
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12390-0
10.1016/j.heliyon.2024.e36359
e36359
Research Article
Numerical simulation of cavitating flow in liquid Nitrogen through a convergent nozzle
Adibi Pouyan adibi@hormozgan.ac.ir
⁎
Bagheri Reza
Hosseini Mohammad
Department of Mechanical Engineering, University of Hormozgan, Bandar Abbas, Iran
⁎ Corresponding author. adibi@hormozgan.ac.ir
15 8 2024
30 8 2024
15 8 2024
10 16 e3635914 11 2023
29 7 2024
14 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
This research has dealt with the simulation of liquid nitrogen cavitation inside a convergent nozzle. This is important in cryogenic industrial applications. So in this study, computational fluid dynamics methods have been used for simulating the cavitation phenomenon. The Two-phase model in this research has been a hybrid/mixed model. Also, k- ε turbulence model has been employed in realizable state. For meshing the nozzle geometry, Gambit software has been used, while for numerical simulation, Ansys Fluent software has been employed. For simulation of cavitation, Schnerr and Sauer cavitation model has been utilized. This research has also examined the effect of changing the nozzle outlet diameter and the impact of changing the pressure difference in the inlet and outlet of the nozzle on the cavitation. As a novelty and unlike what would have been expected based on the Bernoulli effect, the results obtained from the simulation showed that the increase/decrease in the nozzle's outlet diameter resulted in an enhanced/diminished extent of cavitation in the nozzle's outlet region. Also, the increase/decrease of the pressure difference in the input and output of the nozzle would lead to a higher/lower extent of cavitation. This research also found that the effect of altering the nozzle's outlet diameter on the extent of cavitation has been far higher than the effect of changing pressure difference in its inlet and outlet. The results also indicated that upon reduction of the nozzle's outlet diameter from the base state (1.02 mm) by 10, 20, 30, 40, and 50 %, the volume fraction of the vapor diminished by 22.23, 43.029, 60.66, 74.73, and 87.16 % respectively. Finally, with the increase in the nozzle's outlet diameter from the base state (1.02 mm) by 10, 20, 30, 40, and 50 %, the volume fraction of the vapor increased by 26.83, 55.27, 84.47, 117.12, and 149.31 % respectively.

Keywords

Cryogenics
Liquid nitrogen
Cavitation
Nozzle
Vapor void fraction
==== Body
pmc1 Introduction

The cavitation phenomenon occurs because of a pressure drop. Whenever in a hydrodynamic system, the fluid pressure drops almost to its vapor pressure, the fluid starts to boil and form bubbles, which unlike the typical process of boiling, it is because of the pressure drop, not temperature rising (which does not happen in this phenomenon). Thus, the fluid characteristics (such as viscosity and pressure) should not be ignored. In applications where cryogens would be employed as fuel (space applications), overcoming cavitation could be challenging [1]. Fluid pressure is one of the most critical industry issues when using cryogenic fluids. In industries, for some reasons, such as reducing the weight of the storage tank and keeping the rotational speed high, the pressure will be kept low, which increases the risk of cavitation. For this reason, some parameters affecting the fluid flow such as pressure and velocity must be considered.

Today, a combination of cryogenic fluids such as liquid nitrogen/hydrogen/oxygen or propane and iso-butane is used to achieve higher working efficiency. Some properties of cryogenic fluids differ from those of other fluids. For example, the liquid hydrogen and the liquid oxygen boiling point under standard conditions are −119.44 °C and −252.78 °C, respectively. Through cooling and compressing these fluids under specific conditions, they can be stored in small containers [2]. In the rest of this section, previous researches are described.

