Modified Level Set Method Research Articles

Numerical study of a liquid/liquid slug flow in a micro‐reactor

AbstractThe great advantages of micro–reactors are associated with an extremely high surface–to–volume ratio. Hence, micro–reactors permit promising operating conditions, such as almost perfect heat or mass transfer. The hydrodynamics of a liquid/liquid slug flow in a micro–channel is characterized by complex vortex structures in both the disperse and the continuous phase. The disperse phase, in our investigations, is not wetting the walls and, thus, a thin film of the continuous phase persists between the disperse phase and the wall. Due to this phenomenon, a relative movement between disperse and continuous phase is possible and, indeed, observed. Understanding of these complex phenomena allows for a control of the hydrodynamics, and thus, to tailor the heat and mass transport in a desired manner. To study the physics of this complex liquid/liquid system, a modified level–set method in conjunction with an immersed–boundary formulation is engaged. The mesh resolution represents a challenge, as the spatial resolution has to resolve the thin film between the disperse phase and the wall adequately. All simulations are implemented within the software OpenFOAM. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A FAST NUMERICAL METHOD FOR SOLVING A REGULARIZED PROBLEM ASSOCIATED WITH OBSTACLE PROBLEMS

Kirsi Majava and Xue-Cheng Tai [12] proposed a modified level set method for solving a free boundary problem associated with unilateral obstacle problems. The proximal bundle method and gradient method were applied to solve the nonsmooth minimization problems and the regularized problem, respectively. In this paper, we extend this approach to solve the bilateral obstacle problems and employ Rung-Kutta method to solve the initial value problem derived from the regularized problem. Numerical experiments are presented to verify the efficiency of the methods.

Development of level set method with good area preservation to predict interface in two‐phase flows

AbstractA two‐step conservative level set method is proposed in this study to simulate the gas/water two‐phase flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the coupled compact scheme. For accurately predicting the modified level set function, the dispersion‐relation‐preserving advection scheme is developed to preserve the theoretical dispersion relation for the first‐order derivative terms shown in the pure advection equation cast in conservative form. For the purpose of retaining its long‐time accurate Casimir functionals and Hamiltonian in the transport equation for the level set function, the time derivative term is discretized by the sixth‐order accurate symplectic Runge–Kutta scheme. To resolve contact discontinuity oscillations near interface, nonlinear compression flux term and artificial damping term are properly added to the second‐step equation of the modified level set method. For the verification of the proposed dispersion‐relation‐preserving scheme applied in non‐staggered grids for solving the incompressible flow equations, three benchmark problems have been chosen in this study. The conservative level set method with area‐preserving property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam‐break, Rayleigh–Taylor instability, bubble rising in water, and droplet falling in water problems. Good agreements with the referenced solutions are demonstrated in all the investigated problems. Copyright © 2010 John Wiley & Sons, Ltd.

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.

Marangoni effects caused by contaminants adsorbed on bubble surfaces

The influence of Marangoni stresses, caused by contaminants adsorbed on the surface of small air bubbles, rising in water, is examined by numerical simulations. A modified level set method is used to represent the deformable bubble interface, extended by a model for the contaminant transport on the bubble surface. We show that surface tension variations of less than 2% are sufficient to generate Marangoni stresses that are strong enough to change the rising characteristics of a bubble to that of a corresponding solid particle. In such situations, we find that the bubble surface is fully covered with contaminant and the shear stress profile resembles the shear stress profile around a solid sphere.

Modified level set method with Canny operator for image noise removal

The level set method is commonly used to address image noise removal. Existing studies concentrate mainly on determining the speed function of the evolution equation. Based on the idea of a Canny operator, this letter introduces a new method of controlling the level set evolution, in which the edge strength is taken into account in choosing curvature flows for the speed function and the normal to edge direction is used to orient the diffusion of the moving interface. The addition of an energy term to penalize the irregularity allows for better preservation of local edge information. In contrast with previous Canny-based level set methods that usually adopt a two-stage framework, the proposed algorithm can execute all the above operations in one process during noise removal.

Development of a dispersively accurate conservative level set scheme for capturing interface in two-phase flows

A two-step interface capturing scheme, implemented within the framework of conservative level set method, is developed in this study to simulate the gas/water two-phase fluid flow. In addition to solving the pure advection equation, which is used to advect the level set function for tracking interface, both nonlinear and stabilized features are taken into account for the level set function so that a sharply varying interface can be stably predicted. To preserve the conservative property, the mapping given by ξ ̲ = x ̲ - u ̲ t is performed between two coordinates x ̲ and ξ ̲ for the advection–diffusion equation in a flow field with velocity u . To capture the interface, the flux term capable of compressing the level set contours is also adopted in the construction of linear inviscid Burgers’ equation, which is indispensable in the level set method. To resolve the physically sharp interface without incurring contact discontinuity oscillations, a damping term which is nonlinear in terms of the level set function and can render an adequate artificial diffusion to stabilize the contact surface is added into the reinitialization step in the modified level set method. For accurately predicting the level set function, the advection scheme for solving the linear inviscid Burgers’ equation in the advection step of the modified level set method is developed to accommodate the true dispersion relation. The solution computed from the resulting two-dimensional dispersion-relation-preserving advection scheme can minimize the phase error. Less artificial damping is needed to damp the oscillations in the vicinity of contact surface and the interface can be less numerically smeared. For the sake of programming simplicity, the incompressible two-phase flow will be discretized in non-staggered grids without incurring checkerboard oscillations by the developed explicit compact scheme for the approximation of pressure gradient terms. For the verification of the proposed two-dimensional dispersion-relation-preserving scheme and the non-staggered incompressible flow solver, three benchmark problems have been chosen in this study. The proposed conservative level set method for capturing the interface in incompressible fluid flows is also verified by solving the dam-break, bubble rising in water, droplet falling in water and Rayleigh–Taylor instability problems. Good agreements with the referenced solutions are demonstrated for all the four investigated problems.

Modeling the interaction of biological cells with a solidifying interface

In this article, we develop a modified level set method for modeling the interaction of particles with a solidifying interface. The dynamic computation of the van der Waals and drag forces between the particles and the solidification front leads to a problem of multiple length scales, which we resolve using adaptive grid techniques. We present a variety of example problems to demonstrate the accuracy and utility of the method. We also use the model to interpret experimental results obtained using directional solidification in a cryomicroscope.

An Adaptive Remeshing Strategy for Free-Surface Fluid Flow Problems. Part I: The Axisymmetric Case

In this work, an anisotropic adaptive remeshing strategy is used for the numerical simulation of free surface fluid flow problems with surface tension. Carreau and power law models are used for shear-thinning viscosity effects. A modified level set method is also presented for the computation of free surfaces. Numerical examples are then presented: the Young-Laplace relation, the deformation of a drop in an axisymmetric die and the deformation and breaking of drops in elongation and shear flows. Part 1 will focus on axisymmetric geometries while the second part will be concerned with the full three-dimensional case. The main objective of this work is to show that this adaptive methodology can be successfully applied to various types of free surface problems.

A Numerical Method for Bubble Motion with Phase Change

A numerical method for computing the motion of bubbles undergoing liquid?vapor phase change is presented. The method is based on a level set technique for capturing the phase interface, which is modified to include the effect of phase change at the interface as well as to achieve mass conservation during the whole calculation procedure. The modified level set method is applied for numerical simulation of bubble rise and growth in a stationary liquid. The numerical results are found to compare well with the data reported in the literature and the analytical solutions.