Free Surface Boundary Research Articles

An enhanced semi-implicit particle method for simulating the flow of droplets with free surfaces

The accurate modeling of the surface-tension on free surfaces remains a challenging problem in meshfree particle methods. Recently, Matsunaga et al. (2020) have made an important step forward toward addressing this issue by introducing a moving surface mesh to represent the free-surface boundary in 2D. However, the surface mesh makes it difficult to extend this methodology to 3D and to treat topological changes in free surfaces. This study aims to extend this methodology to 3D and restore the capability of the model to simulate topological changes by carefully detecting the free-surface particles. In this situation, the problems regarding fluctuated free-surface boundaries and the dynamic intersection of a free surface with a wall boundary arise. To resolve these problems, a new particle shifting method in the surface normal direction is proposed for the free-surface particles to reduce fluctuations. A new contact angle model is proposed to compute the intersection point of a free surface and a wall boundary. In this manner, an enhanced particle method suitable for simulating the dynamics of liquid droplets with possible topological changes in 3D is developed. Eight numerical examples verified the developed method and demonstrated the enhancement of the new particle shifting technique.

Numerical analysis of hydroelasticity problems by coupling WCSPH with multibody dynamics

This paper aims to extend the implementation of DualSPHysics-Project Chrono coupling to hydroelasticity problems involving free-surface flows and deformable elastic boundaries. In the present model, the fluid sub-domain of the phenomenon is based on the WCSPH model of DualSPHysics. The solid sub-domain, simulated by the Project Chrono engine, is modeled using the set of rigid bodies attached by hinges with torsional and damping stiffness. An analytical solution of Euler-Bernoulli beam theory is employed to adapt the rigid-multibody system to the mechanical characteristics of the elastic sub-domain via the torsional stiffness values of hinges. The stability and accuracy of the present model are tested by three cases with different flow characteristics and mechanical properties for solid sub-domain, involving a newly-designed experimental setup and two benchmark cases in the literature. Obtained results show reasonable accuracy with experimental measurements and other numerical model computations.

Cartesian spatial derivatives of boundary element solutions on the exact free surface of fully nonlinear numerical wave tanks

The convergence and stability of the fully nonlinear simulation in a three-dimensional potential numerical wave tank is vulnerable along the time marching algorithm of the free surface, as integrating the boundary element's solution into the high-order time integration of the free surface can trigger the numerical instability in the material node approach. Although this matter was identified in some of the prior studies on three-dimensional wave tanks, the approximation of the velocity components plays a crucial role in the sustainability of a simulation. Therefore, a methodology for deriving the Cartesian components of the fluid particles’ velocity over the exact free surface is proposed in this paper. Various circumstances come across while determining the velocity components of fluid particles, which are the function of the location of the free surface nodal points. Since plenty of prior techniques have been based on the tangential derivatives of the free surface boundary value from the previous iteration, the disruption in the convergence of the model occurs most often. Hence, achieving a generic algorithm that merely depends on the boundary element solutions, is a highly important achievement for retaining the stability and simultaneously the convergence of the solution.

Vortex-induced vibration of a sphere close to or piercing a free surface

Vortex-induced vibration (VIV) of an elastically mounted sphere placed close to or piercing a free surface (FS) was investigated numerically. The submergence depth ( $h$ ) was systematically varied between $1$ and $-$ 0.75 sphere diameters ( $D$ ) and the response simulated over the reduced velocity range $U^*\in [3.5,14]$ . The incompressible flow was coupled with the sphere motion modelled by a spring–mass–damper system, treating the free-surface boundary as a slip wall. In line with the previous experimental findings, as the submergence depth was decreased from $h^* = h/D =1$ , the maximum response amplitude of the fully submerged sphere decreased; however, as the sphere pierced the FS, the amplitude increased until $h^* = -0.375$ , and then decreased beyond that point. The fluctuating components of the lift and drag coefficients also followed the same pattern. The variation of the near-wake vortex dynamics over this submergence range was examined in detail to understand the effects of $h^*$ on the VIV response. It was found that $h^* = 1$ is a critical submergence depth, beyond which, as $h^*$ is decreased, the vortical structures in the wake vary significantly. For a fully submerged sphere, the influence of the stress-free condition on the VIV response was dominant over the kinematic constraint preventing flow through the surface. For piercing sphere cases, two previously unseen vortical recirculations were formed behind the sphere near times of maximal displacement, enhancing the VIV response. These were strongest at $h^* = -0.375$ , and much weaker for small submergence depths, explaining the observed response-amplitude variation.

