6.28 Overexpanded Nozzle Flow—Shock Losses

There is an aspect of nozzle performance that is hampered by the presence of shocks inside the nozzle. This behavior, which is represented by a severe case of overexpansion, is not readily apparent from Equation 6.66. An overexpanded nozzle has an area expansion ratio A₉/A₈ in excess of that needed for a perfect expansion. This results in an exit static pressure, which falls below the ambient pressure p₉/p₀ < 1. The need for an abrupt adjustment of the static pressure in the emerging supersonic flow causes a shock formation on the nozzle lip. The mild cases of overexpansion are resolved through oblique shock waves on the nozzle lip. A greater mismatch between the static pressures of the jet and ambient calls for a stronger shock, which in essence increases the wave angle with respect to stream. Eventually, a normal shock is formed at the nozzle lip. In case the normal shock at the nozzle lip is found incapable of satisfying static pressure continuity across the jet slipstream, the normal shock is then brought inside the nozzle. At this time, the exit flow is subsonic and a potential boundary layer separation needs to be investigated. Let us first examine the flow environment near the exit of an overexpanded nozzle in supersonic flight. Figure 6.62 shows the external flow around a nozzle boattail and the jet geometry dominated by wave formations. The shear layer separating the inner and outer flowfields is labeled as the jet slipstream. Due to flow curvature near the nozzle exit, local static pressure is higher than ambient pressure, that is, local ambient static pressure near plane 9 is not p₀, which we had assumed all along in our analysis.

Note that we intentionally omitted a drawing of the wall boundary layers on the primary nozzle and divergent flaps to avoid further crowding of Figure 6.62. This figure was produced, with simplifications, to develop an appreciation of the flow complexities in supersonic exhaust flow. Now, let us graphically examine higher levels of overexpansion in the above nozzle. With larger area ratios, assuming an attached flow, the static pressure at the exit plane drops, demanding a normal shock at the lip. Due to the presence of the wall boundary layer and a curved free shear layer (i.e., the jet slipstream), a lambda shock is formed at the lip, which is graphically depicted in Figure 6.63.

Again for simplicity, the inner boundary layer and the vortex sheet emanating from the junction of lambda shock are not graphed in Figure 6.63. Here, we clearly see a curved shock formation, which is predominantly normal to the core flow and dominated by a lambda-shock geometry near the wall and exit shear layer. In the case of a more severe overexpansion, the shock will be drawn inside the nozzle and a subsonic jet will emerge at the exit. Figure 6.64 examines this kind of overexpansion.

From Figure 6.64, we note that a separated boundary layer could alter the picture of exit flow, in particular with the flow direction as well as the effective exit flow area. How much of this complexity can we capture by simple one-dimensional models and correction factors? For example, in our gross thrust coefficient approach to account for viscous losses and nonaxial exit flow, are there any provisions for curved shocks or separated flows, such as demonstrated by some overexpanded nozzle flows? The short answer is no. The long answer starts with a maybe! A plausible approach calls for identifying and bracketing off zones of invalidity from our performance charts that we derived based on simple 1D loss models and assumptions. This approach relies, in part, on the experience base of the manufacturing company and other available experimental data in the literature. For example, in separated nozzle flows, the physical meaning of the velocity coefficient C_V and the angularity loss factor C_A are completely lost, based on our definitions and simplifying assumptions. Consequently, the nozzle gross thrust coefficient C_fg, as described by Equation 6.69, becomes at best inaccurate and at worst meaningless. Here, we need to exert judgment when dealing with overexpanded nozzle flows that include a shock inside the nozzle. Let us reproduce some of the criteria found in the literature for a passage shock that causes flow separation. In Kerrebrock (1992), a reference is made to Zukoski (1990) who suggests a separation point criterion in an overexpanded supersonic nozzle according to

(6.70)

where p_s is the static pressure upstream of the shock, p₀ is the ambient static pressure, and M_s is the gas Mach number upstream of the shock. Over a wide operating range of the nozzle, that is, NPR ranging between 2 and 50, the separation pressure ratio, Kerrebrock notes that p_s/p₀ ranges between 0.4 and 0.625. Mattingly (1996) recommends a p_s/p₀ ≈ 0.37, following Summerfield, Foster, and Swan (1954) and Sutton (1992) who recommend a rough separation criterion of p_s/p₀ ≤ 0.4. In summary, several simple or more sophisticated models for shock separation criterion are introduced, which roughly identify the zones of invalidity of our simple nozzle theory. Figure 6.65 shows the ideal gross thrust coefficient with nozzle area ratio as a function of nozzle pressure ratio. The maximum C_fg = 1 for p₉/p₀ = 1.0.

Note that the nozzle overexpanded mode of operation is more prevalent for a rocket engine than an airbreathing aircraft engine. A rocket engine that operates between the ambient pressures of sea level, ∼14.7 psia (1 atm.), and near vacuum conditions of space, requires a large nozzle area expansion ratio (A₉/A₈) to optimize its performance. A nozzle, which is optimized for high altitude, then operates in an overexpanded mode in lower altitudes. Consequently, the thrust loss and shock interactions inside the nozzle are serious realities of rocket nozzle design. However, the airbreathing engine of an aircraft invariably operates with a smaller area expansion ratio, hence it experiences less severe over-expansion problems. In the hypersonic regime, the airbreathing and a rocket nozzle attain geometric and operational similarities and thus face the same challenges for off-design operations.

Now, we are ready to address the nozzle throat size and scheduling with afterburner operation. Also, we start the discussion on the nozzle exit area scheduling within the flight envelope and its impact on external drag penalties due to boattail and base flowfield.

6.29 Nozzle Area Scheduling, A₈ and A₉/A₈

The exhaust nozzle throat of a modern aircraft engine operates in a choked mode over a wide operating range within the flight envelope. Consequently, the nozzle throat serves as the low-pressure turbine’s backpressure controller or equivalently as the engine aft throttle station. With an afterburner in operation, the exhaust gas temperature increases significantly with a subsequent drop of density. To accommodate a lower density gas, a choked nozzle needs to open up its throat to pass almost the same mass flow rate as with no afterburner operation. The one-dimensional continuity equation for a choked nozzle throat may be written as

(6.71)

We note from Equation 6.71 that the term that undergoes the most dramatic change with the afterburner operation is T_t8 due to heat release in the combustion process. The other terms in the equation undergo smaller changes. For example, the mass flow rate increases by ∼2–4%, the gas constant R hardly changes as the molecular weight of the new mixture has not changed much by the addition of the fuel; the total pressure drops with the afterburner, perhaps by 2–5%; the specific heat ratio γ₈ changes with temperature and may go from ∼1.33 to ∼1.30 or 1.25. In its simplest form, the nozzle throat scheduling with afterburner for a turbojet engine may be written as

(6.72)

If the aim is to be more exact with the nozzle throat area scheduling, then one must do the bookkeeping of all the terms in the continuity equation more carefully, namely,

(6.73)

where the primed terms are the afterburner-on quantities.

6.30 Nozzle Exit Area Scheduling, A₉/A₈

Unlike the choked nozzle throat that required scheduling with an afterburner operation (i.e., a single parameter control), the nozzle exit area scheduling involves a multiparameter optimization study. The reason for this added complexity is rather obvious. Exit area variation of the nozzle impacts external aerodynamics of the nacelle boattail and base flowfields, and hence external drag. Therefore, if we gain an increment of gross thrust due to a better nozzle expansion of say p₉ ≅ p₀, and as a result incur an increment of drag rise due to nacelle boattail and base flow changes, then the nozzle area ratio A₉/A₈ needs to optimize the installed performance, namely,

(6.74)

Here, we are again reminded that the internal optimization of an airbreathing engine inlet and exhaust system is meaningless in isolation. Rather, the engine-installed performance should be optimized at each flight phase within the flight envelope of the aircraft. Traditionally, the nozzle exit area of a variable geometry nozzle assumed only two positions. One closed and the other open, very similar to the nozzle throat with/without afterburner. But, the limitation of only two positions for the nozzle exit area seems unnecessary today especially in light of advanced propulsion control system of modern aircraft. Here, the goal of nozzle area scheduling control is to achieve a continuous improvement of installed performance with altitude, speed, aircraft attitude, and acceleration rather than that achieved by a bang–bang (a two-position, open-closed) controller.

