Increasing Froude Number Research Articles

Propagation of perturbations and meandering in a free surface shallow water jet

A shallow water free-surface turbulent jet is investigated at Froude numbers variable between about 0.15 and 0.6 and a Reynolds number equal to 5000, with a jet height-to-width aspect ratio variable from 0.5 to 1.1. In these experimental conditions, the flow is unstable and develops local transverse instabilities, which are amplified to a global meandering motion, forcing the jet to oscillate orthogonally to its axis. Instantaneous and averaged velocity fields are obtained by means of high-density, correlation-based time-resolved Particle Tracking Velocimetry. From the average point of view, the present jet configuration resembles a confined jet condition similar to that of wall jets, but with additional relevant three-dimensional effects, retaining self-similar properties, with a net co-flow and some ambient fluid entrainment. For increasing Froude numbers, such configuration is moved downstream and the jet spreading is delayed. The high resolution in space and time of present measurements allows to locally detect and following perturbed patterns and to derive amplitude and frequency of oscillation of the global meandering motion, the former increasing with axial distance and decreasing with Froude number, the latter doing just the opposite. The velocity of propagation of perturbed patterns is also investigated in comparison to the jet mean velocity, the local celerity of propagation of fluctuating velocity fields being derived. Results indicate that axial fluctuations propagate downstream similarly to transverse fluctuations along the orthogonal direction, thus showing a close coupling among the two motions and the onset of a self-sustained mechanism driven by the mean flow, responsible for the generation of the macroscopic meandering motion. This effect is in agreement with numerical predictions and is emphasised as the Froude number increases.

Forces on buildings with openings and orientation in a steady post-tsunami free-surface flow

Steady free-surface flows around buildings occurring during flood or tsunami events can produce major damages and a quantification of the post-peak forces is essential for safety and resilience of coastal structures. The loading process is highly affected by the flow Froude number and the drag coefficient, commonly defined for highly subcritical flows, while field observations of tsunami flows reported Froude numbers close to one, visually appearing as a choked regime. The present experimental study addresses the hydrodynamic forces on emerging buildings in a subcritical choked regime, focusing on the effect of openings and orientation. Laboratory experiments indicated a substantial difference in flow depths between the up- and downstream side of the building for increasing Froude numbers. The presence of openings induced a flow through the building lowering the difference in flow depth, limiting the effect of the hydrostatic component of the loading process. The formulation of an empirical resistance coefficient CR allowed a combination of both form drag and hydrostatic forces. Whilst for impervious buildings CR was consistent with reference studies, for buildings with openings CR was directly dependent upon porosity. Contrarily to the unsteady flow conditions, results showed that the sidewalls also played a role in the loading process. Buildings with a rotated orientation resulted in slightly larger surfaces exposed to the flow and larger horizontal forces. Nevertheless, these were applied at lower cantilever arms, thus reducing the tilting moment. Altogether, this study provides experimentally-derived parameters that will support hydraulic engineers in the design of coastal structures.

Numerical simulation of the motion of a Taylor drop in a non-Newtonian fluid

In this study, the motion of a Taylor drop in a non-Newtonian fluid in a vertical tube is investigated numerically. The governing equations are solved using a volume of fluid (VOF) method in a symmetric geometry using OpenFOAM software. The continuous phase fluid behaves as shear-thinning fluid and modeled using Bird–Carreau non-Newtonian model. The effects of the power-law index (n), Froude number (0.05 ≥ Fr ≥ 0.31) and the Eotvos number (9.5 ≥ EoD ≥ 123.7) on the motion and deformation of Taylor drop are studied. The results show that the Froude number increases by increasing the power-law index, leading to an increase in the thickness of the thin layer of the fluid around the drop. For very small power-law indices, n has little effect on the Froude number. The drop deformation in Newtonian fluid (n = 1) is less than that in non-Newtonian ones. It is revealed that the Froude number increases with the Eotvos number. As the Eotvos number decreases, the thickness of the thin layer of the fluid and the drop deformation decrease. The results demonstrate that for EoD > 70, the effect of the Eotvos number on the Froude number is negligible.

