After verification of the model with empirical equations under static condition in Part I, pool boiling under rolling condition is simulated in this section with maximum rolling angle θ _r,m = 30^o and rolling period t _p = 100000δ _t , and the ratio of additional accelerations with gravity acceleration is: a _n /g = 0.06, a _t /g = 0.11. Figure 2 shows bubbles’ unique behaviors during one rolling period (t _p = 3 × 10⁵ δ _t to 4 × 10⁵ δ _t ) at various pool boiling stages under rolling condition. Each column in Figure 2 has the same moment, and the time interval between neighboring column is 0.2t _p = 2 × 10⁴ δ _t . The corresponding fluid field in detailed is showed in the table below: At convection regime (Ja = 0.065), no bubble nucleation occurs on the heater at low heat flux, and the liquid level swings during the rolling period. At higher wall superheat (Ja = 0.142), it comes to nucleate boiling regime, where the bubbles go through the process of nucleation, growth, sliding, coalescence, departure and swinging rising. The obvious non-uniform distribution of bubbles on the wall surface is resulted from bubbles’ nucleation and sliding. The bubbles burst after they rise laterally and periodically to the liquid level under rolling conditions. Then, CHF is obtained at Ja = 0.184, where the surface is partially occupied by small vapor films and local dry-out occurs. The bubbles tend to slide to the heater side with maximum rolling angle and depart from there. When the superheat still grows up to Ja = 0.394, the boiling pattern changes to film boiling where vapor film covers the whole heater surface and restrains the supplement of liquid. Then the vapor column forms and swings under the rolling condition, and big bubbles depart from the heater side at maximum rolling angle.

The comparison of boiling curve with time-averaged heat flux computed with time t from 40000δ _t to 140000δ _t corresponding to one rolling periods between rolling and static conditions is presented in Figure 3, it is shown that the rolling effects slightly increase the average heat transfer at convection and nucleate boiling regimes, and lead to 9.3% higher CHF. Furthermore, the rolling effects continue to strengthen heat flux in transition region. When it comes to film boiling, the heat flux under rolling condition will reduce and be lower than that at static condition. The rolling effects on the whole boiling curve under different boiling regimes will be analyzed as followed.

Transient heat flux varied with time under static and rolling conditions has been made a contrast in Figure 4, and the corresponding rolling angle is also plotted with blue dashed line. The transient heat flux q t ′ is computed as q t ′ = q t q b o = 1 q b o L x ∫ L x − k s ( ∂ T ∂ y ) y = 0 d x (3)

It is shown in Figure 4A that the transient heat flux at convection region decreases in the beginning, and rises to a stable status under static condition, which is in agreement with the results in the literature (Li et al., 2015b; Gong and Cheng, 2017; Zhou et al., 2019). On the contrary, the transient heat flux under rolling condition is higher at first and fluctuates slightly during the rolling period. Since no bubble nucleation exists in the convection regime and the value of heat flux is low, the rolling effects on heat transfer of single liquid phase are small.

As the superheat degree increases, it comes to nucleate boiling region and the transient heat flux is presented in Figure 4B. It is shown that distinct fluctuation of transient heat flux appears under both static and rolling conditions. Especially for pool boiling under rolling condition, high intensity and ordered oscillation of transient heat flux is observed during the rolling period, which is derived from the dynamic evolution of bubbles’ unique behaviors of nucleation, growth, sliding, coalescence and departure. It should be noted that the oscillation period of transient heat flux at nucleate boiling (Ja = 0.142) is almost 0.5t _p. This result can be explained from the periodic inertial force imposed on the bubbles as presented in Figure 1, which shows that centrifugal force F _n, tangential force F _t and Coriolis force F _co have not change unless the rolling system pass through the middle position. In other words, the variation period of additional inertial force exerted on the bubbles is almost half of rolling period, and it leads to the high intensity and ordered oscillation of transient heat flux. Besides, the transient heat flux under rolling condition has higher peek value and lower minimum value than that under static condition, which results in a close time-averaged heat flux.

More intense fluctuation of transient heat flux at CHF is given in Figure 4C. Different from nucleate boiling under rolling condition, similar random oscillations of transient heat flux appear for both static and rolling conditions, which can be attributed to the vigorous bubble behaviors at CHF. Moreover, the transient heat flux of pool boiling under rolling condition is higher than that under static condition during most part of rolling period, and it leads to the higher average heat flux at CHF region.

Relatively periodic variation of transient heat flux at film boiling under both static and rolling conditions is presented in Figure 4D. A stable vapor film forms at film boiling regime and the bubbles periodically depart from the vapor columns, which causes the regular variation of transient heat flux. It should be noted that the transient heat flux under rolling condition is always lower than that under static condition.

At convection heat transfer region, the rolling effects make the heat transfer change from natural convection to forced convection, which can be clearly seen from the temperature field and boundary layer in Figure 5. The tangential inertial force will drive the liquid moving parallelly on the surface as shown in Figure 5A, leading to a thinner boundary layer. And it can be explained from the temperature profile upon the heater surface as presented in Figure 5B, which indicates that temperature gradient (T _w—T)/Δy near the heater surface under rolling condition is larger, with Δy = y - y _heater representing the distance from heater surface. Moreover, the thinner boundary layer caused by tangential inertial force under rolling condition will result in better thermal performance.

