# Is there a mathematical formula to calculate drag force without empirical testing?

Typically people put the interested rocket prototype in a wind tunnel to find out the drag force and use the drag force equation to calculate the drag coefficient of the rocket model. I am wondering whether there is a mathematical formula that we can use to calculate the drag force without empirical measurements. For example, can we simply look at the system of one air molecule and the rocket, calculate the instantaneous change in momentum of the air molecule after collision as the resistive force, and then sum up the forces experienced by all air molecules to get the drag? The motivation behind this question is that I think if we can relatively accurately simulate drag force and find out the drag coefficient of a model real-time while we are doing the prototyping on software, it will accelerate the process of model improvements. Thanks!

• Something close to what you ask is what can be done in simulations. Ansys Fluent, among several other CFD software, can be used to get almost every parameter you are looking for in the flow. These simulations require an in-depth understanding of various mathematical models of the fluid. It is easy to do CFD, but difficult to do good CFD. It is basically just the computer solving equations in iterations, and it remains your responsibility to tell it which ones are the best to use for your scenario. – William R. Ebenezer Sep 17 at 19:22
• "It is easy to do CFD, but difficult to do good CFD." bravo! – Organic Marble Sep 17 at 21:32
• Simulating single molecule colliding with the surface of moving object - for very very rarified air maybe it can work, I'd like to know the answer myself. But for dense enough atmosphere we just can't use this approach. The reason is a molecule bounced from moving object will immediately collide with another molecules and exchange its energy, these molecules will transfer it further and so on. As result in macroscopic level we will have pressure waves around the moving object and the interplay of these waves will define the drag, the lifting force, etc. – Heopps Sep 18 at 8:56

I am wondering whether there is a mathematical formula that we can use to calculate the drag force without empirical measurements.

Yes, much modern rocket design is done with Computational Fluid Dynamics software instead of in wind tunnel testing.

can we simply look at the system of one air molecule and the rocket, calculate the instantaneous change in momentum of the air molecule after collision as the resistive force, and then sum up the forces experienced by all air molecules to get the drag?

That isn't a practical way to do it, because of the staggering number of air molecules involved; I believe modern CFD implementations subdivide the volume around the rocket into (millions of?) small volume regions where the aggregate airflow (pressure, velocity) is simulated, and use heuristics to decide where to subdivide volumes further to get finer-grain simulation where things get turbulent and complicated.

• +1 Before you youngsters and your new-fangled computers took over there were entire books written on estimating drag (Hoerner 1965 Fluid-Dynamic Drag) – Organic Marble Sep 17 at 19:04
• Thanks for your response! Could you elaborate more on how the airflow is simulated within the small volume regions or simply refer me to some other resources? I am actually researching this for a school project and am exploring how we can speed up the calculation of drag force by CFD software. Do you think using quantum algorithms can help? – Xi Liu Sep 17 at 19:11
• There are many references on the Wikipedia page I linked in my answer; they should keep you busy for a while. I don't know much about CFD myself. – Russell Borogove Sep 17 at 21:04
• I was under the impression that CFD algorithms are based off of empirical measurements of simple cases, rather than the implementation of purely mathematical equations. – BillThePlatypus Sep 18 at 14:12
• Note also that the process of subdividing the volume is itself a tricky art form. If you choose the wrong grid to discretize your problem, you will get weird, unphysical, incorrect behavior due to the limitations of the discrete model, like reflections and wakes from surfaces that aren't really there, but are subtly implied by the grid structure you chose. If you have to interpolate the results to a different grid needed for some reason by another part of your toolchain, things get much worse. In summary, +1 to bad CFD is easy, but good CFD is really hard. – Ryan C Sep 18 at 17:28

It is possible to compute rather than measure the drag on an object. However the answer to the specific technique you suggest:

can we simply look at the system of one air molecule and the rocket, calculate the instantaneous change in momentum of the air molecule after collision as the resistive force, and then sum up the forces experienced by all air molecules to get the drag?

is no: you can't do this.

Here is one argument that makes it clear why this is not possible.

One thing that happens when some object is passing through a fluid is that it heats the fluid. A famous example of this is something entering the atmosphere from space: the object entering the atmosphere gets hot, because the fluid (atmosphere) it's passing through is heated adiabatically. A lot of the kinetic energy of the object is dumped as heat into the fluid.

But the temperature of a fluid (or in fact of any object) is a statistical property: it only has meaning when you consider very large numbers of particles of the gas.

This means that you can't simply consider how a single particle of gas bounces off the object and somehow multiply it up to compute the drag. You have to look at how the particles of the fluid interact with each other.

So any mathematical model of drag forces must do one of three things:

• treat the fluid as a huge number of particles with individual velocities and momenta and model the interactions between this huge number of particles, the object and each other;
• based on understanding the statistical mechanics of the first model derive some equations for how fluids behave, and then treat the fluid as a fluid, with properties such as temperature, pressure and so on;
• derive some simpler expressions for drag based on understanding the fluid which will give approximate answers, usually requiring various experimentally-determined parameters to be added.

All of these approaches are 'mathematical formulae', but only the last one comes anywhere close to being something you could, for instance, calculate with the aid of a calculator in a few hours.

The last of these is what was done before computers existed. The expressions you would get would generally only be valid for certain ranges of parameters as they would fail to deal well with phenomena like turbulence and shocks. Anyone using this approach would be well-advised to check their results using a model in a wind-tunnel, and this is what they did.

The first approach is not computationally practical except in very special cases and probably never will be.

The second approach is very widely used. There are some reasons why wind-tunnels are still useful however.

• The calculations are extremely numerically intense: supercomputers were and are built mostly to solve these problems (not only drag problems but general fluid-dynamics problems). This means that, with limited computing power, the calculations can only be done rather approximately (generally this means that the chunk of the fluid – the element, grid-box or cell – that is the smallest unit you can deal with is rather large. This in turn means that various features of the problem can't be modelled properly and must be 'parametrised'. If important behaviour happens below the scale of an element you either simply miss it or do some clever trick of noticing that things are being missed and successively shrinking the element size until you capture the behaviour. This makes the model still more computationally intense: sometimes it's easier, cheaper and quicker to just put something in a wind-tunnel, if you want an answer this year, especially in the era when all you had was a Cray-1 or something.
• Even if the computational model 'works' it will have been built using a necessarily simplified model of the fluid: the equations which govern fluid flow are nonlinear and have all sorts of complicated behaviour like turbulence and shocks which may or may not be properly captured in the model and which can cause tiny numerical errors due to lack of precision in the number representation to blow up. You certainly will want to go and check the model's results against a real fluid, if you can.
• Programs are often just buggy, numerical algorithms are often unstable, computers are often buggy.

All of these things mean that computational approaches, while profoundly valuable, especially where experiment is expensive or forbidden, have limits and actually doing experiments is still useful.

• "You have to look at how the particles of the fluid interact with each other." Hence Computational Fluid Dynamics, which has widely supplemented wind tunnel work, though it hasn't yet entirely replaced it. – Russell Borogove Sep 17 at 18:45
• +1 There was a down vote, I think they missed that you've simply offered an existence proof which is all that's necessary to support "No, you can't do this." – uhoh Sep 17 at 18:46
• The existence proof that refutes "no, you can't do this" is the entire field of CFD. There's no simple formula that works, but you can absolutely do drag simulations without a wind tunnel, which is how I interpret OP's question. – Russell Borogove Sep 17 at 18:46
• @uhoh: yes I think whoever downvoted me is entirely missing my point, which was entirely to offer a sufficient reason why summing over a single molecule/object collision will not work. Oh well. – tfb Sep 17 at 23:27
• @uhoh: Yes, I think that that's close to how I read the question: I'd not thought that the person might be answering 'is this computable at all?'. I specifically was aiming at the suggestion that you could simulate one molecule and then sum. I've reworded the answer extensively now to, I hope, be clearer (and longer!). If you have time have a look to see if you think it's better. – tfb Sep 18 at 11:03

There is a mathematical formula, but it requires knowing the pressure and velocity distribution around the surface of the object:

$$D=\int_{S_{upper}}\left[-pcos(\theta)+\tau_wsin(\theta)\right]dA +\int_{S_{lower}}\left[psin(\theta)+\tau_wcos(\theta)\right]dA$$

where $$S_{lower}$$ and $$S_{upper}$$ refer to the lower and upper surfaces, respectively, and $$\theta$$ is the angle between the local unit normal vector to the surface and the vertical direction.

To get the pressure and velocity under specific conditions, you need one of three things: experiment data, a computational simulation, or an analytical solution to the Navier Stokes equations for your particular case of interest. Analytical solutions are few and hard to come by for the general case with arbitrary flow domain shapes. Experiment data can be costly to obtain and would require using instruments that necessarily disrupt the flow, making it more difficult to accurately measure what you want (although, well designed experiments minimize disruption as much as possible).

Computational simulation, in my opinion, is the best compromise between the other two. It provides a "virtual" experiment using analytical equations that need to be solved iteratively. Nonetheless, as others have stated, it is non-trivial to obtain useful solutions from computational simulation. Experienced CFD analysts spend a lot of time preparing cases carefully, creating appropriate meshes, using/implementing proper gradient schemes, numerical fluxes, time stepping schemes, turbulence models, transition models, shock tailoring, boundary layer treatment, etc... While CFD may be more ubiquitously accessible, it still takes a lot more effort to produce useful solutions.

When computers were still relatively low power and were not yet capable of solving Navier Stokes equations, there were other methods for estimating the flow field parameters. One such method is to first solve the "Euler Equations", which are essentially an inviscid form of the navier stokes equations. Of course, if the fluid is only treated as inviscid, the drag would always be zero. To estimate the viscous solution from the inviscid one, we look at the individual streamlines of the inviscid solution on the body of interest and integrate from start to finish to sort of "build" a boundary layer up iteratively until a convergence criterion is met. Once you've built up enough of a boundary layer thickness, you can use various correlations to obtain the resulting pressure $$p$$ and wall shear stress $$\tau_w$$ that you need to compute drag. This method is often referred to in literature as the "Momentum Integral Equation".