Optimizing a Tail for Low Drag: Part 1

Now that the semester is almost over and I have no plans for the summer, I decided to revisit a modification I’ve cursorily stabbed at before, in a not-very-smart manner: the drag-reducing tail.

I’ll elaborate on these poor attempts as I walk you through this project and how I’m approaching it differently this time. But before all that, first we need to visit some theory and look at how tails are (often incorrectly) characterized in online discussions, what their purpose is, and how best to design one.
First, some basic definitions. A lot of the simplified aerodynamic theory you’ll encounter online is based on analysis of airfoils, which is an important part of aeronautics but not always applicable to cars. One of the reasons for this is that cars are not streamlined shapes. Even the most perfectly shaped production car has a large area of separated flow at its back end:

…which you can see here. My Prius has attached flow over most of its upper and side surface, but that flow separates at the rear—seen clearly in the tufts.

The Volkswagen XL1 is one of the lowest-drag cars ever sold to the public; it still has a large area at its rear that is in separated flow when the car is moving.

This area of separation means cars are bluff shapes rather than streamlined. Generally, bluff bodies have more drag than streamlined ones, as the large turbulent wake that forms after flow separation has fast-moving, chaotic airflow and consequently dissipates a lot of energy. The wake also develops lower pressure than attached flow over the sloping rear part of the car, as I found in my very first pressure measurements on my car (where a lip spoiler at the top of the rear glass, separating the flow there, dropped pressures on the rear window).
So, one approach to achieving low drag from the basic shape of a car is this: keep the flow attached as far back to the rear as possible—and indeed, this is exactly the advice you will read in any textbook on car aerodynamics. Modern cars are pretty good at keeping attached flow all the way back; however, by necessity of balancing maximum interior volume with minimum length, they typically have large wakes (more so now that the market has shifted in large part to crossovers/SUVs with “fullback” tails and away from sedans with smaller “notchback” or “fastback” tails). If you don’t care about adding length to your car, one approach to reducing drag can be to add a tail that has attached flow while tapering to as small an area as possible before separation. (This has to do with the proper definition of a “Kammback” as well—another concept that is almost entirely misunderstood by most people, especially car buffs and auto journalists. That’s a topic of contention I’ll reserve for another day).
Now that we’ve waded into the tail discussion, let’s dispel some myths.
First, there is no perfect “template” that will tell you how to shape a tail on any/every car. This is the big one, as many, many amateur aerodynamicists believe they can simply follow a predetermined shape—often one that originates from one website in particular that keeps popping up in unexpected places—which obeys certain “rules” (such as “the tail should have a maximum taper angle of 22° from horizontal”). In reality, there is no single “perfect” shape (indeed, the early days of automobile streamlining were characterized by the multitude of shapes proposed as the "best" tail for a car).

One of the most popular was this, the Jaray Kombinationsformmade by "stacking" multiple airfoil shapes in different positions and orientations. While lower drag than contemporary cars, this Tatra T87 doesn't come close to the best production cars today, despite its flat underbody and "streamlined" shape (see Julian Edgar's A Century of Car Aerodynamics for more).

The suitability of a shape depends on what you want it to do aerodynamically, which will depend on the shape of the rest of the car that isn’t its tail and your goals. Many of these “rules” of the Template are based on boattail research on axisymmetric (circular cross section) bodies of very large length-to-depth ratio away from the ground i.e. missiles—not cars. Their wholesale application to cars is dubious at best—and unnecessarily limiting and ultimately detrimental to good, thoughtful, informed design at worst.
Second, a fully streamlined tail that tapers to a point with no flow separation does NOT result in zero pressure drag. This is a fallacy that arises from an incomplete understanding of thin airfoil theory (TAT—yes, this area of study is so important we give it an acronym in the aerospace field!). In TAT, potential flow theory is used to predict the lift performance of airfoils to a high degree of accuracy.
Here is the lift curve of the NACA 4412 airfoil, for example. The red line shows predicted lift coefficient as a function of angle of attack from TAT; the blue line shows wind tunnel results reported in Theory of Wing Sections (Dover Publications, 1959). As the blue line tapers off (nearing stall/separation), the plots diverge— a result of the inviscid assumption of TAT. Experimental results at higher Reynolds numbers (i.e. where viscous effects are less dominant) agree more closely with the theory.

This potential theory only applies when the airfoil has no flow separation—which Template proponents correctly transfer to their car analysis—but also depends on several other assumptions (inviscid, incompressible, irrotational, steady flow) that provide the necessary theoretical simplifications to allow solving potential flow problems but do not apply to real flows. Potential flow theory correctly predicts lift over thin airfoils at low angles of attack but does not predict drag because it does not account for the effects of viscosity and rotationality in the fluid (this is named “d’Alembert’s Paradox” after French polymath Jean d’Alembert, who first deduced that potential flow theory will predict drag equal to zero for any body submerged in an inviscid flow—something he puzzled over at the time since experience clearly showed that ships were subjected to hydrodynamic drag despite the theory). And what’s more, simply adding in viscosity and its resultant shear stress on the body still does not result in pressure drag equal to zero as in potential flow! Viscosity and three-dimensional effects alter the pressure distribution over objects in real flows, which means even a fully streamlined shape has not just viscous drag, but pressure drag as well.
Third, no car (or any object with rotating wheels) can be fully streamlined. In addition to the above, another huge caveat with applying TAT to cars is the fact that no car can be fully streamlined: unless you’re planning to turn it into an airplane or magically levitate it, your car must be in contact with the ground on rotating wheels, and these wheels will have areas of separated flow no matter how well they’re incorporated into or shielded by the body.

