A Review of the Developments of the Characteristics and Mechanisms of Airless Spraying on Complex Surfaces

Abstract

The special surface appearance of complex surfaces restricts the coating film quality of spraying. The study of the atomization and film formation characteristics of typical complex surfaces, as well as the spraying mechanism, is essential for planning the spraying robotic trajectory and improving the spraying efficiency. In this paper, modeling and characteristics of the atomization and film formation process, based on CFD numerical simulations in previous studies, are systematically reviewed, focusing especially on airless spraying. In addition, the advantages and disadvantages of the existing research from the perspective of numerical models and methods are discussed. Finally, a further research direction for spraying on complex surface is prospected. Overall, a comprehensive and up-to-date review of spray atomization and film formation characteristics is considered valuable to practitioners and researchers in these fields, and will facilitate the further application of robotic spraying in the mechanical, automotive, marine, aerospace, petrochemical and other industries.

1. Introduction

Within industrial production, surface coatings have certainly asserted themselves as a key field, especially in oil and gas pipeline construction [1,2], automotive manufacturing [3,4] and equipment anticorrosion [5,6,7], with the overarching goal of enhancing protection performance while meeting the functional requirements in practice. Compared to brushing and rolling, coatings obtained by spraying are widely used, owing to their superior coating quality, higher efficiency and adaptability.

The spraying methods include air spraying [8,9,10], electrostatic spraying [11,12], plasma spraying [13,14,15], high-pressure airless spraying [16,17], etc. Airless spraying, owing to its unique spraying principle, always features excellent film quality, spraying efficiency, coating adhesion and environmental friendliness, and it is capable of achieving high-viscosity, high solid component and thick-coat spraying. Moreover, there is no air-assisted in atomization process, thereby avoiding film defects caused by impurities in the compressed air. The airless spraying process is essentially a gas–liquid two-phase turbulence flow movement of paint droplets in the air phase, which can be divided into the paint atomization process and the film formation process according to the time sequence, as is schematically shown in Figure 1.

Nowadays, robotic airless spraying is increasingly replacing manual spraying in coating applications. However, a major problem encountered in robotic spraying is the poor film quality on complex surfaces, namely orange-peel-like surfaces, sagging, graininess and locally uneven surfaces frequently occur. Consequently, it results in film thickness calculation, coating thickness control and trajectory planning challenges for robotic painting.

Essentially speaking, the reason is, first of all, the lack of understanding of the characteristics and mechanisms of the atomization of the spray. Adequate atomization results in smaller and more homogeneous paint droplets, leading to better film uniformity. Secondly, there remains a gap in in-depth research on the characteristics and mechanisms of film formation on complex surfaces, leading to an inability to set the optimal spraying parameters for each kind of surface. For instance, when spraying on a plane, the spraying gun can always be perpendicular to the surface of the workpiece, uniformly moving in a straight line, and ensuring the uniformity and stability of the coating film. Due to the influence of workpiece geometry, the vertical distance from the nozzle to the workpiece, the incident angle of paint droplets relative to the workpiece, and the movement speed of the spraying gun all accordingly change during dynamic spraying, which results in increased difficulty in offline programming of the robotic arm movement. Additionally, complex surfaces seriously impact the diffusion of the paint flow field. Owing to the phenomenon of boundary layer separation, it will produce localized vortex areas and reflux, resulting in localized splashing and excessive deposition when the paint moves to the near-wall zone, thus affecting the uniformity of the film thickness and the transfer efficiency (TE).

For this reason, studies on the atomization and film formation processes during spraying are reviewed in this paper, which highlights previous research work on experimental as well as computational fluid dynamics (CFD) simulation, then expands to include recent works on complex surfaces painting for film formation properties, especially focusing on airless spraying. We also summarize some of the mechanisms behind the breakage and adhesion process of the paint. This paper will conclude with an overview of possible future perspectives of the field. By critically recounting previous studies, this paper aims to provide a comprehensive account of the novelty and remaining unanswered problems of their research. At the same time, it may help understand the current research state and progress of spraying technology, or the research gap existing in the current literature. As a result, the research direction and focus can be better grasped, with the expectation that it will convey referential significance to practitioners and researchers in relevant fields.

As an additional note, the complex surfaces described in this paper are mainly distinguished from plane surfaces, and generally refer to those surfaces with the characteristics of variable or large curvature, or variable normal and irregular surfaces; for instance, arc surfaces, intersect surfaces, ball surfaces, and the like. Figure 2 schematically shows facilities and equipment that contain typical complex surfaces.

