Critical Froude Number Research Articles

Investigating surface and internal waves and viscosity effects in two-layer fluids with a streamlined moving body using fully nonlinear simulation and experiment

Waves generated by a moving submerged moving body were analyzed in a two-layer density fluid. The experiments were performed in a two-dimensional towing tank and compared to the numerical results from a two-dimensional, fully nonlinear numerical towing tank (NTT) based on potential flow. The experimental conditions were determined by the relationship between the depth-dependent Froude number and the critical Froude number in baroclinic mode. The study was performed with various body velocities, changing the upper and lower fluid depth while maintaining the total water depth. An artificial damping coefficient was applied to the entire free surface and interface boundary condition to simulate the viscosity effect of the fluid interface. When the baroclinic mode was dominant, internal precursor soliton waves and depression waves were similar to the experimental results with strong nonlinearity. In contrast, the trailing waves were significantly different from the experimental results without applying an artificial damping scheme. A comparison with the experimental data showed that the effects of the fluid viscosity can be simulated with an appropriate damping coefficient. As the body velocity increases, the depressed or rising waves associated with the baroclinic mode are mixed with the trailing waves of the barotropic mode. As the depth of the upper fluid decreases, the characteristics of the baroclinic mode decrease, and the barotropic mode becomes dominant under the same velocity conditions.

Hydrodynamic performance of a high-Speed ship in waves using a fully nonlinear wave separation method

ABSTRACT A fully nonlinear time-domain simulation of a ship navigating in waves at high speed is realised by proposing a domain-split wave separation method. In the proposed method, a large distorted area of the free surface near the ship is split, on which the disturbed wave is separated by subtracting the incident wave from the total wave. For the rest of the free surface, the disturbed wave is traced by retaining the higher-order components. The present method is capable to simulate the strongly nonlinear interactions between the ship and waves, and can avoid unnecessary spurious waves. The present results are verified through comparison with experimental data. The steady VBM by navigation reaches its peak when the wavelength of the generated wave equals the ship length at the critical Froude number. The hydrodynamic behaviours of the S175 ship navigating in waves at this critical speed are investigated.

Punctuated aggradation and flow criticality in deep water channel systems

AbstractSubmarine channels are conduits for the transfer of material to deep water by sediment gravity flows. Some channels clearly show meandering patterns in planform that have attracted comparisons with fluvial systems. Many submarine channels, however, are aggradational. Transitions from meandering (at grade) channels to aggradational channels have been described in the subsurface, from seismic data. A field example is presented here in which these meandering and aggradational states may alternate several times during the overall development of a fourth‐order sequence before the system is temporarily or permanently abandoned. This implies a change in flow state from one where successive flows behave similarly over extended periods, to one in which the flow parameters are progressively changing. The cause of these cyclic changes is unclear. The generation of sedimentary architectures so strikingly comparable to those of meandering fluvial systems provides strong evidence in favour of stably stratified, essentially two‐layer flows, in which the lower high‐density part is channel‐confined, with a normal (i.e. fluvial‐like) secondary circulation, and the upper, low‐density part extends onto the overbank regions adjacent to the channel, with minimal mixing and entrainment. Such flows are described as subcritical, in line with published experimental and numerical work, allowing that the critical Froude number in such settings may not be unity. The switch to an aggradational state may be linked to changes in flow criticality, but the ultimate driver for these alternations in flow properties remains unknown.

Predicting the vertical density structure of oceanic gravity current intrusions

Understanding the dynamics and structures in the deep ocean is one of the remaining challenges in oceanography and climate sciences. We present results from large-scale laboratory experiments of rotating down-slope gravity currents intruding into a two-layer stratified ambient, performed in the largest rotating tank in the world, the Coriolis Rotating Platform in Grenoble. By means of velocity and density measurements, we show that no mixing occurs once the current has detached from the boundary. The shape of the vertical density profile in the stratified receiving ambient enables to identify two distinct regimes: the first issued by laminar transport through Ekman dynamics, the second by turbulent transport due to intermittent dense water cascading. Vertical density gradients reveal a piece-wise linear dependence on the density anomaly for the turbulent transport, suggesting an advection-diffusion process. For the turbulent regime, the scale height is deduced and an analytical model based on the critical Froude number is proposed to predict its value. Results show that the total thickness of the intruding current is on average 2.5 times the scale height. For laminar intrusions the scale height diverges whereas the thickness of the intrusion is a few times the Ekman layer thickness. Comparing the intrusion scale height with its measured vertical extension has led to a criteria to distinguish between laminar and turbulent regimes, which is corroborated by two additional independent criteria, one based on the sign of the local vorticity and the other based on the local maxima of the vertical density gradient. The model allows us to connect laboratory experiments to deep sea observations, gravity currents and Meddies and emphasizes the importance of laboratory experiments in understanding climate dynamics.

Effect of urban street canyon aspect ratio on fire-induced air pollutants dispersion under cross wind

The incidence of vehicle fires in urban street canyons has continuously increased with the process of urban modernization. The dispersion and accumulation of fire-induced air pollutants seriously affect residents’ safety. The effects of the street canyon aspect ratio and fire source location on fire-induced air pollutants dispersion under cross wind were numerically investigated in the paper. The results show that the street canyon aspect ratio can affect flow patterns and vortex structure and then lead to different accumulation states of air pollutants. Smoke development can be divided into four different regimes according to cross wind velocity. The re-entrainment phenomenon occurs when the fire source is close to the windward building or in the center of the street, but it does not occur when the fire source is close to the leeward building. Smoke tends to accumulate in street canyons when the aspect ratio is smaller and the fire source is closer to the leeward building. The variation of critical velocity with the aspect ratio can be divided into three regions, i.e., initial linear growth region (0.5 ≤ W/H < 1), constant region (1 ≤ W/H ≤ 1.5), and second linear growth region (1.5 < W/H ≤ 2). The dimensionless critical velocity and critical Froude number are used to quantify the re-entrainment phenomenon in different fire scenarios, and prediction models are finally proposed. The main findings can provide a more detailed reference for urban layout and air pollution control to reduce fire-induced air pollutants.

Nonlinear ice sheet/liquid interaction in a channel with an obstruction

The interaction between the flow in a channel with an obstruction on the bottom and an elastic sheet representing the ice covering the liquid is considered for the case of steady flow. The mathematical model based on the velocity potential theory and the theory of thin elastic shells fully accounts for the nonlinear boundary conditions at the elastic sheet/liquid interface and on the bottom of the channel. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. This allows one to formulate the coupled ice/liquid interaction problem and reduce it to a system of nonlinear equations in the unknown magnitude of the velocity at the interface. Case studies are carried out for a semi-circular obstruction on the bottom of the channel. Three flow regimes are studied: a subcritical regime, for which the interface deflection decays upstream and downstream; an ice supercritical and channel subcritical regime, for which two waves of different lengths may exist; and a channel supercritical regime, for which the elastic wave is found to extend downstream to infinity. All these regimes are in full agreement with the dispersion equation. The obtained results demonstrate a strongly nonlinear interaction between the elastic and the gravity wave near the first critical Froude number where their lengths approach each other. The interface shape, the bending moment and the pressure along the interface are presented for wide ranges of the Froude number and the obstruction height.

How hydraulic jumps form downstream of a negative step with channel expansion: experimental and numerical investigations of the transitions between wave jumps and submerged jets

AbstractWeirs and sills, particularly negative steps, play a pivotal role in modulating water flow, inducing hydraulic jumps that efficiently dissipate downstream energy. Beyond their aesthetic appeal, these features hold crucial engineering significance. This study combines physical experiments and numerical simulations downstream of a negative step featuring an abrupt width expansion. The spontaneous alteration of water flow conditions upstream and downstream of the step results in distinct flow regimes. By considering the critical Froude number to sustain an undular jump without wave breaking on a flatbed, we establish a framework for evaluating energy loss. Our analysis successfully delineates the transition limit between wave jumps and submerged jets downstream of a negative step. The co-existence regime of both jumps is explained by the analysis showing that the additional energy loss induced by the negative step is larger for the wave jump compared to the submerged jet. The abrupt width expansion at the negative step significantly reduces the transition depth between the submerged jet and wave jump, attributed to energy loss with intricate three-dimensional vortex motions—exceeding losses incurred by the negative step alone. We delve into the detailed mechanisms of these transitions through a three-dimensional numerical simulation of the energy-loss process and water surface profiles downstream of the step with expansion. The maximum energy loss by the undular jump and the minimum energy loss by the submerged jet are defined by the wave steepness at the limit of maintaining the undular jump and the jet plunging angle capable of sustaining the submerged jet, respectively.

