MAE 222Mechanics of FluidsPrinceton UniversityAssignment # 6March 26, 1997Due on Wednesday, 2PM, April 2, 1997Hope you enjoyed the vorticity movie. We will skip Chapter 6 for now,but will get back to it later in the course. We will meet vorticity then.Chapter 7. Similitude, Dimensional Analysis, and Modeling. This stuffis useful beyond fluid mechanics.§7.1, Dimensional Analysis. What is Similitude? Its a big wordwhich is very, very popular in fluid mechanics. It is a way of iden-tifying different problems being the same problem. For example,we want to know what is the pressure distribution on the wing ofa big, low speed aircraft, and I have this tiny, small wind tunnel inthe MAE222 lab. If these two problems can be shown to be ‘simi-lar,’ then I can use the data from my small tunnel to make designdecisions on the big aircraft. Another example: you have data onthe destructive power of a small atomic bomb; what would be thedestructive power with a bomb with 10 times more energy? Amajor tool in such endeavors is dimensional analysis. The wholeidea of dimensional analysis is exquisitely simple: Lets use dimen-sionless variables! Let the desired answers (in dimensionless form)be expressed in terms of dimensionless parameters! Having agreedto this intuitively simple idea, the big question is: how many di-mensionless parameters (including the desired answers) are therein the problem?§7.2. Buckingham Pi Theorem. Don’t be intimidated by big words.Buckingham Pi Theorem simply tells us the number of dimen-sionless parameters in the problem. Not what the parametersare, but how many. If the problem has k dimensional parame-ters which involve r reference dimensions, then in most cases the1number of dimensionless parametes is k −r, sometimes it is more.Notice the equivocal “in most cases” (as given, you need to be“smart” in picking the reference dimensions). The “real” Buck-ingham Pi theorem is not what is given in this book—it involvesthe concept of rank of a certain rectangular matrix. You will seethe weakness of this version in problem 7.4 assigned to be donethis week. I will talk about the “real” version in class.§7.3-7.4. Determination of Pi Terms. Here is a step by step recipe,with example 7.1. You will encounter Reynolds number on page286. A most important point being made on page 290 is that thePi terms are not unique! The number of Pi terms is fixed; butwhat they are is not unique!!!§7.5. Determination of Pi Terms by Inspection . That is the waymost educated engineers do it.§7.6. Common Dimensionless Groups in Fluid Mechanics. Sincethe Pi terms are not unique, this section tells you what a collectionof Pi terms that most fluid mechanicists have agreed to used—bygiving them names to the first person who identified its usefulness.§7.7 Correlation of Experimental Data. Now that we know howto find the Pi terms, here comes how to use them! To get themost “mileage” out of your data, DO NOT PLOT YOUR DATAIN DIMENSIONAL FORM! Plot them in terms of your Pi’s!!!!!§7.8 Modeling and Similitude. Once you have the relationship be-tween the Pi terms (obtained by experiments on a model or bytheoretical analysis), you can used them to make predictions!!!Here, there are some subtle but intuitive tricks. What if the valueof some of the Pi’s in the real system is not exactly the same as thecorresponding values of Pi’s in the model (or the theory)? Whatif they are just a little bit off? We will talk about this!§7.9. Some Typical Model Studies. Here comes tons of examples.Problems at end of Chapter 7, page 315:• Problem 7.1.• Problem 7.3.2• Problem 7.4. Answer the question “why would it be incorrect toinclude the velocity in the smaller pipe as an additional variable?”the best you can based on the rambling discourse in Step 1 onpage 282. What happens if you did included it?• Problem 7.25. Once we agree that Froude number is the only Piterm for this problem, we can immediately make predictions basedon a 1:30 scale model!• Problem 7.32. Whoever made the measurements for the data onFig. P7.32 was not properly educated! If he/she had only plottedthem in dimensionless form!• Problem 7.39. Obvious the same uneducated person plotted thesedata!Use email or the newsgroup to ask