Stationary Sphere Research Articles

Insights into the physics of dominating frequency modes for flow past a stationary sphere: Direct numerical simulations

Direct numerical simulations are carried out for an incompressible flow past a stationary sphere, in the range of 100 ≤ Re ≤ 1000. It is found that the first instability occurs as the axisymmetric wake undergoes breakage at Re ≥ 250. Adding small perturbations to the flow showed that the preferred direction of breakage of the axisymmetric wake and the corresponding contribution of the y and z-direction lift coefficients are highly sensitive and get randomly affected even due to slightest perturbations that might get induced. The second instability arises at Re = 300 as large-scale hairpin shaped structures are formed and shed periodically at frequency StVS = 0.134. At Re = 350, the highly regular hairpin shedding pattern undergoes a quasiperiodic change. From the Q-criterion isosurface, we observed that the quasiperiodicity is induced due to the formation and shedding of secondary hairpin structures which are alongside the primary ones. These secondary hairpin structures are of discernable orientations and are shed 4 times slower as compared to the primary hairpins at Re = 350. Identification of these secondary hairpin structures confirms the hypothesis of wake modulation. The low-frequency mode (Stm) is captured when energy spectral analysis is performed on the surface integrated instantaneous force coefficients and on the radial velocities. The low-frequency mode further exists at all higher Re, exhibiting a gradual increase in Stm. At Re ≥ 800, shear layer instabilities are manifested, demonstrating a characteristic peak at StKH = 0.32 in the energy spectra, rendering the mean lift coefficients to become zero again.

Generalized yamabe equations on riemannian manifolds and applications to Emden-Fowler problems

In this paper, we establish the existence of solutions and multiplicity properties for generalized Yamabe equations on Riemannian manifolds. Problems of this type arise in conformal Riemannian geometry, astrophysics, and in the theories of thermionic emission, isothermal stationary gas sphere, and gas combustion. The abstract results of this paper are illustrated with Emden-Fowler equations involving sublinear terms at infinity. Two examples reveal the analytic setting of this paper.

Vortex-induced vibration of elastically-mounted spheres: A comparison of the response of three degrees of freedom and one degree of freedom systems

In published literature of the vortex-induced vibration of a sphere, limited attention has been devoted to spheres having three degrees of freedom (DOF) motion and the effect on the vibration response. In this study, the vortex-induced vibration response of a 3-DOF elastically-mounted sphere was examined numerically. The response of a sphere allowed to oscillate in all three spatial directions (3-DOF motion) was compared with the response of a sphere whose motion was restricted to only the transverse direction (1-DOF). Simulations were conducted over the reduced velocity range U∗≡U∕(Dfn)∈[3.5,16], using a sphere of mass ratio m∗≡ρs∕ρ=3, at a fixed Reynolds number of 2000, where U is the free-stream velocity of the flow, fn is the natural frequency of the system, ρs and ρ are the solid and fluid densities, respectively. When the sphere was allowed 3-DOF movement, it was initially excited to vibrate along a linear path in the transverse direction, synchronized with the two-sided shedding of vortex loops behind the sphere. As the simulation time advanced, the sphere trajectory eventually converted into a circular orbit with a spiralling wake behind it. The transition time between these two modes was found to increase with decreasing Reynolds number. The vibration amplitude was significantly smaller when the sphere motion was in all three spatial directions than when it was restricted only to the transverse direction. The maximum vibration amplitudes were approximately 0.6 and 0.8 diameters, for 3-DOF and 1-DOF motion, respectively. In the synchronization regime, the maximum time-averaged drag coefficient was approximately 100% greater than that of a stationary sphere, when the sphere had 3-DOF, similar to previous observations for a tethered sphere. However, the maximum change to the time-averaged drag coefficient was only 76% when the motion was restricted to the transverse direction.

