Example: Water sloshes back and forth in a tank as shown in the figure. The frequency of the sloshing, f, is assumed to be a function of the acceleration of gravity, g, the average depth of the water, h, and the length of the tank, l. Develop a suitable set of dimensionless parameter for this problem using g and l as repeating variables.
Given: f = F(g, h, l)
Find: suitable set of dimensionless parameters for the given problem
Step 1: The variables in the problem are:
Step 2: Expressing these variables in basic dimensions:
Step 3: Determining the number of pi terms:
Step 4: We will choose g and l as our repeating variables because the problem statement told us to; however, we could work the problem using g and h as well. Note: we cannot use f at all because it is the dependent variable, and we cannot use both h and l because the combination of repeating variables must contain all the basic dimensions.
Step 5: We will find the first pi term:
Step 6: We will find the second pi term:
Step 7: Since it's easy to make a mistake, we need to check the pi terms:
Step 8: Next, we will write the pi term containing the dependent variable as a function of the remaining pi terms:
Our variables that we deal with in this process can be classified into three broad categories: those dealing with geometry (lengths, angles), material properties (viscosity, density), or external effects (velocities, external forces).
It is important to note that there is not a unique set of pi terms that arises from a dimensional analysis; however, the required number of pi terms is fixed.
Common Non-dimensional Numbers
Reynolds Number - (rho*V*l)/mu = inertial forces/viscous forces; applicable to all types of fluid dynamics problems
Froude Number - V/sqrt(g*l) = inertial forces/gravitational forces; applicable to flow with a free surface
Euler Number - p/(rho*V*V) = pressure forces/inertial forces; applicable to problems where pressure or pressure difference are important
Mach Number - V/c = inertial forces/compressibility forces; applicable to problems where compressibility of a fluid must be considered