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M Abidat, M K Hamidou, M Hachemi And M Hamel Where ρ is the density and U→ is the mean velocity vector. • Momentum conservation equations : → ∂ρU → → → → (08) →→→ Where u is the fluctuating velocity vector, τ the molecular stress tensor, ρ u⊗ u the Reynolds stress tensor and SM a source term. • Energy conservation equation : ∂(ρH)→→ ∂p ∂t + ∇.( ρ U H - ρ u h - λ ∇ T ) = ∂t (09) In this equation, H is the mean total Enthalpy given by 1→2 H=h+2U +k (10) Where h is the static enthalpy, λ is the thermal conductivity, T is the mean static temperature and p is the static pressure. The addition term k is the turbulent kinetic energy defined as: 1→2 k=2u (11) Temperature, pressure and density are related by the equation of state: p = ρ R T TheturbulenceismodelledbytheRNGk−ε modelwhichisbasedonrenormalizationgroup analysis of the Navier-Stokes equations. The transport equations for turbulence generation and dissipation are the same as those for the standard k-ε model. The model is also based on the eddy viscosity concept which assumes that the Reynolds stresses − ρuiuj can be expressed in terms of the mean velocity gradients and the eddy or turbulent viscosity μt in a manner analogous to the viscous stresses τij for laminar Newtonian flows. (12) ∂t +∇.(ρU⊗U)=∇τ−ρu⊗u +SM ∂Uj ∂Ui2 ∂U τ=μ + −μδ k ij ∂xi ∂xj 3 ij ∂xk ∂Uj ∂Ui 2 ∂U −ρuu =μt + − δ μt k +ρk (13) This model assumes that the eddy viscosity μt is linked to the turbulent kinetic energy k and ij ∂xi ∂xj3ij∂x k its dissipation ε through the following relation: μ =ρC k2 (14) tμε Where C μ = 0.09 and k and ε are defined through the following two equations k − ε model. ∂(ρk) → μ +∇.ρkU=∇.μ+ t ∇k+Pk-ρε (15) ∂t σkPDF Image | ANALYSIS OF RADIAL AND MIXED FLOW TURBINE VOLUTES
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