### Soaring in a Thermal:

Bank Angle, Speed, and Radius

*Larry Bogan - Aug 1998*

#### Simple Circular Motion Model

When a sailplane flies in a perfect circle, it must bank to use the lift force to accelerate it toward the center of the circle (radial acceleration). At the same time the lift force is no longer opposite the pull of gravity (due to the bank) and must fly faster to increase the lift so that the vertical component is just equal that pull of gravity (weight).

Simple physics and trigonometry yields a simple relationship between, bank angle, speed in the circle, and the radius of the circle.
Now for some simple mathematics
But first let's define necessary parameters

- Bank Angle = b
- Lift Force = L
- Weight = W = mg
- g = acceleration of gravity = 32 ft/s
^{2}
- m = mass of the sailplane
- a
_{r} = radial acceleration = v^{2}/R
- v = speed of the glider in the circle of radius, R

Then apply the following:
- Requirement 1 -
**vertical equilibrium**
- The vertical forces on the glider are equal and opposite.

L(vertical) = L cos(b) = mg
- Requiremnt 2 -
**radial acceleration**
- In circular motion the horizontal component of the lift equals the radial acceleration.

L(horizontal)= L sin(b) = m a_{r} = m v^{2}/R

We can eliminate L and m from the two equations and get

sin(b)/cos(b) = v^{2}/gR

since tan(b) = sin(b)/cos(b)

**R = v**^{2}/[g tan(b)]

The radius of the turn increases with the square of the speed of the sailplane and inversely as the tangent of the bank angle. This is why the most effective strategy to get to the core of a thermal is to fly slowing and larger bank angle. NOTICE THAT THE RADIUS DOES NOT DEPEND ON THE MASS OF THE SAILPLANE.

#### Minimum Circling Radius

In an attempt decrease the circling radius by increasing the bank angle, the stall speed increases which increases the lower limit of the speed of flight. This sets a minimum circling radius. I have estimated the minimum turning radii for a sailplane with a 32 kt stall speed in zero bank assuming the wing develops lift in proportion to the square of the air speed. This is shown in the following graph along with the plots of turning radius for various bank angle.