Session B4: CM-1: Continuum Modeling

Development of a Thermal Model for Hypervelocity Impact into Aerogel

The STARDUST mission was the first to return cometary material from a known source. The spacecraft passed through the comet's coma at a relative velocity of 6.5 km/s, collecting the dust particles by impact into very low-density silica aerogel. The low impedance of the aerogel allowed shock stress to remain low, but at the expense of producing very high shock temperatures, which thermally compromised the samples. To support the sample analysis community, we are developing a new model to invert impact track geometries to get detailed thermal histories of some samples during capture. This project requires development of a specialized hydrocode that can deal with large lengthscale disparities and extreme volumetric compressions, while remaining capable of being run efficiently. Also required are material models for the shocked aerogel with accurate thermodynamic, mechanical, and transport properties at a wide range of conditions. We present results for this project and discuss problems that remain.   

Analytical Solution for Isentropic Flows in Solids

In 1860, Riemann gave the equations system and the exact solution for the isentropic flows in ideal gases. This solution is based on the hypergeometric function. For monatomic, diatomic or polyatomic gases, the polytropic exponent is $(2n+1)/(2n-1)$ and the solution is obtained through polynoms of degree $n$. We have to notice that, if Riemann uses the polytropic exponent to show the interest in these practical cases, it is more rigorous to use the fundamental derivative, defined by Thompson in 1971, which is an adimensionnal number associated to the convexity of isentropes. Different authors have given further details for this solution: Hadamard (1903), Courant and Friedrichs (1948), Landau and Lifschitz (1959) and Stanyukovitch (1960). But to our knowledge, nothing has been done to apply it for solids. Properties of shock waves in solid materials can often be described by the equation D=c+s u, where D and u are the shock and particle velocities, and c and s properties of the material. We can notice that s is strongly linked to the fundamental derivative. This means that the assumption of constant fundamental derivative is useful for these solids and that we can apply the exact Riemann solution for them. The hypergeometric function remains very complicated in that case, but can be developed in power series which converges very efficiently. Then, if we just change the coefficients of Riemann polynom for polyatomic gases, we obtain a very good approximation for solids.   

Simulations of pressure-induced phase transitions at the continuum scale

To describe pressure-induced phase transitions, we have a developed continuum model and have performed hydrodynamic simulations that are compared to recent ramped-compression experiments in water. The model incorporates modern nucleation theory through scaled quantities for nucleation and growth. The form of the phase fraction order parameter requires a self-consistent iteration per hydrodynamic time step. We focus our simulations on recent ramp-compression experiments of liquid water, where the system undergoes a pressure-induced phase transition into ice VII, however, other systems will be shown. Also, a comparison with other common continuum models will be presented.   

Numerical study of mixing induced by shock compression

The mixing in a stratified cylindrical shell driven by cylindrical shock compression is numerically studied using the 7th order weighted essentially non-oscillatory shock-capturing method, combined with the 3th Runge-Kutta method. In order to investigate the effect of initial perturbations on the mixing growth, several different spectral shapes were introduced at the outer interface of the shell. In each case, random phases were assigned to each mode and randomized amplitudes were selected from the given spectrum. Three regimes were observed during the evolution history: (1) The shell implodes compressing the inner material. Two mixing zones grow at the shell surfaces due to the interfacial instability. (2) After the shock wave reaches the center and reflects outwards, the shell bounces off and moves back out. (3) At late-time the shell moves less and the two mixing zones grow quite steadily. If the shell is thin enough or the mixing has undergone enough time, the shell is broken up and the two mixing zones join to one. To characterize the mixing evolution, statistical quantities together with the mixing zone width were defined based on the simulation data, such as the actual product, maximum possible product, and mixing fraction. The flow features were analyzed using these measures. It was found that growth of the mixing zone is quite sensitive to the initial perturbation scale. The mixing zone grows more slowly with smaller scale perturbations. The imprint of initial perturbation spectra is obvious in early-time, but vanishes in late-time, except that the initial perturbation scale varies a lot.   

Stability and ambiguous representation of shock wave discontinuity in media with arbitrary thermodynamic properties

The non-linear analysis of the plane shock wave stability in media with arbitrary equation of state has been carried out numerically in a systematic way. The simulation has been conducted in the viscous one- and multi-dimensional formulations. The real and properly constructed model equations of state have been used in calculations. The behavior of neutrally stable shock waves as well as shocks in the region of their ambiguous representation overlapped Hugoniot segments meeting the linear criteria of the shock wave instability [D'yakov, Zh. Eksp. Teor. Fiz. \textbf{27}, 288 (1954)] has been studied. It is shown that unlike linear theory predictions the neutrally stable shock wave perturbations are damped but this process is rather slower than in the case of the absolutely stable shock wave. Within regions of ambiguous representation shocks split with irreversible transition to one of admissible wave configurations. The formation of a cellular detonation-like front structure has been revealed in multi-dimensional simulation (the region of ambiguous shock representation due to instability condition L$>$1+2M).