Transport of non-flushable wipes in sewers and its application in sewer management

Flushing wipes into sewers has caused severe blockage issues in urban sewer systems, especially during the COVID-19 pandemic; however, the transport mechanism of wipes in sewers has not been reported. To address this knowledge gap, the transport of non-flushable wipes with different densities was systematically studied in a circular pipe. The critical shear stress and flow velocity for the incipient motion of wipes were found to increase with increasing wipe density and with decreasing relative wipe size, starting from 0.02 Pa and 0.05 m/s, respectively. Non-dimensional equations were developed for characterizing the incipient motion using two parameters: the critical Shield number and particle Froude number. The mean wipe velocity, mean ambient flow velocity, and wipe density could be well described by a power relationship. The ratio of wipe velocity to local flow velocity increased with the increase of wipe vertical position in the pipe. Different movement modes of wipes were observed and classified by using the ambient cross-sectional average velocity Ua: the sliding mode started first, followed by rolling/saltation at Ua = 0.18 m/s and suspension at Ua = 0.41 m/s. The threshold values for each mode were found to increase with increasing wipe density. Finally, the research results were applied in sewer management.

Gas entrainment in surge tank of CFR600

The Surge Tank in the secondary sodium circuit of China Demonstration Fast Reactor(CFR600)is filled with sodium and argon. The entrainment of gas in the surge tank will be occurred if the operation conditions is not appropriate. Through theoretical analysis and experimental studying it was found that the mechanisms of gas entrainment for surge tank were focused on three points: (1)the vortex of the free surface, (2)the design of the inlet and outlet nozzle, (3)the design of the route for suction- flow. A water experiment was performed using a 1/10 scale model by geometric similarity. It was observed from the experiment that the minimum flow rate required to gas entrainment occurred was about 130% by standard running condition. The experimental critical Froude number is 4.9, when the design value is 2.5. Through the test of the water model we can see that vortex at the free surface of sodium-argon grow up along with the ever-increasing flow velocity. Entrainment of gas was occurred when the intensity of vortex was enough. The results of the experiment confirmed that the vortex is the major factor for gas entrainment of surge tank of CFR600. But anti-vortex baffle can delay the forming process of vortex effective, and raise the critical velocity substantially.

Numerical study on plug-holing of double fires in tunnel with centralized smoke exhaust

In tunnel fires, smoke is the main fatal factor causing casualties. The technical parameters of tunnel smoke extraction are particularly critical. However, the parameters related to smoke control have rarely been experimentally and quantitatively studied in a tunnel caused by two adjacent fire sources. Numerical simulations were performed to study the plug-holing phenomenon in the centralized smoke exhaust tunnel with multi-fire sources. A 1:10 small-scale tunnel (length 25 m; width 1.2 m; height 0.8 m) was built using FDS. The effects of different fire heat release rates Q̇, fire source separation distances, and smoke exhaust velocities (v) on the plug-holing were considered. Firstly, the smoke layer height beneath the exhaust vent gradually increases as the exhaust velocity becomes higher and then remains almost unchanged until the plug-holing phenomenon occurs. Additionally, the smoke layer height becomes lower as the fire source separation distance increases for the fires with constant heat release rate. For all Q̇ values, the critical plug-holing velocity increases with the increasing separation distance S and Q̇. In addition, by modifying the critical Froude number (Frc), a predictive model of the critical plug-holing velocity considering the dimensionless fire source heat release rate and separation distance is obtained. Furthermore, the modified critical Froude number model, accounting for two adjacent fires with different separation distances, is established. The correlation between the new model and the simulation results shows quite good agreement, indicating that the proposed model can provide a satisfactory prediction for the critical smoke exhaust velocity in plug-holing conditions of double fire sources. This work can provide valuable guidance for tunnel design concerning fire safety.

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.

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.

Effects of the gravitational force on the convection turbulence driven by heat-releasing point particles

In this work, we study the convection turbulence driven by heat-releasing point particles, which absorb energy for external sources. Two-way coupling is considered for both momentum and temperature fields, and the particle dynamics includes both the Stokes drag, which is measured by the Stokes number, and the gravity force, which is measured by the Froude number. The gravity effect of particles on the convection turbulence is mainly researched. Two regimes are identified at large and small Froude numbers, respectively. For a large Froude number, the flow can reach a statistically steady state with particles being constantly advected over the whole domain. Within this regime, the transport properties exhibit weak dependence on the Froude number but strong dependence on the Stokes number. When the Froude number is small enough, all particles eventually accumulate toward the boundary layer region near the bottom plate. Scaling laws are derived for the critical Froude number between the two regimes, which agree well with the numerical results.

