Plane Turbulent Surface Jets in Shallow Tailwater

2000 ◽  
Vol 123 (1) ◽  
pp. 121-127 ◽  
Author(s):  
S. A. Ead ◽  
N. Rajaratnam
Keyword(s):  
Laboratory Study ◽  
Momentum Flux ◽  
Return Flow ◽  
Volume Flux ◽  
Limited Extent ◽  
Ambient Fluid ◽  
Forward Flow ◽  
Velocity Scale ◽  
Surface Jet ◽  
Surface Jets

This paper presents a theoretical and laboratory study of plane turbulent surface jets in shallow tailwater. The main objective was to show that when the depth of tailwater is finite, the momentum flux in the forward flow in the surface jet decays appreciably with the distance from the nozzle producing the surface jet. This decay is shown to be due to the entrainment of the return flow which has negative momentum and an increase in the tailwater depth further away from the nozzle produces this return flow. An extensive set of experiments, with different Froude numbers and offset ratios, was conducted to observe and quantify the growth of the surface jet, the decay of the velocity scale, and the momentum flux and the variation of the volume flux. On the whole, the results from this study highlight the effect of the tailwater depth on the behavior of plane turbulent surface jets when the ambient fluid has a limited extent.

The mixing in a room by a localized finite-mass-flux source of buoyancy

2002 ◽  
Vol 471 ◽  
pp. 33-50 ◽  
Author(s):  
C. P. CAULFIELD ◽  
ANDREW W. WOODS
Keyword(s):  
Numerical Simulation ◽  
Mass Flux ◽  
Momentum Flux ◽  
Analytical Description ◽  
Return Flow ◽  
Asymptotic Model ◽  
Buoyant Plume ◽  
Finite Mass ◽  
Ambient Fluid ◽  
Source Fluid

The mixing produced by a turbulent buoyant plume with finite mass flux in a room is examined analytically and numerically. The entrainment of ambient fluid into the ascending buoyant plume leads to a return flow in the room which carries fluid downwards from the top of the room. The cycling of ambient fluid through the buoyant plume and the return flow causes the density to become uniform and gradually evolve towards that of the source fluid. As a result the buoyancy flux associated with the input fluid decreases and the plume motion becomes dominated by the source momentum flux. We develop an asymptotic model of the mixing using buoyant plume theory for a momentum-dominated flow. This provides an analytical description of the evolution of the density in the room which is in excellent accord with a full numerical simulation, and provides an improved description of the experimental filling-box data originally presented by Baines & Turner (1969).

scholarly journals Multiphase plumes in a stratified ambient

2019 ◽  
Vol 869 ◽  
pp. 292-312 ◽  
Author(s):  
Nicola Mingotti ◽  
Andrew W. Woods
Keyword(s):  
Oil And Gas ◽  
Fluid Phase ◽  
Convective Transport ◽  
Maximum Depth ◽  
Buoyancy Flux ◽  
Return Flow ◽  
Ambient Fluid ◽  
Plume Height ◽  
Cylindrical Column ◽  
Fall Speed

