Why Your CFD Is Wrong – Part 1: What Is a Model?

I started writing this article before the end of spring semester, back in May 2025, and quickly found that it grew to monstrous proportions because there's so much to say about the topic of simulation and numerical approximation that it simply can't fit in a concise blog post. I found myself honing in on aerodynamics simulations specifically (perhaps because that was the topic of one of my elective courses) and, after some helpful advice from a friend, decided to split it into several posts.

I intend this series of articles to take the form of three parts and explain, mainly in general terms but with plenty of examples (many drawn directly from the numerical methods and computational aerodynamics coursework I've recently completed), the mathematics behind modeling and simulation: how does it work? How well does it represent real things like airflows? At the end of it, if all goes to plan, you should understand that all computational fluid dynamics (CFD) simulations are wrong and have a basic grasp of why this is. To explain it, we have to start at the very beginning. What is a model?
 
Generically, a “model” is a reproduction, a fake made to approximate something real. You probably had model cars as a child. A lot of people enjoy model cars as adults too (and they might be bigger, like the 1:18 scale Viper below).

The #91 Oreca GTS-R took the overall win at the 2000 24 Hours of Daytona. The final 360 2nd-generation Viper GTS coupes were painted the same red with white stripes in commemoration of this as well as its class wins at Sebring and Le Mans the same year.

The model reproduces, to some varying degree of fidelity, the real thing.

I’m glad I had the chance to live out my teenage fantasy of owning this car, but I would never go back—it was fairly ridiculous. This was number 318 of the 360.

It’s the same in engineering. We use models to try to reproduce real systems so that we can understand them and ultimately design new machines to do things we want. However, unlike a model car, the models we use are not always physical. Instead, as one of my professors put it, “Engineers turn words into math.” We do that most of the time with differential equations.
 
Mathematical Models
 
Differential equations are fundamental to our understanding and description of the physical world. They are mathematical representations of the rates of change of the parameters of a system as they relate to one another. For example, a mass connected to a spring—which can model, say, the body of your car riding on its suspension, or the wing of an airplane subjected to a fluctuating aerodynamic force (“wing flutter”), or the lateral movement of a building in an earthquake—accelerates in proportion to its position:

We use “prime” marks to symbolize general derivatives—so, the left-hand side of this proportionality is the second derivative of the right-hand side. More specifically for an undamped spring-mass system, position is a function of time and so the left-hand side contains a term representing acceleration, which is the second time derivative of position (velocity is how much an object's displacement changes per unit of time; acceleration is how much the velocity changes per unit of time):

…where x here is a function of time and the dots symbolize differentiation with respect to time.
 
The spring-mass system is a fairly straightforward model with wide application. It can stand in for many things, from the vibrations of molecules to the lateral motion of the “sky crane” Mars lander. Computational aerodynamics relies on more complex mathematical models that relate the rates of change of the various properties of an air mass such as its density (ρ), velocity (u, v, and w), pressure (p), friction (τ), heat flux (q), and specific energy (E). This results in complex systems of equations (nonlinear coupled multidimensional partial differential equations, to be specific), by applying conservation of mass:

Conservation of momentum:

And conservation of energy:

These systems, as complicated as they are, still do not capture the full picture of the real flow. At the same time, they are so mathematically complex that we must rely on significant simplifying assumptions to reduce them to something solvable. These assumptions could be anything from considering the flow as one-dimensional to approximating density as constant (most of those pretty CFD pictures of cars you've seen are created by programs that do this) to neglecting viscosity entirely (common in aerospace CFD solvers such as the Athena Vortex Lattice program, since the lift force and induced drag on an aircraft are strongly dependent on pressure and much less dependent on friction).
 
Physical Models
 
The alternative to conceptual models is to build something physical that replicates, in whole or part, the real machine we want to design. These have the advantage of incorporating real effects that mathematical models sometimes cannot. However, most physical models are not completely accurate reflections or duplications of an actual device or machine that performs some function in the real world. Some come close and are good enough to be useful; a lot don’t and are not useful. Examples of the latter are the tiny “desktop” wind tunnels that seem to be all the rage among auto enthusiasts right now.

Before you get too excited, hold up: this does not show the same flow pattern as over the real car. Not even close, especially over the rear window where the model shows complete separation (Image credit: newatlas.com).

