3.1 Flow characteristics

Figure 3 shows the variations of oil, water and total volume flow rates with the oil holdup $\unicode[STIX]{x1D700}_{o}(=V_{o}/V)$ , together with the experimental results of Vanegas Prada & Bannwart (Reference Vanegas Prada and Bannwart2001). As the oil holdup increases (or the water holdup decreases), the oil volume flow rate first increases, reaches a maximum and then decreases. A reasonable agreement between the present and experimental data for $\unicode[STIX]{x1D700}_{o}\leqslant 0.71$ is observed. Note that the experimental data contain fluctuations of total volume flow rate at low $\unicode[STIX]{x1D700}_{o}$ values. Since long wave components of the phase interface appear at low $\unicode[STIX]{x1D700}_{o}$ values (see later figures 12 and 13), the flow-developing length should be long enough. The experimental data were measured at approximately 60 to $120R$ away from an inlet, which might be not long enough to reach a fully developed region, whereas the present simulation ran over about $400R$ at which the flow reached a statistically steady state. However, the agreement becomes poor at high oil holdups such as at $\unicode[STIX]{x1D700}_{o}=0.83$ . This disagreement cannot be disregarded, because maximum oil volume flow rate occurs around $\unicode[STIX]{x1D700}_{o}=0.83$ in our simulation but at $\unicode[STIX]{x1D700}_{o}\approx 0.71$ in the previous experiment. For a high oil holdup, oil frequently sticks to the wall due to narrow water region, which significantly increases the pressure drop (Bai et al. Reference Bai, Chen and Joseph1992; Arney et al. Reference Arney, Bai, Guevara, Joseph and Liu1993). This oil sticking is adhesion rather than a hydrodynamic effect (Arney et al. Reference Arney, Ribeiro, Guevara, Bai and Joseph1996). The present simulation does not consider these chemical procedures, and thus there is a difference in the oil volume flow rate at a very high oil holdup. For this reason, we investigate the flow at $\unicode[STIX]{x1D700}_{o}=0.71$ in detail (rather than at $\unicode[STIX]{x1D700}_{o}=0.83$ ) at which the oil volume flow rate is high in the experiment and is well predicted by the present numerical simulation.

Figure 3.

Variations of the volume flow rates with the oil holdup: present simulation (▴, water; ▪, oil; ●, their sum); Vanegas Prada & Bannwart (Reference Vanegas Prada and Bannwart2001, experiment) at $Re_{\unicode[STIX]{x1D70F}}\approx 720$ (▵, water; ▫, oil; ○, their sum). The data of Vanegas Prada & Bannwart (Reference Vanegas Prada and Bannwart2001) are taken from the PhD thesis of Vanegas Prada (Reference Vanegas Prada1999).

When high viscosity oil alone flows in a pipe ( $\unicode[STIX]{x1D700}_{o}=1$ ) at the same wall friction ( $Re_{\unicode[STIX]{x1D70F},o}=\unicode[STIX]{x1D70C}_{o}u_{\unicode[STIX]{x1D70F},o}R/\unicode[STIX]{x1D707}_{o}=Re_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D707}_{w}/\unicode[STIX]{x1D707}_{o})(\unicode[STIX]{x1D70C}_{o}/\unicode[STIX]{x1D70C}_{w})^{0.5}\approx 0.040$ , where $Re_{\unicode[STIX]{x1D70F},o}$ and $u_{\unicode[STIX]{x1D70F},o}=\sqrt{\overline{\unicode[STIX]{x1D70F}}_{w}/\unicode[STIX]{x1D70C}_{o}}$ are the friction Reynolds number and friction velocity of oil, respectively), the volume flow rate of oil is $q_{o}/u_{\unicode[STIX]{x1D70F},o}R^{2}\approx 0.032$ which is orders of magnitude smaller than that of the core-annular flow, $q_{o}/u_{\unicode[STIX]{x1D70F}}R^{2}\approx 30{-}50$ (figure 3). Therefore, the core-annular flow is required to transport high viscosity oil. On the other hand, when water flows alone ( $\unicode[STIX]{x1D700}_{o}=0$ ), the volume flow rate is $q_{w}/u_{\unicode[STIX]{x1D70F}}R^{2}\approx 56.4$ ( $Re_{\unicode[STIX]{x1D70F}}=685$ : Wu, Baltzer & Adrian Reference Wu, Baltzer and Adrian2012). This value is comparable to the total volume flow rate of the core-annular flow, $(q_{w}+q_{o})/u_{\unicode[STIX]{x1D70F}}R^{2}$ . The total volume flow rate is nearly constant at low to relatively high $\unicode[STIX]{x1D700}_{o}$ values but rapidly decreases at high $\unicode[STIX]{x1D700}_{o}$ values for $\unicode[STIX]{x1D700}_{o}>0.83$ , whereas, with increasing $\unicode[STIX]{x1D700}_{o}$ , the volume flow rate of oil increases and then rapidly decreases, but that of water monotonically decreases. Note that the oil volume flow rate at $\unicode[STIX]{x1D700}_{o}=0.91$ is much lower than its maximum at $\unicode[STIX]{x1D700}_{o}=0.83$ , because the water region is so narrow that the core reaches very close to the pipe wall and suffers from high wall shear stress. Therefore, it is not practical to keep increasing $\unicode[STIX]{x1D700}_{o}$ because the amount of oil delivery decreases from its maximum value.

Figure 4.

Contours of the instantaneous streamwise velocity relative to the core velocity: (a) $\unicode[STIX]{x1D700}_{o}=0.91$ ; (b) $0.71$ ; (c) $0.56$ .

Figure 4 shows the contours of the instantaneous streamwise velocity relative to the core velocity for $\unicode[STIX]{x1D700}_{o}=0.91,0.71$ and $0.56$ , respectively. The core velocity, $u_{core}(=\int \!u\,\text{d}V_{o}/V_{o})$ , is the oil velocity averaged over the volume occupied by oil and is related to the superficial velocity of oil as $u_{core}=j_{o}/\unicode[STIX]{x1D700}_{o}$ . As shown in this figure, the core region contains nearly zero relative velocity, indicating that the flow in the core region is almost a plug flow moving at the speed of $u_{core}$ and has nearly zero velocity fluctuations (e.g. their root-mean-square (r.m.s.) values are less than $0.02u_{core}$ for $\unicode[STIX]{x1D700}_{o}=0.71$ ) due to the high viscosity of oil. As the core region moves at $u_{core}$ and deforms very slowly because of its high viscosity (Bai et al. Reference Bai, Kelkar and Joseph1996), the phase interface behaves like a slowly deforming wavy wall. The water flow in the annulus may be characterized as a Poiseuille–Couette flow such that the top-driven velocity is $u_{core}$ and the mean pressure gradient excluding the buoyancy is $-\text{d}P/\text{d}x_{annulus}=-\text{d}P/\text{d}x-\unicode[STIX]{x1D70C}_{w}g=2\overline{\unicode[STIX]{x1D70F}}_{w}/R-(\unicode[STIX]{x1D70C}_{w}-\unicode[STIX]{x1D70C}_{o})\unicode[STIX]{x1D700}_{o}g$ . Since the non-dimensional pressure gradient, $\unicode[STIX]{x1D6F1}_{c}=(-\text{d}P/\text{d}x_{annulus})(R-h_{avg})/(\unicode[STIX]{x1D70C}_{w}u_{core}^{2})$ , is quite small ( ${\leqslant}7.2\times 10^{-4}$ ), the flow in the annulus is close to Couette flow (Polderman, Velraeds & Knol Reference Polderman, Velraeds and Knol1986). For $\unicode[STIX]{x1D700}_{o}=0.91$ , the Reynolds number based on the mean water height is very low, i.e. $Re_{c}=u_{core}(R-h_{avg})/\unicode[STIX]{x1D708}_{w}\approx 430$ , and thus near-wall streamwise vortical structures are hardly observable. For lower oil holdups ( $\unicode[STIX]{x1D700}_{o}\leqslant 0.83$ ), $Re_{c}\approx 1150{-}3550$ and the flow in the annular region occupied by water becomes turbulent. With decreasing $\unicode[STIX]{x1D700}_{o}$ , the dominant wavelength of the phase interface as well as the average gap size $h_{avg}$ increase. The relative velocity in the water region in a reference frame moving with $u_{core}$ is positive and negative near the phase interface and the wall, respectively, which is also observed in laminar core-annular flows (Bai et al. Reference Bai, Kelkar and Joseph1996; Li & Renardy Reference Li and Renardy1999). This phenomenon is further discussed later in this paper.

Figure 5.

Streamwise distributions of the instantaneous bulk velocities of oil and water ( $\tilde{u} _{o}(x)$ and $\tilde{u} _{w}(x)$ , respectively), superficial velocity of water ( $\tilde{j}_{w}(x)$ ), half of the hydraulic diameter of the phase interface ( $\tilde{h}$ ) and the ratio of instantaneous water area to the pipe cross-sectional area ( $A_{w}/A$ ) for $\unicode[STIX]{x1D700}_{o}=0.71$ at the same instant as in figure 4(b), together with the cross-sectional shapes of instantaneous phase interface at two different streamwise locations: ——, $\tilde{u} _{o}/u_{\unicode[STIX]{x1D70F}}$ ; - - - -, $\tilde{u} _{w}/u_{\unicode[STIX]{x1D70F}}$ ; – - – - –, $\tilde{j}_{w}/u_{\unicode[STIX]{x1D70F}}$ ; – -- – -- –, $A_{w}/A$ ; $\cdots \cdots$ , $\tilde{h}$ .

Figure 5 shows the streamwise distributions of the instantaneous bulk velocities of oil and water ( $\tilde{u} _{o}$ and $\tilde{u} _{w}$ ), superficial velocity of water ( $\tilde{j}_{w}$ ), half of the hydraulic diameter of the core ( $\tilde{h}=2A_{o}/{\mathcal{P}}$ ; $A_{o}$ and ${\mathcal{P}}$ are the instantaneous area and perimeter of the core, respectively) and the ratio of instantaneous water area to the pipe cross-sectional area ( $A_{w}/A$ ) for $\unicode[STIX]{x1D700}_{o}=0.71$ , together with the cross-sectional shapes of instantaneous phase interface at two different streamwise locations, where $\tilde{u} _{o}=\int \!u(1-\unicode[STIX]{x1D713})\,\text{d}A/\!\int (1-\unicode[STIX]{x1D713})\,\text{d}A$ , $\tilde{u} _{w}=\int \!u\unicode[STIX]{x1D713}\,\text{d}A/\!\int \!\unicode[STIX]{x1D713}\,\text{d}A$ , $\tilde{j}_{w}=\int \!u\unicode[STIX]{x1D713}\,\text{d}A/\!\int \!\text{d}A$ and $\tilde{h}=\int _{0}^{2\unicode[STIX]{x03C0}}\!h^{2}\,\text{d}\unicode[STIX]{x1D703}/\!\int _{0}^{2\unicode[STIX]{x03C0}}\!h\,\text{d}\unicode[STIX]{x1D703}$ . As shown in this figure, $\tilde{u} _{w}$ is smaller than $\tilde{u} _{o}$ at all the streamwise locations, indicating that the oil overall drags the water. However, ahead of a crest (local maximum $h$ ), water locally drives oil near the phase interface (see the red contours in figure 4). The instantaneous superficial velocity of water $\tilde{j}_{w}$ varies in the streamwise direction because of non-zero radial velocity fluctuations of the phase interface. With decreasing $A_{w}/A$ , $\tilde{u} _{w}$ and $\tilde{j}_{w}(=\tilde{u} _{w}(A_{w}/A))$ decrease. When the crests of the phase interface are located very near the wall, water is almost trapped between the crests and $\tilde{j}_{w}$ rapidly decreases. This water trapping inside the wave valley between the crests was also noticed in laminar core-annular flows (Bai et al. Reference Bai, Kelkar and Joseph1996; Li & Renardy Reference Li and Renardy1999; Kouris & Tsamopoulos Reference Kouris and Tsamopoulos2002).

Figure 6.

Contours of the instantaneous pressure fluctuations ( $\unicode[STIX]{x1D700}_{o}=0.71$ ) at the same instant as in figure 4(b).

Figure 7.

Instantaneous velocity vectors and contours of the instantaneous pressure fluctuations near the crest of the interface ( $\unicode[STIX]{x1D700}_{o}=0.71$ ): (a) in a reference frame moving with the core velocity; (b) in a stationary reference frame. Here, the time is same as that in figure 4(b).

Figure 6 shows the contours of the instantaneous pressure fluctuations for $\unicode[STIX]{x1D700}_{o}=0.71$ . Since oil moves like a solid body at the speed of $u_{core}$ and drives the water, the pressures ahead of and behind the crests are high and low, respectively. The pressure in the valley region is negative and is more or less uniform. Figures 7(a) and (b) show the instantaneous velocity vectors near the crest of the phase interface in a reference frame moving with the core velocity and in a stationary reference frame, respectively, together with the contours of the instantaneous pressure fluctuations for $\unicode[STIX]{x1D700}_{o}=0.71$ . The discontinuity in the pressure fluctuations across the phase interface is due to the surface tension. In a reference frame moving with $u_{core}$ (figure 7 a), the wall moves upstream, and flow impinges on the front part of the crest, creating high pressure and forming a recirculation bubble there. Also, a positive relative velocity exists above the phase interface in the valley region. In a stationary reference frame (figure 7 b), the velocity field of water is more or less uniform except near the wall and the crest. Near the wall, the boundary layer grows in the streamwise direction, whereas the velocity near the crest rapidly changes in the radial and streamwise directions. The velocity gradient at the phase interface is very small except on the crest. Figure 8 shows the instantaneous phase interface and vortical structures in the annulus for $\unicode[STIX]{x1D700}_{o}=0.71$ . Vortical structures evolve near the wall, but differ from those of a fully developed pipe flow. They exist at the locations ahead of the crests and in the valley regions, but few appear right behind the crests. This is quite interesting because one may expect more vortices behind a crest in the form of wake. However, the gap between the crest and wall is so narrow that the vortices are weaken and flow does not evolve into turbulent shear layer behind the crest.

Figure 8.

Instantaneous phase interface and vortical structures ( $\unicode[STIX]{x1D706}_{2}=-200\,000$ (Jeong & Hussain Reference Jeong and Hussain1995)) in the annulus ( $\unicode[STIX]{x1D700}_{o}=0.71$ ).

Figure 9.

Distributions of the mean water volume fraction in the radial direction.

Figure 10.

Profiles of the mean streamwise velocities of water and oil for different oil holdups: (a) $\overline{u}_{w}$ and $\overline{u}_{o}$ ; (b) $\overline{u}_{w}^{\ast }(=\overline{u}_{w}\overline{\unicode[STIX]{x1D713}})$ . The red line denotes the mean velocity profile of a single-phase pipe flow at $Re_{\unicode[STIX]{x1D70F}}=685$ (Wu et al. Reference Wu, Baltzer and Adrian2012) and the log law of $u^{+}=2.5\ln (R-r)^{+}+5.5$ is from Eggels et al. (Reference Eggels, Unger, Weiss, Westerweel, Adrian, Friedrich and Nieuwstadt1994).

Figure 11.

Radial distributions of shear stresses in the mean streamwise momentum equation (blue lines, Reynolds shear stress; red lines, viscous shear stress; black lines, total shear stress) and the mean water volume fraction (green lines). - - - -, $\unicode[STIX]{x1D700}_{o}=0.83$ ; – - – - –, $0.71$ ; $\cdots \cdots$ , $0.56$ ; ——, single-phase pipe flow at $Re_{\unicode[STIX]{x1D70F}}=685$ (Wu et al. Reference Wu, Baltzer and Adrian2012).

Due to the wavy phase interface, there is an intermittent region near the phase interface where the fluid changes from water to oil in time or vice versa. In this region, the interaction between the water flow and phase interface is quite active, and heavily influences the flow statistics (see figures 10 and 11). Figure 9 shows the mean water volume fraction

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D713}}(r)={\displaystyle \frac{\displaystyle \iiint \unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}{\displaystyle \iiint \text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}}. & \displaystyle\end{eqnarray}$$

Here, $\overline{\unicode[STIX]{x1D713}}=1$ and 0 for water and oil only regions, respectively, and $0<\overline{\unicode[STIX]{x1D713}}<1$ for the intermittent region. Note that $\overline{\unicode[STIX]{x1D713}}$ does not provide a precise position of the phase interface, but indicates the amount of water in volume fraction. As shown, the region with $\overline{\unicode[STIX]{x1D713}}=1$ is located very near the wall, and $\overline{\unicode[STIX]{x1D713}}$ decreases linearly with the distance from the wall except for the region with very low $\overline{\unicode[STIX]{x1D713}}$ values. Note that, with decreasing $\unicode[STIX]{x1D700}_{o}$ , the slope becomes slow and the region of $0<\overline{\unicode[STIX]{x1D713}}<1$ becomes wider.

Figure 10(a) shows the profiles of the mean streamwise velocity of water ( $\overline{u}_{w}$ ) and oil ( $\overline{u}_{o}$ ), respectively, for different oil holdups, together with that of a single-phase pipe flow at $Re_{\unicode[STIX]{x1D70F}}=685$ (Wu et al. Reference Wu, Baltzer and Adrian2012):

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u}_{w}(r)={\displaystyle \frac{\displaystyle \iiint u(x,r,\unicode[STIX]{x1D703},t)\unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}{\displaystyle \iiint \unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}}, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u}_{o}(r)={\displaystyle \frac{\displaystyle \iiint u(x,r,\unicode[STIX]{x1D703},t)(1-\unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t))\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}{\displaystyle \iiint (1-\unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t))\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}}. & \displaystyle\end{eqnarray}$$

Since $\overline{u}_{w}$ and $\overline{u}_{o}$ are the mean velocities obtained with a weight of $\unicode[STIX]{x1D713}$ , both velocity profiles co-exist in the intermittent region ( $0<\overline{\unicode[STIX]{x1D713}}<1$ ). For each $\unicode[STIX]{x1D700}_{o}$ , $\overline{u}_{o}$ is nearly constant ( ${\approx}u_{core}$ ) even very near the wall, but is maximum at the centre and slowly decreases with increasing $r$ (it increases again near $\overline{\unicode[STIX]{x1D713}}=1$ ). The effect of this velocity gradient becomes manifest when it is multiplied by the oil viscosity (see figure 11). With decreasing $\unicode[STIX]{x1D700}_{o}$ (under the condition of constant mean pressure gradient), $\overline{u}_{o}$ increases but converges a nearly constant value, $\overline{u}_{o}/u_{\unicode[STIX]{x1D70F}}\approx 19$ , for $\unicode[STIX]{x1D700}_{o}\leqslant 0.71$ . A similar observation of constant $\overline{u}_{o}$ was made in the experimental study of Tripathi et al. (Reference Tripathi, Tabor, Singh and Bhattacharya2017) for a horizontal core-annular flow. On the other hand, with increasing wall distance, $\overline{u}_{w}$ increases, reaches maximum, and then decreases to $\overline{u}_{o}$ . Thus, the peak velocity of water is faster than the oil core velocity, which results from the positive relative velocity ahead of the crest (see figure 6 a). Also, $\overline{u}_{w}$ values are very different from that of a single-phase pipe flow. Except for $\unicode[STIX]{x1D700}_{o}=0.91$ , the velocity profile shows more upward shift of the log law for higher $\unicode[STIX]{x1D700}_{o}$ . With decreasing $\unicode[STIX]{x1D700}_{o}$ , the upward shift decreases and the velocity profile approaches that of single-phase pipe flow in the log region. This behaviour is natural because the water region increases with decreasing $\unicode[STIX]{x1D700}_{o}$ and thus the water flow becomes more like single-phase pipe flow. The different behaviour for $\unicode[STIX]{x1D700}_{o}=0.91$ is owing to the very thin water region. Figure 10(b) shows the mean streamwise velocity of water weighted by the mean water volume fraction, $\overline{u}_{w}^{\ast }$ :

