# Comparative study of different non-reflecting boundary conditions for compressible flows

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## Abstract

The numerical simulation of hydrodynamic stability and aeroacoustic problems requires the use of high-order, low-dispersion and low-dissipation numerical methods. It also requires appropriate boundary conditions to avoid reflections of outgoing waves at the boundaries of the computational domain. There are many different methods to avoid wave reflection at the boundaries such as the buffer zone and boundary conditions based on characteristic equations. This paper considers the use of a methodology called perfectly matched layer (PML). The PML is evaluated for the simulation of an acoustic pulse in a uniform flow and the Kelvin–Helmholtz instability in a mixing layer using the linear and nonlinear form of the Euler equation. PML results are compared with other non-reflecting boundary condition methods in terms of effectiveness and computational cost. The other non-reflecting boundary conditions implemented were the buffer zone (BZ), widely used in aeroacoustic and hydrodynamic problems, and the energy transfer and annihilation (ETA), a very simple boundary condition to be implemented. The results show that the PML is an effective boundary condition method, but can be computationally expensive. The PML is also more complex to implement and requires careful stability analysis. The other boundary conditions, the BZ and the ETA, are also effective and may perform better than the PML depending on the flow conditions. These two methods have an advantage in terms of robustness and are much simpler to implement than the PML.

## Keywords

PML Non-reflecting boundary conditions High-order numerical methods Euler equations Hydrodynamic stability## Notes

### Acknowledgements

This work was financed by CAPES – Brazilian Higher Education Improvement Coordination within the Ministry of Education.

## References

- 1.Bayliss A, Turkel E (1982) Far field boundary conditions for compressible flows. J Comput Phys 48(2):182–199MathSciNetCrossRefGoogle Scholar
- 2.Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:195–200MathSciNetCrossRefGoogle Scholar
- 3.Berland J, Bogey C, Bailly C (2006) Low-dissipation and low-dispersion fourth-order Runge–Kutta algorithm. Comput Fluids 35(10):1459–1463MathSciNetCrossRefGoogle Scholar
- 4.Bogey C, Bailly C (2004) A family of low dispersive and low dissipative explicit schemes for flow and noise computations. J Comput Phys 194(1):194–214CrossRefGoogle Scholar
- 5.Colonius T (2004) Modeling artificial boundary conditions for compressible flow. Annu Rev Fluid Mech 36:315–345MathSciNetCrossRefGoogle Scholar
- 6.Colonius T, Lele SK, Moin P (1993) Boundary conditions for direct computation of aerodynamic sound generation. AIAA 31(9):1574–1582CrossRefGoogle Scholar
- 7.Edgar NB, Visbal MR (2003) A general buffer zone-type non-reflecting boundary condition for computational aeroacoustics. In: AIAA paper 3300Google Scholar
- 8.Engquist B, Majda A (1977) Absorbing boundary conditions for numerical simulation of waves. Comput Phys 31:629–651MathSciNetzbMATHGoogle Scholar
- 9.Giles MB (1990) Non reflecting boundary conditions for Euler equation calculations. AIAA 28:12CrossRefGoogle Scholar
- 10.Hagstrom T, Haariharan SI, Thompson D (2003) High-order radiation boundary conditions for the convective wave equation in exterior domains. SIAM J Sci Comput 25(3):1088–1101. https://doi.org/10.1137/S1064827502419695 MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Hedstrom G (1979) Nonreflecting boundary conditions for nonlinear hyperbolic systems. J Comput Phys 30(2):222–237MathSciNetCrossRefGoogle Scholar
- 12.Hu FQ (1996) On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J Comput Phys 129(0244):201–219MathSciNetCrossRefGoogle Scholar
- 13.Hu FQ (2001) A stable, perfectly mathed layer for linearized Euler equations in unsplit physical variables. J Comput Phys 173:455–480MathSciNetCrossRefGoogle Scholar
- 14.Hu FQ (2005) A perfectly matched layer absorbing boundary condition for linearized Euler equations with a non-uniform mean flow. J Comput Phys 208:469–492MathSciNetCrossRefGoogle Scholar
- 15.Hu FQ, Hussaini MY, Manthey JL (1996) Low-dissipation and low-dispersion Runge–Kutta schemes for computational acoustics. J Comput Phys 124(1):177–191MathSciNetCrossRefGoogle Scholar
- 16.Israeli M, Orszag S (1981) Approximation of radiation boundary conditions. J Comput Phys 41:115–135MathSciNetCrossRefGoogle Scholar
- 17.Lin KD, Li X, Hu F (2011) Absorbing boundary condition for nonlinear Euler equations in primitive variables based on the perfectly matched layer technique. Comput Fluids 40:333–337MathSciNetCrossRefGoogle Scholar
- 18.Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103(1):16–42MathSciNetCrossRefGoogle Scholar
- 19.Manco JAA (2014) Non-reflecting boundary conditions for high order numerical simulation of compressible Kelvin–Helmholtz instability. Master’s degree in Spatial Engineering and Technology, Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, INPE-17465-TDI/2256Google Scholar
- 20.Manco JAA, Freitas RB, Fernandes LM, Mendonca MT (2015) Stability of compressible mixing layers modified by wakes and jets. Procedia IUTAM 14:129–136CrossRefGoogle Scholar
- 21.Morris PJ, Long LN, Scheidegger TE, Boluriaan S (2002) Simulations of supersonic jet noise. Int J Aeroacoust 1(1):17–41. https://doi.org/10.1260/1475472021502659 CrossRefGoogle Scholar
- 22.Richards S, Zhang X, Chen X, Nelson P (2004) The evaluation of non-reflecting boundary conditions for duct acoustic computation. J Sound Vib 270(3):539–557 (2002 I.M.A. Conference on Computational Aeroacoustics)CrossRefGoogle Scholar
- 23.Tam CKW, Webb JC (1993) Dispersion-relation-preserving schemes for computational acoustics. J Comput Phys 107(184):262–281MathSciNetCrossRefGoogle Scholar
- 24.Thompson KW (1990) Time-dependent boundary conditions for hyperbolic systems ii. J Comput Phys 89:439–461MathSciNetCrossRefGoogle Scholar
- 25.Wasistho B, Geurts BJ, Kuerten JGM (1997) Simulation techniques for spacially evolving instabilities in compressible flow over a flat plate. Comput Fluids 26(7):713–739CrossRefGoogle Scholar