2. Mathematical formulation based on SWT

The continuity and momentum balance equations are formulated for stratified flow of a thin liquid film and gas flowing above the liquid in a 2-D planar geometry. The effect of surface tension is assumed negligible compared to gravity (Kate, Das & Chakraborty Reference Kate, Das and Chakraborty2007; Dhar et al. Reference Dhar, Das and Das2020), since surface tension has been reported to influence jump stability (Bohr et al. Reference Bohr, Putkaradze and Watanabe1997; Watanabe et al. Reference Watanabe, Putkaradze and Bohr2003), which is not the focus of the present paper. We also neglect the entry effect and assume negligible dissipation of streamwise compared to transverse momentum. The effect of gravity in the liquid phase is incorporated by a hydrostatic pressure approximation based on the density of liquid, which couples film height to pressure.

The equations are rendered dimensionless by scaling velocities and lengths from an order of magnitude analysis of the continuity, momentum and mass conservation equations. Following the nomenclatures specified in figure 1(a), the normalising parameters, ${u^n} = q_l^{1/3}{g^{1/3}}$, ${w^n} = q_l^{ - 2/3}{\nu _l}{g^{1/3}}$, ${x^n} = q_l^{5/3}\nu _l^{ - 1}{g^{ - 1/3}}$, ${z^n} = q_l^{2/3}{g^{ - 1/3}}$ and $P_i^n = {\rho _l}q_l^{2/3}{g^{2/3}}$ yield the normalised (non-dimensional) mass and momentum balance equations for the liquid film as

(2.1)\begin{gather}\frac{{\partial {u_l}}}{{\partial x}} + \frac{{\partial {w_l}}}{{\partial z}} = 0,\end{gather}

(2.2)\begin{gather}{u_l}\frac{{\partial {u_l}}}{{\partial x}} + {w_l}\frac{{\partial {u_l}}}{{\partial z}} ={-} \frac{{\textrm{d}h}}{{\textrm{d}x}} - \frac{{\textrm{d}{P_i}}}{{\textrm{d}x}} + \frac{{{\partial ^2}{u_l}}}{{\partial {z^2}}},\end{gather}

and the mass conservation condition for liquid film of depth h flowing in a conduit of height H as

(2.3)\begin{equation}\int_0^h {{u_l}(x,z)\,\textrm{d}z} = 1.\end{equation}

The corresponding mass conservation condition for the gas phase is

(2.4)\begin{equation}\int_h^H {{u_g}(x,z)\,\textrm{d}z} ={\pm} {q_r},\end{equation}

which implies ${V_{av,g}}(H - h) ={\pm} {q_r}$, the ${\pm}$ sign denoting co- and counter-current gas–liquid flow with the average gas velocity as ${V_{av,g}} = (1/(H - h))\int_h^H {{u_g}(x,z)\,\textrm{d}z}$ and the volumetric flow rate ratio per unit width as ${q_r}( = {q_g}/{q_l})$.

Figure 1. Problem domain (a) and the experimental analogue (b) of viscous shear induced hydraulic jump in air–water stratified flow. The liquid film of depth ${h_{l,in}}$ is introduced in the conduit of height H. At streamwise position (x), the film of thickness $h(x)$ flows beneath a gas layer of thickness $[H - h(x)]$. Here, $u(x,z)$ and $w(x,z)$ are the respective streamwise (x) and vertical (z) velocity components for each fluid (of density $\rho$, dynamic viscosity $\mu$ and kinematic viscosity $\nu ( = \mu /\rho )$) flowing at volumetric flow rate q per unit conduit width; P(x) is the pressure at distance x from the entry and g is acceleration due to gravity. Subscripts l and g associated with each notation refer to the liquid and gas phases, respectively, and subscript i denotes the interface. The hydraulic jump confined within control surfaces 1 and 2 is at a distance x ₁ from the entry.

The boundary conditions of the problem are

1. (i) no-slip and no-penetration at the conduit floor and roof:${u_l}(x,0) = {w_l}(x,0) = 0$; ${u_g}(x,H) = {w_g}(x,H) = 0$;

2. (ii) equality of shear stress at the gas–liquid interface (Brauner, Rovinsky & Maron Reference Brauner, Rovinsky and Maron1996): ${\mu _l}(\partial {u_l}/\partial z){|_{z = h}} = {\mu _g}(\partial {u_g}/\partial z){|_{z = h}}$; and

3. (iii) kinematic equality at the interface: ${u_{l,z = h}} = {u_{g,z = h}}$.

Since the simultaneous presence of the nonlinear inertial terms on the left and the second-order viscous terms on the right-hand side makes it difficult to solve (2.2), we adopt the shallow water approximations where the averaged momentum balance equation in the thin film is obtained by vertical averaging of (2.1) and (2.2) over the liquid height. Along with the boundary conditions this gives

(2.5)\begin{equation}{V_{av,l}}\frac{\textrm{d}}{{\textrm{d}x}}(\beta {V_{av,l}}) ={-} \frac{{\textrm{d}h}}{{\textrm{d}x}} - \frac{{\textrm{d}{P_i}}}{{\textrm{d}x}} + \frac{1}{h}{\left. {\frac{{\partial {u_l}}}{{\partial z}}} \right|_{z = h}} - \frac{1}{h}{\left. {\frac{{\partial {u_l}}}{{\partial z}}} \right|_{z = 0}}.\end{equation}

In (2.5), the average liquid velocity ${V_{av,l}} = (1/h)\int_0^h {{u_l}(x,z)\,\textrm{d}z}$ and $\beta = (\int_0^h {u_l^2\,\textrm{d}z} )/(hV_{av,\,l}^2)$. In conjunction with liquid mass conservation condition (2.3) we obtain ${V_{av,l}}h = 1$ and for single phase liquid flow, the third term on right-hand side of (2.5) becomes zero.

For closure of the model, we take our cue from the conventional recourse in single phase flow (Singha et al. Reference Singha, Bhattacharjee and Ray2005; Bonn et al. Reference Bonn, Andersen and Bohr2009; Dhar et al. Reference Dhar, Das and Das2020) where a parabolic velocity profile is assumed before and after the jump. For two phase flow, we consider both the phases to exhibit locally fully developed Poiseuille flow under quasi-steady conditions and express the profiles in scaled coordinates as

(2.6a)\begin{gather}{u_l}(x,z)/{V_{av,l}}(x) = {f_1}(\eta ) = {a_0} + {a_1}\eta + {a_2}{\eta ^2},\end{gather}

(2.6b)\begin{gather}{u_g}(x,z)/{V_{av,g}}(x) = {f_2}(\xi ) = {a_3} + {a_4}\xi + {a_5}{\xi ^2},\end{gather}

where the scaling parameters for the liquid and gas phase are ${V_{av,l}}$ and ${V_{av,g}}$. The corresponding scaling coordinates are $\eta = z/h(x)$ and $\xi = (z - h\,(x))/(H - h(x))$ subject to conditions – (i) $0 \le \eta \le 1$ and $0 \le \xi \le 1$, (ii) $\eta = 0$ at conduit floor and 1 at the gas–liquid interface and (iii) $\xi = 0$ at the gas–liquid interface and 1 at the conduit roof.

The assumption of locally fully developed flow in both the phases, adopted from the analysis of the hydraulic jump in single phase flow, signifies that the time scale of viscous diffusion of momentum in the vertical direction $({t_z})$ is fast compared to the horizontal time scale $({t_x})$. In order to validate the same, we estimate the two time scales ${t_x} \sim {L^{{\ast} 2}}/\nu$ and ${t_z} \sim {h^{{\ast} 2}}/\nu$ expressed in terms of suitable horizontal ${L^\ast }$and vertical length scales ${h^\ast }$. This implies ${({h^\ast }/{L^\ast })^2} \ll 1$ for both the phases. For the order of magnitude of ${L^\ast }$, the channel length ${L_C}$ is considered and for the vertical length scale in the liquid and gas phase, $h_l^\ast{\sim} {h_{l,in}}$ and $h_g^\ast{\sim} H - {h_{l,in}}$ respectively. We compute ${(h_l^\ast{/}L)^2}$ and ${(h_g^\ast{/}{L^\ast })^2}$ for some representative data points of the present flow conditions. The results are presented in table 1. It is evident from the table that the values of ${(h_l^\ast{/}{L^\ast })^2}$ and ${(h_g^\ast{/}{L^\ast })^2}$ are much less than 1 in all cases, thus lending credence to the present assumption of locally fully developed velocity profile for the gas as well as the liquid phase.

Table 1. Values of ${({h^\ast }/{L^\ast })^2}$ for the two phases under different flow conditions.

Further, for the assumption of a locally fully developed velocity distribution to be valid, the change in interface position before and after the jump should be ‘mild’ enough (Sonim Reference Sonim2002). From the present predictions, we observe that the change in the interface height $(\Delta h)$ along the streamwise direction is much less than h.

Considering that the streamwise velocity ansatz must satisfy the mass flux and boundary conditions for each phase, we get

(2.7a)\begin{gather}{f_1}(0) = 0,\quad {f^{\prime}_1}(1) = (h/{V_{av,l}}){u^{\prime}_l}{|_{z = h}},\quad {f_1}(1) = {u_l}(x,h)/{V_{av,l}},\quad \int_0^1 {{f_1}(\eta )\,\textrm{d}\eta } = 1, \end{gather}

(2.7b)\begin{gather}\left. \begin{array}{c@{}} {{f_2}(0) = ({u_g}(x,h))/{V_{av,g}},\quad {{f^{\prime}}_2}(0) = ((H - h)/{V_{av,\,g}}){{u^{\prime}}_g}{|_{z = h}},}\\ {{f_2}(1) = 0,\int_0^1 {{f_2}(\xi )\,\textrm{d}\xi = 1} .} \end{array} \right\}\end{gather}

From ${f_1}(\eta )$ and ${f_2}(\xi )$ in (2.7a) and (2.7b), we obtain the expressions of ${a_0}\hbox{--}{a_5}$ as functions of x only

(2.8a)\begin{gather}{a_0} = 0,\end{gather}

(2.8b)\begin{gather}{a_1} = 3(1 - {h_r}{\mu _r}({\pm} {q_r}{h_r} - 2))/(1 + {h_r}{\mu _r}),\end{gather}

(2.8c)\begin{gather}{a_2} = (3({h_r}{\mu _r}({\pm} 3{q_r}{h_r} - 4) - 1))/(2(1 + {h_r}{\mu _r})),\end{gather}

(2.8d)\begin{gather}{a_3} = (3(1 \pm {q_r}h_r^2{\mu _r}))/({\pm} 2{q_r}{h_r}(1 + {h_r}{\mu _r})),\end{gather}

(2.8e)\begin{gather}{a_4} = (6({\pm} {q_r}{h_r} - 1))/({\pm} {q_r}{h_r}(1 + {h_r}{\mu _r})),\end{gather}

(2.8f)\begin{gather}{a_5} = (3(3 \mp {q_r}h_r^2{\mu _r} \mp 4{q_r}{h_r}))/({\pm} 2{q_r}{h_r}(1 + {h_r}{\mu _r})),\end{gather}

where ${h_r} = h/(H - h)$, the ratio of the flow cross-section occupied by the liquid and the gas phases, and ${\mu _r} = {\mu _g}/{\mu _l}$.

