1. Introduction
Air-breathing propulsion offers a means of reducing the oxidiser burden associated with high-speed atmospheric flight (Bertin Reference Bertin1994; Heiser & Pratt Reference Heiser and Pratt1994; Ladeinde Reference Ladeinde2009; Urzay Reference Urzay2018). Scramjets use atmospheric oxygen and can therefore provide substantially higher specific impulse than rockets during endo-atmospheric operation (Moule et al. Reference Moule, Sabel’nikov, Mura and Smart2014; Vanyai et al. Reference Vanyai, Bricalli, Brieschenk and Boyce2018). Hydrogen-fuelled operation was demonstrated by the NASA X-43A at Mach 9.68 (Moses et al. Reference Moses, Rausch, Nguyen and Hill2004; McClinton Reference McClinton2006), while the X-51A demonstrated sustained hydrocarbon-fuelled operation near Mach 5 (Jackson et al. Reference Jackson, Gruber and Barhorst2009). Together with shock-tunnel studies (Suraweera & Smart Reference Suraweera and Smart2009; Ogawa & Boyce Reference Ogawa and Boyce2012; Smart Reference Smart2015), these programmes provide the experimental basis for extending scramjet concepts to higher flight Mach numbers.
Moving from Mach 10 to Mach 12 increases both compression loss and the temperature sensitivity of the reacting flow. Strong inlet shocks reduce total-pressure recovery and the work available during nozzle expansion (Billig Reference Billig1993; Moule & Smart Reference Moule and Smart2013; Zhang et al. Reference Zhang, Yang, Zhang, Qin and Bao2015). The cycle analysis of Zhang et al. (Reference Zhang, Yang, Zhang, Qin and Bao2015) further showed that increased contraction can raise the combustor temperature into a regime where the thermal limit offsets the benefit of compression. Skin friction also accumulates over the inlet and combustor; for the REST engine studied by Moule & Smart (Reference Moule and Smart2013), this drag removed a substantial fraction of the gross thrust at Mach 12. Aerodynamic considerations therefore favour a short high-pressure flowpath with limited wetted area.
Conversely, combustion considerations impose a conflicting requirement driven by the extreme enthalpy of the airstream. After flow compression, the peak temperature within the combustor in general exceeds 2500 K. Detailed kinetic studies indicate that at these temperatures, the chemical equilibrium shifts towards endothermic dissociation (e.g. H
$_2$
O
$\rightarrow$
OH + H). Dissociation becomes increasingly important as combustion proceeds in a high-temperature environment. This process acts as a substantial heat sink, retaining chemical energy in dissociated radicals rather than depositing it as sensible heat (Zhang et al. Reference Zhang, Yang, Zhang, Qin and Bao2015). Unless these radicals recombine within the nozzle, this energy is not recovered. In addition to this thermal challenge, the mixing limit further complicates the situation. The residence time of the airstream through a typical combustor is of the order of milliseconds, shifting the combustion regime from being kinetically limited to being mixing limited (Moura et al. Reference Moura, Gibbons, Wheatley, McIntyre and Jahn2020). Recent studies of oxygen enrichment and turbulence (Moura et al. Reference Moura, Gibbons, Wheatley and Jahn2021) emphasise that performance depends on a balance in which fuel penetration and micromixing occur efficiently and promptly. Consequently, combustion favours a longer passage that provides sufficient residence time for fuel–air mixing and reaction, in direct competition with the aerodynamic penalty of additional wetted length. For a cavity-stabilised supersonic combustor, Wei et al. (Reference Wei, An, Zhang, Zhou and Ren2023b
) used active-subspace analysis to quantify a related trade-off between total-pressure loss and a flame-stabilisation metric across coupled injector and cavity parameters.
These competing requirements define the combustor-length problem considered here. The selected condition has flight Mach number 12 and dynamic pressure 60 kPa, within established flight corridors for air-breathing ascent vehicles (Curran Reference Curran2001; Smart Reference Smart2007). It represents a regime in which the aerothermal penalty of higher dynamic pressure and the operability limit at lower pressure must both be considered.
A broad survey of this design space requires a reduced-order model. Three-dimensional calculations are needed to resolve local shocks, boundary layers and fuel plumes (Ogawa & Boyce Reference Ogawa and Boyce2012; Moule et al. Reference Moule, Sabel’nikov, Mura and Smart2014; Moura et al. Reference Moura, Gibbons, Wheatley and Jahn2021), but remain costly for two-parameter surveys. Quasi-one-dimensional (Q1D) models provide a complementary tool for preliminary system-level assessment (Birzer & Doolan Reference Birzer and Doolan2009; Chakir et al. Reference Chakir, Collins, Zope and Bhushan2023). They retain the principal effects of compressible heat addition, and can incorporate finite-rate chemistry at much lower cost than three-dimensional computational fluid dynamics (CFD) (Bityurin et al. Reference Bityurin, Lineberry, Litchford and Cole2000; O’Brien et al. Reference O’Brien, Starkey and Lewis2001; Tian et al. Reference Tian, Yang and Le2016).
In practical scramjet applications, engine performance is governed by a strict balance between macroscopic thermodynamic limits and local flow physics. For supersonic mixing-layer flames, inflow-velocity non-uniformity and wall heat loss can jointly alter the auto-ignition-controlled lift-off location and fuel conversion (Wei et al. Reference Wei, An, Wang, Zhang and Ren2023a ). Experiments and hybrid Reynolds-averaged Navier–Stokes (RANS)/large-eddy simulation calculations by Tang et al. (Reference Tang, Sun, Yan, Wang, Yu, Huang, Wang and Zhu2024) demonstrated how shock waves, vorticity dynamics and multi-scale turbulent structures interact with reactant mixing and jet-wake flame stabilisation in an axisymmetric combustor. Complementary large-eddy simulations of a spatially developing supersonic hydrogen mixing layer found that differential diffusion shortened the flame lift-off length, including in cases where turbulence dominated the flame-base region (Wei et al. Reference Wei, Zhu and Wang2025). From a macroscopic perspective, Zhang et al. (Reference Zhang, Yang, Zhang, Qin and Bao2015) used a Brayton-cycle model to evaluate scramjet performance at several high-Mach flight conditions. They showed that inlet irreversibility and the combustor temperature limit, often referred to as the thermal barrier, constrain the attainable specific thrust and fuel impulse. Such cycle analyses provide essential theoretical bounds, but do not resolve the streamwise evolution of mixing and finite-rate heat release.
As a natural progression of our previous work (Yu et al. Reference Yu, Zhang, Liu, Gu and Yue2026), the present Mach 12 Q1D analysis retains streamwise fuel mixing, finite-rate chemical kinetics, wall friction and non-constant-area expansion. It thereby connects the macroscopic thermal limitation to the local competition among dissociation, reaction freezing, residence time and aerodynamic loss. With the inlet contraction, fuel loading and whole-engine exit area fixed, the parameter maps determine whether the largest sampled values of combustion efficiency and fuel-specific impulse occur at the same configuration. The results are interpreted only within the stated parameter ranges, and are not used to claim a universal engineering optimum.
The remainder of this paper is organised as follows. Section 2 describes the Q1D formulation, finite-rate chemistry, validation data, performance measures and parameter space gradients. Section 3 examines the effects of combustor geometry (§ 3.1) and mixing-profile parameters (§ 3.2). Section 4 summarises the resulting physical trade-offs, and the limitations of the sampled design space.
