Page 124 - 48Fundamentals of Compressible Fluid Mechanics

86 CHAPTER 5. NORMAL SHOCK

The “downstream” Mach number reads

(5.55)

Again, the shock is moving to the left but in the moving coordinates. The observer

(with the shock) sees the flow moving from the left to the right. The upstream is on
the left of the shock. The stagnation temperature increases by
(5.56)

The prominent question in this sit-

uation is what the shock wave velocity for
and for a given

specific heat ratio. The “upstream” or the "$#&%
')(
“downstream” Mach number are not known !
a given fluid velocity,
even if the pressure and the temperature c.v.
downstream are given. The difficulty lays in Stationary Coordinates
the jump from the stationary coordinates to
the moving coordinates. it turned out that it
is very useful to use the dimensionless pa-
instead the velocity be- :<;>=9?@;$A&B C 5)6
cause it combines the temperature and ve- *+-,.
locity into one. 798 /$021/43
rameter
, or

The relationship between the Mach
number on two sides of the shock are tied c.v.
Moving Coordinates
through equations (5.54) and (5.55) by
Fig. 5.6: Comparison between stationary
shock and moving shock in a
(5.57) stationary medium in ducts

%

And substituting equation (5.57) into (5.48) results

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(5.58)

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The temperature ratio in equation (5.58)and the rest of the right hand side show

%

%
has
four solutions). Only one real solution is possible. The solution to equation (5.58)

can be obtained by several numerical methods. Note, that analytical solution can
clearly that
has four possible solutions (fourth order polynomial for

be obtained to (5.58) but it seems very simple to utilize numerical methods. The
. For very small values of upstream
typical methods is of “smart” guessing of