Three-dimensional effect of high frequency vibration on convection in silicon melt

Using the Chebyshev spectral method, the effect of high frequency (HF) vibrations on a cubic cell heated from the side and containing a liquid metal is investigated. This study extends the numerical and theoretical two-dimensional results presented in the authors’ past work [Phys. Fluids 31, 043605 (2019)] about the influence of HF vibration direction on the flow structure in a rectangular crucible filled with a liquid metal. The vibration direction can now be three dimensional, and not limited to the main flow plane. In practice, the study considers that the vibration vector is contained in one of the three principal planes of the cavity $(xz,xy$, and $yz$ planes). Two different cases, i.e., under weightlessness and gravity conditions, are considered for each type of vibration to better understand the separate effect of both vibration and buoyancy forces and also their combined effects. Each type of vibration has its own features and affects the flow intensity and patterns differently.

Figure 1

(a) The geometry of the problem and the reference buoyant flow in a parallelepipedic cell of length $L$, width $W$, and height $H$. The cavity can be subjected to a high frequency vibration of amplitude $b$ and angular frequency $\mathrm{\Omega }$. $\mathit{n}$ is the unit vector indicating the direction of the vibration and ${\mathit{g}}_0$ is the static gravity acceleration. Hot and cold walls are colored in red and blue, respectively. Contours of velocity magnitude are plotted in the vertical transverse and longitudinal planes. (b) The coordinates system. The direction of vibration given by $\mathit{n}$ is defined with $\theta$ and $\phi$, the polar and azimuthal angles, respectively.

Figure 2

Comparison of the maximum horizontal (along $x)$ and vertical (along $z)$ velocity components, $u_{\max }$ and $w_{\max }$, obtained by 2D and 3D computations for $xz$ vibrations in all possible directions $(\mathrm{Gr}=1000$ and ${\mathrm{Gr}}_{\mathrm{v}}=10000)$. The 3D computations are performed for a $1\times 1\times 1$ cubic cell and a $1\times 8\times 1$ rectangular duct. Due to symmetry properties, the curves of minimum velocity have exactly the same variations, but with an opposite sign.

Figure 3

Temperature and velocity fields in the reference buoyant case (without vibration) in the differentially heated cubic cell $(1\times 1\times 1$ cavity) for $\mathrm{Gr}=1000\left({\mathrm{Gr}}_{\mathrm{v}}=0\right)$. Left and right boundaries at $x=0$ and $x=1$ are respectively at cold and hot temperatures. Isotherms and velocity vectors are plotted at the three middle planes, $x=0.5,y=0.5,$ and $z=0.5$.

Figure 4

Contours of pulsation velocities $W_x$ (a) and $W_z$ (b) for ${\alpha }_{xz}={60}^{\circ }$ (identical result for ${\alpha }_{xz}={120}^{\circ })$ and contours of the $xz$-vibrational force (main component of ${\mathbf{F}}_{\mathit{Vxz}}/{\mathrm{Gr}}_{\mathrm{v}}$ along ${\mathit{e}}_x)$ for ${\alpha }_{xz}={60}^{\circ }$ (c) and ${\alpha }_{xz}={120}^{\circ }$ (d) in the main $xz$ plane of the differentially heated cubic cell $(1\times 1\times 1$ cavity) for $xz$ vibration with ${\mathrm{Gr}}_{\mathrm{v}}=10000$ and under weightlessness conditions $\left(\mathrm{Gr}=0\right)$.

Figure 5

Horizontal velocity contours, isotherms (white lines), and velocity vectors in the main $xz$ plane of the differentially heated cubic cell $(1\times 1\times 1$ cavity) for $xz$ vibration with ${\mathrm{Gr}}_{\mathrm{v}}=10000$ and under weightlessness conditions $\left(\mathrm{Gr}=0\right)$ for ${\alpha }_{xz}={60}^{\circ }$ (a) and ${\alpha }_{xz}={120}^{\circ }$ (b).

Figure 6

Velocity and temperature fields in the differentially heated cubic cell $(1\times 1\times 1$ cavity) under weightlessness conditions $\left(\mathrm{Gr}=0\right)$ and for ${\mathrm{Gr}}_{\mathrm{v}}=10000$. Vibration is applied in the $z$ direction (i.e., corresponding to ${\alpha }_{xz}={90}^{\circ }$ or ${\alpha }_{yz}={90}^{\circ })$. Left and right boundaries at $x=0$ and $x=1$ are respectively at cold and hot temperatures. Isotherms and velocity vectors are plotted at the three middle planes, $x=0.5,y=0.5,$ and $z=0.5$. A four counter-rotating cells structure is obtained, the axis of the cells being parallel to the $y$ direction. Under weightlessness conditions, an exactly similar four cells structure is also obtained for vibration applied in the $y$ direction, but the cells will have their axis parallel to the $z$ direction.

Figure 7

Maximum velocity components in the differentially heated cubic cell $(1\times 1\times 1$ cavity) as a function of ${\alpha }_{xz}$, the vibration angle in the $xz$ plane (case of $xz$ vibration) for ${\mathrm{Gr}}_{\mathrm{v}}=10000$ and under weightlessness conditions $(\mathrm{Gr}=0$, flows denoted as WF for weightlessness flows). The case without vibration $\left({\mathrm{Gr}}_{\mathrm{v}}=0\right)$ corresponds here to the no-flow situation (blue dashed line). Insets give contours of longitudinal velocity in the middle $yz$ plane $\left(x=0.5\right)$ for ${\alpha }_{xz}={45}^{\circ }$ and ${\alpha }_{xz}={135}^{\circ }$. The three curves are symmetric about ${\alpha }_{xz}={90}^{\circ }$. This corresponds here to similar flows for ${90}^{\circ }-{\alpha }_{xz}$ and ${90}^{\circ }+{\alpha }_{xz}$, but with opposite rotation direction. These curves also give the characteristic velocities as a function of ${\alpha }_{xy}$ for the $xy$ vibration under weightlessness conditions. For that, the curves of $v_{\max }$ and $w_{\max }$ have to be permuted, whereas the curve of $u_{\max }$ is unchanged.

Figure 8

3D flow (fluid particles trajectories) in the differentially heated cubic cell $(1\times 1\times 1$ cavity) under weightlessness conditions $\left(\mathrm{Gr}=0\right)$ when $yz$ vibration is imposed in the direction ${\alpha }_{yz}={45}^{\circ }$ with ${\mathrm{Gr}}_{\mathrm{v}}=10000$.

Figure 9

Maximum velocity components in the differentially heated cubic cell $(1\times 1\times 1$ cavity) as a function of ${\alpha }_{yz}$, the vibration angle in the $yz$ plane (case of $yz$ vibration) for ${\mathrm{Gr}}_{\mathrm{v}}=10000$ and under weightlessness conditions $(\mathrm{Gr}=0$, flows denoted as WF for weightlessness flows). The case without vibration $\left({\mathrm{Gr}}_{\mathrm{v}}=0\right)$ corresponds here to the no-flow situation (blue dashed line). Insets give contours of transverse velocity $v$ and streamlines in the middle $yz$ plane $\left(x=0.5\right)$ and for different values of ${\alpha }_{yz}$. The three curves are symmetric about ${\alpha }_{yz}={90}^{\circ }$. This corresponds here to exactly similar flows for ${90}^{\circ }-{\alpha }_{yz}$ and ${90}^{\circ }+{\alpha }_{yz}$.

