Dynamics of cavitation bubbles in viscous liquids in a tube during a transient process

Highlights

• •During the transient process, cavitation onset is minimally influenced by the liquid viscosity, but decided by acceleration.
• •The bubble dynamics, including the maximum bubble length, bubble lifetime, and collapse speed, all decrease significantly in a specific range of liquid viscosity.
• •The normalized bubble dynamics are solely decided by the combination of the Reynolds number and Euler number.
• •The criteria for occurrence of liquid column separation are proposed.

Abstract

We experimentally, numerically, and theoretically investigate the dynamics of cavitation bubbles in viscous liquids in a tube during a transient process. In experiments, cavitation bubbles are generated by a modified tube-arrest setup, and the bubble evolution is captured with high-speed imaging. Numerical simulations using OpenFOAM are employed to validate our quasi-one-dimensional theoretical model, which effectively characterizes the bubble dynamics. We find that cavitation onset is minimally affected by the liquid viscosity. However, once cavitation occurs, various aspects of bubble dynamics, such as the maximum bubble length, bubble lifetime, collapse time, and collapse speed, are closely related to the liquid viscosity. We further establish that normalized bubble dynamics are solely determined by the combination of the Reynolds number and the Euler number. Moreover, we also propose a new dimensionless number, ${\mathit{Ca}}_2$, to predict the maximum bubble length, a critical factor in determining the occurrence of liquid column separation.

1. Introduction

Cavitation in liquid flows, or hydrodynamic cavitation, is the formation of vapor cavities in the liquid when the static pressure falls below the liquid vapor pressure and the subsequent dynamic oscillations of the cavities [1], [2]. The inception and progression of cavitation contribute to the degradation of the performance of hydraulic systems (hydraulic turbines, pumps, etc.), characterized by pressure oscillations, structural vibrations, cavitation-induced erosion, and premature structural failures [3], [4], [5], [6], [7], [8], [9]. Under steady-state scenarios where the liquid flow’s velocity stays unchanged over time, a reduction in pressure is typically caused by an elevation in flow speed. Thus, the intensity of cavitation is measured by a cavitation number in the form of a pressure coefficient $\mathit{Ca}=\frac{p_r-p_v}{\frac{1}{2}\rho u^2}$, where $p_r$ and $p_v$ are the reference pressure and the saturated vapor pressure, respectively, $\rho$ is the liquid density, and u is the characteristic liquid speed. This definition indicates the static pressure difference relative to the changes in speed (dynamic pressure). The smaller the cavitation number, the more intense the development of cavitation.

In contrast, cavitation can also be observed in transient processes where there are significant accelerations in the liquid, e.g., the sudden valve closing in the pipeline [10], [11], [12], [13], [14], [15], guide vane or runner closing in the hydraulic turbine [16], [17], [18], [19], [20], etc. In such instances, the velocity of the liquid alone cannot induce sufficient pressure changes to initiate cavitation. In extreme cases, cavitation bubbles can grow big enough to block the entire channel, causing separation of the liquid column. When these large cavitation bubbles collapse, or when the liquid column rejoins, enormous pressure fluctuations can occur within the channel, causing catastrophic destruction to the hydraulic systems [21], [22].

From the above description, it can be seen that the occurrence and severity of cavitation generated in transient processes cannot be described by the traditional cavitation number defined by the characteristic velocity. A number of studies indicate that the oscillation period and size of the cavitation bubbles are decided by the initial flow velocity and the pressure difference in circulating water pipeline systems [22], [11], [15]. Considering the dangers and high costs of studying cavitation during transient processes in pipe-valve systems, Pan et al. [23] experimented by impacting tubes against the ground or with a mallet on top and derived a new cavitation onset criteria for the transient processes, i.e., ${\mathit{Ca}}_1=\frac{p_r-p_v}{\rho \mathit{al}}$, where a is the liquid acceleration and l is the length of the liquid column. This new cavitation number is based on the assumption that when there is linear, one-dimensional transient flow in the pipe, the pressure change comes from acceleration, not velocity. By adopting a modified “tube-arrest” method [24], [25], Xu et al. [26] and Wang et al. [27] generated cavitation bubbles by driving a tube filled with water upwards to impact with a buffer. By considering the energy conservation in the transient processes and ignoring the viscous energy dissipation, they discovered that the size of the cavitation bubbles is controlled by another cavitation number, ${\mathit{Ca}}_2$, which is the traditional cavitation number modified by the geometric characteristics of the liquid column.

In previous research on cavitation during transient processes, the impacts of liquid viscosity were often ignored. However, in practical applications, scenarios of cavitation in viscous liquids (Table 1) are also prevalent. Acceleration-induced cavitation can also be found in circulating systems using these viscous liquids, e.g., the transport of crude oil in pipelines [28], [29], diesel injection in the engine of automobiles [30], [31], [32], ethylene glycol in cooling systems [33], etc. In the medical field, it has been discovered that severe impacts from traffic accidents, sports, or explosions can lead to cavitation in the brain, causing damage to soft brain tissue [34], [35], [36]. It was proven that cavitation bubble impingement and secondary localized shock waves are more powerful than blast overpressures [36]. Patients often present a complex response to such injuries, and persistent symptoms can lead to long-term disabilities. Up to now, the influence of liquid viscosity on cavitation onset and bubble dynamics during the transient process remains unclear.

