# Combustion Dynamics of a Liquid Rocket Engine in subcritical Injection Conditions

Giulio Pelenghi, Thomas Schmitt, Nicolas Odier, Jan Martin, Wolfgang

Armbruster, Michael B Örner, Justin Hardi, Annafederica Urbano

► To cite this version:

Giulio Pelenghi, Thomas Schmitt, Nicolas Odier, Jan Martin, Wolfgang Armbruster, et al. Combustion Dynamics of a Liquid Rocket Engine in subcritical Injection Conditions. Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2025), Sep 2025, Trondheim, Norway. hal-05407438

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# Combustion Dynamics of a Liquid Rocket Engine in subcritical Injection Conditions

Symposium on Thermoacoustics in
Combustion: Industry meets Academia
(SoTiC 2025)

Sept. 8 - Sept. 11, 2025

Trondheim, Norway

Paper No.: 37

©The Author(s) 2025

Giulio Pelenghi<fnref n="1" />, Thomas Schmitt<fnref n="2" />, Nicolas Odier<fnref n="3" />, Jan Martin<fnref n="4" />, Wolfgang Armbruster<fnref n="4" />, Michael Börner<fnref n="4" />, Justin Hardi<fnref n="4" />, and Annafederica Urbano<fnref n="1" />

## Abstract

In throttlesable liquid rocket engines using methane and oxygen as propellants, the combustion chamber pressure is expected to vary from subcritical to supercritical with respect to the thermophysical state of the propellants during operation of the engine. The aim of the present work is to investigate the influence that thermodynamic injection conditions could have on combustion stability. A numerical framework based on a multi-fluid approach suitable to simulate both sub- and supercritical pressure flows is used. The research combustor BKN from DLR Institute of Space Propulsion is selected as the experimental test case of the current study. In particular, one unstable supercritical load point and two subcritical load points - one stable and one unstable - are selected. The two unstable cases are obtained through instability triggering. The simulated stable case shows good agreement with the experimental flame structure and the chamber pressure. The unstable cases reproduce the non-linear nature of the experimentally observed instabilities, however they slightly overestimate chamber pressure and oscillation amplitudes. Good agreement is found with the first longitudinal modes' frequencies in all cases. The present work demonstrates the capability of investigating combustion instabilities in liquid rocket engines both for two-phase flow and for supercritical injection conditions, opening the path to a series of investigations on the associated flow physics.

## Keywords

Combustion dynamics, Liquid rocket engine, Deep-throttling, subcritical, Longitudinal combustion instability

## Introduction

Over recent years, the space sector has seen a sharp rise in commercial interest from private companies and investors. This trend is expected to continue, with a forecast average per annum growth of 9% up to 2035 [1]. This places the space launcher segment in a strategic position for current and future developments in the space market. Propulsion systems, liquid rocket engines in particular, represent one of the main focal points of innovation in such an environment.

Three innovation axes are within the scope of the current study: the transition to methane and oxygen as propellants for first-stage propulsion, the need for deep-throttling for vertical re-landing and stage re-use, the use of CFD as a more convenient alternative to experimental tests. The study of combustion instabilities is affected by all the listed points: the change of propellants compared to old engines, and of injection conditions within the same engine (due to deep-throttling), can affect the stability properties of the engine. Of particular interest in this study, are the effects of the combustion chamber's pressure on combustion dynamics.

Liquid rocket engines used in launchers usually work at pressures above the critical pressure of oxygen (> 50.5 bar), these pressure injection conditions can be called *supercritical*. However, if deep-throttling is required, the pressure in the combustion chamber may reach values which are lower than the oxygen's critical pressure, thus leading to the formation of a liquid-vapor interface in the system. Such conditions are termed *subcritical*.

The time delays associated to the evaporation (or pseudo-evaporation) of the injected propellants are theorized to have first-order influence on the combustion dynamics of rocket engines [2]. Therefore, an understanding of these phenomena is necessary in the design of deep-throttlesable liquid rocket engines. This is the scope of the present work.

## State of the Art

Liquid rocket engines have been studied for more than a century now. However, their modeling through computational fluid dynamics has much more recent roots. One of the first key challenges to simulating a Liquid Rocket Engine (LRE), has been the modeling of high-pressure, cryogenic temperature injection conditions (transcritical<fnref n="1" /> injection). The first CFD models for such phenomena began to be studied only in the late 90s/early 2000s, with Oefelein,

**Corresponding author:**

Giulio Pelenghi, Fédération ENAC ISAE-SUPAERO ONERA, Université de Toulouse, France.

Email: giulio.pelenghi@isae-supaero.fr

<footnote n="1">¹Fédération ENAC ISAE-SUPAERO ONERA, Université de Toulouse, France</footnote>

<footnote n="2">²Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C, 91190, Gif-sur-Yvette, France</footnote>

<footnote n="3">³CERFACS, Toulouse, France</footnote>

<footnote n="4">⁴Institute of Space Propulsion, German Aerospace Center (DLR), Hardthausen am Kocher, Germany</footnote>
Yang, and Zong's works [3, 4, 5] on transcritical hydrogen-oxygen test cases. Large Eddy Simulations (LES) were used to model jet injection, first with non-reacting, then with reacting mixtures. These works also saw the introduction of cubic equations of state to model real-gas thermodynamics. The research group continued work on similar setups throughout the 2000s, with a focus on counterflow flamelet modeling [6, 7].

