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Numerical investigation of rising bubbles bursting at a free surface through a multiphase SPH model

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Abstract

As a Lagrangian meshless method, smoothed particle hydrodynamics (SPH) method is robust in modelling multi-fluid flows with interface fragmentations. However, the application for the simulation of a rising bubble bursting at a fluid surface is rarely documented. In this paper, the multiphase SPH model is extended and applied to simulate this challenging phenomenon. Different numerical techniques developed in different SPH models are combined in the present SPH model. The adoption of a background pressure determined based on the surface tension can help to avoid tensile instability and interface penetrations. An accurate surface tension model is employed. This model is suitable for bubble rising problems of small scales and high density ratios. An interface sharpness force is adopted to achieve a smoother bubble surface. A suitable formula of viscous force, which is proven to be able to accurately capture the bubble splitting and small bubble detachment, is employed. Moreover, a modified prediction-correction time-stepping scheme for a better numerical stability and allows a relatively larger CFL factor is adopted. It is also worthwhile to mention that the particle shifting technique, which helps to make the particle distribute in an arrangement of lower disorder, can significantly improve the numerical accuracy. Regarding the treatment of the fluid surface, particles of lighter phase are arranged above the free surface of the denser phase to avoid the kernel truncation in the density approximation. Furthermore, this technique also allows an accurate calculation of the surface tension on the fluid surface. A number of cases of bubbly flows are presented, which confirms the capability of the present multiphase SPH model in modelling complex bubble-surface interactions with the density ratio and viscosity ratio up to 1000 and 100 respectively.

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Abbreviations

\(m\) :

Mass

\(V\) :

Volume

\(\varvec{r}\) :

Position vector

\(\varvec{u}\) :

Velocity vector

\(p\) :

Pressure

\(\varvec{g}\) :

Gravity acceleration

\(W\) :

Kernel function

\(h\) :

Smoothing length

\(\Delta x\) :

Initial particle spacing

\(c\) :

Artificial speed of sound

\(\Delta P\) :

Expected pressure variation

\(U_{\hbox{max} }\) :

Expected maximum velocity

\(H_{ini}\) :

Undisturbed fluid depth

\(\varvec{u}_{\hbox{max} }\) :

Real time maximum velocity

\(\delta \varvec{r}\) :

Particle shifting displacement

\(t\) :

Time

\(\Delta p\) :

Pressure difference

\(\kappa\) :

Interface curvature

\(\hat{\varvec{n}}\) :

Unit surface normal vector

\(R\) :

Initial bubble radius

\(D\) :

Initial bubble diameter

\(W_{f}\) :

Width of the fluid domain

\(H_{f}\) :

Height of the fluid domain

\(p_{b}\) :

Background pressure

\(d\) :

Spatial dimension

\(\Delta t\) :

Time increasement

\(x,\text{ }y\) :

Cartesian coordinates

\(u,\text{ }v\) :

Velocity components

\(C,\text{ }\varphi\) :

Color function

\(\begin{aligned} x^{ * } & = {x \mathord{\left/ {\vphantom {x D}} \right. \kern-0pt} D}, \\ y^{ * } & = {y \mathord{\left/ {\vphantom {y D}} \right. \kern-0pt} D} \\ \end{aligned}\) :

Dimensionless Cartesian coordinates

\(\begin{aligned} u^{ * } & = {u \mathord{\left/ {\vphantom {u {\sqrt {gD} }}} \right. \kern-0pt} {\sqrt {gD} }},\text{ } \\ v^{ * } & = {v \mathord{\left/ {\vphantom {v {\sqrt {gD} }}} \right. \kern-0pt} {\sqrt {gD} }} \\ \end{aligned}\) :

Dimensionless velocity components

\(t^{ * } = t\sqrt {g/D}\) :

Dimensionless time

\(U = \sqrt {gD}\) :

Characteristic velocity

\(Re = {{\rho_{l} D\sqrt {gD} } \mathord{\left/ {\vphantom {{\rho_{l} D\sqrt {gD} } {\eta_{l} }}} \right. \kern-0pt} {\eta_{l} }}\) :

Reynolds number

\(Bo = {{\rho_{l} gD^{2} } \mathord{\left/ {\vphantom {{\rho_{l} gD^{2} } \sigma }} \right. \kern-0pt} \sigma }\) :

Bond number

\(Mo = {{g\eta_{l}^{4} } \mathord{\left/ {\vphantom {{g\eta_{l}^{4} } {\rho_{l} \sigma^{3} }}} \right. \kern-0pt} {\rho_{l} \sigma^{3} }}\) :

Morton number

\({\text{We}} = {{\rho_{l} DU^{2} } \mathord{\left/ {\vphantom {{\rho_{l} DU^{2} } \sigma }} \right. \kern-0pt} \sigma }\) :

Weber number

\(Fr = {U \mathord{\left/ {\vphantom {U {\sqrt {gD} }}} \right. \kern-0pt} {\sqrt {gD} }}\) :

Froude number

\(\rho\) :

Density

\(\Delta \rho\) :

Change of the density

\(\rho_{0}\) :

Reference density at rest

\(\nabla\) :

Gradient operator

\(\sigma\) :

Surface tension coefficient

\(\eta\) :

Dynamic viscosity

\(\omega\) :

Vorticity

\(p_{ave}\) :

Average pressure

\(\omega^{*} = \omega \sqrt {D/g}\) :

Dimensionless vorticity

\(i,\text{ }j\) :

Particle index

\(l\) :

Denser fluid phase

\(g\) :

Lighter fluid phase

\(V\) :

Viscous stress

\(I\) :

Interface sharpness force

\(B\) :

Body force

\(S\) :

Surface tension

\(k,\text{ }l,\text{ }q\) :

Fluid phase index

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Acknowledgements

This work is supported by the fundamental research funds for central universities (HEUCFD1421), the China Scholarship Council (CSC, Grant No. 201506680004), the Natural Science Foundation of China (Grant No. 51609049), the Natural Science Foundation of Heilongjiang (Grant No. QC2016061), the China Postdoctoral Science Foundation (Grant No. 2015M581432) and CNR-INSEAN within the Project PANdA: PArticle methods for Naval Applications, protocol number N. 3263, 21 October 2014.

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Authors and Affiliations

  1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, China

    F. R. Ming, P. N. Sun & A. M. Zhang

  2. CNR-INSEAN, Marine Technology Research Institute, Rome, Italy

    P. N. Sun

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  1. F. R. Ming
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Correspondence to P. N. Sun.

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Ming, F.R., Sun, P.N. & Zhang, A.M. Numerical investigation of rising bubbles bursting at a free surface through a multiphase SPH model. Meccanica 52, 2665–2684 (2017). https://doi.org/10.1007/s11012-017-0634-0

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