2. Historical Overview of Paint Atomization Properties and Mechanisms

2.1. Mechanism Study of Atomization in Airless Spraying

In order to comprehensively analyze research progress in the field of spraying, it is necessary to first overview the theoretical development from a historical perspective. The theory of high-pressure airless spray atomization essentially originates from the liquid jet atomization theory, which refers to the physical process in which a liquid becomes a large number of discrete liquid drops with different morphologies by dispersing and crushing the initial continuous liquid column ejected from the nozzle into the gas environment due to the role of extrinsic energy. During the dispersing and breaking up process, the paint will be subjected to the combined interaction of various forces such as viscous force, inertial force, aerodynamic force and surface tension. In the theoretical investigation of the causes of atomization, the theories of pressure oscillation [18], aerodynamic disturbance [19], turbulence disturbance [20,21], air disturbance [22] and changes in boundary conditions [23] are mainly formed.

Among them, the aerodynamic interference theory is the most fully developed hypothesis, and recognized by most researchers on account of it being a better explanation of the cause of atomization of low-speed liquid jets. Consequently, researchers generalized it and took it as the basic theory for high-speed jet atomization research. According to this theory (also called the liquid surface wave instability breaking mechanism, or linear instability theory), after the paint liquid is sprayed out of the nozzle, unstable waves will be produced between the gas–liquid interface under the influence of itself and the ambient atmosphere; in other words, surface waves in a certain wavelength range are unstable. With the temporal and spatial development, the surface amplitude increases eventually, leading to a breakup of the jet flow.

In the study of jet atomization forms, the main categories are divided into circular jets [24,25,26] and liquid film jets, according to the spray pattern resulting from the shape of the nozzle. The most common jet film shape in airless spraying belongs to the fan-shaped liquid film in liquid film jets. Therefore, a historical overview of the theoretical study of liquid film jets is mainly outlined here.

Liquid film jets are formed when a high-pressure liquid is instantaneously ejected through a slit. After the ejection of the liquid, the amplitude of the surface vibration wave of the liquid film generated by the airflow disturbance increases to a certain extent, such that the top of the jet loses its stability and breaks up into liquid lines, forming the initial atomization. As the liquid continues to move forward in time and space, the size of the initially atomized liquid line exceeds the critical size value for maintaining the steady state, generating secondary atomization and further breakup into microdroplets until the next steady state is reached.

According to the different shapes, they can be roughly classified into planar liquid films, fan-shaped liquid films and annular liquid films, as shown in Figure 3. York et al. [27] investigated the breakage mechanism of planar liquid films, pointing out that liquid film breakage is affected by the wavelength and frequency of the liquid film surface wave, the surface tension and viscosity of the liquid film, the flow rate of ambient atmosphere, the density of the gas or liquid, etc., and concluded that the increase in the amplitude of the surface wave is an important reason for the breakage of the liquid film.

In the early research of fan-shaped liquid film jet atomization, Squire [30] proposed the liquid film stability theory. Subsequently, Hagerty et al. [31] further investigated the stability of liquid films, indicating the symmetric and asymmetric waves of the liquid film, and concluded that the asymmetric waves were responsible for the fragmentation of the liquid film. Fraser et al. [32] found that fragments of liquid are broken off the wavy sheet, which then suffers continuous disintegration by air action, tending to contract into unstable ligaments, and that the mode of disintegration is critically dependent upon the ambient density. The size of droplets broken by surface wave shows variability, and the size distribution of droplets broken by liquid film perforation displays homogeneousness, while the direction of the droplet movement broken by edge stripping remains stable. What is in agreement between Dombrowski [33] and Fraser is that they considered the perforation of the liquid film to be the main reason for the formation of liquid ligaments, and concluded that the higher the surface tension and viscosity, the more difficult the liquid film is to break up, and that the density of the liquid has little effect on the disintegration. The higher the surface tension and viscosity, the more difficult it is to break the film, and the density of the liquid has little effect on breakage. Cao et al. [34,35] utilized the linear instability theory to explain the atomization of a viscous liquid when it is injected into a compressible gas atmosphere.

The aforementioned reports on the dispersion and atomization of jets are given in order to solve the problem of jet stability. Accordingly, starting from the basic control equations and boundary conditions of the jet on this basis, the dispersion relation for the development of small perturbations in the free liquid surface can be inferred using the Normal Model Method, and through theoretical and numerical analyses, explanations of the mechanism and mechanical causes of the jet crushing and atomization are derived. Nevertheless, none of these theories can explain the cause of jet production satisfactorily and independently, and the theories even contradict each other. Despite the underdevelopment of the liquid surface wave instability breakup mechanism, it is the most accepted atomization theory and the mainstream research direction for current jet atomization. Furthermore, coatings are characterized by a variety of features, such as viscosity, volatility and complexity of composition compared to simple liquid jets, which affect the atomization characteristics, and the physicochemical properties of paints also must be taken into account in future atomization studies.

