Nonorthogonal Curvilinear Coordinates Research Articles

Geometric Investigation of Three Thin Shells with Ruled Middle Surfaces with the Same Main Frame

It is proved and illustrated that by taking the main frame of the surface, consisting of three plane curves placed in three coordinate planes, three different algebraic surfaces with the same rigid frame can be designed. For the first time, one three of new ruled surfaces in a family of five threes of ruled surfaces, formed on the basis of some shapes of hulls of river and see ships, which, in turn, are projected in the form of algebraic surfaces with a main frame of three superellipses or of three other plane curves, is under consideration in detail with a standpoint of differential geometry. The geometrical properties of the ruled surfaces taken as the middle surfaces of thin shells for industrial and civil engineering are presented. Analytical formulas for determination of force resultants with using the approximate momentless theory of shells of zero Gaussian curvature given by non-orthogonal conjugate curvilinear coordinates are offered for the first time. The results derived using these formulae will help to correct the results obtained by numerical methods.

The inviscid incompressible limit of Kelvin–Helmholtz instability for plasmas

Introduction: The Kelvin–Helmholtz Instability (KHI) is an interface instability that develops between two fluids or plasmas flowing with a common shear layer. KHI occurs in astrophysical jets, solar atmosphere, solar flows, cometary tails, planetary magnetospheres. Two applications of interest, encompassing both space and fusion applications, drive this study: KHI formation at the outer flanks of the Earth’s magnetosphere and KHI growth from non-uniform laser heating in magnetized direct-drive implosion experiments. Here, we study 2D KHI with or without a magnetic field parallel to the flow. We use both the GAMERA code, which solves the compressible Euler equations, and the STRATOSPEC code, which solves the Navier-Stokes equations under the Boussinesq approximation, coupled with the magnetic field dynamics. GAMERA is a global three-dimensional MHD code with high-order reconstruction in arbitrary nonorthogonal curvilinear coordinates, which is developed for a large range of astrophysical applications. STRATOSPEC is a three-dimensional pseudo-spectral code with an accuracy of infinite order (no numerical diffusion). Magnetized KHI is a canonical case for benchmarking hydrocode simulations with extended MHD options.Methods: An objective is to assess whether or not, and under which conditions, the incompressibility hypothesis allows to describe a dynamic compressible system. For comparing both codes, we reach the inviscid incompressible regime, by decreasing the Mach number in GAMERA, and viscosity and diffusion in STRATOSPEC. Here, we specifically investigate both single-mode and multi-mode initial perturbations, either with or without magnetic field parallel to the flow. The method relies on comparisons of the density fields, 1D profiles of physical quantities averaged along the flow direction, and scale-by-scale spectral densities. We also address the triggering, formation and damping of filamentary structures under varying Mach number or Atwood number, with or without a parallel magnetic field.Results: Comparisons show very satisfactory results between the two codes. The vortices dynamics is well reproduced, along with the breaking or damping of small-scale structures. We end with the extraction of growth rates of magnetized KHI from the compressible regime to the incompressible limit in the linear regime assessing the effects of compressibility under increasing magnetic field.Discussion: The observed differences between the two codes are explained either from diffusion or non-Boussinesq effects.

Surfaces with a main framework of three given curves which include one circle

Superellipses are becoming more and more in demand in various branches of science and national economy due to their versatility. They found the most application in shipbuilding. Suggestions for the use of superellips in architecture and construction have appeared recently. The author proposes explicit and parametric equations of surfaces with a main framework of three predetermined superellips lying in three coordinate planes. These equations describe a large set of analytical shapes suitable for the formation middle surfaces of thin building shells. One of the superellipses is taken in a form of a circle. The shells can be designed on circular and rhombic plans, and also on plans in the shape of superellips of general type with convex and concave sides. All recommended surfaces are illustrated in 24 examples using computer graphics. A network of curvilinear non-orthogonal coordinates is generated on the surfaces using dimensionless independent parameters. The considered surfaces can become a part of the reserve of surfaces for further application in real structures and facilities.

Nonlinear static analysis of thin shallow and non-shallow shells using tensor formulation

In this study, a large-amplitude nonlinear static analysis of thin shallow and non-shallow shells is performed. Simply supported linear homogeneous elastic spherical panels and parabolic conoidal shells subjected to concentrated and distributed external loads are considered. To model the shells, the non-orthogonal tensor theory is applied and, to describe the strain–displacement relations of the shells, two approximations of Koiter’s theory are employed. To obtain the set of nonlinear equilibrium equations, a large-dimensional model is obtained by expanding the field displacements using Fourier series and a reduced order model is generated by using the Karhunen–Loève method. The proper orthogonal modes (POMs) of Karhunen–Loève method are acquired from a numerical model performed via Finite Element Analysis (FEA). The nonlinear static paths are compared with literature and with results obtained from FEA. The tensor formulation applied in this work has proven to be efficient when dealing with both shallow and non-shallow shells which are described by non-orthogonal curvilinear coordinates on nonlinear regime.

