A useable model for laminar free-surface jet evolution during flight, for both horizontal and vertical jets, is developed through joint analytical, experimental, and simulation methods. The jet’s impingement centerline velocity, recently shown to dictate stagnation zone heat transfer, encompasses the entire flow history: from pipe-flow velocity profile development to profile relaxation and jet contraction during flight. While pipe-flow is well-known, an alternative analytic solution is presented for the centerline velocity’s viscous-driven decay. Jet-contraction is subject to influences of surface tension (We), pipe-flow profile development, in-flight viscous dissipation (Re), and gravity (Nj = Re/Fr). The effects of surface tension and emergence momentum flux (jet thrust) are incorporated analytically through a global momentum balance. Though emergence momentum is related to pipe flow development, and empirically linked to nominal pipe flow-length, it can be modified to incorporate low-Re downstream dissipation as well. Jet contraction’s gravity dependence is extended beyond existing uniform-velocity theory to cases of partially and fully developed profiles. The final jet-evolution model relies on three empirical parameters and compares well to present and previous experiments and simulations. Hence, micro-jet flight experiments were conducted to fill-in gaps in the literature: jet contraction under mild gravity-effects, and intermediate Reynolds and Weber numbers (Nj = 5–8, Re = 350–520, We = 2.8–6.2). Furthermore, two-phase direct numerical simulations provided insight beyond the experimental range: Re = 200–1800, short pipes (Z = L/d · Re ≥ 0.01), variable nozzle wettability, and cases of no surface tension and/or gravity.

Jet impingement flow is known to generate one of the highest single-phase heat transfer rates, with potential for micro-electronics cooling applications. Although free-surface jets have been studied extensively, existing models are either too complex for practical use or do not consider all relevant parameters, such as the impinging jet’s velocity profile. Recently the authors have shown that the stagnation zone heat transfer is dictated by the jet’s centerline velocity upon impingement, and that going between the limiting cases (uniform vs. parabolic profiles, laminar flow) corresponds to a two-fold increase in heat transfer. In the present study, which is motivated by cooling at micro-scales (predominantly laminar flows), this simplified analysis is extended leading to a first-order analytical approximation, which is valid not only for the limiting cases but over the entire profile range. Thereby, the development of the jet flow both in the nozzle (pipe-type) and subsequently during its flight (before impingement) is incorporated in this model over a broad range of parameters. For validation of the model, as well as for additional insight into the governing physics, direct numerical simulations were conducted. Through which it is shown that the jet’s velocity profile and its evolution during free “flight” are dependent on the level of the flow’s upstream development in the nozzle, both of which depend on a single self-similar scale: distance travelled normalized by the nozzle diameter and Reynolds number. This one-way coupling requires incorporation of both stages of development for an accurate description, as done in the present model. During pipe-flow, the first-order model employs a more-rapid development rate than during jet-flight (due to the additional pressure-driven flow) — converging to more complex, well-known models, within a few pipe diameters (for Re = 200 to 2300). During flight, the model describes velocity profile relaxation, which is dominated by viscous diffusion and assisted by jet contraction. Jet contraction is dependent on the emerging velocity profile and liquid-vapor surface tension. For most relevant conditions surface tension is negligible, under which the first-order model centerline velocity decay prediction agrees well with both present simulations and previous works. Thereby, the present work lays the foundation for a simpler, more useable model for predicting heat transfer under an impinging free-surface jet, over a wide range of conditions (various liquids, pipe-type nozzles of different lengths, flow-rates and nozzle-to-plate distances), as part of an ongoing study into micro-jet array heat transfer.