Far-Wake of a Sphere in a Stably Stratified Fluid: Characterization of the Vortex Structures

This article discusses the structure of the far-wake of a towed sphere in a saline stratification. We compare very low Froude number experiments to existing results at higher Froude numbers and investigate the vertical structure of the far-wake in terms of their vorticity and density fields. We show that the vertical propagation of vorticity is viscously dominated and propose a simple three-dimensional model for the quasi-equilibrium of the structure in terms of the density field.     

Mixed convection cooling of a heated circular cylinder by laminar upward-directed slot jet impingement

An experimental and numerical study has been carried out to investigate the heat transfer characteristics of a horizontal circular cylinder exposed to a slot jet impingement of air. A square-edged nozzle is mounted parallel with the cylinder axis and jet flow impinges on the bottom of the cylinder. The study is focused on low Reynolds numbers ranging from 120 to 1,210, Grashof numbers up to Gr = 10Re ² and slot-to-cylinder spacing from 2 to 8 of the slot width. The flow field is greatly influenced by the slot exit velocity and the buoyancy force due to density change. A Mach–Zehnder Interferometer is used for measurement of local Nusselt number around the cylinder at 10° interval. It is observed that the average Nusselt number decreases with increasing the jet spacing and increases with rising the Reynolds number. A finite volume method utilizing a curvilinear coordinate transformation is used for numerical modeling. The numerical results show good agreement with the experimental results. The flow and thermal field are seen to be stable and symmetric around the cylinder over the range of parameters studied.     

Drag increment of internal waves generated by horizontally moving spheroid in supercritical regime

A theory on the drag increment of internal waves with a spheroid moving horizontally at a high velocity （or for large internal Froude number） in uniformly vertically stratified fluid （or ocean） is presented in the present paper. A surface source distribution is employed to model a hydrodynamic interaction between the spheroid and the stratified fluid. From theoretical results, it is shown that there exists an asymptote of zero-drag increment in supercritical regimes, where internal Froude numbers are larger than the critical internal Froude numbers. When the spheroid reduces to a sphere, the results in this paper is in good agreement with the previous theoretical and experimental results of the sphere.     

A shearable and thickness stretchable finite strain beam model for soft structures

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In a previous work, an Euler–Bernoulli finite strain beam model with thickness stretching effect was proposed for soft thin structures subject to stiff constraint in the width direction. By extending that model to account for the transverse shear effect, a Timoshenko-type finite strain beam model within the plane-strain context is developed in the present work. With some kinematic hypotheses, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. In the limit of infinitesimal strain, the new model degenerates to the classical extensible and shearable elastica model. The corresponding incremental equilibrium equations and associated boundary conditions are also obtained. Based on the new beam model, analytical solutions are given for simple deformation modes, including uniaxial tension, simple shear, pure bending, and buckling under an axial load. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with immovable ends. The results are also compared with the previously developed finite strain Euler–Bernoulli beam model to demonstrate the significance of transverse shear effect for soft beams with a small length-to-thickness ratio. The developed beam model will contribute to the design and analysis of soft robots and soft devices.     

狭缝节流气体静压轴颈-止推轴承静态特性分析

在气体静压轴颈 -止推轴承的离散化过程中 ,采用有限元方法 ,利用加权余量法将其二阶偏微分方程降低一阶 ,使利用三角单元线性插值函数成为可能 ,简化了压力分布方程式的计算 ,给出了轴承的静态特性 ,分析了结构工艺尺寸、狭缝宽度和轴承间隙等因素对径向和轴向承载性能的影响 ,揭示了轴颈和止推轴承之间的相互影响规律 ,并且研究了轴承内部气膜压力分布的规律 .最后 ,通过实测值对计算结果进行了验证     

Linear logistic regression with weight thresholding for flow regime classification of a stratified wake

A stratified wake has multiple flow regimes, and exhibits different behaviors in these regimes due to the competing physical effects of momentum and buoyancy. This work aims at automated classification of the weakly and the strongly stratified turbulence regimes based on information available in a full Reynolds stress model. First, we generate a direct numerical simulation database with Reynolds numbers from 10,000 to 50,000 and Froude numbers from 2 to 50. Order (100) independent realizations of temporally evolving wakes are computed to get converged statistics. Second, we train a linear logistic regression classifier with weight thresholding for automated flow regime classification. The classifier is designed to identify the physics critical to classification. Trained against data at one flow condition, the classifier is found to generalize well to other Reynolds and Froude numbers. The results show that the physics governing wake evolution is universal, and that the classifier captures that physics.     

