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Abstract

The flow characteristics of slender-column flow in wedge-shaped channel under microgravity condition are investigated in this work. The one-dimensional theoretical model is applied to predict the critical flow rate and surface contour of stable flow. However, the one-dimensional model overestimates the critical flow rate for not considering the extra pressure loss. Then, we develop a three-dimensional simulation method with OpenFOAM, a computational fluid dynamics tool, to simulate various phenomena in wedge channels with different lengths. The numerical results are verified with the capillary channel flow experimental data on the International Space Station. We find that the three-dimensional simulation perfectly predicts the critical flow rates and surface contours under various flow conditions. Meanwhile, the general behaviors in subcritical, critical, and supercritical flow are studied in three-dimensional simulation considering variations of flow rate and open channel length. The numerical techniques for three-dimensional simulation is validated for a wide range of configurations and is hopeful to provide valuable guidance for capillary channel flow experiment and efficient liquid management in space.

Keywords

Numerical study capillary channel critical flow rate three-dimensional simulation microgravity

Introduction

Open capillary channel is a structure that contains a liquid with one or more gas–liquid interfaces, which plays an important role in a number of applications in liquid management in space science, such as heat pipes and propellant management devices (PMD) in on-orbit refueling tanks of satellites.^1,2 Under the condition of microgravity, the gravity is negligible and the flow in the channel is dominated by convection, viscous stress, and surface tension. With the increase in flow rate, the pressure difference across the gas–liquid interface gradually increases, and the free surface has to bend inward to balance it. The subcritical phenomenon is observed when the interface is stable. Once the growth rate of pressure difference exceeds the surface curvature change rate, the surface collapses and gas ingests into the channel. This phenomenon is called choking, and we also call it as supercritical phenomenon, which is undesired for many applications. In critical state, the maximum flow rate is reached before the free surface becomes unstable. Thus, the study of flow rate limitation and behavior of the free surface in capillary channel is critical for efficient tank design and liquid management in certain fields.

Early research on capillary channel flow (CCF) under microgravity conditions began with the liquid flow experiments between parallel plates in drop tower.³ Jaekle¹ presented the governing equation for theoretical analysis but the errors of analytical solution are too high due to the neglect of curvature along the flow direction. Rosendahl et al.^4–6 proposed the steady one-dimensional (1D) momentum equation to investigate forced flow between parallel plates and validated with drop tower and sounding rocket flights. The unsteady flow in parallel plates is studied theoretically, numerically, and experimentally by Grah and colleagues.^7–9 Haake et al.¹⁰ expanded 1D theory to groove-shaped channel and verified the numerical solution in the Bremen drop tower.

The wedge-shaped channel (Figure 1) has also been investigated with numerical and experimental methods. The geometry is verified for its good performance in experiment of positioning the liquid¹¹ in PMD. Klatte et al.¹² developed an iterative procedure with surface evolver (SE) algorithm to calculate the critical flow in wedge channel and validated it with drop tower experiments. Wei et al.¹³ also performed theoretical and experimental study of flow characteristics in wedge channel with a constant geometrical configuration. Both studies on wedge channel did not consider the influence on surface behavior caused by flow rate and open channel length. The CCF experiment abroad the International Space Station (ISS¹⁴) provides a chance to thoroughly validate the 1D theory and other numerical method. With reference to CCF ISS experiments, this article presents a comprehensive numerical study on flow rate limitation for steady flow in wedge channel considering variation of flow rate and open channel length. The CCF experiment on ISS is briefly introduced first. Then, we apply the 1D theoretical model to slender-column flow in wedge channel. The critical flow rate and the surface contour in stable flow can be calculated using Newton method. But the flow separation in supercritical flow cannot be observed. Furthermore, the three-dimensional (3D) simulation is performed with the computational fluid dynamics (CFD) tool to investigate the surface behaviors in subcritical, critical, and supercritical flows. The numerical results are verified with CCF ISS experiments. The descriptions of all symbols used in this work are provided in the appendix. This work aims to study the flow mechanism in wedge channel and provides valuable advices for liquid management in space.

Figure 1.

Flow model in a wedge-shaped channel.