You've probably seen these in videos online or may even have one yourself. I corresponded with someone recently on this topic, writing at one point,
 
“I wonder if there wasn't some influence here of the recent commercialization of a desktop wind tunnel (I'm forgetting the manufacturer now); for the last few months I've been seeing videos of people sticking model cars in these things and 'showing' the airflow, complete with a digital readout of flow 'velocity' ~200+ mph--clearly scaled (incorrectly) to the car size! I think it's so unintuitive for people because they think that a smaller car/model means slower true airspeed = faster modeled speed compared to the size...when in fact the opposite is true. Airspeed must increase to achieve the same Reynolds number, as you know, and then you run the risk of compressibility and transonic effects* ruining things (to get 20mph equivalent on a 1:18 model requires Mach 0.47 in the tiny tunnel--well above the limit for incompressible assumption). Utterly stupid and disingenuous, the way these desktop tunnels are marketed.”
 
(*I've noticed that these are often confused online. Compressibility refers to changes in density in the flow; below Mach 0.3 we usually approximate density as constant when we simulate flow or calculate aerodynamic parameters. Transonic refers to the freestream speed at which the local flow velocity somewhere on the vehicle reaches “sonic condition” or Mach 1, and shock waves begin to develop—which is typically closer to freestream Mach 0.8, depending on the vehicle geometry. On jet airliners, this usually happens on the wing upper surface; look carefully next time you fly and you may notice a “shimmer” there at cruise due to refraction through the shock wave formation).

 
The problem with these desktop tunnels is that the flow patterns they model are nowhere near what happens over the real car due to the difference in Reynolds number and Mach number, the two flow similarity parameters that must match the real car in order for the resulting flow patterns to develop in the same way—a complete failure of the model to replicate the real physical system. While the execution of the tunnel itself may be fine (in other words, its accuracy to the true flow over a scale model RC car may be very close), its dissimilarity to the real flow over full-size cars renders it utterly unsuited for the purpose most people think they can use it for and makes it very easy to mislead enthusiasts, students, and sometimes seasoned engineers who don't know any better. I looked up the website for one of these before writing this; they have ad copy and review excerpts everywhere stating such untruths as, “Lets you visualize what you can’t see, just like the major manufacturers,” and, “Test the aerodynamics of your 3D creations, or even your own car designs, and bring your ideas to life!” The smallest models manufacturers test are 1:4 scale because of those similarity parameters I mentioned above (a physical limitation—there is no way around this), and if your goal is to build a real car these small wind tunnels will tell you nothing about its aerodynamic performance when the design is scaled up.

 
On the other hand, a good model for airflow over a real car is a detailed full-size buck in a large wind tunnel. Such a model is likely to develop airflow patterns and give results similar to the real car out on the road, and wind tunnel design has evolved over the past century to increase this likelihood. Professional wind tunnels now incorporate moving ground surfaces, have open test sections with very low blockage ratios, can attain high flow speeds, and have sensitive measurement equipment for calculation of force and moment coefficients.
 
An even better model? The real car on a real road. This is where a home modifier has an advantage over an OEM developing a new vehicle: we are working with already-existing cars and can test changes to their geometry in the real world fairly easily, although one trade-off is measurement methods that are often less sensitive.
 
The important takeaway here is that there will be inherent limitations any time you model something or look at the results of someone else’s modeling (especially the plethora of amateur CFD images online—we’ll see why in the next posts in this series), and quite often those limitations will not be immediately apparent or intuitive. The goal of any model should be to give us credible information about the real thing it attempts to copy; this stands at odds with the fact that no model is the same as the real thing, and some are drastically different. Remember the adage: “All models are wrong; some models are useful,” and always try to answer the question, “Is this test or simulation going to give me results that are applicable to the real thing?” Don’t ignore the answers to that question; try to figure out how and why the model is wrong so that you can improve it!

A lot of people (online and off) try to crap on real-world testing due to a belief that modeling or simulation, like those desktop wind tunnels or CFD, are somehow more trustworthy because they remove the chaos and noise of real life. In fact, the opposite is true: you should be more skeptical of models and approximations precisely because they cannot account for the messiness and nuance of physical systems—if they can even replicate the fundamentals correctly in the first place.