( ) (

Given the equation

where

Assume everything is known but *M*.
This equation has two solutions.
At a point upstream to the throat the flow is subsonic, so there is the
subsonic solution giving a *M* < 1.
Downstream of the throat the flow is supersonic giving a solution,
*M* > 1.
Karl Kneile devised an efficient method for computing *M*.

The equation will be solved using an iterative process which which requires a starting solution. A very good first approximation can be obtained as follows.

The subsonic case will be done first.

Or,

where

Note that *P* + *Q* = 1 and *EQ* = 1.

Let

Then the derivative is

So

If *F* is approximated by a quadratic, then to the accuracy of the
quadratic

Thus

which has the solution

Now doing the same thing with the supersonic case let

Equation (2), at the start of the subsonic case, can now be written

or,

Rewriting and changing the definitions of the symbols,

where

Following the same process as before (and using the same symbols)

Thus

which yields

Equations (2) and (4) become

=

or

or

where

Note that in both cases (subsonic and supersonic) from Eqs (3) and (5),
that since *A _{t}* (the cross-sectional area at the throat)
is less than

Since in the subsonic case *X* = *M*^{2} and for
the supersonic *X* = 1/*M*^{2}, in both cases it is
desired that *X* be less than one; thus the minus sign is chosen,
giving

From this form it seen that *X* > 0.
Numerically, a better form is

From this form (*r* is positive) it is seen that *X* < 1.

Summarizing,

This gives a good first approximation for an iterative solution to

(

These equations can be rewritten

(

Notice that the above two equations are of identical form except that

Looking first at Newton-Raphson, from Eq. (2) let

Taking the derivative

Also (though not needed for Newton-Raphson),

Newton-Raphson yields

=

= [

= [

= [ (

= (

Recall that

and

so

which is the result for Newton-Raphson.

Better, let

Then

or

Solving for D

where

Making some observations, from Eqs. (6), since *A _{t}* (the
throat cross-sectional area) is less than any other cross-sectional
area,

In the neighborhood of a root, it is necessary that *D* approach
zero as *f* approaches zero in order for the process to converge.
In Eq. (12), choosing *s* to be +1 and letting *f* approach
zero, since sqrt(*f'*^{2}) = abs(*f'*), which is
positive, and since *f'* is negative, it is seen that there is a
zero over zero.
Therefore the limit must be obtained by L'Hospital's rule.
Taking the derivative of the numerator and the denominator with respect
to *f* results in

which in the limit as

Taking the limit as

and since

which is Newton-Raphson and is seen to be zero in the limit as required.

Therefore the root is found by iterating

and the Mach number is given by Eqs. (8). Obtaining the first approximation by Eq. (7), then iterating the above equation, and then comparing the results with a gas table, it was found that only two iterations were necessary to obtain the accuracy of the table.

Last updated 5 May 2003