An aircraft is cruising at a speed of M = 0.77. The freestream temperature is 222.77K , and Density 0.4097 kg/ m^3. The pressure is measured on a point along the wing, and it is found to be 9120 pa .

Question

Calculate the pressure coefficient at this point.

Answer 1

The pressure coefficient \(C_p\) is defined as:

\[ C_p = \frac{p - p_{\infty}}{q_{\infty}} \]

where:

• \(p\) is the local static pressure,
• \(p_{\infty}\) is the freestream static pressure, and
• \(q_{\infty}\) is the dynamic pressure.

To calculate \(C_p\), we first need to determine the freestream static pressure \(p_{\infty}\) and the dynamic pressure \(q_{\infty}\).

Step 1: Calculate the freestream static pressure (\(p_{\infty}\))

First, we need to calculate the dynamic pressure \(q_{\infty}\):

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

Since \(V\) is given in terms of Mach number (\(M\)), we can find the freestream velocity using:

\[ V = M \cdot a \]

where \(a\) is the speed of sound, which can be calculated using the relation:

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

Assuming air as an ideal gas, we take \(\gamma \approx 1.4\) (ratio of specific heats for air) and \(R \approx 287 , \text{J/(kg·K)}\).

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

\[ a = \sqrt{1.4 \cdot 287 \cdot 222.77} \approx \sqrt{ 1.4 \cdot 287 \cdot 222.77 } \approx \sqrt{ 113433.9114 } \approx 336.57 , \text{m/s} \]

2. Calculate velocity \(V\):

\[ V = M \cdot a = 0.77 \cdot 336.57 \approx 259.57 , \text{m/s} \]

3. Calculate the dynamic pressure \(q_{\infty}\):

\[ q_{\infty} = \frac{1}{2} \rho V^2 = \frac{1}{2} (0.4097) (259.57^2) \]

First, calculate \(V^2\):

\[ V^2 \approx 259.57^2 \approx 67454.41 , \text{m}^2/\text{s}^2 \]

Now compute \(q_{\infty}\):

\[ q_{\infty} \approx \frac{1}{2} (0.4097) (67454.41) \approx \frac{1}{2} (27685.499) \approx 13842.75 , \text{Pa} \]

Step 2: Calculate \(p_{\infty}\) using the barometric formula

To calculate \(p_{\infty}\), we can use the hydrostatic equation or the ideal gas law:

\[ p_{\infty} = \rho R T \]

\[ p_{\infty} = (0.4097 , \text{kg/m}^3) \times (287 , \text{J/(kg·K)}) \times (222.77 , \text{K}) \]

\[ p_{\infty} \approx 0.4097 \times 287 \times 222.77 \approx 24567.046 , \text{Pa} \approx 24567 , \text{Pa} \]

Step 3: Now calculate \(C_p\)

Given:

• Local static pressure \(p = 9120 , \text{Pa}\),
• Freestream static pressure \(p_{\infty} \approx 24567 , \text{Pa}\),
• Dynamic pressure \(q_{\infty} \approx 13842.75 , \text{Pa}\).

\[ C_p = \frac{p - p_{\infty}}{q_{\infty}} = \frac{9120 - 24567}{13842.75} \]

Calculating the numerator:

\[ 9120 - 24567 \approx -15447 \]

Now calculate \(C_p\):

\[ C_p = \frac{-15447}{13842.75} \approx -1.115 \]

Final Result:

The pressure coefficient \(C_p\) at this point is approximately:

\[ \boxed{-1.115} \]