2. Methodology and validation
This analysis employs a Q1D solver coupled with finite-rate chemical kinetics. The formulation accounts for variable specific heats and high-temperature dissociation rather than imposing chemical equilibrium. It is evaluated against Mach 10 experimental measurements and three-dimensional calculations (Yu et al. Reference Yu, Zhang, Liu, Gu and Yue2026) before being applied to the Mach 12 parameter study. The comparisons assess its ability to reproduce bulk streamwise trends; they do not establish accuracy for local multi-dimensional structures.
2.1. Governing equations and numerical formulation
In high Mach number scramjets, the flow is predominantly streamwise, indicating that molecular transport phenomena such as viscosity and thermal conduction are typically considered secondary. Fuel is generally introduced in gaseous form, and must effectively mix with the supersonic airstream prior to combustion. The subsequent chemical reactions markedly change the chemical and thermodynamic properties of the working fluid, such as heat capacity ratio and mean molecular weight, which in turn influence both flow dynamics and performance predictions (Yu et al. Reference Yu, Zhang, Liu, Gu and Yue2026). To address these considerations thoroughly, a generalised one-dimensional reactive flow model is utilised. Comprehensive details regarding the mathematical formulation are available in our previous study (Yu et al. Reference Yu, Zhang, Liu, Gu and Yue2026).
The reference state utilised for this analysis corresponds to the airstream conditions present at the entry of the scramjet inlet. In alignment with the conventions (Heiser & Pratt Reference Heiser and Pratt1994), these flow states are designated using the subscript 0. Thus the reference parameters encompass temperature (
$T_0$
), density (
$\rho _0$
) and velocity (
$V_0$
). To render the geometric parameters independent of specific cross-sectional shapes, the characteristic length scale is defined as the square root of the inlet capture area (
$\sqrt {A_0}$
). The non-dimensional variables are defined as
\begin{equation} \tilde {x}=\frac {x}{\sqrt {A_{0}}},\quad \tilde {A}=\frac {A}{A_{0}},\quad \tilde {\rho }=\frac {\rho }{\rho _{0}},\quad \tilde {V}=\frac {V}{V_{0}},\quad \tilde {T}=\frac {T}{T_{0}},\quad \tilde {p}=\frac {p}{\rho _{0}V_{0}^{2}}. \end{equation}
To maintain clarity throughout this discussion, the tilde notation for non-dimensional quantities has been omitted.
The governing equations are arranged into the canonical form discussed by Shapiro (Reference Shapiro1953):
\begin{equation} \begin{aligned} \frac {1}{M^2}\frac {{\rm d}M^2}{{\rm d}x} &= -\frac {2+\left (\gamma -1\right )M^2}{1-M^2}\frac {1}{A}\frac {{\rm d}A}{{\rm d}x} +\frac {1+\gamma M^2}{1-M^2}\frac {1}{c_{\!p}T}\left (\frac {{\rm d}Q}{{\rm d}x}+\frac {{\rm d}H_e}{{\rm d}x}\right ) \\[5pt] &\quad +\frac {\gamma M^2\left [2+\left (\gamma -1\right )M^2\right ]}{2\left (1-M^2\right )}\frac {4f}{D_H} +\frac {\left (1+\gamma M^2\right )\left [2+\left (\gamma -1\right )M^2\right ]}{1-M^2}\frac {1}{\dot {m}}\frac {{\rm d}\dot {m}}{{\rm d}x} \\[5pt] &\quad -\frac {1+\gamma M^2}{1-M^2}\frac {1}{W}\frac {{\rm d}W}{{\rm d}x}-\frac {1}{k}\frac {{\rm d}k}{{\rm d}x}, \end{aligned} \end{equation}
\begin{equation} \begin{aligned} \frac {1}{V}\frac {{\rm d}V}{{\rm d}x} &= -\frac {1}{1-M^2}\frac {1}{A}\frac {{\rm d}A}{{\rm d}x} +\frac {1}{1-M^2}\frac {1}{c_{\!p}T}\left (\frac {{\rm d}Q}{{\rm d}x}+\frac {{\rm d}H_e}{{\rm d}x}\right ) \\[5pt] &\quad +\frac {\gamma M^2}{2\left (1-M^2\right )}\frac {4f}{D_H} +\frac {1+\gamma M^2}{1-M^2}\frac {1}{\dot {m}}\frac {{\rm d}\dot {m}}{{\rm d}x} \\[5pt] &\quad -\frac {1}{1-M^2}\frac {1}{W}\frac {{\rm d}W}{{\rm d}x}, \end{aligned} \end{equation}
\begin{equation} \begin{aligned} \frac {1}{T}\frac {{\rm d}T}{{\rm d}x} &= -\frac {\left (\gamma -1\right )M^2}{1-M^2}\frac {1}{A}\frac {{\rm d}A}{{\rm d}x} +\frac {1-\gamma M^2}{1-M^2}\frac {1}{c_{\!p}T}\left (\frac {{\rm d}Q}{{\rm d}x}+\frac {{\rm d}H_e}{{\rm d}x}\right ) \\[5pt] &\quad -\frac {\gamma \left (\gamma -1\right )M^4}{2\left (1-M^2\right )}\frac {4f}{D_H} -\frac {\left (\gamma -1\right )M^2\left (1+\gamma M^2\right )}{1-M^2}\frac {1}{\dot {m}}\frac {{\rm d}\dot {m}}{{\rm d}x} \\[5pt] &\quad +\frac {\left (\gamma -1\right )M^2}{1-M^2}\frac {1}{W}\frac {{\rm d}W}{{\rm d}x}, \end{aligned} \end{equation}
\begin{equation} \begin{aligned} \frac {1}{\rho }\frac {{\rm d}\rho }{{\rm d}x} &= \frac {M^2}{1-M^2}\frac {1}{A}\frac {{\rm d}A}{{\rm d}x} -\frac {1}{1-M^2}\frac {1}{c_{\!p}T}\left (\frac {{\rm d}Q}{{\rm d}x}+\frac {{\rm d}H_e}{{\rm d}x}\right ) \\[5pt] &\quad -\frac {\gamma M^2}{2\left (1-M^2\right )}\frac {4f}{D_H} -\frac {\left (\gamma +1\right )M^2}{1-M^2}\frac {1}{\dot {m}}\frac {{\rm d}\dot {m}}{{\rm d}x} \\[5pt] &\quad +\frac {1}{1-M^2}\frac {1}{W}\frac {{\rm d}W}{{\rm d}x}, \end{aligned} \end{equation}
\begin{equation} \begin{aligned} \frac {1}{p}\frac {{\rm d}p}{{\rm d}x} &= \frac {\gamma M^2}{1-M^2}\frac {1}{A}\frac {{\rm d}A}{{\rm d}x} -\frac {\gamma M^2}{1-M^2}\frac {1}{c_{\!p}T}\left (\frac {{\rm d}Q}{{\rm d}x}+\frac {{\rm d}H_e}{{\rm d}x}\right ) \\[5pt] &\quad -\frac {\gamma M^2\left [1+\left (\gamma -1\right )M^2\right ]}{2\left (1-M^2\right )}\frac {4f}{D_H} -\frac {\gamma M^2\left [2+\left (\gamma -1\right )M^2\right ]}{1-M^2}\frac {1}{\dot {m}}\frac {{\rm d}\dot {m}}{{\rm d}x} \\[5pt] &\quad +\frac {\gamma M^2}{1-M^2}\frac {1}{W}\frac {{\rm d}W}{{\rm d}x}. \end{aligned} \end{equation}
The Mach number is
$M=V/\sqrt {\gamma \textit{RT}}$
, and the heat-capacity ratio is
$\gamma =c_{\!p}/c_v$
. Both quantities vary as fuel is added and the composition evolves. For species mole fractions
$X_i$
and molecular weights
$W_i$
, the mixture molecular weight is
$W=\sum _i X_iW_i$
.
