Free Surface Profile Research Articles

Bounds for Solutions to the Problem of Steady Water Waves with Vorticity

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the ...

A Numerical Study of Non-hydrostatic Shallow Flows in Open Channels

Abstract The flow field of many practical open channel flow problems, e.g. flow over natural bed forms or hydraulic structures, is characterised by curved streamlines that result in a non-hydrostatic pressure distribution. The essential vertical details of such a flow field need to be accounted for, so as to be able to treat the complex transition between hydrostatic and non-hydrostatic flow regimes. Apparently, the shallow-water equations, which assume a mild longitudinal slope and negligible vertical acceleration, are inappropriate to analyse these types of problems. Besides, most of the current Boussinesq-type models do not consider the effects of turbulence. A novel approach, stemming from the vertical integration of the Reynolds-averaged Navier-Stokes equations, is applied herein to develop a non-hydrostatic model which includes terms accounting for the effective stresses arising from the turbulent characteristics of the flow. The feasibility of the proposed model is examined by simulating flow situations that involve non-hydrostatic pressure and/or nonuniform velocity distributions. The computational results for free-surface and bed pressure profiles exhibit good correlations with experimental data, demonstrating that the present model is capable of simulating the salient features of free-surface flows over sharply-curved overflow structures and rigid-bed dunes.

ISPH modelling of landslide generated waves for rigid and deformable slides in Newtonian and non-Newtonian reservoir fluids

A comprehensive modeling of landslide generated waves using an in-house parallel Incompressible Smoothed Particle Hydrodynamics (ISPH) code is presented in this paper. The study of landslide generated waves is challenging due to the involvement of several complex physical phenomena, such as slide-water interaction, turbulence and complex free surface profiles. A numerical tool that can efficiently calculate both slide motion, impact with the surface and the resulting wave is needed for ongoing study of these phenomena. Mesh-less numerical methods, such as Smoothed Particle Hydrodynamics (SPH), handle the slide motion and the complex free surface profile with ease. In this paper, an in-house parallel explicit ISPH code is used to simulate both subaerial and submarine landslides in 2D and in more realistic 3D applications. Both rigid and deformable slides are used to generate the impulsive waves. A landslide case is simulated where a slide falls into a non-Newtonian reservoir fluid (water-bentonite mixture). A new technique is also proposed to calculate the motion of a rigid slide on an inclined ramp implicitly, without using the prescribed motion in SPH. For all the test cases, results generated from the proposed ISPH method are compared with available experimental data and show good agreement.

Simulation of self-compacting concrete in an L-box using smooth particle hydrodynamics

The three-dimensional Lagrangian-particle-based smooth particle hydrodynamics methodology was used to simulate the flow characteristics of self-compacting concrete (SCC) mixes in an L-box test. A Bingham-type constitutive model was coupled with the Lagrangian momentum and continuity equations to simulate the flow. The simulations of SCC focused on the flow times, the free-surface profile and the distribution of large aggregates (larger than or equal to 8 mm) during the flow. The numerical simulation results were compared with actual L-box tests carried out on several SCC mixes. The comparison revealed that the methodology is very well suited to predicting the flow behaviour of SCC in terms of passing and filling abilities and the distribution of large aggregates.

Strength properties and structure of a submicrocrystalline Al–Mg–Mn alloy under shock compression

The results of studying the strength of a submicrocrystalline aluminum A5083 alloy (chemical composition was 4.4Mg–0.6Mn–0.11Si–0.23Fe–0.03Cr–0.02Cu–0.06Ti wt % and Al base) under shockwave compression are presented. The submicrocrystalline structure of the alloy was produced in the process of dynamic channel-angular pressing at a strain rate of 104 s–1. The average size of crystallites in the alloy was 180–460 nm. Hugoniot elastic limit σHEL, dynamic yield stress σy, and the spall strength σSP of the submicrocrystalline alloy were determined based on the free-surface velocity profiles of samples during shock compression. It has been established that upon shock compression, the σHEL and σy of the submicrocrystalline alloy are higher than those of the coarse-grained alloy and σsp does not depend on the grain size. The maximum value of σHEL reached for the submicrocrystalline alloy is 0.66 GPa, which is greater than that in the coarse-crystalline alloy by 78%. The dynamic yield stress is σy = 0.31 GPa, which is higher than that of the coarse-crystalline alloy by 63%. The spall strength is σsp = 1.49 GPa. The evolution of the submicrocrystalline structure of the alloy during shock compression was studied. It has been established that a mixed nonequilibrium grain-subgrain structure with a fragment size of about 400 nm is retained after shock compression, and the dislocation density and the hardness of the alloy are increased.

