Non-orthogonal Coordinate System Research Articles

ГЕОМЕТРИЯ ПРЯМЫХ КОНОИДОВ С ОРТОГОНАЛЬНОЙ СИСТЕМОЙ КООРДИНАТ.

The geometry and methods of forming right conoids are considered in the article. Right conoids are formed by the movement of a rectilinear generatrix perpendicular to a fixed straight line – the axis of the conoid, along which the rectilinear generatrix moves. When moving, the rectilinear generatrix rotates according to a given law around the directrix curve. Usually, the generatrix line is connected to some reference curve that the generatrix line touches. The coordinate grid along the generatrix lines is also connected with the reference curve. In this case, a non-orthogonal coordinate system of a right conoid is formed. Coefficients of quadratic forms of surfaces with a non-orthogonal coordinate system, usually are complex. Methods for analyzing shells with a non-orthogonal coordinate system of the middle surface of the shell are becoming more complicated. The article considers the possibility of forming right conoids with an orthogonal coordinate system, including with directrix reference curves. A vector equation is given for specifying right conoids with an orthogonal nonconjugate coordinate system. Formulas of coefficients of quadratic forms characterizing the internal geometry and curvature of right conoidal surfaces in space with the help of this vector equation are obtained. Based on the vector equations of the surface assignment, drawings of right conoids with various geometric parameters are constructed in the MathCAD program. The difference in the construction of right conoids with orthogonal and non-orthogonal curved coordinate systems is clearly demonstrated, so with different methods of specifying the angles of inclination of rectilinear generatrix of surfaces.

Improved Virtual Space Vector Modulation Scheme for the Reduced Switch Count Three-Level Inverter With Balanced and Unbalanced Neutral-Point Voltage Conditions

This article presents an improved virtual space vector modulation scheme for the reduced switch count three-level inverter with balanced and unbalanced neutral-point voltage (NPV) conditions. By analyzing the effects of unbalanced NPV conditions on basic voltage vectors, the new virtual vectors are constructed by considering the dc unbalancing coefficient and distribution factor for small vectors simultaneously. To facilitate the calculation of duty cycles, the nonorthogonal coordinate system is employed. In addition, an optimized control scheme is developed to obtain the optimized distribution factor, and the relative durations of P-type and N-type small vectors are adjusted to control the voltages across two dc-link capacitors separately. The satisfactory quality of output currents is maintained with both balanced and unbalanced NPV conditions. The feasibility of the proposed scheme is verified by simulated and experimental results.

Mathematical and Computer Modeling of Convective Heat Transfer in Fuel Cartridges of Fuel Elements with Different Shapes and Packing of Rods

The paper consists of three sections and is of an informational and generalizing nature, indicating promising areas for further research. The first section "R-functions method in mathematical modeling of convective heat transfer in fuel cartridges with fuel elements" is devoted to the use of new constructive tools of the R-functions method for mathematical and computer modeling of fuel elements packings with different types of symmetry, as well as the study of convective heat transfer in fuel elements grids and the effect of the type of packing on the distribution of velocity and temperature. An octahedral cartridge with 37 fuel elements packed according to three patterns (cyclic, checkerboard and in-line) is considered. It is noted that when constructing the equations of a cartridge with bundles of fuel elements using the new method, the number of R-operations and, accordingly, the calculation time are significantly reduced. An analysis of the obtained results allows to conclude that the maximum temperature is obtained with cyclic packing. The scheme of the reactor, the cartridges of which are hexagonal casings, with 91 fuel elements placed in each of them both with checkerboard and cyclic packing, is also considered. In the second section "Thermal-hydraulic calculation of fuel elements cartridges in case of violation of the rods packing symmetry", a hexagonal fuel cartridge with 169 fuel elements and checkerboard packing is considered. An increase in temperature is analyzed in case of violation of the packing symmetry while maintaining the parallelism of the rods, as well as in case of a curvature of one of them. The third section "R-functions, fuel element with polyzonal finning of the shell and heat transfer during fluid motion" is focused on the construction of equations for various finning surfaces of fuel elements and the study of hydrodynamic and temperature fields in case of polyzonal finning of the shell. At the same time, using the apparatus of tensor analysis, a transition to a curvilinear non-orthogonal (helical) coordinate system was made. It is noted that mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult to conduct for one reason or another. In addition, working with mathematical model of the process and the computational experiment make it possible to investigate the properties and behavior of the process in various situations relatively quickly and without significant expenses. The reliability of the methods, results and conclusions is confirmed by comparison with the information given in the references, the results of the analysis of the numerical convergence of solutions and the calculation of the residual.

