Engineers and Physicists talk funny sometimes, and the term”drag crisis” is an example of funny talk. It’s not what you think. It does not refer to the huge increase in home runs in the MLB due to unexplained changes in the drag force on baseballs. It refers to a rather precipitous drop in drag coefficient with speed.
I want to preface this article by admitting it is pretty technical. I’ve made attempts to make it accessible to lay-persons, but that is something I struggle with. If there is a point you would like more clarification on, please let me know.
Let’s start by examining how drag on a sphere varies with velocity. Fluid dynamicists prefer the quantity “Reynolds Number” to velocity. Reynolds number is velocity times the sphere diameter divided by the dynamic viscosity, and a “dimensionless” number such as this allow us to compare the flow over objects of different size in different fluids. For a fixed diameter and fluid (air, in the case of baseball), Reynolds number is proportional to velocity. So bigger Reynolds number just means a faster pitch.
ThatsMaths has a nice page about the drag crisis, and I’m borrowing a couple of figures from that page, which is linked in the captions. I think they borrowed them from NASA. As you can see in the plot below, over a large range of Reynolds numbers, the drag force on a smooth sphere increases with velocity, as you might expect. But there is range of Reynolds numbers where the drag force actually decreases with velocity. This happens when the flow in the boundary layeron the front of the ball becomes turbulent, resulting in a smaller wake.
Fluid dynamicists also prefer the “drag coefficient” to the drag force. The drag coefficient is twice the drag force divided by the velocity squared and area. From the wikipedia page on drag coefficient:
In the case of a baseball, u is the ball speed, air density is about 1.2 kg/m^3 and the reference area is the area of a circle with the ball’s diameter. Note that this means that if the drag force remains constant with increasing speed, the drag coefficient goes down. When the same data are presented in terms of the drag coefficient, the figure below results. The x-axis is logarithmic here and spans a very large range of velocities. Note the sharp drop in drag coefficient near Re = 1 X 10^5. This is termed the “drag crisis.” The drag coefficient can drop by a factor of 5 in this range. Note that this occurs for smaller Reynolds numbers for rough spheres than it does for smooth spheres. Here’s the kicker: Guess where baseballs live? Right smack in the drag crisis (a 90 mph baseball at sea level has a Reynolds number of 1.85 X 10^5). That means that the drag on baseball is touchy.
My friends at the Washington State University Sports Science Lab have examined the drag on several sports balls and have some interesting results. The data come from Jeff Kensrud’s MS thesis (2010), which he was kind enough to share. The WSU results are for pitched balls, although the balls are not spinning. Drag is measured over a short distance by measuring the speed of the ball as it passes a light gate.
These data exhibit a lot of random variation, and in my view, this is due to the small sample time rather than issues in the measurement system. I have added a weighted curve to each of their results to aid in understanding the mean behavior.
Since the pitched velocity range of these measurements is somewhat limited (and indicated by the grey box in the NASA plot above), the Reynolds number range is small and is scaled by the size of the ball (note that the golf ball results are smaller Re values than the baseballs). This impacts several of the interesting cases, in particular, the dimpled ball for low Re and the Smooth ball for high Reynolds number. I have added a dashed line to each of these results that is my prediction of where the data would be if it were acquired for a larger range or Re.
They tested two different baseball models (a modern, low-seam NCAA ball and the MLB ball) in two configurations. Normal is basically a 4-seam orientation while parallel is 2-seam with the two seams facing forward in the direction of travel.
Additionally, I have added some recent wind tunnel results for non-spinning baseballs from Shah (). The advantage of the wind tunnel is that long averages may be acquired, and the data are free of random variations. They investigated 3 2-seam orientations, two of which are pictured at the top of the drag results. Note their configuration 2 is identical to the WSU “parallel” orientation.
My take on these results
First, you should be oriented on which part of the plot is relevant to baseball. A baseball Reynolds number is at least 1 X 10^5 (which is marked with a vertical line) and less than 2 X 10^5 (the next tick). Notice that the drag crisis for smooth balls occurs at Reynolds numbers larger than these values.
These results can be divided into two fundamental categories: “homogenous” balls, where the surface is the same all the way around (smooth spheres, dimpled spheres) and, well, baseballs, under most circumstances. (Kensrud and L. Smith make a similar distinction in their more recent article, Kensrud, J. R., & Smith, L. V. (2018). Drag and lift measurements of solid sports balls in still air. Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology, 232(3), 255-263.) The homogeneous balls, which are all marked with closed circles, exhibit a classical “drag crisis,” or a very rapid drop in drag coefficient with increasing Reynolds number. This behavior starts at higher Re for smooth balls, and the final Cd in these cases is lower, as expected.
For all other cases (marked with open symbols), the reduction in drag coefficient with increasing Reynolds number is much more gradual. This is the interesting stuff. Note that these are most of the baseball cases, with the exception of Shah’s configuration 1.
Configuration 2 is not homogeneous. Two sides of the ball are smooth while the other two have a seam on the front of the ball. I hypothesize that the sides with the seams (top and bottom in the picture) become turbulent at low Re. The smooth sides transition to turbulence at a location that moves forward with increasing Reynolds number, shrinking the wake. Shah draws a similar conclusion.
Configuration 1 has a seam around 3/4 of the front of the ball. Clearly this has a major impact. There is no evidence that the smooth side has an effect here as the two smooth sides do for Configuration 1.
This may all be moot, because for pitched balls, as in the WSU study, the same ball orientation and same ball (labeled MLB Parallel) behaves completely differently, and like a non-homogeneous ball. I have issues with wind-tunnel studies and their potential to produce results that are different from a pitched ball, as noted here. It’s also possible that the WSU balls did not remain in the original configuration for their entire path. The 3rd configuration used by Shah was a miss-aligned ball, and its behavior was the same as Configuration 2 and the MLB Parallel case from Kensrud.
So, who cares about all of this? These balls spin and never remain in one of these configurations, even for knuckleballs. But one thing I take from these results is that baseball drag is incredibly sensitive to subtle aspects of the baseball orientation and construction. Many are surprised by conclusion that the increase in MLB home runs up to 2017 (which seem to be happening again in 2019) are caused by imperceptible changes in the outside of the ball. I have no problem believing that at all. Baseballs are beautiful and mysterious things. They have secrets and we are just beginning to coax them out.