Calculation of Ship Sinkage and Trim Using a Finite Element Method and Unstructured Grids

An unstructured grid-based, parallel free-surface flow solver has been extended to account for sinkage and trim effects in the calculation of steady ship waves. The overall scheme of the solver combines a finite-element, equal-order, projection-type three-dimensional incompressible flow solver with a finite element, two-dimensional advection equation solver for the free surface equation. The sinkage and trim, wave profiles, and wave drag computed using the present approach are in good agreement with experimental measurements for two hull forms at a wide range of Froude numbers. Numerical predictions indicate significant differences between the wave drag for a ship fixed in at-rest position and free to sink and trim, in agreement with experimental observations.     

The vortex ring velocity resulting from an impacting water drop

Experimental observations are reported on the evolution of a vortex ring for the first 70 ms after it is created by the impact of a dyed water drop upon a pool of clear water. The 2.6 mm diameter drops were released from two heights. The drops impacted the pool with Weber numbers of 23.2 and 16.6 and Froude numbers of 25.2 and 18.0. The Reynolds number of the resulting relaminarized vortex rings based on their diameter was in the range of 320 to 390. Precisely controlled multiple exposure photographs were used to measure the position and shape of the vortex ring versus time, and calculate velocity. It is proposed here that the appropriate time scale is the time it takes for the impact crater to reach its maximum depth. Excellent agreement was found when using this scaling to compare both the present data sets and that previously published for a larger drop.This work was supported by the Natural Sciences and Engineering Research Council of Canada Grant No. OGP 00 41747. Mr. B. Faulkner is thanked for help with the strobe-timing     

A numerical investigation of developing flow in an eccentric curved annulus in the presence of gravity

In this article developing incompressible viscous flow in an eccentric curved annulus in the presence of gravity is numerically studied using a second order finite difference method based on the projection algorithm to solve the governing equations including the continuity and full Navier–Stokes equations. The equations written in a bipolar–toroidal coordinate system are discretized in a three dimensional staggered grid. The effects of governing non-dimensional parameters including the eccentricity, non-dimensional curvature ratio, Dean number, Froude number, aspect ratio, and the Reynolds number on the flow field in the entrance and fully developed region are investigated. The numerical results indicate that at the small Froude numbers, the flow field distorts from the symmetrical condition due to the larger body force effect and the axial velocity formation mostly takes place at the lower half of the annulus. In addition, at the constant Froude number, by decreasing the curvature radius, the peak axial velocity and its sharp gradient appear on the outer curvature region due to the larger centrifugal forces and by increasing the eccentricity the flow rate intensifies at the wider region and weakens at the narrower region due to the larger flow resistance. Furthermore, the friction factor increases by decreasing the Froude number and increasing the Dean number.     

Near-field development of gas-phase horizontal laminar jets with positive and negative buoyancy measured with filtered Rayleigh scattering

Near-field mixing characteristics of horizontally issuing jets, alternatively positively and negatively buoyant, are explored. The cross-sectional mass fraction of a buoyant horizontal jet consisting of helium flowing into ambient air is measured using a non-intrusive technique, filtered Rayleigh scattering, for Reynolds numbers ranging from 50 to 1,200, Froude numbers ranging as low as 0.71, and Schmidt numbers on the order of unity for all tests. Several corresponding experiments were carried out using carbon dioxide in place of helium in order to determine whether the direction of the buoyancy changes the characteristic shape of the jet cross-section. Consistent with the literature, mixing rates were consistently higher on the side of the jet where instability, due to density stratification, was present. At jet Froude numbers ranging between 1.5 and approximately 3, the jet cross-section takes a shape consistent with a single plume of fluid being ejected from the core in a vertical direction—upward for a jet with positive buoyancy and downward for a jet with negative buoyancy. Remarkably, for Froude numbers less than unity, the distortion of the jet is quite different in that two separate plumes emanate from each side of the jet while ejection from the center is suppressed. Both the positively and negatively buoyant jet cross-sections exhibited this trait, suggesting that the mechanism that determines the cross-sectional shape of the jet core is only mildly influenced by centripetal effects brought about by streamline curvature. The location of the jet centroid at varied streamwise locations was computed from the mass fraction data, yielding jet trajectory.     

