Cavity and Wake Flows

The phenomenon of wake formation behind a body moving through a fluid, and the associated resistance of fluids, must have been one of the oldest experiences of man. From an analytical point of view, it is also one of the most difficult problems in fluid mechanics. Rayleigh, in his 1876 paper, observed that "there is no part of hydrodynamics more perplexing to the student than that which treats of the resistance of fluids." This insight of Rayleigh is so penetrating that the march of time has virtually left no mark on its validity even today, and likely still for some time to come. The first major step concerning the resistance of fluids was made over a century ago when Kirchhoff (1869) introduced an idealized inviscid-flow model with free streamlines (or surfaces of discontinuity) and employed (for steady, plane flows) the ingenious conformal-mapping technique that had been invented a short time earlier by Helmholtz (1868) for treating two-dimensional jets formed by free streamlines. This pioneering work offered an alternative to the classical paradox of D’Alembert (or the absence of resistance) and laid the foundation of the free-streamline theory. We appreciate the profound insight of these celebrated works even more when we consider that their basic idea about wakes and jets, based on a construction with surfaces of discontinuity, was formed decades before laminar and turbulent flows were distinguished by Reynolds (1883), and long before the fundamental concepts of boundary-layer theory and flow separation were established by Prandtl (1904a). However, there have been some questions raised in the past, and still today, about the validity of the Kirchhoff flow for the approximate calculation of resistance. Historically there is little doubt that in constructing the flow model Kirchhoff was thinking of the wake in a single-phase fluid, and not at all of the vapor-gas cavity in a liquid; hence the arguments, both for and against the Kirchhoff flow, should be viewed in this light. On this basis, an important observation was made by Sir William Thomson, later Lord Kelvin (see Rayleigh 1876) "that motions involving a surface of separation are unstable" (we infer that instability here includes the viscous effect). Regarding this comment Rayleigh asked "whether the calculations of resistance are materially affected by this circumstance as the pressures experienced must be nearly independent of what happens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself." This discussion undoubtedly widened the original scope, brought the wake analysis closer to reality, and hence should influence the course of further developments. An expanded discussion essentially along these lines was given by Levi-Civita (1907) and was included in the survey by Goldstein (1969). Another point of fundamental importance is whether the Kirchhoff flow is the only correct Euler (or outer) limit of the Navier-Stokes solution to steady flow at high Reynolds numbers. If so, then a second difficulty arises, a consequence of the following argument: We know that the width of the Kirchhoff wake grows parabolically with the downstream distance x, at a rate independent of the (kinematic) viscosity u. If Prandtl’s boundary-layer theory is then applied to smooth out the discontinuity (i.e. the vortex sheet) between the wake and the potential flow, one obtains a laminar shear layer whose thickness grows like (ux/U)^-1/2 in a free stream of velocity U. Hence, for sufficiently small u/U the shear layers do not meet, so that the wake bubble remains infinitely long at a finite Reynolds number, a result not supported by experience. (For more details see Lagerstrom 1964, before p. 106, 131; Kaplun 1967, Part II.) The weaknesses in the above argument appear to lie in the two primary suppositions that, first, the free shear layer enveloping the wake would remain stable indefinitely, and second (perhaps a less serious one), the boundary-layer approximation would be valid along the infinitely long wake boundary. Reattachment of two turbulent shear layers, for instance, is possible since their thickness grows linearly with x. By and large, various criticisms, of the Kirchhoff flow model have led to constructive refinements of the free-streamline theory rather than to a weakening of the foundation of the theory as a valuable idealization. The major development in this direction has been based on the observation that the wake bubble is finite in size at high Reynolds numbers. (The wake bubble, or the near-wake, means, in the ordinary physical sense, the region of closed streamlines behind the body as characterized by a constant or nearly constant pressure.) To facilitate the mathematical analysis of flows with a finite wake bubble, a number of potential-flow models have been introduced to give the near-wake a definite configuration as an approximation to the inviscid outer flow. These theoretical models will be discussed explicitly later. It suffices to note here that all these models, even though artificial to various degrees, are aimed at admitting the near-wake pressure coefficient as a single free parameter of the flow, thus providing a satisfactory solution to the state of motion in the near part of the wake attached to the body. On the whole, their utility is established by their capability of bringing the results of potential theory of inviscid flows into better agreement with experimental measurements in fluids of small viscosity. The cavity flow also has a long, active history. Already in 1754, Euler, in connection with his study of turbines, realized that vapor cavitation may likely occur in a water stream at high speeds. In investigating the cause of the racing of a ship propeller, Reynolds (1873) observed the phenomenon of cavitation at the propeller blades. After the turn of this century, numerous investigations of cavitation and cavity flows were stimulated by studies of ship propellers, turbomachinery, hydrofoils, and other engineering developments. Important concepts in this subject began to appear about fifty years ago. In an extensive study of the cavitation of water turbines, Thoma (1926) introduced the cavitation number (the underpressure coefficient of the vapor phase) as the principal similarity parameter, which has ever since played a central role in small-bubble cavitation as well as in well-developed cavity flows. Applications of free-streamline theory to finite-cavity flows have attracted much mathematical interest and also provided valuable information for engineering purposes. Although the wake interpretation of the flow models used to be standard, experimental verifications generally indicate that the theoretical predictions by these finite-wake models are satisfactory to the same degree for both wake and cavity flows. This fact, however, has not been widely recognized and some confusion still exists. As a possible explanation, it is quite plausible that even for the wake in a single-phase flow, the kinetic energy of the viscous flow within the wake bubble is small, thus keeping the pressure almost unchanged throughout. Although this review gives more emphasis to cavity flows, several basic aspects of cavity and wake flows can be effectively discussed together since they are found to have many important features in common, or in close analogy. This is in spite of relatively minor differences that arise from new physical effects, such as gravity, surface tension, thermodynamics of phase transition, density ratio and viscosity ratio of the two phases, etc., that are intrinsic only to cavity flows. Based on this approach, attempts will be made to give a brief survey of the physical background, a general discussion of the free-streamline theory, some comments on the problems and issues of current interest, and to point out some basic problems yet to be resolved. In view of the vast scope of this subject and the voluminous literature, efforts will not be aimed at completeness, but rather on selective interests. Extensive review of the literature up to the 1960s may be found in recent expositions by Birkhoff & Zarantonello (1957), Gilbarg (1960), Gurevich (1961), Wehausen (1965), Sedov (1966), Wu (1968), Robertson & Wislicenus (1969), and (1961)

Three-dimensional oblique water-entry problems at small\ud deadrise angles

This paper extends Wagner theory for the ideal, incompressible normal impact of rigid bodies that are nearly parallel to the surface of a liquid half-space. The impactors considered are three-dimensional and have an oblique impact velocity. A variational formulation is used to reveal the relationship between the oblique and corresponding normal impact solutions. In the case of axisymmetric impactors, several geometries are considered in which singularities develop in the boundary of the effective wetted region. We present the corresponding pressure profiles and models for the splash sheets

Some Problems of Supersonic Cavitation Flows

The basic results obtained by the author in the theory of supercavities in compressible fluid have been presented. The modern numerical methods have been applied in this investigation and the basic correlations for shock in water have been considered. The characteristic features of supersonic cavitation flows and their difference from the analogous flows in the air have been noted. The buckling stability of an elastic body moving at a high speed in water has been investigated

Cavitation Scaling Experiments With Headforms: Bubble Acoustics

Recently Ceccio and Brennen [1][2][3] have examined the interaction between individual traveling cavitation bubbles and the structure of the boundary layer and flow field in which the bubble is growing and collapsing. They were able to show that individual bubbles are often fissioned by the fluid shear and that this process can significantly effect the acoustic signal produced by the collapse. Furthermore they were able to demonstrate a relationship between the number of cavitation events and the nuclei number distribution measured by holographic methods in the upstream flow. Kumar and Brennen [4][5] have further examined the statistical properties of the acoustical signals from individual cavitation bubbles on two different headforms in order to learn more about the bubble/flow interactions. All of these experiments were, however, conducted in the same facility with the same size of headform (5.08cm in diameter) and over a fairly narrow range of flow velocities (around 9m/s). Clearly this raises the issue of how the phenomena identified change with speed, scale and facility. The present paper will describe further results from experiments conducted in order to try to answer some of these important questions regarding the scaling of the cavitation phenomena. These experiments (see also Kuhn de Chizelle et al. [6][7]) were conducted in the Large Cavitation Channel of the David Taylor Research Center in Memphis Tennessee, on similar Schiebe headforms which are 5.08, 25.4 and 50.8cm in diameter for speeds ranging up to 15m/s and for a range of cavitation numbers

Study of fluid transients in closed conduits annual report no. 1

Atmospheric density effect on computation of earth satellite orbit

A simple approach to estimating three-dimensional supercavitating flow fields

A simple method is formulated for predicting three-dimensional supercavitating flow behind cavitators subject to gravitational acceleration and motion of the cavitator. The method applies slenderbody theory in the context of matched asymptotic expansions to pose an inner problem for the cavity evolution downstream from the locus of cavity detachment. This inner problem is solved by means of a coupled set of equations for the Fourier coefficients characterizing the cavity radius and the velocity potential as a function of downstream location and circumferential location, thus resulting in a two-dimensional multipole solution at each station. For the lowestorder term in the Fourier expansion, it is necessary to match the parabolic inner solution to a fully elliptic outer solution. This step allows the application of any one of a number of methods to solve the axisymmetric problem, which serves as the base solution that is perturbed by the three-dimensional effects. The method is an attempt to formalize the Logvinovich principle of independent cavity section evolution. Results flow past a circular disk cavitator are presented for severalvalues of the cavity Froude number.http://deepblue.lib.umich.edu/bitstream/2027.42/84318/1/CAV2009-final145.pd

Cavitation Number as a Function of Disk Cavitator Radius: a Numerical Analysis of Natural Supercavitation

Due to the greater viscosity and density of water compared to air, the maximum speed of underwater travel is severely limited compared to other methods of transportation. However, a technology called supercavitation – which uses a disk-shaped cavitator to envelop a vehicle in a bubble of steam – promises to greatly decrease skin friction drag. While a large cavitator enables the occurrence of supercavitation at low velocities, it adds substantial unnecessary drag at higher speeds. Based on CFD results, a relationship between cavitator diameter and cavitation number is developed, and it is substituted into an existing equation relating drag coefficient to cavitation number. The final relationship predicts drag from cavitator radius fairly well, with an absolute error less than 5.4% at a cavitator radius above 14.14mm and as low as 1.3% at the maximum tested radius of 22.5mm

Cavitation Bubble Dynamics and Noise Production

This paper presents a summary of some recent observations of the interaction between individual traveling cavitation bubbles and the nearly solid surface. These reveal a complex micro-fluid-mechanics associated with the interaction of the bubble with the boundary layer, the surface and the large pressure gradients exterior to the boundary layer which normally occur in the vicinity of a minimum pressure point. The details are important because they affect how the bubble collapses and therefore influence the noise produced and the damage potential of the cavitating flow. We present data showing how the noise from an individual event can be affected by these interaction effects. Since the scaling of cavitation phenomena is always an important issue we also describe the results of some experiments carried out to investigate the scaling of these interaction effects

High-speed photography of supercavitation and multiphase flows in water entry

The supercavitation in water entry and associated multiphase flows were studied using a high-speed camera and single-shot optical device. The formation, growth and collapse of supercavities induced by underwater high-speed projectiles were revealed. The unsteady fluid dynamic processes of the splash, surface deformation, supercavity twisting, down jet in the cavity, etc. were also studied. Both axisymmetrical and three-dimensional supercavities were found in the experiment. The shape of the axisymmetrical supercavity has been comparied with Logvinovich s model. The three-dimensional supercavity is caused by the trajectory deflection of the underwater projectile, which is also related unsteadiness and turbulence of the flow field. When the supercavity is twisted, a grain-like cavitating bubble is formed after the upper and lower pinch-offs. It is newly found that during the surface closure (surface seal), the down-jet is generated simultaneously with the formation of upwards splash dome.http://deepblue.lib.umich.edu/bitstream/2027.42/84316/1/CAV2009-final142.pd

High speed motion in water with supercavitation for sub-, trans-, supersonic Mach Numbers

The results of research for supercavitating motion in water at very high speeds comparable with sonic speed ~1500m/s are presented. At such speeds the water is a compressible fluid and the basic compressible hydrodynamics of supercavitating flows together with practical approaches and experimental data are considered. The theory of ballistic projectiles motion is developed with emphasis on the problems of maximal range, lateral motion prediction and problems of minimal declination, hydro-elastic effects, and resonant oscillation frequencies. One main purpose of the article is an attempt to advance the level of understanding of the problem of very high-speed underwater launch by a comprehensive review of previous research on this topic.http://deepblue.lib.umich.edu/bitstream/2027.42/84267/1/CAV2009-final72.pd