Based on a simple control volume analysis of the flowing gas, streamwise pressure is balanced by shear stress on its top and bottom boundaries. We neglect the pressure drop in the vertical direction across the gas phase. This implies that the pressure in the gas phase at any streamwise position x, equals the interface pressure or P(x) = P_i(x). Using the normalised parameters mentioned above, the interfacial pressure gradient in the streamwise direction can be expressed as

(2.9)\begin{equation}\frac{{\textrm{d}{P_i}}}{{\textrm{d}x}} = \frac{{{\mu _r}}}{{H - h}}\left( {{{\left. {\frac{{\partial {u_g}}}{{\partial z}}} \right|}_{z = h}} + {{\left. {\frac{{\partial {u_g}}}{{\partial z}}} \right|}_{z = H}}} \right).\end{equation}

Further, using the velocity profile expressed in (2.7b), the vertical gradient of streamwise gas velocity can be expressed as

(2.10)\begin{equation}\frac{{\partial {u_g}}}{{\partial z}} = \frac{{{V_{av,\,g}}}}{{H - h}}({a_4} + 2{a_5}\xi ).\end{equation}

This reduces (2.9) to

(2.11)\begin{equation}\frac{{\textrm{d}{P_i}}}{{\textrm{d}x}} = \frac{{2{\mu _r}{q_r}}}{{{{(H - h)}^3}}}({a_4} + {a_5}).\end{equation}

In order to assess the magnitude of contribution of $\textrm{d}{P_i}/\textrm{d}x$ to liquid phase momentum, we use experimental data (figure 5a) of h(x) at different streamwise locations and interpolate h values for small intervals of x using shape preserving cubic interpolation method. Using the h values, we evaluate a ₄ and a ₅ from (2.8e, f) and integrate $\textrm{d}{P_i}/\textrm{d}x$ over the entire flow length to obtain $\Delta {P_i}$, the interface pressure drop across the conduit. The calculations yield $\Delta {P_i}/{P_{exit}}( = 1\,\textrm{atm}) \approx {10^{ - 7}}$ for the experimental data points, thus suggesting negligible pressure drop compared to the exit pressure. Accordingly, we approximate $\textrm{d}{P_i}/\textrm{d}x \approx 0$ and solve the liquid phase momentum equation (2.5).

Using the coefficients ${a_0}\hbox{--}{a_5}$, (2.5) reduces to

(2.12)\begin{equation}\left[ \begin{array}{@{}l@{}} 1 - \dfrac{1}{{{h^3}}}\left( {\dfrac{{a_1^2}}{3} + \dfrac{{{a_1}{a_2}}}{2} + \dfrac{{a_2^2}}{5}} \right)\\ \quad + \left( {\dfrac{{2{a_1}}}{3} + \dfrac{{{a_2}}}{2}} \right)\dfrac{{3{\mu_r}(1 \mp 2{q_r}{h_r} \mp {q_r}{\mu_r}h_r^2)}}{{{{(1 + {h_r}{\mu_r})}^2}}}\dfrac{H}{{{{(H - h)}^2}{h^2}}}\\ \quad + \left( {\dfrac{{{a_1}}}{2} + \dfrac{{2{a_2}}}{5}} \right)\dfrac{{9{\mu_r}({\pm} 2{q_r}{h_r} \pm {q_r}{\mu_r}h_r^2 - 1)}}{{2{{(1 + {h_r}{\mu_r})}^2}}}\dfrac{H}{{{{(H - h)}^2}{h^2}}} \end{array} \right]\frac{{\textrm{d}h}}{{\textrm{d}x}} = \frac{{2{a_2}}}{{{h^3}}}.\end{equation}

We solve (2.12) numerically using fourth-order Runge–Kutta method. Substituting coefficients ${a_0}\hbox{--}{a_5}$ in (2.6a,b), we obtain the local velocity profile of each phase as a function of liquid height h (figure 13).

