Numerical simulation of jets generated by a sphere moving vertically in a stratified fluid

Abstract

The flow past a sphere moving vertically at constant speeds in a salt-stratified fluid is investigated numerically at moderate Reynolds numbers Re. Time development of the flow shows that the violation of density conservation is the key process for the generation of the buoyant jet observed in the experiments. For example, if the sphere moves downward, isopycnal surfaces are simply deformed and dragged down by the sphere while the density is conserved along the flow. (The flow pattern is inverted if the sphere moves upward. Some explanations are given in the introduction.) Then, the flow will never become steady. As density diffusion becomes effective around the sphere surface and generates a horizontal hole in the isopycnal surface, fluid with non-conserved density is detached from the isopycnal surface and moves upward to generate a buoyant jet. These processes will constitute a steady state near the sphere. With lengths scaled by the sphere diameter and velocities by the downward sphere velocity, the duration of density conservation at the rear/upper stagnation point, or the maximum distance that the isopycnal surface is dragged downward, is proportional to the Froude number Fr, and estimated well by πFr for Fr 1 and Re 200, corresponding to a constant potential energy. The radius of a jet defined by the density and velocity distributions, which would have correlations with the density and velocity boundary layers on the sphere, is estimated well by √ Fr/2ReSc and √ Fr/2Re respectively for Fr 1, where Sc is the Schmidt number. Numerical results agree well with the previous experiments, and the origin of the conspicuous bell-shaped structure observed by the shadowgraph method is identified as an internal wave.

Curvilinear Coordinate System for Mathematical Analysis of Inclined Buoyant Jets Using the Integral Method

“…Alternatively, depending on the selection of the solution procedure and in case that the specific particles are in curvilinear motion, the above equations may be preferable to be expressed at a curvilinear coordinate system [3]. Although, several buoyant jet phenomena can be analyzed by the governing equations written in orthogonal Cartesian or cylindrical coordinate systems [4][5][6][7][8], there exist some specific phenomena that the use of a curvilinear orthogonal coordinate system is mandatory [9]. Such phenomena include buoyant jet flows created by inclined discharges, in which the transverse profiles of the mean axial velocity and mean concentration are approximated with Gaussian profiles.…”

Deformation of ambient chemical gradients by sinking spheres

“…For Re → ∞ and Sc → 0, the thickness of the chemical boundary layer is described by δ ρ /l ∼ Pe −1/2 (Schlichting 1968;Yih 1969). Previous studies of spheres sinking through salinity gradients have used this relation to justify the grid spacing in their numerical models but noted that it underestimates the chemical boundary layer thickness for finite Re and Sc (Torres et al 2000;Hanazaki et al 2009aHanazaki et al , 2015Doostmohammadi et al 2014). In this work, we derive a new scaling for δ ρ in the 10 −2 Re 10 2 and Pe > 10 regime.…”

“…A number of studies have depicted disturbance of stratification by falling spheres in order to describe increased drag (Eames & Hunt 1997;Srdić-Mitrović, Mohamed & Fernando 1999;Yick et al 2009;Zhang, Mercier & Magnaudet 2019;Magnaudet & Mercier 2020), settling speed anomalies (Abaid et al 2004;Camassa et al 2009Camassa et al , 2010Camassa et al , 2013Doostmohammadi, Dabiri & Ardekani 2014), internal wave generation (Mowbray & Rarity 1967) and buoyant jets behind the sphere (Ochoa & Van Woert 1977;Torres et al 2000;Hanazaki, Kashimoto & Okamura 2009a;Hanazaki, Konishi & Okamura 2009b;Hanazaki, Nakamura & Yoshikawa 2015). These studies indicate that perturbations of the compositional field by a sphere depend on viscosity, diffusivity and buoyancy, which we parameterize by the Reynolds number Re = Wl/ν, the Péclet number Pe = ReSc = Wl/κ and the Froude number Fr = W/Na, where W is the velocity of the sphere, l is the sphere diameter, a is the sphere radius, ν is the kinematic viscosity, κ is the diffusivity, N is the Brunt-Väisälä frequency and Sc = ν/κ is the Schmidt number.…”

“…The maximum vertical distance an isosurface can be dragged, L Def , is correlated with the extent of gradient 892 A33-4 B. G. Inman, C. J. Davies, C. R. Torres and P. J. S. Franks deformation, as we will show later. Hanazaki et al (2015) employed a dimensional analysis of the buoyancy time scale to estimate this distance as lπFr for Re 200, 1 Fr 10 and Pe > 10 5 . Yick et al (2009) combined experiments and numerical results to empirically estimate the dragged distance as l √…”

“…In § 3.1, we describe the general process of gradient deformation with a representative case and define the quantities δ ρ , L Def , Sh and M that are used in the subsequent analysis to characterize the influence of diffusion ( § 3.2), viscosity ( § 3.3) and buoyancy ( § 3.4). Previous work on spheres sinking through salinity gradients has focused on Re 200 or Re < 10, Fr < 10 and Sc = 7, 700 (Torres et al 2000;Yick et al 2009;Hanazaki et al 2015). The parameter range considered here is 10 −2 Re 10 2 , 10 −1 Fr 10 3 and 10 2 < Pe 10 6 .…”