Large Froude Numbers Research Articles

Convective transport of air bubbles in strong hydraulic jumps

A hydraulic jump is a flow singularity characterised by a significant amount of air entrainment in the shear zone. The air is entrapped at the jump toe that is a discontinuity between the impinging flow and the roller. The impingement point is a source of air bubbles, as well as a source of vorticity. Herein the convective transport of air bubbles in the jump roller is re-visited. Some analytical extension is presented and the theoretical results are compared with some laboratory experiments conducted in a large-size facility operating at large Froude numbers. The turbulent air bubble mixing coefficient was found to increase linearly with increasing distance and be independent of the Froude and Reynolds numbers. Overall the study highlighted some seminal features of the air–water shear layer in hydraulic jumps with large Froude numbers (5.1 < Fr 1 < 11.2). The air bubble entrainment in the mixing zone was a convective transport process, although there was some rapid flow de-aeration for all Froude numbers.

Viscous stability properties of a Lamb–Oseen vortex in a stratified fluid

In this work, we analyse the linear stability of a frozen Lamb–Oseen vortex in a fluid linearly stratified along the vortex axis. The temporal stability properties of three-dimensional normal modes are obtained under the Boussinesq approximation with a Chebychev collocation spectral code for large ranges of Froude numbers and Reynolds numbers (the Schmidt number being fixed to 700). A specific integration technique in the complex plane is used in order to apply the condition of radiation at infinity. For large Reynolds numbers and small Froude numbers, we show that the vortex is unstable with respect to all non-axisymmetrical waves. The most unstable mode is however always a helical radiative mode (m = 1) which resembles either a displacement mode or a ring mode. The displacement mode is found to be unstable for all Reynolds numbers and for moderate Froude numbers (F ~ 1). The radiative ring mode is by contrast unstable only for large Reynolds numbers above 104 and is the most unstable mode for large Froude numbers (F &gt; 2). The destabilization of this mode for large Froude numbers is shown to be associated with a resonance mechanism which is analysed in detail. Analyses of the scaling and of the spatial structure of the different unstable modes are also provided.

Density Effect on Round Turbulent Hypersaline Fountain

An experiment was conducted to examine a round hypersaline fountain. When the discharge was turbulent, and for large densimetric Froude numbers, the results have indicated that the maximum height of the fountain Zm ∕ Lm , in a quiescent and homogeneous surrounding, was a function of relative density difference. The height Zm ∕ Lm decreased from 3.06 at Δρ∕ ρa =0.001 to 2.59 at Δρ∕ ρa =0.1 . The observed density effects on the height of the fountain have indicated that the Boussinesq’s approximation was valid when the relative density difference was typically less than 0.003.

Vorticity determination in a hydraulic jump by application of method of characteristics

The method of characteristics for solving systems of partial differential equations coupled with jump conditions is used in analysing flow downstream of a hydraulic jump instead of the normal analytical approach adopted in Hornung [1]. It is shown that the method of characteristics together with the jump conditions can correctly be used as an alternative method to determine the mean vorticity downstream of the hydraulic jump as a function of the Froude number and height ratio. The mean vorticity does not increase from zero as a function of Froude number minus one but, however, it approaches a constant value at large Froude number. The present work extends the model of Hornung [1] to include non linear velocity profile used in calculating the torque with a view to determining the mean vorticity. The result obtained by this method generalizes that of [1]. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 239-248

Drag increment of internal waves generated by horizontally moving spheroid in supercritical regime

A theory on the drag increment of internal waves with a spheroid moving horizontally at a high velocity (or for large internal Froude number) in uniformly vertically stratified fluid (or ocean) is presented in the present paper. A surface source distribution is employed to model a hydrodynamic interaction between the spheroid and the stratified fluid. From theoretical results, it is shown that there exists an asymptote of zero-drag increment in supercritical regimes, where internal Froude numbers are larger than the critical internal Froude numbers. When the spheroid reduces to a sphere, the results in this paper is in good agreement with the previous theoretical and experimental results of the sphere.

