Low Froude Number Research Articles

Investigation on the Cavity Backwater of Chute Aerators under Various Atmospheric Pressures

A chute aerator is a device that entrains air into water and protects against cavitation erosion. The state of the jet cavity determines the aerator efficiency under different flow conditions. In the case of a low Froude number and low velocity, backwater is generated in the jet cavity. In severe cases, this backwater blocks the air intake holes and affects air intake efficiency. With the development and construction of water conservancy projects, an increasing number of dams have been constructed at altitudes above 3000 m. The influence of cavity backwater depth at reduced atmospheric pressures is unknown and may increase the risk of high-speed aerated flows in high-altitude areas. In this study, the relevant parameters of backwater were measured at various atmospheric pressures, including the jet length, cavity subpressure, backwater depth, and net cavity length. The pressure difference of atmospheric pressure can range from 0 to 94 kPa. The test results indicate that a decrease in atmospheric pressure causes variations in the cavity subpressure. The absolute value of the difference between the inside and outside of the cavity decreases with a decrease in atmospheric pressure. An empirical formula for calculating the subpressure at different atmospheric pressures is proposed for PN < 0.1. The air velocity in the ventilation shaft decreases with a decrease in atmospheric pressure. The effects of variation in the atmospheric pressure on jet length can be ignored because the variation in jet length with different atmospheric pressures was constant. Additionally, the influence of varying atmospheric pressure on the cavity backwater is evident. The backwater depth decreases with a decrease in atmospheric pressure. When the atmospheric pressure decreases from 96 to 6 kPa, the maximum reduction in backwater depth is over 50%. Atmospheric pressure is a parameter that affects cavity backwater. Based on the measured backwater depth data, an empirical formula for calculating the backwater depth at different atmospheric pressures is proposed. This indicates a relationship between the atmospheric pressure and backwater depth under different flow conditions. It was further found that the bottom cavity may require a larger air intake volume at low atmospheric pressures and that it is necessary to optimize the aerator and the ventilation system.

Experimental and theoretical investigation of the cavity dynamics of underwater launched projectiles

During the underwater launch of a projectile using compressed air, a bubble emerges from the launch tube before the projectile. Upon being pierced by the projectile, this bubble has significant effects on the formation and evolution of the projectile’s shoulder cavity. This paper presents experimental and theoretical analyses of the dynamics of such a projectile’s shoulder cavity launched at low Froude numbers. In the experiments, a dynamic sensor and a camera are used to record the internal pressure and profile of the cavity, respectively. The cavity pressure oscillations explain the cavity ripples observed during the experimental sequence. Accordingly, we establish an independent bubble evolution model to predict the internal cavity pressure based on the assumption of spherical expansion; we also separately construct a shoulder cavity evolution model that treats the cavity as a slender body. The matched asymptotic method is used to solve the shoulder cavity evolution model, and quantitative comparisons between the theoretical and experimental results, including the cavity shape and size, show favourable agreement. Finally, the effects of cavity pressure oscillations and of the nose shape and projectile velocity on the cavity’s behaviour are studied.

Hydraulic jumps with low inflow Froude numbers: air–water surface patterns and transverse distributions of two-phase flow properties

Hydraulic jumps are commonly employed as energy dissipators to guarantee long-term operation of hydraulic structures. A comprehensive and in-depth understanding of their main features is therefore fundamental. In this context, the current study focused on hydraulic jumps with low Froude numbers, i.e. Fr1 = 2.1 and 2.4, at relatively high Reynolds number: Re ~2 × 105. Experimental tests employed a combination of dual-tip phase-detection probes and ultra-high-speed video camera to provide a comprehensive characterisation of the main air-water flow properties of the hydraulic jump, including surface flow features, void fraction, bubble count rate and interfacial velocities. The current research also focused on the transverse distributions of air-water flow properties, i.e. across the channel width, with the results revealing lower values of void fraction and bubble count rate next to the sidewalls compared to the channel centreline data. Such a spatial variability in the transverse direction questions whether data near the side walls may be truly representative of the behaviour in the bulk of the flow, raising the issue of sidewall effects in image-based techniques. Overall, these findings provide new information to both researchers and practitioners for a better understanding of the physical processes inside the hydraulic jump with low Froude numbers, leading to an optimised design of hydraulic structures.Article HighlightsExperimental investigation of air-water flow properties in hydraulic jumps with low Froude numbersDetailed description of the main air-water surface features on the breaking rollerTransversal distribution of the air-water flow properties across the channel width and comparison between centreline and sidewall.

Nonlinear effects in steady radiating waves: An exponential asymptotics approach

An asymptotic study is made of nonlinear effects in steady radiating waves due to moving sources in dispersive media. The focus is on problems where the radiated waves have exponentially small amplitude with respect to a parameter μ≪1, as for instance free-surface waves due to a submerged body in the limit of low Froude number. In such settings, weakly nonlinear effects (controlled by the source strength ɛ) can be as important as linear propagation effects (controlled by μ), and computing the wave response for μ, ɛ≪1 may require exponential (beyond-all-orders) asymptotics. This issue is discussed here using a simple model, namely, the forced Korteweg–de Vries (fKdV) equation where μ is the dispersion and ɛ is the nonlinearity parameter. The forcing term f(x) is assumed to be even and its Fourier transform fˆ(k) to decay for k≫1 like Akαexp(−βk), where A, α and β>0 are free parameters. For this class of forcing profiles, the wave response hinges on beyond-all-orders asymptotics only if α>−1, and nonlinear effects differ fundamentally depending on whether α>0, α=0 or −1<α<0. Furthermore, the sign of the forcing amplitude parameter A is an important controlling factor of the nonlinear wave response. The asymptotic results compare favorably against direct numerical solutions of the fKdV equation for a wide range of μ and ɛ, in contrast to the linear wave response whose validity is rather limited.

The Global Existence and Averaging Theorem for the Strong Solution of the Stochastic Boussinesq Equations with the Low Froude Number

The Global Existence and Averaging Theorem for the Strong Solution of the Stochastic Boussinesq Equations with the Low Froude Number

Flow Patterns, Roller Characteristics, and Air Entrainment in Weak Hydraulic Jumps: Does Size Matter?

Abstract A hydraulic jump is a stationary transition from an upstream supercritical to a downstream subcritical flow. Hydraulic jumps with relatively low Froude numbers may be observed downstream of low-head hydraulic structures and their flow properties have not been not well documented. In this study, the hydraulic properties were investigated experimentally in weak hydraulic jumps with an inflow Froude number Fr1 = 2.1 and inflow depths 0.012 m &amp;lt; d1 &amp;lt; 0.130 m. Three novel features of the study were (1) the very wide range of inflow length scales tested systematically, (2) the relatively high Reynolds number Re = 3.05 × 105 achieved in the largest experiment, with the Reynolds number defined as Re = ρ × V1 × d1/μ, and (3) the broad range of inflow conditions. Although no air entrainment was observed at the lowest Reynolds numbers, some distinct air–water flow patterns were observed in the roller region, generally similar to those observed at higher Froude numbers. The ratio of downstream to upstream depths followed closely the analytical solution of the momentum principle irrespective of the inflow depth. On the other hand, noticeable scaling issues were observed in terms of the dimensionless roller length, length of air–water flow region, roller toe fluctuation frequency, and rate of air entrainment, with increasing dimensionless data with increasing inflow depths, hence Reynolds numbers. The present results have some practical implication in terms of physical modeling and upscaling of results for low-head hydraulic structures, including culverts and storm waterways, which typically operate with Reynolds numbers in excess of 105.

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

We present a novel staggered semi-implicit hybrid finite volume / finite element method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh composed of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type finite volume method on the staggered dual grid. Next, a classical continuous finite element scheme provides the free surface elevation at the vertices of the primal simplex mesh. The semi-implicit strategy followed in this paper circumvents the contribution of the surface wave celerity to the CFL-type time step restriction, hence making the proposed algorithm well-suited for the solution of low Froude number flows. It can be shown that in the low Froude number limit the proposed algorithm reduces to a semi-implicit hybrid FV/FE projection method for the incompressible Navier-Stokes equations. At the same time, the conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme can be considered an all-Froude number solver, able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and is also validated via numerical experiments. The numerical solution of several Riemann problems, including flat bottom and a bottom with jumps, attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions obtained for the 3D free surface Navier-Stokes equations and with available experimental reference data.

