Consider an infinitely thin flat plate of chord c at an angle of attack α in a supersonic flow. The pressure distribution
can be approximated as follows:
i) Upper surface: Cp constant at -0.8 from the leading edge to 60% chord, then increasing linearly to +0.1 at the trailing
edge.
ii) Lower surface: Cp constant at –0.4 from the leading edge to 60% chord, then increasing linearly to +0.1 at the trailing
edge.
Estimate the lift coefficient and the pitching moment coefficient about the leading edge due to lift.
first 60%
Cp lower - Cp upper = -.4 +.8 = + .4 = delta Cp front
last 40% = average from 60 to 100% = (.4 +0 )/2 = .2 = delta Cp rear
Cl = 1/c integral 0 to c of delta Cp dx
front part 1/c * .4 (.6 c)
back part 1/c * .2 (.4 c)
sum = .24 + .08 = 0.32 = Cl , lift coef per unit span
for the moment integrate with the distance x from the leading edge in the integral
To estimate the lift coefficient and the pitching moment coefficient about the leading edge due to lift, we can follow these steps:
Step 1: Calculate the average pressure coefficient (Cp) over the upper and lower surfaces by taking the average of their respective values.
For the upper surface:
Cp_upper = (Cp_leading edge + Cp_trailing edge) / 2
= (-0.8 + 0.1) / 2
= -0.35
For the lower surface:
Cp_lower = (Cp_leading edge + Cp_trailing edge) / 2
= (-0.4 + 0.1) / 2
= -0.15
Step 2: Calculate the lift coefficient (Cl) using the average pressure coefficient values.
Cl = 2 * (Cp_upper - Cp_lower)
= 2 * (-0.35 - (-0.15))
= -0.4
Step 3: Calculate the pitching moment coefficient about the leading edge (Cmle) due to lift.
Cmle = (Cp_leading edge * c) + (Cp_trailing edge * c) + (Cp_upper * c/2) + (Cp_lower * c/2)
= (-0.8 * c) + (0.1 * c) + (-0.35 * c/2) + (-0.15 * c/2)
Note: c represents the chord length.
It is important to know the value of the chord length (c) in order to calculate the pitching moment coefficient (Cmle) accurately.
To estimate the lift coefficient and the pitching moment coefficient about the leading edge due to lift, we can use the pressure distribution provided for the upper and lower surfaces of the flat plate.
The lift coefficient (Cl) is a dimensionless coefficient that relates the lift generated by an airfoil to the dynamic pressure and the reference area. It is given by the equation:
Cl = (2 * lift) / (ρ * V^2 * A)
Where:
- lift is the total lift force generated by the airfoil
- ρ is the density of the fluid
- V is the velocity of the fluid relative to the airfoil
- A is the reference area
In this case, since we have an infinitely thin flat plate, the reference area (A) can be taken as the chord length (c).
To calculate the lift, we need to integrate the pressure distribution over the surface of the flat plate. Here's how we can do that:
1. Break down the pressure distribution into two segments: leading edge to 60% chord and 60% chord to trailing edge.
For the upper surface:
- From leading edge to 60% chord, Cp = -0.8
- From 60% chord to trailing edge, Cp increases linearly to +0.1
For the lower surface:
- From leading edge to 60% chord, Cp = -0.4
- From 60% chord to trailing edge, Cp increases linearly to +0.1
2. Calculate the lift by integrating the pressure distribution.
For the upper surface:
- Lift_upper = ∫[(-0.8) * -1] + ∫[(0.1) * -1]
For the lower surface:
- Lift_lower = ∫[(-0.4) * -1] + ∫[(0.1) * -1]
(Note that the negative sign is used because we are considering the pressure acting upwards to generate lift.)
3. Calculate the total lift by summing the lift from the upper and lower surfaces:
Lift = Lift_upper + Lift_lower
4. Plug the values of lift, density (ρ), velocity (V), and chord length (c) into the lift coefficient equation:
Cl = (2 * Lift) / (ρ * V^2 * c)
The pitching moment coefficient (Cm) is a dimensionless coefficient that relates the pitching moment about the leading edge to the dynamic pressure, reference area, and the distance from the leading edge. It is given by the equation:
Cm = (2 * moment) / (ρ * V^2 * A * d)
Where:
- moment is the total pitching moment about the leading edge due to lift
- d is the distance from the leading edge to the reference point (in this case, we'll consider it as 0, so d = 0)
5. Plug the values of moment, density (ρ), velocity (V), chord length (c), and distance (d) into the pitching moment coefficient equation:
Cm = (2 * moment) / (ρ * V^2 * c * d)
Since the distance is considered zero in this case, the pitching moment coefficient about the leading edge due to lift (Cm) will be zero.
By following these steps and plugging in the appropriate values, you can estimate the lift coefficient (Cl) and the pitching moment coefficient about the leading edge due to lift (Cm) for the given pressure distribution.