Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

133

Shock wave at x/l = 0.5

3

2.5

p ROE (BD) ress. dens. ROE (BD) pr ROE (LU) ess. dens. ROE (LU) pr analyt. ess. dens. analyt.

2

p/p 1 ρ ρ/ 1

1.5

1

0.5

0

0

0.25

0.5 x/l

0.75

1

Figure 4.8: Flow in divergent nozzle – presence of a shock wave.

The most significant advantage of the Roe scheme over the procedure exposed in [15] is reflected in the possibility of determination very sharp shock wave without the presence of unwanted oscillation. Modification of Godunov scheme In this chapter, special attention will be paid to the correct implementation of boundary conditions, based on characteristic variables , in order to increase accuracy, stability and the convergence rate of the numerical solution. Three-dimensional Euler equations in non-stationary flow field can be written in Carte sian coordinate system in a conservative form, similar to the system of equations (4.13) ∂ U ∂ t + ∂ f ∂ x + ∂ g ∂ y + ∂ h ∂ z = 0 , (4.63)

where U is the vector of flow variables

U = ( ρ , ρ u , ρ v , ρ w , ρ e t ) T ,

(4.64)

while f , g and h are components of the flux vector

   

    ,

   

   

   

    .

ρ u ρ u 2 + p ρ uv

ρ v ρ uv ρ v 2 + p

ρ w ρ uw ρ vw ρ w 2 + p ρ wH

f =

g =

and h =

(4.65)

ρ uw ρ uH

ρ vw ρ uH

In the relations (4.64) and (4.65) the quantities ρ , u , v , w , p , e t and H denote fluid density, velocity vector projections on three coordinate axes, pressure, total energy and stagnation enthalpy per unit fluid mass, respectively.