CCF experiment on the ISS

The experiments on flow rate limitation in open CCF were extensively performed in drop tower and ballistic rockets.^15–17 But none of the previous experiments like CCF experiments on ISS enabled a broad study with different parameters in order to investigate the stability of CCF. The CCF ISS experiment is a robotic experiment and is operated within Microgravity Science Glovebox (MSG).¹⁴ Till now, the experimental data provide significant reference basis for this area.

The fluid loop in CCF experiment is shown roughly in Figure 2. Along the direction of flow, the liquid is driven by a pump (P) which is connected to the channel’s outlet. Through a flow meter (F), liquid is forced into phase separation chamber (PSC). When gas is ingested into the test channel (TC), the porous screen (S) holds the gas back. In the flow preparation chamber (FPC), the liquid obtains an equal velocity distribution and then goes into the TC. The compensation tube (CT) with a radius of 30 mm maintains a constant pressure in the system and also compensates displaced liquid when the free surface collapses. The knowledge of fluid loop is helpful for numerical study using 1D model and 3D simulation in the following.

Figure 2.

Fluid loop in the CCF experiment on the ISS. Wedge geometry in test channel is considered. The open channel length can be adjusted with movable slider at the outlet.

In this article, we only consider a wedge-shaped channel as shown in Figure 1. The channel has a height b, a width a, and an opening angle $2 α$ . The open channel length l can be adjusted by movable slider (S1) and the flow is driven along the x-direction and the flow rate Q is withdrawn at the outlet. The liquid free surface is exposed to ambient pressure $p_{a}$ . The surface height k is measured from the bottom of the channel to the gas–liquid interface. The experimental parameters are given in Table 1. The test liquid used in the experiments is 3M Novec^TM Engineered Fluid HFE-7500 and its physical attributes are also listed in Table 1. The liquid was chosen according to its feature (the contact angle on glass is $γ = 0 \circ$ ). The behavior of gas–liquid interface in the open section is captured by a high-speed high-resolution (HSHR) camera. These images are important basis to determine the critical flow rate and free surface contours along the open capillary channel.

Table 1.

Channel dimensions and physical attributes of the fluid HFE-7500(in 30°C).

Channel width	a = 8.32 mm
Channel height	b = 30 mm
Wedge angle	2α = 15.8°
Open channel length	0.1 mm < l < 48 mm
Fluid density	ρ = 1610 kg/m⁻³
Surface tension	σ = 16.16 × 10⁻³ kg/s⁻²
Kinematic viscosity	v = 0.715 × 10⁻⁶ m²/s⁻¹

1D theoretical model

The 1D theoretical model could provide approximate flow characteristics for steady capillary flow with little cost. This model has been widely studied as reference for CCF experiments. With an analogy of 1D theoretical model on parallel plates,⁵ this work applied the 1D model to the slender-column flow in wedge-shaped channel. In the 1D model, the flow characteristics are obtained through fast numerical calculations, such as the value of critical flow rate and the free surface contour along the channel in stable flow. The 1D model assumptions, governing equations, and boundary conditions are presented in this section.

Governing equations

As shown in Figure 1, the wedge-shaped CCF is driven by a pump in the x-direction. The flow is assumed to be steady, incompressible, and isothermal. Assuming that the velocity along y- and z-directions is negligible, the 1D form of the momentum equation reads as follows

\underset{convective acceleration}{\underset{⏟}{ρ v \frac{dv}{dx}}} = - \underset{pressure}{\underset{⏟}{\frac{dp}{dx}}} - \underset{viscous pressure loss}{\underset{⏟}{\frac{d w_{f}}{dx}}}

(1)

where $v = Q / A$ is the average velocity along the channel and A is the wetted area. The convective acceleration term in equation (1) is rewritten as follows

v \frac{dv}{dx} = \frac{Q^{2}}{A^{3}} \frac{dA}{dx}

(2)

Free liquid surface in the channel is exposed to ambient pressure $p_{a}$ which is higher than the pressure in liquid p. Thus, the curvature h of the free surface has to balance the pressure difference. According to Young–Laplace equation, the relation can be noted as follows

p_{a} - p = σ h = σ (\frac{1}{R_{1}} + \frac{1}{R_{2}})

(3)

where $R_{1}$ and $R_{2}$ are main radii of the curvature of the interface. As shown in Figure 3, $R_{2}$ is along the x-direction while $R_{1}$ lies in the yoz plane which is perpendicular to $R_{2}$ . The differential form of pressure in equation (1) reads as follows

\frac{dp}{dx} = - σ \frac{dh}{dx}

(4)

Figure 3.