On the free vibration problem of thin shallow and non-shallow shells using tensor formulation

Linear free vibration of thin shells with rectangular base such as parabolic conoids, elliptic paraboloids and hyperbolic paraboloids are investigated. These shells are described by non-orthogonal curvilinear coordinates hence, Koiter theory, which uses a tensor formulation, is applied to accurately describe the strain–displacement relations. Four different boundary conditions are considered: simply supported with movable edges, simply supported with immovable edges; fully clamped with movable edges and fully clamped with immovable edges. Using an energy approach, Rayleigh–Ritz method is applied to obtain a set of ordinary differential equations of motion and Fourier series and Legendre polynomials are used to describe the displacements fields of the shell. Both, shallow and non-shallow shells were studied and obtained results of natural frequencies and vibration modes were compared with those from Maguerre theory and FEM using Abaqus, as can be seen, a good agreement was observed with FEM showing the efficacy of the applied theory. When comparing natural frequencies to those from Marguerre theory, there are substantial differences, indicating that lack of use of the tensor formulation which is not appropriate to shells with non-orthogonal coordinates.

An analysis of annular plate in curvilinear non-orthogonal coordinates with the help of equations of a shell theory

The complete system of equations of a linear theory of thin shells in curvilinear non-orthogonal coordinates proposed in the paper was taken as the basis of the investigation. Earlier, this system was used for static analysis of a long developable helicoid. In the article, this system is applied for the determination of stress-strain state of annular and circular plates under action of the external axisymmetric uniform load acting both in the plane of the plate and out-of-their plane. Presented results for annular plate given in the non-orthogonal coordinates ex-pand a number of problems that can be solved analytically. They can be used as the first terms of series of expansion of displacements of degrees of the small parameter if a small parameter method is applied for examining a long tangential developable helicoid.

CFD Simulation for Turbulent Flow Past NACA4412 Aerofoil

The complex fluid dynamics of different flow situations at low Reynolds number for natural flying objects like birds and insects, needs to be clearly understood for arriving at an optimum design for sizes ranging from the small man-made ones to very large size high speed commercial aircrafts or the fighter aircrafts. Airfoil performance at low Reynolds numbers impacts the performance of a wide range of systems. Computational Fluid Dynamics (CFD) tools have been around for a couple of decades now. With the superfast growth of computing power, speed and accuracy of these mathematical tools have improved to a considerable extent. However, any CFD simulation employing turbulence models needs to be validated against reliable and accurate measurement data obtained from wind tunnels. The present work focuses on 2D numerical simulation of turbulent flow past a symmetric NACA4412 aerofoil, using C- grid topology, for a Reynolds number of 1 million and 3 million. The computation uses the CFD code RANS3D, an implicit, pressure-based finite volume type Reynolds averaged Navier-stokes solver in generalized non-orthogonal curvilinear coordinates.

A three-dimensional mathematical model for simulating ice accretion on helicopter rotors

A new three-dimensional mathematical model is proposed to predict the shape of accreted ice on helicopter rotors. The model assumes that ice accretion on a rotor’s surface occurs by flow and solidification of a thin water film on its surface. The model is developed by analyzing conservation of mass, momentum, and energy in the thin water film on a rotating curved surface. The model is expressed in rotating body-fitted nonorthogonal curvilinear coordinates, and the Coriolis-force term in the momentum equations is neglected because its effect is relatively small compared with that of the centrifugal force. The computational method is presented to solve the model, and the model is validated by comparing its results with published experimental data. The results show that the proposed model accurately predicts both the water-film thickness and the ice-layer thickness on a rotating surface. Additionally, the model has been used to study how the rotor angular speed and MVD (Median Volumetric Diameter) of the droplets affect ice accretion. The results show that an increase in both angular speed and MVD can lead to a thicker layer of ice, while the effect of MVD on ice accretion near the stagnation point becomes unobvious when MVD is larger enough.

