The drag on an airplane wing in flight is known to be a function of density of air (??? viscosity of air (?), free-stream velocity (U?, characteristic dimension of the wing(S) and the shear stress on the surface of the wing (Ts).
Show that the dimensionless drag
can be expressed as a function of the Reynolds number
GIVEN
An airplane wing in flight
Drag on wing
FIND
The relevant variables and their dimensions are shown below
There are 5 variables and 3 primary dimensions. Therefore, the variables can be correlated with 2
dimensionless groups.
Using the Buckingham ? theory
In terms of the primary dimensions
Equating the sum of the exponents of each primary dimension to zero
For M: 0 = a + b + e [1]
For t: 0 = – b – c – 2e [2]
For L: 0 = – 3a – b + c + d – e [3]
Since there are 5 unknowns and only 3 equations, the value two exponents may be chosen for each dimensionless group
For ?1: Let e = 1 and a = –1
From equation [1]: b = 0
From equation [2]: c = –2
From equation [3]: d = 0
For ?2: Let a = 1 and b = – 1
From equation [1]: e = 0
From equation [2]: c = 1
From equation [3]: d = 1
As shown in Equation (5.24)
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