Spatiotemporal Evolution of Gas in Transmission Fluid under Acoustic Cavitation Conditions

Abstract

The presence of gas in transmission fluid can disrupt the flow continuity, induce cavitation, and affect the transmission characteristics of the system. In this work, a gas void fraction model of gas–liquid two-phase flow in a transmission tube is established by taking ISO 4113 test oil, air, and vapor to accurately predict the occurrence, development, and end process of the cavitation zone as well as the transient change in gas void fraction. This model is based on the conservative homogeneous flow model, considering the temperature change caused by transmission fluid compression, and cavitation effects including air cavitation, vapor cavitation, and pseudo-cavitation. In this model, the pressure term is connected by the state equation of the gas–liquid mixture and can be applied to the closed hydrodynamic equations. The results show that in the pseudo-cavitation zone, the air void fraction decreases rapidly with pressure increasing, while in the transition zone from pseudo-cavitation to air cavitation, the air void fraction grows extremely faster and then increases slowly with decreasing pressure. However, in the vapor cavitation zone, the vapor void fraction rises slowly, grows rapidly, and then decreases, which is consistent with the explanation that rarefaction waves induce cavitation and compression waves reduce cavitation.

1. Introduction

Liquid transmission systems are mainly used for the precise transmission of pressure and are widely used in various types of machinery or equipment [1,2]. Complex gas–liquid two-phase flow is often generated in the transmission systems due to factors such as air and high temperature [3,4,5,6,7]. Since the transmission fluid itself dissolves air and has low boiling-point components, cavitation occurs when the transmission fluid pressure induced by the pressure wave is lower than the air separation or saturated vapor pressure [8,9]. This phenomenon is also known as acoustic cavitation. It is caused by air evolution, which forms bubbles or cavities by the low-boiling-point components evaporating around the tiny gas nucleus [10,11,12]. The gas bubbles affect the transmission characteristics; therefore, the ruptured bubbles damage the system, generate noise, and shorten the service life of the machinery [13,14,15,16,17].

The spatiotemporal evolution process of gas in liquid is complex, and it is worth studying with a mathematical model. The gas–liquid two-phase flow model can be divided into the separated flow model and homogeneous flow model. Concerning the separated flow model, Chaudhry et al. [18] divided the continuity equation into liquid and gas phases, considering that the gas and liquid were at the same speed. They coupled the gas–liquid two-phase continuity equation through the source term, and the model validity was verified by the experimental results. However, most researchers used the homogeneous flow model to simulate cavitation [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. Catania et al. [25] developed a new model of positive pressure cavitation based on partial differential equations controlling mass and momentum conservation. The results show that the flow discontinuity caused by the beginning and end of cavitation shows robustness. Furthermore, Catania et al. [26] used a homogeneous flow model to avoid the calculation of slipping velocity between the gas and liquid phases. Thus, the gas and liquid phases were viewed as a single phase. The analytical data were compared with experimental data (the Moehwald-Bosch MEP2000-CA4000 high-performance test bench data). The results show that even in the case of discontinuity caused by cavitation, pressure wave propagation can be simulated with high accuracy [27].

The homogeneous flow model can be divided into the non-conservative and conservative homogeneous flow models. Catania et al. [28] developed a set of fluid dynamics equations for a fuel injection system based on the non-conservative model and simulated the occurrence and time evolution of cavitation. However, Ferrari et al. [29] compared the conservative and non-conservative homogeneous flow models. They ascertained the precision of the conservative homogeneous flow model in cavitation transient flow simulations by comparing the calculated pressure timescales and experimental results for two tube locations in the injection system. The non-conservative homogeneous flow model produced some errors in the analysis.

Catania et al. [30] proposed a comprehensive thermodynamic method to simulate acoustic cavitation considering the temperature change caused by the compression effect and the influence of vapor cavitation in the fuel injection system. The authors compared the isentropic and isothermal evolution of the pure liquid zone to assess the impact of temperature changes on the macroscopic results generated by the local pressure time history and compare them with experiments to verify the effectiveness and robustness at the beginning and end of acoustic cavitation. Ferrari [31] analyzed and discussed the effect of frequency-dependent friction on the simulation of pressure wave dynamics under cavitation conditions and applied it to the momentum conservation equation. Their results showed that the equation provides an outstanding resolution in the time prediction of experimental pressure wave damping.