For a further understanding of the flow characteristics, we rearrange (2.12) as $\textrm{d}h/\textrm{d}x = f(h)$ where $f(h)$ is

(2.13)\begin{equation}f(h) = {{2{a_2}} / {\left[ \begin{array}{@{}l@{}} {h^3} - \left( {\dfrac{{a_1^2}}{3} + \dfrac{{{a_1}{a_2}}}{2} + \dfrac{{a_2^2}}{5}} \right)\\ \quad + \left( {\dfrac{{2{a_1}}}{3} + \dfrac{{{a_2}}}{2}} \right)\dfrac{{3{\mu_r}(1 \mp 2{q_r}{h_r} \mp {q_r}{\mu_r}h_r^2)}}{{{{(1 + {h_r}{\mu_r})}^2}}}\dfrac{{Hh}}{{{{(H - h)}^2}}}\\ \quad + \left( {\dfrac{{{a_1}}}{2} + \dfrac{{2{a_2}}}{5}} \right)\dfrac{{9{\mu_r}({\pm} 2{q_r}{h_r} \pm {q_r}{\mu_r}h_r^2 - 1)}}{{2{{(1 + {h_r}{\mu_r})}^2}}}\dfrac{{Hh}}{{{{(H - h)}^2}}} \end{array} \right]}}.\end{equation}

The exercise reveals several interesting features that are presented in figures 2–4 for ${q_r} = 0$, 25 and (−25) respectively. The figures show that f(h) is +ve for supercritical flow, corresponding to h < 1, and −ve for subcritical flow, corresponding to h > 1, in all cases. The sign of f(h) changes at h ≈ 1, signifying singularity irrespective of the magnitude and direction of the gas flow rate.

Figure 2. Typical plot of (a) f(h) versus h and (b) E_S versus h indicating singularity at h ≈ 1 for q_r = 0 (only liquid flow).

Figure 3. Typical plot of (a) f(h) versus h and (b) E_S versus h indicating singularity at h ≈ 1 for q_r = 25 (co-current flow).

Figure 4. Typical plot of (a) f(h) versus h and (b) E_S versus h indicating singularity at h ≈ 1 for q_r = −25 (counter-current flow).

Figure 5. Co-current flow: observed (data points) and theoretical (solid curves) interface profile and jump location for h_l,in = 0.5 mm and H = 5 mm. In each figure, vertical dotted lines depict the theoretically predicted jump location.

Interestingly, the evolution of specific energy of the liquid film ${E_S}$ with liquid height shows that h ≈ 1 also corresponds to the minimum energy point under a particular flow condition. This is displayed in panels (b) of figures 2–4 where the specific energy of the liquid element at any streamwise position is the summation of pressure, velocity and potential head measured from the conduit floor (Chow Reference Chow1959).

In non-dimensional form using the normalised parameter zⁿ

(2.14)\begin{equation}{E_S} = h + \frac{\alpha }{{2{h^2}}},\end{equation}

where the kinetic energy correction factor $\alpha = (\int_0^h {u_l^3\,\textrm{d}z} )/(hV_{av,l}^3)$ accounts for the effect of the moving gas stream on the liquid phase kinetic energy. Using (2.6a), $\alpha$ can be expressed as

(2.15)\begin{equation}\alpha = \frac{{a_1^3}}{4} + \frac{{a_2^3}}{7} + 3{a_1}{a_2}\left( {\frac{{{a_1}}}{5} + \frac{{{a_2}}}{6}} \right).\end{equation}

In figures 2(b)–4(b), the left and the right-hand arms of the specific energy curves correspond to supercritical and subcritical flows, respectively, and the specific energy curve switches between the two conditions at minimum specific energy. Thus the jump condition does satisfy an energy decreasing condition with increase of h as flow decelerates from supercritical to critical condition.

2.1. Solution methodology

Just as the classical boundary layer theory due to Prandtl encounters singularity at flow separation points, the present theory also becomes singular at the critical points $(\textrm{d}h/\textrm{d}x = \infty )$ where the coefficient of the derivative of liquid height in (2.12) vanishes. A similar situation also occurs in single phase thin film flow (Dhar et al. Reference Dhar, Das and Das2020). Therefore, in order to obtain the interface profile over the entire flow length, (2.12) is solved separately for supercritical flow upstream of the jump and subcritical flow downstream (solid curves in figures 5 and 6). For supercritical flow solution, the inlet boundary condition is h = h_l,in at x = 0. To generate the subcritical flow solution, (2.12) is solved backwards from conduit exit where the liquid falls vertically off the conduit with downward infinite slope of $h(\textrm{d}h/\textrm{d}x ={-} \infty )$. To avoid singularity, (2.12) is solved from slightly before the conduit exit where the liquid height is slightly higher than the height corresponding to infinite liquid slope. This ensures that the downstream interface profile remains relatively insensitive to the assumed liquid height and provides unique solutions upstream and downstream of the jump for each flow condition. A similar approach is commonly adopted for single phase flow as well (Singha et al. Reference Singha, Bhattacharjee and Ray2005; Bonn et al. Reference Bonn, Andersen and Bohr2009; Dhar et al. Reference Dhar, Das and Das2020).

Figure 6. Co-current and counter-current interface profiles from SWT (solid curves) and CFD simulation (dashed-dotted curves) for h_l,in = 0.25 mm and H = 3 mm. The vertical dotted lines depict the SWT predicted jump location (−ve flow rate is for counter-current flow).

Thus, although the solution diverges in the vicinity of the jump, its position can be obtained by connecting the two solutions by a Rayleigh's shock (Rayleigh Reference Rayleigh1914) that conserves mass and momentum flux across the jump. Since ${\rho _g} \ll {\rho _l}$, we neglect the effect of air flow on the jump projected area $({h_2} - {h_1})$. Considering the momentum correction factor $(\beta )$ to be nearly the same at positions 1 and 2 in figure 1(a) (immediately upstream and downstream of the jump), the mass and momentum balance equations express the jump strength $({h_2}/{h_1})$ as

(2.16)\begin{equation}\frac{{{h_2}}}{{{h_1}}} = \frac{1}{2}(\sqrt {1 + 8\beta Fr_1^2} - 1),\end{equation}

where $F{r_1}( = {V_{av,l,1}}/\sqrt {g{h_1}} )$ is the upstream Froude number.

Using (2.16), the jump can be located by connecting the upstream and downstream solutions for h.

Interestingly, (2.16) is a modified form of the classical Bélanger equation (Chow Reference Chow1959; Chanson Reference Chanson2009; Mejean, Faug & Einav Reference Mejean, Faug and Einav2017) which expresses the jump strength for single phase flow. In the present case, the flowing air alters the jump location x ₁ at which (2.16) is satisfied. This is displayed in figures 5 and 6, where the vertical dotted lines pinpoint the jump location for different flow conditions. The approach provides the interfacial profile in a fashion that can be experimentally and numerically probed.

2.2. Limiting condition for only liquid flow

In addition, we note that, in the limit of zero viscosity ratio $({\mu _r} = 0)$, we obtain ${a_1} = 3$ and ${a_2} ={-} 3/2$ from (2.8b) and (2.8c). This simplifies (2.12) to

(2.17)\begin{equation}\left[ {\frac{{{K_1}}}{{{h^3}}} - 1} \right]\frac{{\textrm{d}h}}{{\textrm{d}x}} = \frac{{{K_2}}}{{{h^3}}},\end{equation}

where ${K_1} = 6/5$ and ${K_2} = 3$. We note with interest that Dhar et al. (Reference Dhar, Das and Das2020) had also obtained the same values of the constants ${a_1}$ and ${a_2}$ in their analysis of single phase flow and using the scales determined from order of magnitude analysis of (2.17), ${h^R} = K_1^{1/3}$ and ${x^R} = K_1^{4/3}K_2^{ - 1}$ (2.17) can further be transformed to

(2.18)\begin{equation}\frac{{\textrm{d}h}}{{\textrm{d}x}} = \frac{1}{{1 - {h^3}}}.\end{equation}

Interestingly, (2.18) is the equation obtained by Dhar et al. (Reference Dhar, Das and Das2020) for a single-layer viscous hydraulic jump in a horizontal channel. This confirms that the present approximate theory derived for a two-layer viscous hydraulic jump subscribes to the limiting condition of a single-layer viscous jump in the limit of zero viscosity ratio i.e. zero shear stress at the interface.

3. Experimental validation

Extensive experiments are performed in a test rig (figure 1b) devised to generate simultaneous co-current flow of gas over a thin liquid film such that a planar hydraulic jump occurs over a wide range of flow conditions ($1.25 \le {q_l} \le 1.9\;\textrm{c}{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ and $16.7 \le {q_g} \le 66.8\;\textrm{c}{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$). Special arrangements are made to ensure 2-D laminar flow with minimum interfacial disturbances and negligible entry and exit effects.

Since the experiments are performed to validate the predictions of SWT, the test rig emulates the theoretical domain. It is a rectangular conduit with width (100 mm) much greater than the maximum liquid height to approach the 2-D flow approximation. Its length (90 mm) and height (5 mm) are selected to contain a jump within the conduit over a wide range of experimental conditions. The fluid entry section has a curved and gradually thinning air–water partition which offers minimum resistance to liquid flow and warrants parallel flow with a smooth interface. A multi-orifice liquid distributor is used for uniform liquid velocity along the conduit width. The exit effects are minimised by extending the conduit roof slightly beyond the conduit exit to avoid any external disturbance on air flow. Further, the chamfered exit of the conduit floor is connected to a vertically downward projected face for smooth liquid drainage into the collecting tank. As hydraulic jump characteristics change significantly even with a slight channel tilt, the test rig is mounted on levelling screws to ensure a perfectly horizontal orientation, monitored by a digital protractor.

The electrical conductivity principle (Arakeri & Rao Reference Arakeri and Rao1996; Kate, Das & Chakraborty Reference Kate, Das and Chakraborty2008; Vishwanath et al. Reference Vishwanath, Dasgupta, Govindarajan and Sreenivas2016) is adopted to measure interface profile over the entire conduit length. A thin needle point electrode probe (sensor) is inserted through the conduit roof at different streamwise positions and is traversed vertically to locate the interface with minimum disturbance to the air flow. For a given set of inlet conditions, the experiments are repeated 3 to 5 times.

Figure 5 compares the interface profiles for co-current flow predicted from experiments and theory. We observe an excellent trend matching over the entire range of flow rates (${q_l}$ and ${q_g}$). The interface gradually ascends in the pre-jump region and recedes after the jump in both cases. Even the sharp drooping nature of the interface at the conduit exit is captured by the theory, thus justifying the assumption of $\textrm{d}h/\textrm{d}x ={-} (\infty )$ at the exit. It is interesting to note that, although the theory predicts jump as a position of discontinuity, the predicted interface height differs only by ~10%–15 % from experimental measurements both upstream and downstream of the jump. The deviations are most significant close to the jump region and the differences decrease as one moves away from the jump. Nevertheless, the figure displays a consistent overprediction, more prominent in the post-jump region. This can be attributed to the fact that the mathematical treatment using SWT does not consider viscous losses in the jump region. Such losses in reality reduce the mechanical energy content and lead to a lower post-jump interface height.

4. Validation against CFD analysis

The theoretical results for counter-current flow are validated against numerical simulations performed using the open source code, Gerris. Coflow results for very thin liquid films (h_l,in = 0.25 mm) are also ratified with simulation results.

Gerris models the jump as a two phase flow problem in the computational domain presented in figure 1(a). The details of implementation are provided in Popinet (Reference Popinet2003, Reference Popinet2009), Dasgupta et al. (Reference Dasgupta, Tomar and Govindarajan2015) and Dhar et al. (Reference Dhar, Das and Das2020). Apart from the conventional ‘no slip’, ‘no penetration’ boundary conditions at the solid walls (hatched lines in figure 1a), the free flow boundary condition is imposed at the liquid exit.

Figure 6 shows the results of co- (red curves) and counter- (black) current flow as predicted from theory (solid curves) and CFD (dashed-dotted curves). The theoretical jump locations are denoted by vertical dotted lines (co: red; counter: black). The figure shows that simulations provide the entire range of interface contours, while SWT presents information upstream and downstream of the jump. We note with interest that, despite the approximations involved, the shallow water predictions exhibit a reasonable match with CFD results obtained by solving the full Navier–Stokes equations. This is true for jump location and upstream and downstream profiles for co- as well as counter-flow.