The singularity method in unsteady Stokes flow: hydrodynamic force and torque around a sphere in time-dependent flows

The equations for the hydrodynamic force and torque acting on a sphere in unsteady Stokes equations under different flow conditions are solved analytically by means of the singularity method. This analytical technique is based on the combination of suitable singularity solutions (also called fundamental solutions) such as primary Stokeslets, potential dipoles, or higher-order singularities, to construct the flow field. The different flows considered here include four examples: (1) a rotating sphere in a viscous flow, (2) a stationary sphere in a time-dependent shear flow, (3) a sphere with free rotation in a simple shear flow, as well as (4) a stationary sphere in a time-dependent axisymmetric parabolic flow. Our paradigm is to derive the fundamental solutions in unsteady Stokes flows and to express the solutions as a convolution integral in time using the time–space fundamental solutions. Next the Laplace transform is used to determine the strength of the distributed singularities that induce the velocity field around a stationary or rotating sphere. Then we use the computed strength of the singularities to derive hydrodynamic force and torque. In particular, for the problem of a stationary sphere in unsteady axisymmetric parabolic flow, our solution for the time-dependent force acting on the sphere consists of five force components – the well-known quasi-steady Stokes drag, the added mass term, the Basset historic (memory) force, and two additional memory forces. The first additional memory force due to the rate change of velocity, we find, is similar to the result obtained by Lawrence & Weinbaum (J. Fluid Mech., vol. 171, 1986, pp. 209–218) for the ostensibly unrelated setting of a slightly deformed translating spheroid. The second additional memory force comes from the effect of the rate change of acceleration and is found for the first time in this study to the best of our knowledge.

Flow past a single stationary sphere, 1. Experimental and numerical techniques

Flow separation across bluff bodies has been a fascinating area of fluid mechanics since 1851. A large number of computational and experimental studies have been performed in the last several decades which have enabled the understanding and hence helped in distinguishing the flow regimes based on the characteristics exhibited by different flow parameters. These signify some physical phenomena resulting due to flow separation under different flow conditions. A lot of research from the 70's to 90's was conducted using hot wire techniques for investigating the flow features. These techniques provided three-dimensional insights into the flow and helped in quantitative, reliable and robust corroboration of earlier qualitative findings which were based on inadequate visualization techniques. Moreover, the rapid growth in computing resources in the last couple of decades, have made it possible to use highly efficient numerical techniques to investigate separated flows with much higher accuracy and control. Also, the introduction of sophisticated measurement techniques such as PIV/ PLIF has aided in gaining deeper insights into the physics of sphere wakes. This work aims to present the developments and findings in the study of flow past a stationary sphere with respect to experimental and computational investigations. The statistical and signal processing techniques which form the chief medium to characterize the turbulence in such adverse pressure gradient wake flows, have also been discussed briefly. All the previous studies have been critically analysed presented as a coherent theme. This paper also brings out the knowledge gaps and accordingly recommendations have been made for the future research.

Asymptotics for the level set equation near a maximum

Abstract We give asymptotics for the level set equation for mean curvature flow on a convex domain near the point where it attains a maximum. It is known that solutions are not necessarily C 3 {C^{3}} , and we recover this result and construct non-smooth solutions which are C 3 {C^{3}} . We also construct solutions having prescribed behavior near the maximum. We do this by analyzing the asymptotics for rescaled mean curvature flow converging to a stationary sphere.

Drag upon a sphere suspended in a low magnetic-Reynolds number MHD channel flow

As a first step towards the description of magnetohydrodynamic (MHD) dispersed two-phase flows, a permanent MHD channel flow past a stationary sphere is numerically investigated, highlighting the gradual emergence of atypical MHD flow patterns, among them a genuine ``ghost column.''

Drag force and torque of a stationary sphere in a uniform laminar open channel flow

Drag force and torque of a stationary sphere in a uniform laminar open channel flow

Global Formulation and Motion Planning for a Sphere Rolling on a Smooth Surface

This paper studies the motion planning problem for a rolling sphere on an arbitrary smooth manifold embedded in ℝ3. The sphere is allowed to roll on the surface without slipping or twisting. A mathematical model describing the kinematics of the sphere is developed in a geometric form so that the model is globally defined without singularities or ambiguities. An algorithm for constructing a path between specified initial and final configurations is presented. The algorithm utilizes the nonholonomic nature of the system. The theoretical results are specialized for two specific surfaces defined in ℝ3, namely a flat surface and the surface of a stationary sphere.

