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reconstruction formula includes a limitation step based on the now standard approach proposed in [70] and revisited in [71] which ensures oscillation-free shock-capturing. Fast convergence to steady state is provided by making the scheme implicit following a procedure inspired from [72]: a simple first-order Rusanov-type implicit stage allows the use of large CFL numbers and is solved by an inexpensive point-relaxation technique. The numerical flux through the boundary edges is computed using an inflow / outflow characteristic-based condition to define the ghost-cell states at the far-field boundary and a mirror boundary condition to define the ghost-cell states at the wall. 2.3.3. How a more complex equation of state affects a NS code Using a perfect gas equation of state is straightforward because of the direct relation between the pressure and the internal energy. The van der Waals equation is more complicated of the PGF equation of state, but it’s always possible to write a direct relation between pressure and internal energy (for details see [43]). But, as seen in 2.2.1, VDW equation overestimate the inversion region, and it’s not accurate for describing the variation of Γ. Then the Martin-Hou equation has to be used, but in this case it’s not possible to derive an explicit relation between pressure and internal energy. Generally, an iterative method is necessary to compute thermodynamic properties (for example to compute the primitive variables by starting from the conservative variables). Newton- Raphson Method is used because of its good accuracy and convergence velocity (For details in the implementation see [43]). Generally, if a different equation of state is used, it’s necessary to change all the formulations that adopt implicitly a PFG equation of state. Obviously it’s taken for granted that the computation of thermodynamic properties changes starting from different thermodynamic relations (typically sound of speed and internal energy). For a very detailed description of the main thermodynamic properties with the different equation of state see [43]. For what concerns the solvers used in this work, the numerical schemes formulations used in SGS (2.3.1) and in UGS (2.3.2) are independent from the equation of state. A different formulation for the non-reflecting boundary conditions based on method of characteristic has been used (typically the computation of Riemann invariants changes, because it is usually written by considering a PFG equation of state). 40PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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