An innovative sustainability-oriented multi-criteria decision making framework for prioritization of industrial systems with interdependent factors: Method and a case study of electricity generation

With the development of technology and manufacturing, the conflicts between environment protection, economic interest and social benefit have been awared and paid more attention to in recent years. The sustainability performance has been considered by increasing number of decision makers. In the real-life practice, it will strongly affect the prioritization results if the criteria considered in the sustainability assessment are interdependent. Therefore, a non-orthogonal coordinate system based multi-criteria decision making method which can incorporate the interactions and interdependences among the criteria for sustainability assessment is proposed in this study. An illustrative case about electricity production was used to illustrate the developed sustainability prioritization model. The results obtained by the proposed method are compared with that based on the traditional method, and sensitivity analysis is also carried out. The results reveal that it is prerequisite to incorporate the interactions and interdependences between sustainability-oriented decision-maker, and the developed non-orthogonal coordinates based sustainability assessment framework is feasible for the prioritization of industrial systems with the considerations of the interactions and interdependences among the evaluation criteria.

Parametric effect of the interrupted annular groove fin on flow and heat transfer characteristics of a finned circular tube heat exchanger

The influences of the geometrical parameters of interrupted annular groove fin mainly including the annular groove diameter, the groove arc length, and the fin spacing, on the fin side thermal-hydraulic characteristics of a finned circular tube exchanger were numerically investigated by actualizing the custom FORTRAN programing with SIMPLE algorithm in a non-orthogonal curvilinear co-ordinate system, and the regression formulas of average Nusselt number and friction factor with flow parameters and geometrical parameters were obtained. Compared with the referential plain fin, interrupted annular groove fin could significantly improve thermal performance under the same pumping power constraint, and Nusselt number is closely germane to the secondary flow, which implies that the fin side heat transfer is depended entirely on the secondary flow strength. For Nusselt number, the annular groove diameter and the groove arc length have positive effect, while the fin pitch, the groove circumferential and radial locations have negative effect. The dominant parameters influencing on friction factor in turn are the fin pitch, the groove radial location, and the annular groove diameter. The optimal annular groove diameter is screened, and found that the optimal annular groove diameter is closely related with Reynolds number under the same pumping power constraint, while under the same mass-flow rate constraint that is scarcely related with Reynolds number.

Thermal and flow characteristics of a channel formed by aligned round tube bank fins stamped with curve delta-winglet vortex generators

The primary purpose of present study is to reveal the roles played by CDVGs in vectoring the airflow and breeding longitudinal vortices for enhancing heat transfer. The thermal-hydraulic characteristics of a finned aligned round tube heat exchanger with stamped CDVGs were numerically studied by using the custom code with SIMPLE algorithm in a nonorthogonal curvilinear coordinate system. Four geometric parameters including vortex generator height, radial position, peripheral position and fin pitch, were programmed to ascertain parametric effects on fin side thermal performance, and the regression formulas of Nusselt number and resistance coefficient in the fin side passage with Reynolds numbers and four geometric parameters were secured. The findings suggest that the mean Nusselt number, resistance coefficient and secondary flow strength for the novel fin with CDVGs are all prevail over that in the reference plain fin under a certain Reynolds number. The body fitted CDVGs are ideally situated for finned round tube exchanger in aligned arrangement. The action of CDVGs can not only efficiently lessen the size of recirculation area behind tube but also produce strong longitudinal vortices, accordingly heighten evidently the fin side heat transfer ability in the recirculation zone. Using the studied optimum geometric parameters can make the thermal property factor up to 1.35 under the same pumping power constraint in comparison with the reference plain fin geometry when Reynolds number equals to 3000. This study can be a real application in compact heat exchanger with finned round tubes when heat transfer augmentation is under consideration.

Tensorial elastodynamics for coupled acoustic/elastic anisotropic media: incorporating bathymetry