Cross sections of wedge channel and the radii of curvature $R_{1}$ and $R_{2}$ .

In equation (1), $v, ρ, w_{f}$ can be represented as functions of the free surface height k, which indicates the distance from the free surface to the bottom of channel. The calculation of wetted area A is as follows

A = (R + k) R \sin δ - δ R^{2}

(5)

where the calculation of R is related to the value of the surface height k, which is given by

R = {\begin{matrix} k (\sin (α + γ) + \frac{\cos (α + γ)}{\tan α} - 1) for k < \frac{b \tan α}{\cos δ} \\ \frac{{(b - k)}^{2} + {(b \tan α)}^{2}}{2 (b - k)} for k \geq \frac{b \tan α}{\cos δ} \end{matrix}

(6)

where $(b \tan α) / (\cos δ)$ is the critical value of k, and $δ = (π / 2) - α - γ$ . When the flow rate along the channel is small and the flow is not enough to fill the channel, R is calculated using the upper equation. In this work, the channel is full of liquid so the lower equation is used to calculate R.

For the radius of curvature $R_{1}$ , the surface slope $d_{x} k$ must be considered

R_{1} = \frac{{(R \sin δ)}^{2} + R^{2} {(1 - \cos δ)}^{2} (1 + {(d_{x} k)}^{2})}{2 R (1 - \cos δ) \sqrt{1 + {(d_{x} k)}^{2}}}

(7)

The radius of curvature $R_{2}$ along flow direction writes as follows

R_{2} = \frac{{(1 + {(d_{x} k)}^{2})}^{3 / 2}}{d_{x}^{2} k}

(8)

For the viscous pressure loss in the channel, the first derivative form reads as follows

\frac{d w_{f}}{dx} = \frac{ρ v^{2}}{2} \frac{K_{d_{h}}}{Re d_{h}}

(9)

where $K_{d_{h}}$ is the pressure loss coefficient in fully developed condition defined by Ayyaswamy et al.,¹⁸ which is a function of the wedge angle. $K_{d_{h}} = 44.55$ with open angle $2 α = 15.8 \circ$ . $d_{h} = (4 A) / P$ is the hydraulic diameter (P is the wetted perimeter), and the Reynolds number $Re$ is as follows

Re = \frac{v d_{h}}{ν}

(10)

where $ν$ denotes the kinematic viscosity of liquid.

Boundary condition of inlet pressure

The inlet pressure $p_{0}$ is a necessary boundary condition for the solution of equation (1). With reference of the fluid loop in CCF ISS experiment (Figure 2), the initial pressure at the inlet is determined by the pressure $p_{c}$ in the CT and the pressure loss $p_{n}$ along the channel. According to the Bernoulli equation, the relation reads as follows

p_{c} = p_{n} + \frac{ρ v_{0}^{2}}{2} + p_{0}

(11)

The pressure loss along the triangle-closed channel could be approximately calculated. The details are presented by McComas¹⁹

p_{n} = \frac{1}{2} ρ v_{0}^{2} (\frac{C}{Re} \frac{l_{d}}{d_{h}} + \frac{l_{d}}{X_{e}} K)

(12)

where $C and K$ are pressure loss constants and $X_{e}$ is the length of fully developed flow defined by McComas.¹⁹ The values are given in Table 2. $l_{d}$ is the length of closed conduct connected to the inlet and $l_{d} = 63.3 mm$ in CCF ISS experiment setup. With the calculated pressure loss and equation (11), the boundary condition of the channel upstream is expressed as follows

h (x = 0) = \frac{2}{r_{c}} + \frac{ρ v_{0}^{2}}{2 σ} + \frac{p_{n}}{σ}

(13)

where $v_{0}$ is the average flow velocity at the channel inlet, and $r_{c}$ is the dimensionless radius of the CT. Newton non-linear method is used to solve the governing equation (1). The critical flow rate is calculated when the Jacobian matrix in Newton method is singular. At that time, there will be no solution for the equation. So the governing equation in 1D theoretical model can only calculate the stable cases. Scaling details and dimensionless parameters are given in Table 2, and the pressure loss constants are provided referring to CCF experiments on ISS.

