# Large Eddy Simulation of Supersonic Combustion Using the Eulerian Stochastic Fields Method

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## Abstract

The development of supersonic combustion engines (scramjet-type) presents several challenges. Numerical simulations of scramjet engine combustion have become an attractive alternative to experimental investigations. However, supersonic combustion simulations still have several challenges: such as adequate modelling of shock/boundary layer and turbulence/chemistry interactions. The present work presents two large-eddy simulation probability density function (LES-PDF) novel formulations developed for high-speed applications. One is a conservative joint-scalar approach, where the joint-probability for reactive scalars and energy is solved, while the other is a joint velocity-scalar. The LES-PDF transport equations are solved using the Eulerian stochastic fields technique implemented in a density-based compressible solver. The performance of the models is verified through the simulations of two and three-dimensional supersonic reacting mixing layers and compared against DNS data from the literature. The results show that the joint-scalar formulation is accurate and robust, while the joint velocity-scalar closures require further development.

## Keywords

Supersonic combustion LES PDF## 1 Introduction

Hypersonic aircraft technology can be achieved by the use of air-breathing scramjet engines. In these engines combustion occurs at supersonic speeds, which involves complex physical process such as shock-boundary layer interaction and turbulence-chemistry interaction, which occur under a wide range of scales. Computational Fluid Dynamics (CFD) simulations of such phenomena are still challenging and few numerical model exists [1] that can be used in high-speed and reacting flows.

Probability Density Function (PDF) methods can provide an elegant closure to the highly non-linear source term [2] occurring in such flows. PDF turbulence models do not require a *very* fine mesh, unlike quasi-linear approaches [3], and are therefore promising for turbulence-combustion modelling.

The first compressible PDF formulation was presented by Kollmann and co-workers [4, 5]. In their model the PDF description included variables, such as density, velocity, internal energy, mixture fraction and the relative rate of volume expansion. This representation allowed the fully closure of the reactive and convective terms. Nevertheless, probably due to its mathematical complexity, it was not been further explored.

Alternative compressible PDF models were developed by Delarue and Pope [6]. The authors presented a PDF formulation including velocity, internal energy, pressure and turbulence frequency. Nik et al. [7] expanded the model to account for chemical reactions. This model also provided fully closure for the reactive and convective terms in a Large Eddy Simulation (LES) framework. The presented PDF equation was then solved using a stochastic Lagrangian procedure, where Lagrangian (or notional) particles are tracked and their ensemble average represents the PDF. The process involved interpolation between the Eulerian mesh and the Lagrangian particles which lead to non-uniform spatial resolution of the PDF. To maintain uniform accuracy (and stability) additional approaches were introduced [8].

An alternative approach to the Lagrangian particles is the Eulerian formulation. In this method, first developed by Valiño [9] and Sabel’nikov and Soulard [10], the stochastic equations equivalent to the PDF are solved within a Eulerian framework. This system of equations can be easily implemented into Eulerian CFD solvers. The stochastic Eulerian approach has been used in compressible PDF simulations before [11], however, only mass fractions and enthalpy were considered and sub-grid pressure fluctuations were neglected. This approximation allowed the closure of the reactive term, as employed by [4, 5, 6]. This approach has also been commonly used by several authors [12, 13, 14] within the Lagrangian framework, however, its accuracy has not been quantitatively investigated yet.

This work investigates two new formulations based on a Eulerian compressible LES-PDF modelling, recently published in [15]. One formulation is a joint scalar PDF model [11], employing a filtered pressure closure as suggested by [13], which does not require sub-grid pressure fluctuations modelling. The other formulation is a joint velocity-scalar PDF model, resulting in a more complete approach as it can close both unresolved convective and source terms. Both LES-PDF approaches are solved using the Eulerian Monte-Carlo stochastic fields technique.

The performance of the models is investigated through two and three-dimensional supersonic reacting spatial mixing layers. Mixing layers show different degrees of mixing and turbulent strain, creating different combustion behaviour and therefore are prototype combustion benchmarks. The results are compared with predictions of the Smagorinsky model coupled with the quasi-laminar combustion (neglecting sub-grid fluctuations) and with literature DNS data [16].

