Turbulent buoyant jets into stratified or flowing ambient fluids

Theoretical and experimental studies were made on two classes of buoyant jet problems, namely:

1) an inclined, round buoyant yet in a stagnant environment with linear density-stratification;

2) a round buoyant jet in a uniform cross stream of homogenous density.

Using the integral technique of analysis, assuming similarity, predictions can be made for jet trajectory, widths, and dilution ratios, in a density-stratified or flowing environment. Such information is of great importance in the design of disposal systems for sewage effluent into the ocean or waste gases into the atmosphere.

The present study of a buoyant jet in a stagnant environment has extended the Morton type of analysis to cover the effect of the initial angle of discharge. Numerical solutions have been presented for a range of initial conditions. Laboratory experiments were conducted for photographic observations of the trajectories of dyed jets. In general the observed jet forms agreed well with the calculated trajectories and nominal half widths when the value of the entrainment coefficient was taken to be α = 0.082, as previously suggested by Morton.

The problem of a buoyant jet in a uniform cross stream was analyzed by assuming an entrainment mechanism based upon the vector difference between the characteristic jet velocity and the ambient velocity. The effect of the unbalanced pressure field on the sides of the jet flow was approximated by a gross drag term. Laboratory flume experiments with sinking jets which are directly analogous to buoyant jets were performed. Salt solutions were injected into fresh water at the free surface in a flume. The jet trajectories, dilution ratios and jet half widths were determined by conductivity measurements. The entrainment coefficient, α, and drag coefficient, C_d, were found from the observed jet trajectories and dilution ratios. In the ten cases studied where jet Froude number ranged from 10 to 80 and velocity ratio (jet: current) K from 4 to 16, α varied from 0.4 to 0.5 and C_d from 1.7 to 0.1. The jet mixing motion for distance within 250D was found to be dominated by the self-generated turbulence, rather than the free-stream turbulence. Similarity of concentration profiles has also been discussed.

Development of roll waves in open channels

This study is concerned with some of the properties of roll waves that develop naturally from a turbulent uniform flow in a wide rectangular channel on a constant steep slope . The wave properties considered were depth at the wave crest, depth at the wave trough, wave period, and wave velocity . The primary focus was on the mean values and standard deviations of the crest depths and wave periods at a given station and how these quantities varied with distance along the channel.

The wave properties were measured in a laboratory channel in which roll waves developed naturally from a uniform flow . The Froude number F (F = u_n/√gh_n, u_n = normal velocity , h_n = normal depth, g =acceleration of gravity) ranged from 3. 4 to 6. 0 for channel slopes S_o of . 05 and . 12 respectively . In the initial phase of their development the roll waves appeared as small amplitude waves with a continuous water surface profile . These small amplitude waves subsequently developed into large amplitude shock waves. Shock waves were found to overtake and combine with other shock waves with the result that the crest depth of the combined wave was larger than the crest depths before the overtake. Once roll waves began to develop, the mean value of the crest depths h_nmax increased with distance . Once the shock waves began to overtake, the mean wave period T_av increased approximately linearly with distance.

For a given Froude number and channel slope the observed quantities h^-_max/h_n , T' (T' = S_o T_av √g/h_n), and the standard deviations of h^-_max/h_n and T', could be expressed as unique functions of l/h_n (l = distance from beginning of channel) for the two-fold change in h_n occurring in the observed flows . A given value of h^-_max/h_n occurred at smaller values of l/h_n as the Froude number was increased. For a given value of h /hh^-_max/h_n the growth rate of δh^-_max/h^-_maxδl of the shock waves increased as the Froude number was increased.

A laboratory channel was also used to measure the wave properties of periodic permanent roll waves. For a given Froude number and channel slope the h^-_max/h_n vs. T' relation did not agree with a theory in which the weight of the shock front was neglected. After the theory was modified to include this weight, the observed values of h^-_max/h_n were within an average of 6.5 percent of the predicted values, and the maximum discrepancy was 13.5 percent.

For h^-_max/h_n sufficiently large (h^-_max/h_n > approximately 1.5) it was found that the h^-_max/h_n vs. T' relation for natural roll waves was practically identical to the h^-_max/h_n vs. T' relation for periodic permanent roll waves at the same Froude number and slope. As a result of this correspondence between periodic and natural roll waves, the growth rate δh^-_max/h^-_maxδl of shock waves was predicted to depend on the channel slope, and this slope dependence was observed in the experiments.

