Incompressible Surface Research Articles

Modeling of Free Surface Flows Using Improved Material Point Method and Dynamic Adaptive Mesh Refinement

AbstractThe study uses the material point method (MPM) and dynamic adaptive mesh refinement (AMR) technique to simulate incompressible free surface flows. The MPM can be regarded as a quasi-meshless method in which the background grid acts like a scratch pad while the material point (hereafter referred to as particle) motion overlies it. The states of the particles are updated through the solutions on the background grid. The artificial compressibility coefficient is employed to treat incompressible flows as slightly compressible flows within the MPM framework. Boundary conditions, such as free surfaces and reflective walls (both slip and no-slip conditions), are implemented and tested using the ghost-cell method. A generalized far-field characteristic boundary for wave propagation simulation has been established. Typical water wave propagation and violent wave breaking involving discontinuous free surfaces are simulated as well as compared to the arbitrary Lagrangian-Eulerian (ALE) method. The results in...

Essential surfaces in highly twisted link complements

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram.

AN ESTIMATE OF HEMPEL DISTANCE FOR BRIDGE SPHERES

Tomova (8) gave an upper bound for the distance of a bridge surface for a knot with two different bridge positions in a 3-manifold. In this paper, we show that the result of Tomova (8, Theorem 10.3) can be improved in the case when there are two different bridge spheres for a link in S 3 . Hempel (4) introduced the concept of distance of a Heegaard surface, and it is shown by many authors that it well represents various complexities of 3- manifolds. For example, Hartshorn (2) showed that the Euler characteristic of an incompressible surface in a 3-manifold bounds the distance of its Heegaard splittings, and Scharlemann and Tomova (7) showed that the Euler character- istic of any Heegaard splitting of a 3-manifold similarly bounds the distance of any non-isotopic Heegaard splitting. The above concept and results have been extended to bridge surfaces for knots and links in closed 3-manifolds, and have been studied by several authors. For example, Bachman and Schleimer (1) proved that Hartshorn's results can be extended to the distance of a bridge surface for a knot in a closed orientable 3-manifold, and also Tomova (8) proved that Scharlemann and Tomova's results can be extended to the distance of a bridge surface for a knot in a closed ori- entable 3-manifold. Moreover, Johnson and Tomova (6) proved that Tomova's result can be extended to a bridge surface for a tangle in a compact 3-manifold. Recently Jang (5) showed that for a link in a closed orientable 3-manifold, the result of Bachman and Schleimer (1) can be improved in the case when there exist essential meridional spheres. In this paper, we find a property of essential simple closed curves disjoint from the disk complex of a 3-ball containing trivial arcs (for detail, see Lemma 2.1). This allows us to improve the result of Tomova (8, Theorem 10.3) in the case when there are two different bridge spheres for a link in S 3 .

Derivation of nonlinear shell models combining shear and flexure: application to biological membranes

Biological membranes are often idealized as incompressible elastic surfaces whose strain energy only depends on their mean curvature and pos-sibly on their shear. We show that this type of model can be derived using a formal asymptotic method by considering biological membranes to be thin, strongly anisotropic, elastic, locally homogeneous bodies.

Counting minimal surfaces in quasi-Fuchsian three-manifolds

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many stable ones. In this paper, for any prescribed integer $N > 0$, we construct a quasi-Fuchsian manifold which contains at least $2^N$ such minimal surfaces. As a consequence, there exists some simple closed Jordan curve on $S_{\infty }^2$ such that there are at least $2^N$ disk-type complete minimal surfaces in $\mathbb {H}^3$ sharing this Jordan curve as the asymptotic boundary.

On topologically minimal surfaces of high genus

We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.

Simulating free surface problem using isogeometric analysis

This paper presents the application of isogeometric analysis to simulate the incompressible free surface flows as well as dam break problem. The governing Navier–Stokes equations using a pressure projection method are solved in a Lagrangian form. The least squares method is used in both function approximation and discretization of the governing differential equations in space. Non-uniform rational B-splines are also used as basic functions. One of the challenging free surface problems, namely dam break, is solved to check the accuracy of the proposed method. The isogeometric analysis method enjoys the advantage of producing symmetric, positive definite matrixes for the considered cases. The results of the free surface at different times are compared with the characteristic-based split finite element method. The results show the ability of the isogeometric analysis method to solve complex fluid dynamic problems with moving free surface boundaries. Moreover, the velocity fields and the pressure distribution due to the dam break problem at different times are plotted.

