind the enhancement of sediment scouring and suspension capacities. These results indicate that the redesign of the plates as partially enclosed structures contributes to further improving their sediment reduction performance.

Experimental and numerical investigation of the effects of porous bleed systems on a model scramjet inlet under high backpressure conditions

In this study, we conducted experimental and numerical investigations to assess the effects of backpressure and configurations of porous bleeds on a scramjet inlet's performance at Mach 5. Experiments in a high-enthalpy wind tunnel, using a backpressure controller and a porous bleed with 1 mm holes and 10 mm lateral spacing, were paired with Reynolds-averaged Navier-Stokes simulations to explore the mitigation of flow separation and the prevention of inlet unstart under various backpressure conditions. The simulations provided insights into how backpressure levels and porous bleed geometries impact internal flow structures, as well as the critical backpressure ratio at which inlet unstart occurs. Additional simulations varied the bleed system's hole diameter and spacing, identifying configurations that optimize aerodynamic performance by enhancing total pressure recovery and Mach number at the isolator exit, while reducing bleed mass flow rates. The study also quantified the impact of these configurations on engine thrust, determining that certain bleed system designs significantly improve thrust by efficiently suppressing the interaction between the core flow and corner recirculation zones. This study examined the internal three-dimensional flow structure characteristics under high back pressure and provided insights into porous bleed configuration capable of efficiently suppressing inlet unstart.

Corner Vortex Structures Within Recirculation Zones of Confined Bluff-Body Flows

The vortical structures of recirculation zones in turbulent nonreacting and premixed reacting flows around confined equilateral-triangle bluff bodies are investigated using large-eddy simulations. A three-dimensional flow structure is observed within the recirculation zones of reacting and nonreacting cases; the structure includes both the “spanwise” vortices that extend the length of the recirculation zone and “corner vortex structures” that are situated adjacent to the bluff-body trailing edge and spanwise walls. The corner vortex structures enhance the mixing and residence times of fluid inside the recirculation zone. Fluid is circulated between the spanwise vortices and corner vortex structures. Corner vortex structures always appear downstream (relative to the local flow) of the boundary-layer separation location on the spanwise walls within the recirculation zone, and they are absent when boundary-layer formation on the spanwise walls is prevented. Laminar boundary-layer theory predicts the boundary-layer separation locations within ∼12% of the calculated values from the large-eddy simulations in all but one case. These results suggest that 1) boundary-layer separation on the spanwise walls is a necessary condition for the formation of corner vortex structures, and 2) boundary-layer separation on the spanwise walls is caused by an adverse pressure gradient in the recirculation zone.

Three-dimensional flow structure in a confluence-bifurcation unit

Enhanced understanding of flow structure in braided rivers is essential for river regulation, flood control, and infrastructure safety across the river. It has been revealed that the basic morphological element of braided rivers is confluence-bifurcation units. However, flow structure in these units has so far remained poorly understood with previous studies having focused mainly on single confluences/bifurcations. Here, the flow structure in a laboratory-scale confluence-bifurcation unit is numerically investigated based on the FLOW-3D® software platform. Two discharges are considered, with the central bars submerged or exposed respectively when the discharge is high or low. The results show that flow convergence and divergence in the confluence-bifurcation unit are relatively weak when the central bars are submerged. Based on comparisons with a single confluence/bifurcation, it is found that the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit reign over those of the downstream central bar. Concurrently, the high-velocity zone in the confluence-bifurcation unit is less concentrated than that in a single confluence while being more concentrated than that observed in a single bifurcation. The present work unravels the flow structure in a confluence-bifurcation unit and provides a unique basis for further investigating morphodynamics in braided rivers. Highlights 3D flow structure in a confluence-bifurcation unit (CBU) is numerically investigated. Flow convergence/divergence in CBU is relatively weak when central bars are submerged. Effects of the upstream central bar on CBU flow reign over those of the downstream central bar. The high-velocity zone is less/more concentrated in the CBU than in a single confluence/bifurcation when the central bars are exposed.

