Measuring Changes in Drag
(NB: I conducted this testing early last summer but, due to a busy fall semester, have only got around to posting it now...).
Before I
construct a full tail mockup, I decided it would be worth my while to do some additional testing on the partial buck I already have:
There are
a few reasons I decided this. First, I want to know how much this board and
side plates—effectively, a large spoiler at this point—reduce the drag of the
car. Second, I want to check my process for measuring changes in aerodynamic
drag on this car, which has an electronically controlled throttle that does not
allow for throttle-stop testing.
I’ll
elaborate on that process below. As far as the first reason: not only will
measuring the drag change at this point allow me to better predict the change
from a full tail, but if this spoiler turns out to reduce drag as much as my
design requirement, then I can stop here (if I want; I won’t,
because I want to go through the process of designing the full tail just to see
what’s possible). There are no drawbacks to further testing of this partial tail,
and doing so will add to the body of information I’m accumulating about how
tail extensions behave on this car—information which can only help my decisions
and may prove valuable at some future time.
To measure
the change, I decided to coast down test the spoiler to get an idea of how it
changes total drag—and see how far short it falls of my design requirement of
15% reduction in aerodynamic drag. This is an important step since tuft and
pressure tests, while informative, don’t tell me everything I want to know.
Tufts will show attached flow—but an increase in pressure in front of an
upturned spoiler may cause the flow there to separate.
Pressure measurement allows me to estimate pressure drag on the tail and base, but I cannot map the entirety of the surface so there are
significant gaps in the information. Additionally, the tail may cause pressure
changes upstream, over the existing car surfaces, that alter pressure drag.
Finally, static pressures correlate with shear stress, but I have no way of
measuring shear directly, and although shear stress is a much smaller
contribution to total drag than pressure over bluff bodies like cars, it is
still a contributor. A test of total drag change will account for all of these.
Pressures and tufts were useful in developing the shape up to this point, but
now I need another test to verify or disprove my assumptions.
Coast
Down Test
As I’ve
done before, I did 6 runs in each
configuration (tail extension versus stock spoiler + lip spoiler) at high speed
and videorecorded the speedometer. I
went out to one of my usual test roads at 3am (ugh), when temperatures were
stable for several hours and there was light traffic. Going back through the
videos later, I plotted speed as a function of time with and without the tail
after running a statistical test to check that the differences in average times
from 110-100 kph, 100-90 kph, and 110-90 kph between the two configurations
were statistically significant (they were, indicated by p-values smaller
than 0.05):
|
|
110-100 kph (avg.)
|
100-90 kph (avg.)
|
110-90 kph (avg.)
|
|
Tail
|
8.53 s
|
9.62 s
|
18.15 s
|
|
No Tail
|
8.08 s
|
9.14 s
|
17.22 s
|
|
p-value
|
0.0011
|
0.0016
|
0.00023
|
Now for
the fun part. Instead of just comparing average acceleration time from one
target speed to another, we can actually work out an estimate of the percent change
in drag from this data. In order to do this, we must be aware of some
assumptions: first, that the vehicle’s total mass does not change; second, that
rolling resistance does not change; third, that lift does not change. None of
these is true, but any changes in them should be small—so let’s assume they are
negligible.
Using a
force balance, the equation of motion of the car is easily derived with Newtonian mechanics:
We can plot this
differential equation by solving it numerically with input constants CDA,
m, ρ, and CR, over a small, iterated time increment dt (there is an analytical solution to this differential equation, but the numerical method is much easier to implement for plotting).
Here it is with approximate values for a stock Prius in standard sea level atmosphere:
Since I know my car’s actual
weight (which I’ve previously measured on CAT scales as I’ve removed or replaced
parts for weight reduction, such as the entire back seat) and estimating CR
for low rolling resistance (LRR) tires, the only variable here is CDA—and
I’m not changing A since the tail sits entirely within the car’s
projected area, so changes in total drag scale directly
with changes in CD. Estimating this is as simple as finding values
that, when entered in the differential equation above, produce plots that match
up with the plots of the experimental data.
