Chapter 5
Acoustic Standing Wave
generation using CFD model
Contents 5.1 Introduction . . . 97 5.2 Model Overview . . . 99 5.2.1 General Settings . . . 99 5.2.2 Turbulence Model . . . 100 5.2.3 Material properties . . . 103 5.2.4 Boundary conditions . . . 103 5.2.5 Post-Processing. . . 106 5.3 2D model-cross section . . . 109 5.3.1 Transverse actuation . . . 1095.3.2 Transverse and orthogonal actuation . . . 117
5.4 2D model- ow . . . 123
5.4.1 Results . . . 124
5.4.2 Cumulative eect of time periods . . . 127
5.5 Conclusion . . . 127
5.1 Introduction
Modeling of the pressure eld induced by Acoustic Standing Waves in uids has been since years, an hot topic in Computational Fluids Dynamic. One of the main goal of numerical modeling was and yet is to propose a quantitative tool for calculating the resulting acoustic radiation and streaming forces in particle laden ows to allow for realistic comparison with experimental results, and proposing more accurate device's design. The intrinsic nature of perturbation for the pressure eld induced by MHz ASW makes unavoidable to strongly rely on at least second-order term of the pressure formulation, having as a consequence that the use of simpli ed, analytic, linear model is highly questionable.
border. However in most of the numerical models, the pressure or velocity eld are simply imposed through a re ective boundary condition [41] [36] and not calculated from the motion of the walls as it should be more physically done. Besides, most of them are using a strong inviscid uid assumption, resulting in a restricted description of the phenomena as well as an important validity limitation. This chapter introduces a numerical model developed in the frame of my thesis to reproduce the pressure- eld generated by the actuation of the piezo transducer parts glued to channel walls, as described in the previous Chapter 4.5. Fig. 5.1 displays the dierent con gurations that have been considered. Two dierent 2D models have been investigated, the rst one tackling with the square cross-section of the channel orthogonally to the ow direction, where the borders stand for the channel walls. This model enables the modeling of both the transverse actuation only and the coupled transverse and orthogonal actuation, but it prevents from adding the ow behavior to these results. The second 2D model pictures the longitudinal cross-section of the channel, in the direction of the ow. As a result, the pressure eld based on the ow can be superposed to the pressure eld generated by the motion of the piezo actuators, but in this case, only one actuation mode is allowed. Eventually a 3D model has been developed, that combines the main interests of the two 2D models, but also presents a longer computation time due to the large amount of computational cells it deals with, making it unsuitable for a parametric study purpose.
Fig. 5.1: Schematic view of the dierent geometries that have been considered
5.2. Model Overview 99 ASW is generated by the periodic deformation of the wall modeled by a deforming mesh option controlled by a User De ned Function (UDF). The compressibility of the uid is introduced into the model by means of a state equation linking the density to the pressure and temperature elds. The standard patterns of pressure eld are successfully generated.
5.2 Model Overview
Modeling ASW in a small channel involves the computation of high frequency uctu-ations of transported quantities such as momentum and energy, that are too compu-tationally expensive to simulate directly. The perturbations involved in the present model are determinist and not stochastic, which make a laminar model able to cope with them. The resulting pressure eld coming from a laminar and a turbulent model are the same, except in the boundary layer, where the laminar model is not taking into account the small perturbations generated by the boundary motion, namely in the transient phase. This eect results into an increase of the number of iterations necessary to solve a time step and despite the lower number of equation to solve compared to a turbulent model, the total computation time is analogous between the two models. For the sake of generalization, we used a turbulent model but the relevancy of this choice is still an open question. The laminar model having been described in Chapter 3, the modi cation of the governing set of equations in the case of a turbulent model is described in the following paragraph 5.2.2.
5.2.1 General Settings
The Navier-Stokes equations have been solved thanks to the commercial software FluentT M as described in Chapter3. Parallel processing coupled with a Pressure-based solver and absolute velocity formulation has been chosen as default settings. As the remeshing by layering option was used to reproduce the wall motion, the transient formulation with Implicit Second Order was not permitted and the First Order was thus selected. The non-iterative time advancement was preferred to decrease the time to calculate each time step.
The general form of the continuity equation for a compressible and unsteady ow can be written as:
∂ ρ
∂ t +∇ · (⃗vρ) = 0 (5.1)
The conservation of momentum:
∂
∂ t(ρ⃗v) +∇ · (ρ⃗v⃗v) = −∇p + ∇ · (¯¯τ) (5.2)
Where µ is the molecular viscosity and I is the unit tensor.
The energy E is de ned as the sum of the enthalpy h, the energy coming form the pressure work and terms resulting from the kinetic energy.
E = h−P ρ +
v2
2 (5.4)
The general expression of the Energy Equation, in the case of a single species com-pressible uid solved by FLUENTT M is in the following form:
∂
∂ t(ρE) +∇ · (⃗v(ρE + p)) = ∇ · (kef f∇T + (τef f · ⃗v)) (5.5)
Where the kef f∇T refers to the energy transfer due to conduction (kef f is the eective conductivity) and τ=
ef f · ⃗v refers to the viscous diusion.
This general formulation is adapted following the ow model selected in FLUENTT M, as described by next Chapter 5.2.2. To solve the above set of equations, a Pressure-Velocity coupling using Pressure-Implicit with Splitting of Operators (PISO) scheme was used. Spatial discretization was done using Least Squares cell based for gradient formulation and a Second Order scheme for pressure and Third Order MUSCL(Monotone Upstream-Centered Schemes for Conservation Law) for Momentum, Turbulent kinetic energy, Turbulent Dissipation Rate, Reynolds Stresses and Energy.
5.2.2 Turbulence Model
The ow perturbations carried by ASW are modeled by a Reynolds stress model (RSM) [161], which corresponds to the most elaborate turbulence model providing by FLUENTT M. In 2D models, a total set of 5 equations are solved, which make the RSM more accurate for the prediction of complex ows.
Among the modeling options, a Quadratic pressure strain formulation has been selected because of its superior performance in basic shear ows with axisym-metric expansion/contraction and its higher order scheme also allows for capturing pressure perturbations. The computation of wall boundary conditions is performed using the k-equation of the Reynolds-Stress option. The heating eect due to viscosity has been taken into account in the energy equation.
de-5.2. Model Overview 101 scribed by FLUENTT M as follows:
∂ ∂ t(ρu ′ iu ′ j) | {z }
Local Time Derivative
+ ∂ ∂ xk (ρuku′iu ′ j) | {z } Cij≡Convection =− ∂ ∂ xk [ ρu′iu′ju′k+ p(δkju′i+ δiku′j) ] | {z }
DT ,ij≡Turbulent Diusion
+ ∂ ∂ xk [ µ ∂ ∂ xk (u′iu′j) ] | {z }
DL,ij≡Molecular Diusion(not included)
− ρ ( u′iu′k∂ uj ∂ xk + u′ju′k∂ ui ∂ xk ) | {z } Pij≡Stress Production − ρβ(giu′jθ + gju′iθ) | {z }
Gij≡Buoyancy Production (not included)
+ p ( ∂ u′i ∂ xj +∂ u ′ j ∂ xi ) | {z } ϕij≡Pressure Strain − 2µ∂ u ′ i ∂ xk ∂ u′j ∂ xk | {z } εij≡Dissipation − 2ρΩk(u′ju ′ mεikm+ u′iu ′ mεjkm) | {z }
Fij≡Production by System Rotation
+ S|{z}user
User-De ned Source Term(not included)
(5.6) Of the various terms in these exact equations, Cij, DL,ij, Pij, and Fij do not require any modeling. However, DT,ij, Gij, ϕij , and εij need to be modeled to close the equations. The modeling assumptions required to close the equation set are detailed here.
The Turbulent Diusive Transport DT,ij and the turbulent viscosity µt are modeled as: DT,ij = ∂ ∂ xk ( µt σk ∂ u′iu′j ∂ xk ) (5.7) µt= ρCµ k2 ε (5.8)
Where Cµ = 0.09. The Pressure-Strain Term ϕij is modeled using the optional Quadratic Pressure-Strain Model which has demonstrated to give an improved ac-curacy. ϕij =−(C1ρε + C1∗P )bij + C2ρε ( bikbkj− 1 3bmnbmnδij ) + ( C3− C3∗ √ bijbij ) ρkSij + C4ρk ( bikSjk+ bjkSik− 2 3bmnSmnδij ) +C5ρk (bikΩjk+ bjkΩik) (5.9) Where bij is the Reynolds-stress anisotropy tensor de ned as
The main strain rate Sij and the main rate-of-rotation tensor Ωij are de ned as Sij = 1 2 ( ∂ uj ∂ xi + ∂ ui ∂ xj ) (5.11) Ωij = 1 2 ( ∂ ui ∂ xj − ∂ uj ∂ xi ) (5.12) With the following constant values: C1 = 3.4, C1∗ = 1.8, C2 = 4.2, C3 = 0.8,
C3∗ = 1.3, C4= 1.25, C5 = 0.4
The boundary conditions are speci cally calculated in this model through the fol-lowing k-equation: ∂ ∂ t(ρk) + ∂ ∂ xi (ρkui) = ∂ ∂ xj [ (µ + µt σk )∂ k ∂ xj ] +1 2(Pii+ Gii)− ρε(1 + 2M 2 t) (5.13) Where σk= 0.82.
