Rotating Flows and Fluid Machinery

Introduction and Context

Rotating flows occur whenever fluid motion contains significant angular momentum about an axis. They are central to turbomachines—pumps, compressors, turbines, fans, and hydraulic machines—and also appear in cyclones, stirred reactors, atmospheric vortices, rotating tanks, and geophysical flows. Unlike purely translational flows, rotating flows involve centrifugal acceleration, Coriolis effects in rotating reference frames, angular-momentum transport, and often strong pressure gradients.

The analysis of rotating machinery combines conservation of mass, momentum, and energy with velocity triangles, similarity laws, boundary-layer concepts, and loss modelling. At postgraduate level, it is especially important to distinguish ideal Euler-fluid results from real-machine performance, where viscosity, separation, leakage, secondary flows, compressibility, cavitation, and unsteadiness influence the outcome.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Distinguish solid-body rotation, free-vortex motion, forced vortices, and potential vortices.
2. Derive pressure distributions in rotating fluids and interpret the role of vorticity.
3. Apply the continuity, angular-momentum, and energy equations to turbomachines.
4. Construct and interpret velocity triangles for pumps, turbines, compressors, and fans.
5. Derive Euler’s turbomachinery equation and relate it to machine head or specific work.
6. Evaluate ideal and actual machine performance using efficiencies and loss coefficients.
7. Explain similarity, scaling, cavitation, slip, surge, stall, and other practical limitations.
8. Select appropriate analytical and experimental approaches for rotating-flow problems.

1. Kinematics of Rotating Flow

For a velocity field V, the vorticity is

where ω⃗ is vorticity, with units s⁻¹. The local angular velocity of a fluid element is one-half the vorticity:

This distinction is important. A flow can follow curved streamlines without the fluid elements rotating locally; such a flow is called irrotational if its vorticity is zero.

For axisymmetric flow in cylindrical coordinates, let the radial, tangential, and axial velocity components be V_r, V_θ, and V_z. The circulation around a closed contour is

For a circular contour of radius r with purely tangential velocity,

Thus, constant circulation implies V_θ ∝ 1/r. This is the characteristic velocity law of a free vortex.

1.1 Solid-body or forced-vortex rotation

In solid-body rotation, all fluid elements rotate with the same angular velocity Ω. The tangential velocity is

The vorticity is non-zero and uniform. This flow is called a forced vortex because an external torque is required to maintain it. Examples include fluid in a rotating container after viscous effects have established near-solid-body motion.

1.2 Free-vortex or potential-vortex rotation

In an ideal free vortex,

where C is a constant with units m²/s. Hence

Except at the central singularity, the vorticity is zero. The circulation is constant, and no distributed torque is required in the inviscid core. Real vortices have a finite viscous core and cannot possess infinite velocity at r = 0.

1.3 Combined vortices

Practical swirling flows commonly contain a forced-vortex core and a free-vortex outer region. A simplified combined-vortex model is

and

where r_c is the core radius. Such profiles occur in cyclone separators, stirred vessels, and rotating machinery wakes.

2. Pressure Distribution in Rotating Fluids

Consider steady, axisymmetric rotation with negligible radial velocity. Radial equilibrium requires the pressure-gradient force to balance centrifugal acceleration:

where p is pressure and ρ is density. This equation assumes negligible radial acceleration other than the centripetal requirement and is applicable locally to steady circular motion.

For solid-body rotation, V_θ = Ωr, so

For constant density,

Pressure therefore increases outward. In a rotating tank, the free surface is not horizontal. Combining radial and vertical hydrostatic equilibrium gives

where z is elevation and g is gravitational acceleration. The free surface is a paraboloid of revolution.

For a free vortex, V_θ = C/r, and therefore

Integration gives

The pressure decreases toward the vortex centre. Compressibility must be included when pressure variations are large or when the tangential speed approaches a significant fraction of the speed of sound.

3. Rotating Reference Frames

A rotating coordinate system is useful for analysing impellers and rotors. If the frame rotates with angular velocity Ω⃗, the absolute velocity is

where W⃗ is relative velocity and r⃗ is position from the axis.

The acceleration observed in the rotating frame contains several terms:

The terms are, respectively, relative acceleration, Coriolis acceleration, centrifugal acceleration, and Euler acceleration. For constant-speed rotation, the Euler term vanishes. Coriolis acceleration influences secondary flows and boundary layers in rotating passages, while centrifugal acceleration creates radial pressure gradients.

The relative velocity is related to the blade speed U = Ωr through the velocity triangle:

For turbomachinery analysis, velocities are generally resolved into meridional or axial components and whirl components.

4. Conservation Laws for Turbomachines

4.1 Continuity

For steady flow through a control volume,

where ṁ is mass-flow rate in kg/s, A is flow area in m², and V_n is velocity normal to the area in m/s. For incompressible flow, ρ is constant; for gases, density variation must be considered.

4.2 Angular-momentum equation

The moment of momentum equation about the machine axis is

where T is torque exerted on the fluid, and subscripts 1 and 2 denote inlet and outlet. The corresponding power transferred to the fluid is

where ω is angular speed in rad/s and U = ωr. This result assumes steady flow, a one-dimensional mean-flow description, and torque evaluated about the axis.

4.3 Euler turbomachinery equation

The specific work transferred to the fluid is

For an incompressible pump, the ideal Euler head is

For a turbine, the same angular-momentum principle gives the specific work extracted from the fluid, with the sign interpreted according to the selected control-volume convention. For compressors, w_E is the ideal specific work input.

The equation reveals that energy transfer depends on the change in angular momentum, not simply on the change in speed. A rotor can therefore exert substantial power even when the magnitude of velocity changes modestly, provided the whirl component changes significantly.

4.4 Energy equation and stagnation quantities

For a steady adiabatic machine with negligible potential-energy change,

for work input to a compressor, where h₀ is stagnation enthalpy. For an ideal gas with constant specific heat,

where h is static enthalpy, V is absolute speed, c_p is specific heat at constant pressure, and T₀ is stagnation temperature. In real machines, losses increase entropy and cause the actual work or temperature rise to differ from the ideal value.

5. Velocity Triangles and Blade Design

At a rotor station, the absolute velocity V, blade speed U, and relative velocity W form a vector triangle. If α is the absolute-flow angle measured from the axial or meridional direction and β is the relative-flow angle, then

and

with sign conventions defined by the direction of rotation. The relative-flow angle satisfies

where V_m is the meridional velocity.

For a radial-flow impeller with no prewhirl at inlet, V_θ1 = 0, and the Euler head becomes

If the outlet blade angle is β₂ and the ideal relative exit velocity is aligned with the blade, one commonly obtains a relationship between Vθ₂, U₂, V_m₂, and β₂. The exact form depends on the angle convention and whether slip is neglected; therefore, velocity triangles should always be drawn before applying a formula.

Backward-curved blades generally produce a lower theoretical pressure rise than radial blades but often offer more stable power characteristics and improved efficiency. Forward-curved blades can produce high flow at low rotor speed but may have undesirable power-rise characteristics.

Figure: Vector relationship among absolute velocity V, blade speed U, and relative velocity W.

6. Performance of Pumps, Turbines, and Compressors

6.1 Pump performance

The hydraulic power delivered to an incompressible fluid is

where Q is volumetric flow rate in m³/s and H is developed head in m. Shaft efficiency is

where P_shaft is shaft power. A pump’s total head may be expressed as the Euler head minus hydraulic losses. Mechanical losses include bearing friction, seal friction, and disk friction; volumetric losses include leakage through clearances.

6.2 Turbine performance

For a hydraulic turbine, shaft power is commonly written

The overall efficiency includes hydraulic, mechanical, and volumetric effects. Reaction turbines experience pressure changes through the runner, whereas impulse turbines convert most available pressure head into jet kinetic energy before the runner.

6.3 Compressors and fans

For a compressor, the ideal temperature rise associated with pressure ratio r_p = p₂/p₁ for an isentropic ideal gas is

where γ is the ratio of specific heats. Isentropic compressor efficiency is

for a fixed pressure ratio. This definition assumes negligible changes in kinetic and potential energies between reference stations.

7. Similarity and Scaling Laws

Dimensionless parameters permit performance prediction across geometrically similar machines. Common turbomachine groups are the flow coefficient

and head coefficient

where N is rotational speed in s⁻¹ and D is characteristic diameter. Alternative definitions use angular speed and differ by numerical constants.

For geometrically similar incompressible machines operating at corresponding points,

These affinity laws require comparable geometry, fluid properties, and operating regime. They become unreliable when Reynolds number, surface roughness, compressibility, cavitation number, or efficiency changes substantially.

