Axisymmetric State Research Articles

Theoretical estimates for flat voids coalescence by internal necking

Coalescence of voids by internal necking is in most cases the last microscopic event related to ductile fracture and corresponds to a localized plastic flow between adjacent voids. Macroscopic load associated to the onset of coalescence is classically estimated based on limit analysis. However, a rigorous upper-bound mathematical expression for the limitload required for flat voids coalescence that remains finite for penny-shaped voids/cracks is still unavailable. Therefore, based on limit analysis, theoretical upper-bound estimates - both integral expression and closed-form formula - are obtained for the limit-load of cylindrical flat voids in cylindrical unit-cell subjected to boundary conditions allowing the assessment of coalescence, for axisymmetric stress state. These estimates, leading to finite limit-loads for pennyshaped cracks, are shown to be in very good agreement with numerical limit analysis, for both cylindrical and spheroidal voids. Approximate formula is also given for coalescence under combined tension and shear loading. These coalescence criteria can thus be used to predict onset of coalescence of voids by internal necking in ductile fracture modelling.

Active Brownian motion of emulsion droplets: Coarsening dynamics at the interface and rotational diffusion.

A micron-sized droplet of bromine water immersed in a surfactant-laden oil phase can swim (S. Thutupalli, R. Seemann, S. Herminghaus, New J. Phys. 13 073021 (2011). The bromine reacts with the surfactant at the droplet interface and generates a surfactant mixture. It can spontaneously phase-separate due to solutocapillary Marangoni flow, which propels the droplet. We model the system by a diffusion-advection-reaction equation for the mixture order parameter at the interface including thermal noise and couple it to fluid flow. Going beyond previous work, we illustrate the coarsening dynamics of the surfactant mixture towards phase separation in the axisymmetric swimming state. Coarsening proceeds in two steps: an initially slow growth of domain size followed by a nearly ballistic regime. On larger time scales thermal fluctuations in the local surfactant composition initiates random changes in the swimming direction and the droplet performs a persistent random walk, as observed in experiments. Numerical solutions show that the rotational correlation time scales with the square of the inverse noise strength. We confirm this scaling by a perturbation theory for the fluctuations in the mixture order parameter and thereby identify the active emulsion droplet as an active Brownian particle.

A computational insight into void-size effects on strength properties of nanoporous materials

In this paper, strength properties of nanoporous materials are addressed aiming to establish novel insights into the influence of void-size effects. To this end, a virtual spherically-nanovoided sample of an aluminium single crystal is investigated by adopting a Molecular-Dynamics computational approach. Elasto-plastic mechanical response, under triaxial strain-based conditions and including axisymmetric and shear states, are numerically experienced, identifying the corresponding limit stresses. Computed strength measures are used to furnish estimates of strength domains, described in terms of meridian and deviatoric profiles. The influence of void-size effects on the computed strength properties is clearly quantified for different porosity levels, numerical results confirming a strengthening of the sample when the void radius reduces. Moreover, it is shown that the occurrence and the amount of void-size effects are strongly dependent on the Lode angle, resulting in a shape transition of both meridian and deviatoric strength profiles when the void radius is varied. Finally, present results suggest porosity-dependent threshold values for the void radius above which void-size effects tend to disappear. With respect to the actual state-of-the-art, useful benchmarks for assessing the effectiveness of available theoretical models are provided, resulting in a novel incremental contribution towards the definition of advanced modelling strategies for describing strength properties of nanoporous materials.

Multi-axial response of idealized cermets

The yield response of two idealized cermets comprising mono and bi-disperse steel spheres in a Sn/Pb solder matrix has been investigated for a range of axisymmetric stress states. Proportional stress path experiments are reported, from which are extracted the initial yield surfaces and their evolution with increasing plastic strain. The initial yield strength is nearly independent of the hydrostatic pressure but the strain hardening rate increases with stress triaxiality up to a critical value. For higher triaxialities, the responses are independent of hydrostatic pressure. Multi-axial measurements along with X-ray tomography were used to demonstrate that the deformation of these idealized cermets occurs by two competing mechanisms: (i) a granular flow mechanism that operates at low levels of triaxiality, where volumetric dilation occurs under compressive stress states, and (ii) a plastically incompressible mechanism that operates at high stress triaxialities. A phenomenological viscoplastic constitutive model that incorporates both deformation mechanisms is presented. While such multi-axial measurements are difficult for commercial cermets with yield strengths on the order of a few GPa, the form of their constitutive relation is expected to be similar to that of the idealized cermets presented here.