Table 1.

Typical viscous liquids in applications.

Liquid	Dynamic viscosity ($\mathrm{Pa}·\mathrm{s}$)
Heavy crude oil (20 °C) [28]	0.1–100
Oil-in-water emulsion (20 °C) [29]	0.05–0.2
Diesel (40 °C) [31]	0.3–4.1
Ethylene glycol (20 °C) [33]	0.0198
Extracellular fluid (37 °C) [37]	0.0035–0.008
Cerebrospinal fluid (37 °C) [38]	0.0007–0.001

This study investigates the behaviors of cavitation bubbles in viscous liquids within a cylindrical tube during transient processes using experimental, numerical, and theoretical methods. This paper is structured as follows: in Section 2, we describe in detail the experimental setup and numerical methods; in Section 3, we report the experimental observations of bubbles in water and silicone oil and corresponding simulation results. Combining theoretical analysis, Section 4 discusses the criteria of cavitation onset, and Section 5 provides the scaling law for the bubble dynamics. Finally, conclusions are drawn in Section 6.

2. Methodology

2.1. Experimental setup

A tube-arrest apparatus modified from Xu et al. [26] and Wang et al. [27] is used in the current research, as schematically shown in Fig. 1. An acrylic cylindrical tube (radius $R=7.5,9.5$ mm) filled with liquids to a height l is driven upwards to hit the bottom of a stopper block. Following the impact, the tube comes to a halt while the liquid column continues its upward motion due to inertia. Consequently, the tension within the liquid overcomes its tensile strength, potentially resulting in the formation of cavitation bubbles at the tube bottom. To examine the influence of viscosity on the dynamics of these cavitation bubbles, a series of experiments are conducted utilizing deionized water and two silicone oils with different viscosities as the working fluids (refer to Table 2 for detailed properties). In our experiments, the tube is sealed and a vacuum pump is used to regulate the reference pressure $p_r$ in the tube. The length of the liquid column varies across the range of 10 mm to 600 mm.

Fig. 1.

Schematic of the experimental apparatus and notation (dimensions not to scale).

Table 2.

Properties of the test liquids at room temperature (20 °C).

Liquid	Density ($\mathrm{kg}/{\mathrm{m}}^3$)	Dynamic viscosity ($\mathrm{Pa}·\mathrm{s}$)	Saturated vapor pressure (Pa) [39]
Water	1,000	0.001	2,339
Silicone oil 1 (SO1)	961	0.05	<2,339
Silicone oil 2 (SO2)	972	1	<2,339

We employ a high-speed camera (Phantom V710, Vision Research, USA) to capture the dynamics of the cavitation bubbles, with a frame rate of 20,000–50,000 fps and a maximum resolution of $300\times 800$ pixels. We calibrate the pixel size using the outer diameter of the cylindrical tube. The displacement of the tube $\mathrm{\Delta }x$ is obtained by tracking a certain point on the tube from the high-speed images. The velocity of the tube $u_0$ is then calculated by linear fitting of the displacement curve over $\sim 10$ ms before impact. Self-developed Matlab programs are utilized for identifying the bubble boundaries and calculating the average bubble length L in image processing. For measuring the tube acceleration upon impact, we utilize an accelerometer (357B03, PCB, USA) attached to the tube, sampling at a frequency of 102,400 Hz with an uncertainty of 2$\%$. Synchronization between the accelerometer and the high-speed camera is achieved by using a delay generator (9524, Quantum Composers, USA). In our experiments, the values of $u_0$ and a fall within the range of 0.86–2.17 $\mathrm{m}/\mathrm{s}$ and 261–4,763 $\mathrm{m}/{\mathrm{s}}^2$, respectively.

2.2. Numerical simulation

In our numerical simulations, we concentrate on analyzing the dynamics of cavitation bubbles within viscous liquids throughout transient processes. In contrast to prior investigations conducted by Wang et al. [27] and Ren et al. [40], where the cavitation bubble is modeled by a non-condensable air bubble, we employ a cavitation model to examine the growth and collapse of the cavitation bubbles. These simulations are executed using OpenFOAM, and we provide a brief overview of the methodology below.

Neglecting thermal effects during the bubble growth and collapse, we solve the continuity equation together with the incompressible Navier–Stokes equations for the mixture of liquid and vapor phases:

$$\frac{\partial \rho }{\partial t}+\nabla ·(\rho \mathit{u})=0,$$ (1)

$$\frac{\partial \rho \mathit{u}}{\partial t}+\nabla ·(\rho \mathit{uu})=-\nabla p+\nabla ·\mathit{\tau }+\rho \mathit{g}+\mathit{f},$$ (2)

where $\rho ={\alpha }_l{\rho }_l+{\alpha }_v{\rho }_v$ is the mixture density ($0⩽\alpha \le 1$ is the volume fraction of fluid, and subscripts ‘l’ and ‘v’ represent respectively liquid and vapor phases hereinafter), $\mathit{u}$ the velocity field, $\mathit{\tau }$ the viscous stress tensor, $\mathit{g}$ the gravitational acceleration, and $\mathit{f}$ the source term due to surface tension which is modeled with the continuous-surface-force (CSF) method.

