Velocity determination in conducting fluids via an inverse method

Gunter Gerbeth, Frank Stefani
Forschungszentrum Rossendorf
P.O. Box 510119, D-01314 Dresden, Germany



Usually, MHD is understood as the forward problem of determining velocity and magnetic fields in the sense of solving the coupled system of Navier-Stokes equations and induction equation if some boundary conditions or types of forcing are given. For most laboratory and technical applications with small magnetic Reynolds number Rm the magnetic field is disturbed only slightly by the flow whereas it can modify the latter significantly via Lorentz forces. For higher Rm, however, the fluid flow can change the magnetic field drastically or can even lead to self-excitation of a magnetic field via the dynamo effect.

In the inverse problems approach one strives to get information about the distribution of certain material parameters or temperature and velocity fields inside the fluid from measurements of appropriate quantities at the fluid boundary and/or outside the fluid. We restrict our interest on the spatial reconstruction of the velocity field (or of related mean-field quantities) purely from magnetic fields and electric potentials which can be measured outside the fluid body and at its boundary, respectively.
For the case of small Rm, which is most interesting for a number of technological applications like metal casting and crystal growth, it is necessary to apply external magnetic fields and to measure the flow induced magnetic fields and electric potentials. Analytical results concerning the uniqueness problem of velocity reconstruction /1/, /2/ as well as numerical illustrations /3/ have been published recently. As the inverse problem is ill-posed, appropriate regularization techniques must be applied. For the sake of illustration, consider a simple forward problem. Figures 1 shows a model velocity field in a cube which produces an electric potential at the fluid boundary (Figure 2) and an additional magnetic field outside (Figure 3) if it is exposed to an homogeneous magnetic field pointing in z-direction. Our goal is now set up an experiment where the unknown velocity is reconstructed from the measured electric potentials and induced magnetic fields.

Fig. 1: A model velocity field consisting of a poloidal and a toroidal part.
Fig. 2: If the velocity field is exposed to an homogenious magnetic field pointing in z-direction, an electric potential is induced at the fluid boundary which can be measured by potential probes.
Fig. 3: Outside the fluid, an additional magnetic field is induced which must be determined on the background of the external field.


/1/ Stefani, F., Gerbeth, G.: On the uniqueness of velocity reconstruction in conducting fluids from measurements of induced electromagnetic fields, Inverse Problems, 16 (2000), pp. 1-9

/2/ Stefani, F., Gerbeth, G.: A contactless method for velocity reconstruction in electrically conducting fluids, Mesurement Science and Technology, 11 (2000), pp. 758-765

/3/ Stefani, F., Gerbeth, G.: Velocity reconstruction in conducting fluids from magnetic field and electric potential measurements, Inverse Problems, 15 (1999), pp. 771-786

(14.04.2000) Frank Stefani