Compressor Stalls and Mobius Transformations

Linear fractional transformations, also known as Mobius transformations, arise in 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 θ. Letting g(θ) and h(θ) denote these disturbances, respectively, Moore's equation is

where

λ  =  collection of terms analogous to oscillator mass

m  =  parameter defining lag tendency outside compressor

f   =  stall propagation speed coefficient

K  =  pressure rise parameter at compressor inlet vanes

ψ_c″= second derivative of compressor characteristic (in

          absence of rotating stall) with respect to the average

          flow coefficient, V/U (where V = average axial flow

          speed and U = wheel speed at mean wheel diameter)

δ  =  constant performance increase (perhaps negative) due to

         rotating stall, equal to H − ψ, where H is the upstream

         total to downstream static pressure-rise coefficient

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

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

Although not mentioned in Moore’s paper, it’s interesting that these equations can be mapped to the real and imaginary parts of a sequence of complex numbers z_k for k = 0, 1, 2, … generated by iterating a particular linear fractional (Mobius) transformation. To generate the velocity disturbance components for a given parameter η, set the initial complex number z₀ equal to η/(1+η) and select an arbitrary circumferential step size Δθ.  Then we can iterate the Mobius transformation

and the real and imaginary parts of z_k = x_k + iy_k are proportional to the axial and circumferential velocity disturbance components at the angular position θ = k Δθ.  Specifically, we have

To illustrate, consider the case η = 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 Δθ corresponding to 12 degrees.

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

for which the kth iterate is simply

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

It's interesting to note that the condition for periodicity for iteration (4) is that w 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.)

Although the general form of (4) allows the complex constant w to have a magnitude other than 1, Moore's formula assumes w = 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 modeled by the discrete transformations of the form (4) rather than the continuous relations of (2), we might consider the physical consequences of allowing the magnitude of w to vary slightly from 1.  The figure below shows the result of setting |w| to 1.010.

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

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.

Incidentally, to confirm that (2) actually is a solution of (1), we can substitute the expressions for h and g into (1) to arrive at the equation

where

In order for (5) to be satisfied for all θ each of the coefficients must vanish. Solving the equation c₂ = 0 for δ gives

Substituting this expression for δ into the equation c₁ = 0 and solving for A, we get

Likewise we can substitute the expression (6) for δ into the equation c₀ = 0 and solve for A to give

These two expressions for A must be equal to each other, so we have the equation A₍₁₎ – A₍₀₎ = 0, which we can solve for λ to give the relation

We can now substitute this expression for λn back into either the expression for A₍₀₎ and A₍₁₎ to give

Given values of m, f, ψ_c″, λ, n, and K, we can compute η from equation (7), and then we can compute A from equation (8), and finally we can compute δ from equation (6). By the way, Moore noted that K is actually a function of h so it isn’t really constant, but in this application the quantity 1–K is usually negligibly small, so he deleted the terms involving 1−K. Setting 1−K = 0 in equations (6), (7), (8), and also substituting for A in (6), we get

which agrees exactly with Moore’s equations (2.12), (2.13), and (2.14). Notice that our equations (6), (7), and (8) also match Moore’s equations (2.10), (2.8), and (2.9) respectively, except that there appears to be a typo in (2.10), and possibly an algebraic error in (2.8) and (2.9). Fortuitously, the errors in the latter two equations were in the terms involving 1−K, so they dropped out of the simplified results.

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