Skipping under water: Buoyant sphere hydrodynamics at the air–water interface

We present an experimental study of the hydrodynamics of a buoyant sphere accelerated horizontally along an air–water interface. At low speeds, the sphere floats at the surface, while at higher speeds, the sphere starts oscillating, moving below and toward the free surface akin to underwater skipping. The sphere often breaches and forms an air cavity during its subsequent dive. These underwater air cavities become horizontal and are attached to the sphere surface near the laminar flow separation point (∼π/2). High-speed imaging is used to investigate the effects of changing the pulling angle and counterweight-induced velocity on the hydrodynamics. We examine the transition from underwater skipping oscillations to water exit, particularly above the critical Froude number of 1.2, where buoyant spheres experience complex fluid–solid interactions revealing the influence of the air cavity on drag and lift coefficients and overall sphere hydrodynamics. Finally, we analyze the novel phenomenon of the steady motion of the horizontally pulled sphere with an attached inverted-wing-shaped air cavity.

Modeling of transition from stratified flow to condensation-induced water hammer in horizontal steam-water pipes

Kelvin-Helmholtz (K-H) instability theory is extensively used to predict the transition from stratified flow to slug flow. This study developed a one-dimensional (1-D) model based on K-H theory and considered the variation of axial wave amplitude to predict the transition from stratified flow to condensation-induced water hammer (CIWH). The prediction results of the present model were compared with the experimental data and existing models. The comparison shows that the present model could reasonably predict critical instability, which is the boundary between stratified flow and CIWH. Critical instability is divided into three types according to the wave amplitude of critical instability. Critical instability occurs at the upper limit, middle, and starting point of the exponential growth of wave amplitude under low, medium, and high liquid levels. The effects of major parameters, including liquid level, water temperature, pipe diameter, and pipe length, on the critical inlet steam Froude number and position of critical instability were analyzed. When the liquid level increases, the critical inlet steam Froude number decreases, then increases, and then decreases, and the position of critical instability moves downstream, then upstream, and finally turns to the steam inlet. The effect of water temperature, pipe diameter, and pipe length on the critical instability depends on the liquid level and is complex.

Intermittency and critical mixing in internally heated stratified channel flow

Through direct numerical simulations we investigate the effects of spatiotemporal intermittency as a result of stable stratification in surface heated stratified open channel flow. By adapting the density inversion criterion method of Portwood et al. [J. Fluid Mech., vol. 807, 2016, R2] for our flow, we demonstrate that the flow may be robustly separated into regions of active turbulence for which $Re_B \gtrsim {O}(10)$ and surrounding quiescent fluid, where $Re_B$ is the buoyancy Reynolds number. The intermittency in the flow spontaneously manifests as a deformed horizontal interface between the upper quiescent and lower turbulent flow, characterised by vigorous mixing from ‘overturning’ shear instabilities. The resulting vertical intermittency profile is accurately predicted by a local Monin–Obukhov length normalised by viscous wall units $\varLambda ^+$ such that the flow displays intermittency within the parameter range of $2.5 \lesssim \varLambda ^+ \lesssim 260$ . By considering conditional averages of the ‘turbulent’ and ‘quiescent’ flow separately, we find the ‘turbulent’ flow within this region to be described by constant critical gradient Richardson and turbulent Froude numbers of $Ri_{g,c} \approx 0.2$ and $Fr_c \approx 0.3$ . We find that the turbulent flow continues to display a $\varGamma \sim Fr^{-1}$ relationship when $Fr &lt; Fr_c$ , whereas the quiescent flow shows no correlation between $\varGamma$ and $Fr$ , where $\varGamma$ is the flux coefficient. Hence, we demonstrate directly that for our flow, the emergence of an asymptotic ‘saturated’ $\varGamma$ regime in the limit of a low ‘global’ $Fr$ occurs due to intermittency and increasing contributions to measurements of $\varGamma$ from the quiescent flow.

