Assessment of the mixing-limited hypothesis with first-principles simulation results

Starting with two well-tested, one-dimensional models of non-evaporating, mixing-limited sprays, governing equations for liquid mass and two-phase momentum for each model can be manipulated to reveal the formal similarity between momentum and liquid volume fraction. The consequence of this similarity is that momentum, when properly non-dimensionalized, is equal to the liquid volume fraction at any time and at any axial location within a non-evaporating, mixing-limited spray with a constant rate of injection. An alternative, the more well-known similarity between mass fraction and velocity, is also mathematically evident. We compare predictions of this mathematical analysis to high-fidelity, first-principles simulation results of a non-evaporating spray to assess the validity of the theoretical similarity. The analysis of the simulation not only confirms the mathematical derivations but also points to subtlety in the definition of the spray velocity. In particular, the density-weighted velocity is required to observe similarity. The requirement of density-weighted velocity means that similarity tests require knowledge of both phase velocities. The agreement also works to confirm that the first-principles simulations are indeed mixing-limited, despite the finite nature of domain size and resolution.