Fluid Wedge Research Articles

Water entry of spinning spheres

The complex hydrodynamics of water entry by a spinning sphere are investigated experimentally for low Froude numbers. Standard billiard balls are shot down at the free surface with controlled spin around one horizontal axis. High-speed digital video sequences reveal unique hydrodynamic phenomena which vary with spin rate and impact velocity. As anticipated, the spinning motion induces a lift force on the sphere and thus causes significant curvature in the trajectory of the object along its descent, similar to a curveball pitch in baseball. However, the splash and cavity dynamics are highly altered for the spinning case compared to impact of a sphere without spin. As spin rate increases, the splash curtain and cavity form and collapse asymmetrically with a persistent wedge of fluid emerging across the centre of the cavity. The wedge is formed as the sphere drags fluid along the surface, due to the no-slip condition; the wedge crosses the cavity in the same time it takes the sphere to rotate one-half a revolution. The spin rate relaxation time plateaus to a constant for tangential velocities above half the translational velocity of the sphere. Non-dimensional time to pinch off scales with Froude number as does the depth of pinch-off; however, a clear mass ratio dependence is noted in the depth to pinch off data. A force model is used to evaluate the lift and drag forces on the sphere after impact; resulting forces follow similar trends to those found for spinning spheres in oncoming flow, but are altered as a result of the subsurface air cavity. Images of the cavity and splash evolution, as well as force data, are presented for a range of spin rates and impact speeds; the influence of sphere density and diameter are also considered.

Wave propagation in a fluid wedge over a solid half-space – Mesh-free analysis with experimental verification

Interaction between a bounded ultrasonic beam and a liquid wedge over a solid half-space is studied. A semi-analytical technique called distributed point source method (DPSM) is adopted for modeling the ultrasonic field in a wedge shaped fluid structure over a solid half space. This study is important for analyzing the ultrasonic waves used for the non-destructive inspections of partially immersed structures. It is also useful for studying the effect of underwater ultrasonic or acoustic wave experiments on marine lives near the shore. The problem geometry considers a bounded acoustic beam striking a fluid–solid interface between a fluid wedge and a solid half-space at steady-state. Solution of this problem is beyond the scope of the currently available analytical methods when the beam is bounded. Only numerical method (boundary element method (BEM) or finite element method (FEM)) based packages (e.g. PZFlex) are in principle capable of modeling ultrasonic fields in such structures. At high frequencies FEM and BEM based packages require huge amount of computation memory and time for their executions that DPSM technique can avoid. Effect of the angle of strike and the fluid wedge angle variation on the wave propagation characteristics is studied. Theoretical predictions are compared with some experimental results.

Transmission of sound in a wedge‐shaped layer of fluid with a rigid bottom

AbstractThe theory of generalized rays is applied to the problem of determining the exact shape of the pressure pulse received at very large ranges, in the case of a sound source in a 3° perfect wedge of fluid. The time variation of the pressure at the point source is assumed to be given by a Heaviside unit function H(t). It is found that horizontal refraction and backscattering have a pronounced effect on the overall appearance of the received pulse: at very large down–slope (normal to the apex line) ranges, the shape of the received pulse does not change much, while, at very large cross–slope (parallel to the apex line) ranges, it changes drastically. Thus, in a 3° perfect wedge of fluid, long–range down–slope propagation (exclusively affected by backscattering) seems to be nondispersive, but long–range cross–slope propagation (affected by horizontal refraction and backscattering) seems to be highly dispersive. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Experimental Study of the Lubrication Mechanism for Thrust Slide Bearings in Scroll Compressors

This study focuses on the effect that a pressure difference across the orbiting thrust plate of a thrust-slide bearing has on the improved lubrication of the bearing in scroll compressor applications. A thrust slide bearing model submerged in a refrigerant oil, VG-56, was operated under pressurized conditions using R-22 as the pressurizing gas, where the pressure difference across the friction surface of the thrust bearing was adjusted from 0 to 1.0 MPa and the friction force and friction coefficient at the thrust slide bearing were measured over a range of orbital speeds. As a result, significant improvement in lubrication at the thrust slide bearing due to the pressure difference was identified. Furthermore, careful observation of the wear state of the thrust slide bearing revealed the formation of a fluid wedge between the sliding surfaces due to axial loadings. This fluid wedge appears responsible for the observed improvement in lubrication. In addition, the wedge formation was quantitatively investigated using finite element method (FEM) analysis of the elastic deformation of the thrust plate. The FEM analysis was subsequently validated through strain measurement on the thrust plate.

