Numerical Simulation of Heat Transfer Enhancement by Vibration of an Irregular Pipe

1  Introduction

In order to heat oil-gas-water combinations to the temperatures needed for subsequent processing and transportation, heating furnaces are essential pieces of equipment in oil and gas collection systems. The conventional atmospheric vertical heating furnaces currently employed by Changqing Oilfield have an efficiency of less than 80% and present several operational challenges in field applications [1]. Widely used in oil and gas collecting projects, vacuum phase-change heating furnaces allow water to evaporate below 100°C and achieve over 90% heat transfer efficiency by evaporation-condensation with oil-water mixtures in coils [2,3]. The geometric properties of heat exchange pipes reduce the effective heat exchange area and compromise efficiency during operation by causing condensate to form continuous, uniformly thickened liquid layers along the pipe walls under surface tension and gravity [4–6]. As condensation develops, accumulated condensate leaks into downstream pipelines from the pipe walls, rupturing the gas-liquid boundary layer and resulting in splashing droplets [7,8]. This is a major problem preventing further efficiency gains in vacuum phase-change heating furnaces since it not only makes the distribution of the liquid layer more uneven but also destabilizes the heat transfer performance throughout the pipe bundle.

When mechanical equipment is operating, vibration is an inevitable event. Vibration of the wall surface can have an immediate effect on the efficiency of heat transmission by influencing the flow and thermal boundary layers close to the wall. Numerous studies have demonstrated that vibration can improve heat transmission [9,10]. Beyond conventional mechanical vibration, ultrasonic vibration, as a high-frequency excitation method, has also been extensively applied in the field of external tube heat transfer enhancement, covering scenarios including finned-tube heat exchangers [11], double-pipe heat exchangers [12,13], and indirect water bath heaters [14]. Relevant studies have confirmed that ultrasonic vibration can effectively enhance external tube heat transfer performance. Vibration can raise the heat transfer coefficient by 40%, according to research by Fu and Tong [15] and Bronfenbrener et al. [16]. Heat transfer performance improves at low Reynolds numbers (Re), and the enhancement impact is related to amplitude and frequency. In their experimental investigation of the heat transfer properties of a sinusoidally vibrating circular pipe, Leng et al. [17] found that fluid flow around the pipe at low velocities enhances heat transfer more effectively, with the effectiveness rising with increasing amplitude and frequency. By comparing the cooling effectiveness of micro-fin and flat plate heat sinks at the same air velocity, Jeung [18,19] carried out heat transfer studies on their designed induced vibration micro-fin heat sink and discovered that the micro-fin heat sink could increase cooling efficiency by 5.4% to 11.5%. Vibration enhancement effects become negligible when the pipe’s amplitude stays below 6 mm, according to Yu et al.’s [20] numerical simulation analysis of the heat transfer characteristics of low-velocity air flow around a vibrating cylindrical pipe. Su et al. [21] examined the heat transfer properties of fluid flow around vibrating cylinders in various orientations and found that the direction of vibration does not affect the augmentation of heat transfer, while the effectiveness diminishes as flow velocity increases. The application of vibration-enhanced heat transfer technology to high-efficiency non-circular pipe heat exchangers remains limited because the majority of current vibration-enhanced heat transfer research concentrates on circular pipes or basic flat plate structures. Additionally, there is a dearth of thorough research on the coupling effect of vibration and non-circular pipe structures in condensation scenarios.

Optimizing the condensate surface’s curvature reduces heat resistance in external condensation systems that use non-circular pipes by disrupting and thinning the condensate coating. By reducing both liquid collection and flow resistance, the streamlined shape greatly improves heat transfer efficiency. Non-circular pipes have been extensively studied over the past few decades. Yang and Cha’o-Kuang [22,23], Mosaad [24], and Chang and Yeh [25] theoretically demonstrated that oval pipes have greater condensation heat transfer coefficients (CHTC) than circular pipes with comparable surface areas. Dutta et al. [26] obtained similar results via theoretical evaluation of non-circular pipes with progressively increasing curvature radii along the gravity direction. In their experimental investigation of steam condensation properties on horizontally twisted oval pipes, Zhang et al. [27] discovered that the CHTC enhancement factor (EF) of these pipes was roughly 0.87–1.34 times that of circular pipes. Kerosene-mixing steam condensation in twisted oval pipe bundles yields EF values of 1.5–3 times in non-condensable gas conditions, according to experimental confirmation by Gu et al. [28]. These findings suggest that better tangential gravitational component effects on the condensate film, which decrease its thickness, are the main cause of enhanced condensation heat transmission. Deeb and Sidenkov [29] led numerical and experimental investigations into the fluid flow properties surrounding a single water droplet conduit. According to the study, these droplet pipes cause a delay in the boundary layer’s separation from the pipe wall. Water droplet pipes perform better than circular pipes in lowering resistance and friction coefficients under the same operating conditions. Deeb and Sidenkov [30,31] carried out numerical investigations into the heat transmission and hydrodynamic properties of different water droplet pipe bundle topologies. The findings demonstrated that at 0° and 180° angles of attack, water droplet pipe bundles outperform circular pipe bundles in terms of hydrodynamic resistance. In cross-flow situations, Horvat et al. [32] examined the heat transfer and flow properties of staggered pipe bundles with the same perimeter, such as water droplets, oval, and circular pipes. The results showed that compared to circular pipes, oval and water droplet pipes have lower Stanton numbers and resistance coefficients. According to these investigations, streamlined cross-sections yield lower pressure drops and higher heat transfer coefficients than round pipes. However, the majority of the current research on condensation in non-circular pipes concentrates on static conditions, and it is still unclear how dynamic factors affect liquid film movement and heat transfer mechanisms in non-circular pipes. In summary, Table 1 compares the technical routes and enhancement mechanisms between the present study and previous investigations on non-circular tube heat transfer as well as circular tube heat transfer under vibration, intuitively demonstrating the innovative advantages of tube shape-vibration coupled heat transfer.

This study recommends integrating vibration-enhanced heat transfer methods with non-circular pipe configurations. Leveraging boundary layer theory and Nusselt condensation theory, and accounting for wall thermal resistance, a numerical investigation was conducted on the external flow and heat transfer characteristics of oval and droplet-shaped pipes under vibrational conditions. For identical vibration parameters, the study examines the effects of distinct structural features on the average heat transfer coefficient, wall shear stress, temperature distribution, and near-wall velocity profile in non-circular pipes. Furthermore, it elucidates how varying pipe geometries influence liquid film rupture in external flows under simple harmonic vibration conditions.

