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(indicated in the following by SGS). The governing equations are discretized using a cell- centered finite volume scheme for structured multi-block meshes of third-order accuracy, which allows computing flows governed by an arbitrary equation of state [39]. The scheme is constructed by correcting the dispersive error term of the second-order-accurate Jameson’s scheme [66]. The use of a scalar dissipation term simplifies the scheme implementation with highly complex equations of state and greatly reduces computational costs. In order to preserve the high accuracy of the scheme on non-Cartesian grids, the numerical fluxes are evaluated using weighted discretization formulas, which take into account the stretching and the skewness of the mesh: this ensures truly third-order accuracy on moderately deformed meshes and at least second-order accuracy on highly distorted meshes (see [67] for details). The equations are then integrated in time using a four-stage Runge-Kutta scheme [66]. Local time stepping, implicit residual smoothing and multigrid are used to efficiently drive the solution to the steady state. For external flows, non-reflecting boundary conditions based on a multidimensional method of characteristic are applied at the far-field boundaries; an adiabatic wall condition is imposed at solid boundaries. The accuracy properties of the numerical solver just described have been demonstrated in previous works [39][40], and will not be discussed further. The numerical method is extended to the Navier-Stokes Equations using a classical second-order discretization of the viscous term. 2.3.2. Unstructured solver An unstructured CFD solver (developed in the Sinumef Lab at the Ensam in Paris) has been modified to simulate Dense Gases (see [43])(indicated in the following by UGS). This allows a cross validation concerning the numerical results by comparing the solution to those obtained with the structured code. This unstructured solver (UGS) is also based on a cell-centered finite-volume discretization of (Eq. 5) but formulated on a general unstructured grid dividing the spatial domain into a finite number of triangles or quadrangles; the time rate of change of the cell-averaged state vector w is balanced with the area-averaged (inviscid) fluxes across the cell faces. The fluxes are computed across each cell face using the HLL scheme [68]; second-order spatial accuracy is ensured thanks to a MUSCL-type reconstruction process on the conserved variables [69], where the gradient estimates required at each cell center are obtained through a least-square formula. Since the solver will be applied to the computation of flows containing discontinuities, the 39PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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