Normal Shock Waves

Under certain conditions it is possible to establish a standing
normal shock wave in a duct.  The location of the shockwave depends
on the variation in the cross-sectional flow area of the duct, as
well as on the upstream and downstream boundary conditions.

In general, one-dimensional isentropic flow of a perfect gas is 
characterized by a constant value of the parameter

                     M
            A ----------------   =   constant
               /     1   2 \ 3
              ( 1 + --- M   )
               \     5     /

where A is the cross-sectional flow area, M is the Mach number
(i.e., the ratio of the flow velocity to the speed of sound in the
gas), and we have assumed a ratio of specific heats ("gamma") equal
to 1.4, which is the characteristic value for air.  This relation
holds for both subsonic and supersonic flow.  Thus, given the area
profile of a duct, and the (supersonic) upstream Mach number, we
can compute the Mach number at every point in the duct, based on
the isentropic flow assumption.  Likewise, given the (subsonic)
downstream Mach number, we can compute the Mach at every point 
in the duct, again based on the assumption of isentropic flow.
Ordinarily these two profiles will not intersect, proving that
we cannot maintain those upstream and downstream conditions with
purely isentropic flow.  Consequently, at some point in the duct
there must be a shock wave.

The Mach nuber M_d downstream of a normal shock is related to the
Mach number M_u upstream of the shock (still assuming gamma=1.4)
by the equation
                                2
                     2       M_u   +  5
                 M_d    =   ------------
                                2
                            7M_u   -  1

Therefore, at each point in the duct we can compute the "shock down
profile", which represents the Mach number that would exist just
downstream of a normal shock from the supersonic Mach profile.
The point at which this shock down profile intersects with the
isentropic downstream profile is where the shock wave must occur.

This is illustrated for a generic converging-diverging duct in
the figure below.



The sensitivity of the shock position to changes in the duct area
profile and the upstream and downstream boundary conditions can
easily be inferred from these simple relations.