Dimensional Analysis and Similarity

Introduction - The Purposes and Usefulness of Dimensional Analysis

The full-size wing, or prototype, has some chord length, cp, operates at speed Vp, and generates a lift force, Lp, which varies with angle of attack. In addition, the fluid properties of importance to this flow are the density and viscosity. After the preliminary design, it is usually necessary to perform experiments to verify and fine-tune the design. To save both time and money, these tests are usually conducted with a smaller scale model in a wind tunnel or water tunnel. In the sketch above, a geometrically similar model is constructed. In this case, the model is smaller than the prototype. In some cases the opposite is true; i.e. it may be prudent to build a large model of some small prototype in order to perform more accurate experimental analysis.

which is much simpler than the original functional relationship. In particular, instead of a dependent variable as a function of five independent variables, the problem has been reduced to one dependent parameter as a function of only two independent parameters. Furthermore, each of these three parameters is dimensionless, which makes them completely independent of the unit system used in the measurements.

Dynamic Similarity

If the model and prototype are geometrically similar (i.e. the model is a perfect scale replica of the prototype), and if each independent dimensionless parameter for the model is equal to the corresponding independent dimensionless parameter of the prototype, then the dependent dimensionless parameter for the prototype will be equal to the corresponding dependent dimensionless parameter for the model.

In this manner, we can set the wind tunnel speed properly to match Reynolds number. Then, after measuring the lift on the model wing, Lm, we can properly scale (using the last equation above) to predict the lift, Lp, on the prototype.

The Buckingham Pi Technique

Example: Lift on a wing in incompressible flow

Consider the case of incompressible flow over an airplane wing, as discussed in the previous lecture. Wing lift is known to depend on flow speed, angle of attack, chord length of the wing, and density and viscosity of the fluid. Let's examine this problem with the Buckingham Pi technique of dimensional analysis, following the steps outlined above:

Variable

Description

Dimensions

L

lift force

M(L)(t-2)

V

velocity

L(t-1)

c

chord length

L

density

M(L-3)

viscosity

M(L-1)(t-1)

angle of attack

1 (dimensionless)


Equating exponents of mass: 0 = 1 + c, or 
c = -1.
Equating exponents of time: 0 = -2 - a, or 
a = -2.
Equating exponents of length: 0 = 1 + a + b -3c, or 
b = -2.
Thus,


Likewise, construct the second Pi group using viscosity and the repeating variables:


Equating exponents of mass: 0 = 1 + g, or 
g = -1.
Equating exponents of time: 0 = -1 - e, or 
e = -1.
Equating exponents of length: 0 = -1 + e + f -3g, or 
f = -1.
Thus,


Note that this Pi group has been inverted in order to match the most well known dimensionless group in Fluid Mechanics, the Reynolds number. It would not be mathematically 
incorrect to leave it "upside down," but it is, shall we say, not "socially acceptable" to do so.


Notice that instead of a dependent variable as a function of five independent variables, the problem has been reduced to one dependent parameter as a function of only two independent parameters. The dependent Pi group on the left hand side is a lift coefficient (which normally has a factor of 2 thrown in for convenience), while the first independent parameter on the right is the Reynolds number, as discussed above.

Example: Dimensional analysis of a soap bubble

Consider a soap bubble. It is known that the pressure inside the bubble must be greater than that outside, and that surface tension acts like a "skin" to support this pressure difference. The pressure difference is then a function of surface tension and bubble radius. No other variables are important in this problem. Let's examine this problem with the Buckingham Pi technique of dimensional analysis, following the steps outlined above:

Variable

Description

Dimensions

pressure difference

M(L-1)(t-2)

surface tension

M(t-2)

R

bubble radius

L


Equating exponents of mass: 0 = 1 + a, or 
a = -1.
Equating exponents of time: 0 = -1 + b, or 
b = 1.
Equating exponents of length: 0 = -2 -2a, or 
a = -1.
Fortunately here, the first and third equation yield the same value of exponent a. If they did not, we would suspect either an algebra mistake or a non-physical setup of the problem. Our result is:



Notice that instead of a dependent variable as a function of two independent variables, the problem has been reduced to one dependent parameter as a function of nothing. In cases like this where there is only one Pi group, that Pi must be a constant. (If it is not a function of anything else, it must be a constant!)