Physics of the process and model formula- formula-tion

As we mentioned earlier, the splashing process is extremely complex, and no uni-versal correlations are currently available to model all the types of splashing. As it is done in many other CFD applications, it is standard procedure to introduce some non-dimensional quantities that will help us in defining splashing regimes.

In order to determine the outcome of a droplet splash, the following variables and fluid properties are needed: the initial diameter of the impinging droplet (D₀), the droplet impact velocity (V₀), the viscosity of the liquid (µ), the density (ρ), the surface tension (σ) and the film thickness (h). By applying the Buckingham theorem we can define four non-dimensional numbers:

The Weber number W e = ρU₀²D₀

σ (5.2)

which defines the ratio between inertia and surface tension forces;

The Ohnesorge number Oh = µ

√D0ρσ (5.3)

which defines the ration between the viscous forces and the inertial and surface tension forces;

The dimensionless film thickness h^∗ = h

D₀ (5.4)

which defines how thick is our liquid film compared to the droplet size;

The Bond number Bo = ∆ρgd²

σ (5.5)

which defines the ratio between body forces (gravity) and surface tension.

Splashing is defined as the formation of secondary droplets from the impact of one single droplet into a solid surface that can be dry or wet. The interaction

5.3. PHYSICS OF THE PROCESS AND MODEL FORMULATION

between a droplet and a wall can also feature other outcomes such as rebounding, adhesion or simple spreading. The transition regions between all these regimes are very hard to define, but some empirical correlations are available in the literature.

Also, for these correlations to be used in any CFD applications, they have to be determined for certain specific values.

It has to be mentioned that most of the existing splashing models have been obtained from observation of single droplet impact onto a solid dry or wetted wall and that in reality spray impacts onto surfaces have a different dynamic because impinging droplets are affected by neighbours, and the outcome can be much different. Nonetheless, simulating this kind of process is complicated with the current experimental tools and correlations are much simpler to obtain in the observation of single droplet impact. Also, in the case of CFD applications, the simulation of multi-drop impact is too demanding and practically impossible.

The main steps in the formulation of any splashing model are explained below.

5.3.1 Transition Criteria

First of all, the model has to evaluate if the droplet is going to splash or if other regimes are going to be observed. The threshold between the two regimes is called transition criteria and is usually defined by an expression that includes the Weber number and one between the Ohnesorge and the Laplace number (the Laplace number is related to the Ohnesorge number with the following relationship La = Oh⁻²). Usually a factor K is considered to be the reference point for the transition and is defined the following way:

K = W e · Oh^α (5.6)

Where the value alpha varies between different models but most of the ones available agree with a value of α of around 0.2. In the Bai and Gosman model, which is the one implemented in OpenFOAM, the transition criteria is defined slightly differently respect to the other model and relates the critical Weber num-ber with the Laplace numnum-ber in the following way:

W e_cr = A · La^−0.183 (5.7)

where A is a coefficient that varies with the surface roughness of the wall.

They state that a wetted surface acts like a very rough surface and therefore

5.3. PHYSICS OF THE PROCESS AND MODEL FORMULATION

the value of alpha is the same as a very rough surface (A = 1320). One of the limitations of their model is that it does not take into account the film thickness which plays a fundamental role in the transition criteria and also in the outcome of splashing. In Kalantari model [57], the film thickness is taken into account and the transition criteria is varied accordingly. Four film thicknesses regimes are defined and reported in the table along with the value of K which correspond to the splashing threshold.

Film Thickness Film Thickness Regime Splashing Threshold (K)

h^∗ ≤ 0.1 Wetted K = 1770 ÷ 1840

0.1 < h^∗ ≤ 1 Thin Liquid Film K = 5032h^∗ + 1304 1 < h^∗ ≤ 2 Shallow Liquid Film K = 6100(h^∗)^−0.54

h^∗ > 2 Deep Liquid Layer K = 4050

5.3.2 Post-impingement characteristics

Once the model has evaluated whether the droplet will splash or not, if splashing occurs, it needs to calculate the quantities of the splashed dropped. It is therefore convenient to identify the main quantities that define the status of a droplet.

d

d

u

v

u

v

θ

θ

Figure 5.5: Impinging and Ejecting droplet main parameters

As it can be seen in Figure 5.5, both the impinging and the splashing droplets are characterised by their diameter (d_iand d_s), velocity (which can be decomposed in normal velocity u and tangential velocity v) and impact angle (θi and θs). In order to simplify the calculations, in numerical codes, the solver considers for parcels (which represent a group of droplets) rather than droplets, because it is usually too demanding to reproduce all the droplets in a computational domain.

