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Adjoint-based parametric sensitivity analysis for swirling M-flames

Published online by Cambridge University Press:  21 November 2018

Calum S. Skene  and
Peter J. Schmid Open the ORCID record for Peter J. Schmid [Opens in a new window]
Calum S. Skene*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

A linear numerical study is conducted to quantify the effect of swirl on the response behaviour of premixed lean flames to general harmonic excitation in the inlet, upstream of combustion. This study considers axisymmetric M-flames and is based on the linearised compressible Navier–Stokes equations augmented by a simple one-step irreversible chemical reaction. Optimal frequency response gains for both axisymmetric and non-axisymmetric perturbations are computed via a direct–adjoint methodology and singular value decompositions. The high-dimensional parameter space, containing perturbation and base-flow parameters, is explored by taking advantage of generic sensitivity information gained from the adjoint solutions. This information is then tailored to specific parametric sensitivities by first-order perturbation expansions of the singular triplets about the respective parameters. Valuable flow information, at a negligible computational cost, is gained by simple weighted scalar products between direct and adjoint solutions. We find that for non-swirling flows, a mode with azimuthal wavenumber $m=2$ is the most efficiently driven structure. The structural mechanism underlying the optimal gains is shown to be the Orr mechanism for $m=0$ and a blend of Orr and other mechanisms, such as lift-up, for other azimuthal wavenumbers. Further to this, velocity and pressure perturbations are shown to make up the optimal input and output showing that the thermoacoustic mechanism is crucial in large energy amplifications. For $m=0$ these velocity perturbations are mainly longitudinal, but for higher wavenumbers azimuthal velocity fluctuations become prominent, especially in the non-swirling case. Sensitivity analyses are carried out with respect to the Mach number, Reynolds number and swirl number, and the accuracy of parametric gradients of the frequency response curve is assessed. The sensitivity analysis reveals that increases in Reynolds and Mach numbers yield higher gains, through a decrease in temperature diffusion. A rise in mean-flow swirl is shown to diminish the gain, with increased damping for higher azimuthal wavenumbers. This leads to a reordering of the most effectively amplified mode, with the axisymmetric ($m=0$) mode becoming the dominant structure at moderate swirl numbers.

JFM classification

Mathematical Foundations: Computational methods Instability: Instability Reacting Flows: Reacting Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 859 , 25 January 2019 , pp. 516 - 542
Copyright
© 2018 Cambridge University Press 

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Skene and Schmid supplementary movie 1

The optimal forcing for $\textit{St}=4.21$, $m=0$ ($u'$) shown for 15 time units.

Download Skene and Schmid supplementary movie 1(Video)
Video 3.1 MB

Skene and Schmid supplementary movie 2

The optimal output for $\textit{St}=4.21$, $m=0$ ($E'$) shown for 15 time units.

Download Skene and Schmid supplementary movie 2(Video)
Video 2.1 MB

Skene and Schmid supplementary movie 3

The optimal forcing for $\textit{St}=8.41$, $m=0$ ($u'$) shown for 15 time units.

Download Skene and Schmid supplementary movie 3(Video)
Video 3.2 MB

Skene and Schmid supplementary movie 4

The optimal output for $\textit{St}=8.41$, $m=0$ ($E'$) shown for 15 time units.

Download Skene and Schmid supplementary movie 4(Video)
Video 2.3 MB