# How to calculate Drag Coefficient?

• Victor Holomon
Victor Holomon
As I have looked through the internet and did my research I have found some information on how to calculate the drag coefficient, however none of what I have search for matches what I need. Currently I am doing a physics assignment for grade 12 and I am investigating the physics behind parachutes, for this task I want to mainly focus on how each different factor affects the decent speed of the parachute. So far what I have got are the equation for calculating the drag coefficient, however I am very confused with the equation which can calculate the drag coefficient. Here is how I am confused: according to the following websites:

https://www.grc.nasa.gov/WWW/k-12/VirtualAero/BottleRocket/airplane/rktvrecv.html

https://www.grc.nasa.gov/www/k-12/airplane/dragco.html

http://www.engineeringtoolbox.com/drag-coefficient-d_627.html

The drag coefficient can be calculated with equation(s) shown on the websites. The drag coefficient can be calculated with a velocity as one of the variables, as the velocity changes then the drag coefficient would change too (right?). From my current understanding the velocity is how fast the object/parachute is traveling in the medium (assuming terminal velocity). But the velocity which the parachute is traveling through the medium depends on other factors such as mass (am I right?).

However these websites below tells me that no matter what size or mass these objects have, the drag coefficient would remain the same/relative to the corresponding forms/shapes of the object:

https://www.grc.nasa.gov/www/k-12/airplane/shaped.html

https://en.wikipedia.org/wiki/Drag_coefficient

http://www.aerospaceweb.org/question/aerodynamics/q0231.shtml

Wouldn’t the mass or size of these objects change the velocity which they travel through the medium? Thus changing the drag coefficient according to the equation? I am very confused.

houlahound
The drag would be the same for an object not moving, in fact that's how it is calculated in a wind tunnel.

Victor Holomon
The drag would be the same for an object not moving, in fact that's how it is calculated in a wind tunnel.
What would I do with the velocity variable in the drag coefficient equation?

houlahound
Please post the equation and link. Drag coefficients do not contain velocity.

you might mean drag force which is velocity dependent typically on v or v^2.

houlahound
As I suspected, plug in units both sides. The drag force and drag coefficient are different things.

What are you trying to calculate?

Victor Holomon
As I suspected, plug in units both dived. The drag force and gray coefficient are giggetent things.

What ate youbtrying to calculate?
I am certainly not good at physics.

I am trying to calculate the decent velocity of a parachute if given:
• Area of the Parachute
• Mass on Parachute
• Air Density

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Victor Holomon
In short, I want to use the equation below to workout the decent velocity:
v = sqrt( (8 m g) / (pi r Cd D2) )

Where
• D is the chute diameter in meters
• m is the rocket mass in kilograms
• g is the acceleration of gravity = 9.8 m/s2
• pi is 3.14159265359
• r is the density of air
• Cd is the drag coefficient of the chute
• v is the speed we want at impact with the ground (3 m/s or less)
but how do I work out the Drag Coefficient for a parachute?

houlahound
Personally i would tie one end to a force meter and the other end of the force meter to a fan. You will also need a wind speed meter.

Measure the wind speed and force, calculate the drag coefficient.

Gold Member
• Victor Holomon
Gold Member
At the end of the day, you generally can't "calculate" the drag coefficient except for a few very specific shapes. You have to determine it experimentally in most cases, usually by measuring the force at a known velocity and back-solving the equation. In effect, a drag coefficient attempts to distill a lot of very complex physics into a very simple relation between force and dynamic pressure.

Also, the velocity in question is the relative velocity between the object and the medium through which it travels.

