Low Froude Number Research Articles

Prediction of impulse waves generated by partially submerged landslides with a low Froude number based on prototype physical experiments

Impulse waves generated by landslides are a potential threat to reservoirs. Wave prediction formulas that can quickly assess the hazards and extent of landslide-induced waves are an important means for early warning and disaster prevention and mitigation. Partially submerged landslides often generate landslide waves with low Froude numbers. There is limited research on prediction formulas for such waves, and most studies focused on specific wave propagation stages rather than forming a comprehensive formula system. In this study, three typical low Froude number submerged landslides that occurred in the Baihetan reservoir were selected as prototypes, and a large-scale three-dimensional (3D) physical model experiment field with dimensions of 30 × 29.5 × 1.5 m3 was constructed. A total of 95 experiments were performed. The entire process of impulse wave generation in the reservoir area was investigated by dividing the waves into four successive stages: initiation, rapid circular attenuation propagation, progressive attenuation propagation along the channel, and wave run-up. Based on a large amount of physical experimental data, formulas were derived for the maximum wave amplitude, propagation wave amplitude considering the degree of landslide submergence, and impulse wave run-up considering the shore slope orientation and ravine angle. These formulas were combined to form a comprehensive formula system to calculate the whole process of the impulse waves generated by the landslide in the narrow river channel with a wide influence range. The comprehensive formula system was applied to typical representative landslide experiments, and its accuracy was analyzed; the prediction accuracy ranged from 56% to 89.5%. This study can serve as a reference for assessing the risk of impulse waves generated by landslides in reservoir areas.

Numerical investigation of unsteady shedding behaviors of a reentrant jet supercavity

To investigate the unsteady shedding characteristics of a reentrant jet supercavity with a low Froude number, a high-fidelity numerical model based on the inhomogeneous multiphase model is developed to predict the complex supercavitation flow that occurs during supercavity development. The developed solver is validated quantitatively against experimental results in terms of supercavity geometry and closure mode. This study focuses on the initial generation and development process of a reentrant jet supercavity, revealing three distinct stages: foam cavity, transparent supercavity with rapid growth in dimensions, and fully developed supercavity exhibiting significant deformation. Owing to reverse flow of the gas–water mixture, interfacial instabilities arise from the unsteady cavity shedding, leading to fluctuations in supercavity shape. The types of large-scale cavity shedding observed in this work—wing-like and cloud-like—are caused by the concave deformation resulting from the reentrant jet. As the gas entrainment coefficient increases, the unsteady characteristics of pressure oscillation weaken, and the instance of wing-like cavities decreases. When the gas entrainment coefficient reaches a critical value, the twin-vortex closure mode occurs, resulting in a more stable flow behavior. In sum, we propose a theoretical model that elucidates the strength of the reentrant jet and reveals its unsteady shedding behavior during supercavity development.

The effect of a side wall on cavity dynamics during the water entry of a sphere at low Froude numbers

This paper comprehensively investigates the non-axisymmetric cavity dynamics of a vertically entering sphere under the influence of nearby side-walls through experimental, numerical, and theoretical analyses. Initially, we explore the characteristics of cavity evolutions with the sidewall effect. The emergence of a twin-vortex during cavity pinch-off is observed, and detailed numerical simulations provide insights into its underlying mechanisms. Both the dimensionless distance (λ) and the Froude number Fr significantly influence the pinch-off type. A phase diagram in the λ−Fr parameter space is presented, revealing the interplay between these variables. Moreover, we investigate the sidewall effect on the pinch-off time and location at low Froude numbers. The findings indicate that as λ decreases, both the pinch-off time and depth of the cavity increase. Generally, the wall effect is relatively weak when λ exceeds 4. Additionally, the pinch-off time can be described by τ=kr/g, with the constant k determined by λ. Utilizing 2D cavity theory, we estimate the pinch-off time of the water entry cavity with the sidewall effect, revealing a consistent collapse behavior with the mechanics of a two-dimensional cavity.

