Optimized amplification of interface distortion on liquid jets
Keywords: Mechanical Engineering.

A fragmentation process of the liquid jet begins with the amplification of small disturbances. Understanding the evolution of the initial perturbation is critical in controlling the quality of the atomization outcomes, such as the droplet size distribution and the evaporation rate of the liquid, as these are directly related to the performance of a variety of atomization applications. In this study, we identify a novel mechanism for the effective amplification of the initial perturbation on liquid jets. This mechanism is independent of the exponential instability of the flow and can intensify small perturbations to the material interface by several orders of magnitude. Depending on its operating conditions, the liquid jet can amplify interfacial disturbances at a faster pace than can modal mechanisms such as the Kelvin--Helmholtz (K--H) instability. This study is based on a spatial linear stability theory in a two-fluid formulation that accounts for both viscosity and surface tension effects. The identified mechanism is verified using nonlinear simulations. In simulations, we observe nonlinear effects that create small-scale structures, a result not predicted in linear theory. As a measure of effective atomization, the small-scale structures are quantified by computing the surface area of the liquid jet. A study comparing the surface area between the liquid jet perturbed by the identified mechanism and the random inlet perturbation results higher surface area, which indicates that the identified mechanism drives more severe corrugation on the material interface leading to an effective atomization. A systematic framework for stability analysis of two-phase flows is constructed. The base flow of liquid jets is assumed to be streamwise invariant, stationary in time, axisymmetric, and laminar. Because of these assumptions, the base flow remains as the only function of radial direction. The ansatzes for the disturbance on a liquid jet are exponential functions in corresponding homogeneous directions. For an inhomogeneous direction, i.e., radial direction, disturbances are discretized using the Chebyshev spectral method. With linearized Navier-Stokes equations, the base flow, kinematic/dynamic interface conditions, and boundary conditions, analyzing the stability of liquid jets boils down to a well-posed eigenvalue problem. The eigenvalue problem is solved in the spatial framework where the input is a real perturbation frequency, and the outcomes are complex wavenumbers with corresponding eigenfunctions. A computationally cost-effective method is proposed to evaluate of eigenmode propagation direction. Unlike the temporal eigenvalue problem, solving the spatial eigenvalue problem adds more complexity in that the propagation direction of eigenmodes must be identified. Here, we solve the adjoint eigenvalue problem and invoke bi-orthogonality of the eigenfunctions to evaluate the group velocity of each eigenmode. With this group velocity evaluation metric, downstream propagating modes are identified. The effect of vorticity layer thickness on the downstream propagating unstable mode of liquid jets is explored. In addition, the identification of eigenmodes is illustrated using energy budgets. In contrast to classical modal growth analysis, a new pathway leading to an effective surface distortion of liquid jets is studied in detail. Analysis of the new mechanism is cast as an optimization problem that maximizes surface tension energy gain from the inlet perturbation distribution at the nozzle and discounts the trivial redistribution of perturbation kinetic energy. The identified mechanism is related to the Orr mechanism, and it amplifies distortions to the material interface via a reorientation of perturbations by the mean shear. Analyses of the linearized energy budgets show that energy is extracted from the mean shear by the production term of the streamwise perturbation velocity component and subsequently is transferred to the radial perturbation velocity component, where it is absorbed by the surface-tension potential of the interface. The gain in surface tension energy attributable to the mechanism is shown to scale linearly with the Reynolds number. A critical Weber number is identified as a lower bound beyond which the mechanism becomes active, and a power-law relation to the Reynolds number is established. The realizability of the identified mechanism is verified using nonlinear simulations. Numerical simulations of the optimized jets are conducted with a weakly compressible multiphase solver on an unstructured grid with algebraic VoF as an interface-tracking algorithm. We demonstrate the multiphase Orr mechanism as a realizable process for liquid jet atomization with various flow conditions. The comparison study of liquid jet surface area between the randomly disturbed jet and the optimized jet demonstrates the formation of the small-scale structures, and hence the ligaments, occurring within a relatively short downstream distance in the latter case