Axisymmetric Domain Research Articles

IsoGeometric approximations for electromagnetic problems in axisymmetric domains

We propose a numerical method for the solution of electromagnetic problems on axisymmetric domains, based on a combination of a spectral Fourier approximation in the azimuthal direction with an IsoGeometric Analysis (IGA) approach in the radial and axial directions. This combination allows to blend the flexibility and accuracy of IGA approaches with the advantages of a Fourier representation on axisymmetric domains. It also allows to reduce significantly the computational cost by decoupling of the computations required for each Fourier mode. We prove that the discrete approximation spaces employed constitute a closed and exact de Rham sequence. Numerical simulations of relevant benchmarks confirm the high order convergence and other computational advantages of the proposed method.

Multigrid in H(div) on axisymmetric domains

In this paper, we will construct and analyze a multigrid algorithm that can be applied to weighted H(div) problems on a two-dimensional domain. These problems arise after performing a dimension reduction to a three-dimensional axisymmetric H(div) problem. We will use recently developed Fourier finite element spaces that can be applied to axisymmetric H(div) problems with general data. We prove that if the axisymmetric domain is convex, then the multigrid V-cycle with modern smoothers will converge uniformly with respect to the meshsize.

Superconvergent Algorithms for the Numerical Solution of the Laplace Equation in Smooth Axisymmetric Domains

A fundamentally new—nonsaturable—method is constructed for the numerical solution of elliptic boundary value problems for the Laplace equation in $${{C}^{\infty }}$$-smooth axisymmetric domains of fairly arbitrary shape. A distinctive feature of the method is that it has a zero leading error term. As a result, the method is automatically adjusted to any redundant (extraordinary) smoothness of the solutions to be found. The method enriches practice with a new computational tool capable of inheriting, in discretized form, both differential and spectral characteristics of the operator of the problem under study. This underlies the construction of a numerical solution of guaranteed quality (accuracy) if the elliptic problem under study has a sufficiently smooth (e.g., $${{C}^{\infty }}$$-smooth) solution. The result obtained is of fundamental importance, since, in the case of $${{C}^{\infty }}$$-smooth solutions, the solution is constructed with an absolutely sharp exponential error estimate. The sharpness of the estimate is caused by the fact that the Aleksandrov $$m$$-width of the compact set of $${{C}^{\infty }}$$-smooth functions, which contains the exact solution of the problem, is asymptotically represented in the form of an exponential function decaying to zero (with growing integer parameter $$m).$$

Numerical investigation of heat transfer from a plane surface due to turbulent annular swirling jet impingement

The heat transfer characteristic of an annular turbulent swirling jet impingement on a heated plane surface is investigated numerically. The annular jet configuration causes instabilities and fluctuations in the flow. Depending on the combinations of different parameters, the annular jet impingement may have positive or negative effects on the heat transfer from the impingement surface. Swirl, on the other hand, introduces vorticity and fluid mixing in an impinging jet, which is desirable in some applications. The axisymmetric two-dimensional flow domain on a radial-axial plane is considered in the numerical model. Both non-swirling and swirling jet impingement is studied. The numerical computations are performed using the ANSYS Fluent CFD code. The Realizable k-ε turbulence model with enhanced wall treatment is used in the computation. The computational process is validated against other published data on similar flow configuration for non-swirling annular impinging jets. The flow and geometric parameters are the jet exit Reynolds number, Re (5000 to 25,000), the swirl strength Sw (0–0.77), the jet exit to the impingement surface distance, H (0.5–4.0), and the moderate blockage ratio of the annular jet, BR (0.4–0.6). The thermal-hydraulic field in the domain is computed for various combinations of these parameters. The effects of these parameters on the Nusselt number distribution on the impingement surface are analyzed. For short separation distance (H = 0.5), the swirling motion positively affects the overall heat transfer, and the average Nu is increased as high as 8% for certain combinations of Re, Sw, and BR; compared to the non-swirling annular jet impingement. For higher separation distances, the average and peak Nusselt number is initially reduced and then increased with increasing swirl strength.

Numerical Study of Formation of a Series of Bubbles from an Orifice by Applying Dynamic Contact Angle Model

The dynamic contact angle model is applied in the formation process of a series of bubbles from Period‐I regime to Period‐II regime by using the VOF method on a 2D axisymmetric domain. In the first process of the current research, the dynamic contact angle model is validated by comparing the numerical results to the experimental data. Good agreement in terms of bubble shape and bubble detachment time is observed from a lower flow rate Q = 150.8 cm3/min (Re = 54.77, Period‐I regime) to a higher flow rate Q = 603.2 cm3/min (Re = 219.07, Period‐III regime). The comparison between the dynamic contact angle model and the static contact angle model is also performed. It is observed that the static contact angle model can obtain similar results as the dynamic contact angle model only for smaller gas flow rates (Q ≤ 150.8 cm3/min and Re ≤ 54.77)). For higher gas flow rates, the static contact angle model cannot produce good results as the dynamic contact angle model and has larger relative errors in terms of bubble detachment time and bubble shape.

