Mass flow scaling of gas-assisted coaxial atomizers from laboratory to industrial scale is of major interest for a wide field of applications. However, there is only scarce knowledge and research concerning the effect of atomizer scale-up on liquid breakup and spray characteristics. The main objective of this study is therefore to derive basic principles for liquid jet breakup using upscaled nozzles to increase the liquid mass flow rate $ M \u0307 liq$. For that purpose, atomizers with the same geometrical setup but increased sizes have been designed and experimentally investigated for $ M \u0307 liq = 20$, 50, 100, and 500 kg/h, while the aerodynamic Weber number *We _{aero}* and gas-to-liquid ratio

*GLR*have been kept constant. The primary jet breakup was recorded via high-speed imaging, and the liquid core length

*L*and the frequency of the Kelvin–Helmholtz instability

_{C}*f*were extracted. Applying these results as reference data, highly resolved numerical simulations have been performed to gain a deeper understanding of the effect of mass flow scaling. In the case of keeping

_{K}*We*and

_{aero}*GLR*constant, it has been shown by both experiments and simulations that the breakup morphology, given by a pulsating liquid jet with the disintegration of fiber-type liquid fragments, remains almost unchanged with the degree of upscaling

*n*. However, the normalized breakup length $ L C / d liq$ has been found to be considerably increased with increasing

*n*. The reason has been shown to be the decreased gas flow velocity

*v*at the nozzle exit with

_{gas}*n*, which leads to a decreased gas-to-liquid momentum flux ratio

*j*and an attenuated momentum exchange between the phases. Accordingly, the calculated turbulence kinetic energy of the gas flow and the specific kinetic energy in the liquid phase decrease with

*n*. This corresponds to a decreased

*f*with

_{KHI}*n*or $ M \u0307 liq$, respectively, which has been confirmed by both experiments and simulations. The same behavior has been shown for two liquids with different viscosities and at different

*We*. The obtained results allow a first-order estimate of the liquid breakup characteristics, where the influence of nozzle upscaling can be incorporated into

_{aero}*j*and

*Re*in terms of

_{liq}*n*.

## I. INTRODUCTION

^{1}and the underlying physics of breakup phenomena is of fundamental interest to assess the atomization performance. One of the first morphological classifications of the breakup phenomena was derived by Faragò and Chigier for various liquid jet thicknesses.

^{2}The jet breakup classification was performed using dimensionless numbers such as

*Re*and

_{liq}*We*, according to Eqs. (1) and (2). Therewith, liquid jet diameter

_{aero}*d*, velocity

_{liq}*v*, density

*ρ*, dynamic viscosity

*η*, and surface tension

_{liq}*σ*were applied. The gas and liquid phases are represented by the subscripts

*gas*and

*liq*, respectively,

*We*between 0 and 100.

_{aero}^{2}Whereas the Rayleigh type breakup enables the formation of large droplets, the membrane type breakup leads to gas-filled membranes near the nozzle orifice. The fiber-type breakup is characterized by peeled-off fibers from the liquid jet, while disintegration occurs at the liquid jet core. For a further increase in

*We*, two sub-modes, pulsating and superpulsating, subdivide the fiber-type breakup in terms of droplet number density fluctuations in the resulting spray.

_{aero}^{2}The morphological classification was later expanded by Lasheras and Hopfinger,

^{3}utilizing the momentum flux ratio

*j*[Eq. (3)] for the fiber-type subdivision

*et al.*

^{4}and Sänger

*et al.*

^{5,6}The authors found the membrane-type breakup to be the most common regime for high-viscosity jets in coaxial atomization. Additionally, for the enhanced membrane formation, pulsating and flapping instabilities were identified that lead to liquid jet displacement in the axial and radial directions. Moreover, it has been shown

^{7}that the Kelvin–Helmholtz instability caused by aerodynamic forces represents the main mechanism for liquid destabilization, leading to the formation of initial waves on the intact liquid surface. Thereafter, the Rayleigh–Taylor instability and aerodynamic interactions result in growth and acceleration of these surface perturbations, until primary liquid fragments protrude and disintegrate from the liquid core.

^{8}noted that the discrepancies in geometrical parameters and flow patterns within the nozzles represent the main cause for the inconsistencies of measured breakup morphology in different literature works, which has sparked interest in a more detailed analysis. Investigations on the effect of gas gap width on the primary jet breakup were performed between $ d gas = 0.6 \u2212 2$ mm by Wachter

*et al.*

^{9}In that study, an increase in the gas gap width, which was accompanied by an increase in the gas mass flow, led to a decrease in primary ligament length. This result was explained by an increase in aerodynamic forces and the free jet theory.

^{10}For an increment in the gas gap width, the exiting gas phase from the nozzle orifice remains at higher velocity over a longer distance, as gas entrainment from the surroundings is reduced.

^{10}Investigations on the effect of nozzle geometry, especially the liquid jet diameter, were performed by Kumar

*et al.*

^{11}Here, atomizers with

*d*= 4, 6, and 8 mm and constant

_{liq}*d*= 15 mm were utilized, focusing on the primary breakup with quantitative parameters as an instability frequency and ligament length. For constant

_{gas}*j*= 2.8, while reducing the momentum flow ratio

*J,*

*d*was detected.

_{liq}^{11}The same result has been reported in the experiments,

^{12,13}where a decreased spray angle and a reduced liquid core length were observed at increased

*J*for a two-fluid coaxial atomizer. Leroux

*et al.*

^{14}performed an experimental investigation of nozzle scale-up and its effects on the primary jet breakup, applying three primary jet diameters

*d*= 0.4, 1, and 2 mm and gas gap widths

_{liq}*d*= 3.5, 6, and 8 mm. The comparison of the atomizers was performed for constant

_{gas}*Re*and constant momentum flow ratio

_{liq}*J*. As a result, the breakup morphology of the performed atomization experiments was not comparable, as low

*d*achieved a prompt atomization (or fiber-type breakup with superpulsating mode) and high

_{liq}*d*led to long primary ligaments and large droplets.