On the stability of vortex streets in a stratified fluid

On the basis of the perturbation theory developed previously by the authors for localized hydrodynamic vortices, the influence of a specified jet flow and of the structure of individual vortices on the stability of the Karman street is investigated. It is shown that, for a street of vortices with a power law of decrease in the azimuthal velocity, the jet flow suppresses instability only with respect to perturbations with wavelengths from a certain range determined by the parameters of the flow. At the same time, for streets formed from vortices with a Gaussian profile of the azimuthal velocity, even in the absence of a specified flow, there is a certain region of the street’s parameters in which the street is stable against perturbations of all scales. Thus, for the purposes of modeling quasi-two-dimensional flows in a stratified fluid by a sequence of localized vortices, which is discussed in this study, vortices with a Gaussian profile of the azimuthal velocity turn out to be preferable. The results of this study are consistent with numerous experiments on the structure of a quasi-two-dimensional wake behind a body in a stratified fluid at large Reynolds and Froude numbers.

Non-classical shallow water flows

This paper deals with violent discontinuities in shallow water flows with large Froude number F. On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory, we show that, over a certain time-scale, this discontinuity may be described by a delta shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step function components. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted. For flows on a sloping base, we show that for flow with an aspect ratio of O(F−2) on a base with an O(1) or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a delta shock. The physical manifestation of this discontinuity is a small ‘tube’ of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base

Gradually expanding hydraulic jump in a trapezoidal channel

In this study, the hydraulic jump formed in a gradually expanding stilling basin of trapezoidal cross section is investigated. The expansion of the basin is accomplished by increasing the bed width in the streamwise direction. The range of angle of expansion studied is from 3 to 9?. Experiments have been also done to study the hydraulic jumps for three different basin side slopes. Froude numbers considered ranged from 3 to 10. For a given side slope as the expansion angle increases, the jump within the stilling basin is stable particularly at larger Froude numbers. Divergence of the basin for a given side slope causes reductions in the sequent depth and jump length, and an increase of the energy loss in the jump relative to those observed in the jumps occurring in channels with unchanging rectangular or trapezoidal cross sections. It was also found that a decrease in the slope of the side walls for a given angle of expansion of the basin causes smaller sequent depth and larger jump length and energy loss than those in jumps in channels with the same angle of divergence but with vertical side walls

On ice-induced instability in free-surface flows

The problem of stability of a water-coated ice layer is investigated for a free-surface flow of a thin water film down an inclined plane. An asymptotic (double-deck) theory is developed for a flow with large Reynolds and Froude numbers which is then used to investigate linear two-dimensional, three-dimensional and nonlinear two-dimensional stability characteristics. A new mode of upstream-propagating instability arising from the interaction of the ice surface with the flow is discovered and its properties are investigated. In the linear limit, closed-form expressions for the dispersion relation and neutral curves are obtained for the case ofPr= 1. For the general case, the linear stability problem is solved numerically and the applicability of the solution withPr= 1 is analysed. Nonlinear double-deck equations are solved with a novel global-marching-type scheme and the effects of nonlinearity are investigated. An explanation of the physical mechanism leading to the upstream propagation of instability waves is provided.

Lee waves from a sphere in a stratified flow

Two asymptotic analyses of the generation of lee waves by horizontal flow at velocityUof a stratified fluid of buoyancy frequencyNpast a sphere of radiusaare presented, for either weak or strong stratification, corresponding to either large or small internal Froude numberF=U/(Na), respectively. ForF⋙1, the fluid separates into two regions radially: an inner region of scaleawith three-dimensional irrotational flow unaffected by the stratification, and an outer region of scaleU/Nwith small-amplitude lee waves generated by theO(1) vertical motion in the inner region. ForF⋘1, the fluid separates into five layers vertically: from the lower dividing streamsurface situated at a distanceU/Nabove the bottom of the sphere to the upper dividing streamsurface situated at a distanceU/Nbelow the top, there is a middle layer with two-dimensional horizontal irrotational flow; from the upper dividing streamsurface to the top of the sphere, and from the lower dividing streamsurface to the bottom, there are top and bottom transition layers, respectively, with three-dimensional flow; above the top and below the bottom, there are upper and lower layers, respectively, with small-amplitude lee waves generated by theO(F) vertical motion in the transition layers.The waves are calculated where they have small amplitudes. The forcing is represented by a source of mass: forF⋙1, the surface distribution of singularities equivalent to the sphere in three-dimensional irrotational flow; forF⋘1, the horizontal distribution of singularities equivalent, in the upper (resp. lower) layer, to the flat cut-off obstacle made up of the top (resp. bottom) portion of the sphere protruding above (resp. below) the upper (resp. lower) dividing streamsurface. The analysis is validated by comparison of the theoretical wave drag with existing experimental determinations. ForF⋙1, the drag coefficient decreases as (lnF+7/4-γ)/(4F4), with γ the Euler constant; forF⋘1, it increases as$(32\surd2)/(15\upi)F^{3/2}$. The waves have the crescent shape of the three-dimensional lee waves from a dipole, modulated by interferences associated with the finite size of the forcing. For strong stratification, the hydrostatic approximation is seen to produce correct leading-order drag, but incorrect waves.

