# Analyzing the Interaction of Vortex and Gas–Liquid Interface Dynamics in Fuel Spray Nozzles by Means of Lagrangian-Coherent Structures (2D)

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## Abstract

**:**

## 1. Introduction

## 2. Fundamentals

#### 2.1. SPH—Schemes

#### 2.2. LCS and FTLE

## 3. Computational Setup (2D)

^{−3}. The liquid phase was considered to be Jet-A1 at a temperature of 363.15 K with a dynamic viscosity of 0.0007375 kg m

^{−1}s

^{−1}, a density of 731.5 kg m

^{−3}, and a surface tension coefficient of 0.017922 kg s

^{−2}. As pointed out in previous publications by Dauch et al. [7], the elevation of pressure and temperature leads to smaller liquid structures during disintegration, and might drastically change the local flow field. The values of the Mach number inside the inner and the outer air ducts were ${\mathrm{Ma}}_{\mathrm{inner}}=0.09$ and ${\mathrm{Ma}}_{\mathrm{outer}}=0.12$, respectively. The momentum ratio in the case of the wide slit configuration was approximately MR = 16. The gas flow was highly turbulent—the order of magnitude of the Reynolds number was 100,000.

## 4. Results and Discussion

#### 4.1. Interaction of Vortices and the Phase Interface

**Conventional Visualization**—The reference case is based on the nozzle geometry depicted in Figure 4. Several instances of the velocity field around the gas–liquid interface are illustrated in Figure 5 on the left by means of uniformly distributed glyphs representing the direction of the local velocity vector. The glyphs are located on nodes of a uniform grid, and the SPH solution was interpolated on during post-processing. The three instances A–C represent three steps of a periodically reoccurring flow pattern. At time point ${t}_{A}$, the flow on the prefilmer was balanced and no film instabilities were visible. Due to the onset of an instability at the edge close to the fuel inlet, a film wave was created resulting in a recirculation zone of the gaseous flow directly downstream of the crest of the wave, as presented at instant B. Hence, an inversion of the flow direction became visible in the glyphs’ orientation. At time ${t}_{B}$, even the flow direction inside the film was locally and temporarily inverted upstream. The instability grew as did the size of the wave crest until the liquid was discharged downstream of the prefilmer. Finally, both the accumulated bulk liquid and the collapsing ligaments formed at the gas–liquid interface pushed the vortex structures further downstream, which eventually resulted in a stabilized liquid film, coming back to a similar situation as shown in Figure 5A. Please refer to Video S8 for a more vivid visualization of the liquid interface dynamics.

**Fields of**${\mathbf{FTLE}}^{-}$

**and**$\mathbf{LCS}$—Fields of the backward-in-time ${\mathrm{FTLE}}^{-}$ for the same instances are depicted in Figure 6. As pointed out previously, those fields enable the visual identification of LCS. In the next section, additional information is presented about the sensitivity of ${\mathrm{FTLE}}^{-}$ fields with regard to a variation of $\Delta t$. For now, a finite time of $\Delta t=61.6\pm 2.8\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ was applied across all time steps and geometric configurations in order to demonstrate the capability and strength of ${\mathrm{FTLE}}^{-}$ compared to velocity and vorticity fields. The temporal evolution of the vortices is also demonstrated by the Videos S1 and S2. The database was the same for both videos. However, Video S2 is played in reverse, which makes it easier to identify the root location of the flow instability causing temporal fluctuations.

#### 4.2. Sensitivity of ${\mathrm{FTLE}}^{-}$-Variations of $\Delta t$

#### 4.3. Influence of Nozzle Geometry on Primary Breakup

## 5. Conclusions

- The transport of LCSs can effectively be captured by SPH and identified by means of ${\mathrm{FTLE}}^{-}$ fields at a fixed integration time $\Delta t$.
- Investigating the dynamics of LCSs in the gaseous phase helps to understand the momentum transfer from the gas to the liquid phase, leading to disintegration.
- Suppressing LCSs by a modification of the fuel spray nozzle geometry leads to a suppression of time fluctuations in the fuel flow emerging from the nozzle.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