Broaching-to prevention in real-time using momentary state feedback control focusing on the saddle point in the system

For realizing real-time broaching-to prevention in regular waves, the momentary state feedback control in the vicinity of the saddle point of the dynamical system is newly proposed, based on the previous work (Maniyappan in J Mar Sci Technol, 2020, https://doi.org/10.1007/s00773-020-00743-4). This is an extension of the concept of OGY (Ott, Grebogi, and Yorke) (Ott in Phys Rev Lett 64:1196–1199, 1990) algorithm by applying momentary state feedback using the linear-quadratic regulator (LQR) only when the ship approaches a saddle point. It was numerically confirmed in regular waves that controlling the surging in a wave downslope during broaching-to enables to change the broaching to a periodic motion at lower Froude number. At higher Froude numbers, broaching-to prevention is achieved by undergoing stable surf-riding near the wave trough in lower heading angles. The robustness of the controller towards modeling uncertainties and limitation of the observed variables are also confirmed by using a controller model that ignores the nonlinear terms in the calm-water manoeuvring forces and the longitudinal position of the ship to waves.

Numerical investigation on evaluating nonlinear waves due to an air cushion vehicle in steady motion by a higher order desingularized boundary integral equation method

The problem of nonlinear ship waves generated by an air cushion vehicle is studied and modelled in the framework of potential flow theory. A higher order desingularized boundary integral equation method (HODBIEM) using Rankine source is developed. A stationary iterative method is adopted to solve the nonlinear free surface which is discretized into the inner area disturbed by cushion pressure and the outer undisturbed area. Techniques for numerical implementation are described in detail. To save computing time, grid density and domain size are refined.Both straight-ahead and yawed cases are considered for validations. Wave-making resistance versus various speeds in deep and shallow water are calculated for straight-ahead vehicles. The resistance curves obtained by HODBIEM agree well with results of nonlinear CFD simulations. The linear theory predicts higher peak value of resistance than nonlinear methods. The experimental data fit at low Froude number but deviate from numerical predictions when exceeding primary hump speed. Effects of speed and water depth are analyzed by wave patterns and wave profiles. Resistance, side force and yaw moment are calculated at Fn = 0.533 for drift angle from 0° to 90°. Satisfactory results are given by present method. Details about wave patterns and wave angles are discussed.

Prediction of supercavitation shapes for a wide range of Froude numbers

In this study, we developed an inviscid model to predict the shape of supercavitation in a wide range of Froude numbers (Fr) based on the potential theory suggested by Logvinovich (1973). Visualization experiments were carried out in a cavitation tunnel, and shapes of supercavitation were obtained for 15 < Fr < 30. Existing data on supercavity shapes at high Froude number (100 < Fr < 400) were used to verify the developed model. As a result, the cavity shapes at high Froude number were predicted with high accuracy because the cavity rise is weak due to relatively low buoyancy. In the case of low Froude number (15 < Fr < 30), the cavity shapes were predicted within 5–10% accuracy from the cavitator to 2/3 of the total cavity length due to the strong effect of buoyancy.

STATIC STABILITY OF AERODYNAMICALLY SUPPORTED PLATFORMS MOVING ABOVE WATER

The motion stability is the most important problem of high-speed marine vehicles that utilize aerodynamic support. A simplified analysis and calculations of longitudinal static stability of several basic platforms moving above water are carried out in this study. The analysis is based on the extreme ground effect theory and the assumption of hydrostatic deformations of the water surface. Effects of the underlying surface type, Froude number, and several geometrical parameters on main aerodynamic characteristics, including the static stability margin, are presented. If the underlying surface is water instead of a rigid plane, the static stability worsens for platforms with flat or S-shaped lower surfaces, but it slightly improves for a horizontal platform with a flap. The static stability margin remains positive for S-shaped profiles at sufficiently low Froude numbers, while it is negative for other configurations.

