Biot-Savart Law—Induced Velocities
The Biot-Savart formula reads
where d V i, jk is the induced velocity of the small element dl at a point k along a blade. ~ti, j, k is the distance vector having its origin at the vortex element and its extremity at point k. See Fig. 10.10.
For a three-bladed rotor, the u-component for a blade is given by
Uj = (Гк+1 – Гк)ak, j + Гкйк, j (10.27)
where the first term corresponds to influence of the vortex filaments k of each blade on the control point j on the lifting line and the tilde “~” term corresponds to the influence of the two other lifting line elements at yk on the control point j. Similarly, the w-components are obtained from the circulation with the influence coefficients ck, j and, with straight blades and zero coning angle, Ck, j = 0
Given the induced velocity components, the flow at the blade element can be analyzed and the local lift found. This is an approximation that amounts to neglecting the effect of the neighboring elements and considers each element in isolation, as if it were part of an infinite blade. Mathematically, this corresponds to neglecting the derivatives in the span direction compared to the other derivatives: dy ^ dx, .
This is appropriate, as has been found, for large aspect ratio wings and blades, that is those for which the chord is small compared to the span. This approach is called “strip theory”.