FTLE | Finite-Time Lyapunov Exponent |

ITS | Institut für Thermische Strömungsmaschinen |

IVD | Institut für Visualisierung und Datenanalyse |

LCS | Lagrangian-Coherent Structure |

SPH | Smoothed Particle Hydrodynamics |

Nomenclature | |

$\eta $ | Regularizing parameter |

$\kappa $ | Isentropic coefficient |

$\lambda $ | Eigenvalues |

$\nu $ | Kinematic viscosity |

$\rho $ | Density |

$\tau $ | Shear stress |

$\varphi $ | Flow map |

$\mathbb{C}$ | Right Cauchy–Green tensor |

$\mathbb{F}$ | Gradient of flowmap |

$\mathbb{L}$ | Tensor for Kernel correction |

b | Arbitrary vector |

c | Speed of sound |

$\overrightarrow{f}$ | Volume forces |

h | Smoothing length |

m | Mass |

n | Dimension |

p | Pressure |

Q | Arbitrary matrix |

$\overrightarrow{r}$ | Distance vector |

t | Time |

V | Volume |

$\overrightarrow{v}$ | Velocity |

W | Kernel |

$\overrightarrow{x}$ | Position |

Operators | |

∇ | Gradient |

$\Delta $ | Difference |

⊗ | Dyadic product |

$\langle \rangle $ | SPH discretization |

Subscripts | |

a | Particle a |

b | Particle b |

$back$ | Background |

$ab$ | Difference between particles a and b |

$gas$ | Property of gas |

$liquid$ | Property of liquid |

$nom$ | Reference for EOS |

$opt$ | Optimal value for integration time |

$ref$ | Reference value for integration time |

0 | Reference point in time |

$1,2,3$ | Indices of certain times |

## References

- Lefebvre, A.; Ballal, D.R. Gas Turbine Combustion, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2010. [Google Scholar]
- Jones, W.P.; Lyra, S.; Navarro-Martinez, S. Numerical investigation of swirling kerosene spray flames using Large Eddy Simulation. Combust. Flame
**2012**, 159, 1539–1561. [Google Scholar] [CrossRef] - Knudsen, E.; Shashank; Pitsch, H. Modeling partially premixed combustion behavior in multiphase LES. Combust. Flame
**2015**, 162, 159–180. [Google Scholar] [CrossRef] - Braun, S.; Wieth, L.; Dauch, T.; Keller, M.; Chaussonnet, G.; Höfler, C.; Koch, R.; Bauer, H.J. HPC Predictions of Primary Atomization with SPH: Challenges and Lessons Learned. In Proceedings of the 11th International SPHERIC Workshop, Munich, Germany, 13–16 June 2016. [Google Scholar]
- Koch, R.; Braun, S.; Wieth, L.; Chaussonnet, G.; Dauch, T.F.; Bauer, H.J. Prediction of primary atomization using Smoothed Particle Hydrodynamics. Eur. J. Mech. B/Fluids
**2017**, 61 Pt 2, 271–278. [Google Scholar] [CrossRef][Green Version] - Nagel, W.E.; Kröner, D.H.; Resch, M.M. (Eds.) High Performance Computing in Science and Engineering ’16; Springer: Cham, Switzerland, 2016. [Google Scholar]
- Dauch, T.F.; Braun, S.; Wieth, L.; Chaussonnet, G.; Keller, M.C.; Koch, R.; Bauer, H.J. Computation of Liquid Fuel Atomization and Mixing by Means of the SPH Method: Application to a Jet Engine Fuel Nozzle. In Proceedings of the ASME Turbo Expo 2016: Turbine Technical Conference and Exposition (GT2016-56023), Seoul, Korea, 13–17 June 2016. [Google Scholar] [CrossRef]
- Dauch, T.F.; Braun, S.; Wieth, L.; Chaussonnet, G.; Keller, M.C.; Koch, R.; Bauer, H.J. Computational Prediction of Primary Breakup in Fuel Spray Nozzles for Aero-Engine Combustors. In Proceedings of the ILASS Europe, 29th Annual Conference on Liquid Atomization and Spray Systems, Valencia, Spain, 6–8 September 2017. [Google Scholar] [CrossRef]
- Chaussonnet, G.; Braun, S.; Wieth, L.; Koch, R.; Bauer, H.J.; Sänger, A.; Jakobs, T.; Djordjevic, N.; Kolb, T. SPH Simulation of a Twin-Fluid Atomizer Operating with a High Viscosity Liquid. In Proceedings of the ICLASS 2015, 13th Triennial International Conference on Liquid Atomization and Spray Systems, Tainan, Taiwan, 23–27 August 2015. [Google Scholar]
- Zandian, A.; Sirignano, W.A.; Hussain, F. Understanding liquid-jet atomization cascades via vortex dynamics. J. Fluid Mech.
