Question

Air flowing at \(60 \mathrm{kPa}, 240 \mathrm{K},\) and a Mach number of 3.4 impinges on a two-dimensional wedge of half-angle \(8^{\circ}\) Determine the two possible oblique shock angles, \(\beta_{\text {weak }}\) and \(\beta_{\text {strong }},\) that could be formed by this wedge. For each case, calculate the pressure, temperature, and Mach number downstream of the oblique shock.

Short answer

For the weak oblique shock angle (28.37 degrees), the downstream conditions are: - Mach number: M2 = 2.18 - Pressure: P2 = 289.44 kPa - Temperature: T2 = 403.44 K For the strong oblique shock angle (54.14 degrees), the downstream conditions are: - Mach number: M2 = 0.80 - Pressure: P2 = 822.78 kPa - Temperature: T2 = 842.16 K

Step-by-step solution

Calculate the incoming Mach angle, \(\mu\)

Using the given Mach number \(M_1\), we will find the incoming Mach angle, \(\mu_1\), using the formula: \(\mu_1 = \arcsin{\frac{1}{M_1}}\) Given, \(M_1 = 3.4\). \(\mu_1 = \arcsin{\frac{1}{3.4}} \approx 17.45^{\circ}\)

Find the weak and strong oblique shock angles, \(\beta_{\text {weak }}\) and \(\beta_{\text{strong }}\)

Using the given wedge half-angle \(\theta\), and the incoming Mach angle \(\mu_1\), we find the weak and the strong oblique shock angles, \(\beta_{\text {weak }}\) and \(\beta_{\text{strong }}\), using the theta-beta-Mach relation: \(\theta = 8^{\circ}\) Instead of just using formula, we can employ trial and error to match given Mach number with the Mach number given by the theta-beta-Mach relation for desired weak and strong shock angles. Applying the numerical method, we get: \(\beta_{\text {weak }} \approx 28.37^{\circ}\) \(\beta_{\text{strong }} \approx 54.14^{\circ}\)

Calculate conditions downstream of the oblique shock for both weak and strong shock angles

Using the Normal Shock table and oblique shock properties, we can calculate the conditions downstream of the oblique shock for both the weak and strong cases: 1. For weak oblique shock angle \(\beta_{\text {weak }} = 28.37^{\circ}\): - Calculate the normal Mach number upstream, \(M_{n1}\) \(M_{n1} = M_1 \sin{\beta_{\text {weak }}} = 3.4 \times \sin{28.37^{\circ}} \approx 1.60\) - Find the downstream normal Mach number, \(M_{n2}\), from the Normal Shock table: \(M_{n2} \approx 0.58\) - Calculate the downstream Mach number, \(M_2\) \(M_2 = \frac{M_{n2}}{\sin{(\beta_{\text {weak }} - \theta)}} = \frac{0.58}{\sin{(28.37^{\circ} - 8^{\circ})}} \approx 2.18\) - Determine the pressure and temperature downstream using the Normal Shock table and the following formulas: \(P_2 = P_1 \times \frac{P_{02}}{P_{01}} \approx 60 \text{kPa} \times 4.824 \approx 289.44 \text{kPa}\) \(T_2 = T_1 \times \frac{T_{02}}{T_{01}} \approx 240 \text{K} \times 1.681 \approx 403.44 \text{K}\) 2. For strong oblique shock angle \(\beta_{\text{strong }} = 54.14^{\circ}\): - Calculate the normal Mach number upstream, \(M_{n1}\) \(M_{n1} = M_1 \times \sin{\beta_{\text {strong }}} = 3.4 \times \sin{54.14^{\circ}} \approx 2.768\) - Find the downstream normal Mach number, \(M_{n2}\), from the Normal Shock table: \(M_{n2} \approx 0.41\) - Calculate the downstream Mach number, \(M_2\) \(M_2 = \frac{M_{n2}}{\sin{(\beta_{\text {strong }} - \theta)}} = \frac{0.41}{\sin{(54.14^{\circ} - 8^{\circ})}} \approx 0.80\) - Determine the pressure and temperature downstream using the Normal Shock table and the following formulas: \(P_2 = P_1 \times \frac{P_{02}}{P_{01}} \approx 60 \text{kPa} \times 13.723 \approx 822.78 \text{kPa}\) \(T_2 = T_1 \times \frac{T_{02}}{T_{01}} \approx 240 \text{K} \times 3.509 \approx 842.16 \text{K}\) Now, summarizing our results, we have: - Weak oblique shock angle \(\beta_{\text {weak }} = 28.37^{\circ}\): \(M_2 = 2.18, P_2 = 289.44 \text{kPa}, T_2 = 403.44 \text{K}\) - Strong oblique shock angle \(\beta_{\text{strong }} = 54.14^{\circ}\): \(M_2 = 0.80, P_2 = 822.78 \text{kPa}, T_2 = 842.16 \text{K}\)

Key concepts

Mach Number

The Mach number, denoted as M, is a dimensionless quantity representing the ratio of the speed of an object moving through a fluid to the local speed of sound within that fluid.

For example, in our exercise, air is flowing at a Mach number of 3.4, which indicates that the airspeed is 3.4 times faster than the speed of sound at that particular altitude and temperature. This high Mach number suggests that the flow is in the supersonic regime, which brings about a specific set of fluid dynamic phenomena, including the formation of oblique shock waves when interacting with obstacles like wedges.

The higher the Mach number, the more pronounced these effects, and because of this, comprehending Mach number is paramount in predicting and understanding the complex changes in flow conditions associated with supersonic speeds.

Shock Wave Angle

When a supersonic flow encounters an obstacle, such as a wedge, the flow is deflected, and this deflection results in a discontinuity known as a shock wave. The angle at which this shock wave forms relative to the oncoming flow direction is known as the shock wave angle, \(\beta\).

It's critical to note that for a given deflection angle and Mach number, there can be two shock wave angles - a weak shock (with a smaller angle), which is typically more stable, and a strong shock (with a larger angle), which is less common but can occur under certain conditions. In the exercise, we found two possible shock wave angles, \(\beta_{\text{weak }}\approx 28.37^\circ\) and \(\beta_{\text{strong }}\approx 54.14^\circ\), corresponding to the weak and strong solutions, respectively.

Theta-Beta-Mach Relation

The theta-beta-Mach (\theta-\beta-M) relation is a fundamental expression in gas dynamics that correlates the wedge angle (\theta), the Mach number (M), and the shock wave angle (\beta). This relationship allows us to determine the possible angles of an oblique shock wave that can form when a supersonic flow encounters an angle of deflection.

In practice, calculating these angles can involve a bit of trial and error or numerical methods, since exact solutions are not always straightforward. The exercise has demonstrated the use of such a numerical method to determine the weak and strong shock wave angles that correspond to the given flow conditions and wedge half-angle of \(8^\circ\). It's this complex interplay between \theta, \beta, and M that helps to predict the behavior of the shock waves and their impact on the flow.

Downstream Flow Conditions

After a supersonic flow encounters an oblique shock, the conditions of the flow change dramatically. Downstream of the shock, the flow is characterized by altered pressure, temperature, and Mach number.

These downstream conditions are critical in applications such as aerodynamics and propulsion, where control of these parameters is essential for performance and stability of aircraft and rockets. In our exercise, both the weak and strong oblique shock scenarios are analyzed, yielding different downstream conditions. For instance, after the weak shock, the Mach number decreases to 2.18, whereas after the strong shock, it is significantly lowered to 0.80.

Such calculations for downstream conditions after shock waves are indispensable in designing and analyzing vehicles traveling at supersonic speeds, ensuring that structures can withstand increased pressures and temperatures, and flow behavior is conducive for control and propulsion systems.