In these canonical forms of the governing equations, the influencing terms, such as
${{\rm d}A}/{A}$
and
${{\rm d}Q}/{c_{\!p}T}$
, determine how flow properties vary within the scramjet duct. Detailed discussions of these specific terms are addressed as follows.
The factor
$H_{{e}}$
refers to enthalpy exchange between airstream and injected fuel during the thermal equilibrium process (Shapiro Reference Shapiro1953). The differential form of
$H_{{e}}$
is expressed as
\begin{equation} {\rm d}H_{{e}} = \left [ H_{\textit{total},\textit{fuel}} - \left ( \int _{T_{\textit{ref}}}^{T} c_{\!p,\textit{fuel}} \, {\rm d}T + \frac {1}{2} V^2 \right ) \right ] \frac {{\rm d}\dot {m}}{\dot {m}}, \end{equation}
where
$H_{\textit{total},\textit{fuel}}$
is the fuel total enthalpy at the injection condition,
$c_{\!p,\textit{fuel}}$
is the fuel heat capacity, and
${\rm d}\dot m$
is the mass-flow increment due to fuel injection.
Chemical heat release
$Q$
is evaluated from the composition change produced by reaction at the local reference temperature. Its differential is
\begin{equation} {\rm d}Q = -\sum _{i=1}^{N} h_i\,{\rm d}Y_{i, r}, \end{equation}
where
${\rm d}Y_{i,r}$
is the reaction-induced mass-fraction increment. The species enthalpy
$h_i$
depends on temperature and is obtained from the thermochemical database. In the Q1D formulation, the species equation is written as (Heiser & Pratt Reference Heiser and Pratt1994; Suraweera & Smart Reference Suraweera and Smart2009)
\begin{equation} \frac {\mathrm{d}Y_i}{\mathrm{d}x} = \frac {\dot {\omega }_i W_i}{\rho V} + \frac {\mathrm{d}Y_{i,\textit{mix}}}{\mathrm{d}x}, \end{equation}
where the first term describes finite-rate reaction, and
${\rm d}Y_{i,\textit{mix}}/{\rm d}x$
is the section-averaged composition change prescribed by the mixing closure. Fuel mass and enthalpy addition are retained separately through
${\rm d}\dot m/{\rm d}x$
and
${\rm d}H_e/{\rm d}x$
in the flow equations; they are not added a second time in (2.9). Hydrogen oxidation is represented by GRI-Mech 3.0 (Smith et al. Reference Smith1999).
The skin friction coefficient
$f$
quantifies the drag resulting from boundary layer flow, which cannot be resolved in a one-dimensional framework and thus needs practical models. In this study, we correlate the skin friction coefficient with the reference temperature at the wall and the Reynolds number calculated at that location (Suraweera & Smart Reference Suraweera and Smart2009):
\begin{equation} f = \frac {0.0574}{(\textit{Re}^*)^{1/5}} \times\frac {T}{T^*}, \end{equation}
where
$T^\ast$
is the reference temperature, and
$\textit{Re}^\ast$
is the Reynolds number corresponding to the reference state.
2.2. Mixing model validation
The Q1D model introduces gaseous fuel at prescribed axial positions, where it contributes mass and enthalpy to the stream. Multiple injection stages require a multi-dimensional description of plume interaction, and are not considered here. The present study uses one primary injection region, with gaseous hydrogen supplied at total temperature 300 K.
The mixing of fuel and air in a supersonic, compressible flow environment is an intricate hydrodynamic process involving turbulence and significant three-dimensional effects. Since a comprehensive, first-principles representation of these phenomena is beyond the scope of a Q1D framework, a spatially varying mixing efficiency
$\eta _{\textit{mix}}(x)$
is used to represent the progression of molecular mixing along the combustor. It defines the fraction of injected fuel that is mixed with oxidiser and available for reaction at a given axial location. The empirical closure (Heiser & Pratt Reference Heiser and Pratt1994; Yu et al. Reference Yu, Zhang, Liu, Gu and Yue2026) is
\begin{equation} \eta _{\textit{mix}}(x)= \begin{cases} 0, & x\lt x_{\textit{inj}},\\[3pt] \displaystyle \frac {1-\exp \!\left [-\alpha _{\textit{mix}}(x-x_{\textit{inj}})/L_{\textit{mix}}\right ]} {1-\exp (-\alpha _{\textit{mix}})}, & x_{\textit{inj}}\leq x\leq x_{\textit{inj}}+L_{\textit{mix}},\\[8pt] 1, & x\gt x_{\textit{inj}}+L_{\textit{mix}}. \end{cases} \end{equation}
Here,
$x_{\textit{inj}}$
is the non-dimensional location at which effective mixing begins. This mixing model is described by the mixing length
$L_{\textit{mix}}$
and the near-field mixing-rate factor
$\alpha _{\textit{mix}}$
. The former sets the completion distance, whereas the latter controls the degree to which the profile is front-loaded. For
$x_{\textit{inj}}\leq x\leq x_{\textit{inj}}+L_{\textit{mix}}$
, the initial slope is
\begin{equation} \left .\frac {\mathrm d\eta _{\textit{mix}}}{\mathrm {\rm d}x}\right |_{x_{\textit{inj}}} =\frac {\alpha _{\textit{mix}}} {L_{\textit{mix}}\left (1-\mathrm e^{-\alpha _{\textit{mix}}}\right )}. \end{equation}
The two parameters are independent inputs to the analytic closure, but should not be interpreted as orthogonal hardware controls.
Demonstration of control parameters in the mixing model: (a) mixing length and (b) local mixing rate.
To clarify the physical significance of these parameters, figure 1 evaluates their individual impacts on the spatial development of mixing efficiency. As shown in figure 1(a), a reduction in
$L_{\textit{mix}}$
leads to a more compressed mixing profile along the axial direction. This physically corresponds to conditions involving intense mixing mechanisms that rapidly disintegrate the fuel plume. In contrast, an increase in
$L_{\textit{mix}}$
results in an elongated mixing profile, indicative of a slower exchange of mass between fuel and air, such as that observed with wall-based injection within energetic shear layers.
Simultaneously, the near-field mixing rate factor (
$\alpha _{\textit{mix}}$
) influences the local topology or intensity of the mixing onset; see figure 1(b). Elevated values of
$\alpha _{\textit{mix}}$
result in a convex, front-loaded profile, indicating that the majority of the fuel interacts with the oxidiser immediately downstream of the injection point. Conversely, reduced values produce a more linear mixing efficiency profile, thereby delaying the bulk of fluid interaction.
The two parameters were calibrated against the section-averaged mixing history from a three-dimensional unsteady RANS (URANS) calculation of a representative strutless combustor. A Nelder–Mead simplex search gave
$\alpha _{\textit{mix}}=3.99$
and
$L_{\textit{mix},\textit{actual}}=1.93$
m. The available CFD interval does not extend to complete mixing, and consequently constrains the initial rise of the profile more strongly than its completion distance. These fitted values reproduce the reference history over the available comparison interval, but are not interpreted as uniquely identified injector constants.
