Non-orthogonal Curvilinear Coordinate System Research Articles

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.

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.

To the statement of dynamic problems of the theory of discrete longitudinally reinforced conical shells in a nonorthogonal curvilinear coordinate system

To the statement of dynamic problems of the theory of discrete longitudinally reinforced conical shells in a nonorthogonal curvilinear coordinate system

Numerical simulation to investigate the induced buoyant flow characteristics caused by intensive heat in complex curvilinear geometries

AbstractThe buoyancy‐induced turbulent flows in complex geometries with internal heat source have important application in enclosure fire dynamics, for example, smoke movement in the situation of fire in tunnels. The majority of computational works studying the fire‐induced hot air movement in such cases use a Cartesian coordinate system. Hence, to simulate curved geometries, the Cartesian grid should be refined so that it comparatively matches the geometry. In the present work, a three‐dimensional numerical method using nonorthogonal curvilinear coordinate system was developed. Numerical simulations were performed to study the flow structures and the heat transfer characteristics of fire‐induced buoyant flow and the effects of using curvilinear coordinates on the result's accuracy and the computational cost. Numerical results were compared with the available experimental data and the results obtained by fire dynamics simulator (FDS) software. The results of the present method showed better agreement with the experimental results when compared with the FDS results.

A 2D curvilinear coupled surface–subsurface flow model for simulation of basin/border irrigation: theory, validation and application

Despite the development of pressurized irrigation systems in the second half of the twentieth century, surface irrigation continues to be the most used system in the world. Computer tools are required to support performance improvements leading to its sustainability. A simulation model of basin/border irrigation is presented in this paper combining two-dimensional overland hydraulics based on Saint–Venant equations with three-dimensional infiltration based on the mixed form of Richards’ equation. The coupling of these equations is attained by enforcing continuity of pressure at the soil surface. The model applies a finite-volume approach using the diffusion-wave approximation form of the shallow water equations. Water flow on undulated topographies is simulated using a non-orthogonal curvilinear coordinate system. An underrelaxed-modified Picard iteration algorithm is used for Richards’ equation, and an underrelaxed Picard iteration algorithm is used for Saint–Venant equations. The model was validated using experiments from the literature. Model performance was robust and accurate, even in complex theoretical cases with strong soil-surface undulations. Mass conservation was judged satisfactory in all cases, with the long-term ratio of mass balance error in the order of 10−2 in the most complex simulations. Soil type, mesh non-orthogonality, and the interaction of these variables had a strong effect on CPU time. Additional research will be required to transform this simulation model into an operational tool for surface irrigation. Attention will need to be paid to additional surface irrigation systems, soil types, and the optimization of computational speed.

Numerical modeling of surface wave development under the action of wind

Abstract. The numerical modeling of two-dimensional surface wave development under the action of wind is performed. The model is based on three-dimensional equations of potential motion with a free surface written in a surface-following nonorthogonal curvilinear coordinate system in which depth is counted from a moving surface. A three-dimensional Poisson equation for the velocity potential is solved iteratively. A Fourier transform method, a second-order accuracy approximation of vertical derivatives on a stretched vertical grid and fourth-order Runge–Kutta time stepping are used. Both the input energy to waves and dissipation of wave energy are calculated on the basis of earlier developed and validated algorithms. A one-processor version of the model for PC allows us to simulate an evolution of the wave field with thousands of degrees of freedom over thousands of wave periods. A long-time evolution of a two-dimensional wave structure is illustrated by the spectra of wave surface and the input and output of energy.

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.

Spiralling liquid jets: verifiable mathematical framework, trajectories and peristaltic waves

The dynamics of a jet of an inviscid incompressible liquid spiralling out under the action of centrifugal forces is considered with both gravity and the surface tension taken into account. This problem is of direct relevance to a number of industrial applications, ranging from the spinning disc atomization process to nanofibre formation. The mathematical description of the flow by necessity requires the use of a local curvilinear non-orthogonal coordinate system centred around the jet’s baseline, and we present the general formulation of the problem without assuming that the jet is slender. To circumvent the inconvenience inherent in the non-orthogonality of the local coordinate system, the orthonormal Frenet basis is used in parallel with the local non-orthogonal basis, and the equation of motion, with the velocity considered with respect to the local coordinate system, is projected onto the Frenet basis. The variation of the latter along the baseline is then described by the Frenet equations which naturally brings the baseline’s curvature and torsion into the equations of motion. This technique allows one to handle different line-based non-orthogonal curvilinear coordinate systems in a straightforward and mathematically transparent way. An analysis of the slender-jet approximation that follows the general formulation shows how a set of ordinary differential equations describing the jet’s trajectory can be derived in two cases: $\mathit{We}=O(1)$ and $\unicode[STIX]{x1D716}\mathit{We}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$, where $\unicode[STIX]{x1D716}$ is the ratio of characteristic length scales across and along the jet and $\mathit{We}$ is the Weber number. A one-dimensional model for the propagation of nonlinear peristaltic disturbances along the jet is derived in each of these cases. A critical review of the work published on this topic is presented showing where errors typically occur and how to identify and avoid them.

