How much is the induced drag?

We can quantitatively calculate the induced drag around airplanes using vortex dynamics in 3D. Consider, for example, an airplane with a 10m span wing. The chord length is 1m, and all cross-sections operate at CL=0.5. What is the induced downwash along the wing?

To answer this question, we need first to consider the distribution of the vortices. Let us first set the coordinate system. Let the origin be the center of the wing. Let x be the direction of the freestream. Let y be the spanwise direction. Let z point upward. The wing tips are then LaTeX: (0, -5, 0) and LaTeX: (0, 5, 0) . From our 2D incompressible potential aerodynamics part, we know that the circulation around the airfoil cross-section is $LaTeX: \frac{L}{\rho V_{\infty} b} = \frac12 C_L V_{\infty}\,c = \frac{V_{\infty}}4$ . Because vorticity has no end, the circulation must come out of the wing tips as vortices trailing behind the airplane. When the induced flow is small compared to $LaTeX: V_{\infty}$ , these tip vortices align with the freestream direction (x axis). The one located at LaTeX: y=-5, z=0 has vorticity pointing towards the LaTeX: -x direction; the one located at LaTeX: y=5, z=0 has vorticity pointing towards the x direction.

Now consider the flow induced by these vortices. There are three segments, all with the same vorticity $LaTeX: \frac{V_{\infty}}4$ . The segment on the wing does not affect the wing itself. This is because $LaTeX: d\vec{l}(\vec{x}_0)$ and $LaTeX: \vec{x}-\vec{x}_0$ are aligned and thus $LaTeX: d\vec{l}(\vec{x}_0) \times (\vec{x}-\vec{x}_0)$ term in the Biot-Savart law is zero. The two tip vortices start at the y-z plane in which the wing is located and goes to infinity in the x direction. Their effect on the y-z plane is half that of a 2D vortex that goes from minus infinity to plus infinity in the x direction. We know the induced flow of a corresponding pair of 2D vortices on the wing. A 2D vortex of circulation $LaTeX: \frac{V_{\infty}}4$ located at LaTeX: y=-5, z=0 would create an induced flow field $LaTeX: v=\frac{V_{\infty}}{8\pi} \frac{z}{(y+5)^2+z^2}, w=-\frac{V_{\infty}}{8\pi} \frac{y+5}{(y+5)^2+z^2}$ . A 2D vortex of circulation $LaTeX: \frac{V_{\infty}}4$ located at LaTeX: y=5, z=0 would create an induced flow field $LaTeX: v=-\frac{V_{\infty}}{8\pi} \frac{z}{(y-5)^2+z^2}, w=\frac{V_{\infty}}{8\pi} \frac{y-5}{(y-5)^2+z^2}$ . Considering the tip vortices have half as much effect on the y-z plane and use the superposition principle of incompressible potential flow, the induced flow on the wing (x=z=0) is thus

$LaTeX: w=\frac{V_{\infty}}{16\pi} \left(\frac{1}{y-5}-\frac{1}{y+5}\right)=\frac{V_{\infty}}{16\pi} \;\frac{10}{y^2-25}$

At y=0, 2, and 4, for example, the downwash is 0.8%, 0.9%, and 2.2% of the freestream.

The downwash rotates the effective freestream seen by individual airfoil cross sections. Instead of a freestream pointing towards the x-axis, each airfoil would see a local freestream equal to the local downwash plus the x-directional global freestream. The local freestream points slightly downward. As a result, the local lift vector of the airfoils, perpendicular to the local freestream, no longer points directly toward the z-axis. Instead, the local lift vector points slightly backward, with a component in the x direction. This component of the local lift vector contributes to drag in the global sense, i.e., is aligned with the global freestream. This is the induced drag.

To calculate the induced drag, we can integrate the x-component of the local lift vectors. This is most easily done by assuming $LaTeX: w \ll V_{\infty}$ and using a small angle approximation.

$LaTeX: D_i = \displaystyle\int_{-5}^5 \frac12\rho\, V_{\infty}^2\, C_L \,c \,\frac{w}{V_{\infty}} \, dy$

or

$LaTeX: C_{D\,i} = \frac{1}{10} \displaystyle\int_{-5}^5 C_L \,\frac{w}{V_{\infty}} \, dy$

Note that the ~1% span close to both tips have $LaTeX: w/V_{\infty}\sim O(1)$ . For all practical wings, such a high downwash would not only invalidate the small angle approximation but also decrease the angle of attack enough to invalidate the constant CL assumption. A small fraction of the span near the tips is thus often ignored during the integration. By splitting the wing into 5 equal sections in the spanwise direction, approximating the induced downwash on each section as a constant at y=-4, -2, 0, 2, and 4, respectively, we get

$LaTeX: C_{D\,i} \approx \frac{1}{5} C_L \Big(\frac{w}{V_{\infty}}\Big|_{-4} + \frac{w}{V_{\infty}}\Big|_{-2} + \frac{w}{V_{\infty}}\Big|_{0} + \frac{w}{V_{\infty}}\Big|_{2} + \frac{w}{V_{\infty}}\Big|_{4} \Big)=0.007$

What is Lifting Line Theory and Vortex Lattice Method?

