A Review of Aerospike Nozzles: Current Trends in Aerospace Applications

Abstract

The aerospike nozzle has emerged as a revolutionary technology in rocket propulsion, overcoming the limitations of conventional nozzles by dynamically adjusting the expansion of exhaust gases according to ambient pressure. This results in greater efficiency and fuel savings. Thanks to advancements in materials, computational simulations, and additive manufacturing, aerospike nozzle design has surpassed historical barriers, not only enhancing the performance of multi-stage engines but also enabling the development of new space vehicles such as single-stage-to-orbit (SSTO) systems. These promise a simpler, reusable, and more cost-effective launch architecture. In summary, aerospike technology inspires a new era in space exploration, transforming propulsion efficiency and paving the way for more accessible and sustainable future space transportation systems.

1. Introduction

Since the 1950s, the aerospike nozzle design has captured the attention of the scientific community and aerospace engineers for its potential to revolutionize rocket propulsion. Unlike conventional bell nozzles, which are optimized to operate efficiently at a specific altitude, aerospike nozzles stand out for their ability to automatically adapt to a wide range of atmospheric pressures during flight, from liftoff to ascent into high orbits [1]. This inherent adaptability allows aerospike nozzles to maintain superior efficiency throughout all flight phases, recovering approximately 50% of the specific impulse loss experienced by bell nozzles when compared to an ideal lossless nozzle [2].

In the design and operation of rocket engines, efficiency and safety are two crucial factors that determine the success and feasibility of space missions. Efficiency refers to the engine’s ability to convert the chemical energy of the propellants into the kinetic energy of the exhaust jet, thereby maximizing thrust while minimizing fuel consumption. Safety, on the other hand, encompasses all aspects that ensure the engine operates reliably without the risk of catastrophic failures. Aerospike nozzles represent an advanced technology that integrates these two essential aspects, providing superior efficiency and robust safety compared to conventional nozzles.

One of the main challenges is ensuring that rocket engines operate efficiently under highly diverse conditions, from the dense atmosphere at sea level to the near-vacuum environment of outer space. Conventional nozzles, designed for a fixed pressure, suffer from inefficiencies due to either overexpansion or underexpansion of the exhaust gases [3]. In contrast, aerospike nozzles overcome this obstacle through their adaptive design, although they introduce additional complexities in manufacturing and managing high thermal loads, which require advanced materials and efficient cooling systems such as regenerative cooling [4].

Advancements in this technology not only aim to optimize performance under variable pressure conditions but also pave the way for innovative space vehicles, such as single-stage-to-orbit (SSTO) systems, which can simplify launch architectures and lower operational costs. The key question now is whether these innovations can redefine the current propulsion paradigm, ushering in a new era of space exploration [5,6].

1.1. Background on Aerospike Nozzles

Since their initial theoretical concepts, aerospike nozzles have been the subject of extensive studies and developments aimed at improving the efficiency of rocket engines, particularly for missions requiring optimal performance at various altitudes [7]. During the 1950s and 1960s, experiments were conducted with aerospike nozzles for potential use in the upper stages of Saturn V and, later, in the Space Shuttle main engine [5,7,8,9]. During this period, Rocketdyne carried out extensive research on aerospike performance and thrust vectoring using liquid injection [10,11]. Rocketdyne also explored secondary-injection thrust vector control (SITVC), in which a small mass flow of gaseous or liquid propellant is injected either at the base region or locally along the spike surface to produce a controlled pressure asymmetry and, hence, deflect the thrust by up to 6–8° without moving the engine structure [12,13]. However, the selection of a conventional bell nozzle for the Space Shuttle main engine (SSME) significantly impacted aerospike development momentum until the 1990s. The preference for bell nozzles for the Shuttle program stemmed largely from their higher technological maturity and lower perceived development risk compared to aerospikes at that time [14,15]. The engine procurement of NASA for the SSME, for instance, explicitly required a bell-shaped nozzle to ensure contractors utilized proven designs, thereby avoiding the uncertainties associated with the then-novel aerospike concept, which had unresolved challenges in cooling, materials, and transonic performance without full-scale flight experience [14,15]. Aerospike technology was revived with dedicated programs like the single-stage-to-orbit (SSTO) X-33 vehicle in the 1990s [16] (see Figure 1; image sourced from [17]) once advancements in materials and computational tools began to address some of these historical hurdles [18].

Interest in aerospike nozzles resurged strongly in the following decades, driven by advancements in space technology. During the 1970s and 1980s, the United States, Europe, and Japan conducted research and testing to optimize the geometry of these nozzles and explore their feasibility for various space applications [19]. A significant milestone was the development of the linear aerospike XRS-2200 engine for the X-33 test vehicle, marking a major advancement in integrating this technology into modern propulsion systems [20,21].

In recent decades, international attention on aerospike nozzles has intensified. The European Space Agency has investigated the relative efficiency of different thrust vector control techniques for these nozzles [22,23]. The Technical University of Munich has conducted analyses on the performance of aerospike nozzles, including efficiency losses due to nozzle clustering [24,25]. More recently, research at Peking University has experimentally investigated the performance and pressure gain characteristics of aerospike nozzles integrated with continuous detonation engines, exploring the influence of parameters like area ratio and equivalence ratio on operability and efficiency. This reflects the ongoing interest in optimizing aerospike designs for advanced propulsion systems [26,27].

1.2. Advantages and Disadvantages of Aerospike Nozzles

Aerospike nozzles are distinguished by their efficiency and adaptability benefits thanks to their innovative design, which significantly differs from conventional bell nozzles. Instead of a fixed geometry optimized for a specific atmospheric pressure, aerospike nozzles employ an aerodynamic surface that allows for continuous expansion of exhaust gases. This configuration allows the nozzle to automatically adjust to changing atmospheric pressure, maintaining near-optimal efficiency from liftoff to the vacuum of space [28,29].

A key factor in the efficiency of aerospike nozzles is the reduction in divergence losses. In conventional nozzles, exhaust gases tend to deviate from the ideal trajectory when the atmospheric pressure does not match the exit pressure, resulting in thrust loss. Aerospike nozzles, however, minimize this divergence through their concave shape and central expansion surface, which confines the gases and directs them in a more controlled manner [8].

Aerospike nozzles can provide higher thrust under residual pressure conditions in outer space or on surfaces such as the Moon. While the conventional nozzles used in the Apollo missions had an expansion ratio of 50:1, aerospike nozzles can achieve much higher effective expansion ratios in a vacuum, potentially ranging from 200:1 to 300:1 [29]. This capability is a direct result of the external expansion principle, where the exhaust plume uses the ambient pressure (or its absence in a vacuum) as a ‘virtual’ outer boundary, allowing for near-optimal expansion without the immense physical structure of a conventional bell nozzle required to achieve similar ratios. This translates to an improvement of 5 to 6% in specific impulse ($I_{sp}$) under such conditions [29]. According to the Tsiolkovsky rocket equation, even such modest gains in $I_{sp}$ can dramatically reduce the required propellant mass for a given $\Delta v$ due to the logarithmic nature of the relationship [1,30,31]. This can lead to an approximate reduction of 8 to 9% in propellant mass for missions demanding large velocity changes ($\Delta v$), such as orbital transfer systems and descent and ascent stages in lunar and martian missions [29]. For example, for some upper-stage vehicles, a 10% increase in $I_{sp}$ has been shown to potentially increase payload capacity by roughly 30% [32]. This enhanced efficiency is particularly critical for missions where propellant mass constitutes a large fraction of the total vehicle mass, potentially enabling more ambitious scientific payloads or mission profiles. A recent DLR design study for a $LOX/LC{H}_4$ lunar lander descent stage, for instance, found that an optimized truncated aerospike could provide a payload gain of 4–11% (up to 200 kg extra payload for a ∼2.6 km/s $\Delta v$ maneuver) compared to a bell nozzle, despite the aerospike’s higher anticipated dry mass [33].