The contributors to the aftbody aerodynamics on an engine installation are the contributions from the boattail and base areas of the nacelle. The geometry of nozzle boattail and base areas and their impact on aftbody drag are shown in Figure 6.68 (Bowers and Tamplin, 1985). We note that the base pressure force contribution is in the thrust direction and when integrated with the boattail drag contribution, they yield the “total” pressure drag penalty, as shown in Figure 6.68. Also, a graphic definition of base area is helpful and is shown in Figure 6.69 (from Whitford, 1987).

An installed nozzle performance optimization that accounts for the internal as well as external losses is reported by Younghans and Dusa (1993) and reproduced here in Figure 6.70. First, note that nozzle area ratio A₉ /A₈ that is needed for a perfect expansion does not give the best-installed performance. Second, note that each loss parameter has a different area ratio for its optimization. For example, flow angularity loss vanishes for A₉/A₈ = 1.0. The expansion loss vanishes at full expansion (p₉ = p₀). The friction penalty holds in a somewhat steady manner across the nozzle area ratios. Finally, the internal performance curve is shifted down by the external drag penalties. This leads to a shift in the area ratio for optimized installed performance.

6.31 Nozzle Cooling

The operational temperatures of a modern, multitask exhaust system, with an afterburner, have required an increasing level of active nozzle cooling. In a turbofan engine, the fan stream could provide the required nozzle cooling through an engine bypass system. The supersonic inlet/engine flow match often requires an inlet excess air bypass/bleed system to be integrated into the propulsion system design. The excess air has traditionally been used for engine bay and nozzle cooling purposes. Figure 6.71 shows a schematic drawing of a flow network between an inlet and an exhaust system in a turbofan engine (from Younghans and Dusa, 1993). A typical engine airflow variation with flight Mach number and its comparison with the inlet supply are shown for a Mach 5 aircraft by Dusa (1988).

The primary goal of nozzle cooling is to operate the exhaust system components, that is, the liner, primary and secondary nozzle walls/ramps, seals, and the actuators at a temperature that allows for reliability of operation and good material life expectancy. The material operating temperature should be well below the melting point of the material. A very useful graph from Wazyniak (2000) is reproduced here in Figure 6.72; it shows the design temperature levels of jet engine materials. It is useful to examine the design versus the melting point temperature of different materials in addition to material density, as shown in Figure 6.72. The typical uses of the materials listed are: nickel-based alloys in turbines, titanium in compressors and fan blades, aluminum for aircraft and nacelle skin, intermetallics and nonmetals in the high-temperature environment of afterburners and exhaust systems. The acceptable operating temperature of metals (e.g., stainless steel) extends to ∼1700°F (or ∼927°C), whereas ceramics may operate up to ∼2800°F (or ∼1538°C). Carbon-carbon offers an extended operating temperature of up to ∼3400°F (or ∼1870°C). There are also refractory metals, such as tungsten and molybdenum that offer higher operating temperatures than stainless steel (of up to ∼2880°F). However, refractory metals have exhibited manufacturing-related problems as well as operational problems with embrittlement and oxidation.

Now, let us examine the flight total temperature, or ram temperature, as it increases with flight Mach number. Assuming an adiabatic flow of a calorically perfect gas, the inlet air total temperature follows

(6.75)

Now assuming the supersonic/hypersonic flight occurs in the stratosphere (between 50 and 150 kft) where the static temperature is ∼ − 60°F = constant, then the air ram temperature with flight Mach number varies parabolically according to

(6.76a)

(6.76b)

A graph of the inlet air ram temperature in stratosphere (for constant γ) is shown in Figure 6.73.

We note that the ratio of specific heats γ decreases with temperature, and hence the ram temperature calculation based on cold γ of 1.4 leads to an overestimation of the temperature. To put this in perspective, for a Mach 6 flight in stratosphere, T_t0 calculated based on γ = 1.4 is 2820°F, where as γ = 1.3 yields T_t0 of 2100°F. There is a ∼720°F discrepancy between the two temperatures; although significant, it does not change the arguments about the special cooling needs of high-speed flight. Now, let us accept the goal of ∼3000°F operating temperature of say carbon–carbon, the inlet air becomes too hot to be used as a coolant for flight Mach numbers M₀ = 6⁺. Therefore, the coolant needs to be precooled in a heat exchanger, which adds weight and complexity to the engine. The other alternative is to use a cryogenic fuel that acts as a heat sink for the propulsor walls, before its injection in the combustor and reaction with air or another oxidizer, as in a rocket engine. To use a cryogenic fuel as a coolant is a common and tried approach in rocket propulsion. The method is called regenerative cooling. From the Saturn-V Apollo launch vehicle (all three stages) to the Space Shuttle Main Engine, nozzles and the thrust chambers are regeneratively cooled using fuels and sometimes oxidizers. Cryogenic fuels of interest are liquid hydrogen, H₂, and methane, CH₄. Both of these fuels are available in abundance in nature in various liquid and gaseous forms. The heat sink capacity of these fuels is in part quantified by their latent heat of vaporization, namely, 446 kJ/kg for liquid hydrogen and 510 kJ/kg for liquid methane.

6.32 Thrust Reverser and Thrust Vectoring

Thrust reversers are used to decelerate the aircraft upon landing (for a shorter runway), to reduce wear on the landing gear brakes upon touchdown and landing roll, and to enhance maneuverability of a fighter aircraft in flight. The goals of thrust vectoring nozzles are to enable an aircraft to takeoff and land either vertically or use a very short runway (<500 ft) as well as to achieve supermaneuverability in flight. Thrust vectoring is offered as a means of reducing the size or ultimately eliminating the tail (i.e., tailless aircraft), as well as to provide the aircraft with stall recovery. Additional goals of thrust vectoring are the low-speed flight control where the conventional aerodynamic control surfaces become ineffective and the high angle of attack (the so-called high-alpha) capabilities. The latter two goals fall under the supermaneuverability category. To provide some degree of reverse thrust, we need to divert the exhaust jet partially in the flight direction. This is commonly achieved through a block and turn mechanism designed into exhaust systems of aircraft today. Two such concepts have been the clamshell and cascade thrust reversers. The clamshell thrust reverser is often used for nonafterburning engines to reverse the turbine exhaust gases, and the cascade type is typically used as a fan exit flow reverser in turbofan engines. Schematic drawings of the two types of thrust reverser are shown in Figure 6.74 (from a Pratt & Whitney Aircraft, 1980).

For an elemental mass flow rate that attains an exit velocity from a reverser of V_e magnitude, which exits at an angle of θ with respect to the flight direction, a reverse thrust equal to V_ecos θ is created. This needs to be integrated over the exit of the turning vane cascade to evaluate the reverse thrust. A schematic diagram of a fan reverser is shown in Figure 6.76.

(6.78)

The actual force exerted by the fluid on the blocker and the turning vanes, that is, the reverse thrust force exerted on the aircraft, is established by a momentum balance of the fluid across the blocker and turning vanes, such as

(6.79)

The terms of Equation 6.79 represent momentum and pressure–area terms, similar to the engine net thrust equation. The two positive terms on the RHS of Equation 6.79 dominate the bulk of reverse thrust force generation. The two negative contributions are due to leakage and pressure imbalance in the fan duct, p₁₃, and the ambient pressure p₀. We may estimate these terms to get a feel for the reverser effectiveness. For example, the turning vane cascade can be assumed to operate with an inlet total pressure of the fan, that is, p_t13, and an exit pressure of the ambient air, p₀. We may also assume an adiabatic efficiency of the expansion process in the turning vanes, similar to a nozzle. The vane angle may be assumed to be ∼45°, and the leakage flow may be ∼5% of the incoming flow. To account for the boundary layer blockage in the turning vanes, we may use a discharge coefficient C_D similar to C_D8. We may consider the air speed through the blocker is reduced by a factor, for example, V′₁₃ ∼ 1/2V₁₃ (note that a mistake in here will not be very large as the leakage mass flow rate is a small fraction of the incoming flow). For the fan exit speed V₁₃, we may estimate that based on a fan exit Mach number of ∼0.5. The static pressure p₁₃ may be estimated by the fan total pressure ratio and the M₁₃ of ∼0.5.

