Strong interaction between a buoyancy bubble and a free surface

The growth and collapse of buoyant vapor bubbles close to a free surface in an inviscid incompressible fluid is investigated in this paper. The strong interaction between the deforming bubble and the free surface is simulated numerically by a boundary-integral method (Taib 1985; Blake et al., 1987). Improvements are made in the calculation of the singular integrals, the use of nonuniform boundary elements, and the choice of time-step size. The present numerical results agree better with the experimental observations of Blake and Gibson (1981) than previous numerical predictions for bubbles initiated at one maximum radius from the free surface. There is also concurrence of flow features with the experiments for a bubble initiated as close as half maximum radius from the free surface, where other numerical efforts have failed. The effects of buoyancy on bubbles initiated close to a free surface are also investigated. Vastly different features, depending on the distance of the bubble to the free surface and the buoyancy-force parameter, have been observed.     

Oblique water entry of a cone by a fully three-dimensional nonlinear method

The hydrodynamic problem of a cone entering the water surface obliquely has been analyzed by the three-dimensional (3-D) incompressible velocity potential theory with the fully nonlinear boundary conditions on the moving free surface and body surface boundary. The time stepping method is used in the stretched coordinate system defined as the ratio of the physical system to the distance that the cone has travelled into water. The boundary element method is used to solve the potential at each time step. Both triangular element mesh and quadrilateral element mesh have been used. Discretisation of the body surface and the free surface is applied regularly during the simulation to account for their change and deformation, and data from the old mesh is transferred into the new one through interpolation. Both the dynamic and kinematic free surface boundary conditions are satisfied through the Eulerian form. In particular the free surface elevation and potential variation are traced at a given azimuth of the cylindrical coordinate system, in the direction parallel to the body surface or perpendicular to the free surface to avoid multi-valued function. Detailed convergence study with respect to time step and element size has been undertaken and high accuracy has been achieved. Results for the cone in vertical entry are compared with those obtained from the 2-D axisymmetric method and good agreement is found. Simulations are made for cones of various deadrise angles and different oblique entries and detailed results are provided.     

Flutter and resonances of a flag near a free surface

We investigate the effects of a nearby free surface on the stability of a flexible plate in axial flow. Confinement by rigid boundaries is known to affect flag flutter thresholds and fluttering dynamics significantly, and this work considers the effects of a more general confinement involving a deformable free surface. To this end, a local linear stability is proposed for a flag in axial uniform flow and parallel to a free surface, using one-dimensional beam and potential flow models to revisit this classical fluid–structure interaction problem. The physical behaviour of the confining free surface is characterized by the Froude number, corresponding to the ratio of the incoming flow velocity to that of the gravity waves. After presenting the simplified limit of infinite span (i.e. two-dimensional problem), the results are generalized to include finite-span and lateral confinement effects. In both cases, three unstable regimes are identified for varying Froude number. Rigidly-confined flutter is observed for low Froude number, i.e. when the free surface behaves as a rigid wall, and is equivalent to the classical problem of the confined flag. When the flow and wave velocities are comparable, a new instability is observed before the onset of flutter (i.e. at lower reduced flow speed) and results from the resonance of a structural bending wave and one of the fundamental modes of surface gravity waves. Finally, for large Froude number (low effect of gravity), flutter is observed with significant but passive deformation of the free surface in response of the flag’s displacement.     

Emission of dimers from a free surface of heated water

The emission rate of water dimers from a free surface and a wetted solid surface in various cases was calculated by a simplified Monte Carlo method with the use of the binding energy of water molecules. The binding energy of water molecules obtained numerically assuming equilibrium between the free surface of water and vapor in the temperature range of 298–438 K corresponds to the coordination number for liquid water equal to 4.956 and is close to the reference value. The calculation results show that as the water temperature increases, the free surface of water and the wetted solid surface become sources of free water dimers. At a temperature of 438 K, the proportion of dimers in the total flow of water molecules on its surface reaches 1%. It is found that in the film boiling mode, the emission rate of dimers decreases with decreasing saturation vapor. Two mechanisms of the emission are described.     

Form of a free surface during steady flow of a capillary fluid in a rectangular channel

The two-dimensional problem of the form of a free surface of an ideal incompressible fluid during steady flow from a rectangular channel through a thin slot with simultaneous uniform delivery of fluid through the side walls is examined. Forces of gravity and surface tension are taken into account. The nonlinear problem of the simultaneous determination of the free surface and velocity field of the fluid is solved by the iteration method. Convergence of the iterations to the solution of the problem for small values of the parameters is investigated. The solution of the linearized problem is obtained in a closed form for a small depth of the discharge and small width of the channel, which is compared with the solution of the problem in a complete formulation. Graphs of the free surface of the fluid for different values of the parameters, obtained as a result of numerical solution of the nonlinear problem, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–75, January–February, 1977.     

