Axisymmetric Boundary Research Articles

Global stability analysis of the axisymmetric boundary layer: Effect of axisymmetric forebody shapes on the helical global modes

Global stability analysis of the axisymmetric boundary layer: Effect of axisymmetric forebody shapes on the helical global modes

Axisymmetric BEM analysis of layered elastic halfspace with volcano-shaped mantle and cavity under internal gas pressure

This paper presents an axisymmetric BEM analysis of layered elastic halfspace with volcano-shaped mantle and cavity under internal gas pressure. The problem is of interest to understand the behavior of volcanoes, tectonic earthquakes and other oil and gas reservoirs. The volcano-shaped mantle ground topography and the internal spherical or ellipsoidal cavity are added to the conventional model of layered halfspace. The analysis uses the axisymmetric boundary element method (BEM) associated with the fundamental solution of a multilayered elastic fullspace subject to the concentrated ring-body force. The BEM is further verified for its high efficiency and accuracy by comparing its result with the analytical and FEM solutions for the problem of a homogeneous elastic halfspace whose spherical cavity is under internal pressure. The BEM is used to examine the effect of the volcano-shaped mantle, the cavity shape, the layered material properties and the internal pressure on the elastic behavior of the halfspace under the internal pressure loading within the cavity. The displacements and stresses at the external and internal boundaries and within the layered halfspace are presented and analyzed in detail. The presence of the volcano-shaped mantle can reduce significantly the swelling ground movement induced by the expansive pressure in the cavity in the layered halfspace. The cavity shape can have significant effect to the location and magnitude of the maximum tensile principal hoop stress on the cavity surface induced by its internal pressure.

Modelling of Applied Magnetic Field and Thermal Radiations Due to the Stretching of Cylinder

In this study, a numerical approach was adopted in order to explore the analysis of magneto fluid in the presence of thermal radiation combined with mixed convective and slip conditions. Using the similarity transformation, the axisymmetric three-dimensional boundary layer equations were reduced to a self-similar form. The shooting technique, combined with the Range–Kutta–Fehlberg method, was used to solve the resulting coupled nonlinear momentum and heat transfer equations numerically. When physically interpreting the data, some important observations were made. The novelty of the present study lies in finding help to control the rate of heat transfer and fluid velocity in any industrial manufacturing processes (such as the cooling of metallic plates). The numerical results revealed that the Nusselt number decrease for larger Prandtl number, curvature, and convective parameters. At the same time, the skin friction coefficient was enhanced with an increase in both slip velocity and convective parameter. The effect of emerging physical parameters on velocity and temperature profiles for a nonlinear stretching cylinder has been thoroughly studied and analyzed using plotted graphs and tables.

Analysis and comparison of turbulence model coefficient uncertainty for canonical flow problems

The purpose of this paper is to present results of an uncertainty and sensitivity analysis study of commonly used turbulence models in Reynolds-Averaged Navier–Stokes codes due to the epistemic uncertainty in closure coefficients for a set of turbulence model validation cases that represent the structure of several canonical flow problems. The study focuses on the analysis of a 2D Zero Pressure Gradient Flat Plate, a 2D NASA Wall Mounted Hump, and an Axisymmetric Shock Wave Boundary Layer Interaction, all of which are well documented on the NASA Langley Research Center Turbulence Modeling Resource website. The Spalart–Allmaras (SA), the Wilcox (2006) κ−ω (W2006), and the Menter Shear-Stress Transport (SST) turbulence models are considered in the stochastic analyses of these flow problems, and the FUN3D code was utilized as the flow solver. The uncertainty quantification approach involves stochastic expansions based on non-intrusive polynomial chaos to efficiently propagate the uncertainty. Sensitivity analysis is performed with Sobol indices to rank the relative contribution of each closure coefficient to the total uncertainty for several output flow quantities. The results generalize a set of closure coefficients which have been identified as contributing most to the uncertainty in various output quantities of interest for the set of canonical flow problems considered in this study. Mainly, the SA turbulence model is most sensitive to the uncertainties in the diffusion constant (σ), the log layer calibration constant (κ), and the turbulent destruction constant (cw2). The predictive capability of the W2006 model is most sensitive to the uncertainties in a dissipation rate constant (σw), the shear stress limiter (Clim), and a turbulence-kinetic energy constant (β*). Likewise, the SST turbulence model was found to be most sensitive to the diffusion constants (σw1 and σw2), the log layer calibration constant (κ), and the shear stress limiter (a1).