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

Axial Flutter Effects on the Axisymmetric Turbulent Boundary Layer Along Long Thin Circular Cylinders

Experimental observations of towed sonar arrays as characterized by long thin circular cylinders indicate transverse motions that are clearly identified by low-amplitudes, low-wavelengths, and low-frequencies. Although the cylinder length (L) to radius (a) is commonly large [L/a = O(103)] with high Reynolds numbers [O(104)], the corresponding length scale involving the average skin friction [CfL/a = O(10)] remains within the many experimental determinations of short to moderate length cylinders that experience oscillatory instabilities. Prior to the present investigation, any detrimental effects of these oscillatory instabilities on the thin cylinder flow physics that serve construction of the respective semi-empirical and semi-analytical models remained chiefly unknown. Herein, we began examining those turbulent statistics via fine-scale numerical simulations to critique the pragmatic adequacy of the representative design models. We were concerned in particular about the streamwise effects on the turbulent boundary layer (TBL), skin friction and wall pressure evolutions as well as the radial distributions of the leading normal and shear Reynolds stresses. Fortunately, no major deviations (within 10%) were discovered in the TBL statistics over a characteristic range of Reynolds numbers and TBL thicknesses as compared to the axisymmetric state. However, acute spikes (both subharmonics and harmonics) were detected in the wall pressure autospectra similar to that suspected in the towed cylinder experiments, which were conducted in large tow tanks and lake-type basins. These spikes are of paramount importance and should be explored further because they may lead to signal-to-noise ratios above acceptable limits.

The properties of isolated chiral skyrmions in thin magnetic films

Axisymmetric solitonic states (chiral skyrmions) were first predicted theoretically more than two decades ago. However, until recently they have been observed in a form of skyrmionic condensates (hexagonal lattices and other mesophases). In this paper we report experimental and theoretical investigations of isolated chiral skyrmions discovered in PdFe/Ir(111) bilayers two years ago by Romming et al (2013 Science 341 636). The results of spin-polarized scanning tunneling microscopy analyzed within the continuum and discrete models provide a consistent description of isolated skyrmions in thin layers. The existence region of chiral skyrmions is restricted by strip-out instabilities at low fields and a collapse at high fields. We demonstrate that the same equations describe axisymmetric localized states in all condensed matter systems with broken mirror symmetry, and thus our findings establish basic properties of isolated skyrmions common for chiral liquid crystals, different classes of noncentrosymmetric magnets, ferroelectrics, and multiferroics.

Occurrence of instability through the protostellar accretion disks by landing of low-mass condensations

Low-mass condensations (LMCs) are observed inside the envelope of the collapsing molecular cloud cores. In this research, we investigate the effects of landing LMCs for occurrence of instability through the protostellar accretion disks. We consider some regions of the disk where duration of infalling and landing of the LMCs are shorter than the orbital period. In this way, we can consider the landing LMCs as density bumps and grooves in the azimuthal direction of an initial thin axisymmetric steady state self-gravitating protostellar accretion disk (nearly Keplerian). Using the linear effects of the bump quantities, we obtain a characteristic equation for growth/decay rate of bumps; we numerically solve it to find occurrence of instability. We also evaluate the minimum-growth-time-scale (MGTS) and the enhanced mass accretion rate. The results show that infalling and landing of the LMCs in the inner regions of the protostellar accretion disks can cause faster unstable modes and less enhanced accretion rates relative to the outer regions. Also, more fragmentation of landed LMCs in the azimuthal direction have less chance for instability, and then can produce more values of enhanced mass accretion rate.

CFD simulation of nozzle characteristics in a gas aggregation cluster source

Computational fluid dynamics (CFD) simulation for the sharp-edged nozzle was made using the OpenFOAM source code. Jet dynamics characteristics for the gas stream from a sharped-edged nozzle are then simulated under two-dimensional and axisymmetric steady state condition. Gas and particle velocities in a jet are obtained under different input and boundary conditions to provide an insight into the jet characteristics. For the range of downstream distances considered, the results indicate that a jet is characterized by an initial rapid decay of the axial velocity at the jet center while the cross-sectional flow evolves toward a top-hat profile downstream. Numerical simulation of the free-jet expansion from six types of nozzles indicated that the flow fields had significant differences in the early stages of the expansion. The effects of nozzle geometry on the nanoparticles were investigated. Simulation results indicated that the geometry of nozzle affected the cluster penetration efficiency and beam diameter. The extent to which to the Brownian diffusion can affect the particle extraction from nozzle is investigated. Simulations have shown that the Brownian motion perturbs the clusters from the trajectories dictated by the carrier gas and increases the rate of cluster deposition on the nozzle.

