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A Velocity Decomposition Approach for Moving Interfaces in Viscous Fluids
"... We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the veloc ..."
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We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the velocity into a “Stokes ” part and a “regular ” part. The first part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives secondorder accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the NavierStokes equations with a body force resulting from the Stokes part. The regular velocity is obtained using a timestepping method that combines the semiLagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. For problems with stiff boundary forces, the decomposition approach can be combined with fractional timestepping, using a smaller time step to advance the interface quickly by Stokes flow, with the velocity computed using boundary integrals. The small time steps maintain numerical stability, while the overall solution is updated on a larger time step to reduce computational cost.
COMBINED FIELD FORMULATION AND A SIMPLE STABLE EXPLICIT INTERFACE ADVANCING SCHEME FOR FLUID STRUCTURE INTERACTION
"... Abstract. We develop a combined field formulation for the fluid structure (FS) interaction problem. The unknowns are (u; p;v), being the fluid velocity, fluid pressure and solid velocity. This combined field formulation uses Arbitrary Lagrangian Eulerian (ALE) description for the fluid and Lagrangia ..."
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Abstract. We develop a combined field formulation for the fluid structure (FS) interaction problem. The unknowns are (u; p;v), being the fluid velocity, fluid pressure and solid velocity. This combined field formulation uses Arbitrary Lagrangian Eulerian (ALE) description for the fluid and Lagrangian description for the solid. It automatically enforces the simultaneous continuities of both velocity and traction along the FS interface. We present a first order in time fully discrete scheme when the flow is incompressible NavierStokes and when the solid is elastic. The interface position is determined by first order extrapolation so that the generation of the fluid mesh and the computation of (u; p;v) are decoupled. This explicit interface advancing enables us to save half of the unknowns comparing with traditional monolithic schemes. When the solid has convex strain energy (e.g. linear elastic), we prove that the total energy of the fluid and the solid at time tn is bounded by the total energy at time tn−1. Like in the continuous case, the fluid mesh velocity which is used in ALE description does not enter into the stability bound. Surprisingly, the nonlinear convection term in the NavierStokes equations plays a crucial role to stabilize the scheme and the stability result does not apply to Stokes flow. As the nonlinear convection term is treated semiimplicitly, in each time step, we only need to solve a linear system (and only once) which involves merely (u; p;v) if the solid is linear elastic. Two numerical tests including the benchmark test of NavierStokes flow past a Saint VenantKirchhoff elastic bar are performed. In addition to the stability, we also confirm the first order temporal accuracy of our explicit interface advancing scheme.
unknown title
, 2012
"... Immersed boundary methods for the numerical simulation of incompressible aerodynamic and fluidstructure interactions ..."
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Immersed boundary methods for the numerical simulation of incompressible aerodynamic and fluidstructure interactions
Digital Object Identifier (DOI) 10.1007/s0020501003275 Arch. Rational Mech. Anal. Analysis of a Stochastic Implicit Interface Model for an Immersed Elastic Surface in a Fluctuating Fluid
"... We present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible viscous fluid subject to fluctuation forces. We derive suitable a priori estimates and establish the wellposedness of pathwise solutions and provide ..."
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We present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible viscous fluid subject to fluctuation forces. We derive suitable a priori estimates and establish the wellposedness of pathwise solutions and provide uniform control on the solutions in probability. 1.
Semianalytical
, 2012
"... solutions for twodimensional elastic capsules in Stokes flow ..."
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unknown title
, 2012
"... Immersed boundary methods for the numerical simulation of incompressible aerodynamic and fluidstructure interactions ..."
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Immersed boundary methods for the numerical simulation of incompressible aerodynamic and fluidstructure interactions