The British Harrier, AV-8B, is the first operational fighter aircraft to achieve vertical takeoff through thrust vectoring. The Harrier’s four rotating nozzles turn the engine flow to a near vertical direction to achieve hover, takeoff, and landing. A three-bearing swivel nozzle in Figure 6.77 shows the mechanism for thrust vectoring in an exhaust system (from Dusa, Speir, and Rowe, 1983). The nozzle is of convergent design, hence most suitable for low nozzle pressure ratio (i.e., NPR < 6) operation of a subsonic V/STOL aircraft.

For a supersonic application, a single-expansion ramp nozzle (SERN) offers jet deflection capability through turning of the upper/lower flaps or engaging a separate deflector mechanism. The augmented-deflector exhaust nozzle (Figure 6.78) is capable of 110° thrust vectoring angle. The deflecting flap SERN nozzle is shown in Figure 6.79 (from Dusa, Speir, and Rowe, 1983). The two-dimensional geometry of these nozzles allows for low-drag, body-blended exhaust system integration.

Here, we note that the lower flap also serves as a reverser blocker in the STOL/SERN nozzle configuration, which is shown in Figure 6.79. To get a feel for the cooling requirements of an advanced thrust-vectoring nozzle with an afterburner, we look into the experimental results of ADEN tests (at flight weight, self-cooled with the fan air on a thrust stand at NASA-Lewis). To achieve a surface temperature below the hot streak design limit of 2700°F, a ∼5.7% cooling air was required at maximum afterburner operation of T_t8 = 3028°F and the maximum nozzle pressure ratio of 15 (from Dusa and Wooten, 1984). Now, let us examine the following twin configuration (Figure 6.80).

Another concept for vector thrust–thrust reverser nozzle is shown as a multiflap 2D-CD nozzle in Figure 6.80 (from Younghans and Dusa, 1993). The twin configuration shown uses segmented, that is, split, divergent flaps, which are used to achieve pitch and roll control. The yaw control requires the use of thrust reverser, which causes a reduction in axial thrust. Note that the convergent flaps serve as the blocker in the reverse thrust mode.

6.33 Hypersonic Nozzle

In this section, we explore the challenges of a hypersonic nozzle design. There are several design conflicts in the hypersonic exhaust system that stem from the range of flight speeds, that is, from takeoff to Mach > 5 and the range of altitude, namely, from sea level takeoff to upper atmosphere, or 100⁺ kft. The combination of altitude and Mach number constitutes the flight envelope of an aircraft, and in Figure 6.81 we examine such a flight envelope (from Dusa, 1988). The shaded area in the corner of Figure 6.81 represents the flight envelope of conventional commercial and military aircraft. We examine the representative nozzle pressure ratio for airbreathing hypersonic engines in Example 6.12.

The nozzle area ratio as a function of nozzle pressure ratio is shown in Figure 6.83, using a log-linear scale. We note that a hypersonic airbreathing propulsor will require a very large nozzle area ratio, whose integration into the vehicle is the challenge number one.

To reduce the burden of excessive weight, a hypersonic nozzle will perform an external expansion of the gas, in a geometrically similar manner as a plug nozzle. However, instead of a plug, the nozzle external expansion surface may be the lower surface of the fuselage aftbody. A schematic drawing of such a nozzle is shown in Figure 6.84 where the expansion waves in the exhaust plume are shown as dashed lines and the compression waves as solid lines. Remember from supersonic aerodynamics that the wave reflection from a free boundary, as in the exhaust plume mixing layer, is in an unlike manner and the wave reflection from a solid boundary is in a like manner. Hence, a shock is reflected as an expansion wave from a free boundary and reflects as a shock from a solid surface. A similar statement may be made regarding an expansion wave reflections.

6.34 Exhaust Mixer and Gross Thrust Gain in a Mixed-Flow Turbofan Engine

A mixed-flow turbofan engine offers the potential of gross thrust gain and a reduction in fuel consumption as compared with an optimized separate flow turbofan engine. In this section, we examine the extent of gross thrust improvements and the nondimensional parameters that impact the performance gains.

To demonstrate the gain in the gross thrust realized by mixing a cold and a hot stream, we first treat the problem by assuming an ideal mixer. Expanding the cold, hot, or mixed gases in a nozzle to an exit pressure of p₉, we shall realize a conversion of the thermal energy into the kinetic energy in the exhaust nozzle. The thermodynamics of the expansion process for the three streams is shown in Figure 6.85.

The gross thrust for a separate exhaust turbofan engine, assuming the nozzles are perfectly expanded is

(6.81)

We may write the isentropic pressure ratio for the temperature ratios in Equation 6.81 to get

(6.82)

The mixed exhaust nozzle shall have a gross thrust of

(6.83)

We note that the pressure terms in the brackets are identical hence they cancel when we express the ratio of gross thrusts. Assuming the same gas properties in the three streams, namely, we get

(6.84)

(6.85)

Therefore the rise of the gross thrust as a result of mixing a cold and a hot stream may be expressed in terms of the nondimensional parameters of the respective temperature ratio and the mass flow ratio, namely, the bypass ratio α. This is expressed in the following equation.

(6.86)

6.35 Noise

High-performance gas turbine engines are very noisy machines, by design! Energy transfer in turbomachinery can only take place through unsteady means, as presented in Chapter 8. Also, to maximize energy transfer, modern engines use high-speed wheels of rotating and stationary or counterrotating blade rows. These always involve supersonic tip speeds, which create shocks and their associated discrete tone noise. Since the energy transfer is unsteady, the blade wakes are unsteady with periodic vortex shedding structure. The unsteady wakes of a blade row are then chopped by the next row of blades, as they are in relative motion. In addition, there are ample opportunities for resonance to occur in blade row-blade row interaction. Due to blade elasticity and aerodynamic loads, the blades vibrate in various modes, for example, first bending, second bending, first torsional mode, coupled bending and torsional mode, and higher order modes. The unsteady mechanism of energy release in a combustor creates combustion noise. Finally, the pressure fluctuation in the exhaust jet due to turbulent structure and mixing is the source of broadband jet noise.

Modern fan blades that are thin and incorporate blade sweep, and the inlet fan and core exit ducts that are treated with an acoustic liner, are successful solutions that alleviate the turbine noise problem. For example, the acoustic duct liners, which are Helmholtz resonators embedded in the wall, attenuate the dominant frequency in the transonic fans’ and LPTs’ noise spectra, namely the discrete tone noise at the blade passing frequency. In addition, proper selection of blade numbers in successive blade rows and blade row spacing are the solutions that alleviate, by and large, the intrablade row resonance problem.

6.35.1 Jet Noise

In a classical paper, Lighthill (1952) showed that the acoustic power in a subsonic turbulent jet is proportional to the eight power of the jet speed, V_j, that is,

(6.87)

Therefore, to reduce the broadband noise from an aircraft exhaust, the jet speed needs to be reduced. Interestingly, propulsive efficiency is also improved with reduced exhaust speed, as we discovered in the cycle analysis (Chapter 4). The development of high and ultra-high bypass ratio turbofan engines accomplish both goals, that is, higher propulsive efficiency and reduced jet noise, simultaneously. Also, the concept of Variable Cycle Engine (VCE) promises the capability of higher bypass ratio at takeoff and landing and reduced bypass ratio at cruise condition, which is advantageous in the reduction of the airport’s community noise. Since the kinetic power in a jet is proportional to V_j³, an acoustic efficiency parameter may be defined as the ratio of the radiated acoustic to the jet kinetic power (for derivation see Kerrebrock, 1992), which is then proportional to the fifth power of the jet Mach number, M_j0, which in this derivation is defined as the ratio of V_j and the ambient speed of sound, a₀, that is,

(6.88)

This efficiency parameter may be used to compare different noise suppression concepts. The noise suppressors of the 1950s that were developed originally for turbojets added significant weight to the engine and caused significant loss of gross thrust. The turbofan engines of the 1960s and 1970s alleviated the jet noise problem based on their reduced jet speed and their blade and duct treatments also reduced the turbine noise; but still more stringent noise regulations were on the horizon. These limitations are set by the Federal Aviation Regulation – Part 36 Stage 3 and the new FAA Stage 4 Aircraft Noise Standards (that is used for the certification of turbofan-powered transport aircraft). One thing is for certain and that is the growth in commercial aviation and its impact on society will certainly demand yet more stringent regulatory aircraft noise standards in the future. Beyond a reduction in jet speed, as in high bypass ratio turbofan engines, are there any other means of reducing jet noise without paying a severe penalty on gross thrust? The jet noise reduction depends on many parameters but predominantly on the nozzle pressure ratio (NPR) and the jet Mach number. In this section, the most promising development in transonic jet noise reduction is briefly presented.

