Figure 1 shows the 2D computational grid for the homogeneous mixture model, which contains 1718 quadrilateral meshes.

R_{max} decreases by 0.11% for case D when the grid number is increased to 5087. The bubble is located at the lower-left corner of the computational domain. Since there is only one bubble, a virtual initial bubble number density (

n_{0}) of 10

^{9} m

^{−3} is given inside the region with the radius of

${\left(4\pi {n}_{0}/3\right)}^{-1/3}$ (marked in red in

Figure 1). Additionally, the initial gas volume fraction inside this region is

$4\pi {n}_{0}{R}_{0}^{3}/3$, which makes the total gas volume equal to

$4\pi {R}_{0}^{3}/3$. When using the VOF method, special contractions of the mesh are applied around the bubble, and the grid number is about eight times larger.

R_{max} only increases by 0.019% for case D when the grid size around the bubble is further decreased by 33%.

Figure 2 compares

R and

$\overline{T}$ of case D predicted by the VOF method and homogeneous model. Both

R and

$\overline{T}$ are a little overpredicted by the homogeneous model before bubble rebound, and

${\delta}_{R\mathrm{max}}$ (the relative difference of

R_{max} predicted by the homogeneous model and VOF method) is only 0.18%. The dependences of

${\delta}_{R\mathrm{max}}$ on

R_{0},

t_{w}, and

p_{0} are analyzed based on case D, which are shown in

Figure 3. Cases A, D, and H are used to analyze the influence of

R_{0}, while cases B to G are used to analyze the influence of

p_{0}. It can be seen from

Figure 3a that

${\delta}_{R\mathrm{max}}$ increases with

R_{0}, and

${\delta}_{R\mathrm{max}}$ < 1% when

R_{0} ≤ 100 μm (the nucleus radius is usually smaller than 100 μm). It can be seen from

Figure 3b that

${\delta}_{R\mathrm{max}}$ also increases with

p_{0}, and it reaches 4.1% at 800 kPa, which means the heat transfer is obviously overpredicted. In order to reduce the heat transfer at large

p_{0}, the calculation of

Pe_{T} (Equation (15)) is modified by fixing

p_{0} to the atmospheric pressure. After this modification,

${\delta}_{R\mathrm{max}}$ at 800 kPa decreases to 1.1%, much smaller than the original value. It can be seen from

Figure 3c that

${\delta}_{R\mathrm{max}}$ is small in a wide range of

t_{w} and it is negative at small

t_{w}.

Figure 3b,c also show the relative difference of

R_{max} between the constant-transfer model and VOF method; the results are quite close to the corresponding

${\delta}_{R\mathrm{max}}$, which means that the predictions by the constant-transfer model and homogeneous model are quite close to each other. After the modification to

Pe_{T}, the heat transfer is well predicted by the homogeneous model in wide ranges of

R_{0},

p_{0}, and

t_{w}; the absolute value of

${\delta}_{R\mathrm{max}}$ is below 1.5%.

Then the homogeneous model is used to simulate three gas bubbles trigged by the pressure pulse in Equation (22), which was simulated by Ye et al. [

14] under the isothermal assumption. The parameters are as follows:

R_{0} = 50 μm,

T_{0} = 298 K,

S = 0,

n_{0} = 10

^{9} m

^{−3},

p_{0} = 101325 Pa,

β_{T} = 6.82, and

A_{p} = 1.4; two cases are simulated with

t_{w} = 20 and 200 μs. The computational domain and boundary conditions are shown in

Figure 4. The size of the computational domain is 0.5 mm × 0.5 mm × 50 mm. The pressure pulse is specified at the right face, whereas

p_{0} is specified at the left face, and the rest four are symmetry planes. Three bubbles are placed inside the computational domain in a regular arrangement at the interval of 1 mm. The computational grids are the same as that in Ref. [

14]. Briefly, 1.23 million elements are employed for the VOF method while 500 elements are employed for the homogeneous model.

Figure 5 compares the total bubble volume predicted by the VOF method and homogeneous model. It can be seen that the bubble volume is well predicted by the homogeneous model with the consideration of heat transfer. The maximum bubble volume of the two cases predicted by the homogeneous model are, respectively, 1.5% (

t_{w} = 20 μs) and 0.63% smaller than the corresponding values predicted by the VOF method. The underestimate of the bubble volume is more obvious at smaller

t_{w}, since the bubble radius is underestimated by the constant-transfer model at small

t_{w}, as shown in

Figure 3c.