Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

146

In order to estimate influence of viscosity on the nozzle flow, the flow calculation in the axisymmetric nozzle whose cross-sectional area is defined by law (4.102) was performed, applying the Prandtl turbulent model [43] and [23]. The results for the same flow conditions are obtained using commercial ANSYS package and are illustrated in Figures 4.16 and 4.17. Pressure distribution on the nozzle axis (Figure 4.16) shows very well coincidence with the analytical solution of Euler equations, thus with the method presented in this chapter. At the nozzle inlet a uniform pressure and velocity distribution is given, which is held almost to the location of the shock wave. In the downstream flow, due to present viscous effects, there is a redistribution, which is shown in Figure 4.17.

MACH 1.67991 1.45592 1.23193 1.00794 0.783956 0.559969 0.335981 0.111994

0

2

4

6

8

10

X

Figure 4.18: Mach number at viscous flow.

After crossing the shock wave the fluid flow is subsonic, and due to the increase of pressure in the flow direction, thickness of the boundary layer increases and flow separates from nozzle walls. The distribution of Mach number within the nozzle is given in Figure 4.18. To generalize the procedure for three-dimensional flow, it is necessary to form matri ces A , B and C , defined by expressions (4.68)-(4.70), whose elements will be calculated as follows:

ρ 1 / 2

1 / 2 D u D

ρ 1 / 2

1 / 2 D v D

L u L + ρ

L v L + ρ

˜ ρ 2 = ρ

L ρ D , ˜ u =

˜ v =

,

,

ρ 1 / 2 L

1 / 2 D

ρ 1 / 2 L

1 / 2 D

+ ρ

+ ρ

(4.112)

ρ 1 / 2

1 / 2 D w D

ρ 1 / 2

1 / 2 D H D

L w L + ρ

L H L + ρ

˜ H =

˜ w =

,

,

ρ 1 / 2 L

1 / 2 D

ρ 1 / 2 L

1 / 2 D

+ ρ

+ ρ

where ˜ u , ˜ v and ˜ w are “averaged” velocity projections, while

1 2 ( ˜ u 2 + ˜ v 2 + ˜ w 2 )] .

˜ c 2 = ( γ − 1 )[ ˜ H −

(4.113)

Numerical flux (4.35)-(4.37) in the case of multidimensional flow is calculated on the basis of such “averaged” flow quantities [41]. Second order Godunov scheme Direct replacement of terms in a first order numerical scheme with the corresponding terms of the second order accuracy leads to the difficulties associated with the occurrence of oscillations in vicinity of discontinuous changes of flow quantities.