I'm familiar with the procedure of normalization, but I'm unfamiliar with some of the theory involved.
For instance, using the Navier Stokes equations where the density and viscosity can be treated as constant. I have also neglected gravity.
$$\rho\dot{\mathbf{u}}+\rho\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}=-\nabla p+\mu\nabla^2\mathbf{u}$$
Let me argue that for the flow in question, the relevant scaling terms are the free stream velocity $U$, the distance the flow covers $L$, the time it takes the flow to cover this distance $T=L/U$, and finally the pressure difference between the surface and the free stream $p_0-p_\infty$. That is ... $$\mathbf{u}=\mathbf{u}^*U\\ \nabla=\nabla^* L^{-1}\\ t=t^*\frac{L}{U}\\ p=p^*(p_0-p_\infty)$$ where * is to mean non-dimensional terms.
To make this problem simpler, I will also be assuming all three position, and velocity components to scale with $L$ and $U$ respectively; even though this is unlikely to be physical. Considering the $x$ component ...
$$\rho\dot{u}+\rho u\partial_x u+\rho v\partial_y u+\rho w\partial_z u=-\partial _x p+\mu\partial_x^2 u+\mu\partial_y^2 u+\mu\partial_z^2 u$$ inserting the scaling terms ... $$\dot{u^*}\left(\frac{\rho U^2}{L}\right)+u^*\partial_{x^*} u^*\left(\frac{\rho U^2}{L}\right)+v^*\partial_{y^*} u^*\left(\frac{\rho U^2}{L}\right)+ w^*\partial_{z^*} u^*\left(\frac{\rho U^2}{L}\right)=-\partial _{x^*} p^*\left(\frac{p_0-p_\infty}{L}\right)+\partial_{x^*}^2 u^*\left(\frac{\mu U}{L^2}\right)+\partial_{y^*}^2 u^*\left(\frac{\mu U}{L^2}\right)+\partial_{z^*}^2 u^*\left(\frac{\mu U}{L^2}\right).$$ Usually I would not stop at this point but, my question is about some of the non-dimensional terms. For instance the term $w^*\partial_{z^*}u^*$ has no dimensions and is said to be of order 1. What exactly does this mean? If I continue with the normalization process ... $$\dot{u^*}+u^*\partial_{x^*} u^*+v^*\partial_{y^*} u^*+ w^*\partial_{z^*} u^*=-\partial _{x^*} p^*\text{Eu}+\left(\partial_{x^*}^2 u^*+\partial_{y^*}^2 u^*+\partial_{z^*}^2 u^*\right)\frac{1}{\text{Re}}$$
where Eu and Re are the non-dimensional numbers the Euler and Reynolds numbers. How does the non-dimensional terms being order 1 contribute to the analysis of experimental data?