Air-Film Coupling in Prefilming Airblast Atomisers and the Implications for Subsequent Atomisation

Abstract

Prefilming airblast atomisers are commonly used in gas turbine combustion system fuel injectors. As the film propagates across the prefilmer it interacts with the high velocity gas stream above it. In this paper a numerical investigation into this interaction is presented. A Coupled Level Set & Volume of Fluid method is used to simulate the development of the film along the KIT-ITS planar prefilmer (Gepperth et al., in: 23rd European conference on liquid atomization and spray systems (ILASS-Europe 2010), Brno, Czech Republic, September, 2010). Initial results showed the importance of correctly specifying the contact angle as too high a value leads to the formation of rivulets instead of a continuous film. An analysis of the film and air showed two-way coupling. The presence of the film increases the growth rate of the gas phase boundary layer, and the strength and size of the turbulent structures within it. Surface waves form in the film, initially driven by the turbulent fluctuations, but developing into transverse waves. These waves are shown to be independent, stochastic events instead of a periodic wave system. At the trailing edge of the prefilmer the increased turbulence level in the air, the variations in the film thickness and the associated change in fuel mass flow and momentum will have large implications for the atomisation process and subsequent fuel spray. These will also impact simulation of the atomisation, as the boundary condition complexity is much greater than commonly used, and the variations will require larger domains and longer simulation times to obtain fully converged atomisation statistics.

1 Introduction

For health and environmental reasons increasingly strict legislation surrounding the noise and pollutant emissions produced by aero gas turbines is being introduced, and as such there is a demand to develop increasingly efficient and clean combustion systems. Due to its high energy density, liquid fuel is currently universally used in aircraft engines, and will retain a large market share in the future with the introduction of sustainable aviation fuels (SAFs). The atomisation of the fuel and the resultant spray affect nearly all aspects of a combustion system’s performance. Therefore understanding this process is vital to enable improved designs. Prefilming airblast atomisers are commonly used in gas turbine combustion systems, where high momentum air streams are used to atomize a thin sheet of fuel at the trailing edge of the prefilmer. These are typically part of complex multi-passage, annular injectors, and while experimental and numerical investigations of such geometries are possible (Dauch et al. 2019; Brend et al. 2020), they are challenging due to difficulties in access for experimental measurements and the cost of accurate simulations.

For this reason, simplified geometry is often used to investigate the fundamentals of the atomisation process. These simplified geometries commonly take the form of a planar prefilmer, with parallel airstreams on either side. Examples of this type of prefilmer can be found in Rizk and Lefebvre (1980), Bhayaraju and Hassa (2009) and Gepperth et al. (2010). One of these, the atomisation experiment at Karlsruhe Institute of Technology (Gepperth et al. 2010) (henceforth referred to as the Gepperth prefilmer), has been used extensively to investigate the atomisation mechanisms and the effect of operating conditions (Gepperth et al. 2012, 2014; Chaussonnet et al. 2020). As the name suggests, prefilming airblast atomisers include a prefilmer; a surface upstream of the atomisation edge onto which the fuel is introduced. The fuel on this surface is exposed to the air stream above the prefilmer, allowing the fuel and air to interact prior to the atomisation process.

As computing power has increased, the use of numerical methods to investigate the atomisation process has become more popular, and these simplified geometries are attractive for simulations from a cost point of view. Thin film models can be used to simulate the development of the film across the prefilmer, where a PDE for the film thickness controls the evolution of the film, as explained in more detail in O’Rourke and Amsden (1996) and Hagemeier et al. (2011). This can then be coupled with a phenomenological breakup model which converts the film at the end of the prefilmer into a spray of Lagrangian droplets. Andreini et al. (2013a, b) and Chaussonnet et al. (2016) both present models of this type. In these models the droplet size distribution is generated using a correlation to the Lagrangian film properties, which is then used to generate a spray of droplets. While this approach has several benefits, it also has two key drawbacks. Firstly, the thin film model is able to capture the influence of the air on the film, and will allow waves to form, but coupling is difficult to achieve in the other direction as the change in the geometry of the liquid surface is not seen by the air flow solver. The second drawback is that by definition phenomenological breakup models cannot be used to investigate the breakup process, as they rely on previous knowledge and correlations. To investigate the different mechanisms at play, and to provide understanding to further develop phenomenological models, the atomisation process must be directly simulated.

For this reason there is a need for fully resolved two phase simulations of film development and atomisation. As these simulations are computationally expensive, to reduce cost it is common to simulate just the tip of the prefilmer with film and air conditions at this point set as boundary conditions. To this end, a priori simulations of the whole geometry are performed, often single phase and at lower resolution. This method is known as embedded DNS, or eDNS, and was originally proposed by Sauer et al. (2014). However, a film boundary condition must also be set and this is often set to be a constant velocity and thickness. This method has been used by Warncke et al. (2017) and Wetherell et al. (2020) to model the experiment of Gepperth et al. (2010). It has been shown previously that waves form in the film (Bhayaraju and Hassa 2006; Brend et al. 2020), however, the very short distances between inlet plane and prefilmer tip used in the aforementioned simulations do not provide sufficient time for these waves to form.

By not allowing the waves to develop in the film, simulations using this methodology are limited in their accuracy. The instantaneous ERLIF measurements of Brend et al. (2020) suggest that these waves play an important role in the formation of ligaments at the atomisation edge. This is supported by Bacharoudis et al. (2014) who derive a force balance for liquid films reaching a sharp edge and show that the presence of waves in the film changes the liquid momentum and therefore the behaviour at the edge of the prefilmer. Previous work (Wetherell et al. 2024) has shown that this change in behaviour and in particular the waves in the film introduce temporal and spatial fluctuations into the resulting fuel spray. Variations in spray properties have been shown to impact the heat release and thermo-acoustic response (Treleaven et al. 2019) and the emissions produced (Lefebvre 1995). Understanding the interaction of the air and film across the prefilmer is therefore vital when trying to simulate gas turbine fuel injection and combustion, and also for the design and understanding of combustion system performance.