The vapor-water interface is captured with the volume-of-fluid (VOF) method [41]. The transport equation of the volume fraction of the liquid phase ${\alpha }_l$ reads

$$\frac{\partial {\alpha }_l}{\partial t}+\nabla ·({\alpha }_l\mathit{u})+\nabla ·({\alpha }_l{\alpha }_v{\mathit{U}}_{\mathit{r}})={\alpha }_l{\alpha }_v\left[\frac{\dot{{\rho }_v}}{{\rho }_v}-\frac{\dot{{\rho }_l}}{{\rho }_l}\right]+{\alpha }_l\nabla ·\mathit{u}+\dot{m}\left[\frac{1}{{\rho }_l}-{\alpha }_l\left(\frac{1}{{\rho }_l}-\frac{1}{{\rho }_v}\right)\right],$$ (3)

where ${\mathit{U}}_{\mathit{r}}$ is the relative velocity between the vapor and liquid phases, and $\dot{m}$ indicates the total inter-phase exchange rate due to either evaporation or condensation, which is calculated with the Schnerr-Sauer cavitation model [42]:

$$\left\{\begin{array}{ccc}{\dot{m}}^{-}=\frac{3{\rho }_l{\rho }_v}{\rho }{\alpha }_l{\alpha }_v\frac{1}{R_b}\sqrt{\frac{2}{3}\frac{(p_v-p)}{{\rho }_l}}\hfill & ,\hfill & p<p_v\hfill \\ {\dot{m}}^{+}=\frac{3{\rho }_l{\rho }_v}{\rho }{\alpha }_l{\alpha }_v\frac{1}{R_b}\sqrt{\frac{2}{3}\frac{(p-p_v)}{{\rho }_l}}\hfill & ,\hfill & p>p_v\hfill \end{array}\right)$$ (4)

where $R_b$ is the bubble radius and $p_v$ is the saturated vapor pressure. In our simulation, the initial bubble radius $R_b=1$ $\mu \mathrm{m}$, and the saturated vapor pressure $p_v$ is 2,339 Pa for water and 1,800 Pa for silicone oils.

Considering the axisymmetric structure of the experimental configuration, we employ a wedge-shaped computational domain ($0.1°$ in circumferential direction) to reduce the calculation load, as shown in Fig. 2(a). The domain represents a slice of the cylindrical tube used in the experiments with a tube radius R and a height of liquid column l. The left boundary is the symmetry axis of the cylindrical tube. The right and bottom of the slice are solid walls with a no-slip boundary condition. The top boundary is the free surface of the liquid with $\nabla u=0$ and the pressure fixed to a value consistent with the pressure gauge readings. To align with the experiments, we fit a curve based on the tube displacement observed in the experiments and apply it to the simulation.

Fig. 2.

Numerical setup. (a) The computational domain used in the simulations considering the geometric symmetry. (b) The grid independence test using the maximum bubble length $L_{\max }$. (c) The displacement of the tube in the experiment and its fitting curve applied in the simulation for the validation case.

Following the impact, the liquid velocity u decreases from $u_0$ rapidly to 0. For the experiments in this study, the upper bounds of the classic Reynolds number values for pipe flows $\mathit{Re}=\rho \mathit{ud}/\mu$ can be estimated as 41,230 for water, 792 for SO1, and 40 for SO2. Here, the impact velocity $u_0=2.17\mathrm{m}/\mathrm{s}$, and the inner diameter of the tube $d=19$ mm. For simplicity, we assume that the flow in the tube is laminar during most of the bubble lifetime, and carry out laminar flow simulations. The numerical simulations are validated against experiments as in Section 3.

The simulations are conducted with uniformly distributed grids. An example of the grid independence test is shown with the maximum bubble length $L_{\max }$. As illustrated in Fig. 2(b), $L_{\max }$ exhibits good convergence across different grid spacings, and we adopt a grid spacing of 167 $\mathrm{\mu }\mathrm{m}$ in the simulations. For the validation case, $R=9.5$ mm, $l=700$ mm, $p_r=101,325$ Pa. The displacement of the tube in the experiment and simulation is presented in Fig. 2(c). To maintain numerical stability, the adjustable time step $\mathrm{\Delta }t$ is chosen to satisfy the Courant number $u\mathrm{\Delta }t/\mathrm{\Delta }x\le 0.1$.

3. Experimental and simulation results

In this section, we compare the cavitation bubbles generated at the tube bottom in different viscous liquids through experiments and numerical simulations. Fig. 3 illustrates the results in water [Fig. 3(a)] and SO1 [Fig. 3(b)] while keeping the other parameters constant, i.e. $u_0=1.98$ $\mathrm{m}/\mathrm{s},p_r=60,725$ Pa, $R=9.5$ mm, $l=0.6$ m. Each frame presents the numerical result on the left, including the velocity field, and the experimental observation on the right. The time $t=0$ ms denotes the moment when the bubble grows to its maximum size. The first and last frames denote the moments of bubble inception and collapse in the experiment, respectively. Note that the onset of the bubble occurs immediately after the tube is arrested, both in experiments and simulations.