Continuum modeling of surface roughening in heteroepitaxial structures based on phase field theory

Surface roughening is essential for the design of new nanostructures. In this paper, a continuum model based on phase field theory is developed to study the surface roughening in heteroepitaxial structures. A long-range order parameter is introduced to represent a diffusional free surface profile. By minimizing the system total free energy under constraints, the coupled field equations of quasi-static mechanical equilibrium and diffusion-controlled surface evolution are obtained. In contrast to the existing phase field models of free surface which are mostly based on the analytical Green’s function, the Galerkin finite element formulation is derived in this paper to solve the weak form of the coupled field equations. The advantage is that it is very convenient to deal with the elastic inhomogeneities and complex boundary conditions without additional complications. The proposed model is applied to study the nonlinear surface morphology evolution of heteroepitaxial thin films. The simulation shows that the heteroepitaxial film with an initial random fluctuation develops to form islands and coarsens during annealing. The kinetics of coarsening and the calculated average wavelength are consistent with the existing experimental results. It also shows that the initial island array formed during annealing consists of islands with similar size, shape and spacing. The influences of film thickness and misfit strain on the formation of initial island array are studied.

Numerical simulation of hydrodynamics in an uncovered unbaffled stirred tank

Numerical simulations were carried out to investigate the turbulent flow with free-surface vortex in an uncovered unbaffled stirred tank. An Eulerian–Eulerian multiphase flow model coupled with a volume-of fluid method was applied to capture the gas–liquid interface and describe the flow field in the tank. Turbulence is computed using a Reynolds stress model with seven equations. The reliability and accuracy of the simulations are verified by comparing the predicted free surface profiles and power numbers with experimental data in the literature. The overall agreements between the simulation results and experimental measurements are obtained. The simulation results show that vortex on the liquid free surface in the uncovered unbaffled tank can be divided into central zone and peripheral zone, and the critical radius of the two zones keeps almost unchanged with enhancing of impeller speeds. At the same impeller speed, vortex becomes deeper and larger with increasing of impeller diameter. However, vortex shape changes not very much with increasing of impeller diameter under the same power consumption. Impeller clearance has no significant influence on the vortex shape. Circumferential flow is dominant in the uncovered unbaffled tank, and the flow field can be divided into forced vortex region and free vortex region. Power numbers in uncovered unbaffled stirred tanks decrease slightly with increasing of the Reynolds number, and are far lower than that in baffled stirred tanks.

Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates.

Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.

Numerical modelling of supercritical flow in circular conduit bends using SPH method

The capability of the smoothed-particle hydrodynamics (SPH) method to model supercritical flow in circular pipe bends is considered. The standard SPH method, which makes use of dynamic boundary particles (DBP), is supplemented with the original algorithm for the treatment of open boundaries. The method is assessed through a comparison with measured free-surface profiles in a pipe bend, and already proposed regression curves for estimation of the flow-type in a pipe bend. The sensitivity of the model to different parameters is also evaluated. It is shown that an adequate choice of the artificial viscosity coefficient and the initial particle spacing can lead to correct presentation of the flow-type in a bend. Due to easiness of its implementation, the SPH method can be efficiently used in the design of circular conduits with supercritical flow in a bend, such as tunnel spillways, and bottom outlets of dams, or storm sewers.

