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Register a new userAdd force and/or change underlying projection method to improve accuracy of Explicit Robin-Neumann and fully decoupled schemes for the coupling of incompressible fluid with thin-walled structure
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Yiyi Huang
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This work aims at providing some novel and practical ideas to improve accuracy of some partitioned algorithms, precisely Fernandezs Explicit Robin-Neumann and fully decoupled schemes, for the coupling of incompressible fluid with thin-walled structure. Inspired by viscosity of fluid and justified by boundary layer theory, the force between fluid and structure corresponding to viscosity is increased. Numerical experiments demonstrate improvement of accuracy under such modification. To improve accuracy of fully decoupled schemes further, the underlying projection method is replaced.
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Inspired by Rybaks multiple-time-step technique, jagged-time-step technique is proposed and applied to Fernandezs Explicit Robin-Neumann scheme. For some instances, numerical experiments demonstrate higher convergence orders and accuracy with lower computation cost as time and space get refined.
We consider a fully discrete loosely coupled scheme for incompressible fluid-structure interaction based on the time semi-discrete splitting method introduced in {emph{[Burman, Durst & Guzman, arXiv:1911.06760]}}. The splittling method uses a Robin-Robin type coupling that allows for a segregated solution of the solid and the fluid systems, without inner iterations. For the discretisation in space we consider piecewise affine continuous finite elements for all the fields and ensure the inf-sup condition by using a Brezzi-Pitkaranta type pressure stabilization. The interfacial fluid-stresses are evaluated in a variationally consistent fashion, that is shown to admit an equivalent Lagrange multiplier formulation. We prove that the method is unconditionally stable and robust with respect to the amount of added-mass in the system. Furthermore, we provide an error estimate that shows the error in the natural energy norm for the system is $mathcal Obig(sqrt{T}(sqrt{Delta t} + h)big)$ where $T$ is the final time, $Delta t$ the time-step length and $h$ the space discretization parameter.
Dispersion of low-density rigid particles with complex geometries is ubiquitous in both natural and industrial environments. We show that while explicit methods for coupling the incompressible Navier-Stokes equations and Newtons equations of motion are often sufficient to solve for the motion of cylindrical particles with low density ratios, for more complex particles - such as a body with a protrusion - they become unstable. We present an implicit formulation of the coupling between rigid body dynamics and fluid dynamics within the framework of the immersed boundary projection method. Similarly to previous work on this method, the resulting matrix equation in the present approach is solved using a block-LU decomposition. Each step of the block-LU decomposition is modified to incorporate the rigid body dynamics. We show that our method achieves second-order accuracy in space and first-order in time (third-order for practical settings), only with a small additional computational cost to the original method. Our implicit coupling yields stable solution for density ratios as low as $10^{-4}$. We also consider the influence of fictitious fluid located inside the rigid bodies on the accuracy and stability of our method.
The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a discretization of the Navier-Stokes equation that advects mass and momentum in a consistent manner. Incompressible flow with capillary forces is modeled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul dInterface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressibility is satisfied. To improve the stability of these methods momentum fluxes are advected in a manner consistent with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a 3mm-diameter falling raindrop and for an atomizing flow in air-water conditions.
In this paper we present energy-conserving, mixed discontinuous Galerkin (DG) and continuous Galerkin (CG) schemes for the solution of a broad class of physical systems described by Hamiltonian evolution equations. These systems often arise in fluid mechanics (incompressible Euler equations) and plasma physics (Vlasov--Poisson equations and gyrokinetic equations), for example. The dynamics is described by a distribution function that evolves given a Hamiltonian and a corresponding Poisson bracket operator, with the Hamiltonian itself computed from field equations. Hamiltonian systems have several conserved quantities, including the quadratic invariants of total energy and the $L_2$ norm of the distribution function. For accurate simulations one must ensure that these quadratic invariants are conserved by the discrete scheme. We show that using a discontinuous Galerkin scheme to evolve the distribution function and ensuring that the Hamiltonian lies in its continuous subspace leads to an energy-conserving scheme in the continuous-time limit. Further, the $L_2$ norm is conserved if central fluxes are used to update the distribution function, but decays monotonically when using upwind fluxes. The conservation of density and $L_2$ norm is then used to show that the entropy is a non-decreasing function of time. The proofs shown here apply to any Hamiltonian system, including ones in which the Poisson bracket operator is non-canonical (for example, the gyrokinetic equations). We demonstrate the ability of the scheme to solve the Vlasov--Poisson and incompressible Euler equations in 2D and provide references where we have applied these schemes to solve the much more complex 5D electrostatic and electromagnetic gyrokinetic equations.
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