Free Surface Problems Research Articles

Applicability of the method of fundamental solutions to 3-D wave–body interaction with fully nonlinear free surface

A numerical model for three-dimensional fully nonlinear free-surface waves is developed by applying a boundary-type meshless approach with a leap-frog time-marching scheme. Adopting Gaussian Radial Basis Functions to fit the free surface, a non-iterative approach to discretize the nonlinear free-surface boundary is formulated. Using the fundamental solutions of the Laplace equation as the solution form of the velocity potential, free-surface wave problems can be solved by collocations at only a few boundary points since the governing equation is automatically satisfied. The accuracy of the present method is verified by comparing the simulated propagation of a solitary wave with an exact solution. The applicability of the present model is illustrated by applying it to the problem of a solitary wave running up on a vertical surface-piercing cylinder and the problem of wave generation in infinite water depth by a submerged moving object.

The method of fundamental solutions for free surface Stefan problems

In this paper, free surface problems of Stefan-type for the parabolic heat equation are investigated using the method of fundamental solutions. The additional measurement necessary to determine the free surface could be a boundary temperature, a heat flux or an energy measurement. Both one- and two-phase flows are investigated. Numerical results are presented and discussed.

A robust and efficient hybrid cut-cell/ghost-cell method with adaptive mesh refinement for moving boundaries on irregular domains

A new robust and efficient Cartesian grid method is developed for the moving boundary problem on an arbitrary complex domain. A difficult problem in the application of cut-cell methods is the presence of degenerate cut-cells, which are defined as cut-cells that are intersected by a boundary curve at more than two points. A new strategy is proposed for handling the problem of degenerate cells within the cut-cell methodology. In the proposed approach, gradient fluxes through all sections of the boundary curve intersecting a cut-cell are computed using surface integrals, with the consequence that all cut-cells in the solution domain (whether they are regular or degenerate) are handled using the same procedure. Marker points on the boundary curve are used to define cubic splines, which provide a high-fidelity representation for the boundary curve from which various geometrical properties (e.g., volume, face areas, centroid, etc.) of cut-cells can be determined with high-accuracy. In order to enhance the robustness of the present approach, the concept of a “ghost point” for degenerate cut-cells is introduced, which facilitates the evaluation of gradient fluxes in the fluid domain. This is advantageous as there is no longer any need to deal individually with each (usually small) split cut-cell. In consequence, the method proposed here can be interpreted as a hybrid cut-cell/ghost-cell method, in which the conventional cut-cell method is used to deal with regular cut-cells and the ghost-cell method is used to handle degenerate cut-cells. The accuracy and efficiency of this method can be improved further by application of an adaptive mesh refinement (AMR) method, based on a fully threaded tree data structure, to the cut-cells. A multigrid acceleration technique is used to solve the resulting algebraic system of discretized equations. Six test cases, for which exact solutions are available, are used to demonstrate the accuracy and efficiency of the proposed algorithm for both fixed and moving boundary problems. Using a triangulated surface mesh to represent an internal boundary surface, our proposed approach is generalized to deal with three-dimensional fluid flow problems. A test case of a three-dimensional free surface problem is used to validate the generalization of our Cartesian grid approach for complex boundaries in three dimensions.

Lagrangian Finite–Element Method for the Simulation of K-BKZ Fluids with Third Order Accuracy

A new finite element scheme for the numerical simulation of three-dimensional time-dependent flow of viscoelastic fluids is presented. The viscoelastic fluids are of the K-BKZ integral type and the method is based on a Lagrangian kinematics description of the fluid flow. The spatial coordinate system attached to the particles is discretized by ten-node quadratic tetrahedral elements using Cartesian coordinates and the pressure by linear interpolation inside these elements. The spatial discretization of the governing equations follows the mixed Galerkin finite element method. The time integral is discretized by a quadratic interpolation in time. The convergence of the method in time and space was demonstrated on the free surface problem of a filament stretched between two plates, considering the axisymmetric case as well as the growth of non-axisymmetric disturbances on the free surface. The scheme converges with respect to discretization of the spatial and time dimensions, both with third order.

Quadrature formulas for periodic functions and their application to the boundary element method

Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability.

Water Wave Interaction on a Truncated Vertical Cylinder

The purpose of this paper is to combine and extend existing potential flow theory in order to analyze the linear free surface problem of an oscillating water column (OWC) device and apply it to moon pool design. Analytical results were obtained implementing the previously derived theories, and later compared to experimental results conducted at the University of New Orleans Towing Tank. The model tests consisted of a study of a cylindrical OWC. The theoretical and experimental results of the free surface for the OWC tests agree for the resonant frequency estimation response but they disagree on the amplitude of the response.

