A subsonic flow responds to area changes in a similar manner as an incompressible flow. A supersonic flow behaves in an opposite manner in that when there is an area decrease, Mach Number decreases, while for an area increase, Mach Number increases. It this supersonic case a sonic flow can occur only at a throat, a section where area is the minimum. With this theory it is possible to explore the properties of gas flow through nozzles.

Flow through a Converging Nozzle

The first application of this theory is a converging nozzle connected to a reservoir where stagnation conditions prevail, $P = P_0, T = T_0 , u = 0$. By definition reservoirs are such that no matter how much the fluid flows out of them, the conditions in them do not change. In the reservoir it is assumed that pressure, temperature, density etc. remain the same always.

The pressure level $P_b$ at the exit of the nozzle is referred to as the Back Pressure and it is this pressure that determines the flow in the nozzle.

When the Back Pressure, $P_b$ is equal to the reservoir pressure, $P_r = P_0$ there is no flow in the nozzle. This is condition (1) in Fig. 18. If $P_b$ is reduced slightly to $P_2$ (condition (2)), a flow is induced in the nozzle. For relatively high values of $P_b$ , the flow is subsonic throughout. A further reduction in Back Pressure still results in a subsonic flow, but with a higher Mach Number at the exit (condition (3)). As $P_b$ is reduced we have an increased Mach Number at the exit along with an increased mass flow rate. This situation continues until a Back Pressure value is reached where the flow reaches sonic conditions ( $>M=1$ ) at the exit (4). This value of Back pressure follows from the isentropic equations. For air it is given by

When the Back Pressure is further reduced (5,6 etc.), the Mach Number at the exit tries to increase. It demands an increased mass flow from the reservoir. But as the condition at the exit is sonic, signals do not propagate upstream. The Reservoir is unaware of the conditions downstream and it does not send any more mass flow. Consequently the flow pattern remains unchanged in the nozzle. Any adjustment to the Back Pressure takes place outside of the nozzle. The nozzle is now said to be choked. The mass flow rate through the nozzle has reached its maximum possible value, the choked value. From the Fig. 18 it can be seen that there is an increase in mass flow rate only till choking condition (4) is reached. Thereafter mass flow rate remains constant.

Figure 18 : Flow through a Converging Nozzle

Note that for a non-choked flow, the Back Pressure and the pressure of the flow at the exit plane are equal. But when the nozzle is choked the two are different. The flow will need to adjust out side the exit of the nozzle. Usually this take place by means of expansion waves which reduce the exit pressure until it eventually equals the back pressure at a point down stream.

Flow through a Converging-diverging nozzle

A converging-diverging nozzle is called a de Laval nozzle, it is an essential element of a supersonic wind tunnel. In this application, the nozzle draws air from a reservoir with a fixed stagnation pressure. It is assumed that the back pressure at the end of the diverging section is such that air reaches sonic conditions at throat. The flow Mach Number increases in the diverging section. The area ratio and the back pressure can be set so that a required Mach Number is obtained at the end of the diverging section. Different area ratios give different Mach Numbers.

The effect of Back Pressure on the flow through a converging-diverging nozzle is somewhat more complicated than that for a converging nozzle. Flow configurations for various back pressures and the corresponding pressure and Mach Number distributions are given in Fig. 19.

• Fig 19 (a) Back Pressure is equal to the reservoir pressure, $P_b= P_0$. There is no flow through the nozzle.
• Fig 19 (b) Back Pressure slightly reduced,$P_b<P_0$ . A flow is initiated in the nozzle, but the condition at throat is still subsonic. The flow is subsonic and isentropic through out.
• Fig 19 (c) The Back Pressure is reduced sufficiently to make the flow reach sonic conditions at the throat, $P_b = P_c$. The flow in the diverging section is still subsonic as the back pressure is still high. The nozzle has reached choking conditions. As the Back Pressure is further reduced, flow in the converging section remains unchanged.
• Fig 19 (d) When the Back Pressure is $P_d$ , the flow at the start of the divergent section becomes supersonic. But the Back Pressure is higher than $P_f$, the pressure value required so that a fully established supersonic flow covers the full length of the divergent section. Consequently, the flow meets the external flow at the Back Pressure through a shock in the diverging section. The location and strength of the shock depends upon the Back Pressure. Decreasing the Back Pressure moves the shock downstream.
• Fig 19 (f) Once a Back Pressure $P_f$ is reached, the shock formed has moved to the exit plane. $P_f\/P_0$ is the smallest pressure ratio required for full supersonic flow in the divergent section of this nozzle.
• Fig 19 (g) A further reduction in Back Pressure results in the shocks being formed outside of the nozzle. These are not Normal Shocks. The flow has become two-dimensional and they are Oblique Shocks. The flow in the divergent section has reduced the pressure at the exit to values below the back pressure. Further shocks are required to compress the flow and bring it up to the back pressure. Such a nozzle is termed Overexpanded.
• Fig 19 (i) The Back Pressure is reduced to be equal to the pressure of the gas at the exit of the expansion. In this case no shocks are formed as exit and surrounding pressure are equal.
• Fig 19 (j) The Back Pressure has been reduced to less than $P_i$. The flow adjustment takes place outside of the nozzle, not through shocks, but through Expansion Waves. The flow could not expand to reach the back Pressure, so further expansion is required to finish the job. Such a nozzle is termed Underexpanded.

Figure 19 : Pressure and Mach Number Distribution for the Flow through a Converging-Diverging Nozzle.

Supersonic Nozzle Experiment

A typical method of familiarising students with the behaviour of supersonic gas flow is via the use of a supersonic nozzle experiment. The following section outlines a procedure for carrying out experiments to measure pressure ratio in a nozzle and observe flow patterns.