More recent work by Carmona et al. (2021) used a scaling methodology to reduce the cost of atomisation simulations by ensuring that key non-dimensional numbers were matched (Weber number, momentum flux ratio, density ratio and viscosity ratio). This allowed the simulation of a much longer equivalent prefilmer length, 8.1 mm, and a numerical investigation into the wave formation in the film and the impact this has on the atomisation process. However, when compared to the length of the original experimental geometry, 47.6 mm, the length over which the waves can form and develop is still significantly shorter. The simulation also uses the same approach to boundary conditions as previous work, a fixed film height and velocity, which is unlikely to be the case nearly 40 mm downstream of the fuel slot. The lack of any turbulent fluctuations in the airflow at the inlet may also limit the accuracy of the simulation, as this has previously been shown to impact the atomisation process (Warncke et al. 2017, 2019; Wetherell et al. 2020). Warncke et al. (2017) work also shows that the turbulence changes the film development, even when only 1 mm of prefilming length is simulated. To fully understand the development of the film and how it might impact the atomisation process requires the simulation of the entire geometry.

In our previous work (Wetherell et al. 2024) a simulation of the atomisation experiment of Warncke et al. (2017) was performed in which time varying air and fuel properties close to the prefilmer tip were used as the inlet boundary conditions. These coupled air and fuel properties were extracted from a precursor simulation of the development of the film. The inclusion of these boundary conditions was seen to have a large impact on the atomisation process and led to significantly improved agreement with experiment for droplet size and ligament length measurements. However, the total cost of the two simulations was 2 million CPU hours which would be prohibitively expensive if needing to simulate many operating conditions and geometries. Therefore, there is a need to reduce the cost of such simulations without compromising the accuracy. While there exist multiple methods for the generation of synthetic turbulence at the inlet, see Klein et al. (2003) and Poletto et al. (2013) for examples, to the authors’ knowledge there exists no equivalent method for the generation of unsteady film data. There is therefore a need to further study the interaction of the film and air in greater depth such that further understanding of the formation, development and influence of these waves is gained.

In this paper the simulation of the film development used to provide boundary condition data in Wetherell et al. (2024) is analysed in greater detail. This is a fully resolved two-phase simulation, using a Coupled Level Set & Volume of Fluid (CLSVOF) method. The aim of the paper is to explore the development of the film and quantify the interaction of the air and film. This includes determining the nature of the interaction and whether it is a one- or two-way coupling, and investigating how the waves form and propagate though the film. The behaviour of the air and film immediately upstream of the trailing edge is also analysed to understand the how it differs from that traditionally used in inlet boundary conditions. The contact angle boundary condition needed to allow correct film development as well as the implications for the cost of accurately simulating atomisation in this case are also considered.

The paper is structured as follows. The following section details the numerical method and simulation set-up. The results are split into four sections: the first provides an overview of the whole simulation and the important flow features as well as how the wall contact angle affects the development of the film. The second quantifies the formation and propagation of the surface waves, before Sect. 5 investigates the impact of the film on the gas-phase boundary layer. Finally, the behaviour of the air and film close to the trailing edge is quantified in Sect. 6.

2 Simulation Setup

2.1 Numerical Method

The CLSVOF solver used in this work was original developed by Dianat and co-workers at Loughborough University for simulations of surface contamination (Dianat et al. 2017a, b; Skarysz et al. 2018), and has since been applied to simulations of primary atomisation (Wetherell et al. 2020, 2024). It is a phase-incompressible, pressure-based, cell-centred, finite volume solver built in OpenFOAM v5, a set of open-source libraries for computational fluid dynamics (Greenshields 2017). A single momentum equation is solved for both phases using volume-averaged fluid properties (density and viscosity):

$$\begin{aligned} \frac{\partial \rho U }{\partial t} + \nabla \cdot \rho \hspace{1pt} U \textbf{U} = -\nabla {p} + \nabla \cdot \tau + f_\sigma \end{aligned}$$

(1)

As in common in atomisation simulations, gravitational and buoyancy effects are negligible and are omitted. A Large Eddy Simulation approach is taken, with turbulence closure provided by the WALE (Wall Adaptive Local Eddy-Viscosity) model (Nicoud and Ducros 1999) which has been shown to be able to accurately model boundary layer transition (Mistry et al. 2015). The surface tension body force, $f_{\sigma }$, is calculated using Brackbill et al.’s Continuum Surface Force method (Brackbill et al. 1992), where $f_\sigma = \sigma \kappa \nabla \alpha$. The original formulation used the volume fraction field to estimate the mean surface curvature, $\kappa$. However, in this CLSVOF implementation the curvature is given by

$$\begin{aligned} \kappa = -\nabla \cdot \frac{\nabla \phi }{|\nabla \phi |} \end{aligned}$$

(2)

to take advantage of the continuous nature and more accurate interface representation of the level set field. Advection equations are solved for the liquid volume fraction, $\alpha$, and the level set function, $\phi$

$$\begin{aligned}{} & {} \frac{\partial \alpha }{\partial t} + \nabla \cdot \textbf{U} \alpha = 0 \end{aligned}$$

(3)

$$\begin{aligned}{} & {} \frac{\partial \phi }{\partial t} + \nabla \cdot \textbf{U} \phi = 0 \end{aligned}$$

(4)

No interface compression term is used in the volume fraction equation, instead a geometric Piecewise Linear Interface Calculation (PLIC) reconstruction of the interface is performed to more accurately compute the convection term. As opposed to standard geometric VOF methods, here the gradient of the level set field is used to give the orientation of the interface, further increasing the accuracy of the convection. The full details of the interface reconstruction algorithm can be found in Skarysz et al. (2018).