(3.4) $$\begin{eqnarray}\displaystyle \displaystyle \overline{u}_{w}^{\ast }(r) & = & \displaystyle {\displaystyle \frac{\displaystyle \iiint u(x,r,\unicode[STIX]{x1D703},t)\unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}{\displaystyle \iiint \text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}}\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{\displaystyle \iiint u(x,r,\unicode[STIX]{x1D703},t)\unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}{\displaystyle \iiint \unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}}{\displaystyle \frac{\displaystyle \iiint \unicode[STIX]{x1D713}(x,r,\unicode[STIX]{x1D703},t)\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}{\displaystyle \iiint \text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}}\nonumber\\ \displaystyle \displaystyle & = & \displaystyle \overline{u}_{w}(r)\overline{\unicode[STIX]{x1D713}}(r).\end{eqnarray}$$

As the distance from the wall increases, $\overline{u}_{w}^{\ast }$ increases and then decreases. The reason for the decrease in $\overline{u}_{w}^{\ast }$ is that in this region $0<\overline{\unicode[STIX]{x1D713}}<1$ and $\overline{u}_{w}^{\ast }=\overline{u}_{w}\overline{\unicode[STIX]{x1D713}}$ . This $\overline{u}_{w}^{\ast }$ will be used in developing an oil holdup model (see § 3.3).

Figure 11 shows the radial distributions of the Reynolds and viscous shear stresses and their sum of the core-annular and single-phase pipe flows, respectively, together with the mean water volume fraction. The total shear stress of core-annular flow is larger at all radial locations than that of a single-phase pipe flow. Unlike the single-phase pipe flow, the total shear stress profile is not linear due to the body force term:

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D70F}}(r)=\overline{\unicode[STIX]{x1D70C}u_{x}u_{r}}-\overline{\unicode[STIX]{x1D707}\left({\displaystyle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}r}}+{\displaystyle \frac{\unicode[STIX]{x2202}u_{r}}{\unicode[STIX]{x2202}x}}\right)}={\displaystyle \frac{r}{2}}\left(-{\displaystyle \frac{\text{d}P}{\text{d}x}}\right)+{\displaystyle \frac{1}{r}}\int _{0}^{r}(\overline{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}n_{x}}-\overline{\unicode[STIX]{x1D70C}}g)\,r\,\text{d}r, & \displaystyle\end{eqnarray}$$

where the overbar denotes averaging over the streamwise and azimuthal directions and time. Note that $\overline{\unicode[STIX]{x1D70C}u_{x}u_{r}}=\overline{u}_{x}\overline{\unicode[STIX]{x1D70C}u_{r}^{\prime }}+\overline{\unicode[STIX]{x1D70C}u_{x}^{\prime }u_{r}^{\prime }}\approx \overline{\unicode[STIX]{x1D70C}u_{x}^{\prime }u_{r}^{\prime }}$ because $\unicode[STIX]{x1D70C}_{w}/\unicode[STIX]{x1D70C}_{o}=1.036$ , and the surface tension term is negligibly small (e.g. $|1/r\int _{0}^{r}\overline{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}n_{x}}\,r\,\text{d}r|_{max}<0.012\overline{\unicode[STIX]{x1D70F}}_{w}$ for $\unicode[STIX]{x1D700}_{o}=0.71$ ) as compared to other terms. In a single-phase pipe flow, the Reynolds shear stress is much larger than the viscous shear stress except very near the wall. However, in the core-annular flow, the viscous shear stress is dominant in the water-free region ( $\overline{\unicode[STIX]{x1D713}}=0$ ) and increases in the radial direction, whereas, in the region of $\overline{\unicode[STIX]{x1D713}}\neq 0$ , the viscous shear stress rapidly decreases, reaches local minimum, and increases again in the radial direction. The Reynolds shear stress is nearly zero in the water-free region due to the core flow characteristics, but rapidly increases in the region of $\overline{\unicode[STIX]{x1D713}}\neq 0$ and decreases to zero at the wall. The large magnitudes of the viscous shear stress in the water-free region are due to the high oil viscosity (17600 times that of water), although the velocity gradient in the core region is very small. The viscous (or total) shear stress in this region ( $\overline{\unicode[STIX]{x1D713}}=0$ ) is balanced by the mean pressure gradient and buoyancy force (nearly zero Reynolds shear stress in this region): using (2.16),

(3.6) $$\begin{eqnarray}\displaystyle \displaystyle \overline{\unicode[STIX]{x1D70F}}_{oil}(r)={\displaystyle \frac{r}{2}}\left(-{\displaystyle \frac{\text{d}P}{\text{d}x}}-\unicode[STIX]{x1D70C}_{o}g\right) & = & \displaystyle {\displaystyle \frac{r}{2}}\left[\left({\displaystyle \frac{2\overline{\unicode[STIX]{x1D70F}}_{w}}{R}}+{\displaystyle \frac{1}{V}}\int \unicode[STIX]{x1D70C}g\,\text{d}V\right)-\unicode[STIX]{x1D70C}_{o}g\right]\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{r}{2}}\left[{\displaystyle \frac{2\overline{\unicode[STIX]{x1D70F}}_{w}}{R}}+(\unicode[STIX]{x1D70C}_{w}-\unicode[STIX]{x1D70C}_{o})g(1-\unicode[STIX]{x1D700}_{o})\right].\end{eqnarray}$$

The slopes of the viscous (or total) shear stress in the water-free region predicted by (3.6) for $\unicode[STIX]{x1D700}_{o}=0.83,0.71$ and $0.56$ are $1.15,1.25$ and $1.38\overline{\unicode[STIX]{x1D70F}}_{w}/R$ , respectively, which are accurately predicted by the present simulations (figure 11). With decreasing $\unicode[STIX]{x1D700}_{o}$ , local minimum viscous shear and maximum Reynolds shear stresses in the region of $\overline{\unicode[STIX]{x1D713}}\neq 0$ become smaller and bigger, respectively, and their behaviours very near the wall are similar to those of the single-phase pipe flow. For $\unicode[STIX]{x1D700}_{o}=0.91$ , the Reynolds shear stress is almost zero (non-turbulent annulus) and the viscous shear stress is dominant for both core and annular regions (not shown in the figure).