Magnetohydrodynamic co-rotating flow in a vertical cylindrical container

ABSTRACTThe effect of an external axial magnetic field on the liquid metal flow produced by co-rotation of the top and bottom disks in a vertical cylindrical container with a vertical temperature gradient is numerically analyzed. The governing Navier–Stokes, energy, and potential equations along with appropriate boundary conditions are solved using the finite-volume method. Comparisons with the previous results were performed and found to be in excellent agreement. It was observed that the Reynolds number is increased, and the axisymmetric basic state loses stability for circular patterns of axisymmetric vortices and spiral waves. In the mixed convection case the axisymmetric mode disappears, giving an asymmetric mode m = 1. It was also found that the primary thresholds, Recr corresponding to modes m = 1 and 2, increase with an increase in Hartmann number (Ha). We can therefore conclude that when the magnitude of the magnetic field exceeds a certain value, the instability becomes a steady bifurcation. Finally, stability diagrams were established according to the numerical results of this investigation. These diagrams show the evolution of primary thresholds as a function of Hartmann number for various values of Richardson number.

Failure modes of coatings on steel substrate

Abstract The critical monotonic strain of Ni-W and MoS2(Ti,W) coatings on steel substrates was studied. The idea of axisymmetric bending test (called here as coin bending test) limited to monitoring of the coating failure was used. Experiments revealed mechanism of the coating failure, as cracking initiated from coating surface defects and/or substrate was demonstrated using indentation technique. By pushing the center of the uncoated side of a circular plate, the axisymmetric stress state was generated in the coating. The stress components varied gradually from the greatest value in the center to the smallest value at the edge of the specimen. The changes of the sample surface as a result of loading were monitored step by step via optical microscopy.

Dynamics and equilibria of thin viscous coating films on a rotating sphere

We examine the dynamics of a thin viscous liquid film on the outer surface of a solid sphere rotating around its vertical axis in the presence of gravity. An asymptotic model describing the evolution of the film thickness is derived in the rotating frame based on the lubrication approximation. The model includes the centrifugal and gravity forces and the stabilizing effect of surface tension. Depending on the values of the parameters, the problem admits different types of steady states: one with a uniformly positive film thickness, or those with one or two dry zones on the sphere. We prove that all steady states are energy minimizers and hence global attractors for axisymmetric states.

Stress analysis of rotating cylindrical shell composed of functionally graded incompressible hyperelastic materials

In this paper, rotating thick-walled hollow cylindrical shell composed of functionally graded material are analyzed by using the theory of hyperelasticity. Hyperelastic behavior is modeled by using proposed power law strain energy function with variable material parameters. Material is considered incompressible and material properties are assumed as a function of the radius of the cylinder to a power law function. Material inhomogeneity parameter (n) is a power in the mentioned power law function. Material constant of strain energy function is calculated from experimental data by using Levenberg–Marquardt nonlinear regression method. The analytical solution is obtained for the axisymmetric plane strain state. Following this, profiles of circumferential stretch, radial stress, circumferential stress and longitudinal stress as a function of radial direction are plotted for different values of n. The obtained results show that the material inhomogeneity parameter (n) and structure parameter (β: ratio of outer radius to inner radius) have a significant influence on the mechanical behavior of rotating thick-walled hollow cylindrical shell made of functionally graded materials with power law varying properties. Thus with selecting a proper n and structure parameter (β), engineers can design a specific FGM hollow cylinder that can meet some special requirements.

Asymptotic limits and wrinkling patterns in a pressurised shallow spherical cap

The asymmetric bifurcation problem for a shallow spherical cap is examined. The applied pressure can act either external or internal to the cap and both cases are treated here. Assuming a non-linear axisymmetric basic state, the linearised bifurcation equations for the pressurised shell are investigated in the limit when the thickness of the cap is much less than the maximum rise of the shell mid-surface. Within this regime the wrinkling patterns in both cases are confined to a narrow zone near the edge of the shell, making it possible to solve asymptotically the corresponding equations and derive analytical predictions for both the critical pressure and the corresponding number of wrinkles. Some comparisons with direct numerical simulations are included as well.

Magnetohydrodynamic counter-rotating flow in a cylindrical cavity

Three-dimensional steady combined free and forced convective magnetohydrodynamic (MHD) flow are simulated in a cylindrical cavity filled with a liquid metal and submitted to a vertical temperature gradient and an axial magnetic field. The forcing corresponds to a swirling flow produced by counter-rotation of the top and bottom disks. The governing Navier–Stokes, energy, and potential equations along with appropriate boundary conditions are solved by using the finite-volume method. Comparisons with previous results were performed and found to be in excellent agreement. The effects of magnetic field on flow and temperature fields are analysed. When the Reynolds number is increased, the axisymmetric basic state loses stability and different complex flow appear. Axisymmetric (m=0) and asymmetric m=1 and m=2 azimuthal modes are observed. Azimuthal mode m=3 are found when the Hartmann number, Ha is sufficiently large. In the mixed convection case the m=1 becomes the dominant mode in place of m=0. The stability diagram (Recr–Ha) corresponding to the transition from axisymmetric to non-axisymmetric flow by increasing values of the Richardson number is obtained.

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.

Equilibrium and stability of axisymmetric drops on a conical substrate under gravity.

Motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013], we study a related problem of the determination of the equilibrium and stability of axisymmetric drops on a conical substrate in the presence of gravity. A variational principle is used to characterize equilibrium shapes that minimize surface energy and gravitational potential energy subject to a volume constraint, and the resulting Euler equation is solved numerically using an angle/arclength formulation. The resulting equilibria satisfy a Laplace-Young boundary condition that specifies the contact angle at the three-phase trijunction. The vertical position of the equilibrium drops on the cone is found to vary significantly with the dimensionless Bond number that represents the ratio of gravitational and capillary forces; a global force balance is used to examine the conditions that affect the drop positions. In particular, depending on the contact angle and the cone half-angle, we find that the vertical position of the drop can either increase ("the drop climbs the cone") or decrease due to a nominal increase in the gravitational force. Most of the equilibria correspond to upward-facing cones, and are analogous to sessile drops resting on a planar surface; however we also find equilibria that correspond to downward facing cones, that are instead analogous to pendant drops suspended vertically from a planar surface. The linear stability of the drops is determined by solving the eigenvalue problem associated with the second variation of the energy functional. The drops with positive Bond number are generally found to be unstable to non-axisymmetric perturbations that promote a tilting of the drop. Additional points of marginal stability are found that correspond to limit points of the axisymmetric base state. Drops that are far from the tip are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. We have also found a range of completely stable solutions that correspond to small contact angles and cone half-angles.

Formation and Maintenance of the Tropical Cold-Point Tropopause in a Dry Dynamic-Core GCM

Abstract The formation of the tropical cold-point tropopause (CPT) is examined using a dry primitive equation model driven by the Held–Suarez forcing. Without moist and realistic radiative processes, the dry model successfully reproduces the zonal-mean structure of the CPT. The modeled CPT is appreciably colder (~10 K) than the prescribed equilibrium temperature, and it is maintained by upwelling in the tropical upper troposphere and lower stratosphere (UTLS). A transient simulation starting from an axisymmetric steady state without the CPT shows that the evolution and maintenance of the CPT are closely related to the zonal-mean-flow response to wave driving in the stratosphere. The transformed Eulerian-mean analysis indicates that the wave driving is mostly due to convergence of synoptic-scale waves originating from the midlatitude troposphere and propagating into the subtropical UTLS in this model simulation. The modeled CPT also shows a large sensitivity to increased baroclinicity in the equilibrium temperature. Although planetary-scale waves are not considered in this simulation, the result confirms that wave-driven upwelling in the tropical UTLS is a crucial process for the formation and maintenance of the CPT. In addition, it also implies that synoptic-scale waves may play a nonnegligible role in this mechanism, particularly in the seasons when planetary-scale wave activity in the lower stratosphere is weak.

Vorticity dynamics after the shock–turbulence interaction

The interaction of a shock wave with quasi-vortical isotropic turbulence (IT) represents a basic problem for studying some of the phenomena associated with high speed flows, such as hypersonic flight, supersonic combustion and Inertial Confinement Fusion (ICF). In general, in practical applications, the shock width is much smaller than the turbulence scales and the upstream turbulent Mach number is modest. In this case, recent high resolution shock-resolved Direct Numerical Simulations (DNS) (Ryu and Livescu, J Fluid Mech 756:R1, 2014) show that the interaction can be described by the Linear Interaction Approximation (LIA). Using LIA to alleviate the need to resolve the shock, DNS post-shock data can be generated at much higher Reynolds numbers than previously possible. Here, such results with Taylor Reynolds number approximately 180 are used to investigate the changes in the vortical structure as a function of the shock Mach number, $$M_{s}$$ , up to $$M_{s}=10$$ . It is shown that, as $$M_{s}$$ increases, the shock interaction induces a tendency towards a local axisymmetric state perpendicular to the shock front, which has a profound influence on the vortex-stretching mechanism and divergence of the Lamb vector and, ultimately, on the flow evolution away from the shock.

Experimental and Numerical Investigations to Evaluate Two-Dimensional Modeling of Vertical Drain–Assisted Preloading

AbstractIn this study, the efficiency of proposed formulations for plane-strain modeling of vertical drain–assisted consolidation was evaluated. For this aim, the vertical drain–assisted preloading process was experimentally simulated using a fully instrumented large-scale Rowe cell. Nine pore-water-pressure transducers were installed in various locations to measure the changes in pore-water pressure during the test. Two pressure/volume controllers were connected to an infinite-volume controller to provide continuous water flow. Soft clays with predefined properties were used to form the intact and smear zones. A numerical code was developed by using the finite-difference program FLAC 2D to simulate the consolidation test. A numerical study was conducted to evaluate the efficiency of the proposed solutions for converting the axisymmetric state to a plane-strain condition and was subsequently compared with corresponding numerical analysis. From the results, it is observed that some of the proposed methods ...