Axisymmetric Conditions Research Articles

On the axisymmetric stability of heated supersonic round jets.

We perform an inviscid, spatial stability analysis of supersonic, heated round jets with the mean properties assumed uniform on either side of the jet shear layer, modelled here via a cylindrical vortex sheet. Apart from the hydrodynamic Kelvin-Helmholtz (K-H) wave, the spatial growth rates of the acoustically coupled supersonic and subsonic instability waves are computed for axisymmetric conditions (m=0) to analyse their role on the jet stability, under increased heating and compressibility. With the ambient stationary, supersonic instability waves may exist for any jet Mach number Mj ≥2, whereas the subsonic instability waves, in addition, require the core-to-ambient flow temperature ratio Tj /To >1. We show, for moderately heated jets at Tj /To >2, the acoustically coupled instability modes, once cut on, to govern the overall jet stability with the K-H wave having disappeared into the cluster of acoustic modes. Sufficiently high heating makes the subsonic modes dominate the jet near-field dynamics, whereas the supersonic instability modes form the primary Mach radiation at far field.

Pseudospectral method for gravitational wave collapse

We present a new pseudospectral code, bamps, for numerical relativity written with the evolution of collapsing gravitational waves in mind. We employ the first order generalized harmonic gauge formulation. The relevant theory is reviewed and the numerical method is critically examined and specialized for the task at hand. In particular we investigate formulation parameters, gauge and constraint preserving boundary conditions well-suited to non-vanishing gauge source functions. Different types of axisymmetric twist-free moment of time symmetry gravitational wave initial data are discussed. A treatment of the axisymmetric apparent horizon condition is presented with careful attention to regularity on axis. Our apparent horizon finder is then evaluated in a number of test cases. Moving on to evolutions, we investigate modifications to the generalized harmonic gauge constraint damping scheme to improve conservation in the strong field regime. We demonstrate strong-scaling of our pseudospectral penalty code. We employ the Cartoon method to efficiently evolve axisymmetric data in our 3+1 dimensional code. We perform test evolutions of Schwarzschild perturbed by gravitational waves and by gauge pulses, both to demonstrate the use of our blackhole excision scheme and for comparison with earlier results. Finally numerical evolutions of supercritical Brill waves are presented to demonstrate durability of the excision scheme for the dynamical formation of a blackhole.

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: Theoretical formulation

The present two-part study aims at investigating the specific effects of Mohr–Coulomb matrix on the strength of ductile porous materials by using a kinematic limit analysis approach. While in the Part II, static and kinematic bounds are numerically derived and used for validation purpose, the present Part I focuses on the theoretical formulation of a macroscopic strength criterion for porous Mohr–Coulomb materials. To this end, we consider a hollow sphere model with a rigid perfectly plastic Mohr–Coulomb matrix, subjected to axisymmetric uniform strain rate boundary conditions. Taking advantage of an appropriate family of three-parameter trial velocity fields accounting for the specific plastic deformation mechanisms of the Mohr–Coulomb matrix, we then provide a solution of the constrained minimization problem required for the determination of the macroscopic dissipation function. The macroscopic strength criterion is then obtained by means of the Lagrangian method combined with Karush–Kuhn–Tucker conditions. After a careful analysis and discussion of the plastic admissibility condition associated to the Mohr–Coulomb criterion, the above procedure leads to a parametric closed-form expression of the macroscopic strength criterion. The latter explicitly shows a dependence on the three stress invariants. In the special case of a friction angle equal to zero, the established criterion reduced to recently available results for porous Tresca materials. Finally, both effects of matrix friction angle and porosity are briefly illustrated and, for completeness, the macroscopic plastic flow rule and the voids evolution law are fully furnished.

A study of the surface roughness in elasto-plastic shrink fitted joint

The aim of the present study is to provide a methodology for improving the design of a shrink-fitted assembly by focusing on two objectives: (i) taking into account the surface roughness of components, (ii) introducing a non-classical hardening friction law in which tangential micro-slip displacements are considered. The calculations have been made for elasto-plastic properties of component in axisymmetric conditions. Both the loading and unloading processes have been considered. After finite element simulations, it is observed that overall static behaviour of the assembly is influenced considerably by the surface finish of the mating components. The current numerical results of the shrink fitted assembly received from the coarse mesh could be treated as contact benchmark for further finite element analyses using as closely as possible identical input data.

