To answer these questions, we will use the given equations:
Upper surface velocity: ue(x)V∞ = (xc) - b (Equation 1.22)
Lower surface velocity: ue(x)V∞ = (xc) + b (Equation 1.23)
Now, let's proceed with each question step-by-step.
1) If this airfoil were to be modeled as a simple vortex sheet, what would be the appropriate sheet strength (defined positive clockwise)?
To find the sheet strength, we need to calculate the circulation around the airfoil. The circulation is given by:
Γ = ∫(ue - u∞)dx
Since the upper and lower surface velocities are given by Equation 1.22 and Equation 1.23, we can substitute these values into the circulation equation:
Γ = ∫((xc) - b - V∞)dx (for the upper surface)
Γ = ∫((xc) + b - V∞)dx (for the lower surface)
Integrating both equations will give us the circulation for each surface.
Answer: To find the appropriate sheet strength, you need to evaluate the above integrals.
2) Determine the 2D lift coefficient of this airfoil for b=0.05.
The lift coefficient (cℓ) is given by:
cℓ = (2Γ)/(ρV∞c)
where
Γ = circulation,
ρ = density of the fluid (assumed to be constant), and
c = chord length of the airfoil.
From the previous question, you have the sheet strength (circulation) for each surface. Substitute these values into the lift coefficient equation, along with the given values for ρ, V∞, and c.
Answer: Calculate the lift coefficient using the given equation and values.
3) The dragonfly has a wing chord of 0.01 m, and can fly at 10 m/s. Assume roughly sea level viscosity ν=1.45×10−5m2/s. Determine the momentum thickness θ on the upper and lower surfaces at the trailing edge, again for b=0.05. Specify the results in meters.
The momentum thickness (θ) can be determined using the following equation:
θ = ∫[(u∞ - ue)/u∞]^2dx
where ue(x) is the velocity on the upper or lower surface, and u∞ is the free-stream velocity.
Using the given viscosity (ν), wing chord (c), free-stream velocity (V∞), and surface velocities (given by Equation 1.22 and Equation 1.23), you can calculate the momentum thickness for both the upper and lower surfaces at the trailing edge.
Answer: Calculate the momentum thickness for the upper and lower surfaces using the given equation and values.
4) What is the overall 2D profile drag for this airfoil in this condition?
The 2D profile drag coefficient (cd) can be determined using the following equation:
cd = (4πΓ^2)/(ρV∞^2c)
where Γ is the circulation and the other parameters are as defined earlier.
Using the given circulation from the first question, along with the given values for ρ, V∞, and c, you can calculate the profile drag coefficient.
Answer: Calculate the profile drag coefficient using the given equation and values.
5) What is the 2D lift-to-drag ratio for this airfoil in this condition?
The lift-to-drag ratio (cℓ/cd) can be calculated by dividing the lift coefficient (cℓ) by the profile drag coefficient (cd).
Using the lift coefficient calculated in the second question and the profile drag coefficient calculated in the fourth question, you can find the lift-to-drag ratio.
Answer: Calculate the lift-to-drag ratio by dividing the lift coefficient by the profile drag coefficient.