A PDE-based reinitialisation routine proposed by Sussman (2003) is used on the level set field to ensure it remains a signed distance function, i.e. $|\nabla \phi |=1$. The PDE is given by

$$\begin{aligned} \frac{\partial \psi }{\partial \tau } = S \left( \psi _0 \right) \left( 1 - |\nabla \psi | \hspace{2pt} \right) \end{aligned}$$

(5)

where $\psi _0 = \psi \left( x, \tau = 0 \right) = \phi \left( x,t \right)$ and is solved in pseudo-time, $\tau$. In this case, $\psi$ is a dummy variable used during the solving of the PDE, and is equivalent to the level set field at the start of the reinitialisation routine. $S \left( \psi _0\right)$ is given by

$$\begin{aligned} S \left( \psi _0\right) = \frac{\psi _0}{\sqrt{{\psi _0}^2 + {\left( \hspace{3pt} |\nabla \psi _0|\hspace{2pt} \Delta \hspace{2pt} \right) }^2}} \end{aligned}$$

(6)

The gradient, $\nabla \psi$, is calculated in such a way that the distance function propagates away from the interface, by using information from the cell closest to the interface. This can be compared to upwinding for convective terms. Dianat et al. (2017a) show that 10 iterations and a pseudo-timestep of $0.3\Delta x$ allow the reinitialisation to fully converge, and so are used here. The minimum grid spacing in the domain, $\Delta x$, is calculated using the cube root of the cell volume. The reinitialisation process is performed every timestep.

2.2 Test Case

The geometry selected to investigate the film development in a scaled-up prefilming atomiser is shown in Fig. 1. It is a planar prefilming airblast atomiser designed to be representative of fuel injectors typically found in gas turbines. The geometry is taken from the KIT-ITS atomisation experiment. This has been selected as it has been extensively used to study primary breakup in airblast atomisers, see Gepperth et al. (2010), Gepperth et al. (2012), Gepperth et al. (2013), Chaussonnet et al. (2020) and Gepperth et al. (2014). In the experiment air is supplied by a radial compressor through flow conditioning apparatus into the test section. The airflow is split into two channels, above and below the prefilmer. Fuel is introduced onto the top surface of the prefilmer through a set of 50 equally spaced holes on the surface. The simulation domain has been recreated using Table 1 of Chaussonnet et al. (2020).

Fig. 1

Computational domain used in the present work, based on Karlsruhe prefilmer previously used in Gepperth et al. (2010, 2012, 2013, 2014). All dimensions in mm. Spanwise extent is 25 mm

Minor changes have been made to the geometry to enable the numerical investigation. The main change is the replacement of the 50 discrete fuel injection holes with a full width slot. As a block structured mesh is used this greatly simplifies the block arrangement and improves the quality of the resulting mesh. The width of the slot has been set such that the geometric areas are equivalent, and the velocity calculated to give the correct mass flow. Chaussonnet (2014) has shown that after a settling period the film reaches the same condition irregardless of the injection mechanism. For the conditions used in this paper that time period is 1.25ms, which equates to a distance of 0.6 mm downstream of the fuel slot. For this reason the change from discrete holes to a full-width slot should be of little consequence to the results. The second change is the domain width. In the experiment the prefilmer is 96 mm wide, and the middle 50 mm is wetted with fuel. In this work, only the middle 25 mm of the wetted region is simulated, with periodic boundaries at the edges. This is wide enough to avoid any spanwise correlation due to the periodic boundaries, as is confirmed later in the results. This greatly reduces the cell count and therefore cost of the simulation. The outlet plane of the domain is set at the trailing edge of the prefilmer as the focus of this work is the development of the liquid film and not on the atomisation downstream.

As mentioned previously a purely hexahedral block-structured mesh is used, shown in Fig. 2. A geometric distribution of cells is used near walls to ensure the boundary layer and unsteady film are accurately captured, with a growth ratio of 1.05. The near wall cell distance was determined using a single phase simulation, and is $5~\upmu \hbox {m}$ to give a $y^+$ below one across all surfaces. Away from the walls, the mesh has an approximately isotropic resolution of $100~\upmu \hbox {m}$. This gives approximately 80 cells across the boundary layer thickness of 2.5 mm, 15 cells across the mean film thickness of 100 microns, and 25 cells across the modal wavelength of 2.5 mm. With the accurate geometric advection scheme used in the CLSVOF formulation, these are thought sufficient to describe these phenomena correctly. Previous simulations using this solver have used adaptive mesh refinement (AMR) (Wetherell et al. 2020, 2024), but not in this case. The location of the film is known a priori and there will not be large changes in the topology of the phase interface and so the common reasons for using AMR do not apply. Adequate resolution can be achieved with a static mesh and therefore the cost overheads of AMR can be avoided. The total cell count is 70.1 million.

Fig. 2

Block-structured hexahedral mesh used for simulations

As a static mesh is used, once the flow has developed a fixed timestep gives an approximately constant maximum CFL number. This avoids the additional temporal discretisation error associated with a varying timestep. A fixed time step of $\Delta t = 0.4~\upmu$s is used, giving a maximum CFL number of 0.7 within the domain. This is also required as the geometric VOF algorithm is explicit in the discretisation of the volume fraction convection term. The capillarity constraint on the timestep first proposed by Brackbill et al. (1992)

$$\begin{aligned} {\Delta t_\sigma = \frac{\Delta x}{2c} = \sqrt{ \frac{\rho _g + \rho _f}{4\pi \sigma } \Delta x^3}} \end{aligned}$$

(7)

gives a minimum timestep of $47~\upmu$s and so this case is limited by the high velocity gas phase, rather than the propagation of capillary waves at the surface.

The operating conditions are set to match the conditions used in Warncke et al. (2017) and are summarised in Table 1. The air velocity given in the table is the bulk velocity in the channels above and below the prefilmer. Accounting for the change in area an inlet velocity of 17.63 m/s is used for the inlet to the plenum. Due to the flow conditioning upstream of the test section the turbulence level at the inlet is very low, and so no turbulent fluctuations or velocity profile is applied. For a fuel slot width of 0.6 mm, a fuel inlet velocity of 0.0417 m/s is used to achieve the correct film loading. The upper and lower surfaces of the plenum are set to symmetry planes, while the channel walls above and below the prefilmer, as well as the prefilmer itself are no-slip walls. The total pressure is fixed at the outlet using the prghTotalPressure boundary condition and zero gradient conditions are applied for velocity. This boundary condition for pressure improves the stability of structures moving through the outlet plane by accounting for the different density of the two phases, and has previously been used in Warncke et al. (2017) and Wetherell et al. (2020). A fixed contact angle is applied to wall surfaces for the VOF and LS fields, the value of which is discussed in Sect. 3.1.