Other groups also started developing models capable of simulating transcritical injection: such are the works of Schmitt et al. [8], Lapenna et al. [9], and Müller et al. [10]. In the meantime, first analyses using methane instead of hydrogen as fuel were also being performed [11, 12, 13, 14].

As the models started becoming more widely used, new research directions started to develop from them. On the one hand, full-scale simulations of LRE combustion instabilities started to become feasible [15], on the other hand, the co-existence of two phases at injection also started to be explored [16, 17].

**Combustion Instabilities** Single-injector combustion chambers, while not used in launcher applications, actually offer a good testbed for the study of thermoacoustic instabilities. In the last years, the main project studying this kind of setup in Europe has been REST (Rocket Engine Stability Initiative) [18]. The REST HF-10 test case is a virtual combustion chamber, using a coaxial LOX/CH4 injector, it has been simulated by various labs, with different solvers and different models, to assess the reproducibility of axial combustion instabilities. Parallel to the work conducted in REST, there are also the analyses conducted by Nez et al. [19] and by Laurent et al. [20] on the single-injector MASCOTTE test bench. Particular attention has been given to the doubly transcritical nature of methane-oxygen injection, and to the low-order modeling of the setup, through the computation of its flame transfer function. More recent analyses of MASCOTTE by Schmitt et al. [21] also have managed to compute the setup's flame transfer functions, this time including both oxidizer and fuel modulation. Purdue university's Continuously Variable Resonance Combustor (CVRC) with its varying chamber length has also given rise to CFD comparisons. Harvazinski et al. [22] could recover the correct unstable modes for the short, medium and long configurations of the combustion chamber using DES. Nguyen et al. [23] also managed to reach similar results using an axisymmetric solver, they also showed that the presence of the nozzle instead of a simple pressure boundary condition in the simulation has a noticeable impact on the results. Similar work has also been conducted to study the impact of injector length on the engine's stability [24, 25].

All the listed works deal with supercritical pressure injection conditions and single phase flow in the combustion chamber. Some studies have been conducted at subcritical pressure injection conditions [26]. However, to the author's knowledge, none have employed a method capable of working across all injection pressures: both subcritical and supercritical.

**Two-phase flows** Liquid-vapor phase separation can occur in two cases in LRE conditions: the pressure of the chamber can become lower than the critical pressures of the injected propellants (the engine may operate at these pressures, or it

may encounter them at startup/shutdown or while throttling); or even in transcritical injection cases, if the critical pressure of the species mixtures in the chamber becomes higher than the chamber's pressure.

This second case is especially true in Type-III mixtures (following the definition of van Konynenburg [27]), such as oxygen and hydrogen or oxygen and water (present in the reaction products). Therefore, models capable of hosting two phases, such as multi-fluid methods [28], also started to be employed in transcritical injection simulations.

When cubic equations of state are used, multi-fluid methods need to solve Vapor-Liquid Equilibrium (VLE) problems within the computation's cells. The solution algorithms for VLE can be complex and computationally intensive [29], therefore the available literature currently offers almost only simplified simulations representing LRE conditions (or diesel engine injection).

Pioneering work was done by Qiu and Reitz [30], who applied for the first time VLE solution methods to a RANS multi-fluid simulation. They used the solution algorithm proposed by Michelsen [31, 32] for a binary mixture of nitrogen and dodecane at 60 bar (type-III mixture [33]) in the ECN Spray A configuration. This test-case and method has also been reproduced by other groups, notably in LES by Matheis and Hickel [34] and by Ma et al. [35]. Another test-case more representative of LREs was introduced by Ruiz et al. [36]: a shear injector of oxygen and hydrogen at 100 bars, which has also seen reproductions [35, 37, 38].

All of the above studies include binary mixtures, for which the VLE solution can be tabulated or solved in a relatively stable way. To the authors' knowledge, only two studies have managed to solve the full VLE problem in reacting cases (thus with more than two species): the works of Fathi et al. [39, 40] and the works of Tudisco and Menon [41]. Due to the high computational costs and not fully robust algorithms, other approaches to solve the VLE have been attempted over the years. Traxinger et al. [16] solved the full VLE in 1D counterflow flames simulations, and then used the results in their flamelet model for both hydrogen and methane combustion in the BKH and MASCOTTE test benches respectively. ONERA instead implemented a two-fluid method, in which only oxygen can be liquid, and all gases are assumed to follow ideal gas laws [42]. A similar, more general, approach was developed by Banuti et al. [43]: it allows to model each species with its own equation of state, letting some components behave like real fluids and others like ideal ones. Péden et al. [44] implemented a method capable of solving the VLE analytically at subcritical pressures, using a Noble-Abel Stiffened Gas (NASG) approach on a 4-equation multi-fluid method. Finally, Pelletier et al. [17], Schmitt and Ducruix [45] and Gioud et al. [46, 47] simulated combustion in subcritical rocket engines with an approximate VLE, which uses cubic equations of state, but assumes that the vapor and liquid compositions in each cell are equal to the overall composition of the cell.

## Objectives and Outline

The current paper aims to gain an understanding in the effects of injection conditions on the stability of liquid rocket
engines. In particular, the influence of chamber pressure on phase change and thus on combustion dynamics is studied.

