An improved free surface modeling for incompressible SPH

In Incompressible Smooth Particle Hydrodynamics (ISPH), a pressure Poisson equation (PPE) is solved to obtain a divergence free velocity field. When free surfaces are simulated using this method a Dirichlet boundary condition for pressure at the free surface has to be applied. In existing ISPH methods this is achieved by identifying free surface particles using heuristically chosen threshold of a parameter such as kernel sum, density or divergence of the position, and explicitly setting their pressure values. This often leads to clumping of particles near the free surface and spraying off of surface particles during splashes. Moreover, surface pressure gradients in flows where surface tension is important are not captured well using this approach. We propose a more accurate semi-analytical approach to impose Dirichlet boundary conditions on the free surface. We show the efficacy of the proposed algorithm by using test cases of elongation of a droplet and dam break. We perform two dimensional simulations of water entry and validate the proposed algorithm with experimental results. Further, a three dimensional simulation of droplet splash is shown to compare well with the Volume-of-Fluid simulations. (C) 2014 Elsevier Ltd. All rights reserved.

Free surface vibration in oscillatory convection of half floating zone

Projecting an orthographical grating mask (20pl/mm) on the surface of a small liquid bridge and receiving the reflected distortion image, one can calculate out reversely the shape of free surface of a liquid bridge. In this way we measured the surface shape of a small floating zone and the two-dimensional deformation of its vibration. The mechanism of thermocapillary oscillatory convection and the three-dimensional variation of the free surface are revealed experimentally. The principle for space experiment has been studied in our laboratory.

Free surface oscillation of thermocapillary convection in liquid bridge of half-floating-zone

Free surface deformations of thermocapillary convection in a small liquid bridge of half floating-zone are studied in the present paper. The relative displacement and phase difference of free surface oscillation are experimentally studied, and the features of free surface oscillation for various applied temperature differences are obtained. It is discovered that there is a sort of surface waves having the character of small perturbation, and having a wave mode of unusually large amplitude in one corner region of the liquid bridge.

A Breakdown Criterion of Free Molecular Flows and an Optimum Analysis of EBPVD

Two important issues in electron beam physical vapor deposition (EBPVD) are addressed. The first issue is a validity condition of the classical cosine law widely used in the engineering context. This requires a breakdown criterion of the free molecular assumption on which the cosine law is established. Using the analytical solution of free molecular effusion flow, the number of collisions (N-c) for a particle moving from an evaporative source to a substrate is estimated that is proven inversely proportional to the local Knudsen number at the evaporation surface. N-c = 1 is adopted as a breakdown criterion of the free molecular assumption, and it is verified by experimental data and DSMC results. The second issue is how to realize the uniform distributions of thickness and component over a large-area thin film. Our analysis shows that at relatively low evaporation rates the goal is easy achieved through arranging the evaporative source positions properly and rotating the substrate.

Formation of jets by implusive acceleration of a curved free surface

The sudden axial acceleration of a column of liquid bounded at one end by a concave free surface has been found, experimentally, to produce a jet which issues from the free surface with a speed several times that imparted to the column.

Theoretical approximations to such flows, valid for small time, are formulated subject to the assumption that the fluid is inviscid and incompressible. In a special two-dimensional case, it is found that, for vanishingly small time, the velocity at the point on the free surface from which the jet emanates is π/2 times the velocity imparted to the column. The solutions to several problems in two and three dimensions assuming that the initial curvature of the free surface is small, lead to values for this ratio dependent upon the curvature—the initial velocity in the case of axial symmetry exceeding that of the analogous two-dimensional problem by approximately 25%.

Experiments conducted upon the phenomenon give values systematically in excess of those predicted by the theory, although theory and experiment are in qualitative agreement with respect to the displacement of the free surface. It is suggested that the discrepancy is attributable to effects of finite curvature having been imperfectly accounted for in the axially-symmetric analysis.

Photographic materials on pp. 115, 120, and 121 are essential and will not reproduce clearly on Xerox copies. Photographic copies should be ordered.

