Cavitation Bubble Collapse in Viscous, Compressible Liquids—Numerical Analysis

1965 ◽  
Vol 87 (4) ◽  
pp. 977-985 ◽  
Author(s):  
R. D. Ivany ◽  
F. G. Hammitt
Keyword(s):  
Surface Tension ◽  
High Pressures ◽  
Numerical Solutions ◽  
Compressible Liquid ◽  
Navier Stokes ◽  
Bubble Collapse ◽  
Navier Stokes Equation ◽  
Scaling Parameters ◽  
Collapse Behavior ◽  
Normalized Solution

Collapse of a spherical bubble in a compressible liquid, including the effects of surface tension, viscosity, and an adiabatic compression of gas within the bubble is investigated by numerical solutions of the hydrodynamic equations. A limiting value of shear viscosity causes the bubble collapse to slow down markedly, for both compressible and incompressible liquids, whereas moderate viscosities have very little effect on the rate of collapse. The inclusion of surface tension and viscosity introduces two scaling parameters into the solution, so that a single normalized solution is no longer sufficient to describe collapse behavior. The magnitude of the density changes calculated for the compressible liquid and the extremely rapid changes with time suggest that the usual Navier-Stokes equation of motion may not be appropriate. The possibility of liquid relaxational phenomenon and its contribution to sonoluminescence is considered. Shock waves or damagingly high pressures are not generated during collapse at a distance in the liquid equal to the initial radius from the center of collapse, although they will appear at such a distance if the bubble rebounds.

Application of an Improved Ghost Fluid Method to the Collapse of Non-Spherical Bubbles in a Compressible Liquid

2011 ◽  
Author(s):  
Hiroyuki Takahira ◽  
Yoshinori Jinbo
Keyword(s):  
Surface Tension ◽  
Shock Waves ◽  
Thermal Diffusion ◽  
Level Set ◽  
Compressible Liquid ◽  
Maximum Temperature ◽  
Bubble Collapse ◽  
Ghost Fluid Method ◽  
Toroidal Bubble ◽  
The Impact

The ghost fluid method (GFM) is improved to investigate violent bubble collapse in a compressible liquid, in which the adaptive mesh refinement with multigrids, the surface tension, and the thermal diffusion through the bubble interface are taken into account. The improved multigrid GFM is applied to the interaction of an incident shock wave with a bubble. The multigrid GFM captures the fine interfacial and vortex structures of the toroidal bubble when the bubble collapses violently accompanied with the penetration of the liquid jet and the formation of the shock waves. The multigrid GFM is also applied to the bubble collapse near a tissue surface in which the tissue is modeled with gelatin in order to predict the tissue damage due to the bubble collapse; the motions of three phases for the gas inside the bubble, the liquid surrounding the bubble, and the gelatin boundary are solved directly by coupling the level set method with the improved GFM. Two kinds of level set functions are utilized for distinguishing the gas-liquid interface from the liquid-gelatin interface. It is shown that the impact of the shock waves generated from the collapsing bubble on the boundary leads to the formation of depression of the boundary; the toroidal bubble penetrates into the depression. Also, the surface tension effects are successfully included in the improved GFM. The thermal effects of internal gas on the bubble collapse are also discussed by considering the thermal diffusion across the interface in the GFM. The thermal boundary layers of the toroidal bubble are captured with the method. The result shows that the smaller the initial bubble radius becomes, the lower the maximum temperature inside the bubble becomes because of the thermal diffusion across the interface.

A Sharp Surface Tension Modeling Method for Capillarity-Dominant Two-Phase Incompressible Flows

2007 ◽  
Author(s):  
Zhaoyuan Wang ◽  
Albert Y. Tong
Keyword(s):  
Surface Tension ◽  
Incompressible Flows ◽  
Jump Condition ◽  
Navier Stokes ◽  
Pressure Jump ◽  
Interfacial Flows ◽  
Two Phase ◽  
Navier Stokes Equation ◽  
Numerical Instabilities ◽  
Spurious Currents

A surface tension implementation algorithm for two-phase incompressible interfacial flows is presented in this study. The surface tension effect is treated as a jump condition at the interface and incorporated into the Navier-Stokes equation via a capillary pressure gradient. The interface is tracked by a coupled level set and volume-of-fluid (CLSVOF) method based on the finite-volume formulation on a fixed Eulerian grid. It has been shown in a stationary benchmark test the spurious currents are greatly reduced and the sharp pressure jump across the interface is well preserved. Numerical instabilities caused by the sharp treatment on a fixed grid are avoided. Several dynamic tests are performed to further validate the accuracy and versatility of the present method, the results of which are in good agreement with data reported in the literature.