A note on Navier-Stokes equations with nonorthogonal coordinates

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix. doi:10.1017/S144618111700058X

A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

Development and validation of a Lagrangian method for 3D turbulent flows with curvilinear free-surface

A numerical model was developed based on a moving grid method for simulating three-dimensional turbulent flows affected by curvilinear free surface. Reynolds-Averaged Navier–Stokes equations with the k–e turbulence model were solved in non-orthogonal curvilinear coordinates. In the free surface, the kinematic boundary condition was implicitly imposed in the pressure Poisson equation derived from the momentum and continuity equations. The water surface elevation was calculated at each time step without solving additional equations. The developed numerical model was validated using the experimental data of the strongly curved channels and submerged hydraulic jump. The numerical simulation of the flow field and free surface elevation in all cases were compared with the experimental results. The agreement between the simulated and measured results was satisfactory for three-dimensional turbulent flows in the strongly curved channels and submerged hydraulic jump. The secondary flow and dip phenomena were correctly simulated in strongly curved channels. This numerical model could accurately predict steep water surface gradients in a submerged hydraulic jump. The reasonable results of the numerical simulation demonstrate the capability of the presented Lagrangian method in hydraulic engineering applications. This model provides a suitable method for simulating the free surface in engineering and environmental problems.

Extension to the Myers Model for Calculation of Three-Dimensional Glaze Icing

To simulate three-dimensional glaze ice accretion on a typical aircraft surface, the model of water film flow and ice accretion on a curved surface based on two coupled partial differential equations proposed by Myers et al. is extended in nonorthogonal curvilinear coordinates. Then, the governing equation for water film flow coupling with a Stefan model for glaze icing is built up. An efficient implicit–explicit numerical method is proposed for solving the governing equations. The glaze ice computations on two-dimensional airfoils and three-dimensional typical geometries in in-flight icing conditions are completely accomplished. Validation results of two-dimensional calculation cases with experimental ice shapes are presented. Three-dimensional simulations of glaze ice accretion on a sphere and a swept wing are successfully carried out and compared with ice profiles from icing tunnel tests and simulations based on the classical Messinger model.

High-order finite volume shallow water model on the cubed-sphere: 1D reconstruction scheme

A central-upwind finite-volume (CUFV) scheme for shallow-water model on a nonorthogonal equiangular cubed-sphere grid is developed, consequently extending the 1D reconstruction CUFV transport scheme developed by us. High-order spatial discretization based on weighted essentially non-oscillatory (WENO) is considered for this effort. The CUFV method combines the alluring features of classical upwind and central schemes. This approach is particularly useful for complex computational domain such as the cubed-sphere. The continuous flux-form spherical shallow water equations in nonorthogonal curvilinear coordinates are utilized. Fluxes along the element boundaries are approximated by a Kurganov–Noelle–Petrova scheme. A fourth-order strong stability preserving Runge–Kutta time stepping scheme for time integration is employed in the present work. The numerical scheme is evaluated with standard shallow water test suite, which accentuate accuracy and conservation properties. In addition, an efficient yet inexpensive bound preserving filter with an optional positivity filter is used to remove spurious oscillations and to achieve strictly positive definite numerical solution. To tackle the discontinuities that arise at the edges of the cubed-sphere grid, we utilize a high-order 1D interpolation procedure combining cubic and quadratic interpolations.The results with the high-order scheme is compared with the results for the same tests for various schemes available in literature. Since, the scheme presented here uses local-cell information, it is expected to be scalable to high number of processors count in a distributed node high-performance computer.

Comparison of Robustness and Efficiency for SIMPLE and CLEAR Algorithms with 13 High-Resolution Convection Schemes in Compressible Flows

In this article, a comparison is made between the robustness and efficiency of the CLEAR algorithm and the SIMPLE algorithm on nonorthogonal curvilinear coordinates for compressible flows. Thirteen different high-order convection schemes are employed in the calculations. Subsonic flow, transsonic flow, and supersonic flow in a channel with a circular arc bump and compressible flow in a Laval nozzle are used as test cases. The CLEAR algorithm shows huge potential to compute the transsonic flow in the Laval nozzle and high-speed compressible flows. Results with the ADBQUICKEST scheme, the HLPA scheme, and the MUSCL scheme are stable for both the compressible SIMPLE and CLEAR algorithms for all the mentioned cases.

Performance Analyses of IDEAL Algorithm on Highly Skewed Grid System

IDEAL is an efficient segregated algorithm for the fluid flow and heat transfer problems. This algorithm has now been extended to the 3D nonorthogonal curvilinear coordinates. Highly skewed grids in the nonorthogonal curvilinear coordinates can decrease the convergence rate and deteriorate the calculating stability. In this study, the feasibility of the IDEAL algorithm on highly skewed grid system is analyzed by investigating the lid-driven flow in the inclined cavity. It can be concluded that the IDEAL algorithm is more robust and more efficient than the traditional SIMPLER algorithm, especially for the highly skewed and fine grid system. For example, at θ = 5° and grid number = 70 × 70 × 70, the convergence rate of the IDEAL algorithm is 6.3 times faster than that of the SIMPLER algorithm, and the IDEAL algorithm can converge almost at any time step multiple.