Large Eddy Simulation by particle method coupled with Sub-Particle-Scale model and application to mixing layer flow

The Moving Particle Semi-implicit method (MPS) shows great potential to simulate violent incompressible free surface flows. However, Direct Numerical Simulations of such turbulent flows by particle methods would be extremely time-consuming. Taken ideas from the Sub-Grid Scale (SGS) model in grid-based Large Eddy Simulation (LES), static and dynamic Smagorinsky models are specially designed for Sub-Particle Scale (SPS) model in MPS. These SPS models are discretized by the combination of gradient and divergence models in MPS and the semi-implicit algorithm to implement MPS-LES is developed. The widely-studied mixing layer flows are simulated to quantitatively validate the developed turbulence particle model. Both the turbulent statistical profiles and vortex pairing processes are in good agreements with experimental observations after the SPS model is adopted. Moreover, it is found that the performance of the dynamic Smagorinsky model is better than that of the static model.

A new incompressibility discretization for a hybrid particle MAC grid representation with surface tension

We take a particle based approach to incompressible free surface flow motivated by the fact that an explicit representation of the interface geometry and internal deformations gives precise feedback to an implicit solver for surface tension. Methods that enforce incompressibility directly on the particles are typically numerically inefficient compared to those that utilize a background grid. However, background grid discretizations suffer from inaccuracy near the free surface where they do not properly capture the interface geometry. Therefore, our incompressibility discretization utilizes a particle based projection near the interface and a background MAC grid based projection for efficiency in the vast interior of the liquid domain – as well as a novel method for coupling these two disparate projections together. We show that the overall coupled elliptic solver is second order accurate, and remains second order accurate when used in conjunction with an appropriate temporal discretization for parabolic problems. A similar second order accurate discretization is derived when the MAC grid unknowns are located on faces (as opposed to cell centers) so that Navier–Stokes viscosity can be solved for implicitly as well. Finally, we present a fully implicit approach to surface tension that is robust enough to achieve a steady state solution in a single time step. Beyond stable implicit surface tension for our novel hybrid discretization, we demonstrate preliminary results for both standard front tracking and the particle level set method.

An upwind finite volume method for incompressible inviscid free surface flows

An upwind finite volume method on curvilinear grids has been developed to simulate two-dimensional, inviscid, incompressible free surface flows. In this method, the level set method is used to capture the interface. The level set equation and the flow governing equations (Euler equations) are not solved separately, but incorporated into a system equation. The artificial compressibility method is adopted to solve the resulting hyperbolic equations. The MUSCL-Roe scheme, which is widely used in compressible flow computations, is employed to discretize the convective terms. A fully implicit dual-time stepping method with a highly efficient LU-SSOR algorithm as sub-iteration scheme is applied to advance the solution in time. In the present study, a re-initialization method on curvilinear grids for level set function is presented. An example of re-initialization for level set function on curvilinear grids is demonstrated and the resulting level set function is in good agreement with the exact distance function. Numerical results of a sloshing tank problem and a broken dam problem agree well with the theoretical and experimental data, respectively. The free surface flow over a submerged hydrofoil is also simulated using overset grids to demonstrate the capability of dealing with complex geometry. At last, the simulation of droplet splash problem is performed to show the ability of the present method in handling topological changes of the free surface.

Propeller Effect on 3D Flow at the Stern Hull of a LNG Carrier Using Finite Volume Method

Numerical study and RANS simulations have been applied to investigate the incompressible free surface flow around the stern hull of Liquefied Natural Gas (LNG) ship affected by working propeller behind of her. Experimental works are carried out using LNG ship model in Marine Teknologi Center (MTC) of Univrsiti Teknologi Malaysia (UTM) to verify the computational fluid dynamic (CFD) results. Ansys-CFX 14.0 based on viscous flow finite volume code using the two-phase Eulerian–Eulerian fluid approach and shear stress transport (SST) turbulence model have been used in this study. A tetrahedral unstructured combined with prism grid were used with the viscous flow code for meshing the computational domain of water surface around it. CFD simulation has been verified using available experimental results. Finally, the flow structure, streamlines, velocity and pressure distribution around stern hull and propeller zone are discussed and analysed.

Finiteness of 3-manifolds associated with non-zero degree mappings

We prove a finiteness result for the \partial -patterned guts decomposition of all 3 -manifolds obtained by splitting a given orientable, irreducible and \partial -irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold belong, up to homeomorphism, to a finite collection of compact 3-manifolds. We show also that any closed orientable 3-manifold dominates only finitely many integral homology spheres and any compact orientable 3-manifold dominates only finitely many exteriors of knots in S^3 .