For the past two decades, research and development by NASA’ Glenn Research Center and the aircraft engine and airframe industry on jet noise reduction has led to the development of a new winning technology: the Chevron Nozzle. An excellent review article by Zaman et al. (2011), which chronicles the development of mixing devices, mainly tabs and chevrons, is suggested for further reading.

6.35.2 Chevron Nozzle

Chevrons are triangular serrations at the nozzle exit plane that have the potential of “calming” turbulent jets (Zaman et al., 2011). The calming effect stems from a redistribution of azimuthal vorticity at the nozzle exit that attains certain streamwise components. For optimal redistribution of vorticity, the chevron tips have to slightly penetrate the jet. Excessive penetration gives rise to thrust loss and an insignificant dipping into the jet does not produce the desired sound pressure level (SPL) reduction. Figure 6.87 shows a baseline nozzle that is used to compare the aeroacoustic and propulsive performance of various chevron configurations on the core and/or the fan nozzles in a separate flow turbofan engine.

The most promising of the chevron configurations in static test were then flown to demonstrate takeoff, landing and the flyover noise reduction effectiveness. Naturally, on the propulsion side thrust loss coefficient at cruise conditions is the measure of viability of any noise reduction device.

The benefits in noise reduction of chevron-equipped aircraft in flyovers, that is, community noise, is first measured (using a logarithmic scale known as the Bel or the more commonly used unit of one tenth of a Bel, i.e., the dB) in the frequency range where human ear is sensitive (i.e., between 50 and 10 kHz) and subsequently weighted towards the frequencies where the human ear is most sensitive, that is, 2–5 kHz. There are other corrections that contribute to the calculation of what is known as the Effective Perceived Noise Level (EPNL) in dB; this is the basis of FAR-36 certification limits. Kerrebrock (1992) offers a clear methodology to EPNL calculations and is suggested for further reading on engine noise. There is ample data that support a reduction in turbulent kinetic energy in a jet where streamwise vorticity is injected at the nozzle exit, as produced by chevrons. The significance of the chevron technology is in its light weight and negligible impact (i.e., loss) on nozzle gross thrust, which is measured at 0.25%.

At the outset, the goal of jet noise reduction was to achieve 2.5 EPNL (dB) in flyover, that is, community, noise reduction while limiting the thrust penalty in cruise to less than 0.5%. As evidenced in Table 6.1, where the noise benefit and cruise thrust loss data from Zaman et al. (2011) are shown, there are several chevron configurations that achieve the desired performance.

	Noise benefit	Loss in thrust
Configuration	(EPN dB)	coefficient at cruise (%)
3C12B	1.36	0.55
3I12B	2.18	0.32
3I12C24	2.71	0.06
3T24B	2.37	0.99
3T48B	2.09	0.77
3T24C24	—	0.43
3T48C24	—	0.51
3A12B	—	0.34
3A12C24	—	0.49

The success of the chevron nozzle has made it into a standard component on several new aircraft, such as the Boeing 787.

6.36 Nozzle-Turbine (Structural) Integration

The turbine exhaust case, support struts, and a tail cone flange are shown in Figure 6.88 (from Pratt & Whitney, 1980). The turbine exhaust cone houses the turbine bearings and provides for a smooth flow to the nozzle (or afterburner).

The exhaust cone, jet pipe, and a convergent nozzle are shown schematically in Figure 6.89 (from Rolls-Royce, 2005). In this view of the exhaust system, the turbine rear support struts are also shown.

6.37 Summary of Exhaust Systems

An aircraft exhaust system is a multifunctional component whose primary mission is to convert fluid thermal energy into directed forces in any desired direction (x, y, z) efficiently. These forces are usually in vector thrust mode, including hover, as well as thrust reversing directions. The system weight (and complexity) and cooling requirements are the optimizing parameters for the design and selection of exhaust systems. Advanced nozzle cooling requirements are in the ∼5–10% range. Airbreathing engines with afterburner require a variable geometry nozzle to open the throat with afterburner operation. The nozzle throat scheduling with the afterburner throttle setting is to decouple the afterburner/exhaust system from the gas generator. The nozzle exit area scheduling has to include the net effect of internal performance gains as well as the external boattail drag penalties. The internal performance is optimized when the nozzle is perfectly expanded. The build up of boundary layer in the convergent or primary nozzle is accounted for by a flow (discharge) coefficient at the throat, C_D8. The effect of total pressure loss on the exhaust velocity in the divergent section of the nozzle is accounted for by a velocity coefficient C_V. The effect of nonaxial exhaust flow appears as a reduction in axial force produced by the nozzle. The appropriate accounting of this loss is made using the flow angularity loss coefficient C_A. Lobed mixers in a mixed-stream turbofan engines improve gross thrust by a few percentage points. Off-design operation of supersonic nozzles in overexpanded mode causes shock waves to be formed and a corresponding loss of thrust. Plug nozzles have a superior overexpanded nozzle performance over the convergent–divergent nozzles, but require additional cooling and thus system complexity and weight. In hypersonic flight, the nozzle pressure ratio reaches into several hundred level, somewhat similar to a rocket engine. The required area expansion ratio for efficient expansion of the high-pressure gas demands integration with vehicle. To achieve reduced signature, an exhaust system could employ high-aspect ratio rectangular nozzles with offset to shield the turbine as well as enhanced mixing to reduce the jet temperature. The exit plane of the nozzle placed on the upper surface of the aircraft fuselage or wing provides shielding from ground radar.

Further readings on exhaust systems and installation (references 4, 7, 8, 24 and 26) are recommended.

References

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3. Albers, J.A, et al., “Aerodynamic Analysis of Several High Throat Mach Number Inlets for the Quiet, Clean, Short-Haul Experimental Engine, ” NASA TMX-3183, 1975.
4. Aulehla, F. and Lotter, K., “Nozzle/Airframe Interference and Integration, ” in AGARD Lecture Series, LS-53, Paper 4, 1972.
5. Anderson, B.H., “Three-Dimensional Viscous Design Methodology of Supersonic Inlet Systems for Advanced Technology Aircraft, ” Numerical Methods for Engine-Airframe Integration, Eds. Murthy, S.N.B. and Paynter, G.C., Progress in Astronautics & Aeronautics, Vol. 102, AIAA, Washington, DC, 1986, pp. 431–480.
6. Anderson, J.D., Jr., Fundamentals of Aerodynamics, 4th edition, McGraw-Hill, New York, 2005.
7. Anon., Symposium on “Developments in Aircraft Pro-pulsion, ” Haarlem, the Netherlands, 1987.
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9. Bowers, D.L. and Tamplin, G., “Throttle-Dependent Forces, ” Chapter 5in Thrust and Drag: Its Prediction and Verification, Eds Covert, E.E. et. al., Progress in Astronautics and Aeronautics, Vol. 98, AIAA, New York, 1985.
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13. Dusa, D.J., Speir, D.W., and Rowe, R.K., “Advanced Technology Exhaust Nozzle Development, ” AIAA Paper No. 83-1286, 1983.
14. Dusa, D.J. and Wooten, W.H., “Single Expansion Ramp Nozzle Development Status, ” AIAA Paper No. 84-2455, 1984.
15. Farokhi, S., Sheu, W.L., and WU, C., “On the Design of Optimum-Length Transition Ducts with Offset: A Computational Study, ” Computers and Experiments in Fluid Flow, Eds Carlomagno, G.M. and Brebbia, C.A., Springer Verlag, Berlin 1989, pp. 215–228.
16. Hancock, J.P. and Hinson, B.L., “Inlet Development for the L-500, ” AIAA Paper No. 69-448, June 1969.
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21. Kerrebrock, J.L., Aircraft Engines and Gas Turbines, 2nd edition, MIT Press, Cambridge, Mass, 1992.
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23. Kline, S.J., Abbott, D.E., and Fox, R.W., “Optimum Design of Straight-Walled Diffusers, ” Journal of Basic Engineering, 81, Series D, No. 3, September 1959, pp. 321–331.
24. Kozlowski, H. and Larkin, M., “Energy Efficient Engine Exhaust Mixer Model Technology Report, ” NASA CR-165459, June 1981.
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Problems

• 6.1 In a real inlet, the total pressure loss is 10% of the flight dynamic pressure, i.e., p_t0 − p_t2 = 0.1 q₀ at a flight Mach number of M₀ = 0.85. Calculate
  1. inlet total pressure recovery π_d
  2. inlet adiabatic efficiency η_d.
• 6.2 A subsonic inlet in cruise condition is shown. Calculate
  1. total pressure recovery π_d
  2. area ratio A₂/A₀
  3. static pressure ratio p₂/p₀
  4. adiabatic efficiency η_d
  5. entropy rise Δ_s/R.

  

• 6.3 Consider a subsonic inlet at a flight cruise Mach number of 0.8. The captured streamtube undergoes a prediffusion external to the inlet lip, with an area ratio A₀/A₁ = 0.92, as shown. Calculate
  1. C_p (i.e., the pressure coefficient) at the stagnation point
  2. inlet lip Mach number M₁
  3. lip contraction ratio A₁/A_th for a throat Mach number M_th = 0.75 (assume p_{t, th}/p_t1 = 1)
  4. the diffuser area ratio A₂/A_th if M₂ = 0.5 and p_t2/p_{t, th} = 0.98
  5. the nondimensional inlet additive drag D_add/p₀A₁.

  

• 6.4 The captured streamtube for a subsonic inlet experiences external diffusion, where flight Mach number of M₀ = 0.85 decelerates to M₁ of 0.65 at the inlet lip. Calculate the inlet additive drag nondimensionalized by flight static pressure p₀ and inlet area A₁.
• 6.5 An aircraft flies at an altitude where p₀ is 0.1915 atm and the flight total pressure p_t0 is 1.498 atm. The engine face total pressure is measured to be, p_t2 = 1.348 atm. For this inlet calculate
  1. τ_r, ram temperature ratio
  2. , nondimensional entropy rise in the diffuser
  3. η_d, inlet adiabatic efficiency
• 6.6 A subsonic diffuser has an area ratio of A₂/A₁ = 1.3. The inlet Mach number to the diffuser is M₁ = 0.72 and the total pressure loss in the diffuser is characterized by Δp_t = 0.1 q₁. Assuming the flow in the diffuser is adiabatic, calculate
  1. the diffuser exit Mach number M₂ and
  2. diffuser static pressure recovery C_PR
• 6.7 A subsonic inlet is cruising at M₀ = 0.85 and the capture area ratio A₀/A₁ is 0.90 (as shown). For an altitude pressure of p₀ = 25 kPa, the temperature of T₀ = –25°C, and inlet area A₁ = 3 m², calculate
  1. the inlet Mach number M₁
  2. the inlet additive drag D_add (N)
  3. inlet mass flow rate (kg/s)
  4. the inlet ram drag D_ram (kN)
  5. engine face area A₂, if M₂ = 0.5 (assuming π_d = 0.99)

  

• 6.8 The Mach number at the compressor face is M₂ = 0.65 at takeoff (M₀ ≅ 0.2). Assuming the inlet suffers a 2% total pressure loss at takeoff, calculate the capture-to-engine face area ratio A₀/A₂.

  

• 6.9 A subsonic aircraft cruises at M₀ = 0.85 and its inlet operates with a capture ratio of A₀/A₁ = 0.7. First, calculatethe lip Mach number M₁. Second, assuming an engine becomes inoperative and the inlet lip Mach number drops to 0.3 (the so-called engine wind-milling condition), calculate the additive drag D_add for an inlet area of A₁ = 4 m² and the ambient static pressure of p₀ = 16.6 kPa.
• 6.10 A subsonic inlet is flying at Mach 0.8, with an inlet capture area ratio of A₀/A₁ = 0.7. The inlet lip contraction ratio A₁/A_th is 1.15. Calculate the 1D throat Mach number and comment on the potential shock formation, near the convex surface, at the throat.
• 6.11 A subsonic aircraft flies at M₀ = 0.85 with an inlet mass flow ratio (MFR) of 0.90. Calculate the critical pressure coefficient C_{p, crit} on the nacelle. Also calculate the maximum cowl (frontal) area ratio A_M/A₁ if this inlet is to experience an average surface pressure coefficient corresponding to the critical value, i.e., .
• 6.12 An inlet creates a circumferential distortion at the engine face, as shown. The hub-to-tip radius ratio is r_h/r_t = 0.5. The spoiled sector has a 15% mass flow deficit (per unit area) as compared with a uniform flow. Assuming the static density, temperature, and pressure are uniform at the engine face and the mass flow deficit in the spoiled sector is caused by a velocity deficit (as shown), use Bernoulli equation to estimate the total pressure deficit in the spoiled sector, i.e., [p_t − p]_spoiled/[p_t − p] _uniform. Also calculate
  1. 
  2. 

  

• 6.13 A subsonic inlet has a capture area ratio of A₀/A₁ = 0.8. Assuming the flight Mach number is 0.80 and the inlet area ratio is A₂/A₁ = 1.25, calculate
  1. Mach number at the inlet lip M₁
  2. diffuser exit Mach number if p_t2/p_t1 is 0.95
  3. inlet static pressure ratios p₁/p₀ and p₂/p₁ for p_t2/p_t1 = 0.95
• 6.14 A normal shock inlet operates in the subcritical mode, with the shock in standoff position, as shown. The bow shock is normal to the flow at the inlet centerline and weakens into an oblique shock and eventually a Mach wave away from the inlet.

  Apply conservations of mass and momentum to the control volume a-b-c-d-e-f-a, to approximate inlet cowl external drag force coefficient, in terms of the flight and inlet parameters that are shown, for example V₀, M₀, p₀, p₁, V₁, A₁, A_M, and A_spillage.

  

• 6.15 Consider a variable-geometry, convergent–divergent isentropic inlet that is designed for M_D = 4.0. To swallow the starting shock, i.e., to start the inlet, the throat needs to be opened. Calculate the percentage of throat area opening needed to start this inlet, (ΔA_th/A_th) × 100.
• 6.16 An external compression inlet is in a Mach-2 flow. The shocks are positioned according to the figure shown. Calculate
  1. shock total pressure ratio π_s
  2. A₂/A_th

  You may neglect the frictional losses in the subsonic diffuser.

  

• 6.17 A normal-shock inlet is operating in a supercritical mode, as shown. Flight Mach number is M₀ = 1:6. The inlet capture area ratio A₀ = A₁ = 0:90 and the diffuser area ratio A₂ = A₁ = 1:2. Calculate
  1. M₁
  2. inlet total pressure recovery π_d, i.e., p_t2/p_t0

  

• 6.18 An isentropic convergent–divergent supersonic inlet is designed for M_D = 1.6. Calculate the inlet’s
  1. area contraction ratio A₁/A_th
  2. subsonic Mach number where the throat first chokes
  3. percent spillage at M₀ = 0.7
  4. percent spillage at M₀ = 1.6 (in the unstarted mode)
  5. overspeed Mach number to start this inlet, M_overspeed
  6. throat Mach number after the inlet was started, with still M_overspeed as the flight Mach number

  

• 6.19 Consider an isentropic fixed-geometry C–D inlet, which is designed for M_D = 1.75. The inlet flies at an altitude where ambient (static) pressure is 20 kPa. Calculate
  1. Overspeed Mach number that will start this inlet
  2. The flight dynamic pressure corresponding to the altitude and M_overspeed
• 6.20 An isentropic, convergent–divergent supersonic inlet is designed for M_D = 3.0. Assuming that the throat area is adjustable, calculate the percentage of the design throat area that needs to be opened to swallow the starting shock, i.e.,
• 6.21 A supersonic C–D inlet is designed for a flight Mach number of M₀ = 3.5. This inlet starts by opening its throat (from A_th to A′_th). Neglecting wall frictional losses, calculate
  1. percentage throat opening required
  2. throat Mach number after it starts (with the throat open at A′_th
• 6.22 A Kantrowitz–Donaldson inlet is designed for Mach 2.0. Calculate
  1. the required contraction area ratio A₁/A_th
  2. the inlet total pressure recovery with the best backpressure
• 6.23 A Kantrowitz–Donaldson inlet is designed for M_D = 1.7. Calculate
  1. the inlet contraction ratio A₁/A_th
  2. the throat Mach number after the inlet self started
  3. the total pressure recovery with the best backpressure.

