Multi-Objective Optimization Design of Adaptive Cycle Engine with Serpentine 2-D Exhaust System Based on Infrared Stealth

Abstract

In the overall design process of the turbofan engine, it has become crucial to address the challenge of selecting design parameters that not only meet the flight thrust demand but also enhance engine economy. As the demand for stealth performance in future fighter aircraft increases, it becomes imperative to consider infrared stealth indicators during the design process. The adaptive cycle engine possesses an adjustable thermal cycle, necessitating careful attention to the selection of design parameters to fulfill the requirements. Therefore, this paper proposes a multi-objective optimization design method for the adaptive cycle engine that integrates infrared stealth technology. Initially, the parameter cycle model of the adaptive cycle engine is established based on the principles of aerodynamic and thermodynamic calculations. Subsequently, the model incorporates a serpentine two-dimensional (2-D) exhaust system to achieve infrared suppression. Meanwhile, a method for predicting the infrared characteristics is proposed to calculate the infrared radiation intensity of the engine exhaust system. Finally, the sequential quadratic programming algorithm is applied to comprehensively optimize the engine’s performance. The simulation results reveal that the multi-objective optimization design can effectively select appropriate design parameters to im-prove the engine, thereby reducing fuel consumption while meeting thrust requirements. This approach combines the consideration of infrared stealth technology with the optimization of engine performance, thus contributing to the development of advanced adaptive cycle engines.

1. Introduction

The variable cycle engine has garnered considerable attention in the aviation field due to its ability to modify engine cycle parameters, such as bypass ratio. It has emerged as a prominent research focus worldwide, with numerous countries actively exploring this area. In 2007, the United States engaged GE Company and Rollo Company to conduct a study on the adaptive cycle engine (ACE), resulting in the development of adaptive engine technology. This innovative technology incorporates optimized structures and materials, enabling a remarkable 25% reduction in engine fuel consumption and a 30% increase in aircraft flight range. Furthermore, the adaptive cycle engine delivers higher thrust compared to the military fighter F135, surpassing it by 5–10% [1,2,3]. A key feature of this engine is the inclusion of FLADE (fan on blade), which generates an additional airflow to facilitate greater power extraction and effective thermal management. The third stream would decrease the installed drag, improve the total pressure recovery at the inlet, and reduce the exhaust temperature, which is of great significance for infrared stealth. When the third stream passes through the core of the jet, it takes part of the heat away and dissipates aircraft heat loads, cools turbine cooling air, and provides the third stream of constant pressure ratio for thrust augmentation. Additionally, the adaptive cycle engine incorporates a central cone and inner nozzle that effectively conceal the turbine, greatly enhancing backward radar and infrared stealth capabilities of the engine [4,5,6].

The advancement of adaptive cycle engines has led to widespread interest in their key technologies, such as high comprehensive performance, low signal characteristics, and efficient thermal management, within the aviation industry [7,8]. Notably, the demand for super stealth in the United States’ sixth-generation fighters has intensified the focus on ACE. As aero-engines increasingly require infrared stealth performance, the sixth-generation fighters’ stealth requirements surpass those of their fifth-generation counterparts. To achieve infrared suppression, fighter engines employ various measures, including the use of stealth coating materials, reducing wall temperature through the cool airflow of the third stream, and implementing variable-area bypass injectors and nozzles to suppress infrared radiation [9,10,11,12,13]. Since 2007, the American Air Force Research Laboratory has initiated the ADVENT (adaptive versa engine technology) project. The adaptive cycle engine a three-stream configuration, with its exhaust system featuring a serpentine 2-D nozzle. The cooling airflow from the third stream is mixed with the jet after the throat section, effectively reducing the infrared radiation characteristics of the exhaust system. Consequently, the design of adaptive cycle engines necessitates the consideration of not only thrust performance and fuel consumption but also the infrared radiation intensity emitted by the engine exhaust system [14,15,16,17,18,19]. The traditional overall design method for engines no longer aligns with these requirements, highlighting the urgency to develop a comprehensive design approach for adaptive cycle engines with low infrared characteristics. It is important to note that solely applying infrared stealth material technology to the engineering process will have no impact on the overall engine performance. However, the shielding and cooling technologies employed can significantly compromise the engine’s overall performance. Thus, it becomes essential to incorporate infrared suppression measures into the overall design to ensure optimal engine performance [20,21,22,23,24].

To address this objective, the research on infrared stealth technology for adaptive cycle engines must extend beyond the exhaust system level and encompass the entire design process. Both aircraft and engine performance requirements pose constraints on the implementation of infrared suppression measures [25,26,27]. Hence, it is crucial to consider the impact of these measures on the overall performance of adaptive cycle engines during the design stage, including infrared stealth performance, thrust and fuel consumption rate. If the stealth performance is neglected in the overall design stage of the engine, it will disrupt the engine’s original operating state and diminish the engine performance when implementing the infrared stealth measures [28]. In summary, it is of great significance to study the multi-objective optimization design method for adaptive cycle engine, which includes infrared stealth performance, thrust and fuel consumption rate [29,30,31,32,33,34,35,36].

In this paper, the parameter cycle model of the adaptive cycle engine is established using the modeling method in reference [37]. To suppress the infrared radiation intensity of the engine, a dual-S two-dimensional nozzle is employed [38]. On this basis, the overall performance of the engine during the design stage is calculated, and the multi-objective optimization research is carried out through the optimization algorithm. Compared with the existing literature, this paper places particular emphasis on the parameter cycle analysis of the adaptive cycle engine with a serpentine 2-D exhaust system, with a specific focus on optimizing the overall performance during the engine design stage.

2. Parameter Cycle Analysis of Adaptive Cycle Engine

During the intricate process of aircraft engine design, the selection of engine cycle parameters holds undeniable significance, permeating throughout the entire process. Parametric adjustment constitutes a vital aspect of this process. The objective of parametric cycle analysis is to establish the relationship between engine performance metrics (thrust, and fuel consumption) and cycle design parameters such as combustion temperature, fan pressure ratio, compressor pressure ratio and bypass ratio. During the parameter cycle analysis stage, the size of the engine remains undetermined, which results in the assumption of unit flow rate at the inlet. The unit thrust and fuel consumption rate are selected as the evaluation basis to judge whether the design parameters meet the performance requirements.

