Lanchester’s work introduced a groundbreaking idea: replacing the physical lifting wing with a theoretical model built from a system of vortices. These vortices generated airflow patterns that closely mirrored real aerodynamic behavior and produced a lifting force equal to what an actual wing creates.
This vortex system is made up of three key elements:
Although each part can be studied on its own, they function together as interconnected components of a single system.
At this sharp trailing edge, the fast-moving air would need to change direction abruptly. Such a sharp turn demands extremely high accelerations and results in strong viscous forces, which the air cannot sustain. Instead of following the surface, the airflow separates at the trailing edge and forms a vortex just above it.
As the wing continues to accelerate, the stagnation point gradually shifts toward the trailing edge. With this shift, circulation around the wing increases, and so does the lift. Once the stagnation point reaches the trailing edge, the airflow no longer needs to turn sharply. Instead, it slows smoothly along the surface, comes to rest at the edge, and then accelerates again in a new direction. (Figure 2)
This vortex system is made up of three key elements:
- The starting vortex
- The trailing vortex system
- The bound vortex system
Although each part can be studied on its own, they function together as interconnected components of a single system.
Starting Vortex
When a wing begins to move from rest, lift is not generated instantly. At the very start, the airflow around the rear section of the wing behaves differently, with a stagnation point forming on the upper surface near the trailing edge (Figure 1). |
| Figure 1. Streamlines of the flow around an airfoil with zero circulation, resulting in a stagnation point located on the rear upper surface |
At this sharp trailing edge, the fast-moving air would need to change direction abruptly. Such a sharp turn demands extremely high accelerations and results in strong viscous forces, which the air cannot sustain. Instead of following the surface, the airflow separates at the trailing edge and forms a vortex just above it.
As the wing continues to accelerate, the stagnation point gradually shifts toward the trailing edge. With this shift, circulation around the wing increases, and so does the lift. Once the stagnation point reaches the trailing edge, the airflow no longer needs to turn sharply. Instead, it slows smoothly along the surface, comes to rest at the edge, and then accelerates again in a new direction. (Figure 2)
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| Figure 2. Streamlines of the flow around an airfoil with full circulation, resulting in a stagnation point at the trailing edge. The initial eddy is left far behind and rapidly becomes negligible to flight |
At that moment, a vortex is shed into the wake, the starting vortex (or initial eddy). This vortex has the same strength but opposite rotation to the circulation around the wing. Its creation stabilizes the circulation and establishes the steady lift on the wing.
On the upper surface, air flows inward toward the wing root from the tips, while air from outside the span replaces it. On the underside, the flow either moves inward more weakly or even outward. At the trailing edge, these spanwise flows meet, and the mismatch in velocity causes the air to roll up into numerous small streamwise vortices along the span. These smaller vortices then combine into two powerful trailing vortices, located just inside the wingtips (Figure 3). The strength of each trailing vortex matches the strength of the circulation that effectively replaces the wing itself.
Trailing Vortex System
A lifting wing creates a pressure difference between its surfaces: the pressure on the upper surface is lower than the surrounding atmosphere, while the lower surface has higher pressure, often greater than atmospheric. This imbalance drives complex air movements.On the upper surface, air flows inward toward the wing root from the tips, while air from outside the span replaces it. On the underside, the flow either moves inward more weakly or even outward. At the trailing edge, these spanwise flows meet, and the mismatch in velocity causes the air to roll up into numerous small streamwise vortices along the span. These smaller vortices then combine into two powerful trailing vortices, located just inside the wingtips (Figure 3). The strength of each trailing vortex matches the strength of the circulation that effectively replaces the wing itself.
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| Figure 3. Horseshoe vortex. Because air is largely transparent, such flow structures are generally not visible |
The presence of trailing and starting vortices can be observed in real life. For example, when a fast aircraft pulls out of a dive in humid air, the pressure and temperature drop within the trailing vortices can cause water vapor to condense, creating visible thin streamers behind each wingtip.
