Page 127 - 48Fundamentals of Compressible Fluid Mechanics

5.3. THE MOVING SHOCKS 89

Shock in A Suddenly Open Valve Shock in A Suddenly Open Valve
k = 1 4, M ’ = 0.3 k = 1 4, M ’ = 1.3
y y
1.75 4
M M
x x
M 3.5 M
y y
1.5 T /T x 3 T /T x
y
y
2.5
1.25
2
1.5
1
1
0.5
0.75
0 10 0 5 10 15 20
Number of Iteration Number of Iteration
Wed Aug 23 17:20:59 2006 Wed Aug 23 17:46:15 2006
(a) The number of iterations to (b) The number of iterations to

achieve the results for

achieve the results for
The same question that was prominent in the previous case appears now.
what will be the shock velocity for given upstream Mach number? The relationship
again between the two sides is

(5.63)
%

theatrically equation (5.63) can be solved.

%
%
It is common practice to solve this equation by numerical methods. One of such
method is successive substitutions. This method is done by the following algorithm:
can be transformed to
Since

utilizing the tables or Potto–GDC
(a) Assume that

(c) Utilizing
(b) Calculate the Mach number

against the old one. If it satisfactory stop or
calculate the new “improved”
return to stage (b).

,
(d) Check the new and improved

and

. The results show that the convergence occurs very rapidly
To illustrate the convergence of the procedure consider the case of