Rayleigh presented a mathematical analysis of the growth and destruction of bubbles, which is generally called the Rayleigh–Plesset Equation. This Equation is the most basic analytic relation describing nonlinear and transient bubble dynamics [3]. The initial experimental investigation of the cavitation phenomenon through nozzles revealed a strong dependency on the cavitation number and a weak one on the Reynolds number under turbulent flow conditions. The above results are related to permanent cavitation in axially symmetrical nozzles. Studies on cavitating flow in nozzles consider the cavitation number as a determining factor if the nozzle is surrounded by vapor. The results related to the extension of the cavitation length and the dimensionless cavitation number are based on the Nurick observations [4]. Zhang et al. analyzed the flow of the permanent liquid nitrogen cavitation model on a hydrofoil using Fluent Commercial Code. The pressure and the vapor distribution for an external flow on hydrofoil were presented along with the temperature influence on plate cavitation [5]. Li et al. studied cavitating flows around a NACA†66 hydrofoil using a numerical method and found that higher cavitation may affect turbulence intensity and flow unsteadiness. It was well understood that higher cavitation would lead to higher turbulence intensity and dissipation with fluid viscosity [6]. Sun et al. numerically investigated the cavitation phenomenon over a hydrofoil. The employed hydrofoil was NACA0015 in fluoroketone. The results indicated that cavitating flow on hydrofoil experienced unsteady periodic evolution, and in a common cycle in an unsteady cavitating flow, the shear effects also dominated the cavity growth. The temperature underwent variations near the hydrofoil, which were the result of local evaporation and condensation [7]. An experimental study was done by Esposito et al. to explore cavitation through a cylindrical orifice. The working liquid was nitrogen. Based on the five dimensionless numbers, they proposed a semi-empirical model to investigate the effects of the thermal property of fluid on cavitation [8]. In a numerical study, Ebrahimi et al. investigated cavitation phenomenon in single- and multi-hole plate orifices. They showed that creating more holes in the orifice would prevent noise in the flow and limit cavitation [9]. The cavitation phenomenon in a venturi nozzle was studied by Brennen et al. In their research, water flowed through the horizontal nozzle with a pressure of 400 kPa as the maximum inlet pressure. They showed that the pressure drop in the nozzles is highly significant, and the cavitation depends on the nozzle geometry [10]. Xue et al. studied cryogenic fluid in a nozzle spray. To depict the effect of nozzle geometry on the cavitation, they used computational fluid dynamics (CFD) simulation along with mixture model. The results revealed that the effects of geometrical parameters on the cavitation at the saturation temperature and the subcooled condition differed. Also, by increasing the nozzle angle, the cavitation area diminished [11]. Experimental and numerical investigations on the pressurized liquid nitrogen were done by Jun Ishimoto et al. in order to know cryogenic cavitating flow characteristics. This study focused on a horizontal rectangular nozzle. The onset of cryogenic cavitation was observed and the cavity grows near the wall surface of the inlet throat region [12]. A numerical-experimental study was done by Zhang et al. to investigate cavitation phenomenon of liquid nitrogen in a 2D Laval nozzle, and thermodynamic effect was explained. The results showed a significant correlation between temperature and cavitation in thermo-sensitive cryogenic fluids [13]. Using a dual-step approach, Le et al. proposed a method for design of a two-phase annular nozzle. They introduced non-equilibrium effects through an area factor during the second process and simulated the flashing through the cavitation method in an annular nozzle. They designed both forward and backward flashing two-phase annular nozzle. The forward version exhibited high curvature and non-uniformity. On the other hand, the backward version revealed lower non-uniformity and a marginally higher boundary layer ratio, showing more efficiency for non-equilibrium effects [14]. Rahbarimanesh et al. did a visualization experiment of a cryogenic cavitating flow through a converging-diverging nozzle with liquid nitrogen under subcooled conditions. Their observations indicated that cavitation instability arises when the speed of sound in a gas-liquid two-phase flow intersects with the required velocity for inception and conservation of cavitation [15]. Chen Jiacheng et al. used a high-speed camera to study behavior of liquid nitrogen cavitating flow in a converging-diverging (C-D) nozzle. They used experimental images to investigate the length and area of cavitation [16]. Zhang et al. proposed a thermal cavitation model to study cavitating flows in a cryogenic liquid (the liquid nitrogen was considered as a working fluid). The results showed that the thermal changes could cause a delay in forming and developing the cavitation behavior or even terminate it [17]. Le et al. simulated the hydrogen cavitating flow field using a proposed cavitation model coupled with a proposed thermodynamic model. Their results showed that the turbulence effect is not negligible and needs to be considered as an effective parameter. flows with high Reynolds, it affects the duration of the thermodynamic effect [18]. Chen et al. proposed a numerical model investigating the evolution of cavitating flows (liquid hydrogen) and how thermo-sensitive to quasi-isothermal would affect the cavity area. The results indicated that in the transition from quasi-isothermal to thermo-sensitive (free stream temperature lower than 30 K), initially the cavity grew, while the temperature rose, but then it diminished as transition occurred [19]. Despite extensive research providing deep insights into injection nozzles, due to the cryogenic fluid physical properties cannot be applied to cryogenic systems [20]. A numerical model was performed by Rong et al. for liquid nitrogen cavitating flow in spray nozzles [21]. A numerical study was performed by Rong et al. studied the effects of injection mass flow rate fluctuations on some parameters. The results of this research can be used to design cryogenic spray cooling systems [22]. A computational fluid dynamics code was used by Hong et al. to study the cavitation characteristics of flow in the circular and elliptical nozzles. They studied static pressure of fluid along orifice wall and radial velocity distribution of working fluid [23]. Tairan et al. investigated the instabilities of liquid nitrogen cavitation flow inside a converging-diverging nozzle based on theoretical and numerical methods. They proposed a modified one-dimensional model for examining the dynamic instabilities and pressure fluctuations developed by unsteady cavities. This model should have accurately calculated the pressure fluctuations in the cavitation zone. This model indicated that, neither in cavitation nor non-cavitation regions, the pressure fluctuations depended on the pore size; rather, it depended on the pore volume acceleration [24]. In addition, other researchers have done valuable research, such as Refs. [[25], [26], [27], [28], [29], [30], [31], [32]].

