Numerical simulations of negatively buoyant jets in an immiscible fluid using the Particle Finite Element Method

Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter ambient fluid. The numerical simulation of this kind of buoyancy‐driven flows is challenging as it involves multiple fluids with different physical properties. In the case of immiscible fluids, it requires, in addition, to track the motion of the interface between fluids and accurately represent the discontinuities of the flow variables. In this paper, we investigate numerically the injection of a negatively buoyant jet into a homogenous immiscible ambient fluid using the Particle Finite Element Method and compare the two‐dimensional numerical results with experiments on the injection of a jet of dyed water through a nozzle in the base of a cylindrical tank containing rapeseed oil. In both simulations and experiments, the fountain inlet flow velocity and nozzle diameter have been varied to cover a wide range of Froude Fr and Reynolds Re numbers ( 0.1 < Fr < 30, 8 < Re < 1350), reproducing both weak and strong laminar fountains. The flow behaviors observed for the different numerical simulations fit in the regime map based on the Re and Fr values of the experiments, and the maximum fountain height is in good agreement with the experimental observations, suggesting that particle finite element method is a useful tool for the study of immiscible two‐fluid systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Transient positively and negatively buoyant turbulent round jets

 The transient character of the jet issuing from an upward nozzle centered at the bottom of a vertical cylindrical tank into bulk liquid of a different density was measured using flow visualization and PIV for varying densimetric Froude numbers by varying the jet Reynolds numbers and the ratios of fluid densities. Positively buoyant jets penetrate to the free surface, driven by both momentum and buoyancy in the upward direction. The lighter jet fluid stratifies in a layer above the bulk liquid. Upon starting, a negatively buoyant jet has three stages. First the jet penetrates to its maximum height in the tank. Then the jet penetration decreases due to the downward backflow of heavier fluid surrounding the jet, which reduces the jet’s upward momentum. Finally the jet penetration height fluctuates around a mean value about 70% the maximum height of penetration. For small negative Froude numbers, the flow is fountain-like. The downward flow turns radially outward as it reaches the bottom of the tank and eventually an annular recirculation zone forms at the bottom of the tank with vortical motion opposite the vorticity of the jet. For large negative Froude numbers, the spreading of the jet extends far enough so the annular downward flow is along the walls of the tank resulting in a large annular recirculation zone. The penetration depth, h, and time, t, scale with buoyancy flux, F, and the jet momentum flux, M, as hM ^-3/4∣F∣^1/2 and t∣F∣/M to collapse the transient jet penetration height data onto a single curve over a wide range of Froude numbers for either positively or negatively buoyant jets. Received: 8 June 1998/Accepted: 3 February 1999     

Modelling and experimental investigation of horizontal buoyant gas jets injected into stagnant uniform ambient liquid

In this article, an experimental and theoretical study on the buoyant non-condensable gas jet that is injected horizontally into a high-density liquid ambient at different initial conditions is performed. Direct and instantaneous global measurements of the interface were performed using a high-speed photography. The position and motion of the entire gas jet were captured by a high-velocity camera and the images were processed, averaged and analyzed to extract the jet parameters and interface position. In the mathematical model, the rate of entrainment is assumed to be a function of the jet centerline velocity, the ratio of the mean jet and the ambient densities, while the entrainment coefficient depends on the local Froude number at the jet region. An interfacial shear stress acting at the interface between the jet flow and the water ambient in the opposed direction to the main jet momentum flux is considered. The results showed that the model is able to accurately predict the jet parameters: trajectory, spread, jet angles and penetration lengths as well as the jet regimes. An overall good agreement was obtained between the simulation and experimental results over a large range of Froude numbers and jet diameters. The developed model has proven to be an adequate tool to predict the different jet parameters.     

Asymptotic investigation of turbulent buoyant jets

Results of a theoretical investigation are presented for the averaged characteristics of turbulent slightly swirling jets discharged vertically upward into a stagnant homogeneous medium. A semi-empirical model of turbulence is used to close the system of initial turbulent boundary layer equations. Exact and approximate analytical solutions are suggested. The results of calculations are compared with experimental and numerical data of other authors.     

Instability threshold of a negatively buoyant fountain

Experimental simulations were carried out to investigate the onset of instability in negatively buoyant fountains by injecting glycerin–water mixtures into silicon oil. The transition from a stable to an unstable fountain structure is primarily governed by the Richardson number, and to a lesser extent, Reynolds number, viscosity ratio, Weber number and vent geometry. Transition nominally occurs at a Ri = 1.0. For a fountain issuing from a cylindrical pipe, the major effect of the Reynolds number is in determining whether or not the fountain is laminar or turbulent. The Reynolds number effect can be largely accounted for by basing a corrected Richardson number on the root mean square of the mean velocity. Viscosity ratio deviating from unity has the effect of stabilizing the flow structure and thereby reducing the transition Richardson number. Similarly, interfacial tension stabilizes the flow pattern resulting in a trend of increasing transition Richardson number with increasing Weber number. The results are valid in rectangular vents if the Richardson number and Reynolds number are based on the hydraulic diameter.     