An Exact Axisymmetric Solution in Anisotropic Plasticity

A hollow cylinder of incompressible material obeying Hill’s orthotropic quadratic yield criterion and its associated flow rule is contracted on a rigid cylinder inserted in its hole. Friction occurs at the contact surface between the hollow and solid cylinders. An axisymmetric boundary value problem for the flow of the material is formulated and solved, and the solution is in closed form. A numerical technique is only necessary for evaluating ordinary integrals. The solution may exhibit singular behavior in the vicinity of the friction surface. The exact asymptotic representation of the solution shows that some strain rate components and the plastic work rate approach infinity in the friction surface’s vicinity. The effect of plastic anisotropy on the solution’s behavior is discussed.

A semi-analytical method for vibro-acoustic analysis of submerged ring-stiffened cylindrical shells coupled with arbitrary inner structures

A semi-analytical method is developed for vibration and acoustic analysis of a submerged ring-stiffened cylindrical shell coupled with arbitrary inner structures. The whole structure is firstly disassembled into the stiffened cylindrical shell and inner structures, which are respectively analyzed through analytical substructure method and numerical approach. For the analytical substructure method, the cylindrical shell is decomposed into several shell segments according to positions of stiffeners and external forces, including exciting forces and coupling forces. These shell segments are described by Flügge shell theory and the ribs are governed by the annular plate equations. The solutions for cylindrical shell segments and rings are expanded as wave functions and Bessel functions. For the inner structures with arbitrary irregular shapes, finite element method (FEM) is utilized to establish their motion equations. At each coupling node, an artificial spring is adopted to connect the shell and inner structures, so rigid and elastic coupling conditions can be easily simulated by setting different stiffenesses to these springs. For external fluid field, an axisymmetric boundary element method (BEM) is employed to calculate the acoustic pressure. Combining the displacement continuity and force balance conditions at the joints of adjacent shell segments and introducing the coupling forces derived from inner structures and the fluid loading, the vibro-acoustic equations of the coupled system are obtained. To verify the validity of the present approach, vibro-acoustic responses of the semi-analytical method are compared with the ones calculated through FEM/BEM. Furthermore, effects of the coupling stiffness between the shell and inner structures and external fluid loading on vibro-acoustic responses are also evaluated.

Donaldson Co Inc, USA

A semi-analytical method is developed for vibration and acoustic analysis of a submerged ring-stiffened cylindrical shell coupled with arbitrary inner structures. The whole structure is firstly disassembled into the stiffened cylindrical shell and inner structures, which are respectively analyzed through analytical substructure method and numerical approach. For the analytical substructure method, the cylindrical shell is decomposed into several shell segments according to positions of stiffeners and external forces, including exciting forces and coupling forces. These shell segments are described by Flügge shell theory and the ribs are governed by the annular plate equations. The solutions for cylindrical shell segments and rings are expanded as wave functions and Bessel functions. For the inner structures with arbitrary irregular shapes, finite element method (FEM) is utilized to establish their motion equations. At each coupling node, an artificial spring is adopted to connect the shell and inner structures, so rigid and elastic coupling conditions can be easily simulated by setting different stiffenesses to these springs. For external fluid field, an axisymmetric boundary element method (BEM) is employed to calculate the acoustic pressure. Combining the displacement continuity and force balance conditions at the joints of adjacent shell segments and introducing the coupling forces derived from inner structures and the fluid loading, the vibro-acoustic equations of the coupled system are obtained. To verify the validity of the present approach, vibro-acoustic responses of the semi-analytical method are compared with the ones calculated through FEM/BEM. Furthermore, effects of the coupling stiffness between the shell and inner structures and external fluid loading on vibro-acoustic responses are also evaluated.

Modelling laminar diffusion flames using a fast convergence three-dimensional CVFEM code

Laminar diffusion flames have been employed extensively to investigate the effects of physical and chemical-related parameters on combustion phenomena. Attention has been given to improve combustion models while keeping the two-dimensional (2D) axisymmetric approach and simplified boundary conditions. The flow structure due to the discrete jet nature of the reactants claims a more realistic inlet boundary condition for the burner plate which can only be accomplished if the solution domain is 3D. In this work, we present a novel 3D simulation code for the analysis of non-premixed laminar flames of methane and air, considering detailed inlet boundary conditions. The code is based on the control volume finite element method, written in MATLAB environment. To accomplish that a novel flow-oriented upwind scheme was implemented for the advection terms of the conservation equations. Combustion modelling can be carried out by either an enhanced flame sheet model in which an enthalpy equation is integrated in combination with the mixture fraction equation or detailed reaction schemes. Radiative heat losses were accounted for in the added energy equation. Three-dimensional numerical predictions, based on the flame sheet model, were able to simulate the near-wall gas temperature and inflow velocity field of a set of perforated plate burner with different porosities. It was found that the discrete nature of the injection of fuel and oxidiser, inflow boundary condition, plays a significant role on flame shape and length. Much better agreement between published experimental data and numerical predictions was obtained, particularly regarding flame structure, species mass fractions and temperature distribution of non-premixed laminar flames regardless of the adopted simplified combustion model. It was observed that flame length decreased as the burner porosity increased, up to a certain limit. The modified flow-oriented scheme greatly improved solution stability and computational time of the reacting system predictions.

