My latest work on “Acoustic boundary conditions for can-annular combustors’’ was published last month in the International Journal of Turbomachinery Propulsion and Power, following presentation at the 15th European Turbomachinery Conference in Budapest, Hungary. In this post I will explain the motivation and key results of the paper.

What is the problem?

As discussed in previous posts, I have been working on prediction of sound wave reflection, or ‘acoustic impedance’, from turbines for several years now. When designing a gas turbine combustor, accurate predictions onf reflections are required to check for thermoacoustic instability — if the turbine is too reflective, pressure oscillations will grow in a positive feedback loop to dangerous levels, which must be avoided at all costs!

My previous work has only considered the ‘plane-wave’ case, where the incident sound waves are axisymmetric around the annulus of the turbine. This a good starting point, being easier to analyse, and has direct applications to the annular combustors used in aero engines. Industrial gas turbines, however, use can-annular combustors. In a can-annular combustor, individual burners are placed into discrete cans, which are connected to the turbine annulus via transition ducts. Not only do sound waves propagate towards the turbine in the streamwise direction, but also between cans in the azimuthal direction.

So the task now is to predict the reflection and transmission of sound waves between cans, given that we already know the impedance of the downstream turbine to plane waves.

von Saldern et al. ¹ exploited the azimuthal periodicity of the can-annular combustor, by expressing pressure fluctuations as a Fourier series, to arrive at closed-form analytical expressions for can transfer functions. Their model, however, assumes the downstream turbine is perfectly reflective, and also neglects the mean flow.

This is where my paper comes in. I present a more general derivation of the von Saldern et al. model, and, importantly, use time-marching computational fluid dynamics (CFD) simulations to validate analytical predictions in a representative industrial gas turbine.

Analytical model derivation

The analytical model makes four assumptions

1. The geometry is small compared to the acoustic wavelength. This is true in the low-frequency limit; fortunately, combustion instabilities in large industrial gas turbines tend to occur at sufficiently low frequencies to make this a reasonable approximation.
2. All cans in the annulus are identical. This means we can work in terms of a Fourier series of azimuthal modes.
3. There is no mean flow. The expression for conservation of mass is simplified by neglecting mean flow. It turns out that this simplification is OK in practice.
4. The acoustic impedance of the downstream turbine is known. The turbine impedance could be found using the analytical models I have previously developed ²³, but the paper uses CFD results for a clean validation of the combustor model.

The final expression for acoustic impedance at a given azimuthal mode \(m\) is,

$$ \newcommand{\Zed}{\mathcal{Z}} \newcommand{\Tra}{\mathcal{T}} \newcommand{\Rf}{\mathcal{R}} \newcommand{\can}{\mathrm{can}} \newcommand{\gap}{\mathrm{gap}} \newcommand{\Acan}{A_\can} \newcommand{\Agap}{A_\gap} \newcommand{\plane}{\mathrm{plane}} \Zed_m = \left[\frac{1}{\Zed_\plane} + \frac{1}{\Zed_{\can,m}}\right]^{-1}, \tag{I} $$ where \(\Zed_\plane\) is the (known) plane-wave impedance of the downstream turbine, and the combustor cans impedance, $$ \Zed_{\can,m} = \frac{A_\can}{A_\gap}\frac{\zeta }{4 \sin^2 \frac{m \pi}{N}} \ . \tag{II} $$ Equation (I) shows that the effect of adding the combustor cans is to place an additional impedance in parallel with the downstream turbine impedance. From Eqn. (II), this can contribution is a function of the area ratio \(A_\can/A_\gap\), azimuthal mode number \(m/N\), and an empirical connection impedance parameter \(\zeta\).

The connection impedance is a complex number that quantifies the magnitude and phase delay of the acoustic velocity between two combustor cans, given a pressure difference between those cans. The question now is, how do we assign a value for \(\zeta\,\)?

Calibrating connection impedance

At this point, we can turn to time-marching computational fluid dynamics. The basic approach is to apply time-varying inlet boundary conditions and solve the unsteady Reynolds-averaged Navier–Stokes equations for the flow in the combustor and turbine. Then, extracting the acoustic response from pressure fluctuations observed in the computation allows both connection impedance, \(\zeta\), and the overall impedance \(\Zed_m\) to be found.

The paper shows that the imaginary part of \(\zeta\), quantifying inertia, is closely approximated by a classical analytical expression derived by Lord Rayleigh. This is the form of \(\zeta\) that von Saldern et al. used. However, the simulations in the paper show that \(\zeta\) also has a real part, quantifying acoustic losses, which takes a value of order 0.1 across a range of combustor geometries. Including the real part of \(\zeta\) is required to get good predictions from the analytical model in real turbines.

Validating can transfer functions

Now that we know the value that \(\zeta\) should take, we can validate the analytical model predictions.

We can take the inverse discrete Fourier transform of \(\Zed_m\) to convert Eqn. (I) into a more convenient form of can-to-can transfer functions \(\Tra_k\). The can transfer functions are defined such that, if we inject an acoustic wave of unit amplitude into a single can, then the reflected wave in the same can is \(\Tra_0\), the reflected wave in the adjacent can is \(\Tra_1\), two cans away \(\Tra_2\) and so on. These are the quantities that are compared between analytical model and time-marching simulations for validation.

The agreement shown in the figure is good at low frequencies, within 0.03 magnitude. At higher frequencies, however, the analytical model diverges from the time-marching computations because the compact assumption is no longer valid.

Design space study

Now that we have some confidence in our analytical model, at least at low frequencies, we can use it to explore the design space. Results in the paper show that, for a given frequency, there is a non-zero optimum combustor-turbine gap length for minimum reflection — when the contributions of both terms in Eqn. (1) are in antiphase.

By varying the number of cans, connection impedance, and downstream turbine reflection coefficient, we arrive at a simple design guideline for optimum gap length, $$ \mathrm{Re}(\zeta) \Rf_\plane {A_\can}/{A_\gap} \approx 1.91 \tag{III}\ . $$ Equation (III) allows an engineer to tune the combustor turbine gap to minimise reflections, and make the task of maintaining thermoacoustic stability easier.

Conclusion

This paper wraps up my acoustics postdoc. I think what has made the project successful is the crossover of knowledge between turbomachinery aerodynamics and combustor thermoacoustics. Previously, acousticians would approximate the downstream turbine with a simple choked nozzle or prescribed value of reflection coefficient, turbines not being their area of interest. Now, I have developed models to predict reflections from a given turbine more accurately, and also more is known about how the design of the turbine and combustor affects reflections.

You can read the full paper in the journal; my research page contains a listing of all my publications.