Figure 2 compares the fitted empirical profile with the section-averaged three-dimensional result. The model captures the onset and rapid initial growth of mixing over the available interval. The semi-empirical closure reduces the computational cost while retaining a physically reasonable streamwise evolution of mixing efficiency. It should nevertheless be regarded as an effective section-averaged history rather than a unique representation of a particular injector.
The fitted mixing parameters were first used in a wall-pressure validation against a rectangular direct-connect combustor tested in the JF-24 detonation-driven shock tunnel at the Institute of Mechanics, Chinese Academy of Sciences (Yu et al. Reference Yu, Zhang, Liu, Gu and Yue2026). The model had constant width 210 mm and entrance height 50 mm. A 100 mm constant-area section was followed by a 1100 mm symmetric expansion with total included angle
$1^{\circ }$
, giving total axial length
$L_c=1.2$
m and inlet-to-exit area ratio 1.384. The streamwise origin was located at the model entrance. Fuel was injected symmetrically from the upper and lower walls at
$x=200$
and 260 mm. The test condition represented Mach 10 flight at altitude 33.5 km, angle of attack
$2^{\circ }$
, and dynamic pressure
$P_d=50$
kPa. The section-averaged values at the model inlet were
$M=3.80$
,
$p_t=10.61$
MPa,
$T_t=3575$
K,
$p=64.5$
kPa and
$T=1119$
K.
Wall pressure was measured at the 19 axial stations listed in table 1. The stations span
$0.120\leq x\leq 1.09169$
m. Figure 3 compares the Q1D predictions with the time-averaged measurements at equivalence ratios
$\phi =0.8$
and 1.0. Pressure and distance are presented as
$p/P_d$
and
$x/L_c$
, respectively.
Axial coordinates of the wall-pressure stations measured from the entrance of the Mach 10 validation model.
Table 1. Long description
A table with three columns and 19 rows. The columns are labeled Station, x (mm), Station, x (mm), and Station, x (mm). The table lists the axial coordinates of wall-pressure stations measured from the entrance of a Mach 10 validation model. Row 1: Station 1, x 120.00; Station 8, x 650.46; Station 15, x 930.44. Row 2: Station 2, x 340.55; Station 9, x 690.44; Station 16, x 970.44. Row 3: Station 3, x 380.55; Station 10, x 730.44; Station 17, x 1010.44. Row 4: Station 4, x 420.55; Station 11, x 770.44; Station 18, x 1051.79. Row 5: Station 5, x 460.55; Station 12, x 810.44; Station 19, x 1091.69. Row 6: Station 6, x 500.55; Station 13, x 850.44. Row 7: Station 7, x 610.44; Station 14, x 890.44.
Verification of the mixing efficiency profile against high-fidelity three-dimensional simulation data.
The comparison reveals that the one-dimensional (1-D) model successfully reproduces the primary aerodynamic and combustion features of the flow field. Specifically, the model accurately predicts the location of the rapid pressure rise associated with the onset of intense combustion and heat release. Furthermore, the magnitude and axial location of the peak pressure are captured with reasonable accuracy for both equivalence ratios, indicating that the finite-rate chemistry and heat-release mechanisms are correctly implemented. Following the peak, the model tracks the pressure decay in the expansion section, consistent with the experimental trend. It is noted that the experimental data exhibits local oscillations and scatter that are not present in the smooth one-dimensional profiles. These deviations are physically expected and attributable to complex three-dimensional wave structures, such as oblique shock trains and shock-boundary layer interactions, which are inherent to supersonic combustors but are spatially averaged in a Q1D formulation. Despite these local multi-dimensional effects, the strong agreement in the mean pressure distribution confirms that the current framework captures the dominant aerothermodynamic behaviour and energy conversion processes within the engine.
Validation of the Q1D wall-pressure distribution against time-averaged experimental measurements for (a)
$\phi =0.8$
and (b)
$\phi =1.0$
. Pressure and axial distance are normalised by the simulated flight dynamic pressure
$P_d=50$
kPa and the model length
$L_c=1.2$
m, respectively.
Wall pressure does not fully describe the thermodynamic state of the core flow. A second validation was therefore conducted against a three-dimensional Mach 10 hydrogen calculation of the same rectangular flowpath at the nominal
$\phi =1.0$
condition. The three-dimensional inlet retained its non-uniform distribution, while its cross-sectional averages matched those used by the Q1D model. The computational domain included the combustor and downstream nozzle, with the same prescribed nozzle area distribution as in the Q1D calculation. The quantitative Q1D–CFD validation is restricted to the combustor,
$0\leq x\leq 1.2$
m; the downstream field is shown only to provide context for the nozzle expansion.
The numerical procedure followed Zhang et al. (Reference Zhang, Yue, Yu, Liu and Zhang2026). The Favre-averaged URANS equations were solved using a pressure-based coupled finite-volume method. Convective fluxes were discretised by the third-order QUICK scheme, while pressure and diffusive terms used second-order central differences. Time integration was second-order implicit. The shear stress transport
$k$
–
$\omega$
model included viscous heating, compressibility and low-Reynolds-number corrections, and turbulence–chemistry interaction was represented by the finite-rate/eddy-dissipation formulation. The gas mixture was ideal, with temperature-dependent thermodynamic properties and kinetic-theory transport. A multi-block hexahedral grid was used with a first wall-normal spacing that maintained
$y^+\lt 1$
. Pressure-inlet conditions were imposed at the combustor and fuel inlets, the outlet was supersonic, and all walls were stationary, no-slip and adiabatic. Hydrogen oxidation was described by GRI-Mech 3.0 (Smith et al. Reference Smith1999).
For each plane normal to the streamwise direction, the plotted static pressure is area-averaged, whereas static temperature and velocity magnitude are weighted by the local mass flux:
\begin{equation} P(x)=\frac {1}{A}\int _A p\,\mathrm {\rm d}A, \quad T(x)=\frac {\int _A\rho u_nT\,\mathrm {\rm d}A}{\int _A\rho u_n\,\mathrm {\rm d}A}, \quad V(x)=\frac {\int _A\rho u_n\,|\boldsymbol{u}|\,\mathrm {\rm d}A}{\int _A\rho u_n\,\mathrm {\rm d}A}. \end{equation}
Here,
$u_n$
is the velocity component normal to the sampling plane, and
$|\boldsymbol{u}|$
is the local velocity magnitude. The normalising quantities
$P_3$
,
$T_3$
and
$V_3$
are the corresponding averages at the combustor inlet, evaluated using the same operators. None of these cross-sectional quantities is obtained from the wall-pressure measurements. Figure 4 compares the normalised profiles from the Q1D model and CFD calculation.
The relative difference between the CFD and Q1D profiles is evaluated using the Euclidean
$L_2$
-norm. For a flow variable
$f\in \{P,T,V\}$
,
\begin{equation} \varepsilon _f = \frac {\| f_{\textit{sim}} - f_{\textit{1-D}} \|_2}{\| f_{\textit{sim}} \|_2}, \end{equation}
where
$f_{\textit{sim}}$
and
$f_{\textit{1-D}}$
denote the CFD and Q1D values at the common sampling locations within
$0\leq x\leq 1.2$
m, respectively.
Validation of the Q1D model against cross-sectionally averaged three-dimensional CFD results over the
$0\leq x\leq 1.2$
m combustor at the Mach 10,
$\phi =1.0$
condition. Static pressure is area-averaged, while static temperature and velocity magnitude are mass-flux-weighted according to (2.13).
The Q1D profiles reproduce the principal changes in momentum and thermodynamic state. The largest discrepancies occur in the main heat-release region, where the three-dimensional calculation resolves transverse gradients, boundary layers and shock structures that are absent from the section-averaged formulation. The agreement in the bulk trends supports use of the Q1D model for the subsequent system-level parameter study, while the local differences delimit what can be inferred from the validation.
Figure 5 provides the corresponding centre-plane fields. The temperature field shows the development of a reacting core exceeding 2000 K, while the normalised pressure field resolves the oblique waves generated near the two injection stations. Here,
$p_{\textit{ref}}=P_3=64.5\ \mathrm{kPa}$
is the combustor-inlet static pressure; it is distinct from the flight dynamic pressure
$P_d=50\ \mathrm{kPa}$
used to normalise the experimental wall-pressure data in figure 3. These waves reflect within the weakly diverging combustor, and weaken as the flow enters the downstream nozzle. The Mach number field shows the associated combustor deceleration and subsequent acceleration through the nozzle. These localised structures explain the oscillations present in the wall-pressure measurements, and the differences between the three-dimensional and Q1D profiles.
Centre-plane CFD fields for the Mach 10 hydrogen calculation at
$\phi =1.0$
: (a) static temperature, (b) normalised static pressure
$p/p_{\textit{ref}}$
, where
$p_{\textit{ref}}=P_3=64.5\ \mathrm{kPa}$
, and (c) Mach number. The arrows identify the two fuel-injection stations.
Taken together, the wall-pressure and cross-sectional comparisons show that the Q1D formulation reproduces the principal streamwise pressure and thermodynamic trends over the Mach 10 validation interval. The remaining discrepancies are consistent with the transverse gradients, boundary layers and wave structures omitted by section averaging. The model is therefore used below to examine system-level sensitivities at Mach 12, rather than to predict local multi-dimensional flow features.
2.3. Flowpath geometry and performance measures
The station notation and principal geometric parameters are summarised in figure 6. The combustor extends from the inlet exit (station 3) to the nozzle entrance (station 4). Its dimensional length and area expansion ratio are
\begin{equation} L_{\textit{comb},\textit{actual}}=x_{4,\textit{actual}}-x_{3,\textit{actual}}, \quad \alpha _{\textit{comb}}=\frac {A_4}{A_3}. \end{equation}
The local combustor coordinate originates at station 3, so
$0\leq x_{{actual}}\leq L_{\textit{comb},\textit{actual}}$
and
$L_{\textit{comb}}=L_{\textit{comb},\textit{actual}}/\sqrt {A_0}$
. The downstream nozzle is a separate finite Q1D passage of length 1.5 m. For every combustor configuration, the whole-engine exit-to-capture area ratio is fixed at
$A_e/A_0=2.5$
. With the inlet contraction ratio
$C_R=A_0/A_3$
, the local nozzle area ratio is therefore
\begin{equation} \frac {A_e}{A_4}=\frac {2.5C_R}{\alpha _{\textit{comb}}}. \end{equation}
Varying
$\alpha _{\textit{comb}}$
consequently redistributes the fixed total expansion between the combustor and nozzle rather than changing the engine exit area.
Station and area definitions for the Q1D engine flowpath; the sketch is not to scale.
The geometry survey covers
$0.8\leq L_{\textit{comb},\textit{actual}}\leq 2.0$
m, equivalent to
$2.03\leq L_{\textit{comb}}\leq 5.08$
and
$1.0\leq \alpha _{\textit{comb}}\leq 2.0$
; the mixing inputs are held at
$L_{\textit{mix},\textit{actual}}=2.0$
m and
$\alpha _{\textit{mix}}=10$
. The mixing survey covers
$1.5\leq L_{\textit{mix},\textit{actual}}\leq 3.0$
m, equivalent to
$3.81\leq L_{\textit{mix}}\leq 7.62$
and
$5\leq \alpha _{\textit{mix}}\leq 15$
; the combustor geometry is held at
$L_{\textit{comb},\textit{actual}}=1.0$
m and
$\alpha _{\textit{comb}}=2.0$
. The injection position is fixed at
$x_{{inj,actual}}=0.10$
m and is not a third design parameter. Both surveys use
$M_0=12$
,
$P_d=60$
kPa,
$C_R=20$
,
$\phi =1$
and
$A_e/A_0=2.5$
.
Dimensional quantities are used in the following performance definitions. The exit pressure is obtained from the finite Q1D nozzle solution and is not forced to equal the freestream pressure. The net axial force is
\begin{equation} \begin{aligned} F_N &=(\dot mV+pA)_e-(\dot mV+pA)_0-p_0(A_e-A_0)\\ &=\dot m_eV_e-\dot m_aV_0+(p_e-p_0)A_e, \end{aligned} \end{equation}
where subscripts 0 and
$e$
denote the capture plane and nozzle exit, respectively. The fuel-specific impulse is
\begin{equation} I_{\textit{sp}}=\frac {F_N}{\dot m_f g_0}, \quad g_0=9.80665\ \mathrm{m\,s^{-2}}. \end{equation}
Finite-rate chemistry based on GRI-Mech 3.0 is integrated through both the combustor and nozzle. Neither frozen composition nor chemical equilibrium is imposed at station 4. Reaction freezing occurs when the local chemical time scale becomes long relative to the remaining convective residence time; no axial freezing station is prescribed.
Combustion and thermal efficiencies are defined as
\begin{equation} \eta _{\textit{comb}}=\frac {\dot Q_r}{\dot m_f\,{LHV}}, \quad \eta _{\textit{thermal}} =\frac {\dot m_eV_e^2/2-\dot m_aV_0^2/2}{\dot Q_r}, \end{equation}
where
$\dot Q_r$
is the net chemical heat-release rate evaluated from the 298.15 K reference-state chemical enthalpies. The numerator of
$\eta _{\textit{thermal}}$
is the increase in axial kinetic-energy flux between the engine inlet and exit. Thus
$\eta _{\textit{thermal}}$
measures the fraction of released chemical energy converted into axial kinetic energy; it is not a complete exergy efficiency. Pressure thrust is retained in
$I_{\textit{sp}}$
and is not included in the numerator of
$\eta _{\textit{thermal}}$
.
For differential heat addition
$\mathrm dq$
to a calorically perfect gas, the Rayleigh relations give
\begin{equation} \mathrm ds_Q=\frac {\mathrm dq}{T}, \quad -\left .\mathrm d\ln p_t\right |_Q =\frac {\gamma M^2}{2}\frac {\mathrm dq}{c_{\!p}T_t}. \end{equation}
Equal heat increments are therefore not thermodynamically equivalent when they occur at different local Mach numbers and thermodynamic states. In the reacting Q1D calculation, wall friction, area variation, fuel mass and enthalpy addition, variable thermodynamic properties and composition change are all retained. Equation (2.20) is consequently used only to interpret the influence of heat-release location;
$\eta _{\textit{thermal}}$
quantifies the resulting engine-level conversion into axial kinetic energy.
2.4. Evaluation of parameter space gradients
For a performance metric
$J\in \{I_{\textit{sp}},\eta _{\textit{comb}}\}$
, the gradients in the combustor geometry and mixing parameter spaces are defined as
\begin{equation} \boldsymbol{\nabla} _{\!g}J= \left (\frac {\partial J}{\partial L_{\textit{comb},\textit{actual}}}, \frac {\partial J}{\partial \alpha _{\textit{comb}}}\right )\!, \quad \boldsymbol{\nabla} _{\!m}J= \left (\frac {\partial J}{\partial L_{\textit{mix},\textit{actual}}}, \frac {\partial J}{\partial \alpha _{\textit{mix}}}\right )\!. \end{equation}
The derivatives are evaluated on the two discrete
$20\times 20$
grids using dimensional length coordinates in metres. Second-order central differences are used at interior points, and first-order one-sided differences are used at the boundaries. The vectors are placed at the corresponding non-dimensional abscissae,
$L=L_{{actual}}/\sqrt {A_0}$
, but their components are not rescaled during this linear coordinate mapping. No component-wise or magnitude normalisation is applied. The arrows therefore show the sign and relative strength of the local response under the dimensional-coordinate convention; their angle is coordinate-scale dependent, and is not interpreted as an invariant optimisation path. For presentation, MATLAB’s quiver function is applied with uniform scale factor 0.5 to reduce arrow overlap. Since the numerical ranges of
$I_{\textit{sp}}$
and
$\eta _{\textit{comb}}$
differ, absolute arrow lengths are not compared between performance maps.
3. Results and discussion
The parameter study considers Mach 12 flight at dynamic pressure 60 kPa. Under this condition, net thrust is the difference between a large nozzle impulse and inlet ram drag, and is therefore sensitive to skin friction, heat-release location and chemical non-equilibrium. The following subsections examine two sampled domains: combustor geometry (§ 3.1) and effective streamwise mixing histories (§ 3.2). Specific impulse, combustion efficiency and thermal efficiency are treated as complementary measures. Extrema are reported as the largest sampled values within the stated ranges; no unconstrained global optimum is inferred.
3.1. Effects of combustor geometry on performance
Combustor length determines the residence time available for mixing and finite-rate reaction, but also sets the wetted length over which friction accumulates. The area expansion ratio modifies pressure, temperature, velocity and hydraulic diameter, and thereby changes both chemical conversion and aerodynamic loss. The following results first isolate these effects along one-parameter cuts, then consider the complete
$20\times 20$
geometry grid defined in § 2.3.
3.1.1. The length trade-off: friction versus chemical potential
Figure 7 isolates the effect of combustor length along the upper boundary,
$\alpha _{\textit{comb}}=2.0$
. It shows specific impulse, combustion efficiency and cumulative friction drag as functions of non-dimensional length.
Effect of combustor length on scramjet performance metrics for a fixed expansion ratio (
$\alpha _{\textit{comb}}=2.0$
).
Specific impulse first rises and then falls along this cut. At short lengths, the additional residence time increases molecular mixing and reaction, and the resulting gain in heat release exceeds the accompanying increase in drag. At longer lengths, combustion efficiency approaches a broad maximum, while friction continues to accumulate over the wetted surface. The marginal chemical gain then becomes smaller than the momentum penalty, and
$I_{\textit{sp}}$
decreases. The peak in specific impulse consequently occurs before the peak in combustion efficiency along the
$\alpha _{\textit{comb}}=2.0$
cut. This result illustrates why additional reaction completeness is not necessarily advantageous at Mach 12.
3.1.2. The expansion trade-off: thermodynamic relief versus kinetic limitation
Figure 8 isolates the effect of combustor area expansion at the shortest sampled length,
$L_{\textit{comb}}=2.034$
(
$L_{\textit{comb},\textit{actual}}=0.800$
m).
Effect of combustor expansion ratio on scramjet performance metrics for a fixed combustor length,
$L_{\textit{comb}}=2.034$
(
$L_{\textit{comb},\textit{actual}}=0.800$
m).
Combustion efficiency varies non-monotonically because expansion has opposing equilibrium and kinetic consequences. A moderate decrease in the peak temperature within the combustor suppresses endothermic product dissociation, and increases the net heat release. Stronger expansion, however, lowers both temperature and pressure, and accelerates the flow, reducing reaction rates and residence time. The subsequent decrease in combustion efficiency is consistent with a growing kinetic limitation over the available reaction distance.
The friction measure decreases as
$\alpha _{\textit{comb}}$
increases because expansion increases the hydraulic diameter and lowers density and wall dynamic pressure. Along this fixed-length cut,
$I_{\textit{sp}}$
continues to rise even after combustion efficiency begins to decline. The reduction in aerodynamic loss can therefore compensate for incomplete chemical conversion. Since this trend applies only to one value of
$L_{\textit{comb}}$
, the coupled geometry map is considered next.
3.1.3. Performance landscape in the geometry space
Figure 9 presents the specific-impulse field over the coupled geometry grid. The high-value region forms a shallow peninsula attached to the upper boundary of the sampled domain. The largest sampled value,
$I_{\textit{sp}}=1256.50$
s, occurs at
$L_{\textit{comb}}=2.8375$
(
$L_{\textit{comb},\textit{actual}}=1.1158$
m) and
$\alpha _{\textit{comb}}=2.0$
. It is not located at the upper left corner, where
$I_{\textit{sp}}=1229.05$
s. This difference of 27.45 s is approximately 2.2 % of the sampled maximum, and is difficult to distinguish within the broad red band of the contour. Because the maximum is attached to the
$\alpha _{\textit{comb}}=2.0$
boundary, it is not interpreted as a closed interior island or an unconstrained optimum.
Performance landscape of specific impulse in the geometric design space.
The length dependence reflects the balance identified in figure 7. Short combustors provide insufficient distance for mixing and finite-rate conversion, whereas long combustors accumulate additional wall friction after the heat-release gain has weakened. Increasing
$\alpha _{\textit{comb}}$
within the sampled interval reduces density, wall loading and high-temperature dissociation, so the high-
$I_{\textit{sp}}$
band extends to the upper boundary. The present range does not establish whether
$I_{\textit{sp}}$
would turn over as the kinetic limitation strengthens beyond
$\alpha _{\textit{comb}}=2.0$
.
Figure 10 shows the corresponding combustion-efficiency field. Its largest sampled value,
$\eta _{\textit{comb}}=0.8279$
, occurs farther downstream at
$L_{\textit{comb}}=3.32$
(
$L_{\textit{comb},\textit{actual}}=1.305$
m) and
$\alpha _{\textit{comb}}=2.0$
. The shift towards a longer combustor is consistent with the additional time available for mixing and reaction. At small
$\alpha _{\textit{comb}}$
, the higher temperature promotes product dissociation, whereas at short
$L_{\textit{comb}}$
, the conversion remains limited by the available residence distance.
Performance landscape of combustion efficiency in the geometric design space.
At the
$I_{\textit{sp}}$
maximum,
$\eta _{\textit{comb}}=0.8244$
and
$\eta _{\textit{thermal}}=0.3275$
. At the combustion-efficiency maximum,
$I_{\textit{sp}}=1245.8$
s and
$\eta _{\textit{thermal}}=0.3206$
. Increasing
$\eta _{\textit{comb}}$
by 0.0035 is therefore accompanied by a reduction of 0.0069 in thermal efficiency, while
$I_{\textit{sp}}$
is 0.9 % below its sampled maximum. Both configurations have
$\alpha _{\textit{comb}}=2.0$
, but the latter increases the dimensional combustor length from 1.116 to 1.305 m. The longer flowpath releases a slightly larger fraction of the fuel chemical energy, but converts a smaller fraction of that released energy into axial kinetic energy. The additional residence time also increases wetted length and alters the heat-release history and nozzle-entry state. The small gain in chemical conversion consequently does not compensate for the decrease in thermal efficiency and the associated aerodynamic penalty.