The starting vortex can be demonstrated even more simply. Place a flat board vertically in a tub of water and move it suddenly at a small angle of attack. An eddy will separate from the trailing edge and drift away, this is the starting vortex, produced by the circulation induced around the “wing.”
Now, consider a wing in steady flight. It affects the surrounding airflow, and changes in parameters such as span, planform, twist (aerodynamic or geometric), and flight speed will alter these effects. The bound vortex system must be able to replicate these changes accurately.
Because a real wing sheds trailing vortices, the theoretical bound vortex must do the same. One key feature of real wings is that lift per unit span decreases near the wingtips due to the pressure equalization between upper and lower surfaces. A proper bound vortex system must reproduce this lift variation across the span.
To achieve complete equivalence, the wing is modeled as a large number of spanwise vortex elements of varying lengths. Each element bends backward at its ends to form a pair of trailing vortices. The different element lengths simulate the gradual reduction in lift toward the tips, while the bent ends create the trailing vortex system. Together, they replicate both key aerodynamic behaviors of a real wing.
For simplified analysis, however, the wing can be replaced by a single bound vortex with circulation equal to the mid-span value. Bending this vortex back at both ends produces the trailing vortex pair. While this approach is less detailed, it is sufficient for estimating wing effects at distances greater than about two chord lengths from the center of pressure.
When studying the near-field effects of the wing, this complete vortex system provides the most accurate representation. For estimating far-field phenomena, such as wake effects at larger distances, the model is often simplified. In this case, the wing is represented by a single bound vortex with its trailing pair, forming what is known as the simplified horseshoe vortex (Figure 4).
The starting vortex can be demonstrated even more simply. Place a flat board vertically in a tub of water and move it suddenly at a small angle of attack. An eddy will separate from the trailing edge and drift away, this is the starting vortex, produced by the circulation induced around the “wing.”
Bound Vortex System
The starting vortex and the trailing vortex system are physical phenomena that can be observed under the right conditions. In contrast, the bound vortex system is a theoretical construct, an imagined arrangement of vortices that replaces the physical wing (as in thin-airfoil theory). This model simplifies the wing’s geometry but still reproduces, at least some distance away, the forces, disturbances, and aerodynamic effects of a real wing. This concept lies at the heart of wing theory.Now, consider a wing in steady flight. It affects the surrounding airflow, and changes in parameters such as span, planform, twist (aerodynamic or geometric), and flight speed will alter these effects. The bound vortex system must be able to replicate these changes accurately.
Because a real wing sheds trailing vortices, the theoretical bound vortex must do the same. One key feature of real wings is that lift per unit span decreases near the wingtips due to the pressure equalization between upper and lower surfaces. A proper bound vortex system must reproduce this lift variation across the span.
To achieve complete equivalence, the wing is modeled as a large number of spanwise vortex elements of varying lengths. Each element bends backward at its ends to form a pair of trailing vortices. The different element lengths simulate the gradual reduction in lift toward the tips, while the bent ends create the trailing vortex system. Together, they replicate both key aerodynamic behaviors of a real wing.
For simplified analysis, however, the wing can be replaced by a single bound vortex with circulation equal to the mid-span value. Bending this vortex back at both ends produces the trailing vortex pair. While this approach is less detailed, it is sufficient for estimating wing effects at distances greater than about two chord lengths from the center of pressure.
Horseshoe Vortex
For most aerodynamic purposes, the wing’s vortex model is reduced to the bound vortex within the wing together with the trailing vortex pair that extends from it. These elements form a three-sided structure called the horseshoe vortex (Figure 3).When studying the near-field effects of the wing, this complete vortex system provides the most accurate representation. For estimating far-field phenomena, such as wake effects at larger distances, the model is often simplified. In this case, the wing is represented by a single bound vortex with its trailing pair, forming what is known as the simplified horseshoe vortex (Figure 4).
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| Figure 4. Simplified horseshoe vortex as a crude model for a lifting wing |