# What is the difference between ground effect for an infinite wing and for a finite wing?

A question about ground effect has just reappeared.

I think one needs to differentiate between ground effect for a finite wing (3D flow) and ground effect for an infinite wing (2D flow).

What are the differences?

That wingspan is used as a measure of distance to the ground to indicate the presence or strength of ground effect leaves me with the impression that there is a 3D aspect to ground effect. Ignoring maneuvering a long wing for the moment.

I haven't fully grasped the importance or unimportance of the trailing vortices yet.

• Does your infinite wing match the curvature of the infinite ground? – JAB Dec 25 '17 at 3:03
• @JAB With infinite I mean the dimension along it's length ("span") perpendicular to the airflow, and not the chord or the thickness. – user7241 Dec 25 '17 at 3:06
• @JAB Obviously I go with the flat Earth assumption in both directions. I hope you're not a "flat earther". – user7241 Dec 25 '17 at 4:49

Ground-effect analysis is generally completed using the method of images, where we imagine an opposite geometry providing a flow that interacts with the original geometry.

## The 2D case

Katz and Plotkin have a good explanation; here is their diagram for the 2D case:

For a flat-plate airfoil, the math works out to $$$$C_\mathrm{l} =\frac{\pi \alpha \left(c^2-8 \alpha c h+16 h^2\right)}{2 h (4 h-\alpha c)}$$$$ where $$c$$ is the chord, $$\alpha$$ is the angle of attack, and $$h$$ is the height above the ground.

## The 3D case

In the 3D case, we model the entire wing (or aircraft) as mirrored. The math here gets quite a bit more complicated, but I can summarize Phillips and Hunsaker, who provide the following figure and equation:

For an untapered flat wing with small $$\alpha$$, the lift coefficient in ground effect $$C_\mathrm{L_G}$$ is about $$$$C_\mathrm{L_G}\approx\left(\left(\frac{261.562}{\left(\frac{b}{c}\right)^{0.882}}-\frac{1.944}{\left(\frac{b}{c}\right)^{0.165}}\right) \left(\frac{h}{b}\right)^{0.787} e^{-9.14 \left(\frac{h}{b}\right)^{0.327}}+1\right)C_\mathrm{L}$$$$ where $$C_\mathrm{L}$$ is the lift coefficient in the absence of ground effect, $$h$$ is the height above the ground, $$c$$ is the chord, and $$b$$ is the wingspan.

## Comparing the two

I plotted the 2D equation for an airfoil with $$c=1$$ along with the 3D equation with $$C_\mathrm{l}=C_\mathrm{L}$$ for a wing with $$c=1$$ and $$b=10$$ (for any reasonable aspect ratio, this trend is similar):

As expected, far away from the ground and very close to it the 2D and 3D cases converge. The 3D case gets a bit more help from the ground in the intermediate range. Because we have only considered inviscid flows in the mathematical analysis, we know that the induced drag is the major difference between the 2D and 3D cases.

The (bound) vortices depend on the velocity difference between the upper and lower airfoil surface. the velocity difference depends on the pressure difference (destribution) along the upper/lower airfoil. In case of a 3D flow the pressure difference drops the closer we get to the wing tip -> so the vortices should get weaker/smaller -> the ground effect or the lift in generall decreases ... i would say