Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

140

outflow and inflow become

2   2  

0 

2  

0 

∆ ρ ∆ u

∆ w 1 ∆ w 2

∆ U M

= M M

= M M

=

− 1 2

− 1

− 1

 M − 1 2

 M − 3 2

0 

∆ p / c 2 + ∆ ρ ∆ u + ∆ p / ρ cu

= M M

(4.100)

,

− 1

 M − 3 2

2   0 ∆ u

0   1 2

2   0

0   3 2

2   2  

0   3 2 ,

0 ∆ u − ∆ p / ρ c

∆ w 3

∆ U 1 2

= M 1

= M 1

= M 1

while in the case of a supersonic outflow all three numerical boundary conditions are extrapolated from known values at interior points, which is illustrated in Figure 4.9.a, ie.

2  

∆ ρ ∆ u ∆ p 

  M − 3 2

∆ w 1 ∆ w 2 ∆ w 3

∆ U M

= M M

= M M

L M

=

− 1 2

− 1

− 1 2

− 1

 M − 1 2 2  

(4.101)

  M − 3 2

∆ p / c 2 + ∆ ρ ∆ u + ∆ p / ρ c ∆ u − ∆ p / ρ c

= M M

L M

.

− 1 2

− 1

The system of equations (4.60) can be solved using a standard block-diagonal procedure, based on the inversion of third-order matrices in case of one-dimensional flow. Two dimensional problems or the three-dimensional flow of a non-viscous fluid is solved using a ADI scheme, which is unfortunately difficult to achieve, especially in three-dimensional flow. To increase efficiency of 2D and 3D flow analysis ADI scheme, based on block diagonal matrix inversion, is replaced by LU implicit factorization, explained in the chapter 4.2.2 and literature [20], [21], [31] and [32]. Verification of the presented procedure will be performed by numerical analysis of stationary non-viscous air flow through a nozzle of length ℓ = 10m, whose cross section area varies according to the law S ( x ) = 1 . 398 + 0 . 347 tanh ( 0 . 8 x − 4 ) , (4.102) with supersonic inflow and supersonic/subsonic outflow bpundary. The conditions at the nozzle inlet are defined as follows: ρ 1 = 1 . 22112kg / m 3 , u 1 = 351 . 4351m/s i p 1 = 47 . 87837KPa . (4.103) In the case of a subsonic outflow boundary it is necessary to define only one physical boundary condition, ie. p i = 117 . 7617KPa . (4.104)