A two-dimensional vortex condensate at high Reynolds number

2013 ◽  
Vol 715 ◽  
pp. 359-388 ◽  
Author(s):  
Basile Gallet ◽  
William R. Young
Keyword(s):  
Stokes Equation ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Finite Limit ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Quasilinear Theory ◽  
Zero Integral ◽  
High Reynolds

AbstractWe investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

scholarly journals A PHASE-FIELD DESCRIPTION OF SURFACE-TENSION-DRIVEN INSTABILITY

2004 ◽  
Vol 14 (12) ◽  
pp. 4105-4116 ◽  
Author(s):  
RODICA BORCIA ◽  
DOMNIC MERKT ◽  
MICHAEL BESTEHORN
Keyword(s):  
Surface Tension ◽  
Phase Field ◽  
Linear Functions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Field Function ◽  
Classical Formulation ◽  
Fluid Systems ◽  
Field Formalism ◽  
Cahn Hilliard Equation

In this paper we report on 2D numerical simulations concerning linear and nonlinear evolution of surface-tension-driven instability in two-fluid systems heated from below using classical and phase-field models. In the phase-field formalism, one introduces an order parameter called phase-field function to characterize the phases thermodynamically. All the system parameters are assumed to vary continuously from one fluid bulk to another (as linear functions of the phase-field). The Navier–Stokes equation (with some extra terms) and the heat equation are written only once for the whole system. The evolution of the phase-field is described by the Cahn–Hilliard equation. In the sharp-interface limit the results found by the phase-field formalism recover the results given by the classical formulation.

Influence of temperature dependent physical properties on liquid metal droplet impact dynamics

2021 ◽  
pp. 1-15
Author(s):  
Akash Chowdhury ◽  
Anandaroop Bhattacharya ◽  
Partha Bandyopadhyay
Keyword(s):  
Surface Tension ◽  
Numerical Solutions ◽  
Substrate Surface ◽  
Metal Droplet ◽  
Navier Stokes ◽  
Complex Flow ◽  
Liquid Metal Droplet ◽  
Aluminium Copper ◽  
Energy Equations ◽  
Surface Tension Forces

Abstract The dynamics of a metal droplet impacting on a substrate surface has been studied in the paper numerically. Numerical solutions of the Navier-Stokes and Energy equations show the evolution of the droplet as it spreads upon impact with the substrate while simultaneously undergoing solidification. The interplay of the different forces including inertia, viscous and surface tension, coupled with solidification of the molten material in layers lead to complex flow dynamics. The change in density and viscosity owing to change in temperature resulting from the cooling process, is found to influence the spreading of the droplet significantly. The model was exercised for three different materials viz. aluminium, copper and nickel to determine the final splat radius as well as spreading time. The surface tension forces as well as solidification rates were found to be the dominant factors in determining the above parameters as well as the shape of the splat during spreading. The results were found to be in good agreement with existing analytical model.

Analysis of Capillary Flow in Rounded Corners

Volume 1 ◽  
2004 ◽  
Author(s):  
Yongkang Chen ◽  
Mark M. Weislogel
Keyword(s):  
Numerical Solutions ◽  
Flow Problem ◽  
Capillary Flow ◽  
Outer Region ◽  
Cross Flow ◽  
Numerical Data ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Inner Region ◽  
The Impact

The problem of capillary flow in interior corners that are rounded is re-visited analytically in this work. By the appropriate geometric scaling, and through the introduction of a new parameter that features the roundedness of the corner, the Navier-Stokes equation is reduced to a convenient form for both numerical and analytical solution. The scaling and analysis of the problem is expected to significantly reduce the reliance on numerical data for such problems, and the design process can be both shortened and improved as a result. For capillary flows of perfect wetting fluids in the rounded corner with an advancing tip, a finite interfacial curvature related to the corner roundedness results at the tip. Accordingly, an outer and inner region of the flow is suggested based on the impact of the corner roundedness on the flow. In this study, asymptotic solutions of the geometrical ‘cross-flow’ problem for the outer region are sought under several constraints and are expected to narrowly bracket parallel numerical solutions. A complete understanding of the flow will be obtained only after the cross-flow problem for the inner region is solved. However, for the flow in the outer region a similarity solution is obtained and presented that reveals how roundedness retards the flow.

Convection in Pulsed Laser Formed Melts

1980 ◽  
Vol 1 ◽  
Author(s):  
G. E. Possin ◽  
H. G. Parks ◽  
S. W. Chiang
Keyword(s):  
Surface Tension ◽  
Approximate Solution ◽  
Mass Transport ◽  
Stokes Equation ◽  
Pulsed Laser ◽  
Metal Films ◽  
Navier Stokes ◽  
Mixing Times ◽  
Navier Stokes Equation ◽  
Theoretical Considerations

ABSTRACTIn this paper we treat surface tension driven convection effects in pulsed laser formed melts. Mass transport is determined from an approximate solution of the Navier Stokes equation. It is shown that for small laser spot diameters the characteristic mixing times are on the order of 100's of ns. The dependence of the convection mechanism on material and laser parameters is discussed and extended to thin metal films on Si. Experimental results substantiating the theoretical considerations are presented.

History forces and the unsteady wake of a cylinder

1999 ◽  
Vol 393 ◽  
pp. 99-121 ◽  
Author(s):  
J. R. CHAPLIN
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Stokes Equation ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Asymptotic Properties ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Unsteady Wake

History forces on a stationary cylinder in arbitrary unsteady rectilinear flow are calculated by means of a model based on the asymptotic properties of the steady-state wake. The results capture many features found in numerical solutions of the Navier–Stokes equation for the same flows, though quantitative agreement deteriorates as the Reynolds number increases over the range 2 to 40. The cases studied are the impulsive start, stop, and reverse, and oscillatory flow.

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