Problems with Chorins Method

1
Problems with Chorins Method

• Andy Perrin
• Advisor Dr. H. H. Hu

2
Review and Update

• Want to solve N-S eqn.s by taking some initial
  field variables and updating them. (Called
  explicit method.)
• Homegrown finite difference scheme didnt work,
  so try modifying an existing method.
• Chorin (1967) is closest to what we want, so
  start there.

3
Chorin Method

• Solve incompressible N-S and an artificial
  continuity equation to get an answer thats
  correct in the limit of infinite time.
• ?p/?t gt 0 as t-gt8, giving back incompressible
  continuity eqn.

4
Chorin Method

• Uses Dufort-Frankel scheme a finite difference
  scheme with no dissipation built in.
• Compare to upwind scheme from last time.
• 2nd order in time and space.

5
Determining Convergence

• Simulation has converged when mass conservation
  is true locally (and therefore globally).
• Measure with sum of squares of divergence at each
  point.
• Should tend to zero as time increases.

6
Benchmark Problem

• Like last time parabolic u profile at inlet,
  ?u/?x 0 at outlet.

U
7
Block in Flow
D
3D
U
L15D
8
Block in Flow, Re 1
Pressure field
v-velocity
u-velocity
Streamlines
9
Block in Flow, Re 10
Pressure field
v-velocity
u-velocity
Streamlines
10
Block in Flow, Re 20
Pressure field
v-velocity
u-velocity
Streamlines
11
Block in Flow, Re 40
Pressure field
v-velocity
u-velocity
Streamlines
12
Block in Flow, Re 60?
Pressure field
u-velocity
Streamlines (detail view)
13
Block in Flow, Streamlines
Re 1
Re 10
Re 20
Re 40
14
Block on Wall, Re 40
Block location (Note this is a closeup L25)
P
v
u
Streamlines closeup
15
Convergence History
16
Convergence History
Convergence time does not seem to depend on Re.
Time constants were nearly the same.
17
Lid Driven Cavity Flow
D
D
Lid moves at constant velocity U towards the
right.
18
Lid Driven Cavity Flow, Re 10
detail
19
Lid Driven Cavity Flow, Re 40
20
Lid Driven Cavity Flow, Re 0.01,L/H1
21
Creeping Flow Over Cavity, Re 0.01,L/H3
22
Future Goals

• Try to improve convergence
• Use fake continuity, and choose ? ?(?p/?t) such
  that div(u) -gt 0 more quickly.
• Make more physically correct
• Use a more realistic continuity equation, without
  abandoning any terms.
• Identify/measure various time-scales.