SUMMARYCorrectly implementing the fluid/solid boundary conditions at the seafloor is important for accurate full-wavefield imaging and inversion of marine seismic data. Because bathymetric profiles are rarely flat, the associated undulations influence wave modes interacting with the seafloor and, therefore, the ensuing imaging and inversion results. The conventional method of using single-domain elastic finite-difference (FD) solutions assumes that the fluid/solid contact is welded, which leads to incorrect handling of the boundary conditions and introduces modelling errors. We present a mimetic finite-difference (MFD) approach to solve the equations of anisotropic elastodynamics in a non-orthogonal coordinate system conformal to the bathymetric interface. The vertically deformed coordinate mapping transforms the irregular Cartesian (physical) domain into a regularly sampled generalized computational domain. We partition the medium into the acoustic and elastic subdomains and explicitly satisfy the fluid/solid boundary conditions with a split-node approach involving high-order one-sided MFD operators that achieve uniform spatial accuracy throughout the computational domain. Fully staggered grids (FSGs) are used to solve the velocity–pressure and velocity–stress formulations of the acoustic and anisotropic elastic wave equations, respectively. Numerical examples demonstrate that the proposed MFD+FSG algorithm accurately simulates wavefields even for strongly undulating bathymetric surfaces overlying structurally complex anisotropic media, and produces no spurious numerical artefacts (e.g. staircasing) or unphysical wave modes often caused by improper handling of the strong-contrast bathymetric interface. The wavefields generated by the tensorial MFD scheme closely match those from the more computationally expensive spectral-element method used to validate our implementation. The developed MFD+FSG technique can be effectively employed as the modelling kernel in a variety of coupled acoustic/elastic imaging and inversion applications.

Semi-Analytical Computation of a Quasi-Static Field Induced by a 3-D Eddy Current Probe in Anisotropic Material With Rough Interfaces

This article presents a numerical model dedicated to the computation of a quasi-static field induced by a 3-D eddy current probe in unidirectional composite material presenting delamination. The specimen is homogeneously anisotropic and the flaw is represented by a local deformation in the planar geometry. The curvilinear coordinate method (CCM) is used to solve Maxwell's equations in covariant form. Based on the Fourier modal method, the semi-analytical approach consists of presenting the electromagnetic field components as eigenmode expansion in a new non-orthogonal coordinate system fitting the geometrical perturbation. Then, interface conditions can be analytically included in order to evaluate the unknown amplitude of eigenmodes. To take into account the contribution of lower interfaces, a recursive and stable scattering matrix algorithm is implemented. The numerical validation shows a good agreement between simulation results and finite element (FE) data.

Nonstationary Dynamics of Elliptic Isotropic Conical Shells Under Distributed Loads*

We considered a solution to the problem of the forced vibrations of a truncated elliptic conical shell under a distributed impulsive load. A linear version of the equations of the Timoshenko type theory of conical shells is obtained in a non-orthogonal curvilinear coordinate system. To solve the problem, we elaborated a numerical algorithm based on the finite-difference approximation of the initial equations in the spatial and time coordinates. An example of the dynamical behavior of the conical shell is studied numerically.

A comprehensive mathematical model for nanofibre formation in centrifugal spinning methods

Motivated by our experimental observations of nanofibre formation via the centrifugal spinning process, we develop a string model to study the behaviours of a Newtonian, viscous curved jet, in a non-orthogonal curvilinear coordinate system including both air-drag effects and solvent evaporation for the first time. In centrifugal spinning a polymeric solution emerges from the nozzle of a spinneret rotating at high speeds around its axis of symmetry and thins as it moves away from the nozzle exit until it finally lands on the collector. Except for the Newtonian fluid assumption, our model includes the key parameters of the curved jet flow, e.g. viscous, inertial, rotational, surface tension, gravitational, mass diffusion within the jet, mass diffusion into air and aerodynamic effects, via Rossby ( ), Reynolds ( ), Weber ( ), Froude ( ), Peclet ( ), air Reynolds ( ) and air Peclet ( ) numbers, and the collector radial position ( ). Our results, including comparison to experiments, reveal that the aerodynamic effects must be considered to enable a correct prediction of the jet trajectory and radius. Decreasing not only renders the jet thinning much faster, but also forces the jet to wrap tighter around the rotation axis. Increasing , and leads to a longer jet. Decreasing causes the jet to wrap tighter around the spinneret but it shows trivial effects on the solvent evaporation. Changes in and do not significantly affect the jet trajectory. Finally, we propose simple relations to estimate the jet radius and the jet breakup length.

Explicit and Implicit Finite -Volume Methods for Depth Averaged Free-Surface Flows

In recent years, much progress has been made in solving free-surface flow variation problems in order to prevent flood environmental problems in natural rivers. Computational results and convergence acceleration of two different (explicit and implicit numerical techniques) finite-volume based numerical algorithms, for depth-averaged subcritical and/or supercritical, free-surface, steady flows in channels, are presented. The implicit computational model is a bi-diagonal, finite-volume numerical scheme, based on MacCormack’s predictor-corrector technique and uses the semi-linearization matrices for the governing Navier-Stokes equations which are expressed in terms of diagonalization. This implicit numerical scheme puts primary emphasis to solution convergence using non-orthogonal local coordinate system. The explicit formulation uses volume integrals to solve the governing flow equations. Computational results and convergence performance between the implicit and the explicit finite-volume numerical schemes, for incompressible, viscous, depth-averaged free-surface, steady flows are presented. Implicit and explicit computational results are satisfactorily compared with available measurements. The implicit bi-diagonal technique yields fast convergence compared to the explicit one at the expense of programming effort. Iterations require to achieve convergence solution error of less than 10-5, can be reduced down to 90.0 % in comparison to analogous flows with using explicit numerical technique.

ПОСТАНОВКА ТА ПОБУДОВА ЧИСЕЛЬНОГО АЛГОРИТМУ РОЗВ’ЯЗУВАННЯ ЗАДАЧ ДИНАМІКИ ТЕОРІЇ КОНІЧНИХ ОБОЛОНОК В НЕОРТОГОНАЛЬНІЙ СИСТЕМІ КООРДИНАТ

The paper presents the formulation and numerical algorithm for solving problems of the dynamics of the theory of conical shells in a non-orthogonal coordinate system. The object of the study are conical shells, the equations of which are represented in non-orthogonal coordinate system. Purpose of the work is to formulate and construct a numerical algorithm for solving the problems of the dynamics of conical shells in a non-orthogonal coordinate system. The methods of research include the basic principles of the theory of shells to Tymoshenko's type and numerical methods. The formulation of problems and a numerical algorithm for studying the dynamic behavior of conical shells in a non-orthogonal coordinate system are considered. The results obtained in the work can be used in the design of elements of shell structures in the rocket, aircraft and shipbuilding industries. KEYWORDS: CONIC SHELL, DYNAMIC PROCESSES, NON-ORTHOGONAL COORDINATE SYSTEM, NUMERICAL METHODS

The Variation-Difference Method of Design of Shells with Normal Unconjugated Coordinate System and Design of Some Type of Surfaces for Checking of the Correction of the Program

At the classical literature of the design of the shell there is used the coordinated system of the middle surfaces of principle curvatures or common non-orthogonal coordinate system. The system of equations of the shells of complex forms is complicated and it’s difficult to receive the analytical decision. For analyses of the shells with non-orthogonal coordinate system and the most shells of complex form in the coordinate system of principle curvatures there is used usually finite element method which usually uses only the equation of the surface but don’t take into account the geometrical characteristics of the middle surfaces of the shell. At the chair of strength of materials and constructions of Engineering Academy RUDN there is worked up the complex program VRMSHELL for analyses of the shells of complex forms by variation-difference method. The complex includes the library of curves on the base of which there are calculated the geometrical characteristics of the classes of the surfaces included at the program complex. The complex includes the classes of surfaces as cylindrical, surfaces of rotation, Joachimsthal’s surfaces, Monge’s surfaces. The coordinate system of surfaces of these classes is the coordinate system of principle curvatures.

Experimental validation of wind energy estimation

Wind power assessment in complex terrain is a very demanding task. Modeling wind conditions with standard linear models does not sufficiently reproduce wind conditions in complex terrains, especially on leeward sides of terrain slopes, primarily due to the vorticity. A more complex non-linear model, based on Reynolds averaged Navier-Stokes equations has been used. Turbulence was modeled by modified two-equations k-? model for neutral atmospheric boundary-layer conditions, written in general curvelinear non-orthogonal co-ordinate system. The full set of mass and momentum conservation equations as well as turbulence model equations are numerically solved, using the as CFD technique. A comparison of the application of linear model and non-linear model is presented. Considerable discrepancies of estimated wind speed have been obtained using linear and non-linear models. Statistics of annual electricity production vary up to 30% of the model site. Even anemometer measurements directly at a wind turbine?s site do not necessarily deliver the results needed for prediction calculations, as extrapolations of wind speed to hub height is tricky. The results of the simulation are compared by means of the turbine type, quality and quantity of the wind data and capacity factor. Finally, the comparison of the estimated results with the measured data at 10, 30, and 50 m is shown.