Wall Shear Stress in Pulsating Pipe Flow at Resonance Conditions

Hydraulic vibrations in liquid-filled pipelines may cause unwanted operational problems. Wall shear stress and consequential damping can reduce the harmful effects of vibrations close to resonance. A large-scale pipeline apparatus at Deltares, Delft, The Netherlands, has been used for pulsating pipe flow tests. The apparatus consists of a constant-head tank, a horizontal 206 mm diameter 49 meters long steel pipe and an oscillating valve at the downstream end. Wall shear stress has been measured by a number of hot-film sensors. Tests have been performed with an average flow Reynolds number of about 22,000. Results of a hydraulic resonance case with oscillating frequency fex = 5 Hz are presented. Pipe wall vibrations for this case are small. The shape of the velocity profile at resonance conditions is a typical unsteady-state velocity profile with reverse flow near the pipe wall. The CFD study in an axisymmetric domain was conducted to better understand the pulsating flow phenomena. Different settings of boundary conditions, based on the experimental investigations, were used. The CFD results show the unsteady character of wall shear stress at resonance. This phenomenon has not been observed in the measured results to such an extent. The measured shear stress resembles quasi-steady behaviour.

Development of a semi-empirical 1.5D plasma fluid model for a single microdischarge in atmospheric pressure dielectric barrier discharges

In this work, a semi-empirical 1.5D plasma fluid model (PFM) is proposed to model a single microdischarge (MD) in atmospheric pressure air dielectric barrier discharges (APADBDs). The species continuity equations and the electron energy density equation are solved in one-dimensional domain, while the Poisson equation is solved in the axisymmetric domain to capture the enhancement of the electric field in front of the streamer. The framework of air chemistry is considered and the effect of photoionization is modeled in the axisymmetric domain. The accumulation factor (AF) is introduced and determined by experimental data to model the accumulation of charged particles on the dielectric surface. The simulated results in two gaps are compared with experimental measurements. In the gap of 1.4 mm, the simulated electric current reaches 72 mA, which is close to the typically measured electric current. The simulated maximum wave velocity is around 1.7 × 106 m s−1, which is close to the available experimental data. The change of simulated charge density implies that the average accumulation of charged particles on the dielectric surface during each half period (HP) is around 40 nC cm−2, which is in the same order of magnitude as the average charge density evaluated in the previous measurements as 51.5 nC cm−2. The effect of AF is studied and shows that the AF determines both peak and duration of the electric current. In the gap of 2.0 mm, the simulated current reaches 113 mA, which is close to the typically measured current. Although the gap voltage of the 2.0 mm gap is higher than that of the 1.4 mm gap, the average electric field of the 2.0 mm gap is lower than that of the 1.4 mm gap before breakdown due to larger gap distance. The maximum wave velocity is faster than that simulated in the gap of 1.4 mm due to the longer gap distance for developing higher wave velocity as 2.4 × 106 m s−1. During each HP, the average accumulation of charge density on the dielectric surface reaches around 40 nC cm−2 which is almost identical to that simulated in the gap of 1.4 mm as observed experimentally. In general, the proposed semi-empirical 1.5D PFM captures the dynamics of a single MD in APADBDs.

Numerical study of the melting and resolidification of the substrate during the impact of a ceramic droplet in a plasma spraying process

The substrate melting can significantly improve the properties of plasma spray coatings. Indeed the adhesion of the projected particles to the substrate can be ameliorated by the substrate melting. In this article, a numerical model is developed to study the dynamics of fluids and heat transfer with liquid/solid phase change during impact of a fully melted alumina particle on an aluminum solid substrate, taking into account of the substrate melting. The model is based on solving the Navier-Stokes and energy equations with liquid / solid phase change. These equations are coupled with the fluid of volume method (VOF), to follow the free surface of the particle during its spreading and solidification. The finite volume method is used to discretize the equations in a 2D axisymmetric domain. A comparison with the published experimental results was carried out to validate this numerical model. Simulations were performed for different initial droplet diameters to study its effect on droplet spreading as well as on substrate melting. It has been observed that the substrate melting begins before the droplet spreads completely; the substrate melting reaches its maximum when the droplet is close to its total solidification. Droplet spreading and substrate melting are more important for large sizes droplets.

