PhD Thesis - staffweb - University of Greenwich

APPENDIX 4 : SMARTFIRE TECHNICAL REFERENCE GUIDE∫ div uφ)dV = ∫ ρ(u ⋅ n)φdS≅ ∑ ρf( u ⋅V( ρ n)A φ(9b)SIn this equation the value of ρ f is given the value in the upwind element. Thusffffρf= ρPifff Af( u ⋅ n)> 0.0 and ρ = ρ if ( u ⋅ n)< 0.0(9c)The convected quantity, φ, at the face needs further approximation. One possible approach isto use arithmetic averaging, e.g.φ = α φ + ( 1 −α ) φ(9d)f f P f AAssuming this choice, then the final form of the discretised convection term becomes[ ]∑ ρ ( fu ⋅ n ) fAf αfφP+ ( 1 −α ) fφA(9e)fOther possible choices of the approximation of the convected, φ, are presented in the sectionentitled discretisation schemes.1.2.1.3 Diffusion TermThe diffusion term is approximated using the approximation:∫⎛φ⎞A−φPdiv( Γ = ∫ Γ ⋅ ≅ ∑ Γ⎜⎟φgrad ( φ))dVφgrad( φ)ndS (φ)fAf(9f)f ⎝ AP ⎠S S dwhere the diffusion coefficient is approximated by( Γ )φf=α( Γ ) ( Γ )φAφ( Γ ) + ( 1−α)( Γ )f φ P f φ AP(9g)1.2.1.4 Source TermIn general, since the source term is a function of the dependent variable, φ, a linearized formis used in the final discretised equation to ensure diagonal dominance of the system matrix.This form is as given below:Sφ= S − S φ = V ( S − S φP)(9h)CPPC1.3 Overall Discretisation EquationPAppendix 11.4 Page 144-12 12