Comparison and validation of compressible flow simulations of laser-induced cavitation bubbles

doi:10.1016/j.compfluid.2009.04.004

Computers & Fluids

Volume 38, Issue 9, October 2009, Pages 1850-1862

https://doi.org/10.1016/j.compfluid.2009.04.004 Get rights and content

Abstract

The numerical simulation of compressible two-phase fluid flows exhibits severe difficulties, in particular, when strong strong variations in the material parameters and high interface velocities are present at the phase boundary. Although several models and discretizations have been developed in the past, a thorough quantitative validation by experimental data and a detailed comparison of numerical schemes are hardly available.

Here, two different discretizations are investigated, namely, a non-conservative approach proposed by Saurel and Abgrall [SIAM J Sci Comput 1999;21:1115] and the real ghost fluid method developed by Tang et al. [SIAM J Sci Comput 2006;28:278]. The validation is performed for the case of laser-induced cavitation bubbles collapsing in an infinite medium. For the computations, initial data are deduced implicitly from the experimental data. In particular, the influence of numerical phase transition caused by smearing of the phase boundary is investigated.

Introduction

The investigation of cavitation bubbles is of interest for different real world applications arising, for instance, in engineering, medicine and biology. The processes taking place in the interior and exterior of a collapsing and oscillating bubble are still subject of theoretical and experimental research [2]. However, small time and space scales as well as the complicated dynamics make any theoretical and experimental approach a challenge.

In the present work we will focus on the modeling and simulation of the collapse and rebound of a single vapor-filled spherical bubble in a liquid environment. This configuration has been subject of numerous analytical and experimental investigations. An overview of the field is given in the review article by Lauterborn et al. [14].

Numerical investigations have verified that the fluid state inside the bubble does not stay homogeneous during the collapse [8]. Moreover, shock waves develop in the liquid when the bubble gets maximally compressed, as is confirmed by experiments [2]. Presumably, strong waves are also generated inside the bubble, where they may interact with the phase boundary due to the small radius. Thereby, the frequently made assumption of incompressibility of the liquid and the homogeneity of the bubble medium are inappropriate when considering strong bubble collapse. In addition it could be verified that the modeling of the gas inside the bubble by a perfect gas is only valid at moderate changes in volume [8].

In recent years, several numerical investigations on compressible two-phase fluid flow for even more complex configurations have been published using the stiffened gas law [4], [5], [6], [16], [17], [27]. This model was introduced by Harlow and Amsden [9] and can be considered a combination of the perfect gas law and the barotrop Tait equation supplemented with an appropriate energy law [24]. Here, the material parameters depend on the phase.

For the numerical discretization of the Euler equations equipped with the stiffened gas law, different approaches have been used, for instance, the real ghost fluid method [27], a characteristics-based matching scheme [17] and a stratified flow model [6]. The main objective of this work is to compare two different discretizations, namely, (i) the Saurel and Abgrall (SA) approach [20] and (ii) the real ghost fluid method (rGFM) [27]. Both rely on a finite volume discretization of the Euler equations that only differs in the flux computation at cell interfaces next to the phase boundary. Here, the SA approach uses a two-phase Riemann solver whereas in the rGFM two single-phase Riemann problems with appropriate values for the ghost fluid are solved. The main difference inherent in the discretizations is the choice of the phase indicator function, namely, (i) the gas fraction in the SA approach and (ii) the level set function in the rGFM. The evolution of both phase indicator functions is described by the same non-conservative evolution equation but its discretization differs significantly: The evolution equation for the gas fraction is discretized by a non-conservative upwind scheme adapted to the underlying finite volume discretization of the flow equations. On the other hand, for the discretization of the level set function well-known techniques are applied to the transport equation. In order to ensure the property of a distance function the level set function is reinitialized in a postprocessing step.

The two discretizations are to be compared by means of quasi-1D computations of a spherical bubble in order to investigate the influence of numerical phase transition. We do this by means of the physically relevant configuration of laser-induced cavitation bubbles. One problem is the choice of initial data that cannot be directly deduced from the experiment. Therefore, another focus of the present work is to suggest a strategy how to determine initial data that are implicitly fitted to the time evolution of the measured bubble radius.

