DNS-based Multi-Modal Decomposition of VIV*

D. Lucor1, Harish Mukundan2 & M. Triantafyllou2

1 Laboratoire de Modélisation en Mécanique, UPMC Paris VI.

2 Department of Mechanical Engineering, MIT.

4th Symposium on Bluff Body Wakes & Vortex-Induced Vibrations!!

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* Supported by BP & ONR

Basic Idea:

Calculate/measure riser response at several locations (j=1, 2, … J) - sufficient to resolve all modes - and for several time steps (k=1, 2, …K).

Use the experimental data to accurately calculate all frequencies and modal amplitudes and phases along the riser in the presence of noise and systematic and random errors.

Modal Reconstruction from CFD or Full-Scale/Experimental Data

Formulation

Formulation (continued)

Formulation (continued)

Example 1: Traveling monochromatic wave of given ω

Time

Phase angle:

Modal amplitude:

Span

Span

Unwrapped Phase angle (first 19 odd modes)

Modal amplitude (first 19 odd modes)

and:

l(t, zj) = Re{ N∑

m=1

eiωmtψm(zj)}

. (14)

We derive velocity and acceleration of the structure as:

y(t, zj) = Re{∑N

n=1 iωneiωntφn(zj)}

y(t, zj) = Re{∑N

n=1−ω2neiωntφn(zj)

} (15)

The total lift in phase with velocity Clv is defined as:

Clv(zj) =1T

∫ T0 y(t, zj)l(t, zj)dt√1T

∫ T0 y2(t, zj)dt

(16)

Let us consider the numerator first:

1T

∫ T

0y(t, zj)l(t, zj)dt =

1T

∫ T

0Re

{ N∑n=1

iωneiωntφn(zj)}

Re{ N∑

m=1

eiωmtψm(zj)}

dt (17)

Making use of the following equalities for the complex numbers (z1, z2, z3):

Re{z1}Re{z2} = 14(z3 + z3) with z3 = z1z2 + z1z2

Re{z1} = 12(z1 + z1)

(18)

and dropping the ei(ωn+ωm)t terms which become negligible for large enough T, we are left with:

1T

∫ T0 y(t, zj)l(t, zj)dt = 1

4T

∫ T0

[ ∑Nn=1 iωneiωntφn(zj)

∑Nm=1 e−iωmtψm(zj)

+∑N

n=1 iωneiωntφn(zj)∑N

m=1 e−iωmtψm(zj)]dt

= 12T

∫ T0 Re

{∑Nn=1 iωneiωntφn(zj)

∑Nm=1 e−iωmtψm(zj)

} (19)

All terms vanish except when m = n. In this case:

1T

∫ T0 y(t, zj)l(t, zj)dt = 1

2T

∫ T0 Re

{∑Nn=1 iωnφn(zj)ψn(zj)

}dt

= 12

∑Nn=1 Re

{iωnφn(zj)ψn(zj)

}= 1

2

∑Nn=1 ωn|φn(zj)||ψn(zj)|Re

{iei(θn(zj)−δn(zj))

}= 1

2

∑Nn=1 ωn|φn(zj)||ψn(zj)| sin (δn(zj)− θn(zj))

(20)

where |ψn(zj)| =√

ψ2nRe

(zj) + ψ2nIm

(zj) and δn(zj) = tan−1 ψnIm (zj)ψnRe (zj)

.

Similarly,1T

∫ T

0y2(t, zj)dt =

12

N∑n=1

ω2n|φn(zj)|2. (21)

6

Formulation

Therefore, the total Clv becomes:

Clv(zj) =1√2

∑Nn=1 ωn|φn(zj)||ψn(zj)| sin (δn(zj)− θn(zj))√∑N

n=1 ω2n|φn(zj)|2

. (22)

We now define the component Clvn associated with the mode n such that:

Clvn(ωn, zj) =ωn|φn(zj)||ψn(zj)| sin (δn(zj)− θn(zj))

ωn|φn(zj)| = |ψn(zj)| sin (δn(zj)− θn(zj)) (23)

The total lift in phase with acceleration Cla is defined as follows:

Cla(zj) =1T

∫ T0 y(t, zj)l(t, zj)dt√1T

∫ T0 y2(t, zj)dt

(24)

Following a very similar derivation as previously described, we express the total Cla as:

Cla(zj) =1√2

∑Nn=1−ω2

n|φn(zj)||ψn(zj)| cos (δn(zj)− θn(zj))√∑Nn=1 ω4

n|φn(zj)|2. (25)

