- #1
Saladsamurai
- 3,020
- 7
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
(x2ax3bx4c)x1=constant
and similarly
(x2ax3bx4c)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
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
(x2ax3bx4c)x1=constant
and similarly
(x2ax3bx4c)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