Study of Internal Flow in Open-End and Closed Pressure-Swirl Atomizers with Variation of Geometrical Parameters

Abstract

This study delves into the examination of internal flow characteristics within closed (with nozzles) and open-end pressure-swirl atomizers (lacking nozzles). The number of inlet channels “n” and the opening parameter “C” were manipulated in this study, as they play a pivotal role in understanding various atomizer attributes, such as uniformity of the air-core diameter, the discharge coefficient, spray angle, and more, all of which hold significance in the design of bipropellant atomizers for liquid rocket engines (LREs). To validate our findings, six distinct hexahedral meshes were generated using Ansys ICEM software 2023. Subsequently, we employed Ansys Fluent, considering the RNG k-ε turbulence model and the VOF (volume-of-fluid) multiphase model to identify the liquid–gas interface, to aid in analyzing the uniformity of the air core, which is directly linked to the even distribution of mass, the mixing ratio of propellants, combustion efficiency, and stability. The results indicate that the uniformity of the air core is not solely contingent on an increase in parameter “n” but is also influenced by an increase in the parameter “C”. It is worth noting that the key dimensions of these six atomizers were determined using a mathematical model based on Abramovich and Kliachko theories.

1. Introduction

Pressure-swirl atomizers find widespread application in power generation, gas turbines, and rocket engines [1]. Typically, these atomizers consist of tangential inlet channels, a chamber for swirl generation, a nozzle, and a discharge orifice [2]. The distinguishing feature of a pressure-swirl atomizer is its ability to induce a centrifugal motion within a fluid. This fluid is introduced through periodically distributed channels around the swirl chamber (Figure 1a), resulting in the disruption of the liquid film. The expelled liquid takes on the form of a cone-shaped spray (Figure 1b).

Pressure-swirl atomizers come in various types, categorized based on the angular incidence of their inlet channels (tangential, helical, or conical) [3] and according to nozzle-opening parameter “C”. Pressure-swirl atomizers are classified into two main categories: “open-end” and “closed” [4] (see Figure 2), where the parameter “C” relates the radius of the swirl chamber to the radius of the nozzle outlet orifice (see Equation (1)). In open-end pressure-swirl atomizers, the radius of the swirl chamber (Rs) is equal to the radius of the exit orifice (Rs = ro, where C = 1) [5]. In contrast, the closed atomizer always has a swirl-chamber radius greater than the outlet-orifice radius (Rs > ro, where C > 1).

It is important to note that the variation of the parameter “C” is directly linked to the formation of vortices within the swirl chamber. Consequently, this variation influences other geometric parameters such as the radius of the “air core” (the cylindrical vacuum in Figure 1b), the spray angle (2α), and the film flow area coefficient “φ” (see Equation (2)). As a result, the geometry of the atomizer directly impacts the behavior of the liquid film, with the liquid film diminishing as the pressure drop increases [6].

The bipropellant pressure-swirl atomizer RD-0110 incorporates both types of atomizers, open-end and closed, arranged concentrically to ensure the intersection of the fuel and oxidant spray cones at the outlets of both atomizers (see Figure 3). The utilization of pressure-swirl atomizers played a pivotal role in addressing the combustion-stability challenges in the RD-0110 rocket engine, which were primarily linked to the interaction between the spray cones of the oxidant and fuel atomizers [7].

Considering the categorization of pressure-swirl atomizers based on their geometric characteristics, we analyzed the internal flow behavior while varying the nozzle-opening parameter “C” and the number of inlet tangential channels “n”. This analysis aims to determine their optimal design for specific applications within the combustion chambers of rocket engines.

$$\mathrm{C}=\frac{{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{r}}_{\mathrm{o}}}$$

(1)

$$\mathsf{\phi }=1-\frac{{\mathrm{r}}_{\mathrm{a}}^2}{{\mathrm{r}}_{\mathrm{o}}^2}$$

(2)

2. Mathematical Model for the Internal Flow

2.1. Assuming Ideal Flow

In this study, we utilized the Rivas mathematical model [8], which was employed in the context of a conical pressure-swirl atomizer. This model yielded reasonably accurate results for determining various geometric parameters. It is important to note that this user-friendly model can also be adapted for atomizers with tangential inlet channels by adjusting the angles of incidence for these channels on the swirl chamber. In this adaptation, the helix angle “ψ” [9] is set to 0°, ensuring that it lies on a vertical plane parallel to the xz plane, while the swirl angle “β” is set to 90° and lies on the xy plane (see Figure 4).