ANALYSIS OF OPEN-CHANNEL FLOW WITH A SIDE CAVITY ZONE BY LES TAKING INTO ACCOUNT OF WATER SURFACE VARIATION

Open-channel flows with a relatively large-scale side cavity zone exhibits water surface oscillations in spanwise or streamwise directions. Although the side cavity zone is installed for improving the water-friendly environment, such oscillations, once occurred in the actual river in the urban areas, may cause flooding, resulting in a large damage in the residential area. In order to investigate the flow features associated with such surface oscillation, the authors have developed a 3D LES model that utilizes a density function for detecting air-water interface. With the developed model, it becomes possible to simulate water surface oscillation observed in the experiments. In this paper, the flow features related to the water surface oscillation as well as the local flow features along the interface between the main channel and the side cavity zone are investigated for the subcritical flows with relatively large Froude number.

Exponential asymptotics and gravity waves

The problem of irrotational inviscid incompressible free-surface flow is examined in the limit of small Froude number. Since this is a singular perturbation, singularities in the flow field (or its analytic continuation) such as stagnation points, or corners in submerged objects or on rough beds, lead to a divergent asymptotic expansion, with associated Stokes lines. Recent techniques in exponential asymptotics are employed to observe the switching on of exponentially small gravity waves across these Stokes lines. As a concrete example, the flow over a step is considered. It is found that there are three possible parameter regimes, depending on whether the dimensionless step height is small, of the same order, or large compared to the square of the Froude number. Asymptotic results are derived in each case, and compared with numerical simulations of the full nonlinear problem. The agreement is remarkably good, even at relatively large Froude number. This is in contrast to the alternative analytical theory of small step height, which is accurate only for very small steps.

Plane Impulse Waves in Reservoirs

The characteristics of impulse waves generated in reservoirs by the impact of variable density mass flows were assessed using two-dimensional model experimentation. Particle image velocimetry (PIV) was applied within the wave generation area at the slide impact location and the water wave profiles were measured by seven successive capacitance wave gages. The maximum relative wave amplitude and the normalized wave amplitude of the propagating wave train were correlated with the dimensionless slide quantities and the relative propagation distance, respectively. The impact Froude number was identified as the dominant parameter for slow impacting slides, whereas the water depth and the slide thickness governed the maximum possible wave amplitude for large impact Froude numbers. Four wave types were distinguished due to three different levels of wave nonlinearity associated with variable impact Froude numbers, relative slide densities, and characteristic slide geometries: nonlinear transient bore, transition wave, oscillatory wave, and nonbreaking solitary wave. Moreover, the density effect on the wave generation process was investigated with small impact Froude numbers using sequential PIV velocity vector fields.

A theoretical model of a wake of a body towed in a stratified fluid at large Reynolds and Froude numbers

Abstract. The objective of the present paper is to develop a theoretical model describing the evolution of a turbulent wake behind a towed sphere in a stably stratified fluid at large Froude and Reynolds numbers. The wake flow is considered as a quasi two-dimensional (2-D) turbulent jet flow whose dynamics is governed by the momentum transfer from the mean flow to a quasi-2-D sinuous mode growing due to hydrodynamic instability. The model employs a quasi-linear approximation to describe this momentum transfer. The model scaling coefficients are defined with the use of available experimental data, and the performance of the model is verified by comparison with the results of a direct numerical simulation of a 2-D turbulent jet flow. The model prediction for the temporal development of the wake axis mean velocity is found to be in good agreement with the experimental data obtained by Spedding (1997).