The dissipation tensor εij is modeled as
εij = 2
3δij(ρε) (5.14)
The scalar formulation of the dissipation rate ε is computed as:
∂ ∂ t(ρε) + ∂ ∂ xi (ρεui) = ∂ ∂ xj [ (µ + µt σε ) ∂ ε ∂ xj ] Cε1 1 2(Pii+ Cε3Gii) ε k− Cε2ρ ε2 k (5.15)
Where σε = 1.0, Cε1 = 1.44, Cε2 = 1.92, Cε3 is calculated when the gravitational eld is taken into account and the turbulent viscosity is µt is computed by Eq.5.8. The Near Wall treatment is modeled through the standard wall functions widely used for the modeling of industrial turbulent ows. Reynolds-Stress xed values at the wall adjacent cells are computed from k values using Eq.5.13. Expressed in a local coordinate system, where τ is the tangential coordinate, η is the normal coordinate, λ is the binormal coordinate, the values are:
u′2 τ k = 1.098, u′2 η k = 0.247, u′2 λ k = 0.655,− u′τu′η k = 0.255 (5.16)
The Energy equation Eq.5.5 in FluentT M RSM is computed by analogy with the turbulent momentum transfer:
∂ ∂ t(ρE) + ∂ ∂ xi [ui(ρE + p)] = ∂ ∂ xj [ (k +cpµt P rt )∂ T ∂ xj + ui(τij)ef f ] + Sh (5.17) Where P rt = 0.85 is the turbulent Prandlt number and (τij)ef f is the deviatoric stress tensor representing the viscous heating:
5.2. Model Overview 103 5.2.3 Material properties
The material properties used in the above model are based on the water liquid prop-erties, see Table5.1. However, the compressibility of the uid has been implemented thanks to an Equation of State of water, where the density ρ is expressed as a func-tion of temperature T and pressure P . The variafunc-tion of the density and the speed of sound according to this model are plotted in Fig.5.2.
Table 5.1: Water liquid properties
Property Symbol Value Unity (SI)
Dynamic Viscosity η 0.001003 kg.m−1.s−1
Speed of sound c 1507 m.s−1
Speci c Heat CP 4182 J.kg−1.K−1
Thermal conductivity k 0.6 W.m−1.K−1
Density ρ User-de ned kg.m−3
The density is calculated thanks to the Equation of State developed by Chen in 1976[162] for water for 0◦C < T < 100◦C and 0 bar < P < 1000bar.
ρ = 1
V, V = V0−
V0P
K0+ AP + BP2 (5.19)
Where V0 and V are respectively the speci c volume at P = 0bar and P , K0 is the
secant bulk modulus at P = 0bar,A and B are Temperature dependent parameters:
K0 =−4E − 05T4+ 0.0122T3− 2.2883T2+ 148.09T + 19652 (5.20)
V 0 = 4E− 10T4− 7E − 08T3+ 8E− 06T2− 6E − 05T + 1.0002 (5.21)
A = 9E− 10T4− 8E − 07T3+ 0.0001T2+ 0.0005T + 3.2614 (5.22)
B = 4E− 12T4− 1E − 09T3+ 9E− 08T2− 6E − 06T + 7E − 05 (5.23) The speed of sound is expressed as:
c2= [ ∂ P ∂ ρ ] S (5.24) Where the subscript S indicates isoentropicity. The speed of sound has been set as constant in the modeling work. However, a variable speed of sound, i.e. calculated from the Equation of State 5.19 has been also implemented in some simulations without noticeable dierences respect to the constant case.
5.2.4 Boundary conditions
(a) Evolution of water density with the pressure, for dierent temper-atures
(b) Evolution Speed of Sound with the pressure, for dierent tempera-tures
Fig. 5.2: Density and Speed of Sound dependency with P ,T for liquid water
parts. A velocity boundary condition coupled to Dynamic layering mesh method are applied to these walls. The resulting displacement is a global motion, in which the moving wall it-self experiments an uniform translation while the surrounding adjacent cells recover the consequent distortion by deforming.
