  We have performed numerical simulations of boundary-driven dynamos using a
three-dimensional non-linear magnetohydrodynamical model in a spherical shell
geometry. A conducting fluid of magnetic Prandtl number Pm=0.01 is driven into
motion by the counter-rotation of the two hemispheric walls. The resulting flow
is of von K\'arm\'an type, consisting of a layer of zonal velocity close to the
outer wall and a secondary meridional circulation. Above a certain forcing
threshold, the mean flow is unstable to non-axisymmetric motions within an
equatorial belt. For fixed forcing above this threshold, we have studied the
dynamo properties of this flow. The presence of a conducting outer wall is
essential to the existence of a dynamo at these parameters. We have therefore
studied the effect of changing the material parameters of the wall (magnetic
permeability, electrical conductivity, and thickness) on the dynamo. In common
with previous studies, we find that dynamos are obtained only when either the
conductivity or the permeability is sufficiently large. However, we find that
the effect of these two parameters on the dynamo process are different and can
even compete to the detriment of the dynamo. Our self-consistent approach allow
us to analyze in detail the dynamo feedback loop. The dynamos we obtain are
typically dominated by an axisymmetric toroidal magnetic field and an axial
dipole component. We show that the ability of the outer shear layer to produce
a strong toroidal field depends critically on the presence of a conducting
outer wall, which shields the fluid from the vacuum outside. The generation of
the axisymmetric poloidal field, on the other hand, occurs in the equatorial
belt and does not depend on the wall properties.