Figure 20 : 2-D Supersonic Flow Nozzle

Experimental Equipment.

The set up of the apparatus is shown in Figs 20, 21 and 22. The function of each piece of equipment is noted. A high-pressure gas supply line is attached to a reservior. The reservior is joined to the inlet of a converging-diverging nozzle. Flow is controlled via a manually operated valve on the supply line to the reservior. In this case Back Pressure, $P_b$, is constant and equal to the surrounding atmosphere pressure. Flow is created by increasing the pressure in the reservoir, $P_0$. Once the valve is open to give sufficient upstream stagnation pressure, the nozzle will choke giving Mach 1 flow at the throat.

Downstream of the throat (area $A^\text"*"$) the flow Mach number will depend primarily on the area ratio of the channel ($A\/A^\text"*"$) and a supersonic flow slightly above Mach 2 will be obtained.

Figure 21 : High Pressure Supply System.

A steel wall containing static pressure ports can be attached to the side of the nozzle. Using these ports, static pressure variation along the nozzle length can be determined. The static pressure to stagnation pressure ratio at any point along the channel can be used to predict local Mach number. This result can be compared to area ratio predictions and discrepancies due to boundary layer effects, shock waves or surface imperfections can be evaluated.

A pitot pressure probe can be inserted into the flow at any station and can be used to predict upstream Mach number based on the stagnation pressure ratio change at that station.

Figure 22 : Schematic of Equipment Connections.

Nozzle Geometry

Nozzle static pressure port stations are shown in the following table. They are measured as distances downstream from the throat.

Pressure measurement

Pressure measurement can be carried out using Piezo-electric transducers or conventional manometers.

In this example mercury manometers are used to measure the pressures in the nozzle or upstream reservoir.

Stagnation pressure, $P_0$ is measured as the static pressure of the upstream reservoir where flow velocity is minimal. Since the stagnation pressure in this reservoir is quite high, a multi-column manometer has been used, rather than one very long single column. The mercury columns are joined by incompressible water columns.

The final measure of stagnation pressure can be determined by addition of the mercury column heights and then subtracting the effect of the weight of water.

$$P_0=(h_2-h_1)+(h_4-h_3)+(h_6-h_5)-{ρ_{H_2O}}/ρ_{Hg}((h2-h3)+(h_4-h_5))+Patm\text"    cmHg    "$$

Static pressure along the nozzle is obtained using a multitube manometer. Individual ports are connected in sequence. The axial position of the static pressure measurement points is shown in the previous table

The first static port is located at station 1 which is downstream of the throat of the nozzle (station 0).

Since the pressures drop below atmospheric then the values should be read from the manometer as follows.

Optical Measurement of Shock and Expansion Waves

A Schleiren optical system can be used to visualise flow in the nozzle.

This system uses the deflection of parallel light rays by the flow density gradients to produce visual images of the shock and expansion waves. A point light source shines on a concave mirror with a focal length such that a parallel beam of light is produced to illuminate the nozzle.

Light rays passing through areas of high or low density gradient in the nozzle will be bent away from parallel. A matching pair concave mirror catches the light once it has gone through the nozzle and refocuses it onto a knife edge. The knife edge is set to allow about ½ of the light beam through to a viewing screen or camera.

Light bent by expansions will be stopped at the knife edge. This missing light will cause dark areas on the screen where expansion waves occur. Light bent by compressions (shocks) will completely pass the knife edge and hence brighten these areas on the screen.

The system can thus be used to produce images of supersonic flow. Some examples can be found here in the supersonic flow images page.

Supersonic Nozzle Simulation

Note: If you do not have access to a supersonic nozzle, a high pressure gas line or any of the other equipment mentioned above, then you can use the Supersonic Nozzle Simulation Software to carry out your experimental investigations.

Experimental Procedure

To carry out the experiments, follow the procedure outlined in the steps below.

1. With an empty test section, open the control valve and establish supersonic flow in the channel. Measure the static pressure readings along the length of the nozzle. Measure the total pressure reading for the upstream flow. Measure the atmospheric pressure in the lab. Recorded data can be written in the tables shown in sheets below.
   
2. A pitot pressure probe can be inserted into the flow to measure stagnation pressure behind a normal shock wave. This is another technique for measuring the Mach number of the flow in the nozzle. Record the Stagnation pressure using the probe at station 6 in the nozzle.
   
3. With glass walls on the side of the nozzle and by means of a Schlieren optical system, the shock/expansion wave system produced by objects placed in the flow can be seen. These flow patterns can be recorded digitally. Install a test shape into the test section of the channel. Again open the valve to establish supersonic flow. Record the shock and expansion wave patterns using the Schlieren optical system.
   
4. With an otherwise empty test section, if small step disturbances are introduced at the walls, the resulting weak Mach waves can be observed by the optical system. The angle of these waves will be another measure of the Mach number of the flow in the nozzle.
   

Experimental Results

1. Plot a graph of Mach number versus axial length from throat based on the above experimental measurements. Include all methods, Mach number predicted from nozzle area ratio and Mach number predicted from static/stagnation pressure ratios.
   
2. Comment on the differences in Mach number predicted by these methods. Give explanations of the errors intrinsic to each technique to determine the most accurate method.
   
3. Estimate the flow Mach number in the uniform section of the channel section into which objects are placed.
   
4. Compare oblique shock wave angles and Prandtl-Meyer expansion fan angles for the measured flow with perfect gas theory models.
   

Data Recording

Use the following layout for recording manometer data for each nozzle run.

Static Pressure Recordings.

Static Pressure ports are located every 1/2 inch along the nozzle, starting with port 1 just downstream of the throat.

Record Absolute Ambient Lab Pressure

The Stagnation Pressure of the flow is recorded in the reservoir upstream of the nozzle. Record Stagnation Pressure Data in the following table.