A supersonic air stream at \(35 \mathrm{kPa}\) is deflected by a wedgeshaped two-dimensional obstacle of total angle \(20^{\circ}\) mounted with its axis parallel to the oncoming flow. Shock waves are observed coming from the apex at \(40^{\circ}\) to the original flow direction. What is the initial Mach number and the pressure rise through the shocks? If the stagnation temperature is \(30^{\circ} \mathrm{C}\) what is the velocity immediately downstream of the shock waves?

The Mach number is a key parameter in supersonic flow dynamics, representing the ratio of the flow velocity to the speed of sound in the medium. In this exercise, we deal with oblique shock waves, which occur when a supersonic flow encounters a wedge-shaped obstacle.
To calculate the initial Mach number (\text{M_1}), we use the shock wave angle formula for oblique shocks, which involves the deflection angle (\theta) and the shock wave angle (\beta).
Here is the equation:
\[\tan(\beta - \theta) = \frac{2 \tan(\theta) \times (M_1^2 \times \text{sin}^2(\beta) - 1)}{M_1^2 \times (\text{gamma} + \text{cos} 2\beta) + 2} \]
The given deflection angle (\theta) is 20°, but we use half of this angle (10°) for calculations due to wedge shape constraints. The shock wave angle (\beta) is 40°. With these angles and the specific heat ratio (\text{gamma} = 1.4), we can numerically or iteratively solve for the initial Mach number. In this case, we find that \[M_1 ≈ 2.2 \].

Shock waves result in a sudden increase in pressure, temperature, and density as the flow transitions from supersonic to subsonic speeds. This exercise examines the pressure rise (\text{p2/p1}) across the shock wave.
Using the computed Mach number and the shock wave angle, we can calculate the pressure rise with this formula:
\[\frac{p_2}{p_1} = 1 + \frac{2 \text{gamma}}{\text{gamma}+1}((M_1 \text{sin}\beta)^2 - 1) \]
Given:

• \text{M_1} = 2.2
• \text{p_1} = 35 kPa
• \text{gamma} = 1.4
• \text{sin}\beta = \text{sin}(40°)

By plugging these values into the equation, we can find the pressure ratio across the shock wave. This ratio tells us how much the pressure increases as the flow encounters the shock.

Stagnation temperature (\text{T_0}) is the temperature a flowing gas would attain if it were brought to rest adiabatically. This parameter helps in understanding the thermal state and energy of the flow before and after the shock wave.
In the calculation example, the stagnation temperature is given as 30°C or 303 K. Using the following relations, we can calculate both the upstream (\text{T_1}) and downstream (\text{T_2}) temperatures:
upstream temperature:
\[T_1 = \frac{T_0}{1 + \frac{\text{gamma} - 1}{2}M_1^2} \]
To find the downstream temperature after the shock, we will require the post-shock Mach number (\text{M_2}) and adjust the formula accordingly to get the new temperature.
\text{T_1} and \text{T_2} reflect the temperatures before and after the shock, revealing how the energy is redistributed in the flow due to compression and velocity change.

Supersonic flow dynamics involves the study of flow where the speed exceeds the speed of sound, leading to phenomena like shock waves, changes in pressure and temperature gradients. In supersonic flows:

• Shock waves are crucial as they cause sudden changes in fluid properties.
• Oblique shocks form at angles to the flow direction, usually induced by obstacles or changes in flow direction.

In our exercise, a wedge-shaped obstacle creates shock waves that propagate at a 40° angle to the flow. As the air streams hit the obstacle, transitions occur: pressure rises, speeds change, and temperatures shift. With an initial Mach number of around 2.2, our specific example transitions through significant dynamic changes, crucial for designing vehicles/structures to handle supersonic flows.