**2018**, 843, 293–354. [Google Scholar] [CrossRef][Green Version] - Dauch, T.; Rapp, T.; Chaussonnet, G.; Braun, S.; Keller, M.; Kaden, J.; Koch, R.; Dachsbacher, C.; Bauer, H.J. Highly efficient computation of Finite-Time Lyapunov Exponents (FTLE) on GPUs based on three-dimensional SPH datasets. Comput. Fluids
**2018**, 175, 129–141. [Google Scholar] [CrossRef] - Dauch, T.F.; Okraschevski, M.; Keller, M.C.; Braun, S.; Wieth, L.; Chaussonnet, G.; Koch, R.; Bauer, H.J. Preprocessing Workflow for the Initialization of SPH Predictions based on Arbitrary CAD Models. In Proceedings of the 12th International SPHERIC Workshop, Ourense, Spain, 12–15 June 2017. [Google Scholar]
- Braun, S.; Wieth, L.; Holz, S.; Dauch, T.F.; Keller, M.C.; Chaussonnet, G.; Gepperth, S.; Koch, R.; Bauer, H.J. Numerical prediction of air-assisted primary atomization using Smoothed Particle Hydrodynamics. Int. J. Multiph. Flow
**2019**, 114, 303–315. [Google Scholar] [CrossRef] - Chaussonnet, G.; Braun, S.; Dauch, T.; Keller, M.; Sänger, A.; Jakobs, T.; Koch, R.; Kolb, T.; Bauer, H.J. Toward the development of a virtual spray test-rig using the Smoothed Particle Hydrodynamics method. Comput. Fluids
**2019**, 180, 68–81. [Google Scholar] [CrossRef][Green Version] - Español, P.; Revenga, M. Smoothed dissipative particle dynamics. Phys. Rev. E
**2003**, 67, 026705. [Google Scholar] [CrossRef] [PubMed] - Hu, X.Y.; Adams, N.A. A multi-phase SPH method for macroscopic and mesoscopic flows. J. Comput. Phys.
**2006**, 213, 844–861. [Google Scholar] [CrossRef] - Monaghan, J. Smoothed Particle Hydrodynamics. Ann. Rev. Astron. Astrophys.
**1992**, 30, 543–574. [Google Scholar] [CrossRef] - Szewc, K.; Pozorski, J.; Minier, J.P. Analysis of the incompressibility constraint in the smoothed particle hydrodynamics method. Int. J. Numer. Methods Eng.
**2012**, 92, 343–369. [Google Scholar] [CrossRef][Green Version] - Cleary, P.W. Modelling confined multi-material heat and mass flows using SPH. Appl. Math. Model.
**1998**, 22, 981–993. [Google Scholar] [CrossRef][Green Version] - Cole, R.H.; Weller, R. Underwater Explosions. Phys. Today
**1948**, 1, 35. [Google Scholar] [CrossRef] - Liu, M.B.; Liu, G.R. Smoothed Particle Hydrodynamics (SPH): An Overview and Recent Developments. Arch. Comput. Methods Eng.
**2010**, 17, 25–76. [Google Scholar] [CrossRef] - Gorokhovski, M.; Herrmann, M. Modeling Primary Atomization. Ann. Rev. Fluid Mech.
**2008**, 40, 343–366. [Google Scholar] [CrossRef] - Jeong, J.; Hussain, F. On the identification of a vortex. J. Fluid Mech.
**1995**, 285, 69–94. [Google Scholar] [CrossRef] - Hunt, J.C.R. Vorticity and vortex dynamics in complex turbulent flows. In Canadian Society for Mechanical Engineering; NRC Research Press: Ottawa, ON, Canada, 1987; Volume 11, pp. 21–35. ISSN 0315-8977. [Google Scholar]
- Haller, G. An objective definition of a vortex. J. Fluid Mech.
**2005**, 525, 1–26. [Google Scholar] [CrossRef][Green Version] - Sun, P.N.; Colagrossi, A.; Marrone, S.; Zhang, A.M. Detection of Lagrangian Coherent Structures in the SPH framework. Comput. Methods Appl. Mech. Eng.
**2016**, 305, 849–868. [Google Scholar] [CrossRef] - Bonet, J.; Lok, T.S.L. Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations. Comput. Methods Appl. Mech. Eng.
**1999**, 180, 97–115. [Google Scholar] [CrossRef] - Steinthorsson, E.; Benjamin, M.A.; Barnhart, D.R. Fuel Nozzle for Turbine Combustion Engines Having Aerodynamic Turning Vanes. U.S. Patent No. 6883332 B2, 26 April 2005. [Google Scholar]
- Mansour, A.; Benjamin, M. Pure Airblast Nozzle. U.S. Patent No. 6622488 B2, 23 September 2003. [Google Scholar]
- Zandian, A.; Sirignano, W.; Hussain, F. Mechanisms of Liquid Stream Breakup: Vorticity and Time and Length Scales. In Proceedings of the ILASS Europe, 29th Annual Conference on Liquid Atomization and Spray Systems, Valencia, Spain, 6–8 September 2017. Editorial Universitat Politècnica de València. [Google Scholar] [CrossRef]