Scale Effects Investigation in Physical Modeling of Recirculating Shallow Flow Using Large Eddy Simulation Technique

In this study, the Large Eddy Simulation (LES) model in OpenFOAM was used to investigate the scale effects in the physical modeling of recirculating shallow flow at low Froude numbers. A laboratory test of turbulent flow through a submerged conical island with a Reynolds number of 6,210 was selected. The lab prototype was scaled with factors of 3 and 10 for both undistorted and distorted models. Our study employed the Froude similarity as the gravitational force is more dominant than the others (viscous, drag, and cohesion forces). Because the fluid (water) used for the prototype and model is the same, it is impossible to match the Reynolds, Weber, and Froude numbers simultaneously, resulting in the scale effects. For a scale of 1:1, the LES model could simulate the experimental data by appropriately capturing the vortices behind the conical island. For the undistorted models with scales of 3 and 10, the numerical model captured weaker magnitudes of vortices than the 1:1 scale, indicated by the discrepancies in velocity. In fact, the magnitudes of vortices became weaker with the distorted models. We also observed a significant increment in energy loss behind the conical island (where recirculating flows exist) as the scale increased. However, no significant discrepancies in velocity were observed between the results of the 1:1 scale and the scaled models in front of the conical island, where vortices were absent. These results indicate that the scale effects due to the Froude similarity are quite significant provided that recirculating turbulent flow occurs.

Experimental and numerical investigation of bow wave and wake flows around towed surface-piercing cylinders

An investigation was conducted to study the flow characteristics around towed surface-piercing cylinders and understand the mechanisms behind the bow wave and wake region formation. The experiments took place in a towing tank using three different cylinder models, covering a range of Froude numbers based on cylinder diameter from 0.28 to 1.73. Three-cylinder models are considered M1 is circular, M2 is semicircular ellipse and M3 is elliptical. Numerical simulations were performed using the OpenFOAM framework, considering turbulence effects through the unsteady Reynolds-Averaged Navier Stokes equations with the Shear Stress Transport k-ω turbulence model. The numerical results matched well with the experimental data. The bow wave's general features were similar across all cylinder models, with the height increasing as the Froude number rose. Froude numbers between 0.28 and 0.58 displayed a smooth surge surface, transitioning to a rough surface with roll-up initiation between 0.87 and 1.15. At Froude numbers 1.44 to 1.73, a fountain-like bow wave formed. Models M1 and M2 had the same average bow wave height, approximately 15% higher than that of Model M3. The depression depth or air ventilation at the trailing edge of the cylinders increased with the Froude number. Model M1 exhibited a greater depression depth than Models M2 and M3 at Froude number 1.15, and this difference further increased at Froude number 1.73 due to high vorticity and adverse pressure gradient generated by Model M1. Vortex coherent structures visualization revealed that at a low Froude number of 0.58, only the Karman vortex at the trailing edge was observed. At Froude number 1.15, the deep wake region maintained predominantly two-dimensional vortex structures, while the free surface displayed pronounced three-dimensional vortex structures. Model M1 had a drag coefficient approximately 40% higher than Model M3. These findings are valuable for engineering applications, particularly in the design of submarine masts.

A Horizontal Distribution Model of Static Ice Cover Generated by Static and Dynamic Water Considering the Heat Transfer of Riverbanks

The thermal factor is the main reason for winter ice cover with a low Froude number flow, and the heat transfer to narrow and deep river banks accelerates ice cover formation and ice thickness change. The freezing of water flow to freezing thickening is a nonisothermal-flow phase transition process coupled with the water flow temperature, environment and riverbank. Here, the Nusselt number and viscous dissipation are used to consider the flow velocity influence on icing, and a thermodynamic model of static ice cover horizontal distribution considering riverbed heat transfer is established. The initial ice time, freezing time and static ice cover thickness formed by static and dynamic water calculated by the model were consistent with measured data. The model reflects the horizontal growth process of the static ice cover, which was significant for narrow and deep channels. The horizontal distribution of the static ice cover was thin in the center and thick on both sides. The maximum horizontal thickness difference of −20 °C indoor freezing for 24 h reached 15% of the central ice thickness. Compared with the degree-day method for calculating ice thickness, the numerical model and dimensionless formula better reflect the growth law and horizontal distribution characteristics of static ice cover and provide a theoretical basis for safe water conveyance under ice cover in winter and ice cover formation in reservoirs and lakes in cold regions.

