Values Of Froude Number Research Articles

Prediction of hydrodynamic loading on a mini TLP with free surface effects

This paper describes the extension of a fluid-flow simulations method to capture the free surface evolution around a full-scale Tension Leg Platform (TLP). The focus is on the prediction of the resulting hydrodynamic loading on the various elements of the TLP in turbulent flow conditions and, in particular, on quantifying the effects of the free surface distortion on this loading. The basic method uses finite-volume techniques to discretize the differential equations governing conservation of mass and momentum in three dimensions. The time-averaged forms of the equations are used, and the effects of turbulence are accounted for by using a two-equation, eddy-viscosity closure. The method is extended here via the incorporation of surface-tracking algorithm on a moving grid to predict the free-surface shape. The algorithm was checked against experimental measurements from two benchmark flows: the flow over a submerged semi-circular cylinder and the flow around a floating parabolic hull. Predictions of forces on a model TLP were then obtained both with and without allowing for the deformation of the free surface. The results suggest that the free surface effects on the hydrodynamic loads are small for the values of Froude number typically encountered in offshore engineering practice.

Frictional Drag Reduction Provided with Small Bubbles in Concentric Rotating Cylinders

The frictional drag reduction provided with small bubbles is investigated experimentally using vertical concentric cylinders. The friction reduction ratio is measured using a torque meter mounted on the inner cylinder for a wide range in Reynolds number from wavy vortex flow (WVF) to turbulent Taylor vortex flow (TTV). The present data show around 36% drag reduction in the case of WVF regime at Re =600 and the reduction maintains until Re =4 000. The friction reduction ratioη defined by the unit void fraction obeyed in a linear relation with inverse values of Froude number, and reached up to 10 in the best case, that implies the highest sensitivity for the drag reduction. The bubble distribution measured in the gap showed a peak near the inner cylinder surface resulting in high local shear stress reduction.

On the convective nature of roll waves instability

A theoretical analysis of the Saint‐Venant one‐dimensional flow model is performed in order to define the nature of its instability. Following the Brigg criterion, the investigation is carried out by examining the branch points singularities of dispersion relation in the complex ω and k planes, where ω and k are the complex pulsation and wave number of the disturbance, respectively. The nature of the linearly unstable conditions of flow is shown to be of convective type, independently of the Froude number value. Starting from this result a linear spatial stability analysis of the one‐dimensional flow model is performed, in terms of time asymptotic response to a pointwise time periodic disturbance. The study reveals an influence of the disturbance frequency on the perturbation spatial growth rate, which constitutes the theoretical foundation of semiempirical criteria commonly employed for predicting roll waves occurrence.

Rational regime model of alluvial channel morphology and response

AbstractA theoretical model is developed for predicting equilibrium alluvial channel form. The concept of greatest relative stability, achieved by maximizing resistance to flow in the fluvial system, is presented as the basis for an optimization condition for alluvial systems. Discharge, sediment supply (quantity and calibre) and valley gradient are accepted as independent governing variates. The model is used to define a dimensionless alluvial state space characterized by aspect ratio (W/d), relative roughness (D/d), and dimensionless shear stress (τ*) or, equivalently, channel slope (S). Each alluvial state exhibits unique values of Froude number and sediment concentration. The range of alluvial states for constant values of relative bank strength (parameterized by an apparent friction angle, ϕ′) forms a single plane in the state space (W/d, D/d, τ* or S). The scaling relations produced by the model are consistent with laboratory channels exhibiting a range of bank strengths, and with the behaviour of natural channels. Copyright © 2004 John Wiley & Sons, Ltd.