We report on experiments of turbulent particle-laden plumes descending through a stratified environment. We show that provided the characteristic plume speed $(B_{0}N)^{1/4}$ exceeds the particle fall speed, where the plume buoyancy flux is $B_{0}$ and the Brunt–Väisälä frequency is $N$, then the plume is arrested by the stratification and initially intrudes at the neutral height associated with a single-phase plume of the same buoyancy flux. If the original fluid phase in the plume has density equal to that of the ambient fluid at the source, then as the particles sediment from the intruding fluid, the fluid finds itself buoyant and rises, ultimately intruding at a height of about $0.58\pm 0.03$ of the original plume height, consistent with new predictions we present based on classical plume theory. We generalise this result, and show that if the buoyancy flux at the source is composed of a fraction $F_{s}$ associated with the buoyancy of the source fluid, and a fraction $1-F_{s}$ from the particles, then following the sedimentation of the particles, the plume fluid intrudes at a height $(0.58+0.22F_{s}\pm 0.03)H_{t}$, where $H_{t}$ is the maximum plume height. This is key for predictions of the environmental impact of any material dissolved in the plume water which may originate from the particle load. We also show that the particles sediment at their fall speed through the fluid below the maximum depth of the plume as a cylindrical column whose area scales as the ratio of the particle flux at the source to the fall speed and concentration of particles in the plume at the maximum depth of the plume before it is arrested by the stratification. We demonstrate that there is negligible vertical transport of fluid in this cylindrical column, but a series of layers of high and low particle concentration develop in the column with a vertical spacing which is given by the ratio of the buoyancy of the particle load and the background buoyancy gradient. Small fluid intrusions develop at the side of the column associated with these layers, as dense parcels of particle-laden fluid convect downwards and then outward once the particles have sedimented from the fluid, with a lateral return flow drawing in ambient fluid. As a result, the pattern of particle-rich and particle-poor layers in the column gradually migrates upwards owing to the convective transport of particles between the particle-rich layers superposed on the background sedimentation. We consider the implications of the results for mixing by bubble plumes, for submarine blowouts of oil and gas and for the fate of plumes of waste particles discharged at the ocean surface during deep-sea mining.

Fountains impinging on a density interface

2008 ◽  
Vol 595 ◽  
pp. 115-139 ◽  
Author(s):  
JOSEPH K. ANSONG ◽  
PATRICK J. KYBA ◽  
BRUCE R. SUTHERLAND
Keyword(s):  
Experimental Study ◽  
Relative Density ◽  
Constant Velocity ◽  
Gravity Current ◽  
Return Flow ◽  
Experimental Measurements ◽  
Ambient Fluid ◽  
Initial Momentum ◽  
Front Position ◽  
Homogeneous Environment

We present an experimental study of an axisymmetric turbulent fountain in a two-layer stratified environment. Interacting with the interface, the fountain is observed to exhibit three regimes of flow. It may penetrate the interface, but nonetheless return to the source where it spreads as a radially propagating gravity current; the return flow may be trapped at the interface where it spreads as a radially propagating intrusion or it may do both. These regimes have been classified using empirically determined regime parameters which govern the relative initial momentum of the fountain and the relative density difference of the fountain and the ambient fluid. The maximum vertical distance travelled by the fountain in a two-layer fluid has been theoretically determined by extending the theory developed for fountains in a homogeneous environment. The theory compares favourably with experimental measurements. We have also developed a theory to analyse the initial speeds of the resulting radial currents. The spreading currents exhibited two different flow regimes: a constant-velocity regime and an inertia-buoyancy regime in which the front position, R, scales with time, t, as R ∼ t3/4. These regimes were classified using a critical Froude number which characterized the competing effects of momentum and buoyancy in the currents.

Quantification of Mixing Performance in Parallel Jets

Volume 4 ◽  
2004 ◽  
Author(s):  
Nathan E. Bunderson ◽  
Barton L. Smith
Keyword(s):  
Flow Visualization ◽  
Momentum Flux ◽  
Width Ratio ◽  
Hot Wire ◽  
Ambient Fluid ◽  
Jet Mixing ◽  
Mixing Performance ◽  
Flux Condition ◽  
Two Component ◽  
Hot Wire Anemometry

Experiments of unvented parallel planar jets having variable slot widths and velocities are presented. A flow visualization study shows that, for sufficiently large spacing, the jets “flap” and that this motion is maximized for a matched exit momentum flux condition. The extent of the jet mixing with the ambient fluid is investigated using two-component hot wire anemometry. It is demonstrated that the flapping increases mixing of the jets with the ambient. In addition, it is shown that the mixing increases with distance between the jets and with jet-width ratio.