Boundary conditions at the non moving wall are then discussed. Flow conditions at the inlet and outlet were also considered for the 2D model with a longitudinal cross section and for the 3D model.
5.2. Model Overview 105 5.2.4.1 Moving walls
The motion pro le applied to the modeled piezo and re ector rigid wall is de ned through a User De ned Function by a cosine function. The incident wave displace-ment could be calculated by integrating over time the periodic boundary velocity
Ui. In the model, it is imposed as:
Ux,ti = A0∗ cos(−
π
2 + 2πt
τ N) (5.25)
Where x stands for the direction of the propagation, A0 is the amplitude of the
signal, τ is the period of the signal corresponding to the half-wavelength condition for ASW generation (See Chapter 4) and N is the number of pressure nodes expected. The velocity pro les applied to moving walls over time for several
(a) Velocity pro le at boundary condition
(b) Mesh deforming
Fig. 5.3: Boundary conditions and mesh deformation
hexahedral cells, the dynamic layering mesh method can be used. Among the de ned parameters, the center of gravity was attributed to the barycenter of each de ned wall, the cell height was set to 1µm and the deform adjacent boundary layer with zone option was enabled.
The scattered wave corresponds to the re ection of the incident wave on the opposite wall. Instead of implementing a re ecting boundary condition on the opposite wall, the present model features the scattered wave like the incident wave generated by the motion of the re ecting wall Ur. This motion is described similarly with an additional parity condition associated to the expected number of pressure nodes N in the channel. If N is an even number, Ur
x,t = Ux,ti and if N is an odd number, Ur
x,t =−Ux,ti . 5.2.4.2 Non-moving walls
The mesh deformation generated by the displacement of the moving wall is recovered by adjacent lateral walls that are set as deforming walls, see Fig. 5.3(b).
The eect of ow conditions on the ASW are also investigated by setting mass ow inlet and an out ow condition at the inlet and the outlet of the channel, as in the case of the model of the longitudinal cross-section described in the Chapter5.4. 5.2.5 Post-Processing
The acoustic standing wave phenomenon corresponds to the superposition of multi-ples incident and scattered acoustic waves of the same wavelength, or in a resonator under resonance condition. As a result, the observation of this phenomenon is cumulative time dependent [150] and cannot be obtained by looking at the pressure and velocity eld at instant t.
5.2.5.1 Static Pressure and Velocity elds
The variation in the pressure eld is induced by the viscous compression and expan-sion of the uid that is squeezed under the walls motion. As a result the pressure perturbation travels between the walls at the speed of sound of the medium. The instantaneous Velocity Magnitude Field is displayed on (Fig .5.4a)) while the instantaneous pressure that is experimented by the uid in the frame of our model is shown (in Fig.5.4b). For a speci c simulation time - in this case = 40µs after the initialization of the eld - the pressure exhibits a maximum value at the left wall, and a minimum value at the right wall, while the velocity reaches a maximum value in the center of the section and a minimum value close to the moving walls.
5.2. Model Overview 107 pressure maximum and minimum values are alternated, and pressure nodes, in which the pressure remains roughly constant and close to zero.
5.2.5.2 Mean Pressure and velocity elds
Fig.5.4c) displays the average pressure eld and Fig.5.4d) the average velocity eld that were calculated using:
⟨P ⟩ = 1 T ∫ T 0 P (t)dt (5.26) ⟨V ⟩ = 1 T ∫ T 0 V (t)dt (5.27)
Where T represents the global calculation time, and P (t) and V (t) are the instan-taneous values of the static pressure and velocity eld. As expected for a periodic sinus wave, the average value is zero. As a result, the mean eld values are not representative of the acoustic eect.
