A stabilized formulation for the solution of the incompressible unsteady Stokes equations in the frequency domain

Esmaily, Mahdi; Jia, Dongjie

doi:10.1016/j.jcp.2022.111736

Cited by 3 publications

(2 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In an earlier study, we introduced a pressure-stabilized technique for solving the unsteady Stokes equations expressed in the frequency domain rather than the time domain 43 . The resulting complex-valued stabilization parameter was derived systematically by taking the divergence of the momentum equation and estimating the Laplacian in the diffusion term using a characteristic element size.…”

Section: Formulationmentioning

confidence: 99%

“…More specifically, we replace the inverse of the time step size in the definition of the with a measure of the flow frequency. The motivation behind such formulation lies in the spectral formulation of the unsteady Stokes equations where the time step size dependent parameter in the is replaced by the spectral mode number 43 . The same parameter, when expressed in a spatio-temporal setting, inspires the use of a flow-dependent time scale in the definition, hence motivating the present design.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A time-consistent stabilized finite element method for fluids with applications to hemodynamics

Jia,

Esmaily

2023

Sci Rep

Self Cite

View full text Add to dashboard Cite

Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov–Galerkin stabilization (SUPG) term, which is weighted by $$\tau _{\text{SUPG}}$$ τ SUPG . The conventional formulation of $$\tau _{\text{SUPG}}$$ τ SUPG includes a constant that depends on the time step size, producing an overall method that becomes exceedingly less accurate as the time step size approaches zero. In practice, such method inconsistency introduces significant error in the solution, especially in cardiovascular simulations, where small time step sizes may be required to resolve multiple scales of the blood flow. To overcome this issue, we propose a consistent method that is based on a new definition of $$\tau _{\text{SUPG}}$$ τ SUPG . This method, which can be easily implemented on top of an existing streamline upwind Petrov–Galerkin and pressure stabilizing Petrov–Galerkin method, involves the replacement of the time step size in $$\tau _{\text{SUPG}}$$ τ SUPG with a physical time scale. This time scale is calculated in a simple operation once every time step for the entire computational domain from the ratio of the L2-norm of the acceleration and the velocity. The proposed method is compared against the conventional method using four cases: a steady pipe flow, a blood flow through vascular anatomy, an external flow over a square obstacle, and a fluid–structure interaction case involving an oscillatory flexible beam. These numerical experiments, which are performed using linear interpolation functions, show that the proposed formulation eliminates the inconsistency issue associated with the conventional formulation in all cases. While the proposed method is slightly more costly than the conventional method, it significantly reduces the error, particularly at small time step sizes. For the pipe flow where an exact solution is available, we show the conventional method can over-predict the pressure drop by a factor of three. This large error is almost completely eliminated by the proposed formulation, dropping to approximately 1% for all time step sizes and Reynolds numbers considered.

show abstract

Section: Formulationmentioning

confidence: 99%