On the development of area-conserved level set/immersed boundary method to predict free surface in complex domain

Abstract
In this dissertation, I propose two schemes which accommodate a better dispersion relation
for the convective terms shown in the transport equation, such as in the level set
equation and momentum equations. For the multi-phase tracking problems, the free surface
has been tracked by the level set method (LS) and the conservative level set method
(CLS). In addition, the immersed boundary method (IBM) is developed to simulate mechanical
systems in which structures may interact with the fluid flows. The interaction
between free surface flow and structure is also investigated using a level set/immersed
boundary coupled method (LS-IBM).
(I) Minimized phase error upwinding combined compact difference schemes:
All of schemes are proposed to enhance the convective stability by virtue of the increased
dispersive accuracy and they have been rigorously developed through the dispersion and
dissipation analyses. To verify the proposed method, several problems will be investigated.
The results with good rates of convergence are demonstrated for all the investigated
problems.
(II) Level set method:
I apply a level set method to simulate gas/water interface flow. For the sake of accuracy,
the spatial derivative terms in the equations of motion for an incompressible fluid flow
are approximated by the higher-order accurate minimized phase error upwinding combined
compact difference (UCCD) scheme. This scheme development employs two (or
three) combined equations to calculate the first- and second-order derivative terms (and
third-order derivative terms). For accurately predicting the level set value, the interface
tracking scheme is also developed to preserve the theoretical phase error of the first-order
derivative term shown in the level set equation. For the purpose of retaining the longtime
accurate Hamiltonian in the advection equation for the level set function, the time
derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme.
Also, to keep as a distance function for ensuring the front having a finite thickness for all time, the reinitialization equation is used. For the verification of the proposed scheme
for the pure advection equation, several benchmark problems have been chosen in this
dissertation. The level set method with excellent area preservation property proposed for
capturing the interface in incompressible fluid flows is also verified by solving the dambreak,
Rayleigh-Taylor instability, two-bubble rising in water, and droplet falling in water
problems.
(III) Conservative level set method:
A two-step conservative level set method is proposed to simulate the gas/water two-phase
flow. For the sake of accuracy, the spatial derivative terms in the equations of motion
for an incompressible fluid flow are approximated by the upwinding combined compact
scheme. For accurately predicting the level set function, the advection scheme with minimized
phase error advection scheme is developed to preserve the theoretical phase error
for the first-order derivative terms shown in the pure advection equation cast in conservative
form. For the purpose of retaining its long-time accurate Casimir functionals and
Hamiltonians in the transport equation for the level set function, the time derivative term
is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. To resolve
contact discontinuity oscillations near interface, nonlinear compression flux term and artificial
damping term are properly added to the second-step equation in the conservative
level set method. For the verification of the proposed phase-minimized scheme applied
in non-staggered grids for solving the incompressible flow equations, several benchmark
problems have been chosen in this dissertation. The conservative level set method
with area-preserving property proposed for capturing the interface in incompressible fluid
flows is verified by solving the dam-break, Rayleigh-Taylor instability, bubble rising in
water, and droplet falling in water problems. Good agreement with the referenced solutions
is demonstrated in all the investigated problems.
(IV) Immersed boundary method:
I use the immersed boundary method to solve the flow equations in irregular and timevarying
domains. The artificial momentum forcing term applied at certain points in cells containing both of the fluid and solid allows an imposition of velocity condition to account
for the solid body motion. We develop in this study a differential-based interpolation
scheme which can be easily extended to perform the three-dimensional simulation.
The results obtained from the proposed immersed boundary method agree well with other
numerical and experimental results for the chosen benchmark problems. The accuracy
and fidelity of the Immersed boundary (IB) flow solver developed to predict flows with
irregular boundaries are therefore demonstrated.
(V) Level set/immersed boundary method:
A level set/immersed boundary coupled method combines the level set method and the
immersed boundary method. This method implemented in Navier-Stokes solver allows
simulation of interaction between the fluid flow with free surface and bodies. A solution
algorithm is proposed to prescribe the exact forcing points near the solid boundaries for
providing an accurate numerical solution. The discretized linear system of the Poisson
pressure equation is solved using the divergence-free-condition (DFC) compensated flow
solver. The predicted results are in good agreement with numerical simulations.