Typhoon: A vortex-lattice code for assessing dynamic stability characteristics of hydrofoil crafts

In this paper an open-source implementation of the vortex-lattice method to perform a dynamic stability analysis for hydrofoil crafts is discussed. The difference with existing vortex-lattice codes is the addition of a free-surface boundary condition which is needed to analyse surface piercing foils. This code, called Typhoon, can be used to perform a dynamic stability analysis (DSA) on hydrofoil vessels. The goal of this code is to have an easy-to-use and cheap alternative to compare different designs in early design stages. This paper gives a brief background to all the concepts used, followed by a short theoretical explanation of the vortex-lattice method. The second part of this paper focuses on a practical example of how this code can be used on an example.

Nonlinear time‐domain wave‐structure interaction: A parallel fast integral equation approach

AbstractWe report on the development and validation of a new numerical wave tank (NWT) solving fully nonlinear potential flow (FNPF) equations, as a more efficient variation of Grilli et al.'s NWT [Grilli et al., A fully nonlinear model for three‐dimensional overturning waves over arbitrary bottom. Int J Numer Methods Fluids. 2001; 35:829‐867], which was successful at modeling many wave phenomena, including landslide‐generated tsunamis, rogue waves, and the initiation of wave breaking over slopes. This earlier NWT combined a three dimensional MII (mid‐interval interpolation) boundary element method (BEM) to an explicit mixed Eulerian–Lagrangian time integration. The latter was based on second‐order Taylor series expansions for the mesh geometry and Dirichlet free surface boundary condition for the potential, requiring high‐order derivatives to be computed in space and time. Here, to be able to solve large scale wave‐structure interaction problems for surface‐piercing bodies of complex geometry, of interest for ocean engineering and naval hydrodynamics applications, the NWT is reformulated to use cubic B‐spline meshes and the BEM solution is accelerated with a parallelized fast multipole method (FMM) based on ExaFMM, one of the fastest open source FMM to date. The NWT accuracy, convergence, and scaling are first assessed for simple cases, by comparing results with those of the earlier MII‐NWT as a function of mesh size and other model parameters. The relevance of the new NWT for solving the targeted applications is then demonstrated for surface piercing fixed cylinders, for which we show that results agree well with theoretical and experimental data for wave elevation and hydrodynamic forces.

A comparative study on the stress image and adaptive parameter-modified methods for implementing free surface boundary conditions in elastic wave numerical modeling

The stress image method is a simple and effective approach for implementing the free surface boundary conditions in elastic wave numerical modeling. This method assumes that the normal and shear stresses perpendicular to the free surface are antisymmetric with respect to the free surface. In this way, the values of the normal and shear stresses above the free surface can be updated. However, the stress image method is based on an intuitive viewpoint and lacks a physical foundation. The adaptive parameter-modified method is a recently proposed approach for implementing the free surface boundary conditions. Through adaptive modifications of the density and elastic parameters at the free surface, the implementation of the free surface boundary conditions is achieved. Based on the adaptive parameter-modified method, a new interpretation of the stress image method is used. We determine that the stress image method is equivalent to the adaptive parameter-modified method in terms of the staggered-grid finite-difference scheme for the elastic wave equation in displacement form. This result provides a physical foundation and explanation of the stress image method. Therefore, we can further develop the stress image method from a physical viewpoint. Numerical examples are also developed to perform the theoretical analysis.

A fully nonlinear potential flow wave modelling procedure for simulations of offshore sea states with various wave breaking scenarios

An accurate representation of a given sea state is crucial for the study of hydrodynamic loads on offshore structures. It is straightforward to check the quality of the reproduced regular waves in a numerical wave tank (NWT). However, many more parameters need to be considered to ensure the quality of irregular waves. In this paper, a fully non-linear potential flow (FNPF) wave model is used to reproduce irregular sea states with different severity of wave breaking. The numerical model solves the velocity potential from the Laplace equation and the free surface boundary conditions using a finite difference method on a σ-coordinate grid. A comprehensive procedure is introduced to ensure the quality of the reproduced full-scale sea states. The effect of wave spectrum discretisation techniques and breaking wave algorithms are compared for an optimal performance. The evaluation of the simulation results takes into account the kurtosis and the wave crest distribution in addition to the wave spectra. The vertical arrangement of the σ-coordinate grid plays an important role in representing the dispersion relation and a grid optimisation method is explained. The current study provides a working procedure that reproduces high-fidelity irregular sea states with breaking waves in an efficient FNPF NWT.