Computational fluid dynamics is the tool chosen to carry out this study. Specifically, a multi-fluid method is employed in order to model phase change in both super- and subcritical pressure injection conditions. The research combustor BKN of the DLR Institute of Space Propulsion [48] is chosen as the setup of the simulations, employing methane and oxygen as propellants and hosting a single shear coaxial injector.

To the authors' knowledge, this is the first time an assessment of combustion dynamics in a subcritical rocket engine with a multi-fluid model has been attempted.

The document shall first present the experimental test case selected for the study of subcritical flame stability. Then, the specific models chosen for the CFD study are presented. Finally, preliminary results of the simulations are displayed and compared to the experimental measurements.

## Employed Methodology

In the present work a comparison between numerical simulations and experimental data will be carried out to assess the capabilities of the developed numerical framework in reproducing subcritical CH4/O2 diffusive flames. In the following the experimental test cases considered are first introduced. Then some details on the numerical solver are provided.

### Chosen Test Cases

The Research Combustor Model 'N' (Abbr.: BKN) from the DLR Institute of Space Propulsion has been selected as the experimental reference for the current study [48]. BKN has a cylindrical chamber with film-cooled optical access windows, and tests with a single shear-coaxial injector are the subject of the present study.

A suite of tests using both sub- and supercritical pressures with methane and oxygen injection has been conducted, leading both to unstable and stable combustion conditions. Figure 1 displays the 4 load points (LPs) presented by Martin et al. [48] using liquid oxygen (LOX) and liquefied natural gas (LNG).

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Injection conditions of methane and oxygen in 4 Load Points (LPs) of BKN tests in Ref. [48]. Two-phase domains correspond to different values of species mass fractions, from pure oxygen to pure methane.</figcaption></figure>

The present study mainly focuses on the analysis of the subcritical load points LP3 and LP4. However, supercritical LP1 has also been simulated, in order to allow for comparisons of the differing thermodynamic injection conditions. This brings the total of load points studied to 3. LP4 is stable, while LP1 and LP3 display longitudinal combustion instabilities. All three load points have the same manifold geometry, however, LP1 has a longer chamber compared to LP3 and LP4 (a length from injection plane to nozzle throat of 609 vs 539 mm respectively). The properties of the load points are shown in Table 1.

<table><caption>**Table 1.** Experimental load point data. Taken from [48].</caption><thead><tr><th>Load Point</th><th>LP1</th><th>LP3</th><th>LP4</th></tr></thead><tbody><tr><td>Stable</td><td>NO</td><td>NO</td><td>YES</td></tr><tr><td>Chamber length (mm)</td><td>609</td><td>539</td><td>539</td></tr><tr><td>Pressure p̄ (bar)</td><td>62.9</td><td>41.3</td><td>40.5</td></tr><tr><td>ṁ<sub>fu</sub> (kg/s)</td><td>0.103</td><td>0.068</td><td>0.088</td></tr><tr><td>ṁ<sub>ox</sub> (kg/s)</td><td>0.358</td><td>0.228</td><td>0.215</td></tr><tr><td>ṁ<sub>cool</sub> (kg/s)</td><td>0.338</td><td>0.212</td><td>0.208</td></tr><tr><td>T<sub>fu</sub> (K)</td><td>217</td><td>192</td><td>255</td></tr><tr><td>T<sub>ox</sub> (K)</td><td>104</td><td>105</td><td>109</td></tr><tr><td>T<sub>cool</sub> (K)</td><td>281</td><td>271</td><td>270</td></tr><tr><td>O/F ratio, cooling excluded (-)</td><td>3.48</td><td>3.35</td><td>2.44</td></tr><tr><td>O/F ratio, cooling included (-)</td><td>0.81</td><td>0.81</td><td>0.73</td></tr><tr><td>Momentum flux ratio J (-)</td><td>3.80</td><td>5.14</td><td>19.50</td></tr><tr><td>Power (MW)</td><td>≈4.5</td><td>≈2.9</td><td>≈2.7</td></tr></tbody></table>

### Simulation Setup

The present study employs the AVBP CFD solver, developed by CERFACS [49]. It is an energy-based, unstructured explicit LES solver specialized in internal flows and combustion. The Favre-filtered Navier Stokes equations are discretized using a TTGC scheme [50], an artificial diffusivity sensor similar to the one presented in [51] is employed to reduce pressure noise at liquid-vapor interfaces and flame surfaces.

A multi-fluid model [17] with the SRK cubic equation of state (EoS) [52] has been employed for the simulations of the chosen test cases.

**Real-gas Modeling** The employed EoS takes the form:

$$p(\rho, T, \mathbf{Y}) = \frac{\rho \bar{r} T}{1 - \bar{b} \rho} - \frac{\bar{a}(T) \rho^2}{1 + \varepsilon_1 \bar{b} \rho - \varepsilon_2 \bar{b}^2 \rho^2},$$

where $p$, $\rho$ and $T$ are the fluid's pressure, mass density and temperature values, $\bar{r}$ is the mass-weighted perfect gas constant for the fluid mixture. $\varepsilon_1 = 1$ and $\varepsilon_2 = 0$ for SRK. $\bar{a}(T)$ is the attractive coefficient and $\bar{b}$ is the co-volume parameter, both are averaged quantities for the considered mixture. The van der Waals combining rules [53] are applied to obtain the averages from the single species' properties.