Numerical simulation of the generation of the transient gas in the transmission fluid can be used to understand the complex two-phase flow phenomena in the transmission system and provide theoretical guidance for accurate control of the transmission system performance [32]. Therefore, in this study, the gas–liquid two-phase flow gas void fraction model in the transmission tube is established by considering three kinds of cavitation effects including air cavitation, vapor cavitation, and pseudo-cavitation using the conservative homogeneous flow model to simulate the generation of the transient gas. To the best of the author’s knowledge, for the first time, the fluid state equation of the gas–liquid mixture is used to connect the pressure terms in the transmission system, and a new numerical solution method is proposed to predict the transient gas void fraction change in different cavitation zones by combining Roe scheme decomposition and Steger–Warming flux splitting method in this study.

In addition to the introduction and conclusion, this paper is divided into four sections. Section 2 establishes a prediction model for the gas–liquid two-phase flow gas void fraction in the transmission tube and provides the three types of cavitation and conditions. Section 3 proposes a new numerical solution method using Roe format decomposition and the Steger-Warming flux splitting method. Section 4 presents the basic physical parameters of the ISO 4113 test oil [33], the initial and boundary conditions for the simulations, and discusses the main parameters of the gas–liquid two-phase flow affecting the gas void fraction.

2. Methods

2.1. Conservative Homogeneous Flow Model

Considering the mass conservation of the transmission fluid before and after compression in the transmission tube and the effect of the flow on the shear stress of the tube wall, the compressible gas–liquid two-phase flow is modeled using the mass and the momentum conservation equations. The density ρ and the momentum ρu are used as conservation variables to obtain a set of conservation-type fluid dynamics equations and expressed in the form of a matrix as follows:

$${\displaystyle \frac{𝜕\mathbf{U}}{𝜕t}}+{\displaystyle \frac{𝜕\mathbf{F}(\mathbf{U})}{𝜕x}}=\mathbf{M}$$

(1)

$$\mathbf{U}=\left[\begin{array}{c}\rho \\ \rho u\end{array}\right],\mathbf{F}(\mathbf{U})=\left[\begin{array}{c}\rho u\\ \rho u^2+P\end{array}\right],\mathbf{M}=\left[\begin{array}{c}0\\ -{\displaystyle \frac{f\rho u\left|u\right|}{2\mathrm{d}}}\end{array}\right]$$

(2)

where ρ, u, P, t, and f are the mixture fluid density, transmission fluid flow velocity, transmission fluid pressure, time variable, and transmission fluid flow resistance coefficient, respectively. The flow is laminar, f = 64/Re, if the Reynolds number Re ≤ 2000 and turbulent, f = 0.316/Re^0.25, if the Reynolds number Re > 2000.

The expression for the gas void fraction is introduced to study the change in gas content of the mixture fluid during compression and expansion:

$$\alpha ={\displaystyle \frac{{\rho }_l-\rho }{{\rho }_l-{\rho }_{\chi }}}$$

(3)

where χ is the air (g) or vapor (v), α is the gas void fraction, and ρ_l is the transmission fluid density.

The density of the homogeneous mixture can be expressed as a function of homogeneous bubbly and stratified flows according to the flow type:

$$\rho =\alpha {\rho }_g+(1-\alpha ){\rho }_l$$

(4)

$$\rho ={\displaystyle \frac{{\rho }_v{\rho }_l}{\alpha {\rho }_l+(1-\alpha ){\rho }_v}}$$

(5)

The state equation of an ideal gas is given by the experimental law of gases and Avogadro’s law:

$$P={\rho }_{\chi }{r}_{\chi }T$$

(6)

where r and T are the gas constant and temperature, respectively.

For positive pressure flow, usually in the cavitation zone, the temperature T is a function of the pressure P, T = T(P). In Equation (3), the transmission fluid density is a function of pressure P and temperature T, ρ_l = ρ_l(T,P).

Neglecting the heat exchange at the boundary, the energy conservation equation for the one-dimensional flow of a compressible fluid can be expressed as:

$${\displaystyle \frac{\mathrm{d}h}{\mathrm{d}t}}-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}=Q$$

(7)

where h and Q denote the enthalpy per unit mass of the system and viscous power dissipation per unit mass in the system, respectively.

To express the change of temperature, entropy s, and the first law of thermodynamics are introduced:

$$q\mathrm{d}s=0$$

(8)

$$T\mathrm{d}s=\mathrm{d}h-{\displaystyle \frac{1}{\rho }}\mathrm{d}P$$

(9)

Considering the temperature change caused by the compression effect when the transmission fluid flows with pure liquid [25]:

$$\mathrm{d}h=C_p\mathrm{d}T+{\displaystyle \frac{1}{\rho }}(1-\phi T)\mathrm{d}P$$

(10)

$$\mathrm{d}P={\gamma }_T{ }^2\mathrm{d}\rho +\phi E_T\mathrm{d}T$$

(11)

where C_p is the constant pressure specific heat capacity, γ_T is the isothermal speed of sound, φ is the coefficient of thermal expansion, E_T is the isothermal bulk modulus of elasticity, and q is the mass flow rate of the mixture.

