气动喷嘴综述：航空航天应用的当前趋势

原文

In the combustion chamber, the chemical reaction between fuel and oxidizer generates gases at high pressure (on the order of hundreds of atmospheres) and high temperature (around 3000 K or more). The composition of these gases depends on the propellants used. As they expand through the nozzle, the gases accelerate to supersonic speeds and experience reductions in pressure and temperature, as well as possible changes in their chemical composition.

2.1. Relevance of Thermodynamic Properties for Nozzle Design

The performance of a rocket nozzle critically depends on the thermodynamic properties of combustion gases, which determine fundamental parameters, such as exhaust velocity ( $v_{e}$ ), mass flow rate ( $\dot{m}$ ), and, ultimately, specific impulse ( $I_{s p}$ ). These properties are derived from energy balance and isentropic relations and are integrated with the nozzle’s geometric design to maximize engine performance.

The exhaust velocity is expressed as

$v_{e} = \sqrt{\frac{2 γ R_{u} T_{c}}{(γ - 1) m} [1 - {(\frac{P_{e}}{P_{c}})}^{\frac{γ - 1}{γ}}]},$

(2)

Similarly, the mass flow rate, which remains constant in steady-state conditions, is determined by

$\dot{m} = \frac{P_{c} A_{t}}{c^{*}} = \frac{P_{c} A_{t}}{\sqrt{\frac{R_{u} T_{c}}{γ m} {(\frac{2}{γ + 1})}^{\frac{γ + 1}{γ - 1}}}},$

(3)

where

c^{*}

is the characteristic velocity of the propellant.

Thrust is calculated by combining the change in momentum flow with the effect of the pressure difference between the nozzle exit and the ambient pressure:

$F = \dot{m} v_{e} + (P_{e} - P_{a}) A_{e},$

(4)

where

A_{e}

is the nozzle exit area, and

P_{a}

is the ambient pressure.

From the thrust equation, the specific impulse is defined as

$I_{s p} = \frac{F}{\dot{m} g_{0}} = \frac{v_{e}}{g_{0}} + \frac{(P_{e} - P_{a}) A_{e}}{\dot{m} g_{0}},$

(5)

where

g_{0}

is the acceleration due to gravity at sea level.

For optimal engine performance, it is essential to achieve the following:

Maximize $T_{c}$ : A higher combustion chamber temperature increases $v_{e}$ and, consequently, $I_{s p}$ ;
Minimize m: Reducing the molecular weight improves efficiency by increasing exhaust velocity;
Expand gases until $P_{e}$ = $P_{a}$ : Designing the nozzle so that the exit pressure $P_{e}$ approximates the ambient pressure $P_{a}$ maximizes the thrust coefficient (1).

Integrating these equations with geometric design methods, such as the Method of Characteristics (MOCs), and validating them through computational fluid dynamics (CFDs) simulations enables the development of a functional and efficient nozzle design that maximizes engine performance under real-world conditions.

2.2. Composition and Properties of Rocket Exhaust Gases

Understanding the thermodynamic properties of the exhaust flow is crucial for designing efficient nozzles. The key properties of interest include the following:

Specific heats ( $c_{p}$ and $c_{v}$ ): The amount of heat required to raise the temperature of a unit mass of gas by one degree at constant pressure ( $c_{p}$ ) or constant volume ( $c_{v}$ ), respectively;
Average molecular weight (m): The average mass of a gas molecule, calculated by considering all species present in the mixture;
Specific heat ratio ( $γ = \frac{c_{p}}{c_{v}}$ ): A fundamental parameter in compressible flow analysis that describes gas compressibility and affects the speed of sound in the medium.

These properties depend on temperature, pressure, and chemical composition. At low pressures, an ideal gas behavior is assumed, where enthalpy (h), specific heat at constant pressure ( $c_{p}$ ), and specific heat at constant volume ( $c_{v}$ ) depend only on temperature.

However, in rocket nozzles, where gas temperatures can exceed 3000 K, chemical species dissociation occurs [31,40,43]. High temperatures break chemical bonds, leading to the formation of simpler species and altering the flow properties [31]. For example, carbon dioxide (CO₂) may decompose into carbon monoxide (CO) and atomic oxygen (O), while water vapor (H₂O) may dissociate into molecular hydrogen (H₂), hydroxyl (OH), and atomic oxygen (O), among other species. These endothermic reactions consume part of the energy that would otherwise be converted into kinetic energy, reducing the combustion temperature (

T_{c}

) and decreasing nozzle efficiency.