_{liq}^{14,15}

In order to gain detailed insight into the breakup behavior of liquid jets, numerical simulations have been extensively applied in recent decades.^{16–27} The VOF-LES (volume of fluid–large-eddy simulation) of a high-viscous liquid shown in Zhang *et al.*^{25,27} revealed that the breakup process is triggered by concentric and axisymmetric ring vortices. Moreover, it has been shown that liquid jets break up faster at elevated pressure and increased gas-to-liquid inclination angle. Direct numerical simulation (DNS) of primary atomization of a round liquid water jet injected into a quiescent environment was presented in Ref. 18, indicating that ligament formation is generated via roll-up of the liquid jet tip and disturbances are fed from the liquid jet tip upstream through vortices. The DNS performed by Zandian *et al.*^{19} distinguished three atomization cascades for the primary atomization of a planar liquid jet based on *Re _{liq}* and

*We*. The breakup process of a planar prefilming airblast atomizer has been studied using DNS,

_{aero}^{26}which exhibited reasonably good agreement with the experimental results. Although the resolution is limited by the cell size used in the numerical simulations, large-scale structures of the liquid phase, including destabilization of the intact jet core, as well as the primary ligaments disintegrated from the jet core, can be properly resolved. This is attributed to the fact that the breakup process of the liquid jet is dominated by large-scale, coherent vortices in the gaseous phase close to the liquid core that can be resolved by relatively coarse grids.

^{25}

Taking a closer look at the literature reveals that previous studies mostly have applied gas-assisted atomizers at laboratory scale along with relatively low liquid mass flow rates. Relevant studies combining the effects of single parameters on the resulting the primary breakup, as is necessary for the liquid mass flow scale-up of coaxial atomizers, are scarce. Therefore, the objective of this work is to assess the effect of using upscaled nozzles concerning mass flow scaling on the breakup performance of liquid jets. In a previous study,^{28} the authors performed experimental investigations on the scale-up of coaxial gas-assisted atomizers from laboratory-scale mass flows toward industrial-scale mass flows. For that purpose, a scaling approach based on constant aerodynamic Weber number *We _{aero}* and gas-to-liquid ratio

*GLR*was employed and an empirical model for the generated droplet diameters within the spray located far downstream of the nozzle was derived. The current work represents a follow-on study of the work by Wachter

*et al.*,

^{28}which incorporates time-resolved numerical simulations for a more detailed understanding of the breakup morphology of the liquid jet while upscaling the nozzle or $ M \u0307 liq$. The resolved flow dynamics, i.e., flow velocities and kinetic energies, provide an insight into the three-dimensional, multiphase interactions close to the liquid jet, which allows to reveal the physical mechanisms behind the phenomenological behavior of liquid jets observed in experiments. The experiments are used to guide the simulations with respect to their validity for the observed correlations of breakup morphology with the degree of upscaling. Moreover, additional credibility is given by the experiments considering large nozzle or $ M \u0307 liq$, which is beyond the computational limit.

## II. EXPERIMENTAL CONDITIONS

The gas-assisted coaxial atomizer used in this work is shown in Fig. 1, which has been investigated extensively in the last few years concerning the influences of nozzle design and operating parameters on the breakup of liquid jets.^{5,6,9,23,25,27–31} The diameters of the liquid nozzle and annular gas nozzle are given by *d _{liq}* and

*d*. The thickness of the nozzle wall for the liquid jet is

_{gas}*b*. The setup of the nozzle remains the same, while the outlet area of the nozzle increases proportionally with the liquid mass flow rate $ M \u0307 liq$. The atomizer dimensions corresponding to the respective $ M \u0307 liq$ are given in Table I. As the aerodynamic Weber number

*We*and gas-to-liquid ratio

_{aero}*GLR*represent the most relevant parameters applied for process scaling, an approach that keeps

*We*and

_{aero}*GLR*constant was selected for mass flow scale-up for the investigated nozzles. In this way, the liquid flow velocity

*v*was kept constant with increased $ M \u0307 liq$ or

_{liq}*d*, respectively.

_{liq}$ M \u0307 liq$ (kg/h) . | d (mm)
. _{liq} | b (mm)
. | d (mm)
. _{gas} |
---|---|---|---|

20 | 2.0 | 0.1 | 5.3 |

50 | 3.2 | 0.1 | 9.2 |

100 | 4.5 | 0.1 | 14.1 |

500 | 10.0 | 0.1 | 37.3 |

$ M \u0307 liq$ (kg/h) . | d (mm)
. _{liq} | b (mm)
. | d (mm)
. _{gas} |
---|---|---|---|

20 | 2.0 | 0.1 | 5.3 |

50 | 3.2 | 0.1 | 9.2 |

100 | 4.5 | 0.1 | 14.1 |

500 | 10.0 | 0.1 | 37.3 |

The experiments have been conducted under atmospheric condition, applying water, and a glycerol/water mixture with different physical properties. The dynamic viscosity, surface tension, and density of the liquids are *η _{liq}* = 1 mPa s, $ \sigma liq = 0.0719$ N/m, and $ \rho liq = 998$ kg/m

^{3}for water and $ \eta liq = 100$ mPa s, $ \sigma liq = 0.0649$ N/m, and $ \rho liq = 1220$ kg/m

^{3}for the glycerol/water mixture. Air was used as atomizing gas with a viscosity of $ \mu G = 0.0185$ and a density of $ \rho G = 1.182$ kg/m

^{3}. The operating conditions for different nozzle sizes or $ M \u0307 liq$ are given in Table II.

. | . | 1 (mPa s) . | 100 (mPa s) . | ||||||
---|---|---|---|---|---|---|---|---|---|

$ M \u0307 liq$ (kg/h) . | We
. _{aero} | v (m/s)
. _{gas} | GLR
. | j
. | J
. | v
. _{gas} | GLR (m/s)
. | j
. | J
. |