We now define the component Clan associated with the mode n such that:

Clan(ωn, zj) =−ω2

n|φn(zj)||ψn(zj)| cos (δn(zj)− θn(zj))ω2

n|φn(zj)| = −|ψn(zj)| cos (δn(zj)− θn(zj)) (26)

4 Added mass coefficient

The added mass coefficient Cm is a non-dimensional parameter that is the ratio between the liftin force with the acceleration of the structure and a quantity that involves the acceleration of thestructure itself. We have the lift force in phase with the acceleration which is equal to:

−Cm(π

4ρd2)

√√√√ N∑n=1

ω4n|φn(zj)|2 (27)

If we express Cm as a function of Cla, we have:

Cm = − 2π

Cla√∑Nn=1 ω4

n|φn(zj)|2(28)

Then, we can express the added mass coefficient Cmn associated with each mode n. We define:

Cmn = − 2π

Cla

ω2n|φn(zj)| (29)

7

Formulation (continued)

Fluid-Structure Model used in NEKTAR

Structure: Linear model for beam and/or cable

Flow: 3D Incompressible Navier Stokes

Flow and structural parameters used in the runs

Flow and Structural Parameters used in the runs

FLOW Uniform cases: Case1-3

Linear shear: Case1-3

Exponential shear: Case1-3

GoM shear: Case1-3

Flow Regime Turbulent Regime Turbulent Regime Turbulent Regime Turbulent Regime

Inflow profile Flat Linear Exponential with long tails

Severe Exponential

STRUCTURE Beam/Cable Beam/Cable Beam/Cable Beam/Cable

Boundary conditions

Pinned and hinged ends

Constraints No streamwise motion (x-direction) Free Transverse motion (y-direction)

Aspect Ratio: L/d L/d = 2028 L/d = 2028 L/d = 2028 L/d = 2028 Mass Ratio: m m = 2 m = 2 m = 2 m = 2

Cable Phase Velocity: c

c = 35 c = 35 c = 35 c = 35

Beam Phase Velocity: γ

γ = 450 γ = 450 γ = 450 γ = 450

Damping: ζ ζ = 0.0% ζ = 0.0% ζ = 0.0% ζ = 0.0%

Re=1000

Umin/Umax=1 Umin/Umax=30% Umin/Umax=10%

Re=1000Re=1000

Uniform velocity profile:

Modal amplitude crossflow displacement Modal amplitude lift coefficient

Chosen set of frequenciesωn = 2π(0.2148 0.2295 0.249 0.293 0.317 0.332)

Frequency Frequency

Span

Distribution of total ClV- Full basis (256 modes)- Incomplete basis (≤10 modes)

Linear shear velocity profile:

Modal amplitude crossflow displacement Modal amplitude lift coefficient

Chosen set of frequenciesωn = 2π(0.183 0.193 0.203 0.212 0.222 0.230 0.238)

Frequency Frequency

Span

Distribution of total ClV- Full basis (256 modes)- Incomplete basis (≤10 modes)

Exponential shear velocity profile:

Modal amplitude crossflow displacement Modal amplitude lift coefficient

Chosen set of frequenciesωn = 2π(0.146 0.167 0.179 0.187 0.199 0.204 0.21 0.217 0.225 0.236)

Frequency Frequency

Distribution of total ClV- Full basis (256 modes)- Incomplete basis (≤10 modes)

Linear shear case Exponential shear caseUniform case

RMS values of crossflow response

- Full basis (256 modes)- Incomplete basis (≤10 modes)

Span SpanSpan

Clv modal decomposition

Total Clv

Velocity profile

Total ClvModes overlap!

Velocity profile

No Modes overlap!

Span

Frequency

0.183 0.193

0.203 0.212 0.222 0.230 0.238

Span

Summary

We have used 3D DNS data of the crossflow response of long flexible cylinders subject to VIV with various inflow velocity conditions. These test cases were chosen for their complex multi-frequency response that is difficult to predict a priori and can not be represented by standing waves.

We have proposed a modal decomposition technique that provides the spanwise distributions of the amplitude and the phase of the response. This representation allows the energy to be carried along the length in the structure in the form of traveling waves.

We have tempted to show that a sparse decomposition with a few modes but carrying most of the energy is possible.

Our ultimate goal is to obtain Gopalkrishnan-like maps of lift and added mass coefficients versus modal amplitude and modal frequency.