An improved immersed moving boundary for the coupled discrete element lattice Boltzmann method

In order to improve the accuracy of existing immersed moving boundary method (IMBM) for the coupled discrete element lattice Boltzmann method, a novel weighting function for the fluid–particle interfacial interactions in IMBM is presented in this work. The flow is solved by the three-dimensional lattice Boltzmann method (LBM) D3Q15. The motions of particles are simulated by the discrete element method (DEM). The immersed moving boundary method is applied to resolve the momentum exchange between moving particles and fluids. The solid volume fraction of the fluid cell is calculated by the sphero-polyhedra method. The coupled IMBM–LBM–DEM is programmed based on the open source library Mechsys. The proposed method is validated by simulating flow past a stationary sphere, settling of a sphere and rebound of a spherical particle. These validations demonstrate the accuracy and reliability of this proposed method, which also has a good improvement over the original IMBM. Finally, effects of different parameters in the proposed weighting function are discussed and the suggested values a = 50, b = 0.01 are determined. This paper has implications for improving the ability to capture the fluid–particle interactions for a multiphase flow system.

A novel geometry-adaptive Cartesian grid based immersed boundary–lattice Boltzmann method for fluid–structure interactions at moderate and high Reynolds numbers

A novel computational framework which combines the lattice Boltzmann method (LBM) and an improved immersed boundary method (IBM) based on a dynamic geometry-adaptive Cartesian grid system is introduced for the fluid–structure interaction (FSI) problems at moderate and high Reynolds numbers. In this framework, the fluid dynamics is obtained by solving the discrete lattice Boltzmann equation. The boundary conditions at the fluid–structure interfaces are handled by an improved IBM based on a feedback scheme, which drives the predicted flow velocity (calculated after the LBM stream process without the IBM body force) near the immersed boundaries to match the solid velocity. In the present IBM, the feedback coefficient is mathematically derived and explicitly approximated. The Lagrangian force density is divided into two parts: one is the traction caused by the predicted flow velocity, and the other is caused by the acceleration of the immersed boundary. Such treatment significantly enhances the numerical stability for modelling FSI problems involving small structure-to-fluid mass ratios. A novel dynamic geometry-adaptive refinement is applied to provide fine resolution around the immersed geometries and coarse resolution in the far field. The overlapping grids between two adjacent refinements consist of two layers. In order to enhance the numerical stability, two-layer “ghost nodes” are generated within the immersed body domain which is a non-fluid area. The movement of fluid–structure interfaces only causes adding or removing grids at the boundaries of refinements and consequently a high mesh-update efficiency is guaranteed. Finally, large eddy simulation models are incorporated into the framework to model turbulent flows at relatively high Reynolds numbers. Several validation cases, including an impulsively started flow over a vertical plate, flow over stationary and oscillating cylinders, flow over flapping foils, flexible filaments in a uniform flow, turbulent flow over a wavy boundary, flow over a stationary sphere and a dragonfly in hovering flight, are conducted to verify the accuracy and fidelity of the present solver over a range of Reynolds numbers.

A spectral-element/Fourier smoothed profile method for large-eddy simulations of complex VIV problems

An accurate, fast and robust spectral-element/Fourier smoothed profile method (SEF-SPM) for turbulent flow past 3D complex-geometry moving bluff-bodies is developed and analyzed in this paper. Based on the concept of momentum thickness δ2, a new formula for determining the interface thickness parameter ξ is proposed. In order to overcome the numerical instability at high Reynolds number, the so-called Entropy Viscosity Method (EVM) is introduced in the framework of large-eddy simulation. To overcome resolution constraints pertaining to moving immersed bodies, the Coordinate Transformation Method (Mapping method) is incorporated in the current implementation. Moreover, a hybrid spectral-element method using mixed triangular and quadrilateral elements is employed in conjunction with Fourier discretization along the third direction to efficiently represent a body of revolution or a long-aspect ratio bluff-body like risers and cables. The combination of the above algorithms results in a robust method which we validate by several prototype flows, including flow past a stationary sphere at 200 ≤ Re ≤ 1000, as well as turbulent flow past a stationary and moving cylinder at 80 ≤ Re ≤ 10, 000. Finally, we apply the new method to simulate a self-excited rigidly moving dual-step cylinder and demonstrate that SEF-SPM is an efficient method for complex VIV problems.