**Vapor-Liquid Equilibrium** A Vapor-Liquid Equilibrium (VLE) model has to be implemented in order to obtain the thermodynamic state cells where vapor and liquid coexist. The chosen model has been developed by Pelletier et al. [54]. It solves the equilibrium by assuming that the liquid and vapor compositions within a given computation cells are identical ($\mathbf{Y}_l = \mathbf{Y}_v$). In practice, this corresponds to solving the VLE as if only one species was present, with
its properties being equal to the averaged quantities of the mixture.

The accuracy of this hypothesis in the VLE problem still needs to be fully assessed in literature. Although comparisons in an environment representative of liquid rocket engines still have to be carried out, the following considerations can be made:

* Oxygen and methane form a type-I mixture (following the classification by [27]), this means that the VLE properties of the mixture stay relatively close to the properties of each separate species [38].

* 1D studies by Pelletier et al. [17] have revealed how the accuracy of the hypothesis mainly depends on whether the pressure of the setup is close to the critical pressures of the species. In our cases, a fair distance is present between the measured pressures and the critical properties of the species.

* The hypothesis mainly concerns mixtures. Due to the non-premixed nature of the simulated flame, if there is a species mixture in a given cell, its temperature will quickly rise, preventing the presence of a VLE calculation entirely. This point was tested in particular by Tudisco and Menon [41], at high pressures.

**Thermochemistry** An infinitely fast chemistry model developed by Schmitt et al. [13] is used to model chemical reactions in the system. This model is based on the assumption that, in the studied test cases, the chemical time scale is much smaller than the turbulent mixing time scale. We consider 7 species: CH₄, O₂, H₂O, CO₂, CO, H₂, OH, all with the same Schmidt number equal to 0.75. The mixture fraction Z is defined at the calculation's initial solution, choosing hydrogen as the reference atom:

$$ \tilde{Z} = \frac{W_{CH_4}}{4} \left( 4 \frac{Y_{CH_4}}{W_{CH_4}} + 2 \frac{Y_{H_2O}}{W_{H_2O}} + 2 \frac{Y_{H_2}}{W_{H_2}} + \frac{Y_{OH}}{W_{OH}} \right). $$

It is then transported for the rest of the simulation following [51]. The filtered mass fractions are found through a β-pdf model: $\tilde{Y}_k = \tilde{Y}_k(\tilde{Z}, \tilde{Z}'')$ is tabulated as a function of $\tilde{Z}$ and $\tilde{Z}''$. The filtered variance of the mixture fraction $\tilde{Z}''$ is transported following the method of [51]. The tables themselves are computed beforehand, by performing 1D simulations of counterflow diffusion flames, with an imposed flame stretch of 100 s⁻¹.

The reaction rate is computed a posteriori, using the value of $\tilde{Y}_k^{non-reactive}$ - found from transporting and diffusing the mass fractions without reactions - and the newly calculated value $\tilde{Y}_k^{tables}(\tilde{Z}, \tilde{Z}'')$ from the tables. For a given time step $\Delta t$, we have:

$$ \bar{\omega}_k = \bar{\rho} \frac{\tilde{Y}_k^{tables}(\tilde{Z}, \tilde{Z}'') - \tilde{Y}_k^{non-reactive}}{C_r \Delta t}, $$

where $C_r$ is a relax factor which has to be > 1. In this study we take $C_r = 10$, meaning that we expect the characteristic times of the flow to be higher than 10 time steps $\Delta t$. The filtered heat release rate can then be found from the species reaction rate and their formation enthalpies.

**Geometry** In order to capture the full acoustics, the BKN's full geometry is simulated: this includes the fuel's, oxidizer's

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Schematic showing the simulated domain and the imposed boundary conditions.</figcaption></figure>

and coolant's manifolds as well as the chamber's nozzle. Long (used for LP1) and short (used for LP3 and LP4) chamber geometries are meshed with the same parameters, the only difference being the length of the chamber. The unstructured meshes count around 45 million elements for the long case, and 44 million elements for the short case. The smallest mesh sizes are located at the shear coaxial injector's lip and fuel inlet, with a size of around 40 µm (11 elements fit on the lip's radial thickness). An adiabatic law-of-the-wall model is applied on all of the manifolds' boundaries. As for the chamber walls, a coupled isothermal law-of-the-wall model [55] is used. The temperatures at the boundary are set by averaging the temperatures measured in [48]. A wall temperature of 640 K has been imposed on the walls of LP1, of 600 K on LP3, and of 430 K on LP4. The inlet mass flow rate and temperatures are imposed (see Table 1). Note that the pressure in the chamber is not imposed and will be one of the results of the simulation.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Mesh used to model BKN's Geometry (LP3, LP4 configuration). Position of the used pressure probe highlighted.</figcaption></figure>

## Results

The full geometry has been simulated in the LP1, LP3 and LP4 load points [48]. The section starts by showing the results of these simulations, then, an attempt at triggering the instabilities in LP1 and LP3 is made. Analysis of the resulting behavior follows.

### Stable Simulations

The first output from the simulations are the observations on the injector's thermodynamic behavior, as well as the on the diffusion flame that is generated.

**Instantaneous Solutions** Figure 4 shows two temperature snapshots taken from LP1, LP3 and LP4 respectively. On the same figure the iso-surface of the liquid fraction at $\alpha = 0.5$ is also reported as an indicator of the liquid/vapour interface.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Temperature and liquid mass fraction snapshots for LP1, LP3 and LP4 respectively. Liquid mass fraction iso-surface located at $α = 0.5$.</figcaption></figure>

The light blue iso-surface shows the presence of the diffuse interface, with liquid oxygen in its core. The surface is absent from LP1: since its pressure is supercritical, no phase separation occurs and no interface between liquid and vapor phases is present. LP3 exhibits a longer liquid core compared to LP4, this can be explained by the lower fuel temperature and mass flow, leading to a slower fuel injection velocity and a large difference in momentum flux ratio ($J_{LP3} = 5.14$ vs $J_{LP4} = 19.50$). The separation between the iso-surface and the highest temperature taken as an indicator of the flame position showcases the presence of a time delay between reactant evaporation, mixing and combustion. This delay is known to have a direct influence on the stability of rocket engines [2].

The flames of the three cases clearly show a diffusion (non-premixed) flame structure. The adoption of an infinitely fast chemistry model means that the flame will always be attached to the injector lip, the accuracy of such an assumption needs to be verified [56].

**Pressure Traces** Observation of the pressure traces of the cases (Figure 5) reveals that they all host small-amplitude high-frequency pressure oscillations (in the order of 1% of chamber pressure) at the head of the chamber (the measurement probe is positioned at the same location as DLR's first dynamic pressure sensor $p_{CC,1}$ [48], also visible in Figure 3).

This confirms that the current simulations are not able to fully reproduce the experimentally unstable behavior of LP1 and LP3. On top of these high frequency oscillations, a lower frequency (kept in the filtered signal) is observed. This low-frequency behavior is still under study, but it may reproduce a similar low-frequency mode observed experimentally. A preliminary study of this phenomenon has been conducted in [57].

**Line-of-Sight Comparison** To better study the flame shape, and how it compares to the experiments, a line-of-sight study of LP3 and LP4 is performed. Experimental measurements of OH\* and CH\* emissions for the two LPs obtained via the lateral windows are available [48]. Although the chemistry model used does not host the two radicals, it is still possible to compare the computed flame shape with the radical emission distributions observed experimentally. This exercise can only be qualitative: the correlation between radical emissions and a parameter such as the heat release in diffusion flames at high pressures has been shown to be inconclusive [58]. The choice is made to compare the experimental OH\* emissions with the simulated heat release

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Considered pressure traces of the LP1, LP3 and LP4 cases respectively. Measurement point located at the same position of DLR's first dynamic pressure sensor $p_{CC,1}$ [48].</figcaption></figure>

rate $h_r$, and the OH mass fraction $Y_{OH}$ (Figures 6 and 7). Both experimental and simulated quantities are averaged in time.

The optical access window of BKN has a length of 253.5 mm, which covers around the first half of the combustion chamber, and a height of 38 mm, therefore the experimental comparison is limited to this part of the chamber. The CFD solution is averaged in time, and the chosen quantities are integrated in the direction perpendicular to the chamber axis. All plotted quantities are normalized and use the same color map as the experimental cases. This allows for a clear comparison between the two flames. In the case of $h_r$, the color-scale has been calibrated to show its lower values, to prevent the near-lip region from saturating the field.

The main difference that can be noticed is in the different length between the experimental LP3 and LP4 flames, this is due to the difference in thermoacoustic stability, and is not reproduced by the simulated results. This means that the only relevant experimental comparison can be made with LP4.

Observing $h_r$ in LP4, the global flame shape is somewhat recovered. However, within the flame, a bright spot in $h_r$ is visible in the left-most part, which is absent from the experiment. This could be explained by the fact that the experimental imaging suffers from stratified attenuation between approximately 25 and 125 mm due to soot
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Comparison between experimental (top) and simulated (bottom) LP3 flames. Simulation results have been averaged in time and integrated along the line of sight. $h_r$, iso-level given at 1% of max value, $Y_{OH}$ iso-level given at 10% of max value.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Comparison between experimental (top) and simulated (bottom) LP4 flames. Simulation results have been averaged in time and integrated along the line of sight. $h_r$, iso-level given at 1% of max value, $Y_{OH}$ iso-level given at 10% of max value.</figcaption></figure>

deposits on the inner surface of the optical access window. Furthermore, the line-of-sight post-processing does not model phenomena such as the absorption of radiation by the liquid oxygen jet close to the injector, or the high self-absorption of OH\* at high pressures.

If instead one compares the $Y_{OH}$ field, a closer resemblance between experimental and simulated LP4 results is observed. The simulated values increase further inside the combustion chamber, just like in the experimental case, however, not the entire shape is perfectly reproduced: the first small rise in emissions at 50 mm is anticipated in the simulation, while the maximum value experimentally observed around 150 mm is delayed to around 250 mm in the simulations.

## Instability Triggering

Since both simulated LP1 and LP3 could not correctly reproduce the unstable nature of the experiments, the choice was taken to trigger the instability. This choice was in part justified by the experiments as well: both LP1 and LP3 take place in a chamber which was already unstable [48], meaning that the triggering action happened before the load point was reached.

