Increasing Froude Number Research Articles

Elementary ship models and farfield waves

Kelvin’s classical 1-point ship model, the basic 2-point models of a monohull ship and a catamaran previously studied via elementary geometrical considerations, optimal versions of these two basic 2-point models, and a 4-point model of a catamaran (not considered previously) are analyzed within a common theoretical approach based on potential flow theory, the Green function associated with the Kelvin–Michell linear boundary condition at the free surface, the related Fourier representation of farfield ship waves, and the farfield stationary-phase approximation. The analysis shows that the apparent wake angle associated with the highest waves found inside the cusps of the Kelvin wake as a result of interferences among the divergent waves created by a ship at a high Froude number can be realistically predicted via elementary ship models. In particular, the apparent wake angles for monohull ships and catamarans can be predicted very well via a 2-point model or a 4-point model. Moreover, these models provide basic insight into main features of the farfield waves created by fast ships. Notably, differences between the amplitudes of the bow and stern waves are shown to have no influence on the occurrence of constructive or destructive interferences, although they affect the intensity of wave-interference effects (strong if the amplitudes of the bow and stern waves are commensurate, weak otherwise). Another notable conclusion is that lateral interference effects become more important as the Froude number increases. Indeed, lateral interferences between the twin hulls of a catamaran dominate longitudinal interferences between the fore and aft of the ship at Froude numbers greater than about 1. However, longitudinal interferences between the bow and the stern waves remain dominant for common fast monohull ships.

Experimental study of ventilated cavity flow over a 3-D wall-mounted fence

Ventilated cavity flow over a fixed height 3-D wall-mounted fence is experimentally investigated in a cavitation tunnel for a range of free-stream conditions. The impact of 3-D effects on cavity topology is examined, along with the dependence of the cavitation number and drag on the volumetric flow rate coefficient, fence height based Froude number and vapour pressure based cavitation number. Three different flow regimes are identified throughout the range of cavitation numbers for a particular free-stream condition. Generally, the cavity has a typical re-entrant jet closure the intensity of which is found to increase linearly with increasing Froude number. This increase in re-entrant jet intensity causes an increase in drag with Froude number for constant volumetric flow rate coefficient. At low Froude numbers the closure mechanism transitions from a single to a split re-entrant jet. The parameters used to characterize the cavity topology show a linear dependence on Froude number irrespective of the closure mode. The cavity topology and drag are found to be independent of vapour pressure based cavitation number.

Tidal bore progressing on a small slope

In a natural estuary, a tidal bore may progress on a small sloping bed from the downstream to the upstream. In this study, a simple analytical solution for tidal bore formed in a small slope channel was developed using the finite control volume analysis. New unsteady experiments were conducted to verify the theoretical model. The model predictions generally agree with the observations. A general relation is obtained for the conjugate depth ratio as a function of the Froude number and the channel slope from the experimental data. The results indicate that the conjugate depth ratio increases with an increasing Froude number as well as with a decrease in channel slope. On a negative slope, the Froude number increases as the bore propagates along the channel, and decreases for a positive slope. The theoretically based model is accurate and simple to estimate the celerity of the tidal bore progressing along a small slope channel.

정방형 실린더 주위 유동패턴에 대한 자유수면의 영향

The characteristics of flow past a square cylinder submerged under the free surface have been numerically studied. An immersed boundary method was adopted for implementation of the cylinder cross-section in a Cartesian grid system. Also, a level-set method was used to capture the interface of the two fluids. The case for Reynolds number 150 was examined. At the specific Reynolds number, by varying the gap ratio(0.25, 0.40, 0.55, 0.70, 1.00, 1.50, 2.50, 5.00) the effects of the free surface on the force coefficients and Strouhal number of vortex shedding were identified. The presence of the free surface very close to the cylinder significantly affects the shedding pattern, resulting in considerable deviation of the force coefficients and Strouhal number from those of the single-phase flow. In addition, the influence of Froude number was considered in this study. By increasing Froude number(0.2-0.4), flow topology change was identified at the specific gap ratios(0.40, 0.70, 1.50, 5.00).

Steady performance prediction of a heeled planing boat in calm water using asymmetric 2D+T model

This article presents a simple mathematical model for predicting the running attitude of warped planing boats fixed in a heel angle and free to trim and sinkage. The proposed model is based on asymmetric 2D+T theory utilizing a pressure equation which is previously introduced in the literature to compute the hydrodynamic force acting on a heeled planing hull. Integration of pressure distribution on the asymmetric wedge sections enables the suggested model to compute trim angle, center of gravity rise, resistance, and heeling moment acting on the heeled planing boat in calm water. The hydrostatic force in addition to two drag forces acting on the pressure area and spray area are also taken into account. Finally, a computational algorithm is introduced to find the running attitude of the heeled planing boats. The validity of the proposed model is examined by comparing the obtained running attitudes for two planing hulls series with zero heel angle and computed lift force and heeling moment of a heeled planing boat against available experimental data. Based on the comparisons, favorable accuracy is observed for both symmetrical and asymmetrical conditions. Moreover, it is shown that existence of a heel angle can lead to a decrease in trim angle and resistance, while it intensifies the center of gravity rise of planing boats. It is also observed that as the beam Froude number increases, the heeling moment of the heeled boat reduces.

Numerical Investigation of Two-Phase Flow of Non-Newtonian Fluid Discharged from a Sauce Dispenser

Fluid dispensers are widely used in various food processing applications. The present work investigates a two-phase flow of sauce discharged from a dispenser. Preliminary simulations were conducted to study the effects of non-Newtonian properties on the resulting flow. It was found that the non-Newtonian properties of the sauce tends to distribute the flow more toward the mid-plane of the fluid field. Transient, liquid-air flow simulations were conducted to study the effects of surface tension and geometry of the dispenser on the liquid film in ambient air. The volume of fluid (VOF) method was used to track the interface between phases. Gravity was taken into account since it is comparable with the inertia force. It was found that the breakup time of the liquid film increases as the Weber number decreases and the breakup length of the liquid film increases as the Froude number increases. Practical Applications In food processing applications, various sauces are to be dispensed onto the meal tray. To ensure uniform reheating of the packaged food as well as retaining the moisture content, ideal coverage is required. When sauce leaves the dispenser, it starts to interact with the ambient air and results in a two-phase flow. Understandings of the shape of the liquid film as well as its breakup during the dispensing process are crucial to sauce dispenser design. Through numerical investigations of the dispensing process, the present work proposes optimized dispenser geometry and provides physical insights of the breakup time and the breakup length of the liquid film. Based on these findings, better design of the sauce dispenser can be achieved.

Sinuosity-Driven Water Pressure Distribution on Slope of Slightly-Curved Riparian Zone: Analytical Solution Based on Small-disturbance Theory and Comparison to Experiments

A curved riparian zone can create highly complex flow patterns that have a great effect on erosion, pollutant transport, surface water-groundwater exchange and habitat qualities. The small-disturbance theory has been applied to derive the analytical solutions of pressure distributions along a sinusoidal riverbank. Experiments have also been performed to test the hydrodynamic and geomorphic effects on pressure distribution and to verify the applicability of the derived expressions. The derived expressions were simple, accurate and agreed remarkably well with experimental results for the riparian banks with a low degree of curvature. On the contrary, when a riparian bank had a high degree of curvature, these expressions applying the approach of small-disturbance, could not effectively estimate the pressure distributions for a complex bank boundary or complex flow conditions. Moreover, sensitive analysis has indicated that the disturbed pressures along the riparian banks increased with increasing Froude number Fr, as well as the ratio of bank amplitude to wavelength a/λ. However, a/λ has been found to have more significant influence on pressure variation in subcritical flow.

Motion Prediction of Catamaran with a Semisubmersible Bow in Wave

Abstract Compared with standard vessels, a slender catamaran with a semi-submerged bow (SSB) demonstrates superior seakeeping performance. To predict the motion of an SSB catamaran, computational fluid dynamics methods are adopted in this study and results are validated through small-scale model tests. The pitch, heave, and vertical acceleration are calculated at various wavelengths and speeds. Based on the overset grid and motion region methods, this study obtains the motion responses of an SSB catamaran in regular head waves. The results of the numerical studies are validated with the experimental data and show that the overset grid method is more accurate in predicting the motion of an SSB catamaran; the errors can be controlled within 20%. The movement data in regular waves shows that at a constant speed, the motion response initially increases and then decreases with increasing wavelength. This motion response peak is due to the encountering frequency being close to the natural frequency. Under identical sea conditions, the motion response increases with the increasing Froude number. The motion prediction results, that derive from a short-term irregular sea state, show that there is an optimal speed range that can effectively reduce the amplitude of motion.

Effect of free surface deformation on the extractable power of a finite width turbine array

The effect of free surface deformation on the power extracted by a tidal turbine array partially spanning a wide channel is investigated using a theoretical model. Two predominant flow scales are assumed; turbine-scale flow, and array-scale flow, which are analysed as quasi-inviscid open channel flow problems in which conservation of mass, momentum, and energy are considered, and coupled through kinematic and dynamic boundary conditions. Power extraction may be maximised by determining the optimum inter-turbine spacing, which also enhances efficiency (ratio of power generated to power removed from the flow). Power extraction and efficiency increase as Froude number increases, improving open channel array performance. In the infinitely wide channel limit, the extracted power depends only on Froude number and local blockage (ratio of turbine to local flow passage areas). At zero Froude number, the peak power coefficient increases from the Lanchester-Betz limit (0.593) to 0.798, occurring when the local blockage ratio is approximately 0.4. Froude numbers in the range of 0.1–0.2, typical of prospective tidal energy sites, increase the peak power coefficient by an additional 1%–4.5% when the array occupies a negligible fraction of the channel, increasing further as a greater proportion of the channel is occupied.

The effects of free surface and end cell on flow around a finite circular cylinder with low aspect ratio

Vortex-induced motion is an oscillatory phenomenon which occurs to a floating body with low aspect ratio. The basic phenomenological study about the effects of free surface and end cell on flow around a finite fixed circular cylinder was investigated in this study using particle image velocimetry and hydrodynamic force measurement. It was found from the former experiment that the wake of the cylinder is influenced by the both end cell and free surface. Blowup and back flow are generated from the end cell, and their effects are suspended by free surface. The result of hydrodynamic force measurement showed the effect of Reynolds number, Froude number, and the aspect ratio of the floating body on the hydrodynamic force. Fluctuating components of hydrodynamic coefficients decrease for increasing Reynolds number, Froude number, and the aspect ratio. On the other hand, the mean drag coefficient increases as Froude number increases and decreases as the aspect ratio increases. The interpretation to these results was discussed in comparison with flow structures observed in the experiment. In addition, it was found that the effect of Reynolds number on the mean drag coefficient changes at different aspect ratios. A possible interpretation to this phenomenon was proposed.

Tsunami forcing by a low Froude number landslide

The waves generated by a submarine landslide, of great concern to coastal communities, exhibit strong dependence on the landslide motion along the sea floor. A series of two-dimensional physical experiments investigate the waves generated by a solid block landslide moving along a horizontal boundary, allowing measurement of both onshore- and offshore-propagating waves using the laser-induced fluorescence technique. This technique provides high-quality free surface measurements over the entire length of the experimental flume, and hence a data set that can be used to validate numerical models for this idealised scenario. The landslide motion is provided by a mechanical system, allowing testing of a range of landslide accelerations and terminal velocities. The landslide Froude number governs the overall behaviour of the wave field. At lower Froude numbers, the waves are almost entirely generated by the landslide acceleration and deceleration, and the offshore- and onshore-propagating wave groups contain approximately equal energy. Interactions between the landslide and the offshore-propagating waves become more important as the Froude number increases. Two inviscid-irrotational models demonstrate the importance of dispersive effects for tsunamis generated by a submarine landslide, and correctly predict the behaviour of the entire wave field at low Froude numbers. The predictions in the vicinity of the landslide worsen with increasing Froude number, due to the linear free surface conditions used by the models. Lower Froude numbers appear to be more representative of previous sloping-boundary experimental geometries, although rigid block landslides still represent an idealisation of a field scenario.

The flow pattern in triangular channels along the side weir for subcritical flow regime

Side weirs in triangular channels are used for control and measurement of flow in irrigation networks and drainage systems. The main purpose of this study is investigation of the effects of side weirs on the subcritical flow conditions parameters in the triangular channels that is not 3D simulated numerically, so far. The turbulent RNG k−ε model and the volume of fluid (VOF) scheme were used to simulate turbulent flow field and flow free surface in a triangular channel along the side weir, respectively. Comparison of numerical and experimental results shows that the numerical model simulates the free surface and flow field characteristics with acceptable accuracy. First, the flow pattern within triangular channel along the side weir in subcritical flow regime is investigated numerically. Due to the side weir inlet effects, a sudden drop occurred at the first quarter of the side weir length. Then, the effects of subcritical regime Froude numbers in range of 0.406–0.793 are investigated. At all the Froude numbers, a surface jump occurred along the end quarter of the weir length. The length of this surface jump increased with increasing the Froude number. The angle of spilling jet pattern (φ) was similar for all the Froude numbers. However, the minimum φ decreased upon increasing Froude number. For all the Froude numbers, a secondary flow cell was produced after the side weir, which developed as it moved downstream in the main channel.

