Madhusudan G. Pai

A comprehensive probability density function formalism for multiphase flows

A theoretical foundation for two widely used statistical representations of multiphase flows, namely the Eulerian―Eulerian (EE) and Lagrangian―Eulerian (LE) representations, is established in the framework of the probability density function (p.d.f.) formalism. Consistency relationships between fundamental statistical quantities in the EE and LE representations are rigorously established. It is shown that fundamental quantities in the two statistical representations bear an exact relationship to each other only under conditions of spatial homogeneity. Transport equations for the probability densities in each statistical representation are derived. Exact governing equations for the mean mass, mean momentum and second moment of velocity corresponding to the two statistical representations are derived from these transport equations. In particular, for the EE representation, the p.d.f. formalism is shown to naturally lead to the widely used ensemble-averaged equations for two-phase flows. Galilean-invariant combinations of unclosed terms in the governing equations that need to be modelled are clearly identified. The correspondence between unclosed terms in each statistical representation is established. Hybrid EE―LE computations can benefit from this correspondence, which serves in consistently transferring information from one representation to the other. Advantages and limitations of each statistical representation are identified. The results of this work can also serve as a guiding framework for direct numerical simulations of two-phase flows, which can now be exploited to precisely quantify unclosed terms in the governing equations in the two statistical representations.

Detailed Numerical Simulations of Primary Atomization of Liquid Jets in Crossflow

The problem of primary breakup of liquid jet in crossflow (LJCF) is investigated by means of detailed numerical simulations. A spectrally refined interface tracking technique, which is coupled to a Navier-Stokes/Ghost fluid solver, is employed to track the liquid-gas interface. From the parametric space corresponding to the LJCF, the effect of varying momentum flux ratio, liquid Weber number, and crossflow Weber number on the liquid jet trajectory and the liquid surface wavelengths on the windward side of the liquid jet is investigated. Predicted liquid jet trajectories show good match with published experimental datasets. The numerical simulations predict that the wavelength of the liquid surface disturbances scale with the liquid Weber number rather than the crossflow Weber number for the conditions studied in this work. The numerical simulations also provide preliminary evidence that for the conditions chosen in this study, the smallest liquid length scales are controlled by the liquid Weber number rather than the crossflow Weber number.

Large-eddy simulation study of injector geometry on liquid jet in cross-flow and validation with experiments

In this work, large-eddy simulation of the atomization of a liquid jet in cross-flow is performed. Two different injector geometries are investigated that result in significantly different liquid jets. One of the injectors, referred to as the round-edged injector, produces a laminar flow at the exit plane. The other injector, known as the sharp-edged injector, produces a turbulent flow that enhances the atomization of the liquid jet. The jet penetration, mean droplet size spatial distribution, and mean droplet velocity spatial distribution are compared to experimental results by Gopala, and good agreement is observed. To perform the simulations, we employ a computational methodology that is accurate and robust even when large density ratios and turbulent flows are present. The accurate conservative level set is used to transport the gas-liquid interface. A density correction formulation is used to ensure consistency between the interface transport and momentum transport steps, making a robust scheme for simulating high density ratio flows. A conservative immersed boundary method is used to simulate the injector geometries, which avoids the complexity of generating a body-fitted mesh.

Parametric study of primary breakup of turbulent liquid jets in crossflow: Role of Weber number

A parametric study of primary breakup of turbulent liquid jets in crossflow is performed using detailed numerical simulations. Specifically, the role of crossflow and liquid Weber number in determining the size and shape of the liquid structures that separate from the liquid column is quantified. To quantify the size and shape of the liquid structures, a methodology derived from differential geometry that characterizes three-dimensional structures based on the shape index, curvedness and stretching parameter is applied to the liquid jet. For the range of parameters considered in this study, it is observed that the liquid Weber number determines the shape and number of separated structures, while the crossflow Weber number determines their characteristic thickness. Implications of this study in modeling primary breakup of liquid jets are discussed.