Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

127

where the index ( − ) of eigenvalue λ ( j ) shows that sumation refers to characteristic lines with a negative slopes, associated to negative eigenvalues. Also, a relation can be obtained:

k 2 ∑ j = 1

+( j ) r ( j ) ,

α j λ

F m = F D −

(4.36)

in which the index (+) denotes the sum of the terms associated only with positive eigen values. By combination of equation (4.35) and (4.36) a new form of flow variables flux is obtained F m = (4.37) To prevent "even-odd" decoupling of numerical solution and the presence of unwanted high-frequency oscillations near the shock wave, and also to accelerate convergence, it is necessary to introduce terms of artificial viscosity into appropriate discrete operators. One way, based on Jameson research [18], mentioned in the chapter 4.2.1, refers to the central difference schemes. In the expression for the numerical flux (4.37) artificial viscosity is implicitly included by the second term on the right hand of the mentioned equation, and avoids additional explicit introduction of stabilizing terms. In the case of a one dimensional, non-stationary flow of the ideal gas quantities α j , λ ( j ) and r ( j ) , present in the equations (4.34)-(4.37), are defined by relations 1 2 ( F L + F D ) − 1 2 3 ∑ j = 1 α j | λ ( j ) | r ( j ) .

1 ˜ c 2

1 ˜ c 2

1 ˜ c 2

( ˜ c 2 ˆ ∆ ρ − ˆ ∆ p ) ,

( ˆ ∆ p − ˜ ρ ˜ c ˆ ∆ u ) ,

( ˆ ∆ p + ˜ ρ ˜ c ˆ ∆ u ) ,

α 2 =

α 1 =

α 3 =

λ ( 2 ) = ˜ u , r ( 2 ) = (

λ ( 1 ) = ˜ u − ˜ c , r ( 1 ) = ( 1

λ ( 3 ) = ˜ u + ˜ c , r ( 3 ) = ( 1

(4.38)

2 )

) ,

˜ u + ˜ c ˜ H + ˜ u ˜ c ) ,

1 ˜ u

˜ u − ˜ c ˜ H − ˜ u ˜ c

,

1 2 ˜ u

in which quantities labeled by (~) are defined in [ roe ] by expressions: ˜ ρ 2 = ρ L ρ D , ρ 1 / 2 L H L + ρ 1 / 2 D H D

˜ H =

,

ρ 1 / 2 L

1 / 2 D

+ ρ

ρ 1 / 2

1 / 2 D u D

(4.39)

L u L + ρ

˜ u =

,

1 2

ρ 1 / 2 L

1 / 2 D

˜ c 2 = ( γ − 1 )( ˜ H −

˜ u 2 ) ,

+ ρ

where quantity ˜ H corresponds to the “averaged ” enthalpy of the fluid. The jump of flow variables through cell boundaries is indicated by operator ˆ ∆ in relations (4.38). The problem of applying described scheme is reflected in the possibility of a discon tinuous expansion wave formation in the vicinity of the sonic point where Mach number is M = 1. In order to solve this problem, it is necessary to make a numerical flux correction in the environment of sonic point [41]. Harten and Hyman [13] propose a modification of eigenvalue module | λ ( j ) | in equation (4.37) in the following manner: | λ ( j ) | mod = ( | λ ( j ) | , if | λ ( j ) | ≥ ε , ε , if | λ ( j ) | < ε , (4.40)