ODDLS: A new unstructured mesh finite element method for the analysis of free surface flow problems

AbstractThis paper introduces a new stabilized finite element method based on the finite calculus (Comput. Methods Appl. Mech. Eng. 1998; 151:233–267) and arbitrary Lagrangian–Eulerian techniques (Comput. Methods Appl. Mech. Eng. 1998; 155:235–249) for the solution to free surface problems. The main innovation of this method is the application of an overlapping domain decomposition concept in the statement of the problem. The aim is to increase the accuracy in the capture of the free surface as well as in the resolution of the governing equations in the interface between the two fluids. Free surface capturing is based on the solution to a level set equation. The Navier–Stokes equations are solved using an iterative monolithic predictor–corrector algorithm (Encyclopedia of Computational Mechanics. Wiley: New York, 2004), where the correction step is based on imposing the divergence‐free condition in the velocity field by means of the solution to a scalar equation for the pressure. Examples of application of the ODDLS formulation (for overlapping domain decomposition level set) to the analysis of different free surface flow problems are presented. Copyright © 2008 John Wiley & Sons, Ltd.

Free-surface problems in electrokinetic micro- and nanofluidics

With the development of novel micro- and nanofluidic devices, applications of flows with moving interfaces are in increasing demand. Problems in ultra-small scale with moving boundaries, which have not yet been exploited rigorously, are discussed with focus on electrokinetic flows. A generic formulation for moving interfaces, applicable to micro- and nanoscale electrokinetic flows, is provided, and actual free-surface problems in electroosmosis and electrophoresis are presented.

Viscous flow with free surface motion by least square finite element method

A method for simulation of free surface problems is presented. Based on the viscous incompressible Navier-Stokes equations, space discretization of the flow is obtained by the least square finite element method. The time evolution is obtained by the finite difference method. Lagrangian description is used to track the free surface. The results are compared with the experimental dam break results, including water collapse in a 2D rectangular section and in a 3D cylinder section. A good agreement is achieved for the distance of surge front as well as the height of the residual column.

Numerical simulation of spin coating processes involving functionalised Carbon nanotube suspensions

This paper reports the numerical simulation of spin coating for functionalised Carbon Nanotube (CNT) suspensions. Spin coating is a process commonly used to deposit uniform thin films onto flat substrates by means of high rotation velocity and centrifugal force. The functionalised CNTs modelled in this study were chemically treated in a way such that aggregation was prevented through electrostatic repulsion between CNTs. The functionalised CNTs in the semi-dilute suspensions can be modelled as rigid fibres with their orientation dictated by the flow of the solvent. The evolution of CNT orientations was simulated using a pre-averaged kinetic theory with an appropriate rotary diffusion coefficient accounting for randomising events. A Natural Element strategy with an updated Lagrangian framework was implemented to solve the free-surface problem involving large domain deformation and to avoid numerical problems associated with Finite Element modelling. The model reported herein couples micro-scale CNT orientation with the macroscopic suspension kinematics and it offers important insights in relation to the final properties of spin-coated CNT films as well as the processing behaviour of CNT suspensions.

Improved boundary tracking in meshless simulations of free-surface flows

In this paper we review the use of shape constructors, particularly α-shapes, for the simulation of free-surface flow problems. This technique, in conjunction with meshless methods, allows for the simulation of such problems in an updated Lagrangian approach without the need for an explicit description of the boundary of the domain. At each time step, the shape of the domain is extracted automatically. However, it is well know that α-shape techniques present some drawbacks. The first is the choice of the α parameter, related to the level of detail to which the domain is represented. Also contact detection of free surfaces (auto-contact) or between the free surface and a rigid boundary, for instance, is often detected with an error of the order \({\mathcal{O}(h)}\) , the nodal spacing parameter, in the gap distance. We propose an heuristic technique for the choice of the α parameter and develop a novel methodology for an improved detection of contact or merging flows. The proposed technique is illustrated with the help of some examples in solid and fluid mechanics.

Accurate numerical solutions to stationary free surface problems from capillarity

The paper considers two static problems from capillarity. The first one consists in the determination of the surface of a liquid in a capillary tube and the second in the computation of the shape of a sessile or pendent drop of liquid of a given volume, both configurations being considered in the gravitational field. From a mathematical point of view, both problems are nonlinear two-point boundary value problems which include in their formulations additional geometrical unknowns, i.e., the so-called free boundary value problems. Computational, they are laborious problems owing to the fact that they involve nonnormal Jacobian matrices. The resulting numerical difficulties are considerably reduced by treating these problems by a collocation method in conjunction with the continuation with respect to the parameters. The method is implemented by the MATLAB code bvp4c. The numerical evaluations of the free surfaces are in reasonable accordance with asymptotic estimations.

Numerical simulation of spin coating processes with carbon nanotubes suspensions

In this paper we address the problem of simulating the behaviour of Carbon nanotubes (CNTs) suspensions in spin coating processes. Spin coating is a procedure used to apply uniform thin films to flat substrates by means of a high rotating velocity and the subsequent centrifugal force. In this work we assume a suspension of chemically treated CNTs, such that they do not aggregate. In such a suspension, CNTs can be treated as rigid fibres whose orientation is dictated by the flow of the solvent. In order to treat the associated free-surface problem and to avoid the numerical problems associated to their FE solution, we have implemented a Natural Element strategy in an updated Lagrangian framework. The resulting model is thus composed by a description of the micro scale, related to the orientation of the carbon nanotubes and the assumption of a quadratic closure relation, together with the natural element approximation for the Navier-Stokes problem governing the macroscopic suspension kinematics.

