COMSOL CONFERENCE 2016 MUNICH, 12-14 OCTOBER 1

Numerical Modeling and Validation Concept forAcoustic Streaming Induced by Ultrasonic Treatment

Donato Rubinetti1 Daniel A. Weiss1 Jonas Muller2 Arne Wahlen2

Abstract—Acoustic streaming (AS) denotes the effect where aflow in a fluid is driven by the absorption of a sound wave.In order to have a significant effect in practice, the sound waverequires a high frequency (typically order of ultrasound) and highamplitude. In metal processing industry this treatment is appliedto obtain grain morphology adjustments during the solidificationof metal. Improvement and further development of this techniquefocus on numerical modeling to reduce substantial costs for testrigs and field tests.This study presents a numerical model which takes into accountacoustics and fluid dynamics. An oscillating sonotrode is placedinto the fluid, emitting high-amplitude into the fluid. The distri-bution of the resulting sound field induces a body force describedby an additional source term in the momentum balance of thefluid.The results have been experimentally validated with a small-scalelaboratory model using water, seed oil and glycerin as samplefluids. By adjusting material properties a variety of fluids suchas molten metal can be simulated. The concept presented for thenumerical modeling of AS is numerically stable and appropiate.It can be adapted to related applications involving sound drivenfluid motion.

Keywords—Acoustic Streaming, Attenuation, Pressure Acoustics,CFD, Time-Averaged Variables.

I. INTRODUCTION

Metal processing industry applies the acoustic streamingultrasonic treatment for grain morphology adjustments duringthe solidification of metal. By placing a sonotrode as soundemitter operating at high frequencies (typically ∼20kHz) into aliquid material a steady fluid motion is achieved. Research andfurther development of the AS treatment focus on numericalmodeling which poses a multiphysics problem. It includesthe challenging coupling of high-frequency acoustics andfluid dynamics with timescales significantly below the onesof acoustics. The purpose of this study is to conceive anexperimentally validated numerical model for adaptation toother fluids by changing the material parameters.

II. PHYSICAL MODEL

The physical model is based on the inhomogeneousHelmholtz equation for the acoustic part and on incompressibleNavier-Stokes equations for the fluid dynamics. By segregating

1Institute of Thermal and Fluid EngineeringUniversity of Applied Sciences Northwestern Switzerlande-mail: donato.rubinetti@fhnw.ch, daniel.weiss@fhnw.ch

2Institute of Product and Production EngineeringUniversity of Applied Sciences Northwestern Switzerlande-mail: jonas.mueller@fhnw.ch, arne.wahlen@fhnw.ch

the frequency domains into high-frequency for acoustics andlow-frequency for fluid dynamics compressible and incom-pressible studies are applicable. This circumstance is com-monly misused in literature. The acoustics is calculated withina time-harmonic study which yields complex-valued results.The governing equation for the acoustics writes

−ω2

a2p+ h.o.t = ∇2p+

4

3

iωµ

a2ρ0∇2p (1)

with ω being the angular frequency, a the speed of sound, thedependent variable p for pressure, h.o.t for higher order terms,dynamic viscosity µ and ρ0 being the constant component ofthe fluid density.

The fluid dynamic part of this multiphysics application canbe evaluated with a common CFD tool. To link the acousticpressure field to the flow field an intermediate step is takenthrough the following force term [1], [2]

F =1

2Re

[ρ∗eωu

]+

1

2ρe

(Re

[u∗r∂u

∂r

]+Re

[u∗z∂u

∂z

])(2)

with ρ∗e being the complex conjugate of density perturbation,u the particle velocity and its components ur, uz in cylindricalcoordinates. The model is simplified assuming isothermalbehavior, neglecting cavitation and turbulent effects. The latteris sensitive in terms of physical accuracy due to widely spreadReynolds-numbers. Nonetheless it improves numerical stabilitysignificantly. Density and pressure perturbations are coupledin the acoustics part which requires a compressible fluiddescription. In the stationary flow evaluation compressibilityhas no relevance.

III. NUMERICAL MODEL

The following numerical model is created using COMSOLMultiphysics 5.2. For the acoustics part the Pressure AcousticsFrequency Domain-interface is selected. The fluid flow isdefined by the Laminar Flow-interface.

A. General SetupAn 2D-axisymmetric test case, to match the experimental

setup, is chosen according to figure 1. The simulation isstructured within three studies as follows• Study 1, Acoustics - Calculates the acoustic pressure

field. The governing equation contains an additionaldipole source term for attenuation;

• Study 2, PDE - A Coefficient Form PDE grants accessto higher order derivatives from study 1;

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich

COMSOL CONFERENCE 2016 MUNICH, 12-14 OCTOBER 2

❶ ❷

❸

❹

Fig. 1: Revolved representation of the sample geometry. Thecolored dots indicate the location of tracing massless particles.Boundaries: 1 sonotrode, 2 open interface, 3 + 4 fluid-wallinterface. Units in mm.

Table I: Boundary conditions for the acoustics and fluid flow(CFD) parts of the simulation.

Boundary Part Acoustic BC CFD BC

1 sonotrode acceleration no slip wall2 free surface sound soft free slip wall3 wall sound hard no slip wall4 wall sound hard no slip wall

• Study 3, CFD - The force term according to (2) isimplemented via volume force inducing a stationaryflow.