3.2 Phase interface wave

Figure 12 shows the characteristics of the phase interface. The phase interfaces consist of different streamwise and azimuthal wavenumber components for different oil holdups. For example, the streamwise wavelength of the phase interface increases with decreasing $\unicode[STIX]{x1D700}_{o}$ . Also, the instantaneous phase interface located near the wall has a small wavelength and amplitude, but the one away from the wall has a large wavelength and amplitude, as seen for $\unicode[STIX]{x1D700}_{o}=0.56$ in figure 12(a) (see also Tripathi et al. (Reference Tripathi, Tabor, Singh and Bhattacharya2017) for a horizontal core-annular flow). More detailed analysis of its wave characteristics is given below. As the volume of each fluid is prescribed with given oil holdup, the average location of the phase interface is predetermined as listed in table 1. The maximum radial location of the phase interface ( $h_{max}$ ) is near the wall, but the minimum location ( $h_{min}$ ) decreases with decreasing $\unicode[STIX]{x1D700}_{o}$ (figure 12 b), indicating that the wave amplitude of the phase interface is larger at lower $\unicode[STIX]{x1D700}_{o}$ . This was also observed in the previous experiment by Rodriguez & Bannwart (Reference Rodriguez and Bannwart2006). The probability density function (PDF) of the phase interface location becomes broader and moves towards the pipe centre with decreasing $\unicode[STIX]{x1D700}_{o}$ (figure 12 c), which confirms the behaviours discussed from the instantaneous phase interface and table 1 (i.e. with decreasing $\unicode[STIX]{x1D700}_{o}$ , its mean location moves away for the wall and its fluctuation amplitude becomes larger). Except for $\unicode[STIX]{x1D700}_{o}=0.91$ , the PDFs have flat profiles near the mean locations.

Figure 12.

Characteristics of the phase interfaces: (a) instantaneous shapes; (b) their maximum, average and minimum radii; (c) PDFs of the radial location of the phase interfaces. In (c), thin vertical lines indicate the mean locations of the phase interface for given oil holdups.

Figure 13.

One-dimensional spectra of the phase interface amplitude: (a) streamwise wavenumber; (b) azimuthal wavenumber; (c) frequency.

The wave characteristics of the phase interfaces are investigated from their power spectra. For this purpose, we perform additional computations with the sampling resolution of $\unicode[STIX]{x0394}t=0.0002R/u_{\unicode[STIX]{x1D70F}}$ ( $\unicode[STIX]{x0394}t\approx 0.0026R/u_{core}$ ) during the period of $T=1.6384R/u_{\unicode[STIX]{x1D70F}}\approx 20.12R/u_{core}$ for $\unicode[STIX]{x1D700}_{o}=0.91$ , and with $\unicode[STIX]{x0394}t\approx 0.0038R/u_{core}$ during $T=31.4R/u_{core}$ for $\unicode[STIX]{x1D700}_{o}\leqslant 0.83$ , respectively. Each of these periods is divided into three overlapping segments with 50 % overlap (for more details, see Choi & Moin Reference Choi and Moin1990). The ranges of the streamwise and azimuthal wavenumbers are $0\leqslant k_{x}\leqslant 255/R~(\unicode[STIX]{x0394}k_{x}=1/R)$ for $\unicode[STIX]{x1D700}_{o}=0.91$ and 0.83 and $0\leqslant k_{x}\leqslant 255.5/R~(\unicode[STIX]{x0394}k_{x}=0.5/R)$ for $\unicode[STIX]{x1D700}_{o}\leqslant 0.71$ , and $0\leqslant k_{\unicode[STIX]{x1D703}}\leqslant 255/R$ $(\unicode[STIX]{x0394}k_{\unicode[STIX]{x1D703}}=1/R)$ , respectively, where $k_{x}$ and $k_{\unicode[STIX]{x1D703}}$ are the streamwise and azimuthal wavenumbers. Figure 13 shows one-dimensional wavenumber spectra, $\unicode[STIX]{x1D711}(k_{x})$ and $\unicode[STIX]{x1D711}(k_{\unicode[STIX]{x1D703}})$ , and frequency spectrum, $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D714})$ , of the phase interface amplitude which satisfy the following condition:

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\infty }\unicode[STIX]{x1D711}(k_{x})\,\text{d}k_{x}=\int _{0}^{\infty }\unicode[STIX]{x1D711}(k_{\unicode[STIX]{x1D703}})\,\text{d}k_{\unicode[STIX]{x1D703}}=\int _{0}^{\infty }\unicode[STIX]{x1D711}(\unicode[STIX]{x1D714})\,\text{d}\unicode[STIX]{x1D714}=\unicode[STIX]{x1D701}_{rms}^{2}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}$ is the frequency, and $\unicode[STIX]{x1D701}_{rms}$ is the root-mean-square (r.m.s.) fluctuation of the phase interface amplitude $\unicode[STIX]{x1D701}~(=h{-}h_{avg})$ . The phase interface amplitude has broadband spectra, which is very different from the observations of one-wavenumber peak in laminar core-annular flows (Bai et al. Reference Bai, Kelkar and Joseph1996; Li & Renardy Reference Li and Renardy1999), and of smooth waves from RANS simulations of turbulent core-annular flows (Ghosh et al. Reference Ghosh, Das and Das2010; Ingen Housz et al. Reference Ingen Housz, Ooms, Henkes, Pourquie, Kidess and Radhakrishnan2017; Shi et al. Reference Shi, Gourma and Yeung2017). With decreasing $\unicode[STIX]{x1D700}_{o}$ , low wavenumbers contain more power, as observed by previous experiments (Bai Reference Bai1995; Rodriguez & Bannwart Reference Rodriguez and Bannwart2006), but the powers at intermediate and high wavenumbers are similar. In the case of the azimuthal wavenumber spectra (figure 13 b), the largest power occurs at the smallest wavenumber, $k_{\unicode[STIX]{x1D703}}=1/R$ , whose corresponding wavelength is the pipe circumference. The frequency spectra show a behaviour similar to the streamwise wavenumber spectra, indicating the convective nature of the phase interface.

