Incompressible Surface Research Articles

Simulation of free surface flow around a VLCC hull using viscous and potential flow methods

Numerical simulations have been carried out to determine the incompressible free surface flow around a VLCC hull form for which experimental results are available. A commercial viscous flow finite volume code using the two-phase Eulerian–Eulerian fluid approach and a potential flow code based on the Rankine source method have 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 tetrahedral unstructured grid was used with the viscous flow code for meshing the computational domain, while quadrilateral structural patches were used with the potential flow code for meshing the VLCC hull surface and the water surface around it. The results compare well with the available experimental data and they allow an understanding of the differences that can be expected from viscous and potential flow methods as a result of their different mathematical formulations, which make their complementary application useful for determining the total ship resistance.

Twisted Alexander polynomials and representation shifts

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.

Circular thin position for knots inS3

A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular level surface contains a Seifert surface for the knot. We rearrange the handles in such a way that the regular surfaces are as simple as possible. To make this precise the concept of circular width for E(K) is introduced. When E(K) is endowed with a handle decomposition which realizes the circular width we will say that the knot K is in circular thin position. We use this to show that many knots have more than one non-isotopic incompressible Seifert surface. We also analyze the behavior of the circular width under some knot operations.

ASYMPTOTIC DERIVATION OF THE SECTION-AVERAGED SHALLOW WATER EQUATIONS FOR NATURAL RIVER HYDRAULICS

The section-averaged shallow water model usually applied in river and open channel hydraulics is derived by an asymptotic analysis that accounts for terms up to second order in the vertical/longitudinal length ratio, starting from the three-dimensional Reynolds-averaged Navier–Stokes equations for incompressible free surface flows. The derivation is carried out under quite general assumptions on the geometry of the channel, thus allowing for the application of the resulting equations to natural rivers with arbitrarily shaped cross sections. As a result of the derivation, a generalized friction term is obtained, that does not rely on local uniformity assumptions and that can be computed directly from three-dimensional turbulence models, without need for local uniformity assumptions. The modified equations including the novel friction term are compared to the classical Saint Venant equations in the case of steady state open channel flows, where analytic solutions are available, showing that the solutions resulting from the modified equation set are much closer to the three-dimensional solutions than those of the classical equation set. Furthermore, it is shown that the proposed formulation yields results that are very similar to those obtained with empirical friction closures widely applied in computational hydraulics. The generalized friction term derived therefore justifies a posteriori these empirical closures, while allowing to avoid the assumptions on local flow uniformity on which these closures rely.

Incompressibility and normal minimal surfaces

In this paper we describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces. Let F be a closed orientable incompressible surface in an irreducible 3-manifold M. Then in every triangulation τ of M, F is isotopic to a τ-normal surface F(τ) that is of minimal PL-area (in the isotopy class of F). Using the above scaling refinement we prove the converse. If for every triangulation τ of M, F is isotopic to a τ-normal surface F(τ) that is of minimal PL-area, then we shall show that F is incompressible. Hence we get a characterisation of incompressibility of a surface in terms of existence of a minimal PL-area normal surface.

A volume-of-fluid method for incompressible free surface flows

This paper proposes a hybrid volume-of-fluid (VOF) level-set method for simulating incompressible two-phase flows. Motion of the free surface is represented by a VOF algorithm that uses high resolution differencing schemes to algebraically preserve both the sharpness of interface and the boundedness of volume fraction. The VOF method is specifically based on a simple order high resolution scheme lower than that of a comparable method, but still leading to a nearly equivalent order of accuracy. Retaining the mass conservation property, the hybrid algorithm couples the proposed VOF method with a level-set distancing algorithm in an implicit manner when the normal and the curvature of the interface need to be accurate for consideration of surface tension. For practical purposes, it is developed to be efficiently and easily extensible to three-dimensional applications with a minor implementation complexity. The accuracy and convergence properties of the method are verified through a wide range of tests: advection of rigid interfaces of different shapes, a three-dimensional air bubble's rising in viscous liquids, a two-dimensional dam-break, and a three-dimensional dam-break over an obstacle mounted on the bottom of a tank. The standard advection tests show that the volume advection algorithm is comparable in accuracy with geometric interface reconstruction algorithms of higher accuracy than other interface capturing-based methods found in the literature. The numerical results for the remainder of tests show a good agreement with other numerical solutions or available experimental data. Copyright © 2009 John Wiley & Sons, Ltd.

The Thurston norm via normal surfaces

Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by nonnegative integer weights, 14 for each 3-simplex, that describe how many copies of each oriented normal disc type there are. The Euler characteristic and homology class are both linear functions of the weights. There is a convex polytope in the space of weights, defined by linear equations given by the combinatorics of the triangulation, whose image under the homology map is the unit ball, ℬ, of the Thurston norm. Applications of this approach include (1) an algorithm to compute ℬ and hence the Thurston norm of any homology class, (2) an explicit exponential bound on the number of vertices of ℬ in terms of the number of simplices in the triangulation, (3) an algorithm to determine the fibred faces of ℬ and hence an algorithm to decide whether a 3-manifold fibres over the circle.

Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes (i.e guts) of the complement of an incompressible surface in M.

Closed incompressible surfaces of genus two in 3-bridge knot complements

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.

Meridional almost normal surfaces in knot complements

Suppose $K$ is a knot in a closed 3-manifold $M$ such that $\bar{M-N(K)}$ is irreducible. We show that for any positive integer $b$ there exists a triangulation of $\bar{M-N(K)}$ such that any weakly incompressible bridge surface for $K$ of $b$ bridges or fewer is isotopic to an almost normal bridge surface.

Formula omitted]-manifolds in 4-dimensional graph-manifolds

A standard fact about two incompressible surfaces in an irreducible 3-manifold is that one can move one of them by isotopy so that their intersection becomes π 1 -injective. By extending it to maps of some 3-dimensional Z n -manifolds into 4-manifolds, we prove that any homotopy equivalence of 4-dimensional graph-manifolds with reduced graph-structures is homotopic to a diffeomorphism preserving the structures.

KNOTS WITH INFINITELY MANY INCOMPRESSIBLE SEIFERT SURFACES

We show that a knot in S3 with an infinite number of incompressible Seifert surfaces contains a closed non-peripheral incompressible surface in its complement.

Behavior of the extended complexity of irreducible 3-manifolds

We construct the extended complexity of irreducible 3-manifolds; unlike the usual complexity [1] it is not an integer, but an ordered tuple of five integers. The benefit of extended complexity is that it always decreases when a manifold is cut along some incompressible boundary incompressible surface.

An implicit technique for solving 3D low Reynolds number moving free surface flows

This paper describes the development of an implicit finite difference method for solving transient three-dimensional incompressible free surface flows. To reduce the CPU time of explicit low-Reynolds number calculations, we have combined a projection method with an implicit technique for treating the pressure on the free surface. The projection method is employed to uncouple the velocity and the pressure fields, allowing each variable to be solved separately. We employ the normal stress condition on the free surface to derive an implicit technique for calculating the pressure at the free surface. Numerical results demonstrate that this modification is essential for the construction of methods that are more stable than those provided by discretizing the free surface explicitly. In addition, we show that the proposed method can be applied to viscoelastic fluids. Numerical results include the simulation of jet buckling and extrudate swell for Reynolds numbers in the range [0.01, 0.5].

MORSE POSITION OF KNOTS AND CLOSED INCOMPRESSIBLE SURFACES

In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have the following results. • Smallness of Montesinos tangles with length two and Kinoshita's theta curve. • Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions.

Generalized handlebody sets and non-Haken 3-manifolds

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance between the boundary sets of the handlebodies is zero if and only if the ambient manifold contains a non-separating, two sided incompressible surface. We show that the boundary set is 2-dense in the curve complex, i.e. every vertex is within two edges of a point in the boundary set.

Incompressible surfaces in handlebodies and isotopies

It is shown that every properly embedded incompressible surface in a handlebody can be constructed by a canonical gluing process. A simple condition is given which asserts that the result of the gluing process is an incompressible surface. A new notion of isotopy is introduced in order to distinguish surfaces belonging to distinct isotopy classes. Several examples (known and new) are constructed.

The motions of liquid films in a gas

The motions in a gas of thin films of a viscous incompressible liquid acted upon by capillary forces are considered. The surface tension depends on the impurity concentration of a surface-active material, soluble or insoluble in the liquid, and the liquid is non-volatile. The inertia of the liquid, viscous stresses, the Laplace pressure and the surface-tension gradients, impurity transfer and also the particular properties of super-thin films are taken into account. The motions of the films are described using the model of quasi-steady viscous flow. Systems of equations are obtained in the approximation of an ideal compressible medium and for small Mach numbers. The conditions for the incompressible film surface approximation to hold are obtained. The severe limitations of the gas-dynamic approximation in the case of a soluble impurity due to attenuation of the waves related to diffusion are investigated. A continuum model of the film as a compressible medium with a non-equilibrium pressure is constructed. The asymptotic form of the solutions of unsteady problems of impurity transfer in the limit of weak non-equilibrium is obtained. Integrals of the equations of motion of the films in steady one-dimensional problems are derived. Integral forms of the equations of momentum and its moment for an arbitrary contour of the film are presented, which hold for steady flows in a film and in quasi-statics. The boundary conditions for the solutions of the system of equations of motion of films are given.

Complete flow field computation around an ACV (air-cushion vehicle) using 3D VOF with Lagrangian propagation in computational domain

In this study an algorithm and a 3D solver is developed to solve the flow field around air-cushion vehicles (ACV) in vicinity of free surface. A single set of dimensionless equations is derived to handle both liquid and air phases in viscous 3D incompressible free surface flows in general curvilinear coordinates. The momentum equations are solved using the SIMPLE method in a staggered grid. The Lagrangian approach in the computational domain is also applied in the context of the VOF method to resolve the free surface. To demonstrate the robustness and versatility of the code, an application of this method on the impact problem of a circular cylinder is presented that is compared to experimental, theoretical and other computational results. In a 3D problem, the gravity-fill test is solved in both Cartesian and randomly generated curvilinear grid system. Comparison of results with the available experimental data and other numerical results is presented as well. At last, the code is applied to solve the two and three dimensional air and water flow field around an ACV and the effects of several parameters including the under skirt pressure distribution, initial air gap and effect of Fr. number on the wave elevation and wave making drag are investigated. It is shown that the presented method and the code are, robust and accurate enough to produce complicated 3D flow fields around ACV’s.

An accurate finite element scheme with moving meshes for computing 3D‐axisymmetric interface flows

AbstractAn accurate finite element scheme for computing 3D‐axisymmetric incompressible free surface and interface flows is proposed. It is based on the arbitrary Lagrangian Eulerian (ALE) approach using free surface/interface‐resolved moving meshes. Key features like the surface force, consisting of surface tension and the local curvature, and jumps in the density and viscosity over different fluid phases are precisely incorporated in the finite element formulation. The local curvature is approximated by using the Laplace–Beltrami operator technique combined with a boundary approximation by isoparametric finite elements. A new approach is used to derive the 3D‐axisymmetric form from the variational form in 3D‐Cartesian coordinates. Several test examples show the high accuracy and the robustness of the proposed scheme. Copyright © 2007 John Wiley & Sons, Ltd.