Pinning boundary conditions for phase-field models

Abstract In this paper, we present pinning boundary conditions for two- (2D) and three-dimensional (3D) phase-field models. For the 2D and axisymmetric domains in the neighborhood of the pinning boundaries, we apply an odd-function-type treatment and use a local gradient of the phase-field for points away from the pinning boundaries. For the 3D domain, we propose a simple treatment that fixes the values on the ghost grid points beyond the discrete computational domain. As examples of the phase-field models, we consider the Allen–Cahn and conservative Allen–Cahn equations with the pinning boundary conditions. We present various numerical experiments to demonstrate the performance of the proposed pinning boundary treatment. The computational results confirm the efficiency of the proposed method.

Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

Abstract We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.

Investigation of bubble formation and its detachment in shear-thinning liquids at low capillary and Bond numbers

In the present paper, the formation of an air bubble in a shear-thinning non-Newtonian fluid was investigated numerically. For modeling, an algebraic volume of fluid (VOF) solver of $$\hbox {OpenFOAM}^\circledR $$ was improved by applying a Laplacian filter and was evaluated using the experimental results from the literature. The enhanced solver could compute the surface tension force more accurately, and it was important especially at low capillary and Bond numbers due to the dominance of surface tension force relative to the other forces. The adiabatic bubble growth was simulated in an axisymmetric domain for $$\hbox {Bo}=0.05,0.1,0.5$$ and $$\hbox {Ca}=10^{-1},10^{-2},10^{-3},10^{-4}$$ , and the bubble detachment time and volume were examined. According to the results, for Newtonian fluids, there is a critical capillary number for a given Bond number, and for lower values of this critical number, no difference is observed between the bubble detachment volumes and also bubble detachment times. Similarly, the results indicated that for non-Newtonian fluids, if apparent capillary number (obtained by apparent viscosity) is less than the critical capillary number, the detachment volume is the same as the corresponding Newtonian case. The velocity of the bubble during its formation in Newtonian and shear-thinning fluids was also studied. Moreover, the bubble formation and detachment characteristics such as instantaneous contact angle and necking radius were investigated for Newtonian and non-Newtonian liquids, and the shear-thinning effect was examined as well. The results indicated that changing the ambient fluid to a non-Newtonian liquid has no effect on the trend of the contact angle; however, the minimum contact angle has a higher value when the shear-thinning effect increases. Also, the variation of neck radius with the time until detachment was presented by the power relation $${R}_{\mathrm{n}} =\left( {{t}_{\mathrm {det}} -{t}} \right) ^{{\eta }}$$ , and the effect of shear thinning on its exponent $${\eta }$$ was investigated.

Comparison of interfacial heat transfer correlations for high-pressure subcooled boiling flows via CFD two-fluid model

Heat transfer processes using two-phase boiling flows are often found in the industry, due to the high-efficiency heat removal, with a minimum difference of temperature between the heated surface and fluid. Moreover, the use of computational fluid dynamics to solve safety issues related to nuclear reactors has increased rapidly, but a complete validation is still being carried out. Therefore, this study aims the assessment of different sub-models of the interfacial heat transfer coefficient which is used as a closure relation in the two-fluid multiphase model. As a manner to validate the numerical results, the set of experimental conditions of Bartolomei and Chanturiya (Therm Eng 14:123–128, 1967) were applied to an upwards flow in a circular channel, under high water saturation pressures. The interfacial sub-models were implemented into an axisymmetric simulation domain, using the Eulerian two-fluid approach. Three different saturation pressures and two uniform heat fluxes were considered in the simulation runs. Fixed mass flux and subcooled degree of 900 kg/m2 and 60 K were applied, respectively. A satisfactory agreement was achieved for the estimation of the heated wall temperature, the liquid bulk temperature and the onset of saturated boiling. Different heat transfer closure relations promoted different vapor volume fraction along the channel, indicating the importance of an adequate interfacial heat transfer correlation to predict the flow boiling phenomena.

Passage of a bubble through the interface between a shear-thinning heavier liquid and a Newtonian lighter liquid

Numerical simulations are carried out to study the phenomenon of passage of a bubble through the interface between two initially quiescent liquids. The heavier liquid is a viscous shear-thinning liquid whereas the lighter liquid is a viscous Newtonian liquid. The computational model, which is based on axisymmetric computational domain and VOF (volume of fluid) method, is validated by using the data reported in literature. The validated computational model is then used to perform parametric analysis to understand the effects of bubble diameter, interfacial tension between the two liquids and flow index of the heavier liquid on the drainage time, drainage height, retention time and retention height of the bubble at the liquid–liquid interface. Finally, correlations for retention time and retention height are proposed.

