Theoretical and numerical study on high frequency vibrational convection: Influence of the vibration direction on the flow structure

Abstract

Thermal convection induced simultaneously by horizontal temperature gradient and vibration in a rectangular cavity filled with molten silicon is investigated numerically and theoretically. The time averaged equations of convection are solved in the high-frequency vibration approximation. The Chebyshev spectral collocation method and a Newton-type method based on the Frechet derivative are used in the numerical solution of the streamfunction formulation of the incompressible Navier-Stokes equations. Validation by comparison with previous works has been performed. Different values of the Grashof number Gr and vibrational Grashof number Gr v and all the possible orientations of the vibrations are considered. Numerical results show that depending on the vibration direction, the flow can be amplified or damped, with even the possibility of flow inversion which can occur between critical vibration angles 1 and 2. A general theoretical expression is derived relating these critical angles and the ratio of vibrational to buoyant convection parameters, Gr v /Gr. A very good agreement between the theoretical and numerical results is obtained.

Effect of high frequency vibrations on PV silicon purification

“…With these assumptions, the boundary conditions can be written as follows: Finally, we have to note that the thermal gradient chosen here in accordance with previous simulations of solidification [14][15] (Fig. 1) is opposite to that generally used in thermalvibrational convection studies [24][25]. As a consequence, the optimal vibration direction giving the stronger flow will not be obtained as usually for 45°, but for = 135°.…”

“…The governing equations (1)-(4) subjected to the relevant boundary conditions (5)- (8) are solved with the highly accurate spectral collocation method on Gauss-Lobatto-Chebyshev points, as described in the work of Bouarab et al [25]. The time stepping for these unsteady calculations is based on Euler's method, which, due to the small time steps used for an accurate treatment of the interface conditions, is shown to give similar results as the Crank-Nicolson method.…”

“…As a consequence, the crystal-melt interface can be considered as planar and moving at a constant velocity − deduced from the pulling rate. Under these assumptions and in presence of vibrations, the governing time-dependent equations in the melt, expressed in terms of dimensionless temperature , impurity concentration , and stream functions and associated with the average velocity and the pulsation velocity induced by the vibrations, respectively [18,25], can be written as follows:…”

“…The segregation coefficient is considered as a constant depending on the considered impurity. The boundary conditions for the velocity correspond to no-slip conditions for the average velocity and a nonpermeability condition for the pulsation velocity [18,25]. Concerning the temperature, the boundary conditions are imposed by the constant gradient furnace: the vertical walls have uniform temperatures, ̅ = at the interface and ̅ = + ∇ ̅̅̅̅ at the left crucible wall, and the horizontal walls have a linear temperature variation between these two limit temperatures.…”

“…To give order of magnitudes, the frequency range to be considered extends from a few hundred Hz up to a few MHz, with oscillation amplitudes below mm down to μm. Various theoretical and experimental studies of thermal-vibrational convection (such convection occurring in heated fluids submitted to high-frequency vibrations), in microgravity as well as in ground conditions and for various configurations, are available in the literature [18][19][20][21][22][23][24][25]. In these references, it has been proved that the vibrations have an impact on both the flow pattern and its convective level.…”