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August 24, 2003, 16:52 
Nondimensionalization

#1 
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Does nondimensionalization really affect the accuracy of solution? I'm working with a nondimensional form of NS eq. anyway, but, why should I?
To me, dimensional form of eq. with consistent units is just same as a nondimensional one.. with different reference scale. If that makes the difference on the accuracy, doesn't it mean there is an optimal choice of reference scale? 

August 25, 2003, 05:36 
Re: Nondimensionalization

#2 
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There are two reasons to use a nondimensional form: i) That way it is easier to compare numerical and real life experiments, as it is now just a matter of scaling. ii) In numerical calculations, it is desirable to have quantities of about the same size, so that, for example you don't have to multiply a huge dimensional pressure variable with a small velocity. The nondimensionalization should leave you with quantities which are all O(1).
Yours, Philipp 

August 25, 2003, 22:50 
loss of precision?

#3 
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Thank you, Philipp.
Reason i) was why I used the nondimensional form. But, as for ii), I don't think I'm loosing much by multiplying a high pressure, say 10^6Pa by a small velocity~10^3m/s. If the exponents are within machine limit, the multiplication shouldn't be a big deal as far as the precision is concerned. Isn't the substraction between 'near equal numbers', which causes the loss of precision? 

August 28, 2003, 07:45 
Re: loss of precision?

#4 
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You are right that the multiplication is not an issue. The reason is a bit more subtle and I fear cannot be put into quantitative theorems. No matter how you solve your PDE, in the end you have a matrix. Even with an explicit method, you end up with one, altough you don't have to store it. And for these matrices it seems to be advantageous to have entries wich are of roughly the same order.
Or that's what I was told 

August 28, 2003, 11:22 
any discussion in text books?

#5 
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The discussions are interesting. But can you provide some textbooks or pubilcations about the issue, saying that nondimensionalization gives better accuracy?
Thanks H 

August 28, 2003, 21:15 
Still not sure..

#6 
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Yes, I was looking for texts or lecture notes, also.
I thought about the issue in terms of matrix, too. Suppose I am deaing with compressible inviscid flow. The first thing that came into my mind was the Jacobian matrix of convective flux vector. The condition number of the matrix is (u+c)/u and it's nondimensional whether 'u' was nondimensionalzed or not priory. So, this is no big deal. For xmomentum, x term is (rho*u*u + p) so we have addition, which can be a source of problem. For simplicity, suppose u=0.01m/s and rho~1kg/m^3, p~10^5pa. 1*0.01^2+100000=100000.0001. In single precision, the velocity contribution will be lost. Nondimensionalization I used was: rho' = rho/rho" u' = u/c" p = p/ (0.5*rho"*c"*c") where LHSs (with ') are nondiemensional vars and rho" is ref. density. c" is sonic speed. suppose c"~333m/s and rho''=rho such that u' ~ 3e5, rho'=1, p'=1/gamma~0.7. Then, rho'*u'*u'= 9e10 and p'=0.7. Actually, the sum is worse in term of # of 0s between the most and least significant terms than the one I got with dimensional form. So, the choice of ref values in my nondimensionalization were really bad. They were meant to be used for high speed flow. So, such a small velocity could be easily lost. Well, I guess there will much more subtle things I haven't considered yet than what I have mentioned above. So please give me any pointer where I should look. Thanks in advance! 

August 29, 2003, 02:08 
Re: Still not sure..

#7 
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You can't simulate such a low Mach number (0.01/333 = 3*10**5) with a standard compressible code. You've got a stiff system and must preconditionne it.


August 29, 2003, 05:55 
Re: Still not sure..

#8 
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I'm sorry, I don't know any references in textbooks.
Regarding the condition number: you probably used the minmax formula for the condition number, which is only valid for SPD matrices. We have a unsymmetric, indefinit and nonnormal matrix. This formula is not valid. Furthermore, Eigenvalues and condition number don't describe the behaviour of nonnormal matrices too well. In fact, for nonnormal matrices, theorems that describe their behaviour are few and very technical (see the pseudospectra concept of N. Trefethen). Lastly, you look at the wrong matrix. We have to look at the matrix obtained after discretization, meaning that is is very big and has unknown eigenvalues which have not much to do with the physical eigenvalues u, u+c, uc. So where does the nondimensionalization come in? As I said, the matrix is more "balanced" after the nondimensionalization. And experience from iterative solution methods suggests that this is beneficial. Sorry, I don't know anything more specific. Regarding your example with the addition: I thought of that one, too, but this error seems to be physical. The momentum coming from velocity IS small in comparison to the forces from the pressure. Even if you use the low Mach nondimensionalization, where you obtain a factor of 1/M/M in front of the pressure forces, the velocity term in the momentum equations will be small. 

August 29, 2003, 12:18 
Thanks, Mr. Birken

#9 
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I will look into the pseudospectra stuff you mentioned. That's a new thing for me. Thanks! I agree with you that it is not the Jacobian matrix that ultimately determines the accuracy of solution. But, I was not sure where to start. When I talked about the Jacobian, I was visualizing a single cell mesh in 1D and the Euler equations in explicit form in terms of characteristic vars. It's then a simple system of ODE and I thought the Jacobian of the flux vector was what I should look.
As for the low speed issue in compressible flow as commented by Kevin, even high speed case can have stagnation points so that the speed will be 0 and close 0. Thanks anyway for your comment. I guess I will have to put aside the issue so I can move on, but, I think I will return to it. I've been using the nondimensionalization blindly cause it's more or less customary to do so. Now, I can't stand that I'm doing it without knowing the reasons why I have to do so. 

August 29, 2003, 12:38 
Re: Thanks, Mr. Birken

#10 
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I think the reason why nondimesionalization is needed has been expleined in a lot of CFD textbooks. You could have a check on them.
Regards Li 

August 29, 2003, 13:28 
Which textbook?

#11 
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Would you tell me which book you are refering to? Thanks


August 29, 2003, 19:40 
Re: Nondimensionalization

#12 
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The nondimensionalization (or physical similarity) is not only convenient but also critical. It gives a much better understanding on the physics of a problem, in terms of the relative importance among different components. It is also the starting point of the perturbation theory. If you publish papers in theoretical journals like JFM, you¡¯d better do this. In numerics, it can at least provide a universal guidance on choosing the CFL, i.e. the sizes of grids and time marching steps, if the roundoff error is not an issue.
For a particular problem, if you can find an optimal choice of reference scale, you will be famous and your name might appear in textbooks. In conclusion, try to use the dimensionless form. You will be amazed by yourself how good you are by comfortably exercising this conversion. 

August 30, 2003, 08:53 
Re: Which textbook?

#13 
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I remember that Hoffmann's CFD book has a good description on this issue. I don't have it in hand though. Mainly there are two reasons as stated by Kirken. (1) is that the dynamic similarity can be achieved, (2) the flow variables will fall between the limits such as zero and one hence they are all in the same order, which could avoid possible inaccurate operations.
Regards Li 

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