Nondimensionalization and Scaling

[hanson]

Hi all.
To me, scaling means adopting properly scales in nondimensionalization.
However, as I see in the book "A modern introduction to the mathematical theory of water waves" by R.S.Johnson, the author distinguish the two processes in a way that confuses me much. (Sec. 1.3.1 and 1.3.2)
Can someone who have used this book kindly help clarify the two things?

 

[scotlass]

Hello,

I am not familiar with the book, but I believe non-dimensionalization literally gets rid of all dimensional quantities. So, you might write something like
x=Lx* where L is a typical length scale in the x direction and resplacing all your x' s by x* 's where x* is a non dimesional quantity since you have taken 10metres, say, and divided it by metres to just get 10.

Scaling on the other hand helps you get an idea of the relative sizes of terms so if you had something long in the x direction and short in the y direction you would write something like x=x* and y=epsilon y* where epsilon <<1 and plug this into your equations. You would then be able to see the relative sizes of terms with epsilons, epsilon^2 etc and maybe discard the highest order terms in epsilon if appropriate.

Hope this helps

 

[tacman]

Nondimensionalization and scaling are two important concepts in mathematics and engineering that are often used together. Nondimensionalization is the process of removing units from a mathematical model or equation, making it easier to analyze and compare different systems. This is achieved by dividing all quantities in the model by a relevant characteristic quantity, such as the length or time scale of the system. Scaling, on the other hand, is the process of choosing appropriate scales for the nondimensionalized variables. This involves choosing values for the characteristic quantities that result in a simplified and more manageable equation.

In the context of water waves, Johnson's book distinguishes between the two processes by emphasizing that nondimensionalization is a mathematical procedure, while scaling involves physical considerations. In other words, nondimensionalization is a purely mathematical tool to simplify equations, while scaling takes into account the physical properties of the system being studied.

One way to understand this is by looking at an example. Let's say we have a mathematical model for the height of a water wave, which includes variables such as the wave amplitude, water depth, and gravitational acceleration. Nondimensionalization would involve dividing each of these variables by a relevant characteristic quantity, such as the water depth. This would result in a dimensionless equation, making it easier to analyze and compare different wave systems.

However, choosing the appropriate value for the water depth in this equation would require physical considerations. For example, if we are studying small ripples in a pond, the water depth would be much smaller compared to studying ocean waves. Therefore, the scale for the water depth would be different in these two cases, resulting in different nondimensionalized equations.

In summary, nondimensionalization and scaling are closely related but distinct processes. Nondimensionalization is a mathematical procedure to simplify equations, while scaling involves choosing appropriate scales for the variables based on physical considerations. Understanding the difference between these two processes is important in accurately studying and analyzing complex systems.