Although many successful numerical studies investigated various aspects of cryogenic cavitation, the effects of the geometry changes and the pressure difference variation in cryogenic cavitation in nozzles need more attention. Previous researchers have investigated the results of changing the flow parameters on cryogenics; however, in the present study, an Eulerian– Eulerian approach based on CFD was validated by those of experimental data. Simulations were performed to find out how outlet diameter changes as well as nozzle inlet and outlet pressure difference variations would affect cryogenic fluid cavitation, which were then compared. For this purpose, the liquid nitrogen flow field inside the nozzle (temperature distribution, pressure distribution, and vapor volume fraction distribution) has been comprehensively investigated. To achieve mentioned aims, first theoretical model and numerical simulation of the problem are explained. Then, the fluid flow in the nozzle is studied. In the end, the results and discussion of the study are comprehensively explained. In the conclusion section, some concluding remarks are presented.

2 Theoretical model and numerical simulation

The simulated nozzle is convergent, which is used for cryogenic spray and cooling systems. Among different models that have been proposed to describe the flow of cryogenic fluid as cavitating flow, the mixture model is more suitable than the others. This model treats the flow as a homogenous mixture while the slip between phases is considered negligible. According to Equations (1), (2), (3), (4), (5), (6), (7), the mixture would be analyzed by a set of continuity, energy, momentum, and the volume fraction would be considered for the secondary phase as below [33,34]:

Continuity equation:(1) ∂ρm∂t+∂(ρmuj)∂xj=0

Momentum equation:(2) ∂(ρmui)∂t+∂(ρmuiuj)∂xj=‐∂p∂xi+∂∂xj[μeff(∂ui∂xj+∂uj∂xi−23∂uk∂xkδij)]

where ρm, and μeff are:(3) ρm=ρlαl+ρv(1‐αl)

(4) μeff=μm+μt

Energy equation:(5) ∂(ρmCpT)∂t+∂(ρmujCpT)∂xj=∇.(keff∇T)‐∂(ρmfvL)∂t‐∂(ρmujfvL)∂xj

where Keff and fv are:(6) keff=km+kt=μmCp/Prm+μtCp/Prt

(7) fv=ρv(1‐αl)/ρm

The source term differs from one model to another and the cavitation model employs Rayleigh number to describe bubble growth as well as its collapse in the fluid. In Equation (8), the cavitation model is introduced using Schnerr and Sauer's model [35].(8) SchnerrandSauermodel:{ifP≤PvRe=ρvρlρmα(1−α)3RB2(Pv−P)3ρlifP≥PvRc=−ρvρlρmα(1−α)3RB2(P−Pv)3ρl

RB is equal to 10−6 m.

The finite volume method is selected to discretize the equations. Momentum and pressure-based continuity equations were solved together using the pressure-velocity coupling scheme.

The governing equations were solved employing the following conditions and numerical methods.• A two-dimensional solver considering the steady-state condition,

• CFD simulation convergence criterion employed as 1×10−6,

• Equations discretized using the second-order upwind scheme in FLUENT 15 [34].

2.1 Geometry and mesh generation

The nozzle investigated in this study is illustrated schematically in Fig. 1(a and b).Fig. 1 (a) The geometry of the nozzle with dimensions [35], (b): The length of the nozzle parts.

Fig. 1

Gambit software 2.4.6 was employed to generate the mesh. Since the flow in this problem was along the nozzle axis, the geometry could be generated from a two-dimensional geometry around the axis, so axial symmetry conditions were conducted. To generate the mesh, the computational space of geometry was subdivided into three blocks, as displayed in Fig. 2; the first block was considered for the input area, the second for the convergent area, and the third for the nozzle output area.Fig. 2 Mesh of Nozzle and its various components.

Fig. 2

2.2 Boundary conditions and problem-solving

As mentioned, the geometry of the nozzle was assumed to be two-dimensional and axially symmetric. Fig. 3 reveals the assumed boundary conditions.Fig. 3 Assumed boundary contentions.

Fig. 3

Table 1 presents the boundary conditions for the different cases investigated in this study. It was also simulated for different nozzle outlet diameters.Table 1 Boundary conditions of the nozzle for the different cases.

