A modified hydro-thermo-diffusive theory of shock waves (original) (raw)
Related papers
Arxiv preprint arXiv:1005.1525, 2010
Macroscopic models which distinguish the longitudinal and transverse temperatures can provide improved descriptions of the microscopic shock structures as revealed by molecular dynamics simulations. Additionally, we can include three relaxation times in the models, two based on Maxwell's viscoelasticity and its Cattaneo-equation analog for heat flow, and a third thermal, based on the Krook-Boltzmann equation. This approach can replicate the observed lags of stress (which lags behind the strain rate) and heat flux (which lags behind the temperature gradient), as well as the eventual equilibration of the two temperatures. For profile stability the time lags cannot be too large. By partitioning the longitudinal and transverse contributions of work and heat and including a tensor heat conductivity and bulk viscosity, all the qualitative microscopic features of strong simple-fluid shockwave structures can be reproduced.
Physical Review E, 2010
Guided by molecular dynamics simulations, we generalize the Navier-Stokes-Fourier constitutive equations and the continuum motion equations to include both transverse and longitudinal temperatures. To do so we partition the contributions of the heat transfer, the work done, and the heat flux vector between the longitudinal and transverse temperatures. With shockwave boundary conditions time-dependent solutions of these equations converge to give stationary shockwave profiles. The profiles include anisotropic temperature and can be fitted to molecular dynamics results, demonstrating the utility and simplicity of a two-temperature description of far-from-equilibrium states.
Effects of molecular diffusivity on shock-wave structures in monatomic gases
Physical Review E, 2021
We present a full investigation into shock wave profile description using hydrodynamics models. We identified constitutive equations that provide better agreement for all parameters involved in testing hydrodynamic equations for the prediction of shock structure in a monatomic gas in the Mach number range 1.0 − 11.0. The constitutive equations are extracted from a previously derived thermomechanically consistent Burnett regime continuum flow model. The numerical computations of the resulting hydrodynamic equations along with classical ones are performed using a finite difference global solution (FDGS) scheme. Compared to previous studies that focussed mainly on the density profile across the shock, here we also include temperature profiles as well as nonnegativity of entropy production throughout the shock. The results obtained show an improvement upon those obtained previously in the bi-velocity (or volume/mass diffusion) hydrodynamics and are more accurate than in the hydrodynamic models from expansions method solutions to the Boltzmann equation.
A Full Evaluation of Accurate Constitutive Relations for Shock Wave Structures in Monatomic Gases
2021
We present a full investigation into shock wave profile description using hydrodynamics models. We identified constitutive equations that provide better agreement for all parameters involved in testing hydrodynamic equations for the prediction of shock structure in a monatomic gas in the Mach number range 1.0 − 11.0. Compared to previous studies that focussed mainly on the density profile across the shock, here we also include temperature profiles as well as non-negativity of entropy production throughout the shock. The results obtained show an improvement upon those obtained previously in the bi-velocity hydrodynamics and are more accurate than in the hydrodynamic models from expansions method solutions to the Boltzmann equation.
Journal of Computational Physics, 2008
We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit undercompressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by high-order finite difference schemes, and uncover several new features of the kinetic function associated with with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms. On one hand, the role of the equivalent equation associated with a finite difference scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model-especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized Camassa-Holm model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals fluids, whose flux-function admits two inflection points.
Improved theory for shock waves using the OBurnett equations
Journal of Fluid Mechanics, 2021
The main goal of the present study is to thoroughly test the recently derived OBurnett equations for the normal shock wave flow problem for a wide range of Mach number ($3 \leq Ma \leq 9$). A dilute gas system composed of hard-sphere molecules is considered and the numerical results of the OBurnett equations are validated against in-house results from the direct simulation Monte Carlo method. The primary focus is to study the orbital structures in the phase space (velocity–temperature plane) and the variation of hydrodynamic fields across the shock. From the orbital structures, we observe that the heteroclinic trajectory exists for the OBurnett equations for all the Mach numbers considered, unlike the conventional Burnett equations. The thermodynamic consistency of the equations is also established by showing positive entropy generation across the shock. Further, the equations give smooth shock structures at all Mach numbers and significantly improve upon the results of the Navier–S...
Structure of a Shock-Wave Front in a Liquid
Physical Review Letters, 1979
Solutions of the Navier-Stokes equations for strong shock waves in a dense fluid agree well with recent atomistic simulations using nonequilibrium molecular dynamics.
DSMC Calculations of Shock Structure with Various Viscosity Laws
It has long been known that: the thickness Δ of a plane 1D shock, expressed in terms of the mean free path in the upstream (pre-shock) flo w, is a strong function of shock Mach number; and that the form of this function is sensitive to the form of the viscosity law μ μ T of the gas. On the other hand, the approximate kinetic theory method of Mott-Smith (1) shows that for many different assumed molecular models and viscosity laws, the average number of collisions suffered by a typical molecule as it traverses the shock quickly approaches a limit as the Mach number increases (2). This suggests that a mean free path based on a collision cross-section characteristic of the high speed collisions between up and downstream molecules is the appropriate length scale for normalising shock thickness results. One such length scale is a kinetic length scale L defined for the sonic conditions of the flo w (3), and another is the mean free path in the downstream flo w λ2, where the average collisi...
Local equilibrium in liquid phase shock waves
Physical Review E
We have assessed the assumption of local thermodynamic equilibrium in a shock wave by comparing local thermodynamic data generated with nonequilibrium molecular dynamics (NEMD) simulations with results from corresponding equilibrium simulations. The shock had a Mach number approximately equal to 2 in a Lennard-Jones spline liquid. We found that the local equilibrium assumption holds perfectly well behind the wave front, and is a very good approximation in the front itself. This was supported by calculations of the excess entropy production in the shock front with four different methods that use the local equilibrium assumption in different ways. Two of the methods assume local equilibrium between excess thermodynamic variables by treating the shock as an interface in Gibbs's sense. The other two methods are based on the local equilibrium assumption in a continuous description of the shock front. We show for the shock studied in this work that all four methods give excess entropy productions that are in excellent agreement, with an average variance of 3.5% for the nonequilibrium molecular dynamics (NEMD) simulations. In addition, we solved the Navier-Stokes (N-S) equations numerically for the same shock wave using an equilibrium equation of state (EoS) based on a recently developed perturbation theory. The results for the density, pressure, and temperature profiles agree well with the profiles from the NEMD simulations. For instance, the shock waves generated in the two simulations travel with almost the same speed; the average absolute Mach-number deviation of the N-S simulations relative to NEMD is 2.6% in the investigated time interval.