Under isentropic conditions, Equations (9)–(11) relate the pressure and temperature of the liquid in any process to obtain the following relationship:

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}T}}={\displaystyle \frac{C_p{E}_T}{\phi T{\gamma }_T{ }^2}}$$

(12)

2.2. State Equation of Gas–Liquid Mixtures

The homogeneous flow model requires the addition of the state equation of the gas–liquid mixture to make the set of equations closed. In this study, the fluid is considered as a homogeneous mixture of pure liquid and a certain amount of gas, so the gas–liquid mass flow ratio is introduced as:

$$q=\rho Au$$

(13)

$$\mu ={\displaystyle \frac{q_g}{q_l}}$$

(14)

where μ is the gas–liquid mass flow rate ratio and q_g and q_l represent gas and transmission liquid mass flow rates.

To solve the pressure term for gas–liquid two-phase flow, the gas–liquid mixture fluid state equation needs to be combined. Given a gas–liquid mass flow rate ratio μ, the mixture fluid density ρ, gas density ρ_χ, and gas void fraction α are all functions of pressure P and temperature $T$.

Substituting Equation (3) into Equations (13) and (14):

$$q=\left[\alpha {\rho }_{\chi }+(1-\alpha ){\rho }_l\right]Au$$

(15)

$$q_g=\alpha {\rho }_{\chi }Au$$

(16)

$$q_l=(1-\alpha ){\rho }_{l}Au$$

(17)

Combining Equations (13)–(17):

$$\mu ={\displaystyle \frac{\alpha {\rho }_{\chi }}{(1-\alpha ){\rho }_l}}$$

(18)

$${\rho }_{\chi }={\displaystyle \frac{\rho \mu }{\alpha (1+\mu )}}$$

(19)

Substituting Equations (18) and (19) into Equation (6) yields the state equation of the mixture fluid:

$$\rho ={\displaystyle \frac{{\rho }_l(1+\mu )P}{P+\mu {\rho }_l{r}_{\chi }T}}$$

(20)

The pressure $P$ is related to the mixture fluid density ρ by the mixture fluid state equation, for which an expression for the pressure gradient needs to be given since the flow in the tube generates a pressure drop:

$${\displaystyle \frac{𝜕P}{𝜕x}}={\displaystyle \frac{\mathrm{d}P}{\mathrm{d}\rho }}{\displaystyle \frac{𝜕\rho }{𝜕x}}$$

(21)

2.3. Cavitation Types and Conditions

To obtain an expression for the sound velocity of the mixture fluid, Equation (1) is rewritten as:

$${\displaystyle \frac{𝜕\mathbf{U}}{𝜕t}}+\mathbf{B}{\displaystyle \frac{𝜕\mathbf{U}}{𝜕x}}=\mathbf{M}$$

(22)

$$\mathbf{B}=\left[\begin{array}{cc}0& 1\\ {\gamma }^2-u^2& 2u\end{array}\right]$$

(23)

$$\gamma =\sqrt{{\displaystyle \frac{𝜕P}{𝜕\rho }}}$$

(24)

where B is the Jacobian matrix of conservation hydrodynamic equations.

Two eigenvalues are obtained from Jacobian Matrix B:

$${\lambda }_{1,2}=u\pm \gamma$$

(25)

To calculate the speed of sound, the equation of the bulk modulus of elasticity of the mixture is introduced:

$$E=-V{\displaystyle \frac{\mathrm{d}P}{\mathrm{d}V}}$$

(26)

where V is the fluid volume of the mixture.

According to the expression of sound velocity in Equation (24), the derivative of Equation (4) with respect to pressure P is obtained, and the relationship between the speed of sound and the gas void fraction is obtained:

$${\displaystyle \frac{1}{\rho {\gamma }^2}}={\displaystyle \frac{{\alpha }_v}{E_v}}+{\displaystyle \frac{{\alpha }_g}{E_g}}+{\displaystyle \frac{1-{\alpha }_v-{\alpha }_g}{E_l}}-\rho ({\displaystyle \frac{1}{{\rho }_v}}-{\displaystyle \frac{1}{{\rho }_l}}){\displaystyle \frac{\mathrm{d}{\mu }_v}{\mathrm{d}P}}$$

