Free Surface Profile Research Articles

Transition from undular jump to breaking jump

ABSTRACTIn the present study, a one-dimensional governing equation with a vertical acceleration term is used to analyse the free surface profile of various types of jumps. Initially, the surface profile of various types of jumps are reproduced analytically by using a suitable eddy diffusivity term. The approximations of the surface profile for various jumps are then verified numerically. Based on the comparisons, an empirical relationship between the Froude number and the turbulent diffusivity coefficient is established. The proposed empirical relationship is used in the governing equation to compute the various jumps, where an implicit two dimensional flow is assumed. The obtained results are compared with the experimental data to assure the transition of flow from undular jump to breaking jump. The comparisons of numerical results of wave profiles, pressure distribution, wave length and wave amplitude with the existing experimental data for various Froude numbers indicated the suitability of the proposed empirical relationship. Finally, the transitions of flow from undular jump to breaking hydraulic jump are discussed.

Marangoni stability of a thin liquid film falling down above or below an inclined thick wall with slip

AbstractThe linear and nonlinear instability of a thin liquid film flowing down above or below (Rayleigh-Taylor instability) an inclined thick wall with finite thermal conductivity are investigated in the presence of slip at the wall-liquid interface. A nonlinear evolution equation for the free surface deformation is obtained under the lubrication approximation. The curves of linear growth rate, maximum growth rate and critical Marangoni number are calculated. When the film flows below the wall it will be subjected to destabilizing and stabilizing Marangoni numbers. It is found that from the point of view of the linear growth rate the flow destabilizes with slip in a wavenumber range. However slip stabilizes for larger wavenumbers up to the critical (cutoff) wavenumber. From the point of view of the maximum growth rate flow slip may stabilize or destabilize increasing the slip parameter depending on the magnitude of the Marangoni and Galilei numbers. Explicit formulas were derived for the intersections (the wavenumber for the growth rate and the Marangoni number for the maximum growth rate) where slip changes its stabilizing and destabilizing properties. From the numerical solution of the nonlinear evolution equation of the free surface profiles, it is found that slip may suppress or stimulate the appearance of subharmonics depending on the magnitudes of the selected parameters. In the same way, it is found that slip may increase or decrease the nonlinear amplitude of the free surface deformation. The effect of the thickness and finite thermal conductivity of the wall is also investigated.

Assessment of interfacial turbulence treatment models for free surface flows

The modelling of complex free surface flows is challenging due to the mobility and deformability of the interface and air entrainment characteristics, which are highly affected by turbulence. With the framework of Reynolds averaged Navier–Stokes (RANS) models and the volume of fluid (VOF) method, turbulence quantities at the air–water interface tend to be over-estimated. In this study, interfacial turbulence treatment methods including the buoyancy modification model based on the simple gradient diffusion hypothesis (SGDH) and Egorov’s turbulence damping model are investigated. Furthermore, due to the unconditionally unstable characteristics of the standard k-ε turbulence model, the stabilized k-ε turbulence model is applied as a comparison. The turbulence attenuation performance using different interfacial turbulence treatment methods in the vicinity of the interface is compared and discussed for stratified flows and free overflow weirs for aerated and non-aerated nappe scenarios. The turbulence quantities and free surface profile under different flow conditions are validated against experimental data and an analytical model. The results show that for free surface waves, both the SGDH model and the turbulence damping model give strong improvements in turbulence production compared with the standard k-ε model. The SGDH model augments the turbulence kinetic energy (TKE) in the unstable stratification, leading to unphysical behaviour for the partially dispersed and separated flow.

Steady-state interfacial gravity waves with one-dimensional class-IV triad resonance

Steady-state interfacial waves under a one-dimensional (1-D) class-IV exact triad resonance are investigated in a two-layer fluid with a free surface upper boundary. Four groups (G1–G4) of convergent series solutions are obtained by the homotopy analysis method. Though the harmonic resonance conditions are contained in the class-IV resonance criteria, the influences of 1:2 harmonic resonances on the energy spectrum could be neglected. Unlike former steady-state resonant interfacial wave spectrum where all the components joining the resonance are significant, the energy of one primary component can be ignored for two groups (G2 and G4) of wave solutions obtained in this paper. It is found that a little energy induced from the external environment might greatly change the energy spectrum for G1. However, the energy introduced from the outside cannot vary the wave energy distribution for G2. The reason for the extremely high crests on the instantaneous profiles of free surfaces of G2 and G4 is that the peaks of the class-IV exactly resonant and some trivial components momentarily overlap at some special horizontal positions. The class-IV exact triad resonance curve could be divided into four pieces containing the existence ranges of G3 and G4 and two regions with no solution found. One of the regions without convergent solutions results from an infinite number of singularities or small divisors caused by infinite exact or near resonances. Our results indicate that steady-state interfacial waves with class-IV triad resonance interactions among one surface and two internal wave modes could exist.