Numerical investigation of the aerodynamics of the near‐slot film cooling

Fluid injection from slot or holes into cross‐flow produces highly complicated flow fields. Physical situations encountering the above problem range from turbine blade cooling to waste discharge into rivers. In this paper, the flow field created by a two‐dimensional slot cooling geometry is examined using the finite volume approach with a second‐order upwind differencing scheme. The time‐averaged Navier–Stokes equations were solved on a collocated Cartesian grid with a two‐equation model of turbulence. Attempting to solve the flow field by assuming a uniform velocity profile at the slot exit leads to inaccurate results, while extending the solution domain improves significantly the results, but proves to be costly, both in memory and in computing time (particularly in the case of multiple holes). A pressure‐type boundary condition, based on uniform total pressure, is developed for the slot exit (easily applied to a three‐dimensional geometry), which yields more accurate results than the widely used uniform velocity assumption. It is also found that the implementation of low Reynolds number turbulence models on this geometry provides no significant differences from the standard k–ε model. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element model for free surface flows on fixed meshes

In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element models for such flows, is that we take advantage of the particularities of free surface flow, instead of considering it a particular case of two‐phase flow. The fact that a given free surface implies a known boundary condition on the interface, allows us to solve the Navier–Stokes equations on the fluid domain uncoupled from the solution on the rest of the finite element mesh. This, together with the use of enhanced integration allows us to model low Froude number flows accurately, something that is not possible with typical two‐phase flow models applied to free surface flow. Copyright © 2007 John Wiley & Sons, Ltd.     

Non-Darcian Effects on Buoyancy-Induced Heat Transfer in a Partially Divided Square Enclosure with Internal Heat Generation

The present investigation addresses non-Darcian effects on the buoyancy-induced heat transfer in a partially divided square enclosure with internal heat generation. The generalized model of the momentum equation, which is also known as the Forchheimer–Brinkman extended Darcy model, which takes into account boundary and inertia effects, was used in representing the fluid motion inside the porous layer. The local thermal equilibrium condition was assumed to be valid for the range of the thermophysical parameters considered in the present investigation. The transport equations were solved using the finite element formulation based on the Galerkin method of weighted residuals. The validity of the numerical code used was ascertained by comparing our results with previously published results. Results were obtained in terms of streamlines, isotherms, and Nusselt number for various geometrical parameters specifying the height and width of the partition. In addition, the effects of external and internal Rayleigh numbers and Darcy number were highlighted in the proposed study.     

Convective transport of air bubbles in strong hydraulic jumps

A hydraulic jump is a flow singularity characterised by a significant amount of air entrainment in the shear zone. The air is entrapped at the jump toe that is a discontinuity between the impinging flow and the roller. The impingement point is a source of air bubbles, as well as a source of vorticity. Herein the convective transport of air bubbles in the jump roller is re-visited. Some analytical extension is presented and the theoretical results are compared with some laboratory experiments conducted in a large-size facility operating at large Froude numbers. The turbulent air bubble mixing coefficient was found to increase linearly with increasing distance and be independent of the Froude and Reynolds numbers. Overall the study highlighted some seminal features of the air–water shear layer in hydraulic jumps with large Froude numbers (5.1 < Fr₁ < 11.2). The air bubble entrainment in the mixing zone was a convective transport process, although there was some rapid flow de-aeration for all Froude numbers.     

Influence of skirt radius on performance of circular clarifier with density stratification

A numerical model for predicting the velocity field and suspended solids distribution in a secondary circular clarifier with density difference is evaluated. The density effects are characterized by the inlet densimetric Froude number. This study focuses on the role of the reaction baffle position in the performance of the clarifiers. For a large-radius baffle and low densimetric Froude number an important phenomenon known as the density waterfall occurs in the inlet zone of the clarifiers. This was predicted by the numerical model and confirmed by the physical model tests. This model consists of a series of conservation equations for fluid mass and momentum and sediment concentration. The turbulent stresses are calculated by use of the eddy viscosity concept and the κ–?. turbulence model. The study showed that the density waterfall results in high entrainment and high recirculation. A comparison of the solids concentration distribution for a tank with a small skirt radius to that with a large skirt radius shows that small skirt radius reduces the density waterfall effect and significantly improves the clarifier performance at low densimetric Froude numbers.     

Hydrodynamic characteristics of a vortex source performing translational motion in a multilayer heavy fluid

The problem of translational motion of a vortex source in a three-layer fluid bounded by a bottom from below is considered. The fluid in each layer is perfect, incompressible, heavy, and homogeneous. Based on the previously developed method, formulas for disturbed complex velocities of the fluid in each layer and the wave drag and lift force of the vortex source are obtained. The vortex motion is considered near the interface of two semi-infinite fluid media and in a two-layer fluid with different conditions at the boundary. In all cases, the hydrodynamic characteristics of the vortex source are given as functions of the Froude number. In a number of problems, these characteristics have discontinuities at the transition through the critical Froude numbers. The character of these discontinuities is studied analytically. Omsk Department of Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 140–146, September–October, 2000.     

