Moment Of Fluid Research Articles

Resolving subgrid-scale structures for multiphase flows using a filament moment-of-fluid method

Multiphase flows are present in many industrial and engineering applications as well as in some physical phenomena. Capturing the interface between the phases for complex flows is challenging and requires an accurate method, especially to resolve fine-scale structures. The moment-of-fluid (MOF) method improves drastically the accuracy of interface reconstruction compared to previous geometrical methods. Instead of refining the mesh to capture increased levels of detail, the MOF method, which uses zeroth and first moments as well as a conglomeration algorithm, enables subgrid structures such as filaments to be captured at a small extra cost. Coupled to a finite volume Navier–Stokes solver, the MOF method has been tested on a fixed grid and validated using well-known benchmark problems such as the dam break flows, the Rayleigh–Taylor and Kelvin–Helmholtz instability problems, and a rising bubble. The ability of the novel filament MOF method to capture the filamentary structures that eventually form for the Rayleigh–Taylor instability and rising bubble problems is assessed. Good agreement has been found with other numerical results and experimental measurements available in the literature.

A multi-directional advection based moment of fluid method for phase change problems

A multi-directional face-flux based Moment of Fluid (MoF) advection method is developed to simulate two-phase flows with and without phase change. The material volume and centroid are advected in a face-flux based approach which guarantees the conservation of the advected quantities. The face-flux based advection avoids material overlap and void formation between the advected material volumes. The centroid advection of a material volume is computed by advecting the individual centroids of smaller subvolumes. Hence, the proposed procedure reduces the centroid advection error which is proportional to the size of the material volume. Also, the multi-directional advection enables two-phase simulations over unstructured meshes. A two-phase flow solver with the proposed method is used successfully to study film boiling and condensation phenomena.

A quadtree-based adaptive moment-of-fluid method for interface reconstruction with filaments

Implementation of quadtree adaptive mesh refinement (AMR) to the moment-of-fluid (MOF) method is presented in the context of an interface capturing method. Filaments, thinner than a cell size, are resolved using a computationally efficient technique on an unconstrained quadtree structure. The centroid defect relative to its cell size is used as the refinement criterion, together with an enhanced refinement calculation and subsequently its volume conservation. In addition, different approaches are proposed to ensure mass conservation during the computation. This MOF-AMR framework is validated for a range of benchmark problems which are studied widely in the literature. There is no restriction on the choice of CFL number for the purely Lagrangian advection method considered here and this has advantages when combined with AMR. The current quadtree MOF-AMR method leads to much improved computational efficiency and accuracy relative to its grid size compared with a uniform grid. Higher levels of refinement can be costly, therefore the efficiency of mesh resolution is further discussed in relation to the choice of time step and number of AMR levels.

A moment-of-fluid method for resolving filamentary structures using a symmetric multi-material approach

Multiphase flows have implications in many areas of engineering. The moment-of-fluid (MOF) method is an interface capturing method using both volume fraction and centroid within a cell for interface reconstruction. A symmetric approach to reconstruct thin structures is presented. Also called filaments, these subcell characteristics involve multi-material reconstruction. In addition, a new optimisation algorithm is presented using a bisection method without any necessary initial condition. Using a Lagrangian approach for dynamic cases, no restrictions are imposed on timestep. The new method is validated using several benchmark cases that have been studied extensively in the literature. A near quadratic order of convergence and high accuracy is achieved while maintaining an acceptable runtime.

Comprehensive analysis of magnetized second-grade nanofluid via Fourier's and Cataneo-Christove models past a curved surface

This study presents the computer simulations of the magnetized effects of Cattaneo-Christov double diffusion models on the heat and mass transport behavior of a second-grade nanofluid flowing in two dimensions across a curved stretching and shrinking surface. Moreover, the features of convective heat transfer, slip velocity, viscous dissipation, porosity, and heat source/sink with a chemical reaction are also used. Considering the concentration and thermal relaxation times, the classical models of heat and mass diffusion (Fourier and Fick's laws) are extended to the generalized form in regard to the double diffusion Cattaneo-Christov models. Evaluating the current physical study including the Lorentz forces allows us to better understand how the magnetic field affects the process of double diffusion coupled with Joule heating. The system of highly nonlinear, two-dimensional, coupled partial differential equations caused by the moment of second-grade fluid across a curved sheet is solved using the bvp4c scheme. The key findings proves that the concentration and temperature of classical Fourier and Fick's law causes to rise faster than that of Cattaneo-Christov heat and mass flux model respectively as well as the concentration and temperature of Newtonian fluid causes to rise faster than that of second grade nanofluid. The generalized Fourier and Fick's law respectively generate greater heat and mass transport than the classical Fourier and Fick's law as well as the second-grade nanofluid generate greater heat and mass transport than the Newtonian fluid.

Compatible and energy conserving multi-material arbitrary Lagrangian Eulerian scheme for multi-group radiation hydrodynamics simulations

A compatible, bounds preserving and energy conserving multi-material arbitrary Lagrangian Eulerian (ALE) scheme suitable for an unstructured mesh is developed for multi-group radiation hydrodynamics (RHD) simulation of various high energy density physics (HEDP) phenomena involving large material deformations, such as intense thermal radiation or high-power laser driven systems. In order to study very short time scale HEDP phenomena, the electron and ion energy equations (with the effect of electron-ion energy relaxation) are solved separately with an assumption of single macroscopic fluid velocity for each species (two-temperature hydrodynamics model). The governing multi-material hydrodynamics equations are solved by using a higher-order cell-centered hydrodynamics (CCH) scheme. In multi-material flows, the remapping step introduces mixed computational cells in which multiple materials exist in a single cell. Therefore, material interfaces in mixed cells are required to be reconstructed and tracked. In our implementation, this is achieved by using moment-of-fluid (MOF) method. The information of these interfaces is further used to formulate an interface aware sub-scale dynamics (IASSD) scheme for multi-material closure in mixed computational cells. The IASSD scheme is generalized for the two-temperature hydrodynamics model in which separate equations are used to update the electron and ion energies. The multi-group radiation transport on an unstructured mesh is computed using a cell-centered, monotonic, nonlinear finite volume (NLFV) scheme. The flux-limited diffusion (FLD) approximation is used to recover the free-streaming limit of the radiation propagation in optically thin regions. In the ALE step, the electron, ion and radiation energies due to MG-RHD model are conservatively remapped using a center-of-mass (CM) reconstruction of the fields and a compatible and energy conserving remapping scheme. An exact integration based on the intersection between the new (rezoned) and old meshes is used in the remap step. Similar procedure is followed for the remap of cell mass and cell-centered momentum components. In this paper, the details of a compatible, bounds preserving and conservative ALE scheme developed for MG-RHD are described. The remapping of relevant physical quantities due to multi-physics models implemented in the code such as non-local electron energy transport are also discussed. Several validation test problems including laser driven shock propagation in multi-material geometries are provided to demonstrate the accuracy and performance of the ALE scheme implemented.

A comparative study of split advection algorithms on the moment-of-fluid (MOF) method for incompressible flow

The moment-of-fluid method (MOF) is known as an extension of the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). In this study, several directional splitting advection algorithms are extended from the VOF-PLIC method to the MOF method. These methods, along with some existing directional splitting algorithms, are presented in detail, especially on the geometrical and non-geometrical nature of the advection algorithm. In addition, a simple and efficient analytic form to calculate the volume and centroid of the intersection polyhedron in 3D is proposed. Several numerical tests, including both 2D and 3D tests, are carried out to investigate mass conservation and geometrical error. Numerical results suggest that the mixed EI and LE scheme (EILE2D) (Aulisa et al., 2003) has the overall best performance in 2D, and Weymouth and Yue’s scheme (WY) has the overall best performance in 3D.

A cell-centered discontinuous Galerkin multi-material arbitrary Lagrangian-Eulerian method in axisymmetric geometry

In this paper, a cell-centered discontinuous Galerkin (DG) multi-material arbitrary Lagrangian-Eulerian (MMALE) method is developed for compressible fluid dynamics in axisymmetric geometry. The MMALE method utilizes moment-of-fluid (MOF) interface reconstruction technology to simulate multi-materials of immiscible fluids. It is an explicit time-marching Lagrangian plus remapping type. In the Lagrangian stage, a volume-weighted scheme of updated cell-centered discontinuous Galerkin Lagrangian method is applied to discretize the hydrodynamic equations in pure cells, and Tipton's pressure relaxation closure model is used in mixed cells. A robust MOF interface reconstruction algorithm is used to provide the information on the material interfaces for remapping. In the rezoning stage, Knupp's algorithm is used for mesh smoothing. In the remapping stage, an efficient multi-material cell-intersection-based remapping method is proposed. It can be divided into four stages: polynomial reconstruction, polygon intersection, integration, and detection of problematic cells and limiting. Polygon intersection is based on the “clipping and projecting” algorithm, and detection of problematic cells depends on a troubled cell marker, and a posteriori multi-dimensional optimal order detection (MOOD) limiting strategy is used for limiting. Several numerical tests are given to demonstrate the robustness and accuracy of the method.

液々界面を横切って上昇する気泡運動の三次元数値解析

Ternary fluid flows that single bubbles rise across the interface separating two immiscible liquids (liquid-liquid layer) are investigated through three-dimensional numerical analysis based on the moment-of-fluid (MOF) method. In this study, the effect of the viscosity of both liquids on the bubble rise motion and the behavior of the liquid-liquid interface is discussed. From numerical results, it is clearly shown that the viscosity of the liquids has a significant effect on the motion of a single bubble rising through the liquid-liquid layer. When a bubble rises in the lower layer with the low-viscosity, the bubble rise motion shows dynamic behavior, and the behavior of the liquid-liquid interface is complicated. Also, after passing the liquid-liquid interface, the bubble largely trails the lower layer with the low-viscosity fluid. On the other hand, the dynamic behavior of a bubble cannot be seen when a bubble rises through the liquid-liquid interface between high-viscosity fluids. The MOF method allows us to reproduce the dynamic motion of a single bubble rising across a liquid-liquid layer with a ternary fluid interface.

A New Method for Estimating Bubble Diameter at Different Gravity Levels for Nucleate Pool Boiling

Abstract Nucleate boiling has significant applications in earth gravity (in industrial cooling applications) and microgravity conditions (in space exploration, specifically in making space applications more compact). However, the effect of gravity on the growth rate and bubble size is not yet well understood. We perform numerical simulations of nucleate boiling using an adaptive moment-of-fluid (MoF) method for a single vapor bubble (water or Perfluoro-n-hexane) in saturated liquid for different gravity levels. Results concerning the growth rate of the bubble, specifically the departure diameter and departure time, have been provided. The MoF method has been first validated by comparing results with a theoretical solution of vapor bubble growth in superheated liquid without any heat-transfer from the wall. Next, bubble growth rate, bubble shape, and heat transfer results under earth gravity, reduced gravity, and microgravity conditions are reported, and they are in good agreement with experiments. Finally, a new method is proposed for estimating the bubble diameter at different gravity levels. This method is based on an analysis of empirical data at different gravity values and using power-series curve fitting to obtain a generalized bubble growth curve irrespective of the gravity value. This method is shown to provide a good estimate of the bubble diameter for a specific gravity value and time.