CFD modelling of initial stages of dam-break flow

Cases of dam failures are seen almost every year globally, and the propagation of rapidly varying unsteady flows resulting from those dam failures has significant environmental and economic consequences. A three-dimensional Computational Fluid Dynamics model was created to solve unsteady Reynolds-Averaged equations to investigate the generation and propagation of dam-break flows and reflected flood waves in the presence of a trapezoidal bottom obstacle. The dam-breaks were modelled using OpenFOAM based on the Volume of Fluid method employing the Finite Volume Method, and the performance of various Reynolds-Averaged Navier-Stokes turbulence models, including the realizable k − ε, SST k − ω (Shear Stress Transport k − ω), and v2 − f models, has been evaluated. To assess the capabilities of the different turbulence models, quantitative comparisons of numerical simulations with laboratory experiments were completed, and it was found that the SST k − ω model performed better in predicting the free surface profiles and negative bore propagation speeds than the realizable k − ε and v2 − f models.

Plunging breakers. Part 1. Analysis of an ensemble of wave profiles

An experimental study of the dynamics and droplet production in three mechanically generated plunging breaking waves is presented in this two-part paper. In the present paper (Part 1), the dynamics of the three breakers are studied through measurements of the evolution of their free surface profiles during 10 repeated breaking events for each wave. The waves are created from dispersively focused wave packets generated with three wave maker motions that differ primarily by small changes in their overall amplitude. Breaker profiles are measured with a cinematic laser-induced fluorescence technique covering a streamwise region of approximately one breaker wavelength and over a time of 3.4 breaker periods. The aligned profile data is used to create spatio-temporal maps of the ensemble average surface height and the standard deviation of both the local normal distance and the local arc length relative to the instantaneous mean profile. It is found that the mean and standard deviation maps contain strongly correlated localized features and indicate that the transition from laminar to turbulent flow is a highly repeatable process. Regions of high standard deviation include the splash created by the plunging jet impact and subsequent splash impacts at the front of the breaking region as well as the site where the air pocket entrained under the plunging jet at the moment of jet tip impact comes to the surface and pops. In Part 2 (Erinin et al., J. Fluid Mech., vol. 967, 2023, A36), these features are used to interpret various features of the distributions of droplet number, diameter and velocity.

Numerical investigation of hydraulic jump for different stilling basins using FLOW-3D

Abstract Stilling basin of the Tuansa barrage is a modified form of the United States Bureau of Reclamation (USBR) Type-III basin, which includes impact baffle and friction blocks. Soon after the barrage's operation, baffle blocks were found to be uprooted. Additionally, previous studies also highlighted the issues of rectangular baffle blocks, i.e., less drag, smaller wake zone, and flow reattachment. On the contrary, the use of wedge-shaped splitter blocks is found limited downstream of diversion barrages. Therefore, this study focuses on the hydraulic effects of wedge-shaped splitter blocks on hydraulic jump (HJ) and energy dissipation using FLOW-3D and confirms its suitability downstream of the diversion barrage. The study mainly investigated the free surface profiles, roller lengths, relative energy losses, velocity profiles, and turbulent kinetic energies for three different stilling basins. The results of free surface profiles indicated that new basins stabilised the HJ and produced smaller lengths of HJ. The results also showed that as the flow increased, the roller lengths decreased in the new basins. Similarly, with an increase in the flow, the relative energy loss increased in new basins. Furthermore, within HJ, and at the basins' end, the results showed higher decay of velocity and turbulent kinetic energy in the new basins.