## 2 The LES-PDF Modelling

### 2.1 Joint scalar PDF (SPDF)

*η*,

*Z*

_{α}are the sample enthalpy and mass fractions,

*h*(

*x*,

*t*) and

*Y*

_{α}(

*x*,

*t*) are the real enthalpy and mass fractions, respectively. A non-closed filtered transport equation for LES-PDF can be obtained by deriving a transport equation for \(f^{\prime }\) and applying a spatial filter to it [8]:

*η*,

*Z*

_{α}], in a similar fashion as [11, 13], although it is still accurate closed to regions of small compressibility or low-Mach number flows. The Smagorinsky model [17] is used to close the convective terms. The several remaining unclosed terms are modelled with the IEM micro-mixing model [18]. The joint scalar LES-PDF equation obtained is:

*μ*

_{sgs}/

*μ*(akin to a sub-grid Reynolds number). The time-scale reverts to a molecular one when the local flow is laminar. The model constants \(C_{Y_{\alpha }}\) and

*C*

_{H}are equal to 2 following [20]. Equal diffusivity is assumed (unity Lewis number), where the Schmidt number

*σ*and Prandtl number are equal to unity, while their equivalent sub-grid number,

*σ*

_{sgs}are equal to 0.7. The conventional Smagorinsky model is employed to obtain the sub-grid viscosity,

*μ*

_{sgs}. The equations for the n

^{th}-set of Eulerian stochastic fields for the mass fractions and enthalpy can be obtained following [10]:

*α*and enthalpy, respectively, of the stochastic field

*n*. The Wiener process \(d{W^{n}_{i}}\) is approximated by

*d*

*t*

^{1/2}

*γ*, where

*γ*= {− 1,1} is a dichotomic vector, ensuring \(\left \langle dW_{i} \right \rangle = 0\). Filtered quantities are calculated from the average of the stochastic fields as \( \widetilde {Q} \approx \left \langle Q \right \rangle \).

*R*

_{u}is the gas universal constant,

*M*

*W*

_{α}is the molecular weight of the chemical specie

*α*and

*T*

^{n}is the temperature of the

*n*

^{th}field. This approach has not been tried previously with the stochastic fields approach, only in Lagrangian formulations [13]. Accounting for sub-grid pressure fluctuations may improved accuracy in hydrocarbon fuels-pure oxygen systems [21].

### 2.2 Joint velocity-scalar PDF (VSPDF)

*d*,

*v*

_{i},

*ζ*and

*Z*

_{α}are the sample density, velocity, total energy and mass fraction, while

*ρ*(

*x*,

*t*),

*u*

_{i}(

*x*,

*t*),

*e*

_{t}(

*x*,

*t*) and

*Y*

_{α}(

*x*,

*t*) are the respective real fields. Following the same methods for the SPDF, it is possible to derive the following closed transport equation:

*G*

_{ij}is then defined as:

*k*

_{sgs}and

*𝜖*

_{sgs}are the sub-grid kinetic energy and its dissipation respectively. The IEM micro-mixing model [18] is also chosen to close the remaining unknown terms on the total energy and mass fractions part.

Two further approximations are necessary to derive the stochastic field equations. The first is to substitute the pressure terms by the stochastic pressure field, \(\mathscr{P}(\boldmath {x},t)\), instead of the filtered pressure, \(\overline {p}\), neglecting sub-grid pressure fluctuations. This results in a different momentum equation from the Burgers’ equation, preventing the occurrence of numerical shocks. The stochastic pressure can be directly obtained from stochastic variables \(\mathscr{P} \equiv \mathscr{P}(\varrho ,\mathscr{U}_{i},\mathscr{E}_{t},\mathscr{Y}_{\alpha })\) and present the same properties of the real pressure field.

*𝜖*

_{sgs}, is the following:

*C*

_{𝜖}is equal to 1.05. The micro-mixing constants \(C_{Y_{\alpha }}\) and \(C_{e_{t}}\) are set to 2 and the Langevin constant is set to 2.1, following LES-PDF convention [8]. The sub-grid kinetic energy,

*k*

_{sgs}, can be directly obtained from the stochastic fields information:

*Q*(

*x*,

*t*), it is possible to recover the filtered and the Favre-filtered values:

The novelty of the present VSPDF formulation stems from both the selection of independent variables and its solution method, which has advantages in high speed flows. The VSPDF approach includes a set of variables similar to Kollmann [5] but without a dilatation variable and with density instead of pressure, unlike the formulation of Nouri et al. [24]. This has benefits when combining the method with a density-based solver, which are typically used in high-speed flows. The use of *Eulerian stochastic fields* method to solve the VSPDF has not been used before for fully compressible and reactive flows. Moreover, the addition of an stochastic pressure, unlike other VPDF methods [6, 23] , allowed sub-grid pressure fluctuations as well as increase the stability of the SPDE system (11)-(14).

## 3 Case Description

*N*

_{2}/

*O*

_{2}/

*H*

_{2}

*O*) enters the domain at speed

*U*

_{1}= 1151.6 m/s (Ma = 1.5) and temperature 1475 K (hereafter stream 1). The cold fuel stream (

*N*

_{2}/

*H*

_{2}) inlet speed is

*U*

_{2}= 669.1 m/s (Ma = 1.1) and temperature 545 K (stream 2). The vorticity thickness is defined as:

*y*. The initial vorticity thickness is

*δ*

_{w,0}= 0.198 mm. The composition of the streams are described in Table 1. The convective Mach number for this test case is 0.35, resulting in medium-weak compressible effects. To trigger the instability growth, a perturbation is introduced in the crosswise component of the velocity [16] proportional to the convective velocity

*U*

_{c}= (

*c*

_{1}

*U*

_{1}+

*c*

_{2}

*U*

_{2})/(

*c*

_{1}+

*c*

_{2}). In the two dimensional case, a maximum perturbation of 0.1

*U*

_{c}is used, while in the three-dimensional case the perturbation is periodic in the span-wise direction and its maximum size is

*U*

_{c}.

Mass fractions at the inlet [16]

Specie | Fuel | Oxidizer |
---|---|---|

\(Y_{H_{2}}\) | 0.05 | 0 |

\(Y_{O_{2}}\) | 0 | 0.278 |

\(Y_{H_{2}O}\) | 0 | 0.17 |

| 0 | 5.60 × 10 |

| 0 | 1.55 × 10 |

| 0 | 1.83 × 10 |

\(Y_{HO_{2}}\) | 0 | 5.10 × 10 |

\(Y_{H_{2}O_{2}}\) | 0 | 2.50 × 10 |

\(Y_{N_{2}}\) | 0.95 | 0.55 |

For the two-dimensional case a domain length of *L*_{x} = 350*δ*_{w,0} and *L*_{2} = 80*δ*_{w,0} is used. A mesh size of 36000 cells (360 × 108) is employed, using a stretching grid that increases the mesh elements density towards the centre line. For the three-dimensional simulation, the planar domain is extended to *L*_{3} = 40*δ*_{w,0}. The mesh size for all simulations is of 2.5 million (360 × 108 × 64), also applying a stretching grid with more elements at the centreline. For the current LES, the filter width is approximately four times the Kolmogorov scale Δ ≈ 4*η*_{k}, where *η*_{k} is taken from the DNS [16] at 300*δ*_{w,0} from the inlet. The mesh is relatively fine for a typical LES, with the filter width smaller that the Taylor micro-scale within the inertial sub-range.

Simulations are carried out using the in-house code CompReal. First order zero-gradient boundary conditions are used at the outlet, top and bottom of the domain. Periodic boundary conditions are applied in the span-wise direction for the three-dimensional case. The *H*_{2}/*O*_{2} skeletal mechanism of Yetteret al. [25] is used to describe the hydrogen combustion using 9 species and 19 reactions. At the inlet the flow is supersonic and the boundary conditions are prescribed. In the original DNS [16] the chemical mechanism of O’Conaire et al. [26] was employed. This mechanism has the same number species as Yetter’s [25] but two more steps and has a wider pressure applicability (up to 87 bar). However, in the relatively low range of pressures in the present test-case, the difference between mechanisms is expected to be small and the slightly faster Yetter mechanism is therefore retained.

A hybrid DRP-HLLC numerical scheme is used, where the low-dispersive DRP scheme employs a 13 points stencil and fourth order accuracy in regions with small density/scalar gradients, and the HLLC-TVD second order is used in regions with sharp gradients. A fourth-order central differences discretization is used to obtain the remaining derivatives. All equations are integrated with an explicit third order Runge-Kutta scheme, whereas for the stochastic equations the order is reduced to strong 0.5-order and weak first-order accuracy. A time step of 5.0 × 10^{− 9} secs is employed to ensure that the acoustic Courant number does not increase above 0.2 for both cases. The simulations are performed using 8 stochastic fields in accord with other works in the literature [15, 19, 20, 27, 28]. The results are compared with the two and three-dimensional DNS results of Ferrer [16].

## 4 Results

### 4.1 Two-dimensional case

*real*turbulence and therefore consequences for LES modelling should be handled with caution. Nevertheless, the proposed PDF models represent the unresolved contributions and therefore its relative behaviour can be compared. The instantaneous mass fraction of the radical

*OH*is displayed in Fig. 1, where the mixing layer growth and the combustion process are observed for the different models. The Smagorinsky model predicts a slow growth of the mixing layer, probably due to the large associated turbulence damping. The SPDF model generates adequate mixing levels and increases the growth of vorticity thickness both by increasing combustion level and the pressure correction in Eq. 7. The VSPDF formulation follows similar trend. However, stochastic noise can be observed at the end of the domain. The vortices in the VSPDF model are more distorted than the SPDF and the

*OH*net production is lower.

*x*

_{1}/

*δ*

_{w,0}= 250 the SPDF has a non-negligible difference to the DNS results close to

*x*

_{2}/

*δ*

_{w}= 0.5 (close to the flame front), with the SPDF close second. In the last station, there is a difference between SPDF and VSPDF, which can be attributed to the stronger combustion of the SPDF approach. The Smagorinsky model presents very small momentum changes and remains mostly undisturbed throughout the domain. The traditional Smagorinsky model represents the convective part by adding sub-grid viscosity, which is not enough to simulate the mixing unless more points are used.

*x*

_{1}/

*δ*

_{w,0}= 250 and

*x*

_{1}/

*δ*

_{w,0}= 300. Both PDF approaches show more than four times the amount of fluctuations than the simple Smagorinsky.

*H*

_{2}and

*O*

_{2}profiles of the SPDF and VSPDF models show similar trends, although the hydrogen has been more consumed in the SPDF model. The

*H*

_{2}

*O*profile (indicative of reaction progress) shows that the combustion on the Smagorinsky model still occurs but remains constrained to the centreline. The Smagorinsky predictions of the

*H*

*O*

_{2}radical (indicative of pre-ignition) shows an unphysical double peak, characteristic of a flame front.

The two-dimensional reactive mixing layer showcases the ability of the stochastic models to simulate supersonic flows and use a complex chemical mechanism for the hydrogen combustion. Both PDF approaches show a good agreement with the DNS, while the Smagorinsky model does not generate the correct level of fluctuations.

### 4.2 Three-dimensional case

*OH*mass-fraction distribution. Similar to the two-dimensional case, the comparison to the Smagorinsky model is performed as well. Unlike two-dimensional, the Smagorinsky models shows a “turbulent” structure. The SPDF approach presents a larger defined vortical growth than Smagorinsky, while the VSPDF model has the earliest transition to turbulence and the highest

*OH*concentration (unlike the two-dimensional case).

*x*

_{1}/

*δ*

_{w,0}= 150, when combustion occurs the results diverge. The SPDF and Smagorinsky follow closely to the DNS data. The VSPDF’s growth rate is well above the DNS value from

*x*

_{1}/

*δ*

_{w,0}= 125 onwards. This suggests that the VSPDF sub-grid modelling introduces excessive momentum mixing, which is not damped by the Langevin or micro-mixing.

*x*

_{1}/

*δ*

_{w,0}= 300 together with experimental data for non-reacting cases. As expected, the reacting cases reduce the axial-momentum due to transverse dilatation. The Smagorinsky model has the higher velocity at the end of the domain, mostly because of the delay in the overall combustion rate, and both the SPDF and VSPDF results follow closely the DNS data. It is worth noting that the SPDF model uses the same sub-grid closure for the convective term as quasi-laminar Smagorinsky, but captures better the momentum statistics through improvement of the filtered pressure coupling and the representation of sub-grid combustion processes.

*x*

_{1}/

*δ*

_{w,0}= 300. The Smagorinsky model generates considerably better results compared to its two-dimensional counterpart (see Fig. 3). All models show reasonable qualitative agreement with the DNS data. The VSPDF generates higher fluctuation levels, closer to experimental data for non-reactive mixing layers, while the SPDF shows broader distribution of fluctuations, reaching beyond the vorticity thickness. DNS and model fluctuations levels are (in average) 10 % below experimental non-reacting data, which indicates that the fluctuation levels are not strongly correlated to the combustion in the test case [16].

*H*

_{2},

*O*

_{2}and

*H*

_{2}

*O*, are in good agreement with DNS results. The non-reactive species

*N*

_{2}is a good indicator of the mixing levels, and although all models agree qualitatively with DNS, the SPDF approach diffuses more (also seen in major species profiles). Observing the

*OH*distribution, SPDF and VSPDF models generate a thicker flame than the DNS. The

*H*

*O*

_{2}radical DNS profile could not be reproduced by any model. This may be attributed to the different chemical mechanism used in the simulations and DNS [26].

The mass fractions fluctuations profiles (Fig. 12) show similar behaviour. Predictions for *H*_{2}, *O*_{2}, *H*_{2}*O* and *N*_{2} are higher for all approaches than the DNS values. However, with good qualitative agreement. The SPDF model shows more diffusive behaviour overall, with the highest fluctuations. The SPDF *N*_{2} fluctuation profile, presents two peaks, not seen in the DNS or in the other models. The *OH* and *H**O*_{2} fluctuations are somewhat different from the DNS results, where the models predict lower fluctuation levels. The results, however, are of the same order of magnitude (even in the small *H**O*_{2}) and comparable to the DNS data.

## 5 Conclusions

Two and three-dimensional supersonic reactive mixing layer simulations have been performed using two new LES-PDF formulations using the Eulerian stochastic fields. The models have been compared with a conventional Smagorinsky approach with a quasi-laminar treatment of the chemistry (neglecting sub-grid scalar fluctuations). All models use standard values for the sub-grid micro-mixing and Smagorinsky constants and no attempt has been done to calibrate or adjust the results to the DNS data.

The present SPDF model provides closure to the filtered pressure in the case of reactive ideal gases ensuring a strong coupling between the thermochemical states. The model improves the mixing predictions compared with conventional Smagorinsky model at the same resolution; even if both use the same sub-grid transport model. The SPDF approach show reasonable agreement with DNS data while the Smagorinsky model greatly underestimates the velocity fluctuations. This is probably due to the coupling of the pressure and the chemical state.

The VSPDF model increases the turbulence levels and at the same time reduce the amount of numerical diffusion in the scalars and *a -priori* maintains the closest coupling between thermodynamic and velocity. The results in the planar mixing layer are the closest to experimental data, however, it overestimates the growth in the vorticity layer in the three-dimensional simulation, compared to the SPDF or Smagorinsky approaches. These results suggest that the simplified Langevin model may need to account for sub-grid compressibility effects (for example following the modified model of [6] ). The present work complements the experimental validation in [15], using the benchmark configuration of a supersonic lifted burner [31], and shows that PDF approaches are a promising tool for supersonic combustion.

