The development of laminar thermals and the formation of buoyant vortex rings in thermals are studied by performing direct numerical simulations. The formation number of buoyant vortex rings in thermals is also analysed. We find that the development of thermals can be classified into three modes: the starting vortex ring dominated mode; the mode with the occurrence of a secondary vortex ring with breakup; and the mode with the occurrence of a secondary vortex ring without breakup. For the latter two modes, owing to the stretching of the thermal cap, the fluid at the leading edge rolls up, and a secondary vortex ring occurs, grows and replaces the starting vortex ring. The boundary of non-occurrence and occurrence of the secondary vortex ring is determined in a space of Richardson number (Ri) and injection duration ( ). The final mode occurs only in a small region. For , the secondary vortex ring does not occur even for very long injection duration. The effective Rayleigh number ( ) is proposed to accommodate the cases and , with larger than the critical value (approximates to 1. 95 × 105) for the occurrence of the secondary vortex ring. The formation number of buoyant vortex rings in thermals is beyond the universal formation number of 4 for non-buoyant vortex rings, and increases with the increase of the Richardson number and the injection duration. The switching between the thermal modes by changing the Richardson number and the injection duration has no significant effect on the value of the formation number.

中文翻译：

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热气流中涡环的形成机制

通过直接数值模拟研究了层流热流的发展和热流中浮力涡环的形成。还分析了热气流中浮力涡环的形成数量。我们发现热气流的发展可以分为三种模式：起始涡环主导模式；出现二次涡环破裂的模式；以及出现二次涡环而没有破裂的模式。对于后两种模式，由于热帽的拉伸，前缘处的流体卷起，出现次级涡环，并取代起始涡环。次级涡环不出现和出现的边界由理查森数（Ri）和喷射持续时间（ ）确定。最终模式只发生在一个小区域。对于 ，即使注射持续时间很长，次级涡环也不会出现。提出了有效瑞利数 ( ) 以适应 和 的情况，大于临界值（约 1. 95 × 105），用于出现次生涡环。热气流中浮力涡环的形成数超过了非浮力涡环的通用形成数4，并且随着Richardson数和注入时间的增加而增加。通过改变理查森数和注入持续时间在热模式之间切换对地层数的值没有显着影响。提出了有效瑞利数 ( ) 以适应 和 的情况，大于临界值（约 1. 95 × 105），用于出现次生涡环。热气流中浮力涡环的形成数超过了非浮力涡环的通用形成数4，并且随着Richardson数和注入时间的增加而增加。通过改变理查森数和注入持续时间在热模式之间切换对地层数的值没有显着影响。提出了有效瑞利数 ( ) 以适应 和 的情况，大于临界值（约 1. 95 × 105），用于出现次生涡环。热气流中浮力涡环的形成数超过了非浮力涡环的通用形成数4，并且随着Richardson数和注入时间的增加而增加。通过改变理查森数和注入持续时间在热模式之间切换对地层数的值没有显着影响。并且随着理查森数和注射时间的增加而增加。通过改变理查森数和注入持续时间在热模式之间切换对地层数的值没有显着影响。并且随着理查森数和注射时间的增加而增加。通过改变理查森数和注入持续时间在热模式之间切换对地层数的值没有显着影响。