Specific speed is a classification parameter rather than a universal physical constant. Because several conventions exist, its definition must always be stated. It is used to compare machine types and to guide preliminary selection, not to replace detailed design analysis.

8. Losses and Flow Instabilities

Real rotating machines depart from ideal theory for several reasons:

• Blade-surface friction and boundary-layer growth.
• Incidence losses when inlet flow does not match blade angle.
• Flow separation and stall at excessive loading.
• Tip-clearance leakage and recirculation.
• Secondary flows caused by radial pressure gradients and end-wall effects.
• Wake mixing and non-uniformity at rotor exits.
• Disk friction and bearing losses.
• Surface roughness and manufacturing imperfections.

Slip in centrifugal pumps

The fluid leaving a finite-blade impeller does not exactly follow the ideal blade angle. The reduction in whirl velocity is called slip. Slip factors are empirical or semi-empirical and depend on blade number, blade geometry, loading, and clearance. Applying the ideal Euler equation without a slip correction generally overpredicts head.

Stall and surge

Stall is a local aerodynamic separation phenomenon, often associated with excessive incidence or adverse pressure gradients. Surge is a larger-scale, system-level oscillation involving flow reversal and pressure fluctuations, particularly in compressors. Stall can occur without surge, but severe rotating stall may trigger surge. Stable operating ranges must therefore be identified from maps rather than inferred solely from the design point.

Cavitation

Cavitation occurs when local absolute pressure falls below the liquid vapour pressure. A common pump criterion is net positive suction head available:

where p_abs,s is absolute pressure at the suction reference, p_v is vapour pressure, and V_s is suction velocity. The pump requires an NPSH value specified by the manufacturer, often called NPSH_R. Safe operation requires sufficient margin between available and required NPSH. The exact reference locations must be stated because NPSH definitions vary.

9. Worked Example: Euler Head of a Centrifugal Pump

A centrifugal pump impeller rotates at 1,450 revolutions per minute. The outlet radius is 0.15 m, the outlet absolute velocity has a whirl component of 18 m/s, and the inlet whirl component is zero. Determine the ideal Euler head, neglecting slip and losses.

Angular speed is

The outlet blade speed is

Since V_θ1 = 0,

Therefore,

The ideal Euler head is approximately 41.8 m of fluid. Actual delivered head will be lower because of slip, hydraulic losses, leakage, and mechanical effects. Dimensional checking gives (m/s)(m/s)/(m/s²) = m, as required.

10. Applied Case: Diagnosis of a Centrifugal-Pump System

Background

The following is an illustrative industrial case. A process plant reports that a centrifugal pump delivers less flow than its commissioning value and produces intermittent vibration. The pump handles water at approximately 25 °C and draws from an open storage tank.

Problem

The measured flow is 18% below the expected duty, suction pressure fluctuates, and broadband noise is observed near the impeller. Operators have increased rotational speed, but the symptoms have worsened.

Analysis

The first step is to establish whether the pump is operating at the intended point on its head–flow curve. Increasing speed shifts the curve upward but also increases suction losses and power demand. A suction pressure survey is required, together with verification of suction-pipe diameter, valve position, strainer condition, liquid temperature, and tank level.

The observed noise and fluctuation suggest cavitation or suction recirculation. The available NPSH should be calculated using absolute pressure and the actual suction velocity. The result must be compared with the manufacturer’s required NPSH at the measured flow, including a practical margin rather than relying on equality.

A second possibility is air entrainment caused by a vortex at the tank outlet. This can reduce effective density, disturb the velocity triangle, and create unsteady loading. A blocked suction strainer or partially closed valve can produce a similar pressure deficit.

Approach and solution

The plant team should first return the pump to the approved speed and measure flow, suction pressure, discharge pressure, temperature, and shaft power. The suction line should be inspected for blockage and leaks. If the NPSH margin is inadequate, possible remedies include raising the liquid level, reducing suction losses, lowering liquid temperature, increasing suction-pipe diameter, or selecting a pump with lower NPSH requirement. Speed increase should not be used as a substitute for correcting inadequate suction conditions.

If the system resistance has changed, the system curve should be reconstructed and intersected with the pump curve. If the pump has suffered erosion or wear-ring enlargement, an inspection may be necessary because volumetric leakage can reduce head and efficiency.

Outcome

After cleaning the suction strainer, restoring the tank operating level, and eliminating a small air leak at a flange, the pump returns near its intended duty and vibration decreases. The case illustrates that pump performance is a system property: the machine curve, system curve, suction conditions, and fluid state must be assessed together.

Learning points

1. Increased speed can aggravate cavitation.
2. NPSH must be based on absolute pressure and consistent reference locations.
3. Vibration is a symptom, not a diagnosis.
4. Pump selection and operation require simultaneous consideration of hydraulic, mechanical, and system constraints.

11. Computational and Experimental Methods

Modern rotating-flow analysis commonly combines one-dimensional mean-line design, throughflow methods, computational fluid dynamics, and experimental testing. Reynolds-averaged Navier–Stokes models can estimate performance but depend strongly on turbulence modelling, mesh quality, wall treatment, and treatment of rotating interfaces. Large-eddy simulation can resolve more unsteady structure but requires substantially greater computational resources.

Experimental validation may use pressure taps, torque measurement, laser Doppler velocimetry, particle-image velocimetry, high-speed imaging, strain gauges, and acoustic monitoring. Measurements should be uncertainty-quantified. In rotating systems, phase-locked measurements can distinguish blade-passing phenomena from broadband turbulence.

Artificial intelligence is increasingly used for surrogate modelling, fault detection, digital twins, and predictive maintenance. However, data-driven predictions should not be accepted without physical constraints, range-of-validity checks, sensor validation, and protection against extrapolation. In safety-critical machinery, explainability and traceability are essential.

12. Design Practice, Safety, and Sustainability

Sound design practice requires clearly stated sign conventions, reference stations, fluid properties, operating envelope, and assumptions. Pressure measurements should be distinguished as gauge or absolute. Rotational speed should be specified in revolutions per minute or rad/s, and the conversion should not be omitted.

Machinery must be evaluated for overspeed, pressure containment, fatigue, rotor dynamics, thermal expansion, emergency shutdown, and hazardous-fluid release. Standards and manufacturer requirements govern many details of testing, vibration limits, pressure vessels, and guarding; engineering calculations do not replace applicable regulations.

Energy efficiency has both economic and environmental significance. Avoiding throttling losses, selecting appropriately sized machines, using variable-speed drives where suitable, maintaining clearances, and operating near the best-efficiency point can reduce energy consumption. Nevertheless, variable-speed control may shift a system into a cavitation or instability region, so sustainability measures must be integrated with hydraulic and mechanical analysis.

Chapter Summary

Rotating flows are governed by angular momentum, vorticity, centrifugal pressure gradients, and—in rotating frames—Coriolis effects. Solid-body rotation has tangential speed proportional to radius, whereas a free vortex has tangential speed inversely proportional to radius. Radial equilibrium provides pressure distributions for both forms.

Turbomachinery transfers energy by changing the angular momentum of a fluid. The Euler equation, w_E = U₂V_θ2 - U₁V_θ1, is the central ideal relation for pumps, turbines, compressors, and fans. Velocity triangles connect absolute, relative, and blade velocities and support preliminary blade design. Actual performance is reduced by friction, separation, leakage, slip, secondary flows, mechanical losses, and compressibility.

Similarity laws enable preliminary scaling, but their validity depends on geometric and dynamic similarity. Cavitation, stall, surge, and suction recirculation define important operating limits. Reliable engineering practice combines analytical equations with system-curve analysis, validated computation, measurement, uncertainty assessment, and safety governance.

Key Terms

Vorticity: Curl of the velocity field; twice the local angular velocity of a fluid element.

Solid-body rotation: Rotation in which all fluid elements share the same angular velocity.

Free vortex: Ideal vortex with constant circulation and tangential velocity proportional to 1/r.

Circulation: Line integral of velocity around a closed contour.

Centrifugal acceleration: Apparent outward acceleration in a rotating reference frame.

Coriolis acceleration: Relative-frame acceleration proportional to angular velocity and relative velocity.

Whirl velocity: Tangential component of absolute fluid velocity.

Blade speed: Peripheral speed of a rotor, U = ωr.

Velocity triangle: Vector representation of absolute, relative, and blade velocities.

Euler turbomachinery equation: Relation between specific work and change in angular momentum.

Slip: Reduction of actual impeller-exit whirl relative to ideal blade-following flow.

Hydraulic efficiency: Ratio of useful fluid power to ideal or transferred hydraulic power, depending on convention.

Volumetric efficiency: Measure of the effect of leakage and flow-rate loss.

Flow coefficient: Dimensionless flow-rate parameter used in machine similarity.