## Notes

### Acknowledgements

Yuri P. Almeida acknowledges the financial support provided by the Brazilian National Council for Scientific and technological Development (CNPq) through the Science without Borders programme, grant no. 233815/2014-7.

### Compliance with Ethical Standards

This study was funded by the Brazilian National Council for Scientific and technological Development (CNPq) through the Science without Borders programme, grant no. 233815/2014-7.

### **Conflict of interests**

The authors declare that they have no conflict of interest.

## References

- 1.Berglund, M., Fureby, C.: LES, of supersonic combustion in a scramjet engine model. Proc. Combust. Inst.
**31**(2), 2497–2504 (2007)CrossRefGoogle Scholar - 2.Pope, S.B.: PDF Methods for turbulent reactive flows. Prog. Energy Combust. Sci.
**11**, 119–192 (1985)CrossRefGoogle Scholar - 3.Bouheraoua, L., Domingo, P., Ribert, G.: Large-eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base. Combust. Flame
**179**, 199–218 (2017)CrossRefGoogle Scholar - 4.Eifler, P., Kollmann, W.: AIAA 93-0448. PDF Prediction of Supersonic Hydrogen Flames. In: 31St Aerospace Sciences Meeting & Exhibit. Reno (1993)Google Scholar
- 5.Kollmann, W.: A PDF closure for compressible turbulent chemically reacting flows. Tech. Rep. NAG 3-836 National Aeronautics and Space Administration - NASA (1992)Google Scholar
- 6.Delarue, B.J., Pope, S.B.: Application of PDF methods to compressible turbulent flows. Phys. Fluids
**9**(9), 2704–2715 (1997)MathSciNetCrossRefGoogle Scholar - 7.Nik, M.B., Givi, P., Madnia, C.K., Pope, S.B.: EPVS-FMDF for LES of High-Speed Turbulent Flows. In: 50Th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. Tennessee (2012)Google Scholar
- 8.Haworth, D.C.: Progress in probability density function methods for turbulent reacting flows. Prog. Energy Combust. Sci.
**36**, 168–259 (2010)CrossRefGoogle Scholar - 9.Valiño, L.: A Field Monte Carlo Formulation for Calculating the Probability Density Function of a Single Scalar in a Turbulent Flow. Flow, Turbul. Combust.
**60**, 157–172 (1998)CrossRefGoogle Scholar - 10.Sabel’nikov, V., Soulard, O.: Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars. Phys. Rev. E.
**72**, 1282–1284 (2005)CrossRefGoogle Scholar - 11.Gong, C., Jandi, M., Bai, X., Liang, J., Sun, M.: Large eddy simulation of hydrogen combustion in supersonic flows using an eulerian stochastic fields method. Int. J. Hydrogen Energy
**42**, 1264–1275 (2017)CrossRefGoogle Scholar - 12.Banaeizadeh, A., Li, Z., Jaberi, F.A.: compressible scalar filtered mass density function model for High-Speed turbulent flows. AIAA J.
**49**(10), 2130–2143 (2011)CrossRefGoogle Scholar - 13.Gerlinger, P.: Lagrangian transported MDF methods for compressible high speed flows. J. Comput. Phys.
**339**, 68–95 (2017)MathSciNetCrossRefGoogle Scholar - 14.Möbus, H., Gerlinger, P., Brüggemann, D.: Scalar and joint scalar-velocity-frequency Monte Carlo PDF simulation of supersonic combustion. Combust. Flame
**132**, 3–24 (2003)CrossRefGoogle Scholar - 15.Almeida, Y.P., Navarro-martinez, S.: Large Eddy Simulation of a supersonic lifted flame using the Eulerian stochastic fields method. Proc. Combust. Inst.
**37**, 3693–3701 (2019)CrossRefGoogle Scholar - 16.Ferrer, P.J.M.: Simulation Numérique Directe Dans La Combustion Turbulente Sur Une Couche De Cisaillement. Ph.D. Thesis, Ecole Nationale Supérieure de Mécanique et d’Aérotechique, Poitiers (2013)Google Scholar
- 17.Smagorinsky, J.: General circulation experiments with the primitive equations. Mon. Weather. Rev.
**91**(3), 99–164 (1963)CrossRefGoogle Scholar - 18.Villermaux, J., Devillon, J.C.: Représentation De La Coalescence Et De La Redispersion Des Domaines De SéGréGations Dans Un Fluide Par Un ModèLe D’interáction PhéNoméNologique. In: 2Nd International Symposium on Chemical Reaction Engineering. Netherlands (1972)Google Scholar
- 19.Jones, W.P., Marquis, A.J., Prasad, V.N.: LES of a turbulent premixed swirl burner using eulerian field method. Combust. Flame
**159**(10), 3079–3095 (2012)CrossRefGoogle Scholar - 20.Mustata, R., Valiño, L., Jiménez, C., Jones, W.P., Bondi, S.: A Probability Density function Eulerian Monte Carlo field method for large eddy simulations: Application to a turbulent piloted methane/air diffusion flame (Sandia D). Combust. Flame
**145**, 88–104 (2006)CrossRefGoogle Scholar - 21.Ribert, G., Petit, X., Domingo, P.: High-pressure methane-oxygen flames. Analysis of sub-grid scale contributions in filtered equations of state. J. Supercrit. Fluids
**121**, 78–88 (2017)CrossRefGoogle Scholar - 22.Azarnykh, D., Litvinov, S., Adams, N.A.: Numerical methods for the weakly compressible Generalized Langevin Model in Eulerian reference frame. J. Comput. Phys.
**314**, 93–106 (2016)MathSciNetCrossRefGoogle Scholar - 23.Petrova, N.: Turbulence-Chemistry Interaction Models for Numerical Simulation of Aeronautical Propulsion Systems. Ph.D. Thesis, Ecole Polytechnique, Paris (2015)Google Scholar
- 24.Nouri, A.G., Nik, M.B., Givi, P., Livescu, D., Pope, S.B.: Self-contained filtered density function. Phys. Rev. Fluids
**2**, 094603 (2017)CrossRefGoogle Scholar - 25.Yetter, R.A., Dryer, F.L., Rabitz, H.: A comprehensive reaction mechanism for carbon Monoxide/Hydrogen/Oxygen kinetics. Combust. Sci. Technol.
**79**, 97–128 (1991)CrossRefGoogle Scholar - 26.O’Conaire, M., Curran, H.J., Simmie, J.M., Pitz, W.J., Westbrook, C.K.: A Comprehensive Modeling Study of Hydrogen Oxidation. Int. J. Chem. Kinet.
**36**(11), 603–622 (2004)CrossRefGoogle Scholar - 27.Jones, W.P., Lyra, S., Navarro-Martinez, S.: Large eddy simulation of a swirl stabilized spray flame. Proc. Combust. Inst.
**33**, 2153–2160 (2011)CrossRefGoogle Scholar - 28.Jones, W.P., Navarro-Martinez, S.: Large eddy simulation of autoignition with a subgrid probability density function method. Combust. Flame
**150**, 170–187 (2007)CrossRefGoogle Scholar - 29.Bell, J.H., Mehta, R.D.: Development of a two-stream mixing layer from tripped and untripped boundary layers. AIAA J.
**28**(12), 2034–2042 (1990)CrossRefGoogle Scholar - 30.Spencer, B.W., Jones, B.G.: Statistical investigation of pressure and velocity in the turbulent two-stream mixing layer. AIAA J.
**71**, 613–628 (1971)Google Scholar - 31.Cheng, T.S., Wehrmeyer, J.A., Pitz, R.W., Jarrett, O. Jr, Northam, G.B.: Raman measurement of mixing and finite-rate chemistry in a supersonic hydrogen-air diffusion flame. Combust. Flame
**99**, 157–173 (1994)CrossRefGoogle Scholar

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