Arbitrary Cross-sectional Shape Research Articles

Energy-conserving formulation of the two-fluid model for incompressible two-phase flow in channels and pipes

We show that the one-dimensional (1D) two-fluid model (TFM) for stratified flow in channels and pipes (in its incompressible, isothermal form) satisfies an energy conservation equation, which arises naturally from the mass and momentum conservation equations that constitute the model. This result extends upon earlier work on the shallow water equations (SWE), with the important difference that we include non-conservative pressure terms in the analysis, and that we propose a formulation that holds for ducts with an arbitrary cross-sectional shape, with the 2D channel and circular pipe geometries as special cases.The second novel result of this work is the formulation of a finite volume scheme for the TFM that satisfies a discrete form of the continuous energy equation. This discretization is derived in a manner that runs parallel to the continuous analysis. Due to the non-conservative pressure terms it is essential to employ a staggered grid, which requires careful consideration in defining the discrete energy and energy fluxes, and the relations between them and the discrete model. Numerical simulations confirm that the discrete energy is conserved.

A Method Based on Riccati Equation for the Vibration Analysis of Rods with Variable Cross-Sections

In this study, the longitudinal vibration of rods with variable cross-sections is studied. For the analytical solution of the problem, a new analytical method based on a recently developed method on the Riccati differential equation is utilized. The governing equation is reduced to Hill’s type second-order ordinary differential equation. The transformed equations can readily be solved analytically for various cases according to the method. Seven cases have been considered, and the frequency equations for each case have been obtained. According to the method developed, the problem is solved in the simplest way. By using the present method, the reader can readily decide whether the problem is solved analytically or numerically. The present method can solve the problem of longitudinal vibration of rods having cross-sections of arbitrary shape. Finally, the method is also applied to the longitudinal vibration of stepped rods. Mode shapes are plotted for special values.

Simulation of Balancing Multi-Cavity Molds

Abstract A finite element program is developed for simulating the non-isothermal flow of non-Newtonian liquids in a multi-cavity mold with convergent runners and gates of any arbitrary cross-sectional shape. The program is used to investigate the problem of balancing multi-cavity molds. Results of simulating an actual mold filling show that balancing can be achieved by using a straight runner together with convergent gates having successively increasing angles of convergence. In addition, a strategy for approximating those angles of convergence is suggested.

Two-mirror compact system for ideal concentration of diffuse light.

We introduce a simple, compact two-mirror system for diffuse light concentration. The design principle is based on local conservation of optical brightness. The system design is flexible, and we are able to compute mirror shapes given arbitrary incident beam direction and target cross-sectional shape. As illustration, we showcase our design for flat and cylindrical target geometries, and we also demonstrate that our system is able to concentrate efficiently along one or two dimensions. We perform numeric experiments that confirm our theoretical results and provide diffuse light concentration very close to the thermodynamic limit in all cases we considered.

A new criterion to estimate the capacity of steeply inclined settlers of arbitrary cross-sectional shape

Steeply inclined settler technology has been long used to separate liquid and solid phases in dilute streams in various industries. In the present paper, a new model to account for the capacity of this kind of settler, in the particular case of those consisting of a set of inclined conduits, is proposed. The model combines a mass balance similar to that leading to the Ponder, Nakamura and Kuroda theory of inclined settlers and a particle trajectory approach to derive a simple relation between settling element length, particle settling-mean flow velocity ratio and geometric characteristics. Two new dimensionless parameters are derived to account for shape effects in the mass balance and when comparing settler capacity at equal conduit cross sectional area. Good agreement with previous experimental results and less conservative results than previous models have been found.

OPTIMIZATION OF HEAT CONDUCTION FOR TREELIKE NETWORK WITH ARBITRARY CROSS-SECTIONAL SHAPE

To minimize the thermal resistance of the fractal treelike heat conduction network and develop an optimization principle applicable for the network with an arbitrary cross-sectional shape, this paper first establishes a theoretical model regarding the total thermal resistance of the symmetric treelike network with arbitrary cross-sectional shapes and then studies the effects of the geometric and structural parameters of the network on its total thermal resistance. The numerical simulations are also performed to analyze the influences of the geometric and structural parameters of symmetric treelike networks with circular, rectangular and triangular cross-sectional shapes on the total thermal resistance. Both the theoretical model and the numerical simulation show that the total thermal resistance of the network with an arbitrary cross-sectional shape first decreases and then increases with increasing cross-sectional area ratio but always increases with increasing length ratio of branches at two successive branching levels when the total branch volume is constant. When the cross-sectional area ratio is equal to the reciprocal of the branching number, the treelike network has the minimum total thermal resistance. This scaling law is applicable for the treelike network with an arbitrary cross-sectional shape to achieve the minimum total thermal resistance.

