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1. Semi-Lagrangian Pressure Solver for Accurate, Consistent, and Conservative Volume-of-Fluid Simulations

   Fox, Julian L; Owkes, Mark (May 2024, ILASS-Americas 34th Annual Conference on Liquid Atomization and Spray Systems)

   In this work, a novel discretization of the incompressible Navier-Stokes equations for a gas-liquid flow is developed. Simulations of gas-liquid flows are often performed by discretizing time with a predictor → pressure → corrector approach and the phase interface is represented by a volume of fluid (VOF) method. Recently, unsplit, geometric VOF methods have been developed that use a semi-Lagrangian discretization of the advection term within the predictor step. A disadvantage of the current methods is that an alternative discretization (e.g. finite volume or finite difference) is used for the divergence operator in the pressure equation. Due to the inconsistency in discretizations, a flux-correction to the semi-Lagrangian advection term is required to achieve mass conservation, which increases the computational cost and reduces the accuracy. In this work, we explore the alternative of using a semi-Lagrangian discretization for the divergence operators in both the advection term and the pressure equation. The proposed discretization avoids the need to use a flux-correction to the semi-Lagrangian advection term as mass conservation is achieved through the consistent discretization. Additionally, avoiding the flux-correction improves the accuracy while reducing the computational cost of the advection term semi-Lagrangian discretization. 

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2. Von Neumann Stable, Implicit, High Order, Finite Volume WENO Schemes

   https://doi.org/10.2118/193817-MS  

   Arbogast, Todd; Huang, Chieh-Sen; Zhao, Xikai (April 2019, SPE Reservoir Simulation Conference 2019)

   Simulation of flow and transport in petroleum reservoirs involves solving coupled systems of advection-diffusion-reaction equations with nonlinear flux functions, diffusion coefficients, and reactions/wells. It is important to develop numerical schemes that can approximate all three processes at once, and to high order, so that the physics can be well resolved. In this paper, we propose an approach based on high order, finite volume, implicit, Weighted Essentially NonOscillatory (iWENO) schemes. The resulting schemes are locally mass conservative and, being implicit, suited to systems of advection-diffusion-reaction equations. Moreover, our approach gives unconditionally L-stable schemes for smooth solutions to the linear advection-diffusion-reaction equation in the sense of a von Neumann stability analysis. To illustrate our approach, we develop a third order iWENO scheme for the saturation equation of two-phase flow in porous media in two space dimensions. The keys to high order accuracy are to use WENO reconstruction in space (which handles shocks and steep fronts) combined with a two-stage Radau-IIA Runge-Kutta time integrator. The saturation is approximated by its averages over the mesh elements at the current time level and at two future time levels; therefore, the scheme uses two unknowns per grid block per variable, independent of the spatial dimension. This makes the scheme fairly computationally efficient, both because reconstructions make use of local information that can fit in cache memory, and because the global system has about as small a number of degrees of freedom as possible. The scheme is relatively simple to implement, high order accurate, maintains local mass conservation, applies to general computational meshes, and appears to be robust. Preliminary computational tests show the potential of the scheme to handle advection-diffusion-reaction processes on meshes of quadrilateral gridblocks, and to do so to high order accuracy using relatively long time steps. The new scheme can be viewed as a generalization of standard cell-centered finite volume (or finite difference) methods. It achieves high order in both space and time, and it incorporates WENO slope limiting. 

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3. Implementation of the Accurate Conservative Phase Field Method for two-phase incompressible flows in a finite volume framework, Proc. of the 9th Int. and 49th National Conf. on Fluid Mechanics and Fluid Power (FMFP), December 14-16, 2022, IIT Roorkee, Roorkee-247667, Uttarakhand, India. FMFP2022–1300.

   Jain, S.; Tafti, D. K. (December 2022, Proc. of the 9th Int. and 49th National Conf. on Fluid Mechanics and Fluid Power (FMFP))

   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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4. A 3D OpenFOAM based finite volume solver for incompressible Oldroyd-B model with infinity relaxation time

   Tian, Jing; Cai, Mingchao; Luo, Zhendong; You, Yuncheng; Chen, Goong (November 2019, Communications in Nonlinear Science Numerical Simulation)

   In this paper, we develop a Finite Volume solver for a 3D incompressible Oldroyd-B model with infinity relaxation time. The Finite Volume solver is implemented by using a lead- ing open-source computational mechanics software OpenFOAM. We have imposed the di- vergence free condition as a constraint on the momentum equation to derive a pressure equation and a predictor-corrector procedure is applied when solving the velocity field. Both stability analysis and numerical experiments are given to show the robustness and accuracy of our algorithm. Two concrete examples on a cubical domain and a dumbbell are computed and illustrated. 

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5. On the stability of explicit finite difference methods for advection–diffusion equations

   https://doi.org/10.1002/num.22897  

   Zeng, Xianyi; Hasan, Md Mahmudul (July 2022, Numerical Methods for Partial Differential Equations)

   Abstract In this article we study the stability of explicit finite difference discretization of advection–diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations that is obtained by discretizing the ADE in space and then extends to fully discretized methods in combination with explicit Runge–Kutta methods. In particular, we prove that all stable semi‐discretization of the ADE leads to a conditionally stable fully discretized method as long as the time‐integrator is at least first‐order accurate, whereas high‐order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of the article, the analysis and the stability results are extended to a partially dissipative wave system, which serves as a model for common practice in many fluid mechanics applications that incorporate a viscous stress in the momentum equation but no heat dissipation in the energy equation. Finally, the major theoretical predictions are verified by numerical examples. 

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