Mathematical Modeling and Numerical Simulation

Mathematical Modeling and Numerical Simulation

Mathematical Modeling and Numerical
Simulations of Fluid-Structure Interaction
and Two-Phase Flows Using the Immersed
Boundary Method
Elie Luis M. Padilla, Alice R. da Silva,
Ana Lúcia F. Lima e Silva, Millena M. Villar, Alexandre M.
Roma e Aristeu da Silveira Neto
Federal University of Uberlandia
1. Examples of Industrials Flows
1. Examples of Industrials Flows
2. Methodology
Immersed boundary methodology
Domain
Eulerian
Domain
Lagrangean
2. Methodology being developed
Immersed boundary methodology
 ∂(u i ) ∂ (ui u j )
∂p
∂
+
+
ρ
=−
∂x j 
∂xi ∂x j
 ∂t
  ∂u ∂u j 
 + Fi
 µ  i +
  ∂x j ∂xi 
r r
r r r r
2 r
F ( x ) = ∑ Dij ( x − xk ) f ( xk )∆s ( xk )
k
2. Methodology
Physical Virtual Model - PVM
PVM
Partícula de fluido
r r
f (x k , t )
interface
r
xk
r r
r r
r r r
r r
∂V ( xk , t )
f ( xk , t ) = ρ
+ ρ∇ . V ( xk , t )V ( xk , t ) −
∂t
(
[(
)
)]
r r
r r
r r
T
∇ . µ ∇V ( x k , t ) + ∇ V ( x k , t ) + ∇p ( x k , t )
3. Numerical Simulations
IB with PVM
Poiseuille Flow
0.2
Eulerian force field
Re=250
0.175
0.15
Analítico
Numérico
0.125
f
-5.00
-7.86
-10.71
-13.57
-16.43
-19.29
-22.14
-25.00
-27.86
-30.71
-33.57
-36.43
-39.29
-42.14
-45.00
0.1
0.075
0.6
0.05
parede virtual
0.5
0.025
0.4
y (m)
0
Numérico
Analítico
0.3
parede virtual
1
2
3
400
600
Velocity – analytique – erreur
max = 0,1 %
0.1
0
200
Re
0.2
0
0
4
u (m/s)
5
6
7
Re=250
800
1000
3. Numerical Simulations
Immersed bodies flows
10.0
7.5
Cd
17.0
30
1.0
d
Lima e Silva et al. (2003)
16.5
Sucker (White, 1991)
16.5
0
y
d
Tomtika (White, 1991)
Roshko (1961)
16.0
Sampaio (2000)
x
d
0.1
1.0E+0
7.0
15
7.5
8.0
1.0E+1
1.0E+2
1.0E+3
Re
1.0E+4
1.0E+5
1.0E+6
3. Numerical Simulations
Immersed bodies flows
2
1
0
-1
Cd1
Cd2
-2
0
50
100
150
tU/d
200
250
3. Numerical Simulations
Fluid-Structure Interaction – 3D
3. Numerical Simulations
Taylor-Couette Flow
Ta=100, 42x42x24,
3. Numerical Simulations
Taylor-Couette Flow – Eccentric Mov.
Ta=100, 42x42x24,
3. Numerical Simulations
Taylor-Couette Flow – Eccentric Mov.
3. Numerical Simulations
Taylor-Couette Flow – Eccentric Mov.
3. Numerical Simulations
Taylor-Couette Flow – Eccentric Mov.
4. Two phase flows
4. Two phase flows
4. Two phase flows
4. Two phase flows
Eo=10 e M=10-9
64 x 192
L5
4. Two phase flows
Eo=10
M=10-9
64 x 192
L5
4. Two phase flows
Multi bubbles
Eo=1 e M=10-2 64x128 L5
4. Two phase flows
Multi bubbles
4. Two phase flows