1 Reduction in turbulence -induced non- linear dynamic vibration using tuned liquid damper TLD

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Reduction in turbulence-induced non-linear dynamic vibration using tuned liquid

damper (TLD)

Ananya Majumdar1, Biplab Ranjan Adhikary1 and Partha Bhattacharya1

1Department of Civil Engineering, Jadavpur University, Kolkata-700032, India

ABSTRACT

In the present research work, an attempt is made to develop

a coupled non-linear turbulence-structure-damper model in a

finite volume-finite difference (FV-FD) framework. Tuned

liquid damper (TLD) is used as the additional damping system

along with inherent structural damping. Real-time simulation of

flow-excited bridge box girder or chimney section and the

vibration reduction using TLD can be performed using the

developed model. The turbulent flow field around a structure is

modeled using an OpenFOAM transient PISO solver, and the

time-varying drag force is calculated. This force perturbs the

structure, causing the sloshing phenomena of the attached TLD,

modeled using shallow depth approximation, damping the

flow-induced vibration of the structure. The structural motion

with and without the attached TLD is modeled involving the

FD-based Newmark-Beta method using in-house MATLAB

codes. The TLD is tuned with the vortex-shedding frequency of

the low-Reynolds number flows, and it is found to be reducing

the structural excitation significantly. On the other hand, the

high-Reynolds number turbulent flow exhibits a broadband

excitation, for which by tuning the TLD with few frequencies

obtained through investigations, a good reduction in vibration

is observed.

Keywords: Turbulence-structure interaction, tuned liquid

damper, finite difference, OpenFOAM.

1. INTRODUCTION

Turbulence proves its’ prominent presence in our

surroundings, which can adversely affect the important

structures like bridges, high-rise buildings or chimneys. The

excessive flow-induced vibration even leads to collapse. One of

such an iconic example is Tacoma bridge collapse and a very

recent, the under-construction bridge collapse in Sultangunj,

India on 29th April 2022.

In order to reduce the turbulence-excited structural motion

it is important to properly model the flow field surrounding the

structure. Next, the structural motion is to be estimated with and

without a TLD attached to it.

To obtain the desired reduction in the excitation level of the

structure, several active and passive damping techniques are

used by researchers and engineers. One of the widely used

passive damping instruments is TLD, which is essentially tuned

to the first fundamental frequency [1,2] or first two fundamental

frequencies [3] of the vibrating structure for the best damping

experience. In the case of harmonic excitation and non-

deterministic forcing with a prominent frequency value, the

TLD is tuned to the excitation frequency [4]. In case of

turbulent flow, the behavior of TLD and possible tuning

frequency is not studied yet.

2. LITERATURE REVIEW AND OBJECTIVE

Many researchers have investigated the flow past bluff body

for different Reynolds numbers (Re) using numerical [5] or

experimental techniques [6-8]. The Strouhal number (St) and

drag coefficient (C) are evaluated and reported. Structural

excitation due to any random forcing is estimated using the FD-

based non-linear Newmark-Beta method in time domain [4].

The sloshing behavior of TLD and resulting structural response

reduction is estimated numerically and/or experimentally [1-4]

using harmonic or random earthquake ground acceleration.

However, the coupled non-linear model to capture the

turbulence-induced structural motion and its reduction using

TLD is non-existent.

Therefore, in the present research work, an attempt is made

to develop a numerical FV-FD-based non-linear model that will

estimate the turbulence forcing, and simultaneously at each

time step, it will capture the sloshing-induced base shear

produced by the TLD liquid at the TLD-structure junction,

eventually producing additional damping to the SDOF system.

3. METHODOLOGY

The entire work is subdivided into two subsections, each

consisting of modelling a part of the full solution technique.

a) Modelling the turbulent flow-field past a rigid obstacle

and estimation of the drag force applied to it.

b) Using the time-varying turbulent forcing to estimate

the structural response at each time step with and

without attached TLD.