New developments and explicit results for the thermal constriction resistance of a circular contact under mixed axisymmetric boundary conditions

This article presents a study of the thermal constriction resistance developed in a semi-infinite medium subjected to a circular heat source with mixed boundary conditions. The problem is considered axisymmetric and stationary. Two cases of mixed boundary conditions are considered. The contact area can be subjected to a non-uniform axisymmetric heat flux density while the remainder of the surface is isothermal, or the contact is subjected to a non-uniform axisymmetric surface temperature while the remainder of the surface is adiabatic. The thermal constriction resistance is determined analytically. It takes a simple form and can be used for various problems. To offer a wide choice of applications, a general distribution law for the heat flux density or the temperature of the contact area is considered. Verifications of the proposed solutions are carried out by comparison with some analytical results available in the literature. An analysis of the results according to the kind of boundary conditions and the curvature of the axisymmetric source is also performed.

Stability of interface between liquids with high viscosity contrast in unevenly rotating cavity

The effect of modulation of the cavity rotation rate on the interface of liquids of different densities and high viscosity contrast is studied experimentally. The cavity is a short horizontal cylinder rotating around its axis. The end walls of the cavity form a narrow gap. Under uniform rotation the interface has an axisymmetric shape. With the modulation growth, the axisymmetric boundary loses stability. Instability manifests itself in the appearance of a regular quasistationary relief at the interphase. The relief excitation is associated with the Kelvin-Helmholtz instability. Tangential velocity discontinuity under modulation arises due to various interaction of liquids with the cavity end walls because of viscosity contrast. The viscosity contrast, on the one hand, is responsible for tangential velocity discontinuity; on the other, it has a significant effect on the threshold of Kelvin-Helmholtz instability. It results in decrease of stability threshold and an increase of relief wavelength.

Global stability analysis of axisymmetric boundary layer on a slender circular cone with the streamwise adverse pressure gradient

This paper presents a global stability analysis of the boundary layer developed on a slender circular cone. The base flow direction is towards the apex of a cone, and the pressure gradient is positive (adverse) in the flow direction. The decelerating base flow is non-parallel and non-similar. The increased semi-cone angle increases the adverse pressure gradient in the flow direction. For a given semi-cone angle, transverse curvature increases in the flow direction. However, increased semi-cone angle reduces transverse curvature at any streamwise location. The Reynolds number of the flow is defined based on the displacement thickness at the computational domain’s inlet. The radius of the cone reduces in the flow direction, which results in the increased transverse curvature. The governing stability equations are derived in the spherical coordinates and discretized using the Chebyshev Spectral collocation method. The discretized equations, along with homogeneous boundary conditions, form a general eigenvalue problem, and it is solved using Arnoldi’s iterative algorithm. The global temporal modes have been computed for small semi-cone angles α=2°, 4°, and 6°, azimuthal wave-numbers N=0, 1, 2, and 3 and Re=283, 416, and 610. Thus, the state of base flow is laminar at the inlet (line pq) of the domain pqrs. All the global modes computed within the range of parameters are found temporally stable. Further, the global modes are found less stable at higher semi-cone angles (α) due to the streamwise adverse pressure gradient. The effect of transverse curvature has been found significant at higher semi-cone angles. For a given Re and α, the global modes are found least stable for the helical mode N=1 and most stable for N=3. The eigenmodes’ two-dimensional spatial structure suggests that flow is spatially unstable as the disturbances grow in the streamwise direction.

New Approach of Axisymmetric Compressible Finite-Volume Lattice Boltzmann Method for Numerical Simulation of Supersonic Inviscid Flow

A new approach of the axisymmetric compressible lattice Boltzmann method (LBM) has been developed for the numerical simulation of supersonic inviscid flow using the finite volume method. The circular function idea has been used for capturing the compressibility effect in the supersonic flow field. In this study, the axisymmetric LBM equations based on the circular function have been derived for the first time and presented in detail and appropriate axisymmetric boundary conditions have been applied for the 2-dimensional, 13-velocity, 2-energy-level lattices. For validation of developed code, two supersonic axisymmetric flows have been simulated around the hemisphere nose and 60° blunt body. A comparison of the obtained results with some valid empirical data shows the accuracy of the developed numerical algorithm. Also, the results of 2D and axisymmetric simulations have been studied to evaluate the effects of the new derived axisymmetric formulation on flow macroscopic properties.