Closure to “Exact Solutions for Sequent Depths Problem” by Prabhata K. Swamee and Pushpa N. Rathie

The writers are very grateful to the discussers for their discussions. Following are the replies to the comments by Rajesh Srivastava: 1. Although Eqs. 10 and 11 are explicit and valid for the full range of the independent variable fL, these equations are not exact. In addition to introducing the Lagrange theorem to the engineering community, the paper is aimed at obtaining an exact solution of a system of implicit equations. Since the solution is in the form of an infinite series, its accuracy can be varied by truncating the series at an appropriate term. 2. The writers thank the discusser for conducting a detailed error analysis. A perusal of Fig. 1 of the discussion shows that Eq. 10 exhibits a large error for very large fL corresponding to an extremely large prejump Froude number. For such a large prejump Froude number, the flow becomes

On the two-point correlation of potential vorticity in rotating and stratified turbulence

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Kármán–Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between potential vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law for the third-order velocity structure function in turbulence theory.

Wave drag of an ellipsoid of revolution rapidly moving in a horizontal direction in a uniformly stratified fluid

The problem of the wave drag of ellipsoids moving in a uniformly stratified ideal fluid is considered by means of modeling the bodies by surface distributions of mass sources. Analytical results are obtained using the distributions known from the theory of a uniform fluid, which make it possible to describe the emission of internal waves by rapidly moving ellipsoids of revolution (spheroids) in the limit of large Froude numbers. An asymptotically simplified form of the dependence of the wave drag on the Froude number and the spheroid axis ratio is found. In the particular case of a sphere, the result confirmed earlier by Greenslade by making comparisons with a numerical calculation and experimental data is obtained.

Giant Bubble Pinch-Off

Self-similarity has been the paradigmatic picture for the pinch-off of a drop. Here we will show through high-speed imaging and boundary integral simulations that the inverse problem, the pinch-off of an air bubble in water, is not self-similar in a strict sense: A disk is quickly pulled through a water surface, leading to a giant, cylindrical void which after collapse creates an upward and a downward jet. Only in the limiting case of large Froude numbers does the purely inertial scaling h(-logh)(1/4) proportional tau(1/2) for the neck radius h [J. M. Gordillo et al., Phys. Rev. Lett. 95, 194501 (2005)] become visible. For any finite Froude number the collapse is slower, and a second length scale, the curvature of the void, comes into play. Both length scales are found to exhibit power-law scaling in time, but with different exponents depending on the Froude number, signaling the nonuniversality of the bubble pinch-off.

On the Rayleigh-Bénard problem in the continuum limit: Effects of temperature differences and model of interaction

The transition to convection in the Rayleigh-Bénard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible “slip-flow” problem. Considering a power-law (variable hard-sphere) model of interaction our analysis indicates that for sufficiently large Froude numbers “softer” potentials result in less unstable systems. At small Froude numbers this trend is reversed, i.e., the “softer” interaction potentials correspond to a more unstable response. These results are discussed in terms of the opposing mechanisms of thermal expansion and compressibility. We carry out an asymptotic expansion for small temperature differences and establish the principle of exchange of stabilities for this limit. A singularity appears in this limit when compressibility effects are dominant.

Structure of air–water two-phase flow in helically coiled tubes

Air–water two-phase flow in helically coiled tubes is investigated experimentally to elucidate the effects of centrifugal acceleration on the flow regime map and the spatial and the temporal flow structure distribution. Three kinds of test tubes with 20 mm inner diameters including a straight tube are used to compare the turbulent flow structure. Superficial velocities up to 6 m/s are tested so that the centrifugal Froude number covers a range from 0 to 3. The interfacial structure is photographed from two directions by a high-speed video system with synchronized measurement of local pressure fluctuations. The results reveal that the flow transition line alters due to centrifugal force acting on the liquid phase in the tube. In particular, the bubbly flow regime is narrowed significantly. The pressure fluctuation amplitude gets large relatively to the average pressure loss as void fraction increases. The frequency spectra of the pressure fluctuation have plural peaks in the case of strong curvature, implying that the periodicity of slugging two-phase flow is collapsed by an internal secondary flow activated inside the liquid phase. Moreover, under large Froude number conditions, the substantial velocity of the gas phase that biases to the inner side of the helical coil is slower than the total superficial velocity because the liquid flow is allowed to pass through the outer side and so resembles a radial stratified flow.