Table 1 Fluid properties and operating conditions

The VOF and LS fields are initialised with the fuel slot already full of stationary fuel, and the prefilming surface dry. The simulation must therefore run for an initial period to allow the film to develop across the prefilmer surface, which for these conditions was 120 ms. At this point, data extraction and field averaging is started, and the simulation run for a further 100 ms. The simulations were performed on Sulis at HPC Midlands+ using 1280 cores, for a total cost of 990,000 CPU hours.

2.2.1 Data Extraction

Due to the size of the mesh, storing all fields at the frequency required to resolve the evolution of the film would require unfeasible amounts of storage space. To avoid this, line probes are added to the domain and sampled at a greater frequency than the whole domain is stored. These probes can be split into two groups, boundary layer and film, and their locations are shown in Fig. 3. The boundary layer probes have a length of 3.5 mm and a resolution of $5~\upmu \hbox {m}$, and are equally spaced along the centreline of the prefilmer. The purpose of these probes is to capture the boundary layer that grows above the film, and they are repeated on the lower surface of the prefilmer to allow comparison to the undisturbed boundary layer with no film. The film probes have a length of 2 mm and extract at each cell centre. These probes have a streamwise spacing of 0.8 mm, and are repeated in five sets spaced equidistantly across the prefilmer span, as shown in Fig. 3. These probes are used to capture the development of the film, including the formation and propagation of waves along the prefilmer. For both sets of probes, the velocity, volume fraction and level set fields are extracted every $2~\upmu \hbox {s}$. Once extracted, and prior to any further post-processing or analysis, the velocity field is rotated onto a prefilmer-aligned coordinate system, such that the x-direction is parallel to the prefilmer surface and the y-direction is normal to it. The z-direction is unchanged.

Fig. 3

Location of boundary layer and film data extraction probes. Film probes overlap in Figure due to small axial spacing

Planar data is also extracted from the simulation during run time, on a plane 2 mm upstream of the trailing edge and outlet. This plane is aligned with the original x-direction, and is therefore parallel to the trailing edge of the prefilmer. Again, the velocity, volume fraction and level set fields are extracted every $2~\upmu \hbox {s}$. This allows the behaviour of the film and air at the trailing edge to be assessed, quantified, and compared to the typically used boundary conditions in an atomisation simulation.

3 Flowfield Overview

Figure 4 shows a snapshot of the simulation after 220 ms, at the end of the sampling period, and all of the important flow features are shown. The flow accelerates around the leading edge of the prefilmer but the boundary layer does not separate or transition from laminar to turbulent. Beyond the leading edge, in the straight, constant-height channel, the boundary layer develops along the prefilmer, and still does not transition to turbulence. At the fuel slot, the introduction of the fuel onto the prefilmer surface disturbs the flow, causing the boundary layer to rapidly transition to turbulent and thicken significantly. The figure also shows the surface waves that form in the film.

Fig. 4

Snapshot of simulation after 220 ms of simulated time. $\phi =0$ isosurface used to visualise film, with axial velocity shown on XY slice 2.5 mm away from periodic boundary

The waves in the film shown in Fig. 4 can be more easily seen in Fig. 5a, which colours the $\phi =0$ isosurface by the film thickness. Note, the colour scale has been limited to 300 microns to enhance the contrast across the film, the actual greatest film thickness in the figure is 470 microns. The mean film thickness, measured as the geometric mean of all sampled points in time and space in the simulation, is 100 microns. Here, the peaks and troughs of the wave are clearly shown.   Carmona et al. (2021) suggest that the film can be split into three regions—an initial 2D regime, a transition regime and a 3D established regime, however, this is not observed here. The initial 2D regime is replaced by a highly chaotic wave system, driven by the turbulence in the airflow. The highest wave peaks are seen in this region, and the waves are a mix of transverse and v-shapes. There is no well defined transition regime, instead the interaction of the film and airflow changes the structure such that transverse waves with moderate peak heights dominate, with occasional v-shaped waves with a very high peak. The original work of Carmona et al. did not include turbulent fluctuations at the inlet, allowing the two-dimensional wave formation process followed by the transition to 3D waves as the air and film interact. In this case, the formation process is dominated by the highly turbulent airflow close to the fuel slot, resulting in a fully three-dimensional wave system from the start of the film. This aligns well with the results using short prefilmers of Warncke et al. (2017) and Wetherell et al. (2020), who show that the inclusion of inlet turbulence has a large impact on the film development, even when only simulating 1 mm of the prefilming surface. The formation and propagation of the waves is discussed and quantified in more detail in Sect. 4.

Fig. 5

Important flow features at 220 ms of simulated time

The sudden transition of the boundary layer from laminar to turbulent due to the introduction of the fuel onto the prefilmer is more clearly shown in Fig. 5b, which uses an isosurface of the $\lambda _2$ field (Jeong and Hussain 1995) to visualise the vortical structures. Streaks of vorticity can be seen emanating from the leading edge acceleration, these are consistent with the transition mechanisms for flat plate boundary layers (Schlichting and Kestin 1961). These streaks are transient in nature, changing in size and spanwise position throughout the simulation. The turbulence within these streaks is enhanced by the fuel slot, but the full width of the prefilmer is tripped to turbulence as well. This grows the boundary layer in the gas phase above the film. Section 6 focuses on the impact of the film on this boundary layer, and how it compares to the undisturbed boundary layer that forms on the underside of the prefilmer.

No grid independence study has been performed due to the high cost of a single simulation, and no experimental data exists apart from downstream measurements of the atomisation, so no quantitative assessment of the mesh can be made. However, Fig. 5 qualitatively suggests that both the film and turbulent boundary layer are being well resolved, particularly the large range of structures visible in Fig. 5b. Section 2.2 has already discussed the grid spacing used in relation to the phenomena of interest in this work, such as the boundary layer thickness, wave height and wavelength, and the timestep is sufficiently small that both convective (CFL number) and surface tension (capillarity constraint) effects are well-resolved in time. While limited data is available from the experiment for the development of the film there are some comparisons that can be made to check the agreement of the simulation. The shadowgraphy field of view in the experiments includes the very end of the prefilmer surface, and so the wave structure at the end of the prefilmer is shown, see Fig. 14 in Warncke et al. (2017) or Fig. 3 in   Carmona et al. (2021), which allows qualitative comparisons. The wave structure shown in Fig. 5a shows good agreement with that shown in the shadowgraphy, with an irregular wave system made up of mostly transverse waves, with some more v-shaped waves. This gives confidence that the film development across the prefilmer has been accurately captured. While no quantitative experimental data for the film exists, data extracted from this simulation were used as boundary conditions for an atomisation simulation, which resulted in much better agreement between simulation and experiment  (Wetherell et al. 2024). Confidence is therefore gained that all of the key flow features are being correctly simulated.

3.1 Impact of Contact Angle

All of the walls in the simulation have a constant static contact angle applied to them as a boundary condition for the VOF and LS fields. Initially this value was set to 90${}^\circ$. While known to be inaccurate, this value had been used in previous simulations and achieved good agreement with other experimental and numerical work (Wetherell et al. 2020). However, from an early point in this work it became clear that an incorrect contact angle would prevent any accurate data being collected. Figure 6a shows the simulation after 50 ms of simulated time, and it can clearly be seen that instead of a full width film, a series of rivulets are formed on the surface of the prefilmer due to the high contact angle, which implies a low wettability condition. In previous work this had not been an issue due to the very short prefilmer, which was initialised with a fully covered surface.

Fig. 6

Effect of wall contact angle on film development process. Images after 50 ms of simulated time

Hodgson et al. (2021) have recently developed a new method for the measurement of contact angles, both static and dynamic. Using this method the static contact angle of kerosene on stainless steel was measured as 9${}^\circ$ and a new simulation run with this contact angle, the results of which are shown in Fig. 6b. Here a full width film develops, as opposed to the rivulets seen in the 90${}^\circ$ case. Once the film is fully developed and there are no dry patches then the contact angle will no longer affect the film in this simulation, however, without the correct contact angle a fully wetted prefilmer will likely never have been produced, highlighting the need for accurate boundary conditions for film development simulations. It is the results of the 9${}^\circ$ case that will be presented throughout the remainder of the paper.

4 Surface Waves

Figures 4 and 5a showed that surface waves form in the film due to the interaction of the two phases. These waves can also be visualised using a space-time diagram of film thickness, shown in Fig. 7a. For clarity, a shorter region of the prefilmer and time-span are shown in Fig. 7b. These figures show the film thickness along the prefilmer centreline at all points in time during the sampling period. The waves are shown as diagonal streaks of high film thickness, from the bottom left to the top right of the figure. These figures contain a large amount of information. The wavelength and period of the waves can be measured as the distance between the waves in the horizontal and vertical directions, respectively. The wave speed is the inverse of the gradient of the diagonal lines. The wave heights are the peaks in film thickness. Qualitative analysis is also possible. For example, both figures suggest that there is a dominant wave speed of between 1 and 1.5 m/s, due to the consistent gradient of the streaks. Waves merging together can also be seen in Fig. 7b, for example two small waves are shown merging into a larger wave between $x/L=0.82$ and 0.9, and $t=61$ and 64 ms.

Fig. 7

Space-time diagrams of centreline film thickness, showing propagation of waves along prefilmer surface

4.1 Wave Identification

Before any quantitative data can be extracted, the waves must first be identified using the film thickness data. Figure 8 and Algorithm 1 shows the process by which this is done. At each point in space, Matlab’s findpeaks function (MATLAB 2021) is used to identify the peaks in the film thickness time history. Two criteria are used during the identification process, a minimum peak distance of 0.2 ms and a minimum peak prominence of $25~\upmu \hbox {m}$. Each identified peak is shown in Fig. 8a. The peaks are then connected to identify the peaks that form a contiguous wave. Starting from the first peak, a continuous chain of peaks is identified by finding the closest peak in time that is one spatial location downstream. If this closest peak is less than 0.75 ms after the current peak, it is added to the chain and the search for the next peak in the chain begins. The value of 0.75 ms was chosen as a trade-off between ensuring the next peak in the wave was identified without erroneously connecting peaks from different waves. If no peak is close enough in space and time the algorithm will first see if a peak can be found two spatial locations downstream, and if not checks to see if there is a peak one spatial location downstream, but at an earlier point in time. This allows for sudden jumps in the peak location, such as at $x/L=0.95$ and $t=69$ ms. If no next peak can be found, the algorithm moves onto the next unassigned peak and starts again with the next wave, until all peaks have been assigned to a wave. These connected peaks can be seen in Fig. 8b. The wave height is then calculated as the mean height of all of its identified peaks. The final step in the wave identification process is to fit a straight line through the connected peaks, while also recording the start and end position of the wave in space. These fitted lines are shown in  Fig. 8c. The velocity of a wave is then the inverse of the gradient of its fitted straight line. Only waves made up of four or more peaks are used for all analysis to eliminate very short lived “waves” identified by this algorithm. This is repeated for each of the five sets of streamwise probes explained in Sect. 2.2.1 and shown in Fig. 3.

Fig. 8

Stages of wave identification process

Algorithm 1

Wave Identification Algorithm

The fitted lines are then used to measure the periods and wavelengths in the film, as shown in Fig. 9. The wave periods are measured every 1 mm along the prefilmer, and the wavelengths every 1 ms in time. They are measured as the separation, in time and space respectively, between waves that cross the measurements spatial locations or times. The drawback of this method of wave identification is that it does not lend itself to correlations between the wave properties, as the speed and height are measured on a wave by wave basis, while the period and wavelength are measured using snapshots of the whole film. To remedy this, each measured wavelength and frequency is assigned a wave speed and a wave height, defined as the average of these properties for the two waves used to measure the frequency or wavelength, shown in Fig. 9. This still does not allow the frequency and wavelength to be correlated against each other, however all other properties can be correlated.

Fig. 9

Calculation of wave period and wavelength from identified waves

4.2 Wave Properties

The correlations between wave properties are shown in Fig. 10, along with the Spearman’s rank correlation coefficient (Laerd 2018). The correlation coefficient is a measure of how correlated two data sets are based on their rank in a sorted list, and takes values between $-1$ and 1, where the extremes represent perfect negative and positive correlations. Values between $-$0.5 and 0.5 are generally accepted as being uncorrelated. Figure 10 shows that for all pairs of properties, except wave height and wave speed, the coefficient is well below 0.5 and the properties are therefore uncorrelated. This shows that the waves are not formed by a coherent, periodic process, but instead are formed by a stochastic process driven by the turbulent interaction between the two phases. Wavelength and period are generally considered as the separation in space and time of two similar waves, yet it can be seen in Fig. 7 that the waves are not similar. This makes defining a wavelength or period difficult, and it could be argued that there is no valid wavelength or period to be extracted. Instead, the separation in time should be thought of as how often a wave passes through a point. The correlation between wave height and wave speed is stronger, although still below the threshold of 0.5. The higher the wave the further into the airflow it extends, and therefore is exposed to higher velocity air and a greater force pushing it along the prefilmer. The stochastic nature of the interaction between film and air, particularly close to the fuel slot, means that the correlation between wave height and speed is still relatively weak, however.

Fig. 10

Correlation of wave properties with each other, including Spearman’s Rank coefficient, $\rho$

The lack of similarity between waves is also shown by the distributions of the wave properties, Fig. 11. Figure 7 suggested that there was a dominant wave speed of between 1 and 1.5 m/s, which is confirmed by Fig. 11a. The peak in the PDF is between 1.2 and 1.4 m/s, and the distribution is mostly symmetric about this peak, apart from a small number of fast waves, above 2 m/s. These outliers are short lived waves, existing for only a few millimetres and milliseconds before they catch up to and merge with slower moving waves, and some examples can be seen in Fig. 7b. The wavelength and wave period distributions, Fig. 11b and c are significantly wider than the wave speed distribution, with broader peaks. The wave formation process is stochastic and driven by local instabilities in the interaction between the phases, while, once formed, the waves are driven along the prefilmer by the mean properties of the system. This is because the waves exist for time- and length-scales greater than the turbulent fluctuations in the air, 10 s of milliseconds and 10 s of millimetres, and so the influence of turbulence is averaged out across the lifespan of the wave. This is further supported by the correlation of wave speed and the axial distance along the prefilmer the wave is present for, Fig. 10f, which shows that as the axial life of the wave increases the wave speed converges onto a velocity between 1.2 and 1.4 m/s, the same as the peak in the speed distribution. The wave height distribution, Fig. 11d, shows that the most common wave height is between 160 and 170 microns, although waves above 300 microns high are seen in the simulation which is over three times greater than the mean film thickness of 100 microns. The wave heights are also an average across the life of the wave, and local peaks will be even higher than this.

Fig. 11

Distribution of wave properties, generated using whole film data

Thin film models often assume that the velocity profile within the film is linear. This is confirmed by Fig. 12, which shows the mean velocity profile in the film. Only points in time and space that are within the film are used to calculate the mean profile, hence the reduced number of points and less converged profile further from the wall. The instantaneous fuel mass flow per unit span can be found by integrating away from the wall:

$$\begin{aligned} \frac{{\dot{m}}}{b} = \rho _f \int \limits _{0}^{\inf } U \alpha \; \textrm{d} y = \rho _f \int \limits _{0}^{\inf } my \alpha \; \textrm{d} y \end{aligned}$$

(8)

where b is the prefilmer span, $\alpha$ is the liquid volume fraction, U the streamwise velocity, and m the gradient of the velocity profile in Fig. 12. The consequence of this is that the instantaneous fuel mass flow is proportional to the square of the instantaneous film thickness, and therefore increases in film thickness, caused by the waves, cause disproportional increases in the local mass flow. For example, the largest waves seen in the simulation, which are over three times higher than the mean film thickness, will contain almost an order of magnitude more mass flow. The mean film thickness and film loading, and by extension the film mass flow, have previously been shown to affect the atomisation process (Rizk and Lefebvre 1980), and it is therefore to be expected that instantaneous changes will also affect the atomisation process.

Fig. 12

Mean streamwise velocity profile through film, coloured by percentage of time that liquid reaches that height

5 Boundary Layer Development

The introduction of fuel onto the prefilmer surface at the fuel slot and the unsteady film that forms downstream have a disruptive effect on the boundary layer growing in the gas phase, as was shown in Figs. 4 and 5b. Figure 13 uses isosurfaces of the $\lambda_2$ criterion to compare the turbulent structures on the upper and lower surfaces. On the lower surface, Fig. 13b, streaks of vorticity form downstream of the acceleration around the leading edge. These streaks are transient in nature, and their location, size, and number varies in time. However, due to the low Reynolds number in the constant height channel (27,000 based on the channel height) and the moderate adverse pressure gradients around the leading edge of the prefilmer these streaks are not sustained along the prefilmer length and die out. The same streak formation is seen on the upper surface, Fig. 13a, but instead of the turbulence level decreasing, the introduction of the fuel onto the prefilmer acts as a trip and causes the full width of the boundary layer to become turbulent.

Fig. 13

Comparison of turbulent structures on the upper and lower prefilmer surfaces, shown using an isosurface of the Lambda2 criterion. Same isosurface value of $\lambda ^2=2.5\times 10^8$ used for both surfaces

The effect of this disturbance and transition is shown in Fig. 14, which compares the mean streamwise velocity profiles through the boundary layer at 12 positions along the upper and lower surfaces of the prefilmer. By only including data points with a negative level set value in the time average these plots show only the gas phase velocity. Upstream of the fuel slot, the profiles are almost identical; the small difference is due to the the blockage of the film changing the pressure gradient in the upper channel compared to the lower channel. The change in pressure gradient is also evidenced in Fig. 13, where the vortical streaks forming from the leading edge on the upper surface appear to sustain further downstream than those on the lower surface. After the fuel slot, however, there is a significant change in the profiles. Close to the fuel slot, there is a large region of slow moving fluid, caused by the film acting as a forward facing step and disrupting the boundary layer. The temporal variations in film thickness also change the height of this step, contributing to the disturbance. As the film and air develop over the length of the prefilmer, the difference between the upper and lower profiles reduces as the two phases couple together, however, the extent of the disturbance is seen at greater wall distances due to the thicker, turbulent boundary layer. By the end of the prefilmer the top surface boundary layer thickness (defined as 99% of local freestream) is 2.5 mm compared to 1.5 mm on the lower surface.

Fig. 14

Comparison of gas-phase velocity profiles along upper and lower prefilmer surfaces. Annotations are normalised streamwise position, x/L, with the fuel slot at $x/L=0$. Shaded region shows mean film thickness

The film also changes the level of turbulence within the boundary layer, as shown by the Reynolds stress profiles in Fig. 15. Again, only data points with a negative level set value are used to isolate the gas phase. In the region immediately downstream of the fuel slot a large increase in the $\overline{u'u'}$ profile above the film is seen. This is due to variations in the film thickness changing the location of the slow moving air immediately above the film. This is then seen as a large fluctuation in the streamwise velocity component, hence the increase in the streamwise Reynolds stress value. Similar behaviour is seen upstream of the fuel slot on the upper surface and on the lower surface to a lesser extent, which show a smaller peak in the streamwise component close to the wall. This is due to the disturbances seen for example in Fig. 13a. At the relatively low Reynolds number in this case the disturbances do not develop into fully turbulent flow. As the film and boundary layer develop over the prefilmer, the transition from laminar to turbulence increases the distance from the wall at which the streamwise profile differs from the lower surface, but also reduces the size of the peak relative to the lower surface as the film and air couple together and the disruptive effect of the film lessens.

Fig. 15

Comparison of gas-phase Reynolds stress profiles along upper and lower prefilmer surfaces. Annotations are normalised streamwise position, x/L, where 0 is the fuel slot

The wall-normal and spanwise Reynolds stress components exhibit very different behaviour however. The wall-normal component, Fig. 15b, shows only a minor disturbance immediately after the fuel slot compared to the lower surface. However, as the boundary layer transitions and the turbulence level in the boundary layer increases the difference between the upper and lower surface grows. Similarly, while a larger initial disturbance is seen in the spanwise component, Fig. 15c, compared to the wall-normal, it is still small compared to the streamwise component and then the difference between the profiles grows as the boundary layer develops. The reason for this behaviour is due to the mean velocity in the wall-normal and spanwise components being zero, and hence the varying film thickness does not artificially increase these components as it does for the streamwise one. The wall-normal and spanwise profiles on the lower surface are close to zero at all points along the prefilmer, confirming that the velocity disturbances seen on the lower surface do not develop into turbulent flow.

6 Air/Film Condition at Trailing Edge

Thus far it has been shown that there is a strong coupling between the air flow and the film. The presence of the film causes an increased level of turbulence in the air boundary layer, and also a thicker boundary layer. Surface waves form in the film and these waves are stochastic in nature, and their size, frequency and speed varies greatly. This section explores the effect this has on the condition of the two phase flow at the trailing edge and the implications for the atomisation process itself, as well as simulations attempting to accurately capture it.

Figure 16 shows the axial velocity and film thickness on a plane 2 mm upstream of the trailing edge. Large variations in the film thickness, both in space and time, are shown which are caused by the waves passing through this plane. Large turbulent structures can also be seen in the boundary layer above the film. The time delta between the first and last images is comparable to the typical total simulated time in an atomisation simulation (Warncke et al. 2017; Wetherell et al. 2020; Carmona et al. 2021), and so these variations would be present within the time period of a typical atomisation simulation. The turbulent structures in the boundary layer are also correlated and coupled with the variations in film thickness, which poses the first challenge for simulations of the atomisation process. Synthetic/artificial turbulence generators are a common way of reducing the cost of simulations while ensuring the boundary conditions are accurately specified (Klein et al. 2003; Poletto et al. 2013). These methods often use the mean velocity profiles, combined with turbulence intensity, Reynolds stress, and/or turbulent length scale profiles to generate the turbulence. However, the turbulence in the air is coupled with the film thickness and so matching only the statistical description of the turbulence is insufficient, the instantaneous velocity fields must also align with the two phase flow. To the authors’ knowledge there are currently no synthetic turbulence generation methods that are capable of this instantaneous coupling. This issue is further exacerbated by the inflation of the streamwise Reynolds stresses that was discussed in Sect. 5, which means that even if such a method existed, a way of isolating the actual turbulent fluctuations from the changes in velocity caused by film thickness variations would also be required. The result of this is that, currently, the specification of accurate boundary conditions for an atomisation simulation of this type of prefilmer requires a priori time- and space-resolved information about the film and air upstream of the trailing edge for each individual operating condition and geometry.

Fig. 16

Axial velocity contours and fuel film on plane 2 mm upstream of prefilmer trailing edge, showing coupled, unsteady nature of the two phase flow

As shown in Fig. 16 the waves cause spatial and temporal variations in the film thickness, and these film thickness variations cause variations in the local fuel mass flow. Figure 17 shows the film thickness and fuel mass flow per unit span passing through a plane 2 mm upstream of the trailing edge, normalised by the respective mean values of 100 microns and 0.0385 g/s/mm. The waves are shown in both figures as regions of high film thickness and mass flow. Wetherell et al. (2024) showed that it is these variations in thickness and mass flow due to the waves that have the greatest impact on the atomisation process, with strong correlations between the spray and film properties because of them.

Fig. 17

Space-time plots of film passing through plane 2 mm upstream of prefilmer trailing edge

Section 4 showed that the film mass flow is proportional to the square of the film thickness, due to the linear velocity profile though the film. The result of this is that the difference between the peaks and troughs is more pronounced in Fig. 17b than in Fig. 17a. The difference is actually more pronounced than this, as the colour-scale in Fig. 17b has been capped at ${\dot{m}}/\overline{{\dot{m}}}=8$ to prevent the small number of very high peaks from hiding the majority of the waves; the highest value of ${\dot{m}}/\overline{{\dot{m}}}$ in the figure is 16.1. The result of this is that the waves contain the majority of the fuel mass flow, with little mass flow reaching the trailing edge in the troughs between the waves. This is shown explicitly in Fig. 18, which shows the distribution of fuel mass flow as a function of film thickness. Film thicknesses up to and including the mean film thickness of 100 microns contain only 21% of the total fuel mass flow at the trailing edge. While the peak in the distribution is in the 130–140 micron band, the skewed nature of the distribution means that film thicknesses of up to 140 microns still only account for 45% of the total mass flow due to the large amounts of mass flow carried by the extreme events. For example, Fig. 11d showed that waves greater than 300 microns are relatively rare, however Fig. 18 shows that film thicknesses greater than 300 microns are still responsible for 10% of the total fuel mass flow. The interaction of the film and air and the development of waves in the film is therefore the key method for the delivery of the fuel to the trailing edge of the prefilmer, as it is the waves that are the main source of fuel to the reservoir of fuel behind the trailing edge.

Fig. 18

Distribution of fuel mass flow as a function of film thickness, including cumulative distribution

The variations in film thickness and fuel mass flow shown in Fig. 17 are not removed when averaged across the prefilmer span, as shown in Fig. 19. The total mass flow is calculated using the velocity and volume fraction fields

$$\begin{aligned} {\dot{m}} = \rho _f \int \limits _{-b/2}^{b/2} \int \limits _{0}^{\inf } U \alpha \; \textrm{d} y \; \textrm{d} z \end{aligned}$$

(9)

and is then normalised using the inlet mass flow of 0.9625 g/s. Large variations in the mass flow can be seen, with an RMS value of 0.33 g/s, or 34% of the mean value. The largest variations in mass flow however are more than 80% of the mean value. This means that any assumption of constant mass flow in time is invalid, an assumption that is often made in combustion simulations using Lagrangian sprays (Jones et al. 2012; Prasad et al. 2013; Giusti et al. 2018; Eckel et al. 2019; Foale et al. 2019; Treleaven et al. 2020), and that empirical and phenomenological breakup models need to account for the variations caused by the air-film coupling along the prefilmer.

Fig. 19

Variation in time of normalised total fuel mass flow passing through plane 2 mm upstream of trailing edge

Wetherell et al. (2024) showed that these spatial and temporal variations in the film thickness and fuel mass flow have a direct impact on the atomisation process and resultant spray with much improved agreement seen with experiment for droplet and ligament statistics. This agreement gives confidence that the results seen here are a realistic simulation of the physical film development. However, the result of this unsteadiness is that the collection of fully converged statistics from atomisation simulations with accurate film and air boundary conditions will require a much greater sampling period as the variations in the film at the trailing edge will introduce an additional level of fluctuation into the already stochastic atomisation process. The result of this is that costs required to achieve convergence will increase significantly.

The work of Wetherell et al. (2024) and the results presented in this paper also suggest that large experimental prefilmers, such as the Gepperth prefilmer, are not an ideal validation case for CFD simulations. In this paper, it has been shown that the boundary layer on the underside of the prefilmer is in the early stages of transition at the trailing edge, and that the upper surface boundary layer is heavily influenced by the film. Determining what the gas phase boundary conditions should be on either surface is therefore challenging for both experimental and numerical approaches. Secondly, the delivery of the fuel to the trailing edge of the prefilmer is dominated by the waves that form in the film, and these variations have been shown to be directly linked to the atomisation and spray performance, increasing the required sampling time of the simulation in order to converge the atomisation statistics. This, combined with the large wetted length and resolution requirements, results in a very high computational cost for a single operating condition and geometry. For example, the film simulation presented here and the atomisation simulation presented in Wetherell et al. (2024) combined required almost two million CPU hours. Recreating the experimental results obtained using prefilmers such as this is therefore challenging on multiple fronts, and consideration should be taken regarding its use as a validation case for CFD.

7 Conclusions

In this paper a simulation of the film development over the prefilmer of the Karlsruhe Institute of Technology atomisation experiment (often referred to as the Gepperth prefilmer) has been performed with the aim of quantifying the interaction of the air and film, as previous work has shown that the behaviour of the two phases at the trailing edge plays a vital role in the atomisation process. A previously published and validated CLSVOF solver was used for this purpose. The introduction of the fuel onto the prefilmer disturbs the boundary layer growing on the upper surface of the boundary layer, which is tripped to turbulence by the film. Compared to the boundary layer on the lower surface, which does not fully transition, the boundary layer is significantly thicker and the level of turbulence within the boundary layer is increased.

It has also been shown that the interaction of the air and film causes surface waves to form in the fuel film, whose formation is stochastic and driven by the gas phase turbulence in the region immediately downstream of the fuel slot. The distinct regimes of Carmona et al. (2021) are not seen here as the 2D regime in their work develops due to the lack of turbulence at the inlet of their simulation. Further from the slot it can be seen, perhaps most clearly in Fig. 10f, that the waves trend towards a uniform wave speed indicating the presence of established 3D waves. There is little correlation between wave properties due to the stochastic formation process, apart from a weak correlation between wave height and the propagation speed. Waves of a height more than three times the mean film thickness are seen. The variations in film thickness due to the waves cause large spatial and temporal variations in the fuel mass flow reaching the trailing edge, such that the fuel delivery to the trailing edge is dominated by the waves.

When compared to previous approaches to modelling the atomisation process downstream of this prefilmer, the air-film coupling results in needing a larger domain and longer sampling periods to obtain accurate and converged results due to the increased boundary layer thickness and the variations in fuel mass flow. Our work in Wetherell et al. (2024) shows that this is possible, and that if correctly formed waves can be introduced at the end of the prefilmer then this allows accurate atomisation simulations. However, the wide range of wave properties shown in Sect. 4 make it hard to identify the correct wavelength, size, etc. and therefore simulations of the full prefilmer, such as those in this work, are required. For these reasons, the challenges and cost of performing accurate simulations of this style of prefilmer suggests that consideration should be given to their use as validation cases for CFD.