Table 1Case	Pin(Pa)	Pout(Pa)	ΔP(MPa)	Tin(K)	
1	2101325	101325	2	89.79	
2	1901325	101325	1.8	89.01	
3	1601325	101325	1.5	87.79	
4	1301325	101325	1.2	86.44	
5	1001325	101325	0.9	84.94	
6	901325	101325	0.8	84.32	
7	701325	101325	0.6	83.37	
8	601325	101325	0.5	82.80	
9	501325	101325	0.4	82.11	
10	401325	101325	0.3	81.28	
11	301325	101325	0.2	80.46	
12	267325	101325	0.166	80.22	

Table 2 Different values of the nozzle output diameters relative to the base size (1.02 mm).

Table 2No.	Reduce the nozzle outlet diameter from 10 % to 50 %	No.	Increase the nozzle outlet diameter from 10 % to 50 %	
1	0.918	6	1.122	
2	0.816	7	1.224	
3	0.714	8	1.326	
4	0.612	9	1.428	
5	0.510	10	1.530	

The nozzle output diameter variation involves 10–50 % reduction as well as 10–50 % increment due to its initial value of 1.02 mm. Table 2 reports the examined diameters.

2.3 Validation

In order to ensure the number of grids does not affect the results, different numbers of grids (30,000 (coarse), 70,000 (medium), and 100,000 (fine) cells) were employed. Also, as shown in Fig. 4, the results of mass flow rate at different pressure differences were compared with Xue et al. findings [35]. The minimum and maximum grid size in each direction is mentioned in Table 3.Fig. 4 Mass flow rate at different pressure differences considering different meshes (Present study vs. Numerical [35]).

Fig. 4

Table 3 Minimum and maximum grid sizes (mm) in each direction.

Table 3Mesh type	X direction	Y direction	
Min grid size	Max grid size	Min grid size	Max grid size	
Coarse	0.0203	0.47	0.0068	0.0519	
Medium	0.0122	0.237	0.0051	0.03895	
Fine	0.01016	0.2094	0.00408	0.03116	

Grid residual independency in different meshes was done for the basic model of the nozzle with an outlet diameter of 1.02 mm.

As can be seen in Fig. 4, although at pressure differences lower than 1 MPa, some deviation is seen, a pressure differences greater than 1 MPa, all of them show good accuracy.

In the first stage, we initiated the simulation with the grid with 30,000 cells; at this stage, the results of numerical solution were close to the results of Xue's model [35]. In the second stage, the number of grid cells was increased to 70,000, that the results were very close to the grid with 30,000 cells. In the next stage, the grid with 100,000 cells was extracted for independence from mesh, whereby the results declined to some extent compared to the case of 70,000 cells. However, considering the accuracy of results in the grid with 70,000 cells as well as time and computational cost, and since in the Xue's paper, this grid has been used for the simulation, we also chose this grid with 70,000 cells for the rest of the simulation.

The conservation of the inlet and outlet mass flow rate is considered in Table 4 at a pressure difference of 1.5 MPa, which shows the convergence of the simulation.Table 4 Inlet and outlet mass flow rate for 1.5 MPa pressure difference.

Table 4Reduce the nozzle outlet diameter from 10 %–50 %	Increase the nozzle outlet diameter from 10 % to 50 %	
Diameter (mm)	Inlet mass flow rate (kg/s)	Outlet mass flow rate (kg/s)	Diameter (mm)	Inlet mass flow rate (kg/s)	Outlet mass flow rate (kg/s)	
0.918	0.0273	0.0273	1.122	0.0406	0.0406	
0.816	0.0216	0.0216	1.224	0.0483	0.0483	
0.714	0.0166	0.0166	1.326	0.0567	0.0567	
0.612	0.0123	0.0123	1.428	0.0659	0.0659	
0.510	0.0082	0.0082	1.530	0.0747	0.0747	

As reported in Table 4, the inlet mass flow rates are equal to outlet with perfect accuracy. In other cases, the results show mass conservation. Thus, the simulations are reliable for obtaining physical results.

The contours of the vapor void fraction at pressure differences of 2 MPa and 0.9 MPa are shown in Fig. 5(a) and (b), respectively. As displayed, the contours are the same as Xue et al. [35].Fig. 5 Contour of vapor void fraction at a pressure difference of (a) 2 MPa, (b) 0.9 MPa.

Fig. 5

When the fluid flows in the orifice, nozzle, etc., it is faced with a sudden decrease in cross-sectional area, near which the flow has a radial velocity component inward to the area. This velocity component causes the flow to rotate to the reduced area. The flow separates from the surface at the beginning edge of this region. This separation zone reduces the cross-sectional area of the flow, so that the velocity would increase, but the temperature and pressure would decrease. The pressure and flow behavior through this area are indicated in Fig. 6. If the pressure in this area is lower than the saturated vapor pressure, cavitation will occur. The highest amount of vapor production occurs in the minimum area and adjacent to the wall.Fig. 6 Velocity and pressure changes on the centerline of the flow in Vena Contracta.

Fig. 6

In the present study, due to the smaller cross-sectional area of the nozzle outlet (block 3 in Fig. 2), the fluid-flow isolates from the edge of the nozzle, so the effective passage flow area would diminish.

2.4 Cavitation model verification

This section evaluates the accuracy of the Schnerr-Sauer model for the calculation of cryogenic cavitation is assessed in this section by comparing it with experimental data obtained by Hord for hydrofoil in liquid nitrogen [20], where the standard and realizable k-ε turbulence models are also compared. Based on the experimental data, a two-dimensional axisymmetric model is developed. The tunnel and hydrofoil geometry as well as the boundaries are presented in Fig. 7. The tunnel width and hydrofoil head width are 25.4 mm and 7.92 mm, respectively. The inlet is specified as velocity-inlet and outlet is defined as pressure-outlet.Fig. 7 Hydrofoil dimensions (mm) and applied boundary conditions. (axisymmetric) [35].

Fig. 7

The boundary conditions of the computational case selected from Hord's experiment are listed in Table 5.Table 5 Applied boundary conditions around the hydrofoil [35].

Table 5No.	Vin (m/s)	Tin (K)	Pin (Pa)	Pout (Pa)	
290 c	23.9	83.06	568300	442718.2	

Fig. 8 (a) and (b) illustrate the numerical analysis and the experimental data of the surface pressure and temperature around the hydrofoil wall, respectively. Generally, the results match the experiment with reasonable deviations (Mean errors for pressure and temperature gradient are 13.05 % and 0.74 %, respectively). The flow behavior through the nozzle was studied using the Schnerr-Sauer model and the realizable k-ε turbulence.Fig. 8 The surface pressure and temperature around the hydrofoil, (a) Pressure distributions along the hydrofoil wall for 290° C, (b) Temperature gradient along the hydrofoil .(T=290°C)

Fig. 8

Fig. 9 reveals the Y+ distributions along the hydrofoil wall and nozzle wall. As observed in Fig. 9 (a), the Y+ values of the hydrofoil wall are 30<Y+<60. Also, Fig. 9 (b) indicates that the Y + values of the nozzle wall are 45 <Y+ <86, which are compatible with the criterion of “standard wall function” for simulation of flow near the wall (the Y+ values should be 30 <Y+ <300).Fig. 9 Y+ distributions along the (a) hydrofoil wall (b) nozzle wall.

Fig. 9

3 Results and discussion

The results include the effects of pressure difference and nozzle outlet diameter on the cavitation of liquid nitrogen (see Fig. 10(a)–and (b)). Volume fraction values are used to identify the cavitation. Fig. 10 depicts the mass flow rate for different nozzle pressure differences and nozzle outlet diameters, while the outlet diameter has been changed (10–50 % of decrement or increment).Fig. 10 Mass flow rate at different nozzle pressure differences (a) increasing diameters (b) decreasing diameters.

Fig. 10

As depicted in Fig. 10(a) and (b), the mass flow rate in the same pressure difference has grown with increasing nozzle outlet diameter, while it has diminished with decreasing nozzle outlet diameter. The trend is physical because the mass flow rate depends on the pressure differences as well as the flow cross-section area due to the continuity equation. The higher-pressure difference causes more fluid to flow into the nozzle, so the mass flow rate would rise. As the pressure difference increases, so does the mass flow rate. It should be noted that an increase in mass flow rate occurs at the inlet and the outlet of the nozzle, so the continuity is satisfied.

Fig. 11(a) and (b) illustrate the mean void fraction of vapor in the nozzle vs. the pressure difference, while the nozzle outlet diameter has increased or decreased from 10 to 50 %.

As depicted in Fig. 11(a) and (b), the effect of increasing the pressure difference on the rate of the vapor void fraction changes is negligible; with an increase in pressure difference greater than 0.5 MPa, the vapor void fraction at each outlet diameter was almost constant. When the pressure difference increases, the mass flow rate and exit velocity will be raised, so the liquid has a small quantity of opportunity to vaporize. Hence, the volume fraction is almost constant in each diameter. Thus, the effect of the diameter change on the cavitation is greater than that of the pressure change and this should consider in the design of these nozzles.

With 10, 20, 30, 40, and 50 % increases in the outlet diameter, the vapor void fraction grew by 26.83, 55.27, 84.47, 117.12, and 149.31 %, respectively. Also, upon reduction of the nozzle outlet diameter (with the same percentages), the vapor volume fraction diminished by 22.23, 43.029, 60.66, 74.73, and 87.16 %, respectively (Fig. 11).Fig. 11 Vapor volume fraction at different nozzle pressure differences (a) increasing diameters (b) decreasing diameters.

Fig. 11

Now, to compare the vapor void fraction contours and do a better observation on the effect of nozzle outlet diameter on vapor void fraction, Fig. 12(a)–(l) shows the nozzle outlet diameter change by 50 % increase compared with the basic diameter (i.e., 1.53 mm) at different pressure differences.Fig. 12 Contours of the vapor volume fraction at various pressure differences (nozzle outlet diameter = 1.53 mm). , (a) ΔP=0.166MPa, (b) ΔP=0.2MPa, (c) ΔP=0.3MPa, (d) ΔP=0.4MPa, (e) ΔP=0.5MPa,(f) ΔP=0.6MPa, (g) ΔP=0.8MPa, (h) ΔP=0.9MPa, (i) ΔP=1.2MPa, (j) ΔP=1.5MPa, (k) ΔP=1.8MPa, (l) ΔP=2MPa.

Fig. 12

According to Fig. 12, with increasing pressure difference from 0.166 MPa to 2 MPa, the length of the cavitation region adjacent to the wall has grown. With increasing the pressure difference, exit velocity will be increased, so the local pressure is lowered and the length of the cavitation region adjacent to the wall will grow. The length of that region has the lowest and highest values at 0.166 MPa and 2 MPa, respectively. In addition, the volume fraction of vapor in this region rises with increasing pressure difference, but the thickness of the bubbling region diminishes.

The parameters at the outlet walls (third block walls in Fig. 2) have been investigated while the outlet diameter increased. Fig. 13 (a), (b), and (c) depicts the graph of the pressure, vapor volume fraction, and the flow temperature on the wall for a 1.5 MPa pressure difference while output diameter increased, respectively.Fig. 13 The flow characteristics along the wall for 1.5 MPa pressure difference and outlet diameter increasing (a) Pressure, (b) vapor volume fraction, and (c) Temperature.

Fig. 13

Fig. 13 (a) shows that as the nozzle outlet diameter increases, the pressure on the wall decreases. In other words, as the cross-section of the nozzle outlet expands, the pressure on the wall decreases. From the beginning of the nozzle outlet (x = 35 mm) to the minimum area of effective flow area (x = 35.2 mm) caused by the separation, the lowest value of the pressure has occurred on the wall (fixed diameter). After passing through the minimum area, the effective cross-sectional area grows so that the velocity would diminish (due to mass continuity), and according to the Bernoulli effect, the pressure would increase.

As shown in Fig. 13 (b), as the nozzle outlet diameter increases, so does the volume fraction of vapor on the wall (at a fixed value of x). Increasing the diameter causes the radial component of the flow velocity (which is in the reduced area) to grow. Augmentation of the radial component of the flow increases the thickness of the separation zone as well as its length, which reduces the effective cross-sectional area of the flow relative to the base case. This reduction causes the pressure in the area adjacent to the wall (where cavitation occurs) to drop further whereby the amount of vapor generated by cavitation increases.

From the nozzle outlet area (x = 35 mm) to the minimum effective flow area (x = 35.2 mm), the volume fraction of vapor increased (fixed diameter) since the pressure on the wall in this area has the minimum value. Cavitation near the wall rises as pressure on the wall drops since the liquid nitrogen pressure falls below the saturation pressure. After passing through the minimum area, the vapor volume fraction decreases due to the rising pressure.

According to Fig. 13 (c), as the nozzle outlet diameter increases, the wall surface temperature decreases. It can be concluded from Fig. 13 (b) that lower pressure leads to more cavitation (more generated vapor near the wall) causing heat absorption, which cools the wall. Also, it has been realized that more cavitation leads to a lower wall temperature.

From the beginning of the nozzle outlet (x = 35 mm) to the minimum effective flow area (x = 35.2 mm corresponding to the maximum volume fraction of vapor), the wall temperature has dropped (fixed diameter). After that, it rises steadily to the nozzle outlet due to passing from the minimum area and reduction of cavitation.

For comparison, the contour of vapor void fraction is shown in Fig. 14(a)–(l) for a 50 % reduction in nozzle outlet diameter. The following contours display the volume of vapor inside the nozzle for a diameter of 0.51 mm in various cases of pressure difference.Fig. 14 Contours of the volume fraction of vapor at various pressure differences (nozzle outlet diameter = 0.51 mm), (a) ΔP=0.166MPa, (b) ΔP=0.2MPa, (c) ΔP=0.3MPa, (d) ΔP=0.4MPa, (e) ΔP=0.5MPa, (f) ΔP=0.6MPa, (g) ΔP=0.8MPa, (h) ΔP=0.9MPa, (i) ΔP=1.2MPa, (j) ΔP=1.5MPa, (k) ΔP=1.8MPa, (l) ΔP=2MPa.

