Equations

Euler Equations

IBTFO advances the following set of 2D Euler equations for the conserved state vector: \(\mathbf{U} = (\rho, \rho u, \rho v,\rho E)\)

\[\]

mathbf{U}_t+mathbf{F}_x+mathbf{G}_y = 0,

mathbf{F} &=& (rho u, rho u^2 + p, rho u v,u ( E + p)),

mathbf{G} &=& (rho v, rho u v, rho v^2 + p ,v ( E + p)).

Here \(\rho, u, v\), and \(p\) are the density, x-velocity, y-velocity and pressure, respectively. \(E = e + (u^2+v^2) / 2\) is the total energy with \(e\) representing the internal energy.

In the code, We define the number of equations in ‘EQdefine.H’. We set it in the variable \(U(i,j,k)\) , where k represents different equation.

Some notes:

  • Regardless of the dimensionality of the problem, we always carry all 3 components of the velocity. You should always initialize all velocity components to zero, and always construct the kinetic energy with all three velocity components.

  • There are NADV advected quantities, which range from \({\tt UFA: UFA+nadv-1}\). The advected quantities have no effect at all on the rest of the solution but can be useful as tracer quantities.

  • There are NSPECIES species defined in the chemistry model, which range from \({\tt UFS: UFS+nspecies-1}\).

  • There are NAUX auxiliary variables, from \({\tt UFX:UFX+naux-1}\). The auxiliary variables are passed into the equation of state routines along with the species.