This reduction in propellant mass not only decreases the total vehicle weight but also frees up the margin to incorporate other critical systems or increase payload capacity. Additionally, truncated aerospike nozzles offer the advantage of being more compact than conventional bell nozzles, which is particularly beneficial in space applications where volume and mass are limited.

Despite their advantages, aerospike nozzles present significant technical and economic challenges that have historically limited their widespread adoption. Their design and manufacturing are inherently more complex and costly than conventional nozzles, requiring materials resistant to extreme temperatures and pressures, along with advanced manufacturing techniques [16,19]. These factors, when contrasted with the well-understood and highly reliable bell nozzle, have historically contributed to higher perceived risk and cost [15,34]. Thermal management is crucial, as efficiently handling the intense heat flux on the central spike, particularly with earlier material and cooling technologies, was a major concern [34]. For instance, convective heat-flux densities at the spike tip can exceed 0.7 MW ${\mathrm{m}}^{-2}$ in a 250 kN class module, demanding cooling channels capable of sustaining wall temperatures above 2000 K without loss of structural integrity [35]. Furthermore, the structural mass of the aerospike, including the spike itself and potentially complex cooling systems, can be higher than that of a comparable bell nozzle. This mass penalty is especially acute in linear aerospikes, the long ramp of which accounts for ∼15 % of the dry mass of the XRS-2200 module, and in annular designs where each circumferential chamber/throat requires its own support and cooling hardware [24,25,34]. This added weight can offset some of the $I_{sp}$ benefits, particularly for missions where the $\Delta v$ requirements are not extreme enough for the propellant savings to dominate the mass budget [34,36].

The integration of these nozzles into existing systems and their adaptation to different space missions also pose challenges, requiring further research to improve efficiency, minimize losses due to viscosity and friction, and optimize their design to withstand extreme thermal and mechanical loads [9,10].

1.3. Current State of Research on Aerospike Nozzles

Korte et al. [16] demonstrated that a multidisciplinary design approach, integrating the simultaneous optimization of aerodynamics and structural performance, leads to significant improvements in the performance of aerospike nozzles. By balancing factors such as aerodynamic thrust and structural weight, their designs outperform those obtained through traditional approaches that treat each discipline in isolation. This highlights the importance of considering complex interactions between variables to achieve optimal performance. The coupled optimizer they applied to a 205 kN-class plug cut the vehicle gross-lift-off weight by 4.9% while holding nozzle thrust within 0.1% of the reference and trimming the plug dry mass by 9%. A total of 20 laminate plies (down from 32) still cleared all stress and buckling checks, underscoring that fluid–structure coupling, not simple sequential sizing, is what unlocks genuinely lightweight plugs.

In a different approach, Fick validated an analytical model capable of accurately calculating the thrust and specific impulse of clustered plug-type nozzles, with an accuracy generally within one to two percent. This model enables parametric analysis and systematic performance comparisons between aerospike nozzles and those of conventional rocket engines, making it a valuable tool in the early design stages [24]. By applying the model to a 32-module cluster (205 bar chamber pressure), he showed that, provided inter-module gaps stay below 0.8 $D_t$, the cluster can still realize 99.1–99.3% of the ideal $I_{sp}$; larger gaps or selective module shut-down occur for throttling push losses beyond 6%. Heat flux on the spike was calculated to double relative to an annular plug, signaling the need for high-performance regenerative cooling.

Meanwhile, Ma et al. [26] investigated the influence of various parameters on detonation stability and performance in continuous detonation engines (CDEs) equipped with aerospike nozzles. They concluded that an area ratio below 20% and a reduced mass flow rate favor stable detonation and extend the operational range in terms of equivalence ratio, providing critical insights for optimizing these systems. Quantitatively, the sweet-spot window of 12.8–18.8% delivered characteristic-velocity efficiencies (${\eta }_{c^{*}}$) of 0.90–0.92 and pressure-gain levels of 15–25% over a deflagration baseline. Outside that window (e.g., 37% area ratio), detonation broke down, and $I_{sp}$ fell by >10 s, highlighting how tightly CDE-plug performance hinges on geometric choke pressure.

Moen evaluated the potential of a dual-expander methane aerospike rocket engine (MDEAN) and concluded that, despite expectations of achieving a specific impulse of 383 s and a thrust-to-weight ratio exceeding 108, the results did not show significant improvements over existing technologies. This conclusion underscores the need to continue developing new strategies to meet performance goals [4]. To reach that judgment, the author executed a design-space sweep using the Numerical Propulsion System Simulation (NPSS) code of NASA, orchestrated by ModelCenter. The workflow varied expansion ratio (ER), mixture ratio, throat area, and turbine pressure ratio, producing 541 converged design points. The best compromise for a $25\mathrm{klbf}$ upper-stage engine appears at $\mathrm{ER}=7$ with $I_{sp}=349\mathrm{s}$ and $T/W=120.7$. Pushing ER to 60 allows the target $I_{sp}=383\mathrm{s}$ but increases the dry mass by $\sim 300\mathrm{kg}$ and drives $T/W$ down to 14, offsetting any propellant saving for most orbital-insertion missions. The sweep, therefore, maps a clear Pareto frontier between mass and performance and shows why higher ER alone does not guarantee a superior aerospike stage.

In another study, Ito and Fujii analyzed the effect of base bleed in aerospike nozzles and found that by promoting flow recirculation in the base region, pressure thrust was significantly increased. This finding suggests that controlling the flow in the base region could be an effective strategy for optimizing engine performance [37]. Their RANS parametric study confirms that injecting just 2–4% of the core mass flow as base bleed can raise base pressure by roughly 35% and lift the overall thrust coefficient by 3–5%; beyond that point, diminishing returns set in because the extra propellant penalizes the system $I_{sp}$. Optimal injection is centered on the spike base; off-axis ports showed weaker recirculation and smaller gains.

Finally, Bui et al. [29] conducted flight tests using high-power solid rockets equipped with aerospike nozzles, providing the first transonic performance data under real conditions. These data are essential for understanding the behavior of aerospike nozzles across different flight regimes and refining design and simulation models. Additionally, Gould demonstrated that results obtained through analytical methods, such as the Method of Characteristics, align with higher-fidelity numerical simulations, validating the use of these methods in aerospike nozzle design [3]. The two NASA Dryden flights achieved Mach 1.5 and 30,000 ft, and post-processing gave a nozzle efficiency of 0.96–0.97, perfectly matching Vulcan-CFD predictions. A throat-area machining error of 4% was sufficient to explain the slightly lower chamber pressure relative to a conical baseline, illustrating the tight tolerance budget for annular throats. In parallel, Gould’s design study shows that truncating the spike to 40% of its theoretical length preserves 99% of full-length $I_{sp}$ while halving spike mass, which is an attractive option for mass-constrained small launchers.

