Dynamical variability of axisymmetric buoyant plumes

Abstract

International audienceno abstrac

Energy-consistent entrainment relations for jets and plumes

“…This is indeed consistent -solutions to the PB model are straight-sided, with identical spreading rate regardless of the value of Γ , including forced and lazy plumes (Priestley & Ball 1955, see also § 5). The PB entrainment model provides predictions for α that are in reasonably good agreement with laboratory experiments on forced plumes in an unstratified ambient (Wang & Law 2002;Matulka et al 2014;Ezzamel, Salizzoni & Hunt 2015). In a response to Fox's analysis, Morton (1971) highlighted that for a plume rising in a linearly stratified ambient, both models were equally appropriate; however, near the source and the neutral buoyancy level the MTT approach was preferable because it was able to predict plume necking and spreading respectively.…”

“…The third term is associated with streamwise changes in the shape of the velocity statistics through the profile coefficients β g and γ g . These quantities are primarily associated with the mean flow, but also contain pressure and turbulence contributions, the latter being known to require a large distance to come to a full equilibrium (Wang & Law 2002;Ezzamel et al 2015). The third term is associated with the similarity drift discussed in Kaminski et al (2005) and Carazzo et al (2006), although as previously mentioned the present formulation does not contain contributions of θ m because of the standard top-hat definitions of r m and w m used in this paper (see appendix A).…”

“…In contrast to the relation derived by Kaminski et al (2005), the 'similarity drift' term in α-M (cf. third term), does not depend on the shape of the buoyancy profile via θ m (see also Ezzamel et al 2015), but only on γ m , i.e. the shape of the velocity profile.…”

“…Indeed, many modern particle image velocimetry and laser Doppler anemometry systems can record high-frequency velocity fields, which, if augmented with appropriate buoyancy measurements, provide access to all of the required coefficients in α-M. In complexity, the extension of the Kaminski entrainment relation by Ezzamel et al (2015) sits between α-F and α-M and will not be laboured further here.…”

“…This greatly complicates the use of their entrainment relation, as the relation between α and the Kaminski entrainment coefficient α e is not trivial (see appendix A for details). The framework was further extended with turbulence and pressure contributions by Ezzamel et al (2015), who provide an entrainment relation directly in terms of the Gaussian entrainment coefficient α G . They remark that similarity drift does not feature in α G in the same way as in α e (see also § 3), which, as noted above, is a result of the different scale definitions employed.…”