The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion

C.J. Greenshields, J.M. Reese

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

Abstract

Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007), we found these modifications gave good predictions of the viscous structure of shock waves in argon in the range Mach 1.0-12.0 (while conventional Navier-Stokes equations are known to fail above about Mach 2). However, some areas of concern with this model were a somewhat arbitrary choice of modelling coefficient, and potentially unphysical and unstable solutions. In this paper, we therefore present slightly different modifications to include molecule mass diffusion fully in the Navier-Stokes equations. These modifications are shown to be stable and produce physical solutions to the shock problem of a quality broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. The modifications primarily add a diffusion term to the mass conservation equation, so are at least as simple to solve as the Navier-Stokes equations; there are none of the numerical implementation problems of conventional extended hydrodynamics models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.
LanguageEnglish
Title of host publication2nd European Conference on Aero-Space Sciences (EUCASS)
Number of pages8
Publication statusPublished - Jul 2007
Event 2nd European Conference on AeroSpace Sciences (EUCASS), 2007 - Brussels, Belgium
Duration: 1 Jul 20076 Jul 2007

Conference

Conference 2nd European Conference on AeroSpace Sciences (EUCASS), 2007
Abbreviated titleEUCASS 2007
CountryBelgium
CityBrussels
Period1/07/076/07/07

Fingerprint

hypersonic shock
Navier-Stokes equation
shock waves
Burnett equations
hydrodynamics
nonequilibrium flow
conservation equations
shock
argon
boundary conditions
gradients
fluids
coefficients
predictions
molecules

Keywords

  • fluid dynamics
  • aerospace
  • structures
  • composites
  • design engineering

Cite this

Greenshields, C. J., & Reese, J. M. (2007). The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion. In 2nd European Conference on Aero-Space Sciences (EUCASS)
Greenshields, C.J. ; Reese, J.M. / The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion. 2nd European Conference on Aero-Space Sciences (EUCASS). 2007.
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Greenshields, CJ & Reese, JM 2007, The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion. in 2nd European Conference on Aero-Space Sciences (EUCASS). 2nd European Conference on AeroSpace Sciences (EUCASS), 2007, Brussels, Belgium, 1/07/07.

The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion. / Greenshields, C.J.; Reese, J.M.

2nd European Conference on Aero-Space Sciences (EUCASS). 2007.

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

TY - GEN

T1 - The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion

AU - Greenshields, C.J.

AU - Reese, J.M.

PY - 2007/7

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N2 - Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007), we found these modifications gave good predictions of the viscous structure of shock waves in argon in the range Mach 1.0-12.0 (while conventional Navier-Stokes equations are known to fail above about Mach 2). However, some areas of concern with this model were a somewhat arbitrary choice of modelling coefficient, and potentially unphysical and unstable solutions. In this paper, we therefore present slightly different modifications to include molecule mass diffusion fully in the Navier-Stokes equations. These modifications are shown to be stable and produce physical solutions to the shock problem of a quality broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. The modifications primarily add a diffusion term to the mass conservation equation, so are at least as simple to solve as the Navier-Stokes equations; there are none of the numerical implementation problems of conventional extended hydrodynamics models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.

AB - Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007), we found these modifications gave good predictions of the viscous structure of shock waves in argon in the range Mach 1.0-12.0 (while conventional Navier-Stokes equations are known to fail above about Mach 2). However, some areas of concern with this model were a somewhat arbitrary choice of modelling coefficient, and potentially unphysical and unstable solutions. In this paper, we therefore present slightly different modifications to include molecule mass diffusion fully in the Navier-Stokes equations. These modifications are shown to be stable and produce physical solutions to the shock problem of a quality broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. The modifications primarily add a diffusion term to the mass conservation equation, so are at least as simple to solve as the Navier-Stokes equations; there are none of the numerical implementation problems of conventional extended hydrodynamics models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.

KW - fluid dynamics

KW - aerospace

KW - structures

KW - composites

KW - design engineering

UR - http://arxiv.org/abs/0706.0141

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Greenshields CJ, Reese JM. The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion. In 2nd European Conference on Aero-Space Sciences (EUCASS). 2007