Unit 5: Dimensional Analysis
How does one build a small-scale wind tunnel experiment to determine if
an airplane design on the large scale is practical? Should one use the
same air speeds for the small plane as for the large plane?
One could pose many similar questions about how situations change with
size or other scale.
A related question might be: “How can we combine measured quantities
(e.g. velocity, length, time) to completely characterize a situation?”
A simple example which you will learn about in detail is
fully-developed flow of an incompressible, Newtonian fluid in a pipe.
For this case, we need only specify the Reynolds number (fluid density x
fluid velocity x pipe diameter / fluid viscosity) to fully specify the
flow conditions.
Not only is dimensional analysis useful for designing and analyzing
experiments, it also provides a convenient ways of simplifying the
governing equations and hence simplifies the solution and application of
those equations.
In this unit, you will learn the fundamentals of dimensional analysis.
Unit 5 Time Advisory
This unit will take you approximately 17 hours to complete.
☐ Subunit 5.1: 4 hours
☐ Subunit 5.2: 4 hours
☐ Subunit 5.3: 3 hours
☐ Subunit 5.4: 3 hours
☐ Subunit 5.5: 3 hours
Unit5 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Use the Buckingham Pi Theorem to derive dimensionless numbers for a
given fluid flow problem.
- Define basic dimensionless numbers of fluid flows (e.g. Reynolds
number and Weber number).
- Perform dimensional analysis.
5.1 Introduction to Dimensional Analysis
- Reading: University of Kentucky: Professor J. M. McDonough’s
Lectures Notes on Introduction to Fluid Mechanics: “Lectures on
Elementary Fluid Dynamics”
Link: University of Kentucky: Professor J. M. McDonough’s Lectures
Notes on Introduction to Fluid Mechanics: “Lectures on Elementary
Fluid
Dynamics” (PDF)
Instructions: Please download the PDF file for Lecture Notes of
ME330: Elementary Fluid Dynamics and read pages 83-91. This
reading will introduce you to geometric and dynamic similarities and
scaling of governing equations.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
- Reading: University of Iowa: Professor Fred Stern’s Lectures Notes
on Fluid Mechanics: “Chapter 7: Dimensional Analysis Modeling”
Link: University of Iowa: Professor Fred Stern’s Lectures Notes on
Fluid Mechanics: “Chapter 7: Dimensional Analysis and
Modeling” (PDF)
Instructions: Please click on the hyperlink “Chapter 7” in the “Lecture Notes” section on the left side of the webpage to download the PDF file for Chapter 7. Read page 1 for a brief introduction on the need for dimensional analysis. You may want to save this PDF file as you will review other pages in this chapter throughout this unit.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
5.2 The Buckingham Pi Theorem
- Reading: University of Kentucky: Professor J. M. McDonough’s
Lectures Notes on Introduction to Fluid Mechanics: “Lectures on
Elementary Fluid Dynamics”
Link: University of Kentucky: Professor J. M. McDonough’s Lectures
Notes on Introduction to Fluid Mechanics: “Lectures on Elementary
Fluid
Dynamics” (PDF)
Instructions: Please download the PDF file for Lecture Notes of
ME330: Elementary Fluid Dynamics and read pages 91-96. Using the
Buckingham Pi Theorem introduced in this reading, you will able to
find dimensionless physical parameters without use of governing
equations.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
- Reading: University of Iowa: Professor Fred Stern’s Lectures Notes
on Fluid Mechanics: “Chapter 7: Dimensional Analysis and Modeling”
Link: University of Iowa: Professor Fred Stern’s Lectures Notes on
Fluid Mechanics: “Chapter 7: Dimensional Analysis and
Modeling” (PDF)
Instructions: Please download the PDF file for Chapter 7 and read pages 2-11. This reading will introduce you to the basics of dimensional analysis as well as applications of dimensional analyses in simple fluid flows.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
5.3 Common Dimensionless Numbers
- Reading: University of Kentucky: Professor J. M. McDonough’s
Lectures Notes on Introduction to Fluid Mechanics: “Lectures on
Elementary Fluid Dynamics”
Link: University of Kentucky: Professor J. M. McDonough’s Lectures
Notes on Introduction to Fluid Mechanics: “Lectures on Elementary
Fluid
Dynamics” (PDF)
Instructions: Please download the PDF file for Lecture Notes of
ME330: Elementary Fluid Dynamics and read pages 96-99. You may be
surprised at the large number of useful, dimensionless numbers that
exist in fluid mechanics. The significance of several important
dimensionless numbers will be discussed in detail in this reading.
Note that this reading will cover the material you need to know for
subunits 5.3.1-5.3.5.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
- Reading: University of Iowa: Professor Fred Stern’s Lectures Notes
on Fluid Mechanics: “Chapter 7: Dimensional Analysis and Modeling”
Link: University of Iowa: Professor Fred Stern’s Lectures Notes on
Fluid Mechanics: “Chapter 7: Dimensional Analysis and
Modeling” (PDF)
Instructions: Please download the PDF file for Chapter 7 and read page 12 to learn about common dimensionless parameters for fluid flow problems. Note that this reading will cover the material you need to know for subunits 5.3.1-5.3.5.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
5.3.1 Reynold’s Number
5.3.2 Froude Number
5.3.3 Mach Number
5.3.4 Pressure Coefficient
5.3.5 Weber Number
5.4 Similarity and Model Testing
- Reading: University of Iowa: Professor Fred Stern’s Lectures Notes
on Fluid Mechanics: “Chapter 7: Dimensional Analysis and Modeling”
Link: University of Iowa: Professor Fred Stern’s Lectures Notes on
Fluid Mechanics: “Chapter 7: Dimensional Analysis and
Modeling” (PDF)
Instructions: Please download the PDF file for Chapter 7 and read
pages 15-19. Engineers often need to create sized-down models of
their products in order to safely and adequately test them prior to
production. For example, an airplane wing may be modeled as a much
smaller airfoil, placed in a wind chamber, for testing purposes.
Therefore, we need to know how to precisely “scale up” the results
in order to determine how the real product will react.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
Review Questions for Unit 5
- Assessment: The Saylor Foundation’s “ME201: Unit 5 Assessment”
Link: The Saylor Foundation’s “ME201: Unit 5
Assessment”
(HTML)
Instructions: Please perform this exercise.
You must be logged into your Saylor Foundation School account in
order to access this quiz. If you do not yet have an account, you
will be able to create one, free of charge, after clicking the
link.