A racing car travels at 235 mph along a straightaway.The team engineer wishes to locate an air inlet on the body of the car to obtain cooling air for the driver’s suit. The plan is to place the inlet at a location where the air speed is 60 mph along the surface of the car. Calculate the static pressure at the proposed inlet location. Express the pressure rise above ambient as a fraction of the freestream dynamic pressure.
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:
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PURPOSE OF THE EXERCISE:
Demonstrate how to apply Bernoulli’s equation along a streamline to determine a static-pressure recovery from a reduction in local velocity along a streamlined body. -
ENGINEERING CONTEXT:
In designing a cooling-air inlet for a racing car, the engine is interested in supplying air at a slightly higher static pressure than ambient. Once mastered, this technique allows a designer to locate and size inlets that use the local flow deceleration to “pressurize” the intake air. -
PHYSICAL CONTEXT:
The dominant physical effect is the conversion of dynamic pressure into an increased static pressure via local deceleration of the flow, as captured by the Bernoulli equation for horizontal, inviscid, steady, incompressible flow. -
ASSUMPTIONS:
Assume steady, inviscid, incompressible flow with air density \rho = 1.2\,\mathrm{kg/m^3} and neglect elevation differences; this is reasonable given the high-speed external flow over a smooth car body. -
SOLUTION TECHNIQUE:
A student should first sketch a streamline passing from the undisturbed freestream (velocity U_\infty) to the location of the inlet (where the local velocity U_\mathrm{local} is reduced). Then apply Bernoulli’s equation along that streamline:
p_\infty+\frac{1}{2}\rho U_\infty^2 = p_\mathrm{inlet}+\frac{1}{2}\rho U_\mathrm{local}^2\,.
Solving for the static pressure at the inlet yields
p_\mathrm{inlet} = p_\infty+\frac{1}{2}\rho\left(U_\infty^2-U_\mathrm{local}^2\right)\,.
Given:
• Freestream speed: U_\infty=235\,\text{mph}\approx235\times0.44704\approx104.9\,\mathrm{m/s},
• Local speed at inlet: U_\mathrm{local}=60\,\text{mph}\approx60\times0.44704\approx26.8\,\mathrm{m/s}.
Thus,
p_\mathrm{inlet}=p_\infty+\frac{1}{2}(1.2)\Bigl[(104.9)^2-(26.8)^2\Bigr]\,.
Evaluating,
(104.9)^2\approx11014\,\mathrm{m^2/s^2},\quad (26.8)^2\approx718\,\mathrm{m^2/s^2}\,,
so that
\Delta p=p_\mathrm{inlet}-p_\infty=\frac{1}{2}(1.2)(11014-718)\approx0.6\times10296\approx6178\,\mathrm{Pa}\,.
The freestream dynamic pressure is
q_\infty=\frac{1}{2}(1.2)(104.9)^2\approx0.6\times11014\approx6608\,\mathrm{Pa}\,.
Expressing the pressure rise as a fraction of q_\infty,
\frac{\Delta p}{q_\infty}\approx\frac{6178}{6608}\approx0.935\,.
- REFLECTION:
The analysis shows that by decelerating the external flow from approximately 105\,\mathrm{m/s} to 27\,\mathrm{m/s}, the static pressure increases by about 6178\,\mathrm{Pa} (or about 94% of the freestream dynamic pressure). In engineering terms, this result means that a properly located and designed inlet can provide air at a significantly elevated static pressure suitable for cooling the driver’s suit.
In formulating this response, I reviewed 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:
- Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 2 (HIGHLY RELEVANT)
- Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 5 Part 1 (HIGHLY RELEVANT)
- Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 6 Part 1 (HIGHLY RELEVANT)
- Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 1 (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
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