Impact of Stripe Shape on the Reflectivity of Monolithic High Contrast Gratings

Monolithic high contrast gratings (MHCGs) composed of a one-dimensional grating patterned in a monolithic layer provide up to 100% optical power reflectance and can be fabricated in almost any semiconductor and dielectric material used in modern optoelectronics. MHCGs enable monolithic integration, polarization selectivity, and versatile phase tuning. They can be from 10 to 20 times thinner than distributed Bragg reflectors. The subwavelength dimensions of MHCGs significantly reduce the possibility of ensuring the smoothness of the sidewalls of the MHCG stripes and make precise control of the shape of the MHCG stripe cross-section difficult during the etching process. The question is then whether it is more beneficial to improve the etching methods to obtain a perfect cross-section shape, as assumed by the design, or whether it is possible to find geometrical parameters that enable high optical power reflectance using the shape that a given etching method provides. Here, we present a numerical study supported by the experimental characterization of MHCGs fabricated in various materials using a variety of common surface nanometer-scale shaping methods. We demonstrate that MHCG stripes with an arbitrary cross-section shape can provide optical power reflectance of nearly 100%, which greatly relaxes their fabrication requirements. Moreover, we show that optical power reflectance exceeding 99% with a record spectral bandwidth of more than 20% can be achieved for quasi-trapezoidal cross-sections of MHCGs. We also show that sidewall corrugations of the MHCG stripes have only a slight impact on MHCG optical power reflectance if the amplitude of the corrugation is less than 16% of the MHCG period. This level of stripe fabrication precision can be achieved using the most current surface etching methods. Our results are significant for the design and production of a variety of photonic devices employing MHCGs. The flexibility with regard to cross-section shape facilitates the reliable fabrication of highly reflective subwavelength grating mirrors. This in turn will enable the manufacture of monolithically integrated high-quality-factor optical micro- and nanocavity devices.

General Calculation Method of Normal Stress on Cross Section of Combined Tension (Compression) Bending Deformation

Tension (compression) bending combined deformation is a commonly used combined deformation form in engineering. The current material mechanics textbooks use the superposition method to calculate the stress on the cross section of the combined deformed member, whose characteristics are simple and easy to grasp. However, the premise of superposition principle is that the internal force, stress, strain and displacement have a linear relationship with external force. If the above linear relationship cannot be guaranteed, the superposition principle cannot be applied. In this case, the stress-strain relationship must be used for calculation. Based on the assumption of plane and unidirectional force, in this paper, rods with arbitrary cross-sectional shape under arbitrary load were taken as the research object, and the general calculation methods of the stress on the cross-section during the combined deformation of tension and bending were studied. This paper aimed at expanding students’ thinking mode and improving students’ learning ability in the process of material mechanics teaching.

Testing of generalized Helmholtz equations for gyrotropic waveguides

Universal generalized Helmholtz equations were tested for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with arbitrary magnetization and general Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes with longitudinal magnetization, which are obtained from generalized Helmholtz equations. From the general Helmholtz equations for gyrotropic waveguides are derived the partial Helmholtz equations for gyrotropic waveguides with certain orthogonal cross-sectional shapes (in particular: rectangular, round, elliptical) with a specific (longitudinal, normal, tangent) magnetization. General Helmholtz equations for gyrotropic waveguides with arbitrary orthogonal cross-sectional shapes in longitudinal magnetization and partial Helmholtz equations for gyrotropic elliptic waveguides in longitudinal magnetization are used during testing.