2.2. Experimental Study of Atomization in Airless Spraying

In earlier studies, atomized flow field data was measured by an MgO method [36,37]; however, the measurement results are poor and only a small amount of particle size data of spray can be obtained at a time. Teng et al. [38] investigated the airless spraying atomization characteristics at various pressures using the MgO indentation method. In order to obtain more spray information without affecting the atomized flow field, a series of optical measurement techniques, such as Digital Holographic Microscopy [39], Particle Image Velocimetry [40] and Planar Laser-induced Fluorescence [41], have been applied to the field of atomization research. Subsequently, the droplet concentration in the region of the flow field far from the nozzle, the velocity of the paint droplets and the size of the spray particles generated by the rotary atomizer were successfully obtained [42,43,44,45]. However, owing to the limitations of optical measurements and the complexity of the near-nozzle region, it was still difficult to dive into the near-nozzle region atomization process in a high-speed spray flow field represented by airless spraying. Based on experimentally measured spraying parameters, a series of mathematical models describing the size and velocity of atomized droplets utilizing Stochastic Resonance (SR) matrices similarly allow for simple analysis of the spraying parameters with respect to the atomization effect [46,47,48,49]. However, matching of the surface tension, coating viscosity, etc., with the SR matrixes is not considered, and is not of universality. By setting up a high-speed digital camera visualization system, Naz et al. [50,51,52] visualized and compared the airless, full-cone and hollow-cone jet patterns produced by nozzles with different outlet diameters, and obtained data on the jet dynamics and vortex clouds formation process during atomization of liquids at high temperatures and pressures.

3. Historical Overview of Paint Film Formation Properties and Mechanisms

3.1. Mechanism Study of Paint Film Formation in Airless Spraying

As a subsequent step to the atomization process, the film formation mechanism of sprays has also evolved. Paint film formation can be categorized as spray transfer and droplet deposition. The atomized paint undergoes spatial transfer in the flow field, where the impact with the workpiece and paint deposition occurs at the near-wall surface. Some small droplets drift away from the wall with the flow of the air phase. The spray transfer process can be essentially described as a turbulent flow in which the momentum and mass are transferred to each other in gas–liquid phases. The modes of paint droplet impact on the surface of the workpiece are summarized into the adhesion mode, rebound mode, spread mode and splash mode [53], which are related to the velocity and incidence angles of the droplets, as well as the roughness, temperature and surface moisture of the workpiece. The four impact modes of the droplets are shown in Figure 4. Additionally, the large pressure gradient of the droplets at the wall results in the boundary layer flow. Inside the boundary layer, the fluid immediately adjacent to the wall is completely adhered to the object surface, which is a viscous flow. When outside the boundary layer, the velocity gradient is very small, the viscous force can be ignored, and the flow can be regarded as a non-viscous or ideal flow. In the case of a high Reynolds number flow, the Navier–Stokes equations can be simplified to the boundary layer equations, owing to the rather thin boundary layer, based on the order of magnitude comparisons of scales and rates of change in velocity. In terms of film formation evenness, there is one more factor worth noting: the more a paint wets a surface, the more the amplitude of evenness is increased.

McCarthy [54] and Mirko [55], respectively, found that paint droplets with low surface tension and high viscosity impacting a wall hardly rebounded or splashed and, in some special cases, it can be simply assumed that deposition occurs whenever a paint droplet impacts the target surface, and the thickness of the coating film is only related to the velocity of the near-wall droplets perpendicular to the wall. Santon [56] and O’Rourke [57] proposed a wall-film model that can better predict the process of liquid film formation when a droplet impacts the wall.

3.2. Experimental Study of Paint Film Formation in Airless Spraying

Domnick et al. [58], for the first time, utilized a Phase Doppler Anemometer (PDA) to measure the velocity and droplet diameter during spraying, and explored the effects of these factors on the film thickness distribution by varying the air flow rate, spraying distance, and coating velocity. Similarly, Ye et al. [59] used a PDA and a Spraytec particle sizer based on laser diffraction to measure and calculate the spray velocity distribution by changing the initial and boundary conditions, thereby obtaining the localized coating film thickness on the workpiece surface. Xu [60] carried out an experimental study on the relationship between the spray pressure and the spray mass flow rate of an airless spray gun, and improved the coating film quality by gun trajectory optimization. Plesniak et al. [61] first summarized the TE of airless spraying with the correlation as a function of the actual spray momentum rate (SMR) and Sauter mean diameter (SMD), and Teng et al. [62] likewise reported the main influencing factors and the law of the paint transfer efficiency of airless spray using the electronic weighing method. Chen et al. [63] investigated the relationship between the TE and the curvature of the sprayed surface, and performed trajectory planning for the spraying gun. The TE of more paint spray technology was reviewed by Poozesh et al. [64] a few years ago.

