WIAS Preprint No. 2776, (2020)

Optimal control of a buoyancy-driven liquid steel stirring modeled with single-phase Navier--Stokes equations


  • Wilbrandt, Ulrich
  • Alia, Najib
  • John, Volker
    ORCID: 0000-0002-2711-4409

2010 Mathematics Subject Classification

  • 65M60 76F65


  • ladle stirring, single-phase Navier--Stokes equations, turbulent incompressible flows, optimal control of PDEs, finite element method




Gas stirring is an important process used in secondary metallurgy. It allows to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions which can be detrimental for the end-product quality. In this process, argon gas is injected from two nozzles at the bottom of the vessel and rises by buoyancy through the liquid steel thereby causing stirring, i.e., a mixing of the bath. The gas flow rates and the positions of the nozzles are two important control parameters in practice. A continuous optimization approach is pursued to find optimal values for these control variables. The effect of the gas appears as a volume force in the single-phase incompressible NavierStokes equations. Turbulence is modeled with the Smagorinsky Large Eddy Simulation (LES) model. An objective functional based on the vorticity is used to describe the mixing in the liquid bath. Optimized configurations are compared with a default one whose design is based on a setup from industrial practice.

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