1 ISSN 0018-151X, High Temperature © Pleiades Publishing, Ltd., 2017. Numerical and Theoretical Solutions of a Turbulent Boussinesq Fountain Generated from Maintained Sources 1 I. Ben Abdellaziz*, M. Bouterra**, A. El Cafsi***, A. Belghith****, and B. Kalech***** LETTM, Faculté des Sciences de Tunis, El Manar II, 2092, Tunis, Tunisia *e-mail: benabdellazizimen@gmail.com **e-mail: Mourad.Bouterra@gmail.com ***e-mail: afif.elcafsi@fst.rnu.tn ****e-mail: ali.belghith@fst.rnu.tn *****e-mail: brahim_kalech@yahoo.fr Received January 25, 2016 Abstract⎯This paper provides theoretical and numerical investigation carried out on a vertical up-flow model risen to investigate the jet maximum height based on buoyancy frequency parameter and plume function We study the characteristic solution of the maximum height reached by the fountain through an asymptotic approach. Results show a discontinuity of affinity solution approximation at The fluid became very negligible and ended with a fall back. The entrainment coefficient effect had no significant influence on the topological behavior profiles development of this jet with negative buoyancy force. DOI: 10.1134/S0018151X17040022 INTRODUCTION The current convection generated from the point source (Fig. 1) causes the discharge of two types of jet extension. The variation of the background potential density with height (the density stratification) passes through a number of different dynamical stages or local f low types, as indicated in Fig. 1. The first region is initially characterized by an elevation of a positively vertical buoyant jet (Fig. 1) that mixes with the sur- rounding stratified fluid. That is achieved when the average jet density becomes equal to that of the ambi- ent f luid (neutral buoyancy). In the intermediate con- vective region, air becomes controlled primarily by buoyancy forces. When the direction of motion is reversed and the mixed fluid reaches neutral buoyancy force, it spreads horizontally. Similarly in the third region, an initial vertical negative buoyant jet in a strat- ified (uniform) stable density fluid rises progressively. This jet with negative buoyancy force (Fig. 1) is called a fountain [1]. These naturally bounded fountains are widely encountered in environmental and geophysical flows. Some examples which have grabbed interest include: the replenishment of magma chambers in the Earth’s crust [2, 3], submarine pyroclastic eruptions [4], and volcanic eruptions [5, 6]. The experimental, theoretical, and numerical studies deal mainly with the simplest ambient stratification, namely a linear density gradient. Among the plume (or fountain) variables which are unknown, a coefficient α that characterizes the rate of entrainment [2] stands out. Different types of fountains have been studied such as laminar miscible fountains [7], immiscible fountains [8], and turbulent miscible fountains [9, 10]. In [11], the effect of environ- mental density gradient on the behavior of turbulent fountains have been investigated. Authors compared the theoretical solution of the entrainment equations for the initial fountain height with the experimental results to give for an axisymmetric foun- tain and for a linear one, which is not significantly different from the jet entrainment coefficient of [12]. The modeling of turbulent miscible fountains is generally based on the theory of plumes [13]. It was assumed that the velocity of the ambient fluid entrained at the edge of the plume (or fountain) is proportional to the local vertical velocity. However, in the first study [14], a tur- bulent fountain was in both homogeneous and strati- fied environments. The entrainment equation was used to quantify the increasing radius, the decreasing buoyancy force and the velocity of dense fluid injected upwards into a lighter environment. Under the Bouss- inesq approximation, in [15], authors have theoreti- cally scrutinized the development of the fountain up- flow in a linearly stable stratified environment. They determined its maximal height and characteristics at this height as a function of the release conditions and the stratification strength by an analytical method. To generalize their approach [16], the initial positive buoyant release (plumes) in the case of a linear stable stratification is also studied. They have analytically σ i Γ. i Γ 1. i @ 1 The article is published in the original. α 0.085 0.010 = ± α 0.080 0.008 = ± α 0.076 0.004 = ±