2.1 Dynamic Mass Fraction of Air Components
2.1.1 Dynamic Mass Fraction of Vapor
Singhal et al. [22] proposed a transmission equation for computing the vapor mass fraction while computing the dynamic fraction of vapor in an aerated hydraulic fluid.
$$\frac{\partial }{\partial t}\left( {\rho f_{{\text{v}}} } \right) + \nabla \cdot \left( {\rho {\mathbf{U}}f_{{\text{v}}} } \right) = \nabla \cdot \left( {\Gamma \nabla f_{{\text{v}}} } \right) + R,$$
(1)
where t represents time (s), \({\mathbf{U}}\) the transmission rate of vapor, \(\rho\) the density of the aerated hydraulic fluid (kg/m^{3}), \(f_{{\text{v}}}\) the vapor mass fraction of the aerated hydraulic fluid, and \(R\) the phase change speed between liquid and vapor.
We assume that fluid attributes, such as the density and pressure of the fluid in the control volume, are uniformly distributed. Therefore, Eq. (1) can be simplified to the following form by disregarding the diffusive term, where the position derivative is used as the basis in the transmission equation:
$$f_{{\text{v}}} \frac{{{\text{d}}\rho }}{{{\text{d}}t}} + \rho \frac{{{\text{d}}f_{{\text{v}}} }}{{{\text{d}}t}} = R,$$
(2)
Typically, the density of an aerated hydraulic fluid typically does not change significantly when cavitation occurs and \(\frac{{{\text{d}}\rho }}{{{\text{d}}t}} < < \frac{{{\text{d}}f_{{\text{v}}} }}{{{\text{d}}t}}\). Thus, Eq. (1) can be further simplified as follows:
$$\frac{{{\text{d}}f_{{\text{v}}} }}{{{\text{d}}t}} = R_{{\text{s}}} ,$$
(3)
where \(R_{{\text{s}}}\) is the simplified phase change speed.
From the perspective of bubble collapse and formation in liquid, the bubble dynamics equation derived from the Rayleigh–Plesset equation can be used to calculate the phase transition rate between the liquid and vapor, as follows:
$$R_{{\text{B}}} \frac{{{\text{d}}^{2} R_{{\text{B}}} }}{{{\text{d}}t^{2} }} + \frac{3}{2}\left( {\frac{{{\text{d}}R_{{\text{B}}} }}{{{\text{d}}t}}} \right)^{2} = P  \frac{{4\nu_{{\text{l}}} }}{{R_{B} }}\frac{{{\text{d}}R_{{\text{B}}} }}{{{\text{d}}t}}  \frac{2\sigma }{{\rho_{{\text{l}}} R_{{\text{B}}} }},$$
(4)
where \(R_{{\text{B}}}\) is the bubble radius (m), \(\nu_{{\text{l}}}\) the fluid kinematic viscosity (m^{2}/s), \(\sigma\) the surface tension coefficient of the liquid (N/m), \(P\) the ratio of the difference between the internal pressure of the bubble and the pressure of the fluid to the density of the liquid, and \(\rho_{{\text{l}}}\) the density of the liquid (kg/m^{3}).
P is expressed as
$$P = \left( {\frac{{p_{{\text{B}}}  p}}{{\rho_{{\text{l}}} }}} \right),$$
(5)
where \(p_{{\text{B}}}\) is the internal pressure of the bubble (Pa), and p is the pressure of the fluid (Pa).
The surface tension coefficient \(\sigma\) is computed as follows [29, 30]:
$$\sigma = \sigma_{0} \left( {1  \frac{T}{{T_{{\text{c}}} }}} \right)^{\delta } ,$$
(6)
where \(\sigma_{0}\) is initial surface tension coefficient of the liquid (N/m), \(T_{{\text{c}}}\) the critical temperature of the liquid (K), T the system operating temperature (K), and \(\delta\) a global exponent.
