Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

141

In the presented calculation results, convergence is considered to be achieved if the residual term is reduced in size of fourth order relative to the initial. Courant number CFL , used in numeric analysis, is determined by relation

∆ t ∆ x

CFL = ( u + c ) max

(4.105)

,

in which ∆ t is the time step of integration, while ∆ x determines the spatial dimension of the computational grid.

1.5

2.5

2

an

1

an

1

p/p 1 ρ ρ / 1 p/p 1 ρ ρ / 1

1

an

an , / ρ ρ

an , / ρ ρ

1.5

an

0.5 p/p 1 , / ρ ρ 1 , p/p 1

p/p 1 ρ ρ / 1 p/p 1 ρ ρ / 1

0.5 p/p 1 , / ρ ρ 1 , p/p 1

an

an

0

0

0

0.25

0.5 x/l

0.75

1

0

0.25

0.5 x/l

0.75

1

(a) Subsonic outflow. (b) Supersonic outflow. Figure 4.10: Pressure and density distribution in the nozzle.

In the Figure 4.10 the results of numerical analysis of one-dimensional flow in diver gent nozzle are shown. Described method, based on zero order extrapolation of charac teristic variables , increases accuracy of results compared to results achieved by applying the numerical scheme [15]. Very strong shock wave is present at a distance of x /ℓ = 0 . 5, measured along nozzle axes, in the case of a subsonic outflow boundary, which is shown in Figure 4.10.a. In the case of supersonic outflow boundary numerical solution very well matches the analytical one [17] (Figure 4.10.b). In addition to the improvements shown, modified Roe “averaged” scheme, based on finite volume method and boundary conditions applied to characteristic variables , increases the stability of the numerical scheme with significant increasing of the convergence rate compared to the original method [29], described in the chapter 4.2.3, where the boundary conditions are based on the extrapolation of nonconservative variables . Presented procedure allows the introduction of significantly larger values of Courant number ( CFL ) into a numeric procedure. In order to properly estimate the influence of integration steps on convergence rate, different combinations of ∆ x and ∆ t were used, which determine the value of the Courant number, according to the equation (4.105). Intensive changes are noticed after the first few iteration, followed by a gradually convergence of numerical solution to a stationary one. Convergence rate and the size of the residual term for the subsonic and supersonic outflow boundary are shown in Figures 4.11, 4.12, 4.13 and 4.14, respectively.