Mathematical models for the prediction of temperature distributions resulting from the discharge of heated water into large bodies of water

Mathematical models for heated water outfalls were developed for three flow regions. Near the source, the subsurface discharge into a stratified ambient water issuing from a row of buoyant jets was solved with the jet interference included in the analysis. The analysis of the flow zone close to and at intermediate distances from a surface buoyant jet was developed for the two-dimensional and axisymmetric cases. Far away from the source, a passive dispersion model was solved for a two dimensional situation taking into consideration the effects of shear current and vertical changes in diffusivity. A significant result from the surface buoyant jet analysis is the ability to predict the onset and location of an internal hydraulic jump. Prediction can be made simply from the knowledge of the source Froude number and a dimensionless surface exchange coefficient. Parametric computer programs of the above models are also developed as a part of this study. This report was submitted in fulfillment of Contract No. 14-12-570 under the sponsorship of the Federal Water Quality Administration.

Comparison of model predictions and speed trial results for a group of 32' wooden trawlers

Speed trial results of six 32' wooden stern, trawlers built to the same design are analyzed to determine the EHP ship Froude number curves in each case and are compared with the model test results of the same design and for the corresponding, displacement, conditions.

On the transition from finite-volume negatively buoyant releases to continuous fountains

An experimental investigation to identify the source conditions that distinguish finite-volume negatively buoyant fluid projectile behaviour from fountain behaviour in quiescent environments of uniform density is described. Finite-volume releases are governed by their source Froude number Fr D and the aspect ratio L/D of the release, where L denotes the length of the column of fluid dispensed vertically from the nozzle of diameter D. We establish the influence of L/D on the peak rise heights of a release formed by dispensing saline solution into fresh water for 0Froude-number-dependent, i.e. fountain-like, behaviour. As such, we make the link between finite-volume releases and continuous fountains. The Fr D(L/D) f pairs led us directly to the classification of a Fr D, L/D space from which source conditions giving rise to either negatively buoyant projectiles or fountains may be readily identified. The variation of (L/D) f with Fr D corresponds closely to established fountain regimes of very weak, weak and forced fountains. Moreover, our results indicate that the formation or otherwise of a primary vortex, as fluid is ejected, has a profound influence on the length of the dispensed fluid column that is necessary to achieve rise heights equal to fountain rise heights. © 2012 Cambridge University Press.

Transient flows in enclosures: A generalised approach for modelling the effects of geometry, heat gains and wind

Transient flows in a confined ventilated space induced by a buoyancy source of time-varying strength and an external wind are examined. The space considered has varying cross-sectional area with height. A generalised theoretical model is proposed to investigate the flow dynamics following the activation of an external wind and an internal source of buoyancy. To investigate the effect of geometry, we vary the angle of the wall inclination of a particular geometry in which a point source of constant buoyancy is activated in the absence of wind. Counter-intuitively the ventilation is worse and lower airflow rates are established for geometries of increasing cross-sectional areas with height. We investigate the effect of the source buoyancy strength by comparing two cases: (1) when the buoyancy input is constant and (2) when the buoyancy input gradually increases over time so that after a finite time the total buoyancy inputs for (1) and (2) are identical. The rate at which the source heat gains are introduced has a significant role on the flow behaviour as we find that, in case (2), a warmer layer and a more pronounced overshoot are obtained than in case (1). The effect of assisting and opposing wind on the transient ventilation of an enclosure of constant cross-sectional area with height and constant heat gains is examined. A Froude number Fr is used to define the relative strengths of the buoyancy-induced and wind-induced velocities and five different transient states and their associated critical Fr are identified. © 2010 Elsevier Ltd.

The unidirectional emptying box

A theoretical description of the turbulent mixing within and the draining of a dense fluid layer from a box connected to a uniform density, quiescent environment through openings in the top and the base of the box is presented in this paper. This is an extension of the draining model developed by Linden et al. (Annu. Rev. Fluid Mech. vol. 31, 1990, pp. 201-238) and includes terms that describe localized mixing within the emptying box at the density interface. Mixing is induced by a turbulent flow of replacement fluid into the box and as a consequence we predict, and observe in complementary experiments, the development of a three-layer stratification. Based on the data collated from previous researchers, three distinct formulations for entrainment fluxes across density interfaces are used to account for this localized mixing. The model was then solved numerically for the three mixing formulations. Analytical solutions were developed for one formulation directly and for a second on assuming that localized mixing is relatively weak though still significant in redistributing buoyancy on the timescale of the draining process. Comparisons between our theoretical predictions and the experimental data, which we have collected on the developing layer depths and their densities show good agreement. The differences in predictions between the three mixing formulations suggest that the normalized flux turbulently entrained across a density interface tends to a constant value for large values of a Froude number FrT, based on conditions of the inflow through the top of the box, and scales as the cube of FrT for small values of FrT. The upper limit on the rate of entrainment into the mixed layer results in a minimum time (tD) to remove the original dense layer. Using our analytical solutions, we bound this time and show that 0.2tE ≈tD tE, i.e. the original dense layer may be depleted up to five times more rapidly than when there is no internal mixing and the box empties in a time tE. © 2010 Cambridge University Press.