A hybrid moment of fluid–level set framework for simulating primary atomization

This paper presents a hybrid moment of fluid–level set (HyMOFLS) method of liquid/gas interface reconstruction for the application to simulate primary atomization of liquid fuel. This method combines the moment of fluid (MOF) method and the level set framework in the coupled level set volume of fluid (CLSVOF) method. In this hybrid method, the MOF and CLSVOF methods are used to reconstruct the interface in the under-resolved and the resolved regions of the flow, respectively. An interface surface resolution metric called interface resolution quality (IRQ) is introduced to identify and classify these two flow regions in the computational domain. Such a strategy classifies/tags each computational cell with MOF or CLSVOF method based on a threshold value for the IRQ. The MOF method uses liquid volume fraction as well as centroids of liquid and gas phases for the interface reconstruction. The CLSVOF method uses the level set for describing the interface and liquid volume fraction for conserving the mass. The phase centroids in the HyMOFLS method are computed and transported on-the-fly during the cell tagging process. The transport of the liquid volume fraction, level set, and the phase centroids are performed using a directionally split algorithm. This algorithm is coupled with the Navier-Stokes equations solver that uses ghost fluid method as well as consistent mass and momentum flux computation for the momentum equation. Various numerical tests that exhaustively assess the capabilities, accuracy, and computational time consumption of the HyMOFLS method under multiple flow conditions and configurations are presented. The results from these tests suggest that this hybrid framework is capable of well capturing the liquid/gas interface belonging to thin and under-resolved structures that are often encountered in simulations of turbulent atomization of liquids. Following these tests, a detailed parametric study on the threshold value of IRQ is presented to test its effect on the accuracy of interface reconstruction. Finally, this hybrid framework is employed to simulate turbulent jet injection and pre-filming planar Airblast atomization cases of engineering applications. The proposed HyMOFLS method is found to achieve a good balance between the accuracy and the computational cost of reconstructing the liquid/gas interface for complex and turbulent atomization configurations.

A moment-of-fluid method for diffusion equations on irregular domains in multi-material systems

A new numerical method is developed for the solution of the diffusion problem in a system of several materials. In such a system, the diffusion coefficients are piecewise continuous and jumps in their values can occur across the complex-shaped interfaces between contiguous materials.The boundary conditions along the complex-shaped interfaces can either be a jump condition boundary condition, a Neumann boundary condition, or a Dirichlet boundary condition.The moment-of-fluid (MOF) procedure is employed to reconstruct the interfaces. This procedure enables accurate reconstruction of any number of material interfaces in a computational cell. Furthermore, MOF is a volume preserving reconstruction, as well as capable of capturing thin filamentary regions without the necessity of adaptive mesh refinement. The proposed method is tested on multi-material diffusion problems which demonstrates its potential to enable numerical simulation of complex flows of technological importance relevant to predicting the heat transfer rate in materials and manufacturing processes. Results using the new method are reported on problems in complex (filamentary) domains, and it is found that the method is very efficient at approximating the temperature, the temperature gradient, and the interfacial heat flux, as compared to traditional approaches.

Numerical investigation of coalescence-induced self-propelled behavior of droplets on non-wetting surfaces

We numerically study the self-propelled droplet phenomenon upon droplet coalescence. The numerical method is based on a well-validated multiphase flow solver that solves the three-dimensional Navier-Stokes equations. The liquid-air interface is captured using the moment of fluid along with a direction splitting method applied to advect the interface. And an approximate projection method is used to decouple the calculation of velocity and pressure. The solver is validated by comparing with the experimental results. Our results show that the droplet jumping process can be accurately captured. The simulated droplet deformation also matches the experimental results. To investigate the jumping mechanism, we compare results between two cases with and without a contact substrate. The history of vertical momentum shows that with a contact substrate, the droplet has a longer period of acceleration. The coalesced droplet with a contact substrate also has a smaller surface area which indicates that more surface energy is converted into kinetic energy. The effects of droplet size, surface tension, and droplet density are also studied. The jumping speed generally obeys the capillary scaling law. The effect of approaching speed is also investigated. With lower approaching speed, the surface tension dominates while with higher approaching speed, the inertia force dominates the jumping process.