Table 2.

Scaled details, dimensionless parameters and pressure loss constants are used to solve the 1D model.

Scaled details		Dimensionless parameters		Pressure loss constants
$k' = k / b$	$h' = h l_{c}$	$l_{c} = (α / 2) / (\cos (α + γ))$	$Oh = \sqrt{ρ v^{2} / σ l_{c}}$	$C = 50.754$
$p' = p / p_{c}$	$v' = v / v_{c}$	$p_{c} = ρ / l_{c}$	$Re = v_{c} l_{c} / v$	$K = 2.2303$
$A' = A / A_{0}$	$Q' = Q / Q_{c}$	$v_{c} = \sqrt{p_{c} / p}$	$B_{o} = ρ {gl}_{c}^{2} / σ$	$(X_{e} / d_{h} \underset{)}{Re} = 0.0525$
$R' = R / l_{c}$	$x' = x / l$	$Q_{c} = A_{0} v_{c}$		$r_{c} = 7.345$

1D: one-dimensional.

Detailed descriptions of all symbols in this work are provided in the appendix.

Numerical techniques for 3D simulation

The 1D theoretical model discussed above is a fast approximate method to calculate the surface contour of capillary flow. However, it has a limitation that the 3D effect of flow velocity is neglected and the supercritical flow cannot be obtained. Thus, a 3D computational tool is necessary to study the 3D influence of wedge-shaped capillary flow. In this work, 3D simulation is performed with OpenFOAM. The surface behaviors under different flow rate and open channel length are investigated through numerical simulation.

OpenFOAM is an open-source CFD toolbox²⁰ and is widely used to solve complex fluid flows.^21–23 The solver interFoam is used to compute the two-phase flow in the capillary channel. The finite volume program uses a technique called the volume of fluid (VOF) method to model the interaction between the fluid and the air.

The motion of an incompressible, isothermal flow is governed by the continuity equation

\nabla \cdot v = 0

(14)

where v is the velocity, and the momentum balance equation is as follows

\begin{matrix} \frac{\partial ρ v}{\partial t} + \nabla \cdot (ρ vv) - \nabla \cdot (μ_{eff} \nabla v) - \nabla v \cdot \nabla μ_{eff} \\ = σ κ \nabla α - \nabla p \end{matrix}

(15)

where $μ_{eff}$ is the effective viscosity and $κ$ is the interface curvature, which write

μ_{eff} = μ + ρ υ_{t}

(16)

κ = \nabla \cdot (\frac{\nabla α}{| \nabla α |})

(17)

where $μ$ is the dynamic laminar viscosity and $υ_{t}$ is the turbulence viscosity. $α$ denotes the phase fraction. The phase fraction $α$ in atmosphere has a value 0 while the liquid in the channel is 1.

The governing equations are discretized using the finite volume method. Lots of discretization schemes with different accuracy are provided by OpenFOAM toolbox, and those terms applied for the simulation of two-phase flow are listed in Table 3.

Table 3.

Summary of discretization schemes.

Term	Scheme	Order of accuracy
Gradient term	Gauss linear	Second order
Diffusion term	Gauss linear corrected	Second order
Convective term	Gauss limitedLinearV 1 or Gauss vanLeer	First order
Temporal term	Backward	Second order

A typical structured mesh used for wedge-shaped flow in 3D simulation is shown in Figure 4. The channel section is located on right side. The block above the interface is built to provide a constant ambient pressure $p_{a}$ . The height of the atmosphere block must be high enough in order to guarantee that the position of atmosphere entrance has no significant influence on the flow in the channel. In this work, the height of the block is set as 28 mm. The initial boundary condition of inlet pressure is set as the computed result of 1D theory. The geometrical configurations of the model are set as same as the wedge channel used in CCF ISS experiments.¹⁴

Figure 4.