On the contrary, the centrifugal force F _n is opposite to the gravity as seen in Figure 1, and hence the convection circulation between fluid upside and downside is weakened. It is shown in Figure 6A that convection circulation under static condition goes well and some vortexes have been formed. In contrast, much hot liquid flows upward and little cold liquid has been replenished to the wall under rolling condition. Moreover, we can also see the comparison of convection circulation between rolling condition and static condition from the isotherm diagram in Figure 6B. The isotherm lines always form cycles under static condition. On the other hand, for rolling condition, parallel liquid flows on the heater surface leads to overall thinner superheated layer at the beginning of 0.2t _p. Then, the flow field leans to one side at 0.6t _p and 0.8t _p without full convection cycle. After that, parallel flows account for the majority at t _p and it results in a thinner boundary layer again. To sum up, the tangential inertial force F _t has positive influence on the heat transfer, while the centrifugal force F _n makes negative effects under rolling condition, and heat transfer of convection region in this work will be slightly enhanced by rolling motion.

The difference of time-averaged heat flux between static condition and rolling condition is small at nucleate boiling, which is similar to the result in flow boiling and will be analyzed from the aspects of bubble behaviors, heat transfer near the triple-line and bubble departure. For bubble nucleation under rolling condition, the superheated liquid flows parallelly on the surface, leading to a non-uniform pressure field as shown in Figure 7A. Since bubble nucleation requires the force balance as indicated by Laplace law (p _in - p _out = σ/r), and thus nonuniformity of pressure filed gives rise to easier bubble nucleation at one side with lower external pressure and harder nucleation on the other side of the heater surface. Moreover, bubbles slide under the pressure gradient on the heater surface. Both non-uniform bubble nucleation and bubble sliding under rolling condition result in the asymmetrical bubble distribution as presented in Figure 7B. In contrast, bubble nucleation and distribution under static condition have been very uniform. Although the bubble behaviors exhibit huge difference between rolling condition and static condition, the bubble numbers make little difference as shown in Figure 7B, which means that the intensity of boiling process is similar and hence the time-averaged heat flux of both conditions is close to each other.

On the other hand, evaporation close to the triple-line is important for heat transfer of nucleate boiling. As shown in Figure 8, the surface below bubble base has higher temperature than surface position wetted with liquid because of the low conductivity of vapor, and the triple-line area is distinguished with lower temperature resulted from violent evaporation there. The tangential inertial force caused by rolling motion drives bubbles to slide on the surface, which enhances the evaporation near triple-line and increases the surface area with low temperature. And these results indicate that tangential inertial force imposed on the bubbles according to rolling motion has positive influence on heat transfer of nucleate boiling.

Besides, bubble departure has been deeply influenced by the directional variance of gravity and additional centrifugal force under rolling condition. Normally, buoyancy of a single bubble comes from the integral pressure under gravitational field as shown in Figure 9. Since the gravity is not parallel with the normal direction of heater surface any more under rolling condition, the buoyancy of bubble decreases. Moreover, the additional centrifugal force is at the same direction with bubble departure, which will also reduce the buoyancy. To sum up, the directional variance of gravity and additional centrifugal force has negative influence on heat transfer of nucleate boiling under rolling condition.

CHF is greatly influenced by the vapor film on the surface, and large-scale vapor film will prevent surface rewetting and decrease CHF. It can be seen from Figure 10 with velocity vector diagram, that the surface rewetting is enhanced under rolling condition, since the liquid is driven downward to the heater with fewer small vapor film. In contrast, the bubbles on the heater surface under static condition is dense and form more vapor films, which prevent the rewetting of heater.

The influence of rolling effects on film boiling is shown in Figure 11. As discussed in 3.4, the buoyancy of bubble at departure decreases, leading to lower bubble departure frequency for film boiling under rolling condition. Besides, flow filed with sufficient circulation in the bulk liquid is favorable to bubble departure under static condition as demonstrated by the streamlines in Figure 11A. The eddy grows and exists for a long term, which helps the bubble depart from surface regularly, and there exists two departure bubbles. On the contrary, the long-existed eddy will be destroyed with rolling motion. The vapor wave moves from one side of the heater to the other side, and only one bubble departs from the surface (as seen in Figure 11B). The departure mode under rolling condition might delay a bubble’ leaving, as shown in Figure 11B, a small bubble on the left has to move to the other side for departure. In brief, lower bubbles’ departure frequency caused by decreasing buoyancy and fewer numbers of departure bubbles result in thicker vapor film and heat transfer deterioration in film boiling under rolling condition.

Figure 12 presents the influence of rolling parameters on transient heat flux with different rolling amplitudes and rolling periods. It can be seen from Eq. 1 and 2 that larger rolling amplitude and smaller rolling period will induce larger additional inertial forces, and thus make greater influences on the bubbles’ behavior and pool boiling heat transfer. Compare the fluctuation range of transient heat flux between Figure 12A and Figure 12B, it is found that smaller rolling period results in larger fluctuation of transient heat flux. And the oscillation of heat flux in smaller rolling amplitude is very low as seen in Figure 12C. Besides, we find that the time-averaged heat flux is a bit higher than static condition for cases in Figures 12A,B, while it’s a little lower for Figure 12C. As discussed in Influence of rolling effects on convection region, the rolling motion will force the fluid to flow along heater surface while destroy the natural convection circulation, so the convection heat transfer is strengthened with forced flow and weakened with disappearance of natural convection eddy. That’s to say, the overall heat transfer performance is related to rolling parameters.