3.1 Turbulence past a rigid obstacle

Unsteady or transient simulation for isotropic turbulence is

performed using Large Eddy Simulation (LES), which

essentially works based on filter operation. Eddies larger than a

certain length scale, typically in the order of the grid size, are

fully resolved. The smaller eddies are modelled using a sub-grid

scale model. As the turbulence is considered to be isotropic,

only the size of these smaller eddies becomes important, not the

shape. Once the mean velocity field is computed by the RANS

model, the fluctuating velocity component (u) can

conceptually be estimated by subtracting time-averaged mean

velocity U

, from instantaneous velocity, U. This fluctuating

velocity is then used to calculate the turbulent kinetic energy

(k) per unit mass, as k = 

uu. If all the fluctuating

components in three directions are considered, the Reynolds

stress tensor can be estimated (per unit density) in symmetric

matrix form, and in 3D and 2D domains the resolved kinetic

energy becomes

k =

[(u)



+ (v)



+ (w)



] (1)

k =

[(u)



+ (v)



]

(2)

The amount of the remaining kinetic energy is termed sub-grid

scale kinetic energy, k, and calculated by a sub-grid eddy

viscosity model. In the present study, the Smagorinsky-Lilly

model is used, where an additional stress term () is applied

to break down the eddies larger than the mesh size because

molecular viscosity is not sufficiently strong to do so. This

stress term can be derived by applying filtering operation on

compressible Navier-Stokes equations as follows:



+

(



)





= 0 (3)

(



)

 +

(







)





=













 +

(4)

This sub-grid stress  is modelled using Eq. (5)

 = 2



 (5)



=

2





























(6)

Assuming the profile to be linear within the viscous

sublayer, and obeying the 1/7th power law in the outside region,

the wall function is formulated as described by Germano et al.

(1991).

()=

  <11.8

8.3()/  >11.8

(7)

where non-dimensionalized velocity (U) and wall distance

() are given as

=



,=









(8)

Lilly (1966) proposed a value for the Smagorinsky constant

C as 0.173, considering turbulence to be homogeneous and

isotropic, which is true for a shear-free turbulent event far from

any wall. However, modern CFD codes typically use different

values of C for near-wall turbulent events. In the present wall-

function approach (SLWF) Smagorinsky-Lilly constant, C is

considered as 0.1 to determine the sub-grid scale kinematic

eddy viscosity.

In the Pressure-Implicit with Splitting of Operators (PISO)

algorithm, steady-state flow problems can be solved using the

LES model with a pressure corrector.

The coefficient of drag is given by

=





 2

(9)

FD: Drag Force, A: Projected area

3.2 Sloshing of tuned liquid damper (TLD)

At any point wave height is h. Assuming shallow wave

theory to be valid and no point of time wave is reaching the tank

top, partial differential equations for the sloshing motion of

liquid can be written as [4],



+



+



= 0 

21 (10)





+





+





()+





= 0

(11)

Where, u is the displacement of the structure at any point of

time, thus, displacement of the TLD liquid surface, as it is

attached to the structure.  is the rotational displacement, h

is the wave height at location x and time t, v is the particle

velocity at location x and time t. The boundary conditions are

given by:

(0, )=(,)= 0

(12)

Initial conditions: At the starting of time (t = 0) steady state

condition is considered,

(, 0)=; (. 0)= 0

(13)

Non-dimensional slope of the energy gradient line is written as,

=







(14)

The wall shear stress at the base of the tank,  is given by,

=











, 0.7 (15)











, > 0.7

(16)

 is the forcing frequency, =

;  is the absolute

viscosity of fluid, here water. Sloshing force (F) acting on the

walls of the water tank is given by:

= 0.5(



)+





(17)

,: Wave height w.r.t bottom of the tank at the right and left

wall of the tank. : Density of fluid, here water.

The sloshing frequency of the tank liquid is given as,

= 1

2

tanh(

)



(18)

 = Fundamental natural frequency of liquid sloshing

L = Tank length

3.3 Solving non-linear partial differential equations

The non-linear partial differential equations used to

describe tuned liquid damper system discussed in the previous

section are solved in iterative finite difference technique. Any

general function f can be written as,

=+(1)





+



(19)

The function can be velocity (v), wave height (h), or slope of

the energy gradient line (S). In this numerical approach space

derivatives are estimated as,













2

(20)

for a particular time step.  is the element dimension. ‘i’ is the

node number. Time derivatives are estimated as,

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1Reductioninturbulence-inducednon-lineardynamicvibrationusingtunedliquiddamper(TLD)AnanyaMajumdar1,BiplabRanjanAdhikary1andParthaBhattacharya11DepartmentofCivilEngineering,JadavpurUniversity,Kolkata-700032,IndiaABSTRACTInthepresentresearchwork,anattemptismadetodevelopacouplednon-linearturbulence-str...

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1 Reduction in turbulence -induced non- linear dynamic vibration using tuned liquid damper TLD

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