A Pitot-static tube in a wind-tunnel gives a static pressure reading of \(40.7 \mathrm{kPa}\) and a stagnation pressure \(98.0 \mathrm{kPa}\). The stagnation temperature is \(90{ }^{\circ} \mathrm{C}\). Calculate the air velocity upstream of the Pitot-static tube.

The Bernoulli equation is a fundamental principle in fluid dynamics. It describes the behavior of fluid flow, connecting pressure, velocity, and height in an idealized situation. For compressible flows, like air in a wind tunnel, we often use the compressible Bernoulli equation. It considers changes in density. This equation is crucial in various applications, including the calculation of airspeed using a Pitot-static tube.
When air flows over a surface, its kinetic and potential energy change, affecting pressure. The equation links static pressure (the undisturbed flow pressure) and stagnation pressure (the pressure at zero velocity). By applying Bernoulli's principle to a Pitot-static tube, we can deduce the air velocity.
The compressible form of the equation is:
\[ P_0 = P + \frac{1}{2} \rho v^2 \]
Here, \(P_0\) is the stagnation pressure, \(P\) is the static pressure, \(\rho\) is the air density, and \(v\) is the flow velocity.
It's key to remember this relationship when moving on to solve exercises like Pitot-static tube problems. The pressure difference measured can then be utilized to derive the velocity of the air.

The Mach number is essential in aerodynamics and fluid mechanics. It is a dimensionless quantity representing the ratio of the speed of an object (or flow) to the speed of sound in that medium.
Mathematically, it is expressed as:
\[ M = \frac{v}{a} \]
where \(M\) is the Mach number, \(v\) is the velocity of the object or air, and \(a\) is the speed of sound in the same medium.
In the context of a Pitot-static tube, once we have the stagnation and static pressures, we can use them to find the Mach number through the relationship:
\[ \frac{P_0}{P} = \bigg(1 + \frac{\frac{\text{gamma} - 1}{2}}{2}\bigg( M^2 \bigg )^{\frac{\text{gamma}}{\frac{\text{gamma}} - 1}} \]
Here, \(P_0\) is the stagnation pressure, \(P\) is the static pressure, and \(\text{gamma}\) is the ratio of specific heats for air (approximately 1.4).
Understanding Mach number is critical for compressible flow calculations as it tells us whether the flow is subsonic (M < 1), transonic (M ≈ 1), or supersonic (M > 1). Each regime has different characteristics affecting calculations and interpretations.

The speed of sound in air is a vital constant in aerodynamics and fluid dynamics. It represents how quickly sound waves pass through the air. This speed varies depending on temperature, pressure, and air composition.
The standard formula to determine the speed of sound in air is:
\[ a = \bigsqrt{\text{gamma} \times R \times T} \]
Where \(a\) is the speed of sound, \(\text{gamma}\) is the ratio of specific heats (approximately 1.4 for air at sea level), \(R\) is the specific gas constant for air (287 J/kgK), and \(T\) is the temperature in Kelvin.
In the context of calculating air velocity with a Pitot-static tube, the speed of sound anchors our calculations. Using the stagnation temperature, which is higher than the static temperature due to kinetic energy conversion, we get the speed of sound. This parameter is then crucial for accurately determining the Mach number and subsequently the airspeed.
Always ensure proper conversion of temperature to Kelvin to avoid errors in calculating the speed of sound.

Stagnation temperature is an important concept when dealing with high-speed airflows. It represents the temperature an air parcel would have if it were brought to rest adiabatically, meaning without heat exchange with its surroundings.
You can convert the given temperature from Celsius to Kelvin using:
\[ T_0 = T + 273.15 \]
where \(T_0\) is the stagnation temperature and \(T\) is the temperature in Celsius.
In our example, we convert 90°C to Kelvin to get 363.15 K.
Understanding and calculating the stagnation temperature accurately is crucial for solving problems involving the Bernoulli equation and Mach number. It's often given in problems involving Pitot-static tubes, as it's a reference temperature for determining the speed of sound in air.
Always verify the units you're using, and make sure to convert temperatures to Kelvin to ensure the results are accurate when applying the formulas.