2. Theoretical Basis in Space Propulsion

The design of rocket nozzles aims to expand the combustion gases until the exit pressure matches the ambient pressure, thereby maximizing thrust. However, ambient pressure $P_a$ decreases as the rocket ascends, posing a challenge for conventional nozzles, which operate efficiently only at a specific altitude [30]. By adjusting their flow characteristics to the varying atmospheric pressure, aerospike nozzles offer a significant advantage in this regard.

To quantify this impact, the thrust coefficient ($C_f$) is used, a dimensionless parameter that compares the actual nozzle thrust to the ideal thrust that would be achieved if the gases expanded to zero ambient pressure [38]:

$$C_f={\displaystyle \frac{F}{P_c{A}_t}}=\sqrt{{\displaystyle \frac{2{\gamma }^2}{\gamma -1}}{\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{{\displaystyle \frac{\gamma +1}{\gamma -1}}}\left[1-{\left({\displaystyle \frac{P_e}{P_c}}\right)}^{{\displaystyle \frac{\gamma -1}{\gamma }}}\right]}+{\displaystyle \frac{P_e-P_a}{P_c}}{\displaystyle \frac{A_e}{A_t}}$$

(1)

where F is the thrust, $P_c$ is the chamber pressure, $A_t$ is the throat area, $P_e$ is the exit pressure, $P_a$ is the ambient pressure, $\gamma$ is the specific heat ratio, and $A_e$ is the exit area. Equation (1) illustrates the three possible scenarios:

• Overexpanded nozzle ($P_e<P_a$): The exit pressure is lower than the ambient pressure. The pressure thrust term is negative, reducing the total thrust. Shock waves can form inside the nozzle, negatively affecting performance and potentially causing structural damage.
• Underexpanded nozzle ($P_e>P_a$): The exit pressure is higher than the ambient pressure. The pressure thrust term is positive, increasing the total thrust but at the cost of incomplete flow expansion. Expansion waves form outside the nozzle, indicating that thermal energy is not fully converted into kinetic energy.
• Perfectly expanded nozzle ($P_e=P_a$): The exit pressure matches the ambient pressure. The pressure thrust term is nullified, maximizing $C_f$ for a given area ratio. This condition represents the ideal scenario for a nozzle operating at a fixed altitude.

Due to their design, aerospike nozzles maintain an almost perfectly expanded flow throughout the ascent (see Figure 2), adapting to the varying atmospheric pressure and optimizing $C_f$ across the entire altitude range. The base pressure of the aerospike naturally equalizes with ambient pressure, nullifying the pressure thrust term, unlike conventional nozzles. This adaptability makes aerospike nozzles ideal for single-stage-to-orbit (SSTO) vehicles, opening new possibilities for space transportation.

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.

To predict the thermodynamic behavior in the nozzle, simplified models are employed for preliminary analysis without capturing the full complexity of the real process. The frozen flow model assumes that composition remains constant during expansion, ignoring additional reactions that could release extra energy. Conversely, the chemical equilibrium flow (CEQ) model assumes that species react instantaneously, reaching a local equilibrium that minimizes Gibbs free energy at every point [40].

Although these models are useful for initial analysis, they have limitations. The frozen flow model tends to underestimate performance by neglecting additional reactions, while the CEQ model may overestimate performance by assuming instantaneous reactions without considering finite reaction kinetics [41]. Moreover, gas–wall friction and heat losses prevent achieving ideal isentropic expansion, reducing flow efficiency.

For more realistic predictions, two-dimensional simulations combining computational fluid dynamics (CFDs) with flow chemistry models are used. Specialized tools such as the TDK (Two-Dimensional Kinetics) code allow for the simulation of reactive flows in nozzles, considering finite chemical kinetics, friction, and heat transfer, providing a more accurate analysis of real engine behavior [42].

Finally, the expansion of gases in the aerospike nozzle is crucial for converting thermal energy into kinetic energy and generating the thrust needed for rocket propulsion. The aerospike nozzle’s geometry, which utilizes ambient air as a “virtual wall,” allows expansion to naturally adjust to varying atmospheric pressure during ascent, maintaining engine efficiency over a wide altitude range [38].

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 ($I_{sp}$). 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

$$v_e=\sqrt{{\displaystyle \frac{2\gamma R_u{T}_c}{(\gamma -1)m}}\left[1-{\left({\displaystyle \frac{P_e}{P_c}}\right)}^{{\displaystyle \frac{\gamma -1}{\gamma }}}\right]},$$

(2)

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

$$\dot{m}={\displaystyle \frac{P_c{A}_t}{c^{*}}}={\displaystyle \frac{P_c{A}_t}{\sqrt{\frac{R_u{T}_c}{\gamma m}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}}}},$$

(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

$$I_{sp}={\displaystyle \frac{F}{\dot{m}{g}_0}}={\displaystyle \frac{v_e}{g_0}}+{\displaystyle \frac{(P_e-P_a)A_e}{\dot{m}{g}_0}},$$

(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, $I_{sp}$;
• 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 ($\gamma ={\displaystyle \frac{c_p}{c_v}}$): 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 ($\gamma$) 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 $a=\sqrt{{\displaystyle \frac{\gamma R_{u}T}{m}}}$. A higher speed of sound allows for higher flow velocities at the same temperature.

Table 1. Examples of Propellants and Their Combustion Products.

Propellant	Combustion Products	Characteristics
Liquid Hydrogen (LH2)/Liquid Oxygen (LOX)	Primarily H2O (with dissociation into H, O, and OH at high temperatures)	Low molecular weight in exhaust gases, favoring high exhaust velocity and high specific impulse.
Kerosene (RP-1)/Liquid Oxygen (LOX)	H2O, CO2, CO, and other compounds	Used in engines such as SpaceX’s Merlin; this system is dense, easy to handle, and produces exhaust gases with higher molecular weight.
Methane (CH4)/Liquid Oxygen (LOX)	Primarily H2O and CO2, with traces of CO	A clean and promising fuel for next-generation engines, offering high efficiency and promoting engine reusability.
UDMH/Dinitrogen Tetroxide (N2O4)	Variety of products (including H2O, N2, and CO2, among others)	Hypergolic system with spontaneous ignition, ideal for orbital maneuvers and attitude control systems.
Aerozine-50/Dinitrogen Tetroxide (N2O4)	Mixture of H2O, N2, CO2, and other compounds	Reliable hypergolic propellant, used in upper stages and space applications requiring immediate ignition and high precision.

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;
• $\gamma$ (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-$\omega$, 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.

3. Comparison Between Laval and Aerospike Nozzles

The aerodynamic design of the nozzle is paramount to rocket engine performance. Key configurations include the classic Laval nozzle, known for its simplicity but limited to peak efficiency at a single design altitude, and the aerospike nozzle, which offers superior altitude adaptation through external expansion, albeit with greater design and operational complexities.

