1: procedure marangoni
2: forIF-cell do
call subgrid_modelling→ TΣ, xΣ ⊲ see Alg. 7 call surfgradT→ ∇ΣTΣ ⊲central differences call IFarea ⊲ correct surface area density call force_cellside ⊲ Interpolate MF to cell face
3: end for
4: end procedure
4.4 h e at t r a n s p o rt
The subsequent section is concerned with the discretization and numerical solution of the energy transport equations in temperature form, see Eq. 9. The accuracy of the heat transport, especially in the vicinity of the interface, is of special interest due to the effect of the temperature on the hydrodynamics in form of Marangoni currents, buoyancy and phase change.
Two approaches are discussed in the following. There are additional algorithms already present in FS3D, see [132, 190]1, but these works were disregarded after careful testing, see also Sec. 5.2.
The first approach discretizes the “standard” one-field formulation of the temperature equation originally implemented by Hase [83]. The second is a two-field approach which employs a cut-cell method, first developed in the master thesis [234].
The difference between the two approaches, as the names suggests, is the number of scalar fields present to represent the temperature distribution. While in the well-known one-field approach only one temperature field for both phases is present, the two-field approach transports a field for each phase separately. The description of the methods, their differences, advantages and disadvantages is elaborated in the following. In general, the discussion will follow the structure of addressing the transport equations and discretization first, followed by the convective transport, the diffusive transport and the treatment around the interface. Challenges accompanying each transport are addressed towards the end.
4.4.1 One-field Approach
As explained in Sec. 4.2, Eq. 9 is only valid within each phase. To obtain an expression that is valid within the whole domain, the equations for both phases have to be conditioned with the phase indicator and volume averaged as shown in detail in Appendix A.0.6.
In the one-field formulation, the temperature equation without evaporation takes the form
∂t(ρVcpMTH) +∇ ·(ρVcpMTHuM⊗ uM) =∇ ·(λh∇TH) (46)
1 While Schlottke et al. in [190] pursue a similar approach as the later introduced so-called two-field approach, several difficulties have been neglected, such as the small-cell problem and the new emerging unstructured grid with difficulties as non-orthogonality. Ma and Bothe [132] applied a two-field approach coupled with a ghost-field method that allows to use the present Cartesian grid. However, the necessary extrapolation into the other phase for the ghost-field method misses a guideline by physical principals and can therefore introduce unwanted local errors.
Ωd
Ωc
Af
ρVcpMT
Ff, nf
Figure 19: Control volume with indicated PLIC-plane and position of discretized values. with the averaged temperature obtained by averaging the enthalpy, i.e.
TH =
αρdcdpTd+ (1 − α)ρcccpTc ρVcpM
.
The heat conductivity λh is averaged harmonically, as shown in Eq. 166, as motivated by
Patankar in [159], Sec. 4.2.3. The discretization, as described in Sec. 4.2.2, is done on the Cartesian, staggered grid and leads to solving the semi-discrete (only time discretized) equation (ρVcpMTH) (n+1) c V −(ρVcpMTH) n c V (47) =− 6 X f=1 (ρVcpMTH)fFf · nf Af∆t+λh 6 X f=1 ∇(TH)f · nf Af∆t.
Here nf is the normal vector of the cell face f with the area Af, cf. Fig. 19. The quantities
(φ)c and (φ)f are the discrete values at the cell center of the CV with the volume V and the cell face, respectively.
The advection of the enthalpy is strongly coupled with the volume fraction transport as the same geometrical volume fluxes Ff are used for the convective change according to
6 X f=1 (ρVcpMTH)fFf· nf Af∆t. (48)
This prevents the decoupling of the α-field and the temperature field and ensures the alignment of change in physical quantities to the interface reconstructed from α. Note here that(ρcpMTH)f is needed at the cell face f . This poses a challenge due to the staggered grid
and the cell-centered scalar values(ρVcpMTH)c. The choice of the appropriate face-centered
value (φ)f is a topic of its own and leads to the so-called High-Resolution (HR-)schemes discussed in Sec. 4.4.5.
The overall convective transport is done on the basis of the α-transport with a Strang- splitting scheme in analogy to the α-transport and takes, in x-direction, the form
(ρVcpMTH) ∗ i −(ρVcpMTH) n i ∆t =− (ρVcpMTH)i+1/2Fi+1/2−(ρVcpMTH)i−1/2Fi−1/2 ∆x + 1 2 (ρVcpMTH) n i + (ρVcpMTH) ∗ i ui+1/2− ui−1/2 ∆x , (49) (ρVcpM) ∗ i −(ρVcpM) n i ∆t =− (ρVcpM)i+1/2Fi+1/2−(ρVcpM)i−1/2Fi−1/2 ∆x + 1 2 (ρVcpM) n i + (ρVcpM) ∗ i ui+1/2− ui−1/2 ∆x . (50)
Ωc Ωd (a) Ωd Ωc Ad1 Ac1 Ad 2 Ac2 AΣ Tc Td (b)
Figure 20: Two-field approach for temperature transport: illustration of two temperature fields and placement of discrete values.
The additional transport of the volumetric heat capacity is necessary to access the temper- atures after each one-dimensional step. The same is done during the momentum convection. There the density is transported analogously to access the velocities after each directional step. The procedure is summarized in algorithm 5, including diffusion.
The diffusive term is discretized by central differences and solved implicitly. The diffusive heat flux over the interface is already incorporated in the last term of Eq. 47 due to the volume averaging process and the carefully chosen closure terms.
If phase change is considered, Eq. 46 is extended by a heat sink proportional to the mass flux and evaporation enthalpy ˙m′′′
∆hv, as derived in Appendix A. Different models for the
mass flux are given in Sec. 4.4.4.
If additional terms depending on the IF-temperature are discretized, e.g. Marangoni forces or mass fluxes, IF-values need to be deduced from the cell-centered temperature field. For a sufficient resolution, the assumption TΣ =TH might be justified, but becomes expensive
if the gradients become steep. To overcome this limitation, additional information in the vicinity of the interface is created. Subgrid-models can determine the IF-temperature TΣ at the PLIC-plane center based on the bulk temperatures via explicit use of the energy jump condition normal to the interface, e.g. without evaporation Jλ∇T K · nΣ = 0 and JT K=0. Such subgrid-models are discussed in Sec. 4.4.4.