Method of Characteristics Derivation (continued)

(Source: Notes used by both  Prof. Sankar and Prof. Smith)

Note to the gentle reader from NK: These derivations were converted to html from different documents constructed by different people on different types of computers using different software (Macintosh - Silicon Graphics - PC - Word - WordPerfect -...)  Be kind, please. It takes a while to get all the bugs out. Watch out for differences in nomenclature, and figures which turn out odd, and equation numbers which may apply only to the page where they are used. This is why I have not merged these documents into one set of notes. ( not to mention my need to sleep, watch TV etc. once in a while). Let me know the errors, and I will gratefully correct them, eventually. Thanks.

This is continued from the derivation linked here

We start from the equation:

(1)
The expression under the square root was shown to be positive for supersonic flows, making the slopes real numbers. Thus, in supersonic flows there are two families of lines, called characteristic lines, with real slopes. In subsonic flows, the term inside the square root is negative, and no characteristic lines exist.

In this page, we attempt to simplify the above equation. We use the following transformations:
 
 

(2)
Let us look at the terms inside the square root first. These terms simplify as follows:
 
 

(3)


Using the transformation (2) and equation (3) in equation (1) we get:

(4)


Using the trigonometric identities:

(5)
Equation (4) becomes:

(6)
Using the trigonometric identities:
 
 

(7)
Equation (6) becomes:
 
 

(8)


These two lines are shown graphically below:

Notice that the characteristic lines make equal angles m with respect to the velocity vector. This is exactly what a Mach cone does. Thus, the characteristic lines are simply Mach lines drawn about the velocity vector.

Recall that the characteristic lines were obtained by obtaining an expression for the mixed derivative Fxy=N/D, where N and D were 3x3 determinants, and setting the denominator D to zero (See class notes.) Let us next set the determinant on the numerator N to zero.
 
 

(9)
Evaluating the determinant, we get:





After some rearrangement the above equation yields:
 
 




(10)

 

Using equation (1) in equation (10), we get:
 
 

(11)

 

One can use equation (2) to transform the right hand side of equation (11) in terms of V, M and q. For example, wherever u appears on the right hand side of (11) we will replace it by Vcosq, and so on. We can also write du and dv in terms of V and q as follows:
 
 





Using these transformations, after about two pages of algebra and trigonometric relationships, we get:
 
 

(11)

 

Integrating, we get:

(12)

 

The integral appearing in equation (12) may be familiar. You encountered it (many moons ago) in AE 3004 under the section on Prandtl-Meyer expansion fans (Section 9.6, equation 32). In other words, the integral is simply the Prandtl-Meyer function n(M). Thus, equation (12) becomes:
 
 

(13)
The above equation is called the compatibility relationship, and must be satisfied by the flow angle and the Mach number at every point on the two families of characteristic lines derived earlier. Specifically,

Along the line, called the C- characteristic (see figure 13.4 in the text):
 
 

(14)
Along the line, called the C+ characteristic (see figure 13.4 in the text):
 
 

(15)