Difference between compressible and incompressibe: Time derivative in continuity equation Incompressible $\nabla . \vec{v} = 0$ Compressible: $\frac{ \partial p }{ \partial t }+ \nabla.(\rho \vec{v}) = 0$ hyperbolic character, allowing for pressure (sound) waves Adding a time derivative to continuity equation Use pressure not density because pressure changes much more $\frac{1}{\beta}\frac{ \partial p }{ \partial t }+\frac{ \partial \rho u_{i} }{ \partial x_{i} }=0$ $\beta =$ artificial compressibility parameter $[\mathrm{vel^{2}}]$ - The larger tthe $\beta$, the more incompressible - Calculate lowest acceptable $\beta$ by requiring that sound waves propagate much faster than flow velocity and vorticity speeds to avoid artificial pressure waves Implicit euler in time: $\frac{p^{n+1}-p}{\beta\Delta t}+\left( \frac{ \partial (\rho u_{i}) }{ \partial x_{i} } \right)^{n+1}=0$ Velocity field at $n+1$ not known → coupled system idea 1: expand unknown $u_{i}$ with taylor series in pressure idea 2: ???