A typical structured mesh used for wedge-shaped flow in 3D simulation.

A mesh convergence analysis is performed in this work. We choose three typical meshes with different densities as listed in Table 4, where $Δ x$ is the minimum cell size. Figure 5 shows the changes of the average velocity v at the inlet of the channel with l = 25 mm with three meshes. The flow rate Q increases from 2.7 to 3.2 ml/s during $t \in (55, 65)$ . When $t = 63 s$ , the average velocity v does not increase but decrease because there are bubbles injected in the channel. It is observed that the values of average velocity obtained with M2 and M3 are almost identical and the relative differences are less than 1%. So a converged simulation result is achieved with M2, which is adopted in the following simulations for the computing efficiency.

Table 4.

Properties of meshes with different densities.

Mesh	Cells number	$Δ x_{min} / b$
M1	65,625	0.02083
M2	138,000	0.01667
M3	472,500	0.0119

Figure 5.

Mesh convergence study of the 3D simulation. Results show that the simulation converges after 138,000 cells.

The critical flow rate $Q_{c}$ is the maximum flow rate in stable condition. Once the flow rate exceeds $Q_{c}$ , the free surface collapses and gas is injected into the channel. In this work, an incremental approach is presented to predict the critical flow rate. At the beginning, the flow rate grows from zero to a constant value linearly where the free surface is stable. The velocity in the internal field is also set initially in order to avoid the dynamic effect caused by sudden change in surface. Then, the flow rate is increased by $Δ Q$ in an acceleration time period of $Δ t_{1}$ . Once the new flow rate is reached, the stability of free surface is observed for a time period $Δ t_{2}$ .

The average velocity of cross section at the outlet is chosen to be probe, because it is sensitive to the behavior of the free surface. When the flow rate is small, the free surface is stable and the value of the velocity grows linearly with flow rate increased. Once the critical flow rate is exceeded, the free surface collapses and the average velocity decreases immediately. Critical flow can be determined when the sudden decrease in the average velocity occurs. Assuming that the average velocity changes at the value $v_{c}$ , the critical flow rate is defined as $Q_{c} = A \times v_{c}$ .

The 3D simulation is applied in different capillary channels with a wide range of parameters and its validity will be discussed in the following. The 3D simulation turns out to be helpful to understand the steady flow mechanism in the process of experimental operation.

Numerical results and discussions

In order to verify the effectiveness of the 1D model and the 3D simulation for stability of CCF in wedge-shaped channel under microgravity condition, both the numerical results are compared with the CCF experimental data on ISS.¹⁴ The comparative analysis is presented with variation of flow rate and open channel length. Flow free surface evolution process under various flow conditions is also studied in 3D simulation.

Critical flow rate

In the CCF experiments, an essential purpose is to determine the maximum flow rate before the free surfaces of the liquid becoming unstable. The critical flow rate is also a significant index to verify the numerical model. The numerical results tested in 1D model and 3D simulation are validated with ISS experiments in the following.

In the 3D simulation, the critical flow rate is determined by the incremental approach. Figure 6 shows the average velocity $v (cm / s)$ at the outlet versus time $t (s)$ in channel with l = 25 mm. The flow rate goes from 0 to 2 ml/s in initial 10 s and obtains a curvature on the interface. It is observed that the velocity grows linearly in 0–10 s and keeps constant in 10–20 s. Beginning with 2 ml/s, the flow rate is increased by $Δ Q = 0.5 ml / s$ over the acceleration time period $Δ t_{1} = 10 s$ . Once the new flow rate is reached, the free surface is observed for a period of $Δ t_{2} = 10 s$ to determine its stability. In the process of increase from 3 to 3.5 ml/s, the velocity v does not increase but decrease immediately at the turning point as shown in Figure 6. The flow rate at the vicinity of velocity is defined as critical flow rate. In the case of l = 25 mm, the critical flow rate tested in 3D simulation is 3.1 ml/s. The critical flow rates for other channels with different lengths are calculated using the same method.

Figure 6.

Variation of the average velocity at the outlet in the process of the flow rate acceleration. The value corresponding to the turning point is regarded as critical flow rate. As the channel with l = 25 mm, the critical flow rate obtained in 3D simulation is 3.1 ml/s.

