Plane Bottom Research Articles

Refraction of cnoidal waves

Depth refraction of a first-order cnoidal surface gravity wave in the shallow water zone is calculated for a quasi two-dimensional situation, i.e., a gently sloping bathymetry characterized by straight and parallel sea bed contours (not necessarily a plane bottom), and no phase-independent or phase-averaged wave parameters vary with distance along-shore. The wave system has no energy exchange with the wind or the sea bed. The basic assumption is, that the energy flux is constant between adjacent wave orthogonals. Two non-linear algebraic equations are derived, which govern the refraction of a cnoidal wave train for the quasi two-dimensional situation. The equations are solved numerically. For a deep water angle of incidence α 0 = 60° are angles, wave heights and values of the Ursell parameter versus a dimensionless depth given in figures for all practical ranges of the deep water wave steepness. The refraction solution is compared with the linear solution (Airy theory). The solution is also compared with a solution from another non-linear wave theory, viz. Dean's stream function wave theory. A detailed numerical cnoidal refraction solution is tabulated as a function of the depth.

The effect of polymer additives on the components of the energy balance of a turbulent flow

On the basis of the experimental data obtained, an analysis is made of the effect of polymer additives directly on the generation of turbulent energy, on the dissipation of the energy of the averaged motion, and on the density of the flux of the kinetic energy of the turbulence. The presence of polymer additives in the turbulent flow significantly changes the relationship between the generation of the turbulent energy and the dissipation of the energy of the averaged motion. Under the action of polymer additives, the density of the flux of kinetic energy decreases over the depth of the channel, which, in turn, brings about a decrease in the influx of energy from the averaged motion to the pulsed motion. The following definitions are adopted below: the x1 axis of a Cartesian system of coordinates coincides with the horizontal axis of symmetry of the channel and with the direction of the averaged motion of the liquid; the x2 axis is directed upward; the x3 axis is perpendicular to the lateral wall of the channel; the origin of coordinates, O, coincides with the lower plane (bottom) of the channel. Further, U0 is the mean velocity of the flow of liquid in the channel; Ui is the local component of the averaged velocity (i=1, 2, 3); H is the height of the channel; z=2x2/H; Re is the Reynolds number of the averaged flow;v is the coefficient of kinematic viscosity; ui is the pulsation component of the velocity (i=1, 2, 3); u* is rate of dynamic friction; A=(λ0−λpλ0 −1 is the coefficient of the lowering of the friction resistance with the flow of polymer solutions; λ0 and λp are the coefficients of the friction resistance with the motion of water and polymer solutions in a channel, respectively; c is the weight concentration of the polymer solution (%);τ 0 is the friction stress at the wall; U+ is the velocity of the flow at the axis of the channel.

Tidal Inlets Housekeeping

A tidal inlet on an alluvial shore presents a littoral barrier which must be overcome by flushing and bypassing of material. Presumably, the tidal inlet arranges itself in the most practical way by providing the highest flushing ability at the least loss of energy to bottom friction. This, however, is not the only condition for stability of a tidal inlet. There must be enough tidal flow available to accomplish the task of transporting all the littoral material which entered the inlet. This requires a tidal prism and discharge of a certain size compared to the quantity of littoral drift which is carried to the inlet. Part of this material may be transferred across the inlet entrance on a bar or shoal. In the inlet channel, a bottom shear stress which corresponds to a velocity of about 1 m per sec results. At this velocity, sand bottom with 0.15 mm to 0.3 mm sand changes its geometry by flattening the dunes approaching plane bottom. Both tangential and normal forces contribute to the drag of a duned-bed stream. Because the tangential forces contribute directly to sediment transport, the drag force of the bed on the stream is more effectively used to transport sediment with flatter dunes and plane bottom.