# What makes a flow periodic in Turbomachines?

A quote from my textbook :

Turbomachinery flows are steady only in a time-averaged sense; that is, the flow is periodic, with a period equal to the time taken for a blade to move a distance equal to the spacing between adjacent blades. Despite the unsteadiness, in elementary analysis all variables are assumed to have steady values.

The words unsteady and periodic have different meaning depending on the context they are used in. My answer is mainly with regards to axial or radial turbomachines.

However! There is one thing called the 'Unsteadiness Paradox' 1. It basically boils down to the fact that a turbomachine has to be unsteady in order to work. But the unsteadiness in this case refers to the rotation of the blade-row.

The figure shows a sketch from Greitzer. If one assumes the turbo-machine to be 'black box' which just adds or extracts work than it is not possible to describe the turbo-machines analytically. It only works when adding a stationary and rotating black-box.

Without any simplifications of the flow and the geometry it would not be possible to design a turbomachine and predict its operating characteristics.

Two things have to be simplified:

1. Aerodynamics
2. Geometry

In case of the aerodynamics the unsteadiness (eddies and turbulence) in the flow is usually averaged out. In case of the geometry one assumes periodicity (see the figure below from Oxford University):

For the version (a) only one passage was simulated and simply copied (all passages look the same). In version (b) all passages were simulated showing slight differences under certain operating conditions.

The second simplification to the geometry is the introduction of mixing planes so the rotating and stationary part of the turbomachine can be calculated separately (see figure below from Cambridge University):

The unsteadiness which is introduced by the relative motion of blades and vanes (right) is averaged-(smeared)-out (left).

These simplifications are necessary in order to have fast optimisation and development of the turbomachine and are later corrected and checked by experiments or high fidelity simulations if necessary.

1 Dean, R.C., "On the Necessity of Unsteady Flow in Fluid Machines", ASME J. Basic Eng. March 1959, pp 24-28

Hopefully this picture will help:

You can observe that the stream (blue) is divided by blades. This separation of the initial flow makes the flow unsteady and can be compared to pulses.

Impulse vs reaction turbines comparison:

• Even in case of an axial machine not an impulse one?
– Algo
Commented Apr 24, 2015 at 6:51
• Impulse turbines can be axial too, as in the second illustration, middle section. In a reaction turbine, the rotating blades alternately line up with the stationary blades, or with the gaps between them, introducing the same periodicity. Commented Apr 24, 2015 at 12:12
• @Algo, yes also axial or radial machines are basically unsteady. However, in order to design them a good way to start is to average the unsteadiness out between the stages. Commented Apr 24, 2015 at 14:52

I think what is meant in your textbook could be also induced velocities by a stator (as meaning of "adjacent blades"). Think of a multi-stage turbine with multiple rotor-stator stages: the flow will always be affected by the rate of turn of the turbine, which makes it unsteady (time dependent), because the rate of turn is a function of time.

Even with no stator, the inflow to your turbomachine will almost never be fully homogeneous, resulting in a variation in velocity over the duration of one turn. This means a blade of your turbomachine will experience different forces in one rotation due to altered velocities and pressures.