How do you perform a scale analysis to simplify the governing equations of fluid motion?

1 What is scale analysis?

Scale analysis is a method of estimating the relative importance of different terms in the governing equations of fluid motion, such as the continuity equation, the momentum equation, and the energy equation. By using dimensional analysis, you can nondimensionalize the equations and introduce dimensionless numbers that represent the ratios of different physical quantities, such as the Reynolds number, the Froude number, and the Mach number. By comparing the order of magnitude of these dimensionless numbers, you can determine which terms are negligible and which terms are dominant in the governing equations.

2 Why use scale analysis?

Scale analysis can help you simplify the governing equations of fluid motion and reduce the number of variables and parameters involved in a problem. This can make the problem easier to solve analytically or numerically, and also help you identify the relevant physical phenomena and mechanisms that govern the fluid behavior. Scale analysis can also help you design experiments and models that are similar to the real situation, by ensuring that the dimensionless numbers are matched between the prototype and the model.

3 How to perform scale analysis?

To perform a scale analysis, you need to identify the governing equations of fluid motion that apply to your problem, define the characteristic scales of the problem, nondimensionalize the governing equations, compare the order of magnitude of the dimensionless numbers and terms, and solve the simplified governing equations. This process helps to capture the essential physics of the problem and interpret the results in terms of dimensionless numbers and variables.

4 Examples of scale analysis

Scale analysis can be used to derive equations for a variety of fluid mechanics problems. For example, the boundary layer equations and the Blasius solution for laminar flow over a flat plate can be obtained by neglecting the pressure gradient and normal momentum terms in the momentum equation, and assuming a large Reynolds number and a small boundary layer thickness compared to the plate length. Additionally, the Hagen-Poiseuille equation for laminar flow in a pipe can be derived by disregarding inertia and gravity terms in the momentum equation, and assuming a small Reynolds number and constant pressure gradient along the pipe. Similarly, the energy equation for flow over a weir can be derived by disregarding viscous and compressibility terms in the energy equation, and assuming a large Froude number and small head loss compared to the total head.

5 Limitations of scale analysis

Scale analysis is a useful technique, but there are some limitations that need to be taken into account. For instance, scale analysis is based on order of magnitude estimates, which may not be valid for all conditions. It’s important to check the validity of assumptions and simplifications, and compare results with experimental data or more rigorous solutions. Additionally, scale analysis may not capture all important physical effects and phenomena, such as turbulence, instability, shock waves, separation, and heat transfer. Complex or multidimensional fluid problems may require numerical methods or computational fluid dynamics (CFD) to solve them.