Characteristics of water free-surface with different momentum ratio at 45° confluence

Confluences play a key node role in the river network because they affect flow, sediment transport, water quality, and ecological patterns. Hydrodynamic features of the confluence hydrodynamic zone are complicated due to their three-dimensional (3D) flow structure. The present research goal is to investigate the effects of discharge ratio and momentum ratio on depths and free surface profiles at a 45° confluence. Laboratory experiments and numerical simulations were carried out to investigate the water surface characteristics at confluences under different discharge and momentum ratios. The spatial variation, backwater degree, umbilication degree, and horizontal and longitudinal free-surface slope ratio of the water depth at the confluence were examined in detail. The relationship between the water surface characteristics and the normalized momentum ratios (M*) are constructed via regression analysis. We conclude that (i) the water surface characteristics at the confluence can be grouped into the Backwater zone, Separation zone, Contractile zone and Recovery zone. (ii) The momentum ratio has a more appropriate correlation with the area ratio (the area of low water depth in the separation zone to the area of downstream channel, AL) than the discharge ratio. (iii) The backwater may be closely related to the amount of incoming flow rate. (iv) The logarithmic regression model between M* and area ratio (AL) was established, and it is found that AL increases as (M*) increases. This study provides a broader understanding of controls on spatial gradients of the water depth and provides guidance for the study of mass transport of pollutants at the confluence.

Peridynamic mesh-free simulation of glass and metal beads column collapses

The density of the metal beads is almost 3 times the density of glass beads, and the rheological response is consequently different from glass beads by using the μ(I) rheology. In this study, the μ(I) rheology is first applied to simulate metal beads column collapse, which is implemented in Peridynamics to capture the transition from the fluid-like state to the solid-like state in the granular flow. To achieve this, flow response by the rheological parameters in the μ(I) rheology is extensively investigated by simulating glass beads column collapse. The numerical results are compared to the experimental measurements in terms of the free surface, velocity distribution, and the interface in the collision of two adjacent granular columns. It is found that the mobility of the granular material is significantly affected by static friction μs. The friction μ2 with high inertia in the rheology can affect the movements of free surface points in the simulations. Another parameter of a constant I0 in the rheology can reduce the material's mobility if its value is very small and large values show insignificant effects on the flow characteristics. The parameters of (μs, μ2, I0) = (0.36, 0.42, 0.01) in the rheology model can capture the free surface profiles in good agreement with the experimental measurements for the metal beads column collapse with different aspect ratios by using the numerical method.

Revisiting equivalent tuned mass damper analogy for tuned liquid damper utilizing CFD-FEA framework

The equivalent mechanical model (EMM) of a tuned liquid damper (TLD) is represented through an analogous spring-mass-dashpot system that resembles a tuned mass damper. The present work utilizes the computational fluid dynamics-finite element analyses (CFD-FEA) framework to study the structure-TLD coupled dynamics. A single-storey single-bay building with TLD is taken. The performance of the TLD and the liquid free-surface profile for different structural damping ratios and excitation amplitudes are determined from CFD-FEA. The efficiency of the TLD, obtained from the CFD-FEA and EMM are compared for harmonic and seismic excitations and different TLD configurations. The results indicate that for low excitation amplitude and high structural damping, EMM may be adopted to predict TLD performance. EMM indicates gradual deterioration in the TLD performance as the depth ratio increases, whereas CFD-FEA shows an optimum TLD performance for a depth ratio equal to 0.15. Under seismic excitations, both EMM and CFD-FEA highlight the effectiveness of the TLD, though there is variation in the damper efficacy for individual earthquakes obtained using the two methods. It is also observed that the performance of the TLD along a particular direction does not alter under the bidirectional loading scenario. In case of bidirectional loading, the liquid free-surface displacement is more though it does not enhance the damper performance.

A CNN-supported Lagrangian ISPH model for free surface flow

As a popular method for modeling violent free surface flow, the incompressible Smoothed Particle Hydrodynamics (ISPH) based on the Lagrangian formulation has attracted a great attention worldwide. The Lagrangian ISPH solves the unsteady Navier-Stokes and continuity equations using the projection method, in which the pressure is obtained by solving the pressure Poisson's equation (PPE) that is the most time-consuming part in the ISPH procedure. In this paper, the Convolutional Neural Network (CNN) is combined with ISPH and used to predict the fluid pressure instead of solving the PPE directly. Although limited attempts of using CNN for solving the PPE in Eulerian formulation (referred to as the Eulerian CNN framework) in mesh-based methods are found in the public domain, the present model is the first ISPH model supported by CNN in a Lagrangian formulation. The proposed model overcome several challenges associated with combining CNN with ISPH, including selecting the input parameters, formulating the objective functions, producing the training dataset and dealing with boundary conditions. Two classic free surface problems, i.e. the dam breaking and the wave propagation, are simulated to evaluate the performance of the present model. Quantitative assessments of the numerical error in terms of both the free surface profile and the pressure field are carried out. The assessments show that the new model does not only give results with satisfactory accuracy, but also requires much less computation time for estimating pressure if the number of particles is large, e.g., 100 thousands particles that is usually required in the practical ISPH simulation for free surface flow.

