User:Egm6936.s09/Nondimensionalization (French thesis)

 The "loose" translation of Chapter 5 of the French thesis by E. Rieutord (1985) at the Institut National des Sciences Appliquees, Lyon (INSA de Lyon), France, forms only one of the many possible starting points for this wiki article on dimensional analysis.

By "loose" translation, I meant to take the freedom to improve the way certain ideas and concept were communicated in the thesis, which may or may not be well written (in the sense of being able to convey ideas in the most rigorous and crystal clear possible).

We will not hesitate to add material that would be helpful to help understand this topic. It is possible that the present article may evolve into a tutorial on dimensional analysis, at least for self study.

Tuan, you can contribute to this article by rewriting as many equations from the French thesis as possible in the wiki latex format. At the beginning, it is best to do your writing on wikiversity; in particular, see here. I'll show you how in an e-mail how to create a wikiversity account.

Contributing to writing this article is a good way to learn, not just dimensional analysis, but also latex. At the beginning, you may want to use the following software: Latex equation editor to generate the latex marked-up language. Once you are used to write equations in latex marked-up language directly, you won't need this web site.

This way, I can focus on translating French into English (but we also look at other references as well).

Another program to type wiki articles using firefox, or other browsers, is WikEd.

I identified some web material in English related to dimensional analysis for you to consult, as listed below.

Vql 20:39, 13 February 2009 (UTC)

= References =


 * E. Rieutord, Thesis title??, Institut National des Sciences Appliquees, Lyon (INSA de Lyon), France, 1985.

= Web references =


 * google search for "dimensional analysis"
 * Wikipedia: Dimensional analysis Buckingham's Pi theorem Similitude
 * E. Buckingham, On Physically Similar Systems; Illustrations of the Use of Dimensional Equations, Physical Review, Vol.4, No.4, pp.345-376, 1914.
 * P.W. Bridgman (Nobel laureate), Dimensional analysis (free-access internet book), by Yale University Press, 1922.
 * H. Hanche-Olsen, Buckingham's Pi theorem, 2004.
 * Measurement theory and dimensional analysis: Methodological impact on the comparison and evaluation process, E. Coatanea, J. Vareille, B. Yannou.
 * Conceptual modelling of life cycle design: A modelling and evaluation method based on analogies and dimensionless numbers, E. Coatanea, doctoral dissertation, Helsinki University of Technology, 2005.
 * Dynamos, Section 4, p.16, by T. Alboussiere, edited by P. Cardin and Cugliandolo, Elsevier, 2008.
 * Talk slides: 1. History, Equations 2. Energetics 3. Dimensionless parameters 4. MHD (Magneto-Hydro-Dynamics) effects

= Dimensional analysis -- similitude and model testing =

Mathematical analysis by itself is insufficient to solve all problems encountered in fluid mechanics and in other areas, despite enormous progress made in recent years in numerical analysis and in computational technology (large, powerful computers).

Experiments continue to be a most reliable, fastest, and less onerous method to solve a given problem. A recourse to experiments is also useful even when a theoretical solution had been found so to validate not just such solution, but also notably the hypotheses made in the problem statement.

On the other hand, when a problem possesses some degree of complexity, a judicious choice of the problem parameters and the analysis of the effects of such choice are essential not just in theoretical analysis, but also in the use of the experimental results.

To satisfy the above two requirements -- selection of a most rational formulation for theoretical analysis and experimental study or experimental verifications -- similitude theory and dimensional analysis are useful tools.

Let's begin with dimensional analysis, which is the study of the general relations that exist between different quantities that characterize a physical phenomenon.

Dimensional analysis
Dimensional analysis is based on a simple fundamental principle that a relation between different physical quantities should be dimensionally homogeneous, i.e., independent of the unit system being used.

Vashy-Buckingham Pi theorem
The above mentioned fundamental principle is given by the Vashy-Buckingham theorem, which can be stated as follows:

 The above statement of the Vashy-Buckingham theorem is rather obscure! It did not define quatities such as $$\displaystyle g_i$$ and $$\displaystyle \Pi_j$$.

