LES Modeling of the DLR Generic Single-Cup Spray Combustor: Validation and the Impact of Combustion Chemistry

Abstract

Turbulent spray combustion in a generic kerosene-fueled single-cup combustor at typical idle and cruise conditions of an aeroengine are studied with Large Eddy Simulations (LES) using Lagrangian spray and finite-rate chemistry combustion modeling. Three reaction mechanisms of varying complexity are used to model the combustion chemistry. The choice of turbulence-chemistry interaction model is shown to affect the results significantly. The impact of the choice of chemical reaction mechanism and the difference in operating conditions are gauged in terms of time-averaged flow, spray, and combustion characteristics as well as unsteady behavior. Good agreement between LES predictions and experimental results are generally observed but with a notable dependence on the choice of chemical reaction mechanism. The mechanism specifically targeting Jet A displays the best agreement. The choice of reaction mechanism is further demonstrated to influence the flow and thermoacoustics in the combustor, resulting in different thermoacoustic modes dominating. The spray cone is found to be too narrow and thin, an inaccuracy which could be remedied by either making the injection method more empirical or by introducing additional models.

1 Introduction and Background

Modern aircraft typically use turbojet or turbofan engines that run on liquid kerosene-based fuel. Improved combustion efficiency, reduced emissions, a demand to use carbon-neutral fuels to meet environmental needs, and the necessity to avoid thermoacoustic instabilities are some of the main drivers in the development of next-generation jet engines. For liquid-fueled combustors, several approaches have been proposed to reach high efficiency, low emissions and enhanced fuel flexibility; these include lean direct ignition, fuel-staging, rich burn-quick mix-lean burn, and lean premixed pre-vaporized combustion. The fuel-air mixing devices in jet engines not only directly influence ignition, efficiency, lean blow-out, and fuel consumption, but also have effects on emissions and exit temperatures (Ranasinghe et al. 2019). Jet engine combustion is generally sensitive to fluctuations of pressure, air and fuel mass flows, and other quantities affecting the heat release, which may create a feedback loop by which pressure, flow, and heat release drive each other into thermoacoustic oscillations that can damage the combustor or turbine. Such instabilities are prone to develop more easily for lower Cetane number fuels, typically including bio-jet-fuels, and it is therefore essential that all aspects of spray combustion are well understood (Yi and Santavicca 2012).

Both experimental and computational studies are needed to advance our understanding of liquid-fueled jet engine combustion. Experimental investigations alone have not yet revealed all details pertaining to liquid fuel injection, spray formation and atomization, vaporization, fuel ambient mixing, turbulent combustion, and emission formation (Sornek et al. 2000). Numerical simulations using Reynolds-Averaged Navier Stokes (RANS) models (Klose et al. 2001) have been useful to help illuminate some of the underlying spray combustion physics, but as RANS is a steady-state approach, unsteady phenomena cannot be accurately captured. Instead, Large Eddy Simulations (LES) (Echekki and Mastorakos 2011) have been developed as a compromise between RANS and Direct Numerical Simulations (DNS) in cost and accuracy. Spray combustion introduces further challenges since phase change and inter-phase interactions need to be properly accounted for. Many of these processes occur at subgrid scales and must therefore be modeled (Patel and Menon 2008). The combustion chemistry plays a crucial role not only for the emission profiles, but also for the thermoacoustics that are triggered by the volumetric expansion of the gas.

Here we report on LES of turbulent spray combustion in a generic combustor operating at typical aeroengine conditions, investigated experimentally by DLR in Behrendt et al. (1999) and Meier et al. (2012). The case has previously been simulated with LES, as described in Jones et al. (2014). That study used the Smagorinsky subgrid model, a global four-step reaction mechanism based on Jones and Lindstedt (1988), the stochastic fields turbulence-chemistry interaction model, a finite volume scheme, and a grid with 2.2 million cells. A Lagrangian formulation was used to model the fuel spray, which was injected into the domain downstream of the burner where experimental data about its angle, velocity and size distribution were available. Good agreement with experimental data was achieved. Another team has also simulated the case with an Eulerian-Lagrangian approach, as described in Puggelli et al. (2016, 2018a, 2018b) and Mazzei et al. (2019). RANS, Scale Adaptive Simulation (SAS), and LES were all used in these studies along with a variety of combustion models including the Flamelet Generated Manifold (FGM), Eddy Dissipation (EDM), and Artificially Thickened Flame (ATF) models. The liquid film near the injection point was found to influence the spray significantly.

Our work draws inspiration from the previous studies, and aims to build on them in three major ways. Firstly, previous reactive simulations have only been carried out for operating conditions corresponding to an idling aeroengine, with the exception of the brief analysis in Mazzei et al. (2019), while our study also includes cruise conditions. This allows for a more thorough validation of the simulation method and its flexibility. Secondly, we choose to shift the focus to the combustion chemistry, as spray modeling has already been examined by Puggelli et al. (2018b). We compare three small comprehensive (77, 277, and 254 reaction steps) pathway-centric mechanisms. The largest two mechanisms come from the same source, but aim to model two different conventional jet fuels. In this way, the sensitivity of the results to the mechanism complexity as well as the fuel are evaluated across two physical cases, providing a good indication of the impact of these models on the overall results. Thirdly, a substantial portion of our analysis is dedicated to the thermoacoustics, and how this affects other aspects of the spray and flame. To our knowledge, such an analysis has not been done for this case before. We have previously reported on our initial results and analysis in Åkerblom (2022) and Åkerblom et al. (2022), where it was shown that the choice of reaction mechanism and fuel can have a non-negligible impact on the LES results, motivating the present, more in-depth work.

The study starts with a description of the simulated case. This is followed by a brief description of the LES methodology, spray model, chemistry model, and numerical schemes. The chemical reaction mechanisms are examined separately, where their predictions in a premixed laminar flame, closed reactor, and non-premixed counter-flow flame are compared. The LES results are then presented and compared, starting with qualitative overviews of the flow and combustion followed by more quantitative comparisons. The sensitivity to the mesh resolution and the choice of turbulence-chemistry interaction model is gauged to allow for an informed choice of resolution and models to be used in subsequent simulations, without conclusively determining the optimal strategy. Once a clear picture of the results has been established, spray and flame shapes are compared with experimental data. Finally, the prominence and characteristics of thermacoustic oscillations are investigated using a quantification of Rayleigh’s criterion, and excited acoustic modes are identified in the simulation results using Proper Orthogonal Decomposition (POD) (Berkooz et al. 1993).

2 Single-Cup Combustor and Experiments

The targeted experiments were performed in the high-pressure single-cup combustor (Behrendt et al. 1999) located at DLR Institute of Propulsion Technology. A schematic of the test rig is presented in Fig. 1. It features a square cross-section of 102 x 102 mm$^2$ and a length of 264 mm. Electrically preheated, compressed air was supplied both to a plenum upstream of the combustor as well as cooling slots upstream of the side walls. Burner and window-cooling air flows were monitored and the burner air-to-fuel ratio was held constant.

The burner features two sets of swirlers inducing co-rotating air flows. Fuel is injected through an annular slot along the inner wall of the burner, close to the downstream exit to the combustion chamber. The fuel enters the burner as a thin liquid film, which is atomized by the swirling air flow at the 2 mm long prefilmer lip.

Two operating conditions (see Table 1) were considered in the experiments: idle and cruise. Idle conditions refers to pressures, temperatures and mass flows typical of an idling jet engine, while cruise conditions corresponds to flight at cruising altitude. Both operating conditions are investigated in the present study, implemented as two sets of boundary conditions.

Table 1 Experimental operating conditions, as reported in Meier et al. (2012)

Multiple diagnostic techniques were applied to this case, including video imaging, OH chemiluminescence, planar Mie scattering for imaging of the liquid fuel phase, Planar Laser-Induced Fluorescence (PLIF) for imaging of both the vapor and liquid phase of the fuel, PLIF of OH for visualization of fuel-air mixing and estimating the temperature distributions in lean parts of the flame, as well as Phase Doppler Anemometry (PDA) for spray analysis including droplet velocities, sizes and volume fluxes..

