Quantitative Flow Field Imaging about a Hydrophobic Sphere Impacting on a Free Surface

This fluid dynamics video shows the impact of a hydrophobic sphere impacting a water surface. The sphere has a mass ratio of m* = 1.15, a wetting angle of 110 degrees, a diameter of 9.5 mm, and impacts the surface with a Froude number of Fr = 9.2. The first sequence shows an impact of a sphere on the free surface illustrating the formation of the splash crown and air cavity. The cavity grows both in the axial and radial direction until it eventually collapses at a point roughly half of the distance from the free surface to the sphere, which is known as the pinch-off point. The second set of videos shows a sphere impacting the free surface under the same conditions using Particle Image Velocimetry (PIV) to quantify the flow field. A laser sheet illuminates the mid-plane of the sphere, and the fluid is seeded with particles whose motion is captured by a high-speed video camera. Velocity fields are then calculated from the images. The video sequences from left to right depict the radial velocity, the axial velocity, and the vorticity respectively in the flow field. The color bar on the far left indicates the magnitude of the velocity and vorticity. All videos were taken at 2610 fps and the PIV data was processed using a 16 x 16 window with a 50% overlap.Comment: American Physical Society Division of Fluid Dynamics 2008 Annual Meeting Replaced previous version because abstract had LaTex markup and was too long, missing periods on middle initial of first two name

Dynamics of Water Entry

The hydrodynamics associated with water-entry of spheres can be highly variable with respect to the material and kinematic properties of the sphere. This series of five fluid dynamics videos illustrates several subtle but interesting variations. The first series of videos contrasts the nature of impact between a hydrophilic and hydrophobic sphere, and illustrates how surface coating can affect whether or not an air cavity is formed. The second video series illustrates how spin and surface treatments can alter the splash and cavity formation following water entry. The spinning sphere causes a wedge of fluid to be drawn into the cavity due to the no-slip condition and follows a curved trajectory. The non-spinning sphere has two distinct surface treatments on the left and right hemispheres: the left hemisphere is hydrophobic and the right hemisphere is hydrophilic . Interestingly, the cavity formation for the half-and-half sphere has many similarities to that of the spinning sphere especially when viewed from above. The third video series compares two millimetric nylon spheres impacting at slightly different impact speeds (Uo = 40 and 45 cm/s); the faster sphere fully penetrates the free surface, forming a cavity, whereas the slower sphere does not. The fourth series shows the instability of an elongated water-entry cavity formed by a millimetric steel sphere with a hydrophobic coating impacting at Uo = 600 cm/s. The elongated cavity forms multiple pinch-off points along its decent. Finally, a millimetric steel sphere with a hydrophobic coating breaks the free surface with an impact speed of Uo = 350 cm/s. The cavity pinches-off below the surface, generating a Worthington jet that pinches into droplets owing to the Rayleigh-Plateau instability.Comment: American Physical Society Division of Fluid Dynamics Gallery of Fluid Motion Video Entry Replaced previous version because abstract had LaTex markup and was too lon

Generation of sentences with their parses : the case of propagating scattered context grammars

Propagating scattered context grammars are used to generate their sentences together with their parses - that is, the sequences of labels denoting productions whose use lead to the generation of the corresponding sentences. It is proved that for every recursively enumerable language L, there exists a propagating scattered context grammar whose language consists of L's sentences followed by their parses

An infinite hierarchy of language families generated by scattered context grammars with n-limited derivations

AbstractThis paper introduces scattered context grammars without erasing productions, in which an application of a production always occurs within the first n nonterminals of the current sentential form. It demonstrates that this restriction gives rise to an infinite hierarchy of language families each of which is properly included in the family of context-sensitive languages. In addition, it proves analogous results for unordered scattered context grammars. Some consequences of these results are derived and open problems formulated

