Maximum Froude Number Research Articles

Analysis of Linear and Monoclinal River Wave Solutions

Linear dynamic wave and diffusion wave analytical solutions are obtained for a small, abrupt flow increase from an initial to a higher steady flow. Equations for the celerities of points along the wave profiles are developed from the solutions and are related to the kinematic wave and dynamic wave celerities. The linear solutions are compared systematically in a series of case studies to evaluate the differences caused by inertia. These comparisons use the celerities of selected profile points, the paths of these points on the x-t plane, and complete profiles at selected times, and indicate general agreement between the solutions. Initial diffusion wave inaccuracies persist over relatively short time and distance scales that increase with the wave diffusion coefficient and Froude number. The nonlinear monoclinal wave solution parallels that of the linear dynamic wave, but is applicable to arbitrarily large flow increases. As wave amplitude increases, the monoclinal rating curve diverges from that for a linear wave, and the maximum Froude number and energy gradient along the profile increase and move toward the leading edge. A monoclinal-diffusion solution for the diffusion wave equations is developed and dynamic wave-diffusion wave comparisons are made over a range of amplitudes with the same case studies used for linear waves. General dimensionless monoclinal-diffusion profiles exist for each depth ratio across the wave, whereas corresponding monoclinal wave profiles exhibit minor, case-specific Froude number dependence. Inertial effects on the monoclinal profiles occur near the leading edge, increase with the wave amplitude and Froude number, and are responsible for the differences between the dimensionless profiles.

Bed-flow instability of the longshore current

An initially uniform longshore current on a plane erodible beach is considered and a linear stability analysis of the bed-flow system is performed in order to investigate the growth of alongshore periodic topographic features such as transverse or oblique bars. γ, numerical model based on the shallow water equations and a simple sediment transport formula is used. For a wide range of parameters instability is found, leading to the growth of large-scale topographic features (lengthscale of the order of the current width) downflow progressing. The growth rates and the dominant unstable mode depend mainly on R = c d/β parameter, where c d is the bottom friction coefficient and β is the beach slope. For a small R, say less than 0.1, instability is very weak, probably negligible. For R between 0.1 and 0.7 instability increases with R, leading typically to a quite simple transverse bars pattern. A further increase in R produces a far more complicated behaviour where complex patterns with downcurrent oriented oblique bars, bumps and holes can be dominant. In this region growth rates may either decrease or increase with R depending on the beach slope and the maximum Froude number of the basic flow, F. Usually, the most complex behaviour is found for gently sloping beaches. The physical mechanism of the instability is found to lie on the disturbances of potential vorticity caused by topographically induced differences in bottom friction. In this sense it is similar to the alternate bars growth in a river rather than the dunes or antidunes occurrence for 1D channel flow. The predictions of the model compare well with the available experimental data. The alongshore wavelength, γ, typically of the order of one to four times the width of the current, is close to four times for the most common values of R. The typical growth time is proportional to γ 2 and for a wavelength of 100 m can be of the order of one day, depending on the sediment transport rate. The migrational speed is inversely proportional to γ, in accordance to earlier field data reported by Sonu (1969) Collective movement of sediment in littoral environment.

Numerical simulation of vorticity waves in the nearshore

A numerical method based on spectral expansions is given for the computation of vorticity waves arising from shear instability of a longshore current. This method allows for any mean flow profile and any beach topography (remaining constant alongshore and with a straight shoreline). The shallow‐water equations are considered without any assumption about the sea surface (such as rigid lid), and dissipative terms accounting for bottom friction and/or eddy viscosity are included. A numerical simulation for some flow profiles that are quite realistic in the surf zone and for several bathymetries is presented. For inviscid flow the predictions of the Bowen and Holman (1989) analytical model for a very simplified geometry are found to give rise to the main features. However, the details in the flow and depth profiles are found to significantly influence the instability curves, especially for a barred beach. For the fastest growing mode, the wavelength is between 1.7 and 2.7 times the width of the mean current l. Frequencies of about 0.09ƒs, where ƒs is the maximum shear at the sea face of the current profile, and an e‐folding time of the exponential growth that is roughly equal to the wave period are obtained. The phase speed is between 0.5 and 0.7 of the mean current peak. Dissipation has a considerable effect on the wavenumber span and the growth rate of the instability, so reasonably constant values of the eddy viscosity and realistic values of the Chezy coefficient can entirely remove the instability. The phase speed of neutral shear waves is analytically found to be equal to the mean flow velocity at the cross‐shore location where the potential vorticity has an extremum. This velocity is found to give an estimate of the phase speed of growing modes. We found that the rigid‐lid assumption tends to overestimate the growth rates by an amount which depends on the maximum Froude number of the mean flow. The instability curves and the dispersion lines for a free surface converge towards the rigid‐lid ones when the Froude number decreases, and the rigid‐lid assumption is therefore valid for a low Froude number.

Laboratory Experiment on Flow Characteristics at a Beach Step

ABSTRACT Flow structure and sediment movement over the beach step, commonly present at the foot of the beach face, were studied in a laboratory wave tank. Two sets of experiments were made: a preliminary run using 2-s waves and a colored mixture of fine, medium, and coarse sand to observe details of sediment movement over the step and the beach face, and a set of runs with periods of 1.5 s, 2.0 s, and 2.5 s and coarse sand, in which the flow field was visualized and measured using small, neutrally buoyant suspended particles. By the flow characteristics over the step, six distinct phases of the swash cycle were identified: (1) uprush of incident wave, (2) flow reversal, (3) backwash, (4) occurrence of critical flow, (5) development of a hydraulic jump, and (6) development of a backwash vortex. The time history of local flow velocity at the top of the step during a swash cycle has a saw-tooth shape, with maximum onshore and offshore speeds of the same magnitude. The velocity on the beach face was quite uniform through the water column during most of the swash cycle. The maximum Froude number over the step during the backwash was about 1.4. A backwash vortex formed after the onset of flow separation at the beach step during the final phase of the swash cycle. The backwash vortex traps suspended fine sediment at the step, and the flow up the step slope induced by the vortex impedes avalanching of coarse sediment down the slope, facilitating high-angle sedimentation. Once caught by the vortex, most of the coarse sediment is eventually deposited either on the step or on the lower part of the beach face, and only a small amount may be deposited farther up on the beach face. The dip of the seaward-dipping stratification in the step (25-32°) is somewhat higher than in the field (about 20°), probably because of the presence of irregular waves in the field together with tidal fluctuation in water level, two factors not included in the experiments.

小口径開口部を通るヘリウム・空気置換流の流量測定と流れの可視化

This paper deals with an experimental investigation on buoyancy-driven exchange flows through horizontal and inclined openings. The method of the mass increment was developed to measure the flow rate in helium-air system and a displacement fringe technique was adopted in Mach-Zehnder interferometer to visualize the flow. As the result, the followings were obtained : Flow visualization results indicate that the upward and downward plumes of helium and air break through the opening intermittently, and they swing in the lateral direction through the horizontal opening. It is clearly visualized that the exchange flows through the inclined openings take place smoothly and stably in the separated passages. The inclination angle for the maximum Froude number decreases with increasing length-to-diameter ratio in the helium-air system, on the contrary to Mercer's experimental results in the water-brine system indicating that the angle remains almost constant.

Limit Slope in Uniform Flow Computations

A method to prepare design charts for uniform flow in open channels is presented by introducing the concept of limit slope which is the smallest critical slope for a given channel shape and roughness. Equations are obtained for limit slope in rectangular, parabolic, and circular channels. The condition for limit slope in trapezoidal channels is given and it is shown that no limit slope exists for z ≥ 0.5. The design chart correlating the Froude number, discharge, depth of flow and bed slope is presented for rectangular, parabolic, and trapezoidal channels. It is shown that for a given bed slope, width and roughness, the maximum Froude number in rectangular channels always occurs at a depth equal to 1/6 the width and a simple expression is given for its value.

Waves on the erodible bed of an open channel

In the first part of this work the stability of the erodible bed of a stream with a free surface is studied within the framework of classical hydraulics, in which the velocity variation with depth is reduced to a single mean velocity and the bed friction is related in a general way to the local depth and mean velocity. Only two-dimensional motions can be studied in this way. In considering bed friction and a difference in phase between deposition and the mean velocity gradient along the channel, this work combines aspects of earlier studies of Exner and Kennedy.In the absence of a phase difference between erosion and mean velocity, the analysis proceeds without linearization. When the development from an equili- brium flow down a uniform slope is considered, the bed waves formed under subcritical flows are found to move downstream, while those under super-critical flows upstream; in both régimes bed waves are damped. In each case the side of a wave facing in the direction of motion is the steeper.When a phase shift is introduced as well, the analysis is carried forward only after linearization. The primary effect of bed slope and friction is a shift in the ranges of phase angle for which growth can take place and a corresponding alteration in the wavelengths for maximum growth. Friction also reduces bed-wave celerity. Consideration is given to the physical processes represented by the artifice of a phase difference between erosion and mean velocity gradient.The second section of this investigation concerns two-dimensional potential flow over a wavy stream bed. This problem is considered from a point of view different from that adopted previously by Kennedy; a modified criterion is proposed for the maximum Froude number at which bed waves will form. It is in better agreement with measured data.In the third part of this paper, the potential analysis is extended to include class of three-dimensional motions. The conditions are found for the formation of dunes (bed waves 180° out of phase with the surface waves above) and anti-dunes (the two in phase). The criterion separating dunes and antidunes for two dimensions is found to give a lower limit on the Froude number for antidunes of the more general three-dimensional class. In the antidune regime the streamwise perturbation to the velocity can change sign between surface and stream bed; the limit for this is determined.The erosion equation relating changes in bed level to local stream speed is generalized to include sediment convection in two dimensions. With this equation, three distinct regimes of bed-wave motion are found: at low Froude numbers, dunes moving downstream; at higher Froude numbers, antidunes moving upstream; and, finally, antidunes moving downstream. The criterion separating the last two r6gimes is just that mentioned above as the limit of flows whose velocity perturbation has the same sign from surface to stream bed. It is argued that this fundamental change in the flow marks also the end of the region of growth of small bed waves. It is found that three-dimensional dunes and upstream-moving antidunes can exist beyond the two-dimensional limit, but that the latter applies for all bed waves such that the ratio of stream depth to wavelength (d/λ) is small. This explains why the modified criterion for two dimensions provides for small d/λ an envelope for data obtained from observations of a wide variety of bed forms, but fails to do so for larger d/λ. The celerity of bed waves beneath three-dimensional flows is discussed. It is suggested that a class of waves occurring in natural streams will move more slowly than would two-dimensional waves in similar conditions.The work concludes with a comparison of several methods of modelling erosive flows.