It is worth mentioning that this model draws its foundations from the Abramovich theory [10,11], which assumes that the liquid is incompressible and frictionless (μ = 0), and the Kliachko theory [12], which considers the viscosity of the liquid. By applying Abramovich’s postulate, the Navier–Stokes equations are simplified to Bernoulli’s equation (see Equation (3)):

$$\frac{u^2}{2}+\frac{p}{\rho }+gz=cte$$

(3)

By applying Bernoulli’s equation (Equation (4)) at the radial position “r_a” which is precisely at the interface between the liquid and gas, the radial component becomes null when we assume constant axial velocity [13]. Following this, applying the equation of angular momentum (Equation (5)), we can establish a relationship for the tangential velocity within the air core, denoted as ${\mathrm{W}}_{{\mathrm{\theta }}_{{\mathrm{r}}_{\mathrm{a}}}}$, in relation to the tangential velocity at the top of the swirl chamber, denoted as (${\mathrm{W}}_{{\mathrm{\theta }}_{\mathrm{i}\mathrm{n}\mathrm{j}}}$):

$$\frac{2\mathsf{\Delta }\mathrm{P}}{\mathsf{\rho }}={\mathrm{W}}_{{\mathrm{z}}_{{\mathrm{r}}_{\mathrm{a}}}}^2+{\mathrm{W}}_{{\mathsf{\theta }}_{{\mathrm{r}}_{\mathrm{a}}}}^2$$

(4)

$${\mathrm{W}}_{{\mathsf{\theta }}_{{\mathrm{r}}_{\mathrm{a}}}}.{\mathrm{r}}_{\mathrm{a}}={\mathrm{W}}_{{\mathsf{\theta }}_{\mathrm{i}\mathrm{n}\mathrm{j}}}.{\mathrm{R}}_{\mathrm{i}\mathrm{n}\mathrm{j}}$$

(5)

Furthermore, we can establish a connection between the pressure drop, the radius of the outlet orifice, and the mass flow rate through the discharge coefficient (Equation (6)). This can be conveniently organized in terms of the film flow area coefficient, denoted as φ. By applying the principle of maximum mass flow rate (dCd/dφ.), we derive the geometric parameter of the conical pressure-swirl atomizer, denoted as “A_c” [8], as shown in Equation (7):

$${\mathrm{C}}_{\mathrm{d}}=\frac{\dot{\mathrm{m}}}{\mathsf{\pi }{\mathrm{r}}_{\mathrm{o}}^2\sqrt{2\mathsf{\rho }.\mathsf{\Delta }\mathrm{P}}}$$

(6)

$${\mathrm{A}}_{\mathrm{c}}=\frac{\mathsf{\pi }.{\mathrm{r}}_{\mathrm{o}}.{\mathrm{R}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}{\mathrm{n}.{\mathrm{f}}_{\mathrm{p}}}\cos \mathsf{\psi }.\sin \mathsf{\beta }$$

(7)

Subsequently, in cases where the inlet channels are tangentially distributed around the swirl chamber, the geometric parameter of the conical pressure-swirl atomizer, denoted as “A_c”, simplifies to the Abramovich number “A” (Equation (8)). In this configuration, the angles “ψ” and “β” are set to 0° and 90°, respectively (see Figure 4).

It should be noted than “A” is a non-dimensional geometrical characteristic parameter of the atomizer or “Swirl number”, which is the ratio of angular momentum to axial momentum [14]. This parameter relates the main dimensions of the atomizer and is therefore essential in the design and dynamics of pressure-swirl atomizers and their use as a means of suppressing various mechanisms of high-frequency instabilities [15] in the combustion chambers of rocket engines:

$$\mathrm{A}=\frac{\mathsf{\pi }.{\mathrm{r}}_{\mathrm{o}}.{\mathrm{R}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}{\mathrm{n}.{\mathrm{f}}_{\mathrm{p}}}=\frac{\sqrt{2}\left(1-\mathsf{\phi }\right)}{\mathsf{\phi }\sqrt{\mathsf{\phi }}}$$

(8)

Additionally, various parameters including the discharge coefficient and the half-angle of the spray cone can be determined as functions of the film flow coefficient “φ”. These relationships are given by Equations (9) and (10), respectively [16]:

$$\mathrm{C}\mathrm{d}=\sqrt{\frac{{\mathsf{\phi }}^3}{2-\mathsf{\phi }}}$$

(9)

$$\sin \mathsf{\alpha }\cong \frac{2\sqrt{2}\left(1-\mathsf{\phi }\right)}{\left(1+\sqrt{1-\mathsf{\phi }}\right)\sqrt{2-\mathsf{\phi }}}$$

(10)

2.2. Assuming Losses for Liquid Viscosity

Next, taking into account the viscosity of the liquid, losses in angular momentum (K), and hydraulic losses (ξ_tot) in the inlet channels (where, ξ_tot = ξ_inj + ξ_λ) [15], both the modified Bernoulli’s equation and the equivalent discharge coefficient are adjusted to Equations (11) and (12), respectively.

