Exponential Time Differencing Gauge Method for Incompressible Viscous Flows

2017 ◽  
Vol 22 (2) ◽  
pp. 517-541 ◽  
Author(s):  
Lili Ju ◽  
Zhu Wang
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Viscous Flows ◽  
Momentum Equation ◽  
Compact Representation ◽  
Kinematic Equation ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Incompressible Viscous Flows

AbstractIn this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Simulation of Incompressible Viscous Flows by Local DFD-Immersed Boundary Method

2012 ◽  
Vol 4 (03) ◽  
pp. 311-324 ◽  
Author(s):  
Y. L. Wu ◽  
C. Shu ◽  
H. Ding
Keyword(s):  
Immersed Boundary Method ◽  
Stokes Equations ◽  
Viscous Flows ◽  
Slip Boundary Condition ◽  
Present Approach ◽  
Immersed Boundary ◽  
Local Domain ◽  
Slip Boundary ◽  
Incompressible Viscous Flows ◽  
Boundary Method

AbstractA local domain-free discretization-immersed boundary method (DFD-IBM) is presented in this paper to solve incompressible Navier-Stokes equations in the primitive variable form. Like the conventional immersed boundary method (IBM), the local DFD-IBM solves the governing equations in the whole domain including exterior and interior of the immersed object. The effect of immersed boundary to the surrounding fluids is through the evaluation of velocity at interior and exterior dependent points. To be specific, the velocity at interior dependent points is computed by approximate forms of solution and the velocity at exterior dependent points is set to the wall velocity. As compared to the conventional IBM, the present approach accurately implements the non-slip boundary condition. As a result, there is no flow penetration, which is often appeared in the conventional IBM results. The present approach is validated by its application to simulate incompressible viscous flows around a circular cylinder. The obtained numerical results agree very well with the data in the literature.

A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN

2006 ◽  
Vol 16 (02) ◽  
pp. 233-263 ◽  
Author(s):  
Z. BELHACHMI ◽  
C. BERNARDI ◽  
S. DEPARIS ◽  
F. HECHT
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
A Priori ◽  
Three Dimensional ◽  
Angular Variable ◽  
A Posteriori ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Good Agreement

We consider the Stokes problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of this problem which combines Fourier truncation and finite element methods applied to each of the two-dimensional systems. We give the detailed a priori and a posteriori analyses of the discretization and present some numerical experiments which are in good agreement with the analysis.

scholarly journals A 3D Chebyshev-Fourier algorithm for convection equations in low Mach number approximation

2009 ◽  
pp. 607-625
Author(s):  
Ouafa Bouloumou ◽  
Eric Serre ◽  
Jochen Fröhlich
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Time Discretization ◽  
Navier Stokes ◽  
Working Fluid ◽  
Uzawa Algorithm ◽  
Low Mach Number ◽  
Preconditioned Iterative

A three-dimensional spectral method based on Chebyshev-Chebyshev-Fourier discretizations is presented in the framework of the low Mach number approximation of Navier-Stokes equations. The working fluid is assumed to be a perfect gas with constant thermodynamic properties. The generalized Stokes problem, which arises from the time discretization by a second-order semi-implicit scheme, is solved by a preconditioned iterative Uzawa algorithm. Several validation results are presented in the case of steady and unsteady flows. This model is also evaluated for natural convection flows with large density variations in the case of a tall differentially heated cavity of aspect ratio 8. It is found that on contrary to convection at small temperature differences (Boussinesq), the 2D unsteady solution at Ra = 3.4 x 105 is unstable to 3D perturbations.

Numerical Analysis of a Stable Discontinuous Galerkin Scheme for the Hydrostatic Stokes Problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Francisco Guillén-González ◽  
M. Victoria Redondo-Neble ◽  
J. Rafael Rodríguez-Galván
Keyword(s):  
Discontinuous Galerkin ◽  
Vertical Velocity ◽  
Large Scale ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Momentum Equation ◽  
Loss Of Ellipticity ◽  
Numerical Tests ◽  
Discontinuous Galerkin Scheme ◽  
Penalty Technique

AbstractWe propose a Discontinuous Galerkin (DG) scheme for the Hydrostatic Stokes equations. These equations, related to large-scale PDE models in Oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the SIP penalty technique, with a particular treatment of the vertical velocity. It is well-known that stability of the mixed formulation of Primitive Equations requires, besides the LBB inf-sup condition, an additional hydrostatic inf-sup restriction relating pressure and vertical velocity. This hydrostatic inf-sup condition invalidates stability of usual Stokes stable continuous finite elements like Taylor-Hood 𝒫2/𝒫1 or bubble 𝒫1b/𝒫1. Here we prove stability for our 𝒫k/𝒫k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.

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