Mentor
As I have looked through the internet and did my research I have found some information on how to calculate the drag coefficient, however none of what I have search for matches what I need. Currently I am doing a physics assignment for grade 12 and I am investigating the physics behind parachutes, for this task I want to mainly focus on how each different factor affects the decent speed of the parachute. So far what I have got are the equation for calculating the drag coefficient, however I am very confused with the equation which can calculate the drag coefficient. Here is how I am confused: according to the following websites:

https://www.grc.nasa.gov/WWW/k-12/VirtualAero/BottleRocket/airplane/rktvrecv.html

https://www.grc.nasa.gov/www/k-12/airplane/dragco.html

http://www.engineeringtoolbox.com/drag-coefficient-d_627.html

The drag coefficient can be calculated with equation(s) shown on the websites. The drag coefficient can be calculated with a velocity as one of the variables, as the velocity changes then the drag coefficient would change too (right?). From my current understanding the velocity is how fast the object/parachute is traveling in the medium (assuming terminal velocity). But the velocity which the parachute is traveling through the medium depends on other factors such as mass (am I right?).

However these websites below tells me that no matter what size or mass these objects have, the drag coefficient would remain the same/relative to the corresponding forms/shapes of the object:

https://www.grc.nasa.gov/www/k-12/airplane/shaped.html

https://en.wikipedia.org/wiki/Drag_coefficient

http://www.aerospaceweb.org/question/aerodynamics/q0231.shtml

Wouldn’t the mass or size of these objects change the velocity which they travel through the medium? Thus changing the drag coefficient according to the equation? I am very confused.
You are confusing the two sides of the equation. The drag force and drag coefficient depend on the velocity but, from the other side of the force balance equation, the force depends on the weight (mass) of the object. When you set the two sides of the equation equal to one another, you can solve for the velocity. But, as far as the drag force is concerned, you only need to know the velocity.

David Lewis
...as the velocity changes then the drag coefficient would change too (right?)

It depends. On a small parachute, CD can vary significantly with respect to speed because Reynolds number goes up in direct proportion to speed (and in most cases, CD goes down as Re goes up). Whereas the CD of large parachutes tends to be fairly invariant with respect to Re.

Gold Member
At the end of the day, you generally can't "calculate" the drag coefficient except for a few very specific shapes. You have to determine it experimentally in most cases, usually by measuring the force at a known velocity and back-solving the equation. In effect, a drag coefficient attempts to distill a lot of very complex physics into a very simple relation between force and dynamic pressure.
I'm wondering if there are any online or free tools that would allow calculation of drag force or drag coefficient, for any arbitrary shape that I define. I find some interesting free CFD tools available, such as Flowsquare (2D), simFlow (3D), and OpenFoam (3D), but these seem to be more for visualizing flows and vortices than actually calculating practical things like drag.

Gold Member
I'm wondering if there are any online or free tools that would allow calculation of drag force or drag coefficient, for any arbitrary shape that I define. I find some interesting free CFD tools available, such as Flowsquare (2D), simFlow (3D), and OpenFoam (3D), but these seem to be more for visualizing flows and vortices than actually calculating practical things like drag.

The only one of those that I have heard of is OpenFoam, and it is a lot more powerful than just visualization. It is a C++ library used to generate your own custom solvers and could absolutely calculate approximate drag assuming you know how to program a solver to do it.

ahiddenvariable
Experience has shown that the drag force on an object in a fluid stream is proportional to the dynamic pressure, 1/2*rho*V^2, and the drag coefficient is the empirically determined constant of proportionality. Experiment has shown that this proportionality constant is usually well defined for a specific geometry, although it does depend upon the Reynold's Number of the flow in complicated ways for each geometry. Use of this quantity does, however, greatly simplify drag calculations, as all you need as inputs are the specific geometry, e.g., sphere, cylinder, etc., and Reynold's number. Numerical values can be found in most aerodynamic textbooks. In other words, the drag coefficient is not calculated. It is tabulated for specific geometries and Reynold's numbers.

