Christine M. Hrenya, PhD, University of Colorado (Boulder)
The overall goal of the current effort is gain insight into the interplay of various mechanisms giving rise to instabilities in solids flows. The quantitative prediction of such instabilities is also at the forefront of this work, since previous contributions have demonstrated the ability of such models to capture the qualitative nature of instabilities, but a quantitative assessment is lacking.
Here, we have focused on a simple system in order to isolate physics of interest and proceed in a sequential manner in building up the complexity of the system. In particular, we focus on the homogeneous cooling system (HCS), which is a three-dimensional system with periodic boundaries, zero mean flow, and no external forces. The initial state of the system is a given total kinetic energy applied randomly to particles; this kinetic energy then dissipates with time due to dissipative collisions, and thus the name “homogeneous cooling”.
Two mathematical tools are used to investigate this system, namely molecular dynamics (MD) simulations and continuum, or kinetic-theory-based, models. The former serves as ideal data since it involves the tracking of each particle with time according to Newton’s law of motion. The latter, although it involves additional constitutive relations, is important for the understanding and design of practical systems since MD is too computationally expensive to do so.
In the first year of the grant, we found that the kinetic-theory-based (KT) models did an excellent job of predicting the onset of the first instability observed in HCS, namely the velocity vortex instability. The particles were monodisperse spheres with inelastic but frictionless contacts. The KT predictions took the form of the critical domain length scale non-dimensionalized by the particle size, L_vortex/d, which was determined as part of a previous linear stability analysis. In the second and final year of the grant, three major findings came out of the work. These findings related to an extension to polydisperse flows, the effect of frictional dissipation and the KT to predict the second instability, or clustering. Each of these is discussed in turn below.
First, a binary mixture of particles which differed in size and/or material density was considered. MD simulations were carried out to determine L_vortex/d for these flows, and this data serves as a testbed for the KT predictions. The KT predictions were obtained from a linear stability analysis of the polydisperse theory applied to HCS. Similar to the monodisperse case, the binary predictions are in excellent agreement with the MD simulations. This comparison is perhaps one of the most stringent tests of these class of theories to date since it involves polyidspersity, instabilities, moderate concentrations, and a range of inelasticities.
Second, MD simulations were used to investigate the effect of frictional dissipation on instability formation, as previous MD and KT efforts alike focused solely on inelastic dissipation. As mentioned above, dissipation is the source of hydrodynamic instability, and all previous efforts supported the notion that increased dissipation leads to a greater prevalence of instabilities. Our results show differently, however, for the case of highly frictional systems. In particular, for a system with a given level of inelasticity, also adding a high amount of friction actually attenuatesthe formation of instabilities relative to the frictionless case (i.e., critical length scales increased). In other words, adding more dissipation serves to decrease the instabilities. A more detailed analysis of the simulations show that this counterintuitive and first-time observation is attributed to the coupling of translational and rotational motion that occurs in a frictional contact.
Finally, our focus turned toward clustering instabilities, which for the case of HCS is always preceded in time by the vortex instability. MD simulations were used to obtain L_cluster/d in addition to the L_vortex/d previously obtained. Again, monodisperse, frictionless systems were examined. Here, our predictions from linear stability analysis consistently over-predict the value of L_cluster/d. This result is not so surprising due to the nonlinear mechanisms (viscous) associated with the onset of clustering, which are clearly not included in a linear stability analysis. Accordingly, transient simulations were performed using the governing, continuum equations (i.e., computational fluid dynamics solution). The resulting L_cluster/d is in excellent agreement with the MD simulations. The high level of agreement is actually a bit suprising given the assumptions used to derive the KT model. Specifically, the KT equations are a Navier-Stokes order description, meaning that an assumption of low Knudsen numbers, or equivalently “small” gradients in hydrodynamic fields, is implemented. However, at the onset of clusters, velocity vortices are well formed. Such localized vortices are characterized by large gradients in the velocity field. This hypothesis was confirmed via the generation of Knudsen field maps for the domain, which showed large areas of the domain characterized by Knudsen numbers of at least the order of unity. This finding demonstrates the ability of Navier-Stokes-order models well outside of their expected range of validity, analogous to their molecular counterparts, which is encouraging given the non-trivial complexity of higher-order descriptions.