Numerical Solution of the Linearized Euler Equations Using Compact Schemes

Kris Van Den Abeele, Jan Ramboer, Ghader Ghorbaniasl, Christian Lacor

Onderzoeksoutput: Conference paper

Samenvatting

Computational Aeroacoustics (CAA) is a research field that focuses on the simulation of
sound. This is achieved by numerically solving the governing equations of sound. In its
most general form, these equations are the Navier-Stokes equations. However, for the
simulation of the propagation of sound from a known source, a simplified form can be
used without significant loss of accuracy. The linearized Euler equations are often used
for this purpose. During the present study a code for solving the linearized 2D Euler
equations was developed, as well as a code for calculating rotor noise. The calculation of
the rotor noise is done by solving a set of linearized 3D Euler equations that have been
reduced to 2D.
The discretization used is the Finite Volume method, with a fourth-order accurate compact
scheme [1] for the calculation of the fluxes through the cell faces. The time integration
is achieved with a fourth-order accurate low storage Runge Kutta algorithm.
In CAA the boundary conditions are very important, because the generation of nonphysical
reflections at the boundaries of the computational domain must be minimized.
Two different sets of non-reflecting boundary conditions were implemented and tested
during this study. The first is a set of radiation and outflow boundary conditions that
are based on an asymptotic solution [2]. The second set of boundary conditions is based
on the characteristic solutions of the linearized Euler equations. Although these sets of
boundary conditions produce fairly good results, some high-frequent reflections are generated
by them nonetheless. To dissipate these reflections Artificial Selective Damping [2]
is implemented in the code.
The code for solving the linearized Euler equations was tested by means of two numerical
experiments from the second workshop on benchmark problems for CAA [3]. Furthermore,
the sound radiation from a dipole and a quadrupole source [5] was calculated using
this code as well. Good results were achieved, especially with the boundary conditions
based on an asymptotic solution.
Finally, a benchmark problem from the third workshop on benchmark problems for CAA
[4] was solved on a non-uniform cartesian mesh using the code for the calculation of rotor
noise. For a rotor with subsonic as well as supersonic tip speed results were found that
showed a good match with the exact solution.

References
[1] C. Lacor, S. Smirnov, M. Baelmans, A finite volume formulation of compact central
schemes on arbitrary structured grids, J. Comput. Phys. 198 (2004) 535-566.
[2] C. K. W. Tam, Numerical Methods in Computational Aeroacoustics, AIAA Short
Course Computational Aeroacoustics: Methods and Applications, Lahaina, Hawaii,
10-11 June 2000.
[3] Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems
(1996), NASA Conference Publication 3352, Eds. J.C. Hardin, J.R. Ristorcelli and
C.K.W. Tam, J.C. See also NASA CP 3300.
[4] Third Computational Aeroacoustics (CAA) Workshop on Benchmark Problems
(2000), NASA CP 2000-209790.
[5] C. Bailly and D. Juv´e, Numerical Solution of Acoustic Propagation Problems Using
Linearized Euler Equations, AIAA Journal, Vol. 38, No. 1, 2000, pp. 22-29.
Originele taal-2English
TitelProceedings International Conference for Computational Fluid Dynamics (ICCFD)
StatusPublished - 12 jul 2006
EvenementUnknown - Stockholm, Sweden
Duur: 21 sep 200925 sep 2009

Conference

ConferenceUnknown
Land/RegioSweden
StadStockholm
Periode21/09/0925/09/09

Bibliografische nota

Proceedings International Conference for Computational Fluid Dynamics (ICCFD)

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