Fig. 1

Schematic of the combustor test rig, adapted from Meier et al. (2012). Preheated air passes from the plenum to the combustion chamber via the burner, where swirlers induce rotational motion and fuel is injected. The windows are cooled by a film of the same preheated air supplied to the plenum

3 LES Modeling and Numerical Methods

The LES model is based on implicitly filtered transport equations for the gas-phase mass (Eq. 1a), momentum (Eq. 1b), energy (Eq. 1c), and species mass fractions (Eq. 1d), closed by constitutive equations for a linear viscous ideal mixture with Fickian diffusion and Fourier heat conduction (Menon and Fureby 2010). Filtered quantities are denoted by overbars, or tildes in the case of density-weighted filtering. Density, velocity, pressure, enthalpy, and species mass fractions are denoted $\overline{\rho }$, $\tilde{\textbf{v}}$, $\overline{p}$, $\tilde{h}$, and $\tilde{Y}_i$, respectively. For compact notation, the energy variable is defined as $\tilde{H}_K=\tilde{h}+\frac{1}{2}\vert \tilde{\textbf{v}}\vert ^2$. The filtered viscous stress tensor is given by $\overline{\varvec{\tau }}=2\overline{\mu }\tilde{\textbf{D}}-\frac{2}{3}\overline{\mu }(\nabla \cdot \tilde{\textbf{v}})\textbf{I}$, and the strain rate tensor is given by $\textbf{D}=\frac{1}{2}(\nabla \textbf{v}+\nabla \textbf{v}^T)$. The species’ diffusivities $\overline{D_i}$ and their thermal diffusivites $\alpha _i$ are computed from the filtered viscosity $\overline{\mu }$ using constant Prandtl and Schmidt numbers following Giacomazzi et al. (2007), and the overall filtered thermal diffusivity $\overline{\alpha }$ in the energy equation is based on a mass-weighted average of individual species diffusivities.

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial \overline{\rho }}{\partial t}+\nabla \cdot (\overline{\rho } \tilde{\textbf{v}})=\overline{\dot{\rho }_{l}},&{}\quad (\hbox {1a}) \\ \frac{\partial \overline{\rho }\tilde{\textbf{v}}}{\partial t}+\nabla \cdot (\overline{\rho } \tilde{\textbf{v}}\otimes \tilde{\textbf{v}})=-\nabla \overline{p}+\nabla \cdot \overline{\varvec{\tau }}+\overline{{\textbf{f}}_{l}}-\nabla \cdot \textbf{B},&{}\quad (\hbox {1b}) \\ \frac{\partial \overline{\rho } \tilde{H}_K}{\partial t}+\nabla \cdot (\overline{\rho } \tilde{H}_K \tilde{\textbf{v}})=\frac{\partial \overline{p}}{\partial t}+\nabla \cdot (\overline{\varvec{\tau }}\tilde{\textbf{v}})+\nabla \cdot (\overline{\alpha }\nabla \tilde{h})+\overline{\dot{q}_{c}}+\overline{\dot{q}_{l}}-\nabla \cdot \textbf{b}_\textbf{E},&{}\quad (\hbox {1c})\\ \frac{\partial \overline{\rho } \tilde{Y_i}}{\partial t}+\nabla \cdot (\overline{\rho } \tilde{Y_i} \tilde{\textbf{v}})=\nabla \cdot (\overline{D_i}\nabla \tilde{Y_i})+\overline{\dot{\omega }_{i}}+\overline{\dot{m}_{il}}-\nabla \cdot \textbf{b}_\textbf{i}.&{}\quad (\hbox {1d}) \end{array}\right. \end{aligned}$$

The subgrid stress and flux terms $\textbf{B}$, $\mathbf {b_E}$, and $\mathbf {b_i}$ are explicitly modeled using Boussinesq’s hypothesis as described by the equations,

$$\begin{aligned} \left\{ \begin{array}{ll} \textbf{B}=2\overline{\rho }\bigg (-\nu _{sgs}\text {dev}(\tilde{\textbf{D}})+\frac{1}{3}k\textbf{I}\bigg ), &{}\quad (\hbox {2a}) \\ \mathbf {b_E}=-\overline{\rho }\frac{\nu _{sgs}}{\mathrm {Pr_{t}}}\nabla \tilde{H}_K, &{}\quad (\hbox {2b}) \\ \mathbf {b_i}=-\overline{\rho }\frac{\nu _{sgs}}{\mathrm {Sc_{t}}}\nabla \tilde{Y_i}{,} &{}\quad (\hbox {2c}) \end{array}\right. \end{aligned}$$

where the deviatoric part of the strain rate tensor is denoted $\text {dev}(\tilde{\textbf{D}})$. Here, the conservative estimates $\mathrm {Pr_t}\approx \mathrm {Sc_t}\approx 1$ are used for the turbulent Prandtl and Schmidt numbers, respectively. The subgrid viscosity $\nu _{sgs}$, which is needed to close the subgrid model equations, can be computed using a variety of models. Here we use the One Equation Eddy Viscosity Model (OEEVM) (Yoshizawa and Horiuti 1985), in which the subgrid turbulent kinetic energy k is solved for using the transport equation,

$$\begin{aligned} \frac{\partial \overline{\rho } k}{\partial t}+\nabla \cdot (\overline{\rho } k \tilde{\textbf{v}})=-{\textbf {B}} \cdot \tilde{\textbf{D}}+\nabla \cdot (\overline{\rho }\nabla k)-C_\epsilon \frac{\overline{\rho }k^{3/2}}{\Delta }. \end{aligned}$$

(3)

When k has been obtained, the subgrid viscosity is computed as $\nu _{sgs}=C_k\sqrt{k}\Delta$, where the LES filter width $\Delta$ is taken to be the cubic root of the local mesh cell volume. The model coefficients are set to $C_k=0.094$ and $C_\epsilon =1.048$ based on an assumed inertial subrange $k^{-5/3}$ spectrum.

Because turbulent diffusion is expected to dominate over laminar thermal and molecular diffusion, the transport properties of individual species may be simplified. Here, identical coefficients derived from dry air are assigned to all species in the computation of viscosity. The Schmidt number is assumed to be unity following Karalus et al. (2021), which describes a similar case.

The source terms $\overline{\dot{\rho }_l}$, $\overline{\textbf{f}_l}$, $\overline{\dot{q}_l}$, and $\overline{\dot{m}_{il}}$ represent mass, momentum, energy, and species mass transfer between the gas phase and the liquid phase, i.e. the fuel spray. Because the liquid phase mostly consists of a dilute cloud of microscale droplets, it is practical to apply the Discrete Droplet Model (DDM), where the spray is modeled as a cloud of discrete Lagrangian particles that individually follow a set of governing equations (Faeth 1987). Each particle represents a characteristic group of droplets with identical properties and thermodynamic states (Dukowicz 1980). In the targeted case, the liquid exists as a continuous film at the prefilmer lip, undergoing primary breakup as it enters the combustion chamber. The result is a dense cloud of large droplets, in which droplets affect each other’s transport rates and collisions play an important role. The dense cloud then undergoes secondary breakup into smaller droplets, forming a dilute cloud where droplet-droplet interactions are of lesser importance. A stochastic model for droplet collisions, based on O’Rourke (1981), is applied to increase the accuracy of the spray model in the dense regime. The liquid film regime is however neglected, as the spray is considered fully atomized at injection. Secondary droplet breakup is modeled with the Reitz-Diwakar model (Reitz and Diwakar 1987), in which breakup occurs in the bag and strip** modes. The droplets are subjected to gravity and spherical drag, where $C_D$ is computed following Crowe (1998). The heat transfer coefficient is computed from the parcel Reynolds and Prandtl numbers using the Ranz-Marshall model (Ranz and Marshall 1952). The vaporization rate is computed from the Reynolds and Schmidt numbers as outlined in Crowe (1998). Subgrid fluctuations in the gaseous phase are not considered in the evaporation model, which may lead to an overly concentrated spray. Surface reactions are likewise neglected, as the spray and flame are not expected to overlap. The same thermodynamic and transport properties, based on a generic kerosene mixture, are used regardless of reaction mechanism. For further details on the spray model, we refer to Åkerblom et al. (2022).

