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The phase field method provides a simple mass conserving method for solving two-phase immiscible - incompressible Navier-Stokes Equations. The relative ease in implementing this method compared to other interface reconstruction methods, coupled with its conservativeness and boundedness makes it an attractive alternative. We implement the method in a parallel structured multi-block generalized coordinate finite volume solver using a collocated grid arrangement within the framework of the fractional-step method. The discretization uses a second-order central difference method for both the Navier-Stokes and the phase field equations. A TVD-based averaging technique is used for calculating density at cell faces in the pressure correction step to handle high-density ratios. The simulation framework is verified in standard test cases: Zalesak Disk, a droplet in shear flow, Solitary Wave Runup, Rayleigh Taylor Instability, and the Dam Break Problem. A second-order rate of convergence and excellent phase volume conservation is observed.
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A note on the accuracy of the generalized‐α scheme for the incompressible Navier‐Stokes equations
Abstract We investigate the temporal accuracy of two generalized‐ schemes for the incompressible Navier‐Stokes equations. In a widely‐adopted approach, the pressure is collocated at the time steptn + 1while the remainder of the Navier‐Stokes equations is discretized following the generalized‐ scheme. That scheme has been claimed to besecond‐order accurate in time. We developed a suite of numerical code using inf‐sup stable higher‐order non‐uniform rational B‐spline (NURBS) elements for spatial discretization. In doing so, we are able to achieve high spatial accuracy and to investigate asymptotic temporal convergence behavior. Numerical evidence suggests that onlyfirst‐order accuracyis achieved, at least for the pressure, in this aforesaid temporal discretization approach. On the other hand, evaluating the pressure at the intermediate time step recovers second‐order accuracy, and the numerical implementation is simplified. We recommend this second approach as the generalized‐ scheme of choice when integrating the incompressible Navier‐Stokes equations.
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- Award ID(s):
- 1663671
- PAR ID:
- 10454460
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 122
- Issue:
- 2
- ISSN:
- 0029-5981
- Page Range / eLocation ID:
- p. 638-651
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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