(27)

When the local pressure inside the transmission tube is lower than the saturated vapor pressure, vapor cavitation occurs. Considering the case of pure vapor cavitation (undissolved air content is zero), Equation (26) is simplified to:

$${\displaystyle \frac{1}{\rho {\gamma }^2}}={\displaystyle \frac{{\alpha }_v}{E_v}}+{\displaystyle \frac{1-{\alpha }_v}{E_l}}-\rho ({\displaystyle \frac{1}{{\rho }_g}}-{\displaystyle \frac{1}{{\rho }_l}}){\displaystyle \frac{\mathrm{d}{\mu }_v}{\mathrm{d}P}}$$

(28)

Since only a small portion of the transmission fluid undergoes vapor cavitation, the heat generated by the phase change of the liquid can be ignored here. On the premise that the cavitation zone is an isothermal flow, the pressure is the saturated vapor pressure at that temperature, so E_v and E_l are constants. The mass fraction of vapor is given, and the derivative of μ_v with respect to P tends to infinity, i.e., γ = 0.

When the transmission fluid pressure is greater than the saturated vapor pressure and less than the air separation pressure, air cavitation will occur, and in a high-pressure environment, the air separation pressure will increase accordingly. Meanwhile, considering the case of air pseudo-cavitation (the air content decreases as the fluid pressure increases), Equation (27) is simplified as follows:

$${\displaystyle \frac{1}{\rho {\gamma }^2}}={\displaystyle \frac{{\alpha }_g}{E_g}}+{\displaystyle \frac{1-{\alpha }_g}{E_l}}$$

(29)

A simple pure liquid flow will emerge when the transmission fluid pressure is greater than the air separation pressure:

$$\gamma ={\gamma }_l$$

(30)

Expressions for the speed of sound for pure liquid flow, air cavitation, and vapor cavitation are derived from Equations (28)–(30):

$$\gamma =\left\{\begin{array}{c}\hspace{1em}{\gamma }_l\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P\ge P_j\hfill \\ \sqrt{{\displaystyle \frac{E_g{\rho }_l{\gamma }_l{ }^2}{\rho \left[\alpha {\rho }_l{\gamma }_l{ }^2+(1-\alpha )E_g\right]}}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{P}_v\le P<P_j\\ \hspace{1em}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P<P_v\hfill \end{array}\right.$$

(31)

2.4. Numerical Solution Method

According to the Harten–Lax theorem, the condition for quasi-linear hyperbolic conservation equations to satisfy Roe decomposition is that there is Matrix B(U_L, U_R) such that F(U_R) − F(U_L) = B(U_L, U_R)(U_R − U_L), and Matrix B should have real eigenvalues and eigenvectors. Thus, Equation (22) is transformed from the nonlinear conserved hyperbolic Equation (1) to the linear conserved hyperbolic equation utilizing a Jacobian coefficient Matrix B. At the same time, the decomposed coefficient Matrix $\overline{\mathbf{B}}$ can be obtained by a similar transformation.

$$\overline{\mathbf{B}}=\overline{\mathbf{R}}\overline{\mathbf{\Lambda }}\overline{\mathbf{L}}$$

(32)

$$\overline{R}=\left[\begin{array}{cc}1& 1\\ {\displaystyle \frac{1}{\overline{u}+\overline{\gamma }}}& {\displaystyle \frac{1}{\overline{u}-\overline{\gamma }}}\end{array}\right], \overline{\Lambda }=\left[\begin{array}{cc}\overline{u}+\overline{\gamma }& 0\\ 0& \overline{u}-\overline{\gamma }\end{array}\right], \overline{L}={\displaystyle \frac{1}{2\overline{\gamma }}}\left[\begin{array}{cc}\overline{\gamma }-\overline{u}& 1\\ \overline{u}+\overline{\gamma }& -1\end{array}\right]$$

(33)

where the matrices R and L are the right and left eigenvector matrices of the coefficient matrix B, respectively. Λ is the eigenvalue diagonal matrix of the coefficient Matrix B.

