A Brinkman penalization method for compressible flows in complex geometries

• Published in: Journal of computational physics Vol. 227; no. 2; pp. 946 - 966
• Main Authors: Liu, Qianlong, Vasilyev, Oleg V.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 10-12-2007; Elsevier
• Subjects: Amplitude and phase errors; Asymptotic analysis; Brinkman penalization; Complex geometry; Compressible flow; Compressible Navier–Stokes equations; Computational techniques; Exact sciences and technology; Immersed boundary method; Mathematical methods in physics; Physics; Porous media; Theory of acoustics; Porous media; Complex geometry; Theory of acoustics; Compressible Navier–Stokes equations; Immersed boundary method; Brinkman penalization; Asymptotic analysis; Compressible flow; Amplitude and phase errors; Continuity equations; Strong convergence; Porous medium; Complexity; Convergence rate; Mathematical models; Calculation; Analytical solution; Compressible Navier-Stokes equations; Digital simulation; Permeability; Calculation methods; Incompressible flow; Energy equation; Boundary layers; Viscous flow; Navier-Stokes equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2007.07.037

To simulate flows around solid obstacles of complex geometries, various immersed boundary methods had been developed. Their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non-body conformal Cartesian grids. The Brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations. Its main idea is to model solid obstacles as porous media with porosity,
ϕ, and viscous permeability approaching zero. It has the pronounced advantages of mathematical proof of error bound, strong convergence, and ease of numerical implementation with the volume penalization technique. In this paper, it is extended to compressible flows. The straightforward extension of penalizing momentum and energy equations using Brinkman penalization with respective normalized viscous,
η, and thermal,
η
T, permeabilities produces unsatisfactory results, mostly due to nonphysical wave transmissions into obstacles, resulting in considerable energy and mass losses in reflected waves. The objective of this paper is to extend the Brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media. In addition to penalizing momentum and energy equations, the continuity equation for
porous media is considered inside obstacles. In this model, the penalized porous region acts as a high impedance medium, resulting in negligible wave transmissions. The asymptotic analysis reveals that the proposed Brinkman penalization technique results in the amplitude and phase errors of order O((
ηϕ)
1/2) and O((
η/
η
T)
1/4
ϕ
3/4), when the boundary layer within the porous media is respectively resolved or unresolved. The proposed method is tested using 1- and 2-D benchmark problems. The results of direct numerical simulation are in excellent agreement with the analytical solutions. The numerical simulations verify the accuracy and convergence rates.