Lagrangian formulation in OpenFOAM

The Lagrangian-Eulerian approach in OpenFOAM is available for both solid and fluid particles. Even though the two formulations are quite similar, we are going to concentrate mainly on liquid droplets because of the interest in our application.

As in every lagrangian formulation, the representation of the dispersed phase is done using a certain number of parcels, which consist in much reduced number respect to the actual number of droplets present in the domain, this because representing each droplet would be too demanding and we can obtain accurate results even with such approximation. Each parcel contains an amount of particles n_s. The equations that describe the Eulerian phase for a compressible flow are the following: These are the Mass, Momentum and Energy Equation with an additional term in each of them to take into account for the Lagrangian phase (α_{ef f} is the thermal diffusivity that takes into account for the laminar and turbulent terms).

In the mass conservation equation, the source term is present because due to evaporation of parcels there can be some mass introduced into the carrier phase or subtracted from it due to evaporation and/or condensation. In the momentum equation, the source term represents the influence of the parcels on the velocity field due to drag and lift forces, and other forces that may act on the droplets.

At last, we find the source term in the energy equation due to heat transfer and phase change on the parcels that affect the Eulerian phase. These three terms can be different from zero but can also assume a null value if we consider only one-way coupling where the dense phase affects the dispersed one while the latter has negligible or no influence on the Eulerian phase.

The Lagrangian phase is solved by the use of a different set of equations.

These take into account the fact that the particles are considered spherical, and that the mass changes due to evaporation/condensation. We can then write the

3.3. SIMULATION OF MULTIPHASE FLOWS

equations that describe the evolution of the particles:

dm_d

where the subscript d refers to the parcel. These three equations are solved for each parcel. For this reason, the solution of a given cloud can become really expensive if we increase the number of parcels to be closer to the real solution.

The sum over the vectors of the forces acting on the droplet can be simplified if we assume that the primary forces are gravity and aerodynamic drag. Gravity is simple to define wile for the drag force we need a bit more detailed analysis. As commonly done in Aerodynamics theory, a drag coefficient C_D can be determined for the parcel, using the following equation:

C_D = F_D

ρF

2 |u_F − u_d|²A_d, (3.31) with F_D being the aerodynamic drag acting of the parcel, ρ_F the density of the surrounding fluid, u_F and u_d are the velocities of the fluid and the droplet respectively, and Ad = d²_Pπ/4 is the cross-section of the spherical droplet.

The mass of the droplet can be expressed as:

m_d= ρ_d1

6πd³ (3.32)

therefore calculating the drag force from Equation 3.31 and multipling and dividing by m_d we obtain:

F_D = 3 4

m_dρ_F

ρ_dd C_D|u_F − u_d|(u_F − u_d) (3.33) The calculation of the drag force relies on the drag coefficient only because all the other quantities are given.

The drag coefficient is calculated as a function of the Reynolds number (the ratio between inertia and viscous forces):

3.3. SIMULATION OF MULTIPHASE FLOWS

Re = ρ_F|u_F − u_d|d

µ_F (3.34)

Results obtained from experiments of flow around a sphere show that the drag coefficient does not have a simple correlation with the Reynolds number and also CD depends on the surface shape of the sphere, as shown in Figure 3.2.

When the sphere surface is not smooth, experimental observations showed that the transition from laminar to turbulent flow occurs at lower Reynolds number respect to a smooth sphere.