B. Boundary Conditions

Table I lists the boundary conditions used in the simulation.Study 2 is an intermediate step to store higher order derivativeswith one simplified partial differential equation. As it has nophysical meaning no boundary conditions are needed.The sonotrode acceleration az in axial direction is given byequation (3) [3]

az = 4π2f2A (3)

with f being the frequency and A the amplitude - typicallyin orders of 10 − 50µm. The sound soft BC physicallyimposes zero impedance, while sound hard BC imposes infiniteimpedance. The CFD boundary conditions are divided into freeslip walls for the open boundary and no slip wall, where theflow velocity is constrained to zero.

Fig. 2: Computational mesh consisting of hexa- and tetra-elements including boundary layers.

Fig. 3: Resulting acoustic velocity RMS field

C. Mesh

All studies are carried out with the same mesh as illustratedin figure 2. Due to remarkable gradients of the source termsin the sonotrode region boundary layers are required. The gridis predominantly mapped to minimize numerical diffusion.

D. Results

1) Acoustics: The frequency domain results of the acousticsare shown in figure 3. The sonotrode acceleration leads toa sharp acoustic particle velocity rise. The force term (2) isbased on the acoustic velocity field. In a separate test-case thefunctionality of the attenuation term defined as dipole sourceqd

qd =4

3

iωµ

a2ρ0∇p (4)

within the governing equation (1) has been analytically verifiedwith Stokes’ law of sound attenuation [4].

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich

COMSOL CONFERENCE 2016 MUNICH, 12-14 OCTOBER 3

Fig. 4: Resulting stationary velocity field of the study 3.

Fig. 5: Velocities of the massless particles located as shown infigure 1.

2) CFD: Figure 4 shows the resulting stationary velocityfield for an aluminium melt at an amplitude of 30µm andfrequency of 20kHz. The flow pattern is driven by an axial jetemanating from the tip of the actuating sonotrode. As this jet isdeflected from the bottom wall it creates a vortex in the regionof lower corners. Underneath the sonotrode the momentumreaches its maximum, whereas in the open interface zone theflow is close to standstill. Figure 5 compares the velocitiesof three tracing particles initially dispersed. Particles locatedunder the sonotrode experience a remarkable acceleration,increasing the amount of cycles.

IV. EXPERIMENTAL VALIDATION

A. Experimental setup and procedureThe experimental setup is shown in figure 6. The aluminium

sonotrode is dipped into the fluid-filled crucible. The experi-ments are carried out with a frequency of 20kHz and with an

High-Speed

Camera

diode laser

40 mW

532 nm wavelength

lasersheet

polycarbonate crubile

fluorescent particles

(20-40 μm)

tested fluids:

water

seed oil

glycerine

Aluminium sonotrode

10s ultrasonic treatment

frequency 20 kHz

amplitude 10-30 μm

Fig. 6: Experimental setup

amplitude of 10, 20, 30µm. Water, seed oil and glycerin areused as test fluids.A high-speed camera in combination with a diode laser andlasersheet allows the tracking of the fluorescent particles. ByParticle Image Velocimetry the corresponding velocity field isderived. The ultrasonic treatment is limited to 10 seconds forthe presented setup.

B. Results comparisonFigure 7 compares the velocity fields of the seed oil test

case. It is obvious that the jet occurs in simulation andexperimental results. Even though the experimental data doesnot appear completely symmetric, on the right-hand side ofthe crucible the streaming pattern is recognizable. In this area,the direction and location of the induced flow matches thepredicted behavior by the simulation. Figure 8 compares thevelocities along the rotational axis. Experiment and simulationare in good agreement in proximity of the sonotrode tip. Thesimulation reaches its peak velocity within 10mm from thesonotrode tip which is more than twice the experiments maxi-mum velocity. However, with increasing axial distance the gapbetween simulation and experiment results decreases showingthe same characteristic decline behavior. The deviation canbe explained primarily by the lack of accuracy of the opticalmeasurements and by simplifications taken in the simulation.

V. CONCLUSION

The presented model is numerically stable and appropiatefor testing parameter variations, geometry modifications andmaterial adjustements. Simplifications and stabilizations meth-ods allow the coupling of acoustics and fluid dynamics, whichoperate on timescales orders of magnitude from each other.Due to highly transient behavior and short treatment timeexperimental data proves to be delicate to obtain. To achievestationary conditions an extended ultrasonic treatment suchas industrial application is beneficial. A validated numericalmodel to predict behavior of the flow characteristics fordifferent fluids and for parameter variations turns out to bea reasonable approach.

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich

COMSOL CONFERENCE 2016 MUNICH, 12-14 OCTOBER 4

Fig. 7: Comparison of the velocity fields (m/s) of the experi-ment and simulation for seed oil actuated at an amplitude of30µm and frequency of 20kHz.

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80

Velo

city M

agnitude [m

/s]

Axial Distance [mm]

Experiment

Simulation

Sonotrode tip Crucible bottom

Fig. 8: Comparison of the velocity magnitude along therotational axis.

REFERENCES

[1] Muller, P. et al. (2012): A numerical study of microparticle acoustophore-sis driven by acoustic radiation forces and streaming-induced drag-forces. Lab Chip, The Royal Society of Chemistry, Vol. 12, p. 4617-4627.

[2] COMSOL (2016): Acoustic Streaming in a Micro-tunnel Cross Section. https://www.comsol.com/model/acoustic-streaming-in-a-microchannel-cross-section-17087 . Accessedon September 7, 2016.

[3] Gallego, J. et al. (2015): Power Ultrasonics, Applications of HighIntensity Ultrasound. Woodhead Publishing, Cambridge.

[4] Rossing, T.D. (2014): Springer handbook of acoustics. 2nd ed. Springer,Berlin.

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich