Problem In Cylindrical Coordinates Research Articles

The polar-Cartesian mild-slope equation

A novel mild-slope equation is derived based on a manipulation of the cylindrical and Cartesian coordinate reference systems. The vertical profile of the velocity field is constructed by solving an approximate problem in cylindrical coordinates. This allows us to address the local derivatives on the bottom profile along a constant-slope line. This formulation is as opposed to the Cartesian-based mild-slope equations in terms of which the profile is constructed by assuming a constant depth. An angular profile is derived for the three-dimensional case on a sloping plane beach. For the two-dimensional case, a mild-slope polar-Cartesian equation is derived, for which an improved linear dispersion relation is reconstructed. This is accomplished due to the inclusion of first-order derivatives of the local bottom profile. The coefficients of the polar-Cartesian mild-slope equation contain the derivatives of the bottom profile up to third order as opposed to second-order derivatives in the Cartesian-based equations. The equation is derived by applying the variational principle to the Cartesian Lagrangian when formulated as a function of the profile in polar coordinates. It is then compared with existing models of the mild-slope equation for simulations of two-dimensional test cases and a quasi-three-dimensional case, which have known analytical solutions. Our modified equation exhibits better matching to the exact solutions for a majority of the investigated cases.

A second-order reduced homogenization for nonlinear structures with periodic configurations in cylindrical coordinates

A high-efficiency second-order reduced homogenization (SORH) method is established for the inelastic problem of periodic heterogeneous structures in cylindrical coordinate. The composite structures investigated in this paper are periodical in radial, axial and circumferential directions. The new second-order linear/nonlinear local solutions at microscale are given by introducing various multiscale auxiliary functions. And, the nonlinear high-order homogenization solutions are obtained at macroscale using a modified asymptotic expansion method. The significant novelties of the work are (i) an effective reduced model on the basis of transformation field analysis given to solve the nonlinear multiscale problems in cylindrical coordinates with less computational amount and (ii) a new second-order nonlinear multiscale algorithm established using asymptotic expansion technique for simulating the heterogeneous cylindrical structure. Finally, by three typical numerical examples including nonlinear elastic and elasto-plastic periodic problems, the validity and accuracy of the proposed algorithms are verified.

Axisymmetric and in-plane transient responses of multilayered transversely isotropic poroelastic media

Based on the integral transform theorem and the extended precise integration method (EPIM), this paper investigates the axisymmetric and in-plane transient dynamic responses of multilayered transversely isotropic (TI) poroelastic media. First, the governing equations are transformed into ordinary differential equations by virtue of the integral transform technique. It can be found that the transformed equations of the axisymmetric problem are similar to those of the in-plane problem. Then, we introduce the boundary conditions and apply the EPIM to obtain the transient dynamic solution in the transformed domain. The actual solution is further gotten by the numerical transform inversion. Various suddenly applied load forms are considered in this work, including concentrated point loads, line loads, uniform strip loads and circular loads. Several numerical examples are provided to verify the proposed theory and to investigate the differences of the transient problems in cylindrical coordinates and Cartesian coordinates. The influences of transverse isotropy and load forms on transient behaviors are also analyzed.

Derivation and validation of a simplified analytical mass transfer model of the laminar co-flow tube for nucleation studies