Range and cross-range propagation effects in a liquid wedge overlaying an elastic bottom

The penetrable-wedge model, a constant density isospeed layer of fluid with a pressure-release horizontal surface and a sloping elastic bottom, provides an extensive insight into the role of the ocean bottom in acoustic propagation from an underwater source. In particular, it is a realistic model of a rock-bottom ocean near a shoreline that accounts for horizontal refraction and allows for a ground wave. This paper discusses some new results for small and large range propagation for two penetrable-wedge models: one where the shear wave speed in the bottom is lower than the sound speed in the fluid and the other where the shear wave speed is higher. The operational representation of the ray-integral solution for the acoustic field from a point source in a penetrable wedge of fluid is further developed to the stage at which one can compute the exact, other than the omission of diffraction at the wedge apex, pressure response curve, as recorded at a receiver, due to an arbitrary time variation of the pressure at the source in a 3° wedge. The three-dimensional propagation effects are examined for range transmission when the receivers are located up-slope and down-slope of the source, and for cross-range transmission when the receivers are located cross-slope of the source.

Theoretical Model of Lubrication Mechanism in the Thrust Slide-Bearing of Scroll Compressors

This study presents a theoretical model of the remarkably good lubrication characteristics of the thrust slide-bearings found in scroll compressors experimentally by Ishii et al. (2007). It has been suggested that the thrust plate undergoes elastic deformation due to axial loading. As a result of this deformation, a fluid wedge is believed to form at the periphery of the thrust plate. The fluid wedge angle between the sliding surfaces was calculated with a finite element method analysis and then used in the average Reynolds equation from Patir and Cheng (1978, 1979) to analyze the fluid lubrication on rough sliding surfaces. The plastic and elastic contacts between the orbiting and fixed thrust plates were analyzed using the solid contact theory from Greenwood and Williamson (1966). The derived theoretical model permitted the calculation of the oil film pressure, the solid contact force, the fluid frictional force, and the solid shearing drag force. These results were used, in turn, to determine the resultant friction coefficient at the thrust slide-bearing. Finally, it is shown that the predicted theoretical results agree well with the experimental lubrication test results of Ishii et al. (2007). The proposed theoretical development appears to model accurately the essential mechanism for the remarkably good mixed fluid-and-solid lubrication in a thrust slide-bearing.

Cavity formation in the wake of a spinning sphere impacting the free surface

A spherical object impacting the free surface generatesboth a splash curtain and a subsurface air cavity. In the caseof a spinning sphere, the angular momentum combined withforward motion causes the sphere to move along a curvedpath, similar to a curve ball pitch in baseball. The hydrody-namics of a billiard ball diameter 5.7 cm impacting the freesurface with a downward vertical velocity of 7.5 m/s and aclockwise angular velocity of 232 rad/s are revealed throughhigh speed video imaging. The curved trajectory of thesphere is evident in Fig. 1. Figure 2 reveals the evolution ofthe splash curtain above the free surface as the sphere im-parts momentum into the fluid. Initially, at impact, the mo-mentum transfer forms a radial jet just above the free sur-face, until vertical growth outpaces radial expansion formingthe splash curtain Fig. 2 a . The curtain eventually col-lapses inward, forming a dome. As the sphere spins, it drawsfluid from the left side of the cavity causing the dome tobreakdown asymmetrically Fig. 2 b , in contrast to theaxis-symmetric dome in the case of surface impact by a non-spinning sphere. A bird’s eye view of the surface Fig. 3shows a wedge of fluid being pulled away from the left wallalong the equator of the sphere. The loss of fluid at the cavitywall hinders symmetrical growth of the splash curtain andresults in an asymmetrical collapse of the splash dome Fig.2 c . As the ball moves through the fluid, the wedgestretches and travels toward the right side of the cavity Fig.3 . Eventually the wedge impacts the opposite side of thecavity. This impact forces air to be ejected from the cavity;the ejected air forms a line of bubbles that can be seen atright in Fig. 1. As the ball travels along its curved trajectory,the cavity continues to elongate but no longer grows radially,eventually resulting in cavity pinch-off.Figure 1 was photographed by Brant Avondet and TaddTruscott.

