Download Modeling of Oscillating Flow Past a Vertical Plate

The 13th International Offshore and Polar Engineering Conference
ISOPE-2003, Hawaii, May25-30, 2003
Modeling of Oscillating Flow Past
a Vertical Plate
Spyros A. Kinnas, Yi-Hsiang Yu, Hanseong Lee, Karan Kakar
Ocean Engineering Group, Department of Civil Engineering
The University of Texas at Austin, Austin, TX, USA
Overview
Introduction
Governing Equations, Boundary
Conditions and Numerical Formulation.
Comparison of results from the 2D
unsteady Euler, Navier-Stokes Solvers
with measurements.
Visualization of separated flow field.
Summary and future work.
Introduction
Motivation & Objectives
 Develop and validate a
method for the prediction
of the effects of bilge
keels on FPSO Hull
motion.
 Understand the physics of the unsteady separated
flow about the bilge keels of a hull subject to roll
motions, and predict the corresponding
hydrodynamic coefficients.
Motivation & Objectives
In this work, we apply and validate our model to
a bilge keel (vertical plate) subject to horizontal
sinusoidal inflow.
Literature Review
Numerical Work:
-
Gentaz et al. 1997
Korpus and Falzarano 1997
Vassalos et al. 2000
Yeung et al. 1992, 1996, 1998, 2000, 2002
Experiments on Vertical Plate:
-Keulegan and Carpenter 1958
-Sarpkaya and O’Keefe 1995
Numerical Formulation
Governing Equation
Non-Dimensional
Navier-Stokes equations
U=-U m Cos(
Plate
1 U F G


Q
KC t x y
u 
u 2  1  2u / x 
U   , F    
,


v
 
uv  Re u / y  v / x 


uv 1 u / y  v / x 
 p x  f x 
G

, Q
,
 p y  f 
v Re  2v / y 
y 

 
 
 
 2
 
KC: Keulegan-Carpenter number, KC 
Re: Reynolds number, Re  U m h

2
t)
T
U mT
h
h
GRID FOR OSCILLATING FLOW PAST A FLAT PLATE
Direction of oscillating flow
Top
u
 0,
y
v  0,
p
0
y
Inflow
Outflow
u  U m cos(t )
v  0,
u  U m cos(t )
2 p
0
x 2
v  0,
Plate
v
p
u  0,
 0,
0
x
x
Bottom
u
 0,
y
v  0,
p
0
y
Grid size: 201x101
Domain size: -8 to 8 in x-direction and 0 to 8 in y-direction
Um, T: Amplitude, and period of oscillating flow
h: height of the plate
2 p
0
x 2
Numerical Method
Finite Volume Method
U
Sij   ( Fdy  Gdx)  QSij where Sij is the area of the cell
t
edges
Ni’s Lax-Wendroff Method for time
n
  2 U  (t )2
 U 
n
Ui  j  
,
 t   2 
 t  i  j
 t  i  j 2
n
Uinj1
n 1
n
U1  U1 

c  A , B ,C , D
( U1 )cn
Artificial dissipation (viscosity) ~ only in the case of
Euler solver
U1n 1  U1n 

c  A B C  D
( U1 )cn  t (2  4 )
2   2 (ii U1n   jj U1n ),
4   4 ( iiii U1n   jjjj U1n )
SIMPLE Method for pressure correction
Euler and Navier-Stokes Solver
Choi & Kinnas 2000, Choi PhD thesis 2000
-Development of Euler Solver and Navier-Stokes solver.
Kakar MS thesis 2002
- Application of unsteady 2D Euler solver of flow over
bilge keels for FPSO hull motion.
 Euler Solver ~ Applied artificial viscosity
 Navier-Stokes Solver ~ Viscous terms
Force Calculation on the plate
Morison’s equation as follows [Sarpkaya and
O’Keefe (1995)]
2 Fˆ
2 2
 Cd  cos  cos  Cm
sin
2
K
 hwU m
3 2 Fˆ ( ) cos 
Cd   
d Drag coefficient
2
0
4
 hwU m
Cm 
2 KC
3