Gold Member
The only one of those that I have heard of is OpenFoam, and it is a lot more powerful than just visualization. It is a C++ library used to generate your own custom solvers and could absolutely calculate approximate drag assuming you know how to program a solver to do it.
After further reading, simFlow turns out to be a GUI wrapper for OpenFoam. Man, that really bogs down my laptop, this stuff is computation-intense. I downloaded it because I found a demo showing how to get the drag coefficient for a 3-dimensional shape.
ahiddenvariable said:
Experiment has shown that this proportionality constant is usually well defined for a specific geometry, although it does depend upon the Reynold's Number of the flow in complicated ways for each geometry.
My Reynolds number is in the 50K-100K range. This appears to be taken care of automatically in the software, given the velocity input and size of the object. I'm not interested in drag coefficients for standard objects like spheres and cylinders, I need to know the drag coefficient for a custom geometry. For my purposes, a 2D software package would work, but the only 2D one I've found is just for visualization of flows.

ahiddenvariable
My Reynolds number is in the 50K-100K range. This appears to be taken care of automatically in the software, given the velocity input and size of the object. I'm not interested in drag coefficients for standard objects like spheres and cylinders, I need to know the drag coefficient for a custom geometry. For my purposes, a 2D software package would work, but the only 2D one I've found is just for visualization of flows.
Okay, I understand what you want to do. It's not easy to make such calculations on arbitrary 3-D shapes, but some are harder than others. There are complications that arise with transitions from laminar flow to turbulent flow, and these transitions at the surface of the body can cause huge changes in the global flow patterns. For instance, for spheres and cylinders, there form Von Karman vortex streets, or the shedding of vortices into the wake in alternative fashion from one side of the body to the other. It seems to me that any software package that can calculate the drag in such instances would have to be quite sophisticated. In addition, it may require much user input to guide the software during the calculation. I myself am not very familiar with any of the most recent software packages, but I'd guess that they are quite sophisticated. I thus suggest that you need to focus on a specific geometry for a specific range of Reynold's number. Do you have a particular shape in mind? What is it? In order to calculate a Reynold's number, you must have some idea of the overall size of the object.

Gold Member
Do you have a particular shape in mind? What is it? In order to calculate a Reynold's number, you must have some idea of the overall size of the object.
Yes, I know exactly the size of the object and the velocity range it would experience, which would determine the Reynolds number. The object is a water rocket, made from a soda bottle. Given that I have a nicely shaped nose cone, I want to get the drag coefficients for the model using different taper shapes at the tail end. The length is about 0.5 meters, the maximum velocity is about 70 m/s, giving a Reynolds number of about 2×106. (I wrote 50K-100K in a previous post, but that was just for the fins; my error.)

The drag coefficient of a bullet (a hemisphere on a cylinder) is 0.295 according to this NASA page: https://spaceflightsystems.grc.nasa.gov/education/rocket/shaped.html

That same page also says (at the bottom) that the drag coefficient of a model rocket is 0.75. Now, why would it be higher than a bullet shape? It seems to me that, for the same cross-sectional area and body length, a bullet with a pointy ogive-shaped nose would have a lower drag than a round-nose bullet. If they're including the drag of the fins too, then that might explain the value of 0.75, in which case 0.75 would be too high for the drag coefficient of just the rocket body.

The page also says the drag can be reduced by adding a tapered fairing on the tail end.

So I wanted to see for myself, using some CFD software, what the drag coefficient actually comes out to be, not for idealized shapes, but for my specific shape, with different variations in nose and tail geometries. Because a rocket body is radially symmetric, I thought perhaps a 2D software package would be sufficient, but the only one I found that can generate drag coefficients is simFlow+OpenFoam (see the demo I linked in my previous post above). I suspect my computer may not be up to the task -- not to mention a really steep learning curve.

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Gold Member
It seems to me that any software package that can calculate the drag in such instances would have to be quite sophisticated.

Quite. They also do a very poor job of doing this. You've just touched on one of the biggest weaknesses of modern CFD. It is impossible to predict the transition point in a boundary layer in general, so CFD software has to resort to (usually) rather poor correlation methods to try to determine the transition point. Otherwise, the user has to specify a point. Afterward, the software has to use one of several turbulence models to model quantities of interest since the equations and flow are too complex to calculate directly for all but the simplest of problems.

That same page also says (at the bottom) that the drag coefficient of a model rocket is 0.75. Now, why would it be higher than a bullet shape?