Longitudinal velocity patterns and bed morphology interaction in a rill

AbstractPresent‐day understanding of rill dynamics is hampered by a lack of detailed data on velocity distributions in rills. The latter are difficult to collect with traditional techniques due to the very low water depths and the relatively high flow velocities in rills. The objectives of this paper were to investigate the feasibility of miniaturized acoustic Doppler velocimeter (mADV) measurements in rill flow and to explore longitudinal variations in flow velocities and their relationship with rill bed morphology. Detailed data on longitudinal flow velocity were required to achieve these objectives.A 1·8 m long rill was formed freely in a flume at 5° slope and 0·001 m3 s−1 discharge. Rill topography was characterized by an alternation of steps and pools. The flume surface was then fixed to preserve rill roughness. A topographical scanning of the entire flume surface was made. Velocity was measured with a mADV along the rill, and at different depths. Flow depth in a longitudinal direction was also measured using an elevation gauge.A strong relationship exists between rill topography and flow hydraulics. Over steps, flow was unidirectional and rapidly accelerating until a threshold Froude number (Fn) value between 1·3 and 1·7 was reached and a hydraulic jump occurred leading to the formation of a pool. In the pool, the flow pattern was multidirectional and complex. The flow was subcritical when leaving the pool and accelerated over the next step until the threshold Froude number value was again reached. Energy loss in the rill was concentrated in the pools, mainly due to the action of a hydraulic jump. This mechanism of energy dissipation appeared to be an essential factor in rill formation and bedform evolution. Copyright © 2004 John Wiley & Sons, Ltd.

SYMMETRIC SURFACE WAVES OVER A BUMP

We study the surface waves of an incompressible fluid passing over a small bump. A forced KdV equation for surface wave is derived without assuming that flow is uniform at far upstream. New types of steady solutions are discovered numerically. Two new cut o values of Froude number are found, above the larger of which two symmetric solutions exist and under the smaller of which two dierent symmetric solutions exist.

Flow resistance in gravel‐bed channels with large‐scale roughness

AbstractA previously published mixing length (ML) model for evaluating the Darcy–Weisbach friction factor for a large‐scale roughness condition (depth to sediment height ratio ranging from 1 to 4) is briefly reviewed and modified (MML). Then the MML model and a modified drag (MD) model are experimentally tested using laboratory measurements carried out for gravel‐bed channels and large‐scale roughness condition. This analysis showed that the MML gives accurate estimates of the Darcy–Weisbach coefficient and for Froude number values greater than 0·5 the MML model coincides with the ML one. Testing of the MD model shows limited accuracy in estimating flow resistance.Finally, the MML and MD models are compared with the performance of a quasi‐theoretical (QT) model deduced applying the P‐theorem of the dimensional analysis and the incomplete self‐similarity condition for the depth/sediment ratio and the Froude number. Using the experimental gravel‐bed data to calibrate the QT model, a constant value of the exponent of the Froude number is determined while two relationships are proposed for estimating the scale factor and the exponent of the depth/sediment ratio. This indirect estimate procedure of the coefficients (b0, b1 and b2) of the QT model can produce a negligible overestimation or underestimation of the friction factor. Copyright © 2003 John Wiley & Sons, Ltd.

Computing 3-D atmospheric trajectories for complex orography: application to a case study of strong convection in the western Mediterranean

A three-dimensional Lagrangian model (TRACOMX) has been developed to simulate large-scale atmospheric transport over complex terrain. Trajectories are integrated in time steps, assuming constant velocity during the time interval, non-interaction with the environment and non-diffusive processes. The vertical velocity is computed, at high levels, by three-dimensional relaxation from the Q-vector formulation of the omega equation. At low levels, the vertical velocity is computed by means of a ‘specific treatment’ of the orography consisting of the application of the kinematic boundary condition to the cells intersected by the orography, and taking into account the kinetic energy of the flow, the atmospheric stability and the presence of horizontal thermal gradients. These effects are parameterized by means of a multiplicative factor obtained from the Froude number value. The trajectory computation is achieved by successive approximations, using an iterative method, to the final position for each trajectory vector. The model has been applied to a case study of strong convection over NE Spain and the results have been compared with those obtained from an isentropic method.

Effect of a Localized Random Forcing Term on the Korteweg-De Vries Equation

In this work, we numerically investigate the influence of a white noise-type forcing on the phenomenon of forced generation of solitons by a localized moving disturbance. Our numerical method is based on finite elements and least-squares. We present numerical experiments for different values of noise amplitude and Froude number, which describe some damping effects on the emission of solitons.

Gravity current structure of the Chesapeake Bay outflow plume

Measurements made across the edge of the Chesapeake Bay outflow plume in September 1996 are examined for evidence of the frontal head structure expected for a buoyant gravity current. Shipboard acoustic Doppler current profiler and hydrographic data from two distinct, 2‐hour‐long sets of frontal transects are analyzed. One set of observations was acquired just south of the bay mouth (near Cape Henry, Virginia) during late flood tide in water depth of 8 m. These measurements show a nearly steady frontal structure and a frontal head that is approximately 5‐m deep and 20‐m wide. The second data set was acquired about 20‐km southeast of the bay mouth, where the plume bulges horizontally outward over the continental shelf. These measurements show a 9‐m‐deep frontal head (in water depth of 18 m) that ultimately dissipates during early flood tide. In both sets of observations, relative inflow of plume water into the frontal head is confined to a shallow surface layer estimated to be no more than about 2‐m deep. Referenced to this layer, frontal propagation is supercritical (Froude number greater than 1). Also measured was an expected sinking motion (of the order of 3 cm/s) along the front. Overall, the results are consistent with previous measurements made across smaller buoyant plumes and in the laboratory, and they are compatible with the use of fixed values of Froude number and entrainment parameter in two‐layer plume models. Additional study of the plume front during ebb tide is recommended.

Mesoscale Simulation of Supercritical, Subcritical, and Transcritical Flow along Coastal Topography

Abstract A mesoscale atmospheric model is used to address the characteristics of stratified flow bounded by a side wall along a varying coastline. Initial Froude number values are varied through alteration of marine inversion strength, permitting examination of supercritical, subcritical, and transcritical flow regimes encountering several coastal configurations. Consistent with shallow water models, sharp drops in boundary layer depth and flow acceleration occur in flow rounding convex bends; however, significant flow response occurs in the stratified layer aloft, which is unexplained by conventional shallow water theory. The strongest flow acceleration occurs in the transcritical case while, regardless of inversion strength, the deformation of the isentropes aloft shows general structural similarity. Advection of horizontal momentum is an important component of the horizontal force balance. A simulation having several coastline bends exhibits a detached, oblique hydraulic jump upwind of a concave bend t...

Free-surface flow over a semi-circular obstruction

The fully non-linear free-surface flow over a semi-circular bottom obstruction was studied numerically in two dimensions using a mixed Eulerian–Lagrangian formulation. The problem was solved in the time domain that allows the prediction of a number of transient phenomena, such as the generation of upstream advancing solitary waves, as well as the simulation of wave breaking. A parametric study was performed for a range of values of the depth-based Froude number up to 2.5 and non-dimensional obstacle heights, α up to 0.9. When wave breaking does not occur, three distinct flow regimes were identified: subcritical, transcritical and supercritical. When breaking occurs it may be of any type: spilling, plunging or surging. In addition, for values of the Froude number close to 1, the upstream solitary waves break. A systematic study was undertaken to define the boundaries of each type of breaking and non-breaking pattern and to determine the drag and lift coefficients, free-surface profile characteristics and transient behavior. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical simulation of circular disks entering the free surface of a fluid

In this paper numerical simulations of the irrotational fluid flow associated with the entry of circular disks of a given mass into a semi-infinite fluid domain in the limits of very low to moderate Froude numbers are reported. This work is motivated by an experimental study performed by Glasheen and McMahon who investigated the low-Froude-number water entry of circular disks and found a linear relationship between the cavity seal depth and the Froude number and also showed that a single value of a modified drag coefficient is sufficient to predict the drag force on the disk. The numerical calculations performed in this paper confirm these experimental findings for steady cavity regimes and identify the ranges of Froude number and dimensionless mass values for which these results hold. Excellent agreement between the numerical computations and analytical velocity predictions, as well as the experimental cavity seal depth measurements, are obtained although the agreement between the measured and the computed drag coefficient values is not as good. The cavity seal depth and the drag coefficient are also found to depend on the disk mass and the numerical results in this paper show that for any disk of dimensionless mass ℳ there exists a value of the Froude number for which the cavity dynamics are steady. Also, a very low-Froude-number regime in which gravitational forces are dominant and for which the cavity dynamics are qualitatively different than for low-to-moderate-Froude-number cases is also numerically explored in this paper. Finally, a bifurcation in the cavity seal mechanism from deep seal to surface seal was found at a Froude number F=105.

The growth, saturation, and scaling behaviour of one- and two-dimensional disturbances in fluidized beds

It is well-known that fluidized beds are usually unstable to small perturbations and that this leads to the primary bifurcation of vertically travelling plane wavetrains. These one-dimensional periodic waves have been shown recently to be unstable to two-dimensional perturbations of large transverse wavelength in gas-fluidized beds. Here, this result is generalized to include liquid-fluidized beds and to compare typical beds fluidized with either air or water. It is shown that the instability mechanism remains the same but there are big differences in the ratio of the primary and secondary growth rates in the two cases. The tendency is that the secondary growth rates, scaled with the amplitude of a fully developed plane wave, are of similar magnitude for both gas- and liquid-fluidized beds, while the primary growth rate is much larger in the gas-fluidized bed. This means that the secondary instability is accordingly stronger than the primary instability in the liquid-fluidized bed, and consequently sets in at a much smaller amplitude of the primary wave. However, since the waves in the liquid-fluidized bed develop on a larger time and length scale, the primary perturbations need longer time and thereby travel farther until they reach the critical amplitude. Which patterns are more amenable to being visually recognized depends on the magnitude of the initially imposed disturbance and the dimensions of the apparatus. This difference in scale plays a key role in bringing about the differences between gas- and liquid-fluidized beds; it is produced mainly by the different values of the Froude number.

Energetics of grid turbulence in a stably stratified fluid

A biplane grid with a mesh spacing of 10.8 cm was towed horizontally in a towing tank to generate turbulence in a non-stratified fluid and in stratified fluids with different constant density gradients. Turbulence velocity components and density fluctuations were measured using an array of cross-film and conductivity probes. Based on the mesh size of the grid, the nominal values of the (internal) Froude numbers were ∞, 80 and 40, and the corresponding Reynolds number was 4.3 × 104. The decay rates of the (turbulence) kinetic, potential and total energies and the dissipation rates of the kinetic and potential energies were calculated from the experimental data. For each of these quantities, the decay may be represented as a function of the downstream distance raised to a given power. The kinetic energy and its dissipation rate are lower for the stratified cases than for the non-stratified case but are almost compensated for by the corresponding potential energy and its dissipation rate. Our results are consistent with those of direct numerical simulations and agree reasonably well with those obtained in stratified wind and water tunnels. However, the results differ from laboratory results obtained using an optical method to measure the turbulent motion of tracer particles in the wake of a vertically towed grid; these latter results show an abrupt reduction in the decay rate of the turbulence kinetic energy after one Brunt–Vaisala period. A similar trend is also observed in results obtained in facilities with fairly high background turbulence or internal waves. This discrepancy is discussed and an explanation is presented. Furthermore, it is demonstrated that strongly stratified thin sheets with density gradients larger than that of the undisturbed fluid may be generated by local but incomplete mixing. The persistence of such thin sheets is proportional to the Schmidt number (≈ 500) in stratified salt water or the Prandtl number (≈ 0.71) in thermally stratified air.

Flushing Criteria in Estuarine and Laboratory Experiments

Laboratory experiments have shown the range of densimetric Froude numbers for which a depression that is filled with saline water and located at the bottom of a watercourse will be flushed by the overflow of fresh water. If this densimetric Froude number, based on the water speed, the density difference between the saline water in the depression and the overflowing fresh water, and the depth of the depression, is greater than 1.5 or 2, then the saline fluid is purged very quickly in what is called the one-shot mode. (The range of Froude number depends on the cavity geometry and the manner of the initiation of the overflowing current). For lower values of Froude number the density interface also exhibits an internal wave that sweeps along the cavity ejecting fluid from it, and any liquid of high salinity that remains in the cavity is subsequently removed by turbulent entrainment. Data from a field test of an estuary in Chile show the occurrence of an internal wave that was formed at the upstream sill during ebb tide, the amplitude of which compares favorably with the results collected in the laboratory.

Super-critical free-surface flow over a trapezoidal obstacle

A numerical method based on series truncation is presented to solve the problem of an irrotational, inviscid, incompressible and steady flow over a two-dimensional trapezoidal obstacle, with obstacle height W, lying on the bottom of the running stream. The suggested numerical method for the solution of the fully nonlinear problem is presented for which the flow is super-critical both upstream and downstream. It is found that solutions exist for symmetric trapezoidal for arbitrary size. The results are plotted for different trapezoidal shapes and different values of Froude number F > 1. The effect of the Froude number, the bottom height and the shape of the bottom are discussed.

Energetics of grid turbulence in a stably stratified fluid

A biplane grid with a mesh spacing of 10.8 cm was towed horizontally in a towing tank to generate turbulence in a non-stratified fluid and in stratified fluids with different constant density gradients. Turbulence velocity components and density fluctuations were measured using an array of cross-film and conductivity probes. Based on the mesh size of the grid, the nominal values of the (internal) Froude numbers were ∞, 80 and 40, and the corresponding Reynolds number was 4.3 × 104. The decay rates of the (turbulence) kinetic, potential and total energies and the dissipation rates of the kinetic and potential energies were calculated from the experimental data. For each of these quantities, the decay may be represented as a function of the downstream distance raised to a given power. The kinetic energy and its dissipation rate are lower for the stratified cases than for the non-stratified case but are almost compensated for by the corresponding potential energy and its dissipation rate. Our results are consistent with those of direct numerical simulations and agree reasonably well with those obtained in stratified wind and water tunnels. However, the results differ from laboratory results obtained using an optical method to measure the turbulent motion of tracer particles in the wake of a vertically towed grid; these latter results show an abrupt reduction in the decay rate of the turbulence kinetic energy after one Brunt–Väisälä period. A similar trend is also observed in results obtained in facilities with fairly high background turbulence or internal waves. This discrepancy is discussed and an explanation is presented. Furthermore, it is demonstrated that strongly stratified thin sheets with density gradients larger than that of the undisturbed fluid may be generated by local but incomplete mixing. The persistence of such thin sheets is proportional to the Schmidt number (≈ 500) in stratified salt water or the Prandtl number (≈ 0.71) in thermally stratified air.

Free-surface flow over a polygonal and smooth topography

The study of two-dimensional, irrotational, inviscid, incompressible steady state motion generated by a polygonal and a smooth obstruction, is made in terms of the linearized theory. The bottom is represented in integral form using Fourier's double integral theorem. Then following Lamb [1], a linear free-surface profile is obtained for the supercritical and subcritical cases. The results are plotted and discussed for the two cases of the flow for different shapes of the bottom and different values of Froude number,F.

Time-domain solution for wave—current interactions with a two-dimensional body

The effects of a current on the radiation and diffraction of regular waves around a two-dimensional body are examined by a time-domain method. The Froude number is assumed to be small so that the steady wave system associated with the current is insignificant. The boundary-value problem is then separated into a steady current problem with a rigid-wall condition applied at the still water level and an unsteady wave problem with modified free surface boundary conditions accounting for the disturbed current field. The boundary conditions for the unsteady problem are satisfied to first order in wave amplitude by a time-stepping procedure, and the field solution at each time step is obtained by an integral equation based on Green's theorem. For the case of a semi-circular cylinder with axis at the still water level, the wave amplitudes, the first-order oscillatory forces and the second-order steady forces at various values of Froude number are presented, and the effects of the current on the solution are discussed. Comparisons are also made with previous numerical results and good agreement is indicated.