Modelling drag force of big spheres in air.

[Vigardo]

Hi all, I´m a researcher in computational biology thinking on topics quite far from my field. Would you help me? Thanks in advance!

I want to calculate the drag force in air of big rigid spherical and cylindrical objects at speeds form 0 to 100 Km/h. Their diameter would range from 1 to 20m.

Does anybody know where would I find this information?
Would you recommend me some good book?

 

[Angry Citizen]

Look at the Anderson book, Fundamentals of Aerodynamics. Look for the section in chapter 6 on spherical bodies, or if you know the coefficient of drag for your shape, you could check out the information on how to calculate D from Cd. Incompressible flow is a good assumption for the range you want. It also assumes inviscid flow, so if you have a viscous fluid, it won't be a great approximation.

May I ask what you're wanting to model?

 

[Vadar2012]

Since you're into computational stuff. You could write a code to calculate the drag using Newton's Method. All you do is enter the inital conditions and angle of attack and it will return the drag, drag coefficient etc. This method uses very simple assumptions, but it would give nice estimates with your range. It works by cacluating the pressure and integrating it over the defined surace. You have to enter in the geometry as points in global coordinates. I use this to give rough approximations before running CFD simulations.

 

[boneh3ad]

Since you're into computational stuff. You could write a code to calculate the drag using Newton's Method. All you do is enter the inital conditions and angle of attack and it will return the drag, drag coefficient etc. This method uses very simple assumptions, but it would give nice estimates with your range. It works by cacluating the pressure and integrating it over the defined surace. You have to enter in the geometry as points in global coordinates. I use this to give rough approximations before running CFD simulations.

Because, you know, angle of attack actually means something to a sphere...

If he is using simple shapes like this at velocities this comparatively low, there is no reason not to use use the simple C_D values that you can find all over the place.

 

[Vigardo]

Now I´m reading Anderson´s book. I´ll let you know if I don´t understand something.
Thanks all for your help!

 

[bigfooted]

If he is using simple shapes like this at velocities this comparatively low, there is no reason not to use use the simple C_D values that you can find all over the place.

A sphere of 20m going 100 km/hr? The Reynolds number for this would be of the order of $10^7$!

The explanation and values that you can find all over the place for calculating drag coefficients are for example here:
http://web2.clarkson.edu/projects/crcd/me437/downloads/0_2Pastsphere.pdf[/URL]
[URL="here(Clarkson University)"]http://web2.clarkson.edu/projects/crcd/me437/downloads/1_2Drag.pdf[/URL]
This will get you started hopefully.

Most people use something like this for the drag coefficient of solid spheres in air:
$C_D=\frac{24(1+0.15Re^{0.687})}{Re}$ with the Reynolds number based on diameter: $Re=\frac{\rho V D}{\mu}$

Most research involves non-deformable objects so if you're working with hot-air balloons you should take into account the deformation to get a better prediction of the drag coefficient

 

[Vigardo]

Thanks a lot Bigfooted, I´ll use your equations as soon as possible!
In principle, the balloon would be rigid, so the approximations should be ok.

And Angry Cityzen, I´m modelling a dirigible or balloon like object.

 

[Vadar2012]

Because, you know, angle of attack actually means something to a sphere...

If he is using simple shapes like this at velocities this comparatively low, there is no reason not to use use the simple C_D values that you can find all over the place.

Yeah, but where's the fun in that. It seemed to me like he wanted to actual model it somehow himself. The object might not of been fully spherical, and he might want to model other objects... Picky picky...

 

[boneh3ad]

A sphere of 20m going 100 km/hr? The Reynolds number for this would be of the order of $10^7$!

The explanation and values that you can find all over the place for calculating drag coefficients are for example here:
http://web2.clarkson.edu/projects/crcd/me437/downloads/0_2Pastsphere.pdf[/URL]
[URL="here(Clarkson University)"]http://web2.clarkson.edu/projects/crcd/me437/downloads/1_2Drag.pdf[/URL]
This will get you started hopefully.

