Tuesday, November 29, 2011

MrsDrPoe: Dimensional Analysis, Part III

Welcome to the final Thesis Tuesday on Dimensional Analysis.  As promised last week, we're going to go through an example problem today so you can see the Buckingham Pi Theorem in action.

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: 
frequency, f
tank length, l
gravity, g
water height, h

Step 2: Expressing these variables in basic dimensions:
f = 1/T
l = L
g = L/(T*T)
h = L

Step 3: Determining the number of pi terms:
k = 4 (f, g, l, h) and r = 2 (L, T) so k-r = 2

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:
Pi1 = f*g^(a1)*l^(b1)
(1/T)*(L/T*T)^(a1)*L^(b1) = T^(0)*L^(0)
for L to cancel: a1 + b1 = 0
for T to cancel: -1 -2*a1 = 0
So: a1 = -1/2 and b1 = 1/2
Pi1 = f*sqrt(l/g)

Step 6: We will find the second pi term:
Pi2 = h*g^(a2)*l^(b2)
(L)*(L/T*T)^(a2)*L^(b2) = T^(0)*L^(0)
for L to cancel: 1 + a2 + b2 = 0
for T to cancel: -2*a2 = 0
So: a2 = 0 and b2 =-1
Pi2 =h/l

Step 7: Since it's easy to make a mistake, we need to check the pi terms:
Pi1 = f*sqrt(l/g) = (1/T)*sqrt((L*T*T)/L) = 1
Pi2 = h/l = L/L = 1

Step 8: Next, we will write the pi term containing the dependent variable as a function of the remaining pi terms:
Pi1 = F(Pi2) or f*sqrt(l/g) = F(h/l)

Not too bad, huh?

Additional Considerations

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


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