Conical Coordinates Research Articles

Supersonic flow of a Chaplygin gas past a delta wing

We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges. The flow under study is described by the three-dimensional steady Euler system. In conical coordinates, this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type. The type of this equation depends fully on the solutions of the problem itself, and thus it cannot be determined in advance. We overcome the difficulty by establishing a crucial Lipschitz estimate, and finally prove the unique existence of the solution via the method of continuity.

Field theory in normal conical coordinates

In the scientific literature, the field theory is most fully covered in the cylindrical and spherical coordinate systems. This is explained by the fact that the mathematical apparatus of these systems is most well studied. When the source of field has a more complex structure than a point or a straight line, there is a need for new approaches to their study. The goal of this research is to adapt the field theory related to curvilinear coordinates in order to represent it in the normal conical coordinates. In addition, an important part of the research is the development of a geometrical modeling apparatus for scalar and vector field level surfaces using computer graphics. The paper shows the dependences of normal conical coordinates on rectangular Cartesian coordinates, Lame coefficients. The differential characteristics of the scalar and vector fields in normal conical coordinates are obtained: Laplacian of scalar and vector fields, divergence, rotation of the vector field. The example case shows the features of the application of the mathematical apparatus of geometrical field modeling in normal conical coordinates. For the first time, expressions for the characteristics of the scalar and vector fields in normal conical coordinates are obtained. Methods for geometrical modeling of fields using computer graphics have been developed to provide illustration in their study.

ICarPS: a computational tool for identifying protein carbonylation sites by novel encoded features.

Protein carbonylation is one of the most important oxidative stress-induced post-translational modifications, which is generally characterized as stability, irreversibility and relative early formation. It plays a significant role in orchestrating various biological processes and has been already demonstrated to be related to many diseases. However, the experimental technologies for carbonylation sites identification are not only costly and time consuming, but also unable of processing a large number of proteins at a time. Thus, rapidly and effectively identifying carbonylation sites by computational methods will provide key clues for the analysis of occurrence and development of diseases. In this study, we developed a predictor called iCarPS to identify carbonylation sites based on sequence information. A novel feature encoding scheme called residues conical coordinates combined with their physicochemical properties was proposed to formulate carbonylated protein and non-carbonylated protein samples. To remove potential redundant features and improve the prediction performance, a feature selection technique was used. The accuracy and robustness of iCarPS were proved by experiments on training and independent datasets. Comparison with other published methods demonstrated that the proposed method is powerful and could provide powerful performance for carbonylation sites identification. Based on the proposed model, a user-friendly webserver and a software package were constructed, which can be freely accessed at http://lin-group.cn/server/iCarPS. Supplementary data are available at Bioinformatics online.

The ellipticity principle for the 3-D steady potential flow in conical coordinates

In this paper, we consider the 3-D steady potential flow for a compressible gas with pressure satisfying $p&apos;(\\rho)=\\rho^{\\gamma-1}$, where $\\rho$ is the density and $\\gamma\\geq-1$ is a constant. The potential equation is of mixed type in conical coordinates, and the type is completely determined by a pseudo-Mach number $L$ with $L<1$ (resp. $L>1$) corresponding to elliptic (resp. hyperbolic) regions. We first establish an ellipticity principle: for $\\gamma\\geq-1$, the equation is elliptic in the interior of a parabolic-elliptic region; in particular, when $\\gamma>-1$, there exists a domain-dependent function $\\delta_0>0$ such that $L\\leq~1-\\delta_0$. Then applying this ellipticity principle, we further prove that for $\\gamma\\geq-1$, the equation is uniformly elliptic in any subregion that is strictly away from the degenerate boundary.

Simulation of Heat Transfer in Husk Furnace with Cone Geometry Based on Conical Coordinate System

Simulation of Heat Transfer in Husk Furnace with Cone Geometry Based on Conical Coordinates has been performed. This simulation aimed to study the heat distribution of temperature based on conduction and convection mechanism on conical coordinate system. Fluid dynamics inside the cone of husk furnace was obtained by solving the Navier - Stokes equations with laminar flow approach. The initial temperature in all parts of the cone is room temperature, except at the bottom of the cone is 700 °C. Through numerical calculation of heat conduction and convection equation by FDM method, we got that the velocity of fluid flow at the center cone is 13.69 m/s for 45 s, 11.90 m/s for 60 s, and 7.25 m/s for 120 s, with unfixed temperature condition in the cone.

GPU-accelerated 3D mipmap for real-time visualization of ultrasound volume data

Ultrasound volume rendering is an efficient method for visualizing the shape of fetuses in obstetrics and gynecology. However, in order to obtain high-quality ultrasound volume rendering, noise removal and coordinates conversion are essential prerequisites. Ultrasound data needs to undergo a noise filtering process; otherwise, artifacts and speckle noise cause quality degradation in the final images. Several two-dimensional (2D) noise filtering methods have been used to reduce this noise. However, these 2D filtering methods ignore relevant information in-between adjacent 2D-scanned images. Although three-dimensional (3D) noise filtering methods are used, they require more processing time than 2D-based methods. In addition, the sampling position in the ultrasonic volume rendering process has to be transformed between conical ultrasound coordinates and Cartesian coordinates. We propose a 3D-mipmap-based noise reduction method that uses graphics hardware, as a typical 3D mipmap requires less time to be generated and less storage capacity. In our method, we compare the density values of the corresponding points on consecutive mipmap levels and find the noise area using the difference in the density values. We also provide a noise detector for adaptively selecting the mipmap level using the difference of two mipmap levels. Our method can visualize 3D ultrasound data in real time with 3D noise filtering.

Transient two-phase boundary layer modeling for hollow cone sprays

This paper presents a spray model suited for dense sprays. It captures the transient evolution of the two-phase jet characteristics resulting from hollow cone injection. The model is designed for fast model response as needed in engine system simulation. It is based on the description of the gas phase boundary layer surrounding the dense spray. Mass and momentum equations are solved for both the dispersed liquid and the continuous gas phase. Spatial gradients are resolved along one dimension, namely the main injection direction. The conservation equations are expressed in conical coordinates. The model’s response is studied qualitatively and global characteristics such as the penetration behavior are compared to both experimental and CFD data.

Near‐field acoustic holography in conical coordinates.

Near‐field acoustical holography (NAH) techniques can be optimized if the method capitalizes on the geometry of the noise source under investigation. Helmholtz‐equation least squares method (HELS) uses the solutions of Helmholtz equation in spherical coordinates as basis functions for the pressure field. HELS is an efficient NAH technique if the source and the measurement surfaces are spherical in nature. For nonspherical cases, such as radiation from a plate or bar, it takes a large number of functions to represent the field. In these cases, there is also a question about where to place the origin of the wave functions. In search of a HELS‐type method that could be applied to nonspherical sources, a study into the features of conical coordinates has been conducted. Because the Helmholtz equation is separable in conical coordinates, the solutions can be used, in a manner similar to HELS, as basis functions to represent the pressure field. For conical coordinates, the basis functions are spherical Hankel functions and Lame functions. This HELS‐type formulation in conical coordinates could be a natural choice for NAH on conical sources and may be appropriate for NAH on jet noise. [Work supported by Blue Ridge Research and Consulting and Air Force Research Laboratory.]

Magnetothermoelastic transient response of multilayered conical shells

This paper considers problems of the three-dimensional axisymmetric quasi-static coupled magnetothermoelasticity for the laminated circular conical shells subjected to magnetic and temperature fields. The temperature and pressure relation are assumed for the inner boundary. The formulation begins with the basic equations of magnetothermoelasticity in curvilinear circular conical coordinates. Laplace transform and finite difference methods are used to analyze problems. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions in a transient and steady state. Moreover, the computational procedures established in this thesis, can solve the generalized magnetothermoelasticity problem for multilayered conical shells with nonhomogeneous materials.

Magnetothermoelastic analysis of multilayered conical shells subjected to magnetic and vapor fields

This paper considers problems of the three-dimensional axisymmetric quasi-static coupled magnetothermoelasticity for the laminated circular conical shells subjected to magnetic and vapor fields. The water vapor temperature and pressure relation are assumed for the inner boundary. The water vapor temperature and pressure data were obtained from a thermodynamic steam table. The formulation begins with the basic equations of magnetothermoelasticity in curvilinear circular conical coordinates. Laplace transform and finite difference methods are used to analyze problems. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions in a transient and steady state. Moreover, the computational procedures established in this thesis, can solve the generalized magnetothermoelasticity problem for multilayered conical shells with nonhomogeneous materials.

Thermoelastic transient response of multilayered conical shells subjected to vapor field

The problems of the three-dimensional axisymmetric quasi-static coupled thermoelastic for the laminated circular conical shells subjected to vapor field. The water vapor temperature and pressure relation assumed for the inner boundary. The water vapor temperature and pressure data were obtained from a thermodynamic steam table. The formulation begins with the basic equations of thermoelasticity in curvilinear circular conical coordinates. Laplace transform and finite difference methods are used to analyze problems. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions in a transient and steady state. Moreover, the computational procedures established in this thesis, can solve the generalized thermoelasticity problem for multilayered conical shells with nonhomogeneous materials.

Jacobi elliptic coordinates, functions of Heun and Lamé type and the Niven transform

Lame and Heun functions arise via separation of the Laplace equation in general Jacobi ellipsoidal or conical coordinates. In contrast to hypergeometric functions that also arise via variable separation in the Laplace equation, Lame and Heun functions have received relatively little attention, since they are rather intractable. Nonetheless functions of Heun type do have remarkable properties, as was pointed out in the classical book “Modern Analysis” by Whittaker and Watson who devoted an entire chapter to the subject. Unfortunately the beautiful identities appearing in this chapter have received little notice, probably because the methods of proof seemed obscure. In this paper we apply the modern operator characterization of variable separation and exploit the conformal symmetry of the Laplace equation to obtain product identities for Heun type functions. We interpret the Niven transform as an intertwining operator under the action of the conformal group. We give simple operator derivations of some of the basic formulas presented by Whittaker and Watson and then show how to generalize their results to more complicated situations and to higher dimensions.

A generalized thermoelasticity problem of multilayered conical shells

The formulation begins with the basic equations of thermoelasticity in curvilinear circular conical coordinates. A method based on a hybrid Laplace transformation and finite difference method is developed to obtain the two-dimensional axisymmetric quasi-static coupled thermoelastic problems of laminated circular conical shells. It was shown that the solutions are rapidly convergent. Solutions for the temperature, displacement and thermal stress distributions in both transient and steady state are obtained. The present method can obtain stable solutions at a specific time; thus it is a powerful and efficient method to solve the coupled transient thermoelastic problems of a circular multilayered conical shell.

Ground Testing Facility for Modeling Real Projectile Flight Heating in Earth Atmosphere

An analytical method for obtaining the heating of a hypersonic projectile with a conical forebody and data obtained from a special ballistic range to validate the method are presented. The method is a classical boundary-layer technique for turbulent flow. It involves a coordinate transformation that enables flat plate solutions to be adapted to the conical coordinates. The heating to the projectile nose cone was then obtained analytically with a semi-infinite slab, according to the given boundary conditions for flow velocity and gas temperature. The use of a transformation technique enables scaling of high-pressure conditions in the ballistic range, the so-called aerothermoballistic high-pressure test range, to actual atmospheric flight conditions. This facility allows duplication of the heating scenario during some seconds of atmosphere flight, with a flight in compressed gas resulting in the same heat flux for a flight time of some milliseconds. The transformation was validated using data from the range, obtained from the infrared surface emission at two wavelengths. Herewith the cone surface temperature is measured by detecting the intensity of the infrared emission coming from the cone surface

Triaxial haloes and particle dark matter detection

This paper presents the properties of a family of scale-free triaxial haloes. We adduce arguments to suggest that the velocity ellipsoids of such models are aligned in conical coordinates. We provide an algorithm to find the set of conically aligned velocity second moments that support a given density against the gravity field of the halo. The case of the logarithmic ellipsoidal model –the simplest triaxial generalization of the familiar isothermal sphere– is examined in detail. The velocity dispersions required to hold up the self-consistent model are analytic. The velocity distribution of the dark matter can be approximated as a triaxial Gaussian with semiaxes equal to the velocity dispersions. There are roughly 20 experiments worldwide that are searching for evidence of scarce interactions between weakly interacting massive-particle dark matter (WIMP) and detector nuclei. The annual modulation signal, caused by the Earth's rotation around the Sun, is a crucial discriminant between WIMP events and the background. The greatest rate is in June, the least in December. We compute the differential detection rate for energy deposited by the rare WIMP–nucleus interactions in our logarithmic ellipsoidal halo models. Triaxiality and velocity anisotropy change the total rate by up to ∼40 per cent, and have a substantial effect on the amplitude of the annual modulation signal. The overall rate is greatest, but the amplitude of the modulation is weakest, in our radially anisotropic halo models. Even the sign of the signal can be changed. Restricting attention to low energy events, the models predict that the maximum rate occurs in December, and not in June.

