Q:

Dimensional Analysis: How to choose the repeating variables?

The aerodynamic lift (force) $F$ that acts on an aircraft to support its weight when at cruise conditions depends on the following quantities:

the speed of the aircraft relative to the air (the "air-speed") $v$,
the speed of sound in the air $c$,
the area of the wings $A$,
the density of the air $p$.

So in this question how do I choose the repeating variables to use Buckingham pi theorem in order to find the dimensionless variables? I am aware that at least there would be two dimensionless variables.

A:

Application of the Buckingham Pi Theorem for your problem goes as follows.
1.) List and count the $n$ variables in the problem. Here we have a lift force $F$, speed of the aircraft $V$, sound speed of air $c$, aircraft wing area $A$, and freestream density $\rho$. This gives us $n = 5$ variables. 
2.) List the dimensions of each variable according to their fundamental dimensions. For this problem we have: 
$$ F: MLT^{-2} \qquad V: LT^{-1} \qquad c: LT^{-1} \qquad A: L^2 \qquad \rho: ML^{-3}$$
where $M$ is a dimension of mass, $L$ is a dimension of length, and $T$ is a dimension of time.  
3.) Find the parameter $j$, which is almost always equal to the number of fundamental dimensions present in the problem. This will not always work, but for most aerodynamic problems, this is the rule of thumb. Here we have only $M$, $L$, and $T$ showing up in the problem. So we expect, $k = n-j = 5-3 = 2$ distinct non-dimensional Pi groups. 
4.) Now we have to select our repeating variables. We must select $j$ number of variables, and we have determined for this problem that $j = 3$. So we need a selection of 3 variables from the ones listed in step (1), but the selected variables must not form a non-dimensional group. It is very common practice and a rule of thumb in aerodynamics related problems to always select $A$, $V$, and $\rho$ as our repeating variables. Note, if you have a reference length instead of an area, you would select that in place of the area $A$. 
5.) Now we need to combine the remaining variables $F$ and $c$ with our selected repeating variables ($A$, $V$, and $\rho$) to form our $k = 2$ non-dimensional Pi groups. This is the fun part (depending on your definition of fun). We start by forming a system of equations subject to,
$$ \Pi_1 = A^x V^y \rho^z F = (L^2)^x (L T^{-1})^y (M L^{-3})^z (M L T^{-2}) = M^0 L^0 T^0$$
which yields the system of equations,
\begin{array}{lc}
L: & 2x+y-3z+1=0 \\
M: & z+1=0 \\
T: & -y-2=0 \end{array} 
The solution to this system yields, $x = -1$, $y = -2$, and $z = -1$, hence our first Pi group $\Pi_1$ is given by,
$$ \Pi_1 = \frac{F}{\rho V^2 A} $$
This is obviously a lift coefficient, which usually is scaled by 2 to make use of the freestream dynamic pressure, or equivalently, 
$$ \boxed{ \Pi_1 = C_L = \frac{F}{\frac{1}{2}\rho V^2 A} }$$
Now in a similar manner, the second non-dimensional Pi group $\Pi_2$ is given by,
$$ \Pi_2 = A^x V^y \rho^z c = (L^2)^x (L T^{-1})^y (M L^{-3})^z (L T^{-1}) = M^0 L^0 T^0$$
which yields the system of equations,
\begin{array}{lc}
L: & 2x+y-3z+1=0 \\
M: & z=0 \\
T: & -y-1=0 \end{array} 
Hence, our second Pi group $\Pi_2$ is given by,
$$ \Pi_2 = \frac{c}{V} $$
This is obviously an inverse Mach number, which usually is taken as the reciprocal and given as,
$$ \boxed{ \Pi_2 = M = \frac{V}{c}} $$
The selection of the repeating variables really comes with practice. You need at least $j$ repeating variables, but as previously mentioned $j$ is not always equal to the number of fundamental dimensions in the problem. Sometimes you have to be clever with which variables you pick, but in aerodynamics related problems we usually always form non-dimensional groups from repeating variables consisting of a reference length, velocity and density, or a reference area, velocity and density. Hope this helps!