Numerical study of submerged body tail linearization in a strongly stratified fluid on the characteristics of generated internal waves

Underwater motion generates internal waves in a density-stratified seawater environment. These waves can persist for several days in oceanic regions, making them detectable by synthetic aperture radar (SAR). To clarify the effect of tail linearization on the internal wave characteristics of an object, this paper builds a strongly stratified fluid flume based on the delayed detached eddy model (DDES), through which we numerically simulate the internal waves characteristics of two models, namely, a tail-planar type and a cylindrical rear spherical type. The numerical results show that the turbulent wake of the cylindrical rear spherical type diverges to both sides under the low and high Froude (Fr) number, leading to an increase in the amplitude of the generated volume effect internal wave and wake angle, but the wavelength of the volume effect internal wave is almost the same for both. Notably, at high Fr numbers, the influence of the trailing effect internal wave of the object with a linearized tail is significantly reduced. The tail-planar type of internal waves shows violent fluctuations over long distances and high-intensity violent changes in locally concentrated regions. In contrast, the internal waves of the object with tail linearization show slight fluctuations over long distances.

Drag decomposition and realistic prediction of transom-stern monohull ships

A practical decomposition of the pressure drag for the transom-stern monohull ships is proposed here, where the pressure drag is decomposed into four different components, including the hydrodynamic drags CS∗, CTS∗ for the side surface or the wetted transom-stern surface, the hydrostatic drags CTSdry, CTSwet for the unwetted or wetted transom-stern surface. The influence of the transom-stern on these drag components are researched here, and practical approaches to predict these drag components are subsequently proposed. This research finds that accurately predicting the unwetted height ηdry at low Froude numbers, which is usually thought unimportant in the conventional researches, is actually important for the drag prediction.

Experimental study on energy dissipation and tailwater wave in two-stage stilling basin with supercritical inflow of low Froude number

This study investigates the geometric parameters of a two-stage stilling basin; 14 sets of tests were carried out in a horizontal flume with a supercritical inflow of relatively low Froude number (Fr1 = 1.1 ∼ 1.85). The flow pattern, relative amplitude of the tailwater, and energy dissipation ratio were investigated. The two-stage stilling basin led to a hydraulic jump at each stage. Compared to the corresponding classical jump, the energy dissipation ratio in the two-stage stilling basin could be increased by 30%–45%. Herein, a novel method is proposed to calculate the energy dissipation ratio and relative amplitude of the tailwater. The investigation demonstrates the potential of two-stage stilling basin with low-Froude-number flow.

The resistance of a trans-critically accelerating ship in shallow water

ABSTRACT The acceleration resistance of a vessel advancing in shallow water is investigated. Four acceleration intensities and two water depths are modelled using the CFD and potential flow methods. The results show a pronounced peak in resistance exists near the critical depth Froude number, but its location and magnitude are sensitive to the acceleration intensity and water depth. Excellent agreement between the results obtained from the CFD and potential flow methods is found in the low and high depth Froude number ranges regardless of water depth or acceleration, indicating that linear and unsteady methods can provide robust results at a low cost in those ranges. The magnitude of the resistance peak and its position are sensitive to nonlinear effects, evidenced by slight disagreements between the two adopted methodologies. The variation in the results produced by the two solvers is found to be sensitive to the parameters investigated.

Rigorous Derivation of the Oberbeck–Boussinesq Approximation Revealing Unexpected Term

We consider a general compressible viscous and heat conducting fluid confined between two parallel plates and heated from the bottom. The time evolution of the fluid is described by the Navier–Stokes–Fourier system considered in the regime of low Mach and Froude numbers suitably interrelated. Surprisingly and differently to the case of Neumann boundary conditions for the temperature, the asymptotic limit is identified as the Oberbeck–Boussinesq system supplemented with non-local boundary conditions for the temperature deviation.

Shallow waters resistance of an accelerating ship

This paper investigates the acceleration resistance of a vessel in shallow waters using potential flow and CFD methods. Results indicated a pronounced resistance peak near the critical depth Froude number. The peak’s location and magnitude were sensitive to acceleration intensity and water depth. Excellent agreement between potential flow and CFD methods was found in low and high depth Froude number ranges, suggesting their effectiveness and cost-efficiency. However, non-linear effects affected the resistance peak’s magnitude and position, leading to slight disagreements between the methods. The solvers’ variation was observed to be sensitive to the investigated parameters.