Stream surface theory of bird-like flapping flight

Through analyzing the motion characteristics of bird-like flapping flight, it is considered that the wing angular acceleration is equal to zero at the point of maximum angular speed. Thus, the flapping flight is equivalent to a uniform rotating motion which can be analyzed by using the stream surface theory of turbomachinery during a micro period of time. In this article, the N-S equations of the motion are expanded in a non-orthogonal curvilinear coordinate system, and simplified on stream surfaces of the flapping flight model. By using stream function method, the three-dimensional unsteady flow equations are simplified as a two-order partial differential equation with variable coefficients eventually and the equation’s iterative solving method on S1 and S2 stream surfaces of the flapping flight model is presented. Through expanding the relatively steady equations of flapping flight at an arbitrary time point of a stroke on meridional plane of the flapping flight model, it can use a relatively steady motion to approximate the real flapping flight at that time point, and analyze the flow stability influenced by the wing’s flexibility. It can be seen that the wing flexibility is related to the higher pressurization capacity and the flow stability, and the pressurization capacity of flexible wing is proportional to the angular speed, angular distortion rate and radius square.

Interplanetary particle transport simulation for warning system for aviation exposure to solar energetic particles

Solar energetic particles (SEPs) are one of the extreme space weather phenomena. A huge SEP event increases the radiation dose received by aircrews, who should be warned of such events as early as possible. We developed a warning system for aviation exposure to SEPs. This article describes one component of the system, which calculates the temporal evolution of the SEP intensity and the spectrum immediately outside the terrestrial magnetosphere. To achieve this, we performed numerical simulations of SEP transport in interplanetary space, in which interplanetary SEP transport is described by the focused transport equation. We developed a new simulation code to solve the equation using a set of stochastic differential equations. In the code, the focused transport equation is expressed in a magnetic field line coordinate system, which is a non-orthogonal curvilinear coordinate system. An inverse Gaussian distribution is employed as the injection profile of SEPs at an inner boundary located near the Sun. We applied the simulation to observed SEP events as a validation test. The results show that our simulation can closely reproduce observational data for the temporal evolution of particle intensity. By employing the code, we developed the WArning System for AVIation Exposure to Solar energetic particles (WASAVIES).

Numerical modeling of 3D fully nonlinear potential periodic waves

A simple and exact numerical scheme for long-term simulations of 3D potential fully nonlinear periodic gravity waves is suggested. The scheme is based on the surface-following nonorthogonal curvilinear coordinate system. Velocity potential is represented as a sum of analytical and nonlinear components. The Poisson equation for the nonlinear component of velocity potential is solved iteratively. Fourier transform method, the second-order accuracy approximation of vertical derivatives on a stretched vertical grid and the fourth-order Runge–Kutta time stepping are used. The scheme is validated by simulation of steep Stokes waves. A one-processor version of the model for PC allows us to simulate evolution of a wave field with thousands degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of nonlinear 2D surface waves, generation of extreme waves, and direct calculations of nonlinear interactions.

The effect of adiabatic obstacles on natural convection inside a horizontal elliptical porous annulus

This study numerically examines the influence of installing two identical radial adiabatic obstacles on natural convection within a porous layer of constant thickness. This porous layer surrounds an isothermal hot elliptical cylinder and is surrounded by an isothermal cold one. The obstacles' length, angular position and radial position were changed in various amounts. A non-orthogonal curvilinear coordinate system is adopted, in order to treat the presence of the obstacles inside a two dimensional elliptical annulus. Consequently, Finite Differences Techniques and ADI with Relaxation Method were used to approximate the computational regime of the non-Darcian-Boussinesq equation, governing the flow inside the computational porous domain. An efficient code of high accuracy was constructed to solve the approximated governing equations. The results showed that the installation of long obstacles in the meeting place of two convection cells causes an increase in heat transfer rates by up to 16%. Whereas the installation of short obstacles in a place lying close to the center of the major convection cell causes a reduction in heat transfer by up to 19% .

Diffusion Experiments with a Global Discontinuous Galerkin Shallow-Water Model

Abstract A second-order diffusion scheme is developed for the discontinuous Galerkin (DG) global shallow-water model. The shallow-water equations are discretized on the cubed sphere tiled with quadrilateral elements relying on a nonorthogonal curvilinear coordinate system. In the viscous shallow-water model the diffusion terms (viscous fluxes) are approximated with two different approaches: 1) the element-wise localized discretization without considering the interelement contributions and 2) the discretization based on the local discontinuous Galerkin (LDG) method. In the LDG formulation the advection–diffusion equation is solved as a first-order system. All of the curvature terms resulting from the cubed-sphere geometry are incorporated into the first-order system. The effectiveness of each diffusion scheme is studied using the standard shallow-water test cases. The approach of element-wise localized discretization of the diffusion term is easy to implement but found to be less effective, and with relatively high diffusion coefficients, it can adversely affect the solution. The shallow-water tests show that the LDG scheme converges monotonically and that the rate of convergence is dependent on the coefficient of diffusion. Also the LDG scheme successfully eliminates small-scale noise, and the simulated results are smooth and comparable to the reference solution.

Application of Multi-Block method for simulating shallow free surface flows in complex geometries

In the present study a numerical method is developed for simulating shallow free surface flows in complex geometries. A Multi-Block (MB) method is employed in conjunction with non-orthogonal curvilinear coordinate system. This technique gives the numerical method a high flexibility to tackle flow domains with any complex boundary which may be encountered in nature. The MB method utilizes patched grid with continuous grid lines across block boundaries. In this paper the MB methods is applied to 2-D shallow flows. However, this technique is extendable to 3-D models too. The model is based on collocated grid arrangement and the control volume method is used for solution of equations. The method is implicit and uses a SIMPLEC-like algorithm to find the water surface elevation. The flexibility and accuracy of the model is presented in few test cases with complex block arrangement and circulating flows.

NUMERICAL STUDY OF INCOMPRESSIBLE FLOW AND HEAT TRANSFER IN NON-CIRCULAR DUCTS WITH CUSP CORNERS

A numerical finite-volume calculation method was used for application to fully-developed flow and heat transfer in straight ducts with 2-Cusp, 3-Cusp and 4-Cusp cross-sectional area. The method was formulated with reference to a non-orthogonal curvilinear coordinate system which was fitted exactly into duct shape. In turbulent flow, the Reynolds stresses were calculated by using the coupled algebraic stress model of Launder and Ying (1973) with a (k ) transport model. This turbulent stress model enabled secondary flows to be included in the non-circular duct calculations. In heat transfer solution, three thermal boundary conditions were investigated. Predictions were compared with available numerical and experimental data. The turbulent flow predictions of secondary and axial velocity, wall shear stress and Nusselt number were in reasonable agreement with experiment for the fully-developed flow cases. It is concluded that, although the maximum secondary flow was found to be (1.5%-2.5%) of the mean axial flow, the absence of this flows have significant influence on the flow and heat transfer parameters. Moreover, these flows will increase when the corners of a duct become more acute. Heat transfer results showed that the peripheral temperature variation around the duct has a significant effect on the heat transfer results, and confirmed the inadequacy of the hydraulic radius concept in the cases of the very non-circular ducts when comparison is made with Blasius correlation.

A semianalytical finite element method for elastic guided waves propagating in helical structures

Steel multiwire cables are widely used in civil engineering as load-carrying members. The basic element of these cables is usually a simple straight strand made of a straight core and one layer of helical wires. Several difficulties arise in the understanding of guided ultrasonic waves in such structures, partly due to the helical geometry and the interwire coupling effects. In the context of nondestructive evaluation, this paper aims at theoretically investigating the propagation of elastic waves in helical waveguides. A numerical method is chosen based on a semianalytical finite element technique that relies on a specific nonorthogonal curvilinear coordinate system. This system is shown to be translationally invariant along the helix centerline so that a spatial Fourier transform can be explicitly performed along the axis and the problem is reduced to two dimensions. A single helical wire is first considered. The convergence and accuracy of the proposed method are assessed by comparing finite element results with reference solutions. The method is then extended to study a seven wire cable with simplified contact conditions. Dispersion curves are presented for wave numbers as well as energy velocities. Significant differences with the infinite straight cylinder are observed.

Elastic waves in helical waveguides

The goal of this paper is to theoretically investigate the propagation of elastic waves in helical waveguides. In the context of non-destructive evaluation for structural health monitoring, this study is motivated by the need for inspecting helical structures such as cables or springs. A numerical method is chosen based on a semi-analytical finite element technique. The proposed method relies on a non-orthogonal curvilinear coordinate system that is translationally invariant along the helix centreline, so that a Fourier transform is explicitly performed and the problem is reduced to two dimensions. Some useful expressions are also derived for the averaged energy and flux in order to directly compute the energy velocity. The convergence and accuracy of the proposed method are then assessed by comparing finite element results with reference solutions. A dispersion analysis inside a 7.5° helical wire, typically encountered in civil engineering cables, is realised including attenuation due to material damping. Some dispersion curves are finally presented for a wide range of lay angles and for several centreline radii. Significant differences with the infinite cylinder are observed.