Recovery of steady rotational wave profiles from pressure measurements at the bed

We derive equations relating the pressure at a flat seabed and the free-surface profile for steady gravity waves with constant vorticity. The resulting set of nonlinear equations enables the recovery of the free surface from pressure measurements at the bed. Furthermore, the flow vorticity (unknown a priori) is determined solely from the bottom pressure as part of the recovery method. This approach is applicable even in the presence of stagnation points and its efficiency is illustrated via numerical examples.

Measuring and investigating the effect of sloshing on the performance of an OWC wave energy converter by means of a novel experimental approach

Renewable energy development could be an appropriate method to tackle the problems caused by fossil fuels. In this regard, an Oscillating Water Column (OWC) with a scale of 1:10 has been studied in this research. Sloshing is an unwanted phenomenon in OWC. Therefore in order to study sloshing, A Novel experimental approach for measuring free surface profile inside OWC by considering the effect of wave characteristics and sloshing was investigated experimentally in this paper. Eight wave gauges were installed at the top of the converter to study the hydrodynamic behaviour. The results showed that an increase in frequency raises the possibility of sloshing. However, up to the frequency of 0.37, the TFS (Top Free Surface) angle increase is insignificant. As frequency approaches 0.37, suddenly TFS angle rises, and sloshing occurs at frequencies of 0.37–0.80. Also, dead volumes are formed on TFS (some positive and negative volumes formed on free surface which cancel each other). Therefore if TFS angle is more than 3.85°, the sloshing is happened, and the converter's performance decreases severely. In the best conditions, efficiency and power are 53.7% and 192.2 W, respectively, and in sloshing area, efficiency and power are 8.3% and 35 W, respectively.

Analysis of a stable bathtub vortex in a rotating container

Rotating flows with free-surface vortices can be found in many engineering applications, such as pump and turbine intakes, vessels, and nuclear reactors. The need to address rather different flow regions existing in such flows, such as Ekman and Stewartson layers and the line vortex zone, in a coupled manner, makes modeling of free-surface rotating flows very challenging. In this work, the flow field of a free-surface vortex, created in a rotating cylinder with a drain hole in its bottom, is investigated numerically and analytically. Above the drain hole of the cylinder, a free-surface vortex, accompanied by axial velocity, is created. This axial velocity profile is governed by the Ekman boundary layer far from the axis and by the drainage in its proximity. The experiments of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)] on the so-called bathtub vortex are numerically modeled with the finite volume method. The simulations are validated with the available measurements from the experiments. Using the simulation results, self-similar and non-self-similar models, describing the velocity fields in the Ekman boundary layer, are compared and tested. It is shown that the self-similar model is more accurate than the non-self-similar model. It is also demonstrated that the analytical model of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)], when modified as suggested in the present study, is capable of predicting the free-surface profile for low rotation rates. However, for high rotation rates, only the numerical simulation can predict the relation between the flow field within the liquid and the free-surface profile.

Anisotropic spall failure of additively manufactured 316L stainless steel

The dynamic high strain rate tensile (spall) failure of additively manufactured (AM) materials is an area of great interest but much less understood than material failure under other loading regimes. The research presented here examines the effects of microstructure anisotropy in 316L stainless steel fabricated by laser powder bed fusion (LPBF) on its spall failure and properties. Plate-on-plate impact experiments were performed using an 80-mm gas gun to investigate the spall behavior of AM fabricated sample discs with impact along the perpendicular and parallel orientations relative to build directions. Experiments were also performed on wrought 316L stainless steel as baseline material with uniform equiaxed grain microstructure for comparison with the AM material of same composition. The spall experiments involved free-surface velocity profiles measured using Photon Doppler Velocimetry (PDV) to capture the spall pull-back signals, combined with post-mortem analysis of the microstructure of soft-recovered impacted samples. In the case of AM samples, spall failure initiated preferentially at melt pool boundaries in both impact directions. The spall plane location and spall strength, however, were both heavily influenced by the orientation of the impact direction relative to the build direction in the AM samples. The spall behavior is also observed to be more complex in the samples impacted perpendicular to the build (through-thickness) direction, due in large part to the orientation of the microstructural features such as melt pool boundaries, laser scan tracks, and general grain morphology.