The lifting line theory extends the method we used to calculate the induced drag above. It accounts for the variation of chord length and lift coefficient across the span of the wing by allowing different circulation at different spanwise locations. Instead of a single pair of vortices coming from the wing tip, the change in vorticity from one section of the wing to another necessitate vortices coming from between the wing tips. The following figure shows all these vortices, which are assumed to align with the global freestream. The circulation of each vortex is computed from the difference in circulation between adjacent wing sections. The effect of all the vortices is then added together to compute the downwash on different sections of the wing. The downwash distribution is then used to calculate the direction of the local freestream seen by each cross-section. The component of the local lift vector in the direction of the global freestream is then integrated to compute the induced drag.

The vortex lattice method is a more advanced method that considers the effect of the induced flow on the downstream location of the vortices themselves. The vortices are often approximated as the superposition of many vortex "rings" in the form of quadrangles. The induced flow is computed on the four corners of each quadrangle, which deforms the quadrangles as they move downstream.

GetImage (1).png

The method can also take into account the movement of the wing as well as time variation in the lift. These advantages allow the vortex lattice method to simulate flows around helicopters, propellers, and wind turbines, as shown in the following picture. The vortex lattice method still assumes incompressible potential flow and the resulting superposition principle. Unlike the lifting line theory, however, it no longer assumes steady flow nor that the induced flow is small compared to the freestream.

GetImage (2).png

How do helicopters generate thrust?

Realistic helicopter flows are complex, as seen in the following video: https://www.youtube.com/watch?v=Gss_-bwoOIc Because vortices stretch each other, the vorticity gets tangled into stronger but thinner filaments. This eventually leads to turbulence. The vortex lattice method often requires artificial diffusion to prevent excessive computation costs due to these effects. We can, however, understand the gists of flows around hovering helicopters by incorporating simple conservation laws.

GetImage (3).png

The simplest hovering helicopter model blurs the individual blades into a rotor disk that pushes the flow downward. As flow crosses the rotor disk, the velocity cannot change suddenly due to the conservation of mass. But its pressure increases by the action of the rotor disk. We can thus not apply Bernoulli across the rotor disk. Tip vortices shed from the blade tip are blurred into a vortex sheet that separates the air that passed the rotor disk from the air that did not. The nonzero vorticity in the vortex sheet prevents the application of Bernoulli across it. Flow velocity is zero away from the disk, except for in an area directly below the rotor disk. Flow speed at the rotor disk is v, and far below the rotor disk has velocity w. We denote the rotor disk area as A and the area of the flow tube far downstream below the rotor disk as A'.

Conservation of mass in a control volume between the rotor disk and far downstream bounded by the vortex sheet:

$LaTeX: v\,A = w\,A'$

Conservation of momentum in a large control volume (different from the one above) whose boundary is very far away from the rotor:

$LaTeX: T = \rho\,w^2 A'$

Here T is the upward rotor thrust, which gives a downward force to the flow. Because the boundary is far from the rotor in all directions, we can ignore the pressure influence.

Conservation of energy in the same large control volume, the power output by the rotor must be equal to the energy outflow from the control volume:

$LaTeX: T\,v = \frac12\rho\,w^3 A'$

Using the first equation, we can cancel all A' in the equations and get

$LaTeX: T = \rho\,w\,v A$ and $LaTeX: T\,v = \frac12\rho\,w^2\,v\, A$

Plugging the equation on the left above into the one on the right, we get LaTeX: w=2v . Thus,

$LaTeX: T = 2\rho\,v^2 A$ , or $LaTeX: v = \sqrt{\frac{T}{2\rho\,A}}$

Plugging v and w into the rotor power equation, we get

$LaTeX: T\,v = \sqrt{\frac{T^3}{2\rho\,A}}$

Recall fractional sensitivity analysis, a 1% increase in power would only generate 0.67% more thrust. A 1% increase in either air density or rotor area would lead to 0.5% decrease in the required power without changing the thrust.

An interesting consequence of the relation above is when a small power LaTeX: Tv needs to generate large thrust . If air density $LaTeX: \rho$ cannot be changed significantly, the only engineering option is to design a very large rotor area A. An example is the human-powered helicopter. See how large A is in the video: https://www.youtube.com/watch?v=emK-qIbuJ-k

The required power we derived is an ideal scenario. It assumes no viscous effect and that all kinetic energy downstream of the rotor is in a pure downward motion. In a realistic helicopter, viscous forces and angular momentum in the downwash make the required power even larger for the same thrust. The ratio of the ideal required power to the actual power is known as the figure of merit:

FoM $LaTeX: = P \Big/ \sqrt{\frac{T^3}{2\rho\,A}}$