On the influence of inlet perturbations on slug dynamics in horizontal multiphase flow—a computational study

The experimental data of a horizontal gas–liquid slug flow are used to verify the numerical model and the developed inlet-perturbation described later in the text. The experiment was performed by TÜV SÜD National Engineering Laboratory (NEL) as part of the project Multiphase flow metrology in oil and gas production [34]. The experimental set-up is illustrated in figure 1. It consists of a straight horizontal pipe with an inner diameter of D = 0.097 18 m and a length of approximately 10 m, followed by a transparent Perspex viewing section with a length of 0.5 m, where the slug flow was recorded from the side by a high-speed RGB-camera (Casio EX-10) with a frame rate of 240  fps. The spatial resolution of the system was 512 × 384 pixel.

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The fluid properties and superficial velocities of the experimental slug flow are summarized in table 1.

Table 1. Fluid properties and superficial velocities of the considered slug flow.

	Paraflex oil	Nitrogen
Density in $\mathrm{k}\mathrm{g}\enspace {\mathrm{m}}^{-3}$	815.83	10.8
Dyn. viscosity in Pa ⋅ s	7.836 × 10−3	1.75 × 10−5
Surface tension in $\mathrm{N}\enspace {\mathrm{m}}^{-1}$	0.028 58
Superficial vel. in $\mathrm{m}\enspace {\mathrm{s}}^{-1}$	1.873	0.624

For a better assessment of the development of slug flow, a longer pipe (L = 16 m ≈ 165D) than in the experiment is considered in the numerical simulation, see figure 2. If no perturbation is used, such a long pipe is needed for the development of slug flow in the numerical simulation because of the simplified modeling of the inlet boundary conditions.

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In the following, we first introduce the mesh used for the simulations. Then, we define the random perturbation as well as the other initial and boundary conditions. Finally, the numerical discretization schemes are summarized.

2.2.1. Mesh design

In order to ensure mesh independent results, four o-grid type meshes were generated using the utility blockMesh, supplied with OpenFOAM^®. The corresponding number of cells per diameter and per one meter in the direction of the flow are summarized in table 2. Note that, the meshes m3 and m4 were designed as only one half of the geometry with a symmetry plane in the y-axis, see figure 2. Since the flow structures observed in the simulation of the complete geometry show a symmetrical behavior, this simplification is supposed to not influence the resulting flow pattern.

Table 2. Mesh parameters.

Number of cells in mesh	m1	m2	m3	m4
Per diameter	32	46	60	70
Per meter in length	80	100	150	200

Figure 3 shows the velocity profiles and liquid phase fraction at x = 5 m averaged over a time interval of 5 s. As a compromise between time expenses and accuracy, mesh m3 was selected for further investigations. In total, this mesh consisted of approximately 2.8 million cells. The first quadrant of the pipe cross section of mesh m3 is illustrated in figure 4.

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2.2.2. Boundary conditions and random perturbation

The inlet cross section was bisected horizontally as indicated in figure 2. In the lower half, the liquid volume fraction, α, was set to 1. In the upper half, it was set to 0. For t = 0, it was assumed that the pipe is only filled with gas. Altogether, this leads to the following initial and boundary conditions for α:

$$\begin{equation}\alpha \left(t,x=0\right)=\begin{cases}0,\hfill & y{ >}0,\hfill \\ 1,\hfill & y{\leqslant}0,\hfill \end{cases}\quad \alpha \left(t=0,x\right)=0.\end{equation} \tag{ 1 }$$

The boundary conditions for the velocity are as follows. At the inlet (x = 0), we prescribed twice the superficial velocities for the x-component, U_x , in each phase:

$$\begin{equation}{U}_{x}\left(t,x=0\right)=\begin{cases}2{U}_{\mathrm{s}\mathrm{g}},\hfill & y{ >}0,\hfill \\ 2{U}_{\mathrm{s}\mathrm{l}},\hfill & y{\leqslant}0.\hfill \end{cases}\end{equation} \tag{ 2 }$$

Here, U_sg and U_sl denote the gas and liquid superficial velocities, respectively. For the y- and z-components, U_y and U_z , random perturbations were defined as follows:

$$\begin{equation}{U}_{y}\left(t,x=0\right)=\begin{cases}2{U}_{\mathrm{s}\mathrm{g}}{{\Delta}}_{y}\left(t\right),\hfill & y{ >}0\hfill \\ 2{U}_{\mathrm{s}\mathrm{l}}{{\Delta}}_{y}\left(t\right),\hfill & y{\leqslant}0\hfill \end{cases},\end{equation} \tag{ 3 }$$

$$\begin{equation}{U}_{z}\left(t,x=0\right)=\begin{cases}2{U}_{\mathrm{s}\mathrm{g}}{{\Delta}}_{z}\left(t\right),\hfill & y{ >}0\hfill \\ 2{U}_{\mathrm{s}\mathrm{l}}{{\Delta}}_{z}\left(t\right),\hfill & y{\leqslant}0\hfill \end{cases},\end{equation} \tag{ 4 }$$

where Δ_y (t) and Δ_z (t) denote the displacement of the velocity vector in y- and z-direction in polar coordinates, i.e.,

$$\begin{equation}{{\Delta}}_{y}\left(t\right)=r\left(t\right)\mathrm{cos}\left(\varphi \left(t\right)\right),\hspace{25.0pt}{{\Delta}}_{z}\left(t\right)=r\left(t\right)\mathrm{sin}\left(\varphi \left(t\right)\right).\end{equation} \tag{ 5 }$$

The random parameters r(t) and ϕ(t) are assumed to be uniformly distributed:

$$\begin{equation}r\left(t\right)\sim \mathcal{U}\left[0,{R}_{\text{pert}}\right],\hspace{25.0pt}\varphi \left(t\right)\sim \mathcal{U}\left[0,2\pi \right].\end{equation} \tag{ 6 }$$

Here, $\mathcal{U}\left[a,b\right]$ denotes the uniform distribution over the interval $\left[a,b\right]$. The parameter R_pert determines the maximal possible amplitude of the perturbation. In the following, R_pert is set to 0.025, 0.05, 0.1, 0.2, and 0.4. This corresponds to a perturbation of the velocity vector in y- and z-direction of maximally 2.5, 5, 10, 20, and 40 per cent of the velocity in flow direction.

At the outlet, a constant pressure boundary condition was used. The walls were treated as hydraulically smooth with classical no-slip boundary conditions.

2.2.3. Numerical schemes

In the following, we introduce the methods used for the analysis of the results. First, it is described how the liquid level time series were extracted from the experimental video observations as well as from the simulation results. Afterwards, we introduce the slug characteristics that are used in the paper. Finally, the frequency analysis and its parameters are summarized.

2.3.1. Extraction of liquid level time series

For the validation of the simulated slug flow with the experimental slug flow, the time series of the vertical position of the gas-liquid interface at a certain x-position are considered. This non-dimensional parameter represents the dynamics of the gas-liquid flow pattern and has a range of [0, 1] with respect to the inner pipe diameter D. It is hereinafter also referred to as the liquid level time series at a certain x-position, denoted by h_l(x, t).

For the experimental data, the liquid level time series are extracted from high speed video observations. Here, the video observations represent a two-dimensional projection of the three-dimensional flow from the side, i.e., in the x–y plane. To extract the liquid level time series from the video, an image processing routine is used, which was developed in our earlier work [38]. Since the flow is observed through a transparent Perspex pipe wall with a thickness of 0.0214 m, the liquid level observed from outside is distorted in y-direction compared to the real liquid level inside the pipe. Therefore, a correction was applied, which is based on Snell-Descartes law and trigonometry. Details on this correction algorithm can be found in our earlier work [39]. The slug flow was recorded for approximately 122 s. Note that, only 94% of the inner diameter of the pipe are visible in the experimental observation, due to the construction of the Perspex viewing section. The lowest and highest 3% of the pipe cannot be seen because of tie bars that were needed for the installation of the Perspex pipe section. Nevertheless, the liquid level time series extracted from this observation represents the dynamics of the gas-liquid interface and reveals temporal characteristics of the flow structures, such as slugs or waves.

From the simulation results, the liquid level time series at a fixed position x are approximated for all t ∈ [t₁, t₂] by the mean value of the liquid volume fraction α over the vertical line in this cross section through the middle of the pipe:

$$\begin{equation}{h}_{\mathrm{l}}\left(x,t\right)\approx {h}_{\mathrm{l}}^{\text{sim}}\left(x,t\right)=\frac{1}{D}{\int }_{-D/2}^{D/2}\alpha \left(t,x,y,z=0\right)\mathrm{d}y.\end{equation} \tag{ 7 }$$

For the analysis of the simulation results, we considered a time interval of 50 s (t₁ = 10 s, t₂ = 60 s). Note that, the chosen time interval is large enough that there is no dependency of the results on time. This was checked by subdividing the considered time interval into two smaller intervals of half size and comparing the results on these intervals with each other. It was found that, for the two intervals [10 s, 35 s] and [35 s, 60 s], the absolute error of the mean values was lower than 0.0022 for all x-positions (x ∈ {6 m, 8 m, ..., 14 m}) and all perturbation amplitudes (R_pert ∈ {0, 0.025, 0.05, 0.1, 0.2, 0.4}). For the standard deviations, the absolute error was below 0.0066 for all cases.

2.3.2. Slug characteristics

Slug flow is characterized by a continuous liquid phase with coherent blocks of aerated liquid called slugs, which are separated by volumes of gas and moving on top of a slowly flowing liquid layer downstream the pipe at approximately the same velocity as the gas [12, 20, 40], as shown in figure 5. Slugs are typically quantified by their length and time scales, such as the slug body length L_b, the slug unit length L_s, the time of a slug body passing by at a fixed position T_b and the time of a slug unit passing by at a fixed position T_s [41, 42]. The slug unit with corresponding length and time scales is illustrated in figure 5. Note that L_b is related to T_b and L_s is related to T_s with the use of corresponding translational velocities. In the following, T_b will be referred to as slug body time and T_s will be called slug unit time. Based on the slug unit time, T_s, the slug frequency is given by ${f}_{\mathrm{s}}={T}_{\mathrm{s}}^{-1}$. Then, the mean or averaged slug frequency can be calculated by

$$\begin{equation}\bar{{f}_{\mathrm{s}}}=\frac{1}{{\bar{T}}_{\mathrm{s}}}\quad \text{with}\;\bar{{T}_{\mathrm{s}}}=\frac{1}{{N}_{\mathrm{s}}}\sum _{i=1}^{{N}_{\mathrm{s}}}{T}_{{\mathrm{s}}_{i}},\end{equation} \tag{ 8 }$$

where N_s denotes the number of slugs in the considered time interval [43].

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For the analysis of the slug flow in this contribution, the focus is on the time scales and interface dynamics, derived from the liquid level time series. To detect the slug fronts and slug rears in the liquid level time series, a threshold for h_l is set, as illustrated in figure 5. This threshold needs to be chosen carefully and individually for all time series. It should not be too high, otherwise larger slugs are separated by their entrained gas bubbles. However, it should be high enough to avoid the detection of large amplitude waves.

2.3.3. Frequency analysis

Figure 6 shows snapshots of the simulated slug flow at time t = 6.5 s for the different amplitudes. These snapshots are gas volume fraction fields in a longitudinal section through the middle of the pipe (z = 0) between x ≈ 9 m and x ≈ 14 m. Without perturbation (R_pert = 0), one cannot observe any slugs in the shown section of the pipe (see top picture in figure 6). In this case, the slugs are formed further downstream than 14 m. For a perturbation amplitude of R_pert = 0.025, slug flow can be observed approximately 10.75 m downstream of the inlet. For higher amplitudes, the onset of slug initiation occurs further upstream. For R_pert = 0.025, R_pert = 0.05, and R_pert = 0.1, narrow slug precursors are visible between x ≈ 9.5 m and x ≈ 11 m and larger slugs can be observed further downstream in the pipe. This illustrates how the slugs grow when travelling downstream the pipe. Based on this slug growth, it can be concluded that the slugs appearing larger in the snapshots were formed further upstream the pipe compared to the ones that appear smaller. Altogether, one observes that, the higher the amplitude of the random perturbation at the inlet, the earlier slugs occur in the pipe.

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Figure 7 shows the mean value (a) and standard deviation (b) of the liquid level time series at different cross sections between x = 6 m and x = 14 m for different perturbation amplitudes. In figure 7(a), one observes a decrease in the mean liquid level from approximately 0.80 to 0.78 for all amplitudes. This decrease indicates the development of the slug flow and can be explained as follows. As long as there are no slugs in the pipe, the gas phase flows with a much higher velocity than the liquid phase. When slugs are formed, they block the gas flow. Thus, the gas phase is decelerated. Furthermore, the liquid slugs are accelerated by the faster gas flow. Therefore, due to conservation of mass, the mean liquid level decreases during slug flow. The larger the perturbation amplitude, the earlier the decrease of the liquid level occurs. This shows that the random perturbation enhances the development of slug flow in the pipe. For the simulation without perturbation (R_pert = 0), no slug flow is observed between x = 6 m and x = 14 m. Therefore, we additionally extracted the liquid level time series for x = 15 m, where slugs can be observed in the snapshots of the gas volume fraction fields (not shown). As expected, the mean liquid level decreases for R_pert = 0 at x = 15 m indicating the onset of slug flow.

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Figure 7(b) shows an increase in the standard deviation of the liquid level for all perturbation amplitudes. The higher the perturbation amplitude, the faster this increase occurs. Since the standard deviation still changes at position x = 14 m, it can be concluded that the slug flow in the pipe is still evolving.

In the following, a frequency analysis of the liquid level time series is performed in order to determine the dominant frequencies of the interface dynamics. Considering the evolution of the flow pattern downstream along the pipe, one observes the following stages: (1) stratified flow, (2) development of waves, (3) wave growth and slug precursors, and (4) slug flow. The position, where the flow changes from one pattern to another, is influenced by the amplitude of the perturbation prescribed on the inlet. The higher the perturbation amplitude, the earlier these changes occur. Figure 8 illustrates this observation. It shows the PSD of the liquid level time series at position x = 10 m for two different perturbation amplitudes, R_pert = 0.025 (left picture) and R_pert = 0.2 (right picture), which correspond to two different stages of the evolving flow pattern. For R_pert = 0.025, the flow is still wavy at x = 10 m, whereas for R_pert = 0.2, slugs can already be observed at this position. For wavy flow, the corresponding amplitudes of the PSD are more than an order of magnitude smaller (see left picture in figure 8) than for slug flow (see right picture in figure 8). This significant difference can be explained by the fact that for slug flow there is more power present in the time signal than for wavy flow. On the other hand, considering the frequencies that correspond to the highest peaks in the PSD, one observes a decrease for higher perturbation amplitudes. Therefore, the occurrence of slug flow in the pipe can also be detected by considering the most dominant frequencies of the FFT/PSD spectra at different downstream positions and comparing them with each other.

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Figure 9 shows the dominant frequencies derived by FFT (a) and pwelch (b) of the liquid level time series at different cross sections between x = 6 m and x = 14 m for the perturbation amplitudes R_pert ∈ {0.025, 0.05, 0.1, 0.2, 0.4} and between x = 6 m and x = 16 m for R_pert = 0. In figure 9(a), a drop of the frequencies from approximately 4 Hz to 7 Hz down to 2 Hz and less can be observed for all perturbation amplitudes. Since the corresponding amplitudes of the FFT are much lower for the higher frequencies than for the lower ones, it can be concluded that the higher frequencies obtained closer to the inlet indicate wavy flow, whereas the lower frequencies observed further downstream correspond to slug flow. Furthermore, all positive perturbation amplitudes show a 'convergence' of the most dominant frequency towards a value of approximately 1.3 Hz. The same behavior can be observed for the most dominant frequencies obtained with the function pwelch, see figure 9(b). This shows that, in sufficient distance from the inlet, there is no dependence of the parameters on the initial amplitude of the perturbations. This means that our random perturbation does not prescribe a certain frequency on the flow as it is the case for sinusoidal perturbations, see appendix A.

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For R_pert = 0, no drop of the most dominant frequencies can be observed between x = 6 m and x = 14 m. Therefore, liquid level time series were additionally extracted for x = 15 m and x = 16 m, see red dashed lines in figures 9(a) and (b). The drop of the most dominant frequencies between x = 14 m and x = 16 m indicates the onset of slug flow in the pipe.

The most dominant frequencies obtained by applying FFT and pwelch to the experimental liquid level time series are plotted as diamonds in figures 9(a) and (b), respectively. One observes good agreement between experiment and simulation for the three highest perturbation amplitudes R_pert = 0.1, R_pert = 0.2, and R_pert = 0.4. Thus, the dynamics of the interface between the phases are reproduced well by the numerical simulation.

Comparing figure 8 with the bottom pictures in figure A2 (see appendix A), one can see a qualitative difference between the spectra of the random perturbations and the spectra of the sinusoidal perturbations. While the spectra of the sinusoidal perturbations have just single peaks at the agitation frequency and its multiples, the spectra of the random perturbation show several peaks in a broader range of frequencies. This shows that, the sinusoidal perturbations force a periodic slug flow, whereas the random perturbations lead to a more intrinsic behavior of the flow.

