Numerical Modeling of Fluid Flow and Solidification Characteristics of Ultrasonically Processed A356 Alloys

Abstract

The main objectives of this study is to determine the fluid flow characteristics and cavitation phenomena during molten A356 alloys processed via high power ultrasonic treatment (UST) as well as the resulting solidification microstructure of the ultrasonically processed A356 alloys. A previously developed UST modeling approach is applied to obtain the simulation results in this study. The modeling approach consists of two main models: (1) A CFD UST cavitation model is applied to simulate acoustic streaming, cavitation and bubble dynamics during molten alloy processing and (2) A stochastic mesoscopic modeling to predict microstructure evolution during alloy solidification. The first model can predict the amount of the cavitated phase that is used to determine the number of potential nuclei formed due to the UST processing. The second model uses the predicted number of nuclei from the first model as an input to predict the microstructure evolution during the solidification of cast alloys. In addition, the UST model predicts the fluid flow characteristics that include flow profile, velocity vectors and magnitude, pressure contours, and streamlines under two different gravity conditions. The fluid flow characteristics are analyzed to ensure that proper ultrasonic stirring and cavitation of the melt are achieved.

1. Introduction

Ultrasound technology is a powerful technology to affect a material. Ultrasonic processing can significantly affect heat and mass transfer in liquids, modify the structure and properties of solids and interfere with the interactions of solids and liquids. The primary causes are due to ultrasonic cavitation, acoustic streaming and movement of dislocations associated with propagation of ultrasound waves in media. Ultrasonically-induced cavitation consists of the formation of small cavities (bubbles) in the molten metal followed by their growth, pulsation and collapse. These cavities are created by the tensile stresses that are produced by acoustic waves in the rarefaction phase.¹⁾

Relevant applications include structural refinements of solidifying metals with applications to continuous casting as well as to remelting processes (vacuum arc remelting, electroslag remelting, electron beam remelting and plasma arc melting) for steel and Ti alloys, specialty steels and superalloys. Additional potential applications include die-casting, semi-solid metal processing, and forging of high fractions of fine spherical grains. Some critical technologies that have immediate application to manufacturing industry are presented in.^{1,2,3,4,5,6,7,8,9)}

The Computational Fluid Dynamics (CFD) multiscale modeling approach is based on the numerical acoustic solution of Lilley model, compressible fluid flow, and heat transfer equations, and stochastic mesoscopic modeling of the microstructure evolution. The Noltingk–Neppiras model, which is a general case of a gas vapor–filled cavity in an acoustic field, is used to predict the cavity time evolution in an acoustic field. The ultrasonic cavitation model is thus capable to model acoustic streaming, cavitation and bubble dynamics during molten metal and solidification processing of alloys. The stochastic mesoscopic model uses the predicted number of nuclei from the ultrasonic cavitation model as an input to predict the final solidification microstructure.

The objective of this study is to determine in detail the fluid flow characteristics and ultrasonic cavitation effects during molten metal processing of A356 alloys as well as the to predict the solidification microstructure of the ultrasonically processed A356 alloys.

2. Description of the Multiscale Modeling Approach

The modeling approach consists of two main models: (1) A CFD ultrasonic cavitation model is applied to simulate acoustic streaming, cavitation and bubble dynamics during molten alloy processing and (2) A stochastic mesoscopic modeling to predict microstructure evolution during alloy solidification. These models are summarized below.

2.1. CFD Ultrasonic Cavitation Model

Fully couple energy, mass, acoustic, and compressible fluid flow models were previously developed for this ultrasonic analysis.^10,11,12) Lilley model based on Lighthills’s acoustic analogy¹⁰⁾ was adapted in this modeling study. The UST model is coupled with a stochastic mesoscopic model to predict the microstructure evolution during solidification of cast A356 allloy. The Noltingk–Neppiras model, which is a general case of a gas vapor–filled cavity in an acoustic field, is used to predict the cavity time evolution in an acoustic field. The UST modeling approach is thus capable to model acoustic streaming, cavitation and bubble dynamics during molten metal and solidification processing of alloys.

Figure 1 describes the geometry of the A356 liquid pool. Since the time-domain computations were used for resolving the acoustic field, a very small time step of the order of 1e-07 s was used. The ultrasonic probe has a diameter of 50 mm, an amplitude, A = 10 microns, and a frequency, f = 18 kHz.^8,9) The applied effective power of the ultrasonic generator is 1.7 kW. To simulate the movement/oscillation of the ultrasonic probe (actuator) as a rigid body, a dynamic mesh procedure was used in Fluent.¹⁰⁾ A user defined function (UDF) was developed and used in Fluent to provide the proper sine motion profile of the ultrasonic probe.

Fig. 1.

Model geometry.

