# Wind Reversal in Bubbly Natural Convection
## 2 Methods
### Dispersed Phase  
The bubbles were considered to be point-like.  
The single bubble is located at the coordinate \(\mathbf{x}_{i}\) and the force \(\mathbf{f}_{i}\) that it induce on the continuum phase reads as follows:  
\[\mathbf{f}_{i}\;=\;\frac{4}{3}\pi R_{bi}^{3}\rho\left(\left.\frac{D\mathbf{u} }{Dt}\right|_{\mathbf{x}_{i}}-\mathbf{g}\right) \tag{3}\]  
in which \(R_{bi}\) is the radius of the \(i\)-th bubble.  
The continuum phase energy equation is:  
\[\rho c_{p}\frac{DT}{Dt}\;=\;k\
abla^{2}T+\sum_{i}Q_{i}\delta(\mathbf{x}- \mathbf{x}_{i}) \tag{4}\]  
where \(k=\kappa\rho c_{p}\) is the liquid thermal conductivity and \(Q_{i}\) is the energy source or sink due to phase change of the \(i\)-th bubble.  
We take into account the thermal balance between each bubble and the surrounding fluid by means of a heat transfer coefficient \(h_{bi}\)  
\[Q_{i}=4\pi R_{bi}^{2}h_{bi}(T_{sat}-T_{i}) \tag{5}\]  
where \(T_{i}=T(\mathbf{x}_{i},t)\) is the liquid temperature evaluated at the position of each bubble center. In the previous equation we assume that the phase change is slow and the temperature at the bubble surface is kept at the saturated conditions.  
The equation of motion for each bubble balances added mass, drag, lift and buoyancy,  
\[C_{A}\frac{d\mathbf{v}}{dt}\;=\;(1+C_{A})\frac{D\mathbf{u}}{Dt}- \frac{3C_{A}}{R_{b}}(\mathbf{v}-\mathbf{u})\frac{dR_{b}}{dt}\] \[-\frac{3}{8}\frac{C_{D}}{R_{b}}|\mathbf{v}-\mathbf{u}|(\mathbf{v}- \mathbf{u})-\mathbf{g}\] \[+C_{L}\left(\boldsymbol{\
abla}\times\mathbf{u}\right)\times( \mathbf{v}-\mathbf{u}) \tag{6}\]  
in which \(C_{A}\), \(C_{L}\), and \(C_{D}\) are the added mass, lift, and drag coefficients, respectively. The bubble radius \(R_{b}\) is calculated by balancing the latent heat associated to evaporation or condensation with the heat exchanged with the liquid  
\[\frac{dR_{b}}{dt}\;=\;\frac{h_{b}}{L\rho_{V}}(T-T_{sat}) \tag{7}\]  
in which \(\rho_{V}\;=\;\rho_{V}(T_{sat})\).  
We take \(C_{A}=1/2\), the standard potential-flow value for a sphere and we model the drag coefficient as suggested by [29]:  
\[C_{D}\;=\;\frac{16}{R\epsilon_{b}}\left[1+\frac{R\epsilon_{b}}{8+\frac{1}{2}( R\epsilon_{b}+3.315\sqrt{R\epsilon_{b}})}\right] \tag{8}\]  
in which \(R\epsilon_{b}=2R_{b}|\mathbf{v}-\mathbf{u}|/\upsilon\) is the bubble Reynolds number.  
The equation of the dispersed phase and continuum phase are solved numerically according to the following procedure:  
Upon integration of the fluid momentum and energy equations over a computational cell, the effect of the particles is localized at their position and this effect must be replaced by an equivalent one located at the computational nodes. This task is achieved by a second-order-accurate interpolation. The interpolation preserves the resultant and the couple of the particle forces, as well as the total amount of heat that each particle exchanges with the fluid. The numerical integration of the dispersed phase follows three steps:  
(i) the equation of the thermal balance at the vapor surface is solved by means the 2 nd order Adam-Bashfort scheme and the radius of the bubble is updated; (ii) the momentum equation is solved using the 2 nd order Adam-Bashfort scheme too, whereas the implicit 2 nd order Crank-Nicholson method has been used for the drag force (iii) the bubble position is updated by Adam-Bashfort 2 nd order. The implicit 2 nd order scheme for the bubble momentum equation makes the solution stable with the smallest bubbles since they have a fast time evolution.  
When a bubble reaches the top of the cell, it is removed from the calculation and a new bubble is re-injected at a random position on the bottom plate with a velocity equal to that of the surrounding fluid.  
Each bubble is injected with its initial diameter as it is not estimated using any nucleation model. The mesh independence study and both the experimental and numerical validations were reported in [PERSON] et al. [26]. More details were reported by the authors--in the numerical code by [PERSON] et al. [28].