Evolution Of Free Surface Research Articles

Further enhancement of the particle shifting technique: Towards better volume conservation and particle distribution in SPH simulations of violent free-surface flows

The non-conservation of total fluid volume, caused by the accumulating errors on the potential energy of the fluid, has been a serious numerical issue in a Weakly-Compressible SPH (WCSPH) simulation when transitional Particle Shifting Techniques (PSTs) are employed to prevent disordered particle distribution and the tensile instability from negative pressures. This paper is dedicated to, within the framework of WCSPH, developing an enhanced version that remedies the aforementioned deficiency of the traditional PSTs, and meanwhile improves the quality of the particle distribution in the vicinity of a free-surface region. To this end, a Corrective Cohesion Force (CCF) between a target particle and its interacting particles is introduced to provide adaptive compensation corresponding to the particle repositioning. Four classical benchmarks are implemented to validate the effectiveness and stability of the present PST. It is demonstrated that the new PST incorporating with the CCF shows satisfactory performance to improve the conservation of total fluid volume, and to obtain more uniform particle distribution in the proximity of the free-surface. In addition, the newly-developed PST also maintains the accuracy and stability inherited from the traditional versions, suggesting that it can be treated as an ideal alternative with regard to the traditional PSTs in a WCSPH simulation, especially for a long-term-duration case with violent free-surface evolutions.

Measurements of unconfined fresh concrete flow on a slope using spatial filtering velocimetry

Fluidity of fresh concrete is an important performance, and it can be evaluated by concrete flow process. Before the experiments, the theory of the lateral spread and accumulation were assumed. Then, the velocity and spread characteristics of concrete with different water content on an inclined plane were studied. Spatial filtering velocimetry (SFV) was used to measure the free-surface spreading evolution of concrete of a fixed volume slowly released onto a slope. The velocity gradients along the flow direction were calculated in the steady zone and the results showed that the velocity gradient was a reasonable indicator to assess concrete fluidity. Based on these analysis and results, the flow depth during the flow process was estimated too.

Formation and evolution of roll waves in a shallow free surface flow of a power-law fluid down an inclined plane

A laminar flow of a thin layer of mud down an inclined plane under the action of gravity is considered. The instability of a film flow and the formation of finite amplitude waves are studied within the framework of both two-dimensional governing equations of a power-law fluid and its depth-averaged hyperbolic simplification. The conditions of the existence of roll waves for these models are formulated in terms of the Whitham criterion. A free surface evolution and the development of roll waves are numerically calculated. The amplitude of roll waves obtained by the 2D equations is slightly larger than that for the 1D model. Moreover, for certain flow parameters, the small perturbations of the basic solution grow for the 2D equations and decay for the depth-averaged model. A two-parameter class of exact piecewise-smooth solutions of the 1D model is obtained and a comparison with a numerical solution is made. In the region of these parameters, diagrams of the roll waves existence are constructed.

Interfacial phase change effect on a viscous falling film having odd viscosity down an inclined plane

We investigate the phase change effect at the thin film interface with broken time-reversal-symmetry falling down an inclined flat wall. We consider the impact of evaporation and as well as condensation at the fluid-vapor interface separately. The non-zero odd part of the Cauchy stress tensor with the odd viscosity coefficient, which arises due to broken-time-reversal-symmetry, adds an attractive characteristic in the flow by stabilizing the effect of evaporation/condensation. We study the long-wave instabilities of the ununiform film by deriving a non-linear evolution equation in the classical long wave expansion method framework. The one-equation model can track the free surface evolution and accounts for viscosity, gravity, surface tension, and phase change (evaporation/condensation) effects. Linear stability analysis confirms that the odd viscosity has no significant impact on the instability threshold of an evaporating film as a contrast of condensate film, where the critical Reynolds number increases with the odd-viscosity parameter μ. While studying the weakly non-linear waves, we use the method of multiple scales to obtain the Complex Ginzburg Landau equation that shows both supercritical and subcritical solutions are possible for evaporating and condensate film. Interestingly, while one subcritical region is visible for an evaporating film, two subcritical unstable regions are found for condensate film. The numerical solution of the free-surface equation illustrates the finite-amplitude behavior that tends to dry out for evaporating film and rapidly increase film thickness for condensate film. The presence of odd viscosity slows down both the film thinning process for evaporation and the film thickening process for condensation. Furthermore, odd-viscosity cannot prevent film rupture in case of evaporation.

