Adaptive Mesh Refinement for Two-Phase Viscoelastic Fluid Mixture Models (2409.19974v2)

Abstract: Multiphase flows are an important class of fluid flow and their study facilitates the development of diverse applications in industrial, natural and biomedical systems. Simulating such flows requires significant computational resources, making it prudent to devise an adaptive mesh refinement (AMR) method to mitigate this burden. We use a mathematical model that takes a continuum mechanical approach to describe multiphase mixture flows. The resulting system of equations poses numerical challenges due to the presence of multiple non-linear terms and a co-incompressibility condition, while the resulting fluid dynamics necessitate the development of an adaptive mesh refinement technique to accurately capture regions of interest while keeping computational costs low. We present an accurate, robust, and efficient computational method for simulating multiphase mixtures on adaptive grids, and utilize a multigrid solver to precondition the saddle-point system. We demonstrate that the AMR solver asymptotically approaches second order accuracy in $L^1$, $L^2$ and $L^\infty$ norms for all solution variables of the Newtonian and non-Newtonian models. All experiments demonstrate the solver is stable provided the time step size satisfies the imposed CFL condition. The solver can accurately resolve sharp gradients in the solution and, with the multigrid preconditioner, the solver behavior is independent of grid spacing. Our AMR solver offers a major cost savings benefit, providing up to 10x speedup in the numerical experiments presented here, with greater speedup possible depending on the problem set-up.

• The paper introduces a novel AMR method that achieves up to a 10x reduction in computational cost for simulating two-phase viscoelastic flows.
• The methodology employs second-order finite differencing and a multigrid preconditioned Krylov solver, ensuring stable, second-order accurate solutions under CFL conditions.
• Numerical experiments, including the four-roll mill test case, validate enhanced resolution, efficient computation, and accurate stress capture in complex fluid dynamics.

Adaptive Mesh Refinement for Two-Phase Viscoelastic Fluid Mixture Models

The research article, "Adaptive Mesh Refinement for Two-Phase Viscoelastic Fluid Mixture Models," presents a comprehensive paper on the utilization of Adaptive Mesh Refinement (AMR) for accurately and efficiently simulating multiphase viscoelastic fluid flows. This work spans multiple domains including industrial processing, natural phenomena, and biomedical systems, offering a robust computational methodology relevant to a wide range of multiphase systems.

Key Contributions:

The paper addresses the significant challenges posed by simulating multiphase flows which often involve substantial computational resources. Traditionally, such simulations on uniform grids incur high computational costs, especially when high-resolution grids are required to capture sharp gradients or intricate dynamics in localized regions. To mitigate this burden, the authors develop an AMR methodology, which dynamically refines the computational grid in regions of interest, thereby enhancing resolution where needed and maintaining coarser grids elsewhere to save computational effort.

Mathematical and Computational Framework:

The authors employ a continuum mechanical approach to model each phase of the multiphase fluid as occupying the same spatial region with distinct velocity fields. The focus is on a two-phase system comprising a viscous or viscoelastic network and a viscous solvent. The governing equations consist of advection equations for the volume fractions of the network and solvent, momentum equations for each phase, and a co-incompressibility condition. To handle the viscoelastic nature of the network, additional equations govern the evolution of the viscoelastic stress tensor.

The computational method leverages second-order finite differences for spatial discretization and a combination of implicit trapezoidal and Adams-Bashforth schemes for time integration. The resulting system of equations, which forms a saddle point problem, is solved using a multigrid preconditioned Krylov method. The multigrid preconditioner implemented is geometric, and the authors demonstrate its robustness and independence from grid spacing through various numerical experiments.

Convergence and Stability Analysis:

The convergence analysis conducted on both uniform and adaptively refined grids highlights that the solver achieves asymptotically second-order accuracy in $L^1$, $L^2$, and $L^\infty$ norms for all solution variables. Stability is ensured provided the time step size adheres to the Courant-Friedrichs-Lewy (CFL) condition. This guarantees that the simulations remain accurate and stable over time, which is crucial for modeling complex dynamics in multiphase flows.

Numerical Experiments:

The paper includes detailed numerical experiments to validate the convergence and efficiency of the AMR methodology. Test problems with known solutions are employed to verify second-order accuracy for both prescribed and evolved volume fractions. The four-roll mill problem, a classical test case involving complex fluid dynamics, is utilized to illustrate the capabilities of the solver in capturing intricate flow features on adaptive grids. Further, simulations involving a viscoelastic network under transient forcing highlight the solver's effectiveness in capturing relaxation dynamics and stress development.

Computational Efficiency:

One of the significant findings is the substantial reduction in computational costs achieved by using AMR. The article reports up to a 10x speedup in numerical experiments when comparing the AMR approach to uniform grid simulations. This reduction in computational time makes the methodology highly attractive for problems requiring high spatial resolution in localized regions.

Implications and Future Work:

The implications of this research are multifaceted. Practically, the methodology can be applied to enhance the simulation of industrial processes, natural phenomena, and biomedical systems, potentially leading to better design and understanding of these systems. Theoretically, the work advances the computational techniques available for multiphase fluid dynamics, offering a solid foundation for further refinement and application of AMR in even more complex scenarios.

The authors suggest that future work could focus on developing efficient projection or block factorization preconditioners to further ease the computational load. Such advancements could extend the applicability of their methodology to larger and even more complex multiphase systems.

In summary, this paper presents a crucial advancement in the numerical simulation of multiphase viscoelastic fluid flows using adaptive mesh refinement. It establishes a robust, accurate, and efficient computational framework with wide-ranging applications and opens avenues for future enhancements in the field of computational fluid dynamics.