The convection velocity can be obtained from the power spectrum (Wills Reference Wills1970; Choi & Moin Reference Choi and Moin1990):

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle U_{conv}(k_{x})=-{\displaystyle \frac{\unicode[STIX]{x1D714}(k_{x})}{k_{x}}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}(k_{x})$ is obtained from

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \left.{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}(k_{x},\unicode[STIX]{x1D714})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}}\right|_{\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}(k_{x})}=0, & \displaystyle\end{eqnarray}$$

and $\unicode[STIX]{x1D6F7}(k_{x},\unicode[STIX]{x1D714})$ is the two-dimensional power spectrum. A quadratic polynomial is used to find $\unicode[STIX]{x1D714}(k_{x})$ in (3.9) (Kang, Moin & Iaccarino Reference Kang, Moin and Iaccarino2008). Figure 14 shows the variation of the convection velocity normalized by $u_{core}$ with the streamwise wavenumber. The scatter of the data shown in this figure is due to the limited statistical sample available for locating the maxima of the spectrum within a set of discrete frequencies and wavenumbers. The convection velocities at low wavenumbers are also contaminated by a specific window function (Hanning window) or a sampling period used (see Choi & Moin Reference Choi and Moin1990). The convection velocity increases with increasing $k_{x}$ , but maintains nearly a constant value $(=0.98{-}0.99u_{core})$ , slightly less than $u_{core}$ , at high wavenumbers. The overall convection velocity $U_{conv}$ of the most energetic wave is also obtained from

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \left.{\displaystyle \frac{\unicode[STIX]{x2202}W(U^{\ast })}{\unicode[STIX]{x2202}U^{\ast }}}\right|_{U^{\ast }=U_{conv}}=0, & \displaystyle\end{eqnarray}$$

where

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle W(U^{\ast })=\int \unicode[STIX]{x1D6F7}(k_{x},-U^{\ast }k_{x})\,\text{d}k_{x}. & \displaystyle\end{eqnarray}$$

The overall convection velocities of the phase interface are calculated to be $U_{conv}/u_{core}=0.984,0.989,0.989,0.990$ and $0.985$ , respectively, for $\unicode[STIX]{x1D700}_{o}=0.91,0.83,0.71$ , 0.62 and $0.56$ , indicating that its most energetic motions travel downstream nearly at the core velocity.

Figure 14.

Convection velocities as a function of the streamwise wavenumber.

3.3 Oil holdup model

The holdup is defined as the volume fraction of a particular fluid in a multiphase pipe flow. In experiments, the flow rates are controlled, but the holdups are not easily measurable. Thus, holdup models have been widely investigated (Oliemans Reference Oliemans1986; Bai et al. Reference Bai, Chen and Joseph1992; Arney et al. Reference Arney, Bai, Guevara, Joseph and Liu1993). In this study, we develop a new model to predict the oil holdup based on the observation that the flow in the annulus is divided by two characteristic regions: a boundary layer flow near the wall and a nearly constant velocity flow near and above the phase interface (see figure 7 b).

Figure 15.

Displacement thickness $\unicode[STIX]{x1D6FF}^{\ast }$ ( $\unicode[STIX]{x1D700}_{o}=0.71$ ): (a) schematic diagram; (b) instantaneous streamwise distributions of the displacement thickness and $R-\tilde{h}$ in wall units with local friction velocity, $\tilde{u} _{\unicode[STIX]{x1D70F}}$ , at the same instant as in figure 4(b). In (b), ——, $\unicode[STIX]{x1D6FF}^{\ast }\tilde{u} _{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}_{w}$ ; – - – - –, $(R-\tilde{h})\tilde{u} _{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}_{w}$ .

Since the flow field near the wall shows boundary layer characteristics except near the crest, we define a displacement thickness, $\unicode[STIX]{x1D6FF}^{\ast }(x,t)$ , as in figure 15(a), which satisfies

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{2\unicode[STIX]{x03C0}}\!\!\int _{h}^{R-\unicode[STIX]{x1D6FF}^{\ast }}u_{core}\,r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}=\int _{0}^{2\unicode[STIX]{x03C0}}\!\!\int _{h}^{R}u\,r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}. & \displaystyle\end{eqnarray}$$

By rearranging (3.12), $\unicode[STIX]{x1D6FF}^{\ast }(x,t)$ is given as

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D6FF}^{\ast }(x,t)}{R}}=1-\sqrt{1-{\displaystyle \frac{1}{\unicode[STIX]{x03C0}R^{2}}}\int _{0}^{2\unicode[STIX]{x03C0}}\!\!\int _{h}^{R}\left(1-{\displaystyle \frac{u}{u_{core}}}\right)\,r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}}. & \displaystyle\end{eqnarray}$$

Similarly, we define the overall displacement thickness $\overline{\unicode[STIX]{x1D6FF}^{\ast }}$ as (using (3.1) and (3.4))

(3.14) $$\begin{eqnarray}\displaystyle \displaystyle {\displaystyle \frac{\overline{\unicode[STIX]{x1D6FF}^{\ast }}}{R}} & = & \displaystyle 1-\sqrt{1-{\displaystyle \frac{1}{\unicode[STIX]{x03C0}R^{2}L_{x}T}}\int _{0}^{T}\!\!\int _{0}^{L_{x}}\!\!\int _{0}^{2\unicode[STIX]{x03C0}}\!\!\int _{h}^{R}\left(1-{\displaystyle \frac{u}{u_{core}}}\right)\,r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t}\nonumber\\ \displaystyle \displaystyle & = & \displaystyle 1-\sqrt{1-{\displaystyle \frac{1}{\unicode[STIX]{x03C0}R^{2}L_{x}T}}\int _{0}^{T}\!\!\int _{0}^{L_{x}}\!\!\int _{0}^{2\unicode[STIX]{x03C0}}\!\!\int _{0}^{R}\left(1-{\displaystyle \frac{u}{u_{core}}}\right)\unicode[STIX]{x1D713}\,r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}\,\text{d}x\,\text{d}t},\nonumber\\ \displaystyle \displaystyle & = & \displaystyle 1-\sqrt{1-{\displaystyle \frac{2}{R^{2}}}\int _{0}^{R}\left(\overline{\unicode[STIX]{x1D713}}-{\displaystyle \frac{\overline{u}_{w}^{\ast }}{u_{core}}}\right)\,r\,\text{d}r},\end{eqnarray}$$