Here, ξ_inj represents the loss coefficient in the inlet channel geometry, which is a function of the tilt angle “γ” ($\mathsf{\gamma }=\arctan (\frac{{\mathrm{l}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}{{\mathrm{R}}_{\mathrm{s}}})$) and can be determined from the data presented in Figure 5.

On the other hand, ξ_λ accounts for the losses due to flow friction along the channel walls (Equation (13)) and is connected to the Blasius friction coefficient (λ = 0.3164 Re^−0.25) [15]:

$${\mathrm{U}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\sqrt{\frac{2\mathsf{\Delta }\mathrm{p}}{\mathsf{\rho }}}=\sqrt{{\mathrm{W}}_{{\mathrm{z}}_{\mathrm{a}}}^2+{\mathrm{W}}_{{\mathsf{\theta }}_{\mathrm{a}}}^2+{\mathsf{\xi }}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\mathrm{U}}_{\mathrm{i}\mathrm{n}}^2}$$

(11)

$$\mathrm{C}{\mathrm{d}}_{\mathrm{e}\mathrm{q}}=\frac{1}{\sqrt{\frac{1}{{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}^2}+\frac{{\left(\mathrm{A}\mathrm{K}\right)}^2}{1-{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}}+{\mathsf{\xi }}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\left(\frac{\mathrm{A}{\mathrm{r}}_{\mathrm{o}}}{{\mathrm{R}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}\right)}^2}}$$

(12)

$${\mathsf{\xi }}_{\mathsf{\lambda }}=\mathsf{\lambda }\frac{{\mathrm{l}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}{2{\mathrm{r}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}$$

(13)

In conclusion, by applying the Abramovich theory to a real flow (maximum flow), we derive the equivalent geometric parameter according to Kliachko, denoted as A_eq [18], expressed in terms of the equivalent film flow area coefficient “φ_eq” (Equation (14)). This leads to the determination of the equivalent discharge coefficient (Equation (15)) and the equivalent half-spray angle (Equation (16)) as functions of φ_eq [19]:

$${\mathrm{A}}_{\mathrm{e}\mathrm{q}}=\mathrm{A}\mathrm{K}=\frac{\sqrt{2}\left(1-{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}\right)}{{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}\sqrt{{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}}}$$

(14)

$$\mathrm{C}{\mathrm{d}}_{\mathrm{e}\mathrm{q}}=\frac{1}{\sqrt{\frac{2-{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}}{{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}^3}+{\mathsf{\xi }}_{\mathrm{t}\mathrm{o}\mathrm{t}}{\left(\frac{\mathrm{A}{\mathrm{r}}_{\mathrm{o}}}{{\mathrm{R}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}\right)}^2}}$$

(15)

$$\sin {\mathsf{\alpha }}_{\mathrm{e}\mathrm{q}}\cong \frac{2\mathrm{C}{\mathrm{d}}_{\mathrm{e}\mathrm{q}}{\mathrm{A}}_{\mathrm{e}\mathrm{q}}}{\left(1+\sqrt{1-{\mathsf{\phi }}_{\mathrm{e}\mathrm{q}}}\right)\sqrt{1-{\mathsf{\xi }}_{\mathrm{t}\mathrm{o}\mathrm{t}}\mathrm{C}{\mathrm{d}}_{\mathrm{e}\mathrm{q}}^2{\left(\frac{\mathrm{A}{\mathrm{r}}_{\mathrm{o}}}{{\mathrm{R}}_{\mathrm{i}\mathrm{n}\mathrm{j}}}\right)}^2}}$$

(16)

3. Numerical Simulation

To conduct the numerical simulations, six distinct geometries accompanied by their hexahedral meshes were created using ICEM CFD software 2023. Variations were made in the nozzle-opening parameter “C” and the number of inlet channels ”n”. Detailed information regarding these parameters and the principal dimensions is provided in

Table 1. Dimensions and characteristics of meshes.

Meshes	Cells and Nodes	Inlet Channel Diameter, 2rinj (mm)	Swirl-Chamber Diameter, 2Rs (mm)	Outlet Diameter, 2ro (mm)	Inlet Channel Length, linj (mm)
Mesh a (C = 1; n = 2)	5,118,846; 5,195,386	1.00	8.00	8.00	5.00
Mesh b (C = 1; n = 4)	4,022,457; 4,088,728	1.00	8.00	8.00	5.00
Mesh c (C = 1; n = 6)	5,115,184; 5,189,524	1.00	8.00	8.00	5.00
Mesh d (C = 2; n = 2)	5,027,856; 5,098,288	1.00	8.00	4.00	5.00
Mesh e (C = 2; n = 4)	4,567,680; 4,632,935	1.00	8.00	4.00	5.00
Mesh f (C = 2; n = 6)	5,407,524; 5,483,524	1.00	8.00	4.00	5.00

. Ensuring mesh independence, it is essential to consider the dimensionless wall distance, denoted as ‘y+’ (as per Equation (17)), to identify the appropriate region for addressing turbulence phenomena.