2  Model Description

2.1 Physical Model

The dynamic detachment of liquid films on single non-uniform pipes is the main subject of this investigation. The simulation assumes that the pipe is positioned horizontally under periodic horizontal vibration in a saturated steam environment. The reference design was a 50 mm diameter steel round pipe, which is frequently used in oilfield heating furnaces. The simulation domain was a 2D 300 mm × 300 mm area, as shown in Fig. 1. In numerical studies on condensation heat transfer outside horizontal tubes, two-dimensional (2D) modeling has emerged as the predominant approach [10]. By implementing near-wall mesh refinement, 2D modeling not only meets the computational accuracy requirements for liquid film flow and phase-change heat transfer but also substantially reduces the calculation time. With the deviation between its simulation results and experimental data kept below 10%, this modeling method is proven to possess adequate reliability for addressing such problems. In the physical model, oval and droplet pipes have the same perimeters and share the same condensation heat transfer area as circular pipes. Fig. 2 depicts six non-circular pipe configurations. Oval pipes 1–3 exhibit gradually increasing eccentricity (e) values, which are indicative of enhanced cross-sectional flatness. The aspect ratio D1/L grows successively for droplet pipes 1–3, as shown in Fig. 2, indicating an increasing percentage of tail length L to semi-circular diameter D1. Seven curvature configurations listed in Table 2 were subjected to numerical research. Table 3 illustrates that the computational domain adopts a pressure outlet boundary condition at the top and a velocity inlet boundary condition at the bottom. To match the thickness of the first wall-layer mesh, the starting liquid film thickness is chosen at 0.05 mm. The prescribed steam inlet velocity is 0.1 m/s. The primary focus of this work is the dynamic separation of liquid layers on single non-uniform pipes. In a saturated steam environment, the simulation assumes that the pipe is horizontally positioned and subject to periodic horizontal vibration. An e = 0 steel round pipe with a 50 mm diameter, commonly seen in oilfield heating furnaces, served as the reference design. The two-dimensional simulation area was 300 mm by 300 mm. Oval and droplet pipes share the same condensation heat transfer area and perimeter as circular pipes in the physical model. Fig. 2 shows six atypical pipe configurations. The eccentricity (e) values of oval pipes 1–3 are increasing progressively, indicating improved cross-sectional flatness. An angle pipe is vibrating horizontally on a regular basis. The dynamic mesh model incorporates vibration characteristics using user-defined functions (UDFs), as shown in Eq. (1), and uses rigid body motion for the pipe wall and deformable motion for the lateral boundaries. Initialization involves setting the steam volume fraction to 1 and initially saturating the entire domain. A partial initiation results in a wall liquid film volume percent of 1.

Figure 1: Physical model of a heat exchange pipe.

Figure 2: Schematic diagram of the cross-sectional structure of the special-shaped tube.

Mesh generation for the heat exchange circular pipe network is illustrated in Fig. 3. Given the small size of the liquid film thickness in the computational domain, using coarse grids to simulate thin liquid film fluctuations may result in discontinuous segmentation [33]. Additionally, pipe vibrations can cause abrupt flow field variations near the pipe wall. To accurately simulate liquid film flow and detachment processes, it is essential to densify the grid near the pipe wall by employing minimized triangular grids. Through sensitivity testing, the minimum grid size was determined to be 0.05 mm.

Figure 3: Diagrammatic representation of the heat exchange circular pipe grid division.

The physical model was simplified and predicated on the following assumptions to examine the impact of various pipe types on the process of liquid film detachment from the pipe’s exterior under the condition of micro vibration:

a)   Flow and Phase Change Assumptions: Both gas and liquid phases are assumed to be incompressible fluids. Phase transition occurs only at the gas-liquid interface and is driven by the interfacial temperature difference. Non-condensable gases in the vapor are neglected, and only the two-phase coupling between saturated vapor and condensate is considered.

b)   Geometric and Boundary Assumptions: The outer wall of the pipeline is assumed to be smooth, with the effect of surface roughness on liquid film flow ignored. The left and right boundaries of the computational domain and the pipe wall are set as isothermal walls, and the no-slip velocity boundary condition is adopted.

c)   Initial Condition Assumptions: The initial liquid film uniformly covers the outer wall of the pipeline, with a thickness of 0.05 mm that matches the size of the first grid layer near the wall to ensure the accuracy of liquid film capture. The computational domain is initially in a saturated vapor environment, where the vapor volume fraction is 1 and no initial disturbance exists.

d)   Vibration Simplification Assumptions: The pipeline is assumed to perform only horizontal simple harmonic vibration with constant amplitude and frequency. Additional motions such as torsion and inclination during vibration are neglected to focus on the direct disturbance effect of horizontal vibration on the liquid film.

2.2 Mathematical Model

The pipe’s simple harmonic vibration is computed as follows:

x=Acos(ωt)=Acos(2πft)(1)

where A is the amplitude (in meters), ω is the angular frequency of vibration (in radians per second), f is the vibration frequency (in Hertz), t is the vibration time (in seconds), and x is the displacement of the circular pipe during vibration (in meters). This study simulates gas-liquid two-phase flow using the VOF (Volume of Fluid) model. Developed by Nichols et al. [34], this technique uses fluid volume computations to track interface motion. It efficiently resolves complicated free boundary conditions by monitoring volume fractions across computing domains and solving momentum equations [35]. Liquid is the primary phase, and gas is the secondary phase in the model, which tracks phase volumes. Continuity equations define the volume proportion of each phase.

Continuity equation:

∂∂t(αlρl)+▽⋅(αlρlul)=Sl(2)

∂∂t(αgρg)+▽⋅(αgρgug)=Sg(3)

The total integral satisfies the following conditions:

αl+αg=1(4)

In the formula, Sg and Sl represent the mass sources of the gas and liquid phases, respectively; αg and αl denote the volume fractions of the gas and liquid phases; ρg and ρl are the densities of the gas and liquid phases (kg/m3); ug and ul are the velocities of the gas and liquid phases (m/s).

Momentum equation:

∂∂t(ρu)+▽⋅(ρuu)=−▽p+▽⋅[μ(▽u+▽uT)]+ρg+Fσ(5)

u is the velocity of the gas-liquid two-phase (m/s); g is the gravitational acceleration (m/s2); Fσ is the surface tension of the phase interface (kg/(m2∙s2)); P is the pressure (Pa); μ denotes the dynamic viscosity, kg/(m∙s).

Energy equation:

∂∂t(ρE)+▽⋅[u(ρE+p]=▽⋅(λ▽T)+Sh(6)

φ=αlφl+(1−αl)φg(7)

where: λ is the thermal conductivity (W/m·K); E is the energy per unit mass (J/kg); T is the temperature (K); φl and φg are the thermal properties of the gas and liquid phases, respectively.