 |
| The experimental data show exactly what we expect based on the model, with some slight variation due to the real-world conditions. Clearly, the tail has reduced the drag of the vehicle; this is evidenced by the shallower slope of the lines in the second graph. |
However, since I did
runs with and without the tail buck attached to the car, I don’t even need to
do that. Instead, I can calculate the percent change in drag directly using
only the measured acceleration:
(You can derive this
yourself from the first equation above). Look at what this equation tells us:
assuming constant mass of the car (one gallon of gas—about how much the car
burned during testing—equals approximately 6 lb, less than 0.2% of the car’s
total weight) and constant lift (in reality, it is likely to vary slightly with
and without the tail spoiler), the change in total drag depends only on the
measured acceleration at a given speed, gravitational acceleration, and
coefficient of rolling resistance. Longitudinal acceleration is known from my measurements
and gravity is an established constant; thus, the only assumed value here is
rolling resistance. The smaller the estimated CR, the more
pessimistic will be the estimated change in aerodynamic drag—so I will use a
fairly low value for LRR tires of 0.008.
Rather than estimate
acceleration from the plots, I’ll use the average elapsed time that the
speedometer displays the target speed to linearly approximate the slope of the
velocity curve with respect to time at that speed. Looking at the plots, this is a reasonable assumption: the curves are virtually indistinguishable from straight lines to the eye. This gives an estimated acceleration
at each integer speed between 110 kph and 90 kph. How much does this large,
sloped spoiler reduce drag, and is it anywhere close to my 15% requirement?
Well, taking the average acceleration of each run and then averaging these values (the acceleration is the slope of each line above) gives acceleration with no tail of -0.329 m/s2 and with the tail, -0.312 m/s2.
 |
| Averaging the values of all 6 runs in each configuration makes the difference more apparent. The acceleration in each configuration is the slope of the line of vehicle speed as a function of time; however, because there are two resisting forces at work here (aerodynamic and rolling drag), the difference in aerodynamic drag is not simply the difference in acceleration. Rolling drag must be accounted for, even if it is assumed constant as I've done in this analysis. |
By the relation above, this gives a change in aerodynamic drag of around -7%. That isn’t nearly -15%, but it
is almost half of my goal. It is also a bit less than what I estimated the
change in drag to be using the areas of the spoiler and base and the centerline
pressures I measured over them in my last test, around -8%. This means that my
estimation of drag change from (limited) pressure data is at least in the
ballpark and has been corroborated by further testing (this is important! If multiple tests using different methods give the same result, that result is more likely to be true).
Conclusion
As in my previous tests
in this tail development process, these numbers are estimates, not exact
figures. There is some variation in the difference of calculated
acceleration at any given speed due to the natural variability of the system,
so my intention here is just to get an idea of how much this test buck
reduces drag. If I were to put this car in a wind tunnel, it might spit out a
different number (however, note that figures generated in wind tunnels are also
averaged—a consequence of the inherently chaotic nature of turbulent flow! Good
wind tunnel reports will show average forces and force coefficients as well as
standard deviations). In the real world, there are unaccounted forces here from
the road surface, motion of the car’s suspension, differences in mass, rolling
resistance, and aerodynamic lift that I assumed negligible, slight variations
in temperature, atmospheric pressure and density, wind, road grade, passing cars (there were a few out at 4am), etc.
There’s no easy way to account for all those in a real-world test like this,
which is just one reason why it’s both incredibly stupid and very dishonest to
calculate absurdly specific figures and act like they’re the
be-all, end-all truth from real testing—and even more so from theoretical
assumptions such as conversion factors from SI to imperial units or vice versa, or from the artificial environments of wind tunnel testing or, even worse, CFD (which many, many online aerodynamicists place an inordinate amount of confidence in and probably shouldn't). In aerodynamics, increasing specificity is often less accurate, not more! Since these things characterize aerodynamic parameters with a single, specific number, amateurs often don't realize that they are averages and approximations. When you test aerodynamic changes to your car, look at trends and behavior, especially over several different tests, and see if they corroborate each other.
All that said, the one
thing I can take away from this testing is this: while a sloped spoiler reduces overall drag, it does not appear to
do so effectively enough to meet my design requirements. So, my next move will
be to mock up a full tail and test that, then decide how I want to proceed.
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