In this paper, an adaptive cycle engine is established as shown in Figure 1. It becomes evident that the airflow passing through the FLADE ultimately mixes with the mainstream flow and is expelled through the tail nozzle, which constitutes a mixed exhaust configuration. The relevant section number is given in the figure, where 0 is the freestream, 2 is the fan entry, 12 is the FLADE entry, 13 is the FLADE exit, 21 is the fan exit, 24 is the CDFS (core-driven fan stage) exit, 25 is the high-pressure compressor entry, 3 is the high-pressure compressor exit, 225 is the secondary bypass exit, 125 is the CDFS bypass exit, 15 is the front mixer entry, 4 is the burner exit, 41 is the high-pressure turbine entry, 45 is the low-pressure turbine entry, 5 is the low-pressure turbine exit, 6 is the core stream mixer entry, 16 is the front mixer exit, 64 is the mixer exit, 8 is the exhaust nozzle throat and 9 is the exhaust nozzle exit.

Utilizing the principles of aerothermodynamics, a variable specific heat model for the adaptive cycle engine (ACE) is developed. This model employs NASA Glenn thermochemical data and the Gordon McBride equilibrium algorithm [37] to simulate the air and combustion gas as ideal gases at the inlet and outlet of each engine component.

To establish the parameter cycle model for the ACE with variable specific heat, it is necessary to determine the engine design variable parameters. The most significant engine mass flow ratios are shown as follows.

$$\alpha =\frac{Flade inlet airflow}{Fan inlet airflow}=\frac{m_{13}}{m_{21}}$$

(1)

$${\alpha }_1=\frac{secondary bypass airflow}{CDFS inlet airflow}=\frac{m_{225}}{m_{24}}$$

(2)

$${\alpha }_2=\frac{bypass total airflow}{high compressor inlet airflow}=\frac{m_{15}}{m_{25}}=\frac{m_{15}}{m_c}$$

(3)

In addition to the above selected bypass ratio, the modeling process incorporates additional input parameters. These parameters encompass the pressure ratios and polytropic efficiencies of each compression component, the temperature of the combustion chamber, the Mach numbers at the CDFS bypass exit and low-pressure turbine exit, and more. The calculation process for some components of ACE (inlet, fan, compressor, combustion chamber, high-pressure turbine, low-pressure turbine, main mixing chamber, and nozzle) are similar to those of the turbofan engine, which can be found in reference [37]. This paper conducts formula derivation for special components such as the CDFS and front mixing chamber, and the same calculation process as the turbofan engine will not be repeated.

Since the high-pressure rotor shaft is connected with the CDFS, high-pressure compressor and high-pressure turbine, the power balance equation for the high-pressure rotor can be expressed as follows.

$$m_{41}(h_{t41}-h_{t44}){\eta }_{mH}=m_{24}(h_{t24}-h_{t21})+m_{25}(h_{t3}-h_{t25})+P_{TOH}/{\eta }_{mPH}$$

(4)

where η_mH represents the adiabatic efficiency of the high-pressure spool, P_TOH represents the high turbine shaft power takeoff, η_mPH represents the adiabatic efficiency of the power takeoff shaft from the high-pressure spool.

According to the above equation and the equations obtained from reference [37] in Chapter 4, the following holds.

$$m_{41}=m_c((1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+{\epsilon }_1);h_{t41}=h_0{\tau }_{\lambda }{\tau }_{m1};h_{t44}=h_0{\tau }_{\lambda }{\tau }_{m1}{\tau }_{tH};$$

(5)

$$m_{24}=m_c\frac{(1+{\alpha }_2)}{(1+{\alpha }_1)};h_{t24}=h_0{\tau }_r{\tau }_f{\tau }_{CDFS};h_{t21}=h_0{\tau }_r{\tau }_f;$$

(6)

$$m_{25}=m_c;h_{t3}=h_0{\tau }_r{\tau }_f{\tau }_{CDFS}{\tau }_{cH};h_{t25}=h_0{\tau }_r{\tau }_f{\tau }_{CDFS};$$

(7)

$${\tau }_{m1}=\frac{(1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+{\epsilon }_1{\tau }_r{\tau }_f{\tau }_{CDFS}{\tau }_{cH}/{\tau }_{\lambda }}{(1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+{\epsilon }_1}$$

(8)

The total enthalpy ratio of the high-pressure turbine can be determined by substituting the above equations into Equation (4).

$${\tau }_{tH}=1-\frac{{\tau }_r{\tau }_f{\tau }_{CDFS}({\tau }_{cH}-1)+\frac{(1+{\alpha }_2)}{(1+{\alpha }_1)}{\tau }_r{\tau }_f({\tau }_{CDFS}-1)+\frac{(1+{\alpha }_2)}{{\eta }_{mPH}}{C}_{TOH}}{{\eta }_{mH}{\tau }_{\lambda }\{(1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+{\epsilon }_1{\tau }_r{\tau }_f{\tau }_{CDFS}{\tau }_{cH}/{\tau }_{\lambda }\}}$$

(9)

where τ_r represents the adiabatic freestream recovery enthalpy ratio, τ_f represents the total enthalpy ratio of the fan, τ_CDFS represents the total enthalpy ratio of the CDFS, τ_cH represents the total enthalpy ratio of the high-pressure compressor, τ_λ represents the enthalpy ratio of the burner, τ_m1 represents the enthalpy ratio of the cooling air to the high-pressure turbine, C_TOH = P_TOH/m₀h₀, β represents the bleed air fraction, ε₁ is the cooling air of the high-pressure turbine, ε₂ is the cooling air of the low-pressure turbine, and f represents the fuel-to-air mass flow ratio.