20 | 250 | 88 | 0.36 | 2.99 | 17.90 | 83 | 0.33 | 3.25 | 18.89 |

50 | 250 | 70 | 0.36 | 1.89 | 14.24 | 66 | 0.33 | 2.05 | 15.02 |

100 | 250 | 59 | 0.36 | 1.35 | 12.00 | 56 | 0.33 | 1.48 | 12.74 |

500 | 250 | 40 | 0.36 | 0.62 | 8.14 | ⋯ | ⋯ | ⋯ | ⋯ |

20 | 500 | 124 | 0.50 | 5.95 | 35.03 | 117 | 0.47 | 6.45 | 37.92 |

50 | 500 | 98 | 0.50 | 3.71 | 27.68 | 93 | 0.47 | 4.08 | 30.14 |

100 | 500 | 83 | 0.50 | 2.66 | 23.45 | 79 | 0.47 | 2.94 | 25.61 |

500 | 500 | 56 | 0.50 | 1.21 | 15.82 | ⋯ | ⋯ | ⋯ | ⋯ |

20 | 750 | 151 | 0.61 | 8.82 | 52.04 | 143 | 0.57 | 9.64 | 56.21 |

50 | 750 | 120 | 0.61 | 5.57 | 41.36 | 114 | 0.57 | 6.13 | 44.81 |

100 | 750 | 101 | 0.61 | 3.94 | 34.81 | 96 | 0.57 | 4.34 | 37.74 |

500 | 750 | 68 | 0.61 | 1.79 | 23.44 | ⋯ | ⋯ | ⋯ | ⋯ |

20 | 1000 | 174 | 0.70 | 11.71 | 68.81 | 165 | 0.66 | 12.83 | 75.10 |

50 | 1000 | 138 | 0.70 | 7.36 | 54.58 | 131 | 0.66 | 8.09 | 59.63 |

100 | 1000 | 117 | 0.70 | 5.29 | 46.27 | 111 | 0.66 | 5.81 | 50.52 |

500 | 1000 | 79 | 0.70 | 2.41 | 31.24 | ⋯ | ⋯ | ⋯ | ⋯ |

. | . | 1 (mPa s) . | 100 (mPa s) . | ||||||
---|---|---|---|---|---|---|---|---|---|

$ M \u0307 liq$ (kg/h) . | We
. _{aero} | v (m/s)
. _{gas} | GLR
. | j
. | J
. | v
. _{gas} | GLR (m/s)
. | j
. | J
. |

20 | 250 | 88 | 0.36 | 2.99 | 17.90 | 83 | 0.33 | 3.25 | 18.89 |

50 | 250 | 70 | 0.36 | 1.89 | 14.24 | 66 | 0.33 | 2.05 | 15.02 |

100 | 250 | 59 | 0.36 | 1.35 | 12.00 | 56 | 0.33 | 1.48 | 12.74 |

500 | 250 | 40 | 0.36 | 0.62 | 8.14 | ⋯ | ⋯ | ⋯ | ⋯ |

20 | 500 | 124 | 0.50 | 5.95 | 35.03 | 117 | 0.47 | 6.45 | 37.92 |

50 | 500 | 98 | 0.50 | 3.71 | 27.68 | 93 | 0.47 | 4.08 | 30.14 |

100 | 500 | 83 | 0.50 | 2.66 | 23.45 | 79 | 0.47 | 2.94 | 25.61 |

500 | 500 | 56 | 0.50 | 1.21 | 15.82 | ⋯ | ⋯ | ⋯ | ⋯ |

20 | 750 | 151 | 0.61 | 8.82 | 52.04 | 143 | 0.57 | 9.64 | 56.21 |

50 | 750 | 120 | 0.61 | 5.57 | 41.36 | 114 | 0.57 | 6.13 | 44.81 |

100 | 750 | 101 | 0.61 | 3.94 | 34.81 | 96 | 0.57 | 4.34 | 37.74 |

500 | 750 | 68 | 0.61 | 1.79 | 23.44 | ⋯ | ⋯ | ⋯ | ⋯ |

20 | 1000 | 174 | 0.70 | 11.71 | 68.81 | 165 | 0.66 | 12.83 | 75.10 |

50 | 1000 | 138 | 0.70 | 7.36 | 54.58 | 131 | 0.66 | 8.09 | 59.63 |

100 | 1000 | 117 | 0.70 | 5.29 | 46.27 | 111 | 0.66 | 5.81 | 50.52 |

500 | 1000 | 79 | 0.70 | 2.41 | 31.24 | ⋯ | ⋯ | ⋯ | ⋯ |

As the experimental conditions cover a large range of $ M \u0307 liq$ up to 500 kg/h, two different spray test rigs were utilized. The spray test rig (ATMO), which is described in detail in Wachter *et al.*,^{32} was applied for liquid mass flows at the lab scale between $ M \u0307 liq = 20 \u2013 100$ kg/h. The burner test rig (BTR), which is described in further detail elsewhere,^{28} was employed for the investigations of the nozzles featuring $ M \u0307 liq$ up to an industrial scale of 500 kg/h. As the BTR test facility was not equipped with a suction system, experiments with glycerol/water mixtures were not applicable for $ M \u0307 liq = 500$ kg/h. More details of the nozzle system considering mass flow upscaling can be found in Wachter *et al.*^{28}

*GLR*and

*We*, are kept constant, other characteristic parameters such as

_{aero}*J*and

*j*are inevitably changed while upscaling the nozzle due to a decrease in

*v*according to the chosen scaling approach (see Table II). In the following, the scaling factor

_{gas}*n*representing the ratio of the value used and the reference (smallest considered) liquid mass flow rate $ M \u0307 liq , 0 = 20$ kg/h,

*v*remains constant, the outlet area of the liquid stream

_{liq}*A*increases proportionally with

_{liq}*n*( $ A liq \u221d n 1$), leading to an increased

*d*with

_{liq}*n*by $ d liq \u221d n 1 / 2$ [see Eq. (6)]. Due to $ W e aero = const .$, the nozzle exit velocity of the gas

*v*decreases with

_{gas}*n*by $ v gas \u221d n \u2212 1 / 4$ [see Eq. (7)]. Furthermore, the constant

*GLR*results in a linear increase in $ M \u0307 gas$ with

*n*, so that the outlet area of the gas flow scales with

*n*by $ A gas \u221d n 5 / 4$ [see Eq. (8)]. The modifications of these basic parameters with

*n*while applying the provisions of constant

*v*,

_{liq}*We*, and

_{aero}*GLR*are summarized in Eqs. (6)–(8)

*j*,

*J*, and

*Re*with

_{liq}*n*

*j*and

*J*and an increase in

*Re*with

_{liq}*n*at constant

*v*,

_{liq}*We*and

_{aero,}*GLR*. Accordingly, an increase in the liquid jet core length

*L*with

_{C}*n*is expected based on the correlations derived from previous experiments of coaxial liquid jets for

*L*as functions of

_{C}*We*and

_{aero}*j*.

