1) Dimensionless numbers allow for comparisons between very different systems. Let's say you're designing a stirrer for a syrup vat and you want to test a prototype. It would make sense to make a miniature of the system but you know that size makes a difference. Well if you also decrease the viscosity of the liquid you're stirring, you can make sure the Reynolds number is the same between both processes and can take your conclusions from the miniature and apply them to the big system.

2) Dimensionless numbers tell you how the system will behave. The classic example of this yet again involves the Reynolds number to predict the onset of turbulence in a system. Critical values for the Reynolds number for many different systems are tabulated and so you can easily predict the onset of turbulence. Other examples include using the Rayleigh numberto predict whether a fluid's heat transfer will happen mostly through natural convection or through conduction. Similarly, the Péclet number will tell you whether transport will happen through advection (active convection) or diffusion.

3) Many useful relationships exist between dimensionless numbers that tell you how specific things influence the system. A classic example here is determining how a thermal boundary layer scales with the velocity of the flow. It turns out that you can use the Peclet number again to represent the flow and the boundary layer thickness, δ scales as δ∼Pe−1/3. Given a few data measurements you could readily extrapolate temperature gradients into a fluid.

4) Dimensionless numbers allow you to solve a problem more easily.Many solution techniques require you to non-dimensionalize your problem before moving forward because the choice of scale matters.A Similarity solution is possible only when you can map one scale onto another (say a timescale onto a length scale). Other techniques for approximating solutions such as Asymptotic analysis often ask questions about what happens when a parameter is either very large or very small. These parameters are most conveniently dimensionless numbers so that one can compare many different scenarios at once.

5) When you need to solve a problem numerically, dimensionless groups help you to scale your problem. Computers can't deal with wide ranges of numbers especially when adding small numbers to very big numbers. In numerical methods, problems that have certain terms that are much larger than others are said to have a large Condition number and are consequently both difficult to solve and difficult to solve accurately. By scaling the problem to appropriate scales, you can make many terms be of the same order so that it the effects of numerical errors are minimized when calculating the residual.