Analytical Solution for Spheroidal Drop under Axisymmetric Linearized Boundary Conditions

A liquid spheroidal drop freely suspended in another fluid is considered under arbitrary axisymmetric boundary conditions, which are linearized with respect to the velocity field and can result, in particular, from an axisymmetric external flow and an electric field being applied either separately or in combination. All nonlinear effects including inertia and surface charge convection are assumed to be negligible, whereas the drop and the ambient fluid are assumed to be leaky dielectrics and to have different viscosities. Central to the analysis are the reformulated stress boundary conditions and representation of the velocity field inside and outside the drop in terms of non-Stokes stream functions. In the prolate spheroidal coordinates, the stream functions are expanded into infinite series of spheroidal harmonics, and the reformulated velocity and stress boundary conditions yield a first-order difference equation for the series coefficients, which admits an exact nonrecursive solution. “Steady” spheroidal drops then correspond to minimums of a kinematic condition error that admits a simple efficient approximation. Under the simultaneous presence of aligned linear flow and a uniform electric field with corresponding capillary numbers $Ca$ and $Ca_E$, a spherical drop is stationary when ${Ca}=k\,{Ca}_E$ and becomes prolate/oblate when ${Ca}\gtrless k\,{Ca}_E$, where $k$ is proportional to the Taylor discriminating function and depends on ratios of viscosities, dielectric constants, and electric conductivities of the two phases. A spheroidal drop is “steady” when ${Ca}=k_1\,{\rm Ca}_E+k_2$, where $k_1$ and $k_2$ depend on the spheroid's axes ratio $d$ and approach $k$ and 0, respectively, as $d\to1$. The results show that when the Taylor deformation parameter $D$ is in the range $[-0.5,0.4]$, this relationship can be used for finding any of the three $Ca$, $Ca_E$, and $D$ when the other two are given.

Three Dimensional Finite Element Analysis of Deep Excavations’ Concave Corners

The behavior of deep excavation’s retaining structures may become rather complex near the corners. The vicinity of concave corners is an especially problematic zone, and causes numerous problems in everyday practice. Conventional 2D analyses considering axisymmetric or plane strain conditions cannot be used in such cases due the spatial nature of the problem. A set of three dimensional finite element analyses have been performed aiming better understanding of soil and structure behavior near excavations’ concave corners and evaluating the influence of corner angle. The bending moment arising in anisotropic diaphragm walls near the concave corners are summarized, and compared to 2D plane strain and axisymmetric results.

MW급 풍력발전기 동력전달용 테이퍼 연결장치 설계에 관한 연구

This study focuses on the design of the tapered joints of a wind power turbine. The main variables of the tapered joint are the transmitted torque, shaft diameter, contact area of the tapered ring, and tightening torque of the bolts, which applies a compressive pressure from the hub to the shaft. The stress distribution of the taper fit was calculated under axisymmetric plane strain conditions because of the small taper angle. The axial displacement of the clamp can be calculated from the radial elastic deformation and the taper angle. The stress field of each ring is obtained from the cylinder stress equation. To verify the accuracy of the calculation, finite element (FE) analysis was performed, and the results of the calculation and FE analysis were compared. The hoop stress of the tapered surface showed a discrepancy of approximately 10, but the trends of the stress distributions of each component and the relative movement obtained by FE analysis were in good agreement with the analytical calculation results.

Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials

The present paper develops stress analysis of functionally graded hyperelastic thick spherical shell subjected to internal and external pressure. Hyperelastic behavior is modeled by using modified neo-Hookean strain energy function with variable material parameters. The material constants of strain energy function are graded along the radial direction based on a power law function. Material inhomogeneity parameter (m) is a power in the mentioned power law function. Material constant of strain energy function calculated from experimental data by using Levenberg-Marquardt nonlinear regression method. Stress components and stretches of spherical shell have been obtained for axisymmetric radial condition. Following this, profiles of extension ratio and stress components are plotted as a function of radius of sphere in the undeformed configuration for different material inhomogeneity parameter (m). The obtained results show that the inhomogeneity properties of FGMs have a significant influence on the displacement, stretch and stresses distribution along the radial direction. Sensitivity of stress components and displacement field to applied pressure and outer to inner radius ratio of vessel (structure parameter) are also investigated and leads to the fact that outer to inner radius ratio has a great effect on the deformation field and stress components and is a useful parameter from a design point of view which can be tailored to specific applications to control the stress. Thus with selecting a proper m value and structure parameter (B/A), engineers can design a specific FGM hollow sphere that can meet some special requirements.