It is worth mentioning that we found the change rate of free surface is related to the acceleration time period $Δ t_{1}$ . The faster flow rate increases, the more roughly the free surface changes. So the computational process for critical flow rate has been repeated for five times with different acceleration time periods $Δ t_{1} = (5, 10, 15, 20, 25)$ . The relative difference, $ε$ , between each solutions and the average value of five solutions (for the critical flow rate) is less than 5%. In this work, $Δ t_{1} = 10 s$ is chosen for analysis considering the computing efficiency.

The predictions of critical flow rate calculated in 1D model and 3D simulation are verified with results in CCF ISS experiments. As shown in Figure 7, a clear boundary between the stable (subcritical) and unstable (supercritical) regime is observed. With increasing open channel length, the critical flow rate seems to decrease hyperbolic for channels with l < 15 mm and the decline appears to be linear for longer channel l > 15 mm. For channels with short length, the viscous pressure loss is small and the flow is dominated by the convective pressure, which results in higher values of critical flow rate. The comparisons in Figure 7 show that for channels with short free surface length (l ≤ 10 mm), the discrepancies between the numerical results (1D model and 3D simulation) and experimental data are obvious. The relative difference between 1D model and experimental data is up to 20% while the relative difference between 3D simulation and experimental data is less than 10%. In contrast, both the relative differences decrease with the increase in the free surface length and are less than 5% for long channel (l ≥ 20 mm). The predictions of critical flow rate in 3D simulation are found to agree well with the experimental data in the entire measurement range. The analysis and comparisons show that the mechanism of steady flow is well understood and the 3D simulation could provide a good and low-costing prediction for CCF experiments.

Figure 7.

Critical flow rate versus open channel length for experimental data and numerical predictions (1D model and 3D simulation) for slender-column wedge-shaped channel.

Critical free surface contours

The 1D model and 3D numerical technique can also be used to calculate the free surface contour for flow rate $Q = Q_{c}$ . Figure 8 shows the comparisons between experimental data and numerical results of the free surface contour in 1D model and 3D simulation at the critical flow rate $Q_{c}$ . The curvature of the free surface reaches its high value at critical flow rate. k denotes the surface height from the bottom of the channel to the surface. At both the inlet $x = 0$ and outlet $x = l$ , the free surface is pinned at k = 30 mm. Three representative surface contours, respectively, in short (l = 5 mm), medium (l = 25 mm), and long (l = 45 mm) channels are chosen for comparison.

Figure 8.

Three representative free surface contours height k along x-direction in critical state for experimental data and numerical predictions (1D model and 3D simulation). They are obtained in channels with l = 5 (top), l = 25 (middle), and l = 45 (bottom), respectively.

As shown in Figure 8, for short channels, the contour of surface is almost symmetrical with respect to center of channel and the numerical results of 1D model and 3D simulation are almost the same as the experiment. With the increase in open channel length, the surface contour becomes asymmetrical gradually, and deviation between 1D theory and CCF experiments occurs. The lowest point of free surface $P_{\min} (x^{*}, k^{*})$ is the typical feature of free surface contour, where the $x^{*} \in [0, l]$ and $k^{*} \in [0, b]$ denotes shortest height from the bottom of channel to the surface. As the length of channel increased, the deformation of the free surface also increases resulted from the movement of $P_{\min}$ toward the outlet of channel. The free surface deformation also indicates the presence of the irreversible pressure loss. In channels with l ≥ 10 mm, $k^{*}$ seems to tend toward the value of 25. Through numerical prediction for sufficiently long channel, $k^{*} = 25 mm$ is found to be a minimum value that is based on the channel geometry.

For further analysis on the surface contour deviations between the numerical results and experiments, note that $ε_{x^{*}}$ and $ε_{k^{*}}$ are the relative differences between the positions of the lowest points $P_{\min} (x^{*}, k^{*})$ obtained in 1D model, 3D simulation, and those in ISS experiments, where $ε_{x^{*}} = | x^{*} - x_{e}^{*} | / x_{e}^{*}$ and $ε_{k^{*}} = | k^{*} - k_{e}^{*} | / k_{e}^{*}$ . $P_{\min} (x_{e}^{*}, k_{e}^{*})$ is the lowest point obtained in ISS experiments. Table 5 shows the relative differences of $P_{\min}$ in 1D model and 3D simulation with different open channel lengths. It is obvious that the x-axis difference $ε_{x^{*}}$ in 1D model increases with the increase in open channel length and reaches 20.89% when l = 45 mm. The same trend is observed in 3D simulation but its $ε_{x^{*}}$ is less than 5%. The height difference $ε_{k^{*}}$ in 1D model is within 2% while $ε_{k^{*}}$ in 3D simulation is less than 1%. The surface contours obtained in 3D simulation are in better agreement with the experimental results than those of 1D model.

Table 5.

Relative differences of the lowest point between 1D model, 3D simulation, and ISS experiments with different open channel lengths.

	$l (mm)$	5	10	15	20	25	30	35	40	45
1D model	$ε_{x^{*}} (%)$	9.18	11.34	16.67	17.50	19.09	19.03	20.24	20.56	20.89
1D model	$ε_{k^{*}} (%)$	0.46	1.29	1.95	0.39	0.79	0.12	1.58	0.74	1.91
3D simulation	$ε_{x^{*}} (%)$	3.6	3.78	4.01	3.51	3.38	4.33	4.72	4.96	5.0
3D simulation	$ε_{k^{*}} (%)$	0.16	0.23	0.5	0.54	0.20	0.37	0.65	0.32	0.72

1D: one-dimensional; 3D: two-dimensional.

As discussed above, the 1D theory generally predicts the critical flow rate and the contours of critical free surface profiles for wedge-shaped channels. However, it overestimates critical flow rate for short channels l ≤ 10 mm, and the numerical contours of free surface differ from the results of experiment typically in long channels. These deviations expose disadvantages of 1D model. During the experiments on ISS, the velocity on the wall is accelerated with flow transforming from the closed duct to the open channel, the process will result in extra pressure loss. Moreover, the separation of the free surface causes an increase in the pressure loss. Both extra pressure losses are not involved in the governing equations of 1D model. This could explain the overestimation of critical flow rate for short channel and differences in the shape of liquid level. Meanwhile, those influences are considered in 3D simulation, and the numerical results are in good agreement with the experimental data. It is concluded that the 1D theory can be effective for fully developed flow and the 3D simulation could provide a good approximation for open CCF.

Free surface evolution process in 3D simulation

Based on the validity of numerical techniques for 3D simulation discussed above, free surface behaviors under various flow conditions are studied in the 3D simulation. Figure 9 shows the representative free surface evolutions in subcritical, critical, and supercritical flow, respectively, in channel with l = 25 mm. The free surface evolution is captured in time period of 10 s after the flow rate was exceeded. It is obvious that the surface behaviors vary with the flow rate under the same open channel length. As introduced above, the lowest point of free surface $P_{\min}$ is located in the center of channel for shorter channel and is typical for convective dominated flow. The changes of position of $P_{\min}$ over time at three different flow rates are shown in Figure 10.

Figure 9.

Free surface development processes of subcritical, critical, and supercritical flow in 3D simulation. These phenomena are observed at flow rate of Q = 2 ml/s, Q = 3.1 ml/s, and Q = 4 ml/s, respectively. In this case, the length of interface l = 25 mm: (a) subcritical flow, Q = 2 ml/s; (b) critical flow, Q = 3.1 ml/s; and (c) supercritical flow, Q = 4 ml/s.

Figure 10.

Position of the spatial coordinates of the lowest point $P_{\min} (x^{*}, k^{*})$ as a function of time $t (s)$ in subcritical, critical, and supercritical flow, respectively. Point $x^{*} = 25$ on the left ordinate axis corresponds to the channel outlet. For subcritical flow, the lowest point oscillates around the center of the channel, the coordinate $x^{*} = 15$ . In critical flow, $x^{*}$ develops forward along x-direction and eventually close to the outlet in supercritical flow. The right ordinate axis represents the corresponding $k^{*}$ : (a) subcritical flow, Q = 2 ml/s; (b) critical flow, Q = 3.1 ml/s; and (c) supercritical flow, Q = 4 ml/s.

