Glauert's Induced Velocity Equation

In this article we describe Glauert’s equation for estimating the induced velocity of a moving rotor. The equation provides an estimate of the induced velocity in general, trimmed flight (not just hover like the momentum theory result we described in an earlier article).

Unlike momentum theory, Glauert’s model does not follow from first principles. Instead, it’s a heuristic that seems to work reasonably well in many scenarios.

Equation

Glauert’s equation is provided below. Notice that the equation is transcendental in $v$—you can’t solve for $v$ directly. We’ll discuss how to estimate $v$ in the next section.

$$ \begin{equation} T =2v \rho A \sqrt{u^2 + (w+v)^2} \label{eq:glr} \end{equation} $$

The variables used are described below and in Figure 1.

$T$ - rotor thrust
$v$ - the rotor induced velocity we are trying to calculate (opposite in direction to thrust)
$A$ - the area of the rotor
$u$ - the “edgewise” or tangential speed of the rotor through the air (roughly the forward speed of the helicopter)
$w$ - the speed of the rotor in the direction of thrust, relative to the surrounding air excluding the induced velocity

Figure 1 - The free stream flow u,w and induced flow v associated with a helicopter in trim flight.

Solving Glauert’s Equation

Glauert’s equation is typically solved by iteratively (1) guessing the induced velocity, (2) plug the guess into the equation above to see how well it works, and (3) update your guess based on the error. A particularly good way to do this, used by most practitioners, is using Newton’s method. Most people use the hover induced velocity $\sqrt{T/2 \rho A}$ from momentum theory for the initial guess.

The plot below shows how $v$ changes as a function of $u$ and $w$ for a 20’ rotor producing 10,000 lbs of thrust. The $v$ for each combination of $u$ and $w$ was estimated using Newton’s method.

Plot of v as a function of both u and w for a rotor with 20' radius producing 10,000lb of thrust at standard air density — Figure 2 - Plot of $v$ as a function of both $u$ and $w$ for a rotor with 20' radius producing 10,000lb of thrust at standard air density.

Assumptions

There are a couple important assumptions to be aware of. First, this equation assumes a uniform distribution of induced velocity over the disk. For a practical rotor, this is never the case. This is more so in forward flight, where the front (leading edge) of the rotor actually experiences less induced velocity than the rear (trailing edge) because the air has had more time to be accelerated by the time it reaches the back. This is explained in our article on transverse flow. To work around this, engineers often add a "linear inflow" correction factor (like Pitt-Peters or Drees models) to account for this gradient.

Next, the equation assumes that the flow is an equilibrium state instant. Imagine making a step change to the collective (feathering of all blades). This increases thrust, but the flow around the rotor can’t instantly “snap” to the equilibrium flow associated with this new state. Instead, the flow gradually adapts over seconds. Glauert’s equation does not attempt to model these dynamics.

Despite these limitations, Glauert’s equation is heavily used because it’s (1) simple, (2) captures basic phenomena like Effective Translational Lift (ETL), and (3) provides an excellent "first guess" for more complex models.

Why is Induced Velocity Smaller with Forward Speed?

You may have noticed above that induced velocity actually decreases with forward speed. The reason is similar to why induced velocity is smaller for larger rotors. Thrust can be estimated as the added downward momentum to the surrounding air. The more volume (and hence mass) of air that’s pushed down, the less the velocity required to achieve a given momentum, and hence thrust. When a rotor moves forward, it effectively acts on a larger volume of air per unit time, meaning more mass and less required velocity.

Of course, this is assuming the thrust remains constant. In reality, main rotor thrust must both counter the helicopter’s weight and overcome aircraft drag. At high speeds drag becomes large and hence thrust must increase well beyond the weight of the helicopter, which can then drive the induced velocity to increase rather than decrease.

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