In Figure 6, the six meshes are illustrated (Figure 6a–f). Notably, a cylindrical region (Figure 6g) was introduced within the domain to aid in the development and visualization of the spray angle. Additionally, proper refinement of boundary layers on the walls was taken into consideration (Figure 6h).

It is noteworthy that all numerical simulations were performed utilizing a 12th Generation Intel^® Core™ i9-12900KF processor operating at 3.20 GHz with 64.0 GB of RAM.

$$y^{+}=\frac{\rho U_{\tau }y}{\mu }$$

(17)

Subsequently, while considering a range of [10,11,12,13,14,15,16,17,18,19,20,21,22] m/s for the free stream velocity (U_∞) and a hydraulic diameter for the inlet channels (D_h = 2r_inj = 1 mm for all geometries), the skin friction coefficient “C_f” can be determined using Equation (18) [20].

By calculating the wall shear stress (τ_w) and the velocity friction (U_τ) using Equations (19) and (20), respectively, and taking into account the nearest cell-to-wall distance (y = 0.0016 mm for all meshes), these values can be inserted into Equation (17). This gives us the range of y+ values depicted in Figure 7:

$${\mathrm{C}}_{\mathrm{f}}=0.078{\mathrm{R}\mathrm{e}}^{-0.25}$$

(18)

$${\tau }_{\mathrm{w}}=0.5{\mathrm{C}}_{\mathrm{f}}\mathsf{\rho }{\mathrm{U}}_{\mathrm{\infty }}^2$$

(19)

$${\mathrm{U}}_{\tau }=\sqrt{\frac{{\tau }_{\mathrm{w}}}{\mathsf{\rho }}}$$

(20)

As depicted in Figure 7, the range of y⁺ was determined to be (0.99; 1.98), which corresponds to the viscous sublayer region (indicated by the red box in Figure 8). In this region, it is advisable to use the “enhanced wall function” approach (y⁺ < 5) [21].

Using this criterion, convergence was successfully achieved employing the k-epsilon RNG model. It is noteworthy that the turbulence intensity (I = 0.16 Re^−1/8) [19] and the hydraulic diameter (D_h) were employed as a specification method, particularly for low Reynolds models (Re < 3 × 10⁶) [22].

The numerical simulation encompasses four different cases of inlet pressure ΔP (300, 350, 400, and 450 kPa) for each mesh. These pressure variations allow the determination of the inlet velocity, which, in turn, facilitates the calculation of the turbulence intensity (I), where Re represents the Reynolds number within the inlet channel.

In all cases, a no-slip condition was applied to the walls of the atomizers. The chosen turbulence model was RNG k-epsilon [23]. At the outlet of the atomizers, a gauge pressure of zero was assumed. The working fluid used in this simulation was water and, as it exhibits incompressible flow behavior, the continuity equations (Equation (21)) and momentum equations (Equation (22)) are independent. In this simulation, the SIMPLE algorithm, which stands for “Semi-implicit method for pressure-linked equations,” was utilized for correcting the pressure–velocity field [24].

$$\frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}+\nabla .\left(\mathsf{\rho }\mathrm{u}\right)=0$$

(21)

$$\frac{\partial }{\partial \mathrm{t}}\left(\mathrm{u}\right)+\nabla \left(\mathrm{u}\mathrm{u}\right)=\frac{1}{\mathsf{\rho }}\left[-\nabla \mathrm{p}+\frac{1}{3}\mathsf{\mu }\nabla \left(\nabla .\mathrm{u}\right)+\mathsf{\mu }{\nabla }^2\mathrm{u}\right]+\mathrm{f}$$

(22)

In the described equations:

“u” represents the velocity vector.

“μ” denotes the absolute viscosity of the liquid.

“f” represents the vector of volumetric forces.

“p” stands for pressure.

“ρ” is the density of the liquid.

For determining the location of the liquid/gas interface, the VOF HRIC (high-resolution interface capture) [25] multiphase model was employed. This model facilitates the calculation of the liquid film thickness, the air core, and the spray angle. The VOF model, which stands for “Volume of Fluid,” was initially proposed by Hirt and Nichols [26]. It operates under the assumption that the volume of one phase cannot be occupied by another. This model relies on the concept of volumetric phase fractions, where the sum of all the parts must add up to unity (Equation (23)), with “η” representing the volume fraction of the fluid:

$${\eta }_{liquid}+{\eta }_{gas}=1$$

(23)

4. Results

In the first part of the study, the results of the mathematical model are presented, assuming an ideal liquid where the atomizers operate at the same injection pressure as the design point (ΔP = 400 kPa). Using Equations (8)–(10), we can determine the geometric parameters as functions of the film flow area coefficient “φ” (as shown in

Table 2. Geometrical parameters obtained with the mathematical model assuming ideal liquid (for all cases: ΔP = 400 kPa).