0.1 0.5 1.0 1.5

10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

Cd

Re

Rough Smooth

Figure 3.2: Drag Coefficient for a sphere as a function of Reynolds number for rough and smooth surface

Mainly three regimes are observed: the laminar, transitional and turbulent regimes. For each regime, a different correlation is used. For small values of the Reynolds number (Re < 0.5), i.e. a laminar regime, the drag coefficient is calculated as:

CD = 24

Re (3.35)

This is also called Stokes regime because he found an analytical solution for the drag. For the transitional regime (0.5 < Re < 1000) several correlations have been proposed, and in OpenFOAM the one obtained by Schiller and Neumann

3.3. SIMULATION OF MULTIPHASE FLOWS

[88] can be found:

C_D = 24

Re(1 + 1

6Re^2/3) (3.36)

At last we find the turbulent regime (Re > 1000) which is characterized by an almost constant value of the drag coefficient, which is expressed by:

C_D = 0.424 (3.37)

For higher values of the Reynolds number (Re > 10⁵) the drag coefficient has a drop and eventually recovers. These values are usually not experienced in spray applications and therefore not modelled. It was also mentioned that the shape of the sphere could affect the value of the drag coefficient, and usually, for rough spheres, this results in an earlier drop in drag respect to the smooth sphere. This is why golf balls have small holes on the surface because the drag is highly reduced at a typical speed and they can cover longer distances. In the case of a liquid droplet, the surface can be considered smooth and therefore we can use the standard correlations.

For differently shaped droplets (non-spherical) the drag coefficient is more complicated and different assumptions have to be made.

After the calculation of the drag coefficient, the solver has to calculate the drag force and makes use of it in the momentum equation to solve the motion of the droplet. This would be enough to describe the status of a solid droplet where heat transfer does not play an important role, but in the case of a liquid droplet, boiling and evaporation can take place. Therefore we need to introduce two more equations, the energy and mass conservation principles. The first one takes into account for all the types of heat exchange that can affect the temperature of the droplet and calculates the temperature change accordingly, while the second one gives the change in mass of the droplet which can be positive (condensation) or negative (evaporation).

The equation that calculates the change in temperature for the droplet is the following:

m_dc_pdT_d

dt = ˙Q + h_{f g}dm_d

dt + A_dσ(T_a⁴− T_d⁴) (3.38) where h_{f g} is the specific enthalpy of condensation/vapourisation, σ is the Stefan-Boltzmann constant and  is the emissivity of the droplet surface for the

3.3. SIMULATION OF MULTIPHASE FLOWS

liquid analysed.

On the left-hand side of this the equation we find the temperature change of the droplet weighted by its mass and heat capacity, while on the right-hand side the three main contributors to the temperature change due to conduction-convection, latent heat and radiation. The first term, the convective heat transfer ( ˙Q) is usually calculated by using the heat transfer coefficient:

Q = htc(T˙ _s− T_d)A; A = 4πr²(Droplet surface area) (3.39)

htc = N u · λ

d (3.40)

where N u is the Nusselt number, λ is the thermal conductivity of the liquid droplet, and d is the diameter of the droplet. The value of the Nusselt number is calculated with the correlation obtained by Ranz-Marshall [89]:

N u = 2 + 0.6Re^1/2P r^1/3 (3.41) where P r = c_pµ/λ is the Prandtl number, defined as the ratio between mo-mentum and thermal diffusivity (P r = ν/α).

The second term is the temperature change due to evaporation/condensation, because of latent heat. When a droplet evaporates, there is heat released due to the evaporation process, and this causes a temperature drop for the droplet.

The third and last term in the temperature equation is heat exchange due to radiation. This term can assume essential values when a combustion process is taking place, while it is mostly negligible for any other type of application.

One of the assumptions of the models presented here is about the shape of the droplets. All the droplets are considered to be spherical in order to simplify the problem and make some assumptions on parameters such as the aerodynamic drag. This highly depends on the shape of the droplet, and a maximum defor-mation of the droplet can lead to an increase in drag by a factor of 4 [90], which obviously is not negligible when a single droplet is considered. The change in the shape also depends on the Reynolds number, because an increase in the latter leads to an increase in the shear stress on the droplet surface, therefore modifying its shape. From a breakup point of view, this is somehow linked to the deforma-tion of the droplet, because during the breakup process the droplet experiences a time-dependent deformation until it reaches a value where the surface tension