A robust MoF method applicable to severely deformed polygonal mesh

The MoF (Moment of Fluid) method is an accurate approach for interface reconstruction in numerical simulation of multi-material fluid flow. So far, most works focus on improving its accuracy and efficiency, such as developing analytic reconstruction method and deducing the iteration schemes based on high order derivatives of the objective function. In this paper, we mainly concern on improving its robustness, especially for severely deformed polygonal meshes, in which case the objective function has multiple minimum value points. By using an efficient method for solving multiple roots of the nonlinear equation in large scope, a new algorithm is developed to enhance robustness of the MoF method. The main idea of this algorithm is as follows. The first derivative of the objective function is continuous, so the minimum value points of the objective function must be the zero points of the first derivative. Instead of finding the zero points of the first derivative directly, we turn to calculating the minimum value points (also zero points) of the square of the first derivative, which is a convex function on a neighborhood of each zero point. Applying the properties of convex function, the neighbor of each extreme minimum point of it can be obtained efficiently. Then each zero point of the square of the first derivative can be obtained using the iterative formula in its neighbor. Finally, by comparing the values of the objective function at these zero points of the first derivative, the global minimum value point of the objective function can be found and is the desired solution. The new algorithm only uses the first derivative of the objective function. It doesn't need an initial guess for the solution, which has to be carefully chosen in previous works. Numerical results are presented to demonstrate the accuracy and robustness of this new algorithm. The results show that it is applicable to severely deformed polygonal mesh, even with concave cells.

Establishing mesh topology in multi-material cells: Enabling technology for robust and accurate multi-material simulations

Real world problems are typically multi-material, combining materials such as gases, liquids and solids that have very different properties. The material interfaces may be fixed in time or can be a part of the solution, as in fluid-structure interactions or air-water dynamics, and therefore move and change shape. In such problems the computational mesh may be non-conformal to interfaces due to complexity of these interfaces, presence of small fractions of materials, or because the mesh does not move with the flow, as in the arbitrary Lagrangian–Eulerian (ALE) methods. In order to solve problems of interest on such meshes, interface reconstruction methods are usually used to recover an approximation of material regions within the cells. For a cell intersecting multiple material regions, these approximations of contained subregions can be considered as single-material subcells in a local mesh that we call a minimesh. In this paper, we discuss some of the requirements that discretization methods have on topological information in the resulting hierarchical meshes and present an approach that allows incorporating the buildup of sufficiently detailed topology into the nested dissections based PLIC-type reconstruction algorithms (e.g. Volume-of-Fluid, Moment-of-Fluid) in an efficient and robust manner. Specifically, we describe the X-MOF interface reconstruction algorithm in 2D, which extends the Moment-Of-Fluid (MOF) method to include the topology of minimeshes created inside of multi-material cells and parent-child relations between corresponding mesh entities on different hierarchy levels. X-MOF retains the property of being local to a cell and not requiring external communication, which makes it suitable for massively parallel applications. We demonstrate some scaling results for the X-MOF implementation in Tangram, a modern interface reconstruction framework for exascale computing.

Mechanism of the interannual oscillation in length of day and its constraint on the electromagnetic coupling at the core–mantle boundary

A significant 6 yr oscillation exists in the length of day (LOD) on the interannual scales. There are mainly two models currently to explain this oscillation, i.e., mantle–inner core gravitational (MICG) coupling mode and the fast torsional waves within the fluid outer core. The former has been doubted, while the source of the excitation of the latter is not yet understood. Therefore, the mechanism of the 6 yr oscillation is still not clear. Here, by considering the mantle and inner core angular momentum, we investigate the MICG coupling mode and its natural period (T0). Given that the strength of gravitational core–mantle coupling (Γ¯) within a recently constrained range is quite weaker than that estimated previously, the mechanism of the 6 yr oscillation still can be attributed to MICG coupling mode (i.e., T0 equals to 6 yrs), but, we require the inertia moment of fluid within the tangent cylinder involved in the 6 yr oscillation to be smaller than 1.23×1035 kgm2. This interpretation can be used to constrain the electromagnetic (EM) coupling at the inner core boundary (ICB). In order to study quantitatively the constraints on the EM coupling at the core–mantle boundary (CMB) from the observed 6 yr oscillation with a quality factor Q (∼51.6), we further develop the mathematical expression between the Q value based on the observations and EM coupling at the CMB. According to the Γ¯ value from recent estimates and assuming that the 6 yr oscillation is in a free decay, we can obtain the radial magnetic field at the CMB is 0.52 mT∼0.62 mT when conductance at the CMB is 108 S.

