Visualizing low-flow stall, recirculation, and abrupt transition to attached flow
At low flow the pump can operate in a stalled, recirculating state. A part of the fluid leaving the rotor does not become useful discharge flow; it circulates internally from the high-pressure region back toward the suction side.
The motor can therefore consume shaft power while the measured through-flow remains low. Energy is not lost physically, but it is converted into recirculation, swirl, turbulence, vibration, and heat instead of useful flow.
When the internal stall pattern collapses, the passages reattach and losses drop. The same smooth speed increase can then produce an abrupt flow jump. With hysteresis enabled, the return to stall during ramp-down occurs at a lower speed.
Physical interpretation, simplified simulation model, practical implications, and supporting references
Pump characteristics are often drawn as smooth and single-valued curves. Real pumps can behave less cleanly at low flow. During start-up or speed ramp-up, axial and centrifugal pumps may enter a low-flow stalled state, where increasing speed does not immediately produce a proportional increase in measured through-flow.
The reason is internal hydraulics. At low through-flow, blade incidence becomes unfavourable, local separation appears, and recirculation develops near the rotor, impeller eye, casing wall, or discharge region. A part of the pump work is then spent on internal circulation rather than useful discharge flow.
When a critical condition is reached, the stalled structure can collapse. Flow reattaches, hydraulic losses decrease, and the measured flow can jump abruptly to a higher-flow branch even though pump speed changes smoothly.
The flow jump does not violate energy conservation. Before the jump, a larger part of the shaft power is dissipated in non-useful hydraulic motion:
After reattachment, the internal loss fraction drops and more of the same shaft power becomes useful hydraulic output. The observed discontinuity is therefore a redistribution of energy from losses to through-flow.
The transition point can depend on operating history. During ramp-up, the pump may remain stalled until the speed and flow exceed an attachment threshold. During ramp-down, an already attached flow field can remain attached down to a lower threshold.
In the simulator, the hysteresis slider controls the separation between these thresholds. With hysteresis set to zero, the two threshold pairs collapse toward a single transition region, but the dynamic stall variable can still create a short transient delay.
Axial pumps are sensitive to low axial velocity because the blade inlet angle depends strongly on the ratio between axial flow velocity and blade speed. When axial velocity decreases, the relative inlet velocity no longer matches the blade angle and separation can begin.
At low flow, inlet backflow and rotating stall can appear. In the axial cross-section, blue arrows represent net forward flow, while red loops represent recirculation from the downstream high-pressure side back toward the suction side.
In a centrifugal pump, fluid enters axially through the impeller eye and is accelerated radially outward into the volute. At low flow, the impeller still adds angular momentum, but the system does not accept the design discharge flow. This can produce inlet recirculation, discharge recirculation, blade-passage separation, and pressure pulsations.
The simulator combines these local mechanisms into one scalar recirculation variable, R. This is not a CFD model; it is a compact teaching model that shows how low-flow recirculation can produce a non-smooth pump response.
The simulator uses dimensionless variables. Speed is denoted by N, flow by Q, head by H, stall level by S, and recirculation fraction by R. The model is intentionally simple enough to be transparent, but it retains the main feedback loop between speed, flow, stall, recirculation, losses, and system resistance.
The numerical solution uses explicit Euler integration with sub-steps of at most 0.02 s. Reset starts from zero speed and zero through-flow, so the acceleration from standstill remains visible.
The Q–N plot is a pedagogical visualisation of the quasi-steady response. It uses a low-flow branch, a high-flow branch, and a steep logistic transition between them.
A small oscillatory perturbation is added near the transition region to make the unstable zone visually recognisable. It should be interpreted as an illustration of hydraulic instability, not as a calibrated pump curve.
| Parameter | Default | Description |
|---|---|---|
| \(a\) | 1.20 | Head coefficient for speed-squared term |
| \(b\) | 0.85 | Head reduction coefficient for flow-squared term |
| \(L_0\) | 0.05 | Base hydraulic loss fraction |
| \(L_s\) | 0.45 | Stall-related loss coefficient |
| \(L_r\) | 0.35 | Recirculation-related loss coefficient |
| \(H_{static}\) | 0.05 | Static system head |
| \(K_s\) | 0.75 | Quadratic system resistance coefficient |
| \(\tau_Q\) | 0.8 s | Flow response time constant |
| \(\tau_N\) | 1.2 s | Speed response time constant |
| \(\tau_S\) | 0.25 s | Stall-state response time constant |
| \(N_{attach}\) | 0.72 | Nominal attachment speed threshold |
| \(N_{stall}\) | 0.62 | Nominal stall-return speed threshold |
| \(Q_{attach}\) | 0.42 | Nominal attachment flow threshold |
| \(Q_{stall}\) | 0.28 | Nominal stall-return flow threshold |
| \(R_{max}\) | 0.75 | Maximum recirculation fraction |
| \(\eta_{max}\) | 0.85 | Maximum qualitative efficiency indicator |
In an installation, this type of behaviour can appear as a fault that is difficult to interpret if only a smooth pump curve is assumed. Typical symptoms include:
A practical diagnosis should compare speed, flow, differential pressure, vibration, and motor current. The key indicator is a smooth actuator or speed signal combined with a non-smooth hydraulic response.
This simulator is a teaching tool. It is not a vendor pump model, a CFD calculation, or a plant-specific performance guarantee.
For plant decisions, this type of simulator should be combined with vendor curves, measured data, minimum-flow protection review, and, where needed, CFD or physical testing.
The simulator also estimates shaft power and motor current for a simplified European industrial motor supply. The default electrical basis is three-phase 400 V / 50 Hz line-to-line supply.
During stalled low-flow operation, the motor current can be high compared with the useful hydraulic output because shaft power is dissipated internally. After the flow jump, current may settle at a higher absolute level but with more useful flow per ampere.
| Quantity | Typical value | Note |
|---|---|---|
| \(U_{LL}\) | 400 V | Line-to-line voltage for common European low-voltage three-phase supply |
| \(U_{LN}\) | 230 V | Line-to-neutral voltage, approximately \(400/\sqrt{3}\) |
| Frequency | 50 Hz | Nominal European grid frequency |
| \(\cos\varphi\) | 0.80–0.92 | Typical induction-motor full-load power factor |
| \(\eta_{motor}\) | 0.88–0.96 | Typical industrial motor efficiency range |
| Starting current | 5–8 × \(I_n\) | Typical direct-on-line starting range; VFD start is lower |