Flow behaviour of turbulent nozzle jets issuing from bevelled collars

An experiment was conducted to investigate turbulent, low-speed air jets issuing from bevelled and non-bevelled circular collared-nozzle configurations. The collar-to-nozzle expansion ratio used was three and Reynolds number was approximately 20,000. Detailed mean flow velocity fields and velocity spectra of the resultant jet flows at different collar lengths and bevel angles were evaluated using hot-wire anemometry along both axial and radial directions of the jets. Centreline velocity decay was shown to be augmented when either collar length or bevel angle was increased, with the collar length playing a more dominant role. Results also showed that bevelled collared-jets vectored towards the longer collar-length region, the extent to which was enhanced when the collar length or bevel angle was increased. The study demonstrated that a bevelled collar exit could be used as an additional control device on top of the collar length to achieve finer jet flow control in terms of jet momentum vectoring and asymmetric jet spread.     

BEHAVIOR OF OBSTRUCTED SQUARE BUOYANT VERTICAL JETS IN STATIC AMBIENT （Ⅱ）——ANALYSIS ON BEHAVIOR OF FLOW FIELD

Based on a series of numerical calculations, the behavior of flow field in obstructed square buoyant vertical jet is summarized and analyzed. Based on the axial line velocity distribution, the flow after the disc can be divided into three regions, i.e., recirculation region, transitional region and self-similar region The characteristic of self-similarity of upright velocity was validated. The three regions can also be distinguished based on the axial velocity. The axial velocity in self-similar region was found to obey the same law and the formula presented by introducing the velocity expression used by Chen and Rodi. The isolines of pressure on cross-sections of different heights were displayed and the production, expansion, breaking and disappearing of negative pressure regions were found.     

Mixing enhancement in axisymmetric turbulent isothermal and buoyant jets

Measurements of the velocity and concentration in axisymmetric, turbulent, isothermal and buoyant jets have been performed with laser-Doppler velocimetry and planar and point laser-induced fluorescence to quantify the mixing enhancement achieved by periodic forcing when the jet exit has a fully-developed turbulent pipe flow, a situation less well-studied than the case of laminar initial conditions. It was found that forcing at Strouhal numbers around 0.6 enhances mixing in the developing region of the jet and this enhancement increased with increasing amplitude of excitation, consistent with results of initially-laminar jets. The initial turbulence intensity did not have any effect, but an increase in the initial lengthscale of the turbulence, controlled by a perforated plate inside the nozzle, caused faster mixing. In agreement with previous experiments, the initial conditions of the jet did not affect the far-field rate of decay, but the jet-fluid concentration there was significantly reduced by forcing due to the increased mixing during the early stages of development, an effect that can be described by a smaller virtual origin in decay laws of jet decay. These results are independent of the Froude number because the initial conditions have an influence only in the early stages where the flow is still momentum dominated.List of Symbols A normalised excitation amplitude, defined by A = u'/U ₀ - D nozzle diameter - f jet-fluid concentration - F mean f - f r.m.s. f - Fd Froude number, defined by Fd=U ₀ ² /(gDT ₀) - g acceleration of gravity - I fluorescent intensity - I _inc incident light intensity - I _ref light intensity of the reference flow - K decay constant - L _hf concentration halfwidth - M mixing enhancement, defined by U _cl/U _cl,st=0 at x/D=5 - r radial coordinate - Re Reynolds number, defined by Re=U ₀ D/v - [Rh] concentration of Rhodamine B - St Strouhal number, defined by St=D/U ₀ - T ₀ temperature of jet fluid - T temperature of outer fluid - T ₀ temperature difference (= T ₀–, T ) - u r.m.s. axial velocity - u r.m.s. of the sinusoidal velocity fluctuation due to forcing - U mean axial velocity - U _cl mean axial centreline velocity - U _cl,st=0 mean axial centreline velocity for an unforced jet - U _max U at the centre of the nozzle exit - U ₀ bulk velocity at nozzle exit - x streamwise coordinate - X ₀ virtual origin Greek coefficient of thermal expansion - kinematic viscosity of the jet fluid - forcing frequency The experiments described here have been performed together with Mr. J. Sakakibara. Acknowledgments are also due to Prof. H. Longmire, of the University of Minnesota, for helpful discussions on forcing. This work was done while E.M. visitied Keio University with the financial assistance of TEPCO.     

