1. Introduction
Vibration issue in piping is one of the major causes of downtime, leakage, explosion, and fire in industrial plants. Turbulence within flowing fluid, especially two-phase flows, may results in fluid-induced excitation. Among various flow regimes, slug flow is notorious to cause fluid-induced vibration (FIV) due to its oscillating changes in pressures and phase change hence resulting in a variation of piping amplitudes [
1]. In addition, experimental and numerical investigations on dynamic behaviour of slug flow in a horizontal piping system were performed by Wang et al. [
2]. They concluded that slug transitional velocity affects the rate of system properties such as mass, stiffness, loading, and damping. As a result of this, an intense vibration occurred after the slug leaves. This demonstrates that the multi-phase fluid flowing inside a piping system can lead to fluid-induced vibration, and hence needs to be carefully investigated and monitored to avoid catastrophic damage at the piping system.
Computational fluid dynamics (CFD) provides an easier and more versatile method for designing and using large-scale computational models to predict complex two-phase flow. In addition, it offers time saving and reduced cost in characterizing the behaviour of the flow [
3]. There are several studies on slug flow that utilized CFD as the main tool of the research. Lun et al. [
3] were amongst the pioneers in using commercial CFD software to simulate horizontal two-phase flow. CFD for multiphase flows has grown greatly in popularity over the last two decades. Numerous articles have been produced on numerical simulations of gas-liquid flow for horizontal pipelines [
4,
5,
6,
7].
The general governing equations for CFD are mass conservation law and Navier-Stokes equations as expressed in Equations (1) and (2), respectively.
where
u is velocity,
is density,
is pressure and
is shearing stress and
is body force (e.g., gravity). The “.” denotes a dot product between vector while symbol
is the vector of spatial partial derivatives.
An appropriate turbulence equation must be selected to better predict the evolution of turbulence within the flow. The types of turbulence equations are categorized into large-eddy simulation (LES), hybrid approaches and Reynold-Averaged Navier-Stokes (RANS). The general form of turbulence model is expressed in Equation (3).
The turbulence model, which is formulated with five different terms, is normally applied in CFD simulations to increase accuracy, and reduce computational time. Given
as turbulence variables, the unsteady term represents time dependence of turbulent variables while the convection term is a rate of change of turbulent variables due to convection by the mean flow. The transport of turbulent variables due to the summation of material viscosity and eddy viscosity is known as diffusion term. The production term is a production rate of turbulent variables from the mean flow gradient while dissipation term refers to the rate of turbulence variables due to viscous stresses.
Table 1 shows the parameters employed for these terms for selected turbulence models. Note that Boussinesq hypothesis is employed in Spalart Allmaras (SA), k-epsilon, and k–omega models and it offers a relatively low-cost computation for the turbulence viscosity (ν
t = μ
t/
).
Spalart Allmaras is the latest turbulence equation based on one-equation eddy viscosity model developed in 1992. Hence, there is a renewed kind of interest for researchers in CFD field towards SA. It has the advantage of having economic solutions for attached flows and moderately separated flows. It solves a modelled transport equation for the kinematic eddy (turbulent) viscosity [
8]. It is also reliable in converging the simulation’s iterative process, hence providing a good result to the simulation [
9]. Besides SA, detached-eddy-simulations (DES), which is a hybrid model of Reynolds-averaged Navier-Stokes (RANS) and large-eddy simulation (LES) can also be used to describe a more accurate flow. It uses RANS method for near wall computation and dynamic LES for a region a bit far from the wall. DES employs the distance to the closest wall as the definition for the length scale which plays a major role in determining the production level and destruction of turbulent viscosity.
There is a number of researchers investigating slug flow behaviour who performed their analysis by employing k-epsilon model as turbulence equation of choice [
10,
11,
12]. This k-epsilon assumes that the flow is fully turbulent, and the effects of molecular viscosity are negligible [
13]. The turbulence equation is popular, and it has been the industry standard for many years because there is such a rich database of results (i.e., simply because it has been around since the early 1970s). The k-epsilon, however, tends to face convergence issue in solving the governing equation. In comparison to k-omega, k-epsilon model yields better predictions for void fraction close to the wall [
13]. Unlike k-epsilon, Spalart Allmaras is claimed to provide more stability in computing the governing equation, so the tendency to face convergence issue is decreased [
9]. The reason behind this is because Spalart Allmaras model adds only one more equation to represent turbulence viscosity transport, whereas k-epsilon and k–omega models add two more, thus adding complexity/sophistication to the distribution of eddy viscosity. Based on Ref. [
14], among the various turbulence modeling methods including the time average (e.g., RANS), space average (e.g., LES) and hybrid model (e.g., DES), the DES approach is the most practical and a fairly accurate turbulence model. Within the context of slug flow simulation, the impact of the selection of turbulence equation on the resultant flow characteristics has not been elucidated yet.
