An Arbitrary Lagrangian–Eulerian framework for the consistent analysis of entropy wave generation

Moritz Merk, Alexander J. Eder, Wolfgang Polifke

Research output: Contribution to journalArticlepeer-review


Entropy waves are generated in many technically relevant flow processes such as combustion, mixing, or convective heat transfer. When accelerated, entropy waves generate acoustic waves that contribute to the overall sound emission and can lead to self-excited thermoacoustic instabilities, especially at low frequencies. In order to reduce or prevent these undesirable byproducts of the flow, an understanding of the generation mechanisms of entropy waves is key. This study derives the analytical source terms of entropy disturbances for moving sources in general three-dimensional reactive flows. In this general setup, the consistent derivation of the generation mechanisms requires the tracking of the moving source for which an Arbitrary Lagrangian–Eulerian (ALE) framework is utilized. The derived differential equations provide a fundamental understanding of the underlying source mechanisms. In addition, the general three-dimensional differential equations are reduced to a quasi-one-dimensional jump condition to unify the analysis of the entropy wave generation. This unified framework is used for an in-depth analysis of a premixed flame, where all source terms that generate entropy disturbances are analyzed and their relative importance are quantified. The dominant contribution of unsteady heat addition per unit mass to the generation of entropy waves is reaffirmed for lean premixed flames. Finally, by comparison with the entropy generation mechanisms of a heated gauze at rest, it is emphasized once more that a heat source at rest is an invalid model for a premixed flame. Novelty and significance statement The analytical terms of entropy wave generation are derived consistently for moving sources in a general three-dimensional reactive flow. It is shown and emphasized that a consistent derivation requires the tracking of the local entropy sources. Therefore, an Arbitrary Lagrangian–Eulerian framework is used. The derived equations provide a fundamental insight into the generation of entropy disturbances in reactive flows, e.g. premixed flames. Furthermore, the derived equations can be seen as a starting point to consistently extract sources of entropy waves from numerical simulations.

Original languageEnglish
Article number113334
JournalCombustion and Flame
StatePublished - Apr 2024


  • Arbitrary Lagrangian–Eulerian framework
  • Entropy wave generation
  • Moving sources
  • Quasi-one-dimensional jump condition
  • Three-dimensional reactive flows


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