Based on the derivation process of the full cavitation model and the bubble dynamics equation, as well as disregarding the viscosity term in the equation, the net phase transition rate can be expressed as [15]
$$R = \frac{{3\alpha_{{\text{v}}} }}{{R_{{\text{B}}} }}\frac{{\rho_{{\text{v}}} \rho_{{\text{l}}} }}{{\rho^{2} }}\left[ {\frac{2}{3}P  \frac{2}{3}\frac{2\sigma }{{\rho_{{\text{l}}} R_{{\text{B}}} }}  \frac{2}{3}R_{{\text{B}}} \frac{{{\text{d}}^{2} R_{{\text{B}}} }}{{{\text{d}}t^{2} }}} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} ,$$
(7)
where \(\alpha_{{\text{v}}}\) is the volumetric vapor fraction of the aerated hydraulic fluid, and \(\rho_{{\text{v}}}\) is the density of the vapor (kg/m^{3}).
By substituting R into Eq. (7) and disregarding the secondorder derivative of the bubble radius, the following formula is obtained:
$$\frac{{{\text{d}}f_{{\text{v}}} }}{{{\text{d}}t}} = \frac{{3\alpha_{{\text{v}}} }}{{R_{{\text{B}}} }}\frac{{\rho_{{\text{v}}} \rho_{{\text{l}}} }}{{\rho^{2} }}\left[ {\frac{2}{3}P  \frac{2}{3}\frac{2\sigma }{{\rho_{{\text{l}}} R_{{\text{B}}} }}} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} ,$$
(8)
where \(\alpha_{{\text v}}\) can be expressed as a function of \(f_{{\text v}}\) as follows:
$$\alpha_{{\text{v}}} = f_{{\text{v}}} \left( t \right)\frac{\rho }{{\rho_{{\text{v}}} }},$$
(9)
By assuming that all bubbles have the same radius and that the balance between aerodynamic drag and liquid surface tension determines the bubble radius, \(R_{{\text{B}}}\) can be calculated as follows [30]:
$$R_{{\text{B}}} = \frac{0.061W\sigma }{{2\rho_{{\text{l}}} v_{{{\text{rel}}}}^{2} }},$$
(10)
where \(W\) is the Weber number.
For an aerated hydraulic fluid containing bubbles, \(v_{{{\text{rel}}}}^{{}}\) is the relative velocity of the aerated hydraulic fluid, whose value is relatively small, i.e., 5%–10% of the mean velocity of the fluid. By substituting Eqs. (9) and (10) into Eq. (8), the following equation is obtained:
$$\frac{{{\text{d}}f_{{\text{v}}} }}{{{\text{d}}t}} = \frac{{6v_{{{\text{rel}}}}^{2} }}{0.061W\sigma }\frac{{\rho_{{\text{l}}} \rho_{{\text{l}}} }}{\rho }\left {P^{\prime}  \frac{2}{3}\frac{{4v_{{{\text{rel}}}}^{2} }}{0.061W}} \right^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} f_{{\text{v}}} ,$$
(11)
Here,
$$P^{\prime} = \frac{2}{3}\frac{{p_{{\text{v}}}  p}}{{\rho_{{\text{l}}} }},$$
(12)
where \(p_{{\text{v}}}\) is the saturated vapor pressure (Pa).
Using the analysis methods presented in Refs. [22, 23], \(v_{{{\text{rel}}}}^{{2}}\) can be expressed as a function of two components: one is \(\sqrt k\), which is the delegate of the flowing state of the fluid; and the other is a component simplified with the parameters in Eq. (11), such as the Weber number and the surface tension coefficient to the vapor condensation coefficient \(a_{11}\) for characterizing the speed of vapor condensation. Thus, when the pressure of the aerated hydraulic fluid remains higher than the saturation vapor pressure, the coagulation of vapor can be described as
$$\frac{{{\text{d}}f_{{\text{v}}} }}{{{\text{d}}t}} = a_{11} \sqrt k \frac{{\rho_{{\text{l}}} \rho_{{\text{l}}} }}{\rho }\left[ {\left {P^{\prime}  \frac{4}{9}a_{11} \sigma \sqrt k } \right} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} f_{{\text{v}}} .$$
(13)
Subsequently, the formula for turbulent energy can be used to calculate the turbulent energy of the aerated hydraulic fluid, as follows:
$$k = \frac{3}{2}\left( {\mu l} \right)^{2} ,$$
(14)
where \(\mu\) is the mean velocity (m/s), and \(l\) is the turbulence intensity.
The condensation of vapor is associated with the vapor mass fraction, and during vaporization, residual liquid is regarded as the source of vapor. Therefore, analogous to Eq. (13) is a polynomial function of pressure. The segmented expression for the air mass fraction is
$$\frac{{{\text{d}}f_{{\text{v}}} }}{{{\text{d}}t}} = a_{12} \sqrt k \frac{{\rho_{{\text{v}}} \rho_{{\text{l}}} }}{\rho }\left[ {\left {P^{\prime}  \frac{4}{9}a_{12} \sigma \sqrt k } \right} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} \left( {1  f_{{\text{v}}}  f_{{{\text{g}}0}} } \right),$$
(15)
where \(a_{12}\) is the vaporization coefficient of the vapor, and \(f_{{{\text{g0}}}}\) is the initial air mass fraction in the aerated hydraulic fluid.
2.1.2 Dynamic Mass Fraction of Air
The air pressure required to release air in an aerated hydraulic fluid is typically higher than the saturated vapor pressure required for vaporization. Cavitation typically involves a significant amount of air released from an aerated hydraulic fluid in a short duration and a rapid collapse of gas or vapor bubbles when the pressure increases. Because air constitutes the air phase in an aerated hydraulic fluid, the Rayleigh–Plesset equation can be applied to describe the release and dissolution of air. In addition to Eqs. (11) and (15), the following assumptions can be applied as bases for obtaining the air release and dissolution equation:
1. \(f_{{{\text{gH}}}}\) is the theoretical target value of the air mass fraction (steadystate value) when the pressure function time is sufficiently long. Meanwhile, when the transient pressure remains below the saturation vapor pressure, all of the dissolved air should be released. When the transient pressure remains above the air apart pressure, all free air should dissolve, which results in a \(f_{{{\text{gH}}}}\) of zero. Subsequently, using the improved Henry’s law, when the transient pressure is within the ranges of the saturation vapor and air apart pressures, the target value of \(f_{{{\text{gH}}}}\) is a polynomial function of pressure. The segmented expression of the air mass fraction is
$$f_{{{\text{gH}}}} = \left\{ \begin{gathered} f_{\text g0} {,\, }p \le p_{{\text{v}}} , \hfill \\ f_{{{\text{g}}0}} \left( {1  10k_{{\text{g}}}^{3} + 15k_{{\text{g}}}^{4}  6k_{{\text{g}}}^{5} } \right){,\, }p_{{\text{v}}} < p \le p_{{\text{s}}} , \hfill \\ \, 0{,\, }p > p_{{\text{s}}} , \hfill \\ \end{gathered} \right.$$
(16)
where \(p_{{\text{v}}}\) is the saturated vapor pressure (Pa), \(p_{{\text{s}}}\) is the air apart pressure (Pa), and \(k_{{\text{g}}}^{{}}\) is expressed as
$$k_{{\text{g}}}^{{}} = \frac{{p  p_{{\text{v}}} }}{{p_{{\text{s}}}  p_{{\text{v}}} }},$$
(17)
2. \(f_{{\text{g}}}\) is the instantaneous air mass fraction. When \(f_{{\text{g}}}\) is lower than \(f_{{{\text{gH}}}}\), air should be released from the liquid; otherwise, it should dissolve gradually.
When \(f_{{\text{g}}} \le f{}_{{\text {gH}}}\), the air release equation is
$$\frac{{{\text{d}}f_{{\text{g}}} }}{{{\text{d}}t}} = a_{21} \sqrt k \frac{{\rho_{{\text{l}}} \rho_{{\text{l}}} }}{\rho }\left[ {\left {P^{\prime\prime}  \frac{4}{9}a_{21} \sigma \sqrt k } \right} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} \left( {f_{{{\text{gH}}}}  f_{{\text{g}}} } \right),$$
(18)
where \(a_{21}\) is coefficient of air release.
