Inspired by the recent realization of a two-dimensional (2-D) chiral fluid as an active monolayer droplet moving atop a 3-D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2-D linear, incompressible and isotropic fluid, having a viscous shear stress, an active chiral driving stress and a Hall stress allowed by the lack of time-reversal symmetry. The droplet interacts with itself through its driven internal mechanics and by driving flows in the underlying 3-D Stokes phase. We pose the dynamics as the solution to a singular integral–differential equation, over the droplet surface, using the mapping from surface stress to surface velocity for the 3-D Stokes equations. Specializing to the case of axisymmetric droplets, exact representations for the chiral surface flow are given in terms of solutions to a singular integral equation, solved using both analytical and numerical techniques. For a disc-shaped monolayer, we additionally employ a semi-analytical solution that hinges on an orthogonal basis of Bessel functions and allows for efficient computation of the monolayer velocity field, which ranges from a nearly solid-body rotation to a unidirectional edge current, depending on the subphase depth and the Saffman–Delbrück length. Except in the near-wall limit, these solutions have divergent surface shear stresses at droplet boundaries, a signature of systems with codimension-one domains embedded in a 3-D medium. We further investigate the effect of a Hall viscosity, which couples radial and transverse surface velocity components, on the dynamics of a closing cavity. Hall stresses are seen to drive inward radial motion, even in the absence of edge tension.
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An efficient tree-topological local mesh refinement on Cartesian grids for multiple moving objects in incompressible flow
This paper develops a tree-topological local mesh refinement (TLMR) method on Cartesian
grids for the simulation of bio-inspired flow with multiple moving objects. The TLMR nests
refinement mesh blocks of structured grids to the target regions and arrange the blocks
in a tree topology. The method solves the time-dependent incompressible flow using a
fractional-step method and discretizes the Navier-Stokes equation using a finite-difference
formulation with an immersed boundary method to resolve the complex boundaries. When
iteratively solving the discretized equations across the coarse and fine TLMR blocks, for
better accuracy and faster convergence, the momentum equation is solved on all blocks
simultaneously, while the Poisson equation is solved recursively from the coarsest block
to the finest ones. When the refined blocks of the same block are connected, the parallel
Schwarz method is used to iteratively solve both the momentum and Poisson equations.
Convergence studies show that the algorithm is second-order accurate in space for both
velocity and pressure, and the developed mesh refinement technique is benchmarked and
demonstrated by several canonical flow problems. The TLMR enables a fast solution to an
incompressible flow problem with complex boundaries or multiple moving objects. Various
bio-inspired flows of multiple moving objects show that the solver can save over 80%
computational time, proportional to the grid reduction when refinement is applied.
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- Award ID(s):
- 1931929
- NSF-PAR ID:
- 10473217
- Publisher / Repository:
- ELSEVIER
- Date Published:
- Journal Name:
- Journal of Computational Physics
- Volume:
- 479
- Issue:
- C
- ISSN:
- 0021-9991
- Page Range / eLocation ID:
- 111983
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Summary A fourth‐order finite‐volume method for solving the Navier–Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth‐order quadrature rules for evaluating face‐averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge–Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth‐order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non‐rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries. Copyright © 2016 John Wiley & Sons, Ltd.
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SUMMARY Combining finite element methods for the incompressible Stokes equations with particle-in-cell methods is an important technique in computational geodynamics that has been widely applied in mantle convection, lithosphere dynamics and crustal-scale modelling. In these applications, particles are used to transport along properties of the medium such as the temperature, chemical compositions or other material properties; the particle methods are therefore used to reduce the advection equation to an ordinary differential equation for each particle, resulting in a problem that is simpler to solve than the original equation for which stabilization techniques are necessary to avoid oscillations.
On the other hand, replacing field-based descriptions by quantities only defined at the locations of particles introduces numerical errors. These errors have previously been investigated, but a complete understanding from both the theoretical and practical sides was so far lacking. In addition, we are not aware of systematic guidance regarding the question of how many particles one needs to choose per mesh cell to achieve a certain accuracy.
In this paper we modify two existing instantaneous benchmarks and present two new analytic benchmarks for time-dependent incompressible Stokes flow in order to compare the convergence rate and accuracy of various combinations of finite elements, particle advection and particle interpolation methods. Using these benchmarks, we find that in order to retain the optimal accuracy of the finite element formulation, one needs to use a sufficiently accurate particle interpolation algorithm. Additionally, we observe and explain that for our higher-order finite-element methods it is necessary to increase the number of particles per cell as the mesh resolution increases (i.e. as the grid cell size decreases) to avoid a reduction in convergence order.
Our methods and results allow designing new particle-in-cell methods with specific convergence rates, and also provide guidance for the choice of common building blocks and parameters such as the number of particles per cell. In addition, our new time-dependent benchmark provides a simple test that can be used to compare different implementations, algorithms and for the assessment of new numerical methods for particle interpolation and advection. We provide a reference implementation of this benchmark in aspect (the ‘Advanced Solver for Problems in Earth’s ConvecTion’), an open source code for geodynamic modelling.
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Abstract We present a formulation and numerical algorithm to extend the scheme for gray radiation magnetohydrodynamics (MHD) developed by Jiang to include the frequency dependence via the multigroup approach. The entire frequency space can be divided into an arbitrary number of groups in the lab frame, and we follow the time-dependent evolution of frequency-integrated specific intensities along discrete rays inside each group. Spatial transport of photons is done in the lab frame while all the coupling terms are solved in the fluid rest frame. Lorentz transformation is used to connect different frames. The radiation transport equation is solved fully implicitly in time while the MHD equations are evolved explicitly so that time step is not limited by the speed of light. A finite volume approach is used for transport in both spatial and frequency spaces to conserve the radiation energy density and momentum. The algorithm includes photon absorption, electron scattering, as well as Compton scattering, which is calculated by solving the Kompaneets equation. The algorithm is accurate for a wide range of optical depth conditions and can handle both radiation-pressure- and gas-pressure-dominated flows. It works for both Cartesian and curvilinear coordinate systems with adaptive mesh refinement. We provide a variety of test problems including a radiating sphere, shadow test, absorption of a moving gas, Bondi-type flows, as well as a collection of test problems for thermal and bulk Compton scattering. We also discuss examples where frequency dependence can make a big difference compared with the gray approach.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jcp.2023.111983