5.2.5.3 Time-Averaged pressure and velocity eld
Static pressure and velocity eld can only give a representation of the system at a speci c time t while the mean pressure and velocity remain nil due to the exact superposition of waves under ASW condition. However, as described in Chapter4.2, the acoustic radiation force experimented by particles under ASW depends on the second order terms of the perturbation of the pressure and velocity. These terms are taken into account in the so-called time-averaged pressure and velocity. The time-averaged values of the pressure and velocity are calculated in FluentT M using the Root-Mean-Square (RMS) of pressure - PRM S - and velocity - VRM S- elds that could be de ned as:
⟨ P2⟩= 1 T ∫ T 0 P (t)2dt = PRM S2 (5.28) ⟨ V2⟩= 1 T ∫ T 0 V (t)2dt = VRM S2 (5.29)
Where T represents the global calculation time, and P (t) and V (t) are the instan-taneous values of the static pressure and velocity eld. At the center of section, in the area of the so-called pressure node, RMS velocity eld, displayed on Fig. 5.4e), exhibits a maximum, while the RMS pressure eld, displayed on Fig.5.4f) exhibits a minimum. These elds are representative of the acoustic eect and will be used to de ne the acoustic potential and the radiation force that are responsible for particle migration.
5.2.5.4 Acoustic potential
Fig. 5.4: Comparison of the instantaneous Field value, Mean value and Root Mean Square value for pressure and velocity in the case of the only transverse actuation de-picted by Fig. 5.5
are implemented thanks to a Custom Field Function.
Acoustic potential is also a function of the particle radius, to facilitate fur-ther comparisons between experiments and numerical results, we have calculated a normalized one by the radius acoustic potential Urad:
Urad = 3
4πr3U (5.30)
5.3. 2D model-cross section 109
5.3 2D model-cross section
This 2D model, as displayed on Fig.5.5, is a cross-section model orthogonal to the ow direction. This model presents a total number of 68416 cells, their size range from 10µm to 2.5µm close to the wall where a re nement is required to ensure a proper mesh deformation.
Fig. 5.5: Schematic view of the dierent actuation modes that have been investigated using the 2D-cross section model
Investigations were lead using two dierent actuation modes, as displayed on Fig.5.5. The rst one is taking advantage of the transverse actuation only which is the main convention for half-plane acoustic waves in a half-in nite parallel planes theoretical device. In the frame of this thesis, a square cross section channel has been used, in which the incident wave re ections from orthogonal boundary walls can not be further neglected. That is the main point of the the second actuation mode model where the motion of both transverse and orthogonal wall was implemented. 5.3.1 Transverse actuation
computed periods.
5.3.1.1 Acoustic wave propagation
One of the interesting result obtain with our model was the possibility to follow the propagation of the acoustic wave through the medium in the transient phase preceding the establishment of a standing wave inside the channel. This analysis allowed for checking that the waves generated by the motion of the walls are not attenuated by the numerical viscosity of the chosen discretization schemes and correctly propagated with a velocity corresponding to the speed of sound in the medium. A scheme of the wave propagation under transverse actuation con guration is proposed on Fig.5.6.
Fig. 5.6: Schematic view of the wave propagation under transverse actuation in the 2D-cross section model
The initial transient phase of ASW generation is followed by plotting the static pressure for 10 successive time interval of the rst period τ of the acoustic wave propagation. Fig. 5.7 displayed with red lines the generation of a 1-node acoustic wave, while Fig. 5.8 and Fig. 5.8 displayed respectively with green lines and blue lines the generation of a 2-node and 3-node ASW. In this gures, the instantaneous static pressure is represented with a plain line, while the previous steps are represented with dashed lines.
In the case of the 1-node ASW generation, the re ector wall motion has been run at t = λ
4 after the incident wave has reached it. On the contrary, for
5.3. 2D model-cross section 111 (a) Pressure at t = 1 10τ (b) Pressure at t = 2 10τ (c) Pressure at t = 3 10τ (d) Pressure at t = 4 10τ (e) Pressure at t = 5 10τ (f) Pressure at t = 6 10τ (g) Pressure at t = 7 10τ (h) Pressure at t = 8 10τ (i) Pressure at t = 9 10τ (j) Pressure at t = τ
(a) Pressure at t = 1 10τ (b) Pressure at t = 2 10τ (c) Pressure at t = 3 10τ (d) Pressure at t = 4 10τ (e) Pressure at t = 5 10τ (f) Pressure at t = 6 10τ (g) Pressure at t = 7 10τ (h) Pressure at t = 8 10τ (i) Pressure at t = 9 10τ (j) Pressure at t = τ
5.3. 2D model-cross section 113 (a) Pressure at t = 1 10τ (b) Pressure at t = 2 10τ (c) Pressure at t = 3 10τ (d) Pressure at t = 4 10τ (e) Pressure at t = 5 10τ (f) Pressure at t = 6 10τ (g) Pressure at t = 7 10τ (h) Pressure at t = 8 10τ (i) Pressure at t = 9 10τ (j) Pressure at t = τ
5.3.1.2 Acoustic Standing Wave generation
As described in the previous section, the ASW phenomena corresponds to the constructive superposition of incident and scattered waves in the channel. Fig.5.10, Fig.5.11 and Fig.5.12displayed respectively the static pressure along the width of the channel for an 1-node, 2-node and 3-node ASW.