Adaptive fully nonlinear potential model for the free surface under compressible air pressure of oscillating water column devices

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

On surface flow induced by internal waves generated by a slender body moving at low internal Froude number in a sharply stratified fluid

The studies on internal waves(IWs) generated by a moving body in a stratified fluid generally focused on wave amplitude phase characteristics in underwater space. The surface flow induced by internal waves(IWs) generated by an underwater moving body in stratified fluid is generally weak, and is rarely studied. In the present paper, the surface flow induced by body-generated IWs of a slender body(SUBOFF model) moving at Fri = 0.547～2.189 and high Reynolds number (Re = 2.0 × 106～8.0 × 106) in a sharply stratified fluid with finite depth is investigated numerically and experimentally. For the theoretical calculation, the Euler equations and the free surface boundary conditions are linearized, and the source-sink distribution method is used to calculate the velocity field of body-generated IWs of SUBOFF model using the potential flow theory combined with the associated eigenvalue problem on vertical velocity. Meanwhile Particle Image Velocimetry (PIV) is used to measure the surface flow induced by body-generated IWs of SUBOFF model under the same conditions (object size, fluid stratification and movement parameters) at a large stratified fluid tank (25m × 3m × 1.5m) and validate the theoretical calculation method. The wave patterns, the maximum peak-to-peak velocity value of surface flow and its variation with Fri are analyzed.

Slow Rhythmical Brain Activity During Coordination of Cognitive Processes

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

Viscoelastic wave finite-difference modeling in the presence of topography with adaptive free-surface boundary condition

Viscoelastic wave finite-difference modeling in the presence of topography with adaptive free-surface boundary condition

The Peregrine Breather on the Zero-Background Limit as the Two-Soliton Degenerate Solution: An Experimental Study

Solitons are coherent structures that describe the nonlinear evolution of wave localizations in hydrodynamics, optics, plasma and Bose-Einstein condensates. While the Peregrine breather is known to amplify a single localized perturbation of a carrier wave of finite amplitude by a factor of three, there is a counterpart solution on zero background known as the degenerate two-soliton which also leads to high amplitude maxima. In this study, we report several observations of such multi-soliton with doubly-localized peaks in a water wave flume. The data collected in this experiment confirm the distinctive attainment of wave amplification by a factor of two in good agreement with the dynamics of the nonlinear Schrödinger equation solution. Advanced numerical simulations solving the problem of nonlinear free water surface boundary conditions of an ideal fluid quantify the physical limitations of the degenerate two-soliton in hydrodynamics.

The MEMOS-U macroscopic melt dynamics code—benchmarking and applications

The MEMOS-U code, a significantly upgraded version of MEMOS-3D, has been developed to address macroscopic metallic melt motion in large-deformation long-displacement regimes, where melts spill onto progressively colder solid surfaces, that are ubiquitous in contemporary tokamaks and expected to be realized in ITER. The modelling of plasma effects, appearing via the free-surface boundary conditions, is discussed along with the sensitivity to external input. The crucial roles of convective and thermionic cooling are exemplified by simulations of ELM-induced tungsten leading edge melting. Key melt characteristics, revealed by previous MEMOS-U modelling of grounded sample exposures, are confirmed in new simulations of the recent floating sample exposures in ASDEX-Upgrade.

MESOSCALE DEFORMATION-INDUCED SURFACE PHENOMENA IN LOADED POLYCRYSTALS

The paper reviews the results of numerical analyses for the micro-and mesoscale deformation-induced surface phenomena in three-dimensional polycrystals with the explicit account for the grain structure. The role of the free surface and grain boundaries in the appearance of the grain-scale stress concentrations and plastic strain nucleation is illustrated on the examples of aluminum polycrystals. Special attention is paid to the discussion of mesoscale deformation-induced surface roughening under uniaxial tension.

Boundary condition enforcement for renormalised weakly compressible meshless Lagrangian methods

This paper introduces a boundary condition scheme for weakly compressible (WC) renormalised first-order accurate meshless Lagrangian methods (MLM) by considering both solid and free surface conditions.A hybrid meshless Lagrangian method-finite difference (MLM-FD) scheme on prescribed boundary nodes is proposed to enforce Neumann boundary conditions. This is used to enforce symmetry boundary conditions and the implied Neumann pressure boundary conditions on solid boundaries in a manner consistent with the Navier-Stokes equation leading to the accurate recovery of surface pressures. The free surface boundary conditions allow all differential operators to be approximated by the same renormalised scheme while also efficiently determining free surface particles.The boundary conditions schemes are implemented for two renormalised MLMs. A WC smoothed particle hydrodynamics (SPH) solver is compared to a WC generalised finite difference (GFD) solver. Applications in both 2D and 3D are explored. A substantial performance benefit was found when comparing the WCGFD solver to the WCSPH solver with the WCGFD solver realising a maximum speedup in the range of three times over WCSPH in both 2D and 3D configurations. The solvers were implemented in C++ and used the NVIDIA CUDA 10.1 toolkit for the parallelisation of the solvers.

Biaxial modulus in fiber-textured thin films: Coinciding Voigt and Reuss bounds and planes of isotropy

The anisotropy of crystals necessitates the use of effective biaxial moduli for practical applications of fiber-textured thin films and coatings. The coincidence of the widely used Voigt and Reuss bounds of the effective moduli gives exact relations between the moduli of the polycrystal and the single crystal. It was known that only planes normal to three-, four-, and sixfold axes have coinciding bounds.1 The present study reports that planes without symmetry can also have coinciding bounds. A general mathematical framework to identify such planes in any crystal system is established. It is shown that the Voigt assumption of identical strain states in each grain cannot be made, as has been done in previous studies, for a general {hkl} fiber-textured thin film where the free surface boundary conditions are assumed. The conditions under which the Voigt average can be defined are also identified. For the textures where the Voigt average is invalid, a new method to estimate the polycrystalline average of the biaxial modulus under equibiaxial strain is defined. Another aspect of interest is that, the biaxial modulus, although generally anisotropic, is isotropic along certain planes. This is known for planes with three-, four-, and sixfold symmetry. However, here we demonstrate the existence of isotropy in planes without symmetry. A method to identify such planes under both equibiaxial strain and equibiaxial stress states are presented. Novel three-dimensional surfaces representing the averages of the biaxial moduli as functions of the orientation of the fiber axis are illustrated.

An iterative time-marching scheme for the investigation of hydrodynamic interaction between multi-ships during overtaking

An iterative time-marching scheme is developed to investigate the hydrodynamic interactions between multiple ships. Such an unsteady interactive effect could be magnified in restricted waterways, e.g., a channel or harbor area. To the author’s knowledge, nearly all the research on the ship-to-ship interaction neglecting the free surface effects. The free surface is usually treated as a rigid wall. This assumption is only reasonable when the speed of the ships is very low in deep water condition, due to the hydrodynamic interaction between the ships is mainly induced by near-field disturbances. However, when the moving speeds are moderately higher, especially with a small lateral separation between ships, the far-field effects arising from the ship waves become important. The main objective of the present paper is to develop an iterative time-matching algorithm to solve the hydrodynamic interaction between high-speed ships taking into account the nonlinear free surface boundary condition in time domain.

Higher order phenomena connected with the two-dimensional breaking wave impact on a vertical impermeable wall with air entrapment

This study investigates the higher order hydrodynamic phenomena of a breaking wave impact on a vertical impermeable wall. During the collision, a small amount of air is entrapped between the water and the wall. Here, we idealize the problem under consideration and we assume that the hydrodynamic pressure in the air-pocket is zero and that the brief collision does not change significantly the volume of the entrapped air. In addition, no compression effects are taken into account. We use a perturbation method along with the small-time approximation to cope with the kinematic and the dynamic boundary conditions of the free-surface. This approach allows us to satisfy the free-surface boundary conditions on the still water level and to avoid satisfying them on the a-priori unknown free-surface elevation induced by the collision. The breaking wave is idealized in a logical manner, allowing the consideration of the problem in an analytical context. In this way, we treated two separate mixed boundary value problems (BVP). Clearly, the solution of the higher order problem dictates the solution of the leading order problem. The higher order problem leads to a substantially complicated Sturm–Liouville problem, with mixed conditions and consequently, requires to be treated in a sophisticated manner. The analysis is performed in the two dimensions.

Steady-state harmonic resonance of periodic interfacial waves with free-surface boundary conditions based on the homotopy analysis method

Abstract