A laminar co-flow tube is a device recently developed by the present team to study the formation of aerosol particles in atmospherically relevant vapor mixtures. The device employs isothermal mutual diffusion of vapors in two merging laminar gas flows, each saturated with a different vapor. The main quantity of interest, namely, the nucleation rate at which aerosol particles are formed at given temperatures and vapor concentrations, is not measured directly. Therefore, a mass transfer model is needed to evaluate the local nucleation rate and local concentrations obtained from measured integral quantities the flow rates and vapor concentrations at the inlets and the number concentration of generated aerosol particles at the tube outlet. This article presents the derivation and testing of an analytical simplified mass transfer model in which the diffusion problem is solved with approximations for simplification. The purpose of the simplified model is to provide a tool suitable for experimental design and evaluation of results that does not require lengthy computations and transparently shows the effects of the main parameters. By idealizing the velocity field in the co-flow section as plug flow and neglecting axial dispersion, the governing equations of mass transport reduce to a 1D unsteady diffusion problem in cylindrical coordinates moving with the fluid. Such a simplified model has an analytical solution in the form of an infinite series. The simplified mass transfer model is tested for particle formation in the sulfuric acid-water system. New experimental data for particle formation at temperatures of 15 ∘C and 27 ∘C at a relative humidity of 0.38 and various sulfuric acid concentrations are provided as a basis for testing. The results of the model are compared with numerical Computational Fluid Dynamics (CFD) simulations of the full mass and momentum transfer problem. The appropriate choice of plug-flow velocity for the analytical model is discussed. In this connection, hydrodynamics in the undeveloped region of the co-flow are described in detail. Based on the hydrodynamic entrance length, two essential fluid flow regimes are identified. The model results and CFD simulations show a nucleation zone with a shape resembling a candle flame located in the central part of the laminar co-flow tube, with a maximum nucleation rate at the tube axis. The main quantitative characteristics of the particle formation evaluated with the help of the simplified model and with CFD simulations are compared. It turns out that, especially when choosing the plug-flow velocity appropriately, the difference between the numerical CFD model and the simplified analytical model is less than the experimental uncertainty of the measured data. This justifies the application of the simplified mass transfer model for the purpose of quantitative evaluation of experiments with the laminar co-flow tube. This article presents new details of the mass transfer processes and evaluation procedures for the novel experimental device exemplified with a new set of experimental data for particle formation in a sulfuric acid-water system.

Asymptotic analysis of a two-phase Stefan problem in annulus: Application to outward solidification in phase change materials

Stefan problems provide one of the most fundamental frameworks to capture phase change processes. The problem in cylindrical coordinates can model outward solidification, which ensures the thermal design and operation associated with phase change materials (PCMs). However, this moving boundary problem is highly nonlinear in most circumstances. Exact solutions are restricted to certain domains and boundary conditions. It is therefore vital to develop approximate analytical solutions based on physically tangible assumptions, e.g., a small Stefan number. A great amount of work has been done in one-phase Stefan problems, where the initial state is at its fusion temperature, yet very few literature has considered two-phase problems particularly in cylindrical coordinates. This paper conducts an asymptotic analysis for a two-phase Stefan problem for outward solidification in a hollow cylinder, consisting of three temporal and four spatial scales. The results are compared with the enthalpy method that simulates a mushy region between two phases by numerical iterations, rather than a sharp interface in Stefan problems. After studying both mathematical models, the role of mushy-zone thickness in the enthalpy method is also unveiled. Moreover, a wide range of geometric ratios, thermophysical properties and Stefan numbers are selected from the literature to explore their effects on the developed model with regards to interface motion and temperature profile. It can be concluded that the asymptotic solution is capable of tracking the moving interface and evaluating the transient temperature for various geometric ratios and thermophysical properties in PCMs. The accuracy of this solution is found to be affected by Stefan number only, and the computational cost is much less compared with the enthalpy method and other numerical schemes.

Reduction of plane-radial freezing - thawing problems to plane-parallel ones

Construction in northern conditions is complicated by the processes of seasonal freezing and thawing of soils. This factor is of particular importance in the area of permafrost distribution. The paper deals with the problems that arise when laying communication networks. Any engineering solution must be justified by a heat engineering calculation. In one case, it is necessary to exclude the thawing of water or sewer networks, in the other case – to prevent the destructive effect of the heating main on the permafrost soil. The most common mathematical model of the heat transfer process in the presence of phase transitions is the Stefan problem. When calculating the processes of freezing-thawing near the pipes of communications, it is rational to set the problem in cylindrical coordinates, using the property of plane-radial symmetry. But the Stefan problem has an exact solution only in Cartesian coordinates, for the case of plane-parallel symmetry. For this case, many approximate dependencies are also obtained. The paper presents a method that allows using the results of solving the plane-parallel Stefan problem to solve the plane-radial problem with the same values of the input parameters. Approximate values of the coordinate of the phase transition front as a function of time for the plane-parallel and plane-radial areas are obtained using the method of sequential change of stationary states. A one-to-one relationship is established between the obtained values. If there is a solution obtained by any method for a plane-parallel area, and then using this dependence, it can be applied to a plane-radial area. This technique is suitable not only for the Stefan problem, but also for many nonlinear heat conduction problems. The results given in the paper are of practical importance: using the above method, almost any solution obtained for one type of symmetry directly extends to other types of symmetry.

