# ME201: Fluid Mechanics

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.

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.

• 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.

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.

• 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.

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.

• 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.

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.

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