Terms Of Mass Conservation Research Articles

A numerical model for simulating the interaction of dam-break waves and floating structures

Using OpenFOAM, a numerical model is proposed to simulate the interaction of dam-break waves and floating structures. A geometric volume-of-fluid (VOF) method (isoAdvector) is combined with overset grid techniques. To avoid undesired air bubbles appearing in bulk water regions, a source term is introduced to the VOF equation. Attempts are made to alleviate mass conservation defects of the overset grid. It is found that: (i) Among the mesh interpolation schemes considered, the kernel scheme is the most preferable because it reaches a better balance between accuracy and efficiency; (ii) Extending donor cell stencils is not recommended because it magnifies the mass conservation defect of the overset grid; (iii) A strong coupling strategy is recommended because it alleviates the “water leak” phenomenon. Then, to validate the numerical model, two cases are simulated. The first case is a three-dimensional floating body case. The result demonstrates that this model performs better in terms of mass conservation than the model proposed by Wang et al. (2021). The second case is a two-dimensional experiment of the interaction of a dam-break wave and a floating box. The result demonstrates that this model has better temporal and spatial convergence than the default overset grid solver in OpenFOAM.

Experimental and numerical analysis on low-temperature off-design organic Rankine cycle in perspective of mass conservation

Experimental and numerical studies on the low-temperature heat recovery organic Rankine cycle operating in off-design conditions are conducted for the in-depth understanding of the system's underlying mechanisms in terms of mass conservation. Experimental data sets were obtained from a 1 kW lab-scale organic Rankine cycle test bed using R245fa as the working fluid. The effects of several boundary conditions, including charged mass, are thoroughly examined under low-temperature heat source within the range of 65–95 °C. Numerical models of the heat exchangers are developed by applying the discretization method to predict the captured mass inside the phase-changing components and validated within 5% error range. By the integration of experimental and numerical methods, unprecedented and critical results covering the pressure formation process, mass distribution, and liquid receiver modeling are derived from the analysis which could not be discovered through previous approaches. The unconventional thermodynamic state of the working fluid inside the liquid receiver is revealed in detail and a passive design is suggested for the liquid receiver model. An improved solver architecture is proposed for the complete development of a fully deterministic off-design organic Rankine cycle simulation model, where the reality-based logics obtained from the key findings are projected into the novel model.

Multiphase and multiphysics modeling of dendrite growth and gas porosity evolution during solidification

Gas porosity is one of the most detrimental defects that can considerably deteriorate mechanical properties of casting parts. Simulating the interaction between gas porosity and solidifying microstructure is challenging due to multiphase and multiphysical characteristics. In this work, a solid-liquid-gas multiphase-field lattice-Boltzmann model is developed to describe the complex multiphase interaction during solidification. The model relaxes the assumption of pure metal system, simplified bubble shape and pure diffusion condition. It can consider solid growth, bubble motion, interface deformation, component transport, melt flow, and partition of both alloy solute and dissolved gas species. The model is validated in terms of mass conservation, mapping operation to the two-phase model, Laplace pressure condition, and bubble dynamics. The effectiveness of the model is further evaluated by comparing with other four models and experiments. The model is successfully used to describe the interaction between gas porosity and magnesium dendrite. The dependence of phase fractions on the characteristic parameters including melt undercooling, interface mobility coefficient and internal bubble pressure is quantified to explore the multiphase equilibrium. The proposed model can be regarded as complementary to the previous models and it is suitable for addressing the problems involving the solid-liquid-gas multiphase and multiphysical characteristics.

Coupling Molecular Dynamics and Direct Simulation Monte Carlo using a general and high-performance code coupling library

A domain-decomposed method to simultaneously couple the classical Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC) methods is proposed. This approach utilises the MPI-based general coupling library, the Multiscale Universal Interface. The method provides a direct coupling strategy and utilises two OpenFOAM based solvers, mdFoam+ and dsmcFoam+, enabling scenarios where both solvers assume one discrete particle is equal to one molecule or atom. The ultimate goal of this work is to enable complex multi-scale simulations involving micro, meso and macroscopic elements, as found with problems like evaporation.Results are presented to show the fundamental capabilities of the method in terms of mass and kinetic energy conservation between simulation regions handled by the different solvers. We demonstrate the capability of the method by deploying onto a large supercomputing resource, with attention paid to the scalability for a canonical NVT ensemble (a constant number of atoms N, constant volume V and constant temperature T) of Argon atoms. The results show that the method performs as expected in terms of mass conservation and the solution is also shown to scale reasonably on a supercomputing resource, within the known performance limits of the coupled codes. The wider future of this work is also considered, with focus placed on the next steps to expand the capabilities of the methodology to allow for indirect coupling (where the coarse-graining capability of the DSMC method is used), as well as how this will then fit into a larger coupled framework to allow a complete micro-meso-macro approach to be tackled.

A mass-conserved multiphase lattice Boltzmann method based on high-order difference**Project supported by the National Natural Science Foundation of China (Grant Nos. 11862003 and 81860635), the Key Project of the Natural Science Foundation of Guangxi Zhuang Autonomous Region, China (Grant No. 2017GXNSFDA198038), the Project of Natural Science Foundation of Guangxi Zhuang Autonomous Region, China (Grant No. 2018GXNSFAA281302), the Project for Promotion

The Z–S–C multiphase lattice Boltzmann model [Zheng, Shu, and Chew (ZSC), J. Comput. Phys. 218, 353 (2006)] is favored due to its good stability, high efficiency, and large density ratio. However, in terms of mass conservation, this model is not satisfactory during the simulation computations. In this paper, a mass correction is introduced into the ZSC model to make up the mass leakage, while a high-order difference is used to calculate the gradient of the order parameter to improve the accuracy. To verify the improved model, several three-dimensional multiphase flow simulations are carried out, including a bubble in a stationary flow, the merging of two bubbles, and the bubble rising under buoyancy. The numerical simulations show that the results from the present model are in good agreement with those from previous experiments and simulations. The present model not only retains the good properties of the original ZSC model, but also achieves the mass conservation and higher accuracy.

Research on dam-break flow induced front wave impacting a vertical wall based on the CLSVOF and level set methods

The problem of a dam-break flow induced front wave impacting a vertical wall placed at the downstream end was numerically investigated by using the Level Set (LS) method and Coupled Level Set/Volume Of Fluid (CLSVOF) method. For the LS approach, the advection equation of the level set function to capture the interface is discretized by the fifth-order, weighted, essentially non-oscillatory (WENO5) scheme. For the CLSVOF approach, the Tangent of Hyperbola for Interface Capturing/Weighed Line Interface Calculation (THINC/WLIC) scheme is adopted to capture the interface and improve mass conservation. The interface normal vector in the WLIC reconstruction procedure can be estimated from the LS function instead of VOF function. The Navier–Stokes solver which uses an explicit Adams–Bashforth algorithm is performed on a staggered Eulerian grid. The numerical experiments are carried out in a rectangular flume with a smooth horizontal wet bed with four different tailwater levels. This study investigates the formation and propagation of the negative wave resulting from the reflection of flood waves against the downstream end wall. The numerical results are compared with laboratory experiment results studied in Ref (Kocaman and Ozmen-Cagatay, 2015). and a reasonable agreement is achieved. Numerical results show that the CLSVOF solutions are better than LS solutions in terms of mass conservation.

Numerical study and evaluation of solid-liquid extraction of Montan wax in stirred tanks on different scales

Abstract In this paper, mathematical models for the solid-liquid extraction of Montan wax from brown coal are formulated, considering the external mass transfer in liquid phase and the diffusion inside the brown coal particles (effective diffusion) in terms of mass conservation. For parameter identification and evaluation purposes, the Montan wax extraction from brown coal is conducted with small-scale and scaled-up immersion apparatuses. The effective diffusion coefficient D e for the solute transport of Montan wax from brown coal in toluene is obtained with the proposed correlation D e = D AB · e p n e ( n e = 2 .9) where e p denotes the particle porosity. The effects of the important conditions such as temperature, solvent-solid ratio, the stirring speed and the particle size in terms of extraction rate are investigated by simulation. A sensitivity analysis is performed for the extraction yield with respect of mass transfer coefficient k f and effective diffusion coefficient D e .

A comparative study of local and nonlocal Allen-Cahn equations with mass conservation

The local and nonlocal Allen-Cahn equations (ACEs) have received increasing attention in the study of the complicated interfacial problems. In this paper, we conduct a comparison between local and nonlocal ACEs with the property of mass conservation in the framework of lattice Boltzmann (LB) method. To this end, we first propose two simple multiple-relaxation-time LB models for local and nonlocal ACEs, and through the Chapman-Enskog expansion, the local and nonlocal ACEs can be recovered correctly from the developed LB models. Then we test these two LB models with several examples, and the numerical results show that the developed LB models are accurate and also have a second-order convergence rate in space. Finally, a comparison between the local and nonlocal ACEs is also performed in terms of mass conservation, stability and accuracy. The results show that both local and nonlocal ACEs can preserve mass conservation of system and each phase. And additionally, it is also found that the local ACE is more accurate than nonlocal ACE in capturing the interface profile, but the latter is more stable than the former.

A Flexible Coupled Level Set and Volume of Fluid (flexCLV) method to simulate microscale two-phase flow in non-uniform and unstructured meshes

An innovative Flexible Coupled Level Set (LS) and Volume of Fluid (VOF) algorithm (flexCLV) to simulate two-phase flows at the microscale on unstructured and non-uniform meshes is proposed. The method combines the advantages of the VOF method in terms of mass conservation and the LS method in terms of accuracy of the surface tension implementation and can handle both 2D and 3D domains discretized by either structured hexaedra or unstructured tetrahedral grids with high aspect ratio elements, thus guaranteeing flexibility and robustness. The method is implemented within the VOF-based OpenFOAM’s solver interFoam, which is retained as the base algorithm for the interface advection, while the surface tension force is calculated by using the level set function reconstructed from the VOF’s fraction. The method is first validated in static flow conditions by simulating a circular bubble at equilibrium and then in dynamic flow conditions by studying a freely bubble rising in both 2D and 3D domains discretized by both structured and unstructured meshes. The proposed flexCLV algorithm is then used to simulate the dynamics of confined bubbles in circular microchannels in the low capillary number regime. 2D and 3D mesh grids with high aspect ratio elements are utilized to discretized the liquid film at the tube’s walls. The numerical results are compared with the available literature and simulations performed with the original interFoam solver in terms of bubble shape and velocity, thickness of the liquid film and amplitude of the bubble tail oscillations. Results compare very well with the experimental measurements and demonstrate the superior accuracy of the coupled flexCLV method with respect to the original VOF method when surface tension and accurate interface representation play a fundamental role. Importantly, the present study also provides a precious insight on the time-dependent patterns appearing on the bubble surface in the visco-inertial regime, which could be here investigated in detail.

A comparative study of reinitialization approaches of the level set method for simulating free-surface flows

Unstructured grids were used to compare the performance of a direct reinitialization scheme with those of two reinitialization approaches based on the solution of a hyperbolic Partial differential equation (PDE). The problems of moving interface were solved in the context of a finite element method. A least-square weighted residual method was used to discretize the advection equation of the level set method. The benchmark problems of rotating Zalesak’s disk, time-reversed single vortex, and two-dimensional sloshing were examined. Numerical results showed that the direct reinitialization scheme performed better than the PDE-based reinitialization approaches in terms of mass conservation, dissipation and dispersion error, and computational time. In the case of sloshing, numerical results were found to be in good agreement with existing experimental data. The direct reinitialization approach consumed considerably less CPU time than the PDE-based simulations for 20 time periods of sloshing. This approach was stable, accurate, and efficient for all the problems considered in this study.

A mass-conserving lattice Boltzmann method with dynamic grid refinement for immiscible two-phase flows

A mass-conserving lattice Boltzmann method (LBM) for multiphase flows is presented in this paper. The proposed LBM improves a previous model (Lee and Liu, 2010 [21] ) in terms of mass conservation, speed-up, and efficiency, and also extends its capabilities for implementation on non-uniform grids. The presented model consists of a phase-field lattice Boltzmann equation (LBE) for tracking the interface between different fluids and a pressure-evolution LBM for recovering the hydrodynamic properties. In addition to the mass conservation property and the simplicity of the algorithm, the advantages of the current phase-field LBE are that it is an order of magnitude faster than the previous interface tracking LBE proposed by Lee and Liu (2010) [21] and it requires less memory resources for data storage. Meanwhile, the pressure-evolution LBM is equipped with a multi-relaxation-time (MRT) collision operator to facilitate attainability of small relaxation rates thereby allowing simulation of multiphase flows at higher Reynolds numbers. Additionally, we reformulate the presented MRT-LBM on nonuniform grids within an adaptive mesh refinement (AMR) framework. Various benchmark studies such as a rising bubble and a falling drop under buoyancy, droplet splashing on a wet surface, and droplet coalescence onto a fluid interface are conducted to examine the accuracy and versatility of the proposed AMR-LBM. The proposed model is further validated by comparing the results with other LB models on uniform grids. A factor of about 20 in savings of computational resources is achieved by using the proposed AMR-LBM. As a more demanding application, the Kelvin–Helmholtz instability (KHI) of a shear-layer flow is investigated for both density-matched and density-stratified binary fluids. The KHI results of the density-matched fluids are shown to be in good agreement with the benchmark AMR results based on the sharp-interface approach. When a density contrast between the two fluids exists, a typical chaotic structure in the flow field is observed at a Reynolds number of 10000, which indicates that the proposed model is a promising tool for direct numerical simulation of two-phase flows.

MERIDIONAL FLOW IN THE SOLAR CONVECTION ZONE. II. HELIOSEISMIC INVERSIONS OF GONG DATA

Meridional flow is thought to play a very important role in the dynamics of the solar convection zone; however, because of its relatively small amplitude, precisely measuring it poses a significant challenge. Here we present a complete time-distance helioseismic analysis of about two years of ground-based GONG Doppler data to retrieve the meridional circulation profile for modest latitudes, in an attempt to corroborate results from other studies. We use an empirical correction to the travel times due to an unknown center-to-limb systematic effect. The helioseismic inversion procedure is first tested and reasonably validated on artificial data from a large-scale numerical simulation, followed by a test to broadly recover the solar differential rotation found from global seismology. From GONG data, we measure poleward photospheric flows at all latitudes with properties that are comparable with earlier studies, and a shallow equatorward flow about $65$\,Mm beneath the surface, in agreement with recent findings from HMI data. No strong evidence of multiple circulation cells in depth nor latitude is found, yet the whole phase space has not yet been explored. Tests of mass flux conservation are then carried out on the inferred GONG and HMI flows and compared to a fiducial numerical baseline from models, and we find that the continuity equation is poorly satisfied. While the two disparate data sets do give similar results for about the outer $15\%$ of the interior radius, the total inverted circulation pattern appears to be unphysical in terms of mass conservation when interpreted over modest time scales. We can likely attribute this to both the influence of realization noise and subtle effects in the data and measurement procedure.

P1/P0+ elements for incompressible flows with discontinuous material properties

We present a 3-noded triangle and a 4-noded tetrahedra with a continuous linear velocity and a discontinuous linear pressure field formed by the sum of an unknown constant pressure field and a prescribed linear field that satisfies the steady state momentum equations for a constant body force. The elements are termed P1/P0+ as the “effective” pressure field is linear, although the unknown pressure field is piecewise constant within each element. The elements have an excellent behavior for incompressible viscous flow problems with discontinuous material properties formulated in either Eulerian or Lagrangian descriptions. The necessary numerical stabilization for dealing with the inf–sup condition imposed by the incompressibility constraint and high convective effects (in Eulerian flows) is introduced via the Finite Calculus (FIC) approach. For the sake of clarity, the element derivation is presented first for the simpler Stokes equations written in the standard Eulerian frame. The extension of the formulation to the Navier–Stokes equations written in the Eulerian and Lagrangian frameworks is straightforward and is presented in the second part of the paper.The efficiency and accuracy of the new P1/P0+ triangle is verified by solving a set of incompressible multifluid flow problems using a Lagrangian approach and a classical Eulerian description. The excellent performance of the new triangular element in terms of mass conservation and general accuracy for analysis of fluids with discontinuous material properties is highlighted.

Simulating One‐Dimensional Unsaturated Flow in Heterogeneous Soils with Water Content‐Based Richards Equation

Changes in soil hydraulic properties with depth are common in field soils and need to be described in numerical simulations. Water content is undesirable to calculate Darcian flux since it is not continuous across boundaries between different soil layers. A general form of the one‐dimensional water content–based Richards equation, which was first derived by Hills et al., is adopted here to simulate one‐dimensional unsaturated flow in gradational soils by adding a correction term. For cell‐centered grid, several algorithms are presented to find the correction term for the two nodes in the vicinity of the layer interface. For vertex‐centered grid, the dyadic values of water content at the interface are represented by one sole water content value through the introduction of composite soil water curve. The proposed algorithms are implemented in an iterative model, and then the model is tested with several synthetic cases in gradational soil and layered soil. The water content–based Richards equation leads to superior numerical performance in terms of mass conservation, accuracy, and efficiency over the mixed form Richards equation, especially for the flow in relatively dry soil and with coarse grid. In layered soil, it is found that the algorithm based on vertex‐centered grid is the most robust among all the methods discussed here.

A Q2Q1 finite element/level‐set method for simulating two‐phase flows with surface tension

SUMMARYA Q2Q1 (quadratic velocity/linear pressure) finite element/level‐set method was proposed for simulating incompressible two‐phase flows with surface tension. The Navier–Stokes equations were solved using the Q2Q1 integrated FEM, and the level‐set variable was linearly interpolated using a ‘pseudo’ Q2Q1 finite element when calculating the density and viscosity of a fluid to avoid an unbounded density/viscosity. The advection of the level‐set function was calculated through the Taylor–Galerkin method, and the direct approach method is employed for reinitialization. The proposed method was tested by solving several benchmark problems including rising bubbles exhibiting a large density difference and the surface tension effect. The numerical results of the rising bubbles were compared with the existing results to validate the benchmark quantities such as the centroid, circularity, and rising velocity. Furthermore, we focused our attention mainly on mass conservation and time‐step. We observed that the present method represented a convergence rate between 1.0 and 1.5 orders in terms of mass conservation and provided more stable solutions even when using a larger time‐step than the critical time‐step that was imposed because of the explicit treatment of surface tension. Copyright © 2011 John Wiley & Sons, Ltd.

Modelling sediment transport over partially non‐erodible bottoms

SUMMARYIn‐depth‐averaged and cross‐section‐averaged morphodynamic models, based on explicit time integration, it may happen that the computed bed level becomes lower than the top level of a non‐erodible layer (e.g. concrete, bedrock or armoured layer). This is a standard pitfall, which has been addressed in different ways. In this paper, we present an original approach for avoiding computation of non‐physical bed levels, using an iterative procedure to correct the outward sediment fluxes. The procedure is shown to be computationally efficient while it achieves a high accuracy in terms of mass conservation. We compare our original approach with the existing Struiksma's method and with a reformulation of the problem in terms of mathematical optimization of a linear or nonlinear objective function under linear constraints.The new procedure has been incorporated into an existing finite volume morphodynamic model. It has been validated with several 1D benchmarks leading to configurations with sediment transport over non‐erodible bottom. The computation time has been verified not to increase by more than 15% compared with runs without non‐erodible bottom. Copyright © 2011 John Wiley & Sons, Ltd.

Reaction diffusion of MoSi 2 and Mo 5SiB 2

To establish quantitative basis for oxidation-protective coating of Mo–Si–B ternary alloys by MoSi 2, phase transformations in MoSi 2 vs. Mo 5SiB 2 diffusion couples have been studied. Two layers are formed on reaction diffusion at temperatures between 1400 and 1600 °C: a single-phase layer of Mo 5Si 3 and a two-phase layer consisting of Mo 5Si 3 and MoB. The growth obeys the parabolic low for both the layers, and the rate constants of the two layers are found to be approximately equal. The interdiffusion coefficient in the T 1 layer has also been evaluated. The microstructural evolution in the diffusion zone is modeled in terms of mass conservation, as well as that of a Mo–9Si–18B two-phase alloy coated with MoSi 2 reported previously [Intermetallics 12 (2004) 407].