A Volume of Fluid Based Method for Fluid Flows with Phase Change

This paper presents a numerical method directed towards the simulation of flows with mass transfer due to changes of phase. We use a volume of fluid (VOF) based interface tracking method in conjunction with a mass transfer model and a model for surface tension. The bulk fluids are viscous, conducting, and incompressible. A one-dimensional test problem is developed with the feature that a thin thermal layer propagates with the moving phase interface. This test problem isolates the ability of a method to accurately calculate the thermal layers responsible for driving the mass transfer in boiling flows. The numerical method is tested on this problem and then is used in simulations of horizontal film boiling.     

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 front-tracking/ghost-fluid method for fluid interfaces in compressible flows

A front-tracking/ghost-fluid method is introduced for simulations of fluid interfaces in compressible flows. The new method captures fluid interfaces using explicit front-tracking and defines interface conditions with the ghost-fluid method. Several examples of multiphase flow simulations, including a shock–bubble interaction, the Richtmyer–Meshkov instability, the Rayleigh–Taylor instability, the collapse of an air bubble in water and the breakup of a water drop in air, using the Euler or the Navier–Stokes equations, are performed in order to demonstrate the accuracy and capability of the new method. The computational results are compared with experiments and earlier computational studies. The results show that the new method can simulate interface dynamics accurately, including the effect of surface tension. Results for compressible gas–water systems show that the new method can be used for simulations of fluid interface with large density differences.     

Modeling merging and breakup in the moving mesh interface tracking method for multiphase flow simulations

The three-dimensional, moving mesh interface tracking (MMIT) method coupled with local mesh adaptations by Quan and Schmidt [S.P. Quan, D.P. Schmidt, A moving mesh interface tracking method for 3D incompressible two-phase flows, J. Comput. Phys. 221 (2007) 761–780] demonstrated the capability to accurately simulate multiphase flows, to handle large deformation, and also to perform interface pinch-off for some specific cases. However, another challenge, i.e. how to handle interface merging (such as droplet coalescence) has not been addressed. In this paper, we present a mesh combination scheme for interface connection and a more general mesh separation algorithm for interface breakup. These two schemes are based on the conversion of liquid cells in one phase to another fluid by changing the fluid properties of the cells in the combination or separation region. After the conversion, the newly created interface is usually ragged, and a local projection method is employed to smooth the interface. Extra mesh adaptation criteria are introduced to handle colliding interfaces with almost zero curvatures as the distance between the interfaces diminishes. Simulations of droplet pair collisions including both head-on and off-center coalescences show that the mesh adaptations are capable of resolving very small length scales, and the mesh combination and mesh separation schemes can handle the topological transitions in multiphase flows. The potential of our method to perform detailed investigations of droplet coalescence and breakup is also displayed.     

A High-Order Discontinuous Galerkin Method for 2D Incompressible Flows

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.     

Multigrid Solution of the Incompressible Navier–Stokes Equations for Density-Stratified Flow past Three-Dimensional Obstacles

The steady incompressible Navier–Stokes equations in three dimensions are solved for neutral and stably stratified flow past three-dimensional obstacles of increasing spanwise width. The continuous equations are approximated using a finite volume discretisation on staggered grids with a flux-limited monotonic scheme for the advective terms. The discrete equations which arise are solved using a nonlinear multigrid algorithm with up to four grid levels using the SIMPLE pressure correction method as smoother. When at its most effective the multigrid algorithm is demonstrated to yield convergence rates which are independent of the grid density. However, it is found that the asymptotic convergence rate depends on the choice of the limiter used for the advective terms of the density equation, and some commonly used schemes are investigated. The variation with obstacle width of the influence of the stratification on the flow field is described and the results of the three-dimensional computations are compared with those of the corresponding computation of flow over a two-dimensional obstacle (of effectively infinite width). Also given are the results of time-dependent computations for three-dimensional flows under conditions of strong static stability when lee-wave propagation is present and the multigrid algorithm is used to compute the flow at each time step.     

A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows

We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. This equation is necessary to close the overall system. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. We propose robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. It allows the simulation of interface problems between pure fluids as well as multiphase mixtures. Several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible. The method provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.     

使用界面追踪对膜态沸腾的数值研究

提出一种计算带相变的自由界面的数值算法.基于分段线性界面重构(PLIC)的VOF(volume-of-fluid)方法用于追踪自由界面,并对汽液交界面上的相变导致的不连续速度场给出处理方法.此方法容易实施且被证明是有效的.流场的求解使用SIMPLE方法,表面张力使用连续表面力模型(CSF)进行计算.在三维直角坐标系下,模拟了水平壁面上的膜态沸腾,在二维适体坐标系下,模拟了竖直圆头柱体表面的自然对流膜态沸腾.计算结果与理论关系式符合较好.     

The Local Front Reconstruction Method for direct simulation of two- and three-dimensional multiphase flows

We present a new interface reconstruction technique, the Local Front Reconstruction Method (LFRM), for incompressible multiphase flows. This new method falls in the category of Front Tracking methods but it shares automatic topology handling characteristics of the previously proposed Level Contour Reconstruction Method (LCRM). The LFRM tracks the phase interface explicitly as in Front Tracking but there is no logical connectivity between interface elements thus greatly easing the algorithmic complexity. Topological changes such as interfacial merging or pinch off are dealt with automatically and naturally as in the Level Contour Reconstruction Method. Here the method is described for both two- and three-dimensional flow geometries. The interfacial reconstruction technique in the LFRM differs from that in the LCRM formulation by foregoing using an Eulerian distance field function. Instead, the LFRM uses information from the original interface elements directly to generate the new interface in a mass conservative way thus showing significantly improved local mass conservation. Because the reconstruction procedure is independently carried out in each individual reconstruction cell after an initial localization process, an adaptive reconstruction procedure can be easily implemented to increase the accuracy while at the same time significantly decreasing the computational time required to perform the reconstruction. Several benchmarking tests are performed to validate the improved accuracy and computational efficiency as compared to the LCRM. The results demonstrate superior performance of the LFRM in maintaining detailed interfacial shapes and good local mass conservation especially when using low-resolution Eulerian grids.     

3D Analysis of Crystal/Melt Interface Shape and Marangoni Flow Instability in Solidifying Liquid Bridges

Solidification of gallium (Pr=0.02) in liquid bridges in zero-gravity conditions is investigated by numerical solutions of the three-dimensional and time-dependent flow-field equations. A single region (continuum) formulation based on the enthalpy method is adopted to model the phase-change problem. This paper analyzes the influence of the azimuthally asymmetric and steady first bifurcation of the Marangoni flow on the shape of the solid/melt interface during the crystal growth process. The numerical results show that this interface is distorted in the azimuthal direction. The distortion is related to the sinusoidal three-dimensional temperature disturbances due to the instability of the Marangoni flow. The three-dimensional flow field organization, related to the wave number, changes during the solidification process; this behavior is explained according to the variation of the aspect ratio of the solidifying liquid bridge. A correlation law is found for the azimuthal wave number of the instability as function of the melt zone aspect ratio.     

Phase-Accuracy Comparisons and Improved Far-Field Estimates for 3-D Edge Elements on Tetrahedral Meshes

Edge-element methods have proved very effective for 3-D electromagnetic computations and are widely used on unstructured meshes. However, the accuracy of standard edge elements can be criticised because of their low order. This paper analyses discrete dispersion relations together with numerical propagation accuracy to determine the effect of tetrahedral shape on the phase accuracy of standard 3-D edge-element approximations in comparison to other methods. Scattering computations for the sphere obtained with edge elements are compared with results obtained with vertex elements, and a new formulation of the far-field integral approximations for use with edge elements is shown to give improved cross sections over conventional formulations.     

The extended finite element method for two-phase and free-surface flows: A systematic study

In immiscible two-phase flows, jumps or kinks are present in the velocity and pressure fields across the interfaces of the two fluids. The extended finite element method (XFEM) is able to reproduce such discontinuities within elements. Robust and accurate interface capturing schemes with no restrictions on the interface topology are thereby enabled. This paper investigates different enrichment schemes and time-integration schemes within the XFEM. Test cases with and without surface tension on moving or stationary meshes are studied and compared to interface tracking results when possible. A particularly useful setting is extracted which is recommended for two-phase flows. An extension of this formulation for the simulation of free-surface flows and of floating objects is proposed.     

Divergence- and Curl-Preserving Prolongation and Restriction Formulas

We present new second-order prolongation and restriction formulas which preserve the divergence and, in some cases, the curl of a discretized vector field. The formulas are suitable for adaptive and hierarchical mesh algorithms with a factor-of-2 linear resolution change. We examine both staggered and collocated discretizations for the vector field on two- and three-dimensional Cartesian grids. The new formulas can be used in combination with numerical schemes that require a divergence-free solution in some discrete sense, such as the constrained transport schemes of computational magnetohydrodynamics. We also obtain divergence-preserving interpolation functions which may be used for streamline or field line tracing.     

An interaction potential based lattice Boltzmann method with adaptive mesh refinement (AMR) for two-phase flow simulation

The lattice Boltzmann method (LBM) for two-phase flow simulation is often hindered by insufficient resolution at the interface. As a result, the LBM simulation of bubbles in bubbling flows is commonly limited to spherical or slightly deformed bubble shapes. In this study, the adaptive mesh refinement method for the LBM is developed to overcome such a problem. The approach for this new method is based on the improved interaction potential model, which is able to maintain grid-independent fluid properties in the two-fluid phases and at the interface. The LBM–AMR algorithm is described, especially concerning the LBM operation on a non-uniform mesh and the improved interaction potential model. Numerical simulations have been performed to validate the method in both single phase and multiphase flows. The 2D and 3D simulations of the buoyant rise of bubbles are conducted under various conditions. The agreement between the simulated bubble shape and velocity with experiments illustrates the capability of the LBM–AMR approach in predicting bubble dynamics even under the large bubble deformation conditions. Further, the LBM–AMR technique is capable of simulating a complex topology change of the interface. Integration of LBM with AMR can significantly improve the accuracy and reduce computation cost. The method developed in this study may appreciably enhance the capability of LBM in the simulation of complex multiphase flows under realistic conditions.     

A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation

In this paper, we propose a new surface-tension formulation for multi-phase smoothed particle hydrodynamics (SPH). To obtain a stable and accurate scheme for surface curvature, a new reproducing divergence approximation without the need for a matrix inversion is derived. Furthermore, we introduce a density-weighted color-gradient formulation to reflect the reality of an asymmetrically distributed surface-tension force. We validate our method with analytic solutions and demonstrate convergence for different cases. Furthermore, we show that our formulation can handle phase interfaces with density and viscosity ratios of up to 1000 and 100, respectively. Finally, complex three-dimensional simulations including breakup of an interface demonstrate the capabilities of our method.     

Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.     

Unstructured Adaptive Grid Flow Simulations of Inert and Reactive Gas Mixtures

Unstructured adaptive grid flow simulation is applied to the calculation of high-speed compressible flows of inert and reactive gas mixtures. In the present case, the flowfield is simulated using the 2-D Euler equations, which are discretized in a cell-centered finite volume procedure on unstructured triangular meshes. Interface fluxes are calculated by a Liou flux vector splitting scheme which has been adapted to an unstructured grid context by the authors. Physicochemical properties are functions of the local mixture composition, temperature, and pressure, which are computed using the CHEMKIN-II subroutines. Computational results are presented for the case of premixed hydrogen–air supersonic flow over a 2-D wedge. In such a configuration, combustion may be triggered behind the oblique shock wave and transition to an oblique detonation wave is eventually obtained. It is shown that the solution adaptive procedure implemented is able to correctly define the important wave fronts. A parametric analysis of the influence of the adaptation parameters on the computed solution is performed.     

Tagging of the Magnetization with the Transition Zones of 360° Rotations Generated by a Tandem of Two Adiabatic DANTE Inversion Sequences

A new technique is presented for generating myocardial tagging using the signal intensity minima of the transition zones between the bands of 0° and 360° rotations, induced by a tandem of two adiabatic delays alternating with nutations for tailored excitation (DANTE) inversion sequences. With this approach, the underlying matrix corresponds to magnetization that has experienced 0° or 360° rotations. The DANTE sequences were implemented from adiabatic parent pulses for insensitivity of the underlying matrix to B₁ inhomogeneity. The performance of the proposed tagging technique is demonstrated theoretically with computer simulations and experimentally on phantom and on the canine heart, using a surface coil for both RF transmission and signal reception. The simulations and the experimental data demonstrated uniform grid contrast and sharp tagging profiles over a twofold variation of the B₁ field magnitude.     

Cogwheel phase cycling

A new method for constructing phase cycles is described. The new schemes apply to experiments involving several consecutive coherence transfer steps. The radiofrequency phases of two or more irradiation blocks are incremented simultaneously, as opposed to the traditional "nested" scheme, in which the block phases are incremented independently. In many cases, the "cogwheel" phase cycles achieve the same selectivity as traditional phase cycles, using fewer steps. Significant time savings are achievable in a wide range of NMR experiments.     

Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method

In many realistic fluid-dynamical simulations the specification of the boundary conditions, the error sources, and the number of time steps to reach a steady state are important practical considerations. In this paper we study these issues in the case of the lattice-BGK model. The objective is to present a comprehensive overview of some pitfalls and shortcomings of the lattice-BGK method and to introduce some new ideas useful in practical simulations. We begin with an evaluation of the widely used bounce-back boundary condition in staircase geometries by simulating flow in an inclined tube. It is shown that the bounce-back scheme is first-order accurate in space when the location of the non-slip wall is assumed to be at the boundary nodes. Moreover, for a specific inclination angle of 45 degrees, the scheme is found to be second-order accurate when the location of the non-slip velocity is fitted halfway between the last fluid nodes and the first solid nodes. The error as a function of the relaxation parameter is in that case qualitatively similar to that of flat walls. Next, a comparison of simulations of fluid flow by means of pressure boundaries and by means of body force is presented. A good agreement between these two boundary conditions has been found in the creeping-flow regime. For higher Reynolds numbers differences have been found that are probably caused by problems associated with the pressure boundaries. Furthermore, two widely used 3D models, namelyD₃Q₁₅andD₃Q₁₉, are analysed. It is shown that theD₃Q₁₅model may induce artificial checkerboard invariants due to the connectivity of the lattice. Finally, a new iterative method, which significantly reduces the saturation time, is presented and validated on different benchmark problems.