Self-similar relaxation dynamics of a fluid wedge in a Hele-Shaw cell

Let the interface between two immiscible fluids in a Hele-Shaw cell have, at t = 0, a wedge shape. As a wedge is scale-free, the fluid relaxation dynamics are self-similar. We find the dynamic exponent of this self-similar flow and show that the interface shape is given by the solution of an unusual inverse problem of potential theory. We solve this problem analytically for an almost flat wedge, and numerically otherwise. The wedge solution is useful for analysis of pinch-off singularities.

Surface Tension‐Driven Flow in a Slender Wedge

We consider an inviscid fluid, initially at rest inside a wedge, bounded by one free surface and one solid surface. When $t=0$, we allow the contact angle to change discontinuously, which leads the free surface to recoil under the action of surface tension. As noted by Keller and Miksis [SIAM J. Appl. Math., 43 (1983), pp. 268–277], a similarity scaling is available, with lengths scaling like $t^{2/3}$. We consider the situation when the wedge is slender, with angle $\epsilon \ll 1$, and the contact angle changes from ε to λε. The leading order asymptotic problem for $\lambda = O(1)$, a pair of nonlinear ordinary differential equations, was considered by King [Quart. J. Mech. Appl. Math., 44 (1991), pp. 173–192], numerically for $\lambda = O(1)$ and asymptotically for $|\lambda-1| \ll 1$. In this paper, we begin by considering this system when $1 \ll \lambda \ll \epsilon^{-1}$, and use Kuzmak’s method to construct the asymptotic solution. When $\lambda =O(\epsilon^{-1})$, the slope of the free surface becomes of $O(1)$, and it is no longer possible to reduce the problem to ordinary differential equations alone. However, we can approach this problem in a similar manner, even though the underlying oscillator is the solution of a nonlinear boundary value problem for Laplace’s equation, and construct an asymptotic solution. In fact, the solution takes the form of a modulated set of waves on fluid of finite depth, with the underlying analytical solution given by Kinnersley [J. Fluid Mech., 77 (1976), pp. 229–241]. The case $\lambda\epsilon = 90^\circ$ is the solution for the inviscid recoil of a wedge of fluid with two free surfaces and semiangle $\epsilon \ll 1$, which was discussed by Billingham and King [J. Fluid Mech., 533 (2005), pp. 193–221]. We also show that no non‐self‐intersecting solution is available for $\lambda\epsilon > 90^\circ$ as $\epsilon \to 0$, and compare our asymptotic solutions with numerical, boundary integral solutions of the full, nonlinear free boundary problem.

Wet comminution of raw salt using high-pressure fluid jet technology

Based on the impact mechanism and fluid wedge effect of the high-pressure fluid jet, a new approach to raw salt comminution has been worked out. The fluid jet comminution process has a unique characteristic of providing a fine appearance for the product particle as well as a high washing efficiency. Dissolution of raw salt and loss of product are avoided owing to saturated sodium chloride solution used as high-pressure medium. Compared with mechanical comminution, the product processed by this technique has higher quality and better appearance. The working condition is also improved since the comminution is operated under wet conditions. Also, this approach has the potential to be used to comminute the thermally sensitive, inflammable and explosive materials.

The on-line measurement of lubricant film thickness for condition monitoring

The ultrasonic reflectivity of a lubricant layer between two solid bodies depends on the ultrasonic frequency, the acoustic properties of the liquid and solid, and the layer thickness. In this paper, ultrasonic reflectivity measurements are used as a method for determining the thickness of lubricating films in bearing systems. An ultrasonic transducer is positioned on the outside of a bearing shell such that the wave is focused on the lubricant film layer. The transducer is used to both emit and receive wide-band ultrasonic pulses. For a particular lubricant film the reflected pulse is processed to give a reflection coefficient spectrum. The lubricant film thickness is then obtained from either the layer stiffness or the resonant frequency. The method has been validated using fluid wedges at ambient pressure between flat and curved surfaces. Experiments on the elastohydrodynamic film formed between a ball sliding on a flat were performed. Film thickness values in the range 50-500 nm were recorded which agreed well with theoretical film formation predictions. Similar measurements have been made on the oil film between the balls and outer raceway of a deep groove ball bearing.