Similarly, the total enthalpy ratio of the low-pressure turbine can be derived as follows.

$$m_{45}(h_{t45}-h_{t5}){\eta }_{mL}=m_{13}(h_{t13}-h_{t12})+m_{21}(h_{t21}-h_{t2})+P_{TOL}/{\eta }_{mPL}$$

(10)

$$m_{45}=m_c((1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+{\epsilon }_1+{\epsilon }_2);h_{t45}=h_0{\tau }_{\lambda }{\tau }_{m1}{\tau }_{tH}{\tau }_{m2};h_{t5}=h_0{\tau }_{\lambda }{\tau }_{m1}{\tau }_{tH}{\tau }_{m2}{\tau }_{tL};$$

(11)

$$m_{13}=m_c\frac{\alpha (1+{\alpha }_2)}{(1+\alpha )};h_{t13}=h_0{\tau }_r{\tau }_{flade};h_{t12}=h_0{\tau }_r;$$

(12)

$$m_{21}=m_c\frac{(1+{\alpha }_2)}{(1+\alpha )};h_{t21}=h_0{\tau }_r{\tau }_f;h_{t2}=h_0{\tau }_r;$$

(13)

$${\tau }_{m2}=\frac{(1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+{\epsilon }_1+{\epsilon }_2{\tau }_r{\tau }_f{\tau }_{CDFS}{\tau }_{cH}/({\tau }_{\lambda }{\tau }_{m1}{\tau }_{tH})}{(1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+{\epsilon }_1+{\epsilon }_2}$$

(14)

$${\tau }_{tL}=1-\frac{\frac{1+{\alpha }_2}{1+{\alpha }_1}{\tau }_r\left\{({\tau }_{cL}-1)+{\alpha }_1({\tau }_f-1)\right\}+\alpha \left(1+{\alpha }_2\right){\tau }_r({\tau }_{flade}-1)+(1+\alpha )(1+{\alpha }_2)\frac{C_{TOL}}{{\eta }_{mPL}}}{{\eta }_{mL}{\tau }_{\lambda }{\tau }_{tH}\{(1-\beta -{\epsilon }_1-{\epsilon }_2)(1+f)+({\epsilon }_1+\frac{{\epsilon }_2}{{\tau }_{tH}})\frac{{\tau }_r{\tau }_{cL}{\tau }_{CDFS}{\tau }_{cH}}{{\tau }_{\lambda }}\}}$$

(15)

where τ_FLADE represents the total enthalpy ratio of FLADE, η_mL represents the adiabatic efficiency of the high-pressure spool, η_mPL represents the adiabatic efficiency of the power takeoff shaft from the low-pressure spool, C_TOL = P_TOL/m₀h₀, P_TOL represents the low turbine shaft power takeoff, and τ_m2 represents the enthalpy ratio of the cooling air to the low-pressure turbine.

Furthermore, to obtain the parameters resulting from the mixture in the front bypass mixing chamber, the concept of the bypass ratio α′_f is introduced and defined as follows.

$${{\alpha }^{\prime }}_f=\frac{m_{225}}{m_{125}}=\frac{(1+{\alpha }_2){\alpha }_1}{{\alpha }_2-{\alpha }_1}$$

(16)

Thus, the enthalpy of mixing two air streams in the front bypass mixing chamber can be calculated.

$${{\alpha }^{\prime }}_f=\frac{m_{225}}{m_{125}}=\frac{(1+{\alpha }_2){\alpha }_1}{{\alpha }_2-{\alpha }_1}$$

(17)

$$h_{t15}=\frac{h_{t125}+{{\alpha }^{\prime }}_f{h}_{t225}}{1+{{\alpha }^{\prime }}_f}$$

(18)

The modeling of front bypass mixing chamber can refer to the modeling principles of the turbofan engine’s mixer in reference [37]. By employing this design approach, it becomes possible to calculate the relevant parameters for the 125 section and 15 section of the mixing chamber.

Based on the structure of the engine, it is evident that the flow within the FLADE duct would eventually mix with the mainstream flow and exit through the nozzle. For this reason, the ratio of FLADE duct flow to the main flow is defined as α′. The relevant section parameters of the nozzle are calculated through this definition.

$${\alpha }^{\prime }=\frac{m_{13}}{m_8}=\frac{\alpha (1+{\alpha }_2)}{f(1-\beta -{\epsilon }_1-{\epsilon }_2)+1+{\alpha }_2-\beta }$$

(19)

According to the above cycle analysis method, the overall performance parameters of ACE could be obtained, which include total fuel gas ratio f_o, unit thrust F/m_0, and fuel consumption rate sfc.

$$f_o=\frac{f\left(1-\beta -{\epsilon }_1-{\epsilon }_2\right)}{(1+\alpha )(1+{\alpha }_2)}$$

(20)

$$\frac{F}{m_0}=\frac{a_0}{g_c}\left\{\left[(1+f_o)-\frac{\beta -\alpha \left((1+{\alpha }_2)\right)}{(1+\alpha )(1+{\alpha }_2)}\right]\left(\frac{V_9}{a_0}+\frac{R_9{T}_9{a}_0}{R_0{T}_0{V}_9}\frac{1-\frac{P_0}{P_9}}{{\gamma }_0}\right)-M_0\right\}$$

(21)

$$sfc=\frac{f_o}{F/m_0}$$

(22)

where g_c is Newton’s gravitational constant; V₉, T₉, R₉, P₉ are the velocity, static temperature, gas constant and static pressure at the exit of nozzle; and M₀, γ₀, T₀, R₀, a₀ are the Mach number, specific heat ratio, gas constant and sound velocity of free flow.

Based on the derived equations and analysis, the design model of ACE is established, and the engine design parameters are shown in

Table 1. Adaptive cycle engine design parameters.