New radial basis function network method based on decision trees to predict flow variables in a curved channel

Open channel bends have fascinated engineers and scientists for decades while providing water for domestic, irrigation and industrial consumption. The presence of curvature in a channel impacts the flow pattern, velocity and water surface profile. Simulating flow variables such as velocity and water surface depth is one of the most important matters in the design and application of open channel bends. This study investigates a new neural network method using the radial basis function (RBF) based on decision trees (DT-RBF) to predict velocity and free-surface water profiles in a 90° open channel bend. In this study, 506 flow depth and 520 depth-averaged velocity field data obtained at 5 different discharges (5, 7.8, 13.6, 19.1 and 25.3 l/s) in a 90° sharp bend were used for training and testing purposes. The obtained results showed that the proposed DT-RBF models were more accurate than RBF models in estimating flow depth and depth-averaged velocity in the bend. The RBF root-mean-square error (RMSE), mean absolute error (MAE) and relative error (δ) were reduced by 20, 24 and 23.5%, respectively, when using the hybrid DT-RBF model to estimate the depth-averaged velocity. For water surface prediction, the RMSE, MAE and δ decreased by 33, 27.5 and 37%, respectively, when using the proposed DT-RBF hybrid model. For the longitudinal profiles of water surface profile prediction at the outer edge, MAE (0.018) improved to MAE (0.0084) with DT-RBF. It was found that the hybrid decision tree-based method significantly improved RBF neural network performance in forecasting the velocity and free-surface water profiles in a 90° open channel sharp bend.

Influence of Rack Slope and Approaching Conditions in Bottom Intake Systems

The study analyzes the flow over bottom racks made of longitudinal T-shaped bars. A clear water flow is considered in a laboratory flume. Free surface profiles, wetted rack lengths, and discharge coefficients are measured, changing parameters such as longitudinal slope, void ratio, and approaching flow. The present work complements existing experimental studies, considering the influence of the approaching flow conditions. The velocity field measured with Particle Image Velocimetry (PIV) technique and the pressure field with Pitot tubes are quantified. Numerical simulations (CFD) are used to complement laboratory data. The energy head along the rack is calculated and compared with the hypothesis of horizontal energy level with minimum energy at the beginning of the rack. A discharge coefficient adjustment that considers the slope, the void ratio, and the position along the rack is proposed and presented with the results of other works. Theoretical proposals to calculate the pressure field along the flow are compared with measurements in the laboratory. The relation between the static pressure head in the space of bars and the discharge coefficient is used as an alternative method to define the discharge.

Numerical Modeling and Simulation of Wave Impact of a Circular Cylinder during the Submergence Process

Wave slamming loads on a circular cylinder during water entry and the subsequence submergence process are predicted based on a numerical wave load model. The wave impact problems are analyzed by solving Reynolds-Averaged Navier-Stokes (RANS) equations and VOF equations. A finite volume approach (FV) is employed to implement the discretization of the RANS equations. A two-dimensional numerical wave tank is established to simulate regular ocean waves. The wave slamming problems are investigated by deploying a circular cylinder into waves with a constant vertical velocity. The present numerical method is validated using other numerical or theoretical results in accordance with varying free surface profiles when a circular cylinder sinks in calm water. A numerical example is given to show the submergence process of the circular cylinder in waves, and both free surface profiles and the pressure distributions on the cylinder of different time instants are obtained. Time histories of hydrodynamic load on the cylinder during the submergence process for different wave impact angles, wave heights, and wave periods are obtained, and results are analyzed in detail.

Thin-film flow in helically wound shallow channels of arbitrary cross-sectional shape

We consider the steady, gravity-driven flow of a thin film of viscous fluid down a helically wound shallow channel of arbitrary cross-sectional shape with arbitrary torsion and curvature. This extends our previous work [D. J. Arnold et al., “Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] on channels of rectangular cross section. The Navier-Stokes equations are expressed in a novel, non-orthogonal coordinate system fitted to the channel bottom. By assuming that the channel depth is small compared to its width and that the fluid depth in the vertical direction is also small compared to its typical horizontal extent, we are able to solve for the velocity components and pressure analytically. Using these results, a differential equation for the free surface shape is obtained, which must in general be solved numerically. Motivated by the aim of understanding flows in static spiral particle separators used in mineral processing, we investigate the effect of cross-sectional shape on the secondary flow in the channel cross section. We show that the competition between gravity and inertia in non-rectangular channels is qualitatively similar to that in rectangular channels, but that the cross-sectional shape has a strong influence on the breakup of the secondary flow into multiple clockwise-rotating cells. This may be triggered by small changes to the channel geometry, such as one or more bumps in the channel bottom that are small relative to the fluid depth. In contrast to the secondary flow which is quite sensitive to small bumps in the channel bottom, the free-surface profile is relatively insensitive to these. The sensitivity of the flow to the channel geometry may have important implications for the design of efficient spiral particle separators.

