DIMENSIONAL ANALYSIS
AND MODELING
Chapter 7
Professor K.T. Cho
Method of Repeating Variables
Method of Repeating Variables
Step 1: List the parameters in the problem and count their total number, n.
Step 2: List the primary dimensions of each of the n parameters.
Step 3: Set the reduction, j. As a first guess, set j as the number of primary
dimensions. If this is wrong, reduce j by 1 and try again. Calculate k, the
expected number of s, k = n – j.
Step 4: Choose j repeating parameters.
Step 5: Construct the k s (may need to reset j), and manipulate as necessary.
Step 6: Write the final functional relationship and check your algebra.
Example problem
Example problem
Repeating parameters?
Here are some guidelines:
1. Never pick dependent variable (one in the left in the functional relation)
2. The repeating variables cannot by themselves form a dimensionless group.
3. All the primary dimensions must be represented by the j repeating parameters.
4. Do not pick variables that are already dimensionless (e.g., angles).
5. Do not pick two variables with the same dimensions.
6. Pick “common” variables, since the repeating variables end up appearing in
more than one . (That’s why we call them “repeating variables” in the first place!)
7. Whenever possible, choose simple variables (e.g., pick {L} instead of {mL2t/T} if
appropriate.)
Method of Repeating Variables
Summary
Examples for selecting repeating parameters
Analysis of a ball falling in a vacuum
If primary dimension is L,t,
j is 2. Repeating variables?
Lift on wing
𝐹𝐿 = 𝑓(𝑉, 𝐿𝑐 , 𝜌, 𝜇, 𝑐, 𝛼)
If primary dimension is m,L,t,
j is 3. Repeating variables?
Examples for selecting repeating parameters
Couette flow
Pipe flow
𝑢 = 𝑓(𝜇, 𝑉, ℎ, 𝜌, 𝑔, 𝑦)
If primary dimension is m,L,t
j is 3. Repeating variables?
Δ𝑃 = 𝑓(𝑉, 𝜀, 𝜌, 𝜇, 𝐷, 𝐿)
If primary dimension is m,L,t
j is 3. Repeating variables?
Flat-plate flow
𝛿 = 𝑓(𝑥, 𝑉, 𝜌, 𝜇)
If primary dimension is m,L,t
j is 3. Repeating variables?
Example Problem
Example Problem
Example Problem
Although the Darcy friction
factor for pipe flows is most
common, you should be
aware of an alternative, less
common friction factor called
the Fanning friction factor.
The relationship between the
two is f = 4Cf .