(Image credit: ecomodder.com).

Fourth, the method of images does not correctly predict the aerodynamic forces acting on cars. You’ll see Template promoters online claim that their “perfect” shape derives from the “ideal” teardrop cut in half and brought into contact with the ground, doubling its drag and producing no lift (see the image above). This is another misunderstanding of potential flow theory. We can predict a flow field by superposing basic flows in a velocity potential function or stream function, from which flow velocity can be derived and then pressure from that velocity field.
For example, here's the flow field around a Rankine oval, formed by the superposition of a source and sink of equal strength and equidistant from the origin, plus uniform flow from the left. Notice that this potential flow solution exhibits no separation, where in reality a shape like this will have a large wake. Another important difference from real flows: this inviscid solution, with no circulation, predicts aerodynamic force equal to 0!

In order to solve a potential flow problem for an airfoil close to the ground (e.g. for an airplane in takeoff roll), one must superpose its mirror image at the same distance across the ground plane, which can then be represented by a horizontal streamline, to correct the velocity field.
Now, with the presence of another oval below it, the dividing streamline at y = 0 is flat, representing the ground. The flow around the upper oval is now "corrected" by the ground constraint and can be analyzed as if everything below y = 0 were not there.

As in non-mirrored potential flow, this method accurately predicts lift for thin airfoils at small angles of attack but cannot account for drag due to the inviscid model. And even using mirrored models in a real, viscous flow does not give accurate results. This point is worth emphasizing: placing two car models in a wind tunnel (as has been done in the distant past)—one upside down, touching at the tires’ normal point of contact with the road—is only an approximation and does not accurately replicate the ground plane, does not account for viscous effects at the ground plane, and does not simply "double the drag" and "cancel out lift" of the car (testing in this manner, only the aerodynamic forces on the upper car are measured, not both! The mirror car's only purpose here is to correct the velocity field around the real car).

Further, as Barnard points out, the “perfect” teardrop cut in half and sliding along the actual ground (rather than a dividing streamline) will have significant lift—besides being unrepresentative of a real car rolling on wheels with some finite gap between the underside of the body and ground. To reduce lift of a longtail streamlined car, the underside of the body must be shaped carefully (for more on that, see the proceedings of the 1976 GM conference on aerodynamics, which includes what must have been a lively discussion after one paper presentation on the interaction between body camber, drag, and lift).

Fifth, cars are NOT infinite wings; three-dimensional effects must be taken into consideration when shaping a tail. Template believers often make one of two mistakes: treating a tail design as if its profile extends to infinity (i.e. conceptualizing the tail as a 2D section of a wing), or treating it as a simple half-body of revolution and claiming there will be no ill effects such as vortices. Neither of these is true. Cars generate strongly three-dimensional flow fields, more so than finite wings; add in that they are characterized by flow separations (unlike wings), and you get a shape that generates complicated vortex systems which themselves alter the flow characteristics. Consequently, when designing a tail you will have to take this into consideration—and not in the arbitrary, “just overlay the Template on the side of the car!” way (see Point 1 above).
Tail Theory
Now that we have that out of the way, what is the best way to design a tail? If not by following a template, how should one go about figuring out what shape to build?
Well, someone asked me recently about the tails I built before and if I had any advice for anyone wanting to make their own. I thought about it quite a lot (which resulted in my decision to try building a tail again) and came up with these recommendations:
1) Start with cheap/quick mockups (cardboard, coroplast, or plywood) to test angles for the top, sides, and bottom:
a. Tuft test: range of angles for each of top, side, bottom--identify max angle that has attached flow for each
b. Pressure measurement: at range of angles up to max angle with attached flow--see what the behavior of pressure is with angle (we're looking for the combination of greatest pressure + largest taper angle + shortest length + smallest wake for maximum drag reduction)
2) After completing the above, I would then mock up a full tail using the angles found in (b), and then tuft and measure pressures on top, bottom, sides to see how close they are to the individual components; adjust angles if something is different e.g. separated flow and retest
3) In combination with (2), I would probably also coastdown test at this point to estimate total drag change from acceleration change
4) Retest with any changed parameters e.g. curvature of panels, radiusing edges between top, sides, bottom--could incorporate this in (2)
5) If the mockup is strong enough, crosswind test and measure steering angle, also tuft test/pressure test in crosswind
6) Once that's all done and I have a good design, build the real thing and do a final tuft test to verify it behaves the same as the mockup, then log fuel economy for the next several months to a year
It’s been several days since I wrote this, and in cogitating further I think this is honestly a pretty good plan (with the addition of some other steps, which I will elaborate on as I go along). So, I’ll take my own advice and make this my summer project. If you decide to follow a similar plan, start with one step I’ve already completed: tuft test your car as-is to identify and correct any areas of separated flow. If you don’t have attached flow already to the location of your proposed tail, you’ll need to address that first.


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