Abstract
This work deals with a new methodology for the implementation of high-resolution (HR) schemes employed to advect the volume fraction in the volume of fluid (VOF) method, in which the numerical stability and convergence depend heavily on the numerical advection scheme and implementation method. The proposed method is based on the normalized weighting factor (NWF) method, which linearizes the normalized interpolation profile and rewrites the face value directly using the donor, acceptor, and upwind nodes. However, unlike the NWF, which is fully implicit and results in pentadiagonal linear systems, the new modified normalized weighting factor (MNWF) method only forms the implicit terms with the contribution of the donor and acceptor nodes, while the contribution of the upwind node explicitly forms part of the source term. Therefore, the method results in a tridiagonal linear system. The comparison of the new method with the deferred correction (DC), downwind weighting factor (DWF), and the RNWF methods shows that the MNWF requires about 5%–25% fewer iterations than DC and RNWF, and around 10%–85% less than DWF. Thus, a similar order of accuracy of the results can be o btained with less computational time.
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05 February 2022
A Correction to this paper has been published: https://doi.org/10.1007/s42757-022-0132-z
Abbreviations
- u :
-
velocity vector (m·s−1)
- p :
-
pressure field (Pa)
- A :
-
volume fraction
- P :
-
density (kg·m−3)
- M :
-
dynamic viscosity (kg·m−1·s−1)
- V :
-
volume of the control volume (m3)
- S :
-
surface of the control volume (m2)
- T :
-
time (s)
- C x :
-
convective flux (m3·s−1)
- A P, A F, A E, A EE, A W, A WW :
-
coefficients of the algebraic equation (m3·s−1)
- b P :
-
source term of the algebraic equation (m3·s−1)
- P, F, E, EE, W, WW:
-
center and neighbor control volumes
- D, A, U :
-
donor, acceptor, and upwind control volumes
- ᾶ :
-
normalized volume fraction
- λ :
-
blended function
- θ f :
-
the angle between the flow direction and the grid lines at face f
- Co :
-
local courant number
- UD:
-
Upwind scheme
- DD:
-
Downwind scheme
- DC:
-
Deferred Corrector method
- DWF:
-
Downwind Weighting Factor method
- NWF:
-
Normalized Weighting Factor method
- RNWF:
-
Reviewed Normalized Weighting Factor method
- MNWF:
-
Modified Normalized Weighting Factor method
- [ℓ, m], L :
-
factors for the NWF formulation
- n+1, n, n−1:
-
superscript to indicate the next, current, and last time-step
- f :
-
subscript that denotes variable approximated at the face of the control volume
- ζ :
-
convergence criterion
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Acknowledgements
This work is supported by the “Excellence Initiative” of the German Federal and State Governments within the Graduate School of Computational Engineering at Technische Universität Darmstadt.
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Mariño-Salguero, J., Schäfer, M. A Modified Normalized Weighting Factor method for improving the efficiency of the blended high-resolution advection schemes in the context of multiphase flows. Exp. Comput. Multiph. Flow 3, 208–225 (2021). https://doi.org/10.1007/s42757-020-0074-2
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DOI: https://doi.org/10.1007/s42757-020-0074-2