The triggering procedure followed the one used by Urbano et al. [15]: the first longitudinal mode of the chamber was superimposed to the pressure field. The simulation with the modified field was then allowed to continue without

any further alterations. After an initial transient, a relatively stable limit cycle is reached. Using once again the same dynamic pressure sensor used by DLR, the pressure signals are traced. They are displayed in Figures 8a and 8b.

<figure><img src="image_11.png" /><figcaption>(a) Simulated triggered LP1 pressure trace.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Simulated triggered LP3 pressure trace.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) Experimental LP3 pressure trace.</figcaption></figure>

<caption>**Figure 8.** Simulated and experimental pressure traces. Measurement point located at the same position of DLR's first dynamic pressure sensor $p_{CC,1}$ [48].</caption>

The traces show in both cases the presence of a non-linear instability. Observation of the pressure fields reveals that it consists in the propagation of a high-pressure front back and forth through the length of the chamber. This behavior is in accordance with the experimental pressure traces observed by DLR. As an example, Figure 8c shows the measurements on LP3.

**Flame Dynamics** The newly-found limit cycles call for an analysis of the flame dynamics.

As a starting point, the pressure trace measured at the head of the combustion chamber (first axial position of the dynamic pressure sensor in the experimental test case), is analyzed. In particular, its spectral energy density is
estimated using Welch's method [59]. Figures 9 and 10 illustrate the results before and after the triggering.

<figure><img src="image_14.png" /><figcaption>**Figure 9.** PSD of LP1 pressure signals before and after triggering, measured at the first dynamic pressure probe at the wall.</figcaption></figure>

In both triggered cases, a clear rise in power is observed. The first and following modes are visible. These modes are also visible (in a less prominent way), in the signals before the triggering. A quantitative comparison between the measured frequencies of the first modes and their respective experimental values are visible in Table 2.

<figure><img src="image_15.png" /><figcaption>**Figure 10.** PSD of LP3 pressure signals before and after triggering, measured at the first dynamic pressure probe at the wall.</figcaption></figure>

An interesting observation lies in the fact that the modes of the triggered cases show higher frequencies. This behavior can be attributed to the higher pressure oscillations promoting mixing between propellants, leading to a higher combustion efficiency, higher chamber temperatures, and therefore higher speeds of sound.

In order to better understand the instability, the first mode of LP1 and LP3 is analyzed using Dynamic Mode Decomposition (DMD) [60]. The pressure's module and phase can be seen in Figure 11.

The DMDs clearly display the first longitudinal mode of the chamber. Some slight coupling with the longitudinal modes of the oxygen injector is also visible. There are still some artifacts on the module and phase fields (e.g., the phase change halfway through the chamber, the visible pressure propagation fronts, which may be reduced by conducting

a DMD over a longer time span). The noise given by the dense oxygen entering the chamber is likely due to its low compressibility: a pressure wave traveling from the chamber to the injector will see its amplitude increased.

In order to better assess the instabilities, it was chosen to calculate the Rayleigh index of the triggered LP1 and LP3 load points along the chamber axis. This is done for the first mode, using the DMD results for pressure and heat release. An integral along the transversal direction of the chamber allows to obtain the desired result:

$$RI^{mode,1}(x) = \iint |p^{mode,1}(x, y, z)| |h_r^{mode,1}| \cos(\phi_p^{mode,1} - \phi_{h_r}^{mode,1}) dy dz.$$

The DMD of the non-triggered LP3 is performed to recover its first mode as well. The resulting Rayleigh indexes are compared in Figure 12.

Observing the two red lines, the figure clearly displays how the same load point is capable of hosting different stability conditions, depending on the presence or not of a triggering action. We can conclude that the observed behavior is a non-linear instability. If instead the magnitudes of the triggered LP1 and LP3 indexes are considered, we can notice that LP3 displays a higher level of instability: this is in agreement with the higher pressure amplitudes reported in Table 2.

**Integrated Values** Leaving aside flame dynamics along the chamber axis, it is interesting to also observe how other global parameters change in time. Figure 13 displays the heat release integrated on the chamber volume together with the usual dynamic pressure probe measurements. The results clearly show a positive Rayleigh index (pressure rises with heat release and vice-versa), in agreement with the previously observed 1D Rayleigh index. It is important to highlight that while the heat release behavior reflects the entire chamber (it is integrated over the whole volume), the pressure is actually only measured at the start of the chamber (Figure 3). These plots therefore also confirm that the biggest contribution to the thermoacoustic instabilities happen within the first part of the chamber.

In a more interesting note, for the first time a coupling between a thermoacoustic instability and phase-change in subcritical LP3 can be observed. It appears the quantity of liquid present in the chamber anticipates the pressure and heat release by around a quarter of a period. One could speculate that more liquid in the chamber would (generally) mean a larger contact area between the cryogenic oxygen and the fuel, resulting (after a certain delay due to oxidizer evaporation and diffusion), into a higher heat release rate.

This three-way coupling behavior between chemistry, thermodynamics and acoustics is one of the main novelty factors of the present paper. It warrants an in-depth investigation that could result in new insights into the fundamental mechanisms governing combustion stability under subcritical conditions.