Shallow angle water entry of ballistic projectiles

The water-entry of ballistic projectiles is investigated using high-speed digital imaging to capture the subsurface cavity dynamics. Specially designed 0.22 caliber projectiles are fired into water at shallow angles to the free surface (5º to 15º) at Mach numbers between 0.3 and 1.0. Redesigned projectile tip geometries allowed projectiles to successfully enter the water and travel large distances underwater, due to the subsurface vapor-cavity that forms after impact, dramatically decreasing drag on the projectile. Projectile dynamics, critical entry angle and cavity formation are discussed for various bullet geometries, and results show that successful water-entry is a function of tip shape and length-to-diameter ratio. The data conclusively show that bullets with lower length-to-diameter ratios tumble inside the vapor cavity, while higher length-to-diameter ratios can lean against the cavity walls inducing a planing force pushing them back inside the cavity and mitigating the tumbling behavior. Experimental cavity observations of vapor-cavity formation is compared to a modified version of Logvinovich's [1] theoretical model, which includes an updated formulation of the model and an angle of attack correction. Despite the unsteady nature of this problem, this improved steady state model fits well with experimental data and serves as an accurate design tool for naval engineers.http://deepblue.lib.umich.edu/bitstream/2027.42/84284/1/CAV2009-final100.pd

Vortical patterns behind a tapered cylinder oscillating transversely

Visualization studies of the flow behind an oscillating tapered cylinder are performed at Reynolds numbers from 400 to 1500. The cylinder has taper ratio 40:1 and is moving at constant forward speed U while being forced to oscillate harmonically in the transverse direction. It is shown that within the lock-in region and above a threshold amplitude, no cells form and, instead, a single frequency of response dominates the entire span. Within certain frequency ranges a single mode dominates in the wake, consisting of shedding along the entire span of either two vortices per cycle (`2S' mode), or four vortices per cycle (`2P' mode); but within specific parametric ranges a hybrid mode is observed, consisting of a `2S' pattern along the part of the span with the larger diameter and a `2P' pattern along the part of the span with the smaller diameter. A distinct vortex split connects the two patterns which are phaselocked and have the same frequency. The hybrid mode is periodic, unlike vortex dislocations, and the location of the vortex split remains stable and repeatable, within one to two diameters, depending on the amplitude and frequency of oscillation and the Reynolds number

Forces on oscillating uniform and tapered cylinders in a crossflow

Forces are measured at both ends of rigid cylinders with span 60 cm, performing transverse oscillations within an oncoming stream of water, at Reynolds number Re ~3800. Forced harmonic motions and free vibrations of uniform and tapered cylinders are studied. To study free motions, a novel force-feedback control system has been developed, consisting of: (a) a force transducer, which measures forces on a section of a cylinder moving forward at constant speed; (b) a computer using the measured force signal to drive in real time a numerical simulation of an equivalent mass-dashpot-spring system; (c) a servomotor and linear table which impose, also in real time, the numerically calculated motion on the cylinder section. The apparatus allows very low equivalent system damping and strict control of the parametric values and structure of the equivalent system. Calculation of the cross-correlation coefficient between forces at the two ends of the uniform cylinder reveals five distinct regimes as a function of the nominal reduced velocity Vrn: two regimes, for low and high values of Vrn, and far away from the value of VrS corresponding to the Strouhal frequency, show small correlation; two regimes immediately adjacent to, but excluding, VrS show strong correlation, close to 1; surprisingly, there is a regime containing the Strouhal frequency, within which correlation is low. Free vibrations with a 40:1 tapered cylinder show that the regime of low correlation, containing the Strouhal frequency, stretches to higher reduced velocities, while lock-in starts at lower reduced velocities. When comparing the amplitude and phase of the lift coefficient measured for free and then for forced vibrations, we obtain close agreement, both for tapered and uniform cylinders. When comparing the cross-correlation coefficient however, we find that it is much higher in the forced oscillations, especially for the uniform cylinder. Hence, although the force magnitude and phase may be replicated well in forced vibrations, the correlation data suggest that differences exist between free and forced vibration cases