Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

119

ie.

1 2 , U − c ( ξ 2 1 2 , V − c ( η 2

1 2 ) ,

λ A = ( U , U , U , U + c ( ξ 2 λ B = ( V , V , V , V + c ( η 2 λ C = ( W , W , W , W + c ( ζ 2

2 y + ξ

2 z )

2 y + ξ

2 z )

x + ξ

x + ξ

1 2 ) ,

2 y + η

2 z )

2 y + η

2 z )

(4.20.4)

x + η

x + η

1 2 ) .

1 2 , W − c ( ζ 2

2 y + ζ

2 z )

2 y + ζ

2 z )

x + ζ

x + ζ

In the expressions (4.20.4) U , V and W are contravariant coordinates of velocity vector, previously determined by transformations (4.15.3), while c denotes local speed of sound. Solutions of the system of equations (4.20) can be determined in two passes: I + β ∆ t ( δ − ξ A + + δ − η B + + δ − ζ C + ) n ∆ q ∗ n = − ∆ t R n , I + β ∆ t ( δ + ξ A − + δ + η B − + δ + ζ C − ) n ∆ q n = ∆ q ∗ n . (4.20.5) The process of solving of the first system of equation (4.20.5) takes place in the direction of increasing index ( i , j , k ) by replacing the known, previously calculated values of unknown quantities ∆ q ∗ n , while the second one is solved in the direction of decreasing index ( i , j , k ) . The two-factor LU implicit scheme requires correct implementation of boundary conditions in the implicit term on the left hand of equation (4.20.5). One way comes down to use of boundary conditions based on determination values of characteristic variables [50], which on the body surface are determined by the following expressions: 2 p b = p 1 + ρ 1 c 1 ( ζ t + ζ x u 1 + ζ y v 1 + ζ z w 1 ) , ρ b = ρ 1 +( p b − p 1 ) / c 2 1 ,

u b = u 1 − ζ x ( ζ t + ζ x u 1 + ζ y v 1 + ζ z w 1 ) , v b = v 1 − ζ y ( ζ t + ζ x u 1 + ζ y v 1 + ζ z w 1 ) , w b = w 1 − ζ z ( ζ t + ζ x u 1 + ζ y v 1 + ζ z w 1 ) ,

(4.20.6)

where ζ t , ζ x , ζ y and ζ z are:

ζ y

ζ t

ζ t =

ζ y =

,

,

1 2

1 2

( ζ 2

x + ζ

2 y + ζ

2 z )

( ζ 2

x + ζ

2 y + ζ

2 z )

(4.20.7)

ζ x

ζ z

ζ x =

ζ z =

,

.

1 2

1 2

( ζ 2

x + ζ

2 y + ζ

2 z )

( ζ 2

x + ζ

2 y + ζ

2 z )

In the relations (4.20.6) the index 1 denotes values of flow quantities inside cells in direct contact with the body surface, while index b correspond to values of the mentioned quantities exactly on the body surface. At the outer boundaries of the physical domain, flow quantities are determined by the far field values, for the case of subsonic flow, when the Mach number of undisturbed flow M ∞ does not exceed 1. In supersonic flow ( M ∞ ≥ 1 ) , values at the outer boundary are determined by extrapolation of flow variables values known within cells on "outflow" boundary. By analyzing the equations (4.20) and (4.20.1) it can be concluded that on the body surface information on the values of flow quantities within the body are needed, thus coming into conflict with the physical flow pattern. In oreder to overcome the mentioned problem, it is necessary to perform a modification of the scheme for all grid cells with index k = 1 C + i , j , k − 1 / 2 ∆ q i , j , k − 1 = EC − i , j , k − 1 / 2 ∆ q i , j , k , (4.20.8) 2 On the meaning of introduction characteristic variables will be said in chapter 4.2.3.