Wind tunnel speed

Homework Help
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An open-circuit wind tunnel draws in air from the atmosphere through a well-contoured nozzle. In the test section, where the flow is straight and nearly uniform, a static pressure tap is drilled into the tunnel wall. A manometer connected to the tap shows that static pressure within the tunnel is 45 mm of water below atmospheric. Assume that the air is incompressible, and at 25C, 100 kPa (abs). Calculate the air speed in the wind-tunnel test section.

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This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:

  1. PURPOSE OF THE EXERCISE:
    Use Bernoulli’s equation to relate a measured static pressure depression to the freestream velocity in a wind tunnel, reinforcing the connection between pressure differences and kinetic energy.

  2. ENGINEERING CONTEXT:
    Static pressure taps in wind tunnels are used to monitor test section conditions; by correctly linking the pressure dip (45 mm of water) to dynamic pressure, one can accurately calibrate and interpret wind tunnel speeds, a key procedure in aerodynamic testing.

  3. PHYSICAL CONTEXT:
    The dominant physical effect is the inviscid conversion of kinetic energy into pressure (and vice versa) along a streamline, as captured in Bernoulli’s equation for steady, incompressible flow.

  4. ASSUMPTIONS:
    Assume steady, incompressible, inviscid flow and that gravity can be neglected in the test section (since the pressure tap is on the horizontal wall); this is reasonable because the pressure difference is small and the streamline is nearly horizontal.

  5. SOLUTION TECHNIQUE:
    The static pressure in the test section is

p = p_\mathrm{atm} - \Delta p\,,

with the pressure depression given by the hydrostatic conversion

\Delta p = \rho_\mathrm{water}\,g\,h\quad\text{with }h=0.045\,\mathrm{m}\,.

Taking \rho_\mathrm{water}=1000~\mathrm{kg/m^3} and g=9.81~\mathrm{m/s^2}, compute

\Delta p = 1000\times9.81\times0.045\approx441.5\,\mathrm{Pa}\,.

Invoking Bernoulli’s equation between a region far upstream (at ambient static pressure) and the test section, the dynamic pressure is

\frac{1}{2}\rho_\mathrm{air}U^2=\Delta p\,.

Assuming \rho_\mathrm{air}\approx1.2~\mathrm{kg/m^3}, solve for U:

U=\sqrt{\frac{2\Delta p}{\rho_\mathrm{air}}}=\sqrt{\frac{2(441.5)}{1.2}}\approx\sqrt{735.8}\approx27.1~\mathrm{m/s}\,.
  1. REFLECTION:
    The analysis shows that the measured 45 mm water head corresponds to a dynamic pressure that yields a wind-tunnel speed of approximately 27 m/s. This result demonstrates the practical use of Bernoulli’s equation in experimental aerodynamics and highlights the importance of proper instrumentation to quantify the energetic state of the flow.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 2 (HIGHLY RELEVANT)
  2. Chapter 5 How Stuff Works: How a Pitot–static tube works (HIGHLY RELEVANT)
  3. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 5 Part 1 (HIGHLY RELEVANT)
  4. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 1 (HIGHLY RELEVANT)
  5. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (RELEVANT)

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