3.2 Interface capturing
3.2.2 Sharp interface reconstruction
Using the VOF method one can distinguish between cells that contain pure liquid or vapor (F =1and F = 0, respectively) and cells that contain the liquid-vapor interface (0 < F < 1). A typical volume fraction field is shown in the top part of Figure 3.2. Thus, the location of the interface can be determined with an accuracy of around the size of a cell. The accurate calculation of the local rate of phase change requires very accurate information about the location of the interface as will be shown in section 4.1.
Two approaches are possible to achieve sufficiently accurate information on the interface position. First, the mesh resolution and thereby the cell size can be chosen such that the accuracy is sufficient. The obvious disadvantage of this straightforward approach is that the high mesh resolution goes along with an extremely high number of cells. Therefore, this approach is usually limited by the computational re-sources. Second, the location of the interface within a cell can be determined by a local reconstruction of the interface. In addition to the information that the interface lies in a particular cell, the reconstruction provides more detailed data on where the interface is located within the cell. This augments the level of accuracy without changing the mesh resolution.
The VOF method that is implemented inOpenFOAMdoes not include any reconstruction of the inter-face. Therefore, this feature had to be developed and implemented within the framework of the present thesis. Typically, the reconstruction of the interface in VOF methods is done during the transport of the volume fraction field, i.e. for the calculation of the fluxes of the volume fraction through the faces of a cell. A commonly used approach is the Piecewise Linear Interface Calculation (PLIC). An overview of reconstruction methods is given by Rider and Kothe [89]. The reconstruction algorithm usually requires
3.2 Interface capturing 24
interface position that matches
= 0.3 in cell F
1
1 1
1
1
0.8
1 1
1
1
0.1
0.9 0.6
1
1
0
0.1 0
0.3
0.6 0
0 0
0
0
normal vector calculated from gradient of volume fraction
possible interface positions
0.225 0.1 0.575
0.725
interpolated values of at cell corners
F interpolated
locations on edges ( = 0.5)F
contour based
reconstruction algorithm reconstruction algorithm
in VOF methods
Figure 3.2:Comparison between classical reconstruction in VOF methods (left) and the reconstruction approach developed and used in the present thesis (right).
the interface normal vector~nas an input parameter. It is typically calculated from the gradient of the volume fraction field.
~n= ∇F
|∇F| (3.14)
The interface normal vector describes the orientation the interface in each cell. However, the position of the interface within the cell is not yet known. Therefore, the interface is moved within the cell in order to match the volume fraction of the cell (see bottom left part of Figure 3.2). In the present thesis, the reconstruction of the interface is not used for the flux calculation but only for the calculation of phase change and curvature. Therefore, the reconstruction algorithm follows a different approach. Instead of matching the position of the interface to the volume fraction, the interface is reconstructed as the F = 0.5-contour (see bottom right part of Figure 3.2) similar to the work of Alke and Bothe [1] who model the enrichment of a surfactant at a fluidic interface. The computational effort for such a reconstruction of an iso-surface of the volume fraction is comparably smaller than a classical reconstruction algorithm like PLIC. The reconstruction of the iso-surface is called contour based reconstruction in the following.
The first step of the contour based reconstruction is the interpolation of the cell values of F to the corners of the cells. The intersection between the interface and a cell has the shape of a general polygon
3.2 Interface capturing 25
Sp
Sp
Edges of cell (visible) Edges of cell (hidden) Cutting lines between interface and cell faces Reconstructed interface
Figure 3.3:Illustration of the general polygon created by cutting a general polyhedral cell with the interface.
(see Figure 3.3). The corner points of this cutting polygon are located at the positions where theF = 0.5-contour cuts the edges of the cell. These positions are found by linearly interpolating the position where the volume fraction equals the contour value (F =0.5) along an edge. The edge of the cell is defined by its end points~x1and~x2and by the interpolated volume fractionsF1 andF2at the end points. If the term (F −0.5)changes its sign between the two end points of an edge, a corner point of the cutting polygon exists on this edge. Its exact position is calculated from a linear interpolation.
~xp=~x1+ ~x2−~x10.5−F1
F2−F1 (3.15)
Applying this equation to all edges of a cell gives the number and positions of the corner points of the cutting polygon. To properly define the cutting polygon, the corner points must be arranged in the right order, i.e. clock-wise with the face area vector pointing into the liquid phase. This is done by looping over the faces and edges of the cell in a particular way. Details on the procedure can be taken from López and Hernández [65] who describe the geometrical aspects of interface reconstruction on general polyhedral meshes in the framework of a classical reconstruction algorithm [34, 66], i.e. matching the volume fractions in each cell. Once the corner points are defined and sorted in the right order, the surface area vector of the cutting polygon can be calculated.
