Given a rotor in vertical climb with the following characteristics: σ=0.1,θ tw​ =−9 degrees, κ=1.15,C do​ =0.01 (for simple performance estimate), C l α​ ​ =5.7 (per radian) a. For the case where the rotor is in hover condition and the collective pitch angle is 10 degrees, compute the total inflow ratio λ=(V+ v)/ΩR at r/R=0.7 using B.E.M. theory. b. What will be the values of λ and λ i​ at r/R=0.7 for climb at λ c​ = V/ΩR=0.01 using the same pitch angle. c. Will C T​ be the same or different in cases a and b above? Explain theoretically. d. Comment on the reason of trend of values of λ i​ from hover to λ c​ = 0.01

The question is asking for calculations and explanations related to rotor dynamics, specifically in the context of the Blade Element Momentum (BEM) theory. Here are the step-by-step solutions:

a. The total inflow ratio λ can be calculated using the BEM theory. First, we need to calculate the thrust coefficient (C_T). The formula for C_T in hover is:

C_T = 2 * T / (ρ * A * (ΩR)^2)

where T is the thrust, ρ is the air density, A is the rotor disk area, and ΩR is the rotor tip speed. However, we don't have the thrust. Instead, we have the solidity ratio (σ), the twist angle (θ_tw), the lift curve slope (C_lα), and the collective pitch angle (θ_0). We can use these to calculate C_T as:

C_T = σ * C_lα * (θ_0 - θ_tw / 2) / 2

Substituting the given values, we get:

C_T = 0.1 * 5.7 * (10 - (-9) / 2) / 2 = 0.1 * 5.7 * 14.5 / 2 = 0.41325

Now, we can calculate λ using the equation:

λ = sqrt(C_T / 2)

which gives λ = sqrt(0.41325 / 2) = 0.45 (approximately)

b. For the climb case, the total inflow ratio λ is the sum of the induced inflow ratio λ_i and the climb inflow ratio λ_c. We can calculate λ_i using the equation:

λ_i = sqrt(C_T / 2 - λ_c^2)

Substituting the given λ_c = 0.01, we get:

λ_i = sqrt(0.41325 / 2 - 0.01^2) = 0.44 (approximately)

Then, λ = λ_i + λ_c = 0.44 + 0.01 = 0.45

c. The thrust coefficient C_T will be different in the hover and climb cases. In hover, all the thrust is used to counteract gravity. In climb, some of the thrust is used to provide upward velocity. Therefore, C_T is generally higher in hover than in climb for the same power.

d. The induced inflow ratio λ_i decreases from hover to climb. This is because in hover, the entire inflow is induced by the rotor drawing air through the rotor disk. In climb, some of the inflow is due to the upward motion of the helicopter, reducing the amount of inflow that needs to be induced by the rotor.