  

• 6.24 Calculate the contraction ratio A₁/A_th and the maximum total pressure recovery of a self-starting C–D inlet designed for M_D = 3.2.
• 6.25 A normal-shock inlet is operating in a supercritical mode, with the shock inside the inlet. If the flight Mach number is M₀ = 1.6 and the shock is located at A_s/A_t = 1.2, calculate
  1. Mach number ahead of the shock wave, M_x
  2. percentage total pressure gain if the inlet were to operate in the critical mode
• 6.26 A variable geometry isentropic supersonic inlet is designed for M_D = 1.6. Calculate
  1. percentage flow spillage at M₀ = 0.8
  2. percentage flow spillage at M_D = 1.6 before the inlet is started
  3. percentage throat area increase needed to start this inlet
  4. throat Mach number after the shock is swallowed, M_th
  5. inlet total pressure recovery with the best backpressure (with open throat)
• 6.27 An isentropic, fixed-geometry inlet, is designed for M_D = 1.5. If this inlet is to be started by overspeeding, calculate the necessary Mach number for overspeed.
• 6.28 A fixed-geometry, convergent–divergent, internal-compression inlet is designed for M_D = 2.0 and a self-starting capability. Calculate
  1. A₁/A_th
  2. M_th
  3. inlet total pressure recovery for the “best” backpressure
• 6.29 A normal-shock inlet is flying at a Mach number of 1.8. However, due to a nonoptimum backpressure, the normal shock is inside the duct where A_s/A_i = 1.15. Calculate the percentage loss in the total pressure recovery due to this back pressure.
• 6.30 A variable-geometry supersonic convergent–divergent inlet is to be designed for an isentropic operation (in the started mode) at M_D = 2.6. Calculate
  1. the inlet design contraction ratio A₁/A_th
  2. the percentage opening of the throat (A′_th − A_th)/A_th needed to start the inlet
  3. the throat Mach number in the open position, M′_th
• 6.31 A variable-geometry, internal-compression, C–D inlet is designed for M = 2.0. Calculate percentage opening of the throat required to swallow the starting shock.
• 6.32 A 2D fixed geometry convergent–divergent diffuser is shown. Mach number at the entrance is M₁ = 3.0 and the static pressure is p_t = 10 kPa. A shock occurs at an area ratio A_s/A_th = 1.25 downstream of the throat, as shown. Calculate
  1. M_x
  2. P_y
  3. P_ty
  4. A₂/A₁ if the exit Mach number is 0.5, i.e., M₂ = 0.5

  You may assume the throat is choked and neglect wall friction.

• 6.33 A normal-shock inlet operates in a Mach 1.86 stream with ambient static pressure of p₀ = 30 kPa. Neglecting total pressure loss in the subsonic diffuser, calculate
  1. inlet total pressure recovery with the shock at the lip, i.e., the best backpressure
  2. inlet total pressure recovery when the shock is inside the duct at A_x/A_t = 1.2, i.e., the supercritical mode
  3. inlet total pressure recovery in subcritical mode with 10% spillage, i.e., A_slage/A₁ = 0.1
  4. flight dynamic pressure q₀
• 6.34 Consider an external compression inlet with two ramps operating in a Mach-2.5 stream of air. Calculate the total pressure recovery of the inlet shock system for the case of the best backpressure for the two ramp angles of 8° and 12°, respectively, and compare it to the normal-shock inlet at the same Mach number and with the best backpressure.

  

• 6.35 A supersonic flow is to be decelerated over two oblique shocks and one normal shock, similar to Problem 6.34 (i.e., M₀ = 2.5). The Mach number downstream of the normal shock is 0.7. Assuming the maximum total pressure recovery is obtained when the first two oblique shocks are of equal strength, calculate the necessary ramp angles.
• 6.36 A supersonic nozzle operates in an underexpanded mode with an area ratio A₉/A_th = 2. We know that the exhaust plume turns outward by 15°, as shown. Assuming the flow inside the nozzle is isentropic and γ = 1.4, calculate
  1. exit Mach number M₉ and
  2. the nozzle pressure ratio p_{t, noz}/p₀ (note that p₁₀ = p₀)
  3. the Mach number after the expansion waves M₁₀
  4. the NPR if the nozzle was perfectly expanded
  5. the nozzle pressure ratio if a normal shock appears at the exit

  

• 6.37 A convergent–divergent nozzle discharges to ambient air and has an exit-to-throat area ratio of A_e/A_th = 2.4. The total pressure at the entrance to the nozzle is p_t = 100 kPa. A normal shock occurs at the nozzle exit, as shown. Calculate
  1. the ambient pressure p₀ (note that p₀ = p_y)
  2. the exhaust temperature T_y if the temperature at the throat is known T_th = 350°C
  3. mass flow rate through the nozzle if the throat area is A_th = 0.25 m²

  Assume the medium is air with γ = 1.4, c_p = 1.004 kJ/ kg · K.

  

• 6.38 A supersonic nozzle has an exit-to-throat area ratio of A_e/A_th = 3.5 and an upstream stagnation pressure of p_t = 19 × 10⁵ Pa, which remains constant with altitude. Calculate
  1. the altitude at which the nozzle is perfectly expanded
  2. the altitude at which the exhaust plume is turned 10° outward
  3. the altitude at which the exhaust flow is turned 4° inward

  Assume: γ = 1.4 and c_p = 1, 004 J/kg · K.

  

• 6.39 A convergent–divergent nozzle operates at a pressure altitude of p_amb = 1 kPa. The nozzle total pressure and temperature are p_t = 101 kPa and T_t = 2000 K, respectively. Calculate
  1. nozzle area expansion ratio A₉/A_th, for perfect expansion
  2. nozzle exit Mach number M₉ for perfect expansion
  3. nozzle exit velocity V₉

  Assume isentropic flow of air (with γ = 1.4 and R = 287 J/kg · K).

• 6.40 Consider an overexpanded nozzle as shown. Assuming the oblique shock at the exit makes a 40° angle with respect to the exit flow and the nozzle area ratio is 2.4, i.e., A₉/A_th = 2.4, calculate
  1. M₉
  2. jet turning angle θ
  3. NPR, i.e., p_t/p_amb

  

• 6.41 A convergent–divergent nozzle has an area ratio A₉/A_th = 6.79 and a stagnation pressure of p_t = 38.13 atm. First, calculate the pressure altitude (i.e., p₀) for which the nozzle is perfectly expanded. If this nozzle operated at a higher altitude, i.e., discharge to a lower backpressure atmosphere, it will operate as an underexpanded nozzle with an attendant expansion fan at the exit lip. Now, if the backpressure, i.e., p₀ = 1/2p_{0–perfect–expansion}, calculate
  1. Mach number after the (first) expansion fan, M₁₀
  2. the jet turning angle θ
  3. the tail wave angle of the first expansion fan with respect to the local flow

  

• 6.42 The exit flow from a two-dimensional C–D nozzle is shown to turn inward an angle of 15°.

  Calculate

  1. exit Mach number, M₉
  2. wave angle at the nozzle exit lip
  3. nozzle total pressure p_t (kPa)

  

• 6.43 A convergent nozzle experiences π_cn of 0.98, the gas ratio of specific heats γ = 1.30, and the gas constant is R = 291 J/kg · K. First, calculate the minimum nozzle pressure ratio that will choke the expanding nozzle, i.e., NPR_crit. This nozzle operates, however, at a higher NPR than the critical, namely, NPR = 4. 2 and with an inlet stagnation temperature of T_t7 = 939 K. Assuming this nozzle operates in p₀ = 100 kPa ambient static pressure, calculate
  1. the exit static pressure and temperature p₉ and T₉, respectively
  2. the actual exit velocity V₉ in m/s
  3. nozzle adiabatic efficiency η_n
  4. the ideal exit velocity V_9s in m/s
  5. percentage gross thrust gain, had we used a convergent–divergent nozzle with perfect expansion
  6. nozzle discharge coefficient C_D8
  7. draw a qualitative wave pattern in the exhaust plume

  

• 6.44 A convergent–divergent nozzle has a conical exhaust shape with the half-cone angle of α = 25°. Calculate the divergence loss C_A for this nozzle due to nonaxial exhaust flow. Assuming the same (half) divergence angle of 25°, but in a 2D rectangular nozzle, calculate the flow angularity loss and compare it to the conical case.
• 6.45 The bypass ratio in a turbofan engine is α = 3.5. We intend to mix the cold fan and the hot core flows in a mixer to enhance the engine gross thrust. Assuming the hot core temperature of the gas is T_h = 3 T_c, calculate
  1. the mixed-out temperature of the gas T_m/ T_c
  2. the percentage increase in ideal gross thrust as a result of mixing the cold and hot streams in the mixer

  You may assume a reversible adiabatic mixer flow with γ_h = γ_c = 1.4.

• 6.46 Consider a fixed-geometry supersonic nozzle with the following inlet and geometrical parameters: T_t7 = 2500 K, p_t7 = 233.3 kPa, , A₉/A₈ = 7.45, γ = 1.4, and c_p = 1004 J/kg · K. Assuming isentropic flow in the nozzle, calculate
  1. ambient pressure p₀, if this nozzle is perfectly expanded
  2. ambient pressure if a normal shock appears at the nozzle exit
  3. nozzle throat and exit areas A₈ and A₉, respectively, in m²
  4. the nozzle gross thrust with a NS at the exit (part b)
  5. the nozzle gross thrust in the severely overexpanded case of the N.S. being inside the nozzle at A_s/A₈ = 4.23. Assume the flow remains attached after the shock

  6. in reality the boundary layer in part e separates and reduces the exit flow area by the amount of “blockage”.

     Assuming the effective exit area is A_9e = A_s calculate the gross thrust for this more realistic scenario.

  7. compare the gross thrust with/without separation. Why did separation help?

  

• 6.47 A convergent nozzle discharges into an open duct, as shown (i.e., the ejector concept). Through flow entrainment in the jet, air is drawn in from the upstream open end of the duct. The two flows mix in the duct and produce a mixed-out exit flow at the ambient static pressure p₀. Neglecting wall friction in the duct flow, calculate:
  1. the jet exit area A_j (in cm²) and the ejector duct area A in cm²
  2. the airflow drawn in from the duct inlet, i.e., the secondary flow rate
  3. the exit mixed-out gas velocity V₂
  4. the percentage increase in mass flow rate
  5. the percentage decrease in gas static temperature with the ejector
  6. gross thrust with and without the ejector

  Hint: Apply conservation of mass, momentum, and energy to the control volume between 1 and 2. Flow between 0 and 1 is assumed to be isentropic.

  

• 6.48 The thermodynamic states of gas in an exhaust nozzle are shown on the T–s diagram.

  Calculate

  1. exit Mach number M₉
  2. nozzle adiabatic efficiency η_n

  Assume γ = 1:30.

  

• 6.49 A convergent–divergent nozzle has an area ratio of A₉/A_th = 5.9. Assuming the flow is adiabatic and frictionless with γ = 1.40, calculate
  1. the nozzle pressure ratio p_t/p₀ for perfect expansion
  2. the nozzle pressure ratio p_t/p₀ if there is an oblique shock with β = 30° at the exit lip, as shown
  3. the nozzle pressure ratio p_t/p₀ if there is an expansion wave at the lip that turns the flow 15° outward, as shown.

  

• 6.50 A scramjet flies at Mach 6 with an inlet total pressure recovery of 50%. Assuming the combustor experiences a total pressure loss of 42% (from its inlet condition), calculate the NPR, assuming γ = 1 .30 and is constant.
• 6.51 Calculate the ratio of gross thrust produced by a convergent–divergent nozzle to the gross thrust produced by a convergent nozzle when they both operate with the same nozzle pressure ratio of NPR = 10 and γ = 1 .30.
• 6.52 An external compression inlet operates in supercritical mode, as shown. The normal shock is at A_x/A_th = 1.2.

  Calculate:

  1. Throat Mach number, M_th
  2. Normal shock Mach number, M_x
  3. Total pressure ratio across the shock system

  (Assume losses are only due to shocks)

  

• 6.53 A convergent–divergent exhaust system has an exit-to-throat area ratio, A₉/A₈ = 3.0.

  Assuming the flow in the nozzle is isentropic, calculate:

  1. nozzle exit Mach number, M₉
  2. nozzle exit pressure, p₉ (kPa)
  3. the ratio of momentum thrust to pressure thrust at the nozzle exit, for p₀ = 5 kPa

  

• 6.54 Consider two supersonic inlets, first is a self-starting (K–D) inlet, and the second is a NS inlet. Both inlets are designed for M_D = 1.7. Calculate:
  1. the inlet contraction ratio, A₁/A_th, for the self–starting (K–D) inlet
  2. throat Mach number in the K–D inlet after the inlet started
  3. the total pressure recovery of K–D inlet with best back pressure
  4. % gain in total pressure recovery in K–D versus NS inlet (both at best back pressure)

  

• 6.55 A normal-shock inlet operates in air at Mach 1.5 in subcritical mode, as shown. The Mach Number at the cowl lip is M₁ = 0.62.

  Calculate

  1. inlet spillage mass fraction,
  2. inlet total pressure recovery, π_d

  (Neglect viscous losses in the subsonic diffuser)

  

• 6.56 A subsonic diffuser has an area ratio, A₂/A₁ = 2.02 and operates in air with γ_c = 1.4 and R = 287 J/kg.K. Its inlet condition is measured to be: p_t1 = 100 kPa, T_t1 = 288 K and M₁ = 0.5.

  Assuming the flow in the diffuser is inviscid, calculate

  1. inlet dynamic pressure, q₁, in kPa
  2. exit static pressure, p₂ in kPa
  3. diffuser static pressure recovery, C_PR

  

• 6.57 There is a subsonic inlet in cruise condition (as shown). We know the flight condition is: M₀ = 0.8, p₀ = 25 kPa, T₀ = 245 K.

  The inlet mass flow rate is and the inlet capture ratio is A₀/A₁ = 0.837.

  For γ = 1.4, and R = 287 J/kg.K, calculate

  1. impulse at station 0, I₀, in kN
  2. impulse at the inlet lip, I₁, in kN
  3. additive drag force in kN
  4. ram drag in kN

  

• 6.58 A subsonic diffuser has an inlet Mach number of M₁ = 0.4. The inlet pressure and temperature are p₁ = 100 kPa and T₁ = 15°C, respectively. The diffuser static pressure recovery is C_PR = 0.75 and the exit Mach number is M₂ = 0.1.

  For γ = 1.4, and R = 287 J/kg.K, calculate

  1. inlet dynamic pressure, q₁, in kPa
  2. exit static pressure, p₂, in kPa
  3. exit total pressure, p_t2, in kPa
  4. diffuser area ratio A₂/A₁

  

• 6.59 A subsonic inlet is in M₀ = 0.86 cruise with external diffusion characterized by the inlet mass flow ratio parameter, MFR=0.89, which is the same as the inlet capture area ratio, A₀/A₁. Assume an isentropic flow with γ = 1.4 and c_p = 1004 J/kg.K, to calculate
  1. the Mach number at the inlet lip, M₁
  2. the external static pressure ratio, p₁/p₀, i.e., as the result of external diffusion
  3. the external velocity ratio, V₁/V₀, i.e., as the result of external diffusion
• 6.60 A supersonic inlet creates two oblique shocks and a normal shock, as shown. The normal component of Mach number to these shocks is 1.3, for each of the three shocks.

  For γ = 1.4, calculate

  1. the total pressure recovery of the shock system
  2. Mach number at the entrance to the subsonic diffuser, M₃

  

• 6.61 An isentropic C–D inlet is designed for Mach 4.0. Calculate the percentage opening of the throat area that is needed to start the inlet. Assume γ = 1.4.
• 6.62 A normal shock inlet operates in subcritical mode, as shown. Calculate the percentage spillage for this inlet if M₁ = 0.60. Assume γ = 1.4 (Note: percentage spillage = 100 * (A₁ − A₀)/A₁)

  

• 6.63 A convergent nozzle operates at a nozzle pressure ratio of 3.0. Assuming γ = 1.3 and π_cn = 0.98,

  Calculate

  1. exit Mach number, M₈
  2. ratio of exit to ambient static pressure, p₈/p₀

  

• 6.64 A convergent-divergent nozzle has an area ratio of A₉/A₈ = 4.2. The nozzle pressure ratio is NPR = 20 with γ = 1.3. Assuming that the flow in the CD nozzle is isentropic, calculate
  1. exit Mach number, M₉
  2. ratio of exit static to inlet total pressure, p₉/p_t7
  3. what kind of wave will form at the nozzle exit? Why?
  4. what is the static pressure ratio across the wave?

  

• 6.65 A convergent nozzle is shown. The ambient static pressure is p₀ = 30 kPa. Assuming the flow in the nozzle is isentropic with γ = 1.3 and R ≈ 287 J/kg.K, calculate
  1. nozzle exit Mach number, M₈
  2. exit static pressure, p₈ in kPa
  3. exit velocity, V₈, in m/s

  

• 6.66 A subsonic inlet has an average throat Mach number of and an average total pressure of 75 kPa. The engine face area is 125% of the throat area, i.e., A₂ = 1.25 A_th. For the engine face Mach number of M₂ = 0.5, calculate
  1. engine face total pressure, p_t2, in kPa
  2. static pressure at the throat, p_th, in kPa
  3. dynamic pressure at the throat, q_th, in kPa

  

• 6.67 A convergent nozzle has an inlet total pressure, p_t7 = 150 kPa and inlet total temperature of T_t7 = 660 K with γ₇ = 1.33 and C_p7 = 1, 156 J/kg.K. The ambient pressure is p₀ = 100 kPa. Assuming that the convergent nozzle total pressure ratio is π_cn = 0.98, calculate
  1. nozzle exit Mach number, M₈
  2. nozzle exit velocity, V₈, in m/s
• 6.68 A subsonic inlet is in flight, as shown. Calculate
  1. the Mach number at the inlet lip, M₁
  2. the Mach number at the engine face, M₂

  

• 6.69 A convergent–divergent nozzle has a throat discharge coefficient of C_D8 = 0.98. The velocity coefficient in the divergent section is C_V = 0.96.

  Assuming the ratio of specific heats in the nozzle is γ = 1.30, and nozzle is of conical shape with the semi-vertex angle of α = 22.5°, calculate

  1. gross thrust coefficient, C_fg
  2. percentage increase in gross thrust if the nozzle were bell-shaped, i.e., exhaust velocity was axial

  Note that the nozzle is perfectly expanded.

  

• 6.70 An external-compression supersonic inlet creates oblique and normal shocks in flight, as shown. Calculate the inlet total pressure recovery due to shocks.

  

• 6.71 A convergent–divergent nozzle operates at NPR of 14.86 in an altitude where the ambient pressure is p₀ = 30 kPa. Assuming the flow in the nozzle is isentropic with γ = 1.3, calculate
  1. nozzle area ratio, A₉/A₈, if the nozzle is perfectly expanded 5
  2. exit Mach number, M₉

  

• 6.72 An external-compression supersonic inlet has a single ramp and it operates in critical mode, as shown. The flight speed is V₀ = 750 m/s where the ambient speed of sound is a₀ = 300 m/s and the ambient (static) pressure is 20 kPa. Assuming that the air flow rate captured by the inlet is , calculate
  1. ram drag in kN
  2. inlet total pressure recovery (due to shocks)

  

• 6.73 Consider a subsonic inlet in cruise condition as shown. Assuming the mass flow rate (of air) is 100 kg/s (as shown), calculate:
  1. the ram drag, D_r, in kN
  2. freestream capture area, A₀ in m²
  3. the static pressure at the inlet lip, p₁ in kPa
  4. the static temperature at the inlet lip, T₁ in K
  5. the inlet capture area, A₁ in m²
  6. the additive drag, D_ad, in N

  

• 6.74 An overexpanded supersonic convergent–divergent nozzle has an inlet total pressure of p_t7 = 213 kPa and an area ratio of A₉/A₈ = 3.02. The ratio of specific heats is γ = 1.3 and c_p = 1, 250 J/kg.K. Neglecting losses in the nozzle and assuming ambient static pressure is p₀ = 25.2 kPa, calculate
  1. nozzle exit Mach number M₉
  2. nozzle exit (static) pressure, p₉
  3. the oblique shock wave angle, β in degrees

  

• 6.75 A subsonic pitot inlet is cruising at M₀ = 0.84 at an altitude where p₀ = 30 kPa and T₀ = 240 K. The inlet capture ratio is A₀/A₁ = 0.88. The inlet lip area contraction ratio is A₁/A_th = 1.05. The area ratio between the throat and the engine face is A_th/A₂ = 0.85. Assuming the flow is reversible and adiabatic inside the inlet, and γ = 1.4, calculate
  1. Mach number at the inlet lip, M₁
  2. Mach number at the throat, M_th
  3. Mach number at the engine face, M₂
  4. overall static pressure recovery coefficient, C_PR between the engine face and flight, i.e.,

  

• 6.76 A convergent–divergent nozzle operates with a normal shock at its exit plane, as shown. The nozzle total Pressure at its inlet is p_t7 = 200 kPa and the nozzle area ratio is A₉/A₈ = 3.0. Assuming γ = 1.3, and The flow in the nozzle is isentropic, calculate
  1. the exit Mach number, M₉, before the NS
  2. the ambient pressure, p₀, in kPa

  

• 6.77 A Kantrowitz–Donaldson inlet is designed to self-start at M_D = 1.5. Assume the medium is air with γ = 1.4.

  Calculate the

  1. inlet contraction ratio, A₁/A_th
  2. Mach number at the inlet throat, M_th at the design Mach number when the inlet is started
  3. best inlet total pressure recovery at M_D = 1.5.

  

• 6.78 A subsonic inlet has a throat area, A_th = 1 m², with the average axial Mach number at the throat of M_th = 0.7. The corresponding flight condition is: M₀ = 0.86, p₀ = 30 kPa, T₀ = –50°C, γ = 1.4, R = 287 J/kg.K. Calculate
  1. the captured stream area, A₀, in m²
  2. the mass flow rate, , in kg/s

  You may treat the flow to the inlet throat to be isentropic.

• 6.79 A supersonic inlet uses isentropic pre-compression for a total of 17° turn, as shown.

  Calculate

  1. Mach number upstream of the normal shock, M₁
  2. inlet total pressure recovery, p_t2/p_t0
  3. static pressure on the cowl lip after the cowl shock, p₃ in kPa (assume lip is very thin)

  

• 6.80 Calculate nondimensional additive drag, D_add/p₀A₁, for a subsonic inlet at cruise condition, M₀ = 0.8, A₀/A₁ = 0.835.
• 6.81 A subsonic inlet has a contraction area ratio of 1.112, i.e., A_HL/A_th = 1.112. The (mass) average throat Mach number is 0.74. Neglecting the frictional losses (on total pressure) between the highlight and the throat, calculate the Mach number at the highlight.
• 6.82 Calculate and compare the nondimensional additive drag, D_add/p₀A₁, on a subsonic inlet with a cruise Mach number of M₀ = 0.86 that is designed with either of the two inlet capture ratios:
  • Case I: A₀/A₁ = 0.82
  • Case II: A₀/A₁ = 0.76

  

• 6.83 A 2D external-compression inlet is shown in supersonic flight. The external cowl is composed of three flat panels, as shown. The Cartesian coordinates of the panels are:
  • x₁ = 0.0 m, y₁ = 0.00 m
  • x₂ = 1.0 m, y₂ = 0.25 m
  • x₃ = 2.0 m, y₃ = 0.35 m
  • x₄ = 3.0 m, y₄ = 0.35 m

  Assuming the panels’ static pressure is:

  • Panel #1: p ≈ 50 kPa
  • Panel #2: p ≈ 30 kPa
  • Panel #3: p ≈ 25 kPa

  Calculate

  1. cowl wave drag (per unit width of the inlet) in kN/m
  2. can you estimate M₀? (Hint: static pressure doubled on panel #1 from free stream, and the flow turning angle to the first panel is also known.)

  

• 6.84 A three-ramp external compression inlet is shown. The three oblique shocks and the terminal, i.e., the normal shock, are all of equal strength. Assuming that normal component of Mach number to each shock is 1.3, calculate
  1. the total pressure recovery of the shock system
  2. the inlet total pressure recovery
  3. estimate the flight Mach number, M₀, if this inlet represents “the optimum” four-shock external compression inlet.

  

• 6.85 It is desirable to design a two-ramp external compression inlet for a design-point Mach number of M₀ = 2.125 that creates three shocks of equal strength. We have calculated the normal components of Mach number in this design to be 1.3, as shown. Validate that the two successive ramp angles are 10.8° and 11.8° respectively. Also, calculate the (shock) total pressure recovery for this inlet at the design point.

  [Note that the figure is not-to-scale]