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

Question

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!

Answer 1

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.

Answer 2

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.

Answer 3

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".