Time intervals displayed are t = nτ,t = nτ + 1
4τ, t = nτ + 1
2τ and t = nτ + 3 4τ
where τ corresponds to the period of the acoustic wave and n is an integer number of computed periods. For t = nτ and t = nτ + 1
2τ, the acoustic pressure exhibits
an expected nil value (small perturbation comes from the dierences in averaging the computed time-steps) while the maximum values are displayed at t = nτ + 1
4τ
and t = nτ +3 4τ.
It is worthy to remind that the pressure nodes correspond to the spatial lo-cation where the Static pressure is steadily nil while the RMS pressure exhibits a minimum value. By considering that simulations are performed in a geometry reproducing the one used in experiments - i.e. a uidic square channel of 2mm side-the geometrical position where this condition is met coincides with x = 0mm for a 1-node ASW, x = −0.5mm and x = 0.5mm for a 2-node ASW, and x = −0.67mm,
x = 0mm and x = 0.67mm for a 3-node ASW (with the origin of the x-axis x = 0 set at the center of the channel cross-section, and the left and right wall are
respectively at x = −1 x = +1).
(a) Pressure at t = nτ (b) Pressure at t = nτ +1 4τ
(c) Pressure at t = nτ +1
2τ (d) Pressure at t = nτ + 3 4τ
5.3. 2D model-cross section 115
(a) Pressure at t = nτ (b) Pressure at t = nτ +1 4τ
(c) Pressure at t = nτ +1
2τ (d) Pressure at t = nτ + 3 4τ
Fig. 5.11: Instantaneous Pressure evolution of the 2-node wave period (f=800KHz); Y axis: Pressure (Pa) and X axis: Channel width (m)
(a) Pressure at t = nτ (b) Pressure at t = nτ +1 4τ
(c) Pressure at t = nτ +1
2τ (d) Pressure at t = nτ + 3 4τ
5.3.1.3 Comparison with experimental results
(a) Acoustic potential normalized by the radius under a 1-node transverse actuation
(b) Particles position in the cross-section under a 1-node ASW (f=400KHz)
(c) Acoustic potential normalized by the radius under a 2-node transverse actuation
(d) Particles position in the cross-section under a 2-node ASW (f=800KHz)
(e) Acoustic potential normalized by the radius under a 3-node transverse actuation
(f) Particles position in the cross-section under a 3-node ASW (f=1.2MHz)
5.3. 2D model-cross section 117 The resulting Acoustic Potential under this transverse actuation presents the stan-dard parallel lines patterns as displayed in Fig. 5.13. The minimum values are organized into 1, 2 and 3 lines alongside the moving walls, depending on the reso-nant mode. These patterns are compared to the reconstructed positions of particles recorded during the experiments (presented in the Chapter4). Particles are indeed located in area of minimum potential (displayed in the numerical model in blue). However it is clear from a visual comparison that this model is not able to reproduce the speci c matrix-like experimental patterns.
5.3.2 Transverse and orthogonal actuation
The patterns of particle positions in the cross section of the channel that have been experimentally observed presented a peculiar distribution in form of spots or ribbons. The formation of this matrix-like structure has never been reported to our knowledge in the literature. To better understand how these structures could arise, we did the hypothesis that the formation of spots was due to the preponderantly scattering role of the orthogonal side walls of the channel that would re ect the incident wave in the same way that the transverse re ector. This hypothesis has been tested by implementing a 2D model that enables the independent motion of the 4 outer boundary walls. The patterns generated by the conjugate action of transverse and orthogonal actuation modes could be then investigated as depicted in Fig. 5.5. Velocity motion has been set to orthogonal walls similarly to the previous transverse actuation, as de ned in the Chapter5.2.4.1. Generation of the ASW under a transverse and orthogonal actuation is re-ported, and the resulting acoustic potential eld is displayed for several resonance modes. Eventually the investigation of an asymmetrical stimulation using a dierent amplitude for the orthogonal and the transverse actuation was leaded and the acoustic potential was compared to the experimental positions of particles.