Thermomechanical surface instability at the origin of surface fissure patterns on heated circular MDF samples

SummaryWhen a flat sample of medium density fibreboard (MDF) is exposed to radiant heat in an inert atmosphere, primary crack patterns suddenly start to appear over the entire surface before pyrolysis and any charring occurs. Contrary to common belief that crack formation is due to drying and shrinkage, it was demonstrated for square samples that this results from thermomechanical instability. In the present paper, new experimental data are presented for circular samples of the same MDF material. The sample was exposed to radiant heating at 20 or 50 kW/m2, and completely different crack patterns with independent eigenmodes were observed at the two heat fluxes. We show that the two patterns can be reproduced with a full 3‐D thermomechanical surface instability model of a hot layer adhered to an elastic colder foundation in an axisymmetric domain. Analytical and numerical solutions of a simplified 2‐D formulation of the same problem provide excellent qualitative agreement between observed and calculated patterns. Previous data for square samples, together with the results reported in the present paper for circular samples, confirm the validity of the model for qualitative predictions and indicate that further refinements can be made to improve its quantitative predictive capability.

Modeling of Material Removal Rate and Surface Roughness Generated during Electro-Discharge Machining

This study reports on the numerical model development for the prediction of the material removal rate and surface roughness generated during electrical discharge machining (EDM). A simplified 2D numerical heat conduction equation along with additional assumptions, such as heat effect from previously generated crater on a subsequent crater and instantaneous evaporation of the workpiece, are considered. For the material removal rate, an axisymmetric rectangular domain was utilized, while for the surface roughness, a rectangular domain where every discharge resides at the end of previous crater was considered. Simulated results obtained by solving the heat equation based on a finite element scheme suggested that results are more realistic by considering instantaneous evaporation of the material from the workpiece and the effect of residual heat generated from each spark. Good agreement between our model and previously published data validated the newly proposed models and demonstrate that instantaneous evaporation, as well as residual heat, provide more realistic predictions of the EDM process.

CFD modeling of liquid film reversal of two-phase flow in vertical pipes

The phenomenon of two-phase flows is characterized by its wide range of presence in nature and industrial applications. In gas and oil production, the simultaneous two-phase flow of gases and liquids often occurs in gas wells. The produced gases flow rate decreases over the years until reaching a critical flow rate when they lose their ability to lift the associated liquids upward and the liquid film reverses direction, which triggers liquid loading. Liquid loading causes the produced liquids to accumulate in the bottom of the wellbore, which causes a high back pressure that reduces the well production rate till production is ceased eventually. The critical gas velocity exists in the churn flow regime which is mainly characterized by the oscillatory behavior of the liquids flow field. This study employs CFD techniques to model the churn flow in a 3-inch-diameter vertical pipe near the critical gas flow rate for different liquid flow rates. This work utilizes the two-fluid Eulerian model along with the RNG (Re-Normalization Group) k-ε turbulence model to investigate the behavior of the flow field in a two-dimensional axisymmetric computational domain. Simulations were carried out using the commercial software ANSYS Fluent 18.2 with air and water as the two working fluids. The model results showed a good agreement with the experimental data and proved the mesh and time independence of the model. Oscillatory behaviors of the liquid film flow rate, shear stress, and pressure were observed along with the formation of interfacial waves. Detailed information about the velocity, shear stress, and pressure behaviors is presented. Accordingly, recommendations are suggested for future considerations.

EVALUATION OF STATIC PRESSURE BEHAVIOR IN AN OSCILLATING WATER COLUMN WAVE ENERGY CONVERTER

The international scenario of non-renewable resources scarcity coupled with increasing energy demand are incentives for the diversification of the world's energy matrix with a focus on renewable energy sources. Among these sources, energy from sea waves is especially attractive because its global resource is estimated around 2 TW, comparable to the average electrical power consumed worldwide each year. There are currently several technologies proposed for the sea wave energy conversion into electricity. Among them it stands out the Oscillating Water Column (OWC) converter, which basically consists of a hydropneumatic chamber and a turbine duct where a turbine is installed. Its chamber is opened below the sea water free surface while the turbine duct outlet is free to atmosphere. Inside the chamber the water free surface oscillating movement produced by the incident waves causes the air to flow through the turbine duct and to activate the turbine, so the OWC principle of operating can be approximated to a cylinder-piston system. Therefore, one of the methodologies used in the computational modeling to simulate the operating principle of this device is the Piston Methodology, which simplifies the problem analysis considering only the air flow through the OWC converter. Among the phenomena that occur within the OWC device, the static pressure behavior is arguably one of the most important because it is through it that it is possible to estimate the hydropneumatic power and the converter efficiency. Thus, the objective of this work is to evaluate the static pressure behavior within the OWC, using the Piston Methodology, by imposing a monochromatic wave boundary condition in an axisymmetric domain. Among the obtained results it was inferred that the static pressure, in this case, depends directly on the flow acceleration and it is strongly influenced by the vorticity generated in domains with a change of area.