In the ultrasonic cavitation model, a two-phase cavitation model is used that consists of using the standard viscous flow equations governing the transport of phases (Eulerian multi-phase) and the k-ε turbulence model. In cavitation, the liquid-bubble mass transfer is governed by the cavity (bubble) transport equation:^11,12)   

$$\frac{\partial }{\partial t}\hspace{0.17em}\left(f_b\mathrm{\hspace{0.17em} }{\rho }_b\right)\mathrm{\hspace{0.17em} }+\mathrm{\hspace{0.17em} }\nabla \cdot \mathrm{\hspace{0.17em} }\left(f_b\mathrm{\hspace{0.17em} }{\rho }_b\mathrm{\hspace{0.17em} }{\overrightarrow{V}}_b\right)\mathrm{\hspace{0.17em} }=\mathrm{\hspace{0.17em} }{R}_G\mathrm{\hspace{0.17em} }-\mathrm{\hspace{0.17em} }{R}_C$$(1)

where b subscript denotes the cavitation bubble phase, f_b is the bubble volume fraction, ρ_b is the bubble density, ${\overrightarrow{V}}_b$ is the bubble phase velocity, R_G and R_C are the mass transfer source terms related to the growth and collapse of the cavitation bubbles, respectively.

In Eq. (1), the interphase mass transfer rates per unit volume (R_G and R_C) account for the liquid and bubble phases in cavitation. They are calculated using the growth of a single bubble based on the Rayleigh-Pleasset model.¹¹⁾ The model assumes no barrier for nucleation; thus, the bubble dynamics can be obtained from the general Rayleigh-Plesset equation as follows:^11,12,13)   

$$R_b\frac{d^2{R}_b}{d{t}^2}+\frac{3}{2}{\left(\frac{d{R}_b}{dt}\right)}^2=\frac{p_b\mathrm{\hspace{0.17em} }-\mathrm{\hspace{0.17em} }p}{{\rho }_L}\mathrm{\hspace{0.17em} }-\mathrm{\hspace{0.17em} }\frac{2\mathrm{\hspace{0.17em} }{\sigma }_L}{{\rho }_L\mathrm{\hspace{0.17em} }{R}_b}\mathrm{\hspace{0.17em} }-\mathrm{\hspace{0.17em} }\frac{4\mathrm{\hspace{0.17em} }{\nu }_L}{R_b}\mathrm{\hspace{0.17em} }\frac{d{R}_b}{dt}\mathrm{\hspace{0.17em} }$$(2)

where R_b is the bubble radius, σ_L is the surface tension coefficient of the liquid phase, ρ_L is the liquid density, ν_L is the kinematic viscosity of the liquid phase, p_b is the bubble surface pressure, and p is the local far-field pressure.

2.2. Stochastic Mesoscopic Model

The mesoscopic model developed previously in^14,15,16,17) is adapted to include the effect of UST on solidification structure evolution and therefore on grain refinement of alloys. The mesoscopic model includes computations of the grain size and columnar-to-equiaxed transition (CET), as well as of segregation. It also includes computations at the dendrite tip length scale (mesoscopic scale) for prediction of columnar and equiaxed dendritic morphologies and of microsegregation patterns.

Equations (1) and (2) are used to estimate the number of nuclei in the presence of UST. The evolution of the cavitation phase is also simulated with the UST model. Equations (1) and (2) assumed that the cavitation-induced heterogeneous nucleation is the main mechanism for grain refinement. This is also in accordance with the work reported in Refs. 6) and 12), that is, the cavitation-induced heterogeneous nucleation plays a more significant role than the “sluggish” dendrite fragmentation process in the formation of equiaxed grains during alloy solidification.

3. Simulation Results

3.1. Ultrasonic Cavitation

The simulation study consists of modeling the flow characteristics for the geometry presented in Fig. 1 under two different gravity conditions:

Case 1: gravitational acceleration oriented downward. This is the normal case when the ultrasonic case is inserted into the melt through the top of the furnace.

Case 2: gravitational acceleration oriented upward. This is a special case that required that the ultrasonic probe to be inserted into the melt through the bottom of the furnace.

It is assumed that the furnace type is resistive and therefore the fluid flow is mostly due the ultrasonic stirring and some natural convection because of ultrasonic heating effect. The ultrasonic probe (actuator) is modeled as a rigid body using the dynamic mesh in Fluent. The sine motion of the ultrasonic probe is dictated by the given power and frequency of the ultrasonic generator. The applied effective power of the ultrasonic generator is 1.7 kW and its frequency is 18 kHz, similar to the ones used in the experiments in Refs. 8) and 9). For case 1, the location of the outlet pressure boundary conditions is at the top left and right corners (close to the ultrasonic probe). For case 2, the location of the outlet pressure boundary conditions is at the bottom left and right corners (opposite to the ultrasonic probe). The size of the outlet is 5 mm for both cases.