3.1. Operating Principles and Efficiency

3.1.1. Laval Nozzle

Named after the Swedish engineer Gustaf de Laval, this type of nozzle operates based on the isentropic expansion of exhaust gases through a convergent–divergent geometry. The efficient conversion of the gas’s thermal energy into kinetic energy is achieved by varying the cross-sectional area along the nozzle.

Within Laval nozzles, several geometric configurations are distinguished based on the shape of the divergent section, a critical element for flow expansion. The most notable among these are illustrated in Figure 3.

• Conical nozzles: In this design, the divergent section is a linear extension of the throat. Although its simplicity facilitates manufacturing, the abrupt transition in the exit area limits the optimization of flow expansion, increasing the risk of flow separation, especially when there is a mismatch between the exit pressure and the ambient pressure.
• Bell nozzles: These nozzles incorporate curvature in the divergent section, allowing for a gradual area profile. The bell geometry promotes a more uniform isentropic expansion, minimizing the formation of internal shock waves and reducing the tendency for flow separation, resulting in a more efficient conversion of enthalpy into kinetic energy.

This study focuses exclusively on the analysis of bell nozzles, as their curved geometry allows for a superior expansion profile compared to conical nozzles, reducing losses associated with flow separation. Additionally, this configuration is the most commonly used in modern space propulsion, and its technological development is highly mature, ensuring high reliability.

The operation of the bell nozzle is described in Figure 4:

• Convergent section: The convergent section compresses and accelerates the subsonic flow from the combustion chamber, channeling the gas energy to transform it into velocity.
• Throat: The throat, with the minimum area, is the point where the flow reaches sonic velocity (Mach 1). The throat area, determined by the mass flow rate and the propellant properties, sets the maximum flow limit.
• Divergent section (bell):The divergent section, with a curved profile, enables controlled flow expansion, reducing gas pressure and density while increasing flow velocity. This geometry facilitates smooth and efficient expansion, minimizing the generation of internal shock waves.

Ideal performance is achieved under isentropic flow conditions. In this adiabatic and reversible process, the conversion of enthalpy into kinetic energy is maximized, generating the highest possible theoretical thrust [54].

3.1.2. Aerospike Nozzle

The aerospike nozzle is characterized by the absence of an external rigid wall, allowing the flow expansion to naturally adjust to variations in atmospheric pressure. Instead of expanding the gases through a solid divergent wall, the aerospike utilizes a central spike or expansion ramp [27]. This configuration is the key to its main advantage: altitude compensation [46]. At low altitudes, the high atmospheric pressure compresses the rocket plume against the ramp, increasing the pressure on its surface. As the rocket ascends and atmospheric pressure decreases, the flow expands away from the spike [38]. This dynamic adjustment enables more consistent performance throughout the ascent, improving overall engine efficiency [55].

3.1.3. Efficiency in Rocket Engines

The performance of a rocket engine is characterized by two key non-dimensional parameters: the thrust coefficient $C_f$, and the specific impulse $I_{sp}$, which together measure how efficiently propellant mass is converted into thrust (see Equations (1) and (5)).

A bell (Laval) nozzle reaches peak efficiency only at its design altitude, where the exit pressure equals the ambient pressure. Below that altitude, it over-expands, and above it, it under-expands, so both $C_f$ and $I_{sp}$ drop off rapidly away from the design point. In contrast, an aerospike nozzle entrains the free stream along its external plume, allowing the flow to re-equilibrate continuously with the surrounding pressure; its thrust coefficient, therefore, remains almost flat with altitude, limiting the loss in specific impulse at sea level to a few tens of percent of the vacuum value.

Indeed, static-fire tests of the XRS-2200 linear aerospike quantify this behavior: the engine produced $I_{sp}=339\mathrm{s}$ at sea level—78% of its 436 s vacuum value—whereas a bell nozzle optimized for 30 km would forfeit roughly 35–40% under identical conditions because of severe over-expansion [56]. Furthermore, CFD-validated altitude sweeps by Johnson [57] corroborate the same “flat” aerospike curve vs. the peaked bell curve across 0–50 km, demonstrating that this advantage is empirical, not merely schematic. Figure 5 visually summarizes these empirically supported trends, clearly illustrating how the aerospike sustains near-optimal performance throughout the ascent, while the Laval nozzle is efficient only in a narrow altitude band centered on its design point.

The performance and efficiency of rocket nozzles depend largely on the operating environment and factors such as thermal management, design complexity, and integration with the vehicle.

Table 2. Comparison of operational characteristics: Laval nozzle vs. aerospike nozzle.

Aspect	Laval Nozzle	Aerospike Nozzle
Atmospheric Ascent	Optimized for a specific altitude. At low altitudes, high pressure can cause overexpansion and flow separation; at high altitudes, underexpansion occurs, reducing efficiency [2].	Automatically adapts to variations in ambient pressure, maintaining efficient expansion throughout ascent and optimizing specific impulse [3].
Vacuum Operation	Designed with a high expansion ratio to maximize performance in a vacuum; its simplicity and lower weight make it favorable [2].	Offers potentially superior vacuum performance due to higher achievable effective expansion ratios and continuous adaptation, leading to significant $I_{sp}$ gains (e.g., 5–6% reported) [29]. While system complexity and inert mass can be higher compared to simple bell nozzles [7], the substantial propellant savings can be mission-enabling for high-$\Delta v$ applications [31,32,33].
Thermal Management	The convergent section and throat experience high temperatures; uses cooling systems (regenerative or ablative) and high-temperature-resistant materials [30].	The central spike is subjected to high thermal loads, requiring more complex cooling systems and advanced materials, increasing design complexity [24].
Thrust Vector Control	Typically relies on mechanical systems (gimbals); mature technology but adds weight and complexity [27].	Can achieve vectoring through flow variation or secondary fluid injection, offering potential for lighter designs but posing challenges in precise control [27].
Design and Manufacturing Complexity	Simple and well-understood design; easy to manufacture using conventional techniques.	Three-dimensional complex design requiring advanced manufacturing techniques, increasing development and production costs.
System Weight and Efficiency	Generally lighter, which can compensate for performance losses at certain altitudes.	Additional structures and cooling systems increase weight; the trade-off must be evaluated. New materials and manufacturing techniques may tip the balance [27,34,36].
Technological Maturity	Proven technology with long operational history.	Significantly less technological maturity and limited flight experience [14,15,27]. Requires further research and flight demonstrations.
Integration with Vehicle Design	Easily integrates into existing vehicles.	May require significant modifications with complex aerodynamic and structural considerations.

compares Laval and aerospike nozzles in different operational contexts.

4. Types of Aerospike Nozzles

The design of aerospike nozzles involves choosing from several configurations, each offering unique advantages and drawbacks concerning efficiency, weight, and manufacturability. This section focuses on the three main geometries: linear, toroidal, and annular, outlining the core design principles that dictate their application and performance.

4.1. Linear Aerospike Nozzles

Also known as flat-spike nozzles, these feature a straight expansion ramp forming a two-dimensional wedge. Combustion occurs in a series of chambers arranged along the base of the ramp, as illustrated in Figure 6. The RS-2200, developed by Rocketdyne for the X-33 program, is a notable example of this design [58].

