### Abstract

Language | English |
---|---|

Pages | 407-429 |

Number of pages | 22 |

Journal | Journal of Fluid Mechanics |

Volume | 580 |

DOIs | |

Publication status | Published - Aug 2007 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Fluid Mechanics*,

*580*, 407-429. https://doi.org/10.1017/S0022112007005575

}

*Journal of Fluid Mechanics*, vol. 580, pp. 407-429. https://doi.org/10.1017/S0022112007005575

**The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations.** / Greenshields, Christopher; Reese, Jason M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations

AU - Greenshields, Christopher

AU - Reese, Jason M.

PY - 2007/8

Y1 - 2007/8

N2 - 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.

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.

KW - fluid mechanics

KW - mechanical engineering

KW - navier-stokes

KW - shock waves

UR - http://arxiv.org/abs/physics/0611275

U2 - 10.1017/S0022112007005575

DO - 10.1017/S0022112007005575

M3 - Article

VL - 580

SP - 407

EP - 429

JO - Journal of Fluid Mechanics

T2 - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -