Nonzero Froude Number Research Articles

Nonpositive curvature of the quantomorphism group and quasigeostrophic motion

In this paper, we compute the sectional curvature of the group whose Euler-Arnold equation is the quasi-geostrophic (QG) equation in geophysics and oceanography, or the Hasegawa-Mima equation in plasma physics: this group is a central extension of the quantomorphism group Dq(M). We consider the case where the underlying manifold M is rotationally symmetric, and the fluid flows with a radial stream function. Using an explicit formula for the curvature, we will also derive a criterion for the curvature operator to be nonpositive and discuss the role of the Froude number and the Rossby number on curvature. The main technique to obtain a usable formula is a simplification of Arnold's general curvature formula in the case where a vector field is close to a Killing field, and then use the Green's function explicitly together with a criterion for nonnegativity of a general bilinear form. We show that nonzero Froude number and Rossby numbers typically both tend to stabilize flows in the Lagrangian sense, although there are counterexamples in general.

Hydrodynamic Study on Water Column Oscillation of Varying Cross-Sectional Moonpool and Its Effect on Resistance of a Drill Ship

A moonpool is meant for access to the underwater part of the ship from onboard. It is a vertical opening along the depth having an effect on the performance of the floating platform. Inside the moonpool, water motions in horizontal plane is called sloshing and in vertical planes it is called piston mode. Moonpool causes deck wetness and sometimes results in the downtime of the platform. It is the necessity of the operator to be at the safe conditions of platform facing varied environmental conditions. In the present study, vessel response in the region of moonpool resonance was investigated with different shapes of moonpool and comparison is made with Molin's (2001, “On the Piston and Sloshing Modes in Moonpools,” J. Fluid Mech., 430, pp. 27–50.) theoretical and Fukuda's (1977, “Behavior of Water in Vertical Well With Bottom Opening of Ship and Its Effects on Ship-Motion,” J. Soc. Nav. Archit. Jpn., 1977(141), pp. 107–122.) empirical formulas. It is seen that there is a shift in the frequency of resonance based on moonpool shapes. The effect of moonpool on the ship motion with forward speed is also attempted in this paper. Proven packages are used to calculate the calm water resistance of the ship with moonpool of various cross section. Wave making coefficient of the ship is modified due to opening to accommodate the moonpool. The openings to accommodate moonpool causes further entry of water both zero and nonzero Froude number especially in the presence of waves.

Effect of Pretension on Moored Ship Response

Moonpools are vertical wells in a floating body used onboard many types of vessels like Exploration and drilling vessels, Production barges, Cable-laying vessels, Rock dumping vessels, Research and offshore support vessels. Moonpool gives passage to underwater activities for different types of ships as per their mission requirements. It is observed that inside a moonpool considerable relative motions may occur, depending on shape, depth of the moonpool and on the frequency range of the waves to which the ship is exposed. The vessel responses are entirely different in zero and non-zero Froude number. Former situation is paid attention in this study as the mission requirement of the platform is to be in the particular location for long period of operation. It is well known that there are two modes of responses depending on the shape of the moonpool viz., piston mode for square shape and sloshing mode for rectangular shapes with different aspect ratios of opening like 1:1.5 and 1:2 ratios. Circular shaped moonpool is also tested for measuring the responses. The vessel moored using heavy lines are modelled and tested in the wave basin. The pretensions of the lines are varied by altering the touchdown points and the dynamic tensions on the lines are measured. The different modes of oscillations of water column are measured using wave gauge and the vessel response at a particular situation is determined. RAOs calculated for various situations provide better insight to the designer.

Model tests on the moored vessel with different moonpool shapes

Moonpools are vertical wells in a floating body used onboard many types of vessels like cable-laying vessels and offshore support vessels. Moonpool gives passage to underwater activities for different types of ships as per their mission requirements. It is observed that inside a moonpool considerable relative motions may occur, depending on shape, depth of the moonpool and on the frequency range of the waves to which the ship is exposed. The vessel responses are entirely different in zero and non-zero Froude number. Former situation is paid attention in this study as the mission requirement of the platform is to be in the particular location for long period of operation. It is well known that there are two modes of responses depending on the shape of the moonpool viz., piston mode for square shape and sloshing mode for rectangular shapes with different aspect ratios of opening like 1:1.5 and 1:2 ratios. Circular shaped moonpool is also tested for measuring the responses. The vessel moored using heavy lines are modeled and tested in the wave basin. The moored lines are provided with pre-tension and the dynamic tensions on the lines are measured. The different modes of oscillations of water column are measured using wave gauge and the vessel response at a particular situation is determined. RAOs determined for various situations provide better insight to the designer. The experiments done in the wave basin may also be compared with a software package meant for handling moored floating bodies.

3 차원 시간영역 근사비선형 2 차경계요소법에 의한 선체의 대진폭 운동 및 파랑하중 계산

A three-dimensional time-domain calculation method is of crucial importance in prediction of the motions and wave loads of a ship advancing in a severe irregular sea. The exact solution of the free surface wave-ship interaction problem is very complicated because of the essentially nonlinear boundary conditions. In this paper, an approximate body nonlinear approach based on the three-dimensional time-domain forward-speed free-surface Green function has been presented. The Froude-Krylov force and the hydrostatic restoring force are calculated over the instantaneous wetted surface of the ship while the forces due to the radiation and scattering potentials over the mean wetted surface. The time-domain radiation and scattering potentials have been obtained from a time invariant kernel of integral equations for the potentials which are discretized according to the second-order boundary element method (Hong and Hong 2008). The diffraction impulse-response functions of the Wigley seakeeping model advancing in transient head waves at various Froude numbers have been presented. A simulation of coupled heave-pitch motion of a long rectangular barge advancing in regular head waves of large amplitude has been carried out. Comparisons between the linear and the approximate body nonlinear numerical results of motions and wave loads of the barge at a nonzero Froude number have been made.

The surface layer for free-surface turbulent flows

Direct numerical simulation (DNS) is used to examine low Froude number free-surface turbulence (FST) over a two-dimensional mean shear flow. The Navier–Stokes equations are solved using a finite-difference scheme with a grid resolution of 1283. Twenty separate simulations are conducted to calculate the statistics of the flow. Based on the velocity deficit and the vertical extent of the shear of the mean flow, the Reynolds number is 1000 and the Froude number is 0.7. We identify conceptually and numerically the surface layer, which is a thin region adjacent to the free surface characterized by fast variations of the horizontal vorticity components. This surface layer is caused by the dynamic zero-stress boundary conditions at the free surface and lies inside a thicker blockage (or ‘source’) layer, which is due to the kinematic boundary condition at the free surface. The importance of the outer blockage layer is manifested mainly in the redistribution of the turbulence intensity, i.e. in the increase of the horizontal velocity fluctuations at the expense of the vertical velocity fluctuation. A prominent feature of FST is vortex connections to the free surface which occur inside the surface layer. It is found that as hairpin-shaped vortex structures approach the free surface, their ‘head’ part is dissipated quickly in the surface layer, while the two ‘legs’ connect almost perpendicularly to the free surface. Analysis of the evolution of surface-normal vorticity based on vortex surface-inclination angle shows that both dissipation and stretching decrease dramatically after connection. As a result, vortex structures connected to the free surface are persistent and decay slowly relative to non-connected vorticities. The effects of surface and blockage layers on the turbulence statistics of length scales, Reynolds-stress balance, and enstrophy dynamics are examined, which elucidate clearly the different turbulence mechanisms operating in the respective near-surface scales. Finally we investigate the effect of non-zero Froude number on the turbulence statistics. We show that the most significant effect of the presence of the free surface is a considerable reduction of the pressure–strain correlation at this surface, compared to that at a free-slip at plate. This reduction is finite even for very low values of the Froude number.

