Question

The subject of the potl fluid flow over Joukowski airfoils was covered in Chapter 4. Once the center of a circle is moved into the second quadrant of the -plane and the cicle is allowed to pass one of the 2 critical points (Kutta Condition), a Joukowski airfoil is created on the transformed z-plane with appropriate thickness and camber. Consequently, the coresponding hydrodynamic force can also be determined. Consider exteuding this basic knowledge to the design of blades of propellers. Each of the blades of a propeller can be considered as a collectiou of different airfoils. Now consider Figures la and Ib shown below, Propellers are designed to exert a thrast force (Fsk) to drive the aircraft forward in the direction of axis of rotation. Deooting the following angular speed of the propeller shaft 2). forward speed in the direction of the axis of rotation torque exerted by the eugine the power of the engine and propeller are 3Ω and Fnv. respectively. Efficiency is then: Calling line OE in Fig. la to be the axis of the blade, the cirele formed by line OE is called the propeller ase. Somewhere along line OE at radial distance (r) from the shaft of the propeller, consider airfoil AB (Fig. la) with its side view shown in Fig. Ib. 0 (la) fL (Lb) Tbr mabar airfoil velocity (WO) dat is relied so v nd will give rise so difsereazial dng ove (Di that eau be found inge viscous tlow theoey, diteretial li soce that we have discussod ia this elass. (a) What is the differential forward thrust force (aF)in direction of V? (b) What is the differential I torque of the s,me) that opposes the rotation of the blade? (c) What are the expressions for the total thrust force and torque of the propeller?

Answer 1

(a & b) The blade twist angle is a function of the geometry of the blade. The relative fluid angle is the sum of the blade pitch angle beta and the angle of attack alpha. Therefore, we can express angle of attack as a relationship to the angle of relative fluid as The following expressions can also be determined: tan phi = U(1 - a)/(1 + a^1) Ohm r = 1 - a/(1 + a^1) lambda_r Also, for the relative fluid velocity, denoted by U_ rel we can write, U_ rel = U(1 - a)/sin phi Differential Lift force (dF_L), dF_L = C_1 1/2 p U^2_rel cdr Differential drag force (dF_D), dF_D = C_d 1/2 pU^2_rel cdr Differential normal force (dF_N), contributing to thrust, is Given by dF_N = dF_L cos phi + dF_D sin phi Differential tangential force (dF_T), contributing to useful torque, is Given by, dF_T = dF_L sin phi - dF_D cos phi