The outline is as follows. In Section 2 we summarize the stiffened gas model. This model is discretized by the SA approach as well as the rGFM summarized in Section 3. Then, in Section 4, we propose a strategy for the computation of the initial data fitted to experimental data that are provided by the collapse of a laser-induced spherical bubble. Finally, in Section 5, we present several numerical computations for two test cases corresponding to a small and a large equilibrium radius of the bubble. In particular, we compare the two different discretizations and validate them by the experimental data.

Section snippets

Mathematical model

A compressible fluid flow is characterized in continuum mechanics by the fields of density ρ, velocity v, internal energy e and pressure p. The balances of mass, momentum and energy for inviscid flow lead to the Euler equations. In spherical coordinates and employing rotational symmetry, they read $\begin{matrix} \frac{\partial}{\partial t} (r^{2} ρ) + \frac{\partial}{\partial r} (r^{2} (ρ v_{r})) = 0, \\ \frac{\partial}{\partial t} (r^{2} ρ v_{r}) + \frac{\partial}{\partial r} (r^{2} (ρ v_{r}^{2} + p)) = 2 pr, \end{matrix}$ $\frac{\partial}{\partial t} (r^{2} ρ E) + \frac{\partial}{\partial r} (r^{2} (ρ v_{r} (E + p / ρ))) = 0,$ where $E = e + \frac{1}{2} v_{r}^{2}$ is the total energy and v_r the radial velocity.

In order to close the system, we have to provide a pressure

Discretization

In order to discretize the inviscid flow equations (1) and the evolution equations of the phase indicator function (3), (6), respectively, we will employ the Saurel–Abgrall approach [20] and the real ghost fluid method [27]. These two approaches have in common that the flow equations for the conserved quantities are approximated by a second-order finite volume scheme in space and an upwind discretization is used for the non-conservative transport equations of the phase indicator. Due to the

Computation of initial data

In order to start the computation we need initial data not only for the liquid state but also for the medium inside the bubble. However, it is not yet possible to measure the pertaining variables inside the bubble. Therefore, we propose an indirect way to obtain the initial pressure and temperature in the bubble from the experimentally measured radius vs. time curve. The solution of a second-order ODE (the Keller–Miksis model) describing the radial bubble dynamics is fitted to the measured data

Results

We investigate the collapse of a spherical bubble using the real ghost fluid method and the Saurel–Abgrall approach. This configuration can be considered as a benchmark problem to validate modeling, discretization as well as implementation, because the results can be compared with experiments and simplified 1D models. Since the flow field is inherently one-dimensional, we would like to emphasize that in 1D there are other models and discretizations available that might give better results for

Conclusion

The SA approach and the rGFM have been applied to investigate the spherical bubble collapse of laser-induced cavitation bubbles. Initial data are derived by first fitting the equilibrium radius in the Keller–Miksis model to minimize the least-squares error of the model solution relative to the experimental data on the bubble dynamics. Then the gas state for an arbitrary bubble radius is deduced applying an ideal and adiabatic gas law.

The CFD computations verify that the SA approach results in a

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^☆: This work has been performed with funding by the Deutsche Forschungsgemeinschaft in the DFG-CNRS-Forschergruppe FOR 563 “Micro-Macro Modeling and Simulation of Liquid–Vapor Flows”.

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Comparison and validation of compressible flow simulations of laser-induced cavitation bubbles☆

Abstract

Introduction

Section snippets

Mathematical model

Discretization

Computation of initial data

Results

Conclusion

J Comp Phys

Comput Fluids

J Comp Phys

J Comp Phys

J Comp Phys

J Comp Phys

J Comp Phys

J Comp Phys

J Comp Phys

Collapse and rebound of a laser-induced cavitation bubble

Phys Fluids

Practical computation of axisymmetrical multifluid flows

Int J Finite Volumes