5.3.2.1 ASW generation
(a) Potential Energy (3-node ASW; t = 60µs) (b) Kinetic Energy (3-node ASW; t = 60µs)
(c) Acoustic Potential (3-node ASW; t = 60µs)
Fig. 5.14: Acoustic potential resulting from the superposition of the Kinetic Energy and the Potential Energy (3-node ASW; t = 60µs, f=1.2MHz)
ASW generation over time is displayed on Fig. 5.15 under transverse and or-thogonal actuation with the same amplitude. Minimum values for acoustic poten-tial generated using this transverse and orthogonal model present weavy lines that would provide a better representation of the experimental patterns with respect to the case where only transverse actuation is imposed.
5.3. 2D model-cross section 119
(a) Acoustic potential eld at t = 2µs (b) Acoustic potential eld at t = 4µs
(c) Acoustic potential eld at t = 10µs (d) Acoustic potential eld at t = 20µs
(e) Acoustic potential eld at t = 40µs (f) Acoustic potential eld at t = 60µs
Fig. 5.15: 3-node ASW generation in the cross section of the channel under transverse and orthogonal actuation (f=1.2MHz)
5.3.2.2 Asymmetrical actuation
(a) Acoustic potential eld with asymmetrical stimulation amplitude Aortho= 0
(b) Acoustic potential eld with asymmetrical stimulation amplitude Aortho= 14Atrans
(c) Acoustic potential eld with asymmetrical stimulation amplitude Aortho=12Atrans
(d) Acoustic potential eld with symmetrical stimulation amplitude Aortho= Atrans
Fig. 5.16: Comparison of the normalized by the radius acoustic potential under a 3-node ASW in the cross section of the channel for dierent amplitude ratios of trans-verse and orthogonal actuation (t = 40µs, f=1.2MHz))
5.3.2.3 Comparison with experimental results
The comparison between the numerically computed acoustic potential with the experimental particles position is displayed as a superposition of the two results are plotted in the cross section. For a better observation, a gray color scale has been used, where the minimum values of the acoustic potential, corresponding to the nodes, are displayed in white and the particles are displayed as black dots.
5.3. 2D model-cross section 121
Fig. 5.17: Aortho= 14Atrans
In the case of a 2-node, Fig. 5.18 shows a good match between experimental (particle diameter range 75 − 90µm, P= 31.4mW.cm−2, ow rate=0.2ml.min−1, f=800kHz) and numerical data was found for a symmetrical actuation Aortho =
Atrans.
In the case of a 3-node ASW, Fig.5.19displayed the numerical pattern for dierent
Fig. 5.18: Aortho= Atrans
(a) Superposition of Acoustic po-tential eld with symmetrical stim-ulation amplitude Aortho = 0with
particle positions under a 3-node ASW
(b) Superposition of Acoustic potential eld with symmet-rical stimulation amplitude
Aortho = 14Atrans with particle
positions under a 3-node ASW
(c) Superposition of Acoustic potential eld with symmet-rical stimulation amplitude
Aortho = 12Atrans with particle
positions under a 3-node ASW
(d) Superposition of Acoustic po-tential eld with symmetrical stim-ulation amplitude Aortho = Atrans
with particle positions under a 3-node ASW
Fig. 5.19: Comparison of the normalized by the radius acoustic potential with the exper-imental particles positions
Eventually, in the case of a 4-node ASW, the best match between experimental patterns (particle diameter range 75 − 90µm, P(too low to be calculated), ow rate=1ml.min−1) was found for A
ortho= 14Atrans for as described on Fig. 5.20.
5.4. 2D model- ow 123
Fig. 5.20: Aortho= 14Atrans
5.4 2D model- ow
This model has been developed to represent the transverse section of the channel, orthogonal to PZT actuation, as described by Fig.5.21. Using this con guration, it is possible to model a ow orthogonal to the acoustic standing wave and to observe how the velocity and pressure eld are impacted by the ow. When comparing the
Fig. 5.21: Schematic view of the dierent actuation modes that have been investigated using the 2D-cross section model
unsteady modeling.
In the present model for instance, the time step applied by the solver during the resolution was set to 2ns (2D model) or 1ns (3D model) for an accurate resolution of the equations coming from acoustic actuation. The typical duration of the compu-tation of one time-step takes 2s. The required time to compute a 10s sequence using the 2D model with a personal computer would be then 1010s (more than 300years!).