An improved 2D MoF method by using high order derivatives

The MoF (Moment of Fluid) method is one of the most accurate approaches among various interface reconstruction algorithms. Alike other second order methods, the MoF method needs to solve an implicit optimization problem to obtain the optimal approximate interface, so an iteration process is inevitable under most circumstances. In order to solve the optimization efficiently, the properties of the objective function are worthy of studying. In 2D problems, the first order derivative has been deduced and applied in the previous researches. In this paper, the high order derivatives of the objective function are deduced on the convex polygon. We show that the nth (n≥2) order derivatives are discontinuous, and the number of the discontinuous points is two times the number of the polygon edge. A rotation algorithm is proposed to successively calculate these discontinuous points, thus the target interval where the optimal solution is located can be determined. Since the high order derivatives of the objective function are continuous in the target interval, the iteration schemes based on high order derivatives can be used to improve the convergence rate. Moreover, when iterating in the target interval, the value of objective function and its derivatives can be directly updated without explicitly solving the volume conservation equation. The direct update makes a further improvement of the efficiency especially when the number of edges of the polygon is increasing. The Halley's method, which is based on the first three order derivatives, is applied as the iteration scheme in this paper and the numerical results indicate that the CPU time is about half of the previous method on the quadrilateral cell and is about one sixth on the decagon cell.

Concerning the Role of Supercritical Carbon Dioxide in SN 1 Reactions.

A series of SN 1-type reactions has been studied under various conditions to clarify the role of supercritical carbon dioxide (scCO2 ) as reaction medium for this kind of transformations. The application of scCO2 did not result in higher yields in any of the experiments in comparison to those under neat conditions or in the presence of other inert compressed gases. High-pressure UV/Vis spectroscopic measurements were carried out to quantify the degree of carbocation formation of a highly SN 1-active alkyl halide as a function of the applied solvent. No measureable concentration of carbocations could be detected in scCO2 , just like in other low polarity solvents. Taken together, these results do not support the previously claimed activating effect via enhanced SN 1 ionization due to the quadrupolar moment of the supercritical fluid.

Moment-of-fluid analytic reconstruction on 2D Cartesian grids

Moment-of-Fluid (MoF) is a piecewise linear interface reconstruction method that tracks fluid through its volume fraction and centroid, which are deduced from the zeroth and first moments. We present a method that replaces the original minimization stage by an analytic reconstruction algorithm on bi-dimensional Cartesian grids. This algorithm provides accurate results for a lower computational cost than the original minimization algorithm. When more than two fluids are involved, this algorithm can be used coupled with the minimization algorithm. Although this paper deals with Cartesian grids, everything remains valid for any meshes that are made of rectangular cells.

An improved 3D MoF method based on analytical partial derivatives

MoF (Moment of Fluid) method is one of the most accurate approaches among various surface reconstruction algorithms. As other second order methods, MoF method needs to solve an implicit optimization problem to obtain the optimal approximate surface. Therefore, the partial derivatives of the objective function have to be involved during the iteration for efficiency and accuracy. However, to the best of our knowledge, the derivatives are currently estimated numerically by finite difference approximation because it is very difficult to obtain the analytical derivatives of the object function for an implicit optimization problem. Employing numerical derivatives in an iteration not only increase the computational cost, but also deteriorate the convergence rate and robustness of the iteration due to their numerical error. In this paper, the analytical first order partial derivatives of the objective function are deduced for 3D problems. The analytical derivatives can be calculated accurately, so they are incorporated into the MoF method to improve its accuracy, efficiency and robustness. Numerical studies show that by using the analytical derivatives the iterations are converged in all mixed cells with the efficiency improvement of 3 to 4 times.