Scale analysis (mathematics)
Orders of approximation|
Scale analysis · Big O notation
Curve fitting · False precision
Approximation · Generalization error|
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.
Example: vertical momentum in synoptic-scale meteorology
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:
- R = 6.378 × 106 m;
- Ω = 7.292 × 10−5 rad·s−1;
- ν = 1.46 × 10−5 m2·s−1;
- g = 9.81 m·s−2.
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:
Rules of scale analysis
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:
Rule1- First step in scale analysis is to define the domain of extent in which we apply scale analysis. Any scale analysis of a flow region that is not uniquely defined is not valid.
Rule2- One equation constitutes an equivalence between the scales of two dominant terms appearing in the equation. For example,
In the above example, the left-hand side could be of equal order of magnitude as the right-hand side.
Rule3- If in the sum of two terms given by
the order of magnitude of one term is greater than order of magnitude of the other term
then the order of magnitude of the sum is dictated by the dominant term
The same conclusion holds if we have the difference of two terms
Rule4- In the sum of two terms, if two terms are same order of magnitude,
then the sum is also of same order of magnitude:
Rule5- In case of product of two terms
the order of magnitude of the product is equal to the product of the orders of magnitude of the two factors
here O(a) represents the order of magnitude of a.
~ represents two terms are of same order of magnitude.
> represents greater than, in the sense of order-of-magnitude.
Scale analysis of fully developed flow
Consider the steady laminar flow of a viscous fluid inside a circular tube. Let the fluid enter with a uniform velocity over the flow across section. As the fluid moves down the tube a boundary layer of low-velocity fluid forms and grows on the surface because the fluid immediately adjacent to the surface have zero velocity. A particular and simplifying feature of viscous flow inside cylindrical tubes is the fact that the boundary layer must meet itself at the tube centerline, and the velocity distribution then establishes a fixed pattern that is invariant. Hydrodynamic entrance length is that part of the tube in which the momentum boundary layer grows and the velocity distribution changes with length. The fixed velocity distribution in the fully developed region is called fully developed velocity profile. The steady-state continuity and conservation of momentum equations in two-dimensional are
These equations can be simplified by using scale analysis. At any point x ~ L in fully developed zone, we have y ~ D and u ~ U∞. Now, from equation(1), the transverse velocity component in fully developed region simplified using scaling as
In fully developed region the section of the duct flow far from inlet such that scale of transverse velocity component v is negligible(L>>d).So, in fully developed flow limit, continuity equation requires
Based on equation(5), the y momentum equation eq3 reduces to
this means that P is function of x only. From this, the x momentum equation becomes
Each term should be constant, because left side is function of x only and right is function of y. Solving equation(7) subject to the boundary condition
this results in the well-known Hagen–Poiseuille solution for fully developed flow between parallel plates.
where y is measured away from the center of the channel. The velocity is to be parabolic and is proportional to the pressure per unit duct length in the direction of the flow.
- Barenblatt, G. I. (1996). Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press. ISBN 0-521-43522-6.
- Tennekes, H.; Lumley, John L. (1972). A first course in turbulence. MIT Press, Cambridge, Massachusetts. ISBN 0-262-20019-8.
- Bejan, A. (2004). Convection Heat Transfer. John Wiley & sons. ISBN 978-81-265-0934-8.
- Kays, W. M., Crawford M. E. (2012). Convective Heat and Mass Transfer. McGraw Hill Education(India). ISBN 978-1-25-902562-4.
|The Wikibook Partial Differential Equations has a page on the topic of: Scale Analysis|