In the capillary channel with length l = 25 mm, the subcritical flow is found at the flow rate of 2 ml/s. Evolution of the free surface during the simulation is shown in Figure 9(a). The liquid–gas interface can be observed with the description of phase fraction and is drawn in black color. In the stable case, the surface stops moving inward and keeps a certain curvature. Figure 10(a) shows the change of position of $P_{\min}$ over time. With time increasing, $k^{*}$ continues downward slowly, while its $x^{*}$ seems to oscillate around the value of 15 mm and not develops forward along the x-direction. In subcritical flow, the free surface is almost symmetrical with respect to center of channel.

In the test with $Q = 3.1 ml / s$ , as shown in Figure 9(b), the curvature of free surface increases in critical flow, and the surface becomes asymmetrical with the slightly shift of $P_{\min}$ toward the outlet. But gas ingestion does not occur. Figure 10(b) shows that $k^{*}$ develops downward faster and lower than change of $k^{*}$ in subcritical flow, and $x^{*}$ also moves toward the outlet besides oscillating. The change of $x^{*}$ indicates that the surface behavior in critical flow is influenced by the flow evolution and oscillation effect at the same time. The critical flow is also the maximum stable flow.

When the critical flow rate is exceeded, the free surface collapses and supercritical flow is observed. The process of bubbles involvement at flow rate $Q = 4 ml / s$ is shown in Figure 9(c). Meanwhile, Figure 10(c) shows that $k^{*}$ decreases rapidly and exceeds the minimum value observed in critical flow as Figure 10(b). $x^{*}$ does not oscillate but moves fast close to the outlet before bubbles are produced. Once $P_{\min}$ moves beyond the outlet of the interface, and bubbles involves eventually, which is called flow separation. And after that the free surface will not recover. The constant flow rate is set to avoid the dynamic effect in 3D simulation. The simulated phenomena coincide with the experimental images of steady flow in capillary channel on ISS.

Conclusion

The mechanism of open CCF has been a hot topic and is widely investigated experimentally and numerically. The CCF experiments on ISS developed a broad study on the stability of open channel flow under microgravity conditions with a large range of parameters. For slender-column flow in wedge-shaped channel, the 1D theoretical model with combination of the Young–Laplace equation and the momentum equation is presented to study the flow behavior in this work. Only the velocity in x-direction along the channel is considered in 1D model. We calculate the critical flow rate and the height of fluid level using Newton method. The numerical results are compared with experimental data which shows that the 1D model generally predicts the shape of flow rate and the value of critical flow rate. Nevertheless, the neglect of extra pressure loss in 1D model results in not only higher predictions of the critical flow rate for short channels but also more stable free surface contour in subcritical flow.

Since the limitation that the supercritical flow cannot be obtained through 1D model, the CFD tool is employed for further investigation of general behavior of gas–liquid interfaces in open capillary channels in this work. It helps to understand surface behaviors at different flow rates. Critical flow can be determined when sudden decrease in velocity at the outlet occurs with the step-increasing flow rate. The simulated results have been validated by comparison with experimental data on ISS. Meanwhile, the predicted contours of free surface are in good agreement with experiments in investigated channels with various lengths. Furthermore, surface behaviors under various flow conditions are perfectly reproduced in 3D simulation, The flow mechanisms in subcritical, critical and supercritical flow are studied in detail respectively. The numerical techniques for 3D simulation can be employed to evaluate the general behavior of the gas–liquid interface which is beyond the physical constraints of the CCF experiment.

Footnotes

Appendix 1

Academic Editor: Ishak Hashim

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant Nos 61303071 and 61120106005) and Open fund from State Key Laboratory of High Performance Computing (Nos 201303-01, 201503-01, and 201503-02).

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Numerical study on flow rate limitation of open capillary channel flow through a wedge

Abstract

Keywords

Introduction

CCF experiment on the ISS

1D theoretical model

Governing equations

Boundary condition of inlet pressure

Numerical techniques for 3D simulation

Numerical results and discussions

Critical flow rate

Critical free surface contours

Free surface evolution process in 3D simulation

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References