An airplane is moving through the air at a velocity v=200m/s. Streamlines just over the top
of the wing are compressed to 80% of their original area, and those under the wing are not
compressed at all.
a) Determine the velocity of the air just over the wing.
b) Find the difference in the pressure between the air just over the wing, P and that under
the wing, P′.
c) Find the net upward force on both wings due to the pressure difference, if the area of the
wig is 40m2 and the density of the air is 1.3kg/m3.
a) To determine the velocity of the air just over the wing, we can use the principle of continuity. According to this principle, the mass flow rate of a fluid remains constant along a streamline.
The equation for the principle of continuity is:
A1v1 = A2v2
where A1 and A2 are the areas of the wing cross-section at points 1 and 2 respectively, and v1 and v2 are the velocities of the air at those points.
In this case, we know that the area just under the wing remains unchanged, so we can substitute A1 with the original area. Let's assume the original area is A0. We also know that the area just over the wing is compressed to 80% of its original area, so A2 = 0.8A0.
Substituting these values into the continuity equation, we have:
A0v1 = 0.8A0v2
Since A0 is common to both sides, we can cancel it out:
v1 = 0.8v2
So the velocity of the air just over the wing is 0.8 times the velocity of the air just under the wing.
b) To find the difference in pressure between the air just over the wing (P) and that under the wing (P'), we can use Bernoulli's equation. Bernoulli's equation relates the pressure, velocity, and height of a fluid along a streamline.
The equation for Bernoulli's equation is:
P + 0.5ρv^2 + ρgh = constant
where P is the pressure, ρ is the density of the fluid, v is the velocity of the fluid, g is the acceleration due to gravity, and h is the height of the fluid.
Assuming the height and acceleration due to gravity are constant along the wing, we can simplify the equation to focus on the pressure and velocity terms:
P + 0.5ρv^2 = constant
Let's consider two points along the wing: just over the wing and under the wing. We can denote the pressure just over the wing as P and the pressure under the wing as P'.
Using Bernoulli's equation at each point, we have:
P + 0.5ρv1^2 = constant ... (1)
P' + 0.5ρv2^2 = constant ... (2)
Since the constant value is the same at both points, we can equate the two equations:
P + 0.5ρv1^2 = P' + 0.5ρv2^2
Substituting the relationship v1 = 0.8v2 from part (a), we can solve for the difference in pressure (P - P'):
P - P' = 0.5ρv2^2 - 0.5ρ(0.8v2)^2
Simplifying the equation, we get:
P - P' = 0.36ρv2^2
So the difference in pressure between the air just over the wing and under the wing is 0.36 times the density of the air times the square of the velocity of the air just under the wing.
c) To find the net upward force on both wings due to the pressure difference, we can use the equation:
Force = Pressure difference * Area
Substituting the given values, we have:
Force = (P - P') * A
Since we have already found the difference in pressure (P - P') in part (b), we can plug in that value, along with the area of the wing (A) to compute the net upward force. The density of air is given as 1.3 kg/m^3.
Force = (0.36ρv2^2) * A
Finally, substituting the density of air (ρ = 1.3 kg/m^3) and the area of the wing (A = 40 m^2) into the equation, we can calculate the net upward force on both wings.