A modified integral model for negatively buoyant jets in a stationary ambient

A simple modification is introduced into the integral model (IM) CorJet in an effort to predict better the characteristics of negatively buoyant jets (NBJ) discharged in a stationary ambient. Although this modification was developed for the CorJet model, it can be applied to every IM which employs the entrainment hypothesis. The detrainment of fluid from the main flow is taken into account by inserting a coefficient “p” into the conservation equations of volume, buoyancy and tracer mass flux. This coefficient expresses the ratio of the specific mass flux of the detrained fluid to the net specific mass flux entrained to the NBJ. The value of p is assumed constant along the jet trajectory and up to the maximum jet height, becoming zero thereafter. Results show that the modified CorJet model (MCM) predicts reasonably well experimental data from the literature and data from experiments performed in this work. The optimal value of p and therefore the detrained fluid from the main NBJ flow was found to decrease as the jet initial densimetric Froude number increases.

Trim effect on the resistance of sailing planing hulls

Sailing skiffs are light-weight high-performance small boats of growing interest in competitive sailing. The present paper presents towing tank tests performed on the Aura skiff, which was a candidate for the 2016 Olympic games. Resistance, sink and trim were measured for different longitudinal positions of the crew weight and for Froude numbers (based on the boat׳s length over all) ranging from 0.30 to 1.03. For each test, detailed analysis of the measurement uncertainty was performed. The measured resistance was found in good agreement with the resistance computed with established empirical formulations developed for planing hulls. It was found that the optimum crew position moves from forward to aft when the Froude number increases. An incorrect longitudinal centre of gravity led to a maximum resistance penalty at Froude number around 0.4. These trends are in agreement with the sailor׳s experience and with measurements performed by other authors on large vessels.

Numerical investigation of the effect of Froude number on flow pattern around a submerged T-shaped spur dike in a 90$^{\circ}$ bend

A spur dike is a structure that projects from a stream bank into the river channel and causes a redirection of flow away from the bank toward the tip of the spur dike. Construction of a spur dike against the flow causes significant changes in flow pattern in the channel. In this study, the flow pattern around a 25% submerged T-shaped spur dike in a 90$^{\circ}$ bend with rigid bed and various Froude numbers 0.2, 0.34, 0.45, and 0.6 was examined using FLOW-3D software. Numerical results were compared with experimental results and analyzed. The results indicated that in cross section around the spur dike, when Froude number increases, the formed vortexes decreased by 35%. Longitudinal velocity and lateral velocity increased twice as much as their former values. The highest amount of vertical velocity was observed between the wing of the spur dike and the external bend, and closer to the external bend, with an increase in Froude number, vertical velocity components variations decreased by 33%. Moreover, the results indicated that when the Froude number increases, the width of the separation zone increases, but the change in the length of the separation zone in different levels is negligible.

Roll-Coupling Effect on Ship Maneuverability

Roll-coupling effect on ship maneuvering motions is not negligible for high speed vessels, large container ships, passenger ships etc. To perceive the roll-coupling effect on the ship maneuverability, this paper proposes new simplified formulae for course stability criterion and steady turning index in the framework of the linear motion theory. Then, an approximation is employed where the depth wise acting point of lateral force related to roll moment is constant in any oblique motion mode. The course stability criterion including the roll-coupling effect is written aswhere C′ denotes the usual course stability criterion, and the linear hydrodynamic derivatives and l′ the depth wise acting point of the lateral force. The formula is simple and useful for checking the course stability taking the roll-coupling effect into account. To obtain the hydrodynamic derivatives, oblique towing test and circular motion test with various heel angles were conducted for models of a container ship, a pure car carrier and a ferry. In general, course stability deteriorates with increasing Froude number Fn or decreasing of metacentric height GM′, whereas turning performance improves and the time until response reaches steady turning state increases.

Forward Speed Calm Water Roll Decay for Surface Combatant 5415: Global and Local Flow Measurements