In the following, we investigate the influence of the perturbation amplitude on slug characteristics. For this, we first determine the time scales T_s and T_b (see section 2.3.2) for the different perturbation amplitudes at three different positions in the pipe (x = 10 m, x = 12 m, and x = 14 m). Furthermore, these time scales are also calculated for the experimental liquid level time series, which were derived from video observations at x = 10 m. As already mentioned in section 2.3.2, the threshold for determining slugs has to be adapted to each case individually to avoid splitting up slugs and/or classifying large amplitude waves as slugs. Table 5 summarizes the thresholds used for the analysis of the simulation data. Note that, the thresholds are slightly decreasing for increasing perturbation amplitude and increasing distance from the inlet. This can be explained as follows. For more evolved slugs that have already increased in size, the risk of counting one large slug as two increases, but the risk of recognizing a large amplitude waves as a slug decreases. For the experimental data, a threshold of 0.85 was used.

Figure 10 shows the mean values for the two characteristic time scales, slug unit time T_s (a) and slug body time T_b (b). When observing a fixed x-position in the pipe, the first one describes the length of the time interval between two consecutive slug fronts passing by. Hence, its inverse can be associated with slug frequency. The second one gives the length of time that the observed pipe cross section is fully filled with liquid during one slug passing by. Both, mean slug unit time and mean slug body time, become larger for higher perturbation amplitudes. This can be explained as follows. The higher the perturbation amplitude, the earlier slugs occur in the pipe. If we now consider a fixed position in the pipe, the slugs that were formed further upstream in the pipe already grew in size. Hence, in these cases, the time scales T_s and T_b are larger.

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The diamonds in figure 10 show the mean slug unit time (a) and mean slug body time (b) derived from the corresponding experimental liquid level time series. A comparison between experiment and simulation shows that the mean slug unit time is overestimated by the numerical simulation, whereas the mean slug body time is underestimated. Altogether, it can be concluded that the slug flow at x = 10 m is still evolving, both in experiment and simulation. It is still subject of current research to correctly predict the dynamic phenomena of the formation and evolution of slug flow by numerical simulations. Furthermore, since the boundary conditions on the inlet are usually not modeled in detail, one cannot expect that a specific position in the experiment fits exactly to the corresponding position in the simulation. On the other hand, if fully developed slug flow is considered, a detailed modeling of the inlet conditions is probably not necessary. In this case, a random perturbation can be used to enhance the formation of slugs in the pipe.

Figure 11 shows the mean slug frequency $\bar{{f}_{\mathrm{s}}}$, see equation (8), for the different perturbation amplitudes in comparison with experimental data. This parameter can also be derived by counting the number of slugs in the considered time interval and dividing the result by the length of the time interval:

$$\begin{equation*}\bar{{f}_{\mathrm{s}}}=\frac{\text{number}\;\text{of}\;\text{slugs}\;\text{in}\;\left[{t}_{1},{t}_{2}\right]}{{t}_{2}-{t}_{1}}.\end{equation*}$$

Note that, for the smallest perturbation amplitudes, R_pert = 0.025 and R_pert = 0.05, the mean slug frequency at x = 10 m is not representative because too less slugs have been observed in the time interval of 50 s, which was considered in the numerical simulations. For these cases, slug flow is just about to start. This indicates that the positions considered in the experiment and in the simulation do not match. Since the boundary conditions have not been modeled in detail, the evaluation of the numerical simulations probably needs to be shifted to a position further downstream in the pipe to match the experimental data.

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Figure 12 shows histograms of the inverse slug unit time, ${T}_{\mathrm{s}}^{-1}$, for three perturbation amplitudes (R_pert = 0.1, R_pert = 0.2, and R_pert = 0.4) in comparison with experimental data. Note that, for the two lowest perturbation amplitudes, R_pert = 0.025 and R_pert = 0.05, slug flow is just evolving and, therefore, it does not make sense to consider histograms due to the low number of slugs observed at this stage. In all the pictures in figure 12, the blue scale on the left indicates how often a certain time difference between two consecutive slugs occurs. Thus, the sum over all counts gives the total number of slugs observed in the considered time interval. Since the experimental liquid level time series are obtained from a time interval of more than 2 min, the counts in figure 12(d) are much higher than for the simulation data (a)–(c), where only 50 s have been considered. Therefore, we also plotted the fitted probability density functions (pdfs) (red lines in figure 12), which are independent of the total number of counts. Their shapes show good agreement between experiment and simulation, especially for the case R_pert = 0.1.

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