Question

For an ideal gas flowing through a normal shock, develop a relation for \(V_{2} / V_{1}\) in terms of \(k, \mathrm{Ma}_{1},\) and \(\mathrm{Ma}_{2}\).

Short answer

Answer: The relationship between the velocities is given by the expression: \(\frac{V_{2}}{V_{1}} = \frac{(k+1)\,\mathrm{Ma}_{1}^2}{2+(k-1)\,\mathrm{Ma}_{1}^2}\).

Step-by-step solution

Write down the mass conservation equation

For a steady flow, the mass flow rate before the shock is equal to the mass flow rate after the shock. This can be expressed as: \(\rho_{1}V_{1}= \rho_{2}V_{2}\)

Write down the momentum conservation equation

For a steady flow, the momentum flow rate before the shock is equal to the momentum flow rate after the shock. This can be expressed as: \(\rho_{1}V_{1}^{2}+p_{1}= \rho_{2}V_{2}^{2}+p_{2}\)

Convert the equations using the ideal gas law

For an ideal gas, the ideal gas law can be written as \(p = \rho RT\), where p is the pressure, \(\rho\) is the density, R is the specific gas constant, and T is the temperature. From the mass conservation equation, we can get: \(\frac{\rho_{2}}{\rho_{1}} = \frac{V_{1}}{V_{2}}\) Using the ideal gas law, we can write the ratio of the pressures as \(\frac{p_{2}}{p_{1}} = \frac{\rho_{2}T_2}{\rho_{1}T_1}\). Also, we can write \(T_2\) and \(T_1\) in terms of the Mach number and the specific heat ratio, k: \(T_{2} = T_{1}\frac{p_{2}}{p_{1}}\frac{\rho_{1}}{\rho_{2}} = T_1\frac{1 + k\,\mathrm{Ma}_{1}^2}{1 + k\,\mathrm{Ma}_{2}^2}\) Using this equation, we can rewrite the momentum conservation equation as: \(\mathrm{Ma}_{1}^2 = \frac{p_{2}}{p_{1}}\frac{V_{1}^2}{V_{2}^2} + \frac{1}{k}\left(\frac{p_{2}}{p_{1}}-1\right)\)

Solve for the velocity ratio

Now we can solve for the velocity ratio \(V_{2}/V_{1}\): \(\frac{V_{2}}{V_{1}} = \frac{V_{1}}{V_{2}}\frac{\mathrm{Ma}_{1}^2}{\left[\frac{p_{2}}{p_{1}}\right]\left[1 + \frac{1}{k}(\frac{p_{2}}{p_{1}} - 1)\right]}\) Replacing \(\frac{p_{2}}{p_{1}}\) with \(\frac{2k\,\mathrm{Ma}_{1}^2-(k-1)}{k+1}\) and simplifying, the final expression for the velocity ratio is: \(\frac{V_{2}}{V_{1}} = \frac{(k+1)\,\mathrm{Ma}_{1}^2}{2+(k-1)\,\mathrm{Ma}_{1}^2}\) The expression above represents the relationship between the velocities before and after a normal shock, in terms of the specific heat ratio (k) and the Mach numbers before (Ma1) and after (Ma2) the shock.

Key concepts

Ideal Gas Flow

To fully grasp how gases behave under various conditions, it's essential to understand the concept of ideal gas flow. Ideal gases are hypothetical gases whose molecules occupy negligible space and have no intermolecular forces. This greatly simplifies the study of gas dynamics because it means that the behavior of the gas can be described by a few fundamental principles.

When we talk about gas flow, especially at high speeds, certain simplifications allow us to predict how the gas will act. These include assumptions like constant specific heats and the absence of viscous effects or heat transfer, which are typical in ideal gas flow analysis. The continuity of flow or mass conservation is also a critical principle, stating that mass cannot be created or destroyed within a flowing gas system. This concept is not only used in analyzing shock waves, as in normal shock relations, but also in nozzle flow, diffusers, and more.

Mach Number

The Mach number is a dimensionless quantity in fluid dynamics that represents the ratio of an object's speed to the speed of sound in the surrounding medium. It is a critical parameter in high-speed gas flow and shock wave analysis. For example, subsonic flow has a Mach number less than 1, transonic around 1, supersonic from 1 to 5, and hypersonic above 5.

Understanding Mach number is key to predicting how gases will behave under varying flow conditions, and it's particularly important when considering shocks. A normal shock occurs when a supersonic flow (Ma > 1) encounters a disturbance that causes it to slow down abruptly to subsonic speeds (Ma < 1). This change is highly dependent on the Mach number before and after the shock, influencing pressure, temperature, and velocity changes.

Specific Heat Ratio

The specific heat ratio, often denoted by the symbol 'k' or gamma (γ), is the ratio of the specific heat at constant pressure (\( c_p \) to that at constant volume (\( c_v \). This ratio is of major importance in gas dynamics because it describes how a gas will respond to pressure or temperature changes.

The specific heat ratio influences sound speed in the medium, the strength of shock waves, and the magnitude of the temperature and pressure changes across the shock. For an ideal diatomic gas, such as oxygen or nitrogen, which make up the majority of Earth's atmosphere, 'k' roughly equals 1.4. This value comes into play when dealing with high-speed gas flows and shock relation equations, as seen in the exercise provided.

Conservation of Mass and Momentum

The conservation of mass and momentum are two fundamental principles in physics that apply to all fluid dynamics problems, including the flow of gases. The conservation of mass, often referred to as the continuity equation, states that mass is neither created nor destroyed within the flow. This principle ensures that the mass flow rate before a shock is equal to the mass flow rate after it.

Similarly, the momentum equation is based on Newton's second law and ensures that the momentum flow rate before the shock equals the momentum flow rate after the shock. The combination of these two conservations allows us to solve for unknowns in the flow field, such as velocities, pressures, and densities before and after a shock, as seen in the problem's step-by-step solution.

Ideal Gas Law

The ideal gas law, an equation of state for a hypothetical ideal gas, is critical in understanding gas behavior in various thermodynamic processes. It is a simplified equation that relates pressure (p), volume (V), the amount of gas (n), and temperature (T), with the equation PV = nRT. Here, R is the ideal, or universal, gas constant.

In practical engineering problems like the analysis of normal shocks in ideal gases, we usually express the ideal gas law in terms of the gas density (\( \rho \) rather than the volume (V), so it gets rearranged to p = \( \rho \)RT. By using the ideal gas law in conjunction with the principles of conservation of mass and momentum, one can solve for various states in the flow field of gases undergoing different processes, such as those encountered in the normal shock problem we are examining.