Of course, the net mass flow rate, , must be constant along the nozzle, so
Figure 14.1 shows , , , , and , plotted as functions of the local Mach number, , for an ideal gas with . Here, use has been made of Equations (14.71), (14.67), (14.68), (14.66), and (14.72), respectively. Inspecting the curves, we can see, somewhat surprisingly, that the cross-sectional area function, , attains a minimum value when . In fact, the figure indicates that, for subsonic flow ( ), a decreasing cross-sectional area of the nozzle in the direction of the gas flow leads to an increasing flow speed, and decreasing pressure, density, and temperature. However, for supersonic flow ( ), this behavior is reversed, and an increasing cross-sectional area of the nozzle leads to an increasing flow speed, and decreasing pressure, density, and temperature. We conclude that local Mach number of gas flowing through a converging nozzle (i.e., a nozzle whose cross-sectional area decreases monotonically in the direction of the gas flow) can never exceed unity. Moreover, the maximum Mach number (i.e., unity) is achieved on exit from the nozzle, where the cross-sectional area is smallest. On the other hand, the local Mach number of gas flowing through a converging-diverging nozzle (i.e., a nozzle whose cross-sectional area initially decreases in the direction of the gas flow, attains a minimum value, and then increases) can exceed unity. For this to happen, the flow conditions must be arranged such that the sonic point corresponds precisely to the narrowest point of the nozzle, which is generally known as the throat. In this case, as the gas flows through the converging part of the nozzle, the local cross-sectional area, , travels down the left-hand, subsonic branch of the curve shown in Figure 14.1, while the flow speed, , simultaneously increases. After passing through the throat at the sonic speed, the gas flows through the diverging part of the nozzle, and the cross-sectional area travels up the right-hand, supersonic branch of the curve, while the flow speed continues to increase. The type of converging-diverging nozzle just described is known as a de Laval nozzle, after its inventor, Gustaf de Laval (1845-1913).
Consider a de Laval nozzle whose gas supply is derived from a large reservoir. Assuming that the gas in the reservoir is essentially at rest, it follows that the temperature, pressure, and density of the gas in the reservoir correspond to the stagnation temperature, pressure, and density-- , , and , respectively. Equation (14.63)-(14.65) then specify the temperature, pressure, and density of the gas at the throat of the nozzle-- , , and , respectively--in terms of the temperature, pressure, and density of the gas in the reservoir.
Suppose that a de Laval nozzle exhausts gas into the atmosphere, whose pressure is . Now, for the case of incompressible flow, the pressure of the gas exhausted from a nozzle, (say), must match the ambient pressure, . The reason for this is that any mismatch between the exhaust and ambient pressures is instantly communicated to the whole fluid by means of sound waves that travel infinitely fast (because an incompressible fluid corresponds to the limit ). Of course, the sound speed is finite in a compressible gas. However, in subsonic compressible flow, upstream communication is still possible, because the local sound speed exceeds the local flow speed. However, in supersonic compressible flow, upstream communication is impossible, because sound waves cannot catch up with the flow. Consequently, in the case of a nozzle with a subsonic exhaust speed, we would generally expect the exhaust pressure to match the ambient pressure. However, in the case of a nozzle with a supersonic exhaust speed, the exhaust pressure can be significantly different to the ambient pressure.
Equation (14.61) yields
Equations (14.64) and (14.67) yield
If the exhaust pressure of a de Laval nozzle is higher than the ambient pressure, , then the gas is said to be under-expanded. In this case, a pattern of standing shock waves, called shock diamonds, forms in the exhaust plume external to the nozzle. On the other hand, if the exhaust pressure is lower than the ambient pressure, , then the gas is said to be over-expanded. In this case, a static shock front forms inside the diverging part of the nozzle. (See Section 14.8.) As the gas passes through the front, its speed drops abruptly from a supersonic to a subsonic value, whereas the pressure, density, and temperature all increase abruptly. As the subsonic gas flows through the remainder of the nozzle, its velocity decreases further.