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Next: The Maximum Conditions Up: Solution of the Governing Previous: Informal Model   Index

Formal Model

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 it simplifies and clarifies the solution. It can also be noted that in many cases the dimensionless equations set is more easily solved.

From the continuity equation (5.1) substituting for density, ρ, the equation of state yields

Squaring equation (5.8) results in
Multiplying the two sides by the ratio of the specific heat, k, provides a way to obtain the speed of sound definition/equation for perfect gas, c2=kRT to be used for the Mach number definition, as follows:
Note that the speed of sound on the different sides of the shock is different. Utilizing the definition of Mach number results in
Rearranging equation (5.11) results in
Energy equation (5.3) can be converted to a dimensionless form which can be expressed as
It can also be observed that equation (5.13) means that the stagnation temperature is the same, T0y= T0x . Under the perfect gas model, ρU2 is identical to k P M2 because
Using the identity (5.14) transforms the momentum equation (5.2) into
Rearranging equation (5.15) yields
The pressure ratio in equation (5.16) can be interpreted as the loss of the static pressure. The loss of the total pressure ratio can be expressed by utilizing the relationship between the pressure and total pressure (see equation (4.11)) as
The relationship between Mx and My is needed to be solved from the above set of equations. This relationship can be obtained from the combination of mass, momentum, and energy equations. From equation (5.13) (energy) and equation (5.12) (mass) the temperature ratio can be eliminated.
Combining the results of (5.18) with equation (5.16) results in
Equation (5.19) is a symmetrical equation in the sense that if My is substituted with Mx and Mx substituted with My the equation remains the same. Thus, one solution is
It can be observed that equation (5.19) is biquadratic. According to the Gauss Biquadratic Reciprocity Theorem this kind of equation has a real solution in a certain range5.3which will be discussed later. The solution can be obtained by rewriting equation (5.19) as a polynomial (fourth order). It is also possible to cross-multiply equation (5.19) and divide it by (Mx2 - My2) results in
Equation (5.21) becomes
The first solution (5.20) is the trivial solution in which the two sides are identical and no shock wave occurs. Clearly, in this case, the pressure and the temperature from both sides of the nonexistent shock are the same, i.e. Tx = Tx ; Px = Px . The second solution is where the shock wave occurs.

The pressure ratio between the two sides can now be as a function of only a single Mach number, for example, $ M_x$ . Utilizing equation (5.16) and equation (5.22) provides the pressure ratio as only a function of the upstream Mach number as

$\displaystyle {P_y \over P_x} & = {2k \over k+1 } {M_x}^2 - {k -1 \over k+1}

The density and upstream Mach number relationship can be obtained in the same fashion to became

The fact that the pressure ratio is a function of the upstream Mach number, $ M_x$ , provides additional way of obtaining an additional useful relationship. And the temperature ratio, as a function of pressure ratio, is transformed into
In the same way, the relationship between the density ratio and pressure ratio is
which is associated with the shock wave.

Figure: The exit Mach number and the stagnation pressure ratio as a function of upstream Mach number.
Image shock

next up previous index
Next: The Maximum Conditions Up: Solution of the Governing Previous: Informal Model   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21