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

fit the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten. this paper presents an extension of it previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier-Stokes equations. Second, it reports numerical simulation results for 1D hock tube problem, 2D impinging jet and 2D/3D broken clam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems. (C) 2007 IMACS. Published by Elsevier B.V. All rights reserved,

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

fit the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten. this paper presents an extension of it previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier-Stokes equations. Second, it reports numerical simulation results for 1D hock tube problem, 2D impinging jet and 2D/3D broken clam flows. Furthermore, these results are compared with existing analytical and experimental data. and third, it presents the application of the numerical method for solving 3D free surface flow problems. (C) 2007 IMACS. Published by Elsevier B.V. All rights reserved,

Dynamic interactions between solids and viscous liquids with free surface = Interacciones dinámicas entre sólidos y líquidos viscosos con superficie libre

En esta tesis se investiga la interacción entre un fluido viscoso y un cuerpo sólido en presencia de una superficie libre. El problema se expresa teóricamente poniendo especial atención a los aspectos de conservación de energía y de la interacción del fluido con el cuerpo. El problema se considera 2D y monofásico, y un desarrollo matemático permite una descomposición de los términos disipativos en términos relacionados con la superficie libre y términos relacionados con la enstrofía. El modelo numérico utilizado en la tesis se basa en el método sin malla Smoothed Particle Hydrodynamics (SPH). De manera análoga a lo que se hace a nivel continuo, las propiedades de conservación se estudian en la tesis con el sistema discreto de partículas. Se tratan también las condiciones de contorno de un cuerpo que se mueve en un flujo viscoso, implementadas con el método ghost-fluid. Se ha desarrollado un algoritmo explícito de interacción fluido / cuerpo. Se han documentado algunos casos de modo detallado con el objetivo de comprobar la capacidad del modelo para reproducir correctamente la disipación de energía y el movimiento del cuerpo. En particular se ha investigado la atenuación de una onda estacionaria, comparando la simulación numérica con predicciones teóricas. Se han realizado otras pruebas para monitorizar la disipación de energía para flujos más violentos que implican la fragmentación de la superficie libre. La cantidad de energía disipada con los diferentes términos se ha evaluado en los casos estudiados con el modelo numérico. Se han realizado otras pruebas numéricas para verificar la técnica de modelización de la interacción fluido / cuerpo, concretamente las fuerzas ejercidas por las olas en cuerpos con formas simples, y el equilibrio de un cuerpo flotante con una forma compleja. Una vez que el modelo numérico ha sido validado, se han realizado simulaciones numéricas para obtener una comprensión más completa de la física implicada en casos (casi) realistas sobre los había aspectos que no se conocían suficientemente. En primer lugar se ha estudiado el el flujo alrededor de un cilindro bajo la superficie libre. El estudio se ha realizado con un número de Reynolds moderado, para un rango de inmersiones del cilindro y números de Froude. La solución numérica permite una investigación de los patrones complejos que se producen. La estela del cilindro interactúa con la superficie libre. Se han identificado algunos inestabilidades características. El segundo estudio se ha realizado sobre el problema de sloshing, tanto experimentalmente como numéricamente. El análisis se restringe a aguas poco profundas y con oscilación horizontal, pero se ha estudiado un gran número de condiciones, lo que lleva a una comprensión bastante completa de los sistemas de onda involucradas. La última parte de la tesis trata también sobre un problema de sloshing pero esta vez el tanque está oscilando con rotación y hay acoplamiento con un sistema mecánico. El sistema se llama pendulum-TLD (Tuned Liquid Damper - con líquido amortiguador). Este tipo de sistema se utiliza normalmente para la amortiguación de las estructuras civiles. El análisis se ha realizado analíticamente, numéricamente y experimentalmente utilizando líquidos con viscosidades diferentes, centrándose en características no lineales y mecanismos de disipación. ABSTRA C T The subject of the present thesis is the interaction between a viscous fluid and a solid body in the presence of a free surface. The problem is expressed first theoretically with a particular focus on the energy conservation and the fluid-body interaction. The problem is considered 2D and monophasic, and some mathematical development allows for a decomposition of the energy dissipation into terms related to the Free Surface and others related to the enstrophy. The numerical model used on the thesis is based on Smoothed Particle Hydrodynamics (SPH): a computational method that works by dividing the fluid into particles. Analogously to what is done at continuum level, the conservation properties are studied on the discrete system of particles. Additionally the boundary conditions for a moving body in a viscous flow are treated and discussed using the ghost-fluid method. An explicit algorithm for handling fluid-body coupling is also developed. Following these theoretical developments on the numerical model, some test cases are devised in order to test the ability of the model to correctly reproduce the energy dissipation and the motion of the body. The attenuation of a standing wave is used to compare what is numerically simulated to what is theoretically predicted. Further tests are done in order to monitor the energy dissipation in case of more violent flows involving the fragmentation of the free-surface. The amount of energy dissipated with the different terms is assessed with the numerical model. Other numerical tests are performed in order to test the fluid/body interaction method: forces exerted by waves on simple shapes, and equilibrium of a floating body with a complex shape. Once the numerical model has been validated, numerical tests are performed in order to get a more complete understanding of the physics involved in (almost) realistic cases. First a study is performed on the flow passing a cylinder under the free surface. The study is performed at moderate Reynolds numbers, for various cylinder submergences, and various Froude numbers. The capacity of the numerical solver allows for an investigation of the complex patterns which occur. The wake from the cylinder interacts with the free surface, and some characteristical flow mechanisms are identified. The second study is done on the sloshing problem, both experimentally and numerically. The analysis is restrained to shallow water and horizontal excitation, but a large number of conditions are studied, leading to quite a complete understanding of the wave systems involved. The last part of the thesis still involves a sloshing problem but this time the tank is rolling and there is coupling with a mechanical system. The system is named pendulum-TLD (Tuned Liquid Damper). This kind of system is normally used for damping of civil structures. The analysis is then performed analytically, numerically and experimentally for using liquids with different viscosities, focusing on non-linear features and dissipation mechanisms.