Case	2α (°)	Mass Flow, (kg/s)	$\mathbf{Dimensionless} \mathbf{Mass} \mathbf{Flow}, \stackrel{-}{\mathit{m}}$	A	φ	Cd
a (C = 1; n = 2)	138	0.0340	0.358	27.85	0.125	0.032
b (C = 1; n = 4)	128	0.0630	0.663	14.38	0.186	0.059
c (C = 1; n = 6)	120	0.0950	1.000	9.18	0.239	0.088
d (C = 2; n = 2)	126	0.0240	0.253	12.78	0.199	0.066
e (C = 2; n = 4)	114	0.0400	0.421	6.78	0.282	0.114
f (C = 2; n = 6)	106	0.0540	0.568	4.68	0.341	0.155

).

The relationship between mass flow and injection pressure can be established using Equation (6). Consequently, the atomizer group’s maximum mass flow rate corresponds to geometry or case “c”, due to case “c” having the maximum outlet radius (r_o = R_s) and the largest number of inlet channels (n = 6).

To facilitate the analysis, we take the mass flow rate of case “c” as a reference. Then, we proceed to determine the dimensionless mass flow using Equation (24), where “${\dot{m}}_{case}$” represents the mass flow rate for each atomizer case:

$$\overline{m}=\frac{{\dot{m}}_{case}}{{\dot{m}}_{max}}$$

(24)

Figure 9 depicts the dimensionless mass flow ($\overline{m}$) as a function of the number of inlet channels “n” and the parameter “C”. It is evident that the highest mass flow among the six cases is attained when the atomizer is open (C = 1) and equipped with the maximum number of channels (nmax = 6). Furthermore, it is apparent that for closed atomizers (C > 1), the dimensionless mass flow tends to decrease.

Figure 10 displays the behavior of the film flow area coefficient (φ), which tends to increase with a higher number of inlet channels “n”. Importantly, it is worth noting that “φ” becomes even larger as the nozzle-opening parameter “C” increases. This implies that when C > 1, the radius of the air core “ r_a ” tends to increase, resulting in a reduction in the liquid film thickness.

In Figure 11, it is evident that when the parameter “C” increases (C > 1), the geometric parameter “A” tends to decrease. It can be confirmed that the reduction in “A” becomes more pronounced with an increase in the number of inlet channels “n”.

Figure 12 demonstrates that the spray angle decreases notably as the number of inlet channels increases. It can be concluded that, among this family of six atomizers, the design of the open-end atomizers exhibits a larger spray angle.

Indeed, as observed in Figure 11, when both the parameter “C” and the number of inlet channels “n” increase, the geometric parameter “A” tends to reach a minimum value. Referring to Figure 13, it is apparent that as “A” decreases, the discharge coefficient “Cd” and the coefficient “φ” increase. This indicates that when “A” approaches zero, the air core disappears (r_a = 0), and the film flow area coefficient reaches its maximum value (φ = 1), effectively causing the pressure-swirl atomizer to behave like a jet atomizer. In contrast, when the parameter “A” increases, the spray angle cone (2α) also increases.

By referencing Figure 13, and considering the information from Figure 9, Figure 10, Figure 11 and Figure 12, it becomes clear that these observations and trends are consistent with the equations of the non-viscous mathematical model used in this study.

In the next phase of this study, the mathematical model considers the viscosity of the liquid through Equations (11)–(16). It is important to note that the same injection pressure (ΔP = 400 kPa) is maintained for all six cases, and the respective mass flow rates for each of them, as determined in Table 2, are imposed.

Then, the nominal parameters (mass flow rate “$\dot{\mathrm{m}}$” and injection pressure “ΔP”), when related to the liquid viscosity (“Re” in the inlet channels), allow the determination of equivalent geometric parameters (A_eq, φ_eq, Cd_eq, and 2α_eq). These, when coupled with hydraulic losses “ ξ_tot ” and “K” (losses due to the reduction of angular movement), ultimately lead to the calculation of the primary dimensions of the atomizer (2r_inj, 2R_s, and 2r_o).

It is important to mention that the results shown in

Table 3. Equivalent parameters and final dimensions of atomizers, assuming liquid viscosity (for all cases ΔP = 400 kPa).

