On the development of a scheme with the optimized numerical dispersion relation equation for simulating a truly multidimensional incompressible fluid flow
Date Issued
2011
Date
2011
Author(s)
Pai, Che-An
Abstract
This thesis is aimed to develop a flow solver for properly simulating the
incompressible fluid flow at high-Reynolds numbers.
The strategy of getting a higher simulation quality is to retain some
rich geometrical properties embedded in its limiting Euler flow equations.
Following the theory of Clebsch velocity decomposition,
I can rigorously decompose the velocity vector as the sum of the velocity components that
account, respectively, for the flow potential,
rotation, and dissipation.
Simulation of the Navier-Stokes equations cast in velocity-pressure primitive variables can be therefore fractionally
split into three corresponding steps.
The equations are coupled to each other.
In the pure advection solution step,
we employ the mid-point symplectic time integrator to approximate the temporal derivative term so as to
retain the discrete Hamiltonian and Casimir properties embedded in the lossless Euler equations.
For the sake of reducing numerical dispersion error,
the upwinding spatial scheme with the smallest difference between the numerical and exact dispersion relation equations
for the time-dependent pure advection equation is developed in wavenumber space.
In the diffusion step,
I approximate the time derivative term shown in the time-dependent parabolic equation using the time-stepping scheme
that can be
different from the one used in the first solution step for the calculation of Euler solutions.
I then update the velocity vector in projection step that is solved subject to the divergence-free
constraint condition.
The proposed method has been validated through some chosen benchmark tests.
The predicted results for the incompressible
Navier-Stokes equations are also justified
by solving two problems at high-Reynolds numbers.
Subjects
Clebsch velocity decomposition
symplectic
dispersion relation equations
Hamiltonian and Casimir properties
Type
thesis
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