Scaling arguments for the fluxes in turbulent miscible fountains

For established axisymmetric turbulent miscible Boussinesq fountains in quiescent uniform environments, expressions are developed for the fluxes of volume, momentum and buoyancy at the outflow from the fountain: the outflow referring to the counterflow at the horizontal plane of the source. The fluxes are expressed in terms of the fountain source conditions and two dimensionless functions of the source Froude number, Fr0: a radial function (relating a horizontal scale of the outflow to the source radius) and a volume flux function (relating the outflow and source volume fluxes). The forms taken by these two functions at low Fr0 and high Fr0 are deduced, thereby providing the outflow fluxes and outflow Froude number solely in terms of the source conditions. For high Fr0, the outflow Froude number, Frout, is shown to be invariant, indicating (by analogy with plumes for which the 'far-field' Froude number is invariant with source Froude number) that the outflow may be regarded as 'far-field' since the fluxes within the fountain have adjusted to attain a balance which is independent of the source conditions. Based on Frout, the fluxes in the plume that forms beyond the fountain outflow are deduced. Finally, from the results of previously published studies, we show that the scalings deduced for fountains are valid for 0.0025 ≲ Fr0 ≲ 1.0 for low Fr0 and Fr0≳ 3.0 for high Fr0. © 2014 Cambridge University Press.

Study on methane hydration process in a semi-continuous stirred tank reactor

The methane hydration process is investigated in a semi-continuous stirred tank reactor. Liquid temperatures and reaction rates without stirrer are compared with those occurring with stirrer, while at the same time better stirring conditions of the methane hydration process are given by the experiments. Some basic data of fluid mechanics, for example, stirring Reynolds number, Froucle number and stirrer power, are calculated during the methane hydration process, which can be applied to evaluate stirrer capacity and provide some basic data for a scaled up reactor. Based on experiment and calculations in this work, some conclusions are drawn. First, the stirrer has great influence on the methane hydration process. Batch stirring is helpful to improve the mass transfer and heat transfer performances of the methane hydration process. Second, induction time can be shortened effectively by use of the stirrer. Third, in this paper, the appropriate stirring velocity and stirring time were 320 rpm and 30 min, respectively, at 5.0 MPa, for which the storage capacity and reaction time were 159.1 V/V and 370 min, respectively. Under the condition of the on-flow state, the initial stirring Reynolds number of the fluid and the stirring power were 12,150 and 0.54 W, respectively. Fourth, some suggestions, for example, the use of another type of stirrer or some baffles, are proposed to accelerate the methane hydration process. Comparing with literature data, higher storage capacity and hydration rate are achieved in this work. Moreover, some fluid mechanics parameters are calculated, which can provide some references to engineering application.

黄土坡面水力学特征参数与土壤侵蚀量间关系研究

通过室内人工降雨模拟试验,研究黄土坡面水流的水力学特征参数与土壤侵蚀量间的关系。结果表明:随着放水流量的增加,出口断面平均流速、雷诺数在逐渐增大,弗劳德数在逐渐减小,坡面流流态也从层流状态逐渐过渡到紊流状态,当放水流量在10.1~19.9 L/min范围内时,坡面水流的流态为层流与紊流之间的混和状态;出口断面的平均流速随着放水流量的增加而增大,二者之间呈良好的幂函数关系;黄土坡面的累计泥沙侵蚀总量随着雷诺数的增加而增大,二者呈幂函数关系;单宽土壤侵蚀产沙量与雷诺数之间拟合关系的好坏受雷诺数取值的影响,当雷诺数的值在临界雷诺数(取500)附近时,单宽土壤侵蚀产沙量与雷诺数之间的负相关性良好。研究结果不仅可以揭示土壤侵蚀水动力过程的内在机制,而且对黄土高原地区生态环境建设具有重要的现实意义。