An Investigation of Contaminant Transport and Retention from Storage Zone in Meandering Channels

Contaminant trapping by recirculation zones occurring at the apex of natural meandering channels induces a long tail in the contaminant cloud, thereby complicating the prediction of mixing behaviors. Thus, the understanding of the interaction between solute trapping and recirculating flow is important for responding to and mitigating water pollution accidents. In this research, the EFDC model was employed to reproduce three-dimensional flow structures of recirculating flow at the channel apex and investigate the influence on contaminant mixing. To investigate the contaminant transport characteristics from the storage zone in meandering channels, simulations were conducted using various discharge values to assess the impact of storage zone development on the concentration–time curves. The analysis of the relationship between the storage zone size and mixing behaviors indicates that an increase in discharge could result in a shorter tail and larger longitudinal dispersion even with the larger storage zone size. On the other hand, the enlarged recirculation zone size contributes to reducing transverse dispersion, evidenced by flatter dosage curves under lower flow rate conditions. These findings suggest that the increase in longitudinal dispersion with a larger flow rate is primarily caused by the reduction in transverse dispersion resulting from the formation of the recirculation zone.

Flow three-dimensionality of wavy elliptic cylinder: vortex shedding bifurcation

The three-dimensional flow of a wavy elliptic cylinder and its wake are investigated for wavelength (λ) to hydraulic diameter (Dm) ratio λ/Dm = 3.43, 4.58, and 6.01. The study aims to gain insights into flow separation, flow dynamics, and three-dimensional flow topologies for the selected wavelengths. Three-dimensional flow structures are visualized through time-averaged streamlines with critical point theory. The results for λ/Dm = 3.43 show spiral flow directed from the node to the saddle, counter-rotating vortices, and a wavy vortex structure. On the contrary, the flows for λ/Dm = 4.58 and 6.01 respectively undergo bifurcation in stable recirculation and vortex shedding on a half-span of the cylinder wavelength. The bifurcation results in a spiral flow toward the saddle and the other spiral flow toward the node. The bifurcations of recirculation and vortex shedding are revealed for the first time. The findings contribute to an improved understanding of the insight into intricate flow physics around wavy elliptic cylinders.

Tip effects on three-dimensional flow structures over low-aspect-ratio plates: mechanisms of spanwise fluid transport

Three-dimensional flows over low-aspect-ratio rectangular flat plates ( ${A{\kern-4pt}R} = 1.00$ – $1.50$ ) are investigated using tomographic and planar particle image velocimetry techniques. The chord-based Reynolds number is $5400$ , and the angle of attack is fixed at $6^\circ$ . This study reveals for the first time the interplay between spanwise fluid transport and downwash, both originating from the tip effects. Spanwise fluid transport promotes the formation and subsequent coherent development of leading-edge vortices, whereas downwash stabilizes the flow. Specifically, two mechanisms related to spanwise fluid transport are revealed. First, the spanwise fluid transport enhances the intensity of the reversed flow, promoting the shear layer roll-up and vortex shedding. Second, the near-wall spanwise flow interacts with the shed C-shape vortices, thereby strengthening the vortex heads. In particular, through these interactions, spanwise fluid transport can sustain the coherence of the C-shape vortices until the vortex heads split in a regular fashion. Consequently, the C-shape vortices are transformed into novel Þ-shape vortices for the plates of ${A{\kern-4pt}R} \leq 1.25$ , which supplements the previously discovered transformation from C-shape to M-shape vortices for larger ${A{\kern-4pt}R}$ plates. Downstream of this novel vortex-splitting transformation, two fundamental processes contribute to the formation of hairpin vortices. The above comprehensive understanding of complete vortex evolution routine provides valuable insights into the tip effects on the formation of three-dimensional flows over low- ${A{\kern-4pt}R}$ plates.

Numerical Calculations for Curved Open Channel Flows with Advanced Depth-Integrated Models

This paper examines the effect of coupling several equations for vertical velocity profiles in a depth-integrated model to clarify the roles of three-dimensional flow structures minimizing the errors from the two-dimensional calculation model. In addition, this paper develops a numerical discretization method for the dispersion terms in the horizontal momentum equations. It is revealed that the use of the 2DC model underestimates the water surface elevation through the comparisons with the experimental datasets, advanced 2DC and 3DC results. The prediction of water surface elevation is considerably improved by taking into account of secondary flow effect with vorticity equations. It is clarified that the accuracy in predicting the water surface elevation is increased with increasing the degree of the function for vertical velocity profiles in advanced 2DC models or the number of vertical grids in 3DC model, while non-hydrostatic pressure and variation in vertical velocity have a second importance.