Vortex Induced Vibration and Fluid Structure Interaction of Viscoelastic plate

1. OUTLINE
1
3.
Results
2.
Study of
viscoelastic
splitter plate
behind the
cylinder
4.
Conclusions
and Future
Work
1.
Work
Presented in
APS I and II

2. 2
Work Presented in
APS-I and II
1

3. 3
Study of vortex-induced-vibration(VIV) of
a viscoelastic splitter plate behind a
cylinder
WORK PRESENTED IN APS-I

4. 4
 Vortex induced vibration (VIV) is suppressed when the splitter plate
length is large
 The VIV decreases with decreasing of flexibility of splitter plate at
critical length.
 The investigations on the effect of VIV using splitter plate
composed of viscoelastic material is still untouched.
 To study the effect on VIV using linear viscoelastic model (standard linear
solid) of splitter plate
 To perform the simulation using nonlinear viscoelastic models (Reese-
Govindjee Model)

5.  Linear Viscoelastic models
5
WORK PRESENTED IN APS-I I
 Maxwell Body:
 Voigt Body:
 Standard Linear Solid:
η E
1 0 1
d d
p q q
dt dt
 
 
  
1 0 1
p , 0,
q q
E


  
1 0 1, 1
p 0,q E q 
  
1
1 0 1, 1
2 2
p , (1 )
E
q E q
E E


   

6.  Study of static and dynamic behaviour of
Viscoelastic material
 Validation of creep and relaxation using structure
solver (Tahoe) with derived analytical solution
 Comparison of 2D Elastic and Viscoelastic splitter
plate behind cylinder
6
WORK PRESENTED IN APS-I I

7.  Applications
 Literature Review
 Objectives of the present work
 Finite Strain Viscoelastic
7
Study of Viscoelastic splitter plate behind the cylinder
2

8. Snoring
Printing process
8
Auregan et al. 1995
Watanabe , 2002
FPCL, JHU
Phonation
Piezoelectric Energy
Harvesting
InTechOpen

9. • Chen et al. (2011). Analyzed the flutter . Either increasing the
structure-to-fluid ratio or decreasing dimensionless bending
stiffness ( )causes the system transits from periodic to
chaotic.
• Chen et al.(2014). Increase of either elastic or viscous component
of flag stabilizes the system.
• Tang et al. (2007) considered Kelvin-Voigt model , flutter amplitude
decreases with the increase of the material damping.
• Purohit et al. (2016). At lower flexibility forcing field of flowing field
dominates the vibration plate, as the plate stiffer plate response is
effected by elastic inertial force.
9
2 3
EI
u L


10. • The investigations of effect of flag material finite strain
viscoelastic model is scarce.
• Few studies has been done using Kelvin –Voigt model. Standard
linear solid model is very few.
• To study the stability of viscoelastic splitter plate attached behind
the cylinder.
• To study the effect on vibration amplitude with viscoelastic
parameters
10

11. Based on the multiplicative decomposition of deformation gradient into elastic and inelastic part
In thermodynamic equilibrium spring of maxwell element is relaxed , ,hence
Stress
For equilibrium and non equilibrium parts of material neo-Hookean model has been implemented
11
Second Piola-Kirchho stress tensor, 2
is strain energy density function, E Lagrangian strain tensor, C right Cauchy-Green deforformation tensor
ij
ij ij
ij ij
W W
S
E C
W
 
 
 
2
2 /3 2 2 2 2 2 2 2 2 2 2 2 2
1 1 2 3 2 1 2 2 3 1 3 3 1 2 3
2 2 2
1 2 3
W= ( 3) ( 1 1ln( )
4
, shear and bulk modulus, , invarients as I , ,
, , is Eigen values (principal stretch ratios) of Green deform
i
i i
k
I J J
k I J I I I I

            
  

   
       
2 2 2
1 2 3
ation tensor Cij, J=  
2 2
1 2
1 1
Split of free energy, ( ) ( )
2 2
e eq neq e
E E
   
      
0
e
  ( )
eq 
  
2
The ansatz, ( ) ( ), elastic right Cauchy-Green tensor (loosly speaking strain E )
eq neq e e
C C C
    
1
2 +2F F = , F deformation gradient of dashpot in fig
eq neq T
eq neq i i i
e
S S S
C C C
 
  
  
  

12. 12
Results
3
 Free vibration of cantilever beam under time varying sinusoidal l
 Vortex induced vibration of viscoelastic splitter plate behind a cy

13. 13
 P1=7.125 N, P2=6.84 ( 50 nodes with 0.285 per node, p1 25, p2 24 nodes)
 L=10 m , f(t) = sin(0.2t), density 10 kg/m3
 Viscoelastic parameters
 Kneq, Keq are bulk modulus corresponding to E2 and E1
 Sneq, Seq are shear modulus for E2, E1 in N/m2
 Relaxation time (sec)
 A: tip (point A) vibration amplitude
 Standard linear solid
2
/ E
 


14. cas
e
Keq Kneq Seq Sneq R.
time
A
1 2333.33 2333.33 500 500 0.1 9
2 2333.33 2333.33 500 500 1 8
3 2333.33 2333.33 500 500 2 6.6
4 2333.33 2333.33 500 500 10 4.7
5 2333.33 2333.33 500 500 100 4.5
6 2333.33 233333.33 500 50000 0.1 2.85
7 2333.33 233333.33 500 50000 1 0.3
8 2333.33 233333.33 500 50000 10 .07
9 233333.33 2333.33 50000 500 1 0.5
Variation of amplitude with relaxation time

15. Variation of amplitude with relaxation time

16. cas
e
Keq Kneq Seq Sneq R.
time
A
1 2333.33 2333.33 500 500 0.1 9
2 2333.33 2333.33 500 500 1 8
3 2333.33 2333.33 500 500 2 6.6
4 2333.33 2333.33 500 500 10 4.7
5 2333.33 2333.33 500 500 100 4.5
6 2333.33 233333.33 500 50000 0.1 2.85
7 2333.33 233333.33 500 50000 1 0.3
8 2333.33 233333.33 500 50000 10 .07
9 233333.33 2333.33 50000 500 1 0.5
× 100
Effect of Kneq and Sneq on amplitude
Amplitude decreases with increase of
relaxation time

17. cas
e
Keq Kneq Seq Sneq R.
time
A
1 2333.33 2333.33 500 500 0.1 9
2 2333.33 2333.33 500 500 1 8
3 2333.33 2333.33 500 500 2 6.6
4 2333.33 2333.33 500 500 10 4.7
5 2333.33 2333.33 500 500 100 4.5
6 2333.33 233333.33 500 50000 0.1 2.85
7 2333.33 233333.33 500 50000 1 0.3
8 2333.33 233333.33 500 50000 10 .07
9 233333.33 2333.33 50000 500 1 0.5
× 100
Effect of Keq and Seq on amplitude

18.  Amplitude decreases with increase of modulus
 Structure vibrates with the forcing frequency
18

19. 19
case Keq Kneq Seq Seq R. time A
1 2333.33 2333.33 500 500 0.1 2.17
2 2333.33 2333.33 500 500 1 2.25
3 2333.33 2333.33 500 500 2 2.98
4 2333.33 2333.33 500 500 10 2.89
5 2333.33 2333.33 500 500 20 2.96
6 2333.33 2333.33 500 500 100 2.97
 Amplitude almost constant
 Structure vibrates with forcing frequency
 (w=0.2, f=.05)

20.  Time to achieve steady state decreases with increase of Relaxation time.
 ( relaxation time =100 has different behavior)
20

21.  Re=100,

21
case Keq Kneq Seq Seq Relaxation
time
A
1 2333.33 2333.33 500 500 0.1 1.02
2 2333.33 2333.33 500 500 1 0.18
3 2333.33 2333.33 500 500 2 0.53
4 2333.33 2333.33 500 500 10 0.6
5 2333.33 2333.33 500 500 100 0.9
6 233.33 233.33 50 50 1 chaotic
7 2333.33 233333.33 500 50000 1 No vibration
8 2333.33 23333.33 500 5000 1 No vibration
/ 10, 1400, 0.4
s f E
  
  
Amplitude is not monotonic with Modulus as here fluid and
structure both play role

22. 22
tau=2
tau=100
tau=1
tau=0.1
Time for steady state is not
monotonic, increases and then
tau=10

23. 23

24. Symmetry breaking bifurcation which oscillates in upper or lower part of cylinder wake
24
Case 6, chaotic Case 7,

25.  Amplitude increases with increase of relaxation time
25
case Keq Kneq Seq Sneq R. time A
1 2333.33 2333.33 500 500 0.1 .017
2 2333.33 2333.33 500 500 0.5 0.00005
3 2333.33 2333.33 500 500 1 0.19
4 2333.33 2333.33 500 500 2 .47
5 2333.33 2333.33 500 500 10 0.00005
6 2333.33 2333.33 500 500 20 0.6
7 2333.33 2333.33 500 500 100 0.7
8 233.33 233.33 50 50 1 chaotic
9 233.33 23333.33 50 5000 1 No vibration
10 2333.33 233333.33 500 50000 1 No vibration
11 4666.66 4666.66 1000 1000 1 No vibration
12 2333.33 23333.33 500 5000 1 No vibration
13 2333.33 4666.66 500 1000 1 0.48

26. 26
Steady state time increases with relaxation then further increasing the
relaxation time steady state time decreases

27. 27
Damped to small amplitude vibration

28. 28
Conclusions and Future Work
5

29. For constant modulus
 amplitude decreases with increase of viscosity
 Steady state time decreases with decrease of viscosity
 Steady state time increases with relaxation then further increasing the
relaxation time steady state time decreases
 Amplitude increases with increase of relaxation time
29

30.  The four dimensionless governing parameters for viscoelastic FSI are structure-to-
fuid density ratio, Young modulus (equilibrium and non equilibrium) , viscosity of
structure
 Current study structure-to-fuid density ratio kept constant. Its effect also need to be
investigated.
 Solving an FSI problem contrasting the behavior of hyperelastic vs. viscoelastic
solid using interaction of two splitter plates mounted on cylinders in cross flow.
As in hyperelastic vortex of one plate helps to enhance the amplitude and vice
versa. Its effect when damping is present in structure needs to be studied.
30

31.  THANK YOU
31