The meaning of circulation

470 International Class

Circulation is one of the most misunderstood terms in sailing. Yet, circulation is essential to understand lift and why sails, wingsails, keels, rudders, foils, etc., work. We can safely say that sailing is possible, at least when beating windward, thanks to circulation.

One of the meanings attached to the word circulation is the movement of “something” in a circle or a circuit. The “something” can be either blood, money, water, air, …, but the important thing is that, under this meaning, we understand circulation as a movement to and fro or around something. This meaning is quite popular, and this is how the word circulation is often understood.

However, the word circulation is also used in maths, physics, and more concretely in fluid dynamics. Here, circulation has an entirely different meaning, which sometimes leads to confusion. In these contexts, circulation does not require the flow to move in circles. Circulation can actually happen in a flow moving around and also in a flow moving in parallel straight paths.

In sailing, we are interested in fluid dynamics, specifically in aerodynamics and hydrodynamics. How should we then understand the word circulation? Here it goes:

Circulation is a mathematical tool that captures the degree to which a vector field aligns with the path defined by a closed curve. That’s all. Still the same, but in other words, circulation measures the extent to which a path defined by a closed curve goes along with a vector field she is in.

Yet, the explanation given above can be very confusing. Let’s deep dive into this concept of circulation by approaching its definition one step at a time. For doing so, we are going to predict “by hand” the circulation around an example curve in an example vector field.

Bear with us. It’s going to be rewarding.

Vector field

In a vector field, we assign a vector to each point of the space. Vectors are characterized by a direction and a magnitude. Usually, an arrow represents the direction, and the length of the arrow represents the magnitude.

Gravity is an example of a vector field. We can assign to any point in the space above the earth a vector whose direction points downward, and its magnitude is the value of the acceleration of gravity at that point.

Another example is the velocity field in a flow. Here we can assign a vector to each fluid element in the flow: the arrow will point to the direction the individual fluid element is moving, and the length of the arrow will represent the speed at which it is doing so.

Figure 1 shows the graphical representation of a two-dimensional vector field. We will use this vector field to explain how circulation is calculated.

Velocity field
Figure 1

Closed curve

A closed curve is a “closed” and “simple” path. “Closed” means it ends up exactly at the point at which it started. “Simple” means it has no self-intersection points (i.e., it encloses a single region).

An important thing to note is that closed paths have positive and negative directions. Imagine you were able to walk over the path defined by the curve. By definition, you will be moving in a positive direction if the area enclosed by the curve is always on your left. This is because, by convention, the positive direction around a closed curve is always counterclockwise.

Closed curve
Figure 2

Points of interests

Let’s now include our closed curve above into our example vector field, as shown in Figure 3:

Closed curve within the velocity field
Figure 3

Out of all the possible points in the vector field, we are only interested in those laying over the curve. In Figure 4, we arbitrarily chose twenty of those points and drew their associated vectors. Note the direction and magnitude of each of them.

Velocity vectors over closed curve
Figure 4

Curve – vector field alignment

In Figure 4, we don’t find any vector that fully aligns with the curve. However, we notice that some of them have a more tangential direction to the curve than others. But, how much is a vector more or less tangential to the curve than another one? For answering this question, we will project each vector over a tangent line to the curve that passes over each of the points, as shown in Figure 5:

Velocity vectors tangential projection
Figure 5

The projection is itself another vector (see Figure 6) whose direction and magnitude will depend on the magnitude of the original vector and the angle between the original vector and the tangent line.

Another way to understand this is by thinking of the original vector as the addition of two orthogonal vectors where one of them is aligned with the tangent line (see Figure 5). The projection will be the component that lays over the tangent line.

Figure 6

Circulation calculation

We have said that circulation is a mathematical tool that measures the alignment between a vector field and a closed curve. In fact, we are going to use the projected vectors obtained above to determine the degree of that alignment.

