1. ES 202 Fluid and Thermal Systems Lab 1: Dimensional Analysis

2. Road Map of Lab 1 Announcements Guidelines on write-up
Fundamental of dimensional analysis
difference between “dimension” and “unit”
primary (fundamental) versus secondary (derived)
functional dependency of data
Buckingham Pi Theorem
alternative way of data representation (reduction)
active learning exercises

3. Announcements Lab 2 will be at Olin 110 (4th week)
Lab 3 will be at DL 205 (8th week)
You are not required to hand in the in-class lab exercises.

4. About the Write-Up
Raw data sheet and write-up format for Lab 1 can be downloaded at
Due by 5 pm one week after the lab at my office (O-219)

5. Dimension Versus Unit Dimensions (units) Length (m, ft)
Mass (kg, lbm) MLT system
Time (sec, minute, hour)
Force (N, lbf) FLT system
Temperature (deg C, deg F, K, R)
Current (Ampere)

6. Primary Versus Secondary
In the MLT system, the dimension of Force is derived from Newton’s law of motion.
In the FLT system, the dimension of Mass is derived likewise.
Quantities like Pressure and Charge can be derived based on their respective definitions.
Do exercises on Page 1 of Lab 1

7. Dimensional Homogeneity
The dimension on both sides of any physically meaningful equation must be the same.
Do exercises on Page 2 of Lab 1

8. Data Representation Given a functional dependency
y = f (x1, x2, x3, …………..., xk )
where y is the dependent variable while all the xi’s are the independent ones. Both y and the xi’s can be dimensional or dimensionless.
One way to express the functional dependency is to view the above relation as an n-dimensional problem: to plot the dependency of y against any one of the xi’s while keeping the remaining ones fixed.

9. Buckingham Pi Theorem If an equation involving k variables is
dimensionally homogeneous, it can be
reduced to a relationship among k - r
independent products (P groups),
where r is the minimum number of
reference dimensions required to
describe the variables.

10. Alternative Way of Data Representation
It is advantageous to view the same functional dependency in a smaller dimensional space
Cast
y = f (x1, x2, x3, …………..., xk-1 )
into
P 1 = g (P 2, P 3, P 4, …………..., P k-r )
where P i’s are non-dimensional groups formed by combining y and the xi’s, and r is the number of reference dimensions building the xi’s

11. What is the Procedure?
Come up with the list of dependent and independent variables (the least trivial part in my opinion)
Identify the number of reference dimensions represented by this set of variables which gives the value of r
Choose a set of r repeating variables (these r repeating variables should span all the reference dimensions in the problem)
All the remaining k - r variables are automatically the non-repeating variables

12. Continuation of Procedure
Form each P group by forming product of one of the non-repeating variables and all the repeating variables raised to some unknown powers. For example,
P = y x1a x2b x3c
By invoking dimensional homogeneity on both sides of the equation, the values of the unknown exponents can be found
Repeat the P group formulation for each of the non-repeating variables

13. Properties of P Groups
The P groups are not unique (depend on your choice of repeating variables)
Any combinations of P groups can generate another P group
The simpler P groups are the preferred choices

14. Motivational Exercise
Drag on a tennis ball
work out the whole problem
what if it is not spherical, say oval?
what if it is not placed parallel to flow direction but at an angle?

15. Any Advantages?? Absolutely “YES”
You may reduce a thick pile of graphs to a single xy-plot
For examples:
4 variables in 3 dimensions can be reduced to 1 P group which is equal to a constant (dimensionless)
5 variables in 3 dimensions can be reduced to 2 P groups taking the general form
P 1 = f (P 2 )

16. Drag Coefficient for a Sphere
taken from Figure 8.2 in Fluid Mechanics by Kundu

17. More Exercises
Sliding block
Pendulum

18. What is the Key Point?
There are more than one way to view the same physical problem.
Some ways are more economical than others
The reduction of dimensions from the physical dimensional variables to non-dimensional P groups is significant!

19. Reflection on the Procedures
The most important step is to come up with the list of independent variables (Buckingham cannot help in this step!)
Once the dependent and independent variables are determined (based on a combination of judgment, intuition and experience), the rest is just routine, i.e. finding the P groups!
However, Buckingham cannot give you the exact form of the functional dependency. It has to come from experiments, models or simulations.

20. Complete Similarity Model versus prototype (full scale)
Geometric similarity
Kinematic similarity
Dynamic similarity

21. The Dimensionless world is simpler!!
Central Theme
The Dimensionless world is simpler!!

23. More Examples Sliding block Pendulum Nuclear bomb
Terminal velocity of a falling object
Pressure drop along a pipe