Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

135

explicitly written in form

   u ·

   , e B =

   v · ρ · · · v · · · · · v · 1 / ρ · · · v · · · ρ c 2 · v

   and

ρ u

· ·

·

· · 1 / ρ

e A =

· · u · · · · · u · · ρ c 2 · · u

(4.72)

   w · · · w · · · · w · · · · 1 / ρ · · · ρ c 2 w ρ · · · w

   ,

e C =

whereby it is obvious that their structure is far away simpler than the structure of Jacobi matrices A , B and C , specified by relations (4.68)-(4.70). Characteristic variables in one-dimensional flow One-dimensional flows play a very important role in calculation and analysis of solutions of Euler equations. They are simple enough to allow detailed analysis of nonlinear effects of wave propagation, allowing in many cases local application of one-dimensional flow properties in multiple spatial dimensions flows. Local application of a one-dimensional concept in defining boundary conditions is a very important consequence of introduction of one-dimensional characteristics. The most general case is described as quasi one-dimensional flow in a nozzle of variable cross section area S ( x ) , where x is the distance measured along the nozzle axis. Conservative form of Euler equations becomes [42] ∂ ∂ t ( ρ S )+ ∂ ∂ x ( ρ uS ) = 0 , ∂ ∂ t ( ρ uS )+ ∂ ∂ x ( ρ u 2 + p ) S = p d S d x , (4.73)

∂ ∂ t

∂ ∂ x

( ρ e t S )+

( ρ uHS ) = 0 .

By introducing basic ( nonconservative ) variables ρ , u and p , the system (4.73) is trans formed into ∂ρ ∂ t + u ∂ρ ∂ x + ρ ∂ u ∂ x = − ρ u S d S d x ,

∂ u ∂ t

∂ u ∂ x

∂ p ∂ x ∂ u ∂ x

1 ρ

+ u

= 0 ,

(4.74)

+

ρ uc 2 S

∂ p ∂ t

∂ p ∂ x

d S d x

+ ρ c 2

+ u

= −

,

where c is the local speed of sound. By defining the source vector e Q , e Q =   − ρ u 0 − ρ c 2 u   1 S d S d x ,

(4.75)