On standing gravity wave-depression cavity collapse and jetting

2019 ◽  
Vol 866 ◽  
pp. 112-131 ◽  
Author(s):  
D. Krishna Raja ◽  
S. P. Das ◽  
E. J. Hopfinger
Keyword(s):  
Wave Amplitude ◽  
Bond Number ◽  
Working Fluid ◽  
Cavity Depth ◽  
Depth Limit ◽  
Cavity Collapse ◽  
Jet Velocity ◽  
Surface Jet ◽  
Singular Wave ◽  
Surface Jets

Parametrically forced gravity waves can give rise to high-velocity surface jets via the wave-depression cavity implosion. The present results have been obtained in circular cylindrical containers of 10 and 15 cm in diameter (Bond number of order $10^{3}$) in the large fluid depth limit. First, the phase diagrams of instability threshold and wave breaking conditions are determined for the working fluid used, here water with 1 % detergent added. The collapse of the wave-depression cavity is found to be self-similar. The exponent $\unicode[STIX]{x1D6FC}$ of the variation of the cavity radius $r_{m}$ with time $\unicode[STIX]{x1D70F}$, in the form $r_{m}/R\propto \unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6FC}}$, is close to 0.5, indicative of inertial collapse, followed by a viscous cut-off of $\unicode[STIX]{x1D6FC}\approx 1$. This supports a Froude number scaling of the surface jet velocity caused by cavity collapse. The dimensionless jet velocity scales with the cavity depth that is shown to be proportional to the last stable wave amplitude. It can be expressed by a power law or in terms of finite time singularity related to a singular wave amplitude that sets the transition from a non-pinching to pinch-off cavity collapse scenario. In terms of forcing amplitude, cavity collapse and jetting are found to occur in bands of events of non-pinching and pinching of a bubble at the cavity base. At large forcing amplitudes, incomplete cavity collapse and splashing can occur and, at even larger forcing amplitudes, wave growth is again stable up to the singular wave amplitude. When the cavity is formed, an impulse model shows the importance of the singular cavity diameter that determines the strength of the impulse.

The Importance of Nonlinear Cross-Shelf Momentum Flux during Wind-Driven Coastal Upwelling*

2004 ◽  
Vol 34 (11) ◽  
pp. 2444-2457 ◽  
Author(s):  
Steven J. Lentz ◽  
David C. Chapman
Keyword(s):  
Boundary Layer ◽  
Wind Stress ◽  
Momentum Flux ◽  
Coastal Upwelling ◽  
Bottom Boundary Layer ◽  
Return Flow ◽  
Flux Divergence ◽  
Bottom Boundary ◽  
Shelf Circulation ◽  
The Cross

Abstract A simple theory is proposed for steady, two-dimensional, wind-driven coastal upwelling that relates the dynamics and the structure of the cross-shelf circulation to the stratification, bathymetry, and wind stress. The new element is an estimate of the nonlinear cross-shelf momentum flux divergence due to the wind-driven cross-shelf circulation acting on the vertically sheared geostrophic alongshelf flow. The theory predicts that the magnitude of the cross-shelf momentum flux divergence relative to the wind stress depends on the Burger number S = αN/f, where α is the bottom slope, N is the buoyancy frequency, and f is the Coriolis parameter. For S ≪ 1 (weak stratification), the cross-shelf momentum flux divergence is small, the bottom stress balances the wind stress, and the onshore return flow is primarily in the bottom boundary layer. For S ≈ 1 or larger (strong stratification), the cross-shelf momentum flux divergence balances the wind stress, the bottom stress is small, and the onshore return flow is in the interior. Estimates of the cross-shelf momentum flux divergence using moored observations from four coastal upwelling regions (0.2 ≤ S ≤ 1.5) are substantial relative to the wind stress when S ≈ 1 and exhibit a dependence on S that is consistent with the theory. Two-dimensional numerical model results indicate that the cross-shelf momentum flux divergence can be substantial for the time-dependent response and that the onshore return flow shifts from the bottom boundary layer for small S to just below the surface boundary layer for S ≈ 1.5–2.

