Effect of Cross-Section Change on Mach Number

3.7 Flow at High Subsonic and Transonic Mach Numbers

3.7.2 Effect of Cross-Section Change on Mach Number

(3.72)

(3.73)

(3.74)

(3.75)

(3.76)

(3.77)

(3.78) flow function

4

until

4

max has been reached. Further increase in pressure ratio results in a choking state, where the flow function remains constant.

(3.79)

(3.80)

Fig. 3.22 Left: Subsonic nozzle with dA <0, dV > 0, dp < 0, Right: Subsonic diffuser with dA > 0, dV < 0, dp > 0.

Fig. 3.23 Left: Supersonic nozzle with dA >0, dV > 0, dp < 0, Right: Supersonic diffuser with dA > 0, dV < 0, dp > 0.

Introducing Eq. (3.79) into (3.77) results in:

With Eqs. (3.79) and (3.80) we have established a relationship between the velocity change, the pressure change, and the Mach number. For M < 1 we obtain qualitatively the same behavior as in incompressible flow: a decrease in the cross- sectional area leads to an increase in velocity, leading to a subsonic nozzle flow, Fig.

3.22(left).

On the other hand, an increase in the cross-sectional area (dA > 0) corresponds to a decrease in velocity (dV < 0), resulting in a subsonic diffuser flow, Fig. 3.22 (right).

For M = 1, we obtain dA/dx = 0 and dp = 0. For M > 1 Eq. (3.79) and (3.80) show that if the cross-sectional area (dA/dx > 0) increases, the velocity must also increase (dV/dx > 0), or if the area decreases, so does the velocity. As a result, we have a supersonic nozzle and diffuser shown in Fig. 3.23.

Equations (3.78) and (3.79) indicate that in order to reach the sonic speed M = 1, the area change dA/dx must vanish. This implies that the cross section must have a minimum. Considering the configurations shown in Fig. 3.22 and Fig. 3.23, channels can be constructed that satisfy the requirements stated in Eq. (3.79) and (3.80). A

Fig. 3.24 (a) A generic Laval nozzle operating at the design pressure ratio, (b) Stator row of a control stage of a steam turbine.

M>1

Normal shock

M<1 Oblique shok

M=1

Fig. 3.25: Flow through a supersonic inlet: deceleration from M>1, to M=1 at the throat and M <1 downstream of the throat.

generic convergent-divergent nozzle known as the Laval nozzle shown in Fig. 3.24 (a), is an example. A similar configuration is used in the so called control stage of steam turbines, Fig. 3.24 (b). This channel can accelerate the flow from a subsonic to a supersonic Mach range.

The condition for a supersonic flow to be established is that the pressure ratio along the channel from the inlet to the exit must correspond to the nozzle design pressure ratio, which is above the super critical pressure ratio. In this case, the flow is accelerated in the convergent part, reaches the Mach number M = 1 in the throat, and is further accelerated in the divergent portion of the nozzle. If the channel pressure ratio is less than the critical pressure ratio, the flow in convergent part is accelerated to a certain subsonic Mach number, M<1, and then decelerates in divergent parts. A supersonic diffuser, on the other hand, is characterized by convergent divergent channels with an inlet Mach number M>1.

Figure 3.25 shows schematically the flow through an inlet diffuser of a supersonic aircraft. Passing through an oblique shock, the flow enters the inlet and is decelerated from M > 1 to M = 1 at the throat, where the sonic velocity has been reached. Further deceleration occurs at the divergent part of the inlet.

0 1 2 3 4 5

M

0 0.2 0.4 0.6 0.8 1

A*/A

ρ/ρt

p/p_t T/T_t

Fig. 3.26 Area ratio and the thermodynamic property ratios as a function of Mach number for = 1.4.

Equation (3.79) and its subsequent integration, along with the flow quantities listed in Table 4.1, indicate the direct relation between the area ratio and the Mach number. These relations can be utilized as a useful tool for estimating the cross section distribution of a Laval nozzle, a supersonic stator blade channel, or a supersonic diffuser. If, for example, the Mach number distribution in the streamwise direction is given, , the cross section distribution is directly calculated from Table 4.1. If, on the other hand, the cross section distribution in the streamwise direction is prescribed, then the Mach number distribution, and thus, all other flow quantities can be calculated using an inverse function. Since we assumed the isentropic flow with calorically perfect gases as the working media, important features such as flow separation as a consequence of the boundary layer development under adverse pressure gradient, will not be present. Therefore, in both cases, the resulting channel geometry or flow quantities are just a rough estimation, no more, no less. Appropriate design of such channels, particularly transonic turbine or compressor blades, require a detailed calculation where the fluid viscosity is fully considered.

In order to represent the thermodynamic variables as functions of a Mach number, we use the continuity and energy equations in conjunction with the isentropic relation, and the equation of state for the thermally perfect gases with p =

N

RT. The isentropic flow parameters as a function of a Mach number are summarized in Table 3.1, which contains two columns. The first column gives the individual parameter ratios at arbitrary sections, whereas the second one gives the ratios relative to the critical state. The gas dynamics relations presented in Table 3.1 are depicted in Fig. 3..26.