Remember that the drag coefficient is different from the drag itself. It can also be a strong function of Reynolds number depending on the application, so there is more at work here than just the shape of the nose. For example, a bullet moves considerably faster than your rocket, so even with a lower drag coefficient, its drag may still be higher once you multiply by the dynamic pressure. Like I mentioned before, the drag coefficient distills a lot of complicated physics down into a single number and it therefore isn't always intuitive.

So I wanted to see for myself, using some CFD software, what the drag coefficient actually comes out to be, not for idealized shapes, but for my specific shape, with different variations in nose and tail geometries.

That's all well and good, but you have to realize that drag is one of the most difficult things for software to calculate accurately given all of the physics involved. Some of the free packages may be able to handle it in a rudimentary sense, but if you want to actually capture all of the effects, you will need something a bit more robust, which can get pricey. Otherwise, you can use other free options such as OpenFOAM or just program a solver yourself in C++, but this is going to requires a very intricate knowledge of both fluids and numerical methods, which are both non-trivial. The computation time is going to be rather large, to boot.

Because a rocket body is radially symmetric, I thought perhaps a 2D software package would be sufficient, but the only one I found that can generate drag coefficients is simFlow+OpenFoam (see the demo I linked in my previous post above).

Any random 2D code will not be able to handle an axisymmetric shape. Axisymmetry does simplify the equations considerably, but they are not the same as the 2D equations you would expect from Cartesian coordinates. They are still only 2D, but they include radial terms that come from the fact that they are rooted in a cylindrical (or spherical) representation of the Navier-Stokes equations. You need to find a solver that can handle axisymmetric flows.

I suspect my computer may not be up to the task -- not to mention a really steep learning curve.

There is a very, very good chance you are correct here. Most CFD is performed on high-performance clusters or even supercomputers because it can be so computationally intensive. You can do simpler simulations on your home computer, but if you want to include the effects of things like viscous drag (skin friction) and wakes, then you have to resort to larger computations. The learning curve is absolutely going to be steep as well. There's a reason that Boeing pays people the big bucks to run these codes.

If you are interested in perhaps a more approximate method, look for a code that only solves the Euler equations. This would considerably reduce your computational complexity and time. The consequence is that the result would no longer take into account viscosity (the Euler equations are inviscid flow equations). This would give you the roughly correct relationship between the shape and the drag as a result of inviscid/pressure effects.

Gold Member
Remember that the drag coefficient is different from the drag itself. It can also be a strong function of Reynolds number depending on the application, so there is more at work here than just the shape of the nose. For example, a bullet moves considerably faster than your rocket, so even with a lower drag coefficient, its drag may still be higher once you multiply by the dynamic pressure.
Well, the text of that page https://spaceflightsystems.grc.nasa.gov/education/rocket/shaped.html implies that all the drag coefficients shown are apples-to-apples comparisons of shapes that NASA tested at similar conditions in a wind tunnel. So I assume the "bullet" just refers to the shape (hemisphere-capped cylinder), not the speed at which a bullet actually flies.

And that's why I don't understand why they say a bullet shape has a lower drag coefficient than a rocket shape. It's almost like they missed a zero there, and meant 0.075 instead of 0.75. Or maybe it was a typo and they meant 0.25 (less than a bullet shape) instead of 0.75. That's a mystery to me, and I was hoping to resolve it using CFD, since I don't have a wind tunnel handy. Ironically, NASA Ames, which has wind tunnels and is probably the place that did the tests and wrote that page, is just a mile from where I live.

Some of the free packages may be able to handle it in a rudimentary sense, but if you want to actually capture all of the effects, you will need something a bit more robust, which can get pricey.
simFlow has a free version. You need to install OpenFoam to use it. simFlow is basically a graphical front-end for OpenFoam. But it sure bogs down my computer when I try to run it.

Thanks for explaining why 2D code probably won't work.