Visualizations of density fluctuations before and after the passage of bluff body objects in a laboratory stratified tank

abstract One of the main causes of the formation of internal waves is the ocean currents, from the seabed's mounds. For this reason, numerous empirical studies have been carried out in laboratories on the flow of streams on artificial mounds or the passage of objects in a stratified medium. The purpose of this study is to identify the changes in the density of the layers by internal waves due to the movement of the physical model in a stratified water with different Froude numbers. This study is done in, a large stratified tank that density changes due to movement of spherical and airfoil bodies are recorded by an array of fast response sensors. Lee waves and the blocking area are well illustrated before and after waves, created by passing objects, are transmitted through the tank, and are visualized using optical shadowgraph technique. The results showed that the mean amplitude of waves increases with increasing Froude numbers. The average wave's periods before the object reaches the lowering point of 0.4D/U (D being the thickness of the moving object and U is its speed) and then increases, and the mean period after the arrival of the object decreases with the Froude number.

Linear instability of tilted parallel shear flow in a strongly stratified and viscous medium

A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier in J Fluid Mech 699:174–197, 2012). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr–Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a horizontal inflectional instability and should remain two-dimensional at small tilting angles as long as the Reynolds number is sufficiently low. To support this claim, the asymptotic regime where this analysis is strictly valid is subsequently discussed in relation to previous work on the proper vertical length scale. The absolute and convective instability analysis of parallel wake is further performed, showing qualitatively good agreement with the experimental result. The low-Froude-number mode is found to be stabilised on increasing Froude number, as in the experiment. It is shown that the emergence of small vertical velocity at finite Froude number, the size of which is proportional to the square of Froude number, plays the key role in the stabilisation by modifying the inflectional instability and paradoxically creating stabilising buoyancy effect with the increase of Froude number.

Pinch-off location and time during 2D droplet impact onto a wetted stationary cylinder using the lattice Boltzmann method

This paper focuses on pinch-off location and time during 2D droplet impact onto a wetted stationary cylinder using the lattice Boltzmann method. A general off-lattice boundary condition is implemented for the curved boundary. The boundary condition is examined by comparing the steady-state velocity profile of the Taylor–Couette flow with its analytical solution. Additionally, the spreading length of a droplet impacting a wetted stationary cylinder is shown to be in good agreement with the numerical data of the literature. In the results section, the effects of Reynolds, Weber, and Froude numbers, as well as the droplet/cylinder diameter ratio, on the pinch-off length and time are investigated. The simulations revealed that as the Froude number increases, the pinch-off time increases with a slope 1.39 times more than that of the pinch-off length decrease. Moreover, it is shown that the critical Weber number for onset of dripping process increases with increasing the Froude number or decreasing the Reynolds number.

LIDAR Observations of Free-Surface Time and Length Scales in Hydraulic Jumps

A hydraulic jump is a complex, rapidly varied flow phenomenon occurring in the transition from supercritical to subcritical flows. Hydraulic jumps are characterized by instationary motions including jump toe oscillations and free-surface fluctuations. A light detection and ranging (LIDAR) instrument was used to continuously record the time-varying free-surface features of fully aerated hydraulic jumps with fully developed inflow conditions and high spatial and temporal resolution. The free-surface features were analyzed with two processing methods considering a freely moving hydraulic jump and a case where each instantaneous hydraulic jump profile was shifted to the mean jump toe position. The results provided detailed dimensionless distributions of the mean profiles, as well as free-surface fluctuations and jump toe oscillations that increased with increasing Froude numbers. The high spatial resolution data provided continuous free-surface time and length scales along the hydraulic jumps. Largest free-surface scales were observed close to the jump toe linked with longitudinal oscillations of the hydraulic jumps, and the dimensionless free-surface scales increased with the Froude number. The present study provides new insights into free-surface turbulent structures, highlighting the strong relationship between jump toe movements and free-surface fluctuations in hydraulic jumps.

Flow Expansion and Deflection Downstream of a Symmetric Multi-gate Sluice Structure

A sluice structure with multiple gates is often followed by an enlarging channel downstream. Experiments are conducted for different enlargement ratios in cross-section and Froude numbers at the gates. A large-scale PIV system is adopted to capture the surface flow field for examination of the flow features. The study shows that, despite the centrally placed sluice structure with symmetric outflow, a hydraulic jump occurs and the main flow downstream exhibits, in terms of expansion and deflection, a high degree of asymmetry and formation of large circulation zones. The degree of deflection increases significantly along the longitudinal direction for all the enlargement ratios except the smallest. The toe of the hydraulic jump is controlled at the upper edge of the sloping surface downstream of the gates. In light of outflow width, Froude number and enlargement ratio, the flow is classified into three regions. Immediately downstream, the effect of outflow width gradually decreases with increasing Froude number. Further downstream, the degree of deflection augments with increasing enlargement ratio, with circulation zones. At a given cross-section, the degree of expansion is positively related to the enlargement ratio. The study is expected to provide guidance for examination of similar issues of flow pattern and erosion protection design.

Flow Expansion and Deflection Downstream of a Symmetric Multi-gate Sluice Structure

A sluice structure with multiple gates is often followed by an enlarging channel downstream. Experiments are conducted for different enlargement ratios in cross-section and Froude numbers at the gates. A large-scale PIV system is adopted to capture the surface flow field for examination of the flow features. The study shows that, despite the centrally placed sluice structure with symmetric outflow, a hydraulic jump occurs and the main flow downstream exhibits, in terms of expansion and deflection, a high degree of asymmetry and formation of large circulation zones. The degree of deflection increases significantly along the longitudinal direction for all the enlargement ratios except the smallest. The toe of the hydraulic jump is controlled at the upper edge of the sloping surface downstream of the gates. In light of outflow width, Froude number and enlargement ratio, the flow is classified into three regions. Immediately downstream, the effect of outflow width gradually decreases with increasing Froude number. Further downstream, the degree of deflection augments with increasing enlargement ratio, with circulation zones. At a given cross-section, the degree of expansion is positively related to the enlargement ratio. The study is expected to provide guidance for examination of similar issues of flow pattern and erosion protection design.

Manipulation of bubble migration through thermal capillary effect under variable buoyancy

The spontaneous bubble migration and the active manipulation have widely applications in outer space. In this paper, the bubble migration driven by the thermal capillary effect is numerically investigated under variable buoyancy. The numerical model is built up through the transient two-dimensional axisymmetric model with a level set method. It is found that the magnitude and direction of bubble migration velocity is determined by the competition of upward buoyant effect and downward thermal capillary effect. These two effects can be controlled by changing the ratios of density, viscosity, thermal conductivity of bubble over surrounding liquid, as well as Reynolds number, Froude number, Peclet number and Marangoni number. With increasing Froude number or decreasing density ratio, the thermal capillary effect becomes dominant over the buoyant effect, the bubble will migrate from upward to downward. The Marangoni number has negligible effect on the bubble migration at low Reynolds numbers, while at high Reynolds numbers, increasing Marangoni number can reduce the upward migration velocity due to stronger thermal capillary effect. At low Reynolds numbers, with the increasing ratios of viscosity and thermal conductivity, the downward driving force is increased, while at high Reynolds numbers, the ratios have negligible effect on the bubble migration. These findings may provide the guidance for actively manipulating the bubbles under variable buoyancy.

Physical study of the 3-dimensional characteristics and free-surface properties of a breaking roller in bores and surges

Breaking surges and bores are observed during flood events, tidal bores and tsunamis propagating in rivers. The sudden increase in water depth generates an aerated and recirculating region, called the roller, whose turbulent behaviour is poorly understood. Based on ensemble-average analyses with multiple repetitions, this experimental work processed high-speed videos from multiple locations to characterise the spatial and temporal dynamics of the bore’s roller. The results showed different air entrainment mechanisms for increasing Froude numbers, providing adapted formulae to predict the extension of the shear layer and the air-water boundary. Seen from above, the roller toe perimeter had an indented profile rapidly evolving in time, revealing a certain level of periodicity. A statistical analysis in terms of position of the roller toe, longitudinal fluctuations and instantaneous celerities allowed for a detailed characterisation of the turbulent surface fluctuations and three-dimensional properties of the breaking bore roller, leading to a better understating of the governing process.

Visualization Tests on Variation of Scour Front under a Pipeline in Steady Currents

The intersection of the span shoulder and the scour hole under a pipeline, also called the scour front, can affect the scour hole extension rate and the bearing capacity of the span shoulder, which is associated with pipeline spanning and failure. Studies on scour front evolution are limited due to technical restrictions. In this study, visualization experiments on three-dimensional scour under a pipeline in steady currents were performed to reveal the evolution of the scour front at the beginning of scour process, and the parametric effects on the scour front pattern. The angle of scour front witnessed a steady drop at the start of the scour development process and fluctuated gently thereafter. This can be attributed to the decrease in the blockage effect over time. A gravitational movement was observed on the downstream end of the scour front, which is associated with the variation of fluid pressure along the scour front. The scour front pattern was described by the angle between the scour front and the streamwise direction. Parametric analysis showed that the averaged scour front angle climbs as the Froude number increases and the pipeline embedment drops. This phenomenon can be expounded by the intensified flow deflection near the span shoulder due to the enhancement of the pipeline blockage effect, caused by the ascending Froude number and descending pipeline embedment. The results of this study can be adopted in future mechanism investigations on the scour hole propagation rate and the failure analysis on the span shoulder.

Equation for ship wave crests in the entire range of water depths

An equation for ship wave crests y/x in the entire range of water depths is developed using the linear dispersion relation. In deep water, the developed equation is reduced to the equation of Kelvin (1906). The locations of ship wave crests in the x - and y -directions are obtained using a dimensionless constant C. The wave ray angle θc at the cusp locus is determined using the condition that θc is maximal at the cusp locus and the cusp locus angle is determined as αc=−tan−1(y/x)max. Numerical experiments are conducted using the FLOW-3D to simulate ship wave propagation. The cusp locus angles of the FLOW-3D are similar to both those of the present theory and Havelock (1908) theory in the entire range of the Froude number. Both the present theory and the FLOW-3D yield that, with the increase of ship speed, the Froude number increases and does the wavelength. For the Froude number equal to or greater than unity, the wavelength becomes infinitely large and the transverse waves disappear. The wavelengths of the FLOW-3D are slightly smaller than those of the present theory because the FLOW-3D considers the decrease of wavelength due to energy dissipation which happens because of viscosity of water and turbulence of high-speed particle velocities.

The Effect of Froude Number on Flow Field of U-Shaped Channel Along a Side Weir in Supercritical Flow Regime

In this study, turbulence of the flow field as well as the flow free surface in a U-shaped channel located along a side weir under supercritical flow conditions are simulated using the RNG k-e turbulence model and VOF scheme. Comparison of the numerical and laboratory results shows that the numerical model simulates the free surface and characteristics of the flow field with acceptable accuracy. Then, the effects of the upstream Froude number of the side weir on the flow pattern of the main channel are investigated. For all Froude numbers, in the vicinity of the inner bank a free surface drop occurs at the beginning of the side weir upstream, and a surface jump is formed at the final quarter of the side weir length. Across the surface jump, the kinematic energy of water increases and the potential energy decreases. The dividing stream surface and the stagnation zone dimensions increase as the Froude number increases.

Pressure distributions of stepped spillways with different horizontal face angles

Stepped spillways are commonly used in hydropower engineering, landscape irrigation and aeration tanks due to their high energy dissipation and superior aeration. This study used the k–ε turbulence model to study the pressure distributions of three types of stepped spillways with different horizontal face angles in fully developed skimming-flow regions. The horizontal surfaces of V-shaped steps exhibited a pressure peak that decreased from both sidewalls to the axial plane, while those of inverted V-shaped steps exhibited the opposite pattern. There was barely any variation from those of traditional stepped spillways. Negative pressures on the vertical surfaces of V-shaped, inverted V-shaped and traditional stepped spillways occurred near the sidewalls, near the axial plane and along the entire cross-section, respectively. For all the stepped spillways studied, the minimum pressure decreased with increasing Froude number and increasing absolute values of the horizontal face angles.

Analysis of Lift and Drag of Mono-foil Hysucat due to Longitudinal Foil-placement Variation

CFD simulations were conducted to study the effects of a hydrofoil and its relative placement in the longitudinal direction on the total resistance of a mono-foil hysucat (hydrofoil supported catamaran). Three foil positions were considered: (i) precisely below the vessel’s center of gravity, (ii) 3 chord-lengths aft from position 1 and (iii) 6 chord-lengths aft from position 1. At relatively low speed (volumetric Froude number FnV 1.8), the hydrofoil results in a decrease of the total resistance (up to 34.86%). The resistance coefficient first increases, takes a maximum value and then decreases with increasing Froude number. The maximum value is observed at FnV ≈ 1.4 (or Fn ≈ 0.5), consistent with previous observations. The most optimum foil placement is that precisely below the center of gravity of the vessel.

Analysis of a self-propelling sheet with heat transfer through non-isothermal fluid in an inclined human cervical canal.

The present theoretical analysis deals with biomechanics of the self-propulsion of a swimming sheet with heat transfer through non-isothermal fluid filling an inclined human cervical canal. Partial differential equations arising from the mathematical modeling of the proposed model are solved analytically. Flow variables like pressure gradient, propulsive velocity, fluid velocity, time mean flow rate, fluid temperature, and heat-transfer coefficients are analyzed for the pertinent parameters. Striking features of the pumping characteristics are explored. Propulsive velocity of the swimming sheet becomes faster for lower Froude number, higher Reynolds number, and for a vertical channel. Temperature and peak value of the heat-transfer coefficients below the swimming sheet showed an increase by the increment of Brinkmann number, inclination, pressure difference over wavelength, and Reynolds number whereas these quantities decrease with increasing Froude number. Aforesaid parameters have shown opposite effects on the peak value of the heat-transfer coefficients below and above the swimming sheet. Relevance of the current results to the spermatozoa transport with heat transfer through non-isothermal cervical mucus filling an inclined human cervical canal is also explored.

Three-dimensional lattice Boltzmann simulations of high density ratio two-phase flows in porous media

A three-dimensional multiphase lattice Boltzmann model is implemented to study the spontaneous phase transport in complex porous media. The model is validated against the analytical solution of Young’s and Laplace’s laws. Afterward, three-dimensional porous layers are randomly generated to investigate droplet penetration into a substrate, liquid transport in a porous channel as well as extraction of a droplet from a porous medium. Effects of several geometrical and flow parameters such as porosity, density ratio, Reynolds number, Weber number, Froude number and contact angle are considered. A parametric study of the influence of main non-dimensional parameters upon the impact of liquid drops on permeable surface is performed. Results show that while increasing Froude number causes spreading of the droplet on the surface, increasing Reynolds number, Weber number, porosity and liquid-air density ratio will enhance the penetration rate into the surface. Furthermore, increasing the contact angle decreases both the spreading and the penetration rate at the same time. In the same way, for the liquid transport in a porous channel, it is found that increasing the porosity and Reynolds number will result in increasing penetration rate in the channel. For the extraction of a droplet from a porous medium, it is shown that by increasing the gravitational force and/or porosity the droplet extracts faster from the substrate.

Predicting the vortex shedding frequency at the interface of the lateral cavities

The vortex shedding frequency in the mixing layer between a cavity and a main stream has been examined experimentally in absence of large oscillation of the free surface inside the cavity, called seiching. It was observed that the vortex shedding frequency follows a monotonically increasing trends with possible jumps from one to another with increasing Froude number of the main stream. These trends are obtained by solving a model based on the Rossiter approach. This model considers that the vortices shed in the mixing layer create some water depth variation at the impingement corner of the cavity. This water depth variation in turn generate pressure waves that propagates upstream and influence the vortex shedding process. Finally the measured vortex shedding frequencies correspond quite accurately to the frequencies predicted by the model. This highlights the existence of a resonant phenomena between vortices in the mixing layer of a lateral cavity and gravity waves even without any seiching phenomena.

An investigation on the outer bank cell of secondary flow in channel bends

Beside the curvature-induced secondary flow in curved channels, often a counter-rotation secondary flow cell is observed near the outer bank, called outer-bank cell. In the current study, experiments were conducted for three low subcritical Froude numbers typical for natural streams that complement the Froude number range investigated in previous studies. It was found that for increasing Froude numbers, the main cell of secondary flow expanded and strengthened. Simultaneously, the outer-bank cell reduced in size and retreated towards the boundary. A term-by-term analysis of the energy production and the vorticity equations was performed to substantiate this observation. In the outer-bank zone, for increasing Froude numbers, a reduction of the energy flux from turbulence to the mean flow was discernible; which confirms that the turbulence-induced processes affect the outer-bank cell. The analysis of the vorticity equation showed the balance between generative and degenerative terms in the streamwise vorticity equation.