As the gases expand and cool in the nozzle, the dissociated species may recombine, releasing energy and further altering flow properties. This recombination process can help recover some of the energy absorbed during dissociation, but its effectiveness depends on chemical kinetics and the time available for reactions to occur. The balance between dissociation and recombination is influenced by local temperature and pressure conditions, which are crucial in the design and analysis of nozzles.

To more accurately predict the flow behavior under these conditions, chemical kinetics models that account for dissociation and recombination, such as the chemical equilibrium flow (CEQ) model, are employed. These models consider that chemical reactions occur rapidly and reversibly, continuously adjusting to local chemical equilibrium. Compared to frozen flow models, equilibrium models provide more realistic estimates of engine performance and efficiency, though they can be more complex to implement. As the flow expands in the nozzle and pressure and temperature decrease, recombination of dissociated species can occur. This exothermic process releases additional energy, contributing to overall performance by increasing the energy available to accelerate the flow.

The average molecular mass (m) and the specific heat ratio ( $γ$ ) of the exhaust gases depend on their chemical composition. Fuels that produce gases with lower m tend to generate higher exhaust velocities ( $v_{e}$ ), as the speed of sound (a) in the gas increases with decreasing m, according to the relation $a = \sqrt{\frac{γ R_{u} T}{m}}$ . A higher speed of sound allows for higher flow velocities at the same temperature.

Table 1 presents examples of common propellants and their combustion products:

Chemical equilibrium codes, such as CEA (Chemical Equilibrium Analysis), allow for the simulation of combustion and gas expansion in nozzles [44]. These codes (software) compute the exhaust gas composition at given temperatures (

T_{c}

) and pressures (

P_{c}

) and evaluate their thermodynamic properties as functions of temperature and pressure. By modeling combustion processes at both constant volume and constant pressure, the constant pressure process is particularly relevant for the design of combustion chambers and nozzles, as combustion in rocket engines occurs essentially at constant pressure [45].

A CEA output file provides detailed information on combustion and expansion. Some of the most relevant parameters include the following:

Mass fractions: The proportion of each species present in the mixture, allowing for an exact determination of the flow’s chemical composition;
$T_{e}$ and $P_{e}$ : Temperature and pressure at the nozzle exit, which are essential for performance analysis and nozzle design;
m (Molecular mass of the mixture): Influences gas density and speed of sound, affecting exhaust velocity;
$γ$ (Specific heat ratio): Affects isentropic relations and flow expansion;
a (Local speed of sound): Important for determining the Mach number and analyzing flow regimes in different nozzle sections.

CEA is a fundamental tool for predicting the ideal thermodynamic behavior of combustion gases under specific initial conditions, as it calculates composition and properties based on chemical equilibrium and isentropic flow assumptions. In propulsion system design and analysis, CEA is used in various ways. For example, Moen [4] integrated it into a broader simulation framework (using NPSS and ModelCenter) for a Methane Dual Expander Aerospike Nozzle (MDEAN) rocket engine. He used the rocket problem function of CEA to calculate the thermochemical properties of methane/oxygen combustion products—assuming frozen flow and specifying the chamber state (P, T) and the Oxidizer-to-Fuel ratio (O/F). This data was essential for the engine model, aiding in initial parametric studies (such as Isp vs. O/F) and ensuring consistency between thermodynamic reference states from CEA and other data sources like the National Institute of Standards and Technology (NIST). Ma et al. [26] employed CEA as a reference tool to evaluate the performance of continuous detonation engines (CDEs) with aerospike nozzles. They iteratively calculated the ideal performance (Isp,

c^{*}

, thrust coefficient (CF)) of a traditional deflagration rocket engine under experimental conditions, thereby establishing an ideal baseline to define normalized performance metrics and quantify the gains or losses of their experimental CDEs compared to ideal isobaric combustion. Whitmore & Armstrong [46] applied CEA to calculate the theoretical combustion temperature and characteristic velocity for their GOX/ABS hybrid propulsion system as a function of the O/F ratio. These calculations allowed them to compare the performance potential of different motor configurations and generate the necessary thermodynamic property tables for analyzing experimental data, such as iteratively determining the actual combustion efficiency observed in tests.

However, this idealized approach based on equilibrium and isentropic assumptions does not account for critical phenomena in nozzle design—such as shock waves, flow separation, thermal effects, and non-equilibrium reactions—which can significantly impact real performance. Therefore, while CEA provides a valuable first approximation of thermodynamic properties and flow behavior, its results must be validated and complemented with computational fluid dynamics (CFDs) simulations and more advanced combustion models that incorporate chemical kinetics and transport effects.

