  In this paper, a numerical investigation of power-law fluid flow in the
trapezoidal cavity has been conducted by incompressible finite-difference
lattice Boltzmann method (IFDLBM). By designing the equilibrium distribution
function, the Navier-Stokes equations (NSEs) can be recovered exactly. Through
the coordinate transformation method, the body-fitted grid in physical region
is transformed into a uniform grid in computational region. The effect of
Reynolds (Re) number, the power-law index $n$ and the vertical angle {\theta}
on the trapezoidal cavity are investigated. According to the numerical results,
we come to some conclusions. For low Re number Re=100, it can be found that the
behavior of power-law fluid flow becomes more complicated with the increase of
n. And as vertical angle {\theta} decreases, the flow becomes smooth and the
number of vortices decreases. For high Re numbers, the flow development becomes
more complex, the number and strength of vortices increase. If the Reynolds
number increases