Free Surface Height Research Articles

Planing of a flat plate at high Froude number

The planing of a plate of width L with velocity U at high Froude number [U / (gL)1/2≫ 1] on the surface of an incompressible fluid is considered. The solution of this problem by Green [A. E. Green, Proc. Cambridge Phil. Soc. 32, 67 (1936)] which omits gravity (g = 0), is inadequate because it yields a one-parameter family of solutions rather than a single solution. In addition, it yields a free surface height which tends to — ∞ far from the plate. These two defects are removed by treating Green's solution as the leading term in an inner expansion of the flow. An appropriate outer expansion is constructed by using the linear theory of surface waves with a suitable pressure distribution moving along the free surface. Matching of the two solutions determines the free parameter in the inner expansion, and yields the correct behavior of the free surface far from the plate. From the solution, graphs of the lift coefficient of the plate are obtained. An important feature of the solution is the inclusion of the effect of the impact on the free surface of the jet thrown up by the plate.

Wind-Driven Model for Evaporite Deposition in a Layered Sea

Deposition of evaporites in a salinity- and density-layered sea is examined by theoretically predicting the large-scale circulation in a simplified model sea. In the constant depth model sea, vertical stratification is represented by two homogeneous layers; the upper layer is normal or slightly concentrated sea water, and the considerably denser lower layer is saturated with respect to halite. Each layer has a constant density and volume for all time. The horizontal transport form of the time-dependent equations of motion and continuity for each layer are solved numerically by a finite-difference method. A forward time extrapolation of four transport components, upper layer depth, and free surface height is performed on a square lattice covering the model basin. Interface stresses acting between the two layers and bottom stresses acting between the lower layer and the bottom allow consideration of vertical turbulent friction. Forcing is by a constant and uniform wind stress acting on the free surface. Rotation and hydrostatic pressure gradient terms are also included in the equations of motion. Several 140-hr (5.8 days) calculations were made for the Upper Silurian evaporites in Michigan Basin, using reasonable values of physical parameters. For an initially static sea 120 m deep with a 5-m-deep upper layer, mild easterly paleowinds cause upwelling of the lower layer at the north boundary. As the wind continues to blow, the region of upwelling slowly increases in area and migrates around the margin to the east boundary. In an evaporite sea, salt would precipitate in the exposed lower-layer brine because of evaporation. The salt could precipitate at the free surface as hopper crystals, internally, or at the bottom. By assuming a modest evaporation rate of 10 cm per yr, model calculations show that the total Upper Silurian salt volume is easily accounted for by an average of fewer than two 140-hr wind events per year. Model predicted upwelling at the north and east boundaries indicates that hopper crystals should be concentrated around the northeast margin of Michigan Basin. Other regional sedimentological and chemical variations related to circulation patterns, such as lithology and reef type, are suggested by the model calculations.

ON THE BEHAVIOR OF A CAPILLARY SURFACE IN A WEDGE

Estimates above and below are obtained for the height of the equilibrium-free surface of a liquid when the liquid partially fills a cylindrical container whose cross section contains a corner with interior angle 2alpha. The surface is characterized by the condition that its mean curvature be proportional to its height above a reference plane (or, in the case of zero gravity, that the mean curvature be constant), and by the requirement that it meet the container wall with prescribed contact angle gamma. It turns out that the qualitative behavior of such a surface near the vertex changes markedly, according as alpha + gamma < (1/2)pi, or alpha + gamma >/= (1/2)pi. In the former case, the surface is either unbounded or fails to exist, while in the latter case every such surface is bounded. Some experimental comparisons are indicated, and an application to the problem of describing the mechanism of water rise in trees is discussed.The above results describe a limiting case among corresponding properties that hold for surfaces defined over domains with smooth boundaries. This extension is indicated, as well as a formal extension to n-dimensional surfaces; here the interest centers on the fact that it is the mean curvature of an (n-1)-dimensional boundary element that controls the local behavior of the n-dimensional solution surface.

Liquid sloshing due to a time-dependent discontinuous boundary

The problem of sloshing of a liquid due to a time-dependent discontinuous boundary is studied in detail for axisymmetric motion. Solution of the transient problem for small motions is given when the velocity at the boundary fluctuates with a known arbitrary function of time. The complete solution for the initial-value problem is obtained for a sinusoidal time variation of the velocity. Numerical examples are worked out to evaluate the velocity and pressure distributions on the bottom of the tank and the free-surface wave heights.