Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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while ξ t , η t and ζ t are determined in the following way:

ξ t = − x τ ξ x − y τ ξ y − z τ ξ z , η t = − x τ η x − y τ η y − z τ η z , ζ t = − x τ ζ x − y τ ζ y − z τ ζ z ,

(4.15.4)

bearing in mind the well-known law of coordinate transformation

ξ = ξ ( x , y , z , t ) , η = η ( x , y , z , t ) , ζ = ζ ( x , y , z , t ) and τ = t . (4.15.5) Time derivatives ξ t , η t and ζ t in the expressions (4.15.4) are equal to zero for the case of a stationary, fixed grid. Transformation Jacobian J = ∂ ( x , y , z ) / ∂ ( ξ , η , ζ ) in relations (4.14.1), (4.15.1) and (4.15.2) is calculated from the expression J = x ξ ( y η z ζ − z η y ζ ) − y ξ ( x η z ζ − z η x ζ )+ z ξ ( x η y ζ − y η x ζ ) . (4.15.6) In the system of equations (4.13), ie. (4.13.1) unknown quantity to be determined is a vector of flow variables q . Because the quantities F , G and H are nonlinear functions of variable q , defining their values at time n + 1 is done by local linearization with respect to the previous moment determined by time index n F n + 1 = F n + D F D q n ∆ q n , G n + 1 = G n + D G D q n ∆ q n , H n + 1 = H n + D H D q n ∆ q n , (4.16) where is ∆ q n = q n + 1 − q n . (4.17) In the equations (4.16) matrices [ D F / D q ] n , [ D G / D q ] n and [ D H / D q ] n , marked with A , B and C , respectively, are defined as follows: A , B , C =    k t k x k y k z 0 k x φ 2 − u θ k t + θ − k x ( γ − 2 ) u k y u − k x ( γ − 1 ) v k z u − k x ( γ − 1 ) w k x ( γ − 1 ) k y φ 2 − v θ k x v − k y ( γ − 1 ) u k t + θ − k y ( γ − 2 ) v k z v − k y ( γ − 1 ) w k y ( γ − 1 ) k z φ 2 − w θ k x w − k z ( γ − 1 ) u k y w − k z ( γ − 1 ) v k t + θ − k z ( γ − 2 ) w k z ( γ − 1 ) θ ( φ 2 − ω ) k x ω − ( γ − 1 ) u θ k y ω − ( γ − 1 ) v θ k z ω − ( γ − 1 ) w θ k t + γθ    , where k = ( ξ , η , ζ ) for matrices A , B and C , respectively, while the quantities φ 2 , θ and ω are defined by expressions: φ 2 = 1 2 ( γ − 1 )( u 2 + v 2 + w 2 ) ,

(4.18)

θ = k x u + k y v + k z w , ω = γ e / ρ − φ 2 .

Finite volume method, applied in solution of Euler equations, implies discretization of the computational domain on an appropriate number of cells, which in this approach have a hexahedron form. After the discretization was performed, for each cell a system of equations, that approximate