不同坡度草地含沙水流水力学特性及其拦沙机理

参照黄土区侵蚀降雨和坡面片蚀产沙特征,采用恒流泥沙输送装置模拟坡面上方来水来沙,探讨不同坡度草地含沙水流的水力学特性及其对上方来沙的拦蓄机理。结果表明,草地坡面的水流弗劳德数随坡度增大而增加,而阻力系数与坡度呈反势。按明渠水流的一般标准,不同坡度草地水流均为层状缓流。草地坡面拦沙效应随坡度增大而减小,且径流前期的减沙作用较后期更为显著。不同坡度草地坡面的出流泥沙平均直径和大颗粒(>10μm)泥沙含量均显著小于上方来沙,这说明草地的拦沙效应主要体现在对大粒径泥沙的拦蓄上。

Drum granulation of NPK fertilizers

Nitrate-based NPK fertilizer was granulated in a bench scale drum granulation unit. The initial fertilizer possessed a particle size distribution similar to those in industrial granulation units. In this work, three factors are identified affecting the degree of fertilizer granulation, these are solution to solid phase ratio, the binder viscosity and the optimal rotation speed of the drum. Experimental results indicate that a critical solution to solid phase ratio is required for an increase in granulation in terms of mass median diameter. The saturated solution viscosity in this system was measured and correlated well to binder viscosity granulation theory with the critical Stokes number calculated at 700. The optimum rotation speed for flighted and unflighted drums correlated with the Froude number relationship for full scale granulation units. (C) 2000 Elsevier Science S.A. All rights reserved.

A Description of the Stationary Wave Pattern of a Marine Craft in Shallow and Intermediate Water Depths

A stationary phase model is used to study supercritical waves generated by high speed ferries. Some general relationships in terms of wave angle, propagation direction, dispersion relationship and depth wavelength relationship are explored and discussed. In particular, it is shown that the wave pattern generated by high speed craft at supercritical speeds depends mainly on the relationship of water depth and ship speed and that the wave patterns are similar in terms of location of crests and troughs for a given depth Froude number. In addition it is found that the far field wave pattern can be described adequately using a single moving point source. The theoretical model compares well with towing tank measurements and full scale data over a range of parameters and hull shapes. The paper also demonstrates that the far field wave pattern at supercritical speeds should be non-dimensionalised by water depth and not hull length unlike it is usually done for subcritical speeds.

Properties of Hydraulic Jump over Apparent Corrugated Beds

The characteristics of hydraulic jumps were investigated for three shapes of artificial apparent corrugated beds in a horizontal rectangular flume. Rectangular, triangular, and circular-shaped tire waste corrugated beds were used. Froude number ranged from 2.75 to 4.25. The experimental observations included water surface profiles, bed shear stress, and the hydraulic jump length. Results showed that the shape of the corrugation had relatively insignificant effects on hydraulic jump properties for small Froude numbers. The rectangular, triangular, and circular-shaped corrugated beds reduced the hydraulic jump length by up to 7, 10, and 11%, respectively. The corrugated bed also reduced the tailwater depth by up to 11.5% compared with the smooth bed. The apparent conditions of corrugated bed reduced the hydraulic jump relative length and height by about 0.4 and 0.5, respectively. The circular-shaped tire waste was found to be more effective in reducing the length and depth of the hydraulic jump.

Spatial variability of macroinvertebrate assemblages and the influence of hydrological and environmental variables along the Sigi River, Tanzania-East Africa

The study of investigating the spatial and temporal variability of macroinvertebrate and their relation to hydrology, hydraulic and environmental factors was done along the Sigi River during two sampling periods in the dry (March) and wet (May) periods of 2012. The river was demarcated based on slope ranges and five river zones were identified as mountains streams (MS), upper foothills (UF), lower foothills (LF), rejuvenated foothills (REJ) and mature lower river (MR). Samples of macroinvertebrate were collected from the five river zones and measurements of hydrological (discharge), hydraulics (Depth, velocity and Froude number) and Environmental (pH, Temperature, substrate, conductivity) parameters were done in each zone. In characterizing the macroinvertebrate assemblages along the Sigi River diversity indices (number of taxa, total abundances, Margalef richness index and ShannonWiener index) were calculated and the most representative species for the spatial and temporal variation were identified. Melanoides and Afronurous showed differences in abundance in two samplings periods while Cleopatra, Potamonautes, Ephemerythus, Neoperla, Caenis, Ceratogomphus and Cheumatopsyche showed significant difference among the river zones. Spearman rank correlation and Distance Linear Model (DistLM) used to revealed physical factors governing the macroinvertebrate assemblages distribution. The study demonstrated that the variation of physical factors like discharge, temperature, conductivity and pH have an important role in the spatial distribution of macroinvertebrate assemblages along the river and the life cycle of macroinvertebrate (Afronurus) is important in determining the temporal variability.