Numerical PMM test in shallow water using CFD method

This study aims to the estimation of the ship manoeuvrability in shallow water by using the numerical PMM approach, which analyses the hull hydrodynamic responses and manoeuvring performance from the viewpoint of coefficients. The test model is a scaled KVLCC2 tanker operating at a low Froude number of 0.0514. Static drift test and pure yaw test are considered, and the water depths vary from deep water to very shallow water of 20% under-keel clearance. The numerical tank is built based on an OpenFOAM+ package. The free surface is modelled by volume-of-fraction method, and the k-ω SST model associating with wall functions is applied for turbulence. After the convergence tests and validation works, the numerical tank is employed to investigate the wave-depth effect on hydrodynamic forces and manoeuvring coefficients. The present results show that the forces of static drift test, particularly sway force, can be significantly changed as the water depth decreases.

The effect of reservoir geometry on the critical submergence depth in hydroelectric power plants intake

The most important design index of water intakes is critical submergence depth of the intake. The depth at which the air core formed by the vortex is about to enter the intake. The emergence of a vortex and air entry into the intake increase head loss and decrease discharge coefficient. The reservoir geometric asymmetry, presence of unevenness in the bottom of the reservoir and angle of approach flow are among the factors that influence formation of the vortex and critical submergence depth. In this research, a physical model has been used to investigate the effect of reservoir geometry on the critical submergence depth. This model is designed in such a way that it can produce the strongest type of vortices with air core and with different strengths. The results showed that by creating asymmetry in the flow approaching the water intake (with side blockage in upstream), the presence of even 10% side blockage can have a great effect on the formed vortex and increase the critical submergence depth by about two times. To create uneven conditions on the reservoir bottom, blockages were created on the bottom of the reservoir upstream of the intake. The results showed that the blockage up to half of the height below the intake caused an increase of about 10–25% of the critical submergence depth, in low and high Froude numbers, respectively. However, in blockages more than half of the height below the intake, this effect increases about 60% of the critical submergence depth. In addition, the effect of the slope of the intake head wall on the order (type) of the vortex and its stability and instability was studied, and it was found that the order of the vortex decreases with the increase in the slope of the head wall toward the vertical position. Also, by increasing the slope of the head wall, the vortices form in an unstable manner. The vertical head wall can act as an anti-vortex structure and cause a reduction in critical submergence depth.

“Scour bed morphology caused by partially free and submerged flow at pipe culvert”

ABSTRACT Scour is the most dangerous hazard that can affect waterways, since it is responsible for more than sixty percent of the failures that have been reported, including the scour that is caused by floods at water structures. The influence of partially free and submerged flow at a pipe culvert on scour bed morphology was experimentally studied. In addition, three different diameters of the pipe culverts that were used with four Froude numbers for seven different water depths upstream of the culverts were included. The results show that at low Froude number for different heights of water, upstream and downstream are slightly different regarding the height of the water. To illustrate, at high Froude numbers, the height of the water has a very slight difference upstream, whereas there is a large and noticeable decrease in water heights and an increase in velocity downstream. When the height of water increases, the scour depth increases upstream and downstream. The increase downstream is more noticeable. That is because the scour increases when the submergence ratio and water velocity increase, in addition to the water depths with high submergence ratios cause a larger scour hole.

Singular limit of planar magnetohydrodynamic equations with vacuum free boundary

In this paper, we study the low Mach and Froude number limit of global strong solutions to the fluid‐vacuum free boundary problem of the planar magnetohydrodynamic (MHD) equations with degenerate viscosity coefficients and well‐prepared initial data. Only the initial energy at the basic level is required to be small. The main difficulties are the degeneracy of the system yielded by vanishing of the density close to the free boundary, the singular terms due to the large coefficients inversely proportional to the Mach number and the Froude number, and the strong coupling of the magnetic field and the velocity. The uniform estimates of solutions with respect to the Mach number and the Froude number are established, which is the first result on the MHD equations with vacuum free boundary. At the same time, we obtain the convergence rates for the strong solutions global in time with well‐prepared initial data as the Mach number and the Froude number vanish.

