The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations

Christopher Greenshields, Jason M. Reese

Research output: Contribution to journalArticle

60 Citations (Scopus)

Abstract

Brenner (Physica A, vol. 349, 2005a, b, pp. 11, 60) has recently proposed modifications to the Navier-Stokes equations that are based on theoretical arguments but supported only by experiments having a fairly limited range. These modifications relate to a diffusion of fluid volume that would be significant for flows with high density gradients. So the viscous structure of shock waves in gases should provide an excellent test case for this new model. In this paper we detail the shock structure problem and propose exponents for the gas viscosity-temperature relation based on empirical viscosity data that is independent of shock experiments. We then simulate monatomic gas shocks in the range Mach 1.0-12.0 using the Navier-Stokes equations, both with and without Brenner's modifications. Initial simulations showed that Brenner's modifications display unphysical behaviour when the coefficient of volume diffusion exceeds the kinematic viscosity. Our subsequent analyses attribute this behaviour to both an instability to temporal disturbances and a spurious phase velocity-frequency relationship. On equating the volume diffusivity to the kinematic viscosity, however, we find the results with Brenner's modifications are significantly better than those of the standard Navier-Stokes equations, and broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. Brenner's modifications add only two terms to the Navier-Stokes equations, and the numerical implementation is much simpler than conventional extended hydrodynamic models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.
LanguageEnglish
Pages407-429
Number of pages22
JournalJournal of Fluid Mechanics
Volume580
DOIs
Publication statusPublished - Aug 2007

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Shock waves
Navier-Stokes equation
Navier Stokes equations
shock waves
Viscosity
Hydrodynamics
Viscosity of gases
shock
viscosity
Phase velocity
Burnett equations
kinematics
gas viscosity
Gases
hydrodynamics
nonequilibrium flow
Mach number
monatomic gases
Experiments
Boundary conditions

Keywords

  • fluid mechanics
  • mechanical engineering
  • navier-stokes
  • shock waves

Cite this

Greenshields, Christopher ; Reese, Jason M. / The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations. In: Journal of Fluid Mechanics. 2007 ; Vol. 580. pp. 407-429.
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The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations. / Greenshields, Christopher; Reese, Jason M.

In: Journal of Fluid Mechanics, Vol. 580, 08.2007, p. 407-429.

Research output: Contribution to journalArticle

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AU - Reese, Jason M.

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AB - Brenner (Physica A, vol. 349, 2005a, b, pp. 11, 60) has recently proposed modifications to the Navier-Stokes equations that are based on theoretical arguments but supported only by experiments having a fairly limited range. These modifications relate to a diffusion of fluid volume that would be significant for flows with high density gradients. So the viscous structure of shock waves in gases should provide an excellent test case for this new model. In this paper we detail the shock structure problem and propose exponents for the gas viscosity-temperature relation based on empirical viscosity data that is independent of shock experiments. We then simulate monatomic gas shocks in the range Mach 1.0-12.0 using the Navier-Stokes equations, both with and without Brenner's modifications. Initial simulations showed that Brenner's modifications display unphysical behaviour when the coefficient of volume diffusion exceeds the kinematic viscosity. Our subsequent analyses attribute this behaviour to both an instability to temporal disturbances and a spurious phase velocity-frequency relationship. On equating the volume diffusivity to the kinematic viscosity, however, we find the results with Brenner's modifications are significantly better than those of the standard Navier-Stokes equations, and broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. Brenner's modifications add only two terms to the Navier-Stokes equations, and the numerical implementation is much simpler than conventional extended hydrodynamic models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.

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