Analysis of Variables Influencing Scour on Large Sand-Bed Rivers Conducted Using Field Data

Throughout the lifespan of a bridge, morphological changes in the riverbed affect the variable action-imposed loads on the structure. This emphasizes the need for accurate and reliable data that can be used in model-based projections targeted for the identification of risk associated with bridge failure induced by scour. The aim of this paper is to provide an analysis of scour depth estimation on large sand-bed rivers under the clear water regime, detect the most influential (i.e., explanatory) variables, and examine the relationship between them and scour depth as a response variable. A dataset used for the analysis was obtained from the United States Geological Survey’s extensive field database of local scour at bridge piers, i.e., the Pier-Scour Database (PSDB-2014). The original database was filtered to exclude the data that did not reflect large sand-bed rivers, and several influential variables were omitted by using the principal component analysis. This reduction process resulted in 10 influential variables that were used in multiple non-linear regression scour modeling (MNLR). Two MNLR models (i.e., non-dimensional and dimensional models) were prepared for scour estimation; however, the dimensional model slightly overperformed the other one. According to the Pearson correlation coefficients (r), the most influential variables for estimating scour depth were as follows: Effective pier width (r = 0.625), flow depth (r = 0.492), and critical and local velocity (r = 0.474 and r = 0.436), respectively. In the compounded hydraulic-sediment category, critical velocity had the greatest impact (i.e., the highest correlation coefficient) on scour depth in comparison to densimetric Froude and critical Froude numbers that were characterized by correlation coefficients of r = 0.427 and r = 0.323, respectively. The remaining four variables (local and critical bed shear stress, Froude number, and particle Reynolds number) exhibited a very weak correlation with scour depth, with r &lt; 0.3.

μ(J)-Rheology-based depth-averaged dynamic model for roll waves in granular–fluid avalanches

A depth-averaged two-dimensional mathematical model of granular-fluid avalanches is proposed based on the μ(J) rheology. In the model, a depth-averaged deviatoric viscous stress of granular-fluid mixture is formulated and the corresponding depth-integration term is linearly proportional to the flow depth, which is different from the 1.5th-power law for dry granular avalanches. A basal resistance on the granular-fluid mixture, composed of a Coulomb friction term and a viscous shear term, is adopted in the model. Based on the model, a critical Froude number of 1/2 is obtained for the linear instability of steady-uniform granular-fluid mixture flows. The model is capable of capturing the cutoff frequency for the spatial growth rate of perturbation in the mixture flows. The proposed model is applied to simulate the evolution of imposed perturbations in steady granular-fluid avalanches on an inclined chute. The intensification of imposed perturbations and the generation of final stable roll waves are captured. A parameter sensitivity analysis is conducted to investigate the effects of the depth-averaged in-plane deviatoric viscous stress and the basal resistance on the final stable roll waves. Further, a preliminary analytical solution for the crest/trough heights of the stable roll waves is proposed to verify the numerical findings.

Experimental Study on Geometrical Characteristics of a Square Turbulent Buoyant Jet Flame

This paper gives the experimental results for the buoyant jet diffusion flame in square burners. The flame height and lift-off were measured and discussed. The results show that the normalized flame height and lift-off height of square flames are lower than that of round flames with the same experimental conditions, which indicates enhanced air entrainment in square flames. The model of flame height based on the flame Froude-number, which integrates a correction factor to account for the enhancement of air entrainment, reasonably collapses the square flame data under different experimental cases. The critical Froude number for the transition from buoyancy to momentum controlled regime of a square jet flame is highly dependent on the fuel type and burner size. The lift-off height of the square jet flames increases linearly when the initial velocity increases, but also depends on the fuel type and burner size. A unified correlation using a dimensionless flow number derived from the mixedness-reactedness flamelet model is established, which reasonably predicts the lift-off distances of square jet flames.

Wave Radiation from Bottom Vibrations in Open-Channel Flow with Uniform Vorticity

The linearized water-wave radiation problem for a 2D oscillating source in an inviscid shear flow with a free surface is investigated analytically. The singular source is located at the bottom, with constant fluid depth. The velocity of the basic flow varies linearly with depth, assuming uniform vorticity. The far-field surface waves radiated out from the 2D bottom source are calculated, based on Euler’s equation of motion with the application of radiation conditions. The present analysis extends the work by Tyvand and Sveen to include a nonzero surface flow. Doppler effects will arise in a system at rest with the undisturbed free surface. Resonance with zero group velocity will occur at a critical Froude number for the surface flow. In the presence of a basic surface flow, there are in total four radiated waves when the surface flow is subcritical. With supercritical flow, there are only two emitted waves, with downstream propagation. The critical Froude number depends on the surface velocity, the shear rate and the oscillation frequency of the bottom source.