2
0
Fˆ ( )sin
d Inertia coefficient
2
 hwU m
Results
Convergence Study
At KC=1
Comparison of different numbers of grid in vertical direction
The Convergence study of different numbers of grid
in vertical direction (log scale)
200x050
2
1.75
200x050
200x075
200x100
200x150
15
grid on the plate surface
grid on the plate surface
grid on the plate surface
grid on the plate surface
1.5
1
0.75
0.5
0.25
0
-2
-1
0
1
2
200x050
0.06
200x075
2
10
0.1
0.09
0.08
0.07
1.25
1.75
0.05
1.5
1.25
1
5
0.04
0.75
0.5
0.25
0
-2
-1
0
0
1
2
200x100
2
Error
Non-Dimensional Force, F/  U 2 H 2
20
10
15
20
30
0.03
1.75
1.5
-5
200x075
0.02
1.25
1
0.75
0.5
-10
0.25
0
-2
-1
2
-15
0
1
2
y = -2.3x + 8.
(in log scale)
200x150
200x100
1.75
0.01
1.5
1.25
-20
1
0
0.25
0.5
Time Period
0.75
1
0.75
10000
0.5
0
-2
12500
15000
Numbers of Grid
0.25
-1
0
1
2
17500
20000
Convergence Study
At KC=1
Comparison of different size of time step with KC=1
Comparison of different numbers of grid in horizontal direction
20
dT=0.001
dT=0.0005
15
2
Non-Dimensional Force, F/  U H
15
2
250x100
200x100
100x100
2
Non-Dimensional Force, F/  U H
2
20
10
5
0
-5
-10
-15
-20
0
0.25
0.5
Time Period
0.75
1
10
5
0
-5
-10
-15
-20
0
0.25
0.5
Time
0.75
1
Force History comparison
Force over 1 time period acting on flat plate
--- using Euler solver and Navier-Stokes solver
15
2
Non-Dimensional Force, F/  U H
2
20
Navier-Stokes Solver
Euler Solver
10
5
0
-5
-10
-15
-20
0
0.25
0.5
Time Period
0.75
1
Drag coefficient for a range of
KC (0.5 < KC < 10)
30
Euler
Navier-Stokes
Sarpkaya(1995)
25
Cd
20
15
10
5
0
2
4
Keulegan-Carpenter Number
6
8
10
Inertia coefficient for a range
of KC (0.5 < KC < 10)
4
3.75
3.5
3.25
3
Euler
Navier-Stokes
Sarpkaya(1995)
2.75
2.5
Cm
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
1
2
3
4
Keulegan-Carpenter Number
5 6 7 8 910
Streamline & vorticity contour at 0*T/4 for KC=1
2
2
ES
u = -U m
1.8
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.5
Y
1.25
1
0.75
0.5
0.25
0
-1
-0.5
0
0.5
NS
u = -U m
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.6
1.4
1.2
Y
1.75
1
0.8
0.6
0.4
0.2
0
-1
1
-0.5
0
0.5
1
X
X
Streamline & vorticity contour at 1*T/4 for KC=1
2
1.8
ES
u=0
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.5
Y
1.25
1
0.75
0.5
0.25
0
-1
-0.5
0
X
0.5
1
NS
u=0
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.6
1.4
1.2
Y
1.75
2
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
X
0.5
1
Streamline & vorticity contour at 2*T/4 for KC=1
2
ES
u = Um
1.8
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.6
1.4
Y
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
u = Um
NS
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.6
1.4
1.2
Y
1.8
2
1
0.8
0.6
0.4
0.2
0
-1
1
-0.5
X
0
0.5
1
X
Streamline & vorticity contour at 3*T/4 for KC=1
2
1.8
ES
u=0
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.6
1.4
Y
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
X
0.5
1
NS
u=0
30.00
27.00
24.00
21.00
18.00
15.00
12.00
9.00
6.00
3.00
0.00
-3.00
-6.00
-9.00
-12.00
-15.00
-18.00
-21.00
-24.00
-27.00
-30.00
1.6
1.4
1.2
Y
1.8
2
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
X
0.5
1
KC=1
KC=10
Summary
 A finite volume method was applied to the
solution of oscillating flow around a vertical flat
plate. The same plate and tunnel dimensions as
in the experiment of Sarpkaya and O'Keefe
have been used.
 It was found, that either the Euler or the NavierStokes solver produced force coefficients and
separated flow fields which were close to each
other and also close to the measurements,
especially for lower values of the KeuleganCarpenter number.
Summary
 For higher KC numbers, the viscous terms start to
be more important, and the effects of turbulence
may need to be taken into account.
 The Euler solver has already been extended in the
case of FPSO hull sections, with and without
bilge keels, and some preliminary results have
been presented in Kinnas, et al., 12th Offshore
Symposium, Texas section of the SNAME, 2003.
Future Work
Further investigations with different Re numbers
and KC numbers.
Continue with the extension in the case of FPSO
hull sections to study the effects of bilge keels.
The End