Numerical and experimental study of acoustically enhanced heat transfer from a single particle in flue gas

ABSTRACTTo investigate the heat transfer of pulverized coal particles under the effect of an acoustic field (without a superposed steady component), we derived and solved the two-dimensional, unsteady, laminar conservation equations for mass, momentum, and energy transport in the flue gas phase of a power plant boiler. The local, space-averaged, and space- and time-averaged Nusselt (Nu) numbers for the coal particles are discussed for the full audible frequency range. Experiments were also conducted to study the heat transfer, in particular the heat exchange at the surface, considering the effect of the acoustic wave around a stationary copper sphere. The sphere was exposed to different sound pressure levels of acoustic waves in air, and the temperature of the sphere was recorded. The range of the studied sound frequencies is between 500 Hz and 3 kHz. Both, numerical and experimental investigations indicate that the acoustic field can increase the heat transfer significantly. Higher sound pressure levels cause higher heat-transfer rate. However, there is a special frequency, at which Nu reaches a maximum, due to the flow separation caused by flow acceleration. This special frequency increases with increasing sound pressure level.

Experimental study of the flow in the wake of a stationary sphere immersed in a turbulent boundary layer

In many applications, finite-sized particles are immersed in a turbulent boundary layer (TBL) and it is of interest to study wall effects on the instantaneous shedding of turbulence structures and associated mean velocity and Reynolds stress distributions. Here, 3D flow field dynamics in the wake of a prototypical, small sphere (D+=50, 692lReDl959) placed in the TBL's outer, logarithmic, and buffer layer, were measured using time-resolved tomo-PIV. Increasing wall proximity increasingly tilted the mean recirculating wake away from the wall implying a negative lift force. Mean velocity deficit recovery scaled with the mean wake length with minor effects of wall proximity. Farthest from the wall, streamwise Reynolds normal stresses encircled the mean wake as an axisymmetric tubular shell, while transverse and wall-normal stresses extended off its tip as axisymmetric tapered cones. Wall proximity removed axisymmetry and attenuated values near the wall. Reynolds shear stresses were distributed as antisymmetric lobes extending off the mean wake displaying increasing values with reducing sphere-wall gap. Instantaneous snapshots revealed a wake densely populated by archlike vortices with shedding frequencies lower than for a sphere in uniform flow except in the buffer layer. Tilting of the wake away from the wall resulted from self-induced motion of shed hairpinlike vortices whose symmetry plane was increasingly wall-normal oriented with reduced sphere-wall gap.

An h-adaptive implicit immersed boundary-lattice Boltzmann flux solver based on JASMIN AMR package

In order to efficiently simulate flows with complex geometries, an implicit immersed boundary-lattice Boltzmann flux solver (IB-LBFS) combined with an h-adaptive mesh refinement (AMR) technology is proposed and implemented on JASMIN infrastructure. In the present implementation, the original velocity-splitting IB-LBFS is modified by a momentum-splitting during the factional step. Four benchmark problems are used to validate the present method, including the flow over a stationary circular cylinder, the sedimentation of a two-dimensional elliptical particle, the flow around a stationary sphere and the sedimentation of a single sphere. The simulated results are in good agreements with previously published data, which demonstrates the accuracy and the capability of the proposed method in simulating flow problems with stationary or moving boundaries.

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

Fundamental solutions (Green’s functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier–Stokes–Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.

Modified discrete source method as applied to the simulation of flows over a periodically irregular surface and a body of revolution

The incompressible flow over a stationary periodically irregular surface and a compact body of revolution is computed by applying a modified discrete source method. The method is tested as applied to the flow over a stationary sphere. Numerical results are presented for flows over a sinusoidal surface, a cycloidal surface, a spheroid, a Chebyshev particle, an “analytical” cylinder, and a half ball.

Low-cost, portable, robust and high-resolution single-camera stereo-DIC system and its application in high-temperature deformation measurements

A low-cost, portable, robust and high-resolution single-camera stereo-digital image correlation (stereo-DIC) system for accurate surface three-dimensional (3D) shape and deformation measurements is described. This system adopts a single consumer-grade high-resolution digital Single Lens Reflex (SLR) camera and a four-mirror adaptor, rather than two synchronized industrial digital cameras, for stereo image acquisition. In addition, monochromatic blue light illumination and coupled bandpass filter imaging are integrated to ensure the robustness of the system against ambient light variations. In contrast to conventional binocular stereo-DIC systems, the developed pseudo-stereo-DIC system offers the advantages of low cost, portability, robustness against ambient light variations, and high resolution. The accuracy and precision of the developed single SLR camera-based stereo-DIC system were validated by measuring the 3D shape of a stationary sphere along with in-plane and out-of-plane displacements of a translated planar plate. Application of the established system to thermal deformation measurement of an alumina ceramic plate and a stainless-steel plate subjected to radiation heating was also demonstrated.

Pairwise-interaction extended point-particle model for particle-laden flows

In this work we consider the pairwise interaction extended point-particle (PIEP) model for Euler–Lagrange simulations of particle-laden flows. By accounting for the precise location of neighbors the PIEP model goes beyond local particle volume fraction, and distinguishes the influence of upstream, downstream and laterally located neighbors. The two main ingredients of the PIEP model are (i) the undisturbed flow at any particle is evaluated as a superposition of the macroscale flow and a microscale flow that is approximated as a pairwise superposition of perturbation fields induced by each of the neighboring particles, and (ii) the forces and torque on the particle are then calculated from the undisturbed flow using the Faxén form of the force relation. The computational efficiency of the standard Euler–Lagrange approach is retained, since the microscale perturbation fields induced by a neighbor are pre-computed and stored as PIEP maps. Here we extend the PIEP force model of Akiki et al. [3] with a corresponding torque model to systematically include the effect of perturbation fields induced by the neighbors in evaluating the net torque. Also, we use DNS results from a uniform flow over two stationary spheres to further improve the PIEP force and torque models. We then test the PIEP model in three different sedimentation problems and compare the results against corresponding DNS to assess the accuracy of the PIEP model and improvement over the standard point-particle approach. In the case of two sedimenting spheres in a quiescent ambient the PIEP model is shown to capture the drafting-kissing-tumbling process. In cases of 5 and 80 sedimenting spheres a good agreement is obtained between the PIEP simulation and the DNS. For all three simulations, the DEM-PIEP was able to recreate, to a good extent, the results from the DNS, while requiring only a negligible fraction of the numerical resources required by the fully-resolved DNS.

Time-domain electromagnetic scattering by a sphere in uniform translational motion

Scattering by a uniformly translating sphere of a pulse that modulates the amplitude of a linearly polarized plane wave was formulated using the frame-hopping method involving a laboratory inertial reference frame and the sphere's comoving inertial reference frame. The incident signal was defined in the laboratory frame and transformed to the comoving frame with the Lorentz transformation, thereby altering the incident signal's spectrum, direction of propagation of the carrier plane wave, and the direction of the incident electric field, depending on the sphere's velocity. In the comoving frame, the incident signal was Fourier-transformed to the frequency domain, and the scattered field phasors were computed in all directions using the constitutive parameters of the material of the sphere at rest. The scattered signal in the comoving frame was obtained using the inverse Fourier transform. Finally, the scattered signal in the laboratory frame was obtained by inverting the original Lorentz transformation. The backscattered signal was found to depend strongly on the sphere's velocity, when the sphere's speed is an appreciable fraction of the speed of light in free space. The change in the backscattered signal compared with the backscattered signal from a stationary sphere is the greatest when the sphere's velocity is either parallel or antiparallel to the direction of propagation of the incident signal. The backscattered signal is also affected by motion transverse to the incident signal's direction of propagation; then, the backscattered signal depends on whether or not the motion is aligned with the direction of the incident electric field.