4. CFD Numerical Simulation in Airless Spraying

With in-depth studies of the paint atomization and film formation process, the disadvantages of using experimental methods are gradually revealed. Firstly, expensive and complex optical experiments are not suitable for atomization flow field research with high data requirements. Additionally, the resolution of optical instruments is limited, which makes it difficult to obtain accurate instantaneous spray flow field data. In contrast, the CFD theory has been increasingly developed, and corresponding supporting hardware has been continuously improved, enabling CFD numerical simulations as up-and-coming candidates for the study of airless spraying, owing to their advantages of low costs, comprehensive analytical skills, and a wide application range.

4.1. Classification of Numerical Simulation Models

As mentioned in the introduction part, the airless spraying process is essentially a movement process in multi-phase and turbulent flow. Consequently, the modeling of airless spraying mainly includes multiphase flow models and turbulent flow models.

4.1.1. Modeling of Multiphase Flow

Multiphase flow modeling methods suitable for the airless spraying process have been provided in CFD. In multiphase flow, a general transport equation may be written as

$$\frac{\partial \left(\alpha \rho \phi \right)}{\partial t}+\nabla ·\left(\alpha \rho \overrightarrow{V}\phi \right)=\nabla ·\stackrel{=}{\tau }+S_{\phi }$$

(1)

where $\phi$ is a phase variable, $\alpha$ is the phase volume fraction, $\rho$ is the mixture phase density, $\overrightarrow{V}$ is the phase velocity, $\stackrel{=}{\tau }$ is the diffusion term, and $S_{\phi }$ is the source term.

In the atomization process, the multiphase flow models are customarily classified into the simple engineering model method, the direct numerical simulation (DNS)-like method and the multi-scale simulation method [65]. The simple engineering model method refers to the empirical or semi-empirical numerical model. The advantages of these models are that they are less computationally intensive and have the convenience of calculation coupled with the spray flow field. However, its oversimplification results in poor generalizability and accuracy, and human factors have a great impact on the calculation results. In the DNS-like method, the phase interface between the gas and liquid is directly identified using interface capture technology, which is of great significance to study the spray mechanism [66,67]. This method is distinguished from the DNS method, since the simulations such as interface identification and turbulence are still closed with models. The DNS-like method can be categorized into the Lagrange tracing method and the Euler tracing method, based on different interface identification methods. The former is aimed at discrete phase particle motion, which can obtain detailed information such as the average velocity and trajectory of a single mass point, but requires higher computer numeracy. The latter identifies the overall characteristics of the continuous phase from the perspective of the entire flow field using time-averaged Navier–Stokes (N-S) equations. The multi-scale simulation method was created to compensate for the deficiency of the simple engineering model method and the DNS-like method. The large-scale liquid micelle was simulated by the Euler tracing method, which can determine the fine motion process of large fluid micelles. In contrast, the small-scale droplets were tracked by the Lagrange method, which thus greatly reduced the calculations.

While studying the coating film formation process, two approaches that belong to the DNS-like method are the most commonly used, and are summarized as the Euler–Lagrange approach [68,69,70,71,72] and the Euler–Euler approach [73,74]. The gas phase is described as a continuous phase and the paint droplets as a discrete phase in the Euler–Lagrange approach, while both the droplets and the air are considered to co-exist and interpenetrate in space and are considered as a continuous phase in the Euler–Euler approach. A more rigorous discussion of the advantages and limitations of the Euler–Euler approach and the Euler–Lagrange approach is elaborated in Section 6.1.

4.1.2. Modeling of Turbulence

Modeling of turbulence is another difficulty in airless spraying research. Existing numerical simulation methods for turbulence include direct numerical simulation (DNS), Large eddy simulation (LES), and Reynolds-averaged Navier–Stokes simulation (RANS). DNS is the simplest method to deal with turbulence, and instead of modeling the turbulence, the N-S equations for the gas–liquid phase flow are directly solved by numerical calculations. There is no doubt that this method requires an excellent computer hardware support and long computation time, and cannot be practically applied to the simulation of complex turbulent motions yet. LES divides the turbulence into large-scale and small-scale eddies by a spatially filtering function and treats the large eddies more exactly than the small ones. It is a good compromise between the low-accuracy RANS and computationally expensive DNS, while computationally incidental to numerical dissipation. RANS is the most widely applied numerical simulation method for turbulence modeling in engineering. It models the flow by statistically averaging the mass, momentum, and energy transport equations, and solves the N-S equations by combining the unclosed turbulent stresses with the turbulent viscosity [75]. The more commonly used models included in RANS that are applicable to spray simulations are the Standard k-ε model, the Renormalization group (RNG) k-ε model and the Realizable k-ε model. A more rigorous discussion of the advantages and limitations of various turbulence models is elaborated on in Section 6.2.