5. Consolidated validation of theory and simulation

We note that the numerical simulations fail to provide results for the entire range of liquid film thickness in the experiments. Therefore, for completeness of the study, we present an additional comparison of the results from SWT and numerical simulations with the experimental data for only liquid flow. The experimental details are provided in Dhar et al. (Reference Dhar, Das and Das2020). The closeness of the predictions of the free surface profile obtained from all the three techniques is evident from figure 7. This lends further credence to the proposed analysis and simulations.

Figure 7. Validation of theoretical and numerical predictions against experimental data for no air flow. The discrete points represent experimental data and dashed-dotted and solid curves correspond to numerical simulation and analytical prediction respectively for ${q_l} = 8.75 \times {10^{ - 5}}\;{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ and L_c = 90 mm. The dotted vertical line indicates the analytically predicted jump location under the specified flow condition.

A close observation reveals that the average mismatches between experimental and simulated results lie within 10 %. This may be attributed to the experimental uncertainties from inherent fluctuations in flow, waves along the conduit walls, unavoidable sidewall effects, etc. These are not considered in the CFD simulations.

6. Parametric variation

Following extensive validation, shallow water results in conjunction with Gerris simulations reveal the influence of the flow parameters (${q_l}$ and ${q_g}$) on the jump nature and characteristics (${x_1}$ and ${h_2}/{h_1}$). The results are presented in figures 8–14. Figures 8 and 9 display the theoretically predicted jump characteristics while figures 10–12 reveal the flow physics as obtained from simulations. Figure 13 compares the velocity profiles obtained from both the approaches. The jump types (wavy, smooth and submerged) are classified based on characteristics revealed from both theory and simulations and the range of existence of each jump type is presented as a phase diagram in figure 14.

Figure 8. Power law dependence of jump location on ${q_l}$ with ${q_g}$ as parameter (−ve flow rates are for counter-current flow).

Figure 9. Influence of ${q_g}$ and ${q_l}$ on (a) jump location and (b) strength (−ve flow rates are for counter-current flow).

Figure 10. Streamwise variation of (a) interface height and (b) Froude number for co-current, counter-current and no flow condition of gas. The W and S marked in (a) denote wavy and smooth jumps, respectively. The inset in figure (b) shows that there is a transition from Fr > 1 to Fr < 1 (i.e. jump) even at high values of co-current gas flow rate.

Figure 11. Typical streamline patterns, interface profiles and streamwise evolution of velocity profiles at ${q_l} = 1.25\;\textrm{c}{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ and ${\pm} {q_g} = 2.75\;\textrm{c}{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ for (a) parallel and (b) opposite flow. Suffixes 1, 2 and 3 display streamwise velocity distribution upstream of jump, at jump and downstream of jump. Markers (hollow circles) in each figure denote interface position. Scaled ordinates non-dimensionalised with inlet liquid velocity u_l,in and height h_l,in to facilitate a comparative study.

Figure 12. Streamline patterns and interface profiles at ${q_l} = 1.25\;\textrm{c}{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ and ${q_g} = (a)\;2.75$ and (b) 5.5 cm² s⁻¹.

Figure 13. Streamwise velocity profiles for (a) co-current and (b) counter-current gas–liquid flow for the gas flow rates used in figure 11 (${q_l} = 1.25\;\textrm{c}{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$ and ${\pm} {q_g} = 2.75\;\textrm{c}{\textrm{m}^2}\;{\textrm{s}^{ - 1}}$). Suffixes 1 and 2 correspond to upstream of jump at x/h_l,in = 40 and downstream of jump at x/h_l,in = 300 respectively. The markers (hollow circles) in the respective velocity profiles denote the interface position.

Figure 14. Phase diagram for jump type as a function of gas and liquid flow rates.