Fig. 3.

Comparison between numerical (left) and experimental (right) results of the cavitation bubble generated at the tube bottom in different viscous liquids: (a) water, (b) SO1. The simulated streamlines and contour plots are superimposed on the left part of each frame. Here, $u_0=1.98$ $\mathrm{m}/\mathrm{s},p_r=60,725$ Pa, $R=9.5$ mm, $l=0.6$ m. The numbers in the upper part of the frames indicate the time in milliseconds. Note that the initial velocity $u_0$ of the experiment in (a) is slightly smaller ($1.94$ $\mathrm{m}/\mathrm{s}$). Movies are provided in the Supplementary data.

First, we discuss the evolution of the bubble in the experiment. In Fig. 3(a), multiple tiny bubbles first appear and grow until filling up the section of the tube (−16 to −14 ms). Subsequently, the bubbles develop into a single cylindrical bubble, and it grows along the tube to the maximum length (−14 to 0 ms). The bubble then undergoes its first collapse (0–18.6 ms). Throughout the cylindrical growth and collapse, the upper surface of the bubble remains flat, indicating a uniform velocity distribution in the liquid along the radial direction. In contrast, the bubble generated in SO1 grows with a convex upper surface (−12 to 0 ms). The curvature of the surface gradually increases until the bubble reaches its maximum length (at $t=0$ ms), which is smaller than that in water. During the collapse, the upper surface deforms and sags downward in the central region (0–18 ms). The shape of the surface thus becomes opposite to that during growth. In the two cases, the difference in the shape of the bubble–liquid interface can be attributed to the liquid viscosity.

It can be seen that the bubble–liquid interface is accurately captured by the simulation, except for the initial moments [frames 1–2 in Fig. 3(a) and (b)]. This can be explained as follows. During the initial bubble growth, it has not yet grown into a cylindrical shape. In case (a), initially many small bubbles grow together. In case (b), the bubble undergoes asymmetric growth and concentrates on the right side of the tube (the left side is not shown in the figure). This is also why we only compare the bubble collapse stage of experiment, simulation, and theory in Section 5. Regarding the flow details in the velocity field, the simulation confirms the uniform velocity distribution along both the radial and axial directions in water, as shown in Fig. 3(a). In contrast, Fig. 3(b) reveals that the boundary layer (blue contour near the tube wall) in SO1 is thicker than that in water, and the liquid velocity increases from the side wall to the center within the boundary layer. During the bubble collapse, the fastest descent occurs at the center of the upper surface of the bubble (see $t=10$ ms). In both cases, the vertical black streamlines in each frame indicate the velocity of the bulk liquid is primarily in the axial direction, except for the region adjacent to the bubble–liquid interface. Moreover, the velocity magnitude of the bulk liquid is close to the impact velocity $u_0$ at the first and final moments and descends to zero when the bubble reaches its maximum size. Based on the above analysis, we can propose a quasi-one-dimensional model for the liquid movement during the transient process. In this model, the liquid column moves along the axial direction of the tube with a radial velocity distribution caused by the liquid viscosity. The average velocity across the section in the beginning can be estimated by the impact velocity $u_0$. This model is crucial for our discussion in the following contexts.

4. Cavitation onset

We discuss the onset criteria of the cavitation bubbles in viscous liquids, based on the quasi-one-dimensional theoretical model as described above. We examine the fluid dynamics within the cylindrical tube as it hits the stopper. For the incompressible and viscous liquid moving in the tube, the continuity equation in cylindrical coordinates (Fig. 1) is

$$\frac{\partial u_x}{\partial x}+\frac{\partial (u_{r}r)}{r\partial r}+\frac{\partial u_{\theta }}{r\partial \theta }=0,$$ (5)

where $x,r,\theta$ represent the axial, radial, and circumferential directions of the tube, respectively. Assuming that the tube and the liquid inside only move along the axial direction (x) of the tube during the impact, i.e., $u_r=u_{\theta }=0$, we can obtain $\partial u_x/\partial x=0$. Denoting $u_x$ as u, the Navier–Stokes equation in x-direction is

$$\frac{\partial u}{\partial t}=-\frac{1}{\rho }\frac{\partial p}{\partial x}-g+\frac{\mu }{\rho }\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}\right),$$ (6)

where $\mu$ is the dynamic viscosity of the liquid. In our experiments, $u\sim O({10}^0)$ $\mathrm{m}/\mathrm{s},t\sim O({10}^{-3})$ s, $\rho \sim O({10}^3)$ $\mathrm{kg}/{\mathrm{m}}^3,p\sim O({10}^5)$ Pa, $x\sim O({10}^{-1})$ m, $\mu \sim O({10}^{-3}-{10}^0)$ $\mathrm{Pa}·\mathrm{s},r\sim O({10}^{-3})$ m. Thus the magnitude order of each term in the equation above can be estimated as (all units are $\mathrm{m}/{\mathrm{s}}^2$): $\frac{\partial u}{\partial t}\sim O({10}^3),\frac{1}{\rho }\frac{\partial p}{\partial x}\sim O({10}^3),g\sim O({10}^1),\frac{\mu }{\rho }\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}\right)\sim O({10}^0-{10}^3)$. It can be seen that the effect of gravity can be neglected.