**Figure 2.**Fuel spray nozzle by Mansour [29].

**Figure 5.**Instantaneous velocity vector and vorticity fields. (

**A**): Balanced Condition ${\mathrm{t}}_{0}$; (

**B**): Vortex Formation ${\mathrm{t}}_{1}$; (

**C**): Flow Reversal ${\mathrm{t}}_{2}$.

**Figure 6.**Instantaneous ${\mathrm{FTLE}}^{-}$ fields. ${\mathrm{FTLE}}^{-}$: Backward in time Finite-Time Lyapunov Exponent. (

**A**): Balanced Condition ${\mathrm{t}}_{0}$; (

**B**): Vortex Formation ${\mathrm{t}}_{1}$; (

**C**): Flow Reversal ${\mathrm{t}}_{2}$.

**Figure 10.**Instantaneous ${\mathrm{FTLE}}^{-}$ fields: (

**left**) Wide slit with cavity; (

**right**) Narrow slit. (

**A**): Vortex detaching from cavity; (

**B**): Vortex passing; (

**C**): Vortex merging boundary layer at phase interface w/o interaction; (

**D**): Vortex merging boundary layer at phase interface with interaction.

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**MDPI and ACS Style**

Dauch, T.F.; Ates, C.; Rapp, T.; Keller, M.C.; Chaussonnet, G.; Kaden, J.; Okraschevski, M.; Koch, R.; Dachsbacher, C.; Bauer, H.-J. Analyzing the Interaction of Vortex and Gas–Liquid Interface Dynamics in Fuel Spray Nozzles by Means of Lagrangian-Coherent Structures (2D). *Energies* **2019**, *12*, 2552.
https://doi.org/10.3390/en12132552

**AMA Style**

Dauch TF, Ates C, Rapp T, Keller MC, Chaussonnet G, Kaden J, Okraschevski M, Koch R, Dachsbacher C, Bauer H-J. Analyzing the Interaction of Vortex and Gas–Liquid Interface Dynamics in Fuel Spray Nozzles by Means of Lagrangian-Coherent Structures (2D). *Energies*. 2019; 12(13):2552.
https://doi.org/10.3390/en12132552

**Chicago/Turabian Style**

Dauch, Thilo F., Cihan Ates, Tobias Rapp, Marc C. Keller, Geoffroy Chaussonnet, Johannes Kaden, Max Okraschevski, Rainer Koch, Carsten Dachsbacher, and Hans-Jörg Bauer. 2019. "Analyzing the Interaction of Vortex and Gas–Liquid Interface Dynamics in Fuel Spray Nozzles by Means of Lagrangian-Coherent Structures (2D)" *Energies* 12, no. 13: 2552.
https://doi.org/10.3390/en12132552