Hydrodynamic instability and wave formation of a viscous film flowing down a slippery inclined substrate: Effect of odd-viscosity

Hydrodynamic instability and formation of waves for thin viscous film flowing down a slippery inclined substrate with broken time-reversal-symmetry have been discussed in this present study. The effect of slip on the substrate is modeled using the Navier slip boundary condition. We have derived a nonlinear evolution model in the framework of the long-wave expansion technique. The one equation model can track the free surface evolution and involve the viscosity, gravity, surface tension, and the effect of the slip of the wall. The odd part of the Cauchy stress tensor with an odd-viscosity coefficient significantly modifies the characteristic of the film flow. We performed the linear stability analysis with the Orr Sommerfeld technique and classical temporal analysis from the model. Comparing the results shows a satisfactory agreement between Orr Sommerfeld and the model when the velocity scale is chosen as twice the free surface velocity. Analyzing the traveling wave solutions and studying different bifurcation analyzes and phase diagrams, we focused on and discussed two significant wave families, namely γ1 (low-wave-speed) and γ2 (high-wave-speed). The spatio-temporal study of the model has been performed numerically for different odd-viscosity and slip parameters. Results show that the odd-viscosity plays a stabilizing role while the slip on the substrate gives a destabilizing effect within the parameter range of our consideration.

Two-dimensional continuum modelling granular column collapse by non-local peridynamics in a mesh-free method with rheology

Abstract

Experimental investigation of the entrapped air on the hydroelasticity induced by liquid slamming under shallow-water condition

The liquid impact induced by slamming in the membrane-type tank is an important issue for the design of Floating Liquefied Natural Gas (FLNG) facilities. The free surface entrapping air may easily happen under shallow-water condition in the tank, which could result in structural local failure. Studying the effect of entrapped air on slamming load is essential to understand the air-free surface-elastic structure interaction. Thus, a series of experiments are designed and conducted in an elastic rectangular tank under shallow-water condition. The evolution of free surface and the development of entrapped air are analyzed quantitatively. Furthermore, both impact load and structural response are measured and studied in the spatial and temporal distribution under shallow-water condition. The results show that the air cavity plays a cushion role between free surface and elastic wall. Therefore, the impact mode is changed from liquid-direct impact to liquid-indirect impact. The existence of the air cavity leads to a significant reduction of the maximum slamming load and structural response but an increase in the pressure impulse in spatial distribution. It is suggested to consider the effect of the entrapped air on the determination of the slamming load for the design of super-large LNG tanks in the offshore engineering.

On the Free Surface Motion of Highly Subsonic Heat-Conducting Inviscid Flows

For the free surface problem of highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables, are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations, for which a geometric approach was introduced by Christodoulou and Lindblad [11]. Due to the loss of the one more derivative and loss of the symmetry of equations which create significant difficulties in closing the estimates in Sobolev spaces of finite regularity, the geometric approach in [11] needs to be substantially developed and extended by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.

Three-Dimensional Simulations of Offshore Oil Platform in Square and Diamond Arrangements

The interaction of the solitary wave with an oil platform composed of four vertical circular cylinders is investigated for two attack angle of the solitary wave β = 0 ° (square arrangement) and β = 45 ° (diamond arrangement). The solitary wave is generated using an internal source line as proposed by Hafsia et al. (2009). This generation method is extended to three-dimensional wave flow and is integrated into the PHOENICS code. The volume of fluid approach is used to capture the free surface evolution. The present model is validated in the case of a solitary wave propagating on a flat bottom for H / h = 0.25 where H is the wave height and h is the water depth. Compared to the analytical solution, the pseudowavelength and the wave crest are well reproduced. For a solitary wave interacting with square and diamond cylinders, the simulated results show that the maximum run-ups are well reproduced. For the diamond arrangements, the diffraction process seems to not affect the maximum run-ups, which approached the isolated cylinder. For the square arrangement, the shielding effect leads to a maximum wave force more pronounced for the upstream cylinder array.

Experimental and numerical modeling of a sloshing problem in a stepped based rectangular tank

In this study, the two-dimensional sloshing of water in a stepped based tank partially filled was analyzed using an arbitrary Lagrangian–Eulerian adaptive fixed-mesh method and including the Smagorinsky turbulence model. The numerical model is verified by contrasting the predictions made by the model with experimental results. The tank was subjected to controlled one-directional motion imposed using a shake table. The free surface evolution was followed using ultrasonic sensors, and a high-speed camera was used to record the experiments. The experimental and numerical analyses include a comparison of the wave height at different control points and snapshots of the free surface evolution for two imposed frequencies. Also, a detailed numerical study of the effects of the frequency of the imposed movement, the step height, and the fluid volume on the wave dynamics was performed. Moreover, the effect of fluid viscosity on the dynamics of the free surface was also studied. In brief, the numerical method proved to be accurate, experimental data were reported, and the effects on the numerical results of different physical and numerical aspects were exhaustively analyzed. The proposed results help to understand the sloshing of stepped geometries.