Analysis of a non-constant gap externally pressurized conical bearing with temperature and pressure dependent viscosity

The present paper analyses an externally pressurized non-constant gap conical bearing rotating with a uniform angular velocity. The lubricant is assumed to be incompressible and its viscosity varies with both pressure and temperature. Although the inertia effect due to lubricant flow has been neglected, rotational inertia is taken into account. The governing system of coupled momentum and energy equations, in conical coordinates, is solved numerically using the finite difference method, to determine various bearing characteristics. The effect of the viscosity-pressure exponent, , on various characteristics of the bearing has been studied. It is seen that variation of does not produce any significant change in the load capacity for convergent and constant gap conical bearings. For the divergent gap also, no significant difference is seen when the gap is slightly divergent. However, for highly divergent gaps, the load capacity increases with an increase in . It is also seen that variation in does not produce any significant change in torque of the bearing.

Asymptotic differential quadrature solutions for the free vibration of laminated conical shells

Three-dimensional (3-D) elasticity solutions for the free vibration analysis of laminated circular conical shells are presented by means of an asymptotic approach. The formulation begins with the 3-D equations of motion in circular conical coordinates. After proper nondimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple time scales is used to eliminate the secular terms and make the asymptotic expansion feasible. The method of differential quadrature (DQ) is adopted for solving the problems of various orders. The present asymptotic formulation is applicable to the analysis of laminated cylindrical shells by vanishing the semivertex angle (α). The natural frequencies, modal stresses of cross-ply cylindrical and conical shells with simply supported – simply supported (S-S) boundary conditions are studied to demonstrate the performance of the present asymptotic theory. It is shown that the asymptotic DQ solutions of the present study converge rapidly. The present convergent results are in good agreement with the accurate solutions obtained from the approximate 2-D shell theories in the cases of thin shells. Furthermore, these present results may serve as the benchmark solutions for assessment of various 2-D shell theories in the cases of moderatively thick shells.

Electromagnetic modes in conical transmission lines with application to the linearly tapered slot antenna

A transmission line analysis of the bow-tie antenna and the linearly tapered slot antenna (LTSA) is presented. These structures belong to the class of conical transmission lines defined here in terms of conical coordinates. A complete set of solutions of the Helmholtz equation is obtained exhibiting TE and TM modes. Modal fields are expressed by Lame (1837) and Bessel-Schelkunoff functions. TE and TM eigenmode analysis is particularized to the bow-tie structure. Bow-tie antenna and LTSA are shown to be dual conical transmission lines by the image method and Babinet's principle. The modes of LTSA are calculated on the basis of the results obtained for the bow-tie structure. The radiation pattern of the LTSA is computed as the integral of a closed-form expression of the dyadic Green's function weighted by the modal electric field distribution over the slot aperture. The obtained dominant mode radiation patterns are validated by measurements from the literature. The radiation patterns of the first two-order modes are calculated and compared.

Analytical solution for the side-fringing fields of narrow beveled heads

By using conical coordinates, exact analytical solutions for three-dimensional side-fringing fields of recording heads that are beveled in the down-track direction are found. These solutions are derived under the assumption of zero gap length. The side-fringing fields for the two limiting cases of infinitesimally narrow heads and semi-infinitely wide heads are presented and compared.

Behavior of pressure and heat transfer in sharp fin-induced turbulent interactions

An experimental study has been performed of sharp fin-induced shock wave/turbulent boundary-layer interactions at Mach 5 under moderately cooled wall conditions. The experimental data include mean surface pressure and heat transfer rate distributions for fin angles of attack of 6, 8, 10, 12, 14, and 16 deg. Data were acquired in spanwise and conical coordinate systems to assess the behavior of the surface properties. Although the surface pressure distributions collapse in conical coordinates, the heat transfer distributions do not. Instead, the heat transfer was found to be nearly constant along rays from a virtual conical origin determined from the pressure distributions.