Development of a Dispersion Relation Equation - Preserving Pure Advection Scheme for Solving the Navier-Stokes Equations with/Without Free Surface

In this article we develop a dispersively more accurate pure advection scheme and apply it to properly simulate incompressible free surface flow equations. According to Clebsch velocity decomposition, the incompressible flow equations are decomposed into the equations accounting respectively for the flow potential, rotation, and dissipation. For the Euler equation, the midpoint implicit symplectic time integrator is applied to approximate the temporal derivative term. For the sake of reducing numerical dispersion error, the upwinding scheme is developed to minimize the difference between the numerical and exact dispersion relation equations for the time-dependent pure advection equation in wavenumber space.

TOTALLY GEODESIC SURFACES AND QUADRATIC FORMS

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

Some applications of Gabaiʼs internal hierarchy

Any Haken 3-manifold (possibly with boundary consisting of tori) can be transformed into a surface×S1 by a series of splitting and regluing along incompressible surfaces. This fact was proved by Gabai as an application of his sutured manifold theory. The first half of this paper provides a few technical details in the proof. In the second half of this paper, some applications of Gabaiʼs theorem to Heegaard Floer homology are given. We refine the known results about the Thurson norm and fibrations. We also give some classification results for Floer simple knots in manifolds with positive b1.

The linearization principle for a free boundary problem for viscous, capillary incompressible fluids

The free boundary problem associated with a viscous incompressible surface wave subject to the capillary force on a free upper surface and the Dirichlet boundary condition on a fixed bottom surface is considered. In the spatially periodic case, a general linearization principle is proved, which gives, for sufficiently small perturbations from a linearly stable stationary solution, the existence of a global solution of the associated system and the exponential convergence of the latter to the stationary one. The convergence of the velocity, the pressure, and the free boundary is proved in anisotropic Sobolev--Slobodetskii spaces, and then a suitable change of variables is performed to state the problem in a fixed domain. This linearization principle is applied to the study of the rest state's stability in the case of general potential forces. Bibliography: 16 titles.

Simulation of Viscous Flow around Surface Ship Using ANSYS-Fluent CFD Code

In this paper, numerical simulations have been carried out to determine the incompressible free surface flow around a Wigley hull form for which experimental results are available. A commercial viscous flow finite volume code using the two-phase Eulerian-Eulerian fluid approach has been used in this study. The simulation conditions are the ones for which experimental results exist. The shear stress transport (SST) turbulence model has been used in the viscous flow code. A structured grid was used with the viscous flow code for meshing the computational domain. The results compare well with the available experimental data, which make their complementary application useful for determining the total ship resistance.

Graph manifolds with boundary are virtually special

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in π1M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a ‘sufficient’ collection of surfaces in M, then π1M is virtually the fundamental group of a special non-positively curved cube complex. We provide a sufficient collection for graph manifolds with boundary, thus proving that their fundamental groups are virtually special, and hence linear.

Detecting surface bundles in finite covers of hyperbolic closed 3-manifolds

The main theorem of this article provides sufficient conditions for a degree d d finite cover M ′ M’ of a hyperbolic 3-manifold M M to be a surface bundle. Let F F be an embedded, closed and orientable surface of genus g g , close to a minimal surface in the cover M ′ M’ , splitting M ′ M’ into a disjoint union of q q handlebodies and compression bodies. We show that there exists a fiber in the complement of F F provided that d d , q q and g g satisfy some inequality involving an explicit constant k k depending only on the volume and the injectivity radius of M M . In particular, this theorem applies to a Heegaard splitting of a finite covering M ′ M’ , giving an explicit lower bound for the genus of a strongly irreducible Heegaard splitting of M ′ M’ . Applying the main theorem to the setting of a circular decomposition associated to a non-trivial homology class of M M gives sufficient conditions for this homology class to correspond to a fibration over the circle. Similar methods also lead to a sufficient condition for an incompressible embedded surface in M M to be a fiber.

On almost-Fuchsian manifolds

An almost-Fuchsian manifold is a class of complete hyperbolic three-manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of ( − 1 , 1 ) (-1,1) . In such a manifold, the minimal surface is unique and embedded, hence one can parametrize these hyperbolic three-manifolds by their minimal surfaces. In this paper we obtain estimates on several geometric and analytical quantities of an almost-Fuchsian manifold M M in terms of the data on the minimal surface. In particular, we obtain an upper bound for the hyperbolic volume of the convex core of M M and an upper bound on the Hausdorff dimension of the limit set associated to M M . We also constructed a quasi-Fuchsian manifold which admits more than one minimal surface, and it does not admit a foliation of closed surfaces of constant mean curvature.