5.3. PHYSICS OF THE PROCESS AND MODEL FORMULATION

For this reason, we have to introduce another parameter, the number of droplets in each parcel n, for both the impinging and splashing droplet.

Starting from the impinging characteristics, the splashing model has to evalu-ate post-impingement quantities such as splashing-to-incident mass ratio, droplet size, velocity, number of secondary droplets and ejection angle.

Considering the splashing-to-incident mass ratio (which is as the name sug-gests the ratio between the total mass of all the splashed droplets and the incident droplet mass), the Bai and Gosman model assumes the following value as observed from experiments:

Other authors use a different approach for the mass ratio, being a function of the Weber number or the K variable. The main problem for applying these models to applications such as the spill of a fuel tank [126] is that the Weber number range investigated in their applications is much smaller than the ones that characterize fuel cascades, where the droplet diameters and velocities are much higher. Therefore a new correlation has to be developed to be applied in higher Weber numbers.

Another parameter that has to be given as an input is the droplet ejection angle. One of the main advantages of having a normal impact is that we do not need to consider the impact angle, therefore, the splashed droplet are symmetric.

Most of the models available in the literature agree that the ejection angle depends strongly on the time of the ejection, meaning that early ejected droplet tend to have a bigger angle and higher velocities, while droplets ejected later are more likely to be slower and with a smaller angle. The experimental results analysed by Bai and Gosman report that droplets ejection angles are likely to lie in the range [5°; 50°], and outside this range, the probability is very low. Other models use a different range and also use the impingement angle in their calculations. It is worth mentioning that the azimuthal angle (being the circumferential direction within the plane tangential to the wall) is chosen to be randomly between 0 and 2π, which is supposed to conserve tangential momentum statistically.

Velocity and size of the ejected droplets are also fundamental parameters to evaluate. Their calculation from the experiments is challenging, and statistical models can be used. Since each splash event produces hundreds or even

thou-5.3. PHYSICS OF THE PROCESS AND MODEL FORMULATION

sands of secondary droplets, focusing the attention on all of them is prohibitive with the current tools. In Bai and Gosman formulation, each splashing droplet is considered to produce p secondary droplets where p is a value greater than one.

All of this secondary droplets contain the same amount of mass m_s/p where m_sis total the mass of the incoming splashing droplets. The value of p is usually chosen to be accurate and not too computationally costly. Experiment observations typ-ically show that the secondary droplets follow characteristic distributions, similar to the general Rosin-Rammler distribution commonly used in spray simulations.

First, the mean diameter is calculated from other parameters in the following way:

where dI is the incident droplet diameter, Nsis the total number of secondary droplets per splash and r_m is the splashing-to-incident mass ratio as mentioned before. Starting from the splashed mean diameter the distribution function of the splashed droplets diameter is calculated as:

f (d) = 1

d¯e^{−d/ ¯d} (5.10)

Along with the droplets sizes, we have to calculate their velocities. It was already mentioned before how to calculate the azimuthal angle and the ejection angle of the droplets, therefore the last quantity that has to be evaluated is the velocity magnitude which is done by energy considerations. The splash kinetic energy is calculated as:

E_KS = E_KI+ E_Iσ− E_D− E_Sσ (5.11) where E_KI is the incident kinetic energy, E_Iσ the incident droplet surface energy, E_Sσ the surface energy of all the splashing droplets and E_D the dissipa-tive energy loss. Even though this equation is physically right, the evaluation of the dissipated energy is complex and some of the formulations available seem to underestimate its value. The calculation of the splash kinetic energy is straight-forward only if one parcel is considered (p = 1), but for more parcels we need to provide an additional equation: