Shock-wave structure and intermolecular collision laws

1974 ◽  
Vol 65 (1) ◽  
pp. 127-144 ◽  
Author(s):  
Shee-Mang Yen ◽  
Winnie Ng
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Spatial Variation ◽  
Numerical Solutions ◽  
Weak Shock ◽  
Velocity Distribution Function ◽  
Collision Integral ◽  
Shock Structure ◽  
Intermolecular Potential ◽  
Macroscopic Properties

The nonlinear Boltzmann equation has been solved for shock waves in a Max-wellian gas for eight upstream Mach numbers M1 ranging from 1·1 to 10. The numerical solutions were obtained by using Nordsieck's method, which was revised for use with the differential cross-section corresponding to an intermolecular force potential following an inverse fifth-power law. The accuracy of the calculations of microscopic and macroscopic properties for this collision law is comparable with that for elastic spheres published earlier (Hicks, Yen & Reilly 1972).We have made comparisons of the detailed characteristics of the internal shock structure in a Maxwellian gas with those in a gas of elastic spheres. The purpose of this comparative study is to find the shock properties that are sensitive as well as those which are insensitive to the change in collision law and to find effective ways to study them.The variation of thermodynamic and transport properties of interest with respect to density and to each other was found to depend only weakly on the change in collision law. The principal effect on the macroscopic shock structure due to the change in intermolecular potential is in the spatial variation of the macroscopic properties. The spatial variation of macroscopic properties may be determined accurately from the corresponding moments of the collision integral, especially in the upstream and downstream wings of the shock wave. The results for the velocity distribution function exhibit the microscopic shock characteristics influenced by a difference in intermolecular collisions, in particular the departure from equilibrium in the upstream wing of the shock and the relaxation towards equilibrium in the downstream wing. The departure of several characteristics of weak shock waves from those of the Chapman-Enskog linearized theory and the Navier-Stokes shock is also insensitive to the change in collision law. The deviation of the half-width of the function ∫fdvyduz from the Chapman-Enskog first iterate at M1 = 1·59 is in agreement with an experiment (Muntz & Harnett 1969).

scholarly journals Nematic Dispersive Shock Waves from Nonlocal to Local

2021 ◽  
Vol 11 (11) ◽  
pp. 4736
Author(s):  
Saleh Baqer ◽  
Dimitrios J. Frantzeskakis ◽  
Theodoros P. Horikis ◽  
Côme Houdeville ◽  
Timothy R. Marchant ◽  
...  
Keyword(s):  
Shock Wave ◽  
Liquid Crystals ◽  
Shock Waves ◽  
Numerical Solutions ◽  
Wave Structure ◽  
Beam Power ◽  
Input Beam ◽  
Shock Wave Structure ◽  
Wave Solutions ◽  
Modulation Theory

The structure of optical dispersive shock waves in nematic liquid crystals is investigated as the power of the optical beam is varied, with six regimes identified, which complements previous work pertinent to low power beams only. It is found that the dispersive shock wave structure depends critically on the input beam power. In addition, it is known that nematic dispersive shock waves are resonant and the structure of this resonance is also critically dependent on the beam power. Whitham modulation theory is used to find solutions for the six regimes with the existence intervals for each identified. These dispersive shock wave solutions are compared with full numerical solutions of the nematic equations, and excellent agreement is found.

On the interaction between shock waves and boundary layers

1951 ◽  
Vol 47 (3) ◽  
pp. 545-553 ◽  
Author(s):  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Reynolds Number ◽  
Shock Waves ◽  
Boundary Layers ◽  
Weak Shock ◽  
Boundary Layer Separation ◽  
Weak Shock Wave ◽  
Layer Separation ◽  
Boundary Layer Equations

AbstractThe effect on the boundary-layer equations of a weak shock wave of strength ∈ has been investigated, and it is shown that ifRis the Reynolds number of the boundary layer, separation occurs when ∈ =o(R−i). The boundary-layer assumptions are then investigated and shown to be consistent. It is inferred that separation will occur if a shock wave meets a boundary and the above condition is satisfied.

scholarly journals On the Effects of Viscosity on the Shock Waves for a Hydrodynamical Case—Part I: Basic Mechanism

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Huseyin Cavus
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Numerical Solutions ◽  
Entropy Change ◽  
Basic Mechanism ◽  
Viscous Stress ◽  
Viscous Stress Tensor ◽  
Supersonic Regime ◽  
Interaction Of Shock Waves ◽  
Case Part

The interaction of shock waves with viscosity is one of the central problems in the supersonic regime of compressible fluid flow. In this work, numerical solutions of unmagnetised fluid equations, with the viscous stress tensor, are investigated for a one-dimensional shock wave. In the algorithm developed the viscous stress terms are expressed in terms of the relevant Reynolds number. The algorithm concentrated on the compression rate, the entropy change, pressures, and Mach number ratios across the shock wave. The behaviour of solutions is obtained for the Reynolds and Mach numbers defining the medium and shock wave in the supersonic limits.

scholarly journals Normal Reflection Characteristics of One-Dimensional Unsteady Flow Shock Waves on Rigid Walls from Pulse Discharge in Water

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Dong Yan ◽  
Jinchang Zhao ◽  
Shaoqing Niu
Keyword(s):  
Shock Wave ◽  
Hydrostatic Pressure ◽  
Shock Waves ◽  
Pulse Discharge ◽  
Weak Shock ◽  
Rigid Wall ◽  
Wave Source ◽  
One Dimensional ◽  
Normal Reflection ◽  
Rigid Walls

Strong shock waves can be generated by pulse discharge in water, and the characteristics due to the shock wave normal reflection from rigid walls have important significance to many fields, such as industrial production and defense construction. This paper investigates the effects of hydrostatic pressures and perturbation of wave source (i.e., charging voltage) on normal reflection of one-dimensional unsteady flow shock waves. Basic properties of the incidence and reflection waves were analyzed theoretically and experimentally to identify the reflection mechanisms and hence the influencing factors and characteristics. The results indicated that increased perturbation (i.e., charging voltage) leads to increased peak pressure and velocity of the reflected shock wave, whereas increased hydrostatic pressure obviously inhibited superposition of the reflection waves close to the rigid wall. The perturbation of wave source influence on the reflected wave was much lower than that on the incident wave, while the hydrostatic pressure obviously affected both incident and reflection waves. The reflection wave from the rigid wall in water exhibited the characteristics of a weak shock wave, and with increased hydrostatic pressure, these weak shock wave characteristics became more obvious.

Propagation of Weak Shock Waves in a Vibrational Nonequilibrium, Nonuniform Gas

1975 ◽  
Vol 42 (3) ◽  
pp. 564-568 ◽  
Author(s):  
D. C. Chou ◽  
S. Y. Maa
Keyword(s):  
Shock Wave ◽  
Wave Propagation ◽  
Shock Waves ◽  
Vibrational Energy ◽  
Weak Shock ◽  
Present Theory ◽  
Shock Wave Propagation ◽  
Perturbation Scheme ◽  
Relaxation Rates ◽  
Vibrational Nonequilibrium

Problems concerned with the propagation of weak planar shock waves in a nonuniform, nonequilibrium gas is theoretically investigated. The medium under consideration is a diatomic thermally perfect gas with excited vibrational energy and is initially inhomogeneous with exponential density and temperature distributions. The systematic characteristic perturbation scheme is employed to render a first-order frozen shock expression. It is shown quantitatively that combined effects of nonequilibrium, nonlinearity, and stratification govern the nature of the shock wave propagation. The uniform gas limit of present theory agrees with previously known results of shock wave propagation in a general relaxing fluid. Numerical examples illustrate the variation of frozen shock strength and speed due to different magnitudes of relaxation rates and inhomogeneity. The interesting competition phenomenon between nonequilibrium effects and nonuniform effects on shock wave propagation is examined.

Direct numerical simulation of isotropic turbulence interacting with a weak shock wave

1993 ◽  
Vol 251 ◽  
pp. 533-562 ◽  
Author(s):  
Sangsan Lee ◽  
Sanjiva K. Lele ◽  
Parviz Moin
Keyword(s):  
Shock Wave ◽  
Kinetic Energy ◽  
Shock Waves ◽  
Shock Front ◽  
Numerical Simulations ◽  
Turbulent Kinetic Energy ◽  
Isotropic Turbulence ◽  
Weak Shock ◽  
Linear Analysis ◽  
Weak Shock Wave

Interaction of isotropic quasi-incompressible turbulence with a weak shock wave was studied by direct numerical simulations. The effects of the fluctuation Mach number Mt of the upstream turbulence and the shock strength M21 — 1 on the turbulence statistics were investigated. The ranges investigated were 0.0567 ≤ Mt ≤ 0.110 and 1.05 ≤ M1 ≤ 1.20. A linear analysis of the interaction of isotropic turbulence with a normal shock wave was adopted for comparisons with the simulations.Both numerical simulations and the linear analysis of the interaction show that turbulence is enhanced during the interaction with a shock wave. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. The predictions of the linear analysis compare favourably with simulation results for flows with M2t < a(M21 — 1) with a ≈ 0.1, which suggests that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behaviour not reproduced by the linear analysis. Investigation of the budget of the turbulent kinetic energy transport equation shows that this behaviour can be attributed to the pressure transport term.Shock waves were found to be distorted by the upstream turbulence, but still had a well-defined shock front for M2t < a(M21— 1) with a ≈ 0.1). In this regime, the statistics of shock front distortions compare favourably with the linear analysis predictions. For flows with M2t > a(M21— 1 with a ≈ 0.1, shock waves no longer had well-defined fronts: shock wave thickness and strength varied widely along the transverse directions. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly.

Formation and propagation of a shock wave in a gas with temperature gradients

2002 ◽  
Vol 473 ◽  
pp. 245-264 ◽  
Author(s):  
V. S. SOUKHOMLINOV ◽  
V. Y. KOLOSOV ◽  
V. A. SHEVEREV ◽  
M. V. ÖTÜGEN
Keyword(s):  
Shock Wave ◽  
Temperature Gradient ◽  
Numerical Solutions ◽  
Weak Shock ◽  
Negative Temperature ◽  
Temperature Gradients ◽  
Initial Disturbance ◽  
Weak Shock Wave ◽  
Axial Distance ◽  
Uniform Medium

A theoretical analysis was carried out to study the formation and propagation of a weak shock wave in a gas with longitudinal temperature gradients. An equation describing the formation and propagation of a weak shock wave through a non-uniform medium in the absence of energy dissipation was derived. An approximate analytical solution to the one-dimensional wave propagation equation is established. With this, the thermal gradient effects on the shock-wave Mach number and speed were investigated and the results were compared to earlier experiments. Numerical solutions for the same problem using Euler’s equations have also been obtained and compared to the analytical results. The analysis shows that the time of shock-wave formation from the initial disturbance, for mild temperature gradients, is independent of the gradient. The shock wave forms at a longer axial distance from the initial disturbance when the temperature gradient is positive whereas the opposite is true for a negative temperature gradient.

The Taylor internal structure of weak shock waves

1986 ◽  
Vol 173 ◽  
pp. 625-642 ◽  
Author(s):  
D. G. Crighton
Keyword(s):  
Shock Waves ◽  
Electromagnetic Waves ◽  
Tube Wall ◽  
Weak Shock ◽  
Shock Structure ◽  
Cubic Nonlinearity ◽  
Nonlinear Convection ◽  
Transverse Waves ◽  
Solid Media ◽  
Attenuation And Dispersion

G. I. Taylor's solution in 1910 for the interior structure of a weak shock wave is, with appropriate generalization, an essential component of weak-shock theory. The Taylor balance between nonlinear convection and thermoviscous diffusion is, however, endangered when other linear mechanisms - such as density stratification, geometrical spreading effects, tube wall attenuation and dispersion, etc. - are included. The ways in which some of these linear mechanisms cause the Taylor shock structure to break down when a weak shock has propagated over a large (and in some cases quite moderate) distance will be studied. Different forms of breakdown of the Taylor shock structure will be identified, both for quadratic (gasdynamic) nonlinearity and also for cubic nonlinearity appropriate to transverse waves in solid media or electromagnetic waves in nonlinear dielectrics. From this a description will be given of the fate of a nonlinear wave containing a pattern of weak shock waves, as it propagates over large ranges under the influence of linear and nonlinear mechanisms.

Analysis of Normal Shock Waves in Particle Laden Gas

1964 ◽  
Vol 86 (4) ◽  
pp. 655-664 ◽  
Author(s):  
A. R. Kriebel
Keyword(s):  
Shock Wave ◽  
Particle Size ◽  
Shock Waves ◽  
Solid Propellant ◽  
Weak Shock ◽  
Normal Shock ◽  
Wave Measurements ◽  
Solid Propellant Rocket ◽  
Propellant Rocket ◽  
Motor Exhaust

The internal structure of a normal shock wave in a perfect gas heavily laden with particles having a distribution of sizes is machine computed by numerical integration. The results of a small-perturbation analysis for weak shock waves and one particle size compare well with the machine-computed results for these restricted conditions. Both methods indicate that the thickness of weak shock waves increases in proportion to the particle size squared and inversely with the shock strength. For conditions typical of solid propellant-rocket motor exhaust streams the computed shock-wave thickness is several inches. With such computed results both the amount and the size distribution of suspended particles can be found individually from shock-wave measurements.

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