Study of initial stage of entry of a solid sphere into shallow liquid with Synthetic Schlieren technique

Vertical entry of a 213 mm diameter solid sphere moving with a low speed of 16–40 mm/s onto a shallow liquid layer is investigated experimentally. The focus of the study is on deformation of the free surface before the actual contact between solid and liquid in presence of air cushioning. Owing to local character of air cushioning, the sphere is modeled by a spherical segment. The measurements were conducted with Synthetic-Schlieren technique, adapted to the present experimental configuration. In the studied range of parameters, the value of the Weber number is order O(1), giving rise to capillary oscillations of the free surface below the sphere. The primary solid-liquid contact at higher impact speeds (higher Weber number) occurs due to growth of a circular rim; the radius of contact is predicted satisfactory by existing theories assuming the lubrication/inviscid balance as a key mechanism. At lower speeds (lower Weber number), the contact is significantly delayed, being related with the effects due to surface tension and final horizontal extent of the spherical segment. In case of delayed contact, the evolution of free surface can be studied over a significant time span: the radial profile of the rim evolves into a flat-top structure, reminiscent of axisymmetric bore wave. The front of this wave propagates with constant speed, defined (with a good accuracy) by the condition that group velocity is equal to phase velocity of the waves. This condition is satisfied in i) the long-wave limit and ii) for the capillary waves of lowest admissible phase velocity. The second case is observed in the present experiments. The capillary waves generated by the impact “ride” on the top of the bore wave. The bore wave itself generates capillary waves in front of it, thereby producing a structure reminiscent of dispersive shock waves. Phase velocity and wavenumber of capillary waves are described well by linear theory of gravity-capillary waves in a liquid of finite depth.

A Partitioned Solution Algorithm for Concurrent Computation of Stress–Strain and Fluid Flow in Continuous Casting Process

Control of macrosegregation phenomena and deformation-related defects is a main issue in steel continuous casting. Numerical simulation could help industrial engineers to master these defects. However, as a first step, it is essential to achieve a concurrent computation of fluid flow in the bulk liquid and stress-strain evolution in the already solidified regions. With this aim in view, a new specific partitioned solver has been developed to model the liquid flow, essentially induced by the inlet jet distributed by the submerged nozzle, as well as the thermal deformation of the solid shell. The solver procedure allows simulating the transient regime, up to convergence to the steady-state regime. For this purpose, the computational finite element mesh moves and grows continuously. Within this evolving mesh, three different zones are defined: the solid shell as a pure Lagrangian zone, the liquid nozzle region as a pure Eulerian zone, and an intermediate Eulerian–Lagrangian zone. Conservation equations (energy, mass, and momentum) are solved in a general arbitrary Lagrangian–Eulerian framework, with a level-set formulation to track the free surface evolution at the meniscus. The article is composed of two parts. In the first part, the model is detailed with the resolution steps involved in the coupled resolution approach. In the second part, a simple verification test case is firstly proposed, followed by a more relevant and practical application to model an industrial pilot continuous casting process.

A generalized Hellinger-Reissner variational principle and its PFEM formulation for dynamic analysis of saturated porous media

In this paper, a novel mathematical programming formulation is derived based on the u-p form for the dynamic analysis of saturated porous media. A mixed finite element is used for the interpolation of field variables and after discretization the formulation is remolded into a standard second-order cone programming problem that can be resolved using modern optimization engines. The proposed optimization-based computational scheme is verified against typical benchmarks such as the dynamic consolidation problem and the wave propagation in saturated soils. To tackle issues such as mesh distortions and severe free-surface evolutions resulting from large deformations, the scheme is further implemented into the PFEM framework. The capability of the proposed method for analyzing porous media with large deformations is illustrated by modelling the collapse of a saturated granular column in air and the post-failure process of an embankment due to seepage with results compared to the ones from lab tests and numerical simulations using other approaches such as the material point method and the smoothed particle hydrodynamics method.

Inequality-constrained free-surface evolution in a full Stokes ice flow model (<i>evolve_glacier v1.1</i>)

Abstract. Like any gravitationally driven flow that is not constrained at the upper surface, glaciers and ice sheets feature a free surface, which becomes a free-boundary problem within simulations. A kinematic boundary condition is often used to describe the evolution of this free surface. However, in the case of glaciers and ice sheets, the naturally occurring constraint that the ice surface elevation (S) cannot fall below the bed topography (B) (S-B≥0), in combination with a non-zero mass balance rate complicates the matter substantially. We present an open-source numerical simulation framework to simulate the free-surface evolution of glaciers that directly incorporates this natural constraint. It is based on the finite-element software package FEniCS solving the Stokes equations for ice flow and a suitable transport equation, i.e. “kinematic boundary condition”, for the free-surface evolution. The evolution of the free surface is treated as a variational inequality, constrained by the bedrock underlying the glacier or the topography of the surrounding ground. This problem is solved using a “reduced space” method, where a Newton line search is performed on a subset of the problem (Benson and Munson, 2006). Therefore, the “constrained” non-linear problem-solving capabilities of PETSc's (Portable, Extensible Toolkit for Scientific Computation, Balay et al., 2019) SNES (Scalable Non-linear Equations Solver) interface are used. As the constraint is considered in the solving process, this approach does not require any ad hoc post-processing steps to enforce non-negativity of ice thickness and corresponding mass conservation. The simulation framework provides the possibility to divide the computational domain into different subdomains so that individual forms of the relevant equations can be solved for different subdomains all at once. In the presented setup, this is used to distinguish between glacierised and ice-free regions. The option to chose different time discretisations, spatial stabilisation schemes and adaptive mesh refinement make it a versatile tool for glaciological applications. We present a set of benchmark tests that highlight that the simulation framework is able to reproduce the free-surface evolution of complex geometries under different conditions for which it is mass-conserving and numerically stable. Real-world glacier examples demonstrate high-resolution change in glacier geometry due to fully resolved 3D velocities and spatially variable mass balance rate, whereby realistic glacier recession and advance states can be simulated. Additionally, we provide a thorough analysis of different spatial stabilisation techniques as well as time discretisation methods. We discuss their applicability and suitability for different glaciological applications.

Numerical modelling of solitary wave and structure interactions using level-set and immersed boundary methods by adopting adequate inlet boundary conditions

ABSTRACT Solitary waves are often used to replicate tsunami waves and, therefore, the study of their propagation and interactions with structures could help to understand the impacts of tsunami waves on infrastructures. This paper presents a numerical model that uses the level-set framework for simulating the propagation of solitary waves and their interactions with structures using the immersed boundary method. The model imposes a set of appropriate boundary conditions to prescribe the incoming solitary waves within the sharp-interface immersed boundary method. Turbulence is modelled using the large-eddy simulation. Experimental tests are conducted to measure the free-surface elevation and velocity field evolution of solitary waves, as they propagate downwave interacting with two different wall-mounted structures. Analysis of the computed and experimental results revealed interesting dynamics of primary vortex-tubes forming near the sharp edges of structures. It is shown that the primary vortices eventually split to form pairs of asymmetrical doublets.

Numerical Investigation on the Water Entry of Several Different Bow-Flared Sections

The bow-flared section may be simplified in the prediction of slamming loads and whipping responses of ships. However, the difference of hydrodynamic characteristics between the water entry of the simplified sections and that of the original section has not been well documented. In this study, the water entry of several different bow-flared sections was numerically investigated using the computational fluid dynamics method based on Reynolds-averaged Navier–Stokes equations. The motion of the grid around the section was realized using the overset mesh method. Reasonable grid size and time step were determined through convergence studies. The application of the numerical method in the water entry of bow-flared sections was validated by comparing the present predictions with previous numerical and experimental results. Through a comparative study on the water entry of one original section and three simplified sections, the influences of simplification of the bow-flared section on hydrodynamic characteristics, free surface evolution, pressure field, and impact force were investigated and are discussed here.