Most people use something like this for the drag coefficient of solid spheres in air:
$C_D=\frac{24(1+0.15Re^{0.687})}{Re}$ with the Reynolds number based on diameter: $Re=\frac{\rho V D}{\mu}$

Most research involves non-deformable objects so if you're working with hot-air balloons you should take into account the deformation to get a better prediction of the drag coefficient

It may surprise you to know that 10⁷ is not really that high for Reynolds number for many objects. Unfortunately, it is way too high for the relation you quoted, which is only valid for roughly 1 < Re_D < 1000 and doesn't apply here.

At those Reynolds numbers in question (roughly 1.7 x 10⁶ to 3.4 x 10⁷), you start running up against the limit of available data. At the lower range you have trouble with predicting the drag coefficient because you are in a highly unsteady flow regime where von Kármán vortex shedding is important and the transition point from laminar to turbulent flow is highly variable. At the upper end of that range you would have a little bit easier time because the transition point would be less variable and the wake would be fully turbulent. Surface roughness would be important for both extrema. Your drag coefficient across this regime would fall typically between 0.1 and 0.5, but exactly where it falls would depend on the aforementioned factors.

A good, quick overview can be found at NASA: [url]http://www.grc.nasa.gov/WWW/k-12/airplane/dragsphere.html

 

[Vigardo]

Hi, thanks all again for your precious help and time!

I´ve been checking your equations and computing the Reynolds number for several speeds and diameters of a spherical particle. As you can check in the attached table I made (see below), these numbers surpase quickly the applicability range of the equation cited by "Bigfooted" (Re > 2·10⁵ value). In fact, only some values below 10km/h and 10m diameter fall within this range.
http://www.freeimagehosting.net/t/ax422.jpg

So, what equation would I use around 3.5~7 x 10⁷ Reynolds? Should I use the values extracted from the NASA´s plot cited by "Boneh3ad", i.e. C_d in the range [0.1-0.5] (see below)? Am I runing out of valid available data?
http://www.freeimagehosting.net/t/o95k2.jpg

By the way, perhaps I misunderstooded something, but in the links cited by "Bigfooted" the C_d equation valid range goes up to 2·10⁵, but "Boneh3ad" says: "is only valid for roughly 1 < ReD < 1000 and doesn't apply here". What do you think?

Thanks again!

 

[boneh3ad]

The link cited by bigfooted had a couple of different C_D equations. The one that he cited was for 1 < Re_D < 1000. There is also one for Re_D < 1. For 1000 < Re_D < 10⁵, the drag coefficient is approximately constant. Above Re_D = 10⁵, it is simply more complicated and there is almost certainly not a closed-form solution. Your best bet is to approximate from the plots.

 

[bigfooted]

I agree with boneh3ad that for such high Re it is better to find measurement data (and with high Re I mean you are not anymore in the Stokes or linear regime, and Cd becomes more complicated). Depending on the exact shape of your object and your Re, you are before or after the regime where turbulent flow separation occurs so it might be important to know if your object can be approximated as a sphere. There are many studies on the drag of spheres, but for balloon-shaped objects you can probably find some useful data on e.g. the NASA Technical Report Server.

 

[Vigardo]

Hi all,

1) The equation cited by bigfooted (eq. 35 in that reference) says "0 < Re ≤ 2x10⁵". From what boneh3ad says... should I assume that that equation limits are wrong? Which one is the valid in 1<Re<1000 range? (the eq. number of the reference will be enough, thanks)

2) From what you said, for Re_D greater than ≈2x10⁵, the following drag force (D) formula is not valid anymore, am I right?
D = 0.5 * Cd * ρ * V² * A
By the way,
- is this formula the one I should use below 2x10⁵ Re?
- the "A" is the frontal sphere area, i.e. π·r²? This one may be a bit stupid question, but I need to know it. For me is strange that you need to put the sphere diameter in both the Re number and the drag force (via A)...

3) Where would I find more information for air flows with Re_D ≥2 x10⁵ and basic rigid shapes like spheres and cylinders? Do you remember some book, web, or whatever? (I already tried google without much success...)

If exact Cd data for such a high Re is not available or drag is so shape (or surface roughtness) dependant that can not be determined without some detailed computational study, for me it would be ok just having some "valid range" of Cd to do some approximate stimations. Ok?

Thanks a lot! I really appreciate this!