Large-eddy simulations of flow past a circular cylinder near a free surface

Three-dimensional large-eddy simulations are carried out for flow past a cylinder beneath a deformable free surface at a fixed Reynolds number of Re = 7550. The results are examined for two Froude numbers of Fr=0.2 and 0.6 and a gap ratio of 0.4, to investigate the effects of the distortion of the free surface on the flow fields and hydrodynamics. At the low Froude number of Fr=0.2, the deformation of the free surface is small with little influence on the wake characteristics, and an alternative vortex shedding modulated by the free surface is detected in the wake. As the Froude number increases to 0.6, intense interface distortion occurs, which can be divided into three different regions: a hydraulic jump in the region of overtopping, a well-defined long-wavelength wave generation region in the large-scale recirculation zone near the free surface, and a water level recovery slope further downstream. The sudden change in flow regime from locally supercritical to subcritical allows the occurrence of the hydraulic jump. The induced surface waves behind the cylinder are ascribed to the shedding process of three shear layers, two of which are separated from the cylinder surface and one from the free surface. In addition, a jet-like flow originated from the gap between the free surface and the top of the cylinder occurs, exerting a downward thrust on the cylinder and pushing the wake away from the free surface. The Kármán vortex shedding in the wake is suppressed due to the interruption of the jet-like flow. The fluctuations of the wake turbulence and hydrodynamic forces are also suppressed to a low level.

Radiant heat feedback from a jet flame to the ruptured tank surface

The gas leakage of high-pressure tank can easily induce a jet fire which could heat the accidental tank by radiative heat feedback. In this paper, a new facility consisting of a jet fire apparatus and a hemispherical tank was used to model the jet flame ejected from the ruptured tank. The jet flame geometrical features were systematically measured under different nozzle diameters, exit velocities and jet directions. The radiant heat fluxes were also measured on the ruptured tank surface exposed to the jet fire. Two kinds of thermal radiation models were established to predict the radiant heat flux field on the tank surface. The first model only considers the effect of momentum on the flame geometry as the Froude number is large at the nozzle exit, while the other takes both buoyancy and momentum into account for the low Froude number. The model prediction agrees well with the experimental measurement. Finally, using the thermal radiation model and the heat balance equation, the distribution fields of radiant heat flux and temperature are calculated for the ruptured tank surface.

Numerical Simulations of 2D Hydraulic Jumps by a Parallel SPH Model

Hydraulic jumps are a rapid transition from supercritical to subcritical flow and generally occur in rivers or spillways. Owing to the high energy dissipation rate, hydraulic jumps are widely applied as energy dissipators in hydraulic projects. To achieve efficient and accurate simulations of 2D hydraulic jumps in open channels, a parallel Weakly Compressible Smoothed Particle Hydrodynamics model (WCSPH) with Shepard Density filter was established in this study. The acceleration of the model was obtained by OpenMP to reduce execution time. To further reduce execution time, a suitable and efficient scheduling strategy was selected for the parallel numerical model by comparing parallel speed-ups under different scheduling strategies in OpenMP. Following this, two test cases of uniform flow in open channels and hydraulic jumps with different inflow conditions were investigated to validate the model. The comparison of the water depth and velocity fields between the numerical results and the analytical solution generally showed good agreement, although there was a minor discrepancy in conjugate water depths. The numerical results showed free surface undulation with decreasing amplitude, which is more consistent with physical reality, with a low inflow Froude number. Simultaneously, the Shepard filter was able to smooth the pressure fields of the hydraulic jumps with a high inflow Froude number. Moreover, the parallel speed-up was generally able to reach theoretical maximum acceleration by analyzing the performance of the model according to different particle numbers.

Numerical and experimental time–frequency analysis of internal waves induced by a submerged body

Time–frequency analysis is widely applied in Kelvin ship waves. However, limited studies have been conducted on the time–frequency features of internal waves. In this study, spectrograms used to visualize the frequencies of the internal waves generated by a submerged body are studied using a short-time Fourier transform. A linear dispersion curve and numerical model are introduced considering an idealized situation. The linear dispersion curve indicates wave frequency variation with time. The effects of parameters on the dispersion curve are discussed for a linear density profile and three-layer density profile. The dispersion curve presents upper and lower branches corresponding to the divergent and transverse wave components, respectively. In contrast to surface waves, the upper-branch curve for internal waves continues to increase and finally levels to the middle Brunt–Vaisala (BV) frequency Nm of the density profile, whereas the lower branch curve continues to decrease and finally approaches a constant value. The numerical simulation provides idealized spectrograms of the linear internal waves. The linear dispersion curve provides an excellent prediction of the dominant wave signals. The high-intensity portion in the spectrograms is confined to the lower branch for a low Froude number, whereas for a larger Froude number, the high-intensity portion follows the upper branch but decays rapidly over time. The spectrograms obtained by the model test reveal that the high-color-intensity portion can be categorized into leading, reflected, nonlinear, and low-frequency regions. The experimental results also reveal that the model travel length is a critical factor affecting internal waves.

On surface flow induced by internal waves generated by a slender body moving at low internal Froude number in a sharply stratified fluid

The studies on internal waves(IWs) generated by a moving body in a stratified fluid generally focused on wave amplitude phase characteristics in underwater space. The surface flow induced by internal waves(IWs) generated by an underwater moving body in stratified fluid is generally weak, and is rarely studied. In the present paper, the surface flow induced by body-generated IWs of a slender body(SUBOFF model) moving at Fri = 0.547～2.189 and high Reynolds number (Re = 2.0 × 106～8.0 × 106) in a sharply stratified fluid with finite depth is investigated numerically and experimentally. For the theoretical calculation, the Euler equations and the free surface boundary conditions are linearized, and the source-sink distribution method is used to calculate the velocity field of body-generated IWs of SUBOFF model using the potential flow theory combined with the associated eigenvalue problem on vertical velocity. Meanwhile Particle Image Velocimetry (PIV) is used to measure the surface flow induced by body-generated IWs of SUBOFF model under the same conditions (object size, fluid stratification and movement parameters) at a large stratified fluid tank (25m × 3m × 1.5m) and validate the theoretical calculation method. The wave patterns, the maximum peak-to-peak velocity value of surface flow and its variation with Fri are analyzed.

Hydrodynamic regimes and drag on horizontally pulled floating spheres

We use high-speed imaging to investigate the movement of a floating sphere pulled horizontally along a water surface. The model sphere is 10 cm in diameter and has half of the water density resulting in a half-submerged static sphere. By varying the pulling force, we investigate the flow dynamics in the subcritical Reynolds number range, of Re ≈ 2 × 104 to 2 × 105. We characterize three hydrodynamic regimes with the increase in the pulling force, to which we refer to as: low Froude number, Fr &amp;lt; 0.6, intermediate, 0.6 &amp;lt; Fr &amp;lt; 1.2 and high Froude number, Fr &amp;gt; 1.2 regimes. In the low Fr regime, the sphere moves with little disturbance of the water surface and the drag is close to half of the drag on a fully submerged sphere. In the intermediate Fr regime, a pronounced wave pattern is developed which together with the dipping of the sphere below the water level leads to an increase in the drag force. Based on a potential flow approximation for the downward force on the sphere moving along the surface, we derive a semiempirical relation for the sphere dipping as a function of the Froude number. Finally, in the high Fr regime, the sphere movement switches to a mode of periodic dipping below and surfacing above the water surface. The periodic vertical motion portrays a decrease in the average drag force.

Wet gas pressure drop across multi-orifice plate in horizontal pipe in low gas-phase Froude number region

In comparison with the traditional single-orifice plate, as a throttling element, multi-orifice plates exhibit a low pressure loss, low flow-induced noise, and stable pressure difference response, which are advantageous in wet gas measurements. In this study, the pressure drop of wet gas across two multi-orifice plates (one with circular holes and another with slotted holes) in a horizontal pipe was experimentally investigated in the stratified flow region and intermittent flow region. The two regions are different from the flow pattern but can be collectively referred to as a low gas-phase Froude number (FrG) region. At the same porosity, there is no significant difference between circular and slotted multi-orifice plates in pressure drop characteristics. As the flow pattern transitions from stratified to intermittent, ΦG increases considerably with a decrease in FrG and an increase in liquid-phase Froude number (FrL); this differs from the well-known wet gas pressure drop characteristics of orifice plates as obtained from common wet gas with annular-mist flow. The prediction accuracies of the available pressure drop models for the gas–liquid flow across the orifice plate were analyzed based on the experimental results. The findings indicate that none of the models can provide accurate predictions in the two low-FrG subregions simultaneously. With the modified models, in which FrG and FrL are correlated simultaneously, new correlations are proposed for the low-FrG region. Among these models, the modified Murdock model exhibits the best prediction accuracies, i.e., 15% and 6%, when the upstream is in the flow pattern transition region and stratified flow region, respectively. These results elucidate the mechanism and prediction methods for the application of multi-orifice plates to wet gases as a measuring sensor.

Novel insights into habitat suitability for Amazonian freshwater mussels linked with hydraulic and landscape drivers.

Novel insights into habitat suitability for two Unionida freshwater mussels, Castalia ambigua Lamarck, 1819 (Hyriidae) and Anodontites elongatus (Swainson, 1823) (Mycetopodidae), are presented on the basis of hydraulic variables linked with the riverbed in six 500‐m reaches in an eastern Amazonian river basin. Within the reaches, there was strong habitat heterogeneity in hydrodynamics and substrate composition. In addition, we investigated stressors based on landscape modification that are associated with declines in mussel density. We measured hydraulic variables for each 500‐m reach, and landscape stressors at two spatial scales (subcatchment and riparian buffer forest). We used the Random Forest algorithm, a tree‐based model, to predict the hydraulic variables linked with habitat suitability for mussels, and to predict which landscape stressors were most associated with mussel density declines. Both mussel species were linked with low substrate heterogeneity and greater riverbed stability (low Froude and Reynolds numbers), especially at high flow (low stream power). Different sediment grain size preferences were observed between mussel species: Castalia ambigua was associated with medium sand and Anodontites elongatus with medium and fine sand. Declines in mussel density were associated with modifications linked to urbanization at small scales (riparian buffer forest), especially with percent of and distance from rural settlements, distance to the nearest street, and road density. In summary, the high variance explained in both hydraulic and landscape models indicated high predictive power, suggesting that our findings may be extrapolated and used as a baseline to test hypotheses of habitat suitability in other Amazonian rivers for Castalia ambigua and Anodontites elongatus and also for other freshwater mussel species. Our results highlight the urgent need for aquatic habitat conservation to maintain sheltered habitats during high flow as well as mitigate the effects of landscape modifications at the riparian buffer scale, both of which are important for maintaining dense mussel populations and habitat quality.

Numerical study of surface thermal signatures of lee waves excited by moving underwater sphere at low Froude number

A submerged body moving in density-stratified water, such as in oceans, disturbs the surrounding water layers. Furthermore, the internal waves generate and modulate the flow field on the water surface. Thereafter, the thin water surface layer and underlying water, which differ in temperatures because of factors such as solar radiation and water vapor exchange, mix. Consequently, thermal signatures could be generated on the water surface and detected by high-resolution infrared equipment; hence, the presence of a submerged moving body could also be deduced. To thoroughly examine this phenomenon, a submerged sphere moving horizontally in linearly density-stratified water with a “hot skin” layer on its surface was numerically studied in this work. A suitable solver, i.e., “twoLiquidMixing_skin_Foam”, based on the well-known open-source computational fluid dynamics code, “Open-Source Field Operation and Manipulation”, was developed for the aforementioned problem. The internal waves excited by the sphere were validated by experimental results to illustrate the accuracy of the developed solver. Then, the impact of lee waves on the water surface thermal signatures was investigated in detail for the towed submerged sphere moving at various depths in fluid with different internal Froude numbers.