Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.     

A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations

A spectral element semi-Lagrangian (SESL) method for the shallow water equations on the sphere is presented. The sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals. The equations are written in Cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates. In a previous paper [Int. J. Numer. Methods Fluids 35 (2001) 869] we showed how to construct an explicit Eulerian spectral element (SE) model on the sphere; we now extend this work to a semi-Lagrangian formulation. The novelty of the Lagrangian formulation presented is that the high order SE basis functions are used as the interpolation functions for evaluating the values at the Lagrangian departure points. This makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers. The equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points. By incorporating the Coriolis terms into the Lagrangian derivative, the block LU decomposition of the equations results in a symmetric positive-definite pseudo-Helmholtz operator which we solve using the generalized minimum residual method (GMRES) with a fast projection method [Comput. Methods Appl. Mech. Eng. 163 (1998) 193]. Results for eight test cases are presented to confirm the accuracy and stability of the method. These results show that SESL yields the same accuracy as an Eulerian spectral element semi-implicit (SESI) while allowing for time-steps 10 times as large and being up to 70% more efficient.     

From the paddle to the beach – A Boussinesq shallow water numerical wave tank based on Madsen and Sørensen’s equations

This article describes a one-dimensional numerical model of a shallow-water flume with an in-built piston paddle moving boundary wavemaker. The model is based on a set of enhanced Boussinesq equations and the nonlinear shallow water equations. Wave breaking is described approximately, by locally switching to the nonlinear shallow water equations when a critical wave steepness is reached. The moving shoreline is calculated as part of the solution. The piston paddle wavemaker operates on a movable grid, which is Lagrangian on the paddle face and Eulerian away from the paddle. The governing equations are, however, evolved on a fixed mapped grid, and the newly calculated solution is transformed back onto the moving grid via a domain mapping technique. Validation test results are compared against analytical solutions, confirming correct discretisation of the governing equations, wave generation via the numerical paddle, and movement of the wet/dry front. Simulations are presented that reproduce laboratory experiments of wave runup on a plane beach and wave overtopping of a laboratory seawall, involving solitary waves and compact wave groups. In practice, the numerical model is suitable for simulating the propagation of weakly dispersive waves and can additionally model any associated inundation, overtopping or inland flooding within the same simulation.     

Diffusion of Finite-Mass Particles in a Turbulent Viscous Medium

Based on a Fokker–Planck equation for the Jacobian of mapping of Lagrangian into Eulerian coordinates, derived in this paper, we analyze the process of diffusion of passive-tracer particles in a turbulent viscous medium. Solving this equation allows for studying the effect of finite masses (inertia) of the tracer particles on the appearance of multi-flow motion.     

Arbitrary Lagrangian–Eulerian finite-element method for computation of two-phase flows with soluble surfactants

A finite-element scheme based on a coupled arbitrary Lagrangian–Eulerian and Lagrangian approach is developed for the computation of interface flows with soluble surfactants. The numerical scheme is designed to solve the time-dependent Navier–Stokes equations and an evolution equation for the surfactant concentration in the bulk phase, and simultaneously, an evolution equation for the surfactant concentration on the interface. Second-order isoparametric finite elements on moving meshes and second-order isoparametric surface finite elements are used to solve these equations. The interface-resolved moving meshes allow the accurate incorporation of surface forces, Marangoni forces and jumps in the material parameters. The lower-dimensional finite-element meshes for solving the surface evolution equation are part of the interface-resolved moving meshes. The numerical scheme is validated for problems with known analytical solutions. A number of computations to study the influence of the surfactants in 3D-axisymmetric rising bubbles have been performed. The proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant.     

非旋转性前进波Euler与Lagrange解间的转换

对等深水中非旋转性的前进重力波动场,以求得的Euler与Lagrange两种形式至第三阶的解,按照同一流体质点在相同时间与位置处其流速唯一与质量守恒性及在自由表面水位处Euler形式解与Lagrange形式解为同一值的特性,来推导二者可相互转换.由连续的Taylor级数展开,考虑波动场中各流体质点的运动轨迹与运动周期,将已知的Euler形式解转换成完全未知的Lagrange形式解,解决了以往成果中出现含时间的不合理的共振项,以及无法得到与Euler系统不同的Lagrange形式的流体质点运动频率与平均运动 关键词： 非旋转性前进波 Euler-Lagrange转换 质点运动轨迹 质点运动频率     

Symmetries of Euler Equations in Lagrangian Coordinates

Abstract

The transition from Eulerian to Lagrangian coordinates is a nonlocal transformation. In general, isomorphism should not take place between basic Lie groups of studied equations. Besides, in the case of plane and rotational symmetric motion hydrodynamic equations in Lagrangian coordinates are partially integrated. This fact introduces arbitrary functions, initial data, to the resulting systems and makes cuurently central the problem of group classification. It is stated that under a transition to Lagrangian coordinates, the main group becomes infinite–dimensional as well in space coordinates. The exclusive values of arbitrary functions of Lagrange coordinates (vorticity, momentum), at which the further group widening takes place, are found in [1].     

Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions

Constraints are found on the spatial variation of finite-time Lyapunov exponents of two- and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of separation, along characteristic directions, of neighboring trajectories. The solution of the equations is a coordinate transformation that takes initial conditions (the Lagrangian coordinates) to the state of the system at a later time (the Eulerian coordinates). This coordinate transformation naturally defines a metric tensor, from which the Lyapunov exponents and characteristic directions are obtained. By requiring that the Riemann curvature tensor vanish for the metric tensor (a basic result of differential geometry in a flat space), differential constraints relating the finite-time Lyapunov exponents to the characteristic directions are derived. These constraints are realized with exponential accuracy in time. A consequence of the relations is that the finite-time Lyapunov exponents are locally small in regions where the curvature of the stable manifold is large, which has implications for the efficiency of chaotic mixing in the advection-diffusion equation. The constraints also modify previous estimates of the asymptotic growth rates of quantities in the dynamo problem, such as the magnitude of the induced current. (c) 2001 American Institute of Physics.     

A grid based particle method for moving interface problems

We propose a novel algorithm for modeling interface motions. The interface is represented and is tracked using quasi-uniform meshless particles. These particles are sampled according to an underlying grid such that each particle is associated to a grid point which is in the neighborhood of the interface. The underlying grid provides an Eulerian reference and local sampling rate for particles on the interface. It also renders neighborhood information among the meshless particles for local reconstruction of the interface. The resulting algorithm, which is based on Lagrangian tracking using meshless particles with Eulerian reference grid, can naturally handle/control topological changes. Moreover, adaptive sampling of the interface can be achieved easily through local grid refinement with simple quad/oct-tree data structure. Extensive numerical examples are presented to demonstrate the capability of our new algorithm.     

An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows

A three-dimensional, incompressible, multiphase particle-in-cell method is presented for dense particle flows. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Difficulties associated with calculating interparticle interactions for dense particle flows with volume fractions above 5% have been eliminated by mapping particle properties to an Eulerian grid and then mapping back computed stress tensors to particle positions. A subgrid particle, normal stress model for discrete particles which is robust and eliminates the need for an implicit calculation of the particle normal stress on the grid is presented. Interpolation operators and their properties are defined which provide compact support, are conservative, and provide fast solution for a large particle population. The solution scheme allows for distributions of types, sizes, and density of particles, with no numerical diffusion from the Lagrangian particle calculations. Particles are implicitly coupled to the fluid phase, and the fluid momentum and pressure equations are implicitly solved, which gives a robust solution.     

A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry

We develop a new cell-centered control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics in cylindrical coordinates. The scheme is designed to be able to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. Unlike many previous area-weighted schemes that possess the spherical symmetry property, our scheme is discretized on the true volume and it can preserve the conservation property for all the conserved variables including density, momentum and total energy. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry, accuracy and non-oscillatory properties.     

A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction

We present a new cell-centered multi-material arbitrary Lagrangian–Eulerian (ALE) scheme to solve the compressible gas dynamics equations on two-dimensional unstructured grid. Our ALE method is of the explicit time-marching Lagrange plus remap type. Namely, it involves the following three phases: a Lagrangian phase wherein the flow is advanced using a cell-centered scheme; a rezone phase in which the nodes of the computational grid are moved to more optimal positions; a cell-centered remap phase which consists of interpolating conservatively the Lagrangian solution onto the rezoned grid. The multi-material modeling utilizes either concentration equations for miscible fluids or the Volume Of Fluid (VOF) capability with interface reconstruction for immiscible fluids. The main original feature of this ALE scheme lies in the introduction of a new mesh relaxation procedure which keeps the rezoned grid as close as possible to the Lagrangian one. In this formalism, the rezoned grid is defined as a convex combination between the Lagrangian grid and the grid resulting from condition number smoothing. This convex combination is constructed through the use of a scalar parameter which is a scalar function of the invariants of the Cauchy–Green tensor over the Lagrangian phase. Regarding the cell-centered remap phase, we employ two classical methods based on a partition of the rezoned cell in terms of its overlap with the Lagrangian cells. The first one is a simplified swept face-based method whereas the second one is a cell-intersection-based method. Our multi-material ALE methodology is assessed through several demanding two-dimensional tests. The corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme.     

Efficient schemes to compute diffusive barrier crossing rates

The formulation of the classical barrier-crossing problem is reviewed in the context of numerical simulations, with the focus on barrier crossing problems where the reaction coordinate depends in a non-trivial way on the Cartesian coordinates of many particles. Often it is convenient to measure the barrier height using constrained dynamics. Such a calculation requires a knowledge of the Jacobian for the coordinate transformation between Cartesian and generalized (‘reaction’) coordinates, and it is shown that the calculation of this Jacobian can be simplified. The conventional expression for the crossing rate is found to become computationally inefficient when the barrier crossing is diffusive. An alternative formulation of the barrier-crossing rate is given that leads to much better statistical accuracy in the computed crossing rates.     

Numerical simulations of instabilities in the implosion process of inertial confined fusion in 2D cylindrical coordinates

In this paper, we introduce a multi-material arbitrary Lagrangian and Eulerian method for the hydrodynamic radiative multi-group diffusion model in 2D cylindrical coordinates. The basic idea in the construction of the method is the following: In the Lagrangian step, a closure model of radiation-hydrodynamics is used to give the states of equations for materials in mixed cells. In the mesh rezoning step, we couple the rezoning principle with the Lagrangian interface tracking method and an Eulerian interface capturing scheme to compute interfaces sharply according to their deformation and to keep cells in good geometric quality. In the interface reconstruction step, a dual-material Moment-of-Fluid method is introduced to obtain the unique interface in mixed cells. In the remapping step, a conservative remapping algorithm of conserved quantities is presented. A number of numerical tests are carried out and the numerical results show that the new method can simulate instabilities in complex fluid field under large deformation,and are accurate and robust.     

An immersed boundary energy-based method for incompressible viscoelasticity

Incompressible viscoelastic materials are prevalent in biological applications. In this paper we present a method for incompressible viscoelasticity in which the elasticity of the material is described in Lagrangian form (i.e. in material coordinates), and Eulerian (spatial) coordinates are used for the equations of motion and to enforce the incompressibility condition. The elastic forces are computed directly from an energy functional without the use of stress tensors, and the immersed boundary method is used to communicate between Lagrangian and Eulerian variables. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. For this problem, we study convergence of the velocity field, the deformation map, and the Eulerian force density. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. Finally, the method is applied to a three-dimensional fluid–structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model.     

非旋转性自由表面重力驻波的Euler与Lagrange方法解间的转换

对于等深水中的非旋转性重力驻波流场,本文用Euler与Lagrange两种方法求得其至三阶的解,根据同一粒流体质点在相同时间与位置处其流速值为唯一与质量守恒及在自由表面水位的Euler形式解与Lagrange形式解相同等特性,来推导其间互可转换.由一系列连续的Taylor级数展开,在考虑波动场中各流体质点的运动轨迹与运动周期条件下,将已知的Euler解转换成完全未知的Lagrange形式解.接着再将所得的Lagrange解转换成对应的Euler形式,均可得到完全相同的结果.由此可得知,在考虑波动场各流体质 关键词： 重力驻波 Euler与Lagrange解间的转换 质点运动轨迹     

New Jacobian Elliptic Function Solutions of Modified KdV Equation: II

Recently, we obtained thirteen families of Jacobian elliptic function solutions of mKdV equation by using our extended Jacobian elliptic function expansion method. In this note, the mKdV equation is investigated and another three families of new doubly periodic solutions (Jacobian elliptic function solutions) are found again by using a new transformation, which and our extended Jacobian elliptic function expansion method form a new method still called the extended Jacobian elliptic function expansion method. The new method can be more powerful to be applied to other nonlinear differential equations.     

A Particle-Mesh Method for the Shallow Water Equations Near Geostrophic Balance

In this paper we outline a new particle-mesh method for rapidly rotating shallow water flows based on a set of regularized equations of motion. The time-stepping method uses an operator splitting of the equations into an Eulerian gravity wave part and a Lagrangian advection part. An essential ingredient is the advection of absolute vorticity by means of translated radial basis functions. We show that this implies exact conservation of enstrophy. The method is tested on two model problems based on the qualitative features of the solutions obtained (i.e., dispersion or smoothness of potential vorticity contours) as well as on the increase in mean divergence level.