Inputs	Value	Outputs	Value
H(km)	0	Tt4 (K)	1950
M0	0	Tt7 (K)	2050
T0 (K)	288.15	M6	0.65
P0 (Pa)	101,325.2	P0/P9	1
β	0.01	m0 (kg/s)	50
CTOL	0	ηflade	0.8514
CTOH	0	ηf	0.8399
hPR (MJ/kg)	4,312,400	ηCDFS	0.8500
ε1	0.05	ηcH	0.8601
ε2	0.03	ηtH	0.9005
eFLADE	0.8651	πtH	0.3845
ef	0.8651	ηtL	0.9113
eCDFS	0.8554	πtL	0.4847
ecH	0.8887	M225	0.3029
etH	0.8906	M125	0.70
etL	0.9044	M15	0.3977
ηb	0.995	M16	0.2239
ηAB	0.95	M6A	0.4378
ηmH	0.995	M9	1.5822
ηmPH	1.00	Pt9/P9	3.9666
ηmL	0.995	V9/a0	3.2838
ηmPL	1.00	f	0.036125
πflade	3.8	fAB	0.03854
πf	3.8	fo	0.04803
πCDFS	1.3	ηP	0
πcH	6.0	ηTH	0.31354
α	0.2	F (N)	67,618.5
α1	0.3	F/m0 (N*s/kg)	1352.373
α2	0.7	sfc (kg/(kN*h))	1.2542

. These parameters include flight parameters (H, M₀, T₀, and P₀), aircraft system parameters (β, C_TOL, and C_TOH), fuel heating value (h_PR), component figures of merit (e_FLADE, e_f, e_CDFS, e_cH, e_tH, e_tL, and η_b), and design choices (π_FLADE, π_f, π_CDFS, π_cH, α, α₁, α₂, T_t4, M₆, M₁₂₅, and P₀/P₉), where C_TOL represents the power takeoff shaft power coefficient for the low-pressure spool, C_TOH represents the power takeoff shaft power coefficient for the high-pressure spool, ε₁ represents the cooling air for the high turbine, ε₂ represents the cooling air for the low turbine, η_P represents engine propulsive efficiency, and η_TH represents engine thermal efficiency. The relevant values in Table 1 can be obtained from ref. [37].

Table 2. The design parameters of ACE key sections.

Parameters	Value	Parameters	Value
Pt0 (Pa)	101,325	Pt225 (Pa)	381,185.8
Tt0 (K)	288.15	Tt225 (K)	440.04
m0 (kg/s)	50	m225 (kg/s)	9.61
Pt13 (Pa)	381,185.8	Pt15 (Pa)	418,282.5
Tt13 (K)	447.04	Tt15 (K)	460.82
m13 (kg/s)	8.33	m15 (kg/s)	17.15
Pt21 (Pa)	381,185.8	Pt45 (Pa)	1,024,343.3
Tt21 (K)	447.04	Tt45 (K)	1499.7
m21 (kg/s)	41.67	m45 (kg/s)	24.28
Pt24 (Pa)	495,541	Pt6 (Pa)	417,771.9
Tt24 (K)	487.2	Tt6 (K)	1247.53
m24 (kg/s)	32.05	m6 (kg/s)	25.01
Pt3 (Pa)	2,973,246.1	Pt64 (Pa)	380,006.5
Tt3 (K)	839.9	Tt64 (K)	950.98
m3 (kg/s)	24.51	m64 (kg/s)	42.17
Pt125 (Pa)	495,541	Pt9 (Pa)	359,977.4
Tt125 (K)	487.2	Tt9 (K)	873.35
m125 (kg/s)	7.54	m9 (kg/s)	50.5

provides the total temperature, total pressure, and massflow values for each key section of ACE.

To further study the influence of design parameter variations on engine performance during the parametric cycle analysis of ACE, the relationships of different bypass ratio α₂ and compressor pressure ratio π_cH with unit thrust F/m₀ and fuel consumption rate sfc are depicted in Figure 2. The design point operates at the ground point (H = 0, M = 0), and other design parameters remain constant during the adjustment of the aforementioned two parameters. The figure demonstrates that, in the design state of the adaptive cycle engine, an increase in the bypass ratio α₂ results in a decrease in unit thrust. On the other hand, with the increase in the high-pressure compressor pressure ratio, the fuel consumption rate will decrease, with the overall trend indicating a downward and leftward shift.

3. Infrared Characteristics Prediction of ACE with Serpentine 2-D Exhaust System

Nowadays, the aeroengine not only pursues maneuverability and economy, but also considers stealth performance. Therefore, based on the above design model, the infrared stealth performance of the adaptive cycle engine is studied in this paper. A simplified method for predicting the backward infrared radiation intensity of the exhaust system is presented. Compared with the company’s infrared computing software, the method can greatly shorten the infrared calculation time and improve the real-time performance. Firstly, the simplified diagram of the ACE’s axisymmetric exhaust system is given, which is shown in Figure 3. The station numbers represent the inlet of the third bypass steam (station 13), the exit of the second bypass stream (station 16), the exit of the turbine (station 5), the exit of the main stream (station 6), the exit of the mixer (station 6A), the exhaust nozzle throat (station 8) and the exhaust nozzle exit (station 9). The mathematical model of the engine takes into account the static pressure balance of the mixing section when the third stream meets the mainstream. Furthermore, the adjustment of the rear adjustable ejector depicted in the diagram has an impact on infrared detection. The red line in the figure represents the high-temperature wall and gas detected by the infrared radiation detector.

When the infrared detector is in the backward direction of the exhaust system and the gas coefficient on the jet center line is considered, the calculation equation for the infrared intensity of the exhaust system is as follows.

$$\begin{array}{l}I=\frac{A_c{\epsilon }_c}{\pi }{\displaystyle \underset{\lambda }{\int }{M}_{\lambda }(T_c)}d\lambda +\frac{A_{5e}{\epsilon }_{5e}}{\pi }{\displaystyle \underset{\lambda }{\int }{M}_{\lambda }(T_5)}d\lambda +\frac{A_{16e}{\epsilon }_{16e}}{\pi }{\displaystyle \underset{\lambda }{\int }{M}_{\lambda }(T_{16})}d\lambda +\\ \hspace{1em} \frac{A_{rvabi}{\epsilon }_{rvabi}}{\pi }{\displaystyle \underset{\lambda }{\int }{M}_{\lambda }(T_{rvabi})}d\lambda +\frac{A_n{\epsilon }_n}{\pi }{\displaystyle \underset{\lambda }{\int }{M}_{\lambda }(T_n)}d\lambda +I_{gas}\end{array}$$

(23)

where A_c, ε_c and T_c are, respectively, the projection area, material emissivity and surface temperature of the central cone. A_5e, ε _5e and T₅ are, respectively, the projection area, material emissivity and surface temperature of Section 5. A_n, ε_n and T_n are, respectively, the projection area, material emissivity and surface temperature of the inner wall of the nozzle expansion section. A_16e, ε_16e and T₁₆ are, respectively, the projection area, material emissivity and surface temperature of Section 16. A_rvabi, ε_rvabi and T_rvabi are, respectively, the projection area, material emissivity and surface temperature of section RVABI.