Fig. 14

In this case, the smallest nozzle outlet diameter was investigated. The volume fraction of vapor at pressure difference, equal to 0.166 MPa, had the lowest value compared to the other cases. In other words, the cavitation as well as the cavitation length and the thickness of the cavitation area diminished. It was concluded that the shortest cavitation length and thickness would occur at the lowest nozzle outlet diameter. Since at the minimum diameter, the velocity is maximum and liquid has a small quantity of time to vaporize, so the cavitation length and thickness will be shortest.

The comparisons of pressure, vapor volume fraction, and temperature on the nozzle outlet wall after the convergence zone are shown in Fig. 15 with the pressure being considered 1.5 MPa where the nozzle outlet diameter was reduced.Fig. 15 The flow characteristics along the wall for 1.5 MPa pressure difference and outlet diameter decreasing (a) Pressure (b) vapor volume fraction (c) Temperature.

Fig. 15

As shown in Fig. 15 (a), pressure on the wall increased with the reduction of the nozzle outlet diameter. In other words, by decreasing the cross-sectional area of the nozzle outlet, the pressure on the wall increases. The pressure on the wall has the lowest value from the beginning of the nozzle outlet (x = 35 mm) to the minimum area of effective flow area (x = 35.2 mm, which is caused by the separation flow), while the diameter is considered fixed. After passing through the minimum area, as the practical cross-section level expands (hence diminishing flow rate), the pressure value will increase according to the Bernoulli Effect.

According to Fig. 15 (b), reduction of the diameter causes the radial component of the flow velocity (which is in the reduced area) to decline. The reduction of the radial component of the flow lowers the thickness of the separation zone as well as its length, which increases the effective cross-sectional area of the flow relative to the base case. This increase causes the pressure in areas adjacent to the wall (where cavitation occurs) to grow and the amount of vapor generated by cavitation to decrease.

From the nozzle outlet area (x = 35 mm) to the minimum effective flow area (x = 35.2 mm), the vapor volume fraction experienced an increment because of the pressure on the wall in this area, which had the minimum value. Cavitation in the vicinity of the wall increases as pressure on the wall decreases, since the liquid nitrogen pressure reaches below saturation pressure. After passing through the minimum area, due to the pressure elevation, the volume fraction of vapor decreases.

According to Fig. 15 (c), as the nozzle outlet diameter decreases, the wall surface temperature increases. This is due to the higher pressure on the wall and reduced cavitation.

From the beginning of the nozzle outlet (x = 35 mm) to the minimum effective flow area (x = 35.2 mm corresponding to the maximum volume fraction of vapor), the wall temperature diminishes (while the diameter does not vary). Thereafter, it rises steadily to the nozzle output due to passing from the minimum area and reduction of cavitation.

In Fig. 16, the velocity and volume fraction of vapor are shown for the base case (1.02 mm), a 50 % increase (1.53 mm), and a 50 % decrease (0.51 mm) in the nozzle output diameter. In Fig. 16(a)–(f), the changes in the thickness of the separation zone happen with variations in the outlet diameter. Fig. 17 shows the graph of the volume fraction of vapor for three different outlet diameters. The vapor volume fraction has a proportional relation with the nozzle outlet diameter. In other words, increasing outlet diameter leads to the creation of more cavitation.Fig. 16 1.5 MPa pressure difference, the velocity contour (m) for the nozzle outlet (a) base diameter, (b) a 50 % increase, and (c) a 50 % decrease, the volume fraction of vapor contour for the nozzle outlet (d) base diameter, (e) a 50 % increase, and (f) a 50 % decrease.

Fig. 16

Fig. 17 Vapor volume fraction for three different outlet diameters.

Fig. 17

Finally, the velocity distribution contour at the 2 MPa pressure difference for the nozzle outlet base diameter is shown in Fig. 18.Fig. 18 Velocity contour (m) for 2 MPa pressure difference for the initial nozzle outlet diameter.

Fig. 18

Fig. 18 reveals the separation of the flow on the edge, thereby increasing the velocity and decreasing the pressure and temperature, as seen in Fig. 19 (a), (b), and (c).Fig. 19 The flow characteristics along the wall for the primary case of nozzle outlet diameter at 2 MPa pressure difference (a) Pressure, (b) Temperature, and (c) Vapor volume fraction.