Free vibration analysis of thick disks with variable thickness containing orthotropic-nonhomogeneous material using finite element method

Free vibration analysis of thick functionally graded nanocomposite annular and solid disks with variable thickness reinforced by single-walled carbon nanotubes (SWCNTs) is presented. Four types of distribution of uniaxial aligned SWCNTs are considered: uniform and three kinds of functionally graded (FG) distribution through radial direction of the disk. The effective material properties of the nanocomposite disk are estimated by a micro mechanical model. The axisymmetric conditions are assumed and employing the graded finite element method (GFEM), the equations are solved. The solution is considered for four different thickness profiles, namely constant, linear, concave and convex. The achieved results show that the type of distribution and volume fraction of CNTs and thickness profile have a great effect on normalized natural frequencies.

Asymmetric turbulent boundary layers along long thin circular cylinders at low-Re

Notable deviations of the asymmetric turbulent boundary layer (TBL) statistics from their axisymmetric counterpart along long thin circular cylinders are vitally important to the naval and oceanographic jurisdictions. Although the available experimental evidence backs their concern, the realm of parametric variability (both geometric and kinematic) is extremely limited to draw solid conclusions. We know that only small misalignments which quantify less than one degree of incidence between the freestream and the straight cylinder axis can substantially alter the boundary layer thicknesses, mean axial velocity, and Reynolds stresses. But the statistical database is plainly inadequate to justify modifying the design tools that were founded solely for axisymmetric flow conditions. Herein, we begin rectifying this drawback by numerical means. The investigation centers on low turbulent Reynolds numbers (500 ≤ Rea ≤ 2500) and small angles-of-incidence (0° < α < 9°) to validate and complement the lions-share of the present database (Rea = aUo/ν, where a, Uo, and ν are the cylinder radius, freestream velocity, and kinematic viscosity, respectively). In particular, we numerically resolved the statistical responses of the TBL, mean axial velocity, Reynolds stresses, and skin friction under angles-of-incidence up to the earliest signs of Strouhal-type shedding. Clearly, the first prominent response was the thinning and thickening of the TBL along the respective windward and leeward sides to only a minor misalignment. Tilting the straight cylinder to slightly higher yaw angles transformed the TBL to a transitional boundary layer along the windward side for all simulated Reynolds numbers. For yaw angles α > 2°, all turbulent statistics of the asymmetric boundary layer were measurably dissimilar to those of the axisymmetric state.

Design optimization of a centrifugal compressor vaneless diffuser

Centrifugal compressors are used in many heat pumps and refrigeration systems. Radial vaneless diffuser is a principal component in these compressors. Therefore, the present research aims at improving the centrifugal compressor performance by optimizing the design of the radial vaneless diffuser. Two radial vaneless diffuser geometries were proposed, investigated and numerically optimized. The optimization aimed at minimizing the diffuser loss coefficient and maximizing the pressure coefficient. Simulations were performed by solving the Reynolds averaged Navier–Stokes equation under 2D axisymmetric condition. A genetic optimization algorithm was implemented in order to conclude the optimum diffuser geometry. 2D axisymmetric simulations with air and R134a as working fluids showed that the optimized geometry reduced the diffuser loss coefficient by up to 10% and increased the pressure coefficient by up to 3.8%. Additional 3D simulations with an impeller located before the diffuser were performed. These 3D simulations showed that the optimized diffuser geometry reduced the diffuser loss coefficient by up to 4.7% and increased the pressure coefficient by up to 6.6% under jet-wake and swirl flow conditions.