Expanding U_R − U_L by the eigenvector ${\overline{\mathbf{R}}}_k$.

$${\mathbf{U}}_R-{\mathbf{U}}_L={\displaystyle \sum _{k=1}^2{\overline{\mathbf{R}}}_k{\overline{\mathbf{\sigma }}}_k}$$

(34)

$${\overline{\mathbf{\sigma }}}_k=\overline{\mathbf{L}}({\mathbf{U}}_R-{\mathbf{U}}_L)$$

(35)

Noting $\Delta \rho ={\rho }_R-{\rho }_L$, $\Delta \rho \mathbf{U}={\rho }_R{\mathbf{U}}_R-{\rho }_L{\mathbf{U}}_L$, and to obtain:

$$\Delta \rho ={\overline{\mathbf{\sigma }}}_1+{\overline{\mathbf{\sigma }}}_2$$

(36)

$$\Delta (\rho u)=\overline{u}\Delta \rho +\overline{\gamma }({\overline{\mathbf{\sigma }}}_1-{\overline{\mathbf{\sigma }}}_2)$$

(37)

$$\Delta (\rho u^2+P)=({\overline{u}}^2+{\overline{\gamma }}^2)\Delta \rho +2\overline{u}\overline{\gamma }({\overline{\mathbf{\sigma }}}_1-{\overline{\mathbf{\sigma }}}_2)$$

(38)

Taking ${\overline{\mathbf{\sigma }}}_1=\frac{1}{2{\overline{\gamma }}^2}(\Delta P+\overline{\rho }\overline{\gamma }\Delta u)$ and ${\overline{\mathbf{\sigma }}}_2={\displaystyle \frac{1}{2{\overline{\gamma }}^2}}(\Delta P+\overline{\rho }\overline{\gamma }\Delta u)$, then combining Equations (34)–(38) to obtain:

$$\overline{u}={\displaystyle \frac{\sqrt{{\rho }_R}{u}_R+\sqrt{{\rho }_L}{u}_L}{\sqrt{{\rho }_R}+\sqrt{{\rho }_L}}}$$

(39)

$$\overline{\rho }=\sqrt{{\rho }_R{\rho }_L}$$

(40)

Specifically, $\overline{\gamma }$ is obtained from Equation (31).

In this study, the Roe format decomposition with the Steger–Warming flux splitting method is used to discretize the flux of Equation (22) in the space term, and the Euler forward differencing is used for the time term. Attaining the following differencing format:

$${\mathbf{U}}_i^{j+1}={\mathbf{U}}_i^j-{\displaystyle \frac{\Delta t}{\Delta x}}\left[\widehat{\mathbf{F}}({\mathbf{U}}_i^j,{\mathbf{U}}_{i+1}^j)-\widehat{\mathbf{F}}({\mathbf{U}}_{i-1}^j,{\mathbf{U}}_i^j)\right]+\Delta t{\mathbf{M}}_i^{j+{\displaystyle \frac{1}{2}}}$$

(41)

The numerical fluxes $\widehat{\mathbf{F}}\left({\mathbf{U}}_i^j,{\mathbf{U}}_{i+1}^j\right)$ and $\widehat{\mathbf{F}}\left({\mathbf{U}}_{i-1}^j,{\mathbf{U}}_i^j\right)$ have the functional forms:

$$\widehat{\mathbf{F}}({\mathbf{U}}_i^j,{\mathbf{U}}_{i+1}^j)={\displaystyle \frac{1}{2}}\left[\mathbf{F}({\mathbf{U}}_i^j)+\mathbf{F}({\mathbf{U}}_{i+1}^j)-{\displaystyle \sum _{k=1}^2\left|{\overline{\mathbf{\Lambda }}}_k\right|{\overline{\mathbf{R}}}_k{\overline{\mathbf{\sigma }}}_k}\right]$$

(42)

$$\widehat{\mathbf{F}}({\mathbf{U}}_{i-1}^j,{\mathbf{U}}_i^j)={\displaystyle \frac{1}{2}}\left[\mathbf{F}({\mathbf{U}}_{i-1}^j)+\mathbf{F}({\mathbf{U}}_i^j)-{\displaystyle \sum _{k=1}^2\left|{\overline{\mathbf{\Lambda }}}_k\right|{\overline{\mathbf{R}}}_k{\overline{\mathbf{\sigma }}}_k}\right]$$

(43)

The Steger–Warming flux splitting method can improve the calculation accuracy of the scheme. However, when $\left|u\right|=0$ or on the sound velocity line ($\left|u \pm \gamma \right|=0$), the flow discontinuity differentiable phenomenon occurs, which can lead to numerical solution oscillation. To overcome this drawback, the method is modified as follows.

$${\lambda }_{1,2}={\displaystyle \frac{{\lambda }_{1,2}\pm \sqrt{({\lambda }_{1,2}{ }^2+{\epsilon }^2)}}{2}}$$

(44)

where ε = 2.22 × 10⁻⁴.

