# What's the relationship between mach number and drag force on an airplane?

From what i've read drag coefficient at mach 1 is several times (up to 10x) the drag coefficient below drag divergence mach number. It drops as the mach number increases.

It's easy to find plots representing experimental data for Cd vs M, but I wonder what's the relationship between drag FORCE and mach number. I wasn't able to find a plot for this. I wonder if an aircraft that can reach Mach 1 can automatically reach say mach 1.2, given that even though Cd drops the drag still increases with the square of speed (and engine power required is cubed)

## 3 Answers

The plot you show is typical for airplanes which are not designed for supersonic flight. Yes, there is a maximum in the drag coefficient around or slightly above Mach 1, but in a properly designed airplane this maybe triples the subsonic drag coefficient and shows only a slight reduction as Mach number increases further. Take the F-16, for example: Here, the drag coefficient is nearly constant above Mach 1.1.

Your observation that due to the increase in dynamic pressure the drag force will increase with Mach number is correct. Some early designs with heavy and less powerful jet engines would just break the sound barrier but not even reach Mach 1.2. However, normally the pilot would keep dynamic pressure nearly constant and climb as Mach number increases. This, of course, makes the airplane fly at constant dynamic pressure so the drag force would indeed go down with increasing Mach number, but not as dramatical as in your plot.

F-16 drag coefficient from Ray Whitford's Fundamentals of Fighter Design lecture

• RE Here, the drag coefficient is nearly constant above Mach 1.2 – a reference and/or graph would be a nice addition.
– user14897
Feb 15 at 10:44
• @ymb1 Details are hard to come by. The best I have is a sketch straight from Ray Whitford's Fundamentals of Fighter Design lecture. Feb 15 at 19:20

The amount of drag is related to the coefficient of drag by the following formula:

$$D = C_d (\frac{\rho V^2}{2}) A$$

Where

• $$D$$ is the drag
• $$C_d$$ is the coefficient of drag
• $$\rho$$ is the density of the air the aircraft is moving through
• $$V$$ is the velocity of the aircraft
• $$A$$ is the reference area of the aircraft.

That last value may require some explanation. Drag is typically measured directly, such as in a wind tunnel, rather than being calculated. It's the coefficient that's usually calculated, and to do so, you need to pick an area. You can use the frontal area of the aircraft, the total skin area, or certain other areas. Each choice of reference area will yield a different value for the $$C_d$$, so to get from the coefficient back to the actual drag force, you need to know the reference area that was originally used to go the other way.

Aerodynamic drag is described in the form of a product of:

• Moving body shape parameter $$C_D$$
• Dynamic pressure q.
• Reference area.

For incompressible flow, dynamic pressure can be taken as $$\frac{1}{2} \rho V^2$$, with $$\rho$$ = air density. For compressible flow, dynamic pressure depends on the local speed of sound : q = $$\frac{1}{2} \gamma p M^2$$, with p being static air pressure and $$\gamma$$ = ratio of specific heats = 1.4 for air at sea level.

So the drag equation becomes $$D = C_D. \frac{1}{2} \gamma p M^2. S$$

Engine power cubed is of very little relevance in this situation. High subsonic and transsonic aeroplanes are powered by jet engines, which deliver exhaust thrust.