The plenum decides whether a compressor surges

Matched at one speed and at no other follows a multistage compressor stage by stage and asks whether a steady operating point exists with every stage inside its limits. At part speed there is a speed below which none does: the front stages are pushed towards stall while the rear ones choke. It stops at the edge of that window, because what a compressor does outside it is not a question about a sequence of stages. It is a question about a dynamic system — the compressor, the duct it sits in, the volume it discharges into and the throttle that volume empties through — and its answer is a growth rate rather than a window.

Two answers are possible, and they are nothing alike. In rotating stall a patch of the annulus stops passing flow, the stall cell, and travels round the machine at a fraction of the rotor’s speed, while the average flow through the machine settles to a new, lower, steady value. In surge the average flow itself oscillates: it collapses, often reverses so that hot gas blows out of the intake, recovers, and does it again, at a frequency of a few hertz or less. Rotating stall is a steady state that happens to rotate; surge is an oscillation of the whole system. This essay computes which one a compressor chooses, and finds that the blades have almost nothing to do with it.

A characteristic with a peak #

A compressor’s steady performance is its characteristic: the pressure rise it makes as a function of the flow through it, both made dimensionless with the blade speed. In the blades’ own frame the relative flow obeys a Bernoulli-like balance that a turning frame keeps, and a stage’s pressure rise is the part of the work that balance lets it keep. The work a rotor does is set by the change of swirl across it, and as the flow falls the blades meet the air at a steeper angle, the swirl they give it rises and so does the pressure — until the incidence is too steep for the blade boundary layers, the losses climb faster than the work, and the pressure rise peaks. To the left of the peak the characteristic slopes upwards: less flow, less pressure.

The first figure draws the characteristic used here, a cubic in the flow coefficient $\Phi$, with a peak pressure-rise coefficient of 0.66 at $\Phi = 0.5$. The shape is the one Moore and Greitzer chose for their model, with parameters of the size used throughout the control literature on it; it stands for a low-speed multistage compressor and it is stated, not derived. The throttle downstream passes a flow proportional to the square root of the pressure across it, so each throttle setting is a parabola, and the machine runs where its parabola meets the characteristic. Open, the throttle line crosses just right of the peak, on the falling side, and the machine runs steadily there. Close it past the peak and the crossing moves onto the rising side.

That crossing is an equilibrium, but not one the machine can hold. On the rising side a small drop in flow lowers the pressure the compressor makes; with the plenum’s pressure momentarily unchanged, the flow is decelerated further, and the disturbance feeds itself. What it feeds into depends on which way the disturbance points — whether it is the same all the way round the annulus, or varies round it.

Two ways to leave the peak #

Moore and Greitzer, in 1986, reduced the whole system to three equations. The average flow coefficient $\Phi$ is accelerated by the difference between what the compressor makes and the plenum pressure $\Psi$, against the inertia of the air in a duct of effective length $l_c$ rotor radii. The plenum pressure rises when the compressor delivers more than the throttle passes, at a rate set by the plenum’s volume. And a third variable $J$, the squared amplitude of the first circumferential harmonic of the flow — the size of a stall cell — grows wherever the characteristic slopes upwards and saturates at an amplitude the characteristic’s curvature sets:

$$\frac{dJ}{d\xi} \propto J\left[1 - \left(\frac{\Phi}{W} - 1\right)^2 - \frac{J}{4}\right].$$

Time $\xi$ is measured in rotor radii travelled by the blade tip. The equations for $\Phi$ and $\Psi$ contain one dimensionless group that the compressor itself does not supply:

$$B = \frac{U}{2a}\sqrt{\frac{V_p}{A_c L_c}},$$

with $U$ the blade speed, $a$ the speed of sound, $V_p$ the plenum’s volume and $A_c$, $L_c$ the compressor duct’s area and length. It is Greitzer’s parameter, from 1976, and it compares two things: the pressure the plenum can store, which grows with its volume, and the inertia of the air in the duct that has to be accelerated to change the flow.

The second figure is the experiment. The machine sits on the open throttle line with a small stall cell seeded, the throttle is closed over a hundred radii of rotor travel to a line crossing the characteristic at 86 per cent of the peak’s flow, and the equations are integrated. With B = 0.5 the flow falls from 0.54 and settles at 0.341, and the stall cell grows to an amplitude that swings the local flow round the annulus from 0.81 down to −0.12: over part of the annulus the air is going backwards through the blades while the average is steady. With B = 2 no stall cell survives. The average flow drops, reverses to −0.23, recovers to 0.76, falls again, and repeats with a period of 388 radii — for a rotor half a metre across at two hundred metres per second, about half a second.

Same compressor, same characteristic, same throttle. The only difference is B: a plenum sixteen times larger in volume against the same duct, or the same plenum behind a machine running four times as fast.

What surge is doing #

The third figure draws the same runs as paths in the plane of flow and pressure, which is where surge is easiest to read. The large loop has four legs and two speeds. From the peak the flow collapses almost horizontally — fast, because only the air in the duct has to decelerate, and at nearly constant pressure, because the plenum has not had time to empty. The flow arrives on the reversed-flow part of the characteristic, where air runs backwards through the machine, and the plenum empties through both the throttle and the compressor: pressure falls slowly down the left of the loop. Once it is low enough the reversed branch no longer exists at that pressure and the flow jumps back to the right, again fast, again at nearly constant pressure. Then the compressor refills the plenum and climbs slowly back up to the peak.

So large-B surge is a relaxation oscillation, two fast jumps and two slow legs, and its period is set by the slow legs — the time to fill and empty the plenum through the compressor — rather than by any resonance. The fill time grows as the square of B, and the computed period does too: 388 radii at B = 2, 1,265 at B = 4, a ratio of 3.3 on its way to four. The system’s natural frequency, that of the plenum as a spring and the duct’s air as a mass, is the Helmholtz frequency, and its period is only 201 radii at B = 2 and 402 at B = 4. A resonator of exactly this kind — a volume whose pressure acts on the mass of fluid in a neck — is the surge tank of a hydroelectric tunnel, which swings at its Helmholtz period and decays, because nothing in it has a rising characteristic to drive it. A Helmholtz resonator is a mass in a neck on a spring of trapped gas, and in a compressor the neck is the duct. At the onset of surge the oscillation grows at close to that resonator’s frequency; the rising characteristic feeds it, as a negative damping, in the way the airflow feeds a wing at its flutter speed. Once the oscillation is large it is no longer a resonance at all: the fast jumps and slow legs of the loop take over, and at large B the cycle’s period is its own.

Why B decides #

The onset of surge has a clean linear criterion, and it shows where B enters. Linearise the flow and plenum equations about an equilibrium on the characteristic, with $J = 0$. The equilibrium loses stability through an oscillation — a Hopf bifurcation, the same kind of transition by which a wake begins to shed — when

$$\frac{d\psi_c}{d\Phi} = \frac{1}{4B^2}\,\frac{d\Phi_T}{d\Psi},$$

the characteristic’s slope reaching the throttle’s inverse slope divided by $4B^2$. The computed eigenvalues of the linearised system cross into the right half-plane exactly there, to four parts in $10^{15}$ at three values of B. At small B the right-hand side is large, and the characteristic has to be climbing steeply — well past the peak — before the axisymmetric flow oscillates. At large B it is small, and surge begins almost at the peak itself.

The stall cell’s growth has no B in it at all. It begins as soon as the slope turns positive, at the peak, and grows at a rate set by the characteristic’s shape and the blades’ time lag. So as the throttle closes past the peak the two instabilities race. At small B the stall cell starts growing first and faster, the average flow drops as the cell grows, and by the time the machine reaches the part of the characteristic where surge could begin, the cell has taken it onto the stalled branch, where the characteristic’s slope is negative and the surge instability has nothing to feed on. At large B surge starts first; the flow collapses across the characteristic in a time too short for a cell to form, and the reversed flow at the bottom of the cycle — where the characteristic slopes the other way — kills any cell that has begun.

The fourth figure finds the boundary. For a throttle closed to 86 per cent of the peak flow, bisecting on the outcome puts the change at B = 1.03. Closed further or less far, the critical value moves between about 0.6 and 1.1 without leaving the neighbourhood of one, and it does not move monotonically: it is the outcome of a race between two instabilities growing from the same small disturbance, and the finish line moves as the throttle does. A critical B of order one is what Greitzer found in his experiments on a three-stage compressor in 1976, and it is what makes B useful as a design number: a compression system with B well below one will stall, one with B well above one will surge, and one near one can do either.

The same machine at two speeds #

B contains the blade speed and nothing else about the compressor’s aerodynamics. The characteristic, made dimensionless with the blade speed, is nearly the same curve at every speed — that is what similarity buys in a turbomachine — but B is not: it is proportional to $U$.

The fifth figure puts in hardware: a compressor with an annulus of 0.05 square metres and an effective duct length of 1.5 metres, discharging into plenums of 0.05, 0.5 and 5 cubic metres. With the smallest plenum B reaches only 0.54 at 450 metres per second, and the machine throttled past its peak would stall at any speed it can reach. With the largest it crosses one at 86 metres per second: above that speed it would surge. With the middle one it crosses at 272 metres per second, well inside its running range, and the same machine behind the same throttle stalls below that speed and surges above it.

That is the pattern test rigs have found since the 1950s and that Greitzer’s parameter explained. A laboratory compressor running slowly into a small volume stalls, and its stall cells can be watched with hot wires for hours. The same design built into an engine, at full speed and discharging into a combustor and the volume behind it, surges. Scaling up a compressor while keeping its characteristic does not keep its post-stall behaviour, and a rig whose B does not match the engine’s will show the wrong failure. The rule most often quoted — that low-speed machines stall and high-speed ones surge — is the same statement with B’s dependence on $U$ left implicit.

Stall is easy to enter and hard to leave #

The sixth figure closes the throttle slowly past the peak at B = 0.5 and then opens it again, with a standing disturbance of the size a real machine’s unsteadiness always supplies. Closing, the machine holds the unstalled characteristic a little past the peak’s throttle setting of 0.615, while the cell grows, and drops into stall at 0.608. Opening, it does not come back at 0.615. It stays on the stalled branch, with the flow well below the peak’s, until the throttle reaches 0.641, and then jumps back to the unstalled characteristic at a flow four per cent above the peak’s.

The recovery point is not arbitrary: it is where the throttle line is tangent to the stalled branch, 0.6404, beyond which the two no longer meet and the stalled state ceases to exist. Between 0.608 and 0.641 the machine has two steady states, and which one it is in depends on how it got there. This is the same geometry as the fold a self-heating film loses its steady state at, with the throttle in place of the load.

In an engine the “throttle” is the turbine and nozzle downstream of the compressor, and it cannot always be opened far enough. A stall that the engine’s own operating line cannot back out of is known as a hung or non-recoverable stall, and the only cure is to shut the engine down and restart it. The hysteresis in this figure is the mechanism in its simplest form.

What was checked #

The seventh figure lists the checks. The stalled state the integration settles into at small B was compared with the stalled characteristic, which the model gives in closed form, $\psi = \psi_{c0} + H\big(1 - \tfrac32 u + \tfrac52 u^3\big)$ with $u = \Phi/W - 1$, together with the cell amplitude $J = 4(1 - u^2)$; they agree to rounding. The surge onset was checked by computing the linearised system’s eigenvalues and finding, by bisection, the equilibrium at which their real part vanishes, at three values of B, against the closed-form slope criterion. The critical B was found by bisection on the outcome of the integration itself, which uses neither criterion. And the recovery from stall was checked against the tangency, computed by scanning the stalled characteristic without integrating anything.

What the picture cannot show #

One harmonic, one characteristic. The model keeps only the first circumferential harmonic of the flow, so its stall cell is a sinusoid rather than the sharp-edged cells real machines show, and it assumes the compressor responds to the local flow with a single time lag. Real stall cells come in numbers — one large cell, or several small ones — and part-span cells that affect only the tips; none is in the model.

A stated characteristic, including where it is never measured. The characteristic to the left of the peak, and especially in reversed flow, is extrapolated in almost every real machine, because running a compressor there is destructive. The model’s surge loop passes through the region nobody measures.

Incompressible flow in the compressor. Compressibility enters only through the plenum. In a high-speed multistage machine the density changes through the stages, which is what the matching essay is about, and the post-stall behaviour of individual stages can differ from the machine’s as a whole.

No inlet distortion, no tip clearance, no rotor–stator interaction. Each moves the peak and each can seed a cell; the model’s stall is seeded by a disturbance of stated size.

The convention the numbers depend on #

Flow and pressure-rise coefficients are made dimensionless with the blade speed at the mean radius, and time is in radii travelled by the blade, so a period of 388 is 388 radii of travel. The throttle coefficient $\gamma$ is the constant in $\Phi_T = \gamma\sqrt{\Psi}$; larger is more open. B is Greitzer’s definition, with the compressor duct’s effective length including the lengths upstream and downstream of the blading, and the speed of sound in the plenum. “Deep surge” is used for the cycle in which the average flow reverses, “classic surge” for an oscillation without reversal.

Who found it, and when #

Rotating stall was identified and explained by Howard Emmons and his colleagues at Harvard in 1955, who described the cell propagating from blade passage to blade passage as each stalled passage diverted flow onto its neighbour. Edward Greitzer’s two papers of 1976, on surge and rotating stall in axial compressors, introduced B and showed on a three-stage rig that it separated the two behaviours. Frank Moore and Greitzer’s model of 1986 combined both in the three equations used here, and it became the standard starting point for the active control of stall and surge that followed in the 1990s, in which the throttle, or a bleed valve or an injector, is driven fast enough to hold the machine at its peak.

Still open: holding the machine at the peak #

The peak of the characteristic is where a compressor is most useful — the highest pressure rise it can make — and where every instability here begins. Active control asks whether a fast actuator can hold the machine there: a bleed valve that opens when the flow starts to fall, or air injected at the rotor tips when a stall cell starts to grow. In the Moore–Greitzer model the question has a definite answer, because both instabilities have computed growth rates and the actuator enters the same equations. The next calculation adds a throttle driven by a feedback law on the measured flow and asks for the gain that stabilises the peak, how it scales with B, and whether the hysteresis loop drawn here can be closed — so that a stalled machine recovers without the throttle having to be opened past the tangency.