Page 255 - 48Fundamentals of Compressible Fluid Mechanics

13.4. SOLUTION OF MACH ANGLE 217

issue. He noticed that equation (13.12) the deflection angle is a function of Mach
angle and upstream Mach number, . Thus, one can conclude that the maxi-
mum Mach angle is only a function of the upstream Much number, . This can
be shown mathematical by the argument that differentiating equation (13.12) and
equating the results to zero crates relationship between the Mach number, and

the maximum Mach angle, . Since in that equation there appears only the heat

is a function of of only these parameters. The
differentiation of the equation (13.12) yields

ratio, , and Mach number, ,
?PO
N

(13.32)

c
: E
<
<

"
Because is a monotonous function the maximum appears when has its maxi-

E E

E :

mum. The numerator of equation (13.32) is zero at different values of denominator.
E

"

Thus it is sufficient to equate the numerator to zero to find the maximum. The nom-
inator produce quadratic equation for and only the positive value for is

applied here. Thus, the is

(13.33)

E

!
Equation (13.33) should be referred as Chapman’s equation. It should be noted

?PO

that both Maximum Mach Deflection equation and Chapman’s equation lead to

N

is only a function of of upstream Mach

number and the heat ratio, . It can be noticed that this Maximum Deflection Mach
the same conclusion that maximum
is found than

the Mach angle can be easily calculated by equation (13.8). To compare these two

. Once,
equations the simple case of Maximum for infinite Mach number is examined. A
Number’s equation is also quadratic equation for

simplified case of Maximum Deflection Mach Number’s equation for Large Mach

number becomes
for (13.34)

Hence, for large Mach number the Mach angle is

which make

!

.

With the value of utilizing equation (13.12) the maximum deflection angle

or

+) ),"

has to be made. The general solution of equation (13.31) is

can be computed. Note this procedure does not require that approximation of

+

(13.35)

!

!

E