^{8}

For the detection of the primary jet breakup, a high-speed camera was used with an appropriate light-emitting diode (LED) array in backlight configuration, featuring an illumination of 9 × 4500 lm. For every operating condition, 2000 images were taken at the nozzle orifice. The camera allowed for images with 1 megapixel at a 3600 Hz frame rate. A more detailed description of the setup is given in our previous work.^{31} The detection of the primary ligament length was performed in post-processing by applying the threshold method of Otsu *et al.*^{33} for the glycerol–water experiments with the lowest Weber number. In order to determine the average primary ligament length and eliminate the influence of double detection, every tenth high-speed camera image was analyzed.

*et al.*

^{6}and Kapur

*et al.*

^{34}All measurements were evaluated also in accordance with the Nyquist criterion for frequency analysis.

^{35}The results were compared with atomization instability theory according to Marmottant and Villermaux,

^{7}which defines the KHI frequency

*f*as a function of shear layer thickness

_{KHI}*δ*and Dimotakis vertical wave velocity

_{gas}*v*[see Eqs. (12)–(14)]

_{KHI}^{36,37}

## III. SIMULATION OF MULTIPHASE FLOW

The nozzle setup proposed in Sec. II has been numerically simulated in this work to reveal details of the multiphase interactions during the nozzle scale-up. Due to the use of relatively large $ M \u0307 liq$ and nozzle size, the simulations have been conducted solely for the glycerol/water mixture with $ M \u0307 liq$ = 20, 50, and 100 kg/h. The *GLR* and *We _{aero}* were set to

*GLR*= 0.36 and

*We*= 250. In this way, the numerical simulations reproduce the multiphase flow field in the vicinity of the liquid jet with a reasonably good resolution, which allows a thorough understanding of the physical mechanism responsible for the observed behavior of liquid jet breakup due to nozzle upscaling. In contrast, the experiments have been conducted additionally for water jets, employing a wider range with $ 250 \u2264 W e aero \u2264 1000$ and $ M \u0307 liq$ up to 500 kg/h, which extends the general validity of the obtained results.

_{aero}### A. Mathematical formulations

*f*. An additional equation for the liquid volume fraction

*f*is solved for the VOF method

*f*represents the volume fraction of the liquid phase, creating a virtual single-phase fluid. The relative velocity $ u \u0303 r$ between the liquid and gas phases in Eq. (15), also called the compression velocity, is calculated from

**n**is the surface normal unit vector.

^{38}In this manner,

*f*= 1 indicates the pure liquid phase and

*f*= 0 the pure gas phase. Consequently, the intermediate values of $ 0 < f < 1$ identify the gas–liquid interface. The evolution equation for

*f*has been derived from mass balance equations for both phases, which results in the mentioned extra term. The numerical effect is a compression of the interface, to keep a physical sharp transition of

*f*between 0 and 1, which thus counteracts numerical diffusion of the interface over time. The term is active only within the interface zone with $ 0 < f < 1$ and vanishes at both limits of phase fractions. A detailed description and derivation for Eq. (15) as well as its implementation in OpenFOAM can be found elsewhere.

^{39}

^{40}The LES approach is based on filtering the flow field spatially, which resolves large turbulent vortices and models the effects of unresolved eddies by means of the sub-grid scale (sgs) model. Whereas the large eddies generally depend on the geometry of the bounded flow domain and show a non-isotropic behavior, the small eddies exhibit more universal, isotropic features. Therefore, resolving directly large-scale flow structures and modeling those fine turbulent vortices in analogy to the Reynolds-averaged Navier–Stokes (RANS) approach with an eddy viscosity model is well suited for studying the near-field flow dynamics during the primary breakup of co-axial liquid jets. The cutoff scale in this case represents the filter length, which is related to the grid resolution since LES with implicit filtering is conducted. Below the cutoff scale, flow structures with sizes smaller than it cannot be resolved. In the framework of LES, the compressible Navier–Stokes equations solved in this work are given by

*ρ*is the gas density,

**u**is the velocity vector,

*p*is the pressure, and

**g**is the gravitational acceleration. $ e = \u222b 0 T c v d T + 0.5 | u | 2$ denotes the specific total internal energy with the isochoric heat capacity

*c*and the temperature

_{v}*T*. The shear stress tensor is evaluated based on the gradient of the velocity field $ \tau = \eta ( \u2207 u + \u2207 u T \u2212 2 3 \u2207 \xb7 u \u2009 I )$ with the unit tensor

*I*. $ j \xaf q = \u2212 \lambda \u2207 T$ is the heat flux due to thermal conduction, with

*λ*being the thermal conductivity. The sgs stress tensor $ \tau \xaf sgs$ in Eq. (19) is evaluated by means of sgs turbulence modeling

^{40}

*ν*and the filtered strain rate tensor $ S \u0303 i j$. The sgs heat flux $ j \xaf q sgs$ in Eq. (19) is calculated via a gradient transport approach in a similar way by

_{sgs}*ν*, and the turbulent Prandtl number

_{sgs}*Pr*is set to unity.

_{t}*κ*is the curvature of the interface. Following the continuum surface force (CSF) model by Brackbill

*et al.*,

^{41}this force is evaluated per unit volume in the current work and the curvature of the gas–liquid interface is computed from the divergence of the surface unit normal vector. Albadawi

*et al.*

^{42}proposed a coupled VOF with the level-set method for improved surface tension calculation, which has shown better results than the original VOF method in OpenFOAM when the influence of surface tension dominates. Similar conclusions have been drawn

^{43}for a model concerning sub-grid-scale surface tension for LES, which is best suited for surface tension-dominated flows. In this work, the aerodynamic force dominates the surface tension force due to $ W e aero \u226b 1$, and therefore, the effect of surface tension modeling is subordinate. The material properties of the liquid–gas mixture, i.e., the density and viscosity, are calculated based on the volume-weighted average in terms of

*f,*

### B. Computational setup

The computational domains used for the simulation were constructed according to the experimental design of the nozzle, which is depicted in Fig. 2. It covers a major part of the nozzle geometry in order to resolve the internal flow within the nozzle. The liquid and air inlets start from a length of 8.6*d _{liq}* upstream the nozzle exit plane. The nozzle section is connected to a cone-shaped domain downstream, which has a length of 30

*d*and diameters of 10

_{liq}*d*and 20

_{liq}*d*at the nozzle exit and outlet. The domain length and width have been selected based on a compromise in terms of simulation accuracy and available computational resources, which allows the use of refined grid resolution in the near-nozzle zone and zero-gradient type boundary conditions at the open boundaries. Note that the web thickness

_{liq}*b*(see Fig. 1 and Table I) connecting the central liquid nozzle and annular air nozzle is kept constant during upscaling the nozzle; therefore, the geometries used for different

*n*are not scalable to each other. The

*x*axis indicates the streamwise direction and the origin of the coordinate system, which is located at the center of the nozzle exit.

