Vortex Induced Vibrations of
Multiple Cylinders
Alper Cesur
ALPER CESUR
Division of Fluid Mechanics
Lund Institue of Tecnology, Lund Sweden
OUTLINE
Intoduction
Numerical Method
Problem Setup
Results
Conclusions and Future Work
INTRODUCTION – Main Work
Purpose:
Investigate how the vortex-induced vibrations
are affected by the interaction of elastically
mounted multiple rigid cylinders, confined in
a rectangular channel.
Investigated parameters:
Elasticity
Separation distance
Staggering
INTRODUCTION – Pre Work
Wake development, by Williamson
(1997)
Flow Regimes (for a
stationary cylinder)
• Re < 47: The flow is
steady and twodimensional, comprised
of a recirculation zone.
• For Rec = 47, the
primary wake instability,
known as the unsteady
Karman vortex street
develops.
• For Re > 180, a threedimensional and
irregular wake
develops.
INTRODUCTION
Cons with vortex
formation,
Tacoma bridge (1948)
The character of vortex
formation
NUMERICAL METHOD
Governing equations;
Finite difference approach
on Cartesian staggered
Source term
grid.
Single step defect
correction applied to
improve the accruacy for
convective and remaining
terms to third and fourth
order, respectively.
Multi-grid method for
The equation of motion
for the cylinder:
VIRTUAL BOUNDARY
METHOD
VB represents boundaries
Velocity defectu:f - ub
with momentum sources,
.
1. A 2D grid is generated on
uf
ub
the object surface.
2. Boundary velocity is
calculated in each node.
3. Boundary forces are
determined based on
velocity defect.
4. Resulting forces are
Sources, based on
distributed back to flow
Gaussian average:
solver.
VIRTUAL BOUNDARY
METHOD
Why Virtual Boundary method?
No need of grid regeneration (or overlapping) as for
structured/unstructured grids for moving boundaries.
Can achieve higher computational efficiency and
accuracy.
Cons: Increased computational time due to extra
boundary condition.
PROBLEM SET-UP
Boundary Conditions:
Channel inlet:
Unconfined Cylinder
Flat velocity profile,
Channel Outlet:
Zero velocity gradient
Channel walls:
Periodic at xz-planes
Slip at yz-planes
Cylinder surface:
No slip
4D
y 4D
x
18D
z
20D
PROBLEM SET-UP
Boundary Conditions:
Confined Cylinder
Channel inlet:
Flat velocity profile,
Channel Outlet:
Zero velocity gradient
Channel walls:
No-slip
Cylinder surface:
No-slip
4D
y 4D
x
6D
z
20D
PROBLEM SET-UP
Grid Resolution, Confined
Cylinder
PROBLEM SET-UP
Effect of Added Mass
Out of lock-in, k* = 15
Lock-in, k* = 7
PROBLEM SET-UP & THEORY
What is added or virtual mass??
Inertia added to a system because of the
displasement of the volume of surrounding fluid due
to an accelerating/decelerating body.
Evident at out of lock-in at numerics.
Lock-in??
PROBLEM SET-UP - THEORY
Lock-in:
Also defined as synchronization or resonance
Classical lock-in: The frequency of a body’s
oscillations matches the system’s natural frequency.
(Has been extended)
Characterized by large amplitudes of oscillations
Added mass not noticeable
PROBLEM SET-UP - THEORY
Out of lock-in, k* = 15
Lock-in, k* = 7
A* ≈ 0.027
A* ≈ 0.578
PROBLEM SET-UP – MAIN WORK
A Group of Four Cylinders
• Boundary Conditions: According to Confined Cylinder
• Flow regime: Re = 400
• Grid resolution: d/h = 32
• No damping, i.e. b = 0 in equation of motion for cylinder
4D
7.5D
4D, 5D
Cyl_12
Cyl_11
30D
Cyl_22
Cyl_21
12D
PROBLEM SET-UP – MAIN
WORK
Elasticity/Reduced velocity effect
: 2.0 – 20
Staggering effect
Angle β: 0 – 25°, increment of 5°
Fixed z-position for the downstream
cylinders while a simultaneous displacement
in x-direction
Separation distance
L/D = 4 & 5
Quadratically displacement
β
L/D
x
RESULTS - ELASTICITY
Amplitude Vs U*
Frequency Vs U*
RESULTS - STAGGERING
k* = 15, Out of lock-in
k* = 8, Lock-in
RESULTS – SEPARATION
DISTANCE
CONCLUSIONS
Strong influence of the vortex shedding of the
upstream cylinders upon the downstream cylinders.
An amplitude of at most 80% larger than the
upstyream and single cylinder case.
Wider synchronization range for the multiple cylinders
compared to single cylinder.
The relevance of virtual mass becomes significant for
low reduced velocities, i.e. Out of lock-in.
A small and gradual increase of the oscillation
amplitudes at out of lock-in for all cylinders. At lock-in,
the downstream cylinde oscillations decrease while
the amplitudes remain for the upstream cylinders.
Shorter separation distance yields larger oscillation
amplitudes for the downstream cylinder.
FUTURE WORK
Study the case of multiple cylinders more deeply and
combine the results with two cylinders in tandem.
Continue with deformable structures, using FEM-
method.
Perform experimental simulations for both rigid and
deformable bodies.
THANK YOU