Froude number

The dynamics of the air cavity created by vertical water entry of a three-dimensional body is investigated theoretically, computationally and experimentally. The study is focused in the range of relatively low Froude numbers, Fr ≡ V(gD)−1/2 ≤ O(10) (where V is the dropping velocity of the body, D its characteristic dimension and g the gravitational acceleration), when the inertia and gravity effects are comparable. To understand the physical processes involved in the evolution of cavity, we conduct laboratory experiments of water entry of freely dropping spheres. A matched asymptotic theory for the description of the cavity dynamics is developed based on the slender-body theory in the context of potential flow. Direct comparisons with experimental data show that the asymptotic theory properly captures the key physical effects involved in the development of the cavity, and in particular gives a reasonable prediction of the maximum size of the cavity and the time of cavity closure. Du...

This research is to analyses hydraulic parameters of the spillway design for WADI HORAN Dam. The spillway design of type Ogee overflow and the design based on Water Experiment Station (W.E.S) of the U.S. Corp of Engineers. The inflow are routed in order to decrease the maximum discharge passing on the spillway. The maximum discharge passing are 1400 m 3 /sec with maximum head over the crest equal to 3.4 m and 50 m 3 /sec Attenuation and 2 hrs Reservoir Lag. Froude number is determined in order to select the type of stilling basin, the Froude number equal to 5 that can choose type II stilling Basin. The method used for routing is Inflow-Storage Discharge (I.S.D.) The profile of the downstream and upstream are calculated.

In this research, the performance of support vector machine in predicting relative
energy dissipation in non-prismatic channel and rough bed with trapezoidal
elements has been investigated. To achieve the objectives of the present study,
136 series of laboratory data are analyzed under the same laboratory conditions
using a support vector machine. The present study entered the support vector
machine network without dimension in two different scenarios with a height of 1.50
and 3.0 cm rough elements. Two statistical criteria of Root Mean Square Error and
coefficient of determination are used to evaluate the efficiency of input compounds.
Hydraulically, the results show that at both heights of the rough elements, energy
dissipation increased with increasing Froude number. The results of the support
vector machine show that the height of the roughness element is 1.50 cm in the
first scenario, combination number 6 with R2 = 0.990 and RMSE = 0.0129 for
training mode and R2 = 0.993 and RMSE = 0.032 for testing mode and the height
of the roughness element 3.0 in the second scenario, combination number 6 with
R2 = 0.989 and RMSE = 0.0112 for training mode, R2 = 0.994 and RMSE = 0.0224
for testing mode are select as the best models. Finally, sensitivity analysis is
performed on the parameters and H / y1 parameter is selected as the most effective
parameter.

The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth's rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture the lake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibrium in order to improve significantly the accuracy of numerical simulations of perturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results.

This research is to analyses hydraulic parameters of the spillway design for WADI HORAN Dam. The spillway design of type Ogee overflow and the design based on Water Experiment Station (W.E.S) of the U.S. Corp of Engineers. The inflow are routed in order to decrease the maximum discharge passing on the spillway. The maximum discharge passing are 1400 m 3 /sec with maximum head over the crest equal to 3.4 m and 50 m 3 /sec Attenuation and 2 hrs Reservoir Lag. Froude number is determined in order to select the type of stilling basin, the Froude number equal to 5 that can choose type II stilling Basin. The method used for routing is Inflow-Storage Discharge (I.S.D.) The profile of the downstream and upstream are calculated.

The literature on the structure and behaviour of gravity currents is reviewed, with emphasis on some recent studies, and with particular attention to turbidity currents, though reference is also made to comparable behaviour in pyroclastic flows. Questions of definition are discussed, in particular the distinction between dense currents, which may deposit en masse, and more dilute currents. High-density dispersions may exist as a discrete, independently moving layer beneath a more dilute flow, as the basal part of a continuous density distribution or possibly as a transient depositional layer. Existing theory appears inadequate to explain the behaviour of some high-density dispersions. Surge-type currents are contrasted with quasi-steady currents, which may be generated by a variety of mechanisms including direct feed by rivers in flood. Such fluvially generated currents provide one means of generating currents with reversing buoyancy. Geologically significant turbidity currents are impractical for direct study owing to their large scale and (often) destructive nature. Small-scale laboratory currents offer a wealth of insights into turbidity current behaviour. This paper summarizes recent experimental studies that focus on the physical structure of gravity currents, with emphasis on the velocity and turbulence structure, the vertical density distribution and the stability of stratification. Preliminary quantification of the turbulence structure (including controls on turbulent entrainment, turbulent kinetic energy, Reynolds stresses and turbulence production) has been facilitated by recent technological developments that have allowed the measurement of instantaneous fluctuations in both velocity and concentration. Laboratory models, however, generally involve substantial simplification, and require compromises in some parameters to achieve adequate scaling of the parameters of most interest. Mathematical modelling also provides important insights into turbidity current behaviour. We discuss various approaches to modelling, ranging from simple hydraulic equations to systems of partial differential equations that explicitly treat conservation of momentum, fluid and sediment mass, and turbulent kinetic energy. The application for which the model is designed (i.e. to calculate mean head velocity or to create an instantaneous two-dimensional contour plot of downstream velocity in a current) determines the complexity of the mathematical model required. The behaviour of suspension currents around topography is complex and depends upon the relative height of the topography, and upon the density and velocity structure of the current. Many interactions with topography are well described by the internal Froude number, Fri. Both reflection and deflection of currents may occur on the upstream side of topography, depending upon Fri. On the downstream side of topography, flow separation, lee waves or hydraulic jumps may occur.

The effect of free surface height on the submerged synthetic jet parallel to the surface in quiescent flow is studied experimentally. The purpose of the study is to understand the vortex ring structure of synthetic jet and their effects on the free surface. Flow visualization using Laser Induced Fluorescence (LIF) and measurement of time-averaged streamwise and crossstream velocities carried out using Laser Doppler Velocimetry (LDV) are reported. Due to change in free surface height (H), vortex ring drifts from centerline and surface perturbations are observed. Analysis of the velocity field reveals that the velocity normal to the surface reduces while the velocity parallel to the surface increases as the vortex ring approaches the free surface. Previous studies also suggest similar phenomenon for submerged continuous jets.

In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches ...

In this paper, the effect of quarter cylinder obstacles on hydraulic jump in a rectangular channel has been studied in the laboratory. The quarter-cylindrical obstacles with 2 and 3 cm diameter are considered and the effect of their position and height are studied in the hydraulic jump parameters such as location, energy dissipation and Froude number. The flume used in this research has six meter long, 20 cm wide and 30 cm height. The reason of quarter-cylindrical shape selection for obstacles is reduce the amount of erosion relative to other shapes such as rectangular and trapezoidal obstacles. The obstacles are installed at the end of the channel. Distance between the first obstacle and the beginning of the channel is constant. Totally, 44 tests have been carried out in this research. The results show that increasing distance between the obstacles to a certain amount, increases energy dissipation. But energy dissipation decreases for further distance. Furthermore, optimum s/r incr...

The results of model testing of arch and straight deck bridge constrictions are presented. All tests were carried out in a compound flume that consists of a main channel and two symmetric floodplains set at a fixed bed slope. A simple generalized afflux equation is also proposed. The equation which describes the model characterizes the afflux as a function of Froude number, and blockage ratio in terms of the downstream conditions

We use shallow-water theory to study the self-similar gravity currents that describe the intrusion of a heavy fluid below a lighter ambient fluid. We consider in detail the case of currents with planar symmetry produced by a source with variable inflow, such that the volume of the intruding fluid varies in time according to a power law of the type tα. The resistance of the ambient fluid is taken into account by a boundary condition of the von Kármán type, that depends on a parameter β that is a function of the density ratio of the fluids. The flow is characterized by β, α, and the Froude number [Fscr ]0 near the source. We find four kinds of self-similar solutions: subcritical continuous solutions (Type I), continuous solutions with a supercritical-subcritical transition (Type II), discontinuous solutions (Type III) that have a hydraulic jump, and discontinuous solutions having hydraulic jumps and a subcritical-supercritical transition (Type IV). The current is always subcritical ne...

Dense overflows entrain surrounding waters at specific locations, for example, sills and constrictions, but also along the descent over the continental slope. The amount of entrainment dictates the final properties of these overflows, and thus is of fundamental importance to the understanding of the formation of deep water masses. Even when resolving the overflows, coarse resolution global circulation and climate models cannot resolve the entrainment processes that are often parameterized. A new empirical parameterization is suggested, obtained using an oceanic and laboratory dataset, which includes two novel aspects. First, the parameterization depends on both the Froude number (Fr) and Reynolds number of the flow. Second, it takes into account subcritical (Fr < 1) entrainment. A weak, but nonzero, entrainment can change the final density and, consequently, the depth and location of important water masses in the open ocean. This is especially true when the dense current follows ...

The authors explore the theoretical and empirical relationship between the nonlocal quantities of the entrainment ratio E, the appropriately depth- and time-averaged flux coefficient Γ, and the bulk Froude number Fro in density currents. The main theoretical result is that E = 0.125 Γ Fro2(CU3/CL)/cosθ, where θ is the angle of the slope over which the density current flows, CL is the ratio the turbulent length scale to the depth of the density current, and CU is the ratio of the turbulent velocity scale to the mean velocity of the density current. In the case of high bulk Froude numbers Γ ∼ Fro−2 and (CU3/CL) = Cϵ ∼ 1, so E ∼ 0.1, consistent with observations of a constant entrainment ratio in unstratified jets and weakly stratified plumes. For bulk Froude numbers close to one, Γ is constant and has a value in the range of 0.1–0.3, which means that E ∼ Fro2, again in agreement with observations and previous experiments. For bulk Froude numbers less than one, Γ decreases rapidly with...