Experimental and numerical evaluation of discharge capacity of sharp-crested triangular plan form weirs

Weirs are those structures used extensively for flood passage, flow measurement, flow deviation and controlling the water level in dams, rivers and open channels. In this study, the numerical model was simulated using flow-3D software and the results from experimental and numerical model were compared. Investigations have shown that the results of RNG turbulence model are better compared to k-e and LES models. Then, to model the turbulence, and to determine the location of free-surface profile, RNG model and VOF method were used respectively. Experimental results show proper coordination with estimated results of numerical model. For similar low values of h/p, the performance of 30 degree sharp crested triangular plan form weir for discharging of overflow was more suitable than other sharp-crested triangular plan form weirs.

Dip-coating with prestructured substrates: transfer of simple liquids and Langmuir–Blodgett monolayers

When a plate is withdrawn from a liquid bath, either a static meniscus forms in the transition region between the bath and the substrate or a liquid film of finite thickness (a Landau–Levich film) is transferred onto the moving substrate. If the substrate is inhomogeneous, e.g. has a prestructure consisting of stripes of different wettabilities, the meniscus can be deformed or show a complex dynamic behavior. Here we study the free surface shape and dynamics of a dragged meniscus occurring for striped prestructures with two orientations, parallel and perpendicular to the transfer direction. A thin film model is employed that accounts for capillarity through a Laplace pressure and for the spatially varying wettability through a Derjaguin (or disjoining) pressure. Numerical continuation is used to obtain steady free surface profiles and corresponding bifurcation diagrams in the case of substrates with different homogeneous wettabilities. Direct numerical simulations are employed in the case of the various striped prestructures.The final part illustrates the importance of our findings for particular applications that involve complex liquids by modeling a Langmuir–Blodgett transfer experiment. There, one transfers a monolayer of an insoluble surfactant that covers the surface of the bath onto the moving substrate. The resulting pattern formation phenomena can be crucially influenced by the hydrodynamics of the liquid meniscus that itself depends on the prestructure on the substrate. In particular, we show how prestructure stripes parallel to the transfer direction lead to the formation of bent stripes in the surfactant coverage after transfer and present similar experimental results.

LBM-LES Simulation of the Transient Asymmetric Flow and Free Surface Fluctuations under Steady Operating Conditions of Slab Continuous Casting Process

Transient flow structures in a continuous casting mold can strongly influence the slag entrainment in liquid steel and the bubbles capture in the initial solidified shell, both of which are associated with the quality of the final product. This paper presents a numerical study of the turbulent flow with a top free surface in the continuous casting mold at a meso-scale level by a three-dimensional combined approach of Free Surface Lattice Boltzmann Method and Large Eddy Simulation (FSLBM-LES). The validity of the model is verified by the good agreement between the calculated results and the measurements from various water experiments in terms of the flow velocity and free surface profile. The mathematical model is then used to reveal the transient and spatiotemporal asymmetric characteristics associated with the transient flow field and the free surface fluctuation, although the steady state operation is considered during the continuous casting process. The results show that the locations of the jets of liquid steel from the two out ports of the Submerged Entry Nozzle (SEN) always fluctuate alternatively within a certain range, and periodically deviate from the design angle of the SEN within the same time period. The oscillating behavior of the jets promotes the asymmetric flow patterns and multi-scale vortices at both sides of the SEN. By introducing the Q-criterion in the results analysis, the formation, development, and shedding of the coherent structure (CS) of the turbulent flow are quantitatively characterized. The interaction between the transient flow patterns and the fluctuations of the top free surface as well as the evolution of the transient profile and velocities of the free surface are also demonstrated. The results obtained from the current study suggest that the FSLBM-LES model offers a promising way to study the complex flows and related transfer phenomena in the continuous casting process.