During the brief time span ($\sim {10}^{-1}$ ms) of the impact, the liquid velocity can be considered uniform in the radial direction and is represented by the bulk velocity of the liquid column, i.e., $\frac{{\partial }^{2}u}{\partial r^2}=\frac{\partial u}{\partial r}=0$. In contrast, the liquid undergoes substantial acceleration, indicated by $a=-\partial u/\partial t$, as a result of the impact. Hence, the primary cause of pressure variations within the liquid during the tube impact, even in the case of viscous liquids, is the liquid acceleration:

$$\frac{\partial u}{\partial t}=-\frac{1}{\rho }\frac{\partial p}{\partial x}\text{.}$$ (7)

The integration of Eq. (7) along x-direction from the bottom of the liquid column to the free surface gives

$$p_r-p_b=\rho \mathit{al},$$ (8)

where $p_r$ and $p_b$ are respectively the reference pressure at the free surface and the pressure at the bottom of the liquid column, and l is the length of the liquid column. Since cavitation tends to occur when $p_b<p_v$ ($p_v$ is the saturated vapor pressure), we then obtain the same dimensionless number as Pan et al. [23], Xu et al. [26], and Wang et al. [27]

$${\mathit{Ca}}_1=\frac{p_r-p_v}{\rho \mathit{al}},$$ (9)

and ${\mathit{Ca}}_1<1$ suggests that cavitation is likely to occur. The pressure distribution along x-direction for water and SO1 in Fig. 3 at $t=10$ ms is presented in Fig. 4(a). In the cavitation region, the pressure remains constant at the saturated vapor pressure. At the liquid-bubble interface, the pressure increases abruptly. Then the pressure increases linearly along x-direction to the reference pressure above the free surface (60,725 Pa). This is consistent with the prediction of Eq. (7).

Fig. 4.

The pressure distribution along (a) x-direction and (b) r-direction for water and SO1 in Fig. 3 at $t=10$ ms.

This equation is formulated under the assumption that, during the transient process, the influence of liquid viscosity on cavitation onset can be neglected owing to the absence of relative motion between the liquid column and the inner tube wall. However, once cavitation bubbles begin forming at the bottom of the tube, the liquid column starts to move relative to the tube wall. As shown in Fig. 4, the length of cavitation bubble in water is larger than that in SO1, and the pressure distribution in the boundary layer turns out to be different. Consequently, it becomes important to account for the effects of viscosity on bubble dynamics, and a detailed discussion is provided in Section 5.

To validate the criterion of cavitation onset, we conduct experiments using deionized water ($\mu =0.001$ Pa$·$s) and silicone oils ($\mu =0.05,1$ Pa$·$s) filled to varying heights ($l=10$–200 mm). The reference pressure above the liquid column is set at $p_r=101,325$ Pa. We identify cavitation onset when bubbles larger than 1 pixel are detected in the high-speed image. The experimental validation of Eq. (9) is presented in Fig. 5. The majority of orange markers, representing cavitation onset, are concentrated in the region where ${\mathit{Ca}}_1<1$ (shaded yellow). Conversely, the blue markers, indicating no cavitation, are predominantly clustered in the region where ${\mathit{Ca}}_1>1$ (shaded blue). The results indicate that the onset of acceleration-induced cavitation can be well predicted by ${\mathit{Ca}}_1$ both in water and highly viscous liquids across a wide range of accelerations and liquid column lengths. According to Eq. (9), we can conclude that in a large range of liquid viscosity, cavitation onset is only determined by the liquid density, liquid column length, impact acceleration, and the reference pressure above the free liquid surface.

Fig. 5.

Phase diagram for the onset of acceleration-induced cavitation for liquids of different viscosities. Blue markers denote no cavitation and orange markers denote cavitation onset. Water, SO1, and SO2 are represented by square, circle, and triangle respectively. The black solid line represents ${\mathit{Ca}}_1=1$ based on Eq. (9).

5. Bubble dynamics

Now we investigate the viscous effects on the dynamics of the cavitation bubble generated at the bottom of the tube. During bubble growth and collapse, the liquid column moves relative to the tube, necessitating the consideration of the viscous term in Eq. (6).

Upon integration with respect to r and assuming $u=0$ at $r=R$, where R is the tube radius, we arrive at

$$u=\frac{1}{4\mu }\left(\frac{\mathrm{d}p}{\mathrm{d}x}+\rho \frac{\partial u}{\partial t}\right)(r^2-R^2),$$ (10)

and the average velocity of the section

$$\overline{u}=-\frac{R^2}{8\mu }\left(\frac{\mathrm{d}p}{\mathrm{d}x}+\rho \frac{\partial u}{\partial t}\right)\text{.}$$ (11)

Further integrating Eq. (11) along x-direction from the bottom of the liquid column to the free surface yields

$$p_r-p_b=-\left(\rho \frac{\partial u}{\partial t}+\frac{8\mu \overline{u}}{R^2}\right)l\text{.}$$ (12)

To conveniently describe the dynamics of the bubble, we make an approximation by treating the bubble as a cylindrical shape that grows and collapses at the average velocity $\overline{u}$. With this consideration, given that $p_b=p_v\ll p_r$, we derive an averaged momentum equation of the liquid by approximating $\partial u/\partial t$ with $\partial \overline{u}/\partial t$ as

$$\frac{\partial \overline{u}}{\partial t}+\frac{\overline{u}}{t_f}=-\frac{u_f}{t_f},$$ (13)

where the characteristic time $t_f=\rho R^2/8\mu$ and the characteristic velocity $u_f=p_r{R}^2/8\mu l$. Assuming the initial velocity $\overline{u}{|}_{t=0}$ approximates the impact velocity $u_0$, the analytic solution of Eq. (13) is given by

$$\overline{u}=(u_0+u_f)e^{-\frac{t}{t_f}}-u_f\text{.}$$ (14)

By applying the initial condition $L{|}_{t=0}=0$, we can calculate the average bubble length L by integrating Eq. (14):

$$L=-t_f(u_0+u_f)e^{-\frac{t}{t_f}}-u_{f}t+t_f(u_0+u_f)\text{.}$$ (15)

For an inviscid liquid, Eq. (15) reduces to the same expression of the bubble length as in Xu et al. [26]:

$$L=u_{0}t-\frac{0.5p_r}{\rho l}{t}^2\text{.}$$ (16)

Referring to Table 1, most liquids applied in circulating systems and medical field have a viscosity ranging from 0.001 to 0.3 $\mathrm{Pa}·\mathrm{s}$. We thus focus on the bubble dynamics in 0.001–0.3 $\mathrm{Pa}·\mathrm{s}$ liquids in our current study. We validate Eq. (15) in Fig. 6 by comparing it with experiments depicted in Fig. 3 and simulation results, with the impact velocity $u_0=1.98$ $\mathrm{m}/\mathrm{s}$, the reference pressure $p_r=60,725$ Pa, the tube radius $R=9.5$ mm, and the liquid column length $l=0.6$ m. It should be noted that the experimental validation is only provided for the bubble collapse stage, taking into account the initial irregular growth of the bubble. The theoretical prediction shows good agreement with experimental and simulation results.

Fig. 6.

Temporal evolution of the average bubble length L in different viscous liquids and the experimental validation with a constant initial velocity $u_0=1.98$ $\mathrm{m}/\mathrm{s}$. Note that $u_0$ of the experiment in water is slightly smaller ($1.94$ $\mathrm{m}/\mathrm{s}$). Here $p_r=60,725$ Pa, $R=9.5$ mm, $l=0.6$ m.

In this context, we only discuss the bubble dynamics in the first cycle. Here, $T_b$ is defined as the duration from the bubble inception to its first collapse, and $T_c$ is the duration from its maximum size to the first collapse. Fig. 6 demonstrates a decrease in both $L_{\max }$ and $T_b$ of the bubble with increasing liquid viscosity $\mu$. For a small liquid viscosity, $T_b=2T_c$, the bubble undergoes symmetrical growth and collapse. In more viscous liquids, such as $\mu =0.3\mathrm{Pa}·\mathrm{s}$, the bubble exhibits an asymmetrical L evolution against time t, with the collapse stage lasting longer than the growth stage. This prompts an investigation into the effect of liquid viscosity on bubble dynamics across a wide parameter range, as follows.

We solve Eq. (15) analytically and plot the results in Fig. 7. As can be expected, the maximum bubble length $L_{\max }$, the bubble lifetime $T_b$, and the collapse speed $u_c$ all decrease with the viscosity $\mu$. These parameters decrease more rapidly when $\mu >0.05\mathrm{Pa}·\mathrm{s}$ and are not significantly affected by extremely low liquid viscosities. Since the pressure pulse at the bubble collapse is proportional to its collapse speed [27], we can infer that high liquid viscosity within a certain range can not only reduce the risk of liquid column separation but also decrease the magnitude of the pressure pulse. Furthermore, Fig. 6(b) reveals that the collapse time $T_c$ decreases slower than $T_b$ especially when the viscosity $\mu >0.05\mathrm{Pa}·\mathrm{s}$. This suggests that the collapse stage will account for a larger proportion of the bubble lifetime when the liquid viscosity increases.

Fig. 7.

The effect of liquid viscosity on the bubble dynamics: (a) the maximum bubble length $L_{\max }$; (b) the bubble lifetime $T_b$ and the collapse time $T_c$; (c) the collapse speed $u_c$. Here $p_r=60725$ Pa, $R=9.5$ mm, $l=0.6$ m, $u_0=1.98$ $\mathrm{m}/\mathrm{s}$.

Now we discuss the associated dimensionless numbers controlling the bubble dynamics. For the viscous liquid moving in the cylindrical tube during a transient process, the governing forces are the inertial force, the pressure force, and the viscous force. Thus we define the associated Reynolds number and Euler number as

$${\mathit{Re}}_f=\frac{\rho u_0{l}_f}{\mu },$$ (17)

$$\mathit{Eu}=\frac{\mathrm{\Delta }p}{\rho u_0^2},$$ (18)

where the characteristic length is taken as $l_f=R^2/l$, and the pressure difference $\mathrm{\Delta }p=p_r-p_v\approx p_r$. According to Eqs. (14), (15), we can readily derive the normalized bubble length $L_{\max }/(u_f{t}_f)$, normalized bubble lifetime $T_b/T_f$, normalized collapse time $T_c/T_f$, and normalized collapse speed $u_c/u_f$ as:

$$\frac{L_{\max }}{u_f{t}_f}=\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}-\ln \left(\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}+1\right),$$ (19)

$$\frac{\frac{T_b}{t_f}}{1-e^{-\frac{T_b}{t_f}}}=\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}+1,$$ (20)

$$\frac{\frac{T_c}{t_f}+\ln \left(\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}+1\right)}{1-e^{-\left[\frac{T_c}{t_f}+\ln \left(\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}+1\right)\right]}}=\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}+1,$$ (21)

$$\frac{u_c}{u_f}-\ln \left(\frac{u_c}{u_f}+1\right)=\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}-\ln \left(\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}+1\right)\text{.}$$ (22)

The analytical results, along with the experimental data for validation, are presented in Fig. 8. It is evident that the normalized parameters of the bubble all increase with $8/({\mathit{Re}}_f·\mathit{Eu})$, suggesting that the dynamics of the bubble are solely dependent on the combination of the Reynolds number and Euler number. Specifically, we observe that when $8/({\mathit{Re}}_f·\mathit{Eu})<{10}^0$, the collapse time

$$T_c\approx \frac{8t_f}{{\mathit{Re}}_f·\mathit{Eu}}=\frac{\rho u_{0}l}{p_r},$$ (23)

which is identical to the results of Xu et al. [26] for inviscid liquid [blue dashed line in Fig. 8(b)], and the bubble lifetime is about twice the collapse time [red dashed line in Fig. 8(b)]. When ${10}^0<8/({\mathit{Re}}_f·\mathit{Eu})<{10}^2$, the collapse time becomes smaller than the inviscid results and approaches the bubble lifetime at large $8/({\mathit{Re}}_f·\mathit{Eu})$. For the normalized collapse speed $u_c/u_f$, as depicted in Fig. 8(c), we find that it is approximately equal to $8/({\mathit{Re}}_f·\mathit{Eu})$ when $8/({\mathit{Re}}_f·\mathit{Eu})<{10}^{-1}$. As $8/({\mathit{Re}}_f·\mathit{Eu})$ increases, the collapse speed $u_c$ reaches the upper limit $u_f$, which is the characteristic velocity of Poiseuille flow. It is worth noting that $8/({\mathit{Re}}_f·\mathit{Eu})$ can also be expressed as $u_0/u_f$, indicating that the bubble dynamics is decided by the ratio of the initial velocity to the characteristic velocity. This provides us with another perspective on this problem.

Fig. 8.

Scaling laws controlling the bubble dynamics. (a) normalized maximum bubble length $L_{\max }/(u_f{t}_f)$, (b) normalized bubble lifetime $T_b/T_f$ and collapse time $T_c/T_f$, and (c) normalized collapse speed $u_c/u_f$ versus $8/({\mathit{Re}}_f·\mathit{Eu})$. The solid lines represent the analytical results of Eqs. (19), (20), (21), (22)), and dashed lines represent the fitting results. Open triangle markers denote the experimental data.

When the cavitation bubble grows to a size comparable to the pipe diameter during the transient process, liquid column separation occurs. The rejoining of the liquid column will induce a large pressure pulse and cause severe damage to the pipeline system. We then introduce a new dimensionless number ${\mathit{Ca}}_2$ to predict the maximum bubble length, defined as

$${\mathit{Ca}}_2=\frac{2R}{L_{\max }}=\frac{2R}{u_f{t}_f\left[\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}-\ln \left(\frac{8}{{\mathit{Re}}_f·\mathit{Eu}}+1\right)\right]},$$ (24)

which depends on the tube radius R, the characteristic velocity $u_f$, the characteristic time $t_f$, the Reynolds number ${\mathit{Re}}_f$, and the Euler number Eu. Notably, when $L_{\max }>2R$, liquid column separation is almost guaranteed to occur, making ${\mathit{Ca}}_2<1$ the onset criterion for the large cavitation bubble originating at the tube bottom. For an inviscid liquid, Eq. (24) can be rearranged as

$${\mathit{Ca}}_2=\frac{2R}{l}\frac{p_r}{0.5\rho u_0^2},$$ (25)

which is identical to the results in Xu et al. [26]. The experimental data are plotted with the conventional number ${\mathit{Ca}}_{2,\mathrm{Xu}}$ and our newly established number ${\mathit{Ca}}_2$ in Fig. 9. The new number considering the influence of viscosity is proved to be effective, especially for highly viscous liquids. It is evident that higher values of ${\mathit{Ca}}_2$, indicating increased flow velocity $u_0$, longer liquid column length l, lower reference pressure $p_r$, smaller viscosity $\mu$, and smaller tube radius R, contribute to a higher likelihood of liquid column separation.

Fig. 9.