The source term Sh in the energy equation is derived from the product of the mass transfer rate and latent heat. The phase interface surface tension Fσ is calculated using the continuum surface force (CSF) model [36], which describes constant surface tension as:

Fσ=σρκ▽αl12ρl+ρg(8)

where κ is the surface curvature (1/m), and σ is the surface tension coefficient (N/m).

The surface curvature κ is defined as follows:

κ=▽⋅n=▽αl|▽αl|(9)

In numerical studies, the mass source term is calculated using Lee’s [37] phase transition model. This model posits that phase transitions are driven by the temperature difference between the interface and the saturation temperature, with the transition frequency positively correlated with this temperature difference.

The interfacial liquid-gas mass transfer rate can be obtained from the Lee model.

Sg=rαgρgTs−TgTsTg<Ts(10)

Sl=rαlρlTl−TsTsTl>Ts

where r is the mass transfer coefficient, interpreted as the relaxation time.

The phase interface area density (Ai) is a key parameter for calculating mass transfer coefficients, defined as the phase interface area per unit volume.

Ai=AfaVfa(11)

In this formula, Afa denotes the phase interface area in square meters (m2), while Vfa represents the volume per unit area at the interface in cubic meters (m3). For simple flow conditions such as nucleate boiling and fog condensation, the interfacial area (Ai) can be calculated using the formula.

The phase change factor r is challenging to calculate. By drawing on the evaporation-condensation kinetics theory and Hertz-Knudsen equation [38], the expression is given as follows:

r=6dβM2πRTsLαlρlρl−ρg Evaporate(12)

r=6dβM2πRTsLαlρgρl−ρg Condense

where β is the adaptation coefficient, d is the diameter (m), M is the mass transfer rate per unit area (kg/m2·s), R is the universal gas constant (8.314 J/mol·K), and L is the latent heat (J/kg).

The proportion of sensible heat transmission is quite minimal when the R-value is too low, which makes it appropriate for computations resembling those involving single-phase heat transfer. On the other hand, an unreasonably high r-value speeds up the liquid-to-gas phase transition and produces unreliable computational outcomes. Usually, r is chosen as an empirical constant whose value is contingent upon particular numerical models and research settings. Figs. 4 and 5 shows how the average phase interface temperature and heat transfer coefficient change with r. The heat transfer coefficient varies by less than 1.5% at r = 3 × 104 s−1, which is within the r-value range (104–106 s−1) that Shen et al. [39] advised for water at T = 373 K. At this number, the average phase interface temperature likewise stabilizes, confirming that r = 3 × 104 s−1 is the best option. The periodic average heat transfer coefficient deviation stays below 4% within this range, demonstrating outstanding value stability, according to additional sensitivity research.

Figure 4: Shows the exterior condensation heat transfer coefficient as a function of r.

Figure 5: Shows how the average phase interface temperature changes with r.

2.3 Numerical Method

For the transient simulation of liquid film oscillation on pipe surfaces, the ANSYS Fluent program was chosen. The general model adopted a pressure-based solution that took gravity into account. With PRESTO! for pressure discretization, Geo-Reconstruct for volume fraction discretization, and Quick for gradient discretization, the SIMPLE algorithm combined pressure and velocity fields. The momentum and energy equations were solved using second-order windward methods, while the transient equations were solved using second-order implicit schemes with default relaxation factors.

The velocity of the condensed film in this study is relatively low. To accurately characterize the interaction between the liquid film and gas phase, the SST k-ω turbulence model was adopted in this work. This model is capable of handling flows at low Reynolds numbers and thus is suitable for solving the two-phase flow field of the liquid film under low-velocity conditions [40].

The following equation needs to be solved:

k equation:

∂∂t(ρκ)+∂∂xj(ρκvj→)=∂∂xj[(μ+μtσk)∂k∂xj]+Pk−βρkω+Pkb(13)

ω equation:

∂∂t(ρω)+∂∂xj(ρωvj→)=∂∂xj[(μ+μtσω)∂ω∂xj]+αωκPk−βρω2+Pωb(14)

To capture the gas-liquid interface with high precision, the boundary layer adjacent to the wall was refined using a denser mesh configuration. The thickness of the first mesh layer adjacent to the smooth tube wall was set to 0.05 mm with a mesh growth rate of 1.2. In this numerical study, the y+ values of the tube boundary layer were maintained within the range of 1 to 2. The time step size was determined by comprehensively considering the CFL stability criterion and the requirement for capturing the characteristics of simple harmonic vibration. Based on the minimum mesh size, maximum flow velocity, and a CFL number of 0.3, the time step was finally set to 1 × 10−4 s. The convergence criteria were referenced to the ANSYS Fluent convergence specifications. The residuals of the continuity, momentum, and energy equations were required to be lower than 1 × 10−6, while the residual of the phase volume fraction equation was set to be below 1 × 10−4. Additionally, the relative deviations of parameters such as the wall heat transfer coefficient and liquid film thickness between adjacent time steps were required to be less than 0.1 and remain stable.

To ensure favorable convergence performance, six grid schemes with distinct cell counts and sizes were implemented, with a trade-off between computational accuracy and efficiency. Fig. 6 illustrates the variation in the average condensation heat transfer coefficient on the outer wall of the circular tube with grid size under identical operating conditions, once the condensation heat transfer process attains a steady state (t > 0.5 s). Fig. 7 depicts the variations in the local liquid film thickness and vortex strength at the 225° near-wall position with grid size. When the cell number exceeds 79,681, the variation amplitude of the average condensation heat transfer coefficient is less than ±5%, and the fluctuations in the local liquid film thickness and vortex strength are below 3%, demonstrating grid independence of the numerical results. Thus, the grid scheme with 69,841 cells was selected for subsequent simulations. The identical grid generation strategy was applied to all seven tube types to maintain consistent simulation accuracy.

Figure 6: Grid independence verification.

Figure 7: Local heat transfer coeff. & vorticity vs. grid number.

3  Results and Discussion

This section investigates how vibration affects liquid film separation from circular pipe exteriors. Through the use of variables including temperature, wall shear force, and near-wall velocity, it also examines the effects of oval pipe curvature and the length-to-diameter ratio of water droplets on film rupture. In Fig. 8, the negative x-axis orientation is defined as 0°, with angles increasing clockwise, to standardize the circumferential arrangement of pipes. At 225° along the pipe wall is Point A.

Figure 8: Angle diagram.

