Compressor Stalls and Mobius Transformations

It's fascinating that linear fractional transformations, also known
as Mobius transformations, arise in so many diverse contexts, both 
theoretical and practical.  For example, in the field of gas turbine 
design it's been found that stall cells migrate around the face of a 
compressor following the pattern of an iterated Mobius transformation 
of a point in the complex plane.

In 1984 F. K. Moore published a paper entitled "A Theory of Rotating 
Stall of Multistage Axial Compressors" in the Journal of Engineering 
for Gas Turbines and Power (vol 106, April 1984).  In Part II of this
paper Moore develops an equation for the axial and circumferential
velocity disturbances in the airflow as a function of angular position
a.  Letting g(a) and h(a) denote these disturbances, respectively, 
Moore's equation is

                      1           2   1            2
 M g'(a) - L f h(a) + - (1-K) h(a)  - - F''(q) g(a)  +  d  =  0    (1)
                      2               2
where

    M = collection of terms analagous to oscillator mass
    L = parameter defining lag tendancy outside compressor
    f = stall propogation speed coefficient
    K = pressure rise parameter at compressor inlet vanes
 F(q) = compressor characteristic in absence of rotating stall
    q = average flow coefficient, V/U (where V = average axial 
        flow speed and U = wheel speed at mean wheel diameter)
    d = performance increase (perhaps negative) due to rotating
        stall, equal to H - F(q), where H = upstream total to
        downstream static pressure-rise coefficient

Note that the disturbances g and h are periodic and have vanishing
averages over a cycle: in the case of g because it is defined as a
departure from the average axial velocity, and in the case of h
because net circulation in the entrance flow is assumed to remain
zero.

Moore notes that we need h and g such that h+ig is an analytic
function of exp(ia).  With this restriction, he states that he
believes the following solution of equation (1) is unique

                                ina
                               e
               h + ig  =  A ----------
                                   ina
                            1 + N e

where n is the "wave number", i.e., the number of stall cells, and
the constants A and N are determined by satisfying equation (1).
Obviously if N=0 the relation degenerates to a pure exponential 
so we will be concerned only with the cases when N is non-zero.
Separating the real and imaginary parts of the above expression 
we have the individual velocity disturbance components:

              A(cos(na) + N)                     A sin(na)
    h(a) = --------------------      g(a) = --------------------    (2)
           1 + N^2 + 2N cos(na)             1 + N^2 + 2N cos(na)

These equations can be mapped to the real and imaginary parts of
a sequence of complex numbers z[0], z[1],... generated by iterating
a particular Mobius transformation.

To generate the velocity disturbance components for a given parameter
N, set the initial complex number z[0] equal to 1/(1 + 1/N).  Also,
select an arbitrary circumferential step size d.  Then we can iterate
the Mobius transformation

                              z[k-1]
             z[k] = ---------------------------                  (3)
                      id      /     id\
                     e    +  ( 1 - e   ) z[k-1]
                              \       /

and the real and imaginary parts of z[k] are proportional to the
axial and circumferential velocity disturbance components at the
angular position kd.  Specifically, we have

                 h(kd)/A  =  (1/N) z_real[k]

                 g(kd)/A  =  (1/N) z_imag[k]

To illustrate, consider the case N=0.3.  The figure below shows the
velocity disturbance profiles as given by Moore's formulas (2), and
then superimposed on those curves are the discrete values generated
by iterations of the Mobius transformation (3) with a value of d
corresponding to 12 degrees.

              Figure 1


Of course, the general Mobius transformation is of the form 
z -> (az+b)/(cz+d), but any such transformation is "similar" (i.e.,
con conjugate) to a "bi-polar" transformation

                                    z
                        z ->  -------------                        (4)
                               m + (1-m) z

for which the nth iterate is simply

                       z_0                          1
       z_n  =  -------------------   =   ----------------------
               m^n  +  (1-m^n) z_0        ((1-z_0)/z_0) m^n + 1


By the way, the squared trace of transformation (3) is given by

                 T^2 = -[2 + exp(id) + exp(-id)]

It's interesting to note that the condition for periodicity for
iteration (4) is that m be a root of 1.  This Mobius transformation 
has fixed points at 0 and 1, and the simple linear function that 
transforms the general Mobius transformation to this "bi-polar" LFT 
maps the real axis to the "Riemann line" 1/2 + yi.  (For a more 
detailed discussion of this topic, see Linear Fractional Transformations.

Oddly enough, the general form of (4) allows the complex constant m
to have a magnitude other than 1, but Moore's formula is limited to
having m=1.  Considering that the actual physical mechanism by which
the stall cell propagates from one blade to another around the face
of the compressor would seem to be more naturally modelled by the
discrete transformations of of the form (4) rather than the continuous
relations of (2), we might consider the physical consequences of 
allowing the magnitude of m to vary slightly from 1.  Figure 2 shows
the result of setting |m| to 1.010.

              Figure 2


This shows that the velocity disturbance tends to dampen out over
several cycles.  On the other hand, if we reduce the magnitude of m
to 0.996 we produce the results shown in Figure 3.

              Figure 3


Thus, for a very slight reduction from unity in the magnitude of m
we find that the velocity disturbance cycle becomes unstable.  It
would be interesting to determine whether actual instabilities in
the propagation patterns exhibited by stall cells in real compressors
conform to these profiles.