Question

Air enters a converging-diverging nozzle of a supersonic wind tunnel at \(1 \mathrm{MPa}\) and \(300 \mathrm{K}\) with a low velocity. If a normal shock wave occurs at the exit plane of the nozzle at \(\mathrm{Ma}=2.4,\) determine the pressure, temperature, Mach number, velocity, and stagnation pressure after the shock wave.

Short answer

#Question# The air enters a converging-diverging nozzle at a pressure of 1 MPa and a temperature of 300 K. The Mach number before the shock wave is 2.4. Compute the temperature, pressure, Mach number, velocity, and stagnation pressure after the shock wave. #Answer# To compute the temperature, pressure, Mach number, velocity, and stagnation pressure after the shock wave, follow the steps: Step 1: Calculate post-shock Mach number (M2) using the normal shock relations and given Mach number (M1 = 2.4) and ratio of specific heats (γ = 1.4). Step 2: Calculate post-shock pressure (P2) using the pressure ratio for normal shock, given pre-shock pressure (P1 = 1 MPa), and post-shock Mach number (M2). Step 3: Calculate post-shock temperature (T2) using the temperature ratio, given pre-shock temperature (T1 = 300 K) and post-shock Mach number (M2). Step 4: Calculate post-shock velocity (V2) using the ideal gas law, given post-shock Mach number (M2), ratio of specific heats (γ), gas constant for air (R = 287 J/kg.K), and post-shock temperature (T2). Step 5: Calculate post-shock stagnation pressure (P02) using the stagnation pressure relation for normal shock wave, given pre-shock stagnation pressure (P01 = 1 MPa, since air enters with low velocity) and post-shock Mach number (M2).

Step-by-step solution

Calculate post-shock Mach number

Using the normal shock relations, we first find the post-shock Mach number (\(M_2\)). The formula to calculate \(M_2\) is as follows: $$M_2 = \sqrt{\frac{M_1^2 + 2}{2\gamma M_1^2 - (\gamma -1)}}$$ Where \(M_1 = 2.4\) (Pre-shock Mach number) \(\gamma = 1.4\) (ratio of specific heats for air)

Calculate post-shock pressure

The post-shock pressure (\(P_2\)) can be calculated using the pressure ratio for normal shock: $$\frac{P_2}{P_1} = 1 + \frac{2\gamma}{\gamma+1}(M_1^2 -1)$$ We know, \(P_1 = 1 \mathrm{MPa}\) We can solve this equation to find \(P_2\).

Calculate post-shock temperature

The post-shock temperature (\(T_2\)) can be calculated using the temperature ratio: $$\frac{T_2}{T_1} = \frac{(2+(\gamma-1)M_1^2)}{(\gamma+1)^2 M_1^2}T_1$$ We know, \(T_1 = 300 \mathrm{K}\) We can solve this equation to find \(T_2\).

Calculate post-shock velocity

We can use the ideal gas law to calculate the post-shock velocity (\(V_2\)). Here, we use the Mach number and temperature in the ideal gas equation. $$V_2 = M_2\sqrt{\gamma R T_2}$$ Where \(R = 287 \frac{\mathrm{J}}{\mathrm{kg.K}}\) (gas constant for air)

Calculate post-shock stagnation pressure

To find the post-shock stagnation pressure (\(P_{02}\)), we can use the stagnation pressure relation for normal shock wave: $$\frac{P_{02}}{P_{01}} = \frac{1 + \frac{\gamma-1}{2}M_1^2}{1 + \frac{\gamma-1}{2}M_2^2}(1+\frac{\gamma+1}{2\gamma}(M_1^2-1))$$ We know, \(P_{01} = 1\mathrm{MPa}\) (since the air enters with a low velocity, i.e., \(P_{01} \approx P_1\)) We can solve this equation to find \(P_{02}\).

Key concepts

Normal Shock Relations

In fluid dynamics, especially those involving supersonic flows, a normal shock wave is a boundary across which a sudden and drastic change in flow properties occurs. These shock waves are characterized by an abrupt decrease in velocity and an increase in pressure, temperature, and density of the flow. Understanding normal shock relations is vital for predicting the changes in a flow field as it passes through a shock.

When a normal shock wave occurs, the flow properties post-shock can be determined using certain relationships derived from the conservation of mass, momentum, and energy. These relations are critical when calculating essential parameters like the post-shock Mach number, pressure, temperature, and velocity of the flow, which we will detail in the following sections.

Mach Number

The Mach number, symbolized as Ma, is a dimensionless quantity in fluid dynamics that represents the ratio of the speed of an object moving through a fluid to the local speed of sound. When Ma is greater than 1, the flow is considered supersonic, and shock waves can form. In the context of the exercise, the pre-shock Mach number of the air was given as 2.4, making it a distinctly supersonic flow.

The Mach number allows us to classify the type of flow and anticipate the kind of shock waves that may develop. A normal shock causes the Mach number to drop below 1, which means that the supersonic flow becomes subsonic after passing through the shock. This change significantly affects the downstream flow properties and is central to the design and analysis of supersonic wind tunnels.

Specific Heats Ratio

The specific heats ratio, denoted by \(\gamma\), is the ratio of the specific heat capacity at constant pressure (\(C_p\)) to that at constant volume (\(C_v\)). For a given gas, this ratio is a measure of how much energy is needed to raise the temperature of the gas under constant pressure compared to constant volume. Air, for example, has a specific heats ratio of approximately 1.4. This value is crucial in determining how gases behave under different thermodynamic processes and is frequently used in the equations governing normal shock relations.

Understanding the specific heats ratio is necessary for accurate calculations of changes in gas properties across shock waves, which directly influence the post-shock pressure and temperature, and indirectly, the Mach number and velocity of the flow.

Post-Shock Pressure

The post-shock pressure is the pressure of the fluid after it has passed through a shock wave. In a supersonic wind tunnel, the flow experiences a sharp rise in pressure when encountering a normal shock. The post-shock pressure is determined using the normal shock relations, which incorporate both the pre-shock pressure and the Mach number prior to the shock. These equations predict that the pressure after the shock wave increases as the square of the pre-shock Mach number increases.

In our exercise, the calculation of post-shock pressure involves the specific heats ratio and the pre-shock Mach number, which we've established as essential parameters. This increase in pressure is significant for the aerodynamic forces and the potential for structural loads that must be considered in designing aerospace components like nozzles and wings.

Post-Shock Temperature

Following the normal shock relations, the post-shock temperature is another crucial property that experiences a dramatic rise as the supersonic flow undergoes a normal shock. The temperature behind the shock increases due to the conversion of the flow’s kinetic energy into internal energy, which manifests as raised thermal energy. This transformation follows the principle of conservation of energy and is described quantitatively by the flow's Mach number and specific heats ratio.

The increase in post-shock temperature has several implications, including a change in the speed of sound and hence the Mach number. Moreover, this rise can have effects on the physical properties of the flow, influencing factors like the boundaries of the laminar and turbulent flow regimes and even contributing to thermal stresses on the tunnel's structure.