Besides turbulence equation, the selection of multiphase equation is important for fluid with multiple phases to capture the best representation of liquid and gas distribution within the flow. There are a couple of types of multiphase equation available such Eulerian-Lagrangian, Eulerian-Eulerian, volume of fluid (VOF), and level set. For multi-phase flows, the choices of the multiphase models would be reflected in the mass conservation and momentum conservation equations based on the volume fraction of the phases. In this case, an additional transport equation (volume fraction) is used to track the interface between these two phases (
i,
j) by solving the continuity equation for steady incompressible flow as follows [
15]:
Several CFD packages are available in the market such as Altair® AcuSolve™, ANSYS® and STAR-CCM+. Altair® AcuSolve™ is based on finite element method while ANSYS® and STAR-CCM+ are based on finite volume method (FVM). Finite element method (FEM) is formulated based on variational principles to solve a CFD problem by minimizing an associated error function or residual. In this method, the problem domain is divided into a set of finite elements which are usually (i) triangles or quadrilaterals for those in a two dimension, and (ii) tetrahedra, hexahedra, pyramids, or wedges in three dimensions. On the other hand, the finite volume method is formulated based on approximate solution of the integral form of the conservation equations. The problem is divided into a set of non-overlapping control volumes referred to as finite volumes, cells, or elements. Despite these differences between finite volume and finite element, the outcomes produced by both methods are expected to be similar.
It should be noted that different CFD packages offer different type of multiphase models. This indirectly influences the use of certain set of equation in the development of multiphase flows within the CFD environment. For example, VOF approach is available in ANSYS
® and STAR-CCM+, while Altair
® package only offers level set approach. Several researchers who managed to develop slug flow successfully using VOF model include authors in Refs. [
4,
10,
11,
16,
17,
18,
19]. It was reported that the computational time required to solve the VOF equations is relatively low when compared to that of Eulerian–Eulerian and Eulerian–Lagrangian approaches, making VOF to be utilized more broadly in analyzing two-phase flow [
10,
12,
20]. So far, there is no literature available that reports on the use of level set approach using Altair
® AcuSolve™ in the development of two-phase slug flow.
Altair
® AcuSolve™ offers an improvisational form of standard level set known as back-and-forth error compensation and correction (BFECC). It is applied to the multi-field and field interaction model equation to enhance the definition of the volume fraction between liquid and gas. The first stagger governs the interface’s transport, while the second stagger controls its sharpness in the standard level set form. The level set with BFECC, on the other hand, includes additional stagger iterations to minimize the amount of diffusion in the solution field. In other words, this predictor-corrector type method is based on the observation that if Equation (5) were solved forward in time for one time step using a numerical integrator and then backward in time for one time step with the same method, the difference between the two copies of the solution gives the information about the numerical error which could be used to further improve the accuracy. It improves the space and time accuracy to the second order by adding level set with BFECC to an existing first order semi-Lagrangian scheme, hence developed a better image presentation than a standard level set [
21]. At present, the effectiveness of BFECC implementation with level set multiphase equation on the definition of slug flow morphology has not yet been demonstrated. A passive advection of a scalar field with a given velocity vector field
is modeled by the transport equation as:
Various types of flow regimes, including slug flow, can be induced in CFD environment by setting up appropriate boundary conditions. The obvious setting that affects the resultant flow regime is the input velocity for each phase, as calculated using the following equations:
where
and
are flow rates for gas and liquid, respectively.
In addition to that, for successful formation of slug flow, the inlet initial and boundary conditions require appropriate perturbation. There are several approaches in developing the perturbation. Thomas [
4] developed slug flow in a horizontal piping system by initializing the phasic distribution in the form of sinusoidal form. This technique was also imitated by Ban et al. [
12] to determine a transient liquid level at the pipe’s inlet cross-section. Equation (8) expresses an example of the function imposed on the interface of gas and liquid at the inlet to create a perturbation.
The notation represents the height of the gas-liquid interface from the reference which is the liquid gas interphase. The notation, is the liquid level fluctuation amplitude while notation is the wavelength and is the mixture velocity.
Imposing a certain amount of disturbance at the inlet helps in developing slug within a short computational time without depending solely on the pipe diameter to length ratio for the formation of slug. Although sinusoidal perturbation can generate the desired slug formation, the frequency of sinusoidal perturbation can affect the resultant frequency of the slug. Consequently, this approach is only applicable for the development of slug flow with a predetermined slug frequency. Instead of sinusoidal perturbation, Schmelter et al. [
22] used random perturbation to the interface between liquid and gas phases to avoid premeditating the slug behaviour, especially frequency. The random inlet perturbation can be defined using Equations (9) and (10).
where
denote the displacements of the velocity vector in y- and z-directions in polar coordinate, respectively. Based on the findings by Ref. [
22], it was shown that the higher the perturbation amplitude, the earlier the slugs occur and if the measurement is taken sufficiently far away from the inlet, the flow pattern shows the same dynamics for different perturbation amplitudes indicating that no specific frequency is imposed by the prescribed perturbation in contrast to the outcome of simulations with sinusoidal perturbations.
So far, to the best of the authors’ knowledge, there is no direct comparison in (i) types of inlet perturbations, (ii) choices of multiphase equations, and (iii) turbulence equation on the development of slug flow and their characteristics available in the literature, especially using Altair® computational package. Therefore, this research aims to construct a slug flow model by perturbing each model’s inlets with either sinusoidal and random perturbation and applying different types of multiphase and turbulence equations to the models. The models were then directly compared to characterize the parameters of slug behavior, such as slug body length-to-diameter ratio, slug velocity, and pressure gradient.