Case	Uin (m/s)	Rein	Aeq	φeq	Cdeq	K	ξtot	2αeq	2rinj (mm)	2Rs (mm)	2ro (mm)
a	21.12	30,113.67	27.85	0.1254	0.0234	1.00	0.861	137.45	1.014	8.088	8.088
b	20.51	40,397.42	14.38	0.1858	0.0441	1.00	0.855	127.86	0.989	8.019	8.019
c	19.98	48,967.81	9.18	0.2394	0.0672	1.00	0.846	119.93	1.005	7.984	7.984
d	14.00	20,608.04	10.93	0.2173	0.0674	0.86	0.865	123.11	1.045	8.014	4.007
e	12.59	25,227.42	6.22	0.2949	0.1122	0.92	0.866	112.12	1.006	8.016	4.008
f	11.35	27,823.30	4.42	0.3506	0.1499	0.95	0.864	104.51	1.006	8.054	4.027

were obtained through iterations (the third iteration with the mathematical model), with convergence focused on the continuity equation (U_in). It should be noted that these final dimensions in the six geometries were considered approximately in the six corresponding meshes detailed in Table 1 (with all meshes sharing dimensions: 2r_inj = 1 mm, 2R_s = 8 mm, and l_inj = 5 mm).

Analyzing the data in

Table 4. Comparison of nominal mass flow rate of the mathematical model with respect to the numerical simulation RNG K-ε model (for all cases ΔP = 400 kPa).

Case	Mass Flow Rate, Mathematical Model (kg/s)	Mass Flow Rate, RNG k-ε Model (kg/s)	Deviation (%)
a (C = 1; n = 2)	0.034	0.038	−10.53
b (C = 1; n = 4)	0.063	0.070	−10.00
c (C = 1; n = 6)	0.095	0.098	−3.06
d (C = 2; n = 2)	0.024	0.028	−14.29
e (C = 2; n = 4)	0.040	0.044	−9.09
f (C = 2; n = 6)	0.054	0.057	−5.26

, it is evident that among the group of atomizers, case “c” is the one that consumes the most propellant. This is logical as it has the highest number of channels (n = 6) and is an open-end atomizer. Conversely, case “d” consumes the least amount of propellant (approximately 25% of what case “c” consumes).

In addition, it can be observed that the mathematical model closely approximates the values from the numerical simulation in cases “c” and “f” (both with n = 6). However, there are greater deviations between the mathematical model and the numerical simulation when the numbers of inlet channels are n = 2 and n = 4, with the maximum error occurring in case “d” (−14.29%). Despite these discrepancies, it is noteworthy that the mathematical model is quite accurate for all cases, especially when providing the primary dimensions of the atomizers as shown in Table 3.

In Figure 14, the mass flow rates obtained from the mathematical model, accounting for losses due to liquid viscosity, are compared with the results of the numerical simulation in a steady state for four differential pressures (ΔP: 300, 350, 400, and 450 kPa). It is evident that the mathematical model provides good approximations when compared with the numerical simulation results for cases “c” and “f”.

From this comparison, it can be concluded that the accuracy of the mathematical model depends on the number of inlet channels, and the model tends to perform better when the number of channels exceeds 2 (i.e., n > 2).

Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 display the contours of total pressure in a steady state in two views: (a) vertical plane and (b) horizontal plane (z = −2 mm) for ΔP = 400 kPa. It should be noted that section A-A (Figure 15a) passes through the axes of the inlet tangential channels of the six atomizers and is referenced as “horizontal plane (z = −2 mm)”. In all cases, the pressure losses and vortices within the swirl chamber result from wall friction and variations in the cross-sectional area of the tangential channels as they enter the swirl chamber, and the pressure drop exhibits a radial behavior influenced by the angular momentum equation.

Within the group of open-end atomizers (Figure 15b, Figure 16b, and Figure 17), it is observable that as the number of channels increases, the total pressure distribution is more uniform in proximity to the walls of the swirl chamber. The most notable instance occurs in the open-end atomizer with six tangential channels (Figure 17b).

In closed atomizers, the radial distribution of total pressure in the swirl chamber is more evident due to the equation of angular movement and the increase in the nozzle-opening parameter “C”. This means that the radius of the outlet orifice decreases and the thickness of fluid rotating inside the swirl chamber increases (Figure 18a, Figure 19a and Figure 20a); for this reason, it is evident that the pressure drop in the tangential channels is not immediate (Figure 18b, Figure 19b and Figure 20b). The thickness of the liquid film on the walls of the swirl chamber is greater in closed-end atomizers compared with open-end atomizers (Figure 15a, Figure 16a and Figure 17a).

Figure 21, Figure 22 and Figure 23 present the contours of (a) tangential velocity and (b) axial velocity in a vertical plane in steady state for ΔP = 400 kPa for open-end atomizers. In Figure 22a and Figure 23a, the upper part of the spray cone exhibits a high level of tangential velocity for air, indicating a rotational movement. This rotation is a result of the drag effect at the gas–liquid interface and the low pressures recorded in that region, in accordance with Bernoulli’s principle.

Furthermore, in Figure 21b, Figure 22b and Figure 23b, it is apparent that air particles are drawn into and accelerated within the “air core” due to the centrifugal movement of the liquid. This liquid movement creates low pressures within this region [27].