BEHAVIOR OF OBSTRUCTED SQUARE BUOYANT VERTICAL JETS IN STATIC AMBIENT （Ⅰ）—— VERIFICATION OF MATHEMATICAL MODEL AND NUMERICAL METHOD

Some experiments were made for the buoyant jet from a square orifice with a square disc placed on it in static ambient and concentration along the axis in self-similar area behind disc was measured. And at the same time a three-dimensional mathematical model was established to simulate the whole flowing under different conditions. All the results predicted by the numerical calculation were substantiated by the experiments. The results were compared with experiential formula for obstructed round buoyant vertical jets in static ambient and it was found that the two concentration distributions had good accordance. Star shape of temperature isolines on cross-sections in the near areas from the disc was found and it was a very special figure for obstructed square buoyant vertical jets with a square disc. The shape will transform to concentric circles gradually alike to the round buoyant vertical jet in self-similar area with increasing of the distance from the disc.     

The behaviour of large drops in immiscible liquids

The motion of large drops with Eötvos numbers as high as 1000 has been investigated in large tanks where wall effects are negligible. Shapes observed include ellipsoidal- and spherical-caps with and without skirts, crescents, biconcave disks, toroids and wobbling irregular forms. The ellipsoidal-cap drops are shown to obey an equation based upon an extension of the Davies and Taylor theory for bubbles in liquids, regardless of whether skirts are being formed and shed at the rear.     

Some exact solutions of problems of laminar jets of immiscible fluids

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 94–98, May–June, 1988.     

Asymptotic method for solving problems with radial laminar jets of immiscible liquids

Two problems involving radial laminar jets of immiscible liquids are considered: a free radial slit jet and a jet on a rotating disk. An asymptotic method of solution is proposed that makes it possible to determine the flow parameters far from the source. The difference between these flows and those of homogeneous liquids is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 192–28, May–June, 1980.     

Mathematical model of micropolar fluid in two-phase immiscible fluid flow through porous channel

This paper is concerned with the flow of two immiscible fluids through a porous horizontal channel. The fluid in the upper region is the micropolar fluid/the Eringen fluid, and the fluid in the lower region is the Newtonian viscous fluid. The flow is driven by a constant pressure gradient. The presence of micropolar fluids introduces additional rotational parameters. Also, the porous material considered in both regions has two different permeabilities. A direct method is used to obtain the analytical solution of the concerned problem. In the present problem, the effects of the couple stress, the micropolarity parameter, the viscosity ratio, and the permeability on the velocity profile and the microrotational velocity are discussed. It is found that all the physical parameters play an important role in controlling the translational velocity profile and the microrotational velocity. In addition, numerical values of the different flow parameters are computed. The effects of the different flow parameters on the flow rate and the wall shear stress are also discussed graphically.     

The role of inlet location on the dynamics of thin two-dimensional buoyant surface jets

The transition between the motion of a subsurface buoyant jet and a supersurface buoyant jet has been studied. Measurements indicate that the transition region is sharp and characterized by an intermediate regime where both a wall-hugging flow and the more standard outlet flow are present. The point of transition was found to exhibit significant hysteresis depending on whether the receiving pool height was increasing or decreasing with time. Measurements of the temperature field downstream show that the effects of the different regimes on the density stratification can persist for long distances.     

Long-term tracking of neutrally buoyant tracer particles in two-dimensional fluid flows

An experimental technique has been developed to produce and to track neutrally buoyant particles in a two-dimensional fluid flow. The key aspect of the technique is the ability to track particles for extended intervals (over an hour), which is essential for quantitative studies of transport and mixing. The approach is composed of two stages. In the first stage, digital image processing hardware partially processes the images, reducing the data rate to 50 kbyte/s (typically) and allowing several hours of data to be stored on a conventional computer disk. In the second stage, programs extract particle trajectories from the reduced data. The approach is tested in an experiment on planetary-type flows in a rotating annulus. In an appendix, a technique is discussed for fabricating wax or crayon particles with arbitrary density.This work was supported by the Office of Naval Research Grant No. N00014-89-J-1495- The authors would like to acknowledge useful conversations with and assistance from W. J. Holloway, E. R. Weeks, and H. L. Swinney.     

Buckling of thin fluid jets and threads

The quasione-dimensional equations of the dynamics of thin jets are used as the basis for a theoretical study of flexural loss of stability (buckling) in thin jets and threads of high-viscosity fluids flowing slowly and normally onto a plate. The characteristic equation is obtained within linear analysis. Its solution showed that instability arises only in the presence of a longitudinal compressive force in a jet, and when the distance from the exit of the nozzle creating the jet to the plate exceeds a certain value which is critical for the jet in question. This critical value is calculated. It is shown that the instability is aperiodic in nature if gravity is neglected (horizontal jets).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 37–45, July–August, 1987.