Gradient-augmented Level Set Method Research Articles

Parametric studies on the water impact of one and twin free-falling wedges by a Cartesian grid multiphase flow model

The water impact of one and twin free-falling wedges is numerically investigated by a Cartesian grid multiphase flow model. The effects of the drop velocity and the gap distance on the hydrodynamic behaviors are parametrically investigated. The numerical model involves a radial basis function ghost cell method (RBFGCM) for treating moving bodies and a gradient-augmented level set method (GALS) method for capturing violent free surfaces. A case of twin wedges entering water is simulated to validate the accuracy of the present method. Good convergences are achieved. Then, the water entry of one and twin wedges in free falling is considered. The interaction mechanisms between twin wedges are discussed by comprehensively examining the variation patterns of the slamming load, the moment, the local pressure, and the fluid field. It is found that the second slamming load and the huge pressure pulse occur at the transition stage at narrow gap distances. The hydrodynamic interaction has more significant effects on the local pressure than that on the global load. In addition, distinct hydrodynamic phenomena for twin wedges entering water are observed such as the connection of the pressure contours, the extremely large jet flow, formation of the cavity, and even the ventilation.

Gradient augmented level set method for phase change simulations

A numerical method for the simulation of two-phase flow with phase change based on the Gradient-Augmented-Level-set (GALS) strategy is presented. Sharp capturing of the vaporization process is enabled by: i) identification of the vapor–liquid interface, Γ(t), at the subgrid level, ii) discontinuous treatment of thermal physical properties (except for μ), and iii) enforcement of mass, momentum, and energy jump conditions, where the gradients of the dependent variables are obtained at Γ(t) and are consistent with their analytical expression, i.e. no local averaging is applied. Treatment of the jump in velocity and pressure at Γ(t) is achieved using the Ghost Fluid Method. The solution of the energy equation employs the sub-grid knowledge of Γ(t) to discretize the temperature Laplacian using second-order one-sided differences, i.e. the numerical stencil completely resides within each respective phase. To carefully evaluate the benefits or disadvantages of the GALS approach, the standard level set method is implemented and compared against the GALS predictions. The results show the expected trend that interface identification and transport are predicted noticeably better with GALS over the standard level set. This benefit carries over to the prediction of the Laplacian and temperature gradients in the neighborhood of the interface, which are directly linked to the calculation of the vaporization rate. However, when combining the calculation of interface transport and reinitialization with two-phase momentum and energy, the benefits of GALS are to some extent neutralized, and the causes for this behavior are identified and analyzed. Overall the additional computational costs associated with GALS are almost the same as those using the standard level set technique.

The distortion of the level set gradient under advection

The practice of periodically reinitializing the level set function is well established in two-phase flow applications as a way of controlling the growth of anomalies and/or numerical errors. In the present work, the underlying roots of this anomalous growth are studied, where it is established that the augmentation of the magnitude of the level set gradient (|∇ϕ|) is directly connected to the nature of the flow field; hence, it is not necessarily the result of some type of numerical error. More specifically, for a general flow field advecting the level set function, it is shown that the eigenpairs of the strain rate tensor are responsible for the rate of change of |∇ϕ| along a fluid particle trajectory. This straining action not only affects the magnitude of |∇ϕ|, but the general character of ϕ, and consequently contributes to the growth in numerical error. These numerical consequences are examined by adopting the Gradient Augmented Level Set method. Specifically, it is shown that the local error for ϕ is directly connected to the size of |∇ϕ| and to the magnitude of the second and fourth order derivatives of ϕ. These analytical findings are subsequently supported by various examples. The role of reinitialization is discussed, where it is shown that in cases where the zero level set contour has a local radius of curvature that is below the local grid resolution, reinitialization exacerbates rather than diminishes the degree of error. For other cases, where the interface is well resolved, reinitialization helps stabilize the error as intended.

Level set jet schemes for stiff advection equations: The semijet method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided.

Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quadtree data structures. For adaptive time integration, an embedded Runge-Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compression with respect to the uniform grid discretization are reported.

A narrow-band gradient-augmented level set method for multiphase incompressible flow

A finite-difference based numerical method for the simulation of multiphase incompressible flows is presented. The proposed algorithm adapts the gradient-augmented level set method of Nave et al. [25] for use in multiphase fluid simulations by interpolating the fluid velocity and introducing a simple reinitialization procedure. The method is carried out only along a narrow band surrounding the interface, reducing computational effort while preserving the advantages of the gradient-augmented method. Numerical results show excellent agreement with an analytical solution and available experimental data. A new experimental benchmark is introduced, using data gathered in a wedge-driven wave tank. In addition to the optimally local advection scheme and sub-grid resolution afforded by the new method, numerical comparisons with the standard level set method reveal superior volume conservation.

A gradient augmented level set method for unstructured grids

We present and test a gradient augmented level set method for unstructured grids, which has been designed to allow for easy integration in parallel flow solvers. The means to this end is a Hermite formulation, where the gradient is stored and advected as an independent variable, along with the level set function. The method uses narrow-band storage in conjunction with a Fast Marching Method, which is also designed to take advantage of the gradient augmentation. We demonstrate through standard passive advection test cases that the proposed method is able to compute interface motion with performance comparable to popular high-performing methods on structured grids.

A Semi-implicit Gradient Augmented Level Set Method

Here a semi-implicit formulation of the gradient augmented level set method is presented. By tracking both the level set and it's gradient accurate subgrid information is provided,leading to highly accurate descriptions of a moving interface. The result is a hybrid Lagrangian-Eulerian method that may be easily applied in two or three dimensions. The new approach allows for the investigation of interfaces evolving by mean curvature and by the intrinsic Laplacian of the curvature. In this work the algorithm, convergence and accuracy results are presented. Several numerical experiments in both two and three dimensions demonstrate the stability of the scheme.

A level set projection model of lipid vesicles in general flows

A new numerical method to model the dynamic behavior of lipid vesicles under general flows is presented. A gradient-augmented level set method is used to model the membrane motion. To enforce the volume- and surface-incompressibility constraints a four-step projection method is developed to integrate the full Navier–Stokes equations. This scheme is implemented on an adaptive non-graded Cartesian grid. Convergence results are presented, along with sample two-dimensional results of vesicles under various flow conditions.

A gradient-augmented level set method with an optimally local, coherent advection scheme

The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.