A dragonfly wing airfoil is essentially a thin plate, whose upper and lower surface velocities can be closely approximated by
ue(x)V∞ = (xc)−b(upper surface) (E1.22)
ue(x)V∞ = (xc)+b(lower surface) (E1.23)
Use the following case-sensitive typed names for the various symbols.
Symbol
x
c
b
V∞
Typed
x
c
b
V
Note: For each of the questions in this Problem, there is more than one solution method which can be used. Any of the methods, if applied correctly, will be marked correct.
LIFT (2 points possible)
1) If this airfoil were to be modeled as a simple vortex sheet, what would be the appropriate sheet strength (defined positive clockwise)?
γ(x)=- unanswered
2) Determine the 2D lift coefficient of this airfoil for b=0.05.
Round your answer to two significant digits.
cℓ=- unanswered
You have used 0 of 2 submissions
DRAG (4 points possible)
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.
Round your answer to two significant digits.
upper surface θTE=- unanswered
lower surface θTE= - unanswered
4) What is the overall 2D profile drag for this airfoil in this condition? Round your answer to two significant digits.
cd=- unanswered
5) What is the 2D lift-to-drag ratio for this airfoil in this condition? Round your answer to two significant digits.
cℓcd=- unanswered
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.
To find the appropriate sheet strength for modeling the airfoil as a vortex sheet, we can use the Kutta-Joukowski theorem, which states that the lift generated by an airfoil is equal to the product of the air density, the velocity of the air, and the circulation around the airfoil.
1) The appropriate sheet strength can be found using the formula:
γ(x) = cl * V∞ * c
where cl is the lift coefficient for the airfoil, V∞ is the freestream velocity, and c is the chord length of the airfoil.
To find the lift coefficient for b=0.05, we can use the upper surface velocity equation E1.22:
ue(x) = (xc) - b
2) By substituting the values into the equation, we get:
cl = γ(x) / (V∞ * c)
= [(xc) - b] / (V∞ * c)
Find the appropriate values for x, c, b, and V∞ from the provided symbols and values, and calculate the lift coefficient for b=0.05.
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To determine the momentum thickness on the upper and lower surfaces at the trailing edge, we need to calculate the displacement thickness and the momentum thickness.
3) The displacement thickness can be calculated using the equation:
δ* = ∫ (1 - u(x)/V∞) dx
where u(x) is the velocity of the air at a given position x.
To find the momentum thickness, we can use the equation:
θ = ∫ (u(x)/V∞) (1 - u(x)/V∞) dx
where θ is the momentum thickness.
By integrating these equations over the upper and lower surface velocities given by equations E1.22 and E1.23, we can find the momentum thickness at the trailing edge for b=0.05.
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4) To find the overall 2D profile drag for this airfoil, we need to calculate the drag coefficient using the formula:
cd = D / (0.5 * ρ * V∞^2 * S)
where D is the drag force, ρ is the density of the air, V∞ is the freestream velocity, and S is the reference area.
The drag force can be calculated using the equation:
D = ∫ p(x) * u(x) * dx
where p(x) is the pressure at a given position x.
By integrating this equation over the chord length of the airfoil, we can find the drag force and then calculate the drag coefficient for b=0.05.
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5) Finally, to find the lift-to-drag ratio, we can divide the lift coefficient by the drag coefficient:
cl/cd
By substituting the values for cl and cd calculated in the previous steps, we can find the lift-to-drag ratio for b=0.05.
Remember to round your answers to two significant digits.