Measuring and Modifying Wheel Drag – Part 1: Overview

My background now is in aerospace engineering, where landing gear and wheel drag are sort of an afterthought in the academic curriculum (I mean this seriously—it isn't discussed specifically until the last year of the program I recently completed). That's understandable because it is approachable as a combination of areas of study—materials, structures, mass properties, configuration, performance, and aerodynamics.

 

Hoerner includes a short section on landing gear and wheel drag in his Fluid-Dynamic Drag (yes, with a hyphen if you want to look for a copy. His estate still publishes the book, available by special order). After specifying reference area (the simply projected area of the tire as width times height), Hoerner goes through the drag characteristics of various landing gear configurations, fixed and retractable. He also notes something interesting: wheel drag can account for as much as a 50% increase in zero-lift drag or C_D,0 on commercial and general aviation (GA) aircraft. He suggests streamlining the various landing gear components, such as struts, to reduce this drag.

I visited the US Air Force Museum in Dayton, OH, during spring break last year to look at real-world examples of landing gear configurations, such as this F-16 nose gear—interesting because it is one of the few modern examples of gear that retracts into the engine nacelle.

Another (that did not make it to production) is Boeing's entry into the Joint Strike Fighter competition, the X-32. This aircraft has a bulbous engine inlet and nacelle with plenty of room for the small nose gear. Unfortunately, it also grossly violates the old adage, "If it looks right, it flies right."

Notice that neither of the above aircraft have streamlined landing gear. Why don't we see faired struts and streamlined components, as Hoerner suggested? I suspect it is because, even if zero-lift drag coefficient of the aircraft is increased by upwards of 50% with landing gear deployed, the flight segments in which this is the case represent such a small fraction of total flight time that it's really inconsequential to overall aircraft performance. Further, on landing we want a drag increase (to slow the aircraft), and on takeoff and climb the aircraft will be so near its maximum C_L—with attendant lift-induced drag massively outweighing C_D,0—that the effect of the struts and wheels doesn't really matter.

The relationship between drag and lift is typically shown in a drag polar, with lift plotted as a function of total drag. The curves here are slightly different because the aircraft has a different predicted zero-lift drag coefficient at cruise (M = 0.8) than at loiter (M = 0.65) due to Reynolds number effects.

However, on cars wheel drag is always present and can matter a lot, approaching or even exceeding the 1:3 ratio of aircraft landing gear (for example, the drag coefficient of the Pininfarina/CNR "banana car" more than doubled with the addition of wheels). There are some important differences between aircraft and ground vehicle wheels and tires; let's look at those first.

 

Ground Vehicle Wheel Drag

 

Hoerner gives, for a streamlined wheel and tire with smooth interface between the two, C_D□ = 0.12 (I'm using the square to mean the same as Hoerner: drag coefficient referenced to tire projected area). However, this drag coefficient is for a wheel and tire in freestream flow, away from the ground and with no interference from mounting (e.g. struts).

 

On a car, we have several important differences from this idealized schematic of a tire in free air:

• Tire bulge: where the tire meets the ground, the tread is approximately flat against the road and the sidewall bulges due to the weight of the car above and constraint of the road surface below.
• Ground effect and interference: whenever the car is moving, the tire is in contact with the ground (unless something has gone very, very wrong). Further, the tire rolls along the ground most of the time, and as it rolls the bulge consequently rotates around the tire as well. The ground also imposes a horizontal constraint on the streamline closest to it, which changes the flow field around the car as a whole and around the wheels specifically.
• Complicated "strut": unlike a simple aircraft landing gear strut, the mounting and suspension of a typical car wheel has complicated geometry, as well as being very large (in relation to wheel size). Inboard of the wheel, suspension arms, brakes, dampers, springs, axles, anti-roll bars, and steering gear tie rods all affect the flow field.
• Fairings and bodywork: unlike most aircraft wheels, ground vehicle wheels are typically faired into the body and do not stand proud of it (either free or enclosed in a separated nacelle).
• Tread design and tire shape: finally, where aircraft tires have significant rounding in their shoulders, car tires are a lot closer to rectangular (to maximize contact area) and have complicated, deep grooves and tread designs, as well as sidewall markings and texturing.

We can see all these in the tires on my truck: complicated, deep tread design and high-relief sidewall markings; contact patch flat against the ground; bulge along the bottom of the tire sidewall; complicated brake and suspension components; and the wheel/tire combination (partially) faired into the body.

Analytical Method

 

As in my series on cooling system performance, I'll divide the wheel and driveline system into stations for analysis. However, this time I decided not to do this based on the path of airflow, moving from upstream to downstream. In cooling system analysis, that approach works very well (since the flow is largely confined to a duct) but wheels are more complicated. Consequently, I think the best way to approach wheel drag is by dividing the areas into discrete surfaces for independent analysis before looking at the system as a whole:

 

1) Body sides, fore and aft of the wheel: this encompasses most of the area where we can visualize flow with tufts, as it approaches, crosses, and then leaves the wheel opening cut into the car body side.

2) Body underside: another important flow path, and one that most people don't think about or look at (out of sight, out of mind!). Manufacturers, however, devote considerable resources to the development of fairings, dams, and strakes to manage airflow as it approaches the wheels from the underside of the car.

While I was on the floor taking this photograph of the underside of a Mercedes S-class, the salesperson joked that I was "praying to get the car." Normal people don't have any interest in what happens underneath their vehicles!

3) Wheel housing: above and behind the wheel, and roughly following its cylindrical shape, the internal body surfaces form a housing to accommodate, sometimes faired with plastic panels for both aerodynamic and noise-reduction purposes.