where $T$ is the averaging time. Figure 15(b) shows the streamwise distribution of the instantaneous displacement thickness, $\unicode[STIX]{x1D6FF}^{\ast }\tilde{u} _{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}_{w}$ , for $\unicode[STIX]{x1D700}_{o}=0.71$ , together with that of $(R-\tilde{h})\tilde{u} _{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}_{w}$ , where $\tilde{h}=\int _{0}^{2\unicode[STIX]{x03C0}}h^{2}\,\text{d}\unicode[STIX]{x1D703}/\!\int _{0}^{2\unicode[STIX]{x03C0}}h\,\text{d}\unicode[STIX]{x1D703}$ and $\tilde{u} _{\unicode[STIX]{x1D70F}}=\sqrt{1/(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{w})\int _{0}^{2\unicode[STIX]{x03C0}}(-\unicode[STIX]{x1D707}\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}r|_{wall})\,\text{d}\unicode[STIX]{x1D703}}$ (local friction velocity). The displacement thickness fluctuates around the overall displacement thickness, and its magnitude is much smaller than that of the averaged phase interface height except near the crest where $\unicode[STIX]{x1D6FF}^{\ast }\approx 0.4(R-\tilde{h})$ . The overall displacement thicknesses are $\overline{\unicode[STIX]{x1D6FF}^{\ast }}^{+}=\overline{\unicode[STIX]{x1D6FF}^{\ast }}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}_{w}\approx 6.98,14.85$ and $18.83$ for $\unicode[STIX]{x1D700}_{o}=0.91,0.71$ and $0.56$ in wall units, respectively, which increases with decreasing $\unicode[STIX]{x1D700}_{o}$ , but their magnitudes are small, $\overline{\unicode[STIX]{x1D6FF}^{\ast }}/R<0.027$ , indicating that most of water flows at $u_{core}$ . Equation (3.14) is rearranged using the superficial velocities and oil holdup

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}_{o}=\left(1-{\displaystyle \frac{\overline{\unicode[STIX]{x1D6FF}^{\ast }}}{R}}\right)^{2}{\displaystyle \frac{1}{1+j_{w}/j_{o}}}. & \displaystyle\end{eqnarray}$$

Using (3.15), we confirmed that the overall displacement thicknesses of all experimental data in Vanegas Prada & Bannwart (Reference Vanegas Prada and Bannwart2001) are small, $\overline{\unicode[STIX]{x1D6FF}^{\ast }}^{+}<22$ . Note that the core velocity is nearly constant for $\unicode[STIX]{x1D700}\leqslant 0.71$ and that for $\unicode[STIX]{x1D700}=0.83$ is slightly smaller (see figure 10 a). This explains why the total volume flow rate for $\unicode[STIX]{x1D700}_{o}\leqslant 0.83$ in figure 3 is nearly constant.

Figure 16.

Profiles of the mean water volume fraction, $\overline{\unicode[STIX]{x1D713}}$ , and the mean streamwise velocity weighted by $\overline{\unicode[STIX]{x1D713}}$ , $\overline{u}_{w}^{\ast }$ , for $\unicode[STIX]{x1D700}_{o}=0.71$ .

To estimate the oil holdup, we need to approximate the overall displacement thickness $\overline{\unicode[STIX]{x1D6FF}^{\ast }}$ . Solid lines in figure 16 show the profiles of $\overline{u}_{w}^{\ast }/u_{core}$ and $\overline{\unicode[STIX]{x1D713}}$ obtained from the present simulation for $\unicode[STIX]{x1D700}_{o}=0.71$ . Two profiles collapse in the region of $\overline{u}_{w}\approx u_{core}$ because $\overline{u}_{w}^{\ast }=\overline{u}_{w}\overline{\unicode[STIX]{x1D713}}\approx u_{core}\overline{\unicode[STIX]{x1D713}}$ . Then, substituting (3.14) into (3.15) gives

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}_{o}\approx \left\{1-{\displaystyle \frac{2}{R^{2}}}\int _{r_{c}}^{R}\left(\overline{\unicode[STIX]{x1D713}}-{\displaystyle \frac{\overline{u}_{w}^{\ast }}{u_{core}}}\right)\,r\,\text{d}r\right\}{\displaystyle \frac{1}{1+j_{w}/j_{o}}}, & \displaystyle\end{eqnarray}$$

where $r_{c}$ is defined such that $\overline{u}_{w}^{\ast }=\overline{\unicode[STIX]{x1D713}}u_{core}$ at $r\leqslant r_{c}$ . We model $\overline{\unicode[STIX]{x1D713}}$ as a linear profile whose slope is $m$ and magnitudes are 1 at $r=R$ and 0 at $r=R-1/m$ (figure 9):

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D713}}=\left\{\begin{array}{@{}ll@{}}0\quad & \text{for }0\leqslant r<R-1/m,\\ 1-m(R-r)\quad & \text{for }R-1/m\leqslant r\leqslant R.\end{array}\right. & \displaystyle\end{eqnarray}$$

Here, $m$ is determined by matching the oil volume flow rate, $j_{o}\unicode[STIX]{x03C0}R^{2}=\int _{0}^{2\unicode[STIX]{x03C0}}\int _{0}^{R-1/m}$ $u_{core}\,r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}+\int _{0}^{2\unicode[STIX]{x03C0}}\int _{R-1/m}^{R}u_{core}m(R-r)\,r\,\text{d}r\,\text{d}\unicode[STIX]{x1D703}$ :

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle m={\displaystyle \frac{2}{R(3-\sqrt{12\unicode[STIX]{x1D700}_{o}-3})}}, & \displaystyle\end{eqnarray}$$

where $12\unicode[STIX]{x1D700}_{o}-3\geqslant 0$ (i.e. $\unicode[STIX]{x1D700}_{o}\geqslant 1/4$ ) is usually satisfied for the core-annular flow because the flow pattern changes for very low $\unicode[STIX]{x1D700}_{o}$ values (Bai et al. Reference Bai, Chen and Joseph1992). For the integration in (3.16), $\overline{u}_{w}^{\ast }$ in $r_{c}\leqslant r\leqslant R$ is approximated with the mean velocity of the single-phase turbulent pipe flow (Spalding Reference Spalding1961) because of the high water fraction ( $\overline{\unicode[STIX]{x1D713}}\approx 1$ ):

(3.19) $$\begin{eqnarray}\displaystyle \displaystyle (R-r)^{+} & = & \displaystyle \overline{u}_{w}^{\ast +}+\exp (-\unicode[STIX]{x1D705}B)\left\{\exp (\unicode[STIX]{x1D705}\overline{u}_{w}^{\ast +})-1-\unicode[STIX]{x1D705}\overline{u}_{w}^{\ast +}-{\displaystyle \frac{(\unicode[STIX]{x1D705}\overline{u}_{w}^{\ast +})^{2}}{2}}-{\displaystyle \frac{(\unicode[STIX]{x1D705}\overline{u}_{w}^{\ast +})^{3}}{6}}\right\}\nonumber\\ \displaystyle & & \displaystyle \displaystyle \text{in }r_{c}\leqslant r\leqslant R,\end{eqnarray}$$

where $\unicode[STIX]{x1D705}=0.4$ , $B=5.5$ and $\overline{u}_{w}^{\ast +}=\overline{u}_{w}^{\ast }/u_{\unicode[STIX]{x1D70F}}$ . Also,

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{u}_{w}^{\ast }=u_{core}\overline{\unicode[STIX]{x1D713}}\quad \text{in }R-1/m\leqslant r\leqslant r_{c}. & \displaystyle\end{eqnarray}$$

The location $r_{c}$ is obtained by matching its velocity from (3.19) and (3.20). Finally, we adopt the friction factor model (Arney et al. Reference Arney, Bai, Guevara, Joseph and Liu1993) to evaluate the wall friction:

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D706}_{model}=\left({\displaystyle \frac{2\overline{\unicode[STIX]{x1D70F}}_{w}}{R}}\right){\displaystyle \frac{4R}{\unicode[STIX]{x1D70C}_{avg}(j_{o}+j_{w})^{2}}}=0.316Re_{model}^{-1/4}, & \displaystyle\end{eqnarray}$$

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle Re_{model}={\displaystyle \frac{2\unicode[STIX]{x1D70C}_{avg}(j_{o}+j_{w})R}{\unicode[STIX]{x1D707}_{w}}}\left\{1+\unicode[STIX]{x1D700}_{o}^{2}\left({\displaystyle \frac{\unicode[STIX]{x1D707}_{w}}{\unicode[STIX]{x1D707}_{o}}}-1\right)\right\}, & \displaystyle\end{eqnarray}$$

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{avg}=\unicode[STIX]{x1D70C}_{o}\unicode[STIX]{x1D700}_{o}+\unicode[STIX]{x1D70C}_{w}(1-\unicode[STIX]{x1D700}_{o}), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{model}$ is the friction factor of the turbulent core-annular flow. Given $j_{w}$ and $j_{o}$ , the oil holdup $\unicode[STIX]{x1D700}_{o}$ is obtained by iteratively solving (3.16)–(3.23). As shown in figure 16, the present model accurately predicts the velocity and water volume fraction.

Figure 17.

Variation of the oil holdup with the superficial velocity ratio: (a) $\overline{\unicode[STIX]{x1D70F}}_{w}$ obtained from (3.21)–(3.23); (b) $\overline{\unicode[STIX]{x1D70F}}_{w}$ obtained directly from experimental data (Bai et al. (Reference Bai, Chen and Joseph1992) ( $Re_{\unicode[STIX]{x1D70F}}\approx 100{-}350$ ); Vanegas Prada & Bannwart (Reference Vanegas Prada and Bannwart2001) ( $Re_{\unicode[STIX]{x1D70F}}\approx 600{-}1800$ )) and from present simulation ( $Re_{\unicode[STIX]{x1D70F}}=720$ ). Here, circles are the data from experiments, diamonds are from present numerical simulation, lines are from the previous models and present asymptotic solution and crosses are from the present model. The asymptotic solution is obtained with $\overline{\unicode[STIX]{x1D6FF}^{\ast }}=0$ (i.e. assuming that the bulk velocities of water and oil are same). Higher $\unicode[STIX]{x1D700}_{o}$ values than the asymptotic solution indicate that the bulk velocity of water is faster than that of oil because of an oil dispersion in water or oil adhesion to the wall (Vuong Reference Vuong2009). Our model does not consider this situation.

Figure 17 shows the predictions of the oil holdup from the present model for the previous experimental and present simulation data, together with the predictions from previous models and present asymptotic solution. Two predictions are shown in figures 17(a) and (b), respectively, based on the wall frictions from the present model (3.21)–(3.23) and from experimental data (Bai et al. Reference Bai, Chen and Joseph1992; Vanegas Prada & Bannwart Reference Vanegas Prada and Bannwart2001). The empirical correlations from previous studies and asymptotic solution are (original formulations are modified to compare with the present model)

(3.24) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D700}_{o}=\left\{\begin{array}{@{}ll@{}}\left\{1-0.2{\displaystyle \frac{j_{w}}{j_{o}}}\left({\displaystyle \frac{1}{1+j_{w}/j_{o}}}\right)^{5}\right\}\left({\displaystyle \frac{1}{1+j_{w}/j_{o}}}\right)\quad & \text{(Oliemans 1986)},\\[13.0pt] \left({\displaystyle \frac{1+0.65j_{w}/j_{o}}{1+j_{w}/j_{o}}}\right)\left({\displaystyle \frac{1}{1+j_{w}/j_{o}}}\right)\quad & \text{(Arney et al. 1993)},\\[13.0pt] {\displaystyle \frac{1}{1+1.39j_{w}/j_{o}}}\quad & \text{(Bai et al. 1992)},\\[11.0pt] {\displaystyle \frac{1}{1+j_{w}/j_{o}}}\quad & \text{asymptotic solution}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Here, Oliemans (Reference Oliemans1986) suggested an empirical holdup model based on their experimental data from a horizontal core-annular flow, and Arney et al. (Reference Arney, Bai, Guevara, Joseph and Liu1993) modified it with more experimental data available in the literature. Bai et al. (Reference Bai, Chen and Joseph1992) also proposed an empirical holdup model for a vertical core-annular flow based on their experimental data. Note that the previous empirical holdup models obtained by fitting experimental data depend only on the superficial velocity ratio of oil to water and do not consider the Reynolds number dependence. As shown in this figure, the model of Bai et al. (Reference Bai, Chen and Joseph1992) well predicts their own experimental data, but not the data of Vanegas Prada & Bannwart (Reference Vanegas Prada and Bannwart2001). The present model considers not only the superficial velocity ratio but also the boundary layer characteristics of water flow representing the friction Reynolds number, and thus predicts both experimental data more accurately. The present model prediction in (b) is better than that in (a), indicating that the wall friction estimation is also important in estimating the oil holdup.