Page 114 - 48Fundamentals of Compressible Fluid Mechanics

76 CHAPTER 5. NORMAL SHOCK
5.1 Solution of the Governing Equations

5.1.1 Informal model

Accepting the fact that the shock is adiabatic or nearly adiabatic requires that total
. The relationship of the temperature to the stagna-
tion temperature provides the relationship of the temperature for both sides of the
shock.
energy is conserved,

(5.7)

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All the other derivations are essentially derived from this equation. The
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. Note that
all the other quantities
only issue that is left to derived is the relationship between
and
can be determined at least numerically. As it will be seen momentarily there is

Mach number is function of temperature, thus for known
analytical solution which is discussed in the next section.

5.1.2 Formal Model
The equations (5.1, 5.2, and 5.3) can be converted into a dimensionless form. The
reason that dimensionless forms are heavily used in this book is because by doing
so simplifies and clarifies the solution. It can also be noted that in many cases the
dimensionless questions set is more easily solved.
From the continuity equation (5.1) substituting for density, , the equation
of state yields
(5.8)
Squaring equation (5.8) results

(5.9)

Multiplying the two sides by ratio of the specific heat, k provide a way to obtained

to be used for
Mach definition as following,
the speed of sound definition/equation for perfect gas,
(5.10)

Note that the speed of sound at different sides of the shock is different. Utilizing

the definition of Mach number results in

(5.11)