Engr Help

Dimensional Analysis (Using the method of repeating variables)

Example: Water sloshes back and forth in a rectangular tank as shown in the figure below.

Let ω denote the frequency of sloshing in cycles per second. Assume that sloshing is a function

of the acceleration of gravity, g, in ft/sec², the average depth of water in the tank, h, in feet,

and the characteristic length of the tank, L, in feet. Determine the set of dimensionless

products characterizing sloshing.

Strategy:

1.	Identify variables governing the problem. (Very important and critical first step)
2.	List dimensions for each variable. Use dimensions: F,L, and T
3.	Number of dimensionless products (Pi terms) = N number of variables - number of dimensions
4.	Select the repeating variables, the dependent variable, and the non-repeating variable.
5.	Use the repeating variables to form dimensionless products of the dependent Variable and the non-repeating variables.

Solution:

Variables	ω (dependent)	h non-repeating	g repeating	L repeating
dimensions	1/T	L	L/T²	L

Number, N, of dimensionless products = 4 variables - 2 dimensions = 2

Select repeating variables as follows:

Use g to eliminate the dimension of time, T, and L to eliminate the dimension of length.

Pi terms are: Pi₁ = h/L and Pi₂ = ω² L/g

The result is: ω² L/g = f(h/L)