Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

124

To calculate the values ρ m + 1 / 2 , u m + 1 / 2 , p m + 1 / 2 and e m + 1 / 2 u inside cell with the boundaries x m and x m + 1 , it is assumed that the boundaries of the segment disappear, applying the equations (4.23) on cell bounded with x = x m , x = x m + 1 , t = t 0 and t = t 0 + τ . In calculation process it is considered that the flow variables after the disappearance of the boundaries, due to small time interval τ , are constant on imaginary cell boundaries. Introducing the notations R , U , P and E u for flow variables at the boundaries of the segments, the relation [25] is reached ρ m + 1 / 2 = ρ m + 1 / 2 − τ h [( RU ) m + 1 − ( RU ) m ] , ( ρ u ) m + 1 / 2 = ( ρ u ) m + 1 / 2 − τ h ( P + RU 2 ) m + 1 − ( P + RU 2 ) m , ρ ( e u + 1 2 u 2 ) m + 1 / 2 = ρ ( e u + 1 2 u 2 ) m + 1 / 2 − (4.24) which are supplemented by the state equation of the ideal gas p = p ( ρ , e u ) . The quantities R , U , P and E u are determined from breakdown formulae, described in detail in the literature [22], whose resolving leads to values of the mentioned quantities at the cell boundaries, which at the same time represent breakdown surfaces (lines). Despite the fact that breakdown formulae are relatively complicated [22], [8], flow variables can be considered constant at time boundaries t = t 0 to t = t 0 + τ , provided that the choice of sufficient small time interval τ avoids wave interaction of disturbances emitted from the boundaries x = x m and x = x m + 1 . Thanks to Godunov, Zabrodin and Prokopov [10], compact expressions for calculating the quantities on contact discontinuities p c . d . and u c . d . , after the disappearance of the imagined membrane on boundary x = x m are obtained. Derived relations cover at the same time all possible breakdown cases arising from fictitious boundary removal − τ h RU ( E u + P R + U 2 2 ) m + 1 − RU ( E u + P R + U 2 2 ) m ,

b m p m − 1 / 2 + a m p m + 1 / 2 + a m b m ( u m − 1 / 2 − u m + 1 / 2 ) a m + b m , a m u m − 1 / 2 + b m u m + 1 / 2 + p m − 1 / 2 − p m + 1 / 2 a m + b m ,

p c . d . =

(4.25)

u c . d . =

where is

a m =   1

2 ( γ + 1 ) p c . d . +( γ − 1 ) p m − 1 / 2 ρ m − 1 / 2 γ − 1 2 γ ( γ p m − 1 / 2 ρ m − 1 / 2 ) 1 / 2 1 − p c . d . / p m − 1 / 2 1 − ( p c . d . / p m − 1 / 2 )

1 / 2

, for p c . d . ≥ p m − 1 / 2 , for p c . d . < p m − 1 / 2 ,

(4.26)

,

( γ − 1 ) / 2 γ

where γ is the adiabatic fluid constant. Quantity b m in equations (4.25) is calculated by replacing the indices m − 1 / 2 by the indices m + 1 / 2 in relations (4.26). The equations (4.25) and (4.26) are exact for finite discontinuities at the boundary x = x m and are resolved by an iterative procedure. The iteration process is performed by adopting the initial assumption for p c . d . and by determination the quantities a m and b m from the relation (4.26). The calculated values of a m and b m are then used in the first of the