Axisymmetric Coordinate System Research Articles

Stochastic weighted particle control for electrostatic particle-in-cell Monte Carlo collision simulations in an axisymmetric coordinate system

The non-uniform grids in the axisymmetric coordinate system pose a significant challenge for electrostatic particle-in-cell/Monte Carlo collision (PIC/MCC) simulations because they require numerous macroparticles to manage numerical heating around the mid-axis. To address this, we have developed a stochastic weighted particle control method that selectively samples small-weight particles, effectively controlling the particle number without inducing numerical heating. This method is based on a rejection-acceptance probability merging scheme, which is easy to implement and has a low time complexity. We have also made essential modifications, including a corrected density deposition scheme, an energy conservation scheme, and the introduction of target weights. By applying this particle control method, the number of macroparticles in the simulation can be reduced by more than one order of magnitude, significantly reducing the required computing time and storage. Furthermore, appropriately setting target weights also enables enhanced resolution of dilute regions with an acceptable increase in computational cost.

Analytical solution to the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system

The inverse heat conduction problem involves estimating an unknown cooling or heating action based on internal temperature histories in a body. Such a problem is ill-posed because the estimated results are highly sensitive to input data noise. An analytical solution is developed for the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system using the Laplace transform technique. On the basis of the known temperatures at multiple measuring points distributed on two cylindrical measuring surfaces along the axial direction inside the solid, the expressions for the surface temperature and heat flux are explicitly obtained in the form of a power series of time and eigenfunctions of the space variable. A comprehensive estimation error assessment is conducted using the internal temperature distribution that is obtained using a direct heat conduction analysis under the prescribed stationary and moving boundary conditions. The constant thermal condition is well reproduced using the present inverse solution, except in the vicinity of the discontinuity point. In addition, the effective heat flux and effective heat transfer coefficient are introduced to obtain the best estimation of the transient heat transfer in the major cooling area of the rotating cylinder.

Rheological consolidation analysis of saturated clay ground under cyclic loading based on the fractional order Kelvin model

AbstractBased on Biot porous medium theory, considering the coupled reaction of soil skeleton rheology and pore pressure dissipation, the present work investigates the dynamic consolidation characteristics of saturated clay ground under cyclic loading. First, the rheological behavior of the soil skeleton was described by the fractional order Kelvin model. The dynamic consolidation governing equations for the saturated clay were established theoretically in a three‐dimensional axisymmetric coordinate system. Second, the transform domain analytical solution of the dynamic consolidation of saturated clay was obtained using the Hankel–Laplace coupled transform method, and the solution in the time–space domain was further obtained through numerical inversion. Finally, the rheological consolidation behavior of saturated clay under cyclic loading and the influences of parameters were analyzed. The results show that the rheology of the soil skeleton had an inhibitory effect on pore water permeability; compared with elastic skeleton soil, the rheological clay had a slower settlement rate in the primary consolidation stage, a faster rate in the secondary consolidation stage, and greater long‐term settlement. In addition, under cyclic loading, the pore pressure response in saturated clay lags behind the effective stress, and a larger viscous order is associated with faster development of cumulative settlement.

A numerical study of the droplet impact dynamics on a two-dimensional random rough surface

Considerable efforts had been devoted to investigating numerically the droplet impact dynamics on a superhydrophobic surface, whereas most of these numerical simulations were restricted to the two-dimensional (2D) axisymmetric coordinate system with the one-dimensional (1D) substrate surface. In this work, a three-dimensional (3D) computational fluid dynamics (CFD) model, which intergrew a 2D random rough surface, was proposed to investigate the droplet impact dynamics, and the multi-phase flow issue was solved by the Navier–Stokes equations. It is remarkable that the 3D CFD model revealed several significant dynamic details that were not easily captured in a 2D axisymmetric coordinate system or practical experiments. For instance, the 3D CFD model provided a unique perspective to understand the varying dynamic behaviors of impinged droplet in terms of the velocity streamline and dynamic viscosity analyses. Herein, the dynamic viscosity diagram revealed that the sprawl droplet on the 2D random rough surface was classified as the Cassie state, while as the Wenzel state for the smooth surface, which also explained the better bouncing behaviors of the droplet from the random rough surface. Accordingly, we suggested a visual way to evaluate the solid–liquid contact area surrounded by the triple-phase contact line. The effects of finger protrusion and central cavity growth from the sprawl droplet on the vortex generation were further analyzed on the ground of the velocity amplitude distribution and streamline data. The present work can provide early guidance to inquire into the impact dynamics of droplets on the random rough surface.

Combination of Eulerian and ray-tracing approaches for copper laser welding simulation

Laser welding of pure copper and its alloys is a challenging process with a growing industrial interest due to the latest development in the field of electric mobility. The difficulties are mainly related to the material's high thermal conductivity and a poor absorptivity of few percent at the classical IR laser (YAG). It is also well known that such a configuration can lead to the formation of undesirable defects, such as pores or spatters as a consequence of melt pool instabilities. It has been observed experimentally that the usage of a laser at both high speed and high power tends to limit those instabilities. Although this positive influence has already been observed for equivalent materials, a physical explanation is not yet available. In this perspective, a multiphysical simulation of the process at the melt pool scale is currently being developed by using comsol Multiphysics® software. The latter includes an Eulerian interface tracking method for the liquid-gas interface (phase field) and a ray-tracing description of the laser beam to take into account the well-known beam trapping effect under a keyhole regime. For the sake of time computation, the numerical model is first developed in an axisymmetric coordinate system (r,z) to be representative of a laser spot welding process and to validate the numerical coupling methodology. The model will then be extended to a 3D welding case and used as a predictive tool to make appropriate choices on welding parameters to obtain good quality welds (stable melt pool, low porosity rate, etc.).

Hydrodynamic simulation studies on avalanche and streamer formation in GEM detector

The dynamics of electrons and ions in gaseous ionization detectors have been studied reasonably well with particle simulation models developed using the Garfield++ numerical simulation framework. This is an important area of study since it allows prediction of the detector response in a given experimental situation. In this work, a fluid simulation model has been developed in the COMSOL Multiphysics simulation framework to simulate the avalanche and streamer formation in GEM-based detectors. Possible detector geometries in 2D, 2D axisymmetric and 3D coordinate systems have been explored to find the optimum numerical configuration. Transport of charged fluids has been simulated in the optimized model for various operating voltage ranges suitable for single, double and triple GEM detectors using Ar-CO2 (70-30) as the gas mixture. Simulated gain variations have been compared with experimental observations. Effect of space charge and its relation to streamer formation have been studied.

Axisymmetric free-surface flow simulation using the moving surface mesh particle method and application to drop formation

It is difficult to accurately simulate axisymmetric free-surface flows with surface tension and topological changes, such as drop formation through a nozzle. A promising numerical method that can achieve this goal is the moving surface mesh particle method. It incorporates a moving surface mesh to explicitly represent a free-surface boundary, which enables accurate free-surface tracking and surface tension calculation. In the present study, this method is extended to an axisymmetric coordinate system. In the axisymmetric simulation, particles moving in a Lagrangian fashion result in significantly non-uniform particle distributions, leading to numerical instabilities. To circumvent this issue, a new algorithm is developed for determining the particle movement in an ALE fashion by considering a two-dimensional continuity equation. Moreover, in order to deal with the topological change arising from breakup of liquid domain, a simple algorithm employing surface mesh splitting is developed. The proposed method is verified using fundamental test cases such as axisymmetric cavity flow, axisymmetric patch test, spherical droplet oscillation, and hanging droplet. Therein, precise agreements to reference solutions are confirmed. Furthermore, the proposed method is applied to the numerical simulation of dripping faucet. As a result, complex drop formation phenomena are successfully simulated with reasonable computational cost. The numerical results are also shown to agree well with the experimental measurement in the literature, from which the validity of the proposed method is demonstrated.

Interface reconstruction and advection schemes for volume of fluid method in axisymmetric coordinates

Volume of fluid (VOF) method is a sharp interface method employed for simulations of two phase flows. Interface in VOF is usually represented using piecewise linear line segments in each computational grid based on the volume fraction field. While VOF for cartesian coordinates conserve mass exactly, existing algorithms do not show machine-precision mass conservation for axisymmetric coordinate systems. In this work, we propose analytic formulae for interface reconstruction in axisymmetric coordinates, similar to those proposed by Scardovelli and Zaleski (2000) [12] for cartesian coordinates. We also propose modifications to the existing advection schemes in VOF for axisymmetric coordinates to obtain higher accuracy in mass conservation.

AI OPTIMIZATION OF THE EXOTHERMIC REACTION OF ETHYLENE OXIDE WITH WATER

A computational framework, for the numerical approximation of the exothermic reaction of ethylene oxide (EO) with water, to form ethylene glycol is presented in this paper. Ethylene Glycol also known as Mono-ethylene Glycol (MEG), is a diol with a boiling of 198[Formula: see text]C and conventionally produced through hydrolysis of ethylene oxide which is obtained through the oxidation of ethylene. It is used as an excellent automobile coolant as the 1:1 ratio mixture of MEG with Water boils at 129[Formula: see text]C and freezes at [Formula: see text]C. Other than its use as an antifreeze, it is also used as a reagent during the production of polyester fibers, pharmaceutics, cosmetics, hydraulic fluids, printing inks, explosives, polyesters and paint solvents. The mathematical model presented here, consists of an energy balance and a material balance system, described in an axisymmetric coordinate system. The optimized resulting values using the artificial intelligence approach are summarized in this paper. We derive an analytical solution. The analytical solution for the mathematical model equations is in general not possible for this model but it may be possible to derive an analytical solution to this mathematical model if we consider the equation for the conservation of material (chemical species) as a formulation for plug flow and isothermal conditions. Noteworthy findings are reported in this paper for future research.

Numerical solutions for strain-softening surrounding rock under three-dimensional principal stress condition

Abstract In this study, elastoplastic model for the surrounding rock of axisymmetric circular tunnel is investigated under three-dimensional (3D) principal stress states. Novel numerical solutions for strain-softening surrounding rock were first proposed based on the modified 3D Hoek–Brown criterion and the associated flow rule. Under a 3D axisymmetric coordinate system, the distributions for stresses and displacement can be effectively determined on the basis of the redeveloped stress increment approach. The modified 3D Hoek–Brown strength criterion is also embedded into finite element software to characterize the yielding state of surrounding rock based on the modified yield surface and stress renewal algorithm. The Euler implicit constitutive integral algorithm and the consistent tangent stiffness matrix are reconstructed in terms of the 3D Hoek–Brown strength criterion. Therefore, the numerical solutions and finite element method (FEM) models for the deep buried tunnel under 3D principal stress condition are presented, so that the stability analysis of surrounding rock can be conducted in a direct and convenient way. The reliability of the proposed solutions was verified by comparison of the principal stresses obtained by the developed numerical approach and FEM model. From a practical point of view, the proposed approach can also be applied for the determination of ground response curve of the tunnel, which shows a satisfying accuracy compared with the measuring data.

Experimental and Numerical Study for the Coalescence Dynamics of Vertically Aligned Water Drops in Oil.

In this paper, we propose an experimental and numerical investigation for the impact of the surface tension and the continuous phase viscosity on the dynamics of the liquid bridge during the coalescence process in liquid-liquid systems. A specific configuration of a sessile drop in direct contact with another drop placed over it has been studied. Calculating the redefined Reynolds number Re, it is found that for all studied cases, the coalescence process is dominated by the inertial force. The first step of the work was the validation of the numerical model that has been performed in an axisymmetric coordinate system. This has been done by the comparison between numerical and experimental results obtained in the framework of experimental series realized in parallel for two different liquid-liquid (LL) systems: water drops in silicone oil (SilOil M40.165) and water drops in sunflower oil. A good agreement was found between different results for numerous parameters used for comparisons. It is found that for the first stages of the coalescence (at the start of the drops merging), for a given drop's viscosity, the dynamics of the dimensionless liquid bridge is conducted by the viscosity of the continuous phase where it is illustrated that the more the surrounded viscosity is large, the lower the rate of the liquid bridge growth, the lower the earlier radial velocity of the bridge, and the higher the external capillary pressure generated around the bridge. Moreover, it is depicted that the impact of the surface tension starts appearing after the complete development of the liquid bridge where it is observed that for the same surrounding phase viscosity, the propagation of the capillary wave is faster for a LL system with higher surface tensions than those of lower surface tensions.

Nonlinear ultrasound simulation in an axisymmetric coordinate system using a k-space pseudospectral method.

A full-wave model for nonlinear ultrasound propagation through a heterogeneous and absorbing medium in an axisymmetric coordinate system is developed. The model equations are solved using a nonstandard or k-space pseudospectral time domain method. Spatial gradients in the axial direction are calculated using the Fourier collocation spectral method, and spatial gradients in the radial direction are calculated using discrete trigonometric transforms. Time integration is performed using a k-space corrected finite difference scheme. This scheme is exact for plane waves propagating linearly in the axial direction in a homogeneous and lossless medium and significantly reduces numerical dispersion in the more general case. The implementation of the model is described, and performance benchmarks are given for a range of grid sizes. The model is validated by comparison with several analytical solutions. This includes one-dimensional absorption and nonlinearity, the pressure field generated by plane-piston and bowl transducers, and the scattering of a plane wave by a sphere. The general utility of the model is then demonstrated by simulating nonlinear transcranial ultrasound using a simplified head model.

Near-surface dynamics of a gas bubble collapsing above a crevice

Abstract

Coalescence of vertically aligned drops over a superhydrophobic surface

The coalescence process of two liquid droplets where one is placed initially over the other is investigated. The lower drop is placed over a horizontal surface in a sessile configuration. The liquids of interest selected are water, glycerin, and Cs-alloy. The two liquid drops merge under atmospheric conditions. The substrate is superhydrophobic with respect to the three liquids, the equilibrium contact angle being 150°. For the combined drop, the Bond number is ∼0.2. Numerical simulations have been performed in an axisymmetric coordinate system along with supporting experiments. A variety of contact line models reported in the literature have been adopted and compared. Experiments are carried out for validation against simulation with water as the liquid medium. The coalescence phenomenon is recorded by a high-speed camera. The two drops coalesce spontaneously and generate interfacial shapes, velocity fields, footprint, and wall shear stress in time. In water, the combined drop recoils from the surface before spreading over the surface and approaching equilibrium. This trend, including the instant and height of recoil, is correctly realized in the contact line models. Additionally, two distinct timescales originate during the coalescence process. These are associated with inertia and surface tension at small times and inertia–viscosity for longer durations. The instantaneous footprint radius and the average wall shear stress fall to zero during recoil, increase then to a maximum, and diminish to zero with damped oscillations over the longer timescale. Recoil is seen in water as well as Cs-alloy, but not in glycerin. Despite differences in the instantaneous data, these predictions are broadly reproduced by each of the contact line models.

Axisymmetric lattice Boltzmann model for simulation of ternary fluid flows

In this paper, a lattice Boltzmann model based on the phase-field Cahn–Hilliard-based approach is proposed to deal with ternary fluid flows in the axisymmetric coordinate system. The present model can handle both high density and viscosity ratios for partial and total spreading scenarios. A variety of numerical tests, including a static droplet test, a breakup of a liquid thread test, a spreading of a liquid lens test, and a droplet collision test, were performed to verify the model. It is found that the proposed model produces accurate results which agree well with the available data.

Comparison of two interfacial flow solvers: Specific case of a single droplet impacting onto a deep pool

A numerical study about a droplet impacting onto a deep pool is presented here by employing the Volume of Fluid method to track the interface. In this investigation, two open source solvers, named Gerris and Basilisk, are used to solve the incompressible Navier–Stokes equations with free surface. The results are compared to provide some reference for the researchers involved in this research community. Firstly, the capabilities of the two solvers in simulating the droplet impacting problems are validated against available experimental results. And the adaptive mesh refinement techniques used in the two solvers are also found to be applicable. When simulating the drop impacting problems in an axisymmetric coordinate system, it is found that less meshes are produced in Gerris than that in Basilisk when similar adaptive criterion is adopted in both solvers. However, Basilisk still shows much higher computational efficiency due to its great superiority at parallelization, and thus less CPU time is required for such unsteady problems. When simulating moderate Reynolds number impacting problems, the two numerical solvers show perfect agreement (Re≤5000), however, since higher Reynolds numbers (Re≥5000) require smaller size of mesh in vicinity of the thin impacting region and the concentrated vortex region, it is hard for Gerris to obtain satisfactory results because of its poor computational efficiency. Thus, with such a high parallel capability and computational efficiency, Basilisk is introduced to solve a large number of cases. It is summarized that the flow pattern of the drop impacting problem can be classified to three categories. They are smooth ejecta sheet, main vortex shedding and Von Kármán vortex street. Each flow pattern relates to the vortex evolution inside the liquid closely. Basililsk also shows its potential in solving much more complicated three-dimensional problems. They cannot be completed by Gerris owing to its poor computational efficiency.