Head coefficient: Dimensionless head parameter used in machine similarity.

Cavitation: Formation and collapse of vapour cavities when local pressure falls below vapour pressure.

NPSH: Net positive suction head, a measure of cavitation margin.

Stall: Local flow separation and loss of aerodynamic loading.

Surge: System-level oscillation involving substantial compressor-flow fluctuation or reversal.

Mean-line analysis: Reduced-order turbomachine method based on station-averaged flow quantities.

Review Questions

Short Answer Questions

1. What is the difference between solid-body rotation and free-vortex rotation?
2. Why does pressure increase radially outward in a forced vortex?
3. Define whirl velocity and blade speed.
4. State the Euler turbomachinery equation for specific work transfer.
5. What physical condition causes cavitation in a pump?

Long Answer Questions

1. Derive the pressure distribution for a constant-density fluid in solid-body rotation and explain the shape of its free surface.
2. Explain the derivation and physical meaning of the angular-momentum equation for a turbomachine.
3. Describe velocity triangles and show how they are used in centrifugal-pump analysis.
4. Discuss the principal hydraulic, mechanical, and volumetric losses in a real turbomachine.
5. Explain the limitations of affinity laws when predicting the performance of a geometrically similar machine.

Analytical / Application Questions

1. A pump rotates at 1,800 rpm. Its impeller outlet radius is 0.12 m, outlet whirl velocity is 15 m/s, and inlet whirl is zero. Calculate the ideal Euler head, neglecting losses.
2. A rotating tank contains water and operates at constant angular speed. Explain how pressure measurements can be used to verify solid-body rotation and identify departures caused by viscous or free-surface effects.
3. A compressor shows pressure oscillations and intermittent flow reversal after its speed is increased. Develop a diagnostic procedure that distinguishes rotating stall, surge, inadequate inlet conditions, and instrumentation error.

Multiple Choice Questions with Answers

Question 1

In solid-body rotation, the tangential velocity varies with radius as:

A. V_θ ∝ r

B. V_θ ∝ 1/r

C. V_θ = constant for all r

D. V_θ ∝ r²

Correct Answer: A. V_θ ∝ r

Question 2

For a free vortex with constant circulation, which quantity remains constant?

A. V_θ/r

B. rV_θ

C. r²V_θ

D. V_θ²/r

Correct Answer: B. rV_θ

Question 3

The radial pressure-equilibrium equation for steady circular motion is:

A. dp/dr = ρr/V_θ²

B. dp/dr = ρV_θ²/r

C. dp/dr = ρg

D. dp/dr = 0 for every rotating flow

Correct Answer: B. dp/dr = ρV_θ²/r

Question 4

The ideal specific work transferred in a turbomachine is:

A. U_2V_θ2 + U_1V_θ1

B. U_2V_θ2 − U_1V_θ1

C. U_2V_m2 − U_1V_m1

D. g(H_1 − H_2) for every machine

Correct Answer: B. U_2V_θ2 − U_1V_θ1

Question 5

A pump has Q = 0.020 m³/s, H = 30 m, ρ = 1000 kg/m³, and shaft power 8 kW. Its overall efficiency is approximately:

A. 52.4%

B. 65.0%

C. 73.6%

D. 81.7%

Correct Answer: C. 73.6%

Question 6

For geometrically similar incompressible machines at corresponding operating points, flow rate scales approximately as:

A. N²D²

B. ND³

C. N³D⁵

D. N/D³

Correct Answer: B. ND³

Question 7

NPSH is primarily used to assess the risk of:

A. Rotor fatigue

B. Cavitation

C. Blade slip

D. Bearing misalignment

Correct Answer: B. Cavitation

Question 8

Which acceleration appears because a fluid particle moves relative to a rotating frame?

A. Coriolis acceleration

B. Gravitational acceleration only

C. Euler acceleration only

D. Acoustic acceleration

Correct Answer: A. Coriolis acceleration

Question 9

For a pump with no inlet whirl, the ideal Euler head for a radial impeller is proportional to:

A. U_2V_θ2/g

B. U_1V_m1/g

C. V_m2²/(2g) only

D. ρQ/g

Correct Answer: A. U_2V_θ2/g

Question 10

A compressor instability involving large-scale flow oscillation and possible flow reversal is called:

A. Slip

B. Surge

C. Solid-body rotation

D. Free-vortex motion

Correct Answer: B. Surge