Problem 28 - Aircraft Propulsion

Question

Air enters an insulated duct with a constant area with an average wall friction coefficient of \(C_{f}=0.004\). The inlet Mach number is \(M_{1}=2.0\). There is a choking length \(L_{1}^{*}\), for this duct that corresponds to the inlet Mach number \(M_{1}\). For any duct longer than the choking length \(L_{1}^{*}\), a normal shock appears in the duct. Assuming the length of the duct is \(10 \%\) longer than \(L_{1}^{*}\), i.e., \(L=1.1 L_{1}^{*}\), calculate (a) the shock location along the duct \(x_{s} / D\) (b) percentage total pressure loss in the longer duct (c) percentage loss of fluid impulse Assume \(\gamma=1.4\).

Step-by-Step Solution

Answer

The shock location on the duct is calculated to be 1.1 times the choking length. Next, the total pressure loss with respect to the total duct length is computed. After that, the percentage loss of fluid impulse in the duct is also calculated.

Step 1: Setup

The given information includes: \(\gamma = 1.4, M_1 = 2.0, Cf = 0.004, L = 1.1L_1^*\). Using the definition of choking length \(L^* = \frac{1}{C_fM_1}\int_{M_2}^{M_1}\frac{dM}{M (1-M^2) + \frac{2}{\gamma}(1-\frac{M^2}{M_1^2})}\), where \(M_2\) is 1. Determine the integral \(I= \int_{M_2}^{M_1}\frac{dM}{M (1-M^2) + \frac{2}{\gamma}(1-\frac{M^2}{M_1^2})}\).

Step 2: Determining shock location

Next, define x as a non-dimensional length, \(x^* = x/L^*\). Since the shock occurs at \(x_s = L\), we have \(x_s^* = L/L^* = 1.1 \). This locates the position of the shock in the duct.

Step 3: Compute the Percentage Total Pressure Loss

We compute the total pressure loss by calculating the pressure ratio across the shock \( p_{02}/p_{01} \), where \( p_{01} \) and \( p_{02} \) are the total pressures upstream and downstream of the shock respectively. The total pressure loss can be found using the relation \(\frac{p_{02}}{p_{01}} = (\frac{p_2}{p_1})(\frac{p_{01}}{p_1})(\frac{p_2}{p_{02}})\). Each component can be calculated using the appropriate formulas. The percentage total pressure loss is then \( 100 \times (1 - \frac{p_{02}}{p_{01}}) \).

Step 4: Compute the Fluid Impulse Loss

The impulse loss is the loss of linear momentum, and hence, fluid impulse, across the shock wave. It is calculated by finding the change in quantity \( pV \), where \( p \) is the pressure and \( V \) is the velocity. Compute the percentage fluid impulse loss as \( 100 \times (1 - \frac{p_2V_2}{p_1V_1}) \).

Key Concepts

Mach number

The Mach number is a fundamental concept in fluid dynamics, particularly in the analysis of compressible flows such as those involving shock waves. It is a dimensionless parameter that signifies the ratio of the speed of an object moving through a fluid (or the speed of the fluid itself) to the local speed of sound within that fluid. Mathematically, it is defined as \( M = \frac{V}{a} \), where \(M\) is the Mach number, \(V\) is the object's velocity, and \(a\) is the speed of sound in the fluid.

Specifically, in our exercise, the inlet Mach number \(M_{1} = 2.0\) indicates that the air is entering the duct at a speed twice the speed of sound. Understanding the Mach number is critical in predicting how the fluid will behave. For instance, at Mach 1 the flow is sonic, below Mach 1 it is subsonic, and above Mach 1 it is supersonic, as is the case in this problem. With the supersonic flow, we expect phenomena like shock waves to occur, which can significantly change the properties of the flow.

Pressure loss calculation

Pressure loss in a fluid flow system is a crucial parameter often evaluated to determine the efficiency of the system. When dealing with shock waves, the pressure loss is particularly important because the shock process is inherently a dissipative one, involving a sudden decrease in velocity and a corresponding increase in pressure and temperature.

The pressure loss is typically quantified as a percentage decrease relative to the initial total pressure. In the given exercise, the percentage total pressure loss occurring has been computed using the expression \( 100 \times (1 - \frac{p_{02}}{p_{01}}) \), where \( p_{01} \) and \( p_{02} \) represent the total pressures before and after the shock, respectively. Total pressure is a measure of the energy per unit volume in the flow; thus, a loss in total pressure indicates that energy is being converted into forms (such as heat) that are not useful for propulsive or other work.

Fluid impulse

The concept of fluid impulse involves the momentum aspect of fluid flow—specifically the product of pressure and velocity. The fluid impulse is reflective of the flow's ability to exert force over a period of time, and it is an essential parameter in the design and analysis of propulsion systems, turbines, and other fluid machinery.

In the scenario of a shock wave, the fluid impulse can undergo significant changes due to variations in both pressure and velocity. The calculation for fluid impulse loss is expressed in the form of a percentage and can be identified using the relationship \( 100 \times (1 - \frac{p_2V_2}{p_1V_1}) \), where \(p_1V_1\) and \(p_2V_2\) are the pressure-velocity products upstream and downstream of the shock, respectively. Here, we are interested in assessing the impact that the normal shock has on the overall momentum carried by the flow; this can have important implications for the thrust produced by jet engines or for the efficiency of a duct flow.

Shock wave location

Identifying the location of a shock wave within a flow system is crucial for understanding the flow characteristics and for designing systems that can accommodate or take advantage of the changes in flow properties associated with the shock. In our given problem, the shock wave location is calculated as a non-dimensional length relative to the choking length (a characteristic length defined by the conditions at which the flow becomes sonic).

To determine the location of the normal shock within the duct, the nondimensional position \(x_s^* = 1.1\) is used, implying that the shock occurs at a duct length 10% greater than the choking length, or \(L = 1.1L_1^*\). This location is significant because it dictates where supersonic flow transitions to subsonic, an event accompanied by a rise in pressure and temperature and a drop in velocity. As such, the exact position of the shock wave impacts how the system must be constructed, both to withstand the local effects of the shock and to fulfill the operational goals for the entire flow system.