Finite Element Method in Incompressible Flows

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24 Οκτ 2013 (πριν από 3 χρόνια και 8 μήνες)

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Finite Element Method
in Incompressible
Flows

Sangkyu

“Calvin” Kim

sk2635@cornell.edu

Fluid Motions


Governing
Eqs
.


Mass Conservation:



Momentum Conservation:



Energy Conservation:





Deviatoric

Stress:

Solids and Fluids


Fluids cannot hold
deviatoric

stresses, unless in motion.


Fluids are governed by
non
-
self
-
adjoint

equations, arising
from convective terms.


Galerkin

method no longer applies.

Characteristic
-
Based Split



Introduced by A.J.
Chorin

in 1968.


Perfected by O.C.
Zienkiewicz

and R.
Codina

in 1995.


The most general and powerful algorithm.



Splitting the mass flux into two parts so that the first part
approximates the intermediate value, while the second part
corrects the first guess.


Choose values such that the pressure change, which is
linear on the density change, is self
-
adjoint
.


Aim: to use characteristic
-
Galerkin

method.

Split A

Split B

Spatial Discretization


Step 1


Main goal: Calculate first velocity correction term

Spatial Discretization


Step 2


Main goal: Calculate the pressure correction term

Spatial Discretization


Step 3


Main goal: calculate the second velocity correction term

Program Structure

Problem: Lid
-
Driven Flow

Result


Lid
-
Driven Flow

Result


Lid
-
Driven Flow

Result


Lid
-
Driven Flow

Result


Lid
-
Driven Flow

Result


Lid
-
Driven Flow

Result


Lid
-
Driven Flow

Result


Literature Comparison


Comparison with
Ghia
. et al (1982):



Limitations and Conclusions

Good match with literature overall, however:

Current Limitations:


The program is VERY slow and memory intensive.


Boundary conditions are limited.


2
-
D, and Q4 only

Possible Improvements:


Adaptive meshing


Time step validation


Compressible case


Dual
-
time stepping


3
-
D and more element types


Better commenting



References


Zienkiewicz
, O.C., Taylor, R.L., and
Nithiarasu
, P. Finite Element Method for
Fluid Dynamics. Amsterdam; Boston: Elsevier Butterworth
-
Heinemann,
2005.


Liu
, C
-
B.,
Nithiarasu
, P.
Explit

and Semi
-

Implicit Characteristic Based Split
(CBS) Schemes for Viscoelastic Flow Calculations. University of Wales
Swansea, School of Engineering, 2006.


A.J
.
Chorin
. Numerical solution of
Navier
-
Stokes equations.
Math
.
Comput
.,
22:745
-
762, 1968.


A.J
.
Chorin
. On the convergence of discrete approximation to the
Navier
-
Stokes equations.
Math
.
Comput
.,
23:341
-
353, 1969.


O.C
.
Zienkiewicz

and R.
Codina
. Search for a general fluid mechanics
algorithm. In D.A.
Caughey

and M.M. Hafez, editors,
Frontiers
of
Computational Fluid
Dynamics
,

pages 101
-
113. John Wiley
&
Sons, New York,
1995.


O.C
.
Zienkiewicz

and R.
Codina
. A general algorithm for compressible and
incompressible flow
-

Part I: The split, characteristic
-
based scheme.
I
nternational
Journal for Numerical Methods in
Fluids
, 20:869
-
885
, 1995
.


Ghia
,
Ghia
, and Shin (1982), "High
-
Re solutions for incompressible flow using
the
Navier
-
Stokes equations and a
multigrid

method",
Journal
of
Computational
Physics,
Vol. 48, pp. 387
-
411.