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Numerical Analysis of Viscous Flow Using Composite Grid with Application to Glass Furnaces
The glass for glassware that surrounds us in our daily life:
The raw materials (soda, sand) are put inside the furnace at one end, then they are heated by gas burners and melt gradually, thus forming a glass flow:
Studying the glass flow is essential for understanding melting and mixing processes. Some phenomena in the furnace occur locally but nevertheless affect the global pattern of the flow. We consider bubbling -- a process used to enforce a natural convection in the melt, and stirring -- a process intended to prevent untimely solidification of the melt:
The flow model consists of equations for an incompressible viscous fluid, equations of state and boundary conditions imposed by the geometry of the furnace. The model is too complex to be solved by analytical methods; therefore we need to use numerical methods.
The resolution of the local phenomena in the flow by conventional methods have undesirable effects: the refinement based on non-uniform tensor-product grids leads to excessive memory and computation time usage, the refinement based on a single discretisation on a composite grid leads to a very complex data structure. These difficulties can be avoided by using a technique called Local Defect Correction (LDC). LDC is an iterative procedure that combines the results on a coarse global grid (covering the global domain) and fine local grid (covering the high activity areas).
Since LDC is not a solution method by itself we first build up a general solution algorithm for 3D viscous flow problems. To that end we consider a pressure correction algorithm in conjunction with a collocated grid:
Pressure correction methods are iterative methods aiming at solving viscous flow problems (Navier-Stokes or Stokes equations). For each iteration of the pressure correction algorithm one needs to solve a linear system for the pressure. The system is singular, which reflects the fact that the pressure is included in the flow equation via the gradient only. In practice the system for the pressure turns out to be inconsistent in most cases. We suggest a procedure for mending this inconsistency. The procedure is based on the explicit formulation of the consistency condition which depends on the boundary conditions for the velocity only. By correcting the latter we manage to satisfy the consistency condition.
The solution method is then used in LDC iterations. We consider two applications of the LDC iterations for the flow problems. In the first both the problem on the global grid and the subproblem on the local grid are formulated in cartesian coordinates. The second application of LDC combines the solutions of the problems in the two different coordinates systems: cartesian and cylindrical.
A novelty in LDC for flow problems is a correction of the boundary conditions for the problem on a local grid. These boundary conditions are obtained by means of interpolation from the solution on a global grid. Without correction they give rise to inconsistent systems of equations. We suggest a correction procedure based on a discrete global mass balance. As we show, the correction introduces an error comparable with the interpolation error.
These LDC approaches are applied to two specific high activity problems, namely bubbling and stirring. For the bubbling we use a variant of LDC with global and local grids formulated in cartesian coordinates:
The stirring is solved by LDC combining a cartesian global grid and a cylindrical local grid:
A posteriori error estimates show second order accuracy of the LDC solution. For both examples we observe a fast convergence of LDC iterations.
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