Hydraulic performances of a bulb turbine with full field reservoir model based on entropy production analysis

Bulb turbines are used to generate electric power, usually in low-head hydropower as a renewable source of energy. Flows in the turbines are typically characterized by low Froude number, so the pressure fields are more governed by the hydrostatic pressure gradient relative to the turbine head. As a result, the head loss mechanism is difficult to be identified by comparing local pressure characteristics, especially for vertical distributions. Furthermore, turbine flows are more sensitive to flows in reservoirs, due to shorter penstock lengths and larger runner diameters. In practice, the performances of each turbine obviously differ in a low-head hydropower station. The present paper analyzes hydraulic performance of a bulb turbine prototype, based on the entropy production theory. The full field reservoir modeling is conducted based on the two-phase flow simulation method and the geometric and environmental factors during multi turbine operation (5 units). The results showed the usefulness of the entropy production analysis for analyzing flow interactions between turbine and reservoir flows but also between each turbine unit. And, it showed that the flow interaction could play a noticeably strong role in determining operating conditions as well as the entire available energy in low-head hydropower stations.

Active control of vortex shedding past finite cylinders under the effect of a free surface

This paper presents the analysis of the active flow control promoted by low-aspect-ratio cylinders under the effect of a free surface at a low Froude number, modeled as a slip-allowing plane. To advance the literature in this merit, that is scarce compared with infinitely long and surface-mounted bodies, we carry out Detached-eddy simulations at Reynolds number of 1000 to investigate the active control provided by eight spinning rods surrounding a larger body. One of the ends of this system was immersed in the free stream, while the other was in contact with a free water surface. Our results reveal that when the rods spun with sufficiently large angular velocities, the (non-Kármán) vortex street was progressively organized and the part of the wake associated with the mechanism of vortex formation described by Gerrard [“The mechanics of the formation region of vortices behind bluff bodies,” J. Fluid Mech. 25, 401–413 (1966)] was eliminated. Nevertheless, tip-vortices prevailed throughout the examined range of spinning velocities. We also contrasted drag mitigation with power loss due to viscous traction and found that to reduce the mean drag on the system to a lower value than that of the bare cylinder necessarily required power expenditure. Steady reduction of mean drag and less significant mitigation of root mean square of lift and mean side force were verified to occur for the entire system and for the central body. However, the side force proved less affected by the wake-control mechanism. We demonstrate this to be associated with a novel ring-like vortex that prevailed throughout the simulations. Vortex dynamics and formation of these turbulent structures are explored.

Reconstruction of three-dimensional turbulent flow structures using surface measurements for free-surface flows based on a convolutional neural network

A model based on a convolutional neural network (CNN) is designed to reconstruct the three-dimensional turbulent flows beneath a free surface using surface measurements, including the surface elevation and surface velocity. Trained on datasets obtained from the direct numerical simulation of turbulent open-channel flows with a deformable free surface, the proposed model can accurately reconstruct the near-surface flow field and capture the characteristic large-scale flow structures away from the surface. The reconstruction performance of the model, measured by metrics such as the normalised mean squared reconstruction errors and scale-specific errors, is considerably better than that of the traditional linear stochastic estimation (LSE) method. We further analyse the saliency maps of the CNN model and the kernels of the LSE model and obtain insights into how the two models utilise surface features to reconstruct subsurface flows. The importance of different surface variables is analysed based on the saliency map of the CNN, which reveals knowledge about the surface–subsurface relations. The CNN is also shown to have a good generalisation capability with respect to the Froude number if a model trained for a flow with a high Froude number is applied to predict flows with lower Froude numbers. The results presented in this work indicate that the CNN is effective regarding the detection of subsurface flow structures and by interpreting the surface–subsurface relations underlying the reconstruction model, the CNN can be a promising tool for assisting with the physical understanding of free-surface turbulence.

Exponential asymptotics and the generation of free-surface flows by submerged line vortices

There has been significant recent interest in the study of water waves coupled with non-zero vorticity. We derive analytical approximations for the exponentially small free-surface waves generated in two dimensions by one or several submerged point vortices when driven at low Froude numbers. The vortices are fixed in place, and a boundary-integral formulation in the arclength along the surface allows the study of nonlinear waves and strong point vortices. We demonstrate that, for a single point vortex, techniques in exponential asymptotics prescribe the formation of waves in connection with the presence of Stokes lines originating from the vortex. When multiple point vortices are placed within the fluid, trapped waves may occur, which are confined to lie between the vortices. We also demonstrate that, for the two-vortex problem, the phenomenon of trapped waves occurs for a countably infinite set of values of the Froude number. This work will form a basis for other asymptotic investigations of wave–structure interactions where vorticity plays a key role in the formation of surface waves.