3.1 Model Validation

There are two parts to the model validation in this study. The heat transfer coefficient α for near-steady-state condensation on the exterior of a horizontal cylindrical pipe was first simulated in a vibration-free environment. The outcomes were then contrasted with those determined using the modified Nusselt formula in a steam flow scenario (Fig. 9). The degree of disparity was used to confirm the accuracy of the heat transfer parameter settings in the numerical model. In the first phase of numerical modeling, the external condensation heat transfer coefficient increased sharply as the steam traveling downward quickly condensed on the low-temperature bare pipe wall. As the condensation process went on, the condensed liquid slowly clung to the pipe wall as a whole, creating a liquid layer that quickly raised the thermal resistance of the wall. As a result, heat transfer efficiency immediately decreased until evaporation and condensation reached dynamic equilibrium. A discrepancy exists between the steady-state external condensation heat transfer coefficient of the tube under non-vibration conditions and the calculated value. This deviation stems from the fact that the modified Nusselt formula is a theoretically simplified model, which assumes a laminar and uniformly distributed liquid film along the tube wall while neglecting the local thickness inhomogeneity and micro-fluctuations of the liquid film induced by gravity in actual condensation processes. The maximum deviation is less than 10%, which verifies the reliability of the numerical results obtained from this model.

Figure 9: Shows the variation of the condensation heat transfer coefficient under non-vibration conditions.

The heat transfer coefficient α of horizontal circular pipes under various vibration frequencies was simulated and compared with experimental data from reference [41] in order to validate the dynamic parameter settings of the numerical model (Fig. 10). The reference experiment had the pipe in the middle of a container. Water overflowed to the top of the container after flowing in at a steady rate of 0.02 m/s from the bottom. While the water temperature stayed at 300 K, the pipe wall temperature was kept at 320 K. Convection heat transfer was placed at frequencies between 0.1 and 20 Hz with an amplitude of 1 mm. The deviation between experimental results and numerical simulations is approximately 5% with consistent trends. The discrepancy mainly arises from two aspects. Numerical simulations adopt idealized boundary conditions, whereas unavoidable temperature fluctuations and inlet flow turbulence exist in experiments. Additionally, the simulations fail to fully replicate the surface roughness and installation clearances of the experimental tubes, leading to mismatched vibration energy transfer efficiency between simulated and actual operating conditions. These findings confirm that the numerical method employed is suitable for predicting vibration-enhanced heat transfer performance outside tubes.

Figure 10: Shows the variation under vibration conditions.

3.2 Dynamic Detachment Behavior of the Liquid Film on the Circular Pipe’s Exterior Wall

The circular pipe is subjected to a horizontal simple harmonic vibration with a frequency of 20 Hz and an amplitude of 1 mm. The detachment of the liquid film and the dispersion of the flow field outside the circular pipe are examined in a single vibration cycle.

As seen in Fig. 11, the uniform continuous liquid film breaks under vibration at t = 1/4 T, revealing sizable bare wall surfaces at around 225° in the lower right corner and 75° in the upper left corner of the circular pipe. The residual surface of the liquid film experiences variations and creates widely spaced microcracks. The 225° zone causes the highest liquid film rupture because it is the impact direction of the oblique bypass flow. As a result, the airflow velocity component in this direction is much larger than in other regions, reaching about 0.35 m/s. The rupture in the pipe’s lower right corner keeps growing at t = 3/4 T, which leads to the breakdown of the continuous liquid film. Small droplets move and gather on the pipe’s exterior wall. Horizontal harmonic vibration at t = 1/4 T disturbs the upward airflow, changing its direction and creating vortices close to the wall surface that coincide with the liquid film rupture zones, as shown in Fig. 12 with respect to the flow field distribution. The upward airflow changes from vertical to oblique bypass flow by t = 3/4 T with sustained vibration, indicating that vibration causes significant alterations in the surrounding flow field. By producing a geometric superposition of velocity components, the vibrating pipe produces oblique bypass flow at an angle β with respect to the direction of the incoming flow.

Figure 11: Shows the cloud map of liquid film detachment in a circular pipe during one vibration cycle.

Figure 12: Shows the velocity vector distribution cloud map of the outer surface of the cylindrical pipe during one vibration cycle.

As shown in Eq. (15), the vibration velocity v of the circular pipe is:

v=dxdt=−2πfAsin⁡(2πft)(15)

The inclined flow direction of the vibrating pipe is:

tgβ1=2πfAsin⁡(2πft)/v(16)

The study above shows how vibrations’ horizontal inertial forces change the velocity distribution in the flow field surrounding the cylindrical pipe by upsetting airflow patterns upward. Wall film fluctuates and eventually ruptures as a result of this disruption. A single vibration cycle causes residual droplet movement, droplet detachment, and macroscopic disintegration of the externally vibrated liquid layer. When high-power ultrasonic vibrations are applied to the pipe’s outside, these behaviors physically correspond with the dynamic processes of film rupture, atomization, and agglomeration that Lu et al. [42] observed.

3.3 Oval Pipe Eccentricity’s Effect on the Velocity Field

For this study, three oval pipes with different curvatures were selected as research objects. The eccentricity (denoted as e) of the tubes is defined by Eq. (17), where a larger e value indicates a flatter cross-section of the oval tube. A circular tube was adopted as the reference configuration, and the eccentricities of the oval tubes are e1 = 0.618, e2 = 0.866, and e3 = 0.976 in increasing order of curvature. According to the analysis of liquid film detachment behavior in Section 3.2, the rupture process at Point A exhibits the most prominent time-dependent characteristics and intensity, rendering its liquid film stability the most representative. Consequently, Point A was selected as the key analysis position in this work for an in-depth investigation of subsequent patterns.

e=a2−b2a2(17)

As shown in Fig. 13, the velocity variation trends at Point A of the three oval pipes with different eccentricities are consistent with those of the reference circular pipe. At point A, the velocity rises rapidly to a peak within 0–5 ms before declining. The oval pipes’ velocity peaks at point A are 1.69, 3.14, and 5.79 times higher than the circular pipe’s, respectively. The cross-section of the circular pipe is centrally symmetric. The circumferential flow resistance stays constant and does not significantly deflect when the upward airflow makes contact with the pipe wall. This leads to higher kinetic energy and instantaneous velocity peaks at point A. The impact of the airflow is more evenly distributed as the eccentricity of the oval pipes increases. Improved x-axis steering gradually increases the kinetic energy dispersion, resulting in successively diminishing velocity peaks at point A. The velocity at point A enters a stable phase when t > 5 ms. Table 4 indicates that oval pipe 3’s secondary flow intensity at point A is 7.41 times greater than that of the circular pipe. As a result, secondary flow strength is enhanced, and velocity fluctuations at point A become more noticeable. This suggests that elevated eccentricity magnifies discrepancies in the circumferential pressure gradient [43].

Figure 13: Shows the velocity variation of the oval pipe at point A.

