Shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization. Similar behavior occurs inside of supersonic combustors (scramjets) under startup conditions, but it is challenging to study these conditions experimentally. In order to investigate this phenomenon further, a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces. The flow field is solved via the compressible multicomponent Euler equations (i.e., the five equation model) discretized with the finite volume method on a uniform Cartesian grid. The solver utilizes a total variation diminishing (TVD) third-order Runge-Kutta method for time-marching and second order TVD spatial reconstruction. Surface tension is incorporated using the Continuum Surface Force (CSF) model. Fluxes are upwinded with a modified Harten-Lax-van Leer Contact (HLLC) approximate Riemann solver. An interface compression scheme is employed to counter numerical diffusion of the interface. The present work includes modifications to both the HLLC solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gas-liquid interface. A simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non-dimensionalization of the model. The model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and Mach 1.47 and 3 shock-droplet interaction problems. The general characteristics of secondary atomization at a range of Weber numbers are also captured in a series of simulations.

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The secondary atomization of liquid droplets is a common physical phenomenon in many industrial and engineering applications. Atomization in high speed compressible flows is less well understood than its more frequently studied low Mach number counterpart. The key to understanding the mechanisms of secondary atomization is examination of the breakup characteristics and droplet trajectories across a range of physical conditions. In this study, a planar shock wave impacting a cylindrical water column (ρ l = 1000 kg/m 3 ) is simulated for a range of Weber numbers ranging three orders of magnitude (∼ 10 0 − 10 3 ). Four different incident shock speeds are simulated (Ms = 1.47, 2, 2.5, 3) which induce subsonic, transonic, and supersonic crossflow across the column. The flowfield is solved using a compressible multicomponent Navier-Stokes solver with capillary forces. Fluid immiscibility is maintained with an interface sharpening scheme. Overall, a diverse range of complex interface dynamics are captured across the range of physical conditions studied. Additionally, while the unsteady drag coefficient of the liquid column shows a dependence on the Weber number using the undeformed diameter, calculations using the deformed diameter significantly reduce the dependence, particularly for the supersonic cases, with implications for subgrid droplet modeling in atomization simulations. A preliminary under-resolved three-dimensional simulation of droplet breakup shows reasonable agreement with experimental data, indicating the potential of the numerical approach for future investigations.

Background The respective roles of plant- and animal-sourced foods in sustainable healthy diets for humans remain unclear. Nutritional quality and the monetary cost of diets are key criteria among others for sustainable food production. Objective Linear programming (LP) was used to determine the composition of nutritionally adequate dietary patterns formulated at the lowest cost. The hypothesis tested was that animal-sourced foods would be included in least-cost diets due to their high density of particular essential nutrients. Methods The LP modeling work was based on eating patterns, retail food prices (2020), and the daily energy (11,150 kJ, 2665 kcal) and essential nutrient requirements (29 nutrients in total) of a reference adult in New Zealand (NZ). The LP modeling approach is publicly and freely available to readily illustrate the change in dietary profiles and daily diet cost, in the simulation of changes in energy and nutrient requirements, and price fluctuations within food groups. Results A nutrient-adequate, least-cost dietary pattern formulated from 883 foods, with a daily cost of NZ $3.23, included both animal- and plant-based foods. The nutrients found to be equally first-limiting were biotin, calcium, molybdenum, potassium, selenium, vitamin A, pantothenic acid, and vitamin C. When a dietary scenario with no animal-sourced foods was modeled, by increasing the retail prices of animal-sourced foods by 1.05 to 10.3 times, the daily cost of this plant-only dietary pattern was NZ $4.34. Additional nutrients, such as zinc, vitamin B-12, and vitamin D, were met at their daily minimum required levels. Conclusions Dietary patterns formulated at the lowest cost and meeting the daily minimum requirements for energy and essential nutrients for an adult in New Zealand relied on foods sourced from animals and plants.

A new solution procedure called Diagonal Split Corrected Linked Equations via Operator splitting (DS-CLEO) has been developed for solving the momentum equations in the pressure-based SIMPLER-type formulation of the incompressible Navier-Stokes equations. An explicit momentum predictor step based on a time splitting procedure is shown to possess unconditional stability while only the implicit pressure equation requires a computationally intensive iterative solver. No relaxation of any of the governing equations is necessary to ensure convergence for either steady or unsteady problems. It is demonstrated that pressure-velocity coupling and second order time accuracy is obtained through a set of inner iterations on the pressure and velocity fields using the exact discretized equations. The Diagonal Split procedure offered a 33.9 and 35.9x reduction in runtime compared to SIMPLER for two unsteady problems tested and a minor runtime reduction for steady lid-driven cavity flow.