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The pressure distribution over a section of a two-dimensional wing at 4 degrees of incidence may be approximated as follows: Upper surface: Cp constant at – 0.8 from the leading edge to 60% chord, then increasing linearly to +0.1 at the trailing edge. 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.

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Answer:

The lift coefficient is 0.3192 while that of the moment about the leading edge is-0.1306.

Explanation:

The Upper Surface Cp is given as

[tex]Cp_u=-0.8 *0.6 +0.1 \int\limits^1_{0.6} \, dx =-0.8*0.6+0.4*0.1[/tex]

The Lower Surface Cp is given as

[tex]Cp_l=-0.4 *0.6 +0.1 \int\limits^1_{0.6} \, dx =-0.4*0.6+0.4*0.1[/tex]

The difference of the Cp over the airfoil is given as

[tex]\Delta Cp=Cp_l-Cp_u\\\Delta Cp=-0.4*0.6+0.4*0.1-(-0.8*0.6-0.4*0.1)\\\Delta Cp=-0.4*0.6+0.4*0.1+0.8*0.6+0.4*0.1\\\Delta Cp=0.4*0.6+0.4*0.2\\\Delta Cp=0.32[/tex]

Now the Lift Coefficient is given as

[tex]C_L=\Delta C_p cos(\alpha_i)\\C_L=0.32\times cos(4*\frac{\pi}{180})\\C_L=0.3192[/tex]

Now the coefficient of moment about the leading edge is given as

[tex]C_M=-0.3*0.4*0.6-(0.6+\dfrac{0.4}{3})*0.2*0.4\\C_M=-0.1306[/tex]

So the lift coefficient is 0.3192 while that of the moment about the leading edge is-0.1306.

The lift coefficient and the pitching moment coefficient about the leading edge due to lift are respectively; 0.3192 and -0.13

Aerodynamics engineering

We are given;

Distance of upper surface from leading edge to percentage of chord = -0.8

Percentage of chord for both surfaces = 60% = 0.6

Rate of increase at trailing edge for both surfaces = +0.1

Distance of lower surface from leading edge to percentage of chord = -0.6

angle of incidence; α_i = 4° = 4π/180 rad

Let us first calculate the Cp constant for both the upper and lower surface.

Cp for upper surface is;

Cp_u = (-0.8 × 0.6) - 0.1∫¹₀.₆ dx

Solving this integral gives;

Cp_u =  (-0.8 × 0.6) - (0.1 × 0.4)

Cp_u =  -0.52

Cp for lower surface is;

Cp_l = (-0.4 × 0.6) + 0.1∫¹₀.₆ dx

Solving this integral gives;

Cp_l =  (-0.4 × 0.6) + (0.1 × 0.4)

Cp_l =  -0.2

Change in Cp across the foil is;

ΔCp = Cp_l - Cp_u

ΔCp = -0.2 - (-0.52)

ΔCp = 0.32

Formula for the lift coefficient is;

C_L = ΔCp * cosα_i

C_L = 0.32 * cos (4π/180)

C_L = 0.3192

Formula for the pitching moment coefficient is;

(-0.3 * 0.4 * 0.6) - ((0.6 + (0.4/3)) * 0.2 * 0.4)

C_m,p =  -0.072 - 0.059

C_mp ≈ -0.13

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