Cascade Solver Development
4.3 Development of the cascadeFoam solver
4.3 Development of the cascadeFoam solver
Within the framework of OpenFOAM, the presence of a class able to solve liquid films and consequently the splashing process was implemented in the cascade-Foam solver. Along the development of a splashing model, the possibility to use enhanced buoyancy in the momentum equation was implemented too.
4.3.1 Liquid Film Region
In order to implement and allow the solver to model the splashing, a new region has to be introduced in the domain of solution, so that the droplet interacts with a liquid region representing the film formed by the splashed particles [113]. This is a 2.5D (two and a half dimensional) region where the conservation equations are solved for the thin liquid film. The assumption of thin-film is that [114]:
The wall-normal velocity is equal to zero
The wall-tangential diffusion (momentum, energy) is negligible compared to the wall-normal diffusion
With these assumptions the conservation equations for the film region can be written as:
∂ρδ
∂t + ∇ · (ρδu) = Sρδ (4.5)
∂ρδu
∂t + ∇ · (ρδuu) = −δ∇p + Sρδu (4.6)
∂ρδh
∂t + ∇ · (ρδuh) = Sρδh (4.7)
where δ is the thin film thickness, and S represents the source term for mass momentum and energy arising from the other phases or the solid surface where the film is located. These terms include the interaction with the splashed droplets, evaporation and condensation, heat transfer with the gas phase and with the solid substrate. Other secondary effects such as the capillary effect can be taken into account, but their impact is limited in the current application, while it cannot be avoided in another type of problems [115]. In order to ’couple’ the two regions, the gas and liquid one, the solutions variables have to be mapped to/from the
4.3. DEVELOPMENT OF THE CASCADEFOAM SOLVER
gas phase mesh. A picture showing droplets splashing onto the liquid film region and the film thickness is shown in Figure 4.3.
Figure 4.3: Splashing of droplets onto the liquid region [116]
impinging droplet
splashed droplets
boiling
heat conduction viscous shear
convective heat transfer evaporation
Figure 4.4: Liquid Region
The mechanism of the splashing model is such that any droplet whose path in the computational time step crosses the film region is considered to interact with the liquid. At the start of the simulation, where no liquid is present, particles will interact with the solid surface and a splashing model for dry surface will be applied. All the particles that do not splash will start to form the liquid pool.
The model will then calculate some quantities and evaluate as to whether the droplet will be absorbed, rebounded, spread or splash.
The film region is solved at each time step alongside the gas region, and the
4.3. DEVELOPMENT OF THE CASCADEFOAM SOLVER
timestep depends on which of the two areas requires a smaller timestep with the set Courant number.
The mesh of the liquid phase is obtained by extruding the gas phase mesh on the patch where the film liquid is present. This is because in order to map the two solutions together the cells have to be superimposed one on the other.
The importance of the implementation of the liquid region is primarily because of the splashing model, while a second effect is the evaporation that takes place in this region [117], affecting the vapour production within the domain. For a very cold liquid such as LNG, the heat transfer that takes place when this liquid comes in contact with a solid surface at a normal temperature (20°C) makes the boiling/evaporation process really fast, and the presence of the liquid can have a considerable impact on the vapour formation and consequently the risk analysis of an accidental spill. On the other hand, when considering liquids that have a boiling point well above normal temperature, the evaporation that takes place is very slow and can be negligible. The role of the film is in this case merely for splashing purposes. Also, in the case of a boiling liquid, as it will be demonstrated in Chapter 6, the likelihood of any droplet to get to the ground is very low or equal to zero. Consequently, the film and splashing modelling are not necessary.
For this reason and the other mentioned above, the equations describing the film were not modified, and the standard one available in OpenFOAM were used.
4.3.2 Enhanced Buoyancy
As mentioned earlier, buoyancy plays an important role in the current applica-tion, therefore gravity had to be accounted fully in the cascadeFoam solver. The original version of sprayFoam did not consider the gravity term in the momentum equation, for this reason the latter had to be modified along with the pressure equation. If we define a modified pressure prgh as:
prgh = p − ρgh (4.8)
where h is the value of the coordinate of the axis where gravity is acting (usually the y or z axis). The modified pressure is often more conveniently used to solve the pressure equation in place of the standard pressure, with some terms to be modified respect to the standard equation. When we calculate the gradient of the modified pressure (prgh) in the momentum equation, an additional term arises which has to be removed in order to compute the correct momentum equation.