Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019


The interface was treated using the level set method. This procedure consists on assigning a scalar magnitude, the level


set ϕ, to the distance from the interface. The


interface corresponds to a zero value of the level set. In the present work, a negative sign of ϕ was adopted for the vapor and positive for the liquid. The propagation of the interface is provided by the following equation:


∂ + ⋅∇ = ∂


T u φ


int Where int  φ 0 (18)


u  is the velocity of the interface, which can be


obtained by a mass and heat


interface, given by: int


 = e g = − ρ lv


m h T T uu h


() () −


 transfer balance at the (19)


In order to preserve the property of being distance from the interface, the level set function must be reinitialized at each time step through the following equation:


∂ = ∂


φ signφφ ξ


( 0 )(1−∇ ) (20)


where ξ is an artificial time introduced because Eq. (20) is a temporal equation which must be solved at each time


step, and 0φ the level set field at the start of each


iteration in this artificial time, i.e.: 0


φφ (xx,0) ( ) = 1,00 0,75 0,50 0,25 0,00


Mikic et al. [40] Scriven [39]


Plesset and Zwick [38] Foster and Zuber [41]


Numerical model 0 1 2 t (ms)


Figure 3. Bubble radius applied to a spherical water bubble growing in a 5ºC superheated liquid medium.


The accuracy of the evaporation model was validated with experimental results elsewhere (Lamas,


2012;


Lamas, et al, 2012; and Lamas, et al, 2015). Several authors reported results about spherical bubble growing


3 4 5 2.4 CFD MODEL


The computational mesh is indicated in Figure. 4. The problem was


simulated using the software was chosen because it open software


OpenFOAM (Open Field Operation and Manipulation). This


allows a total


manipulation of the code. A new OpenFOAM solver was programmed for the present study. C++ programming language was employed to write it.


(21)


where ug is the gas velocity, Af the reference area (for a sphere πD2/4), and CD the drag coefficient, obtained by the following expression (Seinfeld, 1986):


C D = + (


24 1 0.15Re Re


0.687 )


ut t ut t dt ∞


∞ ( + ∆ = + ∆ ) () ∞ du t() (24)


Finally, the variation of the free-stream velocity during a time step t∆ is:


(25)


due to evaporation (Plesset & Zwick, 1954; Scriven, 1959; Mikic, et al, 1970; and Foster & Zuber, 1954) and developed correlations findings. As


based on their experimental comparison between these correlations


proposed in the literature and the results proposed by the present numerical model, Figure. 3 shows the evolution of the bubble radius against time for a spherical bubble of water immersed in a liquid medium 5ºC superheated. As can be seen in this figure, the developed numerical model provides reasonable results.


2.3 BOUNDARY CONDITIONS


As mentioned previously, a reference frame which moves with the droplet was employed. A free-stream velocity was


imposed as conditions. The drag force


inlet and outlet boundary decelerates


the relative


velocity between the droplet and gas, while buoyancy accelerates it. This was implemented into the numerical model by adjusting the free-stream velocity at each time step. The acceleration or deceleration is given by:


du∞ t() dt


= − g F l ρπD / 6


drag 3


where u∞ is the free-stream velocity and Fdrag the drag force, given by:


F CA u uρ ∞ + g drag = D f g


() 2


2 (23) (22)


A-338


©2019: The Royal Institution of Naval Architects


R (mm)


Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |  Page 8  |  Page 9  |  Page 10  |  Page 11  |  Page 12  |  Page 13  |  Page 14  |  Page 15  |  Page 16  |  Page 17  |  Page 18  |  Page 19  |  Page 20  |  Page 21  |  Page 22  |  Page 23  |  Page 24  |  Page 25  |  Page 26  |  Page 27  |  Page 28  |  Page 29  |  Page 30  |  Page 31  |  Page 32  |  Page 33  |  Page 34  |  Page 35  |  Page 36  |  Page 37  |  Page 38  |  Page 39  |  Page 40  |  Page 41  |  Page 42  |  Page 43  |  Page 44  |  Page 45  |  Page 46  |  Page 47  |  Page 48  |  Page 49  |  Page 50  |  Page 51  |  Page 52  |  Page 53  |  Page 54  |  Page 55  |  Page 56  |  Page 57  |  Page 58  |  Page 59  |  Page 60  |  Page 61  |  Page 62  |  Page 63  |  Page 64  |  Page 65  |  Page 66  |  Page 67  |  Page 68  |  Page 69  |  Page 70  |  Page 71  |  Page 72  |  Page 73  |  Page 74  |  Page 75  |  Page 76  |  Page 77  |  Page 78  |  Page 79  |  Page 80  |  Page 81  |  Page 82  |  Page 83  |  Page 84  |  Page 85  |  Page 86  |  Page 87  |  Page 88  |  Page 89  |  Page 90  |  Page 91  |  Page 92  |  Page 93  |  Page 94  |  Page 95  |  Page 96  |  Page 97  |  Page 98  |  Page 99  |  Page 100  |  Page 101  |  Page 102  |  Page 103  |  Page 104  |  Page 105  |  Page 106  |  Page 107  |  Page 108  |  Page 109  |  Page 110  |  Page 111  |  Page 112  |  Page 113  |  Page 114  |  Page 115  |  Page 116  |  Page 117  |  Page 118  |  Page 119  |  Page 120  |  Page 121  |  Page 122  |  Page 123  |  Page 124  |  Page 125  |  Page 126  |  Page 127  |  Page 128  |  Page 129  |  Page 130  |  Page 131  |  Page 132  |  Page 133  |  Page 134  |  Page 135  |  Page 136  |  Page 137  |  Page 138  |  Page 139  |  Page 140  |  Page 141  |  Page 142  |  Page 143  |  Page 144  |  Page 145  |  Page 146  |  Page 147  |  Page 148  |  Page 149  |  Page 150  |  Page 151  |  Page 152  |  Page 153  |  Page 154  |  Page 155  |  Page 156  |  Page 157  |  Page 158  |  Page 159  |  Page 160  |  Page 161  |  Page 162  |  Page 163  |  Page 164  |  Page 165  |  Page 166