Two Dimensional Functionally Graded Material Finite Thick Hollow Cylinder Axisymmetric Vibration Mode Shapes Analysis Based on Exact Elasticity Theory

Abstract A thick hollow cylinder with finite length made of two- dimensional functionally graded material (2D-FGM) is considered and its natural modes are determined, based on great importance of mode shapes information in order to understand vibration behaviour of structures. Three dimensional theory of elasticity implemented for problem formulation, since mode shapes of a thick cylinder are three dimensional even with axisymmetric conditions. The axisymmetric conditions are assumed for the 2D-FGM cylinder. The material properties of the cylinder are varied in the radial and axial directions, with power law functions. Effects of volume fraction distribution on the different types of symmetric mode shapes configuration and vibration behaviour of a simply supported cylinder are analyzed. Three dimensional equations of motion are used and the eigen value problem is developed, based on direct variation method.

On the Behavior of a Rotating Functionally Graded Hybrid Cylindrical Shell with Imperfect Bonding Subjected to Hygrothermal Condition

The static behavior of a rotating cylindrical shell with surface bounded sensor and actuator in an axisymmetric hygrothermal condition is analyzed. The shell is simply supported and could be rested on an elastic foundation. The material properties of the shell and piezoelectric sensor and actuator are assumed to be functionally graded in the radial direction. Using the Fourier series expansion method through the longitudinal direction and the differential quadrature method (DQM) across the radial direction, and governing differential equations are solved. The validity of the present work was verified by comparisons with other published works. Numerical results are presented to illuminate the effects of key parameters on the responses of the hybrid shell.

Instability of wind turbine wakes immersed in the atmospheric boundary layer

In this work a technique capable to investigate the near-wake stability properties of a wind turbine immersed in the atmospheric boundary layer is presented. Specifically, a 2D local spatial stability analysis is developed in order to take into account typical flow features of real operating wind turbines, such as the presence of the atmospheric boundary layer and the turbulence heterogeneity of the oncoming wind. This stability analysis can be generally applied on either experimental measurements or numerical data. In this paper it was carried out on wind tunnel experiments, for which a downscaled wind turbine is immersed in a turbulent boundary layer. Through spatial stability analysis, the dominant mode in the near wake, i.e. the most amplified one, is characterized and its frequency matches the hub-vortex instability frequency measured in the wind tunnel. As in the case of [10], where an axisymmetric wake condition was investigated, the hub-vortex instability results in a single-helical mode.

Centrifuge modelling of cone penetration tests in layered soils

Penetration problems are important in many areas of geotechnical engineering, such as the prediction of pile capacity and interpretation of in situ test data. The cone penetration test is a proven method for evaluating soil properties, yet relatively little research has been conducted to understand the effect of soil layering on penetrometer readings. This paper focuses on the penetration of a probe within layered soils and investigates the layered soil effects on both penetration resistance and soil deformation. A series of centrifuge tests was performed in layered configurations of silica sand with varying relative density in a 180° axisymmetric model container. The tests allowed for the use of a half-probe for observation of the induced soil deformation through a poly(methyl methacrylate) window as well as a full-probe for measurement of penetration resistance within the central area of the container. The variations of penetration resistance and soil deformation characteristics as they relate to penetration depth, soil density and soil layering are examined. The results of deformation are also compared with previous experimental data to examine the effect of the axisymmetric condition. The effects of soil layering on both resistance and soil deformation are shown to be dependent on the relative properties between soil layers.

Predictions of Heat Transfer and Flow Circulations in Differentially Heated Liquid Columns With Applications to Low-Pressure Evaporators

Numerical computations are presented for the temperature and velocity distributions of two differentially heated liquid columns with liquor depths of 0.1 m and 2.215 m, respectively. The temperatures in the liquid columns vary considerably with respect to position for pure conduction, free convection, and nucleate boiling cases using one-dimensional (1D) thermal resistance networks. In the thermal resistance networks the solutions are not sensitive to the type of condensing and boiling heat transfer coefficients used. However, these networks are limited and give no indication of velocity distributions occurring within the liquor. To alleviate this issue, two-dimensional (2D) axisymmetric and three-dimensional (3D) computational fluid dynamics (CFD) simulations of the test rigs have been performed. The axisymmetric conditions of the 2D simulations produce unphysical solutions; however, the full 3D simulations do not exhibit these behaviors. There is reasonable agreement for the predicted temperatures, heat fluxes, and heat transfer coefficients when comparing the boiling case of the 1D thermal resistance networks and the CFD simulations.