Fig. 14 demonstrates that at t = 1/4 T, among the normal velocities at point A on the wall of the four pipes, the oval pipe 2 yields the maximum increment, which is 1.23 times that of the circular pipe, 2.37 times that of oval pipe 1, and 14.2 times that of the droplet pipe 3. This observation indicates that the variation in normal velocity has no apparent correlation with the wall eccentricity.

Figure 14: Shows the normal velocity variation of the oval pipe at point A.

Fig. 15 reveals that as the pipe eccentricity increases, the equivalent diameter and central velocity of the near-wall vortices decrease synchronously. The peak vortex velocity in the droplet pipe 3 is merely 0.19 m/s, representing a reduction of 76.4%, 73.6%, and 50.4% compared with those in the circular pipe, oval pipe 1, and oval pipe 2, respectively. This phenomenon is essentially attributed to the intensified asymmetry of the flow channel induced by the increased eccentricity, which strengthens the local confinement effect of the pipe wall on the fluid and thus divides the fluid motion into small-scale local vortices. The intense interfacial shear effect of small-scale vortices accelerates the dissipation of mechanical energy, ultimately leading to a reduction in the overall near-wall velocity. These findings provide critical flow field evidence for elucidating the eccentricity-regulated coupling mechanism between flow and heat transfer.

Figure 15: Outflow field distribution of circular pipe and oval pipe.

3.4 Impact of the Water Droplet Pipe’s Length-to-Diameter Ratio on the Velocity Field

This study examines three droplet pipes with distinct aspect ratios, as detailed in Table 1. As depicted in Fig. 16, during the transient development stage at t < 5 ms, the streamlined cross-sectional profile of the droplet-shaped pipe disrupted the axial uniformity of the flow, which induced premature separation of the wall boundary layer and the formation of local vortices, thereby triggering energy dissipation. Notably, a higher aspect ratio led to more intensive energy dissipation, resulting in a negative correlation between the aspect ratio of the droplet-shaped pipe and its velocity peak. When t > 5 ms, the flow entered a stable regime where dynamic equilibrium was achieved between boundary layer separation and reattachment. The streamlined contour guided the fluid to flow smoothly along the pipe wall, and the flow velocity at point A increased monotonically with the rising aspect ratio of the droplet-shaped pipe.

Figure 16: Shows the velocity variation of the droplet pipe at point A.

As illustrated in Fig. 17, a larger aspect ratio corresponded to a higher degree of local flow channel contraction. As the fluid passed through this contracted region, a more pronounced velocity gradient was generated due to the variation in cross-sectional area, and the inhomogeneous distribution of wall shear stress further amplified the amplitude of normal velocity variation, thus yielding a positive correlation between the velocity variation amplitude and the aspect ratio.

Figure 17: Illustrates the normal velocity variation of the droplet pipe at point A.

As shown in Fig. 18, an increased aspect ratio enhanced the streamlined characteristics of the flow channel, strengthened the directional consistency of fluid movement, and restricted the formation conditions of large-scale vortices. Meanwhile, the streamlined wall accelerated the breakdown and dissipation of vortices, forcing the fragmentation of large-scale vortices into small-scale ones. Accordingly, the size of near-wall vortices exhibited a decreasing trend with the increase in the aspect ratio of the droplet-shaped pipe.

Figure 18: Outflow field distribution of the droplet pipe.

3.5 Influence of Pipe Shape on Wall Temperature

As seen in the previous analysis, the pipeline velocity has reached zero, indicating that the airflow velocity disturbance brought on by the initial vibration at phase t = 1/4 T has diminished. The temperature-regulating effect is more noticeable during this phase because there is less direct flow field disturbance to the liquid sheet. Thus, for our experiment, we choose the usual temperature change at point A in phase t = 1/4 T. All pipelines have achieved the entering gas temperature 1 mm along the normal direction of point A on their walls, as shown in Figs. 19 and 20. This suggests that temperature changes are mostly concentrated in the boundary layer area within 1 mm of the wall, and that the distribution is strongly correlated with the shape of the pipeline.

Figure 19: Shows the normal temperature variation of the oval pipe at point A.

Figure 20: Illustrates the normal temperature variation of the droplet-shaped pipe at point A.

In an oval pipe, as the eccentricity increases, the typical temperature variation rate at point A close to the wall falls. Higher centrifugal force is structurally correlated with larger eccentricity. Although vertical inflow can be guided along axial diffusion by streamlined cross-sections, excessive eccentricity exacerbates wall confinement effects, decreasing the efficiency of heat transfer and delaying the approach of point A to 373.15 K along the wall normal. The temperature response of oval pipe 1 is notably faster than that of oval pipe 3. At point A near the wall, the mainstream steam temperature is reached at a position of 1.04 mm along the wall normal, which is 21.8% longer than that of oval pipe 1. Unlike oval pipes, droplet pipes exhibit an acceleration of the typical temperature variation rate at point A near the wall as the aspect ratio increases. Droplet pipes have poorer x-axis flow field guidance and substantially smaller eccentricity changes at the flow front than oval pipes. Increased aspect ratio speeds up point A’s approach to 373.15 K by destabilizing the thermal boundary layer and intensifying wall boundary layer disturbance, but it also speeds up heat transport. Droplet-shaped pipe 3 (highest aspect ratio) has a significantly faster temperature response than droplet pipe 1, reaching the mainstream steam temperature at point A at a wall-normal distance of 0.64 mm, 21.4% shorter. Circular pipes’ heat transfer uniformity falls between large-aspect-ratio droplet pipes and high-eccentricity oval pipes due to their absence of confinement effects and streamlined guidance. Depending on comparative benchmarks, these lead to relative variances in temperature variation rates.

3.6 Pipe Shape’s Effect on Wall Shear Force

Liquid film separation is directly caused by the wall shear force. The temperature and velocity field evolutions, which together dictate the effectiveness of liquid film rupture and detachment, are closely related to its distribution.

The core mechanism governing the discrepancies in shear stress performance among pipes with different geometric configurations lies in the geometry-dominated coupling effect of velocity, temperature, and viscosity. As illustrated in Fig. 21, the geometric configuration of oval pipe 2 enables the synergistic matching of velocity gradient and temperature distribution. Combined with an optimal fluid viscosity, this pipe achieves a dynamic balance between shear driving force and liquid film resistance, thereby yielding the most favorable shear performance. In contrast, the axisymmetric uniform cross-section of the circular pipe ensures that the flow channel confinement effect is uniformly distributed along the circumferential direction, without localized confinement enhancement induced by sectional contraction or curvature. This results in moderate levels of near-wall velocity gradient, temperature variation amplitude, and fluid viscosity, thus rendering the circular pipe with mediocre shear performance among all tested specimens. For oval pipe 1, despite its substantially higher near-wall velocity variation amplitude compared with oval pipe 3, its excessively high characteristic temperature variation rate triggers a sharp decrease in near-wall fluid viscosity, which in turn induces premature liquid film breakup, disrupts the continuity of shear force transmission, and ultimately leads to significant deterioration in shear efficiency. As for oval pipe 3, its geometric characteristics give rise to insufficient near-wall velocity fluctuation, which fails to generate an effective shear driving force. Coupled with the increased liquid film resistance caused by high fluid viscosity, this structural limitation accounts for its lowest shear stress among all the pipes.