The source terms $\overline{\dot{\omega }_{i}}$ and $\overline{\dot{q}_{c}}=\sum _{i=1}^{N}h_i^\theta \overline{\dot{\omega }_{i}}$ represent heat and mass transfer due to chemical reactions. Here, these terms are modeled using a finite-rate chemistry approach with turbulence-chemistry interactions captured by either the Partially Stirred Reactor (PaSR) (Sabelnikov and Fureby 2013), Eddy Dissipation Concept (EDC) (Magnussen 1981), Fractal (FM) (Giacomazzi et al. 2000), or Eulerian Stochastic Fields (ESF) (Jones and Navarro-Martinez 2007) models. The EDC model was developed for RANS by Magnussen (1981), from the works of Batchelor and Townsend (1949) and Chomiak (1970) and later modified for LES by Fureby (2009). This model reflects the observations that turbulent structures are not uniformly distributed, but concentrated in localized regions, and that the combustion reactions occur primarily within these fine structures (Tanahashi et al. 2000). To account for this, the fine structure volume fraction, $\gamma ^*$, is introduced, resulting in that $\overline{\dot{\omega }_{i}}=\gamma ^*\dot{\omega }_{i}^*+(1-\gamma ^*)\dot{\omega }_{i}^0$, where $^*$ and $^0$ denote fine structures and surroundings, respectively. This may be approximated as $\overline{\dot{\omega }_{i}}\approx \gamma ^*\dot{\omega }_{i}(\overline{\rho },\tilde{Y}_k,\tilde{T})$. A model of the turbulent cascade process was developed by Magnussen (1981) to provide an estimate for the fine structure volume fraction, $\gamma ^*\approx 1.02(\nu /\Delta k^{1/2})^{3/4}$. The FM model (Giacomazzi et al. (2000)) was developed from the EDC model but estimates $\gamma ^*$ using fractal theory such that $\gamma ^*=\gamma _N(\Delta /l_K)^{D_3-3}$, where $\gamma _N=N_{l,K}/N_T$ is the ratio of the number of Kolmogorov scales, $N_{l,K}$, to the overall number of scales, $N_T$, generated locally, whereas $D_3$ is the fractal dimension of the turbulence. Given theoretical estimates for $D_3$, $N_{l,K}$, and $N_T$, in the case of Kolmogorov turbulence, the FM model is closed. For high Re numbers the fine structure volume fraction approaches the limit value $\gamma ^*\approx 0.314$.

The PaSR model (Sabelnikov and Fureby 2013) is based on the same assumptions and approximations as the EDC model, expressing that chemistry occurs in fine structures embedded in an eddy-rich environment. In the PaSR model, however, chemical reactions and mixing are treated separately and associated with the time scales $\tau _c$ and $\tau _m$, respectively. By assuming sequential reaction and mixing steps (in state space), the filtered reaction rates can be shown to become $\overline{\dot{\omega }_{i}}\approx \gamma ^*\dot{\omega }_{i}(\overline{\rho },\tilde{Y}_k,\tilde{T})$, where $\gamma ^*=\tau _c/(\tau _m+\tau _c)$. Here, $\tau _c$ is modeled as $\tau _c\approx \delta _u/s_u$, where $\delta _u$ is the laminar flame thickness and $s_u$ the laminar flame speed. Similarly, $\tau _m$ is modeled as $\tau _m=l_D/v_K=\sqrt{\tau _K\tau _\Delta }$, where $l_D$ is the dissipation length scale, $v_K$ the Kolmogorov velocity scale, $\tau _\Delta =\Delta /v$ the shear time scale, representative of turbulent dissipation, and $\tau _K$ the Kolmogorov time scale, representative of mixing at the smallest scales. In the present work, $\tau _m$ is computed based on local flow conditions while $\tau _c$ is based on stoichiometric conditions.

The ESF model (Jones and Navarro-Martinez 2007; Jones et al. 2015), is a stochastic rather than deterministic method of representing the probability density class of combustion models (Echekki and Mastorakos 2011). In the ESF approach the PDF, P, is represented by an ensemble of N stochastic fields with each field encompassing the $N_s$ scalars, $\xi ^n_\alpha$ for $1\le n\le N$ and $1\le \alpha \le N_s$. In the present work, the $N_s$ scalars $\tilde{\phi }_\alpha$ are the enthalpy and the species mass fractions. Here, the Itô formulation of the stochastic integral is adopted (Jones et al. 2015) and the stochastic fields evolve according to,

$$\begin{aligned}{} & {} \overline{\rho }\textrm{d}\xi ^n_\alpha +\overline{\rho } \tilde{\textbf{v}}\cdot \nabla \xi ^n_\alpha \textrm{d}t =\nabla \cdot (\Gamma \nabla \xi ^n_\alpha )\textrm{d}t +\overline{\rho }\sqrt{\frac{2\Gamma }{\overline{\rho }}} \nabla \xi ^n_\alpha \cdot \textrm{d}\textbf{W}^n\nonumber \\{} & {} \qquad \qquad \qquad \qquad -\frac{1}{2}\overline{\rho }\frac{c_d}{\tau _K} (\xi ^n_\alpha -\tilde{\phi }_\alpha )\textrm{d}t +\dot{\omega }^n_\alpha (\xi ^n_\alpha )\textrm{d}t, \end{aligned}$$

(4)

where $\Gamma$ is the total diffusion coefficient, $c_d$ a dimensionless model coefficient, and $\textrm{d}\textbf{W}^n$ the increments of a Wiener process, independent of spatial location but different for each field. The stochastic term has no influence on the first moments of $\xi ^n_\alpha$. The Wiener process is approximated by time-step increments $\mathbf {\eta }^n\sqrt{\textrm{d}t}$, where $\mathbf {\eta }^n$ is a normalized dichotomic random vector. The stochastic fields given by Eq. (4) form an equivalent stochastic system to the underlying PDF (Gao and O’Brien 1993), smooth over the scale of the filter width. Following Jones et al. (2015) eight fields are solved for each scalar variable.

The LES model equations are solved using a finite-volume code based on the OpenFOAM C++ library (Weller et al. 1998). We employ monotonicity-preserving reconstruction for the convective fluxes, central differencing of the inner derivatives of the diffusive fluxes and backward differencing for time-integration. The code uses a compressible Pressure-based Implicit Splitting of Operators (PISO) scheme (Bressloff 2001) for the pressure–velocity-density coupling. The chemical source terms are incorporated using an operator-splitting approach, with a Rosenbrock solver (Hairer and Wanner 1991) used to solve the resultant stiff ODE system. Stability is enforced using compact stencils and a Courant number less than 0.95.

4 Chemical Kinetics

To compute the net production rates $\dot{\omega }_{i}^*$ in the fine structures, a realistic chemical reaction mechanism for the breakdown and combustion of the fuel is required. Due to the high cost of using detailed chemistry for complex hydrocarbons in LES, the reaction mechanism must also be heavily reduced. In other words, only the most relevant and representative pathways are included and chains of reactions are lumped together into single steps. This kind of reduction naturally involves assumptions and approximations that make the mechanisms less applicable generally, but reduced mechanisms are typically well validated for engine applications. In this study, we employ such reduced mechanisms. We refer to mechanisms that are reduced to only a handful of reactions as global. Larger reduced mechanisms are referred to as pathway-centric.

Four reaction mechanisms of varying complexity are used in this study:

• Z77. A pathway-centric mechanism for generic kerosene fuel with 30 species and 77 reactions. It was proposed in Zettervall et al. (2020) as an efficient mechanism for CFD applications. The surrogate fuel species is C$_{12}$H$_{23}$.

• HyChem JP-5. A pathway-centric mechanism for JP-5 with 50 species and 277 reactions. Presented in Wang et al. (2018) and Xu et al. (2018) as part of the HyChem (Hybrid Chemistry) series of mechanisms, it is currently expensive but feasible to use in LES. We have chosen version 2.5 of the skeletal form of the mechanism, which includes low-temperature chemistry. The surrogate fuel species is C$_{12}$H$_{23}$.

• HyChem Jet A. A pathway-centric mechanism for Jet A with 48 species and 254 reactions. It was also presented in Wang et al. (2018) and Xu et al. (2018) and is a part of the HyChem series. Here we also chose version 2.5 of the skeletal form of the mechanism, which includes low-temperature chemistry. The surrogate fuel species is C$_{11}$H$_{22}$.

• CRECK Jet A. A larger mechanism presented in Ranzi et al. (2014) which supports surrogate mixtures of different jet fuel components. With 231 species and 5591 reactions it is restricted to low-dimensional simulations in the present work.

These four mechanisms were chosen for two primary reasons. Having two mechanisms (Z77 and HyChem JP-5) of different complexity that use the surrogate species C$_{12}$H$_{23}$ allows an investigation into the impact of mechanism complexity on the results. Having two similar mechanisms (HyChem JP-5 and Jet A) for different fuel species reveals the impact of fuel choice.

The general behavior of a reaction mechanism may be understood through its predictions of fundamental flame parameters in low-dimensional simulations. Here, we use ANSYS Chemkin-Pro (Ansys 2018) to compute laminar flame speed ($s_u$), maximum flame temperature ($T_{max}$), ignition delay time ($\tau _{ign}$), and non-premixed extinction strain rate ($\sigma _{ext}$). The first two parameters come from the same set of one-dimensional planar flame simulations, where the unburnt mixture has a temperature of 403 K. Ignition delay time is predicted with zero-dimensional simulations of homogeneous reactors at stoichiometric conditions. Extinction strain rate is predicted using one-dimensional simulations of non-premixed counterflow flames at atmospheric pressure, where fuel/nitrogen mixtures at 473 K meet pure oxygen streams at 300 K.

Predictions of $s_u$, $T_{max}$, and $\tau _{ign}$ are presented in Fig. 2 for three different pressures: 1, 5, and 10 atm. Experimental data is included where available. Note that the experimental $\tau _{ign}$ results were obtained at 12 atm (Xu et al. 2018) and 14.8 atm (Mao et al. 2020). Results from the complex CRECK Jet A mechanism, obtained for a typical multi-component Jet A surrogate, are included for comparison. The HyChem mechanisms give virtually identical predictions for laminar flames, though these are somewhat different from those of Z77. Z77 predicts a higher laminar flame speed, which matches the experimental observations of Kumar et al. (2011) and the CRECK results better. On the other hand, the lower predictions by the HyChem mechanisms match the observations of Xu et al. (2018), which were used as a validation target the development of those mechanisms, better. Given the spread in experimental data and the general similarity of all predictions to those of CRECK, we determine that Z77 and the HyChem mechanisms all predict realistic if slightly different flame speeds. With regard to $T_{max}$, however, the HyChem mechanisms match CRECK very well while Z77 is quite pressure-sensitive, predicting a lower flame temperature at low pressure and a higher one at high pressure. The ignition delay time is approximately proportional to $p^{-1}$, and the NTC (Negative Temperature Coefficient) region, where $\tau _{ign}$ is relatively short at low temperatures, appears to be somewhat reduced at higher pressures. The $\tau _{ign}$ reduction is the most distinct for Z77 and the least distinct for CRECK. Z77 again appears to be the most pressure-sensitive, as it predicts a faster ignition than the HyChem mechanisms at low pressure, and a slower ignition at high pressure. We consider all predictions to match up well with experimental data where available.

Figure 3 shows predictions of extinction strain rate, $\sigma _{ext}$, for various fuel/nitrogen mixtures. All mechanisms behave quite similarly, though HyChem JP-5 consistently predicts a slightly lower $\sigma _{ext}$. This agrees somewhat with the experimental results used to validate these two mechanisms, but both the strain rate itself and the difference between the fuels is overpredicted.

Fig. 2

Laminar flame speed (a), maximum laminar flame temperature (b), and ignition delay time at stoichiometric conditions (c). Left to right: p = 1, 5, 10 atm. Temperature for laminar flame speed: 403 K. Legend, simulations:, Z77 (blue line), HyChem JP-5 (green line), HyChem Jet A (red line), CRECK Jet A (black line). Legend, experimental data: Xu et al. (2018) (Jet A, red squares), Xu et al. (2018) (JP-5, green squares), Hui and Sung (2013) (Jet A, diamonds), Dooley et al. (2012) (Jet A, downward-facing triangles), Kumar et al. (2011) (Jet A, circles), Mao et al. (2020) (n-dodecane, upward-facing triangles)

Fig. 3

Extinction strain rate for a non-premixed counterflow flame at atmospheric pressure. Legend, simulations: Z77 (blue line), HyChem JP-5 (green line), HyChem Jet A (red line). Legend, experimental data: Xu et al. (2018) (Jet A, red squares), Xu et al. (2018) (JP-5, green squares), Liu et al. (2017) (n-dodecane, right-facing triangles)

5 Simulation Setup

The computational domain of the LES is presented in Fig. 4 (along with some key flow features) and consists of a three-dimensional model of the test rig interior, with a fixed-mass-flow inlet plane placed inside the plenum and a wave-transmissive outlet plane placed at the throat of the sonic exit nozzle. The narrow slit where cooling air is supplied is also treated as a continuous fixed-mass-flow inlet. The pressure (controlled by the outlet boundary condition), temperature, and mass flows are chosen to match the conditions presented in Table 1. All walls are treated as adiabatic no-slip surfaces, and a wall model based on Spalding’s law of the wall (Spalding 1961) is used to capture the effects of the walls on the overall flow by adjusting turbulent viscosity, thus avoiding the need for a very fine grid resolution near walls. The liquid phase is injected as a cloud of room-temperature Lagrangian droplet parcels at the prefilmer lip, with an initial velocity matching the surrounding air flow. The droplet diameters are chosen randomly from a Rosin-Rammler distribution, obtained from Jones et al. (2014). The injection rate is $10^9$/s, resulting in approximately 550,000 (idle) and 300,000 (cruise) Lagrangian particles in the simulation at any time.

A baseline block-structured computational grid of 6.0 million hexahedral cells is used to discretize the Eulerian equations, with characteristic dimensions of 0.33 to 0.5 mm in the near-flame region. On average, locally, approximately 85% of the turbulent kinetic energy is resolved. This ratio is higher in the shear layers and flame, which constitute the main regions of interest. Based on Pope’s recommendation that at least 80% of turbulent kinetic energy should be resolved (Pope 2004), the grid resolution should thus be adequate. Realistic results have previously been obtained by Jones et al. (2014) using a 2.2 million cell gridand a minimum cell size of 0.5 mm. In order to measure the mesh sensitivity, the simulation with HyChem Jet A at idle conditions is repeated on a uniformly refined mesh with 48 million cells. (See Fig. 6 and associated discussion.) Based on the similar grid requirements for the three mechanisms in the laminar flame simulations associated with Fig. 2, we assume that the grid sensitivities are similar in turbulent flames as well. The mesh sensitivity is found to be comparable or smaller than the sensitivity associated with choosing a turbulence-chemistry interaction model, as well as the reaction mechanism sensitivity and variation among experimental observations in Figs. 2 and 3.

All simulations are initialized from a simple state where the combustor is filled with burnt gas. When a statistically steady state has been achieved, data is collected over the course of 20 ms at idle conditions. At cruise conditions, a shorter flow-through time and more coherent dynamics make 12 ms sufficient for statistical analysis.

Fig. 4

The computational domain, with non-wall boundaries highlighted in light pink. The time-averaged spray (magenta) and flame are also included. The walls are colored by temperature, and a white vector field shows the direction of the flow in the central plane. Recirculation regions are demarcated by white lines

6 LES Results and Validation

Figure  5 shows snapshots of the spray and flame front in a central slice of the domain, with the flame front defined as the contour of $Q=\textrm{max}(Q)/10$, where Q is the heat release rate. The radial momentum and sudden geometric expansion experienced by the air entering the combustion chamber gives rise to recirculation zones delineated by shear layers. A shear layer is identified as a contour of $\langle \tilde{v}_z\rangle =0$ m/s, where $\langle \tilde{v}_z\rangle$ is the mean axial velocity. All shear layers, as well as the regions between them, are marked in the figure. Axial flow occurs in a large conical volume, the Main Flow Cone (MFC), which originates from the burner and eventually reaches the combustor walls. The flow is reversed close to the centerline, in what is referred to as the Central Recirculation Zone (CRZ). The CRZ ensures that hot combustion products are recirculated towards the spray. The boundary between the MFC and the CRZ is the Inner Shear Layer (ISL), which provides a stable region for the flame front. The outer boundary of the MFC is the Outer Shear Layer (OSL), adjacent to the Outer Recirculation Zone (ORZ). Because hot products are recirculated, cooling films are needed in order to protect the transparent windows in the combustor walls by the ORZ. The cooling films give rise to a small region of axial flow by the back corners of the combustion chamber.

The flames are clearly non-premixed in the sense that there exists a region of pure fuel (the liquid spray) and pure air (the burner air flow). However, because the fuel is injected into the air stream, there may be some mixing between fuel and air before the fuel ignites, giving the flame some premixed characteristics locally. The premixed behavior can be quantified using the dimensionless Takeno Flame Index (TFI) (Yamashita et al. 1996). If the gradients of fuel and oxygen are parallel where combustion occurs, then they are both consumed from one side of the flame; the flame is locally premixed and $\textrm{TFI}=1$. If the gradients are antiparallel, then fuel and oxidizer are consumed from opposite sides; the flame is locally non-premixed and $\textrm{TFI}=0$.

The fuel-containing region in Fig. 5 is colored by TFI, with black representing the non-premixed mode and gray representing the premixed mode. The spray (magenta) is carried by the MFC into the combustion chamber while evaporating, increasing the fuel mass fraction and decreasing the oxygen mass fraction, which results in $\textrm{TFI}=0$. Closer to the flame, the fuel and air have formed a mixture which begins to react, decreasing the mass fraction of both and resulting in $\textrm{TFI}=1$. In other words, the combustion occurs in a premixed mode on the local level.

Fig. 5

Snapshots of the spray (magenta) and flame (red) in a central cross section at idle (a) and cruise (b) conditions. The flame shape is determined by the $\textrm{max}(Q)/10$ contour. Black: non-premixed mode ($\textrm{TFI}=0$). Gray: premixed mode ($\textrm{TFI}=1$). Mixing modes are only highlighted in the presence of fuel. Contours of $\langle \tilde{v}_z\rangle =0$, indicating shear layers, are marked with blue lines

The mesh employed here, and by extension the LES filter width, is not fine enough to resolve the smallest scales of the turbulent flame dynamics. A turbulence-chemistry interaction model should therefore be chosen with care, as it can have a large impact on the overall results. The results of a qualitative sensitivity study containing the PaSR, FM, EDC, and ESF models are presented in Fig. 6. PaSR results obtained with a finer mesh of 48 million cells are shown as well. Only idle conditions with HyChem Jet A is considered, and all results are normalized separately. Experimentally measured, time-averaged OH chemiluminescence, an indicator of heat release, is compared to actual time-averaged heat release in the simulations. It should be noted that these are different quantities, so the comparison has some inherent uncertainty. The experimental results are Abel-transformed to show the mean flame in the central plane, while the simulation results are azimuthally integrated. The azimuthal integration is done separately for two lateral halves of the domain, resulting in slightly asymmetric heat release distributions.

The PaSR, FM, EDC and ESF models all predict roughly the same conical flame shape observed in the experiments, but the flame lift-off is generally overpredicted. If the flame lift-off is defined as the axial position of the steepest gradient of the heat release curve, the PaSR, FM, EDC, and ESF models overpredict it by approximately 30%, 130%, 90%, and 30%, while the spatially and temporally averaged heat release is predicted to be 0.40, 0.26, 0.32, and 0.42 GW/m$^3$, respectively. The fine-structure volume fraction computed by the FM model is $\sim$0.3, close to the high-Re limit of 0.314, which appears to be too low for the present case as it results in a quite lifted and distributed flame. The ESF model predicts a concentrated flame and forms a lower bound for the flame lift-off, indicating that the stochastic treatment is equivalent to a more reactive sub-grid environment. There is indeed a substantial sensitivity to the choice of turbulence-chemistry interaction model. The sensitivity may be reduced by refining the mesh, thus reducing the range of length scales that have to be modeled. The error may be reduced by supplying the models with different parameters, iterating until agreement with experimental observations is reached. Here we chose the model parameters a priori based on the experimental results in Fig. 2, and the result is that PaSR and ESF give the most realistic predictions of flame lift-off. PaSR is therefore used in the subsequent simulations. Conclusively determining which of the models is best suited for the present case would require a separate, dedicated study similar to Fedina et al. (2017), Fureby (2018), or Liu et al. (2022), where PaSR has consistently proven to be a robust choice.

When using PaSR, increasing the mesh resolution evidently affects the results. The flame becomes more compact and takes on a shape strikingly similar to that predicted by ESF with the baseline mesh. This similarity is likely caused by a more concentrated heat release distribution when using the finer mesh, which happens to be very similar to that predicted by ESF when using the baseline mesh.

Fig. 6

Mean flame shape in experiments and in LES using different turbulence-chemistry interaction models, shown in two dimensions (a) and along the axial direction (b). Experimental results obtained from Meier et al. (2012). Legend: experiment (black, squares), PaSR (magenta), FM (red), EDC (green), ESF (blue), PaSR with fine mesh (cyan)

The mean gas-phase temperatures and axial velocities in all simulations are shown in Fig. 7 in a central slice that includes the burner and the first 100 mm of the combustion chamber. Peak axial velocities of $\langle \tilde{v}_z\rangle \approx 100$ m/s are seen in the MFC close to (and inside) the burner. Sharp temperature gradients are seen in the flame region, where the gas is heated and expanded. The reaction mechanisms predict different temperatures in the downstream region, but the velocity field is generally similar. The main outlier is HyChem Jet A at cruise conditions, which predicts a relatively thick CRZ. This is because the flame in that simulation is more spread out over the ISL than in the others, as will be shown later, which directs the MFC outward.

Figure 8 shows snapshots instead of time averages. In addition, a magenta-colored pressure iso-surface is included to reveal a Precessing Vortex Core (PVC), a well-known phenomenon in swirl combustors (Hassa et al. 2002). The PVC is a rotating helical low-pressure region that contains a center of rotation for the velocity. A PVC is typically located at the ISL, and by comparing Fig. 8 to the mean fields in Fig. 7 we see that this holds true in the simulations. The PVC is longer and has a more distinctly helical shape at idle conditions than at cruise conditions. This happens because the flame, which causes rapid volume expansion and thus disrupts the PVC, is located closer to the burner. At cruise conditions, Z77 and HyChem JP-5 appear to predict longitudinally alternating regions of hot and cold gas, as well as a compact instantaneous MFC. This is in contrast to HyChem Jet A, which predicts a non-alternating temperature field and large instantaneous MFC. This difference between the mechanisms is caused by comparatively weak thermoacoustic fluctuations for HyChem Jet A, and is explored further in Sect. 7.

Fig. 7

Time-averaged temperature (left half of panels) and axial velocity (right half of panels). The $\textrm{max}(\langle Q\rangle )/10$ contour, where $\langle Q\rangle$ is the time-average heat release, is shown in black. Left to right: Z77, HyChem JP-5, HyChem Jet A, profiles. The profiles show the radial distribution of temperature (solid lines) and axial velocity (dashed lines) 50 mm downstream of the burner exit (marked by magenta line). Profile legend: Z77 (blue), HyChem JP-5 (green), HyChem Jet A (red). Top to bottom: Idle conditions, cruise conditions

Fig. 8

Instantaneous temperature (left half of panels) and axial velocity (right half of panels). Left to right: Z77, HyChem JP-5, HyChem Jet A. Top to bottom: Idle conditions, cruise conditions

Figure 9 contains a snapshot of the combustion chemistry in each simulation. The Lagrangian parcels (not to scale) are colored by temperature and reveal a highly unsteady spray. Initially parallel to the inner burner wall, the sheet of droplets becomes progressively dispersed by the gas flow and forms a cone as it propagates into the combustion chamber. The liquid fuel is rapidly heated in the hot chamber, causing the droplets to evaporate into gaseous fuel (green). A thin layer of formyl radical (HCO, blue) covers the outside of the fuel cloud. HCO is a very short-lived radical only found in high concentration near the flame front. There is virtually no fuel present beyond this HCO layer. The spray is heated up more rapidly at cruise conditions than at idle conditions, due to the proximity to the flame front and higher temperature of the unburnt gas. The result is a flame located closer to the burner, on average. This is evident in the temperature plots in Figs. 7 and 8 as well.

Naturally, high concentrations of carbon dioxide (CO$_2$, yellow) can be found in the postflame zone downstream of the flame front. There is also a relative abundance of hydroxyl radical (OH, red), which decreases towards an equilibrium concentration in the downstream region. The regions containing both OH and CO$_2$ are thus rendered with an orange tint, with a deeper orange indicating higher OH concentrations. The HyChem mechanisms display higher OH concentrations than Z77, particularly at cruise conditions.

Fig. 9

Snapshots of the fuel spray (colored by temperature) and important gaseous species. Legend: fuel (green), HCO (blue), OH (red), CO$_2$ (yellow). Left to right: Z77, HyChem JP-5, HyChem Jet A. Top to bottom: Idle conditions, cruise conditions

It is clear from Figs. 7 and 8 that the mechanisms predict different exhaust temperatures. To investigate this further, the mass fractions $\tilde{Y}_i$ of gaseous fuel, CO$_2$, CO, OH, and HCO, as well as $\tilde{T}$, are planarly averaged and plotted along the longitudinal direction in Figs. 10 and 11. The asymptotic values are also presented in Tables 2 and 3. At idle conditions, there is a difference of 125 K between Z77 and HyChem Jet A. HyChem Jet A shows 9% and 6% warmer emissions than Z77 at idle and cruise conditions, respectively. There is also a clear trend in the emissions: the higher the temperature, the higher the mass fractions of the primary combustion products CO$_2$ and H$_2$O, and the lower the overall amount of gaseous fuel. Z77 predicts the coldest exhaust gas composition, followed by HyChem JP-5 and then HyChem Jet A. The trend can be explained by a difference in fuel consumption rate between the mechanisms, resulting in different fuel residence times, heat release rates, and emission enthalpies.

The simulations at cruise conditions display the same general trend seen at idle conditions, with two notable exceptions. Firstly, there is no clear inverse correlation between the exhaust temperature and the flame lift-off. The reaction rate is simply too high, regardless of mechanism, for such a difference to be discernible. As will be demonstrated shortly, it is the shape of the flame rather than its overall length that differs between the mechanisms. Secondly, the predictions of Z77 and HyChem JP-5 are much more similar at cruise conditions than at idle conditions; they are virtually identical with regard to temperature and CO$_2$ concentration. This is partly corroborated by the laminar flame simulations presented in Fig. 2; Z77 appears to be significantly more pressure-sensitive in its prediction of $T_{max}$ than the other mechanisms, which follow the same trend regardless of pressure. The pressure at cruise conditions is 1.00 MPa, more than double the 0.40 MPa at idle conditions. This increases the reaction rate in all mechanisms, but the relative increase is greater for Z77 than for the others. The similarity between Z77 and HyChem JP-5 in Fig. 11 implies that these mechanisms perform well at cruise conditions, as they both use the same surrogate fuel species, C$_{12}$H$_{23}$, which should in theory lead to almost identical flames and emissions.

The radicals CO, OH, and HCO are only present in relatively minor concentrations, which means that the magnitudes of their mass fractions are sensitive to the number of species included in the mechanism. With this in mind, each radical’s peak concentration lies inside the flame, the location of which differs slightly between the mechanisms at idle conditions but is mostly the same at cruise conditions. Compared to Z77, the HyChem mechanisms predict substantially more OH in the emissions relative to the peak values inside the flame. This difference is especially dramatic at cruise conditions, where the HyChem mechanisms predict the flame and the emissions to have almost the same OH content while Z77 predicts a significantly lower equilibrium concentration in the emissions than in the flame. This occurs in one-dimensional laminar flame simulations as well, indicating that the reactions influencing equilibrium OH concentrations are configured differently in the HyChem mechanisms than in Z77. HyChem Jet A also predicts twice as much CO in the flame as HyChem JP-5.

Fig. 10

Planarly integrated mass fractions and temperature at idle conditions. Legend: Z77 (blue), HyChem JP-5 (green), HyChem Jet A (red)

Table 2 Exhaust gas composition at idle conditions

Fig. 11

Planarly integrated mass fractions and temperature at cruise conditions. Legend: Z77 (blue), HyChem JP-5 (green), HyChem Jet A (red)

Table 3 Exhaust gas composition at cruise conditions

The mean liquid fuel concentration, mean heat release rate (W/m$^3$), and mean OH mass fraction are plotted in Fig. 12 (idle) and Fig. 13 (cruise). The results are azimuthally integrated and use the same color bar and spatial dimensions as the experimental results presented in Meier et al. (2012). Each simulation result is normalized by the highest value observed at either idle or cruise conditions, and the normalization is done separately for each mechanism. The experimental results being used for comparison are: kerosene-PLIF images of liquid fuel concentration (top row), OH chemiluminescence images that show regions with high heat release rate (middle row), and OH-PLIF images of OH concentration (bottom row). The OH-PLIF images were originally used in Meier et al. (2012) for thermometry, but are also useful for illustrating the shape of the OH distribution.

It can be seen in Fig. 12 that the different reaction rates of the three mechanisms result in different spray penetration depths, flame positions and flame lift-off at idle conditions. Z77 predicts a long spray, with much of the heat release occurring in a toroidal region inside the MFC. The HyChem mechanisms predict shorter spray penetration depths and flame lift-off, with heat release primarily located at the ISL. This is especially true for HyChem Jet A, while HyChem JP-5 predicts a somewhat more lifted flame. The regions just downstream of the flame make up the postflame zone, where the OH mass fraction peaks. The shape of the postflame zone reveals that Z77 predicts a flame primarily in the MFC, HyChem Jet A predicts a flame primarily at the ISL, and HyChem JP-5 predicts significant heat release at both locations. The results at cruise conditions, see Fig. 13, show more similarity between the mechanisms with regard to spray penetration depth and flame lift-off. Z77 and HyChem JP-5 predict very similar flames, while HyChem Jet A shows more heat release close to the centerline and a wider cone angle for the flame, in line with its prediction of a wider CRZ as shown in Fig. 7. As previously seen in Fig. 11, the OH mass fractions predicted by Z77 and the HyChem mechanisms are remarkably different; Z77 shows a sharp decrease downstream of the postflame zone, while the HyChem mechanisms show only a slight decrease.

Fig. 12

Qualitative validation at idle conditions. Left to right: Experimental results obtained from Meier et al. (2012), Z77, HyChem JP-5, HyChem Jet A. Leftmost column, top to bottom: kerosene-PLIF, OH chemiluminescence, OH-PLIF. Remaining columns, top to bottom: time-averaged liquid fuel concentration, time-averaged heat release rate, time-averaged OH mass fraction. The simulations were performed with PaSR and the baseline mesh. The results in each column are normalized by the highest value observed across both idle and cruise conditions

Fig. 13

Qualitative validation at cruise conditions. Left to right: Experimental results obtained from Meier et al. (2012), Z77, HyChem JP-5, HyChem Jet A. Leftmost column, top to bottom: kerosene-PLIF, OH chemiluminescence, OH-PLIF. Remaining columns, top to bottom: time-averaged liquid fuel concentration, time-averaged heat release rate, time-averaged OH mass fraction. The simulations were performed with PaSR and the baseline mesh. The results in each column are normalized by the highest value observed across both idle and cruise conditions

Overall, the agreement with experimental results is reasonable at idle conditions and good at cruise conditions. The spray penetration depth and the flame lift-off are consistently overpredicted at idle conditions, with HyChem Jet A showing the smallest error. This can also be inferred from Fig. 14, which shows the mean heat release rate along the axial direction, using the same approach as in Fig. 10. Because the OH chemiluminescence intensity is not quantitative, the experimental results are simply normalized. A more detailed validation of the spray behavior is provided in Fig. 15, where the Sauter Mean Diameter (SMD) is plotted along the radial direction in three different planes downstream of the burner. The simulated spray is sampled at a frequency of 4 kHz (simulated time), and computed means with $95\%$ confidence intervals larger than 5 $\upmu \hbox {m}$ (which may occur at the edges of the spray) are removed. The liquid volumetric flux at the three planes is presented in Tables 4 and 5 for idle and cruise conditions, respectively.

Fig. 14

Planarly averaged, time-averaged heat release at idle (dashed lines) and cruise (solid lines) conditions. The simulations were performed with PaSR and the baseline mesh. Normalized OH chemiluminescence intensity obtained from Meier et al. (2012) is included for comparison. Legend: experimental results (squares), Z77 (blue), HyChem JP-5 (green), HyChem Jet A (red)

Droplets near the ISL are generally small, which allows them to be carried inward by turbulence, while larger droplets are consistently carried outward by inertia. Because small droplets evaporate more quickly than large droplets, this difference (which can be quantified by the Stokes number) is less relevant further downstream. The simulations agree with the experimental data in that the SMD increases along the radial direction until a plateau is reached. Beyond this plateau, the experiments show a slightly decreasing (10 mm) or constant (15 mm, 20 mm) SMD, whereas the simulations generally show a slightly decreasing SMD along the radial direction. Furthermore, the plateau is reached closer to the centerline in the simulations than in the experiments, especially at idle conditions. This means that the predicted inner angle of the spray cone is too small, which can also be seen in Fig. 12. Note that the angle of the spray cone is not explicitly defined in the simulations; the Lagrangian particles are injected at the prefilmer lip inside the burner, with only the mass flow rate, particle injection frequency, droplet diameter distribution, and initial velocity defined. Though this approach does not rely heavily on assumptions, it may require more sophisticated models in order to accurately capture the initial development of the spray at the prefilmer lip. Another discrepancy is that the SMD is almost consistently underpredicted in the simulations. Only the simulations employing HyChem Jet A show good agreement with experimental data, and only at certain radii. This underprediction is likely linked to the heat release rate, which heavily affects evaporation. The speed of evaporation, which can be inferred from Tables 4 and 5, is correlated with large SMD at idle conditions. The connection holds true at cruise conditions for Z77 and HyChem JP-5, while HyChem Jet A displays disproportionately large SMD. The chosen injection method may be a contributing factor to the underprediction of SMD; large particles are pushed by inertia towards the wall at the prefilmer lip where they tend to slow down, giving them time to evaporate and decrease in size. The initial droplet size distribution used here, which was obtained from Jones et al. (2014), was chosen based on the assumption that evaporation on the prefilmer lip would be negligible. We conclude that the prefilmer lip is a problematic region for the current injection method. The method may be improved by introducing an additional model that captures the effects of a liquid fuel film undergoing primary breakup, or by injecting downstream of the prefilmer lip following Jones et al. (2014).

Fig. 15

Radial distribution of SMD at three different axial locations. Top to bottom: $z=10$ mm, $z=15$ mm, $z=20$ mm. Left to right: idle conditions, cruise conditions. Legend: experimental results (squares), Z77 (blue), HyChem JP-5 (green), HyChem Jet A (red)

Table 4 Liquid volumetric flux at idle conditions. Percentages refer to the amount of flux retained since injection

Table 5 Liquid volumetric flux at cruise conditions. Percentages refer to the amount of flux retained since injection

One factor contributing to the observed differences between the predictions of the three mechanisms may lie in how they are affected by turbulence. We refer to Fig. 3, which contains the extinction strain rates predicted by each mechanism. The predictions suggest that HyChem Jet A and Z77 should be more resistant to turbulence than HyChem JP-5. In order to quantify the turbulence in each LES flame, points that are highly reactive on average are extracted along with their time-averaged turbulent intensity. Points are identified as highly reactive if $\langle Q\rangle >10^8$ W/m$^3$, where $\langle Q\rangle$ is the mean heat release rate. The extracted points are placed in $\langle Q\rangle$-I space, where I is the mean turbulent intensity normalized by the maximum velocity in the plane 50 mm downstream of the burner (see Fig. 7), averaged between the mechanisms. The result is presented in Fig. 16. Note that all interior points have been removed, leaving only the outer edge of the distribution. At idle conditions, the HyChem mechanisms both predict high heat release in points with turbulent intensities up to 0.9, but HyChem Jet A predicts higher heat release overall. The Z77 distribution is completely enveloped by the HyChem mechanisms, as a result of the flame being quenched along much of the highly turbulent ISL. The only clear correlation with the lower-dimensional predictions of $\sigma _{ext}$ presented in Fig. 3 is that HyChem JP-5 is consistently less resistant than HyChem Jet A. As the mechanisms predict similar flame shapes at cruise conditions, they also predict similar distributions in Fig. 16. The outlier is HyChem Jet A, which unexpectedly appears less resistant to turbulence than the other mechanisms. This difference however is not related to turbulence, but to acoustics. As will be demonstrated in the next section, the flame is subject to strong pressure oscillations at cruise conditions. These oscillations are significantly weaker in the simulation with HyChem Jet A, which means that the variance of the velocity, on which the turbulent intensity is based, is lower.

Fig. 16

Each flame’s distribution in Q-I space at idle (a) and cruise (b) conditions. Legend: Z77 (blue), HyChem JP-5 (green), HyChem Jet A (red)

7 Thermoacoustics

In the previous section, it was shown that HyChem Jet A differs from the other mechanisms at cruise conditions with regard to the width of the CRZ (Fig. 7), the nature of the instantaneous temperature and velocity fields (Fig. 8), the shape of the flame (Fig. 13), and SMD (Fig. 15). The acoustic behavior is also quite different from what is observed in the other simulations, which may be the cause of some of these differences. This section explores how each reaction mechanism affects the acoustics of the combustor, in order to provide context for the previously presented results.

There are significant pressure oscillations in the simulations, with amplitudes on the order of $\sim$5 kPa at idle conditions and $\sim$50 kPa at cruise conditions. These occur to varying extents regardless of reaction mechanism, but not when chemistry is disabled altogether. We can infer that the oscillations arise because of coupling between the chemistry and the combustor acoustics. This phenomenon is referred to as thermoacoustic coupling, and can be described using Rayleigh’s criterion, originally formulated in Rayleigh (1878). In brief, the criterion states that a thermoacoustic oscillation requires that pressure and heat release are oscillating in phase. This can be quantified by calculating the rate at which energy is transferred to the acoustic field using an equation derived in Moeck (2010). The rate is given by

$$\begin{aligned} E_a'=\frac{\textrm{d}E_a}{\textrm{dt}} = -\int _{\partial V}^{} \overline{p}'\tilde{\textbf{v}}'\cdot \textbf{n} \textrm{d}A+\frac{\gamma -1}{\gamma P_0}\int _{V}^{} \overline{p}'Q'\textrm{d}V, \end{aligned}$$

(5)

where $\overline{p}'$, $\tilde{\textbf{v}}'$, and $Q'$ are the fluctuating components of pressure, velocity, and heat release. The first integral term describes energy transport across the boundaries of the combustor, and is only non-zero at the outlet as all inlets have constant velocities (in the simulations). The second integral term, where $\gamma$ is the heat capacity ratio and $P_0$ the mean pressure, describes energy transport between the combustion process and the acoustic field inside the combustor. As the integral contains the product of two fluctuation components, it is positive when $\overline{p}'$ and $Q'$ are oscillating in phase. The energy transfer rate to the acoustic field is plotted over the course of 10 ms in Fig. 17. There are no easily discernible patterns at idle conditions, but the integral of each curve is slightly positive. At cruise conditions, there is a distinct periodic variation with high-amplitude peaks and low-amplitude valleys. Note that the graph scale is over 66 times larger at cruise conditions than at idle conditions. The sharpness of the peaks is due to brief heat release events that occur when the pressure is high near the flame. The integral of each curve is clearly positive. HyChem Jet A has significantly lower energy transfer and a more noisy curve, though it also shows periodic oscillations and has a positive integral.

Fig. 17

Acoustic energy transfer rate over 10 ms at idle (a) conditions and cruise (b) conditions (note the different scaling of the y axis). Top to bottom: Z77, HyChem JP-5, HyChem Jet A. The total energy transferred to the acoustic field, $\Delta E_a$, is displayed above each panel

In Lo Schiavo et al. (2020), it was found that thermoacoustic self-excitation in spray flames is contingent on using a film model, where the thickness of the film may oscillate at the same frequency as the excited acoustic harmonic. Here, however, thermoacoustics are present despite the lack of such a model. This may be explained by the fact that the droplets lose a considerable portion of their initial momentum as they pass along the prefilmer lip, thus experiencing a considerable reduction in Stokes number. This allows the volumetric flux of liquid leaving the prefilmer lip to oscillate as a result of the dynamics of the gaseous phase. The amount of fuel along the prefilmer lip thus oscillates at the same frequency, much like a liquid film with an oscillating height.

POD, as described in Berkooz et al. (1993), is a useful mathematical tool for extracting coherent structures, such as pressure waves, from unsteady solutions. First, the whole pressure field is sampled at a frequency of 40 kHz. The resulting series of snapshots is decomposed into a number of orthogonal modes that can be used to reconstruct the pressure field at any given time using a periodically varying coefficient associated with each mode. Each mode has an associated energy, which indicates the importance of the mode to the reconstructed pressure. If any acoustic harmonics are excited, they should correspond to some of the most energetic POD modes. Figure 18 shows the four most energetic pressure fluctuation modes in each simulation. Each mode is presented in two cross sections: one along the longitudinal axis, showing the mode along the length of the combustor, and one along the lateral axes, showing the shape of the mode close the the upstream end of the combustion chamber.

Some of the POD modes are indeed associated with acoustic harmonics, but not all. Each mode can be placed in one of four categories: Drift (D), PVC (P), Longitudinal (L), and Azimuthal (A). Drift (D) refers to modes that are uniformly positive or negative and have coefficients that change linearly with time. These emerge because the mean pressure in the combustor increases or decreases slowly over time in some of the simulations, though not more than $\sim 1\%$ at most between the start and end of the data collection phase. The PVC (P) modes contain the PVC, which is naturally a periodic and coherent structure. The Longitudinal (L) modes correspond to the different longitudinal acoustic harmonics of the combustor. The first (or fundamental) longitudinal acoustic harmonic, which has a wavelength of two combustor lengths, appears in all simulations. Up to three additional longitudinal modes can be seen at cruise conditions. These are the second, third, and fourth longitudinal acoustic harmonics and respectively have the wavelengths L, 2L/3, and L/2, where L is the combustor length. The Azimuthal (A) modes correspond to the first azimuthal acoustic harmonic of the combustor. Upon close inspection, all simulations have azimuthal oscillations; even if the azimuthal acoustic harmonics are not the primary features of their own POD modes, they can be discerned in the PVC modes, which are slightly positive on one lateral side and slightly negative on the other. The temporal evolution of the mode coefficients is shown in Fig. 19, where only Z77 is considered for brevity. The first mode at idle conditions (19a) is longitudinal and oscillates with a clear dominating frequency. The second mode is a drift mode which slowly decreases while also displaying weak coupling to the first longitudinal acoustic harmonic. The third and fourth modes are both azimuthal, oscillating with the same frequency but with a phase shift of 90°. The evolution at cruise conditions, shown in Fig. 19b, is more straightforward as all POD modes correspond to longitudinal acoustic harmonics. The first, second, and third modes correspond to the first, second, and third harmonics, respectively, while the fourth mode also corresponds to the second harmonic.

Our analysis of the four most energetic POD modes strongly suggests that the simulations contain longitudinal and azimuthual pressure waves. The POD modes associated with these two different waves have similar energies at idle conditions, while the longitudinal dominate at cruise conditions. The exception is the simulation with HyChem Jet A, where the third and fourth POD modes are associated with pressure drift and the PVC, respectively. In order to investigate these pressure waves, the pressure evolution is probed at two different locations: one on the centerline 16 mm deep into the combustion chamber, and one at the same depth but shifted 40 mm laterally. The longitudinal component is subracted from the second probe, leaving a signal dominated by azimuthal oscillations. The two signals oscillate periodically in all simulations, with dominant frequencies extracted through FFT analysis and presented in Table 6. The longitudinal pressure wave has a frequency between 1.5 and 1.7 kHz, while the azimuthal wave has a frequency between 4.0 and 4.6 kHz. By assuming a constant sound speed of 900 m/s (dry air at  2000 K) and using twice the combustion chamber length and twice the width as the wavelengths of the fundamental longitudinal and azimuthal acoustic modes, one computes a natural frequency of 1.65 kHz for the longitudinal mode and 4.4 kHz for the azimuthal mode - close to what is observed in the simulations.

Fig. 18

The four most energetic POD pressure modes in each simulation. Top to bottom: idle conditions, cruise conditions. Left to right: Z77, HyChem JP-5, HyChem Jet A. For each simulation, modes are presented in descending order based on energy. Legend: Drift (D), Longitudinal (L), Azimuthual (A), PVC (P)

Fig. 19

Normalized temporal evolution of POD mode coefficients at idle (a) and cruise (b) conditions for pressure in simulations with Z77. Legend: First mode (magenta), second mode (black), third mode (blue), fourth mode (red)

Table 6 Dominant pressure wave frequencies. Azimuthal: $f_a$. Longitudinal: $f_l$

Table 7 contains the mean amplitudes of the longitudinal and azimuthal pressure waves at the probe locations specified previously. Both waves have similar amplitudes at idle conditions, and the amplitude is highest in the simulation with Z77. This explains the prominence of azimuthal POD modes in Fig. 18 for that mechanism. Given that Z77 has been shown to be the most pressure-sensitive of the mechanisms, it is logical that it would predict a stronger variation in heat release, and by extension pressure, in the presence of pressure waves. The longitudinal wave dominates at cruise conditions, with Z77 again showing the highest amplitude for the longitudinal wave, though not for the azimuthal wave. The excitation of the acoustic harmonics is primarily caused by coupling between pressure and heat release. High pressure increases the reaction rate, which causes a rapid pressure increase, sending out a pressure wave. When the pressure peaks again, the process repeats. In this explanation, there is no reason why longitudinal waves should have a higher amplitude than azimuthal ones, or vice versa. To explain why longitudinal waves dominate at cruise conditions, the coupling between the pressure and flow velocity must also be considered. When the pressure is high near the flame, the upstream velocity is reduced, which causes a build-up of unburnt fuel. If the upstream velocity is such that the built-up fuel reaches the flame roughly when the pressure wave returns, the pressure-induced peak in heat release will be amplified by the extra fuel. Because the inflow velocity is 10.7% higher and the distance between the burner and flame is significantly lower (see Fig. 14) at cruise conditions, it is possible that the time scale of transport of built-up fuel into the flame is more accommodating for the excitation of longitudinal pressure waves than at idle conditions. Both the longitudinal and azimuthal waves in the simulation with HyChem Jet A have about half the amplitudes seen with other reaction mechanisms. The different flame shape and higher temperature along with the higher reaction rate in that simulation are likely linked to the strength of the thermoacoustic coupling. It is even possible that the relatively weak pressure waves are the primary reason why HyChem Jet A displays a different flame shape and larger droplets than the other mechanisms, at least at cruise conditions where it defies the correlation between SMD and evaporation rate, but this remains unconfirmed as we are unable to determine in detail how all the relevant parameters (reaction rate, flame shape, spray evaporation rate, temperature, species concentrations, etc) form an equilibrium.

Table 7 Mean pressure wave amplitudes. Azimuthal: $A_a$. Longitudinal: $A_l$. Percentages refer to the ratio between the mean amplitude and the mean combustor pressure

8 Conclusions

LES of spray combustion is performed for a generic gas turbine combustor at aeroengine conditions, employing three different chemical reaction mechanisms. The primary findings are summarized below.

• The simulations capture some of the unsteady features typical of gas turbine combustors, including turbulent recirculation zones and shear layers, precessing vortex cores, and thermoacoustic oscillations.

• The simulations are sensitive to the choice of turbulence-chemistry interaction model, with a notable impact on the shape and location of the flame.

• Compared to experimental data, the flame lift-off and spray penetration depth are overpredicted by all mechanisms at idle conditions but show good agreement at cruise conditions. HyChem Jet A shows the best agreement overall, especially with regard to droplet sizes.

• The injection method used in the simulations is unphysical at the prefilmer lip. Additional models should be introduced, which capture the effects of liquid film break-up. If such models cannot be successfully implemented, the spray should instead be injected downstream of the prefilmer lip.

• The mechanisms predict different reaction rates, affecting the temperature and composition of the emissions. The mechanism with the warmest emissions, HyChem Jet A, predicts $9\%$ (idle) and $6\%$ (cruise) warmer exhaust gas (compared to inlet temperature) than the coldest mechanism, Z77. Due to its sensitivity to pressure, Z77 predicts the largest difference between idle and cruise conditions.

• Longitudinal and azimuthal pressure waves occur in all simulations as a result of thermoacoustic coupling. The two kinds of waves have similar amplitudes at idle conditions, while longitudinal waves dominate at cruise conditions. Predicted amplitudes differ between simulations using different reaction mechanisms. Z77 shows strong oscillations, possibly as a result of high pressure sensitivity. HyChem Jet A shows relatively weak oscillations at cruise conditions, which is likely related to its high heat release and different flame shape.

9 Future Work

Based on the findings presented here, a number of potential improvements to the methodology may be identified. The injection method should either be expanded to properly account for liquid film dynamics, or simplified so that the spray is injected downstream of the film region with boundary conditions based on experimental measurements. For a productive comparison of different jet fuels, separate liquid thermodynamic properties should also be used for each fuel. Considering the substantial sensitivity with regard to the choice of turbulence-chemistry interaction model, we also welcome a deeper and more systematic study to determine the most appropriate model for this case. Having established a baseline methodology using the Z77 and HyChem mechanisms for conventional fuels, we will dedicate a subsequent study to exploring the behavior of alternative jet fuels with considerably different chemical properties.