Global instability in plane liquid sheet flows

The stability of liquid sheet (curtain) flows falling freely in the presence of the gravitational field, and subjected to surface tension at the liquid-gas interfaces, has been the subject of studies since many years, driven from both scientific and industrial interests. As far as the scientific view point is concerned, such a class of heterogeneous plane jet flows (characterized by relatively low velocities and hence low Weber numbers) has been considered as a model problem to apply and verify some of the basic issues of the modern developments made within the fluid dynamic instability framework (Teng, Lin & Chen, JFM 1997; De Luca & Costa, JFM 1997; Schmid & Henningson, JFM 2002). On the grounds of some theoretical and experimental findings presented by De Luca (JFM 1999) it is arguable that, below a certain critical value of the Weber number, the liquid sheet suffers a global instability, due to self-sustained modes, leading to the flow break-up. The presence of such a global instability would arise only when the region of local absolute instability is of sufficiently large extent. This result, although agrees with the basic theory of the linear global instability (Chomaz, Huerre & Redekopp, Phys. Rev. Lett. 1988), is not confirmed by the more recent non linear studies concerning globally synchronized self-sustained unstable structures, carried out by Pier, Huerre & Chomaz (Phys. D, 2001) on infinite domains. To the aim of giving some new insights into the matter, a global instability investigation is being developed, oriented to both linear (mainly based on the eigenvalues analysis) and non linear (mainly based on direct numerical simulations) approaches. The flow is assumed inviscid and, in order to simplify the modelling treatment, the problem is arranged in 1D formulation in the form of an integro-differential system along the streamwise direction (Mehring & Sirignano, JFM 1999; Schmid & Henningson, JFM 2002) taking into account surface tension effects. A spatial collocation numerical technique based on Chebyshev polynomials is employed. In this work linear analysis results are mainly presented, together with a discussion of such results in view of the more recent theories developed in a non linear framework. The frequency of the more unstable global mode is compared with the ones predicted by different theoretical criteria and the influence of the Weber number on the global instability onset is also investigated. Finally, global instability properties are compared with the corresponding local instability properties and a comparison with available experimental results is conducted.