Figure 24, Figure 25 and Figure 26 present the contours of (a) tangential velocity and (b) axial velocity in a vertical plane in steady state for ΔP = 400 kPa for closed atomizers. In Figure 24a, Figure 25a and Figure 26a, the upper part of the spray cone exhibits a high level of tangential velocity for air, also evidencing the presence of drag effects at the gas–liquid interface and the low pressures recorded in that region.

In Figure 24b, Figure 25b and Figure 26b, it is apparent that air particles are drawn into and accelerated within the “air core” due to liquid centrifugal movement creating areas of low pressure within this region. Furthermore, the largest axial component of the air velocity is located further downstream of the conical spray to the air-core region, evidencing its more uniform distribution.

Figure 27, Figure 28 and Figure 29 depict the gas–liquid interface in a steady state for ΔP = 400 kPa for open-end atomizers.

In Figure 27a, the spray generated by the open-end atomizer with two channels (n = 2) is not conical, making it impossible to measure the spray angle. In Figure 27b, it is evident that the liquid film thickness is not constant, leading to the “air core” not taking a circular shape. This demonstrates that the number of channels (n = 2) is associated with the uniformity of the liquid film, which is consistent with the findings of Laurila et al. [28]. This non-uniformity affects the mass flow rate distribution and results in an uneven spray angle, as depicted in Figure 27c,d.

It should be noted that Figure 27c,d represent perpendicular planes (plane “yz” and plane “xz”); these planes facilitate the measurement of the spray angle (2α), where it can be verified that the difference between these angles is very large (116° and 162°), Therefore, numerically, an average spray angle that characterizes this atomizer could not be estimated.

In Figure 28a, an isometric view of the atomizer “b” is presented (C = 1; n = 4), where it can be seen that by increasing the number of inlet channels “n”, the wave amplitude present in the liquid film tends to decrease, causing the spray to begin to take on a more uniform conical shape. In Figure 28b, the liquid film thickness remains “approximately” constant inside the swirl chamber due to the increased number of inlet channels (n = 4). This enhanced uniformity leads to a more stable spray angle, as shown in Figure 28c,d, where the measurements of the spray angles are shown (129° and 120°, respectively), which when averaged result in the characteristic spray angle (2α = 124.5°) of atomizer “b”.

Among the three open-end atomizers, case “c” in Figure 29 exhibits the most stability. In Figure 29a, an isometric view of the atomizer “c” (C = 1; n = 6) is presented, where it can be seen that the wave amplitude present in the liquid is minimal, resulting in the fact that the conical spray is highly uniform. In Figure 29b, the liquid film thickness is entirely uniform due to the greater number of inlet channels (n = 6), aligning with the findings of Alves et al. [29]. Consequently, this uniformity results in better mass flow rate distribution and spray-angle stability, as illustrated in Figure 29c,d.

When analyzing the three cases of ‘closed’ atomizers, specifically in Figure 30, Figure 31 and Figure 32, it is observed that in Figure 30a, the waves in the conical spray have more amplitude than in cases “e” and “f”; however, this design is more stable than atomizer “a”. In Figure 30b, the diameter of the “air core” exhibits slight eccentricity due to the low number of inlet channels (n = 2). However, when compared to atomizer ‘a’ (an open-end atomizer), it performs more effectively and displays a more uniform spray angle, as evident in Figure 30c,d, where 2α ranges from approximately 125° to 123°. An average spray angle of approximately 124° can be considered.

In Figure 31a and Figure 32a, the wave amplitudes of the conical spray are negligible as the number of inlet channels increases. In Figure 31b and Figure 32b, the liquid film thickness remains consistent within the swirl chamber due to the increased number of inlet channels (n = 4 and n = 6). Consequently, both cases of closed atomizers (case “e” and case “f”) exhibit improved uniformity in mass flow rate distribution and spray-angle stability, as depicted in Figure 31c,d and Figure 32c,d respectively.

Table 5. Comparison of spray angle according to the mathematical model with respect to the numerical simulation RNG k-ε/VOF model (for all cases ΔP = 400 kPa).

Case	Spray Angle, 2αeq (°)		Deviation (%)
	Math. Model	RNG k-ε /VOF
a (C = 1; n = 2)	137.45	-------	-------
b (C = 1; n = 4)	127.86	124.5	2.69
c (C = 1; n = 6)	119.93	120.5	−0.47
d (C = 2; n = 2)	123.11	124.0	−0.72
e (C = 2; n = 4)	112.12	117.5	−4.58
f (C = 2; n = 6)	104.51	106.0.	−1.41

illustrates the deviations in the spray-angle measurements between the mathematical model (accounting for losses) and the numerical simulation. It is clear that the mathematical model closely matches the measurements provided by the VOF-HRIC model. This suggests that the increase in the spray angle is primarily dependent on the augmentation of the number of inlet channels around the swirl chamber.

It is worth emphasizing that once the number of channels “n” is established, the spray angle increases in response to the rise in differential pressure.

The combination of open and closed atomizers in liquid propulsion for rocket engines, exemplified by the RD-0110 bipropellant atomizer, is indeed a critical aspect. In this configuration, the RD-0110 employs an open-end atomizer (nozzle-opening parameter C = 1) for the fuel (kerosene T1) and a closed atomizer (nozzle-opening parameter C = 1.67) for the oxidizer (LOX) [7]. Both atomizers have six inlet channels (n = 6), and the number of channels “n” is especially crucial to ensure the uniformity of the liquid film in the annular section, as demonstrated in Figure 29 and Figure 32.

Figure 33 presents a numerical simulation of the RD-0110 atomizer, as conducted in Rivas’ work. In Figure 33a, the total pressure contour of the bipropellant atomizer is displayed, with input conditions of ΔP = 150 kPa for the fuel atomizer and ΔP = 200 kPa for the oxidizer atomizer. In Figure 33b, the behavior of both fluids is depicted using the VOF method. It is evident that the liquid film of the fuel remains uniform and does not come into contact with the external nozzle wall of the cryogenic atomizer (the closed atomizer). This configuration ensures efficient and stable combustion in a bipropellant rocket engine.

5. Conclusions

The objective of this study was to investigate the internal flow of six atomizers, all operating at the same injection pressure (ΔP = 400 kPa) according to their design and sharing common dimensions such as the inlet channel diameter (2r_inj), inlet channel length (l_inj), and swirl-chamber diameter (2R_inj). The analysis becomes particularly interesting when examining how the key parameters vary in response to changes in the nozzle-opening parameter (C) and the number of channels “n”.

One of the noteworthy findings is that certain atomizers may not be suitable for use if uniformity in the liquid film thickness and spray angle is required. For example, the open-end atomizer “a” (C = 1), which has only two inlet channels, results in a noticeable eccentricity in the diameter of the air core. Consequently, the use of this atomizer would lead to an uneven distribution of mass flow rate, significantly affecting the spray angle (2α) and the Sauter mean diameter (SMD). However, this issue is associated with a low number of channels (n = 2) and can be addressed by progressively reducing the outlet diameter (2r_o), essentially increasing the parameter (C). This adjustment stabilizes the liquid film thickness, as observed in the closed atomizer “d” (C = 2) where there is a significant reduction in the eccentricity of the “air core”.

To enhance the uniformity of the liquid film thickness, increasing the number of inlet channels is necessary. A higher number of inlet channels allows the flow to enter at multiple points, ensuring that centrifugal forces act uniformly on the internal flow within the swirl chamber (in accordance with angular momentum conservation). This behavior is demonstrated in atomizers “b”, “c”, “e”, and “f”.

The mathematical model developed by Rivas [3] demonstrates its applicability for various types of pressure-swirl atomizers, both open-end and closed, with tangential inlet channels. This conclusion is supported by the comparison of the mass flow results, which show acceptable error margins when compared with the results of the numerical simulation.

Based on the inviscid analysis of the six atomizers and their number of inlet channels “n”, it can be concluded that the open-end atomizer with six channels (atomizer “c”) always consumes a higher mass flow rate than any closed atomizer with the same number of inlet channels (n = 6 and C > 1), because when the opening parameter (C) increases, the outlet orifice decreases and consequently the mass flow is obstructed

It is important to note that the geometrical parameter (A) tends to decrease as the number of inlet channels increases, irrespective of whether it is an open-end or closed atomizer. This implies that the curves for “C = 1” and “C = 2” may coincide at the same geometrical parameter (A). Consequently, they would share the same spray angle (2α), flow area coefficient (φ), and discharge coefficient (Cd). For example, an open-end atomizer with six channels (C = 1, n = 6) is equivalent to a closed atomizer with three inlet channels (C = 2, n = 3) because they share the same geometrical parameter “A” (as seen in Figure 11). However, it is important to note that these atomizers do not have the same dimensionless mass flow ($\overline{m}$).

The numerical simulation of the internal flow in a steady state successfully converged after approximately 8000 iterations, with each case taking an estimated time of 12 h. This simulation was made possible through the use of the volume-of-fluid (VOF) model in combination with the RNG k-ε turbulence model. It is worth noting that there is room for improvement in the interface resolution, which can be achieved by increasing the number of elements in the spray zone. However, this would come at the cost of higher computational resources and longer simulation times.

In conclusion, thanks to the VOF model, the detail of the spray behavior can be visualized, locating the wave amplitudes present in the conical spray. In open-end atomizers, these wave amplitudes decrease as the number of inlet channels increases (n). Finally, it should be noted that in this work we sought to analyze the uniformity of the mass flow distribution, the stability of the spray angle, and other important atomization parameters, varying the inlet channels and the parameter C.