Drag equation

The equation is easier understood for the idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area. No real object exactly corresponds to this behavior. ${\displaystyle c_{\rm {d}}}$ is the ratio of drag for any real object to that of the ideal object. In practice a rough un-streamlined body (a bluff body) will have a ${\displaystyle c_{\rm {d}}}$ around 1, more or less. Smoother objects can have much lower values of ${\displaystyle c_{\rm {d}}}$. The equation is precise – it simply provides the definition of ${\displaystyle c_{\rm {d}}}$ (drag coefficient), which varies with the Reynolds number and is found by experiment.

Of particular importance is the ${\displaystyle u^{2}}$ dependence on flow velocity, meaning that fluid drag increases with the square of flow velocity. When flow velocity is doubled, for example, not only does the fluid strike with twice the flow velocity, but twice the mass of fluid strikes per second. Therefore, the change of momentum per time, i.e. the force experienced, is multiplied by four. This is in contrast with solid-on-solid dynamic friction, which generally has very little velocity dependence.

The drag force can also be specified as

$${\displaystyle F_{\rm {d}}\propto P_{\rm {D}}A}$$

where P_D is the pressure exerted by the fluid on area A. Here the pressure P_D is referred to as dynamic pressure due to the kinetic energy of the fluid experiencing relative flow velocity u. This is defined in similar form as the kinetic energy equation:

$${\displaystyle P_{\rm {D}}={\frac {1}{2}}\rho u^{2}}$$

The drag equation may be derived to within a multiplicative constant by the method of dimensional analysis. If a moving fluid meets an object, it exerts a force on the object. Suppose that the fluid is a liquid, and the variables involved – under some conditions – are the:

• speed u,
• fluid density ρ,
• kinematic viscosity ν of the fluid,
• size of the body, expressed in terms of its wetted area A, and
• drag force F_d.

Using the algorithm of the Buckingham π theorem, these five variables can be reduced to two dimensionless groups:

• drag coefficient c_d and
• Reynolds number Re.

That this is so becomes apparent when the drag force F_d is expressed as part of a function of the other variables in the problem:

$${\displaystyle f_{a}(F_{\rm {d}},u,A,\rho ,\nu )=0.}$$

This rather odd form of expression is used because it does not assume a one-to-one relationship. Here, f_a is some (as-yet-unknown) function that takes five arguments. Now the right-hand side is zero in any system of units; so it should be possible to express the relationship described by f_a in terms of only dimensionless groups.

There are many ways of combining the five arguments of f_a to form dimensionless groups, but the Buckingham π theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by

$${\displaystyle \mathrm {Re} ={\frac {u{\sqrt {A}}}{\nu }}}$$

and the drag coefficient, given by

$${\displaystyle c_{\rm {d}}={\frac {F_{\rm {d}}}{{\frac {1}{2}}\rho Au^{2}}}.}$$

Thus the function of five variables may be replaced by another function of only two variables:

$${\displaystyle f_{b}\left({\frac {F_{\rm {d}}}{{\frac {1}{2}}\rho Au^{2}}},{\frac {u{\sqrt {A}}}{\nu }}\right)=0.}$$

where f_b is some function of two arguments. The original law is then reduced to a law involving only these two numbers.

Because the only unknown in the above equation is the drag force F_d, it is possible to express it as

$${\displaystyle {\begin{aligned}{\frac {F_{\rm {d}}}{{\frac {1}{2}}\rho Au^{2}}}&=f_{c}\left({\frac {u{\sqrt {A}}}{\nu }}\right)\\F_{\rm {d}}&={\tfrac {1}{2}}\rho Au^{2}f_{c}(\mathrm {Re} )\\c_{\rm {d}}&=f_{c}(\mathrm {Re} )\end{aligned}}}$$

Thus the force is simply ½ ρ A u² times some (as-yet-unknown) function f_c of the Reynolds number Re – a considerably simpler system than the original five-argument function given above.

Dimensional analysis thus makes a very complex problem (trying to determine the behavior of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.

If the fluid is a gas, certain properties of the gas influence the drag and those properties must also be taken into account. Those properties are conventionally considered to be the absolute temperature of the gas, and the ratio of its specific heats. These two properties determine the speed of sound in the gas at its given temperature. The Buckingham pi theorem then leads to a third dimensionless group, the ratio of the relative velocity to the speed of sound, which is known as the Mach number. Consequently when a body is moving relative to a gas, the drag coefficient varies with the Mach number and the Reynolds number.

The analysis also gives other information for free, so to speak. The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project.

To empirically determine the Reynolds number dependence, instead of experimenting on a large body with fast-flowing fluids (such as real-size airplanes in wind tunnels), one may just as well experiment using a small model in a flow of higher velocity because these two systems deliver similitude by having the same Reynolds number. If the same Reynolds number and Mach number cannot be achieved just by using a flow of higher velocity it may be advantageous to use a fluid of greater density or lower viscosity.

• Aerodynamic drag
• Angle of attack
• Morison equation
• Stall (flight)
• Terminal velocity

• Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0-521-66396-2.
• Huntley, H. E. (1967). Dimensional Analysis. Dover. LOC 67-17978.
• Benson, Tom. "Falling Object with Air Resistance". US: NASA. Retrieved June 9, 2022.
• Benson, Tom. "The drag equation". US: NASA. Retrieved June 9, 2022.