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APECSS is a software toolbox to compute pressure-driven bubble dynamics and the resulting acoustic emissions. It is written in C and has been developed with simplicity, versatility and performance in mind. The acronym APECSS stands for "Acoustic Pulse Emitted by Cavitation in Spherical Symmetry".
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@@ -46,5 +59,16 @@ There are several ways in which you can use the APECSS library. You can either i
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- A ````build```` folder containing the ````CMakeLists.txt```` file and a shell script ````compile.sh```` with which this example can be compiled using the command ````./compile.sh````.
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- One or several ````*.apecss```` files in which the options for a specific case are defined.
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## How to cite us
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If you use APECSS for your scientific work, please consider citing the [paper](https://doi.org/10.1063/5.0131930) introducing the theoretical foundation of APECSS
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F. Denner and S. Schenke, Modeling acoustic emissions and shock formation of cavitation bubbles. Physics of Fluids 35 (2023).
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as well as the version of APECSS you've used for your work, e.g.
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F. Denner and S. Schenke, APECSS (v1.2), (2022). https://doi.org/10.5281/zenodo.7465050
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All releases can be found on the [Zenodo page](https://doi.org/10.5281/zenodo.7249297).
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## Acknowledgements
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The development of APECSS has directly benefitted from research funding provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant number 441063377.
{\tt BUBBLE} & {\tt Emissions IC <float>} & Computes the acoustic emissions under the common incompressible assumption.\\
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& {\tt Emissions FTIC <float>} & Computes the acoustic emissions under the assumption of an incompressible fluid but propagating the emissions with the speed of sound.\\
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& {\tt Emissions QA <float>} & Computes the acoustic emissions using the quasi-acoustic model of \citet{Gilmore1952}.\\
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& {\tt Emissions EV <float>} & Computes the acoustic emissions based on the Kirkwood-Bethe hypothesis, with the explicit expression for velocity.\\
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& {\tt Emissions SIV <float>} & Computes the acoustic emissions using the model of \citet{Gilmore1952} based on the Kirkwood-Bethe hypothesis, with the spatially-integrated velocity.\\
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& {\tt Emissions TIV <float>} & Computes the acoustic emissions using the model of \citet{Hickling1963} based on the Kirkwood-Bethe hypothesis, with the temporally-integrated velocity.\\
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{\tt BUBBLE} & {\tt Emissions IC <float>} & Computes the acoustic emissions using the standard incompressible model, Section \ref{sec:emissionsic}.\\
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& {\tt Emissions FSIC <float>} & Computes the acoustic emissions using the finite-speed incompressible model, Section \ref{sec:emissionsfsic}.\\
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& {\tt Emissions QA <float>} & Computes the acoustic emissions using the quasi-acoustic model, Section \ref{sec:emissionsqa}.\\
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& {\tt Emissions EV <float>} & Computes the acoustic emissions based on the Kirkwood-Bethe hypothesis, Section \ref{sec:emissionskb}, with the explicit expression for velocity, see Eq.~\eqref{eq:u_rt}.\\
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& {\tt Emissions SIV <float>} & Computes the acoustic emissions using the model of \citet{Gilmore1952} based on the Kirkwood-Bethe hypothesis, Section \ref{sec:emissionskb}, with the spatially-integrated velocity, see Eq.~\eqref{eq:dudr_rt}.\\
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& {\tt Emissions TIV <float>} & Computes the acoustic emissions using the model of \citet{Hickling1963} based on the Kirkwood-Bethe hypothesis, Section \ref{sec:emissionskb}, with the temporally-integrated velocity, see Eq.~\eqref{eq:dudt_rt}.\\
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& {\tt EmissionIntegration Euler} & Integrates the radial position and, if applicable, the velocity using an Euler scheme.\\
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& {\tt EmissionIntegration RK4} & Integrates the radial position and, if applicable, the velocity using a conventional fourth-order Runge-Kutta scheme. This is the default.\\
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& {\tt KBIterTolerance <float>} & Tolerance $\eta$ for the evaluation of the pressure using a model based on the Kirkwood-Bethe hypothesis in conjunction with the NASG EoS.\\
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\hline
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\end{tabular} \vspace{0.2em}
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The floating-point value associated with the emissions defines the cut-off distance beyond which the emissions are not computed. For the incompressible assumption this value has no meaning, but a value is required as a dummy to facilitate the correct reading of the options.
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The floating-point value given as the final argument of the {\tt Emissions} command defines the cut-off distance beyond which the emissions are not computed. For the standard incompressible assumption this value has no meaning, but a value is required as a dummy to facilitate the correct reading of the options.