Simulation of Unsteady Water Film Flow on Pelton Bucket

In order to simulate the complicated unsteady flow in Pelton bucket, it is necessary to apply the animated cartoon approach. In this paper, a free jet and the inner surface of a bucket is described by boundary fitted grid (BFG) with non-orthogonal curvilinear coordinates. The water flow is discretized in space and time for CFD (computational fluid dynamics). The moving grids of water film are successfully projected onto the bucket’s inner surface by a projection algorithm. The visualization result of the jet landing on bucket’s surface and the unsteady flow in the rotating buckets in 3D verifies the effectiveness of the proposed method.

Extremely long range surface polaritons in a thin corrugated metal film

The essential increase of propagation length of a long range surface plasmon polariton in a thin symmetrically corrugated plasmon-carrying film embedded in a dielectric medium is theoretically predicted. The calculations are based on the differential formalism for the system of Maxwells equations where the solution for electromagnetic fields is written as a superposition of partial plane waves in the presentation of a curvilinear non-orthogonal coordinates system for simplifying the boundary conditions. The spectral and angular dependencies of p-polarized light transmittance/reflectance demonstrate that the in-plane shift between both profiles of corrugated film drastically changes the surface plasmon polariton propagation length from minimum of the asymmetric profile to maximum of the symmetric one. The obtained results were qualitatively explained using the model of weakly coupled photonic wells.

Three-Dimensional Numerical Study of Fluid and Heat Transfer Characteristics of Dimpled Fin Surfaces

In the present study, a code based on the nonorthogonal curvilinear coordinates is developed with a collocated grid system generated by the two-boundary method. After validation of the code, it is used to compare simulated results for a fin-and-tube surface with coupled and decoupled solution methods. The results of the coupled method are more agreeable with the test data. Simulation for dimpled and reference plain plate fin-and-tube surfaces are then conducted by the coupled method within a range of inlet velocity from 1.0 m/s to 5 m/s. Results show that at identical pumping power the dimpled fin can enhance heat transfer by 13.8–30.3%. The results show that relative to the reference plain plate fin-and-tube surface, heat transfer rates and pressure drops of the dimpled fin increase by 13.8%–30.3% and 31.6%–56.5% for identical flow rate constraint. For identical pumping power constraint and identical pressure drop constraint, the heat transfer rates increase by 11.0%–25.3% and 9.2%–22.0%, respectively. By analyzing the predicted flow and temperature fields it is found that the dimples in the fin surface can improve the synergy between velocity and fluid temperature gradient.

Prediction of radiative heat transfer in 2D irregular geometries using the collocation spectral method based on body-fitted coordinates

Recently, an efficient numerical method, which is called the collocation spectral method (CSM), for radiative heat transfer problems, has been proposed by the present authors. In this numerical method there exists the exponential convergence rate, which can obtain a very high accuracy even using a small number of grids. In this article, the CSM based on body-fitted coordinates (BFC) is extended to simulate radiative heat transfer problems in participating medium confined in 2D complex geometries. This numerical method makes simultaneously the use of the merits of both the CSM and BFC. In this numerical approach, the radiative transfer equation (RTE) in orthogonal Cartesian coordinates should be transformed into the equation in body-fitted nonorthogonal curvilinear coordinates. In order to test the efficiency of the developed method, several 2D complex irregular enclosures with curved boundaries and containing an absorbing, emitting and scattering medium are examined. The results obtained by the CSM are assessed by comparing the predictions with those in references. These comparisons indicate that the CSM based on BFC can be recommended as a good option to solve radiative heat transfer problems in complex geometries.

Generalization of the Second Order Vector Potential Formulation for Arbitrary Non-Orthogonal Curvilinear Coordinates Systems from the Covariant Form of Maxwell's Equations

A great number of semi-analytical models, notably the representation of electromagnetic fields by integral equations are based on the second order vector potential (SOVP) formalism which introduces two scalar potentials in order to obtain analytical expressions of the electromagnetic fields from the two potentials. However, the scalar decomposition is often known for canonical coordinate systems. This paper aims in introducing a specific SOVP formulation dedicated to arbitrary non-orthogonal curvilinear coordinates systems. The electromagnetic field representation which is derived in this paper constitutes the key stone for the development of semi-analytical models for solving some eddy currents moelling problems and electromagnetic radiation problems considering at least two homogeneous media separated by a rough interface. This SOVP formulation is derived from the tensor formalism and Maxwell’s equations written in a non-orthogonal coordinates system adapted to a surface characterized by a 2D arbitrary aperiodic profile.