Page 118 - 48Fundamentals of Compressible Fluid Mechanics

80 CHAPTER 5. NORMAL SHOCK
5.1.4 Prandtl’s condition

It can be easily noticed that the temperature from both sides of the shock wave is
discontinuous. Therefore the speed of sound is different in these adjoining medi-
ums. It is therefore convenient to define the star Mach number that will be inde-
pendent of the specific Mach number (independent of the temperature).

(5.32)

The jump condition across the shock must satisfy the constant energy.
(5.33)

Dividing of the mass equation by momentum equation and combining with the

perfect gas model yields
(5.34)

Combining equation (5.33) and (5.34) results in

(5.35)

After rearranging and diving equation (5.35) gives

(5.36)

Or in dimensionless form

(5.37)

5.2 Operating Equations and Analysis
and the Ratio of the total
are plotted as a function of the entrance Mach number. The
In Figure (5.3), the Mach number after the shock,
has minimum value
pressure,
which depends on the specific heat ratio. It also can be noticed that density ratio

working equations are presented earlier. Note that the
(velocity ratio) also have a finite value regardless to the upstream Mach number.
The typical situations in which these equations can be used include also
the moving shocks. The questions that appear are what should be the Mach num-
ber (upstream or downstream) for given pressure ratio or density ratio (velocity ra-
tio). This kind of question requires examining the Table (5.1) for or utilizing