I have a question (maybe more than one) regarding the application of dimensional analysis/Buckingham Pi Theorem to compare a Model to a Prototype.(adsbygoogle = window.adsbygoogle || []).push({});

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

x_{1}=f(x_{2},x_{3},x_{4},x_{5})

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 x_{2},x_{3},x_{4}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

(x_{2}^{a}x_{3}^{b}x_{4}^{c})x_{1}=constant

and similarly

(x_{2}^{a}x_{3}^{b}x_{4}^{c})x_{5}=constant

Nowhere 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 x_{2}and x_{3}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

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# Fluid MechanicsUnderstanding Dim Analysis & Similarity

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