The computational grids for differently scaled nozzles consist of approximately 10.3 × 10^{6} hexahedral elements, which are locally refined close to the primary atomization zone that encloses the intact liquid jet. The mesh topology and the total number of grid cells remain constant while upscaling the nozzle, so that the grid resolution reduces while upscaling the nozzle or mass flow rate. The smallest resolutions for $ M \u0307 liq$ = 20, 50, and 100 kg/h were $ \Delta min , r$ = 25, 32, and 50 $\mu $ m in the radial direction and $ \Delta min , x$ = 50, 64, and 100 $\mu $ m in the streamwise direction. The grid size expands outwardly with a small rate in the radial and streamwise directions. The grid resolution is not fine enough to resolve small-scale liquid ligaments or droplets for the given conditions with high *We _{aero}* and

*Re*. However, particular focus of this work is to study the primary breakup behavior of the intact liquid core, which can be resolved on the current grid. The grid resolution used for the smallest nozzle with

_{gas}*d*= 2 mm is sufficient for resolving the breakup process of the liquid jet according to our previous studies, which shows grid independence for the calculated breakup length with further refined mesh.

_{liq}^{25}For the larger or upscaled nozzles, however, the primary breakup process can be under-resolved. To confirm this behavior, an additional simulation has been performed for the largest nozzle at 100 kg/h using a twice-refined grid (with over 81 ×10

^{6}cells). The refined simulation has been deliberately made to reveal the impact of grid resolution and to point out the difficulties arising in highly resolved simulations for large-scale atomization systems.

The boundaries of the computational domain are indicated in Fig. 2. Therefore, the no-slip condition was used for the nozzle walls, whereas mass flow rates corresponding to the experiments were given for the gas and liquid inlets. The temperature was prescribed at 293 K at all inlets. The volume fraction of the liquid phase *f* was set to zero at the gas inlet and unity at the liquid inlet. The specific boundary condition in OpenFOAM called “fixedFluxPressure” has been used for the pressure at the inlet, which adjusts the pressure gradient such that the mass flux corresponds to the one specified by the velocity boundary condition. In this case, two different interpolation schemes are utilized to determine the pressure gradient: First, the mass flux is interpolated to the boundary faces as part of the general pressure correction step of the PIMPLE algorithm following the Rhie–Chow approach. Second, the mass flux computed from the current density and velocity fields based on the cell centers is linearly interpolated to the boundary faces. The difference between these two mass fluxes serves as predictor to compute new pressure gradients. This is done iteratively until a pressure gradient is found where both interpolations yield the desired inflow mass flux. At the outlet and entrainment boundaries (see Fig. 2), gradients of flow velocities and *f* are set to zero if the flow is directed out of the domain and to a fixed value based on the patch normal fluxes if the flow is directed into the domain. The total pressure, in subsonic formulation, is set to 1 bar. Since these far-field boundaries are sufficiently far away from the jet core, they are not expected to affect the numerical results regarding breakup morphology.

The balance equations (15)–(19) were solved numerically using the finite volume method in the framework of the open-source CFD program OpenFOAM.^{44} An implicit scheme of second-order accuracy (Crank Nicolson) for time integration is used. The van Leer second-order total variation diminishing (TVD) scheme and the unbounded second-order upwind scheme have been used for discretizations of the convection terms in the balance equation of *f* and in the momentum equation. The time step was set to $ \Delta t$ = 0.125, 0.25, and 0.5 $\mu $ s, allowing a maximum CFL (Courant–Friedrichs–Lewy) number below 0.6. Statistical averaging of the flow has been performed for approximately 500 000 time steps, which corresponds to more than 10 flow-through times based on the bulk velocity of the liquid.

## IV. RESULTS

The results of the investigation on the primary breakup of liquid jets are subdivided into two sections. Section IV A contains the experimental results of the high-speed camera images for both investigated liquids, whereas in the second part, the results of the numerical simulations are shown.

### A. Experimental results

Figure 3 presents the high-speed camera images of the jet breakup under various operating conditions ( $ M \u0307 liq$ = 20, 50, 100, and 500 kg/h and *We _{aero}* = 250, 500, 750, and 1000) using water. All operating conditions in

*We*and $ M \u0307 liq$ led to the fiber-type regime breakup. Especially for high

_{aero}*We*and low $ M \u0307 liq$, the superpulsating sub-mode could be detected, characterized by droplet number density fluctuations in the resulting spray. Due to the intensive interaction between gas and liquid phase in combination with low viscosity, the characteristic KHI wave was not detectable in this case.

_{aero}For an increase in *We _{aero}*, a decrease in the droplet size and primary breakup length is observed. This effect can be explained by an increase in aerodynamic forces and higher relative velocities between the exiting gas and liquid phase.

^{28}The scale-up of the nozzles toward increased $ M \u0307 liq$ led to a significant increase in primary ligament length. As shown in Fig. 3, particularly for low

*We*at high $ M \u0307 liq$, the shear forces of the exiting gas phase are insufficient for disintegrating the liquid jet core, which remains undisturbed over a long distance. Nonetheless, for

_{aero}*We*= 1000 and $ M \u0307 liq$ = 500 kg/h, the sheared-off fibers are subsequently disintegrated into fine droplets, while the liquid jet core remains intact. The increment of primary ligament length

_{aero}*L*can be attributed to a decrease in

_{C}*j*and

*J*, as the gas velocity decreases with $ M \u0307 liq$, as shown in Eqs. (9) and (10). Therefore, using constant

*We*and

_{aero}*GLR*is not sufficient to achieve the same atomization quality for upscaled nozzle size or mass flow rate.

Figure 4 depicts the high-speed camera images of the primary jet breakup for the glycerol/water mixture with *η _{liq}* = 100 mPa s. The influences of

*We*and $ M \u0307 liq$ on the breakup behavior remain similar for both low- and high-viscous fluids, where an increase in

_{aero}*L*with increasing $ M \u0307 liq$ and decreasing

_{C}*We*can be detected. However, the droplet number density in the spray is significantly reduced and ligament formation is enhanced especially at low

_{aero}*We*for the high-viscous liquid, which is attributed to the increased viscous force with

_{aero}*η*. The measured

_{liq}*L*for the glycerol/water mixture at different mass flow rates is illustrated in Fig. 7 in Sec. IV B along with the simulation results, where

_{C}*L*increases with the degree of mass flow scaling at constant

_{C}*We*.

_{aero}### B. Simulation results

#### 1. Breakup morphology

In Fig. 5, instantaneous iso-surfaces of the liquid volume fraction *f* = 0.5 are used to visualize the liquid jet for different nozzle scales. In accordance with the experimental results from high-speed imaging shown in Fig. 4, the simulations reveal a pulsating-type breakup of the intact jet core along with disintegration of membrane- and fiber-shaped ligaments for all scaled nozzles. The breakup mechanism remains unchanged with increased nozzle sizes. However, the thickness and length of the primary liquid jet increase with upscaled mass flow rate or nozzle size.

Figure 6 depicts profiles of the time-mean and root mean square (rms) values of *f* along the centerline axis. The breakup of the liquid jet results in a steep decrease in $ f \xaf$ from 1 (intact liquid core) to 0 (air) along the axis, which leads to spatial discontinuities and temporal fluctuations of *f*. According to $ 0 \u2264 f \u2264 1$, the maximum value of *f _{rms}* is 0.5, which occurs under the conditions of $ f \xaf = 0.5$ and assumes a bimodal distribution of

*f*with equally weighted probabilities for

*f*= 0 and

*f*= 1. Consequently, the position of $ f rms , max \u2248 0.5$ is almost co-located with that of $ f \xaf \u2248 0.5$ (see the intersection points between profiles of $ f \xaf$ and

*f*in Fig. 6), which is used to identify the liquid core length

_{rms}*L*from the simulations.

_{C}In this way, the calculated *L _{C}* (triangle symbols) normalized by the corresponding nozzle diameter is shown in Fig. 7 together with

*L*obtained from experiments (square symbols). $ L C / d liq$ increases with

_{C}*n*, which has been confirmed in both experiments and simulations. The solid lines in Fig. 7 indicate fitted lines for the experimental results of $ L C / d liq$ vs

*n*by means of a linear function $ L C / d liq = a \xb7 n + b$, with the fitting coefficients

*a*and

*b*. In this way, the scaling rate

*a*is calculated to

*a*= 1.3 for

*We*= 250 (red squares) and

_{aero}*a*= 1.1 for

*We*= 1000. Note that

_{aero}*L*derived from the experiment is based on line-of-sight imaging shown in Fig. 4, whereas

_{C}*L*is determined from profiles of calculated liquid volume fraction along the centerline axis in the simulations (see Fig. 6). The deviations between the measured and calculated

_{C}*L*are therefore caused by the different evaluation methods. As liquid fragments stripping-off from the jet core obscure the observation of the breakup location along the centerline axis, the measurement records an extended

_{C}*L*based on disintegrated primary ligaments. From the point of view of the numerical simulation, the liquid fragments disintegrated from the jet core cannot be resolved sufficiently due to the limitation given by the grid resolution. Consequently, the same evaluation method for

_{C}*L*cannot be applied for both experiment and simulation.

_{C}*We _{aero}* measures the ratio of aerodynamic or drag force caused by the velocity difference between the phases and the cohesive forces due to surface tension. With increased gas flow velocity or

*We*, the liquid surface is stretched more strongly by the gas flow, which leads to a more intense breakup of the liquid jet. Therefore,

_{aero}*L*at

_{C}*We*= 1000 is smaller compared with

_{aero}*L*at

_{C}*We*= 250 for the measured data, as shown in Fig. 7.

_{aero}As mentioned in Sec. III B, the total number of grid cells is kept constant (about 10 × 10^{6}) while upscaling the nozzle, so that the spatial grid resolution decreases for upscaled nozzles. In order to reveal the impact of grid resolution and to emphasize the difficulties while applying highly resolved simulations for large-scale atomization systems, a twice-refined grid applying a minimum resolution of $ \Delta r , min = 25 \u2009 \mu $m and 81 × 10^{6} cells has been conducted for the 100 kg/h case using the largest nozzle. As shown in Fig. 7, using the fine grid for the 100 kg/h nozzle results in a significant increase in *L _{C}* (indicated by black triangle) compared with that obtained using the reference grid (indicated by “coarse grid”). Therefore, a grid independence of

*L*is not given for this case. According to our previous study,

_{C}^{25}changing $ \Delta r , min$ from $ 25 \u2009 \mu $m (currently used for the smallest nozzle) to $ 12.5 \u2009 \mu $m has led to almost the same

*L*compared with the result derived from the reference mesh, indicating that a grid independence for

_{C}*L*can be achieved by using $ \Delta r , min = 25 \u2009 \mu $m. This resolution, however, would be computationally too expensive for the largest nozzle. Note that the required computing time increases by a factor of 16 by using a twice-refined grid due to eight times increased cell number and reduced simulation time step by half.

_{C}In fact, both VOF and LES methods are essentially limited by grid resolution by definition, which resolve the multiphase flow down to the cutoff scale given by the grid length. The current grid resolution is unable to resolve the droplets within the spray further downstream. However, dominant near-field flow patterns prevailing the multiphase interactions, which cause destabilization of the intact liquid core and disintegration of primary liquid fragments, can be captured.

In the following, the causes leading to the morphological features shown in this section will be elucidated with the help of detailed analysis of the resolved turbulent flow fields and dynamic behavior of the liquid jet.

#### 2. Turbulent flow fields

The breakup of the liquid jet is triggered by aerodynamic forces exerted by the high-speed gas flow on the low-speed liquid jet, which lead to stretching of liquid surface and an increase in the kinetic energy in the liquid phase. Therefore, the flow patterns of the gas close to the liquid jet play a decisive role for the breakup process. The atomization process is enhanced by turbulent fluctuations in the gas flow through multiphase momentum exchange, which are further elucidated in Fig. 8 by means of instantaneous contours of the streamwise velocity *u* on a meridian cutting plane. The liquid jet surface is indicated in Fig. 8 by the iso-contours of *f* = 0.5. The annular gas flow is accelerated when passing through the convergent section within the nozzle, which reaches a maximum flow velocity at the nozzle exit with $ u max \u2248$ 100 m/s. In contrast, the central liquid stream yields a velocity of approximately 1.4 m/s. The large velocity gradients between the central liquid and surrounding gas flows result in a strong momentum transfer from the gas to the liquid phase, where aerodynamic forces exerted by the gas flow on the liquid surface overwhelm the cohesive surface tension force of the liquid phase, leading to breakup of the liquid jet.

A zoomed view of the turbulent flow field close to the nozzle exit is depicted in Fig. 9 for the case with 20 kg/h, which illustrates the breakup mechanism due to the gas–liquid interaction. The liquid jet in Fig. 9 is indicated by the iso-contour of *f* = 0.5, and the arrows denote flow directions of the gas flow. The liquid jet is stretched and deformed by the high-speed airflow in an initial stage, which results in the formation of surface waves on the liquid column. Further downstream, large concentric ring vortices with a length scale of the order of the nozzle diameter are generated, which penetrate into the liquid jet core at its tip and hinder its growth. The strong recirculation of the gas flow can be identified by the blue region near the tip of liquid jet. Afterward, the protruding tip of the liquid jet is elongated by the airflow until first liquid ligaments pinch off from the jet. Further downstream, these primary ligaments break into thin liquid fibers.

Figure 10 illustrates contours of calculated time-averaged velocity (top) as well as their root mean squared fluctuations (bottom) on a cutting plane passing through the centerline axis, with the solid lines representing the time-mean iso-surfaces of $ f \xaf = 0.5$. Both $ u \xaf$ and $ u \u2032$ yield self-similar distributions for the upscaled nozzles. Overall, $ u \u2032$ is large along the shear layers and is at its largest near the tip of the liquid jet, where unsteady breakup of the liquid jet leads to strong fluctuations of the local flow velocity. $ u \xaf$ decreases for upscaled nozzles, which is in accordance with the theoretical analysis shown in Eq. (7). The same behavior can be detected for $ u \u2032$, which decreases with $ M \u0307 liq$ due to the decreased gas flow velocity or the weakened velocity gradient at the gas–liquid interface, respectively. A decrease in $ u \u2032$ indicates less intense turbulent fluctuations, so that the multiphase momentum transfer is attenuated. As a result, atomization performance is worsened while upscaling $ M \u0307 liq$, leading to a decreased breakup length. The results reveal that, in addition to the commonly used time mean flow velocity, the turbulence intensity in terms of $ u \u2032$ represents a reasonable measure for the breakup performance, which dominates the multiphase interactions. An improved atomization performance can be achieved by more intense turbulent fluctuations.

#### 3. Liquid phase kinetic energy

*k*has been evaluated from the simulations

_{L}*k*is calculated from

_{f}*K*and

_{L}*V*in Eq. (23) are evaluated from volume integration of

_{L}*k*and

_{f}*f*over the whole computational domain, which measures the total kinetic energy and volume of the liquid phase. Therefore,

*k*represents an integral quantity for the attained kinetic energy per unit liquid volume. As

_{L}*k*of the liquid stream issuing from the nozzle remains constant while upscaling $ M \u0307 liq$ due to the use of a constant

_{L}*v*, i.e., $ k L , 0 = \rho L u L 2 / 2$, an increase in

_{liq}*k*denotes specifically a more intense multiphase momentum exchange.

_{L}Figure 11 depicts the calculated temporal progression of *k _{L}* (top) and its spectral distributions (bottom) evaluated from FFT (fast Fourier transform) of the time series of

*k*at different mass flows. As shown in the upper part of Fig. 11,

_{L}*k*yields a strong time fluctuation due to the unsteady nature of the breakup process. The time-mean value of

_{L}*k*decreases with the degree of upscaling, which is in accordance with the decreased $ u \xaf$ and $ u \u2032$, as shown in Fig. 10. The difference is larger while comparing

_{L}*k*from 20 and 50 kg/h than that from 50 to 100 kg/h, which is attributed to the more strongly increased

_{L}*n*from 20 to 50 kg/h than that from 50 to 100 kg/h, which leads to a stronger decrease in the characteristic parameters such as

*u*, $ u \u2032$, and

_{gas}*j*as well as

*k*with

_{L}*n*in a non-linear way. The dotted lines shown in Fig. 11 at the top indicate simulation results for the 20 kg/h and 100 kg/h nozzles using twice-refined grids, revealing a decreased

*k*with increased grid resolution. In this case, more small-scale liquid fragments downstream can be resolved and the resolved liquid phase volume

_{L}*V*increases due to the reduced numerical diffusion, leading to a decrease in

_{L}*k*.

_{L}The same behavior with a decrease in *k _{L}* with

*n*is also confirmed for the spectra of

*k*or $ E k L$ shown in the lower part of Fig. 11, which decreases with increased $ M \u0307 liq$ and, therefore, yields a positive correlation with $ u \u2032$. $ E k L$ represents a measure of the fluctuation amplitude of

_{L}*k*at a given frequency, and it yields a broadband distribution in the frequency domain. In addition, $ E k L$ yields a similar shape compared with the spectrum of the turbulence kinetic energy (TKE) for a general turbulent flow. The result reveals that the transfer mechanism of momentum or kinetic energy from the gas to the liquid phase is dominated by turbulent flow fluctuations.

_{L}#### 4. Kelvin–Helmholtz frequency

During the initial stage of the breakup process, surface waves are generated on the liquid jet due to the Kelvin–Helmholtz instability (KHI) caused by the velocity difference between the liquid and gas flows. These interfacial waves develop further periodically and transfer momentum or kinetic energy downstream, until primary liquid ligaments pinch-off from the intact jet core. For the glycerol/water mixture with a high viscosity, the KHI wave formations can be detected clearly from the high-speed images, which are illustrated in Fig. 12 along with snapshots of calculated iso-surfaces of *f* = 0.5 obtained from the simulations for $ M \u0307 liq = 50$ kg/h. The arrows in Fig. 12 indicate instantaneous peaks of the KHI waves.

The characteristic frequencies *f _{KHI}* have been evaluated for varying $ M \u0307 liq$ and

*We*and are depicted in Fig. 13. The measured and calculated

_{aero}*f*yield a reasonably good agreement with the estimation by applying theoretical analysis.

_{KHI}^{7,11,45}An increase in

*We*leads to an increase in

_{aero}*f*, which is due to the faster motion of KHI waves at higher gas velocity. An increase in $ M \u0307 liq$ or nozzle scale causes a decrease in

_{KHI}*f*, which can be explained by the decreased gas velocity [see Eq. (7)] and momentum flux ratio [see Eq. (10)] at constant

_{KHI}*We*. Similar behavior has also been reported in Ref. 46, where

_{aero}*f*increases while upscaling the nozzle at constant

_{KHI}*v*and

_{gas}*v*. The correlation between

_{liq}*f*and

_{KHI}*We*is given in Eqs. (12)–(14) in terms of the gas velocity, which reveals an increase in

_{aero}*f*with

_{KHI}*We*. This correlation results from instability theory according to Dimotakis

_{aero}^{36}and Marmottant and Villermaux,

^{7}which was originally developed for low viscous liquids such as water. For fluids with increased viscosities, a proportionality factor

*X*= 0.1 has been implemented in Eq. (12) as proposed in Sänger

*et al.*,

^{6}which accounts for the damping effect of flow stretch caused by the increased liquid viscosity.

### C. Discussions

The objective of the current work is to study the behavior of primary atomization for upscaled nozzles in terms of increasing the liquid mass flow rate, where the challenge is given by scaling up the nozzle while not changing the atomization or spray quality. Due to the chosen approach of keeping *We _{aero}* and

*GLR*constant,

*u*decreases and

_{gas}*Re*increases for upscaled nozzles, which leads to a worsened atomization performance. Another approach concerning an improved atomization behavior is to increase

_{liq}*u*or

_{gas}*GLR*while upscaling the nozzle. However, this is not desired concerning an optimal product yield in thermo-chemical conversion processes. In fact, as

*GLR*,

*We*,

_{aero}*Re*, and

_{liq}*j*are interrelated with each other via the flow conditions, it is impossible to keep all these parameters constant while upscaling the nozzle. For instance,

*GLR*and

*We*will be changed inevitably with

_{aero}*n*, if the same

*u*or

_{gas}*j*is used. In contrast, keeping

*GLR*and

*We*constant will lead to variations of

_{aero}*u*,

_{gas}*Re*, and

_{liq}*j*, as employed in the current work.

The results reveal that it is not enough to use solely constant *We _{aero}* in order to have the same atomization behavior for upscaled nozzles with proportionally increased mass flows. However, the pulsating-type breakup regime remains unchanged for different nozzle scales, which follows the general classification of breakup regimes proposed by Faragò and Chigier

^{2}based on

*We*and

_{aero}*Re*. As discussed in the review work by Dumouchel,

_{liq}^{8}the breakup length

*L*can be correlated with the other dimensionless parameters like the gas-to-liquid momentum flux ratio

_{C}*j*. In addition, nozzle design parameters, flow patterns within the nozzle emerging from upstream conditions, and even the relative inclination angles can have a strong impact on

*L*.

_{C}^{27}

The current study has been conducted under atmospheric conditions. An alternative approach to increase the mass flow rate is to conduct the atomization process at elevated pressures. The influence of ambient pressure was investigated in previous works.^{9,30,47} In general, the system pressure plays a major role for liquid jet breakup as well as the resulting droplet size. An increased pressure leads to a higher *We _{aero}* and

*j*due to the increased gas-to-liquid density ratio, which results in a reinforced multiphase momentum transfer and a more intensified breakup process.

^{47}This work has shown that at atmospheric pressure or given density ratio, upscaling the nozzle at constant

*We*and

_{aero}*GLR*will cause an increase in

*L*due to the decrease in

_{C}*u*and

_{gas}*j*. At elevated pressure, the density ratio remains constant while upscaling the nozzle, so that the same behavior applies for elevated pressure conditions, too.

## V. CONCLUSIONS

Experiments and accompanying numerical simulations have been carried out in order to study the primary breakup process of liquid jets from coaxial, gas-assisted atomizers in different mass flow scales. An approach to proportionally modulate the nozzle dimensions at constant aerodynamic Weber number *We _{aero}* and gas-to-liquid ratio

*GLR*was chosen. Water and a glycerol/water mixture with viscosity of 1 and 100 mPa $\xb7$ s were used, and the mass flow rate of the liquid $ M \u0307 liq$ has been scaled from 20 to 500 kg/h at

*We*= 250, 500, 750, and 1000 for the experiments. Numerical simulations were performed for $ M \u0307 liq$ = 20, 50, and 100 kg/h at

_{aero}*We*= 250.

_{aero}In summary, the experiments revealed an increase in the breakup length *L _{C}*, a decrease in KHI frequency

*f*, and a decrease in the droplet number density with upscaled $ M \u0307 liq$ or nozzle size at constant

_{KHI}*We*and

_{aero}*GLR*. The same findings have been confirmed by the numerical simulations, and further analysis regarding the dynamic behavior of the turbulent gas flow and liquid jet during the breakup process has indicated a decrease in relative flow velocity, turbulent fluctuations, and specific kinetic energy in the liquid phase with upscaled $ M \u0307 liq$, leading to an attenuated multiphase momentum transfer. Therefore, increasing $ M \u0307 liq$ by upscaling the nozzle geometry worsens the atomization performance, even though

*We*is kept constant. In addition, the kinetic energy from the liquid phase has been found to be characterized by a cascade-like distribution in the spectral domain, similar to that of the turbulent gas flow.

_{aero}In conclusion, special attention should be given to designing high-load gas-assisted, co-axial atomizer based on upscaling the nozzles developed at laboratory scale. The proposed correlations of *L _{C}* with the factor of mass flow scaling

*n*may be used as a first-order estimate for predicting behaviors of the primary breakup process, where the impact of

*n*can be incorporated into

*j*and

*Re*by using Eqs. (9) and (11).

_{liq}## ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support by the Helmholtz Association of German Research Centers (HGF), within the research field MTET (Materials and Technologies for the Energy Transition), subtopic “Anthropogenic Carbon Cycle” (38.05.01). This work utilized computing resources provided by the High Performance Computing Center Stuttgart (HLRS) and the Steinbuch Centre for Computing (SCC) at the Karlsruhe Institute of Technology.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Feichi Zhang:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal). **Simon Wachter:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **Thorsten Zirwes:** Methodology (equal); Software (equal); Validation (equal); Writing – review & editing (equal). **Tobias Jakobs:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). **Nikolaos Zarzalis:** Funding acquisition (equal); Project administration (equal); Supervision (equal). **Dimosthenis Trimis:** Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal). **Thomas Kolb:** Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal). **Dieter Stapf:** Funding acquisition (equal); Project administration (equal); Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Grenzschicht-Theorie: Mit 22 Tabellen*