Question

Find the expression for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave as a function of \(k\) and the Mach number upstream of the shock wave \(\mathrm{Ma}_{1}\).

Short answer

Question: Find the expression for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave as a function of the specific heat ratio \(k\) and the Mach number upstream of the shock wave, denoted as \(\mathrm{Ma}_{1}\). Answer: The expression for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave is given by: \[ \frac{P_{02}}{P_1} = \left(1 + \frac{k-1}{2} \mathrm{Ma}_{1}^2 \right)^{\frac{k}{k-1}} \frac{ [2 k \mathrm{Ma}_{1}^2 - (k - 1)]^{\frac{k}{k-1}} }{[2 + (k - 1) \mathrm{Ma}_{1}^2]^{\frac{k}{k-1}}} \]

Step-by-step solution

Write the normal shock relations

We begin by writing out the normal shock relations. Consider flow across a normal shock wave with Mach number \(\mathrm{Ma}_{1}\). The normal shock relations for the gas pressure, temperature, and density across the shock wave are given by: (a) The pressure relations: \[ \frac{P_2}{P_1} = \frac{2 k \mathrm{Ma}_{1}^2 - (k - 1)}{k + 1} \] (b) The temperature relations: \[ \frac{T_2}{T_1} = \frac{[2 k \mathrm{Ma}_{1}^2 - (k - 1)][k -1 + 2 / \mathrm{Ma}_{1}^2]}{k^2 \mathrm{Ma}_{1}^2} \] (c) The density relations: \[ \frac{\rho_2}{\rho_1} = \frac{k + 1}{2 + (k - 1) \mathrm{Ma}_{1}^2} \] (d) The downstream Mach number relations: \[ \mathrm{Ma}_{2}^2 = \frac{2 + (k - 1) \mathrm{Ma}_{1}^2}{2 k \mathrm{Ma}_{1}^2 - (k - 1)} \] In these relations, the subscript 1 refers to the upstream parameters, and the subscript 2 refers to the downstream parameters.

Write the Equation of State

To relate the pressure and temperature to the density, we use the ideal gas equation of state: \[ P=\rho R T \]

Find the stagnation pressure ratio

We are interested in finding the ratio of the stagnation pressure after the shock wave (\(P_{02}\)) to the static pressure before the shock wave (\(P_1\)). The stagnation pressure is the pressure in the flow when it is brought to rest isentropically. Using the relation between the static and stagnation pressure for the upstream (before the shock) and downstream (after the shock) regions, we obtain: \[ \frac{P_{01}}{P_1} = \left(1 + \frac{k-1}{2} \mathrm{Ma}_{1}^2 \right)^{\frac{k}{k-1}} \] \[ \frac{P_{02}}{P_2} = \left(1 + \frac{k-1}{2} \mathrm{Ma}_{2}^2 \right)^{\frac{k}{k-1}} \] Now, we are looking for \(\frac{P_{02}}{P_1}\) which can be calculated as: \[ \frac{P_{02}}{P_1} = \frac{P_{02}}{P_2} * \frac{P_2}{P_1} \] Substitute the expressions for \(\frac{P_{02}}{P_2}\), \(\frac{P_2}{P_1}\) and \(\mathrm{Ma}_{2}^2\) that we derived in steps 1(d), 1(a), and 2, and simplify: \[ \frac{P_{02}}{P_1} = \left(1 + \frac{k-1}{2} \mathrm{Ma}_{1}^2 \right)^{\frac{k}{k-1}} \frac{ [2 k \mathrm{Ma}_{1}^2 - (k - 1)]^{\frac{k}{k-1}} }{[2 + (k - 1) \mathrm{Ma}_{1}^2]^{\frac{k}{k-1}}} \] This is the expression for the ratio of the stagnation pressure after a shock wave to the static pressure before the shock wave as a function of \(k\) and the Mach number upstream of the shock wave, \(\mathrm{Ma}_{1}\).

Key concepts

Normal Shock Relations

Understanding normal shock relations is fundamental to the study of compressible flow. A normal shock is a sudden increase in pressure, temperature, and density that occurs in supersonic flow as the air is decelerated to subsonic speeds. The key relationships in a normal shock include the ratios of pressures, temperatures, and densities before and after the shock, as well as the Mach numbers on either side.

These ratios are determined by the conservation of mass, momentum, and energy across the shock wave, assuming an ideal gas behavior. For any given upstream Mach number \[\begin{equation}\mathrm{Ma}_{1}\end{equation}\], these shock relations can be used to calculate the downstream thermodynamic properties. In summary, normal shock relations are crucial for engineers and scientists to predict how gases behave when subjected to a shock wave and are employed in designing supersonic aircraft, turbines, and exhaust systems among others.

Mach Number

The Mach number, denoted as \(\mathrm{Ma}\), represents the ratio of the velocity of a fluid to the speed of sound within that fluid. It is a dimensionless quantity and is a critical parameter in the study of compressible flows. When the Mach number is greater than one, the flow is considered supersonic, and shock waves can occur.

A chief use of the Mach number is in the classification of airflow and its effects on various objects, such as airfoils. The Mach number influences how pressure, density, and temperature change across a shock wave. Therefore, knowing the Mach number of a flow helps anticipate changes in fluid properties, which are important in high-speed aerodynamics and propulsion system design.

Ideal Gas Equation of State

The Ideal Gas Equation of State is a model that describes the behavior of an ideal gas under various conditions. It is mathematically expressed as \(P = \rho RT\), where \(P\) is the pressure, \(\rho\) is the density, \(R\) is the specific gas constant, and \(T\) is the temperature. This equation allows us to relate the pressure, volume, and temperature of a gas in a simple form.

In the context of fluid dynamics and thermodynamics, this equation underpins many of the calculations needed to understand the flow properties of gases especially when combined with conservation laws and when making assumptions like constant temperature or pressure. It is a cornerstone of the study of processes involving ideal gases and is widely applied in engineering tasks related to the behavior of gases under different conditions.

Isentropic Flow

Isentropic flow refers to a process in which entropy remains constant. In aerodynamics, this idealization is often applied to the flow of a compressible fluid in which there are no energy losses due to friction, heat transfer, or shock waves.

In isentropic processes, the steady flow relations for pressure, temperature, and density are given by the Poisson equations, which can be derived from the ideal gas law. For a perfect gas, these changes in properties are predictable and can be quantified using relationships that assume a constant specific heats ratio, often denoted as \(k\) or \(\gamma\). The study of isentropic flow is essential because it provides the basis for understanding real flow conditions in nozzles, diffusers, and around airfoils, wherein good design often aims to approximate isentropic behavior as closely as possible for efficiency.