Asymptotic stability of a dual-scale compact method for approximating highly oscillatory Helmholtz solutions

A novel dual-scale compact method for solving nonparaxial Helmholtz equations at high wavenumbers is proposed and analyzed. The approach is based on decomposing the axisymmetric transverse domain and governing equation according to interconnected micro and macro regions to maintain the smoothness of the underlying problem. Dual compact strategies are then implemented for acquiring highly accurate and efficient beam propagation computations. Aiming at the highly oscillatory solution features, the paper provides a rigorous analysis on the numerical stability. It is shown that the dual-scaled compact method is asymptotically stable. The analysis also reveals necessary constraints for the conventional stability. Computer experiments including self-focusing beam propagation simulations are conducted with various domain scaling factors to validate the theoretical results.

Computational evaluations of HYDRUS simulations of drip irrigation in 2D and 3D domains (i-Surface drippers)

HYDRUS is a popular package for simulating water and solutes movement in variably-saturated porous media. It is widely used by designers of drip irrigation systems to simulate the active wetting pattern. However, there are many design parameters of the domain and for the flux calculation; other than the acceptable simulations, some configurations might cause under-estimation or over-estimation of the wetting pattern. Comparative assessments of the surface drip simulation parameters were performed. We evaluated the effect of different domain geometries, element shapes, coordinate systems, emitter discharges, and soil textures on the accuracy and stability of the simulation. The results showed that the 3D simulations are more successful and reliable than the 2D simulations in terms of mass balance error. For the 2D domains, the error increases as the texture goes finer, and for axisymmetric domains than the Cartesian ones. We found a relationship between the flux and the soil hydraulic conductivity that their quotient should not exceed 1.8 and 3.2 for axisymmetric and cartesian coordinates. Our outcomes show that the infiltration stage took more time than the redistribution stage because of the variation in water content that delays the convergence at infiltration.Additionally, we found that the range between the saturated and the residual water content of the soil causes instability of the simulation as it gets higher. The differences between 2D simulations and the equivalent 3D slices were highly dependent on the soil texture, the heavier the larger; and dependent on the simulation phase, where the redistribution phase has fewer differences than the infiltration phase. Finally, we have determined the best matching 2D flux profile to each 3D profile and revealed the source of variation between 2D and 3D profiles where it was mostly-affected by the emitter’s location, the soil texture, and simulation phase. In part 2 of this paper, we expanded the study for subsurface drip simulations and showed the difference between the two systems’ designs.

Numerical study of formation of a series of bubbles at a submerged orifice

• Improved numerical results with higher grid resolution. • New study on the impact of contact angle during bubble formation. • Different periodicity observed with fixed gas flow rate . • Simulation and comparison of inserted tubes. Bubble formation from a submerged orifice is widely applied in bio-process and chemical reaction systems. In this study, the effects of different orifice diameters and contact angles in Period-I and Period-II regimes are studied systematically on a 2D axisymmetric domain. Simulation results are presented from the formation of the first bubble and explained by means of the surrounding fluid field, bubble interaction, and bubble aspect ratio. The orifice diameter is varied from 0.6 mm to 3 mm. The numerical results show that the detachment time of all bubbles remains constant (in time) for smaller orifice diameters ( d a ≤ 1.5 mm), while the detachment time of the first bubble is different from the rest of the bubbles for larger orifice diameters ( d a ≥ 2 mm), which is due to the different surrounding flow field. Contact angles from 60° to 165° are considered for the gas flow rates in the regime of bubble pairing, and it is observed that the bubble detachment time decreases when the contact angle increases, and it converges to a constant value when the contact angle is larger than 135°. In addition, the transition from period doubling to deterministic chaos (in which there is a variable number of bubbles within each period) is observed. A new scenario of inserting a submerged tube upward into the liquid is considered and compared to the previous cases. It is observed that when the tube is vertically inserted into the liquid, the bubble detachment time is even smaller because of higher influence from the surrounding liquid field, leading to a different phenomenon from the non-inserted tube cases.