Analytical Model to Predict the Free Surface Profile of Outgoing Material in Oval-to-round Pass Rolling when Cross-section of Incoming Material is not Circular and Roll Gap Changes

Analytical Model to Predict the Free Surface Profile of Outgoing Material in Oval-to-round Pass Rolling when Cross-section of Incoming Material is not Circular and Roll Gap Changes

A two-particle approach for dry high density-ratio granular collapse

We report on the role of density ratio for granular column collapse experimentally and numerically by demonstrating the fundamental changes using a two-particle approach. The influence of the density ratio for the different profile arrangements on the velocity distribution, interface, and energy transition is described using the Moving Particle Semi-Implicit method (MPS). A constitutive equation utilizing the granular material strength is formulated using the elasto-viscoplastic model under the μ ( I ) rheology framework by replacing the frictional coefficient with the internal friction angle. The dense state of the granular materials results in a linear relationship between the length scale and inertia constant, which makes the shear strength the driving parameter. Experimental data are compared with numerical simulation for granular collapse with varying density profile configurations to validate the method. We found that the density ratio does not influence the velocity and free surface profile. The interface is maintained similarly to a single-phased granular flow except for erosive configurations where a penetrating regime is evident. The rate at which the granular kinetic energy transforms to the potential energy when the lower density material is on top and at the left of the higher granular materials is similar compared to other profile arrangements. The results demonstrated that the density ratio for different profile arrangements could significantly influence the deformation of the granular materials depending on the particle arrangements. The numerical investigation for the granular column collapse agrees well with experimental studies. Collapse mechanisms of varying density ratio configuration. • Flow dynamics and energy evolution in density ratio granular column collapse. • Replacement of the μ (I) rheology model parameters with the material strength parameters. • Effective phase-change for high-density ratio for single and two-phase granular flow. • Relationship between the inertia constant and wave front. • The meshfree particle method, MPS, shows excellent agreement with the experiment.

Adhesive creases: bifurcation, morphology and their (apparent) self-similarity.

An elastic material that experiences strong compression parallel to its free surface can exhibit sharp surface folds. Such creases arise due to an instability where a self-contacting fold appears on the surface, often observed in growing tissues or swelling gels. Self-adhesion of the contact is known to affect the bifurcation behavior and morphology of these structures, yet a quantitative description remains elusive. From numerical simulations and an energy analysis we resolve how adhesion quantitatively affects both morphology and bifurcation behavior. It is found that a reduced energy is able to accurately describe the bifurcation, in terms of an effective scaling that collapses the data very well. The model accurately describes how adhesion hinders crease nucleation. Furthermore, we show that the free surface profiles in the presence of surface tension exhibit self-similarity, and can be collapsed onto a universal curve.

On the Hydraulic Characteristics of Submerged Flow over Trapezoidal-Shaped Weirs

Abstract Subcritical flows over highway and railway embankments, commonly encountered during flood events, can be treated like submerged flows over trapezoidal-shaped weirs. In earlier studies, the equation of the submerged-flow discharge for such types of weirs was developed as a function of the degree of submergence and free-flow discharge. However, the application of this equation in practice requires a pre-determined discharge from experiments performed under free-flowconditions. In this study, a discharge equation was deduced from the streamwise momentum balance equation, which overcomes the drawback of the previous approaches. The results of the validation demonstrated that the proposed equation is capable of predicting the submerged-flow discharge of a trapezoidal-shaped weir within ±6.0% of the measured value. Furthermore, the most prominent features of the submerged overflows were examined by systematically analyzing the experimental data. For such flows, the free-surface and bed-pressure profiles are self-similar only over the upstream face of the weir. Results of this investigation confirmed that the degree of submergence and the slope of the downstream weir face significantly affect the characteristics of the submerged flow, but the effect of the latter on the non-modular discharge is marginal.

Spall strength and equation of states for 2050-T84 Al–Li alloy under shock compression up to 120 GPa

Aluminum-lithium (Al–Li) is an attractive alloy for aerospace and military applications due to its low density, excellent mechanical performance, and corrosion resistance. In this study, a series of plate-impact experiments were conducted using a powder gun and a two-stage light-gas gun to investigate the dynamic response of 2050-T84 Al–Li alloy under shock compression. The Hugoniot elastic limit, spall strength, and Hugoniot state were determined up to 120 GPa according to the measured free-surface velocity profiles and shock wave velocities. The spall strength measured for 2050-T84 Al–Li alloy was about 0.6 GPa, which is lower than that for non-lithium-bearing Al alloys due to the large grain size and different alloying compositions. The high-pressure Hugoniot data agreed well with theoretical calculations based on first-principles molecular dynamics. The dynamic fracture and Hugoniot data obtained for 2050-T84 Al–Li alloy in the present study can guide the design of protective structures for spacecraft or artificial satellites, and lightweight aluminum alloy armor plates.