The flow of hot gases is expelled from the combustion chambers and expands along the linear ramp. Due to the absence of an external wall, the flow interacts with the environment, allowing natural adaptation to atmospheric pressure. At the ramp edges, three-dimensional effects may occur, affecting the flow and posing a design challenge [51].

As summarized in

Table 3. Advantages and disadvantages of lineal aerospike nozzles.

Advantages

Disadvantages

Simple design: The two-dimensional geometry facilitates analysis and design, enabling faster and lower-cost development. This simplicity also allows for small-scale production, making them attractive for proof-of-concept testing or low-power engines. Uniform pressure distribution: The straight ramp promotes a uniform pressure distribution across its surface, reducing thermal and mechanical stresses and extending the nozzle’s lifespan.

High aerodynamic drag: Its larger frontal area compared to other configurations generates significant drag at low altitudes, affecting initial ascent efficiency. Limited scalability: Scaling this design to large sizes presents structural challenges due to the ramp’s weight and length. Edge losses: Flow at the ramp edges experiences turbulence and energy losses due to three-dimensional effects, reducing efficiency.

, linear aerospike nozzles offer manufacturing simplicity and good pressure distribution, though they also face aerodynamic and scalability limitations.

4.2. Toroidal Aerospike Nozzles

Toroidal aerospike nozzles feature a toroidal geometry, resembling a doughnut with a conical central spike. Combustion occurs in a circular chamber at the base of the torus, as seen in Figure 7.

The gas flow expands around the central spike in a three-dimensional geometry. The toroidal shape allows for a more uniform flow distribution around the spike, which can improve efficiency and reduce losses.

To further illustrate the relevance of this configuration,

Table 4. Advantages and disadvantages of toroidal aerospike nozzles.

Advantages

Disadvantages

Reduced frontal area: Its aerodynamic shape lowers drag, improving efficiency, especially at low altitudes. Efficient use of internal volume: The space within the torus can be used to store propellant or other components, optimizing mass and volume distribution in the vehicle.

Complex design and manufacturing: Its three-dimensional geometry demands a more intricate design and manufacturing process compared to linear aerospike nozzles, thereby increasing production costs. Additionally, the pronounced curvature of the spike presents challenges for machining precision and the integration of components. Non-uniform pressure distribution: The toroidal shape may generate pressure variations on the spike, affecting efficiency and impacting engine stability and control.

summarizes the main advantages and disadvantages of toroidal aerospike nozzles, highlighting both their aerodynamic benefits and the engineering challenges they pose.

4.3. Annular Aerospike Nozzles

Annular aerospike nozzles are the most commonly studied and used [27]. In this design, combustion does not occur in a central annular chamber but rather in multiple individual combustion chambers, each associated with an expansion nozzle. These individual nozzles are arranged annularly, surrounding a conical expansion ramp. Each individual nozzle generates an exhaust gas flow that expands around the central ramp, thus creating the aerospike effect, as illustrated in Figure 8.

Table 5. Advantages and disadvantages of annular aerospike nozzles.

Advantages

Disadvantages

Good altitude compensation: Maintains a high $C_f$ over a wide range of altitudes [46,55]. Compact and lightweight design: More compact and lightweight than Laval nozzles with the same expansion ratio, making them ideal for weight-sensitive vehicles. Aerodynamic thrust vectoring: Injecting a secondary fluid near the spike base allows for thrust vectoring without complex mechanical systems, simplifying design.

High heat flux: The throat and ramp are exposed to high temperatures, requiring robust cooling systems. Complex manufacturing: The conical ramp and annular combustion chamber present significant fabrication challenges. Metal 3D printing can help overcome some of these difficulties.

summarizes the advantages and disadvantages of annular aerospike nozzles.

5. Important Aspects in Aerospike Design

5.1. Aerodynamic Design of the Expansion Ramp

The performance of an aerospike nozzle largely depends on the shape of its expansion ramp. This component guides the supersonic flow and defines how the nozzle adapts to variations in atmospheric pressure. The aerodynamic considerations for the expansion ramp design are detailed below (

Table 6. Aerodynamic design factors in the expansion ramp of aerospike nozzles.

Factor	Description
Objective	Maximize isentropic expansion and minimize losses due to shock waves [47].
Altitude compensation	The geometry must adjust to pressure variations to reduce losses from overexpansion and underexpansion [38].
Minimization of drag	At low altitudes, the ramp should reduce aerodynamic drag.
Influence on cooling and manufacturing	A longer, near-ideal ramp increases the surface area exposed to high heat fluxes, which in turn raises coolant mass flow requirements, demands thicker structural walls, and adds weight. Moreover, maintaining tight machining tolerances (≤0.25 mm) is critical to minimize boundary-layer growth and preserve aerodynamic efficiency. However, to reduce complexity and mass, most flight programs adopt a truncated ramp—typically 40–60% of the ideal length. While truncation lowers cooling demands and overall weight, it creates a flat base that can cause significant base drag unless mitigated through secondary-flow injection, aft-body boat tailing, or a contoured “plug-cap” design [37,56]. Therefore, selecting an appropriate truncation ratio involves a trade-off between the vacuum $\Delta I_{sp}$ (typically $-1$% per 20% of ramp length removed), the coolant mass required, and performance penalties from base drag or bleed mass flow.

).

5.2. Material Selection

The choice of material is fundamental to ensuring the structural integrity and performance of the nozzle. Materials must withstand high temperatures, pressures, erosion, and thermal shocks. The main materials used are summarized in

Table 7. Comparison of materials used in rocket engines [61,62,63].

Material	Max Temp. (°C)	Density	Application
Inconel (Ni-Cr)	1000	Medium	High thermal and corrosion resistance.
Titanium alloys	600	Low	Lightweight but limited in temperature resistance.
Molybdenum/Tungsten	>2000	High	Resistant to extreme temperatures but dense and difficult to machine.
Carbon-Carbon (C/C)	>3000	Very low	Excellent for vacuum applications but requires oxidation protection.
Advanced ceramics	1500–2000	Low	High thermal resistance but brittle.

:

5.3. Conventional Manufacturing Processes

The fabrication of aerospike nozzles, with their characteristic complex contours and often integrated cooling channels, posed considerable difficulties and high costs when relying solely on conventional manufacturing techniques. These manufacturing hurdles were a significant factor in their limited historical use, especially when compared to the simpler fabrication of bell nozzles [34]. The advent and maturation of additive manufacturing (see Table 8) now offer pathways to produce these intricate geometries with potentially reduced lead times and costs, thereby improving the economic viability of aerospike engines.

Flight-ready additive manufacturing (AM) of aerospike nozzles still faces four critical hurdles:

• Surface finish: Achieving a surface roughness below $R_a\approx 5\mathsf{\mu }\mathrm{m}$ on the expansion ramp to avoid a $2-3\%$ penalty in divergence efficiency.
• Channel integrity: Guaranteeing the structural integrity of thin-walled regenerative channels (wall thickness $0.3-0.5$ mm), which are built layer-wise and prone to lack-of-fusion defects.
• Residual stress: Managing residual tensile stresses in large Inconel- or CuCrZr-printed plugs, which can exceed $300\mathrm{MPa}$ and distort the plug contour by $>0.2$ mm unless mitigated through in-situ heat treatment or hot-isostatic pressing.
• Inspection techniques: Developing non-destructive inspection (NDI) methods—such as high-energy X-ray CT or phased-array ultrasonics—capable of scanning an entire $600-900$ mm spike and detecting sub-$200\mathsf{\mu }\mathrm{m}$ internal flaws.