I have developed a pretty sophisticated physics-based simulation of a water rocket flight, accounting for everything I can think of, such as adiabatic expansion of the air inside the bottle, moderating effects of water vapor on temperature and pressure, the impulse of excess pressure after the water is gone, temperature-dependent choked airflow, wind resistance, and other factors. The one, single, biggest contributor to the calculation of maximum altitude is drag coefficient. The simulation is amazingly sensitive to it. Reducing CD from 0.295 to 0.25 results in a 10% increase in final altitude.

I'm thinking now, if I've done everything correctly in my simulation, then the best approach would be to actually build and fly the darn rocket using a recording altimeter, and overlay the measured altitude data over my prediction, and adjust the drag coefficient for the best fit. Then repeat for different geometries.

ahiddenvariable
Anachronist (A name I like.): It seems to me that a bullet shape is basically like a simplified rocket shape minus the fins, and as you suggest, the fins can cause significant increased drag. Remember, a large part of fin operation is the drag they create. Because they are thin, they don't have much form drag (pressure times cross flow area), and so skin drag (friction) is probably their major feature. In my experience, the front part of an object is not as important as the downstream parts, in so far as drag is concerned. I'm talking here about subsonic flow. Hence the significant effect on drag by tapering the rear portions. A large diameter end plane with abrupt ending will produce much form drag, because flow separation in that region restricts the recovery of air flow kinetic energy resulting in lower pressure acting on the back of the body. Unfortunately, a large diameter end plane is necessary for a first stage containing an engine. The long tapered shapes are common with bombs. I hope you're not designing a bomb! But a bottle rocket would also allow a large reduction of cross area and still allow much kinetic energy recovery.

If you can find it, software that calculates axisymmetric flow geometries may be helpful. Axisymmetric shapes are 2-D shapes, and they have their own advantages and disadvantages, but my gut feeling is that they are more accurate than software that approximates them with 2-D planar shapes.

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Gold Member
The long tapered shapes are common with bombs. I hope you're not designing a bomb! But a bottle rocket would also allow a large reduction of cross area and still allow much kinetic energy recovery.
No, not building a bomb, just a water rocket. The nozzle of the bottle would be the tail end of the rocket, and if you look at any 2-liter soda bottle, the shape of that end is basically a hemisphere on a cylinder. The whole rocket body could be approximated by a hemisphere cap on both ends of a cylinder. So I'm wondering if I built a cone-shaped fairing on the tail end, would it have a significant or negligible effect on the drag coefficient?
If you can find it, software that calculates axisymmetric flow geometries may be helpful. Axisymmetric shapes are 2-D shapes, and they have their own advantages and disadvantages, but my gut feeling is that they are more accurate than software that approximates them with 2-D planar shapes.
I did find a couple that look like they do this. One is ANSYS Fluent (they have a student edition) and another is EasyCFD (20 day free trial).

Thanks for the help and advice!

David Lewis
It seems to me that, for the same cross-sectional area and body length, a bullet with a pointy ogive-shaped nose would have a lower drag than a round-nose bullet
A paraboloid nose gives the lowest CD in your situation. However, a spherical nose is almost as good. Even a flat nose with rounded corners isn't too bad. The rear end of the rocket, however, has a big influence on drag.

I think the research on the ogive bullet shape was originally an investigation into supersonic drag. The results were used to design the nose of the Bell X-1. Apparently the nose shape is more important at supersonic speeds than the rear.

Gold Member
A paraboloid nose gives the lowest CD in your situation. However, a spherical nose is almost as good. Even a flat nose with rounded corners isn't too bad. The rear end of the rocket, however, has a big influence on drag.
I realize that now. Given the shape of a 2-liter bottle and the constraints of a launcher mechanism that prevents structure from extending much past the nozzle, I'm kind of stuck with what I have. I could construct a short conical taper toward the nozzle retainer collar (which needs to remain uncovered so the launcher can grab it), and this would be slightly gentler than the bottle taper.

But my question to which you responded was wondering, for two shapes with a flat rear end (a bullet and a rocket), measured in similar conditions as that NASA page suggests, WHY would the rocket have a higher drag coefficient than a bullet?

David Lewis
The rocket could be using a different reference area. Rockets typically go by surface area, whereas bullets use frontal area.