The ground response curve of underwater tunnels, excavated in a strain-softening rock mass

This paper presents an elasto-plastic model for determination of the ground response curve of a circular underwater tunnel excavated in elastic-strain softening rock mass compatible with a nonlinear Hoek-Brown yield criterion. The finite difference method (FDM) was used to propose a new solution to calculate pore water pressure, stress, and strain distributions on periphery of circular tunnels in axisymmetric and plain strain conditions. In the proposed solution, a modified non-radial flow pattern, for the hydraulic analysis, is utilized. To evaluate the effect of gravitational loads and variations of pore water pressure, the equations concerning different directions around the tunnel (crown, wall, and floor) are derived. Regarding the strain-softening behavior of the rock mass, the stepwise method is executed for the plastic zone in which parameters of strength, dilatancy, stresses, strains, and deformation are different from their elasto-plastic boundary values as compared to the tunnel boundary values. Besides, the analytical equations are developed for the elastic zone. The accuracy and application of the proposed method is demonstrated by a number of examples. The results present the effects of seepage body forces, gravitational loads and dilatancy angle on ground response curve appropriately.

Field tests and finite element modeling of a Prestressed Concrete Pipe pile-composite foundation

The Prestressed Concrete Pipe (PCP) pile-composite foundation was initially employed in the foundation of a culvert in the ancient Yellow River of China. To analyze the reinforcement effect of the foundation, a composite foundation composed of PCP piles was investigated with field tests and numerical simulations. A static load test was conducted to investigate the stress and deformation of the PCP pile-composite foundation, and a finite element model under three-dimensional (3D) axisymmetric conditions was developed to simulate the stress and deformation of the foundation. The Finite Element (FE) model used a nonlinear constitutive model for the soil, cushion and pile-soil interface. The results of the numerical simulations correspond with the load sharing, stress and displacement results of the static load test and indicate that the soil stress is primarily compressive. The cushion reduces the difference in settlement between the middle region and the side region of the bearing plate and increases the bearing capacity of the PCP pile-composite foundation. The stress of the pile significantly varies in the upper 1 m. The pile-soil load-sharing ratio increases with an increase in external load. This paper is expected to be useful for scientific research and efficient applications of PCP pilecomposite foundations.

Modeling Open-Flow Steam Reforming of Methanol over Cu/ZnO/Al2O3 Catalyst in an Axisymmetric Reactor

This paper describes a CFD study of the steam-reforming process (SRP) of methanol in a short pseudo-contact time reactor of fixed bed type, in axi-symmetric conditions. The SRP is important sake for hydrogen production, and the design /scale-up/control of the industrial processes in the future are supported by a reliable knowledge and prediction of the catalytic reaction. The difficulty of determining the reaction scheme and the associated constants is wellknown, due to the necessity of identifying the reaction kinetics in purely chemical regime, meaning with a perfect homogeneity and flow independence. Practically these ideal conditions, albeit assumed, are not fulfilled so that the intrinsic chemical kinetics is not reached. For the case of SRP, we have attempted here to validate the Peppley’s model by a numerical modelling reproducing exactly the local conditions in the experimental duct, accounting for gradients in the cross section. The numerical results show the same trends than the experimental one, but with a slight shift of 20% as a consequence of the reactor heterogeneity. This result seems acceptable to validate the use of the Peepley’s model for further studies in other types of complex flow reactors.

Phase-field-based simulation of axisymmetric binary fluids by using vorticity-streamfunction formulation

We propose a numerical method for the simulation of axisymmetric binary fluids based on the phase-field theory and vorticity-streamfunction formulation. Unlike most existing methods that solve the incompressible Navier-Stokes equations in the velocity-pressure form, we recast the equations into the vorticity-streamfunction form, which only need to treat one evolution equation of the vorticity for two-dimensional (2-D) or axisymmetric (pseudo 2-D) problems. Besides, this formulation enforces the continuity equation more easily. This method is verified through a series of benchmark problems under axisymmetric conditions, including the Laplace relation for a quiescent drop, coalescence of two drops, and the contact angle of a drop on a smooth wall with given wettability. Good agreements with theoretical relations or another numerical method are achieved for all these problems.