**Line-of-Sight Comparison** As was seen in Figure 6, the short flame shape seen in the experiment could not be replicated by the non-triggered simulation. It is therefore interesting to observe whether the triggered simulations
<figure><img src="image_16.png" /><figcaption>**Figure 11.** First mode DMD for the pressure field of triggered LP1 and LP3 respectively. 2D slices extracted from the 3D DMD analysis.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 12.** Rayleigh index calculated from the first DMD mode of pressure and heat release. To guarantee readability, each signal is normalized to its maximum. In reality the triggered LP1 and LP3 signals respectively are around 20 and 600 times higher than the non-triggered LP3 signal.</figcaption></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>**Figure 13.** Traces in time of triggered LP1 and LP3 respectively. The pressure is measured at the same position of DLR's first dynamic pressure sensor $p_{CC,1}$ [48], heat release is integrated over the chamber volume, liquid mass is reported for subcritical LP3. All quantities are smoothened through a high-frequency filter.</figcaption></figure>

display shorter flame shapes, and how qualitatively close they are to the experimental images.

Much like in the stable cases, the 3D fields of heat release $h_r$ and OH mass fraction $Y_{OH}$ are averaged in time (same time spans as in Figure 8). They are then integrated along the axis of the chamber which includes the viewing windows. The results for the triggered solutions are visible in Figures 14 and 15. Figures 6 and 15 can be compared to see how much the unstable flame has shortened compared to the stable one. The flame length however appears to be shorter than the experimental one, considering both $h_r$ and $Y_{OH}$ fields. An educated guess that the triggered case experiences higher pressure oscillations than the experimental case can be made. This assessment will be verified in the next section.

In the case of the triggered LP1 (Figure 14), the flame shape given by $h_r$ appears to be shorter than the experimental one, while the opposite is true for $Y_{OH}$. As previously discussed, the experimental images of LP3 are somewhat

compromised by the formation of soot on the leftmost part of the viewing window. This however is not the case for the experimental images of LP1, meaning that a clearer flame shape is visible. It is therefore interesting to notice the formation of two “blobs” of higher emissions inside the experimentally captured flame of LP1 (around the 50 and 80 mm marks). A similar structure can also be seen in the
<figure><img src="image_20.png" /><figcaption>**Figure 14.** Comparison between experimental (top) and simulated (bottom) LP1 flames after triggering. Simulation results have been averaged in time and integrated along the line of sight. $h_r$ iso-level given at 1% of max value, $Y_{OH}$ iso-level given at 10% of max value.</figcaption></figure>

simulated $Y_{OH}$ fields (although at higher axial positions, around 75 and 110 mm).

<figure><img src="image_21.png" /><figcaption>**Figure 15.** Comparison between experimental (top) and simulated (bottom) LP3 flames after triggering. Simulation results have been averaged in time and integrated along the line of sight. $h_r$ iso-level given at 1% of max value, $Y_{OH}$ iso-level given at 10% of max value.</figcaption></figure>

## Quantitative Comparisons

Finally, all the conducted analyses in this document can be summed up in a quantitative way, allowing further comparisons between simulated and experimental results. The data is reported in Table 2.

Since the simulation includes the convergent-divergent nozzle, the pressure in the combustion chamber will not be an input of the simulation, but a result of the choking conditions at the nozzle's throat. The same approach can be taken with the heat which is absorbed by the iso-thermal walls: the experimental relative exists and is measured within the chamber's water cooling system. Finally the flame dynamics are compared both considering the frequency and the observed amplitudes of the first mode.

We observe a close match of the average pressure for LP4. A larger error of almost 50 % is found on the wall heat load. These high errors can be due to the employed hypotheses and models, but they could also originate from uncertainties in the experiments: the heat load calculated from the cooling circuit is assumed to be extracted from the quartz windows as well, even if they are not cooled, no thermal losses in the circuit are considered, symmetry along the azimuthal axis is assumed. Finally the computed frequency for the first

longitudinal mode, which is stable in this configuration, is very close to the experimental value. The good agreement observed in the flame shape (see Figure 7) further confirms the solver's ability to accurately reproduce a stable case under subcritical pressure conditions, such as those of LP4.

Let us now compare the experimental and numerical results for LP1 and LP3 before and after the triggering. In both cases, the average chamber pressure increases after triggering. This can be explained by the fact that unstable chambers host pressure oscillations promoting mixing, raising temperatures, changing the chocking conditions at the nozzle. When unstable, the pressure chamber is slightly over predicted for both load points (with an error of 6 % for LP1 and 9 % for LP3). A larger error is observed on the wall heat load which is overpredicted (almost doubled by the numerical case). The same comments on the uncertainty on the wall heat flux raised for LP4 apply here. Interestingly, the simulations preserve the same ordering of heat load magnitudes as observed experimentally: LP4 shows the lowest heat load, triggered LP3 comes next, and triggered LP1 exhibits the highest heat load. It is also interesting to note that a very good agreement is found on the first unstable mode's frequency, both for LP1 and LP3. This frequency was actually underestimated by the non-triggered cases: in the hotter triggered cases the increase in speed of sound provided a shift in the mode frequency. Finally, the pressure amplitudes for the first mode are slightly overestimated with respect to experimental data with a difference of a few percentage points.

## Conclusions

The present work has implemented for the first time a LES study of the flame dynamics within a liquid rocket engine at both sub- and supercritical pressures.

