What factors determine magnitude of "Vind" (induced velocity)?


If we take analogy with induced velocity at wing,isnt "Vind" must be prependicular to Vr?

Induced velocity is almost always neglected

I notice that blade theory very rearly use Vind into consideration.For example effective pitch and advance ratio not include Vind, pretend like blade is "ideal".

I can see that Vind increase axial velocity and decrease circumferential velocity,this reduce real AoA what blade section see.

Is Vind very small compare to V and Vt,so this is reason why it is neglect in most cases?

Θ real angle of advance enter image description here

enter image description here


Using Momentum theory (the french version known as Froude theory is much more detailed) and under several restricting assumption, we can find a relation between thrust ($T$), radius of the propeller ($S=\pi R^2$), flight speed ($V_0$) and induced velocity ($v_i$).

$T=\rho S(V_0+v_i)*2v_i$

A numerical example with $R=2.2m$, $\rho=1.225kg/m^3$ (air density in standard condition), $T=30'000N$ show the evolution of the induced velocity with respect to the flight speed.

Induced velocity <span class=$v_i$ as a function of flight speed $V_0$" />

As you can see the induced velocity is decreasing with flight speed. As we are designing propeller blades to be efficient at cruise speed, the induced velocity becomes in fact quite negligible when you are calculating the ideal twist of your blade.

On the other hand if you want to calculate the thrust available at takeoff you need to take the induced velocity into account as it will greatly decrease your propeller efficiency.

As for the orientation of the induced velocity, it has two component axial ($Uind$) and rotational ($U\phi ind$) velocity that are created by the propeller as clearly shown in your second picture. Therefore it is not exactly parallel to $V_0$. The rotational component is a direct energy loss and is therefore minimized during design phases. In the Froude theory there is an assumption saying it is equals to 0.

  • $\begingroup$ ,Why Vind is not drawn parallel with Vo?What is orientation of Vind in theory? I think in reality Vind comes from same direction as Vo. $\endgroup$ – user52248 Sep 21 '20 at 19:14
  • $\begingroup$ A propeller creates an helicoidal wind, which means it has a rotational component and an axial component. Therefore it's not totally parallel with V0 and this orientation depends on your propeller design. But in the above theory the rotational component is neglected as it is supposed to be very small. In fact this is energy lost as it doesn't provide any thrust, so propeller design try to minimize this value. $\endgroup$ – MaximEck Sep 21 '20 at 19:26
  • $\begingroup$ Vector sum of rotational (Vt) and axial (Vo) velocity is Vr at my first picture.So from wing analogy Vind must be prependicular to Vr.But I think in books they drawn Vind prependicular to Ve(effective).Isnt that strange? $\endgroup$ – user52248 Sep 21 '20 at 19:37
  • $\begingroup$ No Ve is a composition of Vind and Vr none of them are perpendicular. But the drawing is not helping as they put both induced velocity with almost same magnitude. $\endgroup$ – MaximEck Sep 21 '20 at 19:42
  • $\begingroup$ both induced velocity How do you mean both induced velocity,there is only one Vind? $\endgroup$ – user52248 Sep 21 '20 at 19:55

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