The present work is motivated by the pervasive nature of compressible multiphase flow in practical applications. These flows often feature particles (i.e. solid particles, droplets or bubbles) and develop rich dynamics as particles interact with different flow features such as shock waves. These interactions present unique challenges for numerical methods. The underlying primary motivation is to judiciously exploit shock-particle interaction in different flow topology, e.g. in gas-solid and gas-liquid systems, with proper and efficient methods.

In the first part, the interaction of shock wave with a particle cloud in dense gas-solid regime is investigated through a particle resolved direct numerical simulation to quantify the unsteadiness and velocity fluctuations, arising from this interaction, in the particle cloud and the wake behind that. This investigation is performed using a Particle-Resolved Direct Numerical Simulation (PR-DNS) by solving the compressible Navier-Stokes equations coupled with a compressible Immersed Boundary Method (IBM), to account for the particles, in the Parallel Adaptive Wavelet-Collocation Method (PAWCM) framework. The PAWCM is a finite difference framework that uses wavelets to dynamically adapt the grid used to represent the solution, which minimizes the overall computational cost and allows larger simulations to be performed. The quantification is performed in three steps. First the simulation of simplified case of the shock interaction with a transverse array of particles is performed to reveal the source of unsteadiness under the wave-wave and wave wake interaction of the neighboring particles and introduce the dilatation effect arise over the particle wake. Then the interaction of the shock wave with the particle cloud is investigated to replicate the experimental canonical multiphase shock tube problem of Wagner et al. (2011). The budget of the vorticity equation explains the sources of strong unsteadiness in the particle cloud that previously was observed by Regele et. al (2014). In the third step the particle cloud is exposed to a compression wave that gradually introduce the flow. A detailed analysis of the velocity fluctuation and kinetic energy in the fluctuating motion is performed for both cases to ascertain the importance of the velocity fluctuations that arise from the strong unsteadiness in the shock induced case.

In the second part, a finite difference solver is developed for Parallel adaptive Wavelet Collocation method framework to investigate high-speed compressible gas-liquid flows with surface tension effects. This study is motivated by gaining deeper insight into the process of fuel atomization in a supersonic cross flow of supersonic combustors under the startup conditions. The solver is developed based on the five equation interface capturing scheme by solving compressible multiphase/multicomponent Navier-Stokes equations along with an advection equation for the material interface. An interface capturing scheme is applied to counter the numerical diffusion induced by shock capturing scheme and maintain the immiscibility condition at the material interface. The capillary force is modeled using a continuous surface approach. The gas phase is modeled as an ideal gas and the liquid phase is modeled using a stiffened-gas equation of state. Capability of the model is demonstrated by several one and two dimensional benchmark problem.

In the third part a finite volume shock/interface capturing scheme is developed for two phase flows based on the extension of single phase all-speed simple low-dissipation AUSM (SLAU) scheme. SLAU is the latest version of the AUSM-family schemes with a new numerical flux function which features low dissipation without any tunable parameters in low Mach number regimes while maintaining the robustness of AUSM-family fluxes at high Mach numbers with a very simple formulation. To demonstrate the accuracy of the method, it has been tested on the well known two-fluid air/water flow benchmark problems and the results were compared with the two-phase AUSM+ and AUSM+-up schemes. Finally the scheme was applied for the problem of shock particle cloud interaction to solve the phasic averaged governing equations along with the k-ϵ model to attempt modeling the unclosed terms.