Modeling and analysis of thickness gradient and variations in vacuum‐assisted resin transfer molding process

AbstractAs vacuum‐assisted resin transfer molding (VARTM) is being increasingly used in aerospace applications, the thickness gradient and variation issues are gaining more attention. Typically, thickness gradient and variations result from the infusion pressure gradient during the process and material variations. Pressure gradient is the driving force for resin flow and the main source of thickness variation. After infusion, an amount of pressure gradient is frozen into the preform, which primarily contributes to the thickness variation. This study investigates the mechanism of the thickness variation dynamic change during the infusion and relaxing/curing processes. A numerical model was developed to track the thickness change of the bagging film free surface. A time‐dependent permeability model as a function of compaction pressure was incorporated into an existing resin transfer molding (RTM) code for obtaining the initial conditions for relaxing/curing process. Control volume (CV) and volume of fluid (VOF) methods were combined to solve the free surface problem. Experiments were conducted to verify the simulation results. The proposed model was illustrated with a relatively complex part. POLYM. COMPOS., 2008. © 2008 Society of Plastics Engineers

Modified incompressible SPH method for simulating free surface problems

An incompressible smoothed particle hydrodynamics (I-SPH) formulation is presented to simulate free surface incompressible fluid problems. The governing equations are mass and momentum conservation that are solved in a Lagrangian form using a two-step fractional method. In the first step, velocity field is computed without enforcing incompressibility. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. The source term in the Poisson equation for the pressure is approximated, based on the SPH continuity equation, by an interpolation summation involving the relative velocities between a reference particle and its neighboring particles. A new form of source term for the Poisson equation is proposed and also a modified Poisson equation of pressure is used to satisfy incompressibility condition of free surface particles. By employing these corrections, the stability and accuracy of SPH method are improved. In order to show the ability of SPH method to simulate fluid mechanical problems, this method is used to simulate four test problems such as 2-D dam-break and wave propagation.

A numerical method for the prediction of wave pattern of surface piercing cavitating hydrofoils

The iterative boundary-element method, which is originally developed before for submerged cavitating hydrofoils is extended and modified to predict the wave pattern and lift and drag values of surface piercing cavitating hydrofoils (vertical struts) moving with a constant speed on the free surface. The iterative numerical method, which is based on the Green's theorem, allows the separation of surface piercing cavitating hydrofoil (or vertical strut) problem and the free surface problem. Those problems are solved separately, with the effects of one on the other being accounted for in an iterative manner. The wetted surface of the body (hydrofoil or strut) and the free surface are modelled with constant strength dipole and constant strength source panels. In order to prevent upstream waves the source strengths from some distance in front of the body to the end of the truncated upstream boundary are enforced to be zero. No radiation condition is enforced for downstream and transverse boundaries on the free surface. The method is applied to a rectangular non-cavitating hydrofoil with a yaw angle to compare the results with those of experiments and other numerical methods given in the literature. Then, the method is applied to a rectangular cavitating vertical strut and the effects of Froude number on wave pattern and lift and drag values of vertical strut are discussed.

Numerical calculation of free-surface potential flow around a ship using the modified Rankine source panel method

A modified Rankine source panel method is presented for solving a linearized free-surface flow problem with respect to the double-body potential. The method of solution is based on the distribution of Rankine sources on the hull as well as its image and on the free surface. An iterative algorithm is used for determining the free surface and wave resistance using upstream finite difference operator. A verification of numerical modeling is made using the Wigley hull and the validity of the computer program is examined by comparing the details of wave profiles and wave making resistance with Series 60 model.

SPH 기법의 계산인자 민감도에 대한 연구

A smoothed particle hydrodynamics (SPH) method is applied for simulating two-dimensional free-surface problems. The SPH method based on the Lagrangian formulation provides realistic flow motions with violent surface deformation, fragmentation and reunification. In this study, the effect of computational parameters in SPH simulation is explored through two-dimensional dam-breaking and sloshing problem. The parameters to be considered are the speed of sound, the frequency of density re-initialization, the number of particle and smoothing length. Through a series of numerical test. detailed information was obtained about how SPH solution can be more stabilized and improved by adjusting computational parameters. Finally, some numerical simulations for various fluid flow problem were carried out based on the parameters chosen through the sensitivity study.

Application of Smoothed Particle Hydrodynamics Method to Free Surface and Solidification Problems

The smoothed particle hydrodynamics (SPH) method is developed to simulate free surface and solidification problems. A new treatment is included to handle particles near the free and solidification interfaces. The method is applied to a droplet impacting on substrates with different roughnesses. Spreading, solidification, oxide redistribution, and droplet pinch-off are presented in two- and three-dimensional geometry configurations. This work demonstrates the SPH model as a powerful tool to study transport phenomena in problems with free surface deformation and solidification.

On the Euler equations of incompressible fluids

Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.