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February 24, 2001, 03:30 
Nondimensionalization of Governing Eqns.

#1 
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Hello!
The Euler or the NS equations are usually solved in their nondimensionalized form. But I think there are many ways to nondimensionalize the variables. What is the best nondimensionalization? (or is there not the best?) For example, I could define velocities as u/U_inf and v/U_inf (U_inf is freestream), or as u/a_inf and v/a_inf (a is sound speed). Also, there are several choices for pressure. I must choose one, but I'm not sure which one to choose because I can't come up with a paticular reason why I choose what I choose. Any comments are welcomed! Thanks Kevin 

February 24, 2001, 12:32 
Re: Nondimensionalization of Governing Eqns.

#2 
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Kevin:
When I was a grad student I found the texts by A. Bejan to be very helpful in this regard, there are probably many now. One I have used years ago is called "Convection Heat Transfer." It discuses scaling from the standpoint of what makes sense physically. This is important as you have pointed out there are other was to normalize thinsg that are nonphysical wrt the system of interest. But I use dimensional quantities mostly these days, in part because of the proliferation of nonsensical dimensionless groups. 

February 24, 2001, 16:48 
Re: Nondimensionalization of Governing Eqns.

#3 
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(1). The choice is yours. (2). It should not affect your results at all. But make sure that you use only one set of dimensionless group throughout the formulation. Do not use both u/U_inf and u/a_inf in your equations. (3). You can count the distance in miles or inches. You can also count money in dollars or cents. But make sure that you only pick one set in your calculation. 100 (cents) is one hundred times the 1 (dollar), if you use both in your calculation.


February 24, 2001, 21:56 
Re: Nondimensionalization of Governing Eqns.

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I disagree that it will not affect your results, in fact you may not even GET results unless you are working in constant property systems with simple forcing. Sensible nondimensional approches require some estimation as to the important length scales throughout the domain. If the system has a strongly local feedback, a global length scale used as a means of normalization becomes increasingly meaningless.
In theory any normalization should all cancel out, but in my experience with numerical simulations of say, variable viscosity natural convection, choices of nondimensional groups makes a significant difference in finding convergence and importantly in understanding your results. When in doubt, don't bother and at least you will be able to understand your results as they have not been 'laundered' by physically meaningless normalization 'cause if what John says is true, then it won't matter anyway. I suggest you initially follow John's advice and see how far into your carrer that it will serve you well. Maybe a long way, maybe not. All depends on the kinds of systems you wish to address, and how much you hope to learn from the hours of cpu time you expend. 

February 24, 2001, 23:33 
Re: Nondimensionalization of Governing Eqns.

#5 
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Thank you for your advice, George and John. It looks like there are choices that physically make sense and those that do not. I'm happy to know that. Now, I'm gonna look for the book by Bejan.
Thank you! Kevin 

February 25, 2001, 02:30 
Re: Nondimensionalization of Governing Eqns.

#6 
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(1). Very interesting. (2).Equations in nondimensional form should be identical to the original equations. This is because the reference parameters, such as selected length, velocity, temperature, pressure, etc. are constants. (3). If the reference parameters are changing in time and location, then I guess, you sure will run into problems. (4). Anyway, I would be interested in knowing the situation where the selection of nondimensional group is linked to the convergence problem. (5). The only reason to use the nondimensional group is to see the relative importance of variables or terms in the governing equations. So, the proper selection of nondimensional group will improve the understanding of the results.


March 9, 2001, 13:54 
Re: Nondimensionalization of Governing Eqns.

#7 
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Both John and George are correct. Physically, no differences. Numerically, may dependent on how the solver is implemented.


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