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A Mathematical and Numerical Framework for the Simulation of Oscillatory Buoyancy and Marangoni Convection in Rectangular Cavities with Variable Cross Section

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Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 50))

Abstract

It is often assumed that two-dimensional flow can be used to model with an acceptable degree of approximation the preferred mode of instability of thermogravitational flows and thermocapillary flows in laterally heated shallow cavities for a relatively wide range of substances and conditions (essentially pure or compound semiconductor materials in liquid state for the case of buoyancy convection and molten oxide materials or salts and a variety of organic liquids for the case of Marangoni convection). In line with the general spirit of this book, such assumption is challenged by comparing two-dimensional and three-dimensional results expressly produced for such a purpose. More precisely, we present a general mathematical and numerical framework specifically developed to (1) explore the sensitivity of such phenomena to geometrical “irregularities” affecting the liquid container and (2) take advantage of a reduced number of spatial degrees of freedom when this is possible. Sudden variations in the shape of the container are modelled as a single backward-facing or forward-facing step on the bottom wall or a combination of both features. The resulting framework is applied to a horizontally extended configuration with undeformable free top liquid-gas surface (representative of the Bridgman crystal growth technique) and for two specific fluids pertaining to the above-mentioned categories of materials, namely molten silicon (Pr  <  1) and silicone oil (Pr  >  1. The assumption of flat interface is justified on the basis of physical reasoning and a scaling analysis. The overall model proves successful in providing useful insights into the stability behaviour of these fluids and the departure from the approximation of two-dimensional flow. It is shown that the presence of a topography in the bottom wall can lead to a variety of situations with significant changes in the emerging waveforms.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of Strathclyde, James Weir Building, 75 Montrose Street, Glasgow, G1 1XJ, UK

    Marcello Lappa

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  1. Marcello Lappa
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Correspondence to Marcello Lappa .

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  1. School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel

    Alexander Gelfgat

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Lappa, M. (2019). A Mathematical and Numerical Framework for the Simulation of Oscillatory Buoyancy and Marangoni Convection in Rectangular Cavities with Variable Cross Section. In: Gelfgat, A. (eds) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-91494-7_12

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  • DOI: https://doi.org/10.1007/978-3-319-91494-7_12

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