The interaction of background ocean air bubbles with a surface ship

A two-fluid model suitable for the calculation of the two-phase flow field around a naval surface ship is presented. This model couples the Reynolds-averaged Navier–Stokes (RANS) equations with equations for the evolution of the gas-phase momentum, volume fraction and bubble number density, thereby allowing the multidimensional calculation of the two-phase flow for monodisperse variable size bubbles. The bubble field modifies the liquid solution through changes in the liquid mass and momentum conservation equations. The model is applied to the case of the scavenging of wind-induced sea-background bubbles by an unpropelled US Navy frigate under non-zero Froude number boundary conditions at the free surface. This is an important test case, because it can be simulated experimentally with a model-scale ship in a towing tank. A significant modification of the background bubble field is predicted in the wake of the ship, where bubble depletion occurs along with a reduction in the bubble size due to dissolution. This effect is due to lateral phase distribution phenomena and the generation of an upwelling plume in the near wake that brings smaller bubbles up to the surface. © 1998 John Wiley & Sons, Ltd.

The interaction of background ocean air bubbles with a surface ship

A two-fluid model suitable for the calculation of the two-phase flow field around a naval surface ship is presented. This model couples the Reynolds-averaged Navier–Stokes (RANS) equations with equations for the evolution of the gas-phase momentum, volume fraction and bubble number density, thereby allowing the multidimensional calculation of the two-phase flow for monodisperse variable size bubbles. The bubble field modifies the liquid solution through changes in the liquid mass and momentum conservation equations. The model is applied to the case of the scavenging of wind-induced sea-background bubbles by an unpropelled US Navy frigate under non-zero Froude number boundary conditions at the free surface. This is an important test case, because it can be simulated experimentally with a model-scale ship in a towing tank. A significant modification of the background bubble field is predicted in the wake of the ship, where bubble depletion occurs along with a reduction in the bubble size due to dissolution. This effect is due to lateral phase distribution phenomena and the generation of an upwelling plume in the near wake that brings smaller bubbles up to the surface. © 1998 John Wiley & Sons, Ltd.

Prediction of Solid/Free-Surface Juncture Boundary Layer and Wake of a Surface-Piercing Flat Plate at Low Froude Number

Results are reported of a RANS simulation investigation on the prediction of turbulence-driven secondary flows at the free-surface juncture of a surface-piercing flat plate at low Froude numbers. The turbulence model combines a nonlinear eddy viscosity model and a modified version of a free-surface correction formula. The different elements of the model are combined and the model constants calibrated based on the premises that the anisotropy of the normal stresses is mainly responsible for the dynamics of the flow in the juncture region, and an accurate modeling of the normal-stress anisotropy as obtained from the data is a primary requirement for the successful prediction of the overall flow field. The predicted mean velocity, streamwise vorticity, turbulent kinetic energy, and other quantities at the juncture are then compared with data and analyzed with regard to findings of related studies. In agreement with the experimental observations, the simulated flow at large depths was essentially two-dimensional and displayed all the major features of zero pressure gradient boundary layer and wake, including the anisotropy of normal stresses in the near-wall region. In the boundary-layer free-surface juncture region, the major features of interest that were predicted include the generation of secondary flows and the thickening of the boundary layer near the free surface. In the wake free-surface juncture region, even though secondary flows and a thickening of the wake width near the free surface were predicted in accordance with the experimental observations, the overall comparison with the experiment was not as satisfactory as the boundary-layer juncture. This is partly due to the lack of a strong coherent flow structure in the wake juncture and the presence of possible wave effects in the wake in the experiments. An examination of the terms in the Reynolds-averaged streamwise vorticity equation reconfirmed the importance of the anisotropy of the normal Reynolds stresses in the production of streamwise vorticity. The free-surface wave elevations were negligible for the present model problem for the nonzero Froude number studied. Finally, concluding remarks are presented with regards to extensions for practical geometries such as surface ship flows.

A Large-Domain Approach for Calculating Ship Boundary Layers and Wakes and Wave Fields for Nonzero Froude Number

A large-domain approach is developed for calculating ship boundary layers and wakes and wave fields for nonzero Froude number. The Reynolds-averaged Navier–Stokes and continuity equations are solved with the Baldwin–Lomax turbulence model, exact nonlinear kinematic and approximate dynamic free-surface boundary conditions, and a body/free-surface conforming grid. The results are validated through comparisons with data for the Series 60 CB= 0.6 ship model at low and high Froude numbers and results of a precursory interactive approach. Both approaches yield satisfactory results; however, the large-domain results indicate improved resolution of the flow close to the hull and wake centerplane and of the Froude number differences due to near-wall turbulence modeling and nonlinear free-surface boundary conditions. Additional evaluation is provided through discussion of the recent CFD Workshop Tokyo 1994, where both methods were among the best. Last, some concluding remarks are made.