4) Wheel/tire face: finally, the visible part of the wheel and tire (and the area most people target for modification, since they see it every time they approach the car).

 

Ground Effect and Rotation

 

Before we get started, there are some important considerations in wheel and tire flow we have to note because they change the flow significantly from something like a fuselage, wing, or car body outside of ground effect.

 

The presence of the ground imposes a horizontal constraint on the streamline closest to it, which in turn affects the flow field extending upward from the ground. The ground surface tends to increase positive lift on the body or wing as an aircraft or car comes closer to it, an effect that is countered in performance and passenger car design by careful attention to the overall camber and "stance" of the body (e.g. introducing negative camber, varying body inclination and effective angle of attack) and by detail manipulation (things like air dams, smooth undersides, upper body wings or spoilers, and open nozzles and diffusers under the car can be used to counteract unwanted lift).

Take a photograph of your car from a distance (to minimize distortion), then mark a series of points midway between the upper and lower surfaces and overlay them with a curve to plot its camber line (blue); connect the midpoints at the front and rear to plot the chord line (green). Here we see that my car, as is typical, has positive camber (the camber line curves up and over the chord) but negative incidence (the chord line is tilted nose-down relative to the horizontal).

This increase in lift is even more pronounced on a wheel/tire, since the combination rotates forward into the flow. This forward rotation produces circulation opposite that of a wing and, as we expect from that, results in a negative lift force on a rotating wheel in free air. Potential flow models this quite well. Here's a stationary wheel in freestream flow (of the same dimensions as the wheels on my car, at 65 mph), which has no lift:

Add circulation by spinning the wheel forward into the flow, and the streamlines below it compress while the stagnation point moves upward. This induces a negative lift force on the wheel:

Bring the same wheel into ground effect and the inviscid model breaks down. Because potential flow does not account for friction and energy dissipation (so, total pressure is constant everywhere and the Bernoulli equation applies), it still predicts negative lift:

However, the real wheel rotating along the ground will have a flow field that looks more like this:

Scibor-Rylski, AJ, Road Vehicle Aerodynamics, Fig. 3.11.

The real stagnation point moves downward, the streamlines over top of the wheel compress due to the inability of air to pass under the wheel, and the lift force flips sign and becomes positive lift when the wheel is brought into contact with the ground. These lift forces are especially important in motorsports cars with open wheels (e.g. Formula 1, IndyCar, Formula SAE) and not as much in passenger cars.

You can see this car up close, as well as the EcoIllini Supermileage car behind it, on the third floor of the Sidney Liu Mechanical Engineering Building at the University of Illinois.

Enclosing wheels in housings molded into the body tends to change the resultant force on the rotating wheel from mostly lift (note that the force vector on the real wheel schematic above has a small horizontal component) to mostly drag. Further, tire and wheel details are important e.g. deep treads and grooves that allow air to flow between the tire and road affect the development of the force on it. Finally, since the real wheel has a width to height ratio typically w/d ≈ 0.3, three-dimensional effects can strongly influence the flow field around it.

 

Measurement Parameters

 

Before we head out on the road and start measuring stuff, let's work through some concepts to tell us what to measure and how to interpret results.

 

First off, as we will see in a couple of posts, the flow under the car tends to spread out and yaw as it approaches the wheels, especially the front wheels. I'll show you how to find the direction of this flow but that's really an aside; if we are concerned solely with the drag associated with the wheels, we will only really need to measure the axial (longitudinal) component of the velocity there, which we can do by pointing a total probe directly forward, parallel to the centerline of the car and its direction of motion.

 

Why is this? Recall the control volume model (revisit this post to review its derivation), which gives the internal force on a fluid passing through the volume with equal inlet and outlet area:

Local aerodynamic drag (i.e. within whatever control volume we select) goes up as velocity out is reduced (momentum drag) and as outlet static pressure is reduced (pressure drag). Since drag is that component of the force acting along the x direction of the car i.e. front to back and horizontal, it is only proportional to the change in velocity and static pressure along the same axis. So, we only have to measure in x to find changes in drag; any momentum that shows up in y or z must be lost in x (due to conservation), so we don't need to bother measuring those components if we are only interested in drag.

 

Now, what should we actually measure? Velocity is proportional to dynamic pressure q where,

Total pressure is the sum of dynamic pressure q and static pressure p,

Rearrange and substitute these into the first equation above and, making the same assumption of equal inlet and outlet area as well as incompressibility, we get:

So, changes in velocity (momentum) and static pressure are captured by the losses in total pressure and dynamic pressure. Thus, if we measure total pressure change and dynamic pressure change from one spot to another along the car and then measure it again with some geometry change, this should tell us whether local drag has changed if we draw our control volume intelligently. What does this mean? Well, if we draw one side of it inside a solid surface where there is no flow, then there is no friction there—removing an external force on the volume. Similarly, if we draw the other side far from the car body where there is no velocity gradient (i.e. outside of the boundary layer), there is similarly no friction on that wall. By so doing, all friction forces become internal forces, proportional to the changes in pressure and momentum between the inlet and outlet of the volume.

 

Now, to make it easy we can use freestream dynamic and total pressure as our references (measured by probes fixed to a pole at the front of the car, for example. This should be fine for the front wheels, but for the rear I will use a reference just upstream of the rear wheel opening instead). Then, the change in local drag is given by:

Total pressure loss can be measured directly between two total probes; dynamic pressure loss will require taking the difference of two measurements (total pressure minus static pressure at each location). But overall, this should give us an easy and straightforward means of finding changes in wheel drag. Next time, we'll put this into action, starting with the body sides upstream and downstream of the wheel opening.