Figure 21: Shows the shear force distribution near point A of the oval pipe.

As shown in Fig. 22, the increase in aspect ratio not only elevates the velocity gradient of the near-wall fluid but also intensifies the disturbance of the wall boundary layer, thereby enhancing the temperature change rate around Point A. Under the coupling effect of these two factors, the viscosity of the near-wall fluid is adjusted to a level that facilitates shear stress generation. Consequently, the wall shear stress increases with the rising aspect ratio, with the drop-shaped tube 3 featuring the maximum aspect ratio achieving the highest shear stress of 0.067 Pa (see Table 5 for details). Wall shear stress is a critical factor governing liquid film stability. High shear stress tends to tear the liquid film, breaking it into discrete droplets and exposing partial wall areas; in contrast, low shear stress cannot overcome the surface tension of the liquid film, leaving only a residual thin liquid film. As illustrated in Fig. 23, which depicts the liquid film rupture degree around Point A at t = 1/4 T, the rupture extent of each tube is perfectly consistent with the shear stress ranking obtained previously.

Figure 22: Shows the shear force distribution near point A of the droplet-shaped pipe.

Figure 23: Liquid phase volume fraction contour maps for oval and droplet pipes at phase t = 1/4 T.

Moreover, a mixed wetting pattern is observed on the surface of all tubes. Essentially, this phenomenon arises from the combined action of circumferentially uniform shear stress and local temperature gradients, which disrupt the circumferential continuity of the liquid film and lead to a coexistence state of exposed wall regions, discrete droplets, and thin liquid films.

3.7 Heat Transfer Coefficient for Condensation in External Pipes

Fig. 24 shows two different phases in the periodic average heat transfer coefficient. Airflow and horizontal harmonic vibrations quickly break up the liquid film during the first phase (t < 5 ms), increasing the area of contact between the wall and the vapor. Film renewal is accelerated by the stacked shear forces, which cause the heat transfer coefficient to spike sharply to its peak. This change is strongly correlated with changes in flow velocity at point A. The liquid film reaches dynamic equilibrium between rupture and regeneration in the stable phase (t > 5 ms), stabilizing the heat transfer coefficient. A comparison of pipe geometries shows that water droplet pipes have the highest periodic average heat transfer coefficient, which consistently rises with the length-to-diameter ratio due to their streamlined walls. Oval pipes perform marginally better than circular pipes, emphasizing the improved heat transmission effect of non-circular cross-sections.

Figure 24: Average heat transfer coefficient on the pipe surface during one vibration cycle.

At phase t = 1/4 T, droplet pipes perform better than oval and circular pipes, according to the wall heat transfer coefficient at point A. The coefficient increases as the eccentricity and aspect ratio do, as shown in Fig. 25.

Figure 25: Heat transfer coefficient at wall point A during phase t = 1/4 T.

According to our research, because condensation heat transfer involves multi-mechanism coupling, this result does not entirely match the shear stress distribution at point A. Local heat transfer is improved by vibration-driven droplet separation and evaporation, even though oval pipes 2 and 3 show lower shear forces than circular pipes. By reducing droplet coalescence and accelerating droplet migration, near-wall secondary flow further minimizes phase change resistance. while maximizing heat transmission. These mechanisms validate the multi-mechanism synergistic properties of non-circular pipe heat transfer under dynamic operating conditions and are consistent with the phase change heat transfer study findings of Xu et al. [44].

4  Conclusion

In order to computationally simulate the flow and condensation heat transfer processes of liquid films on irregularly shaped pipes (oval and teardrop-shaped) under horizontal simple harmonic vibration, this study uses the finite volume approach. Nusselt’s analytical solution and experimental data from the literature establish the model’s dependability. The study methodically examines the effects of the pipe shape’s geometric features on the heat transfer coefficient, wall shear force, temperature distribution, and external flow field. It displays the improved heat transmission mechanism that arises from the combination of non-circular pipe shapes with dynamic vibration. The following are the main conclusions:

(1) Horizontal simple harmonic vibration can enhance flow field disturbance, inducing an oblique cross-flow at an angle of β1 relative to the incoming flow outside the tube, with the airflow velocity components in the 75° and 225° regions being significantly higher than those in other areas. The liquid film outside the vibrating tube undergoes dynamic renewal processes of breakage, shedding, and migration within one vibration cycle, which provides a crucial flow field regulation mechanism and flow condition support for vibration-enhanced external tube condensation heat transfer.

(2) The aspect ratio of teardrop-shaped tubes and the eccentricity of elliptical tubes exert a significant influence on the flow field morphology from local flow structures to the overall pattern by regulating the cross-sectional variation gradient of the flow channel, boundary layer development, and vortex evolution. An increase in the aspect ratio intensifies the contraction effect and advances boundary layer separation, which not only suppresses the generation of large-scale vortices but also accelerates the wall flow. An increase in eccentricity enhances the asymmetry of the flow field and induces a higher intensity of near-wall secondary flow, accompanied by a shortened boundary layer attachment length and more pronounced local flow separation.

(3) Differences in shear stress performance among elliptical tubes with distinct configurations stem from the velocity-temperature-viscosity coupling effect dominated by geometric parameters. The geometric configuration of elliptical tube 2 enables the synergistic matching of velocity gradient and temperature distribution. Combined with appropriate fluid viscosity, this tube achieves a dynamic balance between shear driving force and liquid film resistance, thereby attaining the optimal shear performance among the elliptical tubes.

(4) An increase in the aspect ratio of teardrop-shaped tubes not only enhances the velocity gradient of the near-wall fluid but also intensifies the disturbance of the wall boundary layer, raising the temperature change rate around Point A. Under the coupling effect of these two factors, the viscosity of the near-wall fluid is adjusted to a level more conducive to shear stress generation, ultimately leading to an increase in wall shear stress with the rising aspect ratio.

(5) Under horizontal simple harmonic vibration, the time-averaged heat transfer coefficient outside the tube exhibited a two-stage variation; the teardrop-shaped tube achieved the optimal heat transfer performance, and the heat transfer coefficient at Point A increased with the geometric parameters of non-circular tubes. The inconsistency between this coefficient and the wall shear stress distribution revealed the multi-mechanism synergy characteristics of condensation heat transfer outside non-circular tubes under dynamic conditions, providing core theoretical support for the engineering design of vibration-enhanced heat transfer technologies.

