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Analysis of the critical pressure of cavitation bubbles

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Abstract

If a liquid pressure drops below the critical pressure of the cavitation nuclei—typically tiny bubble of non-condensable gas—these ones expand rapidly because of intense vaporization and become vapour bubbles. The concentration of cavitation nuclei significantly influences the critical pressure where cavitation is initiated, which is corroborated by previous studies showing that the interaction phenomena between bubbles are important at the early stage of cavitation inception. However, prediction of the critical pressure of cavitation nuclei, considering the interaction of other germs, is still beyond the current state of the art. In this work, a simplified analysis of the theoretical formulation of critical pressure for multiple bubbles, taking into account the interactions between bubbles, is conducted. The critical pressure derived explicitly from the Rayleigh–Plesset equation—using either a quasi-static or a linear dynamic approach—is compared with the inception pressure obtained from the numerical model proposed previously in Adama Maiga (Int J Eng Res 8(4):183–213, 2015). The comparison shows a significant disagreement in case the quasi-static is applied, whereas a fair agreement is obtained if the linear dynamic is used. The results also show that increasing the polytropic exponent \(\gamma\), i.e. the effect of temperature, delays the bubble nucleation. Conversely, the Reynolds and Weber numbers based on the average bubble radius are not influenced by \(\gamma\). The study also confirms that the inter-bubble distance \(D_{ij}\) significantly influences the critical pressure, i.e. the process of bubble nucleation. The critical Reynolds and Weber number, which trigger the activation of the small bubbles, depend significantly on the number of interacting bubbles.

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Abbreviations

\(i\) :

Index \(i\) corresponds to the bubble \(i\) (−)

\(j\) :

Index \(j\) corresponds to the bubble \(j\) (−)

\(N_{b}\) :

Number of bubbles (−)

\(t\) :

Time (s)

\(\Delta t\) :

Time step (s)

\(R_{i}\) :

Radius of bubble \(i\) (m)

\(R_{0i}\) :

Initial radius of bubble \(i\) (m)

\(R_{moy}\) :

Average bubble radius (m)

\(R_{{cs_{i} }}\) :

Critical radius of bubble \(i\) (m)

\(\dot{R}_{i}\) :

Velocity normal to interface of bubble \(i\) (m/s)

\(\ddot{R}_{i}\) :

Interface acceleration (m/s2)

\(q_{i} = 4\pi R_{i}^{2} \dot{R}_{i}\) :

Flow rate at the interface of bubble \(i\) (m3/s)

\(\dot{q}_{i} = \frac{{\partial q_{i} }}{\partial t} = 4\pi \left( {2R_{i} \dot{R}_{i}^{2} + R_{i}^{2} \ddot{R}_{i} } \right)\) :

Time derivative of the flow rate (m3/s2)

\(D_{ij}\) :

Distance between the centers of bubbles \(i\) and \(j\) (m)

\(D_{0}\) :

Constant that links D ij and the initial radii of bubbles \(i\) and \(j\) (−)

\(\sigma\) :

Surface tension (N/m)

\(\mu_{L}\) :

Dynamic viscosity of the liquid (pa s)

\(\rho_{L}\) :

Density of the liquid (kg/m3)

\(p_{ex} \left( t \right)\) :

Ambient pressure (pa)

\(p_{i}\) :

Liquid pressure at the interface of bubble i (pa)

\(p_{0}\) :

Atmospheric pressure (pa)

\(p_{v}\) :

Vapor pressure (pa)

\(p_{{g_{i} }}\) :

Gas pressure in bubble I (pa)

\(p_{{cs_{i} }}\) :

Critical pressure for bubble I (pa)

\(\gamma\) :

Polytropic exponent (−)

\(\varphi = \left( {\vec{y},\overrightarrow {OM} } \right)\) :

Angular position of the distributor (where O is located at the distributor is axis and M is at its periphery) (rad)

\(X_{dis}\) :

Abscissa of M (m)

\(X_{clo}\) :

Distributor travel for \(X_{dis} < 0\) (m)

\(X_{ope}\) :

Distributor travel for \(X_{dis} > 0\) (m)

\(X_{ove}\) :

Overlap length (m)

\(H_{gap}\) :

Gap length (m)

\(H_{hei}\) :

Chamber height (m)

\(L_{dis}\) :

Distributor length in the z direction (m)

\(V_{ch}\) :

Chamber volume (m3)

\(R_{dis}\) :

Distributor radius (m)

\(R_{cra}\) :

Crank shaft radius (m)

\(S_{pis}\) :

Piston section (m2)

\(Q_{dis}\) :

Flow rate in the distributor (m3/s)

\(Q_{pis}\) :

Piston flow rate (m3/s)

\(\omega\) :

Angular velocity (rad/s)

\(\varOmega\) :

Rotational speed (rpm)

\(p_{c}\) :

Chamber pressure (pa)

\(p_{pump}\) :

Pressure at the pump outlet (pa)

\(\delta p_{clo} /\delta p_{ope}\) :

Pressure drop if the distributor is closed/open (pa)

\(\delta p\) :

Pressure drop (pa)

\({\text{Re}}\) :

Reynolds number (−)

\({\text{We}}\) :

Weber number (−)

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Authors and Affiliations

  1. Arts et Metiers ParisTech, 8 bld Louis XIV, 59046, Lille, France

    Mahamadou Adama Maiga & Olivier Coutier-Delgosha

  2. Kevin T. Crofton Aerospace and Ocean Engineering Department, Virginia Tech, 460 Old Turner Street, Blacksburg, VA, 24061, USA

    Olivier Coutier-Delgosha

  3. Mechanic Department, Université de Lille, 59650, Villeneuve d’Ascq, France

    Daniel Buisine

Authors
  1. Mahamadou Adama Maiga
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  2. Olivier Coutier-Delgosha
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  3. Daniel Buisine
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Correspondence to Mahamadou Adama Maiga.

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Adama Maiga, M., Coutier-Delgosha, O. & Buisine, D. Analysis of the critical pressure of cavitation bubbles. Meccanica 53, 787–801 (2018). https://doi.org/10.1007/s11012-017-0778-y

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