Recent work at NASA Marshall on integral-channel nozzles [64], the DLR L-PAK copper plug program [65], and the RAMFIRE aluminum-plug demonstrator [66] of NASA highlights ongoing solutions to each of these challenges and defines the current gate-to-flight qualification of large AM aerospikes.

Table 8. Comparison of manufacturing processes for aerospike nozzles [63,67,68,69].

Process	Advantages	Disadvantages
CNC Machining	High precision and strict tolerances.	Expensive and slow for complex geometries.
Welding and Brazing	Useful for assembling large components.	Can introduce defects and weak zones.
Metal Injection Molding (MIM)	Efficient for mass production.	Limited in size and mechanical properties.
3D Printing(SLM, EBM, DED)	Enables complex geometries and monolithic parts.	Requires post-processing and rigorous quality control.

Note: SLM = Selective Laser Melting; EBM = Electron Beam Melting; DED = Directed Energy Deposition.

Table 8. Comparison of manufacturing processes for aerospike nozzles [63,67,68,69].

Process	Advantages	Disadvantages
CNC Machining	High precision and strict tolerances.	Expensive and slow for complex geometries.
Welding and Brazing	Useful for assembling large components.	Can introduce defects and weak zones.
Metal Injection Molding (MIM)	Efficient for mass production.	Limited in size and mechanical properties.
3D Printing(SLM, EBM, DED)	Enables complex geometries and monolithic parts.	Requires post-processing and rigorous quality control.

Note: SLM = Selective Laser Melting; EBM = Electron Beam Melting; DED = Directed Energy Deposition.

6. Methods for the Design of Aerospike Nozzles

6.1. Application of the Method of Characteristics in the Design of Aerospike Nozzles

The design of aerospike nozzles requires precise adjustment of the exhaust flow expansion to match atmospheric pressure during the rocket’s ascent. Traditionally, the Method of Characteristics (MOCs) has been a fundamental tool for analyzing supersonic flows, as it enables solving hyperbolic PDEs by tracing lines along which flow perturbations propagate.

In the classical MOCs approach, a complete characteristic network is constructed, consisting of two families:

• Direct characteristics ($C^{+}$): Propagate in the direction $\theta +\mu$;
• Inverse characteristics ($C^{-}$): Propagate in the direction $\theta -\mu$.

Here, $\theta$ is the local flow angle relative to the longitudinal axis, $\mu$ is the Mach angle (defined as $\mu =arcsin(1/M)$), and M is the local Mach number. Using compatibility equations at each intersection of these lines, local flow properties (Mach number, pressure, flow angle, etc.) can be determined with high accuracy [54]. However, Lee [70] presents a simplified iterative method for computing plug nozzle contours based on the same principles of isentropic expansion and the Prandtl–Meyer function but avoiding the explicit construction of the full characteristic network. The procedure follows these steps:

• Determination of the exit Mach number ($M_e$): The isentropic area ratio relation is used, and an iterative procedure (e.g., Newton–Raphson) is applied to compute $M_e$ from throat conditions and the required expansion ratio.
• Calculation of the total expansion angle: Once $M_e$ is obtained, the total flow turning angle is determined using the Prandtl–Meyer function:

  $$\nu \left(M\right)=\sqrt{{\displaystyle \frac{\gamma +1}{\gamma -1}}}arctan\left(\sqrt{{\displaystyle \frac{\gamma -1}{\gamma +1}}(M^2-1)}\right)-arctan\left(\sqrt{M^2-1}\right),$$

  (6)

  where $\gamma$ is the specific heat ratio.
• Discretization of expansion: The expansion process is divided into small Mach number increments, starting from $M=1$ (at the throat) to $M=M_e$ (at the exit). At each step, the flow angle increment ($\Delta \theta$) is computed based on the difference in the Prandtl–Meyer function.
• Contour construction: Using mass conservation and the expansion wave geometry, axial and radial coordinates defining the nozzle contour are determined. This procedure iteratively produces the final shape of the plug nozzle.

Figure 9 illustrates the construction of the nozzle contour using the Method of Characteristics (MOC), based on intersecting right- and left-running characteristic lines. The solid black lines represent the expansion characteristics; the dashed blue lines show the reflected characteristics that define the contour of the nozzle. The filled black circles indicate the calculated flow states, and the white circles indicate the boundary or initial conditions used in the method.

We highlight the difference between the full MOCs approach and the simplified method employed in [70]. In the traditional MOCs approach, a network of $C^{+}$ and $C^{-}$ lines is traced, allowing for a detailed resolution of flow perturbations and achieving a highly accurate solution. This method involves the following:

• Tracing both families of characteristics;
• Applying compatibility equations at each intersection;
• Solving a system of equations describing the local flow evolution.

In contrast, the method described in [70] relies on isentropic relations and the Prandtl–Meyer function to directly compute the nozzle contour through discrete Mach number increments. This approach involves the following:

• It does not require explicit construction of the full characteristic network;
• It reduces computational complexity and facilitates implementation (Python, Labview, Matlab, and FORTRAN);
• Provides a good approximation of the actual flow behavior, making it suitable for preliminary design analysis and optimization.

Theoretical and Practical Considerations

The Prandtl–Meyer equations and the Method of Characteristics provide a robust theoretical foundation for the design of aerospike nozzles. However, several considerations must be taken into account:

• Isentropic flow assumption: Both methods assume isentropic flow, implying the absence of viscous and heat losses. In real applications, these assumptions may introduce discrepancies that require correction through numerical methods or CFDs.
• Simplified geometry: The model is formulated for two-dimensional or axisymmetric flows. In the design of complex nozzles, adjustments may be required to capture three-dimensional effects.
• Iterative method vs. full MOCs: While the full MOCs provides greater accuracy, the simplified iterative method presented in [70] is more straightforward and computationally efficient, making it suitable for preliminary design analysis.

6.2. Advanced Fluid Simulation Using Computational Fluid Dynamics (CFDs)

Computational fluid dynamics (CFDs) has become a fundamental tool in the design and optimization of rocket nozzles, enabling the analysis of complex supersonic flow behavior with unprecedented detail. By numerically solving the Navier–Stokes equations, CFD provides deep insight into the fluid field, including shock wave formation, flow expansion, interaction with atmospheric pressure, and the effects of viscosity and turbulence [58,71].

Various studies have demonstrated the capability of CFDs to accurately simulate flow in rocket nozzles. For instance, Stoffel [55] used ANSYS Fluent to simulate the exhaust flow of an aerospike nozzle, while Abada et al. [52] employed this tool to validate the design of a minimum-length nozzle. Khare and Saha [27] highlighted the use of CFD codes to analyze nozzle contour geometry, and Shahrokhi and Noori [50] utilized turbulence models along with the Navier–Stokes equations to simulate the fluid field.