Fig. 19

As the pressure drop causes the local pressure to reach below the saturated vapor pressure, cavitation occurs (Fig. 20) where most of the produced vapor is at the minimum cross-sectional area and adjacent to the wall. Since cavitation occurs near the wall of the outlet area, the flow parameters on this wall have been plotted. Fig. 19 illustrates the lowest local pressure, the lowest local temperature, and the highest vapor void fraction. As seen in Fig. 19, the minimum pressure and temperature and the maximum vapor volume fraction occur at the minimum effective flow cross-section area (i.e. x = 35.2 mm), which agrees with this thermodynamic characteristic of the flow.Fig. 20 Vapor void fraction Contour for the primary case of nozzle outlet diameter at 2 MPa pressure difference.

Fig. 20

When the flow cross-section shrinks, local pressure would drop so that its saturated temperature would decrease. Upon the pressure reduction, the chance of generating more vapor grows, which means a higher vapor volume fraction at the point.

Due to the reduction of cross-sectional area at the nozzle outlet (block 3, Fig. 2), the flow starts to separate from the edge of the nozzle, so the effective flow rate, pressure, and temperature would diminish while the velocity would increase. Fig. 18 displays the flow separation. As the pressure drop causes the pressure to reach below saturated pressure, so cavitation occurs and most of the vapor would be produced at the minimum surface cross-sectional area adjacent to the wall. It has been concluded that when fluid flows through the minimum cross-section, the minimum pressure, minimum temperature, and maximum vapor void fraction would be obtained.

4 Conclusion

In this study, the main goal is to find out how outlet diameter changes, and how spray nozzle inlet as well as outlet pressure difference variation would affect cryogenic fluid (liquid nitrogen) cavitation and to compare them. A numerical investigation based on the multiphase mixture model was done. According to the results, the causes of cavitation formation adjacent to the outlet wall of the nozzle, as well as cause of rise and fall of cavitation due to the increase/decrease of outlet diameter are novel results that have not been presented in any previous publications. The results of the study represent that the elevation of the outlet diameter increased the radial component of the flow velocity (which was toward the reduced area), causing the length and thickness of the separation zone to grow. The thickness growth of the separation zone would lower the effective cross-sectional area of the flow relative to the base case. This reduction causes the pressure to drop near the wall (the area where cavitation occurs), which augments the amount of vapor generated by cavitation, so more cavitation leads to a lower temperature. The vapor was generated in the area adjacent to the walls because the Vena Contracta created by the flow separation corresponded to the maximum velocity, minimum temperature, and pressure. On the other hand, the vapor void fraction was enhanced by increasing the pressure difference. Also, the amount of cavitation in these nozzles had a proportional relation with diameter. The effect of pressure difference changes on the rate of bubble formation was less than the effect of diameter change. To control the cavitation, geometric parameters (output diameter) should be optimized as the main parameter.

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.Nomenclature

C2, C1ε	constant (-)	
CP	specific heat (J.kg-1.K-1)	
E	specific total energy (J.kg-1)	
f	mass fraction (-)	
g	gravity acceleration (m.s-2)	
Gb	generation of turbulence kinetic energy due to buoyancy (W.m-3)	
Gk	generation of turbulence kinetic energy due to the mean velocity gradients (W.m-3)	
H	enthalpy (J.kg-1)	
keff	effective heat conduction coefficient (W.m-1K-1)	
L	latent heat of vaporization (J.kg-1)	
p	pressure (Pa)	
Pr	upstream pressure (Pa)	
Pr	Prandtl number (-)	
Pv(T)	vapor pressure at temperature T (Pa)	
RB	bubble radius (m)	
SE	source term (J.m-3s-1)	
Sk, Sε	source terms (W.m-3)	
t	time (s)	
T	temperature (K)	
v	velocity (m.s-1)	
YM	contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate (W.m-3)	
Greek letters	
ρ	density (kg.m3)	
a	void fraction (-)	
β	coefficient of thermal expansion (K-1)	
δij	Kronecker function (-)	
μ	dynamic viscosity (Pa.s)	
μeff	effective viscosity (Pa.s)	
σk	turbulent Prandtl numbers for k (-)	
σv	cavitation number (-)	
σε	turbulent Prandtl numbers for ε (-)	
Subscripts	
i,j	Direction	
in, out	inlet, outlet	
k	each phase	
l	liquid phase	
m	mixture phase	
t	Turbulence	
v	vapor phase	
Superscripts	
T	Turbulence	

Data availability

Data will be made available on request.

CRediT authorship contribution statement

Pouyan Adibi: Writing – review & editing, Visualization, Supervision. Reza Bagheri: Writing – original draft, Formal analysis, Data curation. Mohammad Hosseini: Validation, Formal analysis.

Declaration of Competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

† National Advisory Committee for Aeronautics.
==== Refs
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