Finite element analysis of flow in a gas-filled rotating annulus

Linearized multidimensional flow in a gas centrifuge can be described away from the ends by Onsager's pancake equation. However a rotating annulus results in a slightly different set of boundary conditions from the usual symmetry at the axis of rotation. The problem on an annulus becomes ill-posed and requires some special attention. Herein we treat axially linear inner and outer rotor temperature distributions and velocity slip. An existence condition for a class of non-trivial, one-dimensional solutions is given. New exact solutions in the infinite bowl approximation have been derived containing terms that are important at finite gap width and non-vanishing velocity slip. The usual one-dimensional, axially symmetric solution is obtained as a limit. Our previously reported finite element algorithm has been extended to treat this new class of problems. Effects of gap width, temperature and slip conditions are illustrated. Lastly, we report on the compressible, finite length, circular Couette flow for the first time.     

Impingement heat transfer at a circular cylinder due to a submerged slot jet of water

An experimental investigation was carried out on the heat transfer due to a submerged slot jet of water impinging on a circular cylinder in crossflow. The cylinder diameter and the slot width are of the same order of magnitude, specifically D_s = 2.0 and 3.0 mm and D_c = 2.5 and 3.0 mm. The experimental apparatus allowed variation of the slot width, the cylinder diameter, and the distance from nozxle exit to heater. Conditions of impingement from the bottom (ascending flow) were taken into consideration as well as impingement from above (descending flow). The Nusselt number was determined as a function of Reynolds and Prandtl numbers in the range 1.5 × 10³ < Re < 2.0 × 10⁴, 2.7 < Pr < 7.0, and 1.5 ≤ z/D_s ≤ 10. The experimental data were correlated with a simple equation that fits 90% of the data with a precision of 20%.     

The Unsteady Wake of a Circular Cylinder near a Free Surface

The behaviour of the wake Strouhal number for flow past a cylinder close to a free surface at both low and moderate Froude numbers is investigated numerically. For the low Froude number case (i.e., gravity-dominated), the results obtained are similar to those for flow past a cylinder close to an adjacent no-slip boundary. As the distance between the wall and the cylinder is reduced, the Strouhal number, as measured from the time varying lift, increases to a maximum at a gap ratio of 0.70. Further gap reduction leads to a rapid decrease in the Strouhal number, with shedding finally ceasing altogether at gap ratios below 0.16. The agreement between the results for a free surface and a no-slip boundary suggests that the mechanism behind the suppression of vortex shedding is common. For flow at a fixed gap ratio and a moderate Froude number, two distinctly different wake states are observed with the flow passing over the cylinder tending to switch from a state of attachment to the free surface, to one of separation from it, and then back again in a pseudo-periodic fashion. Even though there is a significant difference in Reynolds number, the predicted numerical two-dimensional behaviour is found to compare favourably with the experimental observations at higher Reynolds number.     

Two-degree-of-freedom vortex induced vibration of low-mass horizontal circular cylinder near a free surface at low Reynolds number

Two-degree-of-freedom vortex induced vibration (VIV) of a low-mass zero-damping circular cylinder horizontally placed near a free surface at Re = 100 was numerically studied with an adaptive Cartesian cut-cell/level-set method. Two Froude numbers and various normalized submergence depths were considered. The results reveal that the Froude number affects the critical normalized submergence depth and possible physical mechanisms are proposed. The in-line vibration amplitude cannot be neglected. Proximity to a free surface strengthens and suppresses the VIV for low and high Froude numbers, respectively; increases the occurrence of amplitude modulation; and in general enhances the magnitude of the time-averaged lift coefficient, which is always negative. The phase lag of the transverse displacement behind the lift coefficient jumps at some reduced velocity, which strongly depends on the Froude number and normalized submergence depth. Regular trajectories exist only in cases with a small vibration amplitude or a large normalized submergence depth. The vortex structures in any case with large transverse amplitude basically originate from the alternative vortex shedding with the negative vortex weaker than the positive one. For the higher Froude number, an extra free surface positive vortex interacts with the vortices from the cylinder surface. The vibration frequency deviates from the natural structure frequency in fluids in the large-amplitude regime.