Assuming an incompressible liquid ($c_{\ell,\mathrm{ref}} \rightarrow\infty$) with density $\rho_{\ell,\mathrm{ref}}$, the velocity $u(r,t)$ and pressure $p(r,t)$ at a given radial position $r(t)$ are defined as \citep{Neppiras1980}
respectively. The assumption of an incompressible fluid is consistent with the Rayleigh-Plesset models in Eqs.~\eqref{eq:standardRP} and \eqref{eq:modRP}. Note that, because $\mathcal{C} \rightarrow\infty$, these simple incompressible acoustic emissions do not use the Lagrangian wave tracking and no emission nodes are defined and processed, since pressure and velocity are defined instantaneously for all $r$.
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Alternatively, APECSS also supports the assumption that the liquid is incompressible but the information associated with the acoustic emissions still propagates with finite speed $\mathcal{C} = c_{\ell,\mathrm{ref}}$ using the Lagrangian wave tracking approach.
This approach, referred to in APECSS as {\tt FTI} or \text{finite-time incompressible}, accurately recovers the time delay between emitting information at the bubble wall and this information arriving in a certain location.
and $\tau$ is the time at which the acoustic information is emitted at the bubble wall. For $t=\tau$ with $r(t)=R(\tau)$, Eq.~(\ref{eq:u_rt_qa}) reduces to $u(R,\tau)=\dot{R}(\tau)$ and Eq.~(\ref{eq:p_rt_qa}) reduces to $p(R,\tau)=p_\mathrm{L}(\tau)$, thus satisfying the boundary conditions at the bubble wall.
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The quasi-acoustic model is consistent in its modelling assumptions with the Keller-Miksis model, Eq.~\eqref{eq:keller}. The applicability of the quasi-acoustic model is limited to small Mach numbers, $(\dot{R}/c_0)^2\ll1$, as it incorporates a finite propagation speed of the acoustic emissions and the nonlinear pressure contributions resulting from the flow, but since all parts of the wave propagate with speed $c_0$, the quasi-acoustic model can neither describe the nonlinear distortion of acoustic waves nor the formation of shock fronts.
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\section{Finite-speed incompressible model}
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\label{sec:emissionsfsic}
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APECSS also supports the assumption that the liquid is incompressible but the information associated with the acoustic emissions still propagates with finite speed $\mathcal{C} = c_{\ell,\mathrm{ref}}$ using the Lagrangian wave tracking approach, with the radial location given by Eq.~\eqref{eq:r_t_qa}.
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By assuming the specific acoustic impedance of an incompressible liquid, $\rho_{\ell,\mathrm{ref}} c_{\ell,\mathrm{ref}} \rightarrow\infty$, the velocity reduces to
while $p(r,t)$ and $g(\tau)$ are given by Eq.~\eqref{eq:p_rt_qa} and Eq.~\eqref{eq:g_qa}, respectively.
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This approach, referred to in APECSS as {\tt FSIC} or \text{finite-speed incompressible model}, recovers the time delay between emitting information at the bubble wall and this information arriving in a certain location, but treats the flow field as incompressible.
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\section{Emissions based on the Kirkwood-Bethe hypothesis}
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\label{sec:emissionskb}
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Following a similar derivation as for the quasi-acoustic model discussed in Section \ref{sec:emissionsqa}, but assuming a fully-compressible liquid described by a suitable equation of state, the {\it explicit velocity (EV)} is given as
For $t=\tau$ with $r(t)=R(\tau)$, this expression reduces to $u(R,\tau)=\dot{R}(\tau)$, thus satisfying the boundary conditions at the bubble wall.
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\citet{Gilmore1952} proposed instead to solve for the spatial derivative of the velocity along the outgoing characteristic, in APECSS referred to as {\it spatially-integrated velocity (SIV)}, defined as
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\end{equation}
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If either Eq.~\eqref{eq:dudr_rt} or Eq.~\eqref{eq:dudt_rt} is chosen to determine the velocity, this differential equation for the velocity is integrated together with the equation for ${\mathrm{d}r(t)/\mathrm{d}t}$, Eq.~\eqref{eq:drdt}, using the initial condition $u(R,\tau) = \dot{R}(\tau)$. Note that with $\mathrm{d}r(t)/\mathrm{t}$ defined by Eq.~\eqref{eq:drdt} and $g(\tau)$ given by Eq.~\eqref{eq:g_R}, Eqs.~\eqref{eq:dudr_rt} and \eqref{eq:dudt_rt} are interchangeable by the relation ${\mathrm{d}u/\mathrm{d}t} = ({\mathrm{d}u/\mathrm{d}r}) \, ({\mathrm{d}r/\mathrm{d}t})$.
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Regardless of the choice of velocity model, $f(\tau)$ and$g(\tau)$are defined as
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Regardless of the choice of velocity model, the invariant$g(\tau)$is defined as
For a given radial position $r(t)$ and flow velocity $u(r,t)$, irrespective of which model is used to compute the velocity, the enthalpy is then readily evaluated as
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For a given radial position $r(t)$ and flow velocity $u(r,t)$, irrespective of which model is used to compute the velocity, the enthalpy is readily evaluated as
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