$L_{\max }/2R$ versus (a) ${\mathit{Ca}}_{2,\mathrm{Xu}}^{-1}$ and (b) ${\mathit{Ca}}_2^{-1}$ for the experimental results (points) and the theoretical model (solid line). Orange markers denote the occurrence of liquid column separation and green markers denote no liquid column separation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

6. Conclusions

In the present study, we investigate the dynamics of cavitation bubbles generated in viscous liquids in a tube during a transient process via experimental, numerical, and theoretical methods. In the experiments, a modified tube-arrest setup is used to produce cavitation bubbles near the tube bottom and the bubble evolution is recorded with high-speed imaging. The numerical simulations with OpenFOAM are performed and the results agree well with the experimental observations. The velocity fields obtained from simulation provide support for our quasi-one-dimensional theoretical model to describe the bubble dynamics.

Given the negligible relative motion between the liquid column and the inner tube wall during the brief impact, the viscous term in the model can be neglected. We thus propose the dimensionless criterion ${\mathit{Ca}}_1=(p_r-p_v)/\rho \mathit{al}<1$ for cavitation onset, which is the same form as Pan et al. [23], and validate it with experiments using liquids of three different viscosities. This confirms our assumption that the liquid viscosity exerts minimal influence on the cavitation onset during the transient process within a wide viscosity range.

In terms of the temporal evolution of the bubble length, our model aligns well with the experimental and numerical results for liquids with a viscosity up to 0.3 $\mathrm{Pa}·\mathrm{s}$. We then prove theoretically that the bubble dynamics, including the maximum bubble length, bubble lifetime, and collapse speed, all decrease significantly in a specific range of viscosity (e.g., $\mu >0.05\mathrm{Pa}·\mathrm{s}$). We also establish that the collapse time of the bubble constitutes a larger proportion (>0.5$T_b$) with increasing viscosity. Moreover, we find that normalized bubble dynamics are solely decided by the combination of the Reynolds number and Euler number. As the liquid column separation can cause large-scale damage to the structures, we propose a new dimensionless number, ${\mathit{Ca}}_2$, to predict the maximum bubble length based on the theoretical model, which agrees well with experiments. Our findings provide guidance for the design and safe operation of oil pipeline systems, as well as for the study of cavitation-induced brain injury following severe impacts.

CRediT authorship contribution statement

Zhichao Wang: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Writing – original draft, Writing – review & editing. Peng Xu: Investigation, Methodology, Validation. Zibo Ren: Methodology. Liufang Yu: Funding acquisition. Zhigang Zuo: Conceptualization, Funding acquisition, Methodology, Project administration, Supervision, Writing – review & editing. Shuhong Liu: Conceptualization, Formal analysis, Project administration, Supervision, Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Validation of the theoretical assumption of the velocity profile

In the theoretical analysis of the bubble dynamics in Section 5, we made the assumption that the velocity profile of the liquid is quadratic during the bubble lifetime. However, the velocity field depicted in Fig. 3 shows the boundary layer has not fully developed. To validate our previous theoretical assumption, we take the case in Fig. 3(b) as an example and compare the theoretical and numerical velocities in Fig. A.10. For numerical results, we discuss the liquid velocity at the section right above the bubble–liquid interface. As shown in Fig. A.10(a)–(f), the numerical velocity experiences a rapid increase beside the tube wall and remains nearly constant across the radial direction of the tube. The deviation between the numerical and theoretical velocity distributions is minimal, except for the moments near the bubble origination and collapse [Fig. A.10(a) and (f)]. Despite the difference in the maximum value of the velocity profile, the theoretical average velocity throughout the bubble lifetime aligns well with the numerical results, as shown in Fig. A.10(g). This indicates the validity of the simplifications made in our theoretical analysis.

Fig. A.10.

Validation of the theoretical velocity profile for the case in Fig. 3(b). (a)–(f) Theoretical and numerical velocity distributions along the radial direction above the bubble. (g) Comparison of the theoretical and numerical evolution of the average velocity above the bubble. Here the velocity refers to the absolute value of the average velocity both in the simulation and theoretical model.

Appendix B. Typical case of no liquid column separation

The cavitation bubble tends to be smaller in highly viscous liquids and cannot fill up the tube section. A typical case in our experiments [as mentioned in Fig. 9(b)] is presented in Fig. B.11 here as a supplement.

Fig. B.11.

High-speed images of a typical case of no column separation in SO2. The numbers in the upper part of the frames indicate the time in milliseconds. Here, $u_0=1.42$ $\mathrm{m}/\mathrm{s},p_r=101,325$ Pa, $R=7.5$ mm, $l=0.6$ m.

Appendix C. Simulation results of a high-viscosity case

We simulated a 2 $\mathrm{Pa}·\mathrm{s}$ case, and the results are presented in Fig. C.12. It can be seen that the cavitation still occurs in a highly viscous liquid, but the bubble size is much smaller than that in 0.05 $\mathrm{Pa}·\mathrm{s}$ and 0.3 $\mathrm{Pa}·\mathrm{s}$ silicone oils. It confirms our conclusion that the liquid viscosity has minimal effect on cavitation onset, but significantly influences the bubbly dynamics.

Fig. C.12.

Simulation results of a 2 $\mathrm{Pa}·\mathrm{s}$ case.

Supplementary data

The following are the Supplementary data to this article: