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An aircraft is cruising at a speed of M = 0.77. The freestream temperature is 222.77 , and density is 0.4097 kg/m. The pressure is measured on a point along the wing, and it is found to be 9120 Pa .

Calculate the pressure coefficient at this point.
Assume isentropic flow.

1 answer

To calculate the pressure coefficient \( C_p \) at the point along the wing, we can use the following formula:

\[ C_p = \frac{p - p_0}{q_0} \]

where:

  • \( p \) is the pressure at the point on the wing (9120 Pa).
  • \( p_0 \) is the static pressure in the freestream.
  • \( q_0 \) is the dynamic pressure in the freestream.

Step 1: Calculate the Dynamic Pressure \( q_0 \)

The dynamic pressure \( q_0 \) is calculated using the formula:

\[ q_0 = \frac{1}{2} \rho V^2 \]

To find \( V \) (the freestream velocity), we can use the speed of sound \( a \) and the Mach number \( M \):

  1. Calculate the speed of sound \( a \) using the formula:

\[ a = \sqrt{\gamma R T} \]

For air:

  • \( \gamma = 1.4 \) (ratio of specific heats)
  • \( R \) (specific gas constant for air) \( = 287 , \text{J/(kg·K)} \)
  • \( T = 222.77 , \text{K} \)

Substituting the values:

\[ a = \sqrt{1.4 \times 287 \times 222.77} \]

Calculating this value:

\[ a = \sqrt{1.4 \times 287 \times 222.77} \approx \sqrt{111444.77} \approx 333.63 , \text{m/s} \]

  1. Find the freestream velocity \( V \):

\[ V = M \cdot a = 0.77 \cdot 333.63 \approx 256.4 , \text{m/s} \]

  1. Now calculate the dynamic pressure \( q_0 \):

\[ q_0 = \frac{1}{2} \rho V^2 = \frac{1}{2} \cdot 0.4097 \cdot (256.4)^2 \]

Calculating this:

\[ q_0 = \frac{1}{2} \cdot 0.4097 \cdot 65706.76 \approx \frac{1}{2} \cdot 26808.35 \approx 13404.17 , \text{Pa} \]

Step 2: Determine the Static Pressure \( p_0 \)

For isentropic flow, the static pressure can be related to the freestream conditions as follows:

\[ p_0 = p + q_0 \]

Rearranging gives:

\[ p_0 = p + q_0 = 9120 , \text{Pa} + 13404.17 , \text{Pa} \approx 22524.17 , \text{Pa} \]

Step 3: Calculate the Pressure Coefficient \( C_p \)

Now substituting \( p \), \( p_0 \), and \( q_0 \) into the \( C_p \) expression:

\[ C_p = \frac{p - p_0}{q_0} = \frac{9120 , \text{Pa} - 22524.17 , \text{Pa}}{13404.17 , \text{Pa}} \]

Calculating:

\[ C_p = \frac{9120 - 22524.17}{13404.17} \approx \frac{-13404.17}{13404.17} \approx -1 \]

Final Answer

Thus, the pressure coefficient \( C_p \) at the point along the wing is:

\[ C_p \approx -1.0 \]