# Non-reflecting boundary conditions for compressible Navier-Stokes equations

I have some questions about the implementation of non-reflecting OUTFLOW boundary condition for Navier Stokes equations. Following

1. Poinsot, Lele "Boundary Conditions for Direct Simulations of Compressible Viscous Flows"

2. Pirozzoli, Colonius "Generalized characteristic relaxation boundary conditions for unsteady compressible flow simulations"

both authors suggest an equation like

$$\frac{d\mathbf{u}_b}{dt} + \mathbf{d(u_b)} + \mathcal{\mathbf{T(u_b)}} = \mathbf{S(u_b)}\quad\quad (1)$$

where $\mathbf{u}_b = (\rho, \rho u, \rho v, \rho w, \rho e)$ are the conservative variable at the bound b; $\mathbf{d(u_b)}$ is a certain characteristic treatment of x-flux; $\mathcal{\mathbf{T(u_b)}}$ are the transverse therms; $\mathbf{S(u_b)}$ are the source therm.

Now consider a finite difference method and suppose that the bound is located at $n+1/2$ (as usual in this problems); considering furthermore $n+1, n+2,\dots, n+GN$ ghost points. Equation (1) can be applied just in $n+1/2$, so my questions are:

1. How can I discretise the derivatives in $n+1/2$ with some FD scheme in order to solve (1) at this location.
2. How can I treat the ghost nodes?
• The method you cite is for the compressible case, but the Navier-Stokes equations are generally considered incompressible. The method to choose is likely going to be very different: for the compressible case, I'd consider characteristic methods, whereas for the incompressible case I'd just set the normal component of the stress tensor to zero. – Wolfgang Bangerth Feb 25 '18 at 14:45
• Ok. In the case of compressible flows how can I treat ghost points if a charactheristic method is implemented at the bound point? – John Snow Feb 25 '18 at 20:08
• I don't know. I'm neither a finite difference expert nor do I work with these equations. I just wanted to point out the discrepancy. – Wolfgang Bangerth Feb 25 '18 at 22:58
• First of all, the term "Navier-Stokes equations" has been perfectly used in this context. There is no discrepancy. Compressibility condition has nothing to do w/ the momentum conservation (NSE) but mass conservation. Secondly you can extrapolate the solution at the exit boundary for variables that does not need a BC there in the supersonic case. Try different interpolation schemes and derive your own finite difference formula. The polynomial order will depend on the error bound you are considering in the discretisation. What BCs are you considering in which case? – HBR Feb 26 '18 at 7:46
• outflow boundary conditions – John Snow Feb 26 '18 at 9:26