 

[boneh3ad]

The equation cited by bigfooted (eq. 35 in that reference) says "0 < Re ≤ 2x105". From what boneh3ad says... should I assume that that equation limits are wrong? Which one is the valid in 1<Re<1000 range? (the eq. number of the reference will be enough, thanks)

In the first link provided by bigfooted, the exact same equation is cited as being valid for 1<Re_D<1000 range rather than 0<Re_D<10⁵. How do we know this first source is correct instead of the second source? We know that for Re_D>10⁵, the drag coefficient is effectively constant, so some power law formula probably isn't correct. Further, we have an analytical solution for the drag in the Stokes regime, so even though the equation you cite asymptotically approaches the Stokes solution, it isn't going to be exact. If you plot the function itself, you can tell that it doesn't work above 1000. Seriously, go try it.

From what you said, for ReD greater than ≈2x105, the following drag force (D) formula is not valid anymore, am I right?
D = 0.5 * Cd * ρ * V2 * A
By the way,
- is this formula the one I should use below 2x105 Re?
- the "A" is the frontal sphere area, i.e. π·r2? (this one may be a bit stupid question, but I need to know it)

No. That formula is pretty much always valid as long as the C_D you use is valid and rooted in reality for your situation. The problem is you can't always find a nice simple relationship for C_D, especially for more complicated shapes. At that point, you have to rely on empirical data or CFD solutions. Empirical data is usually better in those cases.

Where would I find more information for air flows with ReD ≥2 x105 and basic rigid shapes like spheres and cylinders? Do you remember some book, web, or whatever? (I already tried google without much success...)

You can get some information from books like Anderson's Fundamentals of Aerodynamics but you won't likely find an actual formula for the flows you are looking for most likely unless you find one that is just something like a triple-deck solution that is fit to the data. The best I have seen is essentially a curve fit that works up to 10⁶, and even that is just fitting a curve to experimental data. You can find that here if you are interested, but it still doesn't cover all of your range of sizes.

If exact Cd data for such a high Re is not available or drag is so shape (or surface roughtness) dependant that can not be determined without some detailed computational study, for me it would be ok just having some "valid range" of Cd to do some approximate stimations. Ok?

And this has already been given to you via the NASA plots.

 

[Vigardo]

Ok, I'll take all your answers into account. Thanks a lot for your help!

Regarding CFD and assuming it is a very complicated field... would it be possible for me, a non expert in aerodynamics, to perform some CFD for relatively basic shapes like spheres or cylinders? Can you recommend me some manual or tutorial "for dummies" and some software (free, if possible)?

Thanks a lot!

 

[boneh3ad]

CFD, even for simple shapes, is quite time consuming and has a steep learning curve. It is also expensive since the meshing software is expensive, the solvers are expensive and the post-processors are expensive, plus you need one beastly computer to run it if not a cluster or bigger. As much as I'd like to help you out here, I don't really have that sort of familiarity with any CFD package. I am, at this point in my life, an experimentalist, though I am considering looking for a post-doc in CFD.

 

[bigfooted]

CFD can be quite time consuming, which is why a lot of people are using simple fluid flow models, where the focus is the model for turbulence. Unfortunately these models are not really general in the sense that some models perform better than others for certain cases, but worse for other cases.

Creating a good mesh for your geometry, choosing a good model, and also choosing good solver settings can be quite tricky. This is especially so when trying to get good drag estimates.

I have a quad core at home and sometimes run some small test cases. Creating a 5 million cell mesh and converging a simple 2-equation turbulence model for steady state will take me less than half a day.

As an introduction you could try the book of Ferziger and Peric. Free software: you could try openfoam, but the learning curve is steep.

 

[boneh3ad]

Even CFD won't give a good answer to this problem, and here is why:

The drag in this problem, particularly in the Reynolds number ranges in which the OP is interested, is dominated by the pressure drag that arises from flow separation. The flow separation point is, first, highly unsteady in the lower portion of this Reynolds number range, and second, highly dependent on the state of the boundary layer (laminar vs. turbulent).

CFD has a reasonable shot at solving the unsteady separation point problem if the state of the boundary layer is known. The problem is, there is no truly accurate way to predict the transition point of a boundary layer over a general shape or even over simple shapes like a flat plate or a sphere. Because of that, CFD solutions of this problem would still rely on empirical relations for the transition location. At that point, it is easier to just pick the numbers off of the curve rather than simulate the whole thing.