Question 6.3: Drag force F, on a high speed air craft depends on the veloc...
Drag force F, on a high speed air craft depends on the velocity of flight V , the characteristic geometrical dimension of the air craft l, the density \rho , viscosity \mu , and isentropic bulk modulus of elasticity E_{s} , of ambient air. Using Buckingham’s \pi theorem, find out the independent dimensionless quantities which describe the phenomenon of drag on the aircraft.
The physical variables involved in the problem are F, V, l, \rho, \mu \text { and } E_{s} ; and they are 6 in number. The undamental dimensions involved with these variables are 3 in number and they are, namely, M, L, T. Therefore, according to the \pi theorem, the number of independent \pi terms are (6 – 3) = 3.
Now to determine these \pi terms, V, l and \rho are chosen as the repeating variables. Then the \pi terms can be written as
\pi_{1}=V^{a} l^{b} \rho^{c} F\pi_{2}=V^{a} p^{b} \rho^{c} u
\pi_{3}=V^{a} \rho^{b} \rho^{c} E_{s}
The variables of the above equations can be expressed in terms of their fundamental dimensions as
\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}=\left(\mathrm{LT}^{-1}\right)^{a} \mathrm{~L}^{b}\left(\mathrm{ML}^{-3}\right)^{c} \mathrm{MLT}^{-2} (6.21)
\mathbf{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}=\left(\mathrm{LT}^{-1}\right)^{a} \mathrm{~L}^{b}\left(\mathrm{ML}^{-3}\right)^{c} \mathrm{ML}^{-1} \mathrm{~T}^{-1} (6.22)
\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}=\left(\mathrm{LT}^{-1}\right)^{a} \mathrm{~L}^{b}\left(\mathrm{ML}^{-3}\right)^{c} \mathrm{ML}^{-1} \mathrm{~T}^{-2} (6.23)
Equating the exponents of M, L and T on both sides of the equations we have,
from Eq. (6.21),
c + 1 = 0
a + b – 3c + 1 = 0
– a – 2 = 0
which, give, a = – 2, b = – 2, and c = – 1
Therefore, \pi_{1}=\frac{F}{\rho V^{2} l^{2}}
From Eq. (6.22), c + 1 = 0
a + b –3c –1 = 0
– a – 1 = 0
which give, a = –1, b = –1 and c = –1
Therefore \pi_{2}=\frac{\mu}{\rho V l}
from Eq. (6.23), c + 1 = 0
a + b –3c – 1 = 0
– a – 2 = 0
Which give a = – 2, b = 0 and c = – 1
Therefore, \pi_{3}=\frac{E_{s}}{V^{2} \rho}
=\frac{E_{s} / \rho}{V^{2}}
Hence, the independent dimensionless parameters describing the problem are
\pi_{1}=\frac{F}{\rho V^{2} l^{2}} \quad \pi_{2}=\frac{\mu}{\rho V l} and \pi_{3}=\frac{E_{s} / \rho}{V^{2}}
Now we see that \frac{1}{\pi_{2}}=\frac{\rho V l}{\mu}=\operatorname{Re} (Reynolds number)
And \frac{1}{\sqrt{\pi_{3}}}=\frac{V}{\sqrt{E_{s} /} \rho}=\frac{V}{a}=\mathrm{Ma} (Mach number)
where a is the local speed of sound.
Therefore, the problem of drag on an aircraft can be expressed by an implicit functional relationship of the pertinent dimensionless parameters as
f\left(\frac{F}{\rho V^{2} l^{2}}, \frac{\rho V l}{\mu}, \frac{V}{a}\right)=0Or \frac{F}{\rho V^{2} l^{2}}=\phi\left(\frac{\rho V l}{\mu}, \frac{V}{a}\right) (6.24)
The term F / \rho V^{2} l^{2} is known as drag coefficient C_{D} . Hence Eq. (6.24) can be written as
C_{D}=f(\mathrm{Re}, \mathrm{Ma})