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Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 1

Modelling of acoustic cavitation on a

large scale with OpenFOAM

Sergey Lesnik, Gunther Brenner

Institute of Applied Mechanics / TU Clausthalin cooperation with University Göttingen

German OpenFoam User meetiNg, Online, 22.04.2020

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 2

Acoustic cavitation

Source: Industrial Sonomechanics, LLC

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 3

Acoustic cavitation: multiscale problem

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 4

Motivation

State of the art

fundamental physics of microscopic phenomena well

understood

macroscopic computations: only linear bubble oscillations

with homogeneous distribution

Current ansatz

non-linear cavitation bubble oscillations

spatially inhomogeneous bubble distribution

relatively large geometries (~1-10dm³)

prediction of

ultrasound field

location of cavitation bubble clustering

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 5

Outline

2. Radial Bubble

Dynamics

3. Bubble Motion

1. Ultrasound:

Helmholtz Eqn.

Model Coupling

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 6

Outline

2. Radial Bubble

Dynamics

3. Bubble Motion

1. Ultrasound:

Helmholtz Eqn.

Model Coupling

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 7

Helmholtz equation (HE)

Wave equation in frequency

domain

𝑃ac - complex sound pressure

amplitude

𝑘m - complex wave number of the

gas-liquid mixture

Computation with OpenFOAM

no complex numbers

decompose HE in two equations

solving in segregated manner

leads to divergence in most cases

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 8

HE discretization and solution

Discretized with block-coupled matrix to couple

equations implicitly (foam-extend)

The matrix of discretized HE is highly indefinite

iterative solvers diverge

=> Interface implemented to a direct solver (MUMPS)

MUltifrontal Massively Parallel sparse direct Solver

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 9

Outline

2. Radial Bubble

Dynamics

3. Bubble Motion

1. Ultrasound:

Helmholtz Eqn.

Model Coupling

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 10

Radial bubble dynamics (RBD)

Toegel model: 3 ODEs Keller-Miksis eqn. (𝑅 – bubble radius)

energy transfer (𝜃 – temperature)

mass (vapor) transfer (𝑛 – amount of substance)

Source: University of Göttingen, Drittes Physikalisches Institut

R

Time period

𝑇 = 50µs(𝑓 = 20kHz)

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 11

Coupling non-linear RBD and sound field

Coupling via 𝑘m (Louisnard model)

𝛽 – void fraction / bubble density

ΠVi,Th – integrals over one oscillation period; physically:

energy dissipated per bubble;

ΠVi,Th indirectly dependent on 𝑃ac

100cm³ reactor and 𝛽 = 10−5 => 2.3e+6 bubbles

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 12

Coupling non-linear RBD and sound field

Approach as pre-processing step:

1. choose parameter range for 𝑃ac2. solve RBD (implemented in python)

3. compute integral values and save as interpolation tables

RBD

• 1D Model

• Python

Tabulated

integrals

Helmholtz eqn.

• Finite Volumes

• OpenFOAM

Integrate

over 𝑇

Look up 𝑘m2 𝑃ac

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 13

Coupling non-linear RBD and sound field

Iterative process

highly non-linear, under-relaxation not sufficient

damped Newton-Raphson method implemented

jacobian with numeric differentiation

RBD

• 1D Model

• Python

Tabulated

integrals

Helmholtz eqn.

• Finite Volumes

• OpenFOAM

Integrate

over 𝑇

Look up 𝑘m2 𝑃ac

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 14

Boundary conditions

Sonotrode immersed in a

cylindrical geometry

typical setup also for large scale

reactors

axisymmetric

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 15

Free surface Sound soft 𝑃ac = 0

Sonotrode surface In-phase

displacement 𝑈0

𝛻𝑃ac~𝑈0

Sonotrode wall Anti-phase

displacement 𝑈0

𝛻𝑃ac~ 𝑈0, 𝜙0

Symmetry axis Symmetry axis Empty

Walls Sound hard 𝛻𝑃ac = 0

Free surface Sound soft 𝑃ac = 0

Boundary conditions

Sonotrode immersed in a

cylindrical geometry

typical setup also for large scale

reactors

axisymmetricsound soft

sound hard

symmetry

axis

Geometry Acoustics Numerics

Symmetry axis Symmetry axis Empty

anti-phase

displacement

in-phase

displacement

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 16

Linear vs. non-linear bubble oscillations

Linear Non-linear

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 17

Linear vs. non-linear bubble oscillations

Linear Non-linear

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 18

Outline

2. Radial Bubble

Dynamics

3. Bubble Motion

1. Ultrasound:

Helmholtz Eqn.

Model Coupling

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 19

Bubble motion

Euler-Lagrange (foam-extend)

Force balance

𝑚b, 𝑈b- bubble mass and velocity

Forces:

𝐹G - gravitation

𝐹Am - added mass

𝐹D - drag

𝐹Bj - Bjerknes, due to interaction of non-linear oscillation

and acoustic pressure gradient

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 20

OpenFOAM

Coupling bubble motion

Bjerknes force contains bubble volume term averaged

over 𝑇

RBD

• 1D Model

• Python

Tabulated

integrals

Helmholtz eqn.

• Euler-Euler

Integrate

over 𝑇

Look up 𝑘m2 𝑃ac

Bubble Motion

• Euler-Lagrange

Look up < 𝑉b >𝑇 𝑃ac

Interpolate 𝛻𝑃ac

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 21

1D case

Bjerknes force

Stagnation locations

sound softdisplacement = 5µmx

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 22

1D case inhomogeneous void fraction

Void fraction kept constant at transducer (on the right)

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 23

Coupling Liquid motion

OpenFOAM

RBD

• 1D Model

• Python

Tabulated

integrals

Helmholtz eqn.

• Euler-Euler

Momentum

transfer

Bubble Motion

• Euler-Lagrange

Interpolate 𝛻𝑃ac

Liquid Motion

• Euler-Euler

• URANS

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 24

Liquid and bubble motion

2D axisymmetric wedge case

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 25

Summary

Computation of cavitation flows in large scale reactors

apply different models to different scales

coupling needs caution

Validation of sub-models with the data from

experiments

Nucleation process needs more consideration

where do bubbles nucleate and dissolve?

Sergey Lesnik, Gunther Brenner Modelling of acoustic cavitation on a large scale with OpenFOAM 26

Source code for Helmholtz solver (MUMPS interface):

https://github.com/technoC0re

Questions?