Plane shock waves and Haff’s law in a granular gas

2015 ◽  
Vol 779 ◽  
Author(s):  
M. H. Lakshminarayana Reddy ◽  
Meheboob Alam
Keyword(s):  
Mach Number ◽  
Shock Waves ◽  
Energy Equation ◽  
Navier Stokes ◽  
Maximum Temperature ◽  
Plane Shock ◽  
Density Peak ◽  
Granular Gas ◽  
Planar Shock ◽  
Weak Shocks

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler- and Navier–Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff’s law (${\it\theta}(t)\sim t^{-2}$), but the downstream temperature decays faster than its upstream counterpart. Haff’s law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff’s law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be ‘regularized’ to arrest the maximum density.

On the stability of plane shock waves

1957 ◽  
Vol 2 (4) ◽  
pp. 397-411 ◽  
Author(s):  
N. C. Freeman
Keyword(s):  
Shock Wave ◽  
Mach Number ◽  
Shock Waves ◽  
Plane Shock ◽  
Two Dimensional ◽  
Small Perturbations ◽  
Diverging Channel ◽  
Maximum Stability ◽  
The Stability ◽  
Fourier Type

The decay of small perturbations on a plane shock wave propagating along a two-dimensional channel into a fluid at rest is investigated mathematically. The perturbations arise from small departures of the walls from uniform parallel shape or, physically, by placing small obstacles on the otherwise plane parallel walls. An expression for the pressure on a shock wave entering a uniformly, but slowly, diverging channel already exists (given by Chester 1953) as a deduction from the Lighthill (1949) linearized small disturbance theory of flow behind nearly plane shock waves. Using this result, an expression for the pressure distribution produced by the obstacles upon the shock wave is built up as an integral of Fourier type. From this, the shock shape, ξ, is deduced and the decay of the perturbations obtained from an expansion (valid after the disturbances have been reflected many times between the walls) for ξ in descending power of the distance, ζ, travelled by the shock wave. It is shown that the stability properties of the shock wave are qualitatively similar to those discussed in a previous paper (Freeman 1955); the perturbations dying out in an oscillatory manner like ζ−3/2. As before, a Mach number of maximum stability (1·15) exists, the disturbances to the shock wave decaying most rapidly at this Mach number. A modified, but more complicated, expansion for the perturbations, for use when the shock wave Mach number is large, is given in §4.In particular, the results are derived for the case of symmetrical ‘roof top’ obstacles. These predictions are compared with data obtained from experiments with similar obstacles on the walls of a shock tube.

Interaction of Liquid Droplets With Planar Shock Waves

1990 ◽  
Vol 112 (4) ◽  
pp. 481-486 ◽  
Author(s):  
T. Yoshida ◽  
K. Takayama
Keyword(s):  
Mach Number ◽  
Shock Waves ◽  
Reynolds Numbers ◽  
Liquid Droplets ◽  
Double Exposure ◽  
Cross Sectional ◽  
Planar Shock ◽  
Shock Mach Number ◽  
Holographic Interferogram ◽  
Double Exposure Holographic Interferometry

Interactions and breakup processes of 1.50-mm-diameter ethyl alcohol droplets and 5.14-mm-diameter water bubbles with planar shock waves were observed using double-exposure holographic interferometry. Experiments were conducted in a 60 mm × 150 mm cross-sectional shock tube for shock Mach number 1.56 in air. The Weber numbers of droplets and liquid bubbles were 5.6 × 103 and 2.9 × 103, respectively, while the corresonding Reynolds numbers were 4.2 × 10 and 1.5 × 105. It is shown that the resulting holographic interferogram can eliminate the effect of the mists produced by the breakup of the droplets and clearly show the structure of a disintegrating droplet and its wake. This observation was impossible by conventional optical flow visualization. It is demonstrated that the time variation of the diameter of a breaking droplet measured by conventional optical techniques has been overestimated by up to 35 percent.

A theory of the stability of plane shock waves

1955 ◽  
Vol 228 (1174) ◽  
pp. 341-362 ◽  
Keyword(s):  
Mach Number ◽  
Large Time ◽  
Wave Length ◽  
Plane Shock ◽  
Piston Problem ◽  
Shock Mach Number ◽  
The Stability ◽  
Limiting Case ◽  
Weak Shocks ◽  
Asymptotic Forms

The stability of form of a plane shock, obtained when a ‘corrugated' piston is moved impulsively from rest with constant velocity, is investigated mathematically. Linearization of the problem is accomplished by assuming the corrugations to be small. The solution is built up by methods of Fourier analysis from ‘conefield’ solutions of the analogous ‘wedge’- shaped piston problem, solved by methods due to Lighthill. The plane shock is shown to be stable, perturbations from plane decaying with time in an oscillatory manner like t -½ for large ta 1 /λ (where a 1 is the velocity of sound behind the shock and λ the wave-length of the corrugations). The stability, measured by the amplitude of this oscillation after the shock has traversed a given distance, decreases both as the shock Mach number increases above and decreases below the value 1⋅14. Shocks of this strength exhibit strongest stability. Asymptotic forms for large time are given for both the shock shape and pressure distribu­tion for shocks of moderate strength in §4. A more complicated asymptotic form for the shock shape holds at large Mach numbers (§5) which in the limiting case of infinite Mach number gives the result that the perturbations of shape decay like t -½ only. Complete solutions are obtained for weak shocks in terms of Bessel functions (§6).

scholarly journals Study on Mach stems induced by interaction of planar shock waves on two intersecting wedges

2015 ◽  
Vol 32 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Gaoxiang Xiang ◽  
Chun Wang ◽  
Honghui Teng ◽  
Yang Yang ◽  
Zonglin Jiang
Keyword(s):  
Shock Waves ◽  
Planar Shock

Impacts of grid turbulence on the side projection of planar shock waves

Shock Waves ◽  
2021 ◽  
Author(s):  
G. Fukushima ◽  
S. Ogawa ◽  
J. Wei ◽  
Y. Nakamura ◽  
A. Sasoh
Keyword(s):  
Shock Waves ◽  
Grid Turbulence ◽  
Planar Shock

scholarly journals Computational Evaluation of Shock Wave Interaction with a Cylindrical Water Column

2021 ◽  
Vol 11 (11) ◽  
pp. 4934
Author(s):  
Viola Rossano ◽  
Giuliano De Stefano
Keyword(s):  
Shock Wave ◽  
Water Column ◽  
Wave Interaction ◽  
Navier Stokes ◽  
Plane Shock ◽  
Governing Equations ◽  
Computational Evaluation ◽  
Air Water Interface ◽  
The Mean ◽  
The Impact

Computational fluid dynamics was employed to predict the early stages of the aerodynamic breakup of a cylindrical water column, due to the impact of a traveling plane shock wave. The unsteady Reynolds-averaged Navier–Stokes approach was used to simulate the mean turbulent flow in a virtual shock tube device. The compressible flow governing equations were solved by means of a finite volume-based numerical method, where the volume of fluid technique was employed to track the air–water interface on the fixed numerical mesh. The present computational modeling approach for industrial gas dynamics applications was verified by making a comparison with reference experimental and numerical results for the same flow configuration. The engineering analysis of the shock–column interaction was performed in the shear-stripping regime, where an acceptably accurate prediction of the interface deformation was achieved. Both column flattening and sheet shearing at the column equator were correctly reproduced, along with the water body drift.

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