It is likely that $$\displaystyle g_1, g_2, \cdots, g_q$$ in the above theorem are rational numbers, which form the powers for the independent variables $$\displaystyle x_1, x_2, \cdots, x_q$$. The quantities $$\displaystyle \Pi_{j}$$ are products (and thus the choice of the character $$\displaystyle \Pi$$, which is Greek for "P", and is mnemonic for "Product") of some independent variables, and are the dimensionless quantities.

Vql 20:39, 13 February 2009 (UTC)

 Who was Vashy? google search for "Vashy Buckingham". Many authors referred to the Vashy-Buckingham Pi theorem, but then only cited the paper by Buckingham; there was no citation of any paper by Vashy, e.g., R. Greneche, Y. Ravalard, and D. Coutellier, A Method for Crash Tests on Laminated Composite Scaled-Down Models, Applied Composite Materials, Vol.12, pp.355-368, 2005. What was the contribution of Vashy?

In the above google search, I noticed that many articles written by French researchers referred to the Vashy-Buckingham theorem. In the English literature, the theorem was often referred to as the Buckingham Pi theorem; it is possible that Vashy was a Frenchman who contributed to develop this theorem, but did not publish his (her?) result in an archival journal like Buckingham did. Vql 22:53, 13 February 2009 (UTC)

In practice, we select $$\displaystyle x_1, x_2, \cdots, x_q$$ as the parameters that we consider essential for the problem at hand, and which we want to see appear explicitly in the expression for $$\displaystyle G$$.

At this point, I became a bit disillusioned with the French thesis, and suggest to look at the book by a master:


 * P.W. Bridgman (Nobel laureate), Dimensional analysis (free-access internet book), by Yale University Press, 1922.

and a more recent reference:


 * H. Hanche-Olsen, Buckingham's Pi theorem, 2004.

and perhaps eventually the classic paper by Buckingham himself:


 * E. Buckingham, On Physically Similar Systems; Illustrations of the Use of Dimensional Equations, Physical Review, Vol.4, No.4, pp.345-376, 1914.

I like the general and rigorous mathematical exposition of the VB (short for Vashy-Buckingham) Pi theorem in Hanche-Olsen (2004) and the simple examples that Bridgman (1922) used to begin his exposition of the VB Pi theorem before stating the theorem on p.41.

The above expression in Rieutord (1985), i.e.,

\displaystyle G   = x_1^{g_1} x_2^{g_2} \cdots x_q^{g_q} F ( \Pi_{q+1} \Pi_{q+2} \cdots \Pi_{p}) $$ actually corresponds to Eq.(6) in Hanche-Olsen (2004) and the subsequent arguments:

\displaystyle \begin{align} \Phi [	 (	   x_1^{a_{11}}	    \cdots	    x_m^{a_{m1}}	    \rho_1	 ) ,	 \ldots ,	 (	   x_1^{a_{1n}}	    \cdots	    x_m^{a_{mn}}	    \rho_n	 ) ]     &      =      x_1^{b_{1}} \cdots x_m^{b_{m}} \Phi (	 \rho_1	 ,	 \ldots	 ,	 \rho_n     ) \\     &      =      R_1^{c_1} \cdots R_n^{c_n} \Phi (	 R_1	 ,	 \ldots	 ,	 R_n     ) \\     &      =      R_1^{c_1} \cdots R_n^{c_n} \psi (	 R_1	 ,	 \ldots	 ,	 R_k	 ,	 \Pi_{1}	 ,	 \ldots	 ,	 \Pi_{n-r}     ) \\     &      =      R_1^{c_1} \cdots R_n^{c_n} \Psi (	 \Pi_{1}	 ,	 \ldots	 ,	 \Pi_{n-r}     ) \end{align} $$

and the equation at the top of p.41 in Bridgman (1922):

\displaystyle \alpha =  \beta^{x_1} \gamma^{x_2} \cdots \Phi (     \Pi_2     ,      \Pi_3      ,      \cdots   ) $$

Loosely speaking, the theorem stated in Rieutord (1985) can be restated as follows:

See also Buckingham's Pi theorem (Wikipedia):

Plan: Explain dimensional analysis by starting with the example in Bridgman (1922) but couch it in the language of Hanche-Olsen (2004) thus understand both documents, "one stone for two birds". We will follow the notation in Hanche-Olsen (2004).

See Topic:Dimensional analysis tutorial