The predictions of the flow, pressure, temperature, and ultrasonic cavitation are presented in Figs. 2, 3, 4, 5, 6 for an A356 liquid alloy at time t = 2.0e-04 s, which is after the onset of cavitation Note that the onset of cavitation for this alloy system is around 8.8e-06 s (see Ref. 12)). Figures 2 and 3 show the simulations results for the gravitational acceleration oriented downward (case 1) while Figs. 4 and 5 show the simulations results for the gravitational acceleration oriented upward (case 2).

Fig. 2.

Case 1: (a) Cavitation region (volume fraction) and (b) pressure (Pa) at t = 2e-04 s.

Fig. 3.

Case 1: (a) Velocity magnitude (m/s) and (b) velocity vectors (m/s) at t = 2e-04 s.

Fig. 4.

Case 2: (a) Cavitation region (volume fraction) and (b) pressure (Pa) at t = 2e-04 s.

Fig. 5.

Case 2: (a) Velocity magnitude (m/s) and (b) velocity vectors (m/s) at t = 2e-04 s.

Fig. 6.

Effect of gravitational acceleration orientation: (a) Stream function (kg/s) at time t = 2e-04 s.: (a) case 1 (gravity vector down) and (b) case 2 (gravity vector up).

The predicted ultrasonic cavitation region is presented in Figs. 2(a) and 4(a), where the cavitation phase is hydrogen. The cavitation region is relatively small at time t = 2e-04 s, the acoustic streaming is relatively strong, especially in the ultrasonic probe region (see Figs. 2(a) and 4(a)) and thus the newly created/survived bubbles/nuclei can be transported into the bulk liquid quickly. Note that the legend in Figs. 2(a) and 4(a) shows the volume cavitation region (volume fraction of cavities/bubbles/potential nuclei). The cavitation region is bigger for case 2.

The pressure for nucleation of the bubbles (cavitation threshold pressure or P_c) in A356 alloy (see Figs. 2(b) and 4(b)) is of the same order of magnitude as the experimentally determined pressures to nucleate gas bubbles in other metallic systems (see Ref. 1), p. 123 and Ref. 12)). Note also that the cavitation threshold may decrease with increasing the amount of dissolved gases (such as hydrogen in A356) and especially with the amount of inclusions in the melt (such as Al₂O₃ in A356) (see Ref. 18) and Fig. 2.39 in Ref. 1)).

Figures 3 and 5 show the profiles for both the velocity magnitude and the velocity vectors. As shown in Figs. 3 and 5, the velocity magnitude is significantly high beneath the ultrasonic probe for both cases. It can also be seen from Figs. 3 and 5 that the bulk stirring is significantly stronger for case 2, which is very advantageous for ultrasonic processing of alloys and metal-matrix-nanocomposites.

The overall effect of the gravitational acceleration is clearly shown in Fig. 6, which compared the stream functions for both cases. As it can be seen from Fig. 6 and also from Figs 2, 3, 4, 5, from the flow characteristics perspective, case 2 is more favorable than case 1 to be used for ultrasonic stirring and cavitation processing of alloys and metal-matrix-nanocomposites.

3.2. Prediction of the Solidification Microstructure

Table 1 presents the modeling data used in the microstructure modeling of alloy A356. The predicted solidification structure is shown in Fig. 7, where G_E is the equiaxed grain density. The legend in Fig. 7 shows the 256 color indexes (CI varies from 0 to 255) (legend displays 16 classes, where each class contains 16 different color indexes) used for displaying the preferential crystallographic orientation angle (θ_i). The orientation angles can be extracted from the legend using the following equation: θ_i = π/2*(CI/255)–π/4. Thus, when CI/255 varies from 0 to 1, θ_i ranges from –π/4 to π/4. The simulations of the microstructure shown in Fig. 7 were performed assuming a steel mold and a uniform dispersion of the heterogeneous nuclei in the melt. Also, the effect of the direction of gravitational acceleration is not considered for the simulated results shown in Fig. 7.

Table 1. Data Used in Microstructure Modeling of Alloy A356.

Solidification Kinetics Property	A356
${\mu }_N^C$ [m–1 K–2]	1×103
${\mu }_N^E$ [m–2 K–2]	6×105
Liquid diffusivity, DL [m2 s–1]	3×10–9
Solid diffusivity, DS [m2 s–1]	1×10–12
Liquidus slope [K wt.%–1]	mL = –6.5
Initial Si concentration [wt.%]	Co = 7.0
Si partition coefficient	k = 0.14
Eutectic [wt.%]	12.6
Gibbs-Thomson coefficient, Γ [K m]	0.9×10–7

Fig. 7.

