Scale analysis (mathematics)
| Fit approximation | |
|---|---|
 |
|
| Concepts | |
|
Orders of approximation Scale analysis · Big O notation Curve fitting · False precision Significant figures |
|
| Other fundamentals | |
|
Approximation · Generalization error Taylor polynomial Scientific modelling |
|
Scale analysis (or order-of-magnitude analysis) is a powerful tool used in the mathematical sciences for the simplification of equations with many terms. First the approximate magnitude of individual terms in the equations is determined. Then some negligibly small terms may be ignored.
Consider for example the momentum equation of the Navier–Stokes equations in the vertical coordinate direction of the atmosphere
where R is Earth radius, Ω is frequency of rotation of the Earth, g is gravitational acceleration, φ is latitude, ρ is density of air and ν is kinematic viscosity of air (we can neglect turbulence in free atmosphere).
In synoptic scale we can expect horizontal velocities about U = 101 m.s−1 and vertical about W = 10−2 m.s−1. Horizontal scale is L = 106 m and vertical scale is H = 104 m. Typical time scale is T = L/U = 105 s. Pressure differences in troposphere are ΔP = 104 Pa and density of air ρ = 100 kg·m−3. Other physical properties are approximately:
Estimates of the different terms in equation (1) can be made using their scales:
Now we can introduce these scales and their values into equation (1):
We can see that all terms — except the first and second on the right-hand side — are negligibly small. Thus we can simplify the vertical momentum equation to the hydrostatic equilibrium equation:
Scale analysis is very useful and widely used tool for solving problems in the area of heat transfer and fluid mechanics, pressure-driven wall jet, separating flows behind backward-facing steps, jet diffusion flames, study of linear and non-linear dynamics. Scale analysis is recommended as the premier method for obtaining the most information per unit of intellectual effort, despite the fact that it is a precondition for good analysis in dimensionless form. The object of scale analysis is to use the basic principles of convective heat transfer to produce order-of-magnitude estimates for the quantities of interest. Scale analysis anticipates within a factor of order one when done properly, the expensive results produced by exact analyses. Scale analysis ruled as follows:
...
Wikipedia