Analysis of Buoyant Surface Jets

1976 ◽  
Vol 98 (3) ◽  
pp. 367-372 ◽  
Author(s):  
M. A. Shirazi ◽  
L. R. Davis
Keyword(s):  
Power Plants ◽  
Present Model ◽  
Laboratory Data ◽  
Steam Power ◽  
Thermal Plumes ◽  
Steam Power Plants ◽  
Analysis Computer ◽  
Surface Jet ◽  
Surface Jets ◽  
Best Fit

To obtain improved prediction of heated plume characteristics from a surface jet, an integral analysis computer model was modified and a comprehensive set of field and laboratory data available from the literature was gathered, analyzed, and correlated for estimating the magnitude of certain coefficients that are normally introduced in these analyses to achieve closure. The parameters so estimated include the coefficients for entrainment, turbulent exchange, drag, and shear. Since there appeared considerable scatter in the data, even after appropriate subgrouping to narrow the influence of various flow conditions on the data, only statistical procedures could be applied to find the best fit. This and other analyses of its type have been widely used in industry and government for the prediction of thermal plumes from steam power plants. Although the present model has many shortcomings, a recent independent and exhaustive assessment of such predictions revealed that in comparison with other analyses of its type the present analysis predicts the field situations more successfully.

Particle fountains in a confined environment

2018 ◽  
Vol 855 ◽  
pp. 28-42 ◽  
Author(s):  
Martin C. Lippert ◽  
Andrew W. Woods
Keyword(s):  
Lower Layer ◽  
Theoretical Models ◽  
Liquid Jet ◽  
Confined Space ◽  
Volume Flux ◽  
Ambient Fluid ◽  
Buoyant Jet ◽  
Upward Velocity ◽  
Negatively Buoyant Jet ◽  
Negatively Buoyant

We present new experiments and theoretical models of the motion of relatively dense particles carried upwards by a liquid jet into a laterally confined space filled with the same liquid. The incoming jet is negatively buoyant and rises to a finite height, at which the dense mixture of liquid and particles, diluted by the entrainment of ambient liquid, falls back to the floor. The mixture further dilutes during the collapse and then spreads out across the floor and supplies an up-flow outside the fountain equal to the source volume flux plus the total entrained volume flux. The fate of the particles depends on the particle fall speed, $u_{fall}$ , compared to (i) the characteristic fountain velocity in the fountain core, $u_{F}$ , (ii) the maximum upward velocity in the ambient fluid outside the fountain, $u_{u}(0)$ , which occurs at the base of the fountain, and (iii) the upward velocity in the ambient fluid above the top of the fountain associated with the original volume flux in the liquid jet, $u_{BG}$ . From this comparison we identify four regimes. (I) If $u_{fall}>u_{F}$ , then the particles separate from the fountain and settle on the floor. (II) If $u_{F}>u_{fall}>u_{u}(0)$ , the particles are carried to the top of the fountain but then settle as the collapsing flow around the fountain spreads out across the floor; we do not observe particle suspension in the background flow. (III) For $u_{u}(0)>u_{fall}>u_{BG}$ we observe a particle-laden layer outside the fountain which extends from the floor of the tank to a point below the top of the fountain. The density of this lower particle-laden layer equals the density of the collapsing fountain fluid as it passes downwards through this interface. The collapsing fluid then spreads out horizontally through the depth of this particle-laden layer, instead of continuing downwards around the rising fountain. In the lower layer, the negatively buoyant source fluid in fact rises as a negatively buoyant jet, but this transitions into a fountain above the upper interface of the particle-laden layer. The presence of the particles in the lower layer reduces the density difference between fountain and environment, leading to an increase in the fountain height. (IV) If $u_{fall}<u_{BG}$ then an ascending front of particles rises above the fountain and eventually fills the entire tank up to the level where fluid is removed from the tank. We compare the results of a series of new laboratory experiments with simple theoretical investigations for each case, and discuss the relevance of our results.

scholarly journals A phenomenological model for fountain-top entrainment

2016 ◽  
Vol 796 ◽  
pp. 195-210 ◽  
Author(s):  
Antoine L. R. Debugne ◽  
Gary R. Hunt
Keyword(s):  
Phenomenological Model ◽  
Large Scale ◽  
Predictive Ability ◽  
Dynamical Behaviour ◽  
Volume Flux ◽  
Ambient Fluid ◽  
Physical Processes ◽  
Energetic Balance ◽  
Periodic Fluctuations

In theoretical treatments of turbulent fountains, the entrainment of ambient fluid into the top of the fountain, hereinafter fountain-top entrainment $Q_{top}$ ($\text{m}^{3}~\text{s}^{-1}$), has been neglected until now. This neglect, which modifies the energetic balance in a fountain, compromises the predictive ability of existing models. Our aim is to quantify $Q_{top}$ by shedding light on the physical processes that are responsible for fountain-top entrainment. First, estimates for $Q_{top}$ are obtained by applying, in turn, an entrainment closure in the vein of Morton et al. (Proc. R. Soc. Lond., vol. 234, 1956, pp. 1–23) and then of Shrinivas & Hunt (J. Fluid Mech., vol. 757, 2014, pp. 573–598) to the time-averaged fountain top. Unravelling the assumptions that underlie these approaches, we argue that neither capture the dynamical behaviour of the flow observed at the fountain top; the top being characterised by quasi-periodic fluctuations, during which large-scale eddies reverse and engulf parcels of ambient fluid into the fountain. Therefore, shifting our mindset to a periodical framework, we develop a new phenomenological model in which we emphasise the role of the fluctuations in entraining external fluid. Our model suggests that $Q_{top}$ is similar in magnitude to the volume flux supplied to the fountain top by the upflow ($Q_{u}$), i.e. $Q_{top}\sim Q_{u}$, in agreement with experimental evidence. We conclude by providing guidance on how to implement fountain-top entrainment in existing models of turbulent fountains.

scholarly journals Entrainment in two coalescing axisymmetric turbulent plumes

2014 ◽  
Vol 752 ◽  
Author(s):  
C. Cenedese ◽  
P. F. Linden
Keyword(s):  
Large Scale ◽  
Laboratory Experiments ◽  
Dynamical Evolution ◽  
The Other ◽  
Volume Flux ◽  
Ambient Fluid ◽  
Plume Dynamics ◽  
Complex Models ◽  
Definition Of ◽  
Reduced Surface

AbstractA model of the total volume flux and entrainment occurring in two coalescing axisymmetric turbulent plumes is developed and compared with laboratory experiments. The dynamical evolution of the two plumes is divided into three regions. In region 1, where the plumes are separate, the entrainment in each plume is unaffected by the other plume, although the two plumes are drawn together due to the entrainment of ambient fluid between them. In region 2 the two plumes touch each other but are not yet merged. In this region the total entrainment is a function of both the dynamics of the touching plumes and the reduced surface area through which entrainment occurs. In region 3 the two plumes are merged and the entrainment is equivalent to that in a single plume. We find that the total volume flux after the two plumes touch and before they merge increases linearly with distance from the sources, and can be expressed as a function of the known total volume fluxes at the touching and merging heights. Finally, we define an ‘effective’ entrainment constant, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\alpha _{eff}$, as the value of $\alpha $ needed to obtain the same total volume flux in two independent plumes as that occurring in two coalescing plumes. The definition of $\alpha _{eff}$ allows us to find a single expression for the development of the total volume flux in the three different dynamical regions. This single expression will simplify the representation of coalescing plumes in more complex models, such as in large-scale geophysical convection, in which plume dynamics are not resolved. Experiments show that the model provides an accurate measure of the total volume flux in the two coalescing plumes as they evolve through the three regions.

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