Numerical assessment of the effects of end-restraints and a pre-existing fissure on the interpretation of triaxial tests on stiff clays

Conventional laboratory triaxial tests apply axisymmetric boundary conditions to a cylindrical sample which has an axisymmetric geometry. For a homogeneous sample this implies that the deformed shape of the sample should maintain an axisymmetric geometry during the test. Consequently, the sample should deform in a barrelling mode and if slip planes develop they should define a cup and cone-like failure surface. However, in many triaxial tests such behaviour is not observed, especially as failure is approached when a planar slip surface develops. Such a deformation mode is not axisymmetric. One reason for this behaviour is that a fissure pre-exists in the sample. Employing hydro-mechanically coupled three-dimensional finite-element analyses, this paper investigates the influence of a single fissure in a triaxial sample of stiff clay on its behaviour throughout the test, focusing on the fissure position, orientation, strength and stiffness, in conjunction with the sample's end-restraints (rough or smooth). The effects are quantified in terms of the sample's overall stiffness and strength, indicating that the presence of a fissure can affect the very small strain stiffness, and that it has a significant effect on the strength of the sample, demonstrating that the conventional methods used to interpret laboratory tests may give unconservative results. The results also show a significant effect of the conditions at the top and bottom surfaces of the sample, where in particular the lateral restraint and rough ends introduce ‘bending’ in the sample.

Axisymmetric BEM analysis of one-layered transversely isotropic halfspace with cavity subject to external loads

This paper presents an axisymmetric boundary element analysis of one-layered transversely isotropic material of halfspace extent with either ellipsoidal or spherical cavity. This analysis uses the fundamental solution of a transversely isotropic bi-material fullspace subject to the body force uniformly concentrated at a circular ring. The finite and infinite boundary elements are, respectively, used to discretize the finite and semi-infinite regions of the external boundary and the finite boundary elements are used to discretize the internal boundary. Different types of the integrals in the discretized boundary integral equations are efficiently calculated. An improved traction recovered method is applied to calculate the stresses on the boundary. Using one set of BEM mesh, a total of 12 case studies are calculated for the elastic fields of the one-layered transversely isotropic solid by uniform traction on the external horizontal boundary. The solid has either an ellipsoidal, a spherical, or none cavity in the upper layer. Four combinations of two transversely isotropic metals are adopted for the one-layered solid. One metal is the stiff zinc and the other is the soft magnesium. The displacements for the cases with magnesium as the upper layer are larger than those with zinc as the upper layer. Besides, the effects of the lower solid materials to the displacements are limited. The BEM results also reveal the variations of the displacements and stresses within the halfspace with or without cavities for different cases.

Boundary Layer Flow and Heat Transfer of Al2O3-TiO2/Water Hybrid Nanofluid over a Permeable Moving Plate

Hybrid nanofluid is considered a new type of nanofluid and is further used to increase the heat transfer efficiency. This paper explores the two-dimensional steady axisymmetric boundary layer which contains water (base fluid) and two different nanoparticles to form a hybrid nanofluid over a permeable moving plate. The plate is suspected to move to the free stream in the similar or opposite direction. Similarity transformation is introduced in order to convert the nonlinear partial differential equation of the governing equation into a system of ordinary differential equations (ODEs). Then, the ODEs are solved using bvp4c in MATLAB 2019a software. The mathematical hybrid nanofluid and boundary conditions under the effect of suction, S, and the concentration of nanoparticles, ϕ 1 (Al2O3) and ϕ 2 (TiO2) are taken into account. Numerical results are graphically described for the skin friction coefficient, C f , and local Nusselt number, N u x , as well as velocity and temperature profiles. The results showed that duality occurs when the plate and the free stream travel in the opposite direction. The range of dual solutions expand widely for S and closely reduce for ϕ . Thus, a stability analysis is performed. The first solution is stable and realizable compared to the second solution. The C f and N u x increase with the increment of S. It is also noted that the increase of ϕ 2 leads to an increase in C f and decrease in N u x .