Hydraulic Performance of an Embankment Weir with Rough Crest

AbstractAn experimental study was conducted on a large embankment weir with smooth and rough crests for a range of flow conditions 0.011≤Q≤0.226 m3/s. Close agreement between the rough and smooth crest experiments was observed in terms of flow patterns, free-surface profiles, and pressure distributions. The crest roughness had effects on the velocity distributions, the boundary layer properties, and the shear stress. On the rough crest, the velocity distributions were shifted upwards, increasing the growth rate of the boundary layer. The calculation of the shear stresses with the law of the wall, outer flow region, and momentum integral methods showed consistent results of about τ/(ρ g H)≈0.0015–0.0018 for the smooth crest and of τ/(ρ g H)≈0.0024 for the rough crest. The crest roughness reduced the discharge performance, while upstream and downstream ramps did not have any effects. The present observations highlighted the relevance of roughness and scaling on the hydraulic performances of weirs.

Effects of Moored Boats on the Gradually Varied Free-Surface Profiles of River Flows

AbstractThe last mile of rivers often represents an option worth considering for the development of ports. This is particularly true in coastal areas that are densely developed. The simplest form of a port canal, i.e., boats moored along the banks, is an attractive option for the low investment required and the often acceptable residual wave motion. Some of the major rivers have thousands of boats moored along their banks. The effect of these floaters on the river levels during floods must be taken into account. The authors present a methodology based on the computation of the forces on a single boat or an array of boats that yields a quantification of the effect on the river levels. It essentially provides an iterative procedure for computing the value of the resistance coefficient to be used in hydraulic analysis. Theoretical predictions are satisfactorily confirmed by ad hoc experimental investigation.

A flow field characterization in a circular channel along a side weir

Side weirs have been used for centuries in urban drainage and flood control for their ability to divert high incoming flow rates. The flow along a side weir may be studied by means of different approaches. However, although the question of side weir has been debating for many decades, there are few studies on the local hydraulic characteristics of the flow along these structures in circular channels.An experimental study of the flow field in a circular channel along a side weir using a commercial TSI Particle Image Velocimetry (PIV) system is shown in this paper. Weakly supercritical flows running into different geometric configurations of the weir were investigated, with Froude numbers varying from 1.1 to 1.6. Free surface profiles have been obtained by an image processing technique. An empirical equation has been proposed for their representation. Longitudinal velocity profiles along the side weir can be well predicted by an entropic approach. Local outflow along the side weir may be represented by an asymmetric curve. The peak outflow generally occurs between the 30% and the 50% of the weir length. The elementary discharge coefficient significantly increases from upstream to downstream. An energy head reduction was observed under the investigated configurations. Most of the head variation occurs in the central part of the weir. Finally, the flow power decreases along the weir according to a non-linear function.A thorough knowledge of the flow field features should allow to improve side weir analysis and design, whatever approach is used for the study.

Investigation of molten pool behavior and weld bead formation in VP-GTAW by numerical modelling

A three dimensional model is proposed to characterize the transport phenomena in molten pool and weld bead formation in high speed VP-GTAW. The temperature and velocity distribution, deformation of molten pool free surface and weld bead profile are systematically discussed. The molten pool temperature exhibits a periodically fluctuations (20–40K), and the molten pool shape and velocity remain minor oscillation in the DCEN and DCEP phases. The velocity increases dramatically to 1.012m/s at the initial stage (0–0.01s), then the velocity remains stable as the further development of molten pool volume (0.01–0.4s). Two forms of flow patterns driven by surface tension and electromagnetic force are dominated in the molten pool. The formation mechanisms of weld bead defects (undercut, humping) are analyzed. The simulated weld bead profiles are agree well with the experimental weld appearance. This numerical study may provide a good foundation for optimizing the VP-GTAW process to achieve quality welds.