3.1.4. Geometry-space gradients
Unnormalised finite-difference gradient field of specific impulse in the geometry space.
The gradient calculation is defined in § 2.4. Figure 11 resolves the shallow variation that is compressed by the colour scale in figure 9. On the shorter-length side of the sampled peak, the
$L_{\textit{comb}}$
component is positive; on the longer-length side, it becomes negative. The surrounding vectors therefore turn towards the same upper-boundary high-
$I_{\textit{sp}}$
region near
$L_{\textit{comb}}=2.84$
. The sign reversal is consistent with the contour matrix, and identifies the finite-length peak even though adjacent contour colours are nearly indistinguishable. Vector magnitudes are larger in the lower portion of the map, indicating greater sensitivity to geometry when the combustor approaches a constant-area duct.
Unnormalised finite-difference gradient field of combustion efficiency in the geometry space.
The combustion-efficiency gradients in figure 12 point towards a longer-length region than the specific-impulse gradients. This displacement reflects the additional reaction progress obtained with increasing residence distance. The weakening of the vectors near the high-efficiency band indicates that
$\eta _{\textit{comb}}$
becomes comparatively insensitive there, whereas the specific-impulse field has already turned towards a shorter combustor because friction and kinetic-energy conversion remain important.
3.2. Effects of fuel mixing profiles on performance
The calibration and parameter survey serve different purposes. The fitted values
$\alpha _{\textit{mix}}=3.99$
and
$L_{\textit{mix},\textit{actual}}=1.93$
m reproduce the reference URANS mixing history, whereas the survey samples a bounded set of alternative effective histories. The injection position remains fixed at
$x_{{inj,actual}}=0.10$
m, and neither the total fuel flow nor the equivalence ratio is changed. The range
$1.5\leq L_{\textit{mix},\textit{actual}}\leq 3.0$
m brackets the fitted completion length, while
$5\leq \alpha _{\textit{mix}}\leq 15$
examines profiles that are more front-loaded than the fitted case. All sampled profiles are monotonic and bounded between zero and unity. They are not intended to represent uniquely defined injector configurations: penetration, spacing, orientation and vortex generation can alter the effective history, and a particular
$(\alpha _{\textit{mix}},L_{\textit{mix}})$
pair cannot be mapped to hardware without a corresponding three-dimensional calculation or experiment.
3.2.1. The effects of mixing length
Effect of mixing length (
$L_{\textit{mix}}$
) on scramjet performance metrics along the upper boundary,
$\alpha _{\textit{mix}}=15$
. The curve extends slightly below the lower boundary of the two-parameter survey; only
$3.81\leq L_{\textit{mix}}\leq 7.62$
is included in the reported parameter maps.
Figure 13. Long description
Panel: A line graph showing the effect of mixing length on scramjet performance metrics. The horizontal axis is labeled L mix and ranges from 3 to 8. The vertical axis on the left is labeled I sp (s) and ranges from 1250 to 1400. The vertical axis on the right is labeled with two different metrics: one in red ranging from 0.74 to 0.84 and another in purple ranging from 2.80 to 3.00. The graph includes five lines representing different performance metrics: a black line for I sp, a red line for η comb, a green line for η thermal, a purple line for M min. The black line for I sp peaks around L mix of 4 and then decreases. The red line for η comb increases steadily. The green line for η thermal decreases steadily. The purple line for M min increases steadily.
Figure 13 shows the effect of mixing length along the
$\alpha _{\textit{mix}}=15$
boundary. Within the sampled interval, specific impulse is largest at its shortest length,
$L_{\textit{mix}}=3.81$
, and decreases as molecular mixing and heat release are distributed farther downstream. Combustion efficiency continues to increase over much of the same interval, whereas
$\eta _{\textit{thermal}}$
decreases because a smaller fraction of the released energy appears as axial kinetic-energy flux.
The short-length turnover visible below the map boundary shows that the benefit does not persist as the prescribed history is shortened further. These points are included only to clarify the local trend, and are not used when reporting extrema from the two-parameter survey.
3.2.2. Performance landscape in the mixing parameter space
Figures 14 and 15 present the two performance fields over the complete
$20\times 20$
mixing grid. These maps describe sensitivity to the effective streamwise histories defined by (2.11); they do not constitute a hardware-level injector optimisation.
Performance landscape of specific impulse in the mixing parameter space.
Performance landscape of combustion efficiency in the mixing parameter space.
The largest sampled specific impulse is
$1378.0$
s at
$L_{\textit{mix}}=3.81$
(
$L_{\textit{mix},\textit{actual}}=1.500$
m) and
$\alpha _{\textit{mix}}=15$
. At this point,
$\eta _{\textit{comb}}=0.7981$
and
$\eta _{\textit{thermal}}=0.3817$
. The high-
$I_{\textit{sp}}$
region is concentrated towards short mixing lengths and large front-loading factors because the corresponding histories release energy farther upstream and yield a larger increase in axial kinetic-energy flux. This conclusion is restricted to the sampled range and to the present hydrogen closure.
The largest sampled combustion efficiency, 0.8285, occurs at
$L_{\textit{mix}}=6.22$
(
$L_{\textit{mix},\textit{actual}}=2.447$
m) and
$\alpha _{\textit{mix}}=15$
. The corresponding values are
$I_{\textit{sp}}=1294.5$
s and
$\eta _{\textit{thermal}}=0.3401$
. Increasing the mixing length between the two sampled maxima raises
$\eta _{\textit{comb}}$
by 0.0304, but lowers
$\eta _{\textit{thermal}}$
by 0.0417;
$I_{\textit{sp}}$
is then 6.1 % below its sampled maximum. The additional chemical conversion obtained with the longer mixing length therefore produces a smaller conversion of released energy into axial kinetic energy. Since heat increments introduced at different local Mach numbers and thermodynamic states are not equivalent under Rayleigh heat addition, combustion efficiency alone cannot determine propulsive performance.
3.2.3. Mixing space gradients
Figures 16 and 17 show the unnormalised finite-difference gradients defined in § 2.4.
Unnormalised finite-difference gradient field of specific impulse in the mixing parameter space.
Unnormalised finite-difference gradient field of combustion efficiency in the mixing parameter space.
The specific-impulse vectors in figure 16 point predominantly towards shorter mixing lengths. Their magnitude decreases towards the distributed-mixing side of the map, showing that
$I_{\textit{sp}}$
is most sensitive where a change in the effective history moves a substantial portion of heat release upstream. Near the short-length, high-
$\alpha _{\textit{mix}}$
boundary, the vector rotation reflects the competing effects of early heat addition, dissociation and fuel–air enthalpy exchange.
The combustion-efficiency field in figure 17 has a different direction. At large
$\alpha _{\textit{mix}}$
, its gradients turn towards longer
$L_{\textit{mix}}$
and the high-efficiency band centred near
$L_{\textit{mix}}=6.22$
. The two fields therefore drive the effective history towards different regions:
$I_{\textit{sp}}$
favours the shorter-length boundary, whereas
$\eta _{\textit{comb}}$
favours additional conversion farther downstream. The accompanying decrease in
$\eta _{\textit{thermal}}$
quantifies the thermodynamic cost of that additional conversion.