~Sp=
Np
X
i=1
1 2
~xp,i×~xp,i+1
(3.16)
Herein, the counter(i+1)is set to(i+1) =1ifi=Np in order to accomplish a closed loop over the corner points of the cutting polygon. The contour based reconstruction does not require the interface normal vector as an input parameter as classical reconstruction algorithms in VOF methods do. Here, the interface normal vector does not need to be calculated from Eq. (3.14) but is an output of the reconstruction and can be calculated from the surface area vector of the cutting polygon.
~ n= ~Sp
~Sp
(3.17)
3.2 Interface capturing 26
reconstruction data available
no reconstruction data available
Figure 3.4:Passing of reconstruction data (normal vector ~n and offset λ) from cells containing recon-struction data to cells without reconrecon-struction data. Averaging is applied in cells receiving data from more than one neighbor.
In addition to the normal vector and the surface area of the cutting polygon, the position of the cutting polygon and hereby of the interface is an important input parameter for the calculation of phase change (see section 4.1). The distance of an arbitrary point to the interface in a particular cell can be calculated from the equation describing the plane which represents the cutting polygon.
dint=~n·~x−λ (3.18)
Herein,λis the offset of the segment of the reconstructed interface in a particular cell to the origin of the coordinate system. It can be calculated as the average of the projected distance of each corner point to the origin of the coordinate system.
λ= 1 Np
Np
X
i=1
~n·~xp,i (3.19)
As will be discussed in section 4.1 the reconstruction data should be available not only in cells that contain a part of the interface but in all cells that are close to the interface. However, the reconstruction data is at first only available in the cells that are actually cut by the interface. Therefore, the recon-struction data (normal vector~nand offsetλ) is passed consecutively from these interface cells to their neighboring cells. The process which is also illustrated in Figure 3.4 is run until the reconstruction data is availabe in all cells that lie within a band of five to ten cells around the interface. For each of these cells the distance to the interface can then be calculated according to Eq. (3.18).
The major drawback of this approach is the fact that the position of the interface does not necessar-ily represent the volume fractions of the phases correctly when it is used as an indicator for the phase distribution. In contrast to other interface reconstruction algorithms in VOF methods, it can therefore not be used for the flux calculation during the transport of the volume fraction field. For this purpose, the reconstruction must correctly reproduce the volume fraction in order to achieve a volume and mass conserving transport. However, in the present thesis the reconstruction is used mainly for the calcula-tion of phase change which requires accurate data on the surface area of the interface rather than the
3.2 Interface capturing 27
(a)Classical reconstruction (b)Contour based reconstruction
Figure 3.5:Comparison of reconstructed interface with discontinuities (classical reconstruction approach, e.g. PLIC) and without discontinuities (contour based approach).
volume of each phase. The contour based reconstruction algorithm which is used here reconstructs a surface without discontinuities. Therefore, the calculated surface area is more accurate (see Figure 3.5) compared to a reconstruction algorithm which allows discontinuities in the reconstructed interface.
The calculation of the interfacial area is also possible without reconstruction. As discussed by Hardt and Wondra [28], the volumetric density of the interface area can be calculated as the magnitude of the gradient of the volume fraction. Thus, the total interfacial area can be obtained by a volume integration of this quantity over the whole domain.
Sint= Z Z Z
V
|∇F|dV (3.20)
Results of this approach and results obtained with the interface reconstruction which is presented above (by summing up the area of the cutting polygons obtained from Eq. (3.16)) are shown in Fig-ure 3.6. The results are obtained for a 2D rod with a radius of 2.5 mm within a domain of10×10 mm2 which is discretized in25×25,50×50,100×100and200×200cells. The interfacial area is calcu-lated from Eq. (3.20) (without interface reconstruction) and from the interface reconstruction described above. The results are compared to the analytically calculated interface of a circular rod and the rela-tive error is determined. It can clearly be seen that the interface reconstruction leads to a much better convergence while there is almost no convergence without interface reconstruction. Unfortunately, a comparison of the accuracy of the interface area between the contour based reconstruction method and a PLIC method was not possible as the latter is not implemented inOpenFOAM.