Evaluation of Algebraic Stress Modeling in Free-Surface Jet Flows

This study evaluated three algebraic stress models for predicting turbulent stresses near the free surface in a free-surface jet at nonzero Froude number, by comparing to experiments. The models examined included one with no explicit near-surface modeling, one which specified model coefficients in terms of invariants of the anisotropy tensor, and a third model which employed a surface correction with an ad-hoc damping function. Experiment showed that at low Froude number, the anisotropy near the free surface did not attain the limiting behavior characteristic of two-dimensional turbulence and the anisotropy increased with streamwise distance. At high Froude number the surface can have little effect on the anisotropy. Far from the free surface, all the models performed well. For the model with no explicit free-surface modeling, the turbulence near the free surface was predicted to be isotropic. For the “anisotropy-invariant” model, the predicted anisotropy was too small and confined to locations too near the free surface. The model with the ad-hoc damping function captured the anisotropy near the free surface best, but specification of the decay constant in the damping function is an open question.

Validation of an interactive approach for calculating ship boundary layers and wakes for nonzero froude number

An interactive approach for calculating ship boundary layers and wakes for nonzero Froude number is validated through detailed comparisons with recent extensive experimental data for the practical 3-D geometry of the Series 60 C B = 0.6 ship model at both low (0.16) and high (0.316) Froude number. The former case essentially simulates the zero Froude number condition such that the comparisons with the latter case enables the identification of the salient features of the wave-induced effects. Close agreement is demonstrated between the calculations and the data, which supports the conclusion that the present approach can accurately predict ship boundary layers and wakes, including free-surface effects; however, the present detailed comparisons enables a more critical evaluation. Additionally, comparisons are made with inviscid-flow results, which in combination enable an evaluation of the wave/boundary-layer and wake interaction. Lastly, some concluding remarks are made concerning the limitations of the present approach, prognosis and requirements for improvements, relative merits of interactive vs large-domain approaches, and implications with regard to ship design.

An interactive approach for calculating ship boundary layers and wakes for nonzero froude number

An interactive approach is set forth for calculating ship boundary layers and wakes for nonzero Froude number. The Reynolds-averaged Navier-Stokes equations are solved using a small domain with edge conditions matched with those from a source-doublet Dawson method solved using the displacement body. An overview is given of both the viscous- and inviscid-flow methods, including their treatments of the free-surface boundary conditions and interaction procedures. Results are presented for the Wigley hull, including comparisons for zero and nonzero Froude number and with available experimental data and the inviscid-flow results, which validate the overall approach and enable an evaluation of the wave-boundary layer and wake interaction.

The Second Order Treatment of Ship Surface Condition in the Theory of Wave-Making Resistance of Ships

The linearization of boundary conditions both for ship surface & free surface often gives rise to discussion when we seek the relation between ship forms and their wave-making resistance especially of displacement ships. Inui's improvement for ship surface condition is well known as a double model conception. However it still neglects partially the free surface effect except for exactly corresponding zero Froude number case.An extention of his idea is examined here to build up a more exact theory by introducing a higher or the second order ship surface approximation taking the whole free surface effect into account. His approach is called stream-line persuit method or inverse method because not ship forms but their equivalent source or doublet distributions are determined first with sometimes the aid of variational treatment of wave-making resistance formula expressed by functional singularity distribution.In this paper, two concrete non-zero Froude number hulls corresponding to K0L=14 are obtained to sereve for streamline observation, resistance test and wave pattern comparison. As the result it finds that 2nd order approximation of hulls contributes to the improvement in the theory of wave-making resistance, provided that some disagreement of ship surface condition still remains near the bow.