Meanwhile, \(P^{\prime\prime}\) can be written as
$$P^{\prime\prime} = \frac{2}{3}\left( {\frac{{p_{{\text{s}}}  p}}{{\rho_{{\text{l}}} }}} \right).$$
(19)
When \(f_{{\text{g}}} > f{}_{{\text {gH}}}\), the dissolution of air is expressed as
$$\frac{{{\text{d}}f_{{\text{g}}} }}{{{\text{d}}t}} =  a_{22} \sqrt k \frac{{\rho_{{\text{l}}} \rho_{{\text{l}}}^{{}} }}{\rho }\left[ {\left {P^{\prime\prime}  \frac{4}{9}a_{22} \sigma \sqrt k } \right} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} f_{{\text{g}}} ,$$
(20)
where \(a_{22}\) is the coefficient of air dissolution.
2.2 Dynamic Mass Fraction of Air Components
As the pressure of the aerated hydraulic fluid changes, the density of each component changes accordingly. Based on the air state equation, the air density is written as
$$\rho_{{\text{g}}} = \rho_{{{\text{g0}}}} \left( {\frac{p}{{p_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern\nulldelimiterspace} \lambda }}} ,$$
(21)
where \(p_{0}\) is the standard atmospheric pressure (Pa), \(\rho_{{{\text{g0}}}}\) the air density under the standard atmospheric pressure (kg/m^{3}), and \(\lambda\) the variability index of the air variable course.
The vapor density is expressed as
$$\rho_{{\text{v}}} = \rho_{{{\text{v0}}}} \left( {\frac{p}{{p_{{\text{v}}} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern\nulldelimiterspace} \lambda }}} ,$$
(22)
where \(\rho_{{{\text{v0}}}}\) is the density of vapor under the standard atmospheric pressure (kg/m^{3}).
Eqs. (21) and (22), show that the airphase density in the aerated hydraulic fluid is directly related to the variability index in the air variable course. The course of air expansion and compression is typically regarded as either isothermal or adiabatic, which implies that the variability index is a constant. However, the course of air expansion and compression is typically not completely isothermal or adiabatic in actual circumstances; thus, the variability index must be adjusted based on the operating conditions. The volumetric change rate \(\varepsilon\), which reflects the rate of air volume change, is the main factor affecting the air variability index. A functional relationship is assumed to exist between \(\varepsilon\) and \(\lambda\), i.e., \(\lambda = f\left( \varepsilon \right)\). Based on the theory of the air variable course, the function must satisfy the following conditions:

(1)
When the volumetric changing rate approaches zero, the air variable course becomes isothermal, i.e., \(\mathop {\lim }\nolimits_{\varepsilon \to 0} f\left( \varepsilon \right) = \lambda_{{{\text{iso}}}}\).

(2)
When the volumetric changing rate approaches infinity, the air variable course becomes adiabatic, i.e., \(\mathop {\lim }\nolimits_{\varepsilon \to \infty } f\left( \varepsilon \right) = \lambda_{{{\text{adi}}}}\).

(3)
When \(\varepsilon\) varies from zero to infinity continuously and monotonously, its first and second derivatives exist, and they are continuous with unchanged signs.
By constructing a function that satisfies the conditions mentioned above, \(\lambda\) can be expressed as follows:
$$\lambda = \frac{{\lambda_{{{\text{iso}}}} }}{{1 + {\text{e}}^{\varepsilon /M} }} + \frac{{C_{{\text{n}}} {/}C_{{\text{v}}}  \lambda_{{{\text{adi}}}} }}{{{2}C_{{\text{n}}} {/}C_{{\text{v}}}  1}} + 2\sqrt {\frac{{C_{{\text{n}}} }}{{C_{{\text{v}}} }}\ln \left( {\frac{T}{{T_{0} }}} \right)} ,$$
(23)
where \(\lambda_{{{\text{iso}}}}\) is a multivariable index of the isothermal course, \(M\) a multivariable course coefficient, \(C_{{\text{n}}}\) the molar heat capacity of air (J/mol·K), \(C_{{\text{v}}}\) the molar heat capacity of air (J/mol·K), \(\lambda_{{{\text{adi}}}}\) a multivariable index of the adiabatic course, and \(T_{0}\) the initial temperature (K).
