AE 3005 Handout # 3
Nondimensionalization of the Viscous Flow Equations
Consider the 2-D viscous flow past an airfoil shown below:
At first glance it appears that such a flow will depend on a large number of parameters such as (a) airfoil shape, including its chord c ,(b) its angle of attack, (c) the freestream temperature T∞ (d) the freestream density
r∞ , (e) the freestream velocity V∞, (f) the freestream viscosity m∞, (g) the freestream conductivity k∞, and so on.
We wish to know if these large number of physical and geometric parameters can be grouped into a handful of parameters that can be systematically varied to study their effect on the flow. There are two common ways of identifying these parameters:
(i) Dimensional analysis: Here we attempt to combine the parameters listed about to arrive at a nondimensional form. For example, after some trial and error, we can show that the quantity
r∞Vc/m∞ is a nondimensional quantity. Intuition and experience are needed to realize that this nondimensional parameter, called Reynolds number, is also a useful physical parameter. Dimensional analysis will also produce combinations such as
. This parameter, while nondimensional, is not a significant parameter in incompressible flows, as experience shows.
(ii) Nondimensionalization of Governing Equations: This approach provides a formal manner by which nondimensional parameters of physical significance may be uncovered. This approach is usually used in combination with the dimensional analysis shown above. To illustrate how this approach works, let us consider the 2- flow over an airfoil shown above.
Let us introduce nondimensional quantities identified with a prime (') as follows:
When these nondimensional quantities are used to replace the corresponding physical quantities in the continuity equation the following form results:
This form looks identical to the dimensional form given in Handout #1, and no new nondimensional parameter emerges.
If we repeat this process with the u- momentum equation, and replace terms such as
txy and p with terms such as
we get, after some minor algebra:
If we repeat this process with the v- momentum equation, again the Reynolds number emerges as the nondimensional parameter.
Consider two different flows over the same airfoil, but of different chord. The present normalization shows that these two flows are governed by the same nondimensional form of the governing equations, and will result in the same nondimensional flow quantities such as u', v' etc. if the flows are geometrically similar (same airfoil shape, same
a), and dynamically similar (same Reynolds number).
The parameter Reynolds number may also be interpreted in another way. Consider a typical inviscid term such as p appearing on the left side of the u- momentum equation. This "inertial" force term is roughly of order 
. Consider a viscous stress term such as
. Then, Reynolds number is a measure of the ratio between the inertial forces and viscous forces acting on a fluid element. If this number is large, then inertial forces dominate over viscous forces and vice versa.
In practical aeronautical applications, the Reynolds number is invariably large. For example, the airfoil over a helicopter rotor blade operates at a Reynolds number of 3 to 5 Million, while the airfoil in the Root section of a modern transport aircraft operates at a Reynolds number of 30 to 50 Million. In such flows, viscous forces are small and may be neglected over most of the flow, except in very small regions called boundary layers over the airfoil.
Nondimensionalization of the Energy Equation: If we nondimensionalize the energy equation by the same principles, we get, in addition to the Reynolds number, the parameter 
. This parameter occurs due to the presence of the heat conduction terms and viscous stress work terms in the energy equation. This parameter is called the Prandtl number, and is a property of the fluid. For air, Prandtl number is around 0.72.
In compressible flows, the quantity 
is a nontrivial quanity and must be prescribed. Using equation of state, the definition of the speed of sound and the definition of Mach number, we can relate this quantity to the Mach number and the ratio of specific heats,
Thus, in compressible flows, two flows are identical if they are geometrically similar and if Reynolds number, Mach number, Prandtl number and ratio of specific heats all match.
In incompressible flows, this quantity is not a significant parameter. As hydraulics engineers will attest, raising the static pressure at some point in the flow simply raises the pressure level everywhere by the same constant level, and does not alter the flow behavior.