3. Results

3.1. Physical Properties of Transmission Fluid

Catania et al. [25] used the high-pressure fuel injection system manufactured for medium-sized vehicles, including the fuel pump, pressure tube, and single spring injector, and used ISO 4113 test oil to simulate diesel. A piezoresistive sensor was set near the outlet of the fuel pump and the fuel injector inlet to collect pressure data at both ends through experiments. In this study, the ISO 4113 test oil is used to simulate the bulk modulus of elasticity, density, speed of sound, and coefficient of thermal expansion as the transmission fluid, where these parameters are the functions of temperature T and pressure P.

The initial condition of Equation (1) is:

$$\left\{\begin{array}{c}P(i,0)=6 \mathrm{bar}\\ U(i,0)=0\\ T(i,0)=319.15 \mathrm{K}\end{array}\right.$$

(45)

This study used the pressure boundary conditions shown in Figure 1.

The other parameters used in the simulations are shown in

Table 1. Parameters required for simulation calculation.

Main Parameters	Unit	Value
Tube length	m	1.1
Tube diameter	mm	2
Air gas constant	J/(mol·K)	287
Vapor gas constant	J/(mol·K)	87.34
Atmospheric pressure	bar	1
Air separation pressure	bar	10
Saturated vapor pressure	bar	0.65
Specific heat capacity ratio	-	1.2
Initial gas-to-liquid mass flow rate ratio	%	0.13
Constant-pressure-specific heat capacity	J/(kg·K)	2.06

. The initial gas content is set to 1.425%, including the gas dissolved in the transmission fluid and gas in the form of bubbles. The grid size of 0.01 m × 2.5 μs (4.8 million grids) was used in the simulations. The process of compression and expansion of transmission fluid is simulated by the pressure difference of the inlet and outlet pressure waves.

3.2. Pressure Transmission Characteristics

Figure 2 shows the transient pressure change of transmission fluid. The x-axis represents the axial position of the tube, and the y-axis represents the time. It can be seen from the figure that the initial pressure of transmission fluid is 6 bar, and the maximum pressure is about 500 bar. The pressure wave propagates in the transmission tube. At t = 0.1–0.14 s, x = 0–0.51 m, the transmission fluid pressure drops significantly, and its pressure is around the saturated vapor pressure. The local cavitation effect of vapor is generated near the inlet of the transmission tube, which leads to cavitation, and then the transmission fluid pressure starts to increase. From the perspective of pressure fluctuation, the changing trend of transmission fluid pressure is the same as that of the result by Catania et al. [25], including the zone where cavitation occurs and the propagation trend of the pressure wave, which illustrates the effectiveness of the conservative homogeneous flow model.

Figure 3 shows the change of gas void fraction with pressure, including air and vapor. At the position of x = 0 m, when t = 0.04–0.07 s, the pressure rises and the gas in the transmission fluid dissolves. Within 5 μs, the gas void fraction decreases from 1% to 0.1%. When t = 0.07–0.1 s, the pressure decreases, and the dissolved gas starts to evolve. Within 5 μs, the gas void fraction increases from 0.05% to about 1%. When t = 0.1–0.14 s the pressure is lower than the saturated vapor pressure, a small part of the transmission fluid evaporates, and the maximum air void fraction reaches about 1.3%. Additionally, air cavitation occurs when t is around 0.16 s. The transmission fluid pressure peak value and the vapor cavitation start time are delayed in the tube axial direction. The vapor cavitation duration decreases along the tube axial direction, from 0.04 to 0 s, as found by Ferrari et al. [29]. However, in the pseudo-cavitation zone, the difference is that the gas void fraction rises and drops rate obtained by the conservative solution is rapidly, while the non-conservative solution is slower. This is because the original variable depends on the bubble model to solve the gas void fraction, and the model does not consider the change in the speed of sound under vapor cavitation. The pressure wave propagates along the speed of sound line γ and changes slowly during the rising and falling of the pressure. The solution of the gas void fraction of the conservation variable depends on the conservation variable density ρ and momentum ρu. The pressure wave propagates along the characteristic line u ± γ, while the characteristics of the three cavitation zones have different slopes. At the turning point where air cavitation and pseudo-cavitation effects occur, the gas void fraction changes rapidly.

3.3. Transient Change of Air and Vapor Void Fraction

Figure 4 shows the change in transmission fluid flow velocity. From the figure, the maximum forward flow velocity of transmission fluid is about 50 m/s, and the reverse flow velocity is almost 130 m/s. The pressure difference between the air cavitation and vapor cavitation stage is large, leading to the rapid increase in flow velocity, especially in the vapor cavitation stage.

The cavitation process is divided into the cavitation formation stage, cavitation growth stage, and cavitation end stage. Figure 5, Figure 6 and Figure 7 present the flow states at the various locations of the tube at t = 0.09–0.145 s. The air cavitation and vapor cavitation are expressed separately in the figures, showing the transient changes in flow velocity, pressure, and gas void fraction along the axial direction of the transmission tube at different times in the two cavitation zones.

Figure 5 shows the variation of transmission fluid flow velocity, pressure, and gas void fraction with the tube axial distance in the cavitation formation stage. When t = 0.09 s, the situation before vapor cavitation is described in Figure 5a. At this time, the pressure wave propagates toward the outlet and does not influence the cavitation zone, so the gas void fraction does not change. Vapor cavitation occurs due to the pressure wave effect when t = 0.1 s. Because the local pressure is lower than the saturated vapor pressure, a rarefaction wave is produced and passes through the cavitation zone, making the flow discontinuous. At the same time, the gas void fraction is dominated by air cavitation, and the maximum air void fraction is around 1.38%, while the maximum vapor void fraction is almost 0.008%. At t = 0.1–0.114 s, the vapor cavitation zone along the tube’s axial direction moves toward the tube outlet and the interval increases from 0.05 m to 0.2 m. The trend of movement is consistent with the result by Catania et al. [25]. Figure 5d indicates that the vapor void fraction gradually increases, reaching 0.012%. As the vapor cavitation increases, the reverse flow velocity increases rapidly, from 40 m/s to 130 m/s. At this stage, the vapor void fraction presents a trapezoidal distribution. Therefore, the vapor void fraction generated by cavitation rises slightly.

The variation of transmission fluid flow velocity, pressure, and gas void fraction with tube axial distance in the cavitation growth stage is shown in Figure 6. When t = 0.122–0.128 s, the gas void fraction is still dominated by air cavitation. The maximum air void fraction increases from 1.38% to 1.387%. Along the tube’s axial direction, the vapor cavitation zone continues to move towards the tube outlet, and the interval increases from 0.23 m to 0.293 m. Due to the short time history and small pressure change at this stage, cavitation needs time to grow. The distribution of vapor void fraction is similar to the M-shape distribution. The maximum vapor void fraction increases from 0.013% to 0.308%, so the vapor cavitation is more intensified. The reverse flow velocity continues to increase from 130 m/s to 140 m/s until the rarefaction wave leaves the vapor cavitation zone, and the pressure wave starts to propagate from the tube’s outlet to the tube inlet.

Figure 7 shows the variation of transmission fluid flow velocity, pressure, and gas void fraction with tube axial distance at the end of cavitation.

The maximum vapor cavitation degree is attained at t = 0.134 s, and the maximum air void fraction is about 1.925%. The vapor void fraction presents an inverted V-shaped distribution, and the maximum vapor void fraction is approximately 0.5% in Figure 7a. This is close to the maximum vapor void fraction of 0.52%, reported by Catania et al. [25]. The vapor cavitation degree starts to weaken at t = 0.137–0.14 s. Along the tube’s axial direction, the vapor cavitation zone starts to move towards the tube inlet, and the interval decreases from 0.172 m to 0.061 m. As the figure shows, the vapor cavitation zone finally develops to the middle of the tube, and then enters the cavitation attenuation period. The pressure wave returns to the cavitation zone to generate compression waves. The velocity of the compression wave propagating to the tube inlet decreases with the attenuation of the cavitation degree, and the reverse velocity decreases from 90 m/s to 60 m/s. This increases the pressure in the zone and gradually weakens cavitation. Actually, the pressure wave and the two-phase fluid react upon each other, i.e., the pressure wave can impact on the bubble nucleation, growth, collapse, and coalescence while the cavitation change can also influence the pressure wave propagation accordingly. It is consistent with the description that rarefaction wave induces cavitation and compression wave reduces cavitation.

4. Discussions

4.1. Spatiotemporal Evolution of Gas Void Fraction

To gain deep insight into the cavitation and pseudo-cavitation zones, as well as the spatiotemporal evolution of gas void fraction, the three-dimensional representation and mapping projection of the time direction as the x-axis, along the axial direction of the tube as the y-axis, and the gas void fraction as the z-axis are plotted in Figure 8. As the figure shows, they can be divided into three zones. The dark yellow zone represents the air cavitation zone, mostly occurring in the 0–0.04 s and 0.2–0.24 s time intervals of the whole tube. When x = 0–0.51 m, a few occur in the time intervals of 0.1–0.14 s. Therefore, the pressure drops rapidly below the air separation pressure, and bubbles are generated quickly. The maximum air void fraction is approximately 1.425%. The orange-red zone denotes the vapor cavitation zone. This process mainly occurs in the time intervals of 0.1–0.14 s in x = 0–0.51 m. The cavitation zone moves to the middle of the tube over time and continuously shrinks, and the vapor cavitation degree reaches the maximum when $x$ is almost 0.45 m. Also, the gas void fraction reaches 1.925% at the maximum point. The purple zone represents the pseudo-cavitation zone, occupying most of the space–time area. The void fraction decreases with the increase in pressure, and the lowest void fraction is about 0.05%. These results show that the numerical calculation algorithm is relatively stable and does not produce obvious oscillation.

4.2. Relationship of Fluid Temperature and Gas Void Fraction

Figure 9 shows the three-dimensional representation and mapping projection of the transmission fluid temperature. As shown in the figure, the maximum amplitude of temperature change with time is nearly 6 K, while Catania et al. [25] obtained a temperature change of about 7 K. In this study, the heat generated by the gas–liquid phase change in the cavitation zone and the external exchange heat is not considered, but the maximum amplitude of the simulated temperature change is close to Catania et al.’s results [25]. The mapping projection illustrates that the temperature increases faster in the forward flow than in the reverse flow. Furthermore, the temperature change in the vapor cavitation zone is smaller than the temperature change in the pseudo-cavitation zone.

Figure 10 shows the effect of initial fluid temperature on gas void fraction. The maximum gas void fraction of the whole process is about 1.3% and 1.9% when the initial fluid temperature is 309.15 K and 364.15 K, respectively, indicating that the gas void fraction increases with the increase in fluid temperature. Since the fluid temperature of the transmission system is higher than the room temperature for a long operating time, its performance will deteriorate. Therefore, the transmission fluid temperature should be controlled within a reasonable range considering air void fraction and operating temperature.

4.3. Complexity and Limitations on Gas Spatiotemporal Evolution Modelling

Under acoustic cavitation conditions, solving the equations governing the spatiotemporal evolution of gas in the transmission fluid is somewhat complicated. For instance, cavitation involves complex interactions between liquid and gaseous phases, including bubble nucleation, growth, collapse, and coalescence governed by different physical principles. Cavitation also results in significant heat transfer and phase changes due to local temperature and pressure conditions. Furthermore, the interaction effect between the acoustic waves and fluid forms a dilemma as the acoustic waves can induce cavitation, and cavitation bubbles can, in turn, influence the propagation of acoustic waves. Considering the complexity of gas spatiotemporal evolution, this research for gas spatiotemporal evolution has certain limitations.

However, from the perspective of pressure amplitude and fluctuation in this work, the amplitude and changing trend of transmission fluid pressure are the same as those of the results by Catania et al. [25]. The results of vapor void fraction, cavitation zone movement distance, and duration are also proved by their work [25], including the vapor void fraction distribution presenting an inverted V-shape, the maximum vapor void fraction being about 0.5%, and the rarefaction wave and compression wave effects on cavitation.

This gas void fraction model of gas–liquid two-phase flow in a slender tube, also the related research findings, can be applied in many fields where acoustic cavitation exists, including automotive and mechanical engineering, maintenance and diagnostics, hydraulic systems, pump and turbine design, acoustic engineering, environmental and safety considerations, environmental mitigation, etc.

5. Conclusions

To accurately predict the spatiotemporal evolution of the gas in the cavitating flow of transmission fluid, the fluid state equation of the gas–liquid mixture is combined with the homogeneous flow model to obtain a gas void fraction prediction model. Considering the three cavitation effects, including air cavitation, vapor cavitation, and pseudo-cavitation, the unsteady cavitating flow is simulated using the Roe scheme decomposition and the Steger–Warming flux splitting method. The numerical calculation algorithm is relatively stable without obvious oscillation.

• When the initial gas content is 1.425%, the maximum gas void fraction in the whole cavitation process reaches 1.925%. The gas void fraction increases with the decrease in pressure, and the peak value of pressure fluctuation is delayed.
• In the air cavitation zone, the lower the pressure is, the faster the air void fraction increases. The maximum air void fraction is about 1.425%. In the pseudo-cavitation zone, the gas void fraction decreases with the increase in pressure. In the vapor cavitation zone, cavitation mainly occurs within x = 0–0.51 m. The maximum vapor void fraction is about 0.5%, and the maximum cavitation duration is almost 0.04 s.
• Under the conditions given in this study, the fluid temperature change caused by the compression and expansion of transmission fluid is nearly 6 K.