Three-dimensional Cylindrical Coordinates Research Articles

An improved phase-field algorithm for simulating the impact of a drop on a substrate in the presence of surfactants

This paper is devoted to the numerical study of droplet impact on solid substrates in the presence of surfactants. We formulate the problem in an energetically variational framework and introduce an incompressible Cahn-Hilliard-Navier-Stokes system for the phase-field modeling of two-phase flows. Flory-Huggins potential and generalized Navier boundary condition are used to account for soluble surfactants and moving contact lines. Based on the convex splitting and pressure stabilization technique, we develop unconditionally energy stable schemes for this model. The discrete energy dissipation law for the original energy is rigorously proved for the first-order scheme. The numerical methods are implemented using finite difference method in three-dimensional cylindrical coordinates with axisymmetry. Using the proposed methods for this model, we systematically study the impact dynamics of both clean and contaminated droplets (with surfactants) in a series of numerical experiments. In general, the dissipation in the impact dynamics of a contaminated drop is smaller than that in the clean case, and topological changes are more likely to occur for contaminated drops. Adding surfactants can significantly influence the impact phenomena, leading to the enhancement of adherence effect on hydrophilic surfaces and splashing on hydrophobic surfaces. Some qualitative agreements with experiments are also obtained.

Investigation of the Spiral Wave Generation and Propagation on a Numerical Circular Wave Tank Model

A two-phase incompressible flow model in three-dimensional cylindrical coordinates is applied for oblique wave generation in a numerical circular wave tank. The governing equations are discretized by a finite volume method, and a mass source function is added to reproduce oblique waves through a spiral wave generator positioned at the center of the tank. The volume of fluid method is implemented to track the free surfaces between the air and fluid, and the zonal embedded grid system is adopted to obtain a grid-independent solution in the cylindrical coordinates. A permeable, circumferentially sloping topography (1:7), similar to a natural beach profile, is set up for investigating wave propagation and other characteristics. The simulation and physical experimental results are compared, which show a good agreement in randomly selected water surface elevation profiles and wave heights under the same wave and sloping conditions. The spiral waves are reproduced and propagated in a phenomenon similar to that observed in the physical experiment. The results also show that the wave-breaking positions differ in different wave conditions and suggest a relationship with cross-shore and longshore velocity distribution in terms of incident wave heights and wave-breaking positions in different wave periods on the same sloping topography. Furthermore, this model can be used to investigate the mechanism of longshore current generation and the influences of beach slope on the generated wave propagation in a sloping topography.

Three-Dimensional Analytical Modeling of Axial-Flux Permanent Magnet Drivers

In this paper, the axial-flux permanent magnet driver is modeled and analyzed in a simple and novel way under three-dimensional cylindrical coordinates. The inherent three-dimensional characteristics of the device are comprehensively considered, and the governing equations are solved by simplifying the boundary conditions. The axial magnetization of the sector-shaped permanent magnets is accurately described in an algebraic form by the parameters, which makes the physical meaning more explicit than the purely mathematical expression in general series forms. The parameters of the Bessel function are determined simply and the magnetic field distribution of permanent magnets and the air-gap is solved. Furthermore, the field solutions are completely analytical, which provides convenience and satisfactory accuracy for modeling a series of electromagnetic performance parameters, such as the axial electromagnetic force density, axial electromagnetic force, and electromagnetic torque. The correctness and accuracy of the analytical models are fully verified by three-dimensional finite element simulations and a 15 kW prototype and the results of calculations, simulations, and experiments under three methods are highly consistent. The influence of several design parameters on magnetic field distribution and performance is studied and discussed. The results indicate that the modeling method proposed in this paper can calculate the magnetic field distribution and performance accurately and rapidly, which affords an important reference for the design and optimization of axial-flux permanent magnet drivers.

Jacobian-free Newton Krylov nodal expansion method in three-dimensional cylindrical coordinates

A cylindrical version of Jaocbian-free Newton Krylov nodal expansion methods in cylindrical coordinates (CyliNEM_JFNK) are successfully developed to solve the three-dimensional and multigroup k-eigenvalue reactor problems in cylindrical geometry. Unlike nodal methods in Cartesian geometry, CyliNEM_JFNK adopts the special polynomial-expansion approximation and some separate treatments to solve the transverse-integrated equations in each direction and it is not a simple and trivial extension from Cartesian geometry to cylindrical geometry. To reduce the computational cost and the scale of residual systems constructed in the CyliNEM_JFNK as much as possible, the outgoing interface partial currents, node-averaged fluxes, third and fourth expansion coefficients are only chosen as the final-solving variables to obtain the residual functions by eliminating all the other variables including incoming interface partial currents, first and second expansion coefficients, source and leakage moments. Then the efficient physics-based preconditioners are successfully developed by making full use of the efficient Row-Column iteration and original power iteration framework. The cylindrical version of 3D LWR IAEA benchmark problem and a series of three-dimensional high temperature gas-cooled pebble bed reactor models are chosen to verify the numerical accuracy and computational efficiency of the CyliNEM_JFNK method. Numerical solutions show that the proposed CyliNEM_JFNK method can track the reference solutions very well and obtain higher computational efficiency than the CyliNEM method with efficient Row-Column iteration in three-dimensional cylindrical coordinates, which indicates the advantages of the developed CyliNEM_JFNK method that can serve as a potential nodal diffusion analysis tool for three-dimensional cylindrical advanced reactor problems.

Three-dimensional cylindrical Poisson solver with vacuum boundary conditions

Self-gravity and rotation are two key ingredients in dynamics of astronomical disk systems such as galactic and protostellar disks. Despite its importance, there has not yet been so far an efficient algorithm to solve the Poisson equation in three-dimensional cylindrical coordinates under vacuum boundary conditions. By generalizing the James algorithm to cylindrical coordinates, we develop an accurate (second-order convergence) and efficient (faster than MHD) cylindrical Poisson solver that is scalable up to ∼ 104 cores. We develop a method to calculate the cylindrical discrete Green’s function, which is an essential element of the James algorithm to estabilish its second-order accuracy. We implement our cylindrical version of the James algorithm in Athena++ code and demonstrate its accuracy and efficiency by performing the convergence test and the weak scaling test.

Free Surface Flow Simulation by a Viscous Numerical Cylindrical Tank

In order to numerically investigate the free surface flow evolution in a cylindrical tank, a regular structured grid system in the cylindrical coordinates is usually applied to solve control equations based on the incompressible two-phase flow model. Since the grid spacing in the azimuthal direction is proportionate to the radial distance in a regular structured grid system, very small grid spacing would be obtained in the azimuthal direction and it would require a very small computational time step to satisfy the stability restriction. Moreover, serious mass disequilibrium problems may happen through the convection of the free surface with the Volume of Fluid (VOF) method. Therefore in the present paper, the zonal embedded grid technique was implemented to overcome those problems by gradually adjusting the mesh resolution in different grid blocks. Over the embedded grid system, a finite volume algorithm was developed to solve the Navier–Stokes equations in the three-dimensional cylindrical coordinates. A high-resolution scheme was applied to resolve the free surface between the air and water phases based on the VOF method. Computation of liquid convection under a given velocity field shows that the VOF method implemented with a zonal embedded grid is more advanced in keeping mass continuity than that with regular structured grid system. Furthermore, the proposed model was also applied to simulate the sharp transient evolution of circular dam breaking flow. The simulation results were validated against the commercial software Fluent, which shows a good agreement, and the proposed model does not yield any free surface oscillation.

Electrostatics with reflection symmetry and exact solution of the Dunkl-Laplace equation in cylindrical coordinates

In this paper, electrostatics with reflection symmetry is considered. The Maxwell equation for electrostatics is obtained from deformation of the Maxwell tensor. The general solution of the twodimensional Dunkl-Laplace equation in the polar coordinates is obtained. The general solution of the Dunkl-Laplace equation in three-dimensional cylindrical coordinates is also obtained.

Application of the numerical circular wave tank on the simulations of oblique and multi-directional waves

To simulate the oblique and multi-direction waves, a numerical circular wave tank of implementing a finite volume method in three-dimensional cylindrical coordinates is developed based on the incompressible two-phase flow model in the present study. Both the oblique waves from an arbitrary direction and multi-directional waves reproduced by the integration of oblique waves are investigated using the circular wave tank. By comparing the numerical results with the theoretical ones, it is found that the proposed numerical wave tank can accurately reproduce oblique and multi-directional waves without the influence of the reflection waves. Numerical results of the wave diffraction around a vertical circular cylinder from different incident directions are also presented. The result shows that it is in good agreement with the theoretical and experimental results and it is interesting to note that the influence of the incident direction can be ignored on the performance of the circular wave tank.

Three-Dimensional Investigation of the Stokes Eigenmodes in Hollow Circular Cylinder

This paper studies the influence of boundary conditions on a fluid medium of finite depth. We determine the frequencies and the modal shapes of the fluid. The fluid is assumed to be incompressible and viscous. A potential technique is used to obtain in three-dimensional cylindrical coordinates a general solution for a problem. The method consists in solving analytically partial differential equations obtained from the linearized Navier-Stokes equation. A finite element analysis is also used to check the validity of the present method. The results from the proposed method are in good agreement with numerical solutions. The effect of the fluid thickness on the Stokes eigenmodes is also investigated. It is found that frequencies are strongly influenced.

Three-dimensional modal analysis of an idealized human head including fluid–structure interaction effects

A three-dimensional analytical and numerical method is presented in this article for the analysis of the acoustic fluid–structure interaction systems including, but not limited to, the brain, cerebro-spinal fluid (CSF), and skull. The model considers a three-dimensional acoustic fluid medium interacting with two solid domains. This article deals with the analytical and numerical computation of eigenproperties for an idealized human head model including fluid–structure interaction phenomena. We determine in the present work the natural frequencies and the modes shapes of the system of the brain, cerebro-spinal fluid (CSF), and skull. Two models are presented in this study: an elastic skull model and a rigid model. In the analysis, a potential technique is used to obtain in three-dimensional cylindrical coordinates a general solution for a solid problem. A finite element method analysis is also used to check the validity of the present method. The results from the proposed method are in good agreement with numerical solutions. The effects of the fluid thickness and compressibility on the natural frequencies are also investigated.

The Flow Characteristic Study of Gas Spray Process in Baosteel Spray Forming Equipment

The simulation software of compressible gas flow process in spray forming equipment was developed under three-dimensional cylindrical coordinates. When the gas flow strands sprayed from some definite angles into the atomizer chamber, they converged at the definite height in axial line then expanded outside which showed the jet current characteristic. The substrate stagnated the divergence jet strands of gas and then separated the flow strand into two big recycle zones in whole space. One of the recycle flow zone directed downwards, the other recycle flow zone, directed upwards and reached the inlet area, which is the main reason caused the bonding of small liquid drop in the delivery hole.

Three-dimensional numerical simulations of dust mobilization and air ingress characteristics in a fusion reactor during a LOVA event

Dust mobilization and air ingress characteristics in a fusion reactor during a loss-of-vacuum accident (LOVA) event were analyzed numerically using a newly developed thermal-hydraulic analysis code under the conditions of a compressible flow and three-dimensional cylindrical coordinates. Some physical models on the motion of dust were considered to resolve the dust mobilization characteristics. Air ingress behavior through breaches was calculated by using the compressible Navier–Stokes equations. It was clarified quantitatively from the present numerical results that the predicted average pressures in a vacuum vessel (VV) agree well with the experimental results within an error of 10% for different breach diameters and the dust mobilization depends on the flow configurations of the buoyancy-driven exchange flows through breaches and the dust density, and the rate of dust released from the VV through the breaches to the outside depends on the breach positions on the VV, and furthermore, the present numerical approach is very effective to estimate the thermal-hydraulic phenomena inside the VV after the LOVA event occurs.

Stresses at the intersection of two cylindrical shells

The stress analysis based on the theory of a thin shell is carried out for two normally intersecting cylindrical shells with a large diameter ratio. Instead of the Donnell shallow shell equation, the modified Morley equation, which is applicable to ϱ 0( R T ) 1 2 ⪢ 1 , is used for the analysis of the shell with cut-out. The solution in terms of displacement function for the nozzle with a non-planar end is based on the Love equation. The boundary forces and displacements at the interaction are all transformed from Gaussian coordinates (α, β) on the shell, or Gaussian coordinates (ξ, θ) on the nozzle into three-dimensional cylindrical coordinates (π, θ, z). Their expressions on the intersecting curve are periodic functions of θ and expanded in Fourier series. Every harmonics of Fourier coefficients of boundary forces and displacements are obtained by numerical quadrature. The results obtained are in agreement with those from the finite element method and experiments for d/D ⩽ 0.8.

High order difference methods for heat equation in polar cylindrical coordinates

Fourth-order difference methods for the solution of Poisson equations in cylindrical polar coordinates are proposed. The same technique is then applied to obtain O( k 2 + h 4), two level, unconditionally stable ADI methods for the solution of the heat equation in two-dimensional polar coordinates and three-dimensional cylindrical coordinates. Numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r = 0.