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2. Governing equations and flow solvers 2.1. The Euler and Navier-Stokes equations for single- phase non reacting flows The integral conservation laws for mass, momentum, and energy may be written for a control volume fixed in space as follows (by neglecting the body force): dt wdΩ+ (f − fv)⋅ndS=0 (Eq.5) d∫∫∫ ∫∫ r Ω ∂Ω where w = (ρ ρ v ρE ) is the conservative variable vector, n is the outer normal to dΩ , f =(ρv pI+ρvv ρvH)T and f =(0 τ τ⋅v−q)T are respectively the inviscid and v the viscous part of the flux density, I is the unit tensor, v is the velocity vector, E is the specific total energy, H is the specific total enthalpy, τ = 2μ⎛∇V + ∇V T ⎞ − 2 μ(∇V )I is ⎜⎝ ⎟⎠3 the viscous stress tensor, and q = −k∇T the heat flux vector T being the absolute temperature. The viscosity μ can not be computed through the well-known Sutherland Law. For the computation of viscosity and of the thermal conductivity, the approach widely described in 2.2.2 is used. This system of equation is completed by the thermal and caloric equation of state (respectively of the form p = p(ρ,T ), and e = e(ρ,T )). There will be a detailed description of the equations of state in 2.2.1 2.1.1. Rans Equations Since the full resolution of the NS equation requires the consideration of a big range of length and time scales, the compressible Reynolds-averaged Navier–Stokes equations RANS equation are used to limit the computer costs (For a detailed description of these equations see [63]). The investigation of turbulent dense gas flows past an airfoil is undertaken, based on the following working hypotheses: (a) flow conditions are supposed 32PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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