Acknowledgement: None.

Funding Statement: The authors received no specific funding.

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization, methodology, validation, writing original draft preparation, review and editing, Riyi Lin; formal analysis, data curation, Software, visualization, Bi Pang; Investigation, resources, supervision, Xinwei Wang. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: Requests for data can be made at any time.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

Nomenclature

References

1. Wang HJ. Application of vacuum phase change crude oil heating furnace technology in oilfield gathering station. Energy Conserv Pet Petrochem Ind. 2014;4(4):16–8. (In Chinese). doi:10.3969/j.issn.2095-1493.2014.004.007. [Google Scholar] [CrossRef]

2. Tang Z, Zheng WB, Hu XF. Efficient operation analysis of water jacket heaters in Shengli oilfield’s complex stations. Energy Conserv Pet Petrochem Ind. 2017;7(4):15–7. (In Chinese). doi:10.3969/j.issn.2095-1493.2017.04.005. [Google Scholar] [CrossRef]

3. Zhang YX, Yu DD, Hu CY, Wang YH, Sun D. Optimization design of oilfield regenerative heating furnace burner based on response surface method. J China Univ Petrol Ed Nat Sci. 2017;41(2):156–62. (In Chinese). doi:10.3969/j.issn.1673-5005.2017.02.019. [Google Scholar] [CrossRef]

4. Li HJ, Wang J. Research on the distribution of liquid film and heat transfer performance outside a horizontal drop-shaped tube. J N China Electr Power Univ. 2015;42(5):94–9. (In Chinese). doi:10.3969/j.ISSN.1007-2691.2015.05.16. [Google Scholar] [CrossRef]

5. Zheng SF, Wu ZY, Liu GQ, Yang YR, Sundén B, Wang XD. The condensation characteristics of individual droplets during dropwise condensation. Int Commun Heat Mass Transf. 2022;131:105836. doi:10.1016/j.icheatmasstransfer.2021.105836. [Google Scholar] [CrossRef]

6. Memos G, Kokkoris G, Constantoudis V, Lam CWE, Tripathy A, Mitridis E, et al. The role of shadowed droplets in condensation heat transfer. Int J Heat Mass Transf. 2022;197(2):123297. doi:10.1016/j.ijheatmasstransfer.2022.123297. [Google Scholar] [CrossRef]

7. Zheng T, Liu Y, Ying Y, Zhang S, Xue S, Ma X. Numerical simulation of the dropwise condensation row effect on horizontal tube bundles. Int J Therm Sci. 2025;210:109610. doi:10.1016/j.ijthermalsci.2024.109610. [Google Scholar] [CrossRef]

8. Kang J, Moon J, Ko Y, Lim SG, Yun B. Steam condensation on tube-bundle in presence of non-condensable gas under free convection. Int J Heat Mass Transf. 2021;178(5):121619. doi:10.1016/j.ijheatmasstransfer.2021.121619. [Google Scholar] [CrossRef]

9. Cheng CH, Chen HN, Aung W. Experimental study of the effect of transverse oscillation on convection heat transfer from a circular cylinder. J Heat Transf. 1997;119(3):474–82. doi:10.1115/1.2824121. [Google Scholar] [CrossRef]

10. Sane AG, Esmaeili M, Rabiee AH. Dual benefits of superhydrophobic surfaces: suppression of flow-induced vibrations and enhancement of heat transfer in cylindrical structures. Ocean Eng. 2026;344(10):123643. doi:10.1016/j.oceaneng.2025.123643. [Google Scholar] [CrossRef]

11. Amiri Delouei A, Atashafrooz M, Sajjadi H, Karimnejad S. The thermal effects of multi-walled carbon nanotube concentration on an ultrasonic vibrating finned tube heat exchanger. Int Commun Heat Mass Transf. 2022;135(10):106098. doi:10.1016/j.icheatmasstransfer.2022.106098. [Google Scholar] [CrossRef]

12. Hedeshi M, Jalali A, Arabkoohsar A, Amiri Delouei A. Nanofluid as the working fluid of an ultrasonic-assisted double-pipe counter-flow heat exchanger. J Therm Anal Calorim. 2023;148(16):8579–91. doi:10.1007/s10973-023-12102-7. [Google Scholar] [CrossRef]

13. Jalali A, Amiri Delouei A, Zaertaraghi MR, Amiri Tavasoli S. Experimental investigation on active heat transfer improvement in double-pipe heat exchangers. Processes. 2024;12(7):1333. doi:10.3390/pr12071333. [Google Scholar] [CrossRef]

14. Amiri Delouei A, Naeimi H, Sajjadi H, Atashafrooz M, Imanparast M, Chamkha AJ. An active approach to heat transfer enhancement in indirect heaters of city gate stations: an experimental modeling. Appl Therm Eng. 2024;237(1):121795. doi:10.1016/j.applthermaleng.2023.121795. [Google Scholar] [CrossRef]

15. Fu WS, Tong BH. Numerical investigation of heat transfer from a heated oscillating cylinder in a cross flow. Int J Heat Mass Transf. 2002;45(14):3033–43. doi:10.1016/S0017-9310(02)00016-9. [Google Scholar] [CrossRef]

16. Bronfenbrener L, Grinis L, Korin E. Experimental study of heat transfer intensification under vibration condition. Chem Eng Technol. 2001;24(4):367–71. doi:10.1002/1521-4125(200104)24:43.0.CO;2-P. [Google Scholar] [CrossRef]

17. Leng XL, Cheng L, Du WJ. Heat transfer properties of the vibrational pipe when fluid passes by it slowly. J Eng Thermophys. 2003;24(2):328–30. (In Chinese). [Google Scholar]

18. Go JS. Design of a microfin array heat sink using flow-induced vibration to enhance the heat transfer in the laminar flow regime. Sens Actuat A Phys. 2003;105(2):201–10. doi:10.1016/S0924-4247(03)00101-8. [Google Scholar] [CrossRef]

19. Go JS, Kim SJ, Lim G, Yun H, Lee J, Song I, et al. Heat transfer enhancement using flow-induced vibration of a microfin array. Sens Actuat A Phys. 2001;90(3):232–9. doi:10.1016/s0924-4247(01)00522-2. [Google Scholar] [CrossRef]

20. Yu JC, Li ZX, Xing C. Numerical analysis on convection heat transfer of air flow across a vibrating cylinder. J Eng Thermophys. 2006;27(4):670–2. (In Chinese). doi:10.1080/01495728108961803. [Google Scholar] [CrossRef]