A one-dimensional line element model for transient free surface flow in porous media

A simple and physically sound model has been proposed to investigate the transient free surface flow behavior in porous media. Instead of the continuum-based model, the pore space between solid grains is conceptualized as a network of horizontal and vertical flow paths and then the flow paths are characterized by straight line elements. Based on the Darcy's law and flux equivalence between the continuum-based model and line element model, equivalent hydraulic parameters and continuity equation are derived through the control volume. By introducing a local coordinate system, unified governing equations and equivalent boundary conditions of the line element model are obtained in the form of one-dimensional formulations. The two-dimensional transient free surface flow problem is reduced to a one-dimensional problem of line element network and a finite-element algorithm is developed. The proposed one-dimensional line element model is verified by three illustrative examples, which indicate that the numerical results can satisfactorily match the analytical solutions and experimental observations from different sources. It is found that the solution accuracy has weak sensitivity of time-step size but sensitive to the spacing size, while numerical efficiency can be well guaranteed for different time-step sizes and spacing sizes. Based on the numerical results, the proposed model can give better performance for the evolution of free surfaces than the residual flow procedure, and is efficient to model transient surface flow with a complex drainage system.

Dynamic Behavior in a Storage Tank in Reduced Gravity Using Dynamic Contact Angle Method

The oscillation of liquid/gas free surface in a partially filled storage tank caused by an abrupt drop of gravity level is of critical importance for the fluid management in space. In the present study, the dynamic behavior of free surfaces in a model tank (tube) is numerically investigated using volume of fluid (VOF) method in the context of dynamic contact angle (DCA) model. It is concluded that the dynamic behavior of free surface could be captured pretty well using the selected DCA model, as shown by comparison with the results of Drop Tower Beijing experiment. The temporal evolution of free surface reproduces exactly the characteristics of damping oscillations. The detailed dynamic deflections of meniscus reveal crucial dependency between the oscillation frequency of free surface and the boundary condition in the contact line. The oscillation frequency increases when the range of the moving contact line transfers from the spherical-shaped part to the cylindrical part of the tank and maintains constant when the moving contact line remains always at the cylindrical part of the tank. Meanwhile, the oscillation amplitude decreases in line with the increase of oscillation frequency.

Numerical and experimental analysis of rapid solidification considering undercooling effect during water droplet impact on a substrate

Freezing of water droplets on a cold surface can involve rapid solidification and undercooling. In this numerical and experimental study, the role of rapid solidification considering the undercooling effect during the impact and spreading of a water droplet onto a cold substrate is investigated. Various attendant physical phenomena, such as free surface evolution, fluid flow, heat transfer, rapid solidification and undercooling at the moving solid-liquid front are accounted in the computational model. In the coupled thermo-fluidic-rapid solidification model, the free surface of the liquid droplet is tracked with the help of volume of fluid method, while the solid-liquid front is tracked with the help of rapid solidification kinetics. The rapid solidification model lets freezing to occur at nucleation temperature which is lower than the freezing point. Results of the rapid solidification model are compared with the conventional solidification model where freezing occurs as soon as the temperature reaches the equilibrium freezing point. A realtime imaging setup is used to experimentally measure droplet spreading and freezing behaviour. The numerical predictions are validated with the experimental results. The model successfully captures undercooling and recalescence. Interfacial heat transfer is analyzed with the help of droplet-substrate heat flux. Undercooling and rapid solidification significantly influence the heat flux evolution. In the end, the effect of substrate temperature and droplet impact velocity is described. The understanding developed in this study regarding the role rapid solidification on interfacial heat flux evolution can be useful for research on ice accretion in aircraft.

A conservative level set method on unstructured meshes for modeling multiphase thermo-fluid flow in additive manufacturing processes

Additive manufacturing (AM) is an emerging technology that fuses deposited powder materials together layer-by-layer using a localized heat source to create an arbitrarily complex geometry. This technology is an inherently multiphysics problem, involving the simultaneous evolution of fluid flow, heat transfer, free surface, laser-material interaction and material phases including solid, liquid and gas. Multiphysics and multiphase models are typically used to understand the dominant physics driving AM processes through high fidelity simulations resolving the individual powders as they melt onto the substrate. However, these types of high fidelity models have not been applicable to modeling directed energy deposition (DED) process, where powders are delivered through the ambient vapor onto the substrate by a nozzle and subsequently fused. This paper presents a novel multiphase thermo-fluid formulation for modeling DED processes. A diffuse level set formulation coupled with the Navier–Stokes, energy conservation and radiative transport equation allows us to model complex free surface, fluid flow, thermal and laser interaction evolution. In addition, our formulation enables us to resolve the vapor flow field and its effect on deposited material during AM processes. The accuracy of our proposed method is assessed by comparing with literature solutions and experimental benchmarks. Our proposed formulation is then used to model simple DED processes that enable us to visualize the entrainment of powder particles into the melt pool. This process elucidates the dominant physical forces that can drive powders into the melt pool as well as their effect on the melt pool evolution within DED processes.