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The extended Froude number differs substantially from the classical Froude number for higher surface elevations. The term βh emerges from the change of the geometry of the moving mass along the slope. Dimensional analysis suggests that for shallow flows βh ≪ 1, while u and s g (x d − x) are both of order unity.
Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. [1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed.
This technique can ease the analysis of the problem at hand, and reduce the number of free parameters. Small or large sizes of certain dimensionless parameters indicate the importance of certain terms in the equations for the studied flow. This may provide possibilities to neglect terms in (certain areas of) the considered flow.
Similitude has been well documented for a large number of engineering problems and is the basis of many textbook formulas and dimensionless quantities. These formulas and quantities are easy to use without having to repeat the laborious task of dimensional analysis and formula derivation.
To help visualize the relationship of the upstream Froude number and the flow depth downstream of the hydraulic jump, it is helpful to plot y 2 /y 1 versus the upstream Froude Number, Fr 1. (Figure 8) The value of y 2 /y 1 is a ratio of depths that represent a dimensionless jump height; for example, if y 2 /y 1 = 2, then the jump doubles the ...
The first person to attempt an analytical description of antidunes was Walter B. Langbein in 1942. He applied dimensional analysis to Gilberts' results and came up with transition points using Froude numbers versus velocity and hydraulic radius. [4]
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as x n, while the surface area, being (n − 1)-dimensional, scales as x n−1.
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