CFD simulations are essential for understanding the complex flow physics within aerospike nozzles.

For instance, Gould employed CFD simulations to validate the isentropic flow assumptions made in the initial design and to illustrate viscous effects within the flow, including boundary layer analysis [3]. Ito and Fujii utilized Navier–Stokes simulations to analyze the flow fields of aerospike nozzles, specifically investigating the impact of base bleeding on total thrust and base pressure enhancement using turbulence models such as Baldwin-Lomax [37]. Khan and Khushnood used the ANSYS Fluent package to conduct inviscid flow analyses, exploring various nozzle configurations (contoured, conical, dual-bell, and truncated), comparing the results with theoretical predictions, and observing flow structures such as exhaust plumes and shock waves [47]. In their review, Khare and Saha highlight the use of CFDs to capture flow features under overexpanded regimes and assess the effectiveness of turbulence models, such as the SST k-

ω

, in predicting shock location and flow separation [27]. Nazarinia et al. numerically compared the performance of aerospike nozzles (including variations in truncation and base curvature) with conventional nozzles under different operating conditions (optimal, underexpanded, and overexpanded) using CFDs [48]. He et al. conducted a numerical investigation using RANS to identify the detailed behavior of flow separation—including the progression of shock structures and the influence of gas density—validating the computational methodology against experimental data [49]. Shahrokhi and Noori applied CFDs with the k-

ϵ

turbulence model to study the influence of different plug shapes—generated using B-Spline curves—on total thrust [50]. Besnard and Garvey mention the use of CFDs to predict Mach number distributions compared to static fire tests, emphasizing the need to validate CFD models—especially in the base region—against flight data [51].

Moreover, coupling these calculations with the geometric design of the nozzle using the Method of Characteristics (MOCs) is crucial. The MOCs is a well-established technique that simplifies the hyperbolic Euler equations for supersonic flows by solving them along characteristic lines to generate optimized, shock-free nozzle contours. This method enables profile optimization to avoid internal shock waves and ensure a smooth and efficient gas expansion, thereby maximizing engine performance under realistic conditions—such as achieving uniform and parallel flow at the exit of minimum-length nozzles (MLNs).

Abada et al. [52] specifically applied the MOCs to design the contour of an axisymmetric MLN nozzle, highlighting its capability to incorporate high-temperature gas effects, particularly the variation in specific heats. They utilized the detailed results obtained from the MOCs—such as the wall Mach number and pressure distributions—as a basis for validating CFDs simulations conducted using Ansys Fluent. Their study demonstrated how analytical solutions from the MOCs can effectively support and verify the accuracy of high-fidelity numerical models under real-gas conditions.

Booth et al. [19], on the other hand, used the MOCs in combination with CFDs as an analysis tool to establish and compare inviscid flow characteristics and divergence efficiency across different 2D and 3D thrust cell designs for linear aerospike nozzles. The MOCs provided reference solutions for isentropic conditions, which were then used to assess the accuracy of Euler-based CFDs predictions. This comparative approach allowed for a comprehensive evaluation of thrust performance and flow structure fidelity in various nozzle geometries.

Fernandes et al. [53] not only reaffirmed the use of the MOCs to generate isentropic, shock-free contours for MLNs but also emphasized its utility for analyzing the flow field and calculating thrust coefficients for arbitrary, pre-existing nozzle geometries. They developed a fast shape optimization methodology that integrates the MOCs with Free-Form Deformation (FFD) parameterization techniques, positioning the MOCs as an efficient, low-fidelity tool for preliminary nozzle design and optimization. Their approach enables rapid design iterations, the results of which can inform and refine more computationally expensive CFD simulations in subsequent stages.

The validation of these numerical (CFD) and analytical (MOC) methods with experimental data—such as that obtained from static fire tests and flight experiments—is essential to ensure the accuracy and reliability of performance predictions. Ultimately, integrating CEA, CFDs, and MOCs approaches provides a robust framework for nozzle design, bridging idealized assumptions with the complexities of real-world operation.

气动喷嘴综述：航空航天应用的当前趋势 A Review of Aerospike Nozzles: Current Trends in Aerospace Applications

2.1. Relevance of Thermodynamic Properties for Nozzle Design

2.2. Composition and Properties of Rocket Exhaust Gases

气动喷嘴综述：航空航天应用的当前趋势
A Review of Aerospike Nozzles: Current Trends in Aerospace Applications