International Journal of Photoenergy

Volume 2017 (2017), Article ID 5128345, 8 pages

https://doi.org/10.1155/2017/5128345

## Numerical Simulation of Bubble Free Rise after Sudden Contraction Using the Front-Tracking Method

^{1}School of Mechanical and Electrical Engineering, Nanchang University, Nanchang, Jiangxi 330031, China^{2}Shangrao Normal College, Shangrao, Jiangxi 334001, China

Correspondence should be addressed to Peisheng Li

Received 29 July 2017; Revised 7 September 2017; Accepted 11 September 2017; Published 24 October 2017

Academic Editor: Ben Xu

Copyright © 2017 Ying Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on the front-tracking method (FTM), the movement of a single bubble that rose freely in a transverse ridged tube was simulated to analyze the influence of a contractive channel on the movement of bubbles. The influence of a symmetric contractive channel on the shape, speed, and trajectory of the bubbles was analyzed by contrasting the movement with bubbles in a noncontractive channel. As the research indicates, the bubbles became more flat when they move close to the contractive section of the channel, and the bubbles become less flat when passing through the contractive section. This effect becomes more obvious with an increase in the contractive degree of the channel. The symmetric contractive channel can make the bubbles first decelerate and later accelerate, and this effect is deeply affected by Reynolds number (Re) and Eötvös number (Eo).

#### 1. Introduction

The corrugated tube has been the focus of scholars because the internal structure of the pipe can disturb the movement of the fluid and increase the surface area of the pipe; thereby, it can increase the heat transfer of the pipe. So it is widely used in various fields such as the solar hot water system because there are a number of solar collector tube and heat transfer tubes in this system. In order to improve the heat absorption and heat transfer efficiency, different types of corrugated tubes have been studied. Vicente et al. [1] studied heat transfer of the threaded pipe under different sizes by experimentation, he shows the guidelines to choose which roughness geometry offers the best performance for specific flow conditions. García et al. [2] analyzed the thermal-hydraulic behavior of three types of enhancement technique based on artificial roughness, and the results show that shape of the artificial roughness exerts a greater influence on the pressure drop characteristics than on the heat transfer augmentation. Saha [3] has studied the pressure drop characteristics of pipe with internal transverse rib turbulators on two opposite surfaces and with wire-coil inserts. Pethkool et al. [4] experimentally studied the heat transfer performance of helically corrugated tube and got the empirical formula. Kareem et al. [5] studied the three-start spirally corrugated tube by experiment and simulation, he said that this geometry with a creative spiral corrugation profile can improve the heat transfer significantly with reasonable increase in friction factor. It is obvious that scholars have made a lot of research on corrugated tubes in different structural forms and got some good results. However, most of these studies have been studied for the effects of different pipe shapes on heat transfer and pressure drop with a single fluid, while the study of multiphase flow in the tube has been rarely reported.

In practical cases, such as solar collectors and heat exchangers, the temperature of water in the tube is generally higher than the natural temperature. The gas in the water will be released and the bubbles will form. The interaction of the bubble with the contraction structure can also cause negligible disturbance to the flow in the pipe and affect its heat transfer. Therefore, we believe that the study of bubble flow in the corrugated tube has a certain reference value for improving the heat transfer of the heat exchanger, reducing the pressure drop of the helically corrugated tube and obtaining the empirical formula of thermal performance.

The method of numerical simulation of bubble motion has been well developed [6–11]. In order to describe the movement characteristics of the bubbles in the corrugated tube intuitively, it is convenient to study it with the simulation method. FTM is a classic method used in numerical simulations that has been widely used in multiphase analyses [12–14]. It can accurately capture the complex topological changes of the moving interface, and it is very important in tracking the phase interface during the research process of multiphase flow.

Therefore, in this paper, the FTM was used to simulate the free rise of a single bubble in the transverse ridged tube. However, the cross-sectional shape of the transverse ridged tube is not a regular rectangle like an ordinary pipe, so we set the contraction channel by modifying the boundary position. And we analyzed the coupling effect between the bubble and the solid wall of the channel and studied the influences of sudden contraction on the shape, velocity, and trajectory of the bubble. It is hoped that the analysis of the flow field under coupling effect between the bubble and the solid wall of the channel will provide a theoretical help for the design of the heat transfer pipe in a solar hot water system, so that the internal flow of the transverse ridged tube can be more reasonable and achieve better heat transfer effect.

#### 2. Mathematical Model, Numerical Methods, and Physical Model

##### 2.1. Tracking and Processing the N-S Equations at the Phase Interface

The surface tension is concentrated on the interface when the control equation is applied to the whole calculation area. We consider introducing a source term and is applied by multiplying the volume force and the function. The function is nonzero only on the interface. Using the amendments above, the two-dimensional incompressible flow momentum equation can be expressed as where is the two-dimensional Dirac function, is the position of the surface, the integral is along the interface , is the interface normal, is the interface tension coefficient, is the secondary interface average curvature, and and are noncontinuous density field and viscosity field.

To solve the equation above, we split (4) and remove the pressure introducing the temporary speed . Thus, an equation containing only the convection term and the diffusion term is obtained:

By solving the equation, we obtain the temporary velocity field without pressure, and then add the pressure term to obtain the general velocity discrete equation:

In the formula, is a discrete form of the Hamiltonian operator whose step size is *h* and because of the speed it needs to meet the discrete form of mass conservation:

Then the pressure Poisson equation is obtained:

The CSF (continuous surface force method) method given by Brackbill [15] was used to calculate the surface tension. Take the surface tension on the unit line to study, for the two-dimensional flow is Get the surface tension on the unit interface element: Here, is the tangent vector on the interface.

##### 2.2. Reconstruction of Density and Viscosity

For immiscible and incompressible fluids, the fluid maintains its characteristics at both sides of the interface, so there is a step in the physical properties at the interface. When the interface moves, the density and viscosity distribution of the interface also change, so the Heaviside function is introduced as an indicator to characterize the change: where is the distance from the given point to the interface, and is the absolute value of the thickness of the transition zone between the two fluids. So the density and viscosity distribution is Here, are the density and viscosity of the two fluids.

##### 2.3. Interface Movement

The moving interface is typically combined with a fixed mesh, and the information exchange between the interface and the grid is achieved by the area weight function. With the bilinear interpolation method, the abrupt change density and the surface tension of the interface are changed from the interface to the fixed grid. Finally, we use a simple first-order time integral to obtain the interface position of the next time. Here, is the position of surface and is velocity.

##### 2.4. Physical Model

In the actual production industry, the geometric conditions of the contractive channel are varied, and to facilitate the research, only the unit shown in Figure 1 is extracted as the object of this paper and the upper and lower borders are set to periodic boundary conditions to satisfy the structure of the transverse ridged tube. Since the three-dimensional model will spend much time, so we use the two-dimensional model. The contractive ratio is defined as *E* = *L*/*M* and is used to describe the contractive degree. *D* is the diameter of the bubble.