Table 3.1: Summary of the gas dynamic functions Parameters for any two arbitrary

sections Parameters relative to the critical state

Fig. 3.27 Operational behavior of a generic Laval nozzle, (a) Expansion to the design exit pressure, (b) Overexpansion.

Fig. 3.28 Overexpanded jet

Laval nozzles were first used in steam turbines, but many other applications for these nozzles have been found, for example, in rocket engines, supersonic steam turbines, etc. In the following, we briefly discuss the operational behavior of a generic Laval nozzle, which is strongly determined by the pressure ratio. Detailed discussion of this topic can be found in excellent books by Spurk [4], Prandtl et al.[5]

and Shapiro [6]. Starting with the design operating point, where the exit pressure is set equal to the ambient pressure , curve ±, that corresponds to the design area ratio (Fig. 3.27a).

In this case, the Mach number continuously increases from the subsonic at the inlet to the supersonic at the exit. Increasing the ambient pressure results in an overexpanded jet, because the flow in the nozzle expands above a pressure that does not correspond to its design pressure: point ² with p_e < p_a. At this pressure condition, the flow pattern inside the nozzle does not change as curve ± indicates. However, outside the nozzle, the flow undergoes a system of oblique shocks that emanate from the rim of the nozzle, raising the lower nozzle discharge pressure discontinuously to the ambient pressure. The shock surfaces intersect and are reflected at the jet boundary as steady expansion waves (Fig. 3.28).

Fig. 3.29 Under-expanded jet

A rhombic pattern, characteristic of supersonic jets, arises and this is sometimes visible to the naked eye in exhaust jets of rocket engines. If the ambient pressure is further raised, the shock moves into the nozzle and forms a normal shock wave in the nozzle, curves ³. This discontinuous pressure increase positions itself in the nozzle just so that the required ambient pressure is reached. Behind the shock, the flow is subsonic. The section of the nozzle behind the shock then works as a subsonic diffuser, which theoretically raises the pressure behind the shock to the ambient pressure. However, in practice, a flow separation occurs and the actual gain in pressure is so small that the pressure behind the shock is actually about the same as the ambient pressure. If the ambient pressure is raised even further, it curves, the shock migrates into the nozzle, and it becomes weaker, since the Mach number in front of the shock becomes smaller. If the ambient pressure is increased such that the shock finally reaches the throat of the nozzle, the shock strength drops to zero and the whole nozzle contains subsonic flow, curve ´. If we increase p_a even further, the Mach number has a maximum at the throat, but M = 1 is no longer reached, curve µ. The geometric locations of all pressure discontinuities are also shown in Fig. 3.27, curve ¶.

In under-expanded jets, the pressure at the nozzle exit pe is larger than the ambient pressure p_a (Fig. 3.29).

The pressure is reduced to the ambient pressure through a system of stationary expansion waves. The flow in the nozzle remains unaffected by this. The expansion waves penetrate into themselves and are then reflected at the boundary of the jet as compression waves and these often reform themselves into a shock. In this manner, a rhombi pattern is set up in the jet again similar to the over-expanded jets.

In a convergent nozzle, no steady supersonic flow can be formed in the above stated manner. As long as the ambient pressure pa is larger than the critical pressure p^*, the pressure in the jet p_e is the same as the ambient pressure p_a (Fig. 3.29). If the Mach number, M = 1, is reached at the smallest cross-section, then pe = p^* and the ambient pressure can be decreased below this pressure (p _a < p_e). Next, an after-expansion takes place in the free jet and the pressure at the nozzle exit is expanded to the ambient pressure pa again through stationary expansion waves (Fig.

3.30).

Fig. 3.30 Subsonic nozzle with after expansion

(3.81)

(3.82) 3.7.3 Compressible Flow through Channels with Constant Cross Section This type of flow is encountered in several components of turbomachines such pipes, labyrinth seals, and to a certain degree of simplicity, in combustion chambers and afterburners of supersonic jet engines. In the case of pipes and labyrinth seals, we are dealing with an adiabatic flow process, where the total enthalpy remains constant.

However, entropy increases are present due to the internal friction, shocks, or throttling. Combustion chambers and afterburners can be approximated by a constant cross section pipe with heat addition or rejection. The characteristic features of these devices are that the entropy changes are caused by heat addition, such that the friction contribution to the entropy increase can be neglected. This assumption leads to a major simplification that we may add heat to a constant cross section pipe and assume that the impulse remains constant. The constant total enthalpy is described by the Fanno process, whereas the constant impulse case is determined by the Rayleigh process.

Starting with the Rayleigh process, we will specifically consider the flow in a duct with a constant cross-section, without surface or internal friction, but with heat transfer through the wall. In the absence of the shaft power, Eq. (3.16) is modified to:

In the application of the momentum balance, we assume here that the contribution of the friction forces to the total entropy increase compared to the entropy increase by external heat addition is negligibly small, thus the friction in Eq. (3.4) is F_r = 0. This results in:

(3.83)

(3.84)

(3.85)

(3.86)

(3.87)

(3.88)

(3.89) To find the flow quantities for the Rayleigh process, we present the calculation of a pressure ratio. The other quantities such as velocity ratio, temperature ratio, density ratio, etc., are obtained using a similar procedure. We start with the calculation of the pressure ratio by utilizing the following steps.