Figure 10

Maximum longitudinal velocity $u_{\max }$ (along $x)$ in the differentially heated cubic cell $(1\times 1\times 1$ cavity) as a function of the vibration angle $\alpha$, when the vibration vector is contained in one of the three principal planes $(xz,xy$, and $yz$ planes) for $\mathrm{Gr}=1000$ and ${\mathrm{Gr}}_{\mathrm{v}}=10000$. The results are given with solid lines of different colors. The corresponding results in weightlessness conditions (denoted as WF) are also given as dashed lines for comparison (note that the same curve, plotted as black dashed line, is obtained for $xz$ and $xy$ vibration). Blue dashed line corresponds to the pure buoyant flow (reference case with ${\mathrm{Gr}}_{\mathrm{v}}=0)$. Insets give contours of the longitudinal velocity $u$ in the middle $yz$ plane $\left(x=0.5\right)$ for the reference buoyant case $\left({\mathrm{Gr}}_{\mathrm{v}}=0\right)$ and for the pure vibrational cases (weightlessness cases, $\mathrm{Gr}=0)$ and convectovibrational cases for $xy$ vibration with angles ${\alpha }_{xy}={45}^{\circ }$ and ${135}^{\circ }$.

Figure 11

Maximum vertical velocity $w_{\max }$ (along $z)$ in the differentially heated cubic cell $(1\times 1\times 1$ cavity) as a function of the vibration angle $\alpha$, when the vibration vector is contained in one of the three principal planes $(xz,xy$, and $yz$ planes) for $\mathrm{Gr}=1000$ and ${\mathrm{Gr}}_{\mathrm{v}}=10000$. The results are given with solid lines of different colors. The corresponding results in weightlessness conditions (denoted as WF) are also given as dashed lines for comparison. Blue dashed line corresponds to the pure buoyant flow (reference case with ${\mathrm{Gr}}_{\mathrm{v}}=0)$. Insets give contours of the vertical velocity $w$ and streamlines in the middle $xz$ plane $\left(y=0.5\right)$ for $xz$ vibration with angles ${\alpha }_{xz}={45}^{\circ },{90}^{\circ }$, and ${135}^{\circ }$.

Figure 12

Maximum transverse velocity $v_{\max }$ (along $y)$ in the differentially heated cubic cell $(1\times 1\times 1$ cavity) as a function of the vibration angle $\alpha$, when the vibration vector is contained in one of the three principal planes $(xz,xy$, and $yz$ planes) for $\mathrm{Gr}=1000$ and ${\mathrm{Gr}}_{\mathrm{v}}=10000$. The results are given with solid lines of different colors. The corresponding results in weightlessness conditions (denoted as WF) are also given as dashed lines for comparison. Blue dashed line corresponds to the pure buoyant flow (reference case with ${\mathrm{Gr}}_{\mathrm{v}}=0)$. Insets give contours of the longitudinal velocity $u$ and streamlines in the middle $xz$ plane $\left(y=0.5\right)$ for $yz$ vibration with different angles ${\alpha }_{yz}$.

Figure 13

Velocity and temperature fields in the differentially heated cubic cell $(1\times 1\times 1$ cavity) for $\mathrm{Gr}=1000$ and ${\mathrm{Gr}}_{\mathrm{v}}=10000$ when vibration is applied in the $z$ direction (i.e., corresponding to ${\alpha }_{xz}={90}^{\circ }$ or ${\alpha }_{yz}={90}^{\circ })$. Left and right boundaries at $x=0$ and $x=1$ are respectively at cold and hot temperatures. Isotherms and velocity vectors are plotted at the three middle planes, $x=0.5,y=0.5,$ and $z=0.5$.

Figure 14

Velocity and temperature fields in the differentially heated cubic cell $(1\times 1\times 1$ cavity) for $\mathrm{Gr}=1000$ and ${\mathrm{Gr}}_{\mathrm{v}}=10000$ when vibration is applied in the $y$ direction (i.e., corresponding to ${\alpha }_{xy}={90}^{\circ }$ or ${\alpha }_{yz}=0^{\circ })$. Left and right boundaries at $x=0$ and $x=1$ are respectively at cold and hot temperatures. Isotherms and velocity vectors are plotted at the three middle planes, $x=0.5,y=0.5,$ and $z=0.5$.