Why does Drag force equation contradicts the relationship between Reynolds number and drag coefficient?

Question

The drag force equation is described as:

$$ F_D = \frac{1}{2}\rho v^2 C_D A $$

Where the cross-sectional area of the front of the sphere is = πr^2,

so we can deduce that the radius and drag coefficient graph will look similar to this (the radius is on the x-axis and the drag coefficient is on the y-axis) :

But the Reynolds number equation shows that the radius and Reynolds number are directly proportional to each other.

$$ Re = \frac{\rho vL}{\mu} $$

Therefore the relationship between Reynolds number and drag coefficient should be similar to the relationship between radius and drag coefficient. But the graph below is different than the one above.

What is the actual relationship between the radius of a sphere and its drag coefficient? should it be deduced from the drag force equation or from the relationship between Reynolds number and drag coefficient?

Answer 1

FIRST EQUATION

You cannot obtain a graph of $C_D$ vs $r$ by holding the drag force constant, because the drag force increases with increasing r.

$C_D$ is generally assumed to be a constant, at least over a certain range of values (hence the name drag coefficient). This shows that over a range where C_D is relatively constant, drag force is proportional to cross-sectional area. That’s how the first equation would normally be used: you find or look-up a reasonable C_D for your situation and plug it into the equation with the other values to estimate the drag force (also called the drag).

There are some obvious things in that first equation: the amount of drag goes up as velocity goes up, as density goes up, as area goes up, and as drag coefficient goes up. The viscosity is not in the equation and would affect the drag coefficient directly.

SECOND EQUATION

The second equation is a definition equation. Thats the definition of reynolds number. It’s one of several dimensionless parameters used to characterize a flow situation - the most famous one. You can look at a reynolds number and see if it is a laminar or turbulent situation for example without knowing anything else. The $L$ in the equation is the radius of the sphere in your case.

Normally the drag coefficient is relatively constant across changing reynolds number, but not here. Regardless though, the drag force does not appear in that equation. Drag force uses the first equation. The fact that the coefficient is a function of reynolds number means you could make changes that keep reynolds number constant without changing that coefficient (meaning for example you could double both $\mu$ and L and not change the coefficient, because that change would not change reynolds number). But usually you look-up $C_D$ on there and use it in the first.

GETTING A VALUE TO START

To be extra clear, use Reynolds number and the chart you posted. The x-axis of that chart is exponential, so $C_D$ changes very slowly with reynolds number. You could probably assume $C_D$ is constant, depending upon what reynolds numbers you operate at. Just for example, if you conclude that the entire situation will be operating between $Re = 800$ and $Re = 8,000$... Well, $C_D$ at $800$ is maybe $0.6$ and at $8,000$ is about $0.5$. I would just assume constant of $0.55$, especially at first. Maybe later you could let it change with Re if necessary. Partly because fluid dynamics is not exact anyway.

Answer 2

The drag coefficient depends on an object's shape, the material on its surface, and any particular quirks about how the fluid interacts with the object and itself. It's made to do the best it can at being constant even if you change $A, v^2$ or $\rho$, but in reality it's not quite constant especially if you change one of those variables a lot.

The graph of reynolds number vs. $C_D$ is basically showing the imperfection of trying to represent those complicated relationships between the sphere and drag force as a function of velocity and/or length of the sphere (since $L$ in reynolds number refers to longitudal length and not cross sectional area)