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.

Bow and stern flows with constant vorticity

Free surface flows of a rotational fluid past a two-dimensional semi-infinite body are considered. The fluid is assumed to be inviscid, incompressible, and of finite depth. A boundary integral method is used to solve the problem for the case where the free surface meets the body at a stagnation point. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterized by a train of waves upstream. It is shown numerically that the amplitude of these waves increases as each of the Froude number, vorticity and height of the body above the bottom increases.

Free surface problems for static Coulomb-Mohr granular solids

This paper is concerned with some plane strain and axially symmetric free surface problems which arise in the study of static granular solids that satisfy the Coulomb-Mohr yield condition. Such problems are inherently nonlinear, and hence difficult to attack analytically. Given a Coulomb friction condition holds on a solid boundary, it is shown that the angle a free surface is allowed to attach to the boundary is dependent only on the angle of wall friction, assuming the stresses are all continuous at the attachment point, and assuming also that the coefficient of cohesion is nonzero. As a model problem, the formation of stable cohesive arches in hoppers is considered. This undesirable phenomena is an obstacle to flow, and occurs when the hopper outlet is too small. Typically, engineers are concerned with predicting the critical outlet size for a given hopper and granular solid, so that for hoppers with outlets larger than this critical value, arching cannot occur. This is a topic of considerable practical interest, with most accepted engineering methods being conservative in nature. Here, the governing equations in two limiting cases (small cohesion and high angle of internal friction) are considered directly. No information on the critical outlet size is found; however solutions for the shape of the free boundary (the arch) are presented, for both plane and axially symmetric geometries.

Free surface flow into a horizontal slot with zero gravity

The two-dimensional free surface flow of a finite-depth fluid into a horizontal slot is considered. For this study, the effects of viscosity and gravity are ignored. A generalised Schwarz-Christoffel mapping is used to formulate the problem in terms of a linear integral equation, which is solved exactly with the use of a Fourier transform. The resulting free surface profile is given explicitly in closed-form.

Minimising wave drag for free surface flow past a two-dimensional stern

Free surface flow past a two-dimensional semi-infinite curved plate is considered, with emphasis given to solving for the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the Wiener-Hopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results found using a forced KdV equation are also presented, as are numerical solutions to the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semi-infinite plate, it is shown how the amplitude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely.

Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton-Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.