1: procedure dynCA
2: forall IF-cells at wall do
call check_CL_cell
3: if (IF-cell=CL-cell) then ⊲check if IF-cell is CL-cell
call cl_velocity → Ucl
4: else ⊲ if not, seach for closest CL-cell
call search_CL_cell call cl_velocity → Ucl 5: end if call cl_model → θd 6: end for 7: end procedure
4.5.2.2 Contact Line Velocity - Boundary Condition
Due to the staggered grid, the surface tension force is applied at half a cell width (∆/2) away from the wall. This leads to a method-inherent numerical slip with a slip length of half a cell width, as shown in Fig. 32a, cf. [168]. Hence, the slip length is mesh dependent and equals half the grid size. If an additional slip length is applied, the boundary condition is altered according to Fig. 32b. For comparison, the Navier slip law [149] is applied in the form Uw =lλ d U d x w (102) with the slip length lλand the velocity parallel to the wall U , where x denotes the direction
normal to the wall. Regardless of the applied approach, the stress singularity, cf. Sec. 2.3, is prevented. Many codes use the advantage that an additional slip length of λ 6=0 reduces the mesh dependency of their results. Note that enforcing the no-slip boundary condition can become problematic with increasing grid refinement and a slip law should be applied. However, in this thesis such a fine mesh resolution is never reached.
4.5.2.3 Contact Line Velocity - Mesh Dependence
The contact angle implementation is mesh dependent based on the staggered grid. In addition, the above presented empirical contact angle models are in general valid on a larger scale than the grid resolution. However, it has to be taken into account that the contact angle can change dramatically when going to smaller scales. To counteract the large slip length of ∆/2 and the mesh dependency, a numeric contact angle θnum is introduced,
adapting the idea of Afkhami et al. in [3]. This numeric contact angle is valid on the grid scale and can be calculated from the apparent contact angle via
g(θnum) =g(θapp) +Ca ln
∆/2
r0
, (103)
where g(θ) can be found in [37]. For simplicity, approximations of Eq. 103 can be used as the one given by Voinov [233]. By directly incorporating the cell width ∆ into the applied contact angle, the mesh dependence is reduced, as is shown in Sec. 5.3.2.
4.5.3 Hysteresis
CL-hysteresis poses an additional effort for a VoF framework due to the inherent slip men- tioned in Sec. 4.5.2.1. In FS3D, the value of the CL-speed Ucl is not imposed but received
from the momentum balance. Thus, while being in the hysteresis interval θ ∈[θr, θa] (θa
advancing CA, θr receding CA) a movement of the contact line must be prohibited via
altering the momentum balance based on the IF-orientation.
Three different hysteresis approaches are incorporated in FS3D within the scope of this thesis. The first is applicable also to algebraic VoF codes and is inspired by [45], in which the local cancellation of the velocity around the contact line has been suggested. This is done by adjusting the contact angle in such a way that the acceleration due to the surface tension forces leads to a stationary contact line.
The second approach simply sets a Dirichlet BC for the volume fraction with the values saved from the moment the hysteresis interval is entered [129].
The last and third approach is based on the work of [55]. Here the actual geometrical volume fraction transport is altered in the range of hysteresis to account for a non-moving contact line by reducing the motion of the PLIC-plane to a rotation around the contact line.
In the following, all three approaches are presented in detail. 4.5.3.1 Hysteresis via Surface Tension Alteration
As mentioned above, within this approach, the surface tension is altered via the contact angle in such a way that the velocity is canceled locally in CL-cells. This cancellation bears some complications using a staggered grid. The surface tension is applied at the cell faces where the velocities are placed as well. The curvature and height functions which are directly influenced by the contact angle and CL-speed are, however, cell-centered. Due to the interpolation processes from cell center to cell face and vice versa, there is no easy, straight-forward relation between the contact angle and the final surface tension.
The acceleration due to surface tension is applied last so that the face velocities consist of all accelerations due to convection, viscous forces and gravity. To determine the required contact angle within the hysteresis interval with a secant method as shown in Alg. 9, the
volume fractions α, the densities of both phases and the cell width need to be known in advance.
If the contact angle θ determined with the secant method lies outside the hysteresis interval [θr, θa], the dynamic contact angle is again determined as in Sec. 4.5.
The above approach differs from [45] by additionally setting the face-centered velocities in CL-cells to zero. This assures no CL-motion during the next α-advection which can otherwise still be observed. After careful testing, it can be stated that it is not possible to find an angle that cancels all cell face velocities. In addition, due to the explicit time discretization scheme, velocities resulting from the pressure correction lead to a slowly moving contact line.