  Taylor-Couette flow between rotating cylinders is a classical problem in
fluid mechanics and has been extensively studied in the case of two concentric
circular cylinders. There have been relatively small number of studies in
complex-shaped cylinders with one or both cylinders rotating. In this paper, we
study the characteristics of Taylor cells in an elliptical outer cylinder with
a rotating concentric inner circular cylinder. We numerically solve the
three-dimensional unsteady Navier-Stokes equations assuming periodicity in the
axial direction. We use a Fourier-spectral meshless discretization by
interpolating variables at scattered points using polyharmonic splines and
appended polynomials. A pressure-projection algorithm is used to advance the
flow equations in time. Results are presented for an ellipse of aspect ratio
two and for several flow Reynolds numbers ($Re = \omega r_i (b-r_i))/\nu$,
where $\omega$ = angular velocity [rad/s], $r_i$ = radius of inner cylinder,
$b$ = semi-minor axis, and $\nu$ = kinematic viscosity) from subcritical to
300. Streamlines, contours of axial velocity, pressure, vorticity, and
temperature are presented along with surfaces of Q criterion. The flow is
observed to be steady until $Re = 300$ and unsteady at $Re = 350$.
