# Discretization of Governing Equations

The transport of chemical species is modeled by the mass, momentum, energy and species balance equations. These gas-dynamics and chemistry
governing equations are solved herein using a fully decoupled implicit algo­rithm. Further discussions on the coupled vs. decoupled algorithms for com­bustion problems can be found in [Eberhardt and Brown, 1986, Yee, 1987, Balakrishnan, 1987, Li, 1987]. A correction technique has been developed to enforce the balance of mass fractions. The governing equations are dis­cretized using an implicit, approximate-factorization, finite difference scheme in delta form [Warming and Beam, 1978]. The discretized operational form of both the Reynolds-averaged Navier-Stokes (RANS) and species conservation equations, combined in a Newton-Raphson algorithm [Rai and Chakravarthy,

1986], is:

where A and B are the fhx Jacobian matrices A = dF/dQ, B = dG/dQ. The Y and C matrices are Y = dS/dQ and C = dSch/dQ. Note that the flix Jacobian matrices are split into A = A+ + A-, where A± = PA±P-1. Л is the spectral matrix of A and P is the modal matrix of A. The spectral matrix Л is split into Л = Л+ + Л-, where the components of Л+ and Л – are A- = 0.5(Aj — |AiI) and A+ = 0.5(Ai + |AjI), respectively [Steger and Warming, 1981]. The same flux vector splitting approach is applied to the matrix B. In equation (5), A, V and 5 are forward, backward and central differences operators, respectively. Qp is an approximation of Q”+1. At any time step n, the value of Qp varies from Qn at first internal iteration when p = 0, to Qn+1 when integration of equation (5) has converged. Additional details on the implementation of the inter-cell numerical flixes and on the Roe’s approximate Riemann solver are presented in [Cizmas et al., 2003].