# New paper on acoustic impedance of 3D turbines

I have submitted a new paper entitled “The acoustic impedance of
three-dimensional turbines” to the *Journal of Sound and
Vibration*.
This post contains a short summary of the motivation for the work and important
results. Interested readers can check out the
preprint for full details. [**Update 2022-09-28:** The final published version is now available.]

## Motivation

When designing a gas turbine combustor, an engineer needs to know how reflective the downstream turbine is to sound waves. The amount of reflected sound determines thermoacoustic stability, or, put another way, whether the combustor blows up or not. So there is a need for methods that can predict the reflectivity, or equivalently “impedance”, of a new turbine design. This helps to avoid the hassle of a combustor exploding once built and tested.

The new work follows on from my last paper presented at the 2021 European Turbomachinery Conference. That paper used brute-force computational fluid dynamics (CFD) simulations to validate a simple and elegant mathematical model. Then, I used the analytical model to explore trends in impedance across a range of turbine designs.

In the last paper, only two-dimensional turbines were considered. 2D turbines are mathematically convenient, but of course fictional. Real turbines, as all other machines an engineer might build, are three-dimensional. Moving towards more realistic turbines also gives additional degrees of freedom to the designer.

The research questions are then:

- Does the 2D analytical model work for real 3D turbines?
- What is the effect of 3D design parameters on acoustic impedance?

## A representative multi-stage turbine

To get as close as possible to actual machines, I tried the analytical model on a multi-stage turbine (eight blade rows) representative of large industrial gas turbines. This is a good test that stretches the assumptions of the model, because the turbine has three-dimensional blading, cooling and leakage flows.

Happily, the model is in good agreement with CFD. I also checked out variations in operating condition away from the design point—for example if the power output of the gas turbine is reduced during periods of low electricity demand. The operating conditions made no difference, so we can take impedance as a constant during off-design operation.

## Parametric turbine stages

To investigate three-dimensional geometry effects, I implemented a bit of a
turbine design system. Given a high-level characterisation of the *aerodynamics*
of the turbine, the system generates the *geometry* of annulus line and blade shapes. The
point is to be able to make sets of parametric turbine stages, varying just one
thing at a time while keeping everything else constant. It is quite hard to do
that without designing the turbines effectively from scratch.

The results show:

- Acoustic impedance is insensitive to radial blade height. Broadly, this means that as long as we have the 2D-averaged flow correct, variations along the third radial dimension are not important.
- Changing aerofoil shape to do more turning around the front of the blades increases acoustic impedance. This implies that the turbine designer does have a degree of control over the impedance even when the 2D-averaged flow is fixed.
- We can tweak the acoustic impedance by changing the gap between turbine blade rows. This is because the reflected wave is a sum of two contributions, one from each row.

## Summing up

Overall, the analytical model captures variations in acoustic impedance over a wide range of realistic turbine designs. This suggests that all important physics is contained within the model, and the tool is of practical use for combustion engineers.

There are full references for all my publications on the Research page.