Separability of Stokes Equations in Axisymmetric Geometries

For Stokes flow in non spherical geometries, when separation of variables fails to derive closed form solutions in a simple product form, analytical solutions can still be obtained in an almost separable form, namely in semiseparable form, R-separable form or R-semiseparable form. Assuming a stream function Ψ, the axisymmetric viscous Stokes flow is governed by the fourth order elliptic partial differential equation E4Ψ = 0 where E4 = E2oE2 and E2 is the irrotational Stokes operator. Depending on the geometry of the problem, the general solution is given in one of the above separable forms, as series expansions of particular combinations of eigenfunctions that belong to the kernel of the operator E2. In the present manuscript, we provide a review of the methodology and the general solutions of the Stokes equations, for almost any axisymmetric system of coordinates, which are given in a ready to use form. Furthermore, we present necessary and sufficient conditions that are serving as criterion for identifying the kind of the separation the Stokes equation admits, in each axisymmetric coordinate system. Additionally, as an illustration of the usefulness of the obtained analytical solutions, we demonstrate indicatively their application to particular Boundary Value Problems that model medical problems.

Necessary and Sufficient Conditions for the Separability and the R-Separability of the Irrotational Stokes Equation and Applications

In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric system, namely: simple separation and R-separation. These conditions may serve as a road map for obtaining the corresponding solution space of the irrotational Stokes equation, in any orthogonal axisymmetric coordinate system. Additionally, we investigate how the inversion of the coordinate system, with respect to a sphere, affects the type of separation. Specifically, we prove that if the irrotational Stokes equation separates variables in an axisymmetric coordinate system, then it R-separates variables in the corresponding inverted coordinate system. This is a quite useful outcome since it allows the derivation of solutions for a problem, from the knowledge of the solution of the same problem in the inverted geometry and vice-versa. Furthermore, as an illustration, we derive the eigenfunctions of the irrotational Stokes equation governing the flow past oblate spheroid particles and inverted oblate spheroidal particles.

Dynamics of Growth and Breakup of an Evaporating Pendant Drop

Abstract The growth and pinch-off dynamics of an evaporating pendant drop have been studied through direct numerical simulations. A coupled level-set and volume-of-fluid method is utilized to perform the simulations in an axisymmetric coordinate system. The dynamics of an evaporating pendant drop depends on the combined effects of buoyancy, capillary force, and the evaporation rate at the interface. The volumetric growth-rate of the drop decreases with the increase in degree of superheat of the surrounding medium, thereby enhances the pinch-off time. However, the departure diameter decreases with an increase in superheat. Limiting length and neck radius are not significantly affected due to the variation in degree of superheat. The heat transfer characteristics at different stages during the growth of a pendant drop have been analyzed for various values of superheat.

Group properties of equations describing the motion of a liquid metal with the inclusion of nanopowderinoculators

A group classification of equations describing the motion of a liquid metal with the inclusion of nanopowder inoculators is carried out in relation to the dependence of the viscosity coefficient on temperature and the fraction of the solid phase in the melt. The motion of a liquid metal with the inclusion of nanopowder inoculators will be described by the Navier— Stokes system of equations in an axisymmetric coordinate system taking into account the force of gravity directed against the axial coordinate and the heat transfer equation in a liquid, taking into account crystallization. The viscosity of a liquid is considered dependent on temperature and the fraction of the solid phase in the melt. The widest permissible group is obtained in the case of a constant viscosity coefficient. Also, the expansion of the permissible group occurs with some special types of dependence of the viscosity coefficient on temperature and the fraction of the solid phase in the melt.