Shape transition and hydrodynamics of vesicles in tube flow

The steady motion and deformation of a lipid-bilayer vesicle translating through a circular tube in low Reynolds number pressure-driven flow are investigated numerically using an axisymmetric boundary element method. This fluid-structure interaction problem is determined by three dimensionless parameters: reduced volume (a measure of the vesicle asphericity), geometric confinement (the ratio of the vesicle effective radius to the tube radius), and capillary number (the ratio of viscous to bending forces). The physical constraints of a vesicle -- fixed surface area and enclosed volume when it is confined in a tube -- determine critical confinement beyond which it cannot pass through without rupturing its membrane. The simulated results are presented in a wide range of reduced volumes [0.6, 0.98] for different degrees of confinement; the reduced volume of 0.6 mimics red blood cells. We draw a phase diagram of vesicle shapes and propose a shape transition line separating the parachute-like shape region from the bullet-like one in the reduced volume versus confinement phase space. We show that the shape transition marks a change in the behavior of vesicle mobility, especially for highly deflated vesicles. Most importantly, high-resolution simulations make it possible for us to examine the hydrodynamic interaction between the wall boundary and the vesicle surface at conditions of very high confinement, thus providing the limiting behavior of several quantities of interest, such as the thickness of lubrication film, vesicle mobility and its length, and the extra pressure drop due to the presence of the vesicle. This extra pressure drop holds implications for the rheology of dilute vesicle suspensions. Furthermore, we present various correlations and discuss a number of practical applications.

The influence of magnetized couple stress heat, and mass transfer flow in a stretching cylinder with convective boundary condition, cross-diffusion, and chemical reaction

This paper addresses an analysis for axisymmetric boundary layer couple stress fluid flow by stretching cylinder under the influence of the magnetic field, chemical reaction, Soret, and Dufour. The leading partial differential equations of boundary layer flow are transformed into nonlinear differential equations by applying similarity transformations then solved by the Bvph2.0 based on Homotopy analysis method (HAM). The effect of pertinent parameters shown graphically and observed that velocity decreases when Reynold number and magnetic parameters are increased. The velocity, temperature, and concentrations are enhanced with curvature parameter rise. The dimensionless skin shear stress (skin friction coefficient), heat and mass transfer rates are computed and its convergence analysis also studied. It is assumed that the axial velocity decelerates for the couple stress parameter while the thermal and solutal profile reverse situation is observed.

An Observational Study of the Symmetric Boundary Layer Structure and Tropical Cyclone Intensity

This study analyses Global Positioning System dropsondes to document the axisymmetric tropical cyclone (TC) boundary-layer structure, based on storm intensity. A total of 2608 dropsondes from 42 named TCs in the Atlantic basin from 1998 to 2017 are used in the composite analyses. The results show that the axisymmetric inflow layer depth, the height of maximum tangential wind speed, and the thermodynamic mixed layer depth are all shallower in more intense TCs. The results also show that more intense TCs tend to have a deep layer of the near-saturated air inside the radius of maximum wind speed (RMW). The magnitude of the radial gradient of equivalent potential temperature (θe) near the RMW correlates positively with storm intensity. Above the inflow layer, composite structures of TCs with different intensities all possess a ring of anomalously cool temperatures surrounding the warm-core, with the magnitude of the warm-core anomaly proportional to TC intensity. The boundary layer composites presented here provide a climatology of how axisymmetric TC boundary layer structure changes with intensity.

Study of droplet splashing on a liquid film with a tunable surface tension pseudopotential lattice Boltzmann method

A tunable surface tension pseudopotential lattice Boltzmann method (LBM) with an axisymmetric boundary is applied to study a droplet splashing on a thin film. Our work focused on the crown behavior influenced by the different parameters, including the Reynolds number, Weber number, liquid film thickness, and gravity acceleration. In addition, the total kinetic energy of the crown is proposed to explain the evolution of the crown shape from the perspective of energy. Based on the achieved results, it is found that the crown radius changes with time are unaffected by these parameters and consistent with the power-law in previous studies. However, the height of the crown and the satellite droplet formation process are affected by the referred parameters. Besides, it is found that the energy consumption during the collapse process and splash angle are two key factors affecting the height of the crown. There is a threshold liquid film thickness with the maximum height of the crown, which is between 0.2 and 0.25 times the droplet radius. This study shows that the proposed LBM pseudopotential model is a robust and effective tool for the study of the droplet splashing on the thin film and has the potential to predict the droplet splashing phenomena in the presence of complex boundaries.

On the classification of boundary value problems for elastic homogeneous and piecewise homogeneous spaces with a penny-shaped crack

The classification of the basic axisymmetric boundary value problems of the mathematical theory of elasticity for homogeneous and piecewise homogeneous elastic spaces with a penny-shaped crack is given. Using the Hankel integral transform, governing integral equations (GIEs) of these problems are derived and methods for finding their exact solutions are indicated.