Three-Dimensional Flow Fields

Every day, it seems, I understand something new I had failed to grasp before—especially in aerodynamics. For example, take this statement:
 
"Once again, it is necessary to remember that road vehicles and their air flow patterns are highly three-dimensional" (Barnard 15, emphasis added).
 
This always bothered me a little. Of course the flow over cars is 3-dimensional, I thought; how on earth can something be more or less, let alone highly 3D? Isn’t flow just...3D or not 3D?
 
Well, a few weeks ago in an Incompressible Flow lecture I finally understood it. Now you can too.

Investigating the veracity of Barnard's claim. I've done this before, and have yet to find him wrong.

Flow Fields
 
A field is a region of space where properties vary as a function of position within that space. The flow field around a car is the 3-dimensional space where the seven properties needed to completely characterize a flow (pressure, density, temperature, viscosity, and three components of velocity) vary as a function of position (steady flow) or position and time (unsteady).
 
In car aerodynamics, we make some simplifying assumptions that reduce the number of variables in the flow field. Specifically, we assume density, temperature, and viscosity are constant.* Most of the time, this is a reasonable assumption (e.g. not many cars travel at speeds high enough for changes in density from compression to be significant) but not always (e.g. airflow exiting a heat exchanger and then interacting with external flow may introduce significant temperature differences). But for the most part, in car aerodynamics we’re working mainly with pressure and velocity which, as I’ve written before, are related if we can make certain other assumptions about the flow. These two are usually enough to determine the flow field around a car.

(*Note that an approximation of viscosity as constant does not mean that shear stress, which gives rise to friction drag, is zero or constant! Shear stress is a function of viscosity and the derivative of velocity with respect to displacement—that is, the change in velocity with change in height from the body surface in the boundary layer. Where the flow speed at the upper edge of the boundary layer is faster, the change in velocity per change in height will be greater--and friction drag will be greater).

 
Gradients
 
It might seem obvious that the flow over any part of a car is 3-dimensional; a simple tuft test will show flow following a panel such as the hood, where it has a direction along the car’s longitudinal axis (x), a direction along the car’s lateral axis (y), and a direction along the car’s vertical axis (z).

Look at these tufts on the bumper cover and hood, for example: any tuft in particular has a direction backward, upward, and to the side.

But a flow field, remember, does not refer to the properties (here, the three components of velocity) at one point or even a collection of points. Instead, what we need to characterize a flow field is the change in property with change in position, or gradient. A two-dimensional flow can have velocity components, for example, in all 3 directions but they only change in 2 directions. If the property changes as a function of position in all three directions, it is 3-dimensional. If it changes a lot as a function of position in all three directions, it is highly 3-dimensional.

The gradient of a vector, such as velocity, gives a tensor. Each row represents the change in x, y, and z components of velocity; each column represents the change in x, y, and z components of position. Thus, the first row shows the change in x-velocity per change in x, y, and then z, and so on.

Some simple testing on your own car will illustrate this. I’ll use mine as an example.
 
Pressure Field
 
For this first example, consider a coordinate system aligned not with the car body and road but a specific panel on it—here, the front windscreen. Let’s approximate the windscreen as a plane (a flat surface—a reasonable approximation here, since the windscreen on this car is nearly flat). I’ll measure pressures down an arbitrary line on the windscreen, which returns these values (80 kph, two-way average):

I laid out a set of points on a 200mm grid before going out to measure. You can save time during testing by prepping as much as you can beforehand.

This is a common measurement technique in aerodynamics, measuring properties on a line—usually the centerline—and generating figures like pressure plots:

Centerline pressure profile of a NACA 2415 airfoil. For wings of long span, a 2D approximation like this is often valid for sections of constant chord in the middle of the wing.

But does this give us a full picture of the flow over a car? In a word, no. Rather, depending on how we orient our coordinate axes, this shows us either a 1D or 2D field. The values above from my car, plotted against position along the plane coinciding with the windscreen, gives a function of position in one dimension:

If instead I establish a coordinate system aligned with the road, I get a 2D field: the property I’ve measured changes with both longitudinal position (x) and height (z) rather than direction in a line. Two components, two dimensions. (This will give the same graph as above with pressure as a function of two dimensions instead of just one, with the relationship s = [x² + z²]^1/2).

It's very easy to assume that these pressures don't vary side-to-side since I only have information front-to-back and bottom-to-top. That may or may not be the case, though. If it is, then the flow is 2-dimensional. If it isn't, and the pressures actually change in a side-to-side direction as well as front-to-back and bottom-to-top, then it is 3-dimensional. To fully characterize the pressure field, then, I actually need to take measurements in all three directions—two directions aren’t enough:

I can plot the values of pressure as a function of position, or p(x, y, z), using the measured dimension changes from the car and establishing an arbitrary reference (origin) at the centerline point nearest the bottom of the windscreen:

Now I can see how the pressure changes as a function of position in all three dimensions—in other words, the pressure field over the windscreen. For this field to be “highly three-dimensional,” then, simply means that the gradients are significant in each of three orthogonal directions. Is that true here? Yes: measured pressure varies from +20 Pa to -110 Pa in x and z along the centerline, and from -10 Pa to -80 Pa in y at the middle of the windscreen.
 
This is the important takeaway, and something you can measure on your own car: the flow properties vary dramatically in all 3 dimensions and thus the field is "highly" 3-dimensional.
 
Velocity Field
 
Outside of CFD, it’s much harder to measure the velocity at various points on a real car. But we can measure pressures at several locations on a panel and interpolate the magnitude of velocity from that information, as well as observe the direction of the velocity at these same points with tufts. I’ll do this over the windscreen again so I can easily record the tufts from inside the car and use the pressure measurements from Part I.
 
Velocity directions (think of the tufts as unit vectors—that is, they show a direction but not a magnitude):

Now I can use the Bernoulli relationship to estimate the magnitude of the velocity at each point. I’ll use the barometric pressure and temperature I recorded during testing to determine air density (from the Ideal Gas Law), and freestream velocity is the same as the car’s forward speed (about 22 m/s). Local velocity on the windscreen can be related to gauge pressure and freestream dynamic pressure by

Plotting local velocity against gauge pressure tells us what the flow speed is at any point on the windscreen based on the pressure reading there:

This gives the velocity field over the windscreen:

Note that these velocities are not representative of the flow speed at the body surface itself (which is zero, due to the no-slip boundary condition at a wall); rather, they indicate the speed of the flow at the upper edge of the boundary layer at each measurement location. Where the velocity is higher, friction drag will be higher than at locations with lower velocity due to the larger velocity gradient (du/dy) within the boundary layer. We're able to calculate velocity by measuring pressure due to the fact that pressure at the upper edge of the boundary layer and at the body surface are the same at any given point (i.e. normal vectors within the boundary layer are isobaric).

Again, we see a strongly three-dimensional flow field, with velocity magnitude increasing as we move further back, up, and toward the outside of the windscreen. Additionally, the flow direction changes toward the outside edge, where it has a larger sideways component. If you are careful about recording tuft behavior, you should be able to observe pressure and velocity fields like this over your whole car if you wish.
 
Lesson
 
Fundamentally, velocity and pressure fields over a car are important because they determine its aerodynamic characteristics. Integrated over the entire car, pressure gives a resultant pressure force, and velocity determines the resultant friction force (from shear between fluid “layers” due to viscosity, not to relative movement between air and the surface). Both of these forces can be summed into one aerodynamic force, which can then be divided into its components in the lateral direction (side force), vertical direction (lift), and longitudinal direction (drag).
 
The lesson here is: beware trying to characterize your car’s flow field based on 1D or 2D measurements. They might give you information but only by extrapolation, and that extrapolation could very well be wrong (for example, I had no idea there was such a large pressure gradient side-to-side across the windscreen of my car!). As always, you will have to balance your desire for information from testing with the unpredictability of test conditions in the real world and ability to go quickly to compensate. And now that I’ve understood this, I intend to be more cognizant of it going forward. Live and learn.