Global and local flow measurements for forward speed calm water roll decay are performed in a towing tank for surface combatant model 5415 for both bare hull (BH) and bilge keel (BK) conditions. Roll motion is oscillatory with underdamped exponential decay and linear with respect to initial roll angles less than the average initial roll angle (9°). For larger mean roll angles (> 3°), damped natural frequency is a few percent below the hydrostatic natural roll frequency with larger values for BH than BK condition and increasing Froude number (Fr), whereas for smaller mean roll angles, it sharply increases toward the hydrostatic natural roll frequency. For larger mean roll angles, logarithmic decrement and linear damping linear increase with larger values for BK than BH condition and increasing Fr, whereas for smaller mean roll angles, they sharply increase. For increasing Fr, mean and initial roll angle averaged mean roll angle decreases by 50%; damped natural frequency increases by 6% with larger values for BH than BK condition; and logarithmic decrement/linear damping coefficient increase by a factor of three/four with larger values for BK than the BH condition. Logarithmic decrement and Himeno method linear damping coefficients are qualitatively similar. Nonlinear damping coefficient is two orders of magnitude smaller than linear damping coefficient. Roll reconstruction errors are smallest for Himeno with linear and nonlinear damping. The phase-averaged wave pattern and velocity and axial vorticity fields at x/L = 0.675 initially show larger amplitudes followed by oscillatory exponential decay. Alternating vortex pairs are shed from the bilge keel tip with damped magnitudes for decreasing mean roll angles. The local flow indicates lower frequencies and larger damping than the roll motion.

Forward Speed Calm Water Roll Decay for Surface Combatant 5415: Global and Local Flow Measurements

Global and local flow measurements for forward speed calm water roll decay are performed in a towing tank for surface combatant model 5415 for both bare hull (BH) and bilge keel (BK) conditions. Roll motion is oscillatory with underdamped exponential decay and linear with respect to initial roll angles less than the average initial roll angle (9°). For larger mean roll angles (> 3°), damped natural frequency is a few percent below the hydrostatic natural roll frequency with larger values for BH than BK condition and increasing Froude number (Fr), whereas for smaller mean roll angles, it sharply increases toward the hydrostatic natural roll frequency. For larger mean roll angles, logarithmic decrement and linear damping linear increase with larger values for BK than BH condition and increasing Fr, whereas for smaller mean roll angles, they sharply increase. For increasing Fr, mean and initial roll angle averaged mean roll angle decreases by 50%; damped natural frequency increases by 6% with larger values for BH than BK condition; and logarithmic decrement/linear damping coefficient increase by a factor of three/four with larger values for BK than the BH condition. Logarithmic decrement and Himeno method linear damping coefficients are qualitatively similar. Nonlinear damping coefficient is two orders of magnitude smaller than linear damping coefficient. Roll reconstruction errors are smallest for Himeno with linear and nonlinear damping. The phase-averaged wave pattern and velocity and axial vorticity fields at x/L = 0.675 initially show larger amplitudes followed by oscillatory exponential decay. Alternating vortex pairs are shed from the bilge keel tip with damped magnitudes for decreasing mean roll angles. The local flow indicates lower frequencies and larger damping than the roll motion.

자유표면 밑을 전진하는 원주 주위의 유동에 관한 연구

The flow around a circular cylinder advancing beneath the free surface is numerically investigated using a VOF method. The simulations cover Froude numbers in the range of 0.2~0.6 and gap ratios (h/d) in the range of 0.1~2.0, where h is the distance from the free surface to a cylinder, and d is the diameter of a cylinder at Reynolds number 180. It is observed that the vortex suppression effect and surface deformation increase as the gap ratio decreases or the Froude number increases. The most important results of the present study are as follows. The proximity of the free surface causes an initial increase in the Strouhal number and drag coefficient, and the maximum Strouhal number and drag coefficient occur in the range of 0.5~0.7. However, this trend reverses as the gap ratio becomes small, and the lift coefficient increases downward as the gap ratio decreases.

Rotating shallow water turbulence: Experiments with altimetry

Results from a new series of experiments on turbulent flows in a rotating circular container are presented. Electromagnetic forcing is applied to induce flow in a layer of fluid of constant depth. Continuously forced and decaying flows are investigated. Optical altimetry is used to measure the gradient of the surface elevation field and to obtain the velocity and vorticity fields with high temporal and spatial resolution. Spectral analysis of the flows demonstrates the formation of dual cascade with energy and enstrophy intervals although the corresponding spectral fluxes of energy and enstrophy are not uniform in these intervals. The energy interval is characterized by the slope of ∼−5/3 in terms of wavenumber and is limited in extent by the finite radius of deformation effect. In the enstrophy range, the slope is steeper than −3 due to the presence of long-lived coherent vortices. The spatial patterns of fluxes to large or small scales in the flow indicate that inverse energy transfer and direct enstrophy transfer occur mainly in elongated vorticity patches. Cyclone/anticyclone asymmetry in favor of anticyclones is observed in our flows. Dominance of anticyclones is most clear during the decay phase of turbulence. The anticyclones remain circular, while cyclonic vorticity is stretched into elongated patches. Measurements show that skewness of vorticity distribution increases with increasing Froude number of the flow.