Ultrasonic treatment: prediction of the solidification structure for an A356 alloy (2 mm × 4 mm test geometry): (a) without UST (G_E = 5 × 10⁷ nuclei/m²); (b) with UST and high superheat (G_E = 5 × 10⁸ nuclei/m²); (c) with UST and low superheat (G_E = 5 × 10⁹ nuclei/m²) and (d) with UST, low superheat and 1%SiC (G_E = 5 × 10¹⁰ nuclei/m²).

The nucleation procedure used in the microstructure model is explained in detail in Ref. 14) (Eqs. (4-10) and (4-11), pp. 35–36). The values of nucleation parameters for equiaxed ( ${\mu }_N^E$ ) and columnar ( ${\mu }_N^C$ ) morphologies for the case without the use of UST (Fig. 7(a)) are shown in Table 1. The equiaxed nucleation parameters including nucleation density for the UST cases (Figs. 7(b) and 7(c)) were determined based on UST model (see Eqs. (1) and (2) and Fig. 2(a)). The values of these equiaxed nucleation parameters are at least one order of magnitude higher than those for the case without the use of UST. In these cases, basically there is no substantial nucleation barrier (e.g., instantaneous nucleation that takes places at low bulk undercoolings can be assumed), and the nucleation rate and nucleation density can be calibrated based on the experimental measurements of the grain size.

In these computations (Figs. 7(a) and 7(b)), the pouring temperature of the molten A356 alloy was 750°C (high superheat of about 140°C) and the initial temperature of the steel mold was about 390°C. These conditions are similar to the ones used in the experiments performed at the University of Alabama at Tuscaloosa, Solidification Laboratory.^8,9) The CET is observed in Fig. 7(a), where the UST was not applied. As it can be seen from Fig. 7(b), there is no CET zone under UST condition and the predicted equiaxed grain size is about 200 μm, which is in good agreement with the experimental results presented here and in Refs. 6), 8) and 9). This validates the mesoscopic solidification model.

The simulation results shown in Fig. 7(c) were obtained assuming a lower superheat of about 10°C, which is significantly lower than the standard high superheat case of about 140°C. The decrease in superheat further enhanced both the nucleation rate and the nucleation grain density. The predicted grain size for this case varies from 100–120 μm.

In Fig. 7(d), the simulation results were obtained assuming UST condition, low superheat of about 10°C and 1 vol.% SiC (50 nm diameter) that were distributed uniformly into the melt. The addition of the SiC nanoparticles further enhanced the cavitation-induced heterogeneous nucleation phenomenon, which is the main mechanism for grain refinement in A356 alloy.¹²⁾ Likewise, the number of nucleation sites per unit volume increased (based on the amount and size of SiC nanoparticles in the melt), which also enhanced further the nucleation rate and therefore decreased the final grain size. The value of this nucleation rate is diminished by the wetting angle between the nucleant (e.g., SiC nanoparticles) and the A356 alloy. Similar to the cases presented in Figs. 7(b) and 7(c), there is also no substantial nucleation barrier and therefore the nucleation rate and nucleation density can also be calibrated based on the experimental measurements of the grain size. The predicted grain size for the low superheat cases varies from 50–60 μm, which are in line with the experimental results in Ref. 17).

4. Concluding Remarks and Future Work

A previously developed model (implemented into Ansys’s Fluent- CFD software¹⁰⁾) was adapted in this study to simulate the flow characteristics during ultrasonic processing of molten metallic alloys under two different gravity conditions: (i) gravitational acceleration oriented downward and (ii) gravitational acceleration oriented upward. It was found that the latter case has significantly stronger bulk stirring and a larger cavitation zone that is more favorable for ultrasonic stirring and cavitation processing of alloys and metal-matrix-nanocomposites.

The solidification microstructure of the A356 alloy under UST condition was predicted using a previously developed stochastic mesoscopic model.^14,15,17) The nucleation parameters were determined via the ultrasonic cavitation model. The predicted solidification microstructures compare favorably with the experimental measurements from Refs. 6), 8), 9) and 17).

Additional experiments on ultrasonic stirring of A356 alloys are being conducted at the University of Alabama at Tuscaloosa, Solidification Laboratory, to advance the current understanding of the influence of ultrasonic cavitation and acoustic streaming on the microstructure evolution of alloys under high and low superheat conditions. The developed UST modeling capability will be used to study the globular/dendritic transition in alloys that are solidifying in the presence of ultrasonic stirring at low and high superheat. Also, the present UST model is currently modified (adding a discrete phase model to predict the dispersion of the heterogeneous nuclei and nanoparticles into the melt) to assist in the development and optimization of metal-matrix-nanocomposites.

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