Pseudo-static simplified analytical solution for seismic response of deep tunnels with arbitrary cross-section shapes

Tunnels located in seismic active areas must support not only static loads exerted by the ground under gravity but also seismic loads from earthquake events. Current analytical solutions for seismic analysis of tunnel structures are limited to circular or rectangular tunnels but not available for tunnels with other complex cross-section shapes, such as straight-wall-arch-shaped or semi-rectangular-shaped tunnels. This paper presents a unified simplified analytical solution for deep tunnels with arbitrary cross-section shapes subjected to seismic loading. Since the cross-section dimension of tunnels is normally much smaller than the wavelength of ground peak velocities, the structure can be designed using the pseudo-static approach. The ground and the tunnel liner are assumed to be elastic, homogeneous, and isotropic in plane strain condition. Two different contact conditions, no-slip and full-slip conditions, are considered at the liner-ground interface. The complex variable theory combined with the conformal mapping technique are employed to obtain closed-form solutions for tunnel deformation and stresses. The proposed solution is verified by providing comparisons between its results and those from the finite element program ABAQUS. Furthermore, parametric analyses are carried out to investigate the influence of soil-structure relative stiffness ratio, cross-section shape, height-span ratio and thickness of middle wall on tunnel responses.

The effect of cyclic loading on the nonlinear response of structural concrete members with arbitrary cross-sectional shapes

This paper presents a numerical methodology for analysis of the behaviour of structural concrete sections subjected to cyclic loading. Classical beam theory was utilised to calculate sectional deformations (strain and curvature) under normal force and biaxial moment, while a nonlinear sectional analysis is performed via computer software written in Python by discretising the section of any shape, that is a combination of polygons and circles, into infinitesimal triangles or quadrilaterals. The stress-strain relationship of concrete and steel materials under monotonic loading was also implemented to evaluate the concrete’s secant modulus of elasticity at any strain level. A concrete cyclic model representing stress-strain relationships during unloading is thus proposed in which the secant modulus of elasticity of each mesh element is modified based on the stress level attained at the start of the unloading stage. Cyclic behaviour in tension and compression was investigated, including the transition from compression to tension and vice versa. The cyclic model was then implemented in sectional analysis to allow computation of the behaviour of the section at different loading stages. This study also proposes a new technique to calculate residual deformation during each analysis step and thus to plot the hysteresis behaviour of the section. Software was thus developed to estimate the failure load and produce a biaxial moment-curvature diagram, interaction diagram, and stress-strain diagram across the depth of the section for each load case.

Proposing a general formula to calculate the critical velocities in tunnels with different cross-sectional shapes

Among the parameters affecting the critical velocity in tunnel fires, the tunnel cross-sectional shape could significantly affect the tunnel fire characteristics, mainly due to the wall-bounded physics of the tunnel fire. Previously, the effects of the cross-sectional geometry of the tunnel were calculated using the non-dimensional analysis and hydraulic height of the tunnel. The dimensionless analysis using hydraulic height calculates only the effects of the tunnel sizes and does not capture the effects of the shape of the tunnel cross-section. Developing a 3D computational fluid dynamics tool using the body-fitted grids, the critical velocities are calculated for the 7 different cross-sectional shapes of tunnels, including the curved shape and the rectangular shape cross-sections, to precisely find the effects of the tunnel cross-sectional shape. The shape coefficient of the tunnel cross-section (area of the tunnel cross-section divided by the square of its height) is used as a tuning factor to correlate the results of the different cross-sectional shapes. Also, a modified formula is proposed to calculate the critical velocity of any arbitrary cross-sectional shape of the tunnel. Finally, the proposed formula is validated using the benchmark data. A series of recommendations in tunnel fires are provided using the proposed formula.

An innovative method for surmounting the constrained torsional problems of stocky beams with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties

The constrained torsion problems of stocky beams, being still an unsurmountable challenge in solid mechanics, will be solved by the innovative method proposed, and the computational results of numerical examples signifying that the beam model proposed is reasonable and the numerical analysis methodology of nodal line method is effective. The creative cruxes are as followings:•a type of innovative beam model—stocky beam in solid mechanics analysis is defined, which uses the stress state of the beam to distinguish whether a structural member is a beam or not, and the axis of the stocky beam is the locus of the minimum torsional stiffness centre of all its cross sections;•the warpages of all cross sections of a stocky beam are depicted as a mathematical expression that signifies a family of curved surfaces evolved from Prandtl's membrane shapes;•the constrained torsion problem of a stocky beam can be transformed into the boundary value problem of a set of ordinary differential equations by using the nodal line method developed.

Numerical Simulating Open-Channel Flows with Regular and Irregular Cross-Section Shapes Based on Finite Volume Godunov-Type Scheme

A numerical model based on the Saint-Venant equations (one-dimensional shallow water equations) is proposed to simulate shallow flows in an open channel with regular and irregular cross-section shapes. The Saint-Venant equations are solved by the finite-volume method based on Godunov-type framework with a modified Harten, Lax, and van Leer (HLL) approximate Riemann solver. Cross-sectional area is replaced by water surface level as one of primitive variables. Two numerical integral algorithms, compound trapezoidal and Gauss–Legendre integrations, are used to compute the hydrostatic pressure thrust term for natural streams with arbitrary and irregular cross-sections. The Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and second-order Runge–Kutta methods is adopted to achieve second-order accuracy in space and time, respectively. The performance of the resulting scheme is evaluated by application in rectangular channels, trapezoidal channels, and a natural mountain river. The results are compared with analytical solutions and experimental or measured data. It is demonstrated that the numerical scheme can simulate shallow flows with arbitrary cross-section shapes in practical conditions.

The torsional centre position of stocky beams with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties

The aim of this study focuses on torsional centre position of the cross section of a complex section stocky beam (CSSB), a new type of model for structural members with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties and with length-to-height or length-to-width ratio no more than 3 encountered in modern civil engineering structures. The nodal-line method is utilized to describe the shapes of the out-of-plane warpages of all cross sections of a CSSB under restrained torsional conditions by employing n unknown displacement functions of nodal lines, together with n quadratic interpolation functions inspired and evolved by the shape of Prandtl’ membrane. The governing ordinary differential equations are established by employing the principle of energy in elasticity. Based on the fact that the torsional stiffness of cross section of the same CSSB stays unchangeable with the change of the torque modes exerted to it, and on the computational results of the change rate of torsional angle due to the individual action of each torque corresponding to three torque modes, a formula for calculating two coordinates of the torsional centre position of the cross section is formulated. The computational results of numerical examples are demonstrated to verify the rationality and correctness of the formula.

Bending analysis of laminated beams using isogeometric variational asymptotic method

Variational asymptotic method (VAM) is a robust mathematical method employed to arrive at a reduced order governing equations in solid mechanics. VAM for beam analysis is accomplished in two stages: first stage involves performing a two-dimensional cross-sectional analysis, and the last stage is one-dimensional beam analysis. Cross-sectional analysis stage involves determining the stiffness coefficients for the beam cross section. VAM can be implemented either by developing analytical solutions or by adopting numerical methods. For beams with arbitrary cross-sectional shapes and material heterogeneity, cross-sectional analysis in VAM is generally carried out by adopting a finite element type procedure. In this study, a numerical procedure for the two-dimensional analysis and the one-dimensional beam analysis based on isogeometric framework is presented. NURBS functions are employed to model the geometry of a cross section and the warping displacement field. Warping represents the deformation of cross section of the beam. Stiffness coefficients obtained by cross-sectional analysis are input to the one-dimensional analysis. A NURBS-based one-dimensional method for bending analysis of laminated composite beams is proposed which can handle axial forces, twisting moments, in-plane and out-of plane bending moments. The one-dimensional analysis features four unknown variables, namely axial displacement, twist and two transverse displacements. The one-dimensional beam geometry and the unknown variables are modelled using NURBS functions. The one-dimensional formulation requires $$C^1$$ continuity which can be easily satisfied by NURBS functions. Numerical examples are solved to ascertain accuracy of the proposed methods. Results obtained were compared with exact solution and was found to be accurate. The one-dimensional analysis based on NURBS functions is non-intrinsic. Application of boundary conditions is far simpler to implement in comparison with the intrinsic one-dimensional analysis methods existing in traditional one-dimensional analysis of VAM literature.

TE-wave propagation in graded waveguide structures

We investigate TE-wave propagation in a hollow waveguide with a graded dielectric layer, described using a hyperbolic tangent function. General formulae for the electric field components of the TE-waves, applicable to hollow waveguides with arbitrary cross sectional shapes, are presented. We illustrate the exact analytical results for the electric field components in the special case of a rectangular waveguide. Furthermore, we derive exact analytical results for the reflection and transmission coefficients valid for waveguides of arbitrary cross sectional shapes. Finally, we show that the obtained reflection and transmission coefficients are in exact asymptotic agreement with those obtained for a very thin homogeneous dielectric layer using mode-matching and cascading. The proposed method gives analytical results that are directly applicable without the need of mode-matching, and it has the ability to model realistic, smooth transitions.

Transfer matrix analysis of a duct with gradually varying arbitrary cross-sectional area.

A method is presented to determine the low-frequency sound field in a duct with a cross-section of arbitrary shape and area that varies gradually without discontinuities. The duct is modeled as several fictitious segments, each of which has a cross-sectional area that is a quadratic function of position along its axis. A closed-form solution does not exist for arbitrary quadratic variation. Therefore, a series solution with two coefficients that depend on the boundary conditions and the radius of convergence (ROC) of the series is presented. Losses due to absorption are included. The solution corresponds to standing waves and is recast in terms of traveling waves. A transfer matrix is developed and used to express the pressure and velocity at the throat of a segment in terms of the pressure and velocity at its mouth. The series solution converges very rapidly when the length of the segment is less than half the ROC. The transfer matrix for the entire duct is the product of the transfer matrices of the segments. Numerical results are presented for open-closed ducts with the same length but different variations in area to illustrate the effect of the latter on the lowest resonance frequency and the pressure amplification.

Analytical and finite element solutions of free and forced vibration of unrestrained and braced thin-walled beams

In this article, vibration of thin-walled beams with arbitrary open cross-section shape is investigated. Based on the beam element model accounting for warping and flexural–torsional coupling, analytical solutions for different boundary conditions are derived for higher free vibration modes in bending, torsion and flexural–torsional coupled modes. In the model, the effects of rotational inertial kinematic terms are considered. The finite element approach of the model is also investigated. Three-dimensional beams with seven degrees of freedom per node are adopted in the mesh process. Free vibration and forced vibration analyses are possible. In forced vibration, the behaviour of the beams is studied in the frequency domain using the steady-state method (modal analysis). Damping is considered using the Rayleigh model. The model is validated by comparing the results to benchmark solutions found in the literature and to other recent numerical and experimental results. Additional finite element simulations are performed by means of commercial softwares (Abaqus and Adina). In slender unrestrained beams, the vibration behaviour is predominated by torsion and lateral bending modes. In design, recourse to braces is a good compromise. This solution is discussed, and improvement of the vibration behaviour in the presence of intermediate braces is confirmed. Application of higher vibration modes in building and bridge design is outlined. The effects of the number and distribution of the intermediate braces to improve structural stability against vibration behaviour is outlined.

Particle-laden thin-film flow in helical channels with arbitrary shallow cross-sectional shape

Particle-laden flows in helical channels are of interest for their applications in spiral particle separators used in the mining and mineral processing industries. In this paper, we extend the previous work of Lee, Stokes, and Bertozzi [“Behaviour of a particle-laden flow in a spiral channel,” Phys. Fluids 26, 043302 (2014)] by studying thin-film flows of monodisperse particle-laden fluid in helically wound channels of arbitrary centerline curvature and torsion and arbitrary cross-sectional shape. In the case where the particles are uniformly distributed through the depth of the film, significant analytic progress can be made yielding insight into the influence of channel geometry on particle distribution across the channel cross section: the governing equations reduce to a single nonlinear ordinary differential equation, which is readily integrated numerically to obtain the solution subject to appropriate boundary conditions. Motivated by possible application to the design of spiral separators, we consider the effects of changing the channel centerline geometry, the cross-sectional shape and the particle density on the resulting flows, and the radial distribution of particles. Our results support the findings in the work of Arnold, Stokes, and Green [“Thin-film flow in helically wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] regarding the effect of channel centerline geometry and cross-sectional shape on flows in particle-free regions. In particle-rich regions, similar effects are seen although the primary velocity is lower due to increased effective mixture viscosity. Of key interest is the effect of channel geometry on the focusing of the particles for given fluxes of fluid and particles. We find that introducing a trench into the channel cross section, a feature often used in commercial spiral particle separators, leads to a smaller radial width of the particle-rich region, i.e., sharper focusing of the particles, which is consistent with experiments showing that channel geometry influences particle separation in a spiral separator.