Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

116

LU implicit factorization method In Cartesian coordinate system of three-dimensional non-stationary Euler equations can be written in conservative form, as follows: ∂ t q + ∂ x F + ∂ y G + ∂ z H = 0 , (4.13) where q is a variable flow quantities vector, ie. q = ( ρ , ρ u , ρ v , ρ w , e ) T , (4.14) while F , G and H are flux vector projections on three coordinate axes defined by expressions:

   

    ,

   

   

   

    .

ρ u ρ u 2 + p ρ uv ρ uw u ( e + p )

ρ v ρ uv ρ v 2 + p ρ vw v ( e + p )

ρ w ρ uw ρ vw

F =

G =

and H =

(4.15)

ρ w 2 + p w ( e + p )

In the expressions (4.14) and (4.15) the quantities ρ , u , v , w and p represent the density, velocity vector projections on three coordinate axes and pressure, respectively. The variable e is defined by the expression e = ρ e t , where e t is the total fluid energy determined by the relation (4.5). After introducing of curvilinear coordinate system ( ξ , η , ζ ) , computational grid in physical space, whose surfaces correspond to constant values of curvilinear coordinates, is mapped into a rectangle grid in computational domain. In the transformed space, the equation (4.13) becomes: ∂ τ q + ∂ ξ F + ∂ η G + ∂ ζ H = 0 , (4.13.1) where the quantities q , F , G and H are defined by the following relations: q = J q (4.14.1) and

F = J ( ξ τ q + ξ x F + ξ y G + ξ z H ) , G = J ( η τ q + η x F + η y G + η z H ) , H = J ( ζ τ q + ζ x F + ζ y G + ζ z H ) .

(4.15.1)

After arranging the expression (4.15.1) quantities F , G and H become

   

    ,

   

   

ρ U ρ uU + ξ x p ρ vU + ξ y p ρ wU + ξ z p U ( e + p ) − ξ t p

ρ V ρ uV + η x p ρ vV + η y p ρ wV + η z p V ( e + p ) − η t p

F = J

G = J

and

   

    ,

ρ W ρ uW + ζ x p ρ vW + ζ y p ρ wW + ζ z p W ( e + p ) − ζ t p

H = J

(4.15.2)

where U , V and W denote the contravariant coordinates of the velocity vector, defined by transforma tions:

U = ξ t + ξ x u + ξ y v + ξ z w , V = η t + η x u + η y v + η z w , W = ζ t + ζ x u + ζ y v + ζ z w ,

(4.15.3)