To tackle this issue, the eect of ow velocity on the ASW generation was then lead in two steps. First a mass ow was injected and the corresponding Poiseuille-like pro le was calculated using a steady state laminar model. The acoustic viscous model was then implemented on the previous results. The alteration of pressure and velocity elds when an ASW is generated into the channel could then be computed. 5.4.1 Results
The fully established ow pro le in the channel is rst calculated using the same laminar ow model that the one described in Chapter 3. A mass ow rate is injected at the inlet and the ow pro le is calculated in steady-state conditions. A range of mass ow rate going from 0kg.s−1 to 0.4kg.s−1 have been successively injected at the inlet and the corresponding velocity pro les at t = 40µs in the section are plotted in Fig. 5.22.
Fig. 5.22: X velocity at t = 40µs in the central section of PZT, for a 1-node ASW generated in a channel with dierent ow rates
5.4. 2D model- ow 125
(a) RMS Static Presure in the lenght of the channel
(b) RMS static pressure in the width of the channel
(c) RMS Velocity in the width of the channel
Fig.5.23(a)displays the time-averaged static pressure in the length of the chan-nel. The gray area corresponds to the PZT location, and the dashed lines to the corresponding width of the channel represented in Fig. 5.23 and Fig. 5.22. The comparison of time-averaged pressure and velocity generated under the dierent ow rates in the width of the channel are displayed on Fig. 5.23. When a ow rate is imposed at the inlet of the channel, the pattern and the amplitude of the ASW generated in the width of the channel exhibits the same behavior than in the case there is not ow rate (red line).
The evolution of the static pressure under a 1-node ASW is described in Fig.5.24.
Fig. 5.24: Evolution of Static pressure in a period under a 1-node ASW in the PZT area
For an enhanced clarity of the picture, the section has been cropped to the area between the moving walls. The rst section is displayed after a computation time of 40µs and the time laps between each representation was set to 0.4µs. As a result, a rough period owed by between rst section T0 and the last section T7. From
T0 to T3, the pressure eld shows a minimum value in the half top of the channel
5.5. Conclusion 127 repartition could be observed. The pressure eld is getting null around t=T3, where
a partial inversion of minimum and maximum values is displayed. 5.4.2 Cumulative eect of time periods
The computation of ASW generation in a channel is time-consuming. In the frame of this modeling investigation, the computation was arbitrary suspended after 50µs, as the calculated time averaged pressure and velocity were no longer exhibiting a strong modi cation. Besides, the ASW is a superposition of a periodic phenomena, which means that an increase of the computed time will result in more accurate results, but it will not conduct to a modi cation of the system behavior.
The superposition eect of the acoustic standing waves has been investigated to com-fort this assumption. The time-averaged pressure has shown to reach its maximum value close to the walls. This maximum value has been extracted for a succes-sive number of time periods for a 1-node standing acoustic wave, and its evolution is presented on Fig. 5.25. More than 800 time periods have been computed that corresponds to a total duration of almost 2ms.
Fig. 5.25: Evolution of the time-averaged pressure close the wall with the total number of time periods computed
The cumulative eect of the 100 rst periods on the pressure lead to a time-averaged pressure that ranges from 0 to 0.15MPa. The additional pressure generated by the 700 following time periods is resulted in a 0.06MPa pressure excess.
5.5 Conclusion
the boundary walls of a glass channels. The model was supposed to reproduce at the best the real motion of the walls as induced by the vibrations of the piezoelectric elements in order to be able to autonomously generate the acoustic standing waves and the related potential.
The commercial software FluentT M was used and the basic models enhanced by means of C++ subroutines opportunely describing the motion of the boundaries, the properties of the uid, the tracking of particles and the evaluation of the Root Mean Square quantities needed to evaluate the acoustic potential. Big eorts were spent to nd the right settings for the numerical solution to converge and a relevant amount of CPU time was dedicated to achieve the goal. This model could still be improved and the use of a laminar model with the appropriate settings could eect positively the CPU time.
The generation of a non attenuated, standing wave was the rst important valida-tion of the model. Initially conceived to describe a simple 1D acoustic potential, the model was then enhanced to try to reproduce the experimental observations in a section orthogonal to the sample ow. The starting point was to understand the formation of a matrix-like structure for the focused particles, as reported in Chapter4. Simulations were carried out by considering that the formation of spots and ribbons could be probably due to the coupling of a direct acoustic potential as generated by the piezo-transducers and a secondary potential orthogonal to the former and of unde ned intensity. A parametric analysis has been done by modu-lating the amplitude of the two potentials and checking the compatibility with the experimental results.
