OCR Text |
Show E„ = QXl U*N <xy TXz (5a) \uTXx + VTxy + WTXz + qXr I t F = 9»i \ QVN TyX Tyy i \ UTyX + VTyy + WTyz + fly. (56) / Gv = Qzi QzN TZx rzy (5c) \ UTZX + VTzy + WTZZ + qZe / I Si \ SN &P9, &P9y &P9z \Ap(ugx + vgy + wgz)/ where p, p, w, v, w, and Ys denote the density, pressure, Cartesian velocity components, and species mass fraction, respectively. The total energy is Et = p[e + \(u2 + v2 + w2)] and e is the internal energy. The rate of change of species s due to chemical reactions is denoted by Ss (discussed in more detail later); gXi are body force components and Ap - p - pT with pT equal to some reference density. The viscous stresses are written as: du 2 (du dv dw\ Txx =2pe- o/ieU- + 7- + T-ox 3 V ox oy oz I dv 2 (du dv dw\ Tyy = 2pe- ~ ~Pe (- ^ ~ TZz = 2p( dy dw ~b~z~ 3 \dx (du dy dv dzJ dw\ ~ ^e\dx + dy+ dz) TXy - T,y x /du dv e\dv dx dv\ ) ( du dw \ Tu~ - T,- u = Pe[ dz dv dz dx J dw\ (7) The energy fluxes in the three coordinate directions are given by , AT ^ . n dYa 8=1 N dx dT d\ Qye = ke_+p<£hsDsm-±, (8) 8=1 N ^e=ke-+p2^haD8m-^-, 8=1 Qx3 Qy. Qza = PD8„ = PDS„ = pDs„ d^ 1 dx an 1 dy dYs 1 dz where T, pt, pt and ke = ki + CPm//</Pr, are the temperature, molecular viscosity, turbulent viscosity and effective thermal conductivity, respectively. The effective viscosity is given by pe = pi + pt- The effective thermal conductivity is expressed in terms of a turbulent Prandtl number, Pr*. The sth species diffusion fluxes are written as: (9) The binary molecular diffusivity of species s in the gas mixture is DSm = (1 - X8)/^2J^SXJ/D8J. The sth species molar fraction is Xs and Ds, is the binary mass diffusivity between species s and j. For turbulent flows, the effective diffusivity is expressed in terms of a constant turbulent Schmidt number. Low Speed Preconditioning To maintain accuracy while solving combustion flows with large chemistry schemes, there is a simultaneous need to resolve sharp reaction-induced gradients of flow properties and also the need to resolve more gradually varying regions of the flow such as shear layers and wall boundary layers. In an attempt to achieve these goals, the convective fluxes are treated using the hybrid flux-splitting formulation of Edwards (1997). The numerical diffusion terms in the hybrid upwind scheme are modified for low speed flows in accordance with the preconditioned system (Edwards, 1997). These terms are scaled by a reference velocity rather than the speed of sound as the Mach number becomes very small. In the present preconditioned formulation for reacting flows of multiple components, the species partial pressures, ps are made to be the dependent variables by adding pseudo-time derivatives to the conservation equations. This procedure effectively prevents de-coupling of the velocity and pressure fields by re-scaling the system eigenvalues, making them the same order of magnitude. Following Edwards and Liou (1998) and Edwards and Roy (1998), we use the preconditioning matrix of Weiss and Smith (1994). Equations |