4.2. Research on Atomization Characteristics and Mechanisms

In recent years, numerous researchers have investigated the spraying, jet or fuel atomization process using CFD numerical simulations. Under the background of paint applications in the automotive industry, Li et al. [76] and Andersson et al. [77] investigated the paint droplet particle size distribution in the initial atomized flow field using Taylor Analogy Breakup (TAB) model. Wang et al. [78] revealed the oscillation phenomenon of the nozzle’s internal phase interface and the breakup details of the external liquid sheet on the basis of a large eddy simulation. Moreover, the spatial distribution characteristics of droplet size was obtained through coupling the method of volume of fluid with the discrete phase model (VOF-DPM). Chen et al. [9] established an air spray atomization model based on a large eddy simulation and carried out 2D spray atomization simulations. In order to explore the initial atomization liquid film breaking mechanism of the centrifugal nozzles of a rocket motor, Xiang [79] carried out an numerical simulation to explore the initial atomization liquid film breaking mechanism of the centrifugal nozzle based on the VOF and LES method, and a comparison of the high-speed photographic images with the simulation results shows acceptable errors.

In their research on multi-scale atomization simulations, Chen et al. [8] developed a multi-scale paint atomization model based on the hybrid Euler–Lagrange approach, and the simulation results predicted the atomization flow field parameters, paint atomization shapes, and the changing process from paint to liquid droplets, which is basically consistent with the experimental data. An original multi-scale methodology [80] consisting of a core coupled Level-Set/Volume-of-Fluid method (CLSVOF) for accurate capturing of primary atomization, an adaptive mesh refinement technique (oct-tree AMR) to dynamically optimize the structured Cartesian mesh, and a particle tracking algorithm to capture droplet dynamics has been developed, which can reproduce the whole atomization process and allow a preliminary statistical spray analysis. Similarly, a CLSVOF method was used to capture the trends of the jet trajectory and primary breakup behavior of a liquid column, which revealed that the aerodynamic Weber number significantly influences the primary breakup behavior and the vortex development and morphology behind the liquid column [81].

In general, although researchers have extensively studied the atomization characteristics and mechanisms of air atomizers, centrifugal nozzles, etc., based on CFD numerical simulations, there is almost no research on that of airless sprayers. Admittedly, the pressure and velocity variations in the internal flow field of an airless spraying gun nozzle have been also studied by several groups with the help of CFD numerical simulations, which are of interest for extending the service life of the nozzles [82,83,84,85]. As a result, there is an urgent requirement to study the atomization characteristics and mechanisms of airless sprayers with the help of numerical simulations in order to broaden the research method.

4.3. Research on Film Formation Characteristics and Mechanisms

In the context of coating processes in automotive manufacturing, Pendar et al. [86] focused on reviewing recent findings conducted with improved modern methods in this subject area for higher sustainability and efficiency. Under the research background of marine coating processes, Ye et al. [29] investigated the effects of shoreline winds and spraying distance on the TE and film thickness distribution during airless spraying on ship painting, based on a Euler–Lagrange approach, as is shown in Figure 5. Two years later, this research group performed numerical simulations using CFD and experimental studies to investigate the flow field and droplet trajectories by focusing on the integral velocities and sizes of the droplets using laser Doppler anemometry and Spraytec Fraunhofer-type particle sizers, providing necessary information for understanding the painting process using airless and air-assisted systems [87]. They also presented numerical studies on spray painting processes using different atomizers (including airless sprayers), which found that the impacting droplet characteristics of the atomizers show significant influences on the properties of the forming paint films [88]. Similarly, based on the Eulerian–Lagrange approach, Li [89] and Yi et al. [90,91] modeled the airless spraying on large ship exterior panels, and investigated the effects of the spraying parameters and their liquid film thickness distribution. Selected simulation results are shown in Figure 6. The studies showed that the lower the impact kinetic energy of the droplet when it reaches the surface of the outer plate of the ship, the higher the transfer rate of the coating; and the higher the deposition rate of the atomized particles of the coating spraying adhesion, the better the overall spraying effect. Wang [92] studied the influence of atomization pressure and orifice width on the overall atomization effect of the coating with the help of the Flat-Fan Atomization model in DPM, and explored the influence of spraying parameters on the film quality with the help of experimental methods.

Although the Euler–Lagrange approach is more recommended for paint phase volume fractions below 12% [88,89] when spraying, the modeling of spraying based on the Euler–Euler approach is another novel method. As mentioned in Section 4.1.1, the gas phase and droplet phase are solved using a unified numerical method in Euler–Euler approach, which is less computationally intensive than the Euler–Lagrange one, showing greater applicability for the study of engineering problems.

It is worth mentioning that in ref. [87], they combined full region air flow simulations from the nozzle exit up to the target with experimentally obtained initial conditions for the droplet size distributions and the droplet velocities near the nozzle as a compromise, because the primary phase of liquid disintegration and droplet formation cannot be simulated. This was the main research scheme that most of the researchers used in those years, and even up to now [93,94,95,96]; namely, relevant parameters such as spray particle size and velocity were measured experimentally and then used as initial and boundary conditions for the film formation model. Instead, with the continuous progress of CFD technology, the future use of accomplishing the whole process of numerical simulation of airless spraying from the initial atomization to the adhesion of film formation will definitely be a more advanced, more efficient and more promising research program.

4.4. Challenges and Techno-Economic Analyses for Simulating Airless Spraying

CFD numerical simulations of airless spraying processes demonstrate exceptional versatility and convenience compared to experiments or empirical equations. Primarily, researchers save time and effort in assembling an experimental platform or formulating coatings, therefore cutting down on major expenditure. Furthermore, numerical simulations are highly reproducible, and the flow field information and coating film data of airless spraying under different conditions can be easily obtained by simply changing the boundary conditions or coating parameters.

But there is no room for undue optimism, given that current studies of CFD numerical simulations in spraying process are barely focused on airless spraying, especially in its atomization process. Consequently, in numerical simulation studies of airless spraying, there is no consensus on the multiphase flow models and turbulence models with good prediction effects, leading to difficulty in obtaining simulation results with high accuracy. Therefore, there is a strong need to carry out comprehensive and persistent CFD numerical simulation studies on the atomization and film-formation characteristics of airless spraying to provide theoretical references for the construction process.

5. Numerical Simulations of Film Formation Characteristics on Complex Surfaces

This section will focus on outlining the numerical simulation studies conducted by researchers for various spray technologies, on complex surfaces, from historical to current times.

Before the theory and technology of CFD numerical simulations were developed, researchers mainly utilized explicit deposition models [97,98,99,100] to simulate film thickness in the early days, but a few problems such as oversimplification of the actual situation and low variability of spraying parameters were exposed. What is worse is that the explicit deposition models were developed based on planar spraying, and this made it difficult to adapt it to the study of film formation properties on complex surfaces.

Therefore, some attempts have been made to subdivide simple surfaces into small pieces (i.e., a differential method [101]), where each piece is considered as a plane, but this method is only suitable for predicting the film thickness of simple large-area surfaces with small curvature. Others have also tried to model the non-planar deposition models (i.e., a projection method [102]) to get the film thickness at a point of the surface, while the results are not satisfactory. The film formation model based on explicit deposition models did not take the influence of workpiece geometry on the spray flow field into account, which resulted in the irrational trajectory planning of robotic spraying on complex surfaces.

In view of the poor spraying effects of complex surfaces in engineering applications, researchers have carried out systematic studies of the film formation characteristics of various types of surfaces based on CFD simulations. Ye et al. [103,104] calculated the coating thickness on the curved surface of an automobile body by numerical simulation, and Osman et al. [105,106] discussed the factors affecting the thickness of coatings near a protrusion or indentation on a planar surface for electrostatic spraying, but the effect of surface geometric properties on the coating thickness has not been investigated in their research. The film thickness at the seam of this surface formed by joining two planes (or, as it is called, a V-shaped surface) has been extensively studied. Domnick et al. [42,107] simulated static and dynamic spraying on a V-shaped surface at the rear of a car. Ye et al. [108] carried out a numerical simulation of spray film formation with variable gun pressure on a right-angled slot surface of a car body. Chen et al. [109,110] also studied the film formation characteristics of V-shaped surface spraying with different connecting angles based on the VOF model, and the simulation results obtained were in good agreement with the coating thickness obtained from the experiments.

Since round tubes or rounded surfaces are widely available for spraying construction, Chen et al. [111,112,113] used the Eulerian–Eulerian approach to establish a static and a dynamic spray film formation model, respectively, for arc surfaces consisting of a spray flow field model and a droplet deposition model, and concluded the thickness distribution characteristics of the coating film and the spray flow field characteristics of circular surface spraying. To optimize anti-corrosion spraying process of petroleum pipelines, Yang et al. [114] modeled the airless spraying of arc surfaces based on the Eulerian–Eulerian approach, and analyzed the film formation characteristics of the inner and outer surfaces of circular curved surfaces under static, rotational and translational spraying conditions. Similar studies have been carried out by other researchers, which have contributed to the study of the film formation properties of arc surfaces [115,116,117].

In the study of the spraying of complex shapes with greater curvatures, Chen et al. [118] established a film formation model for spherical surface spraying based on Euler–Lagrange approach, and the numerical simulation results showed that the coating thickness of spherical surface was thinner, but the coating uniformity was better, than that of the plane. Numerical simulations conducted by Poddar et al. [119] on liquid film formation around tubes of horizontal falling film evaporators with different pitches found that the workpiece geometry parameters and spray flow parameters display a large influence on the film thickness. Additionally, numerical simulations of the spray film formation process was carried out for surfaces with Gaussian curvature and mean curvature of less than zero, represented by saddle ridge surfaces [120]. Figure 7 displays the selected simulation results of the study of the typical complex surface spraying.

After the comprehensive overview of the literature, it was found that researchers have carried out a series of valuable studies on the film formation properties of complex surfaces. However, it is apparent that most of the research is still only for pneumatic atomizers. Moreover, the characteristics of spray flow fields and coating films during airless spraying have not been thoroughly studied from the perspective of the essential mechanism of the spray film formation. All in all, numerical simulations of airless spraying for spray flow characteristics and film formation mechanisms still need to be greatly developed, and it is also a research direction in which researchers in related fields should make strong efforts towards.

6. Discussion on Numerical Models for Airless Spraying Simulation

6.1. Discussion on Multiphase Flow Models

In the choice of multiphase flow modeling, the DNS-like method is the most commonly used option, as its well-developed and most validated theory. By reviewing the literature, it was found that that the multi-scale simulation method is more suitable for modeling the atomization process [121]. Continued in-depth analysis reveals that the Euler–Euler approach and the Euler–Lagrange approach achieved the widest acceptance. In this way, which approach is better for modeling airless spraying?

For model comparison, airless spraying film formation characteristics based on the Euler–Euler approach and the Euler–Lagrange approach, respectively, investigated by the same research group are listed here [114,122], as is shown in Figure 8. With the same number of meshes and parameter settings, the model based on the Euler–Euler approach only took half the time of the other, and achieved the same level of solution accuracy. In addition, the simulation results based on the Euler–Lagrange approach show a larger dispersion angle in the short-axis direction compared to the spraying experiments. On the contrary, Li [89] and Yi et al. [90,91], who are also from the same research group, considered it more suitable for modeling by the Euler–Lagrange approach, owing to the low volume fraction of the droplet phase. Moreover, this method can better conform to the particle-loaded flow model and better simulate the impingement behavior of paint droplets on the wall, which cannot be achieved by the Euler–Euler approach.

Admittedly, the Euler–Lagrange approach combines the Eulerian method for simulating the dense spray region near the nozzle and the Lagrange method for simulating the sparse spray region downstream of the nozzle, and can be used to simulate turbulent sprays with a high Reynolds number and a high Weber number. However, it requires a higher mesh quality and a longer computation time. This approach is made considerably simpler when particle–particle interactions can be neglected, and this requires that the dispersed second phase occupies a low volume fraction, even though high mass loading is acceptable. The particle or droplet trajectories are computed individually at specified intervals during the fluid phase calculation. This makes the model appropriate for the modeling of spray atomizers and some particle-laden flows, but inappropriate for the modeling of liquid–liquid mixtures, fluidized beds, or any application where the volume fraction of the second phase cannot be neglected.

Instead, in the Euler–Euler approach the different phases are treated mathematically as interpenetrating continua. Since the volume of a phase cannot be occupied by the other phases, the concept of phasic volume fraction is introduced. These volume fractions are assumed to be continuous functions of space and time, and their sum is equal to one. Conservation equations for each phase are derived to obtain a set of equations that have a similar structure for all phases. These equations are closed by providing the constitutive relations that are obtained from empirical information, or, in the case of granular flows, by the application of kinetic theory. The VOF model in the Euler–Euler approach, as a surface-tracking technique applied to a fixed Eulerian mesh, is designed for two or more immiscible fluids where the position of the interface between the fluids is of interest. It shows application advantages in the prediction of jet breakup, and the steady or transient tracking of any liquid–gas interface. Therefore, it is particularly suitable for modeling the atomization process of airless spraying. The Euler–Euler approach better describes the turbulent mixing process of the paint in the air flow, and detailed information on the spatial distribution of the droplet phase can be given, which is less computationally intensive than the Euler–Lagrange approach. Nonetheless, this approach fails to describe the trajectory and shape of the droplets, and simplifies the behavior of the particles impinging on the walls. In addition, the Eulerian model in the Euler–Euler approach, which is computationally unstable and prone to divergence, requires a higher mesh quality.

6.2. Discussion on Turbulence Models

In the choice of turbulence modeling, even though the development of mathematical models of turbulence is approaching a century, the advantages, disadvantages and applicability of turbulence models are the focus of researchers’ attention and a source of difficulties. Compared to the laminar flow, turbulence is still too poorly understood. No turbulence model can be universally accepted as optimal for all types of spraying predictions. The selection of a turbulence model depends on factors such as the physical properties of the flow, the required computational accuracy, the available computational resources, and time, of course. Thus, the most appropriate turbulence model needs to be selected according to a particular application scenario.