The molar heat capacity of air can be expressed as [31]
$$C_{{\text{n}}} = \frac{{S_{m}  S_{m0} }}{{\ln (T/T_{0} )}},$$
(24)
where \(S_{m}\) is the entropy of air(J/mol·K), and \(S_{m0}\) is the entropy of air at the initial temperature (J/mol·K).
The density of the liquid can be expressed as
$$\rho_{l} = \rho_{{{\text{l0}}}} e^{{\frac{{p  p_{0} }}{{E_{{\text{l}}} }}}} ,$$
(25)
where \(\rho_{{{\text{l0}}}}\) is the density of the liquid at the standard atmospheric pressure (kg/m^{3}), and \(E_{{\text{l}}}\) is the bulk modulus of the liquid at the transient pressure (Pa).
When the pressure and temperature change, the bulk modulus of the liquid changes accordingly. The relationship among the bulk modulus of the liquid, pressure, and temperature can be expressed as follows:
$$E_{{\text{l}}} = E_{{{\text{ref}}}} + m\Delta p + n\Delta T,$$
(26)
where \(E_{{{\text{ref}}}}\) is the bulk modulus of the liquid under the standard atmospheric pressure (Pa), \(m\) the pressure changing coefficient of the bulk modulus, and \(n\) the temperature changing coefficient of the bulk modulus.
Combining Eqs. (21), (22), and (25), the gross density of the aerated hydraulic fluid is expressed as follows:
$$\frac{1}{\rho } = \frac{{f_{\text v} }}{{\rho_{\text v} }} + \frac{{f_{\text g} }}{{\rho_{\text g} }} + \frac{{1  f_{\text v}  f_{\text g} }}{{\rho_{\text l} }},$$
(27)
2.3 Dynamic Mass Fraction of Air Components
The bulk modulus reflects the compression characteristics of the aerated hydraulic fluid and is expressed as follows:
$$E =  V\frac{\partial p}{{\partial V}},$$
(28)
where \(V\) is the total fluid volume (m^{3}).
An aerated hydraulic fluid typically includes three phases: liquid, vapor, and air. In general, an aerated hydraulic fluid contains fewer airphase components; therefore, it is generally assumed that volume changes among the abovementioned components do not occur. Thus, Eq. (28) can be written as follows:
$$E =  \frac{V}{{\frac{{{\text{d}}V_{{\text{l}}} }}{{{\text{d}}p}} + \frac{{{\text{d}}V_{{\text{g}}} }}{{{\text{d}}p}} + \frac{{{\text{d}}V_{{\text{v}}} }}{{{\text{d}}p}}}}.$$
(29)
Based on the equation of the changeful course of air, the volumes of air and vapor are expressed as
$$V_{{\text{g}}} = V_{{{\text{g0}}}} \left( {\frac{p}{{p_{{0}} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern\nulldelimiterspace} \lambda }}} ,$$
(30)
$$V_{{\text{v}}} = V_{{{\text{v0}}}} \left( {\frac{p}{{p_{{0}} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern\nulldelimiterspace} \lambda }}} ,$$
(31)
where \(V_{{{\text{g0}}}}\) and \(V_{{{\text{v0}}}}\) are the air and vapor volumes (m^{3}) under the standard atmospheric pressure, respectively.
The relationship between the liquid volume and pressure is [21]
$$V_{1} = V_{10} e^{{\frac{{p  p_{0} }}{{E_{1} }}}} ,$$
(32)
By calculating the derivative with respect to the pressure using Eqs. (30)–(32) and substituting the obtained derivatives into Eq. (28), the theoretical model of dynamic bulk modulus of an aerated hydraulic fluid can be expressed as
$$E = \frac{1}{A + B + C},$$
(33)
where
$$A = \frac{{\rho f_{\text g} \left( t \right)}}{{\rho_{\text g} \lambda p}},$$
(34)
$$B = \frac{{\rho f_{\text v} \left( t \right)}}{{\rho_{\text v} \lambda p}},$$
(35)
and
$$C = \frac{{\rho {}_{\text g}\rho_{\text v}  \rho \rho_{\text v} f_{\text g} \left( t \right)  \rho \rho_{\text g} f_{\text v} \left( t \right)}}{{\rho {}_{\text g}\rho_{\text v} E_{\text l} }}.$$
(36)