My PhD research was in the development and implementation of high-order realizable discretization schemes for quadrature-based moment methods (QBMM). QBMM is a moment-method for solving kinetic transport equations. Instead of solving the kinetic equation directly, a finite set of moment equations are solved. The moment-closure problem is solved by writing the distribution function as a sum of delta functions. More recently, an extended quadrature-based moment method (EQBMM) has been developed that replaces the delta functions with smooth non-negative functions. Some of the work done as a part of my PhD is shown below.

High-Order Realizable Finite-Volume Scheme for QBMM

Standard high-order finite-volumes schemes cannot be used with QBMM as a they often lead to non-realizable distributions, meaning negative density. This bottleneck with QBMM has existed since its inception. Due to this problem, first-order finite-volume scheme is often used with QBMM leading to highly diffused out solutions. To get rid of this problem, a new class of realizable schemes, called quasi-high-order schemes, was developed that could guarantee both realizability and improved accuracy over first-order finite-volume scheme, simultaneously. The particle number density for an impinging jet problem, using first-order scheme and quasi-second-order scheme are shown below.

Modeling of Dispersed-Phase Flows using QBMM

Lagrangian methods for modeling of dispersed-phase flows are valid for flow regimes over the whole range of Knudsen and Stokes numbers. However, they are computationally intensive as all the particles are tracked individually. An alternative is to use an Eulerian method based on the hydrodynamic model. This is much less expensive compared to the Lagrangian methods. However, the hydrodynamic model is derived with the assumption of low Knudsen number. Also, for large Stokes number flows, the hydrodynamic model does not work as it cannot resolve particle trajectory crossing. Quadrature-based moment method makes no such assumption. It is valid over the whole range of Knudsen and Stokes numbers. A coupled solver was developed for dispersed-phase flows (both gas-particle and liquid-gas flows). For the continuous-phase, both compressible and incompressible solvers were developed and coupled with the particle-phase QBMM solver. The results (instantaneous bubble number density plots at four different times) for a bubble-column flow are shown below. The meandering behavior of the bubble-plume is clearly evident.