In Figure 7, let’s travel over the path defined by the curve in a counterclockwise direction starting at point 1. Each time we cross any of the points, we will note down the magnitude and direction of the projected vector. If the vector points counterclockwise, we will say that it has a positive direction, and its magnitude will be considered positive. Otherwise, if the vector points clockwise, its magnitude will be regarded as negative.

Velocity vectors tangential projections
Figure 7

To display the results, we are going to cut the curve at point 1 and stretch it horizontally (see Figure 8). We will draw the positive values over the line and the negative ones below the line.

Figure 8

Note that for the sake of simplicity, Figure 8 considers that the points we have chosen are located at the same distance from one another. However, this is not what Figure 7 suggests. The actual distance (the arclength) between the points should have been considered for a rigorous calculation.

We have almost finished. Even though we could have added positive and negative values to obtain a final value, there is still one step more: the definition of circulation establishes that we have to multiply the magnitude of the projected vectors by the distance to the following point, as shown in Figure 9.

Figure 9

The circulation, whose value will be either zero, positive or negative, is finally obtained by adding together the area (with its positive or negative sign) of each of the rectangles shown in Figure 9.

It is important to note that the circulation value would have been different if the chosen curve had also been different. In fact, the value of circulation depends on the vector field, the curve’s choice, and the curve’s position within the vector field.

More examples

Let’s look at some additional examples and try to estimate whether there is circulation and if so, determine whether its value is positive or negative.

The closed curved chosen in Figures 10 and 11 is a circle. In Figure 10, the vector field is always perpendicular to the curve. Since the tangential projection of a perpendicular vector is zero, there is no circulation. However, In Figure 11, the vector field is always tangential to the curve. There is circulation, and its value is positive (counterclockwise direction).

Figure 10
Figure 11

Figure 12 displays a vector field where all the points have associated a vector with the same direction and magnitude. Here, we have selected another curve. In this case, there is no circulation since:

  • the points located on the vertical parts will have perpendicular vectors with no tangential components to the path;
  • the upper and bottom sides will cancel each other (negative values at the top and positive at the bottom);
  • the points over the curved corners will also cancel each other (negative values at the top left and right corners, and positive at the bottom left and right ones).
Figure 12

In Figure 13, as we move down in the field, the vectors are shorter and shorter. In this case, although the vertical sides of the curve still will not contribute to circulation, there will be no cancellation between the upper and lower sides, neither between the corners:

  • the vectors located at the upper side of the curve will have more weight than those at the lower side. There will be a negative global contribution (clockwise);
  • the top corners will also contribute more than the bottom ones (negative contribution too since their projections are clockwise).

In this case, even though the vector field does not feature any loop, circulation will exist, and its value will be negative (clockwise).

Figure 13

Circulation and fluid dynamics

In aerodynamics and hydrodynamics, contrary to what we have seen, the clockwise direction is the one that is considered positive. This means that even if the circulation can be calculated in the same way we have proceeded, to follow the “fluid dynamics way” we have to change the sign to the results obtained (from positive to negative, and vice-versa).

Also, the vector field under consideration in fluid dynamics is concretely the velocity field of the flow. Thus, the circulation measures the tendency of the fluid to flow in the direction of the path defined by a closed curve. Its units are square meters per second, i.e., velocity [ m / s ] times distance [ m ], m2 / s.

Moreover, air and water flows are continuous. Instead of a finite number of points (we selected twenty of them for the example above), we will have to handle theoretically infinite of them. Have we had considered this in the example above, Figure 9 would have looked more to Figure 14.

Figure 14

Let’s take a look at a final example. In Figure 15, we have depicted the streamlines of a flow (air or water) over a foil section. Remember from “Flows in sailing” that streamlines are lines always parallel to the velocity field. Also, when adjacent streamlines get closer together, the flows accelerate, and when they get farther apart, the flow slows down. Therefore, we can state that the velocity increases and aligns better with the curve at the top of the foil, in the region close to the leading edge, while it decreases, also in the nose area, at the bottom. Thus, there is circulation, and it is positive (clockwise in fluid dynamics means positive).

Figure 15

Figure 15 features another example where the flow is not moving in circles, but there is circulation.

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