Transition to Tubulence in the Hartmann Layer A. Thess 1, D.Krasnov 1, E. Zienicke 1, O. Zikanov 2, T. Boeck 3 1-Ilmenau University of Technology 2-University of Michigan 3-Laboatoire de Modelisation en Mecanique 1.What is a Hartmann layer? 2.Hartmann‘s experimental work 3.Theoretical background 4.Simulations of transition in the Hartmann layer 5.Experiments on transition in the Hartmann layer 6.Summary GAMM-Tagung, 24 March 2004

1 – What is a Hartmann layer? B v Velocity profile Hartmann number:

1 - Where do Hartmanns layers occur? Continous casting of steel Semiconductor crystal growth Turbulence research Fusion reactors Magnetic field flow wall Hartmann boundary layer

2 - Hartmann‘s experimental work Pipe flow Channel flow Results: pressure drop h (Re, Ha) h B h B

2 - Hartmann‘s experimental work Observation: Transition from turbulent to laminar flow occurs for h ~ Ha²  v ~ Ha Conclusion: The pertinent parameter for transition must be R = with R C  200 Question: How can we compute R C ? R =

3 – Theoretcal background: basic equation Navier-Stokes equation with Lorentz force Decomposition Acceleration Advection Stretching Nonlinearity viscous electomagnetic damping damping Result:

3 – Theoretical background: stability Question: Why does the transition take place at R=200? ? STABLE UNSTABLE R Hartmann } Perturbation amplitude

4 – Simulations: conceptual basis Review: S. Grossmann, Rev. Mod. Phys. Vol 72,2 (2000) pp. 603 – 618 stationary transient growth oflinearly unstable base flow +small 2d perturbations + 3d perturbations t t t E E E

4 – Simulations: energy evolution Direct numerical simulation: Krasnov, Zikanov, Zienicke, Thess (2002) 3d 2d

4 – Simulations: dynamics at Ha=40

5 - Experiments Pablo Moresco & Thierry Alboussiere (Cambridge University) 2002 Experimental setup Result R c =380 Numerical prediction (Krasnov et al 2002) R c =350

6 - Summary Transition scenario similar to ordinary hydrodynamics Numerical simulations provide valuable tool for visualization Recent experiments support numerical results More detailed simulations necessary to determine A(R) New experiments necessary with flow visualization (B=10 Tesla) Krasnov, Zienicke, Boeck, Zikanov, Thess, Numerical investigation of the stability of the Hartmann flow, J. Fluid Mech (2004) in press. Moresco, Alboussiere, Experimental investigation of the stability of the Hartmann flow, J. Fluid Mech (2004) in press.

Acknowledgments Dimitry Krasnov Oleg Zikanov numerical simulation Egbert Zienicke Yuri Kolesnikov Oleg Andreev discussion Rene Moreau Deutsche Forschungsgemeinschaft Thüringer Forschungsministerium } } } financial support