Commit 2d4599dd by Matthieu Schaller

### Wrote the section of the documentation on the EoS.

parent 1a8f00d5
 aa In \swift, all transformations between thermodynamical quantities are computed using a pre-defined equation of state (EoS) for the gas. This allows user to switch EoS without having to modify the equations of SPH. The definition of the EoS takes the form of simple relations between thermodynamic quantities. All of them must be specified even when they default to constants. %####################################################################################################### \subsection{Ideal Gas} \label{sec:eos:ideal} This is the simplest and most common equation of state. It corresponds to a gas obeying a pressure law $P = (\gamma-1) \rho u$, where $u$ is the gas' internal energy, $\rho$ its density and $\gamma$ the (user defined) adiabatic index, often assumed to be $5/3$ or $7/5$. For such a gas, we have the following relations between pressure ($P$), internal energy ($u$), density ($\rho$), sound speed ($c$) and entropic function ($A$): \begin{align} P &= P_{\rm eos}(\rho, u) \equiv (\gamma-1) \rho u \label{eq:eos:ideal:P_from_u} \\ c &= c_{\rm eos}(\rho, u) \equiv \sqrt{\gamma (\gamma-1) u} \label{eq:eos:ideal:c_from_u} \\ A &= A_{\rm eos}(\rho, u) \equiv(\gamma-1) u \rho^{\gamma-1} \label{eq:eos:ideal:A_from_u} \\ ~ \nonumber\\ P &= P_{\rm eos}(\rho, A) \equiv A \rho^{\gamma} \label{eq:eos:ideal:P_from_A} \\ c &= c_{\rm eos}(\rho, A) \equiv \sqrt{\gamma \rho^{\gamma-1} A} \label{eq:eos:ideal:c_from_A} \\ u &= u_{\rm eos}(\rho, A) \equiv A \rho^{\gamma-1} / (\gamma-1) \label{eq:eos:ideal:u_from_A} \end{align} Note that highly optimised functions to compute $x^\gamma$ and other useful powers of $\gamma$ have been implemented in \swift for the most commonly used values of $\gamma$. %####################################################################################################### \subsection{Isothermal Gas} \label{sec:eos:isothermal} An isothermal equation of state can be useful for certain test cases or to speed-up the generation of glass files. This EoS corresponds to a gas with contant (user-specified) thermal energy $u_{\rm cst}$. For such a gas, we have the following (trivial) relations between pressure ($P$), internal energy ($u$), density ($\rho$), sound speed ($c$) and entropic function ($A$): \begin{align} P &= P_{\rm eos}(\rho, u) \equiv (\gamma-1) \rho u_{\rm cst} \label{eq:eos:isothermal:P_from_u} \\ c &= c_{\rm eos}(\rho, u) \equiv \sqrt{\gamma (\gamma-1) u_{\rm cst}} = c_{\rm cst} \label{eq:eos:isothermal:c_from_u} \\ A &= A_{\rm eos}(\rho, u) \equiv(\gamma-1) u_{\rm cst} \rho^{\gamma-1} \label{eq:eos:isothermal:A_from_u} \\ ~ \nonumber\\ P &= P_{\rm eos}(\rho, A) \equiv (\gamma-1) \rho u_{\rm cst} \label{eq:eos:isothermal:P_from_A} \\ c &= c_{\rm eos}(\rho, A) \equiv \sqrt{\gamma (\gamma-1) u_{\rm cst}} = c_{\rm cst} \label{eq:eos:isothermal:c_from_A} \\ u &= u_{\rm eos}(\rho, A) \equiv u_{\rm cst} \label{eq:eos:isothermal:u_from_A} \end{align}
 ... ... @@ -241,7 +241,7 @@ The prediction step is also identical to the \MinimalSPH case with the entropic function replacing the thermal energy. \begin{align} \vec{x}_i &\rightarrow \vec{x}_i + \vec{v}_i \Delta t \label{eq:sph:gadget2:drift__x} \\ \vec{x}_i &\rightarrow \vec{x}_i + \vec{v}_i \Delta t \label{eq:sph:gadget2:drift_x} \\ h_i &\rightarrow h_i \exp\left(\frac{1}{h_i} \frac{dh_i}{dt} \Delta t\right), \label{eq:sph:gadget2:drift_h}\\ \rho_i &\rightarrow \rho_i \exp\left(-\frac{3}{h_i} \frac{dh_i}{dt} ... ...
 ... ... @@ -183,11 +183,13 @@ All kernels available in \swift are shown on Fig.~\ref{fig:sph:kernels}. \includegraphics[width=\columnwidth]{kernels.pdf} \caption{The kernel functions available in \swift for a mean inter-particle separation $\langle x\rangle=1.5$ and a resolution $\eta=1.2348$. The corresponding kernel support radii $H$ (shown by arrows) and number of neighours $N_{\rm ngb}$ are indicated on the figure. A Gaussian kernel with the same smoothing length is shown for comparison. Note that all these kernels have the \emph{same resolution} despite having vastly different number of neighbours.} $\eta=1.2348$ shown in linear (top) and log (bottom) units to highlight their differences. The corresponding kernel support radii $H$ (shown by arrows) and number of neighours $N_{\rm ngb}$ are indicated on the figure. A Gaussian kernel with the same smoothing length is shown for comparison. Note that all these kernels have the \emph{same resolution} despite having vastly different number of neighbours.} \label{fig:sph:kernels} \end{figure} ... ...
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