A homogeneous multi-fluid method with the SRK cubic equation of state is applied to simulate three load points (LP1, LP3 and LP4) of the research combustor of the DLR Institute of Space Propulsion. The LP3 and LP4 load points have methane and oxygen injected at subcritical pressures, while LP1 has a supercritical chamber pressure. LP1 and LP3 are thermoacoustically unstable, while LP4 is stable. Vapor-liquid equilibria are computed in the subcritical cases assuming equal composition in the liquid and vapor phases. First simulations of the 3 load points reveal that all stay thermoacoustically stable. In order to reproduce the unstable behavior of LP1 and LP3, the simulations of these two points undergo a triggering procedure.

The resulting unstable LP1 and LP3, and stable LP4 are in relatively good agreement with experimental parameters. Stable LP4 shows good agreement with the experimental average chamber pressure and flame shape. Unstable cases tend to have slightly higher chamber pressures, and larger pressure oscillations, however they correctly reproduce the experimentally-observed non-linear character of the instabilities. DMD analysis shows that the observed oscillations correspond to the first longitudinal mode of the combustor. All 3 simulations replicate remarkably well the frequencies of the first mode of the chamber. While being in the same order of magnitude, wall heat load is over-estimated by the simulations.
<table><caption>**Table 2.** Notable quantities extracted from the simulation compared to their experimental counterparts.</caption><thead><tr><th>Load Point</th><th>LP1</th><th>Triggered LP1</th><th>LP3</th><th>Triggered LP3</th><th>LP4</th></tr></thead><tbody><tr><td>Experimental <span style="text-decoration: overline;">p</span></td><td>62.9 bar</td><td>62.9 bar</td><td>41.3 bar</td><td>41.3 bar</td><td>40.5 bar</td></tr><tr><td>Simulated <span style="text-decoration: overline;">p</span></td><td>54.8 bar</td><td>66.7 bar</td><td>41.6 bar</td><td>44.9 bar</td><td>40.9 bar</td></tr><tr><td>Relative deviation (<span style="text-decoration: overline;">p</span>)</td><td>12.8 %</td><td>6.0 %</td><td>0.7 %</td><td>8.7 %</td><td>1.0 %</td></tr><tr><td>Experimental <span style="text-decoration: overline;">q</span><sub>wall</sub></td><td>111.3 kW</td><td>111.3 kW</td><td>84.0 kW</td><td>84.0 kW</td><td>34.9 kW</td></tr><tr><td>Simulated <span style="text-decoration: overline;">q</span><sub>wall</sub></td><td>8.4 kW</td><td>195.5 kW</td><td>10.8 kW</td><td>159.4 kW</td><td>50.5 kW</td></tr><tr><td>Relative deviation (<span style="text-decoration: overline;">q</span><sub>wall</sub>)</td><td>92.5 %</td><td>75.6 %</td><td>87.1 %</td><td>89.8 %</td><td>44.7 %</td></tr><tr><td>Experimental <i>f</i><sub>1L</sub></td><td>815 Hz</td><td>815 Hz</td><td>920 Hz</td><td>920 Hz</td><td>830 Hz</td></tr><tr><td>Simulated <i>f</i><sub>1L</sub></td><td>≈750 Hz</td><td>≈840 Hz</td><td>≈825 Hz</td><td>≈934 Hz</td><td>≈825 Hz</td></tr><tr><td>Experimental <span style="text-decoration: overline;">p</span><sub>1</sub>/<span style="text-decoration: overline;">p</span></td><td>12.7 %</td><td>12.7 %</td><td>10.4 %</td><td>10.4 %</td><td>0.2 %</td></tr><tr><td>Simulated <span style="text-decoration: overline;">p</span><sub>1</sub>/<span style="text-decoration: overline;">p</span></td><td>≈0.5 %</td><td>≈10.4 %</td><td>≈0.2 %</td><td>≈12.0 %</td><td>≈0.2 %</td></tr></tbody></table>

The current work would therefore still benefit from a more in-depth analysis of some raised issues, in particular:

* The question of the discrepancies in wall heat load is still an open one: on the one hand different boundary condition approaches could be tried in the simulation. On the other hand more accurate experimental data would also be of help in narrowing down the source of the errors.

* While the unstable simulations managed to reproduce some parameters, there are still some discrepancies in the flame shape and the oscillation amplitudes. Longer simulation times would allow to ascertain whether the current regime is transitory.

* While unstable LP1 and LP3 underwent triggering, LP4 was kept in its stable state. It would be interesting to study whether a triggered LP4 would dissipate its instability and go back to a stable state, or whether it would stay unstable, and thus be considered a metastable load point.

* While the present paper focused on the comparison between simulations and experiments, the physics behind combustion in unstable rocket engines at sub- and supercritical pressures is also an interesting topic. Subcritical conditions in particular are a hardly explored topic in CFD, even if new generation rocket engines are expected to work at these regimes. With a validated model, the thermoacoustic couplings happening in these operating conditions could give novel research insights.

## Acknowledgements

This work was supported by the Chair SaCLaB2, resulting from the partnership between Airbus Defence and Space, Ariane Group and ISAE-SUPAERO that co-fund the PhD thesis of G. Pelenghi, and by the GENIAL project (5700007526) funded by CNES. This work was granted access to the GENCI HPC resources under the allocation A0172B12992.

## Notes

1. In the present paper, by transcritical conditions, we mean injection conditions in which the pressure of the propellant(s) is supercritical, but the temperature of the propellant(s) is below the critical value.

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