In previous work (Israel, 2014) an integral equation analysis has been used to reduce a RANS closure for Rayleigh-Taylor and Kelvin-Helmholtz to a dynamical system, which can be used to examine both the self-similar scaling, and the transient behavior as the flow approaches the self-similar solution. Applying this approach, it appears that the variation of observed growth rates in these two flows is largely attributable to deviations from true self-similarity in both experiments and direct numerical simulation.
Here, the same technique is applied to determining the θ parameter, which is the exponent in the power law growth of the Richtmeyer-Meshkov interface. It is demonstrated that the self-similar solution of the RANS equations is not unique, and consequently θ is not universal. This implies that even given a long time to relax to self-similarity, the measured value of θ will be dependent on the initial conditions through the specifics of the energy deposition in the interface by the shock. This dependence manifests itself through two different mechanisms, both of which will be explored mathematically.
References
Daniel Israel. A dynamical systems approach to the alpha problem for Rayleigh-Taylor. IN IWPCTM 14, 2014.