An improved analytical solution for elastic and elastic-plastic buckling of pipes under hydrostatic pressures

An analytical solution for evaluating the critical pressure of pipes under hydrostatic pressure was proposed. The stability equations and the correct boundary conditions for 3D problem in cylindrical coordinates were established by using energy method. The stability equations were solved together with the derived correct boundary conditions and the solution of the critical pressure for elastic buckling of the pipes was expressed in closed form. By adopting the three-layers model for elastic-plastic buckling analysis, the solution was available for the critical pressure prediction of the pipes in elastic-plastic collapse. The effect of the initial ovality was incorporated in formulation. It shows that the increase in the magnitude of the axial compressive stress results in the further reduction in the critical pressure for fully elastic buckling while the pressures for elastic-plastic collapse are always higher than those of the 2D solutions based on the plane strain deformation. The influence of the initial ovality on the pressure capacity of the elastic-plastic buckling of the pipes was discussed and the results from the present approach were compared with the other solutions and experiments.

Affine reduced-order model for radiation transport problems in cylindrical coordinates

This article focuses on parametric reduced-order models to decrease the computational cost of large-scale neutron transport problems in the atmosphere. The atmospheric humidity content and neutron source spectrum are considered as uncertain parameters throughout the problem. We first present our full-order model, including an axisymmetric SN transport methodology, a geopotential model for computing the thermophysical properties of the atmosphere, and a support vector regression method for rapidly generating one-group cross sections at every point. Then, reduction of the full-order problem using the POD-Galerkin and least-squares POD-Petrov-Galerkin methods is presented. The reduced-order models implemented introduce mean relative errors between 0.06% and 0.4% with acceleration factors between 3.6×103-8.4×104, respectively.

Thermoelastoplastic response of FGM linearly hardening rotating thick cylindrical pressure vessels

An analytical solution is presented to analyze the thermoelastoplastic response of a rotating thick-walled cylindrical pressure vessel made of functionally graded material (FGM). The analysis is based on Tresca's yield condition, its associated flow rule and linear strain hardening material behaviour. The uncoupled theory of thermoelasticity is used, and the plane strain condition is assumed. The material properties except for Poisson's ratio, are assumed to vary nonlinearly in the radial direction. Elastic, partially plastic, fully plastic, and residual stress states are investigated. The heat conduction equation for the one-dimensional problem in cylindrical coordinates is used to obtain temperature distribution in the vessel. It is assumed that‏ the inner surface is exposed to an airstream and that the outer surface is exposed to a uniform heat flux. Tresca's yield criterion and its associated flow rule are used to formulate six different plastic regions for a linearly hardening‏ condition. All these stages are studied in detail. It is shown that the thermoelastoplastic stress response of a rotating FGM pressure vessel is affected significantly by the nonhomogeneity of the material and temperature gradient. The results are validated with those of other researchers for appropriate values of the system parameters and excellent agreement is observed.

On the Telegrapher’s Equation with Three Space Variables in Non-Rectangular Coordinates

This article provides a closed form solution to the telegrapher’s equation with three space variables defined on a subset of a sphere within two radii, two azimuthal angles and one polar angle. The Dirichlet problem for general boundary conditions is solved in detail, on the basis of which Neumann and Robin conditions are easily handled. The solution to the simpler problem in cylindrical coordinates is also provided. Ways to efficiently implement the formulae are explained. Minor adjustments result in solutions to the wave equation and to the heat equation on the same domain as well, since the latter are particular cases of the more general telegrapher’s equation.

A research and modeling of wave processes at the scattering of nonlinear acoustic waves on cylindrical bodies

The article is devoted to the study of the scattering of nonlinear acoustic waves on cylindrical bodies. There was made a review of publications on the scattering of acoustic waves by inhomogeneities of the medium in the form of cylindrical bodies and shells. There were noted features of the small parameter method application in nonlinear acoustics. A three-dimensional model of the geometry of the problem in cylindrical coordinates was presented, nonlinear wave processes occurring between the falling plane and scattered cylindrical waves were described. The inhomogeneous wave equation is solved by the method of successive approximations of series expansion in a small parameter. An asymptotic expression for the acoustic pressure of a difference-frequency wave was obtained. A program for calculating scattering diagrams has been developed, and an algorithm for its operation is given. The acoustic pressure scattering diagram of a differential frequency wave on a rigid cylinder and its three-dimensional model are presented. The radius of convergence of the used method of expansion in a small parameter is determined.

Thermoelastoplastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic-fully plastic condition

In this paper, exact solutions to purely elastic, partially plastic and fully plastic deformation and stresses of rotating functionally graded (FG) cylindrical pressure vessels under thermal load are obtained. The material properties are assumed to vary nonlinearly in the radial direction of the vessel whereas the Poisson's ratio due to slight variations in engineering materials is kept constant. To the best of the researchers' knowledge, in the literature, there is no study carried out exact solution for thermal stresses in elastic-plastic rotating FGM thick cylindrical pressure vessels. The novelty of the current work is to present a complete exact elastoplastic solution for thick cylindrical pressure vessels made of FGM under thermal loading, taking into account all stress regions. Assuming that the inner surface is exposed to an airstream, and that the outer surface is exposed to a uniform heat flux. The heat conduction equation for the one-dimensional problem in cylindrical coordinates is used to obtain temperature distribution in the vessel. Tresca's yield criterion and its associated flow rule are used to formulate six different plastic regions for an ideal plastic material. All these stages are studied in detail. It is shown that the thermo-elasto-plastic response of the functionally graded pressurized cylindrical vessel is affected notably by the radial variation of material properties.

An algebraic approach to a charged particle in a uniform magnetic field

We study the problem of a charged particle in a uniform magnetic field with two different gauges, known as Landau and symmetric gauges. By using a similarity transformation in terms of the displacement operator we show that, for the Landau gauge, the eigenfunctions for this problem are the harmonic oscillator number coherent states. In the symmetric gauge, we calculate the SU(1; 1) Perelomov number coherent states for this problem in cylindrical coordinates in a closed form. Finally, we show that these Perelomov number coherent states are related to the harmonic oscillator number coherent states by the contraction of the SU(1; 1) group to the Heisenberg-Weyl group.

Solving the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families of particular solutions to the Navier-Lame equation in cylindrical coordinates. In this paper, we specifically construct solutions having 2pi-periodic angular parts. These particular solutions can be directly applied to a fundamental set of linear elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis, and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.

Pool boiling heat transfer and quench front velocity during quenching of a rodlet in subcooled water: Effects of the degree of subcooling

ABSTRACTPool boiling heat transfer and quench front propagation were investigated during quenching of cylindrical stainless steel rodlets in subcooled water. The degree of subcooling was varied from 0°C (saturated) to 40°C at an increment of 10°C at atmospheric pressure. The results showed that the increase of degree of subcooling accelerates quenching, with the total quenching time being shortened from 90 second (saturated) to 12 second (subcooled by 40°C). As revealed by the boiling curves that were obtained via solving an inverse heat conduction problem in cylindrical coordinates, boiling heat transfer is enhanced significantly for all boiling modes with raising the degree of subcooling. At the highest degree of subcooling of 40°C, the critical heat flux is improved by nearly 300% as compared to that in saturated water. In addition, the rewetting temperature (i.e., Leidenfrost point) was found to increase as a nearly linear function of the degree of subcooling. The quench front was observed to propagate upward from the bottom of the rodlet, which is accelerated noticeably with increasing the degree of subcooling. The average quench front velocity was shown to agree well with the predicted value of an existing theoretical model that was modified to take the influence of subcooling into consideration.

A new procedure for solving steady-state and transient-state nonlinear radial conduction problems of nuclear fuel rods

Half-boundary method (HBM) is extended to solve one-dimensional radial conduction problems in nuclear fuel rods. The accuracy and efficiency of HBM are first tested by solving three one-dimensional radial conduction problems in cylindrical coordinates. They are steady-state and transient-state problems with linear and nonlinear material properties. HBM shows a comparable accuracy to the analytical solutions, and a higher accuracy than finite volume method (FVM) especially with fewer elements in the discretized model. Besides, the computation time ratio of HBM to FVM decreases as the element numbers increase, and it is only 0.16% when 1000 elements are involved. In order to further validate the practical applicability of HBM, nonlinear heat conduction problems of nuclear fuel rods (NFRs) are solved by HBM. The numerical results of NFRs are carried out both in steady-state and transient-state cases. HBM has a matched result with ANSYS. In addition, the size effect of pellet, helium gap and zircaloy cladding on the highest temperature in NFRs is also studied using HBM. The pellet dimension affects the highest temperature mostly.

