A globally well-posed accurate and efficient finite element CFD algorithm for compressible aerodynamics

For the reliable determination of aerodynamic flows, a minimally dissipative generalized curvilinear-coordinate flux vector splitting implicit algorithm is developed within an optimal finite element integral statement for the two-dimensional (2-D) and axisymmetric compressible Navier-Stokes equations for ideal and reacting gases. Hypersonic-flow dissociated real air is modelled via a neutral mixture of five species, i.e. nitric oxide, molecular and atomic oxygen and nitrogen. The associated thermodynamic state is then determined via an original solution procedure for the classical chemical equilibrium equation system. The analysis develops the curvilinear-coordinate flux vector jacobian matrices and associated characteristic equations. For any equation of state, the eigenvalues are the contravariant components of velocity in combination with the sound speed. Hence, the flow wave propagation in curvilinear space is rigorously represented using a tensor-field resolution of the flux vector into essentially kinematic and kinetic components. The streamwise semi-discretization is achieved using a Lyapunov-stable, Galerkin finite element integral statement for an eigenvalue-based “companion conservation law system", permitting classical test and trial space definitions. Thereafter, the accurate computation of the unsteady flow evolution is attained utilizing an implicit Runge-Kutta time-integration scheme, which is non-linearly stable and 2^nd order accurate. Subsequently, a matrix tensor product factorization permits efficient numerical linear algebra handling for large Courant number. For 2-D and axisymmetric transonic, supersonic, and hypersonic inviscid and viscous flows, including real gas effect simulation, the algorithm is verified to yield reliable determination of velocity, pressure and temperature fields by generating accurate and monotone, essentially non-oscillatory numerical solutions in the presence of weak and strong shocks, attached and detached, and in boundary layer-inviscid flow interactions.