6.2.1. Key Considerations in CFD Simulations of Aerospike Nozzles

The success of CFDs simulations in aerospike nozzle design depends on several critical factors that must be carefully considered:

• Turbulence modeling and selection of physical models: The proper selection of turbulence models is essential to capture the flow phenomena accurately. Reynolds-Averaged Navier–Stokes (RANS) models, such as k-$ϵ$ and k-$\omega$ SST, are commonly used due to their balance between accuracy and computational efficiency. The k-$\omega$ SST model is particularly effective for flows with adverse pressure gradients, typical in nozzles, as it improves the prediction of flow separation along the walls [55].
  However, to resolve small-scale turbulent structures with higher fidelity, Large Eddy Simulation (LES) or detached eddy simulation (DES) can be employed, although they require significantly higher computational resources [50,72]. Although CFDs can incorporate chemical reactions in aerospike nozzles, this is generally unnecessary, as combustion occurs in the chamber rather than in the nozzle itself. Therefore, simulations focus on the expansion and acceleration of exhaust gases through the nozzle [27].
• Computational mesh quality and boundary conditions: The accuracy of CFD simulations heavily depends on the quality of the computational mesh. An inadequate mesh can lead to imprecise results or numerical divergence. Conducting a mesh independence study is essential to ensure that the results are not dependent on mesh size or structure [71]. Boundary conditions must be precisely defined, including inlet pressure, temperature, and mass flow rate. The proper definition of these conditions is crucial for replicating real operational scenarios and obtaining reliable results. Additionally, the validation of simulation results with experimental data is fundamental for ensuring their credibility [73].
• Specific challenges in aerospike nozzle simulations: Aerospike nozzles present unique challenges in CFD simulations due to their geometry and operational characteristics:

  –

  Effects of variable altitude: The performance of aerospike nozzles is influenced by variations in atmospheric pressure with altitude. Simulations must accurately model this effect to predict performance under different flight conditions [31]. This entails resolving the plume–freestream interaction that drives the external expansion process, as well as tracking altitude-dependent shock structures such as the lip shock that forms where the escaping jet meets the ambient flow. During the transonic climb, these shocks can oscillate between attachment and detachment, altering the pressure distribution along the spike and the vehicle after-body. Capturing that unsteady behavior requires at least a dual-sweep (static + dynamic-mesh) approach or a quasi-steady sequence of operating points over $P_a/P_c=1.0-0.01$.

  –

  Interaction with external flows: The interaction between the exhaust flow and the external flow around the vehicle can impact nozzle efficiency. The proper modeling of this interaction is necessary for an accurate performance evaluation [74].

  –

  Flow separation phenomena: Flow separation at the base of the central spike is common in aerospike nozzles and can negatively affect thrust. CFDs models must predict this phenomenon to optimize nozzle design [49]. Predicting the onset, extent, and unsteadiness of separation on truncated plugs—particularly during throttled sea-level operation—is essential because it governs base pressure, overall thrust, and potential side loads. Current practice relies on turbulence closures that cope with adverse pressure gradients, such as the SST $k-\omega$ model or hybrid RANS–LES approaches. He et al. [49] showed that SST $k-\omega$ predicted the separation line on a $L/L_{\mathrm{ideal}}=0.6$ annular plug within one throat diameter, whereas the standard RNG $k-\epsilon$ displaced it by 3–4 diameters. Even so, large, three-dimensional separated regions remain highly unsteady; detached eddy simulation (DES) or wall-modeled LES is often required to capture the low-frequency breathing of the recirculation bubble and the consequent oscillations in base pressure.

6.2.2. Integration of Complementary Methods in Nozzle Design

To achieve a robust and efficient design, it is essential to complement CFDs with analytical and empirical methods, leveraging the strengths of each approach.

• Method of Characteristics (MOCs) and analytical models: As discussed, the MOCs is an analytical technique used for preliminary nozzle design, providing rapid solutions for two-dimensional isentropic flows [53]. Although it does not capture viscous or turbulent effects, it serves as a solid foundation for establishing an initial nozzle geometry, which can then be refined using CFDs simulations.
• Empirical models and experimental validation: Empirical models, based on experimental data and historical correlations, allow design adjustments based on past experiences [75]. Experimental validation is essential for confirming the accuracy of CFDs simulations and analytical models. Recent experimental studies using techniques such as Particle Image Velocimetry (PIV) and Pressure-Sensitive Paint (PSP) have provided valuable data for understanding and validating flow phenomena in hypersonic nozzles [76].
• Comparison and synergy of methods: The integration of different methods enables a multi-perspective approach to nozzle design:

  –

  CFD: Provides detailed modeling of complex three-dimensional phenomena, including viscosity and turbulence effects. However, it requires high computational resources and experimental validation.

  –

  MOCs and analytical methods: Offer rapid solutions and are useful for preliminary design but have limitations in modeling complex phenomena.

  –

  Experimental testing: Provides real-world data to validate models and understand physical phenomena, although it is costly and limited in the conditions it can simulate.

An efficient approach involves using the MOCs to obtain an initial geometry, followed by refinements through CFDs and experimental validation. This combination leverages the strengths of each method to optimize performance while reducing development costs and time [53,77].

7. Future Perspectives

Space propulsion is undergoing a transformative phase driven by innovations in design and technology. This section analyzes emerging trends, identifies current challenges, and outlines research directions that can accelerate the adoption of aerospike nozzles.

7.1. Emerging Trends in Design

• Advanced materials and additive manufacturing: The use of advanced materials—such as ceramic matrix composites (CMCs) and carbon–carbon (C/C) composites—provides high thermal and mechanical resistance. Additive manufacturing (3D printing) enables the creation of complex geometries, integrating, for example, internal cooling channels, which reduce weight and improve thermal efficiency [48,67].
• Computational optimization and machine learning: Computational fluid dynamics (CFDs) simulations facilitate the evaluation of extreme conditions with high precision. Moreover, integrating optimization algorithms and artificial intelligence accelerates the identification of optimal configurations by leveraging large volumes of experimental and simulation data [78,79].

7.2. Current Challenges and Areas for Improvement

• Thermal management and material limitations: Effective thermal management (especially for the central spike, which experiences extreme heat fluxes) has been a persistent and critical challenge directly impacting engine life, reusability, and, ultimately, operational cost [61,80,81]. Historically, limitations in materials capable of withstanding such temperatures for extended durations, coupled with the complexities of integrating robust cooling systems (a major concern in the 1970s) without undue weight penalties, were major deterrents. The development and qualification of advanced alloys (e.g., niobium and molybdenum-based) and innovative cooling strategies (e.g., regenerative, transpiration cooling, or the use of supercritical fluids) are paramount. Success here will not only enhance performance and durability but also potentially simplify designs, reducing manufacturing costs and improving the overall economic case for aerospikes, which is key to justifying investment [15,34].
• Design complexity and manufacturing scalability: The inherent complexity of aerospike nozzles presents significant challenges in modeling and manufacturing, even when employing 3D printing techniques, due to issues related to precision, repeatability, and scalability [68,69]. Manufacturing-informed design has become essential: shape-optimization studies now co-optimize ramp length, truncation ratio, and channel routing so that parts fit within LP-DED build envelopes (≤1.2 m) while respecting a global contour tolerance of 0.25 mm. NASA Marshall’s integral-channel work [64] and the L-PAK copper plug program of DLR [65] report that—even after HIP and XCT—yield for meter-class plugs is still <70%; automated distortion prediction and closed-loop process control are, therefore, required before true serial production.
• Integration with launch vehicles and control systems: Integrating aerospike nozzles into existing launch vehicles requires overcoming structural and aerodynamic compatibility challenges, in addition to developing precise thrust-vector control (TVC) systems [82]. For the X-33 lifting body, plume/airframe CFDs indicated pitching-moment shifts of 6–8% during the transonic climb [56]. Fluidic TVC through differential base-bleed can remove heavy gimbal hardware; wind-tunnel tests by Schoyer et al. demonstrated vector angles of up to $6^{\circ }$ but noted a sharp loss in authority below 30% throttle and a control-response requirement of <50 ms for lateral stability [83]. Proving such fast, high-authority control across the full altitude range remains an open task.

7.3. Future Research Directions

• Development of advanced materials and cooling technologies: Research into new composites—such as hafnium-carbide ceramics and high-entropy alloys—combined with the development of active cooling systems (e.g., using supercritical fluids) is essential for improving thermal management [62,63].
• Enhanced modeling and simulation techniques: High-fidelity DNS and LES provide detailed insight into plume dynamics and heat transfer, but their cost inhibits iterative design. Surrogate modeling and reinforcement learning (RL) are emerging as accelerators: Simpson [84] showed that a kriging surface trained on 420 aerospike CFDs cases predicted thrust-coefficient within 2% at 200× lower cost, while Neelakandan et al. [85] used deep-RL to generate contoured plugs that increased the vacuum $I_{sp}$ by 1.8% and reduced the wetted area by 7% under a prescribed cooling-flux limit. Embedding such AI agents inside multidisciplinary loops could shrink nozzle design cycles from months to days.
• Industrialization of additive manufacturing: Success cases from companies like Aerojet Rocketdyne and Relativity Space demonstrate the potential of 3D printing to reduce production time and costs for rocket components [86]. However, for aerospikes, improving the deposition of refractory materials, ensuring consistent material properties, establishing industrial quality standards, and scaling up for large engine components remain necessary [34]. Achieving cost-effective, serial production via AM is key to making aerospike nozzles a competitive alternative, thereby justifying their development costs. The RAMFIRE project of NASA recently hot-fired a 630 mm-diameter aluminum plug printed in 27 h, and it was post-processed with HIP/laser-peening to hold distortion below 0.15 mm, demonstrating the kind of end-to-end process qualification still required for flight [66].

7.4. Application of Aerospike Engines in SSTO Vehicles

The single-stage-to-orbit (SSTO) concept aims to radically simplify space access. However, achieving a viable SSTO with conventional nozzles is extremely challenging due to inefficiencies caused by varying atmospheric pressure. Aerospike engines, with their altitude compensation capability, present a key technology to make SSTO a reality. Although no SSTO vehicle equipped with aerospike engines has flown yet, the X-33 program in the 1990s represented a major initiative to flight-demonstrate linear aerospike engines (the XRS-2200) on a subscale SSTO demonstrator. The program was ultimately canceled in 2001 before flight tests could occur [87,88]. The primary technical factor leading to the cancellation was the repeated failure of the vehicle’s multi-lobed composite liquid hydrogen tank during testing [88,89,90]. This failure meant the X-33 could not achieve the necessary low mass fraction crucial for SSTO performance using the then-current tank technology. While the XRS-2200 aerospike engines had successfully met many of their performance goals in ground tests, they also faced development challenges, including cooling channel fabrication issues and achieving the target thrust-to-weight ratio. These engine-specific issues, combined with significant delays and cost overruns in other areas like the thermal protection system and avionics, contributed to the program’s vulnerability when the critical tank technology failed. The termination of the X-33, driven by the composite tank’s failure and the prohibitive cost of redesign, thus curtailed a key opportunity for the flight heritage of large-scale aerospike technology [89].

• Pangea Aerospace, ARCA Space, and others: These companies are developing aerospike engines using advanced techniques (3D printing and new materials) that could overcome historical obstacles [91,92].
• Stoke Space and Blue Origin: While not SSTO vehicles, their designs for reusable upper stages equipped with aerospike nozzles highlight the relevance of this technology for efficiency across multiple flight regimes [91].

8. Conclusions

The development of aerospike nozzles is in a continuous state of evolution, driven by advancements in materials (such as CMC and C/C) and manufacturing techniques, with metal 3D printing emerging as a promising solution to address the challenges of complex geometries and extreme operating conditions. These advancements are particularly relevant not only for the ambitious goal of single-stage-to-orbit (SSTO) vehicles but also for enhancing the performance of upper stages and interplanetary vehicles operating primarily in vacuum or near-vacuum conditions, where their high expansion ratio capability can yield significant $I_{sp}$ improvements.

Collaboration between industry, academia, and regulatory bodies is crucial to fostering innovation and establishing the necessary standards, which could promote broader adoption of this technology in space applications and enhance mission efficiency. Additionally, advancements in simulation and modeling (RANS, DNS, and LES and their integration with AI) are paving the way for the practical implementation of aerospike nozzles, positioning them as a potentially revolutionary alternative in the aerospace industry due to their superior efficiency and adaptability.

Nevertheless, the choice between a Laval and an aerospike nozzle depends on multiple factors, including mission profile, design constraints, costs, and acceptable risk levels. While the aerospike nozzle offers theoretical advantages in terms of efficiency and altitude adaptability, its practical challenges—such as complexity, weight, and technological maturity—must be carefully assessed. The historical preference for the Laval nozzle, with its simple design, ease of manufacturing, and extensive operational track record, was often driven by these factors and the higher perceived risk and cost associated with developing aerospike technology with the materials, manufacturing capabilities, and cooling solutions of previous eras. Many of these historical hurdles are now being systematically addressed by ongoing innovation.

Ultimately, the integration of new technologies, combined with sustained investment in research, development, and rigorous flight demonstration, is essential to overcoming existing challenges, building operational heritage, and fully harnessing the potential of aerospike nozzles. The path to wider adoption and the justification of their inherent development costs lies in demonstrably proving their net benefit—through successful ground and flight test campaigns that validate performance, reliability, and operational advantages for specific mission profiles where their unique characteristics offer a compelling advantage over established technologies. The future of aerospike engines is intrinsically tied to the feasibility of SSTO vehicles. If the challenges of weight and cooling can be overcome—with the aid of additive manufacturing, advanced materials, and improved cooling systems—aerospikes could become the technology that finally makes the dream of a single-stage-to-orbit vehicle a reality, revolutionizing access to space. The integration of aerospike propulsion with ultra-lightweight structures, potentially leveraging advancements like cryogenic propellants or staged combustion cycles, represents a key area for enhancing future space transportation systems. Progress in this domain, driven by multidisciplinary and collaborative research, could lead to significant improvements in the efficiency, affordability, and accessibility of space travel.