How is rotor airfoil analysis different than fixed wing airfoil analysis?

What the is difference between rotor airfoil analysis and fixed-wing airfoil analysis?

Intuitively, it seems that the airspeed at the tip of a rotor blade would be more than that at its root. Wouldn't this result in irregular torsion along the blade? Subsequently, wouldn't this affect the airfoil analysis as compared to a fixed wing?

If this is the case, then what software (hopefully free) can carry out this analysis?

You are correct on many fronts. Rotor blades see a varied load along their length, resulting in twisting, bending, and lagging (e.g., the blade being bent against its direction of rotation) loads. It's a mess. And, yes, all those effects feed back into blade aerodynamics, but that's a much more complicated answer that you might be looking for.

What you are describing is the foundation of blade element theory.

Let's consider a helicopter rotor, independent of the helicopter fuselage, and look at one annular ring of it while the rotor is in hover (so we have an axisymmetric flow field):

If we look at that annular ring, it would "cut" through a section of the rotor, which, in 2D, would look something like an airfoil. If we drew a bunch of vectors on it to represent the forces acting on the airfoil and the direction of the relative wind, it does look a heck of a lot like our 2D airfoil free body diagram from fixed-wing aerodynamics. And, truly, blade element theory (BET) considers it as such.

In BET, elements of the blade (such as the 2D airfoil above) operate without impact on the other blade elements. If you can determine the inflow at a particular blade section, you can determine the sectional lift and drag, and, integrating those properties across the blade, you can determine the thrust production and torque requirements that correspond to a particular operating condition. That said, while most of the labels on that diagram are familiar from fixed-wing flight, there are a few that should be different, namely: the induced velocity, inflow angle, and tangential velocity.

As you stated, the horizontal speed of the blade is going to change based on where you are on the blade, radially speaking--which is absolutely correct. If we look at a distribution of elemental velocities around the rotor, it's going to vary as shown here, in a plot from Leishman's Principles of Helicopter Aerodynamics (which has a great description of how to implement all of this, as well as some basic corrections for going into some rudimentary higher-order approximations). Note the hover case on the left hand side, where the velocities are axisymmetric, versus the forward flight case on the left hand side, where the velocities are asymmetric. That second case is a whole 'nother can of worms, and we'll get there in a minute. Stick with the hover case for now.

So knowing your radial location can get you the element's velocity, and knowing your climb rate and help you get how fast flow is moving vertically through the rotor, but flow has to be moving vertically through the rotor even when the aircraft's not climbing -- else our analogy of a rotor being a big fan (which it is) doesn't correspond with reality. Hence, there's some inflow speed, and it makes sense that that inflow speed is correlated to the amount of thrust being produced by a given element of the rotor...and this is the crux of the problem for us, since it has to capacity to dramatically change the angle of attack of the blade section (and thus our lift and drag predictions). So: how do we estimate the inflow velocity?

A simple approximation uses momentum theory (i.e., a quasi-1D analysis on fluid moving through the rotor disk, treating the rotor as a device that simply adds momentum to the flow) to determine the inflow on the rotor along that annular section, assuming that the inflow is axisymmetric. For what it's worth, it's a simple way to capture inflow velocity variation (and thus angle of attack variation as well) across the rotor disk, which is something that momentum theory cannot do alone, and it's decent at what it does. It has been the de-facto standard of the wind energy industry for awhile as more complex CFD computations have caught up to the industry's analysis needs and desires.

However, when flow is no longer axisymmetric, we get into a big mess of approximations to determine what that inflow does look like, since the wake will start to interact with the inflow as well and that, somehow, needs to be accounted for. It is here that we start to depart from analytical expressions and look to computational models using free-wake theory or higher-order models. Where fixed-wing models advance in similarity by modeling the wing's wake, viscous effects, etc, rotary wing models model a rotating wake that is being convected and recirculated by both rotor effects and the freestream.

That said, the expressions for a rotor in hover are relatively easy to code up, depending on how much fidelity you want (e.g., how much you know about the airfoils used on the blade as well as how the blade tapers and twists along its length). Googling around, I found a MATLAB code that might be useful, again, depending on your problem: https://www.mathworks.com/matlabcentral/fileexchange/21994-analysis-of-a-rotor-blade-system-using-blade-element-momentum-theory

• Can you elaborate on how to estimate the inflow...? I'm trying to derive the rotor thrust (in hover conditions) by integration of the elementary lift, but I have no idea on how to calculate the inflow... In momentum theory, I understand the derivation of thrust as a function of power and rotor diameter, but I see no way to get the inflow via momentum theory... Commented Aug 16, 2021 at 13:48

Other responses are complete but i want to add this: To avoid torsion dues to speed difference at root and blade tip ,blades are twiste(The twist of the propeller blade is the angle between the chord of the blade tip profile and the chord of the blade foot profile (drawing below). This twist allows the propeller to operate with a relatively constant angle of incidence over its entire length in cruising flight.

The setting angle θ is defined between the profile reference rope and the plane of rotation.)

the advancing blade and faster than the reversing blade :so to compensate for the asymmetry of the lift:

Each rotor blade is fixed to the rotor hub by means of two articulations, one in the plane of rotation and the other perpendicular to it. The position of the joints is chosen by each manufacturer, mainly with regard to stability and control.

The horizontal joint, also called the hinge, allows the blade to move up and down relative to the plane of rotation individually and also be advanced or delayed relative to its direction of rotation (forward / backward movement). The vertical joint, called the drag hinge, allows the blade to move back and forth. This movement is called offset or drag. Dampers are usually used to prevent excessive back and forth movements around the drag hinge. The purpose of the drag hinge and dampers is to absorb acceleration and deceleration of the rotor blades.

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