Fluid MechanicsUnderstanding Dim Analysis & Similarity

[Saladsamurai]

I have a question (maybe more than one) regarding the application of dimensional analysis/Buckingham Pi Theorem to compare a Model to a Prototype.

The premise of this theorem is more or less to nondimensionalize a function.

In short, the procedure is as follows: Let's say we have some function that contains five variables or

x₁=f(x₂,x₃,x₄,x₅)

These variables are generally of the dimensions [M]=mass [T]=time [L]=length and sometimes [theta]=temp but I will omit this for simplicity.

So here we have m variables where m=5 and n dimensions where n=3, thus from the Pi theorem we have N=m-n=2 non-dimensional parameters that can be formed.The general procedure for finding these non-dimensional parameters is to arbitrarily choose n, 3 in this case, variables that cannot by themselves form a dimensionless power product. In this case we will say that x₂,x₃,x₄ satisfy this condition. These are our "repeating parameters."

Now we find our 2 dimensionless parameters by finding the power product of our repeating variables with each of our remaining 2 variables such that

(x₂^ax₃^bx₄^c)x₁=constant

and similarly

(x₂^ax₃^bx₄^c)x₅=constant

Now here is the question. It seems that sometimes there is more than 1 choice of our 3 repeating variables. That is, there might be two sets of 3 variables whose power product is not dimensionless.

How do we choose? Does it matter? I think that it does not matter in theory, but in practice, some choices may yield more useful relationships than others.

Is that correct?

I am under the impression that if we were to be given values for let's say x₂ and x₃ for both the model and prototype, then these would be obvious choices for 2 of the 3 repeating variables.

Any thoughts?

Sorry for the lengthy post.

Thanks,
Casey

 

[Cyrus]

How do we choose? Does it matter? I think that it does not matter in theory, but in practice, some choices may yield more useful relationships than others.

You are absolutely correct that there is no unique functional form of pi terms. The only unique thing is the number of pi terms you will have. In general, selection of Pi terms are made based on knowledge of the problem. You typically care about Mach number, Reynolds Number, and a few others depending on the type and nature of the flow.

For aerodynamics, we care about the Reynolds number and Mach Number most.

The Strouhal number comes in for unsteadiness, weber for surface tension, and others. Typically, you will know what the Pi terms are based on what you're doing.

Get yourself a copy of this book: Fundamentals of Fluid Mechanics (Okiishi, Young, Munson) and read the section on Pi terms (or the entire book if you want). It's a darn good book that explains concepts well.

 

[Saladsamurai]

You are absolutely correct that there is no unique functional form of pi terms. The only unique thing is the number of pi terms you will have. In general, selection of Pi terms are made based on knowledge of the problem. You typically care about Mach number, Reynolds Number, and a few others depending on the type and nature of the flow.

For aerodynamics, we care about the Reynolds number and Mach Number most.

The Strouhal number comes in for unsteadiness, weber for surface tension, and others. Typically, you will know what the Pi terms are based on what you're doing.

Get yourself a copy of this book: Fundamentals of Fluid Mechanics (Okiishi, Young, Munson) and read the section on Pi terms (or the entire book if you want). It's a darn good book that explains concepts well.

Excellent. I had a feeling that this is just something that will come with practice and is a little hard to generalize.

I will definitely check out that book too Cyrus. My library probably has it. Thanks for the tip.

EDIT: Nope they don't. All of them are by Munson, Roy, et Al. I'll see if I can get it sent from another library.

 

[Cyrus]

Here is a fun real world experiment for you to work out. See if you can have dynamic similitude for a scale wind tunnel model using the Buckingham Pi theorem. You will find the answer to my question will give you a much better understanding of the Pi theorem in practical use: i.e., not just doing BS problems from a book which is what I am sure your instructor will have you do (they all do).

 

[Saladsamurai]

Here is a fun real world experiment for you to work out. See if you can have dynamic similitude for a scale wind tunnel model using the Buckingham Pi theorem. You will find the answer to my question will give you a much better understanding of the Pi theorem in practical use: i.e., not just doing BS problems from a book which is what I am sure your instructor will have you do (they all do).

Hmm. I would like to do this, but I am not exactly what the question is. Maybe we have not covered the subject enough yet for me too. I am used to being given certain information and then asked a very specific question (the BS problems).

That is, are we looking at the power of the fan blade of this thing or something?

Thanks,
Casey

 

[Cyrus]

Just take an airplane (or car, or whatever 'thing' you want), and do dimensional analysis on it.

Given: A wind tunnel and a scale model

Find: The Pi Terms and what you would need for similitude.

That's the real world. Welcome to it.

Remember what I told you: (a) scale model (b) Reynolds number (c) Mach Number.

Those are your pi terms.