3.3. Implications for hydrocarbon fuels
All parameter maps in this study are calculated for gaseous hydrogen. Their numerical boundaries should not be applied directly to hydrocarbon-fuelled engines. In the present calculation, the prescribed mixing history largely determines the axial heat-release distribution. Hydrocarbon fuels add decomposition and multi-step oxidation before the principal heat release. Heavy liquid fuels also require atomisation, heating and vaporisation, and may undergo endothermic pyrolysis. Molecular mixing can therefore precede the principal heat-release region by a substantial distance. The relevant gas-phase and spray-combustion processes are reviewed by Urzay (Reference Urzay2018) and Ren et al. (Reference Ren, Wang, Xiang, Zhao and Zheng2019), respectively.
These additional time scales modify the effect of combustor expansion. Increasing the area lowers static temperature and pressure, and accelerates the gas. The lower temperature can reduce high-temperature dissociation, but also slows fuel decomposition and oxidation, and shortens the high-temperature residence time. Hydrocarbon oxidation introduces CO/CO
$_2$
and partially oxidised intermediates, so the chemical energy retained at the combustor exit and its recovery during expansion need not follow the hydrogen case. Reaction may freeze before fuel conversion is complete. The high-combustion-efficiency region may consequently move towards a longer combustor or a smaller expansion ratio. The displacement in specific impulse cannot be inferred from chemistry alone, because additional conversion competes with the friction of a longer duct and changes in the nozzle-entry state.
The mixing parameters describe an effective streamwise history rather than a particular injector. A prescribed
$\eta _{\textit{mix}}(x)$
represents molecular mixing, and does not include the subsequent chemical delay. If fuel conversion becomes rate-controlling, then the performance field will respond less directly to
$L_{\textit{mix}}$
and
$\alpha _{\textit{mix}}$
than in the hydrogen case. Differences in fuel density, diffusivity, injection momentum and physical state also prevent direct transfer of the fitted history between fuels. A quantitative extension would require a fuel-specific injection model and finite-rate mechanism, including atomisation, vaporisation and pyrolysis where appropriate. The present maps therefore examine the competition among residence time, friction, expansion and heat-release location for hydrogen; they do not define hydrocarbon design limits.
4. Conclusion
This study examined the coupled effects of combustor geometry and effective fuel-mixing history for a hydrogen-fuelled scramjet at Mach 12 and dynamic pressure 60 kPa. The Q1D formulation retains variable thermodynamic properties, finite-rate chemistry, wall friction, area change and a finite reacting nozzle. The principal result is that chemical conversion and propulsive performance favour different locations within both sampled parameter spaces.
In the geometry survey, the largest sampled specific impulse is 1256.5 s at
$L_{\textit{comb}}=2.84$
(
$L_{\textit{comb},\textit{actual}}=1.116$
m) and
$\alpha _{\textit{comb}}=2.0$
. The largest sampled combustion efficiency, 0.8279, occurs at the longer length
$L_{\textit{comb}}=3.32$
(
$L_{\textit{comb},\textit{actual}}=1.305$
m) and the same expansion ratio. The additional conversion is accompanied by a decrease in
$\eta _{\textit{thermal}}$
from 0.3275 to 0.3206, and by additional wetted length; consequently,
$I_{\textit{sp}}$
is 0.9 % below its sampled maximum. The high-
$I_{\textit{sp}}$
region is attached to the upper boundary and is not a closed interior optimum.
The separation is stronger in the mixing survey. The largest sampled
$I_{\textit{sp}}$
is 1378.0 s at
$L_{\textit{mix}}=3.81$
(
$L_{\textit{mix},\textit{actual}}=1.500$
m) and
$\alpha _{\textit{mix}}=15$
, whereas the largest
$\eta _{\textit{comb}}$
, 0.8285, occurs at
$L_{\textit{mix}}=6.22$
(
$L_{\textit{mix},\textit{actual}}=2.447$
m) and the same value of
$\alpha _{\textit{mix}}$
. The longer history increases chemical conversion but reduces
$\eta _{\textit{thermal}}$
from 0.3817 to 0.3401, leaving
$I_{\textit{sp}}$
6.1 % below its sampled maximum. The Rayleigh relation provides a local thermodynamic interpretation of the heat-release location, while the calculated thermal efficiency supplies the engine-level measure of axial kinetic-energy conversion.
The gradient fields support these trends by directing specific impulse and combustion efficiency towards different regions. They are sensitivity measures within the analytic mixing closure, not a one-to-one prescription for injector hardware. A summary of the aerodynamic and thermodynamic mechanisms leading to the performance trade-off is provided in table 2.
Summary of aerothermodynamic mechanisms for performance trade-offs.
Table 2. Long description
A table with three columns and four rows. The columns are labeled Parameter change, Principal benefit, and Competing penalty. The rows detail the effects of changes in parameters on performance. Row 1: Parameter change: L comb increase; Principal benefit: More residence distance for mixing and finite-rate conversion; Competing penalty: More wetted length and friction; lower axial kinetic-energy conversion beyond the I sp peak. Row 2: Parameter change: alpha comb increase; Principal benefit: Lower temperature, wall loading and product dissociation within the sampled range; Competing penalty: Lower pressure and shorter chemical residence time; a reaction-freezing tendency first appears for short combustors, while the I sp trend beyond alpha comb equals 2 remains unresolved. Row 3: Parameter change: L mix decrease; Principal benefit: More upstream heat release and higher eta thermal; Competing penalty: Local temperature rise and dissociation when fuel incorporation becomes highly concentrated. Row 4: Parameter change: alpha mix increase; Principal benefit: More front-loaded molecular mixing and upstream heat release; Competing penalty: No separate upper-penalty is resolved over 5 less than or equal to alpha mix less than or equal to 15; hardware realisability is outside the closure.
The numerical locations reported here apply to gaseous hydrogen. Hydrocarbon fuels introduce longer fuel-conversion time scales, and for liquid fuels, atomisation, vaporisation and pyrolysis. These processes may shift the preferred combustor length and expansion ratio, and their quantitative effects require fuel-specific injection and chemistry models.
The steady formulation is appropriate for the nominal operating condition considered here, but the area-averaged equations cannot resolve transverse structures such as flame-holding recirculation zones, discrete fuel plumes and shock–boundary-layer interactions. They also cannot be used to describe genuinely unsteady or off-design processes, including pressure-wave transients and combustion instability. In addition, the hydraulic-diameter representation does not distinguish rectangular from circular passages, and therefore omits shape-dependent secondary flows and corner effects.
Future work should use fuel-specific three-dimensional calculations and experiments to establish the hardware realisability of selected mixing histories, and to examine shape-specific secondary flows and off-design wave dynamics. Coupling the reduced-order model with trajectory analysis would then allow the sampled engine-level trade-offs to be assessed over an ascent path.
Funding
This work is supported by the Chinese Academy of Sciences Project for Young Scientists in Basic Research (grant no. YSBR-132), the National Natural Science Foundation of China (nos 12372235, 12372236 and 12302392) and the Science Foundation for Young Scientists of the State Key Laboratory of High Temperature Gas Dynamics (grant no. 2023QN12). During the preparation of this manuscript, the authors utilised the large language model Gemini to assist with grammar correction and LaTeX formatting. The authors reviewed and edited all content, and take full responsibility for the final publication.
Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
ϕ=0.8
ϕ=1.0
Pd=50
Lc=1.2
0≤x≤1.2
ϕ=1.0
ϕ=1.0
p/pref
pref=P3=64.5 kPa
αcomb=2.0
Lcomb=2.034
Lcomb,actual=0.800
Lmix
αmix=15
3.81≤Lmix≤7.62
