Comparisons of Measured and Predicted Velocity Distribution Functions in a Shock Wave

Holtz, T.

doi:10.1063/1.1693470

Cited by 9 publications

(1 citation statement)

References 2 publications

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“…Evidence from other phenomena, such as sound propagation, is available (Foch & Ford 1970) and leads to the conclusion that the Burnett equations indeed do offer a better description than the NavierStokes equations, even for the conditions encountered in very low-Mach-number shock waves. Measurements of the molecular velocity distribution function in a M = 1.59 shock wave by Muntz & Harnett (1969) have been analysed by Holtz, Muntz & Yen (1971). The measurements indicate that the Chapman-Enskog first-iterate predictions of the distribution functions fail for ratios of the local shear stress to pressure r/p 2 0.05.…”

Section: Resultsmentioning

confidence: 99%

Testing continuum descriptions of low-Mach-number shock structures

Pham-Van-Diep

Erwin

Muntz³

1991

J. Fluid Mech.

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Numerical experiments have been performed on normal shock waves with Monte Carlo Direct Simulations (MCDS's) to investigate the validity of continuum theories at very low Mach numbers. Results from the Navier—Stokes and the Burnett equations are compared to MCDS's for both hard-sphere and Maxwell gases. It is found that the maximum-slope shock thicknesses are described equally well (within the MCDS computational scatter) by either of the continuum formulations for Mach numbers smaller than about 1.2. For Mach numbers greater than 1.2, the Burnett predictions are more accurate than the Navier—Stokes results. Temperature—density profile separations are best described by the Burnett equations for Mach numbers greater than about 1.3. At lower Mach numbers the MCDS scatter is too great to differentiate between the two continuum theories. For all Mach numbers above one, the shock shapes are more accurately described by the Burnett equations.

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Section: Resultsmentioning

confidence: 99%

Testing continuum descriptions of low-Mach-number shock structures

Pham-Van-Diep

Erwin

Muntz³

1991

J. Fluid Mech.

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Gas dynamics from the point of view of kinetic theory

Guiraud¹

1973

Theoretical and Applied Mechanics

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The internal structure of shock waves

1972

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The non-linear Boltzmann equation has been solved for shock waves in a gas of elastic spheres. The solutions were made possible by the use of Nordsieck's Monte Carlo method of evaluation of the collision integral in the equation. Accurate solutions were obtained by the same method for the whole range of upstream Mach numbers M^ from 1.1 to 10 even though the corresponding degree of departure from equilibrium varies by a factor greater than 1000. Many characteristics of the internal structure of the shock waves have been calcu lated from the solutions and compared with Navier-Stokes, Mott-Smith and Krook descriptions which, except for low Mach numbers, are not based upon the Boltz mann equation itself. Among our conclusions are the following: 1. The reciprocal shock thickness is in agreement with that of the 2 Mott-Smith shock (u-moment) from M^ of 2.5 to 8. The density profile is asymmetric with an upstream relaxation rate (measured as density change per mean free path) approximately twice as large as the downstream value for weak shocks and equal to the downstream value for strong shocks. 2. The temperature density relation is in agreement with that of the Navier-Stokes shocks for the lower Mach numbers in the range of 1"1 to 1.56. The Boltzmann reciprocal shock thickness is smaller than the Navier-Stokes value at this range of Mach number because the viscosity-temperature relation computed is not constant as predicted by the linearized theory. 2 3. The velocity moments of the distribution function are, like the Mott-Smith shock, approximately linear with respect to the number density; however, the deviations from linearity are statistically significant. The four functionals of the distribution function discussed show maxima within the shock. 4. The entropy is a good approximation to the Boltzmann function for all M^. The solutions obtained satisfy the Boltzmann theorem for all Mach numbers. The increase in total temperature within the shock is small, but the increase is significantly different from zero. 5. The ratio of total heat flux q to (associated with the longi tudinal degree of freedom) correlates well with local Mach number for all M^ in accord with a relation derived by Baganoff and Nathenson. The Chapman-Enskog linearized theory predicts that the ratio is constant. The (effective) transport coefficients are larger than the Chapman-Enskog equivalents by as much as a factor of three at the mid-shock position. 6. At M 1=4, and for 40% of velocity bins, the distribution function is different from the corresponding Mott-Smith value by more than three times the 90% confidence limit. The rms value of the percent difference, in distribu tion function is 15% for this Mach number. The halfwidth and several other characteristics of the function Jfdw^dvz differ from that of the Chapman-Enskog first iterate, and many of the deviations are in agreement with an experiment by Muntz and Harnett. 7. The ratio of the collision integral (found from our solution of the Boltzmann equation) to that calculated from Mott-Smit...

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Comparisons of Measured and Predicted Velocity Distribution Functions in a Shock Wave

Cited by 9 publications

References 2 publications

Testing continuum descriptions of low-Mach-number shock structures

Testing continuum descriptions of low-Mach-number shock structures

Gas dynamics from the point of view of kinetic theory

The internal structure of shock waves

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