Formation of a cavity in the initial stage of motion of a circular cylinder in a fluid with a constant acceleration

This paper considers the joint motion of an ideal fluid and a circular cylinder completely immersed in it at small times. It is assumed that the cylinder, which was initially at rest, moves in a horizontal direction with a constant acceleration. The dynamics of the internal and external free boundaries of the fluid at small times is studied. An asymptotic analysis of the form of the internal free surface near the separation points is performed. It is shown that at high acceleration of the circular cylinder, a large cavity is formed behind, with a strong perturbation of the external free surface of the fluid over the surface of the cylinder.     

Oscillations of the fluid flow and the free surface in a cavity with a submerged bifurcated nozzle

The free surface dynamics and sub-surface flow behavior in a thin (height and width much larger than thickness), liquid filled, rectangular cavity with a submerged bifurcated nozzle were investigated using free surface visualization and particle image velocimetry (PIV). Three regimes in the free surface behavior were identified, depending on nozzle depth and inlet velocity. For small nozzle depths, an irregular free surface is observed without clear periodicities. For intermediate nozzle depths and sufficiently high inlet velocities, natural mode oscillations consistent with gravity waves are present, while at large nozzle depths long term self-sustained asymmetric oscillations occur.For the latter case, time-resolved PIV measurements of the flow below the free surface indicated a strong oscillation of the direction with which each of the two jets issue from the nozzle. The frequency of the jet oscillation is identical to the free surface oscillation frequency. The two jets oscillate in anti-phase, causing the asymmetric free surface oscillation. The jets interact through a cross-flow in the gaps between the inlet channel and the front and back walls of the cavity.     

The Onset of Fluid Rotation in a Thermogravitational Boundary Layer with Local Cooling of the Free Surface

Axisymmetric regimes of flows of an inhomogeneous fluid in the boundary layer near a free surface are calculated for a nonuniform temperature distribution on this surface. For the fluid motion equations written in the Oberbeck-Boussinesq approximation, the leading terms of asymptotic expansions of solutions of a steady-state problem are constructed. It is shown that in the presence of local cooling of the free surface and a rising outer fluid stream, as a result of a bifurcation, a pair of rotational regimes may develop in a thin boundary layer near the free surface, with no rotation observed outside this layer. No bifurcation of rotation was detected in the case of local heating of the free surface.     

The nonlinear interaction of vortex rings with a free surface

Nonlinear interactions of vortex rings with a free surface are considered in an incompressible, ideal fluid using the vortex contour dynamics technique and the boundary integral equation method. The flow is axisymmetric and the vorticity is linearly distributed in the vortex. Effects of the gravity and the surface tension as well as the initial geometric parameter of the vortex on the interaction process are investigated in considerable detail. The interaction process may be divided into three major stages: the vortex free-traveling stage, the collision stage, and the vortex stretching and rebounding stage. Time evolutions of both the vortex and free surface under various conditions are provided and analyzed. Two kinds of waves exist on the free surface during interaction. In a special case where the gravity and surface tension are very weak or the vortex is very strong, an electric-bulb-like ‘cavity’ is formed on the free surface and the vortex is trapped in the ‘cavity’ for quite a long time, resulting in a large amount of fluid above the mean fluid surface. The project supported by the National Education Commission of China and NASA under cooperative grant agreement # NCC5-34     

Generation of an average flow by a pulsating stream near a curved free surface

The generation of an average flow near a curved free surface under the action of small-amplitude harmonic translational vibrations is studied. It is found that, in contrast to the case of a flat non-deformable free surface, an average flow is generated in the viscous boundary layer. This flow, propagating in the bulk fluid, can be described by the steady-state Navier-Stokes equations with effective boundary conditions for the shear stresses. It is shown that, in contrast to the flow generation near a solid surface, the flow generated near the free surface depends on the fluid viscosity and the curvature of the surface.     

Hydrodynamics of flow over a transversely oscillating circular cylinder beneath a free surface

In this paper, hydrodynamic force coefficients and wake vortex structures of uniform flow over a transversely oscillating circular cylinder beneath a free surface were numerically investigated by an adaptive Cartesian cut-cell/level-set method. At a fixed Reynolds number, 100, a series of simulations covering three Froude numbers, two submergence depths, and three oscillation amplitudes were performed over a wide range of oscillation frequency. Results show that, for a deeply submerged cylinder with sufficiently large oscillation amplitudes, both the lift amplitude jump and the lift phase sharp drop exist, not accompanied by significant changes of vortex shedding timing. The near-cylinder vortex structure changes when the lift amplitude jump occurs. For a cylinder oscillating beneath a free surface, larger oscillation amplitude or submergence depth causes higher time-averaged drag for frequency ratio (=oscillation frequency/natural vortex shedding frequency) greater than 1.25. All near-free-surface cases exhibit negative time-averaged lift the magnitude of which increases with decreasing submergence depth. In contrast to a deeply submerged cylinder, occurrences of beating in the temporal variation of lift are fewer for a cylinder oscillating beneath a free surface, especially for small submergence depth. For the highest Froude number investigated, the lift frequency is locked to the cylinder oscillation frequency for frequency ratios higher than one. The vortex shedding mode tends to be double-row for deep and single-row for shallow submergence. Proximity to the free surface would change or destroy the near-cylinder vortex structure characteristic of deep-submergence cases. The lift amplitude jump is smoother for smaller submergence depth. Similar to deep-submergence cases, the vortex shedding frequency is not necessarily the same as the primary-mode frequency of the lift coefficient. The frequency of the induced free surface wave is exactly the cylinder oscillation frequency. The trends of wave length variation with the Froude number and frequency ratio agree with those predicted by the linear theory of small-amplitude free surface waves.     

Application of a vortex method to free surface flows

Vortex methods have found wide applications in various practical problems. The use of vortex methods in free surface flow problems, however, is still very limited. This paper demonstrates a vortex method for practical computation of non-linear free surface flows produced by moving bodies. The method is a potential flow formulation which uses the exact non-linear free surface boundary condition at the exact location of the instantaneous free surface. The position of the free surface, on which vortices are distributed, is updated using a Lagrangian scheme following the fluid particles on the free surface. The vortex densities are updated by the non-linear dynamic boundary condition, derived from the Euler equations, with an iterative Lagrangian numerical scheme. The formulation is tested numerically for a submerged circular cylinder in unsteady translation. The iteration is shown to converge for all cases. The results of the unsteady simulations agree well with classical linearized solutions. The stability of the method is also discussed.     

Surface waves generated by a line load on a half-space with depth-dependent properties

The reciprocity theorem of elastodynamics is used in this paper to determine the surface waves that are radiated from a time-harmonic line load applied at the surface of a solid body, whose elastic moduli and mass density depend on the distance from the surface. In a high-frequency approximation, the surface wave velocity and expressions for the displacement and stresses of free surface waves are employed in the reciprocity theorem. The general expressions for the surface wave radiated by the oscillating line load, together with a virtual free surface wave, when employed in the reciprocity theorem, yield relatively simple expressions for the amplitude factor of the radiated surface wave. Results show the amplitude factor as a function of the wavenumber.     

Free surface deformation due to a source and a sink of equal strength in Stokes flow

Two-dimensional Stokes flow due to a source and a sink of equal strength below the free surface is analyzed and free surface shape and cusp formation are discussed. The source-sink pair below the free surface are aligned vertical to the free surface. In the analysis, the Stokes' approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained by using conformal mapping and complex function theory. From the solution, typical free surface shapes are shown and formation of a cusp on the free surface is discussed. As the capillary number increases, the converging free surface shape becomes singular and tends to form a cusp for sufficiently large capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the capillary number vanishes, the solution is reduced to the linearized potential flow solution.     

Effect of a rigid structure on the dynamics of a bubble beneath the free surface

The coupling effect between a pulsating bubble and a free surface near a rigid structure is a complicated physical process. In this study, the evolution of an underwater explosion bubble and the free surface near a rigid structure is modeled by the boundary integral method. An approach of “double-node method” is used to maintain the stability of fluid-structure junction in the simulations, and meshes on the free surface and the structure are transformed to an open domain to ensure the calculation accuracy and efficiency. Validations are conducted against an underwater explosion experiment near a rigid structure. As a result, the simulations trace the jetting behavior of the bubble and the rise of the free surface. Finally, the bubble migration and the height of the free surface for different structure draughts are analyzed.     

Prediction of wave pattern and wave resistance of surface piercing bodies by a boundary element method

An iterative boundary element method, which was originally developed for both two‐ and three‐dimensional cavitating hydrofoils moving steadily under a free surface, is modified and extended to predict the wave pattern and wave resistance of surface piercing bodies, such as ship hulls and vertical struts. The iterative nonlinear method, which is based on the Green theorem, allows the separation of the surface piercing body problem and the free‐surface problem. The free‐surface problem is also separated into two parts; namely, left and right (with respect to x axis) free‐surface problems. Those all (three) problems are solved separately, with the effects of one on the other being accounted for in an iterative manner. The wetted surface of the body (ship hull or strut, including cavity surface if exists) and the left and right parts with respect to x axis of free surface are modelled with constant strength dipole and constant strength source panels. In order to prevent upstream waves, the source strengths from some distance in front of the body to the end of the truncated upstream boundary are enforced to be zero. No radiation condition is enforced for downstream and transverse boundaries. A transverse wave cut technique is used for the calculation of wave resistance. The method is first applied to a point source and a three‐dimensional submerged cavitating hydrofoil to validate the method and a Wigley hull and a vertical strut to compare the results with those of experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

Free surface deformation due to a source or a sink in Stokes flow

Free surface shape and cusp formation are analyzed by considering two-dimensional viscous flow due to a line source or a line sink below the free surface where the strength of source/sink is given arbitrarily. In the analysis, the Stokes' approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are shown and the formation of a cusp on the free surface is discussed. As the capillary number decreases in negative, the free surface shape becomes singular and in a real fluid a cusp should form on the free surface below some negative critical capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the small capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

Formation of the free surface of a viscous fluid volume inside a rotating horizontal cylinder

Two-dimensional viscous flow with a free surface in a horizontal cylinder rotating at a constant speed is investigated numerically using the boundary element method. It is shown that in the initial stage of rotation of the cylinder four different variants of the behavior of the free surface can be realized in the stage of transition from horizontal to steady-state form.