What are Roe States?
I'm trying to do a boundary condition modification on my Beam & Warming Euler implicit scheme. I'm being told to use a form of characteristic boundary conditions based on a flux difference formulation due to Barth, in the form:
Qb = 0.5*(Qinf +Qe) - 0.5*(sgn(A)*(Qinf-Qe))
with sgn(A)= Xsgn(lambda(A))X**(-1) evaluated at a Roe state between node point i and i+1. Do you happen to know what a Roe state is?
Wishing you luck, Joan
Re: What are Roe States?
I have been using boundary conditions on characteristics for quite a while but I never heard of a Roe state. It seems that your equation is a vector equation for the primitive variables, and inf and e apparently denotes the value of the variables outside and inside the computational domain respectively (i.e. exact values imposed from outside and computed values from inside). I am not familiar with the notation you use (sgn, etc..X, I guess lambda is related to the eigen values of the matrix A or so?).
What are your equations exactly? I guess momenta and density (if compressible), do you have an energy equation? what is the equation of state if any? I might try to help with explicitly writting down the characteristic equations for the flow in standard forms.
In practice for most of the problems a linearization of the equations has to be carried out in order to simplify the set of equations and solve for the eigenvector (characteristics) and egeinvalues (propagation velocities of the characteristics). The linearization has to be carried out 'around' (for example) the steady state solution of the flow or an approximation to it (for example a good guess of the solution, or an unperturbed state of the flow). This is what Roe state might refer to, but I am not sure, untill I have not seen the flow equations and the way the characteristics are found.
If this does not help, let me know more about the equations and the flow and then the characteristics can be found.
See also the two review articles:
Givoli, 1991, J. of Comput. PHys., 94, p.1.
Turkel, 1983, Comput. Fluids, 11, p.121
and if you can find it:
Roe, 1986, ICASE report 86-75, NASA Langley, Hampton, VA.
Re: What are Roe States?
Roe state is a special average state at the interface between node point i and i+1. Roe state for primitive variables U including velocity u, v and total enthalpy H, can be evaluated according to the following formula:
U_Roe = ( U_i * SQRT(rho_i) + U_i+1 * SQRT(rho_i+1) )/( SQRT(rho_i) + SQRT(rho_i+1) )
where rho is density. The Roe state for density is obtained by
rho_Roe = SQRT( rho_i * rho_i+1 )
you can also have a look of Roe's original paper on Roe average(state) at J. Computational Phy. Vol. 43, 1981.
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