COMPUTATIONAL ASPECTS OF RADIATIVE TRANSFER EQUATION IN NON-ORTHOGONAL COORDINATES

Non-equilibrium energy transfer takes place for thin films when thermal disturbance is introduced. In this case, phonon transport inside the film governs the heat transport and temperature distribution in the film. In the present study an attempt is made to formulate and illustrate the phonon transfer in micro-scale silicon film of various shapes incorporating the non-orthogonal coordinate system. Successful application of the discrete-ordinates method to the solution of the equation for phonon radiative transport in non-orthogonal coordinates requires the application of various numerical techniques connected to the finite-difference method. The numerical solution of the equation for phonon transfer in non-orthogonal coordinate is introduced via adapting the discrete ordinate method. Phonon intensity distribution in the thin film is presented in terms of equivalent equilibrium temperature. It is found that film shape has significant effect on equivalent equilibrium temperature distribution inside the film. The validation study demonstrates that the code developed solving the equation for phonon transport is also applicable to the phonon transport in non-orthogonal coordinate system.

Non-orthogonal beam coordinate system wave propagation and reverse time migration

Abstract Grid size has a significant influence on the computation efficiency and accuracy of finite-difference seismic modeling and can change the workload of reverse time migration (RTM) remarkably. This paper proposes a non-orthogonal analytical coordinate system, beam coordinate system (BCS), for the solution of seismic wave propagation and RTM. Starting with an optical Gaussian beam width equation, we expand the representation on vertically variable velocity media, which is the most common scenario in seismic exploration. The BCS based on this representation can be used to implement an irregular-grid increment finite-difference that improves the efficiency of RTM. Based on the Laplacian expression in Riemannian space, we derive the wave equation in the BCS. The new coordinate system can generate an irregular grid with increment increasing vertically along depth. Through paraxial ray tracing, it can be extended to non-analytical beam coordinate system (NBCS). Experiments for the RTM on the Marmousi model with the BCS demonstrate that the proposed method improves the efficiency about 52% while maintaining good image quality.

Wind Stress in the Coastal Zone: Observations from a Buoy in Southwestern Norway

Several studies have focused on the investigation of the wind stress in open ocean conditions where coastal processes were negligible. However, the direction and magnitude of the wind stress vector in coastal areas are still not fully known due to the low number of available measurement datasets. Here, we present new observations of the wind stress magnitude and its deviation from the mean wind direction. The data were recorded from a surface buoy during a five-day measurement campaign in southwestern Norway and cover wind speeds up to 10 m s−1 and significant wave heights up to 3.5 m in a coastal area with a steeply sloping sea floor. The adjustment of the wind stress vector due to changes in the wind and the wave conditions is illustrated and discussed by means of seven sample cases associated with both wind-following swell, cross-swell and counter-swell conditions. For this purpose, the stress vector computed in the sonic anemometer’s orthogonal coordinate system is projected into a non-orthogonal wind-swell coordinate system with its components aligned with: (1) the local wind-generated waves propagating in the wind direction; and (2) the swell wave direction. The wind stress direction was found to deviate from the wind direction by more than 20° for 46% of the recorded wind-following swell and cross-swell cases and for 54% of the counter-swell cases. The wind stress magnitude was observed to approach zero during the counter-swell period, which suggest a decoupling between the sea surface and the atmospheric surface layer. This was further investigated by means of an idealized Large Eddy Simulation results. The results in this study provide additional experimental evidence that the wind stress direction in coastal areas with a steeply sloping sea floor is influenced by the swell waves, the wave age and the wave steepness when the wind blows from undisturbed open ocean directions. For landward wind directions, the influence of the land boundary layer can, possibly in combination with atmospheric stability, adjust the magnitude and direction of the wind stress.

Spectral Problem for the Curl of a Vector Field in a Nonorthogonal Coordinate System

We discuss computational aspects of the spectral problem for the curl of a vector field that allow one to find tangent fields to coordinate surfaces of a given curvilinear coordinate system.

To the construction of a numerical algorithm for solving the dynamic problems of the theory of discretely reinforced conic shells in a nonorthogonal coordinate system

To the construction of a numerical algorithm for solving the dynamic problems of the theory of discretely reinforced conic shells in a nonorthogonal coordinate system

Thermal performance of a solar volumetric receiver using the two-energy equation model and radiation boundary condition

This work presents numerical results for the thermal performance of a Solar Volumetric Receiver (SVR). The Thermal Non-Equilibrium Model and Rosseland approximation were used. Radiation boundary condition was implemented at the absorber inlet. The numerical technique employed for discretizing the governing equations was the control volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure-velocity coupling. Effects of inlet velocity (uin), porosity (ϕ), medium permeability (K), and thermal conductivity ratio (ks/kf) on the solid and fluid temperatures were investigated. Reduction of temperatures as porosity increases or thermal conductivity decreases was observed, in addition to an increase in entry length for lower porosities or higher thermal conductivity ratios. Increase in inlet solid temperature as permeability increases was accompanied by a longer entry length and reduced final equilibrium temperature.