Diffusion Penetration Time for Transient Heat Conduction

The time duration for processes involving transient thermal diffusion can be a critical piece of information related to thermal processes in engineering applications. Analytical solutions must be used to calculate these types of time durations because the boundary conditions in such cases can be effectively like semi-infinite conditions. This research involves an investigation into analytical solutions for six geometries, including one-dimensional cases for Cartesian, cylindrical, and spherical coordinates. The fifth case involves a heated surface on the inside of a hole bored through an infinite body, which is a one-dimensional problem in cylindrical coordinates. The sixth case involves two-dimensional conduction from a point heat source on the surface of a slab subjected to insulated boundary conditions elsewhere. The mathematical modeling for this case is done in cylindrical coordinates. For each geometric configuration, a relationship is developed to determine the time required for a temperature rise to occur at a nonheated point in the body in response to a sudden change at a heated boundary. A range of time values is computed for each configuration based on the amount of temperature rise used as a criterion. Plots are given for each case, showing the relationships between the temperature rise at the point of interest and the amount of time required to reach that temperature. It was found that a dimensionless parameter, defined herein as dimensionless penetration time, remains reasonably constant between the various geometries studied. The definition of this term is the dimensionless time required to bring about a desired temperature rise, with the characteristic length being the distance between the point of heating and the point of interest.

Computation of the inductance matrix of axisymmetric windings for very fast transients studies in transformers

Purpose – To study very fast transients in transformers, it is required to compute the inductance matrix of windings at very high frequencies (MHz). The core acts as a flux barrier at very high frequencies, affecting the values of the self and mutual inductances of windings. In the previous work by the authors, analytical methods for computation of the inductance matrix at very high frequencies, using a 2-D planar approximation of the transformer geometry, were presented. The purpose of this paper is to present analytical methods for the same problem in cylindrical coordinates which do not suffer from the previous approximations in geometry. Design/methodology/approach – A method based on the Fourier integral transform is described for the calculation of inductance outside the core window. For the region inside the core window, inductance formulas are extracted using the Fourier series analysis. Findings – The final expressions are accurate and fast convergent. Comparisons with FEM simulations and previous 2-D planar formula prove the excellent accuracy of the proposed inductance formulas. Originality/value – The value of the presented formulas accounts for considering the effect of iron core on inductances in transformer very fast transient analysis.

Basis Functions With Divergence Constraints for the Finite Element Method

Basis functions for solving partial differential equations of vector fields using the finite element method are presented. The basis functions are a combination of cubic Hermite splines and second-order Lagrange interpolation polynomials and allow the divergence to be set as a constraint. The basis functions are tested on 3-D resonant cavities and there are no spurious modes. There is good agreement with analytical solutions in cases where they exist and with calculations using edge elements in other cases. The method is extended to solve problems with singularities at edges and corners. For perfect conductors, this includes mesh refinement in the neighborhood of the edge or corner. For dielectrics, constraints are derived so that the flux of the field is zero through a closed surface that contains the edge or corner. The method is used to solve problems using the electric and magnetic field formulations. Through a change of variables, the method is applied to problems in cylindrical coordinates.

Numerical simulations of fluid pressure in the human eye

In this article we present computational results for the pressure in the human eye. Pressure computations for different flow rates of the aqueous humor, viscosity of the aqueous humor, and permeability of the trabecular meshwork are given. The fluid flow is assumed to be axisymmetric, and modeled as a coupled system of Stokes and Darcy fluid flow equations, representing the fluid flow in the anterior cavity and trabecular meshwork, respectively. Rewriting the problem in cylindrical coordinates reduces the 3-D problem to a problem in 2-D. Computations are also given for varying angles between the base of the iris and the trabecular meshwork.