21. Su YC, Ge PQ, Yan K, Hu RR. Heat transfer characteristic of a vibrating cylinder. J Vib Shock. 2011;30(10):222–3. (In Chinese). [Google Scholar]

22. Yang SA, Cha’o-Kuang C. Role of surface tension and ellipticity in laminar film condensation on a horizontal elliptical tube. Int J Heat Mass Transf. 1993;36(12):3135–41. doi:10.1016/0017-9310(93)90041-4. [Google Scholar] [CrossRef]

23. Yang SA, Hsu CH. Free- and forced-convection film condensation from a horizontal elliptic tube with a vertical plate and horizontal tube as special cases. Int J Heat Fluid Flow. 1997;18(6):567–74. doi:10.1016/S0142-727X(97)00025-8. [Google Scholar] [CrossRef]

24. Mosaad M. Mixed-convection laminar film condensation on an inclined elliptical tube. J Heat Transf. 2001;123(2):294–300. doi:10.1115/1.1338136. [Google Scholar] [CrossRef]

25. Chang TB, Yeh WY. Theoretical investigation into condensation heat transfer on horizontal elliptical tube in stationary saturated vapor with wall suction. Appl Therm Eng. 2011;31(5):946–53. doi:10.1016/j.applthermaleng.2010.11.018. [Google Scholar] [CrossRef]

26. Dutta A, Som SK, Das PK. Film condensation of saturated vapor over horizontal noncircular tubes with progressively increasing radius of curvature drawn in the direction of gravity. J Heat Transf. 2004;126(6):906–14. doi:10.1115/1.1798891. [Google Scholar] [CrossRef]

27. Zhang L, Yang S, Xu H. Experimental study on condensation heat transfer characteristics of steam on horizontal twisted elliptical tubes. Appl Energy. 2012;97(8–9):881–7. doi:10.1016/j.apenergy.2011.11.085. [Google Scholar] [CrossRef]

28. Gu HF, Chen Q, Wang HJ, Zhang Z. A study of shellside condensation of a hydrocarbon in the presence of noncondensable gas on twisted elliptical tubes. J Therm Sci Eng Appl. 2016;8(4):041013. doi:10.1115/1.4034256. [Google Scholar] [CrossRef]

29. Rawad D, Sidenkov DV. Numerical and experimental investigation of heat transfer in cable heated pipeline. In: 2018 IV International Conference on Information Technologies in Engineering Education (Inforino); 2018 Oct 23–26; Moscow, Russia. p. 1–5. doi:10.1109/INFORINO.2018.8581752. [Google Scholar] [CrossRef]

30. Deeb R, Sidenkov DV, editors. Numerical simulation of the heat transfer of staggered drop-shaped tubes bundle. In: Journal of Physics: Conference Series. Bristol, UK: IOP Publishing; 2019. p. 012135. [Google Scholar]

31. Deeb R, Sidenkov DV. Investigation of flow characteristics for drop-shaped tubes bundle using ansys package. In: 2020 V International Conference on Information Technologies in Engineering Education (Inforino); 2020 Apr 14–17; Moscow, Russia. p. 1–5. doi:10.1109/inforino48376.2020.9111775. [Google Scholar] [CrossRef]

32. Horvat A, Leskovar M, Mavko B. Comparison of heat transfer conditions in tube bundle cross-flow for different tube shapes. Int J Heat Mass Transf. 2006;49(5–6):1027–38. doi:10.1016/j.ijheatmasstransfer.2005.09.030. [Google Scholar] [CrossRef]

33. Liu ZY, Wu HY. Numerical study on filmwise condensation heat transfer based on VOF model. J Therm Sci Technol. 2014;13(2):126–30. (In Chinese). doi:10.13738/j.issn.1671-8097.2014.02.002. [Google Scholar] [CrossRef]

34. Nichols BD, Hirt CW, Hotchkiss RS. SOLA-VOF a solution algorithm for transient fluid flow with multiple free boundaries. Los Alamos, NM, USA: Los Alamos National Lab; 1980. [Google Scholar]

35. Hirt CW, Nichols BD. Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys. 1981;39(1):201–25. doi:10.1016/0021-9991(81)90145-5. [Google Scholar] [CrossRef]

36. Brackbill JU, Kothe DB, Zemach C. A continuum method for modeling surface tension. J Comput Phys. 1992;100(2):335–54. doi:10.1016/0021-9991(92)90240-Y. [Google Scholar] [CrossRef]

37. Lee WH. A pressure iteration scheme for two-phase flow modeling. In: Multiphase transport: fundamentals, reactor safety, applications. Washington, DC, USA: Hemisphere; 1980. p. 407–31. [Google Scholar]

38. Tanasawa I. Advances in condensation heat transfer. In: Advances in heat transfer Volume 21. Amsterdam, The Netherlands: Elsevier; 1991. p. 55–139. doi:10.1016/s0065-2717(08)70334-4. [Google Scholar] [CrossRef]

39. Shen Q, Sun D, Su S, Zhang N, Jin T. Development of heat and mass transfer model for condensation. Int Commun Heat Mass Transf. 2017;84(1):35–40. doi:10.1016/j.icheatmasstransfer.2017.03.009. [Google Scholar] [CrossRef]

40. Li Y, Huang H, Duan D, Shen S, Zhou D, Liu S. Non-condensation turbulence models with different near-wall treatments and solvers comparative research for three-dimensional steam ejectors. Energies. 2024;17(22):5586. doi:10.3390/en17225586. [Google Scholar] [CrossRef]

41. Jiang B, Tian MC, Leng XL, Tang YF, Pan JH. Numerical simulation of flow and heat transfer characteristics outside a periodically vibrating tube. J Hydrodyn Ser B. 2008;20(5):629–36. doi:10.1016/S1001-6058(08)60105-5. [Google Scholar] [CrossRef]

42. Lu W, Shang Y, Liu Y, Li D. Enhancing condensation droplets removal on tube through periodic ultrasonic vibration. Int J Refrig. 2025;170:412–22. doi:10.1016/j.ijrefrig.2024.12.012. [Google Scholar] [CrossRef]

43. Jiang B, Li M, Tan Q, Hu A. Analysis of falling film flow and heat transfer on an elliptical tube. Desalin Water Treat. 2019;150:1–8. doi:10.5004/dwt.2019.23617. [Google Scholar] [CrossRef]

44. Xu B, Yang J, Chen Z, Wang X. Dynamic behaviors and heat transfer of HFE-7100 droplet impingement on heated copper surfaces. Int J Heat Mass Transf. 2025;244(1):126965. doi:10.1016/j.ijheatmasstransfer.2025.126965. [Google Scholar] [CrossRef]