A slope-dependent disjoining pressure for non-zero contact angles

A thin liquid film experiences additional intermolecular forces when the film thickness h is less than roughly 100 nm. The effect of these intermolecular forces at the continuum level is captured by the disjoining pressure Π. Since Π dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, Π = Π(h) because thin films are generally uniform. This form, however, cannot be applied to films that end at the substrate with non-zero contact angles. A recent ad hoc derivation including the slope h x leads to Π = Π(h, h x ), which allows non-zero contact angles, but it permits a contact line to move without slip. This work derives a new disjoining-pressure expression by minimizing the total energy of a drop on a solid substrate. The minimization yields an equilibrium equation that relates Π to an excess interaction energy E = E(h, h x ). By considering a fluid wedge on a solid substrate, E(h, h x ) is found by pairwise summation of van der Waals potentials. This gives in the small-slope limit Π = B h 3 (α 4 - h 4 x + 2hh 2 xh xx ), where a is the contact angle and B is a material constant. The term containing the curvature h xx is new; it prevents a contact line from moving without slip. Equilibrium drop and meniscus profiles are calculated for both positive and negative disjoining pressure. The evolution of a film step is solved by a finite-difference method with the new disjoining pressure included; it is found that h xx = 0 at the contact line is sufficient to specify the contact angle.

Wedge solidification with differing types of boundary conditions

Solidification, in the large Stefan number limit, is considered within a wedge of fluid, initially at fusion temperature. The wedge walls have different boundary conditions: one is Neumann and the other Dirichlet. The large Stefan number limit allows a reduction to complex variables, and ultimately a solution utilizing conformal mappings and a connection with Fuchsian differential equations. The physical example is of interest as the one-dimensional solutions corresponding to either Dirichlet or Neumann problems propagate at different velocities, and the connection between these near the corner region can lead to changes in sign of the curvature of the front profile. This manifests itself in the mathematics as an additional singularity in the Fuchsian equation and this enriches the problem over that usually encountered; there are now four singular points rather than three, and unknown parameters appear that must be determined.

Green Water Loading on a FPSO

Abstract Green Water Loading in the bow region of a Floating Production Storage and Offloading unit (FPSO) in head sea waves is studied by numerical means. A 2-D method satisfying the exact nonlinear free-surface conditions within potential-flow theory has been developed as a step towards a fully 3-D method. The flow is assumed 2-D in a plane containing the ship’s centerplane. The method is partly validated by model tests. The importance of environmental conditions, 3-D flow effects, ship motions, and hull parameters are summarized. The wave steepness of the incident waves causes important nonlinear effects. The local flow at the bow is, in general, important to account for. It has become popular to use a dam-breaking model to study the propagation of water on the deck. However, the numerical studies show the importance of accounting for the coupled flow between the deck and outside the ship. When the water is propagating on the deck, a suitable distance from the bow can be found from where shallow-water equations can be used. Impact between green water on deck and a vertical deck-house side in the bow area is studied in details. A similarity solution for impact between a wedge-formed water front and a vertical rigid wall is used. Simplified solutions for an impacting fluid wedge with small and large interior angles are developed, both to support the numerical computations and to provide simpler formulas of practical use. It is demonstrated how the local design of the deck house can reduce the slamming loads. The importance of hydroelasticity during the impact is discussed by using realistic structural dimensions of a deck house. This indicates that hydroelasticity is insignificant. On the contrary, first results from an ongoing experimental investigation document blunt impacts against the deck during the initial stage of water shipping, which deserve a dedicated hydroelastic analysis.

DIFFRACTION BY A HARD HALF-PLANE: USEFUL APPROXIMATIONS TO AN EXACT FORMULATION

In this paper, the problem of diffraction of a spherical wave by a hard half-plane is considered. The starting point is the Biot–Tolstoy theory of diffraction of a spherical wave by a fluid wedge with hard boundaries. In this theory, the field at a point in the fluid is composed eventually of a geometrical part: i.e., a direct component, one or two components due to the reflections on the sides of the hard wedge, and a diffracted component due exclusively to the presence of the edge of the wedge. The mathematical expression of this latter component has originally been given in an explicit closed form for the case of a unit momentum wave incidence, but Medwin has further developed its expression for the more useful case of a Dirac delta point excitation. The expression of this form is given in the time domain, but it is quite difficult to find exactly its Fourier transform for studying the frequency behaviour of the diffracted field. It is thus the aim of this paper to present various useful approximations of the exact expression. Among the approximations treated, three are most accurate for engineering purposes, and one of them is proposed, for its simplicity, as appropriate for most occurring practical situations.

On the use of the method of images to investigate nearshore dynamical processes

This note describes how the method of images may be used to determine the motion and evolution of two related kinds of phenomena within a wedge of inviscid fluid. The image field of a curved vortex within a wedge with vortex lines lying along sectors of circles around the apex of the wedge is that segment of a complete vortex ring which remains outside the wedge and of which the curved vortex forms a part. The image system can be used to describe the motion, interaction and stability of single or multiple vortices within the wedge. Axisymmetric jets form the image system for flow parallel to the edge of the wedge, akin to alongshore currents. Knowledge of the instability of jets provides information about the evolution of waves in the wedge domain. Existing results on the motion and instability of single or multiple co-axial ring vortices and of waves and instabilities in jets may be applied to describe the evolution of low Froude number eddies and waves in alongshore flow over a steadily shelving sea bed.

Merging and wetting driven by surface tension

The surface tension driven merging of two wedge-shaped regions of fluid, and the wetting of a wedge shaped solid, are analyzed. Following the work of Keller and Miksis in 1983, initial conditions are chosen so that the flows and their free surfaces are self-similar at all times after the initial contact. Then the configuration magnifies by the factor t 2/3 and the fluid velocity at the point x /t 2/3 decays like t −1/3, where the origin of x and t are the point and time of contact. In the merging problem the vertices of the two wedges of fluid are initially in contact. In the wetting problem, the vertex of a wedge of fluid is initially at the corner of the solid. The motions and free surfaces are found numerically. These results complement those of Keller and Miksis for the wetting of a single flat surface and for the rebound of a wedge of fluid after it pinches off from another body of fluid.

On large-scale flow structures in a gravel-bed river

Previous studies have suggested the presence of large-scale flow structures in gravel-bed rivers. These structures are pictured as intermittent high-speed wedges separated by regions of lower velocity. However, the characteristics of these structures have not been examined in detail through either visualisation techniques or detailed field measurements. This paper confirms the presence of large-scale flow structures in gravel bed rivers, pictures their sequence and patterns and characterises their mean and individual properties. The analysis relies on a new technique for displaying velocity fluctuations in a space–time matrix that allows one to see the structures as they pass an array of current meters. Streamwise and vertical velocities were measured simultaneously with an array of three electromagnetic current meters. The sampling frequency was 20 Hz. Five velocity profiles of up to 13 1-min series of measurements and one profile of three 20-min measurements were sampled. These data suggest the presence of large wedges of faster fluid joined by regions of slower fluid. Space–time correlation analysis confirmed the presence of vertical coherence of the flow. The average angle of the front of the wedges is 36°. Although individual structures are variable in size and shape, a new detection technique using all three velocity signals simultaneously showed that their average frequency is nine events per minute and their duration is more than 2 s. The high-speed wedges display a complex organisation and do not show a preferred sequence of events as was postulated by previous studies. Because of their duration and size, these high-speed wedges are likely to play a major role in bedload sediment transport.

Self-similar dynamics of a viscous wedge of fluid

The 2D (two dimensional) motion of two symmetric wedges of viscous fluid is determined. The evolution of the fluid is governed by the Stokes equations plus the interfacial boundary conditions including surface tension. A similarity solution is determined and these equations are solved numerically by using a boundary integral method. Solutions for different wedge angles are found.

Surface-tension-driven flow in fat fluid wedges and cones

We consider the evolution under the action of surface tension of wedges and cones of viscous fluid whose initial semi-angles are close to π/2. A short time after the fluid is released from rest, there is an inner region, where surface tension and viscosity dominate, and an outer region, where inertia and viscosity dominate. We also find that the velocity of the tip of the wedge or cone is singular, of O(log(1/t)), as time, t, tends to zero. After a long time, the free surface asymptotes to a similarity form where deformations are of O(t2/3), and capillary waves propagate away from the tip. However, a distance of O(t3/4) away from the tip, viscosity acts to damp out the capillary waves.We solve the linearized governing equations using double integral transforms, which we calculate numerically, and use asymptotic techniques to approximate the solutions for small and large times. We also compare the asymptotic solution for the inviscid fat wedge with a numerical solution of the nonlinear inviscid problem for wedges of arbitrary semi-angle.