The relationship between spectral transmission M_λ(T) and temperature T at wavelength λ is determined by Planck’s law, as shown in the following equation:

$$M_{\lambda bb}\left(T\right)=\frac{c_1}{{\lambda }^5}\cdot \frac{1}{e^{c_2/\lambda T}-1}$$

(24)

$$\begin{array}{l}{c}_1=\left(3.7415\pm 0.0003\right)\times {10}^{-16}\left(W\cdot \mu m^4/m^2\right)\\ c_2=\left(1.43879\pm 0.00019\right)\times {10}^{-2}\left(\mu m\cdot K\right)\end{array}$$

(25)

To further enhance the infrared stealth performance of the adaptive cycle engine, plenty of infrared suppression measures are incorporated during the engine’s parameter cycle analysis stage. In this study, a serpentine 2-D exhaust system based on the existing axisymmetric nozzle is established in the adaptive cycle engine model, which is shown in Figure 4. Its geometric parameters mainly include the following aspects: inlet diameter D, nozzle outlet width W and height H, nozzle length L, offset S and the type of centerline. The more detailed design of the exhaust system can be found in reference [38]. Once equipped with the serpentine 2-D nozzle, the throat area and exit area of the nozzle are fixed and cannot be adjusted. Therefore, during the flight mission, other engine variables need to be adjusted to maintain stable engine operation.

Drawing from the existing literature and data, the empirical relationship between the infrared radiation characteristics of the serpentine 2-D exhaust system and axisymmetric exhaust system is established. Additionally, a thorough analysis of the factors influencing the infrared radiation characteristics of the exhaust system is conducted. By combining these findings, a calculation formula is derived to determine the infrared radiation intensity and its relationship between the serpentine 2-D exhaust system and the axisymmetric exhaust system.

$$I_{2S}=I_0\cdot C_{2S_0}\cdot {\chi }_{AR}\cdot {\chi }_{S/D}\cdot {\chi }_{type}$$

(26)

where I_2S is the infrared radiation intensity of the serpentine 2-D exhaust system, and I₀ is the infrared radiation intensity of the axisymmetric exhaust system with the same size as the serpentine 2-D exhaust system. $C_{2S_0}$ represents the proportional coefficient of infrared radiation intensity between the serpentine 2-D exhaust system and the axisymmetric exhaust system, which is the same size as the standard serpentine 2-D exhaust system. χ_AR, χ_S/D, and χ_type, respectively, represent the ratio coefficients of the serpentine 2-D exhaust system’s infrared radiation intensity with respect to the aspect ratio (AR = W/H), the maximum deviation radius ratio (S/D) and the type of centerline (type).

According to the data of reference [38], the empirical formula between χ_AR, χ_S/D and the aspect ratio AR, the maximum deviation ratio S/D is obtained using the least-squares method.

$${\chi }_{AR}=1.09955-0.02737AR$$

(27)

$${\chi }_{S/D}=14.2966-54.7009(S/D)+53.6484{(S/D)}^2$$

(28)

By employing the center line change rate construction method utilized in the design of the S-shaped exhaust system, the monotonic and continuous smooth curves are constructed. These curves can be categorized into five distinct types based on their slope degree: I, II, III, IV and V. Figure 5 illustrates these polynomial function curves, and each has its own defined domain and value range of [0, 1].

Table 3. Adaptive cycle engine design parameters.

Central Line Type	Scale Factor χtype
I	1.682736648
II	1.280288421
III	1
IV	0.42684665
V	0.268298818

presents the calculated results highlighting the impact of different center lines on the infrared radiation characteristics of the serpentine 2-D exhaust system.

Through the analysis of reference [38], it could be preliminarily known that the main influencing factors of the total pressure ratio are the aspect ratio (AR), the maximum deviation radius ratio (S/D) and length–diameter ratio (L/D). Assuming that all factors are independent, the total pressure ratio of the serpentine 2-D exhaust system is calculated as follows.

$${\pi }_{n,2S}={\pi }_{0,2S}\cdot {\varsigma }_{AR}\cdot {\varsigma }_{S/D}\cdot {\varsigma }_{L/D}$$

(29)

where π_n, _2S is total pressure ratio of the serpentine 2-D exhaust system, and π₀, _2S is the total pressure ratio of the axisymmetric exhaust system with the same size as the serpentine 2-D exhaust system. ς_AR, ς_S/D, ς_L/D, respectively represent the ratio coefficients of the serpentine 2-D exhaust system’s total pressure ratio with respect to the aspect ratio (AR), the maximum deviation radius ratio (S/D) and the length diameter ratio (L/D).

Similarly, the empirical formula of the total pressure ratio is given as follows.

$${\varsigma }_{AR}=0.9993-3.34\times {10}^{-3}AR+1.514\times {10}^{-3}A{R}^2-1.587\times {10}^{-4}A{R}^3$$

(30)

$${\varsigma }_{S/D}=0.9954+9.35\times {10}^{-2}(S/D)-0.205{(S/D)}^2$$

(31)

$${\varsigma }_{L/D}=0.9068+8.245\times {10}^{-2}(L/D)-2.166\times {10}^{-2}{(L/D)}^2+1.844\times {10}^{-3}{(L/D)}^3$$

(32)

To further characterize the effect of equipping the serpentine 2-D exhaust system on engine performance, Figure 6 shows the infrared radiation characteristics and total pressure ratio calculation process.

Figure 7 illustrates the effects of different aspect ratio, maximum offset diameter ratios and length diameter ratios on the performance of the adaptive cycle engine in terms of thrust, fuel consumption and infrared radiation intensity. As could be seen from the figures, with increasing aspect ratio (AR), infrared radiation intensity decreases, while the thrust initially increases and then decreases. Conversely, the fuel consumption rate and thrust exhibit opposite trends. Meanwhile, the thrust and fuel consumption reach their maximum and minimum, respectively, when the aspect ratio is close to 5. In the analysis of the length–diameter ratio (L/D) on engine performance, it is observed that as the length–diameter ratio increases, both engine thrust and infrared radiation intensity initially increase and then decrease, whereas the fuel consumption rate shows the opposite trend. When the length diameter ratio approaches 3.2, the engine achieves maximum thrust and infrared radiation intensity, while the fuel consumption rate reaches its minimum. Additionally, in the analysis of maximum deviation radius ratio (S/D), the engine thrust decreases with the increase in S/D, while engine fuel consumption and infrared radiation intensity increase.

Based on the analysis conducted, it is evident that the installation of the serpentine 2-D exhaust system in the ACE leads to a reduction in thrust compared to the original ACE. This decrease in thrust makes it challenging for the engine to meet the requirements of flight missions. Therefore, the impact of the above parameters on the engine performance ought to be comprehensively considered in the design to meet the engine performance requirements.

4. Multi Objective Optimization Design of Adaptive Cycle Engine

Based on the above-mentioned infrared prediction model of ACE and its exhaust system, a multi-objective optimization is carried out during the overall design stage. In the optimization process, the engine thrust F, fuel consumption sfc and infrared radiation intensity IR should be taken into account, so as to meet the engine thrust demand, reduce the engine fuel consumption and minimize infrared radiation intensity. Therefore, the Sequential Quadratic Programming (SQP) algorithm is used to realize the comprehensive optimization design of ACE. The flow chart of the optimization design is shown in Figure 8. The ACE parameters of variables that can be selected in the optimization design are as follows, where u = [π_FLADE, π_f, π_CDFS, π_cH, α, α₁, α₂, T_t4]. These variables play a crucial role in the optimization process.

The multi-objective optimization problem can be converted into a single-objective optimization problem through the normalization of the objective function and the application of linear weights.

$$\min J=-{\omega }_1\frac{F}{F_i}+{\omega }_2\frac{sfc}{sf{c}_i}+{\omega }_3\frac{IR}{I{R}_i}$$

(33)

In the above equation, sfc_i is the initial value of the fuel consumption rate at the design point, IR_i is the initial value of infrared radiation intensity at the design point and F_i is the initial value of thrust at the design point. ω₁, ω₂ and ω₃ are the weights of the objective function, which represent the importance of the objective functions. The choice of weight coefficients mainly depends on the requirements of the flight mission. In cases where the majority of missions involve high-maneuverability flights, a greater emphasis is placed on achieving high thrust in engine design, resulting in a more pronounced weighting towards ω₁. Conversely, when the majority of flight missions prioritize cruise and stealth operations, engine designs prioritize economy, leading to a more substantial weighting towards ω₂ and ω₃. To ensure the feasibility of solutions in the optimization process, the adjusting variables must satisfy the corresponding constraint u_IR,min ≤ u_IR ≤ u_IR,max.

$$\left\{\begin{array}{c}{({\pi }_{FLADE})}_{\min }\le {\pi }_{FLADE}\le {({\pi }_{FLADE})}_{\max }\\ {({\pi }_f)}_{\min }\le {\pi }_f\le {({\pi }_f)}_{\max }\\ {({\pi }_{CDFS})}_{\min }\le {\pi }_{CDFS}\le {({\pi }_{CDFS})}_{\max }\\ {({\pi }_{cH})}_{\min }\le {\pi }_{cH}\le {({\pi }_{cH})}_{\max }\\ {(\alpha )}_{\min }\le \alpha \le {(\alpha )}_{\max }\\ {({\alpha }_1)}_{\min }\le {\alpha }_1\le {({\alpha }_1)}_{\max }\\ {({\alpha }_2)}_{\min }\le {\alpha }_2\le {({\alpha }_2)}_{\max }\\ {(T_{t4})}_{\min }\le T_{t4}\le {(T_{t4})}_{\max }\end{array}\right.$$

(34)

Since the stable operation of the engine has been given as a precondition in the design process and no other constraints are presented during the design process, the optimization problem for the cycle parameters can be formulated as follows.

$$\left\{\begin{array}{c}\min J\\ u_{\min }\le u\le u_{\max }\end{array}\right.$$

(35)

Furthermore, the serpentine 2-D exhaust system is considered in the model. Due to the change in AR, S/D and L/D would affect the infrared radiation of the engine exhaust system. It is necessary to consider the three parameters in the engine design optimization, which are expressed as u_IR = [AR, S/D, L/D]. The adjusting variables also need to meet the corresponding constraint u_IR,min ≤ u_IR ≤ u_IR,max.

$$\left\{\begin{array}{c}{(AR)}_{\min }\le AR\le {(AR)}_{\max }\\ {(S/D)}_{\min }\le S/D\le {(S/D)}_{\max }\\ {(L/D)}_{\min }\le L/D\le {(L/D)}_{\max }\end{array}\right.$$

(36)

The parameter optimization problem for the adaptive cycle engine can be expressed as follows.

$$\left\{\begin{array}{l}\hspace{1em} \min J\\ u_{\min }\le u\le u_{\max },u_{IR\min }\le u_{IR}\le u_{IR\max }\end{array}\right.$$

(37)

5. Simulation Results and Analysis

Based on the parameter cycle analysis of ACE and the multi-objective optimization design method, the multi-objective optimization of design parameters is carried out, which considers unit thrust, specific fuel consumption and infrared radiation. To ensure stable operation of the engine at the design point and other operating conditions, it is necessary to limit the design parameters within reasonable ranges. Meanwhile, to prevent the high temperature of the high-pressure turbine due to insufficient bleed air or insufficient thrust caused by excessive bleed air, limits are imposed on the maximum and minimum bleed air coefficients. The upper and lower limits of ACE’s design parameters are given in

Table 4. Upper and lower limits of design parameters.

Design Parameters		Values
Definition	Symbols	Lower	Upper
Compression component	πFLADE	2.4	2.6
	πf	3.7	3.9
	πCDFS	1.2	1.4
	πcH	5	7
Bypass ratio	α	0.15	0.25
	α1	0.25	0.35
	α2	0.65	0.75
Combustion	Tt4 (K)	1850	1950
Exhaust system geometry	AR	3	6
	S/D	0.3	0.5
	L/D	1	3

.

According to the selection range of the design parameters, the multi-objective optimization numerical simulation is carried out. This study analyzes the design performance of the standard adaptive cycle engine and evaluates the engine’s performance under various optimization objectives. The optimization objectives include three single optimization objectives, aiming for high unit thrust, low specific fuel consumption, and low infrared radiation intensity, as well as a combination of all three objectives for multi-objective optimization.

Through the above SQP algorithm, five groups of different weights are set to optimize the engine. In the multi-objective optimization, two optimization schemes are selected, which are ω₁ = 0.4, ω₂ = 0.2, ω₃ = 0.4 (engine design target based on infrared stealth) and ω₁ = 0.7, ω₂ = 0.1, ω₃ = 0.2 (engine design target based on mobility). The optimization results and algorithm iterations are shown in Figure 9. The figure demonstrates that the SQP algorithm achieves efficient optimization with a short step size and exhibits excellent convergence properties.

Furthermore, the following conclusions could be drawn from the graph analysis. In single-objective optimization, the design method can iteratively achieve optimization results that satisfy the constraints. However, when focusing on high unit thrust optimization, increasing the thrust may also lead to a higher fuel consumption rate and infrared radiation intensity. On the contrary, optimizing for a low fuel consumption rate and low infrared radiation intensity will result in a significant decrease in unit thrust. Therefore, it is necessary to consider multi-objective optimization and select an appropriate weight value to determine the design result of ACE.

Compared with single-objective optimization, multi-objective optimization can comprehensively consider the overall performance of engine thrust, fuel consumption and infrared radiation intensity. In the design target of economy, it can not only reduce the fuel consumption of the engine, but also reduce the thrust loss, which meets the thrust demand of the engine. In the design target of mobility, although the thrust is lower than that of the high unit thrust optimization target, the fuel consumption and infrared radiation intensity are significantly lower than that of the single-objective optimization.

In order to clearly see the difference of the five optimization results, the corresponding optimized design parameters and engine performance results are given in

Table 5. Optimized design parameters and engine performance.

Optimization Design Parameters	Weight Value
	ω1 = 0 ω2 = 0 ω3 = 0	ω1 = 1 ω2 = 0 ω3 = 0	ω1 = 0 ω2 = 1 ω3 = 0	ω1 = 0 ω2 = 0 ω3 = 1	ω1 = 0.4 ω2 = 0.2 ω3 = 0.4	ω1 = 0.7 ω2 = 0.15 ω3 = 0.15
πFLADE	2.4	2.4	2.4	2.4	2.4	2.4
πf	3.9	3.9	3.9	3.9	3.9	3.9
πCDFS	1.2	1.4	1.4	1.4	1.315	1.2
πcH	5.01	7	7	7	7	5.01
α	0.25	0.25	0.25	0.25	0.25	0.25
α1	0.25	0.35	0.35	0.343	0.25	0.25
α2	0.65	0.75	0.75	0.65	0.65	0.65
Tt4	1950	1850	1850	1876	1950	1950
AR	4.94	4.94	6	6	6	4.94
S/D	0.3	0.3	0.5	0.5	0.5	0.3
L/D	3	3	3	2.543	2.945	3
Fs	941.3	830.6	826.2	872.9	925.2	941.3
sfc	0.616	0.529	0.531	0.551	0.574	0.616
IR	277.3	145.9	18.67	20.86	25.51	277.3

. Compared with the standard point, although Fs is promoted by 7% in the high unit thrust target optimization, the fuel consumption is significantly increased and the infrared radiation intensity is increased by more than three times. The other single-objective optimizations also have a similar extreme result. In the engine design target based on infrared stealth, although the unit thrust of the engine slightly decreases by 0.7%, the fuel consumption and infrared radiation are significantly reduced by 8.6% and 74%, respectively. In the engine design target based on mobility, the unit thrust of the engine is increased by 5.24%, the fuel consumption rate is decreased by 4.79% and the infrared radiation is reduced by 68.27%, which means all engine performances are improved. It can be seen that compared with single-objective optimization, multi-objective optimization would consider the overall performance during the engine design, which has a better optimization design effect than single-objective optimization.

In order to validate the operability of the designed ACE in the large envelope range, the off-design point performance ought to be analyzed. The designed ACE is applied to different mission segments to study the engine’s performance deviations from the design point. The technical indicators of six flight segments are provided in

Table 6. Flight mission parameters.

	Takeoff	Subsonic Cruise	Supersonic Cruise	Horizontal Acceleration	Dash	Land
Altitude(m)	609.6	12,823.6	11,300	11,300	11,300	3048
Flight M	0.1	0.9	1.5	1.365	1.5	0.3973
Tt4(K)	1900	1660	1850	1900	1900	1750
Mode	High	High	Low	Low	High	High
Range(km)	0.222	218.573	171.736	4.834	46.3	0
Time(s)	7	823	377.5	11.7	101.8	1200

, which include takeoff, subsonic cruise, supersonic cruise, horizontal acceleration, dash and landing. The data in the table include altitude, flight Mach number, combustor temperature, engine mode, range and time. The mode refers to the mode of ACE under high bypass ratio (‘High’ in Table 6) and low bypass ratio (‘Low’ in Table 6). During the simulation, ACE optimized for high unit thrust (ω₁ = 1, ω₂ = 0, ω₃ = 0), ACE mainly focused on aircraft infrared stealth (ω₁ = 0.4, ω₂ = 0.2, ω₃ = 0.4) and ACE prioritizing mobility (ω₁ = 0.7, ω₂ = 0.1, ω₃ = 0.2) are compared to analyze the engine performance at each mission point.

Table 7. Three types of ACEs engine performance.

Engine Number	Engine Performance	Flight Mission Number
		1	2	3	4	5	6
Basic ACE	Fs (N*s/kg)	751.89	555.68	563.94	513.93	533.5	630.64
	F (N)	35,440	7948	16,699	15,518	16,999	22,965
	sfc (kg/(s*N))	0.0741	0.0826	0.0971	0.0971	0.0988	0.0765
	IR (W/sr)	82.674	49.773	85.006	78.299	85.596	61.837
ACE 1 (ω1 = 0.7, ω2 = 0.1, ω3 = 0.2)	Fs (N*s/kg)	767.473	575.543	581.847	535.717	554.693	648.097
	F (N)	36,240	8292	17,261	16,304	17,824	23,596
	sfc (kg/(s*N))	0.07207	0.07933	0.09303	0.09255	0.09443	0.073791
	IR (W/sr)	25.097	15.294	25.723	24.175	26.3	18.847
ACE 2 (ω1 = 0.4, ω2 = 0.2, ω3 = 0.4)	Fs (N*s/kg)	725.913	538.005	546.025	503.47	521.62	607.95
	F (N)	34,185	7671.101	16,075.58	15,131	16,555.58	21,945.89
	sfc (kg/(s*N))	0.06968	0.07763	0.090439	0.0902	0.092049	0.071835
	IR (W/sr)	20.659	11.961	21.115	19.989	21.85	15.177
ACE 3 (ω1 = 1, ω2 = 0, ω3 = 0)	Fs (N*s/kg)	782.68	587.97	603.47	561.04	578.99	662.03
	F (N)	36,799	8304.6	17,669	17,025.6	18,425.3	23,955.2
	sfc (kg/(s*N))	0.077443	0.083885	0.09797	0.0979	0.0997	0.0793
	IR (W/sr)	278.72	170.96	282	267.92	289.08	211.4

shows the engine performance data of four types of ACE in each mission segment, including unit thrust, thrust, specific fuel consumption and infrared radiation intensity. The numbers 1 to 6 represent the above six mission segments. It could be seen from the table that the engine optimized for high unit thrust (ω₁ = 1, ω₂ = 0, ω₃ = 0) has excellent performance in thrust and unit thrust of each mission section. However, its fuel consumption and infrared radiation intensity show significant increases compared to the reference engine. The thrust performance of infrared stealth ACE (ω₁ = 0.4, ω₂ = 0.2, ω₃ = 0.4) has decreased in flight missions, but it achieves noteworthy reductions in both infrared intensity and fuel consumption. On the other hand, the high mobility ACE (ω₁ = 0.7, ω₂ = 0.1, ω₃ = 0.2) shows commendable thrust performance and low infrared characteristics in flight missions, accompanied by a decrease in fuel consumption.

To further distinguish the advantages and disadvantages of each engine performance, Figure 10 shows the percentage change in the total performance between standard ACE and ACE with three optimal designs under different mission points. The black bar represents the ACE mainly based on maneuverability (the first type), the red bar represents the ACE mainly based on aircraft infrared stealth (the second type), and the yellow bar represents the high unit thrust ACE (the third type). In Figure 10a,b, the unit thrust and thrust of the first and third type has increased by about 2.5% and 6%, respectively, in all mission segments, while the second type of ACE decreases in the whole mission segments. From Figure 10c, it could be seen that except for the third type, the specific fuel consumption of the other two ACEs has decreased in all mission segments, and the second type decreased most obviously by about 6%. In Figure 10d, the contrast diagram of three ACEs’ infrared radiation intensity is presented. The figure clearly demonstrates that the first type of exhaust system exhibits a notable increase of 250% in infrared radiation during each mission phase, whereas the other two types show a reduction in infrared radiation.

The above optimization-designed ACEs are based on the ground point, where H = 0 and M₀ = 0. In this paper, the influence of the high-altitude design point on the performance at different mission segments is further considered. Figure 11 presents the total performance comparison between the standard ACE and the ACEs resulting from the three optimal designs at the high-altitude design point (H = 11.3 km, M = 1.5), revealing distinct differences from the ground point. From Figure 11a,b, it is evident that the optimized ACEs consistently exhibit higher unit thrust and engine thrust throughout the mission segments. Figure 11c depicts the change in ACEs’ specific fuel consumption rates, which show an overall increase. This implies that the engine would consume more fuel at the high-altitude design point. The results for infrared radiation in Figure 11d align with those of the ground point. In conclusion, compared to the ground design point, the high-altitude design point allows for further optimization of the engine’s thrust, albeit at the expense of increased fuel consumption.

6. Conclusions

This paper is concentrated on the multi-objective optimization design of the adaptive cycle engine with a serpentine 2-D exhaust system. With the analysis and simulation results of the optimized ACE in its mission phase, the following conclusions could be drawn.

(1)

The variable specific heat parameter cycle model of adaptive cycle engine is established and the infrared prediction model is combined with the engine model. The model could be used to study the engine’s overall performance and infrared characteristics at the design point.

(2)

According to the above modeling method, an infrared stealth adaptive cycle engine model is established by adopting a serpentine 2-D nozzle. The infrared radiation characteristics and total pressure characteristics of the exhaust system are studied, and the impact of the serpentine 2-D nozzle on the overall performance of the engine is analyzed. In the engine design model, the serpentine 2-D exhaust system is used to reduce the ACE’s infrared radiation intensity and improve the infrared stealth of the engine.

(3)

The SQP algorithm is used to optimize the engine overall performance. The simulation results show that the multi-objective optimization could make the engine have better overall performance than the single-objective optimization. Furthermore, the performance of each mission segment could be further optimized at the high-altitude design point.