Helmholtz’s decomposition for compressible flows and its application to computational aeroacoustics

Stefan Schoder (), Klaus Roppert and Manfred Kaltenbacher
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Stefan Schoder: TU Graz, Institute of Fundamentals and Theory in Electrical Engineering
Klaus Roppert: TU Graz, Institute of Fundamentals and Theory in Electrical Engineering
Manfred Kaltenbacher: TU Graz, Institute of Fundamentals and Theory in Electrical Engineering

Partial Differential Equations and Applications, 2020, vol. 1, issue 6, 1-20

Abstract: Abstract The Helmholtz decomposition, a fundamental theorem in vector analysis, separates a given vector field into an irrotational (longitudinal, compressible) and a solenoidal (transverse, vortical) part. The main challenge of this decomposition is the restricted and finite flow domain without vanishing flow velocity at the boundaries. To achieve a unique and $$L_2$$ L 2 -orthogonal decomposition, we enforce the correct boundary conditions and provide its physical interpretation. Based on this formulation for bounded domains, the flow velocity is decomposed. Combining the results with Goldstein’s aeroacoustic theory, we model the non-radiating base flow by the transverse part. Thereby, this approach allows a precise physical definition of the acoustic source terms for computational aeroacoustics via the non-radiating base flow. In a final simulation example, Helmholtz’s decomposition of compressible flow data using the finite element method is applied to an overflowed rectangular cavity at Mach 0.8. The results show a reasonable agreement with the source data and illustrate the distinct parts of the Helmholtz decomposition.

Keywords: Helmholtz’s decomposition; Compressible flows; Aeroacoustic; FEM; 76-10; 68-04 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00044-w

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