A previously mentioned study [9] selected large eddy simulations for spray turbulence prediction. However, it are found that its computational volume rises sharply with an increase in the number of paint droplets, and thus they considered it not suitable for carrying out the paint atomization simulation with variable parameters. The RANS method is the most widely used in spraying simulations. Both the Realizable k-ε and RNG k-ε models have shown substantial improvements over the Standard k-ε model, where the flow features include strong streamline curvature, vortices, and rotation. The Spalart–Allmaras (S-A) model, the k-ω model, and the Reynolds Stress Model (RSM) in Fluent, which have not been mentioned in preceding sections, belong to this modeling approach. It is the statistical averaging of the control equations that makes it unnecessary to calculate turbulent pulsations at all scales, thus reducing spatial and temporal resolution and computational effort.

Compared to the standard k-ε model, the RNG model adds an additional term to its equation and the effect of swirl on turbulence that was considered in the RNG, and improves the accuracy for rapidly strained flows and swirling flows, respectively. In the k-ω models, production terms have been added to both the k and ω equations, which have improved the accuracy of the model for predicting the free shear flows. Indeed, the main disadvantage is that RANS can only provide average information on turbulence, and the greatest weakness is that it fails to feature universality. Specifically speaking, the k-ε model is insensitive to adverse pressure gradients and boundary layer separations, and the standard k-w model is more sensitive to free streams.

Guan et al. [123] conducted a comparison study of turbulence models in numerical simulations, including the S-A model, standard k-ε model, RNG k-ε model, Realizable k-ε model, standard k-ω model, SST k-ω model and RSM model, for multiple jet impinging injectors. It was found that the RNG k-ε model overpredicts the decay of the main jet, while the standard k-ω model underpredicts the decay of the main jet. Consequently, it is recommended to use the Realizable k-ε model to simulate the turbulent flow of multiple jet impinging injectors. A study [10] also showed that the distribution of particles in the flow field is more homogeneous and more consistent with the shape of the spray when the Realizable k-ε model is applied to calculate the gas-phase flow field, but the simulation results that were predicted by the RNG obtained a more consistent velocity decay with the experimental results. Hu et al. [124] simulated the ax-symmetric submerged turbulent gas-jet in infinite space with three k-ε turbulent models, and the results show that the simulation of the standard k-ε model is the most realistic.

7. Conclusions and Suggestions

To sum up, this paper offers an in-depth exploration of the mechanism, numerical simulation and experimental research into the airless spraying of atomizing and film formation processes, highlighting the critical importance of CFD numerical simulations for investigating spray flow field characteristics and coating film quality on complex surfaces. The following conclusions and suggestions can be drawn:

• The complexity of the measuring means and expensiveness of measuring equipment makes it difficult to study the spraying process experimentally, so numerical simulations of airless spraying are a much more straightforward and feasible approach. From historical perspectives to current applications, CFD numerical simulations have demonstrated exceptional versatility and convenience, compared with experiments or empirical equations, and will continue to be a focus of research in this field.
• Numerical simulations have evolved to the point where numerous models for multiphase flows, turbulence, and wall-impingement have been proposed to describe these behaviors. The accuracy of the spraying simulation of the same model for different processes shows variation; so does the accuracy of the spraying simulation of the same process predicted by different models. Overall, all multiphase flow and turbulence modeling approaches feature their advantages and disadvantages, and the researcher should choose an appropriate numerical model according to the focus of their research.
• Current studies of CFD numerical simulations are mainly focused on pneumatic atomizer and electrostatic rotary bell atomizers, whereas little research has been performed on airless spraying, especially its atomization process. Due to the difference in the atomization principle of the paint, the paint flow and deposition characteristics show variability, leading to a large gap in the accuracy and reliability of the various multiphase flow models or turbulence models in the numerical simulation during the spraying processes. Therefore, there is a strong need to carry out CFD numerical simulation studies on the atomization and film formation characteristics of airless spraying to provide theoretical references for the construction process.
• At present, more numerical simulation studies are focused on static spraying, or simply extend to constant parameter uniform-speed dynamic spraying research, lacking in-depth research on variable trajectory and variable parameter spraying, which is exactly what is needed in the actual spraying processes. All in all, it is necessary to conduct systematic studies of dynamic variable parameter airless spraying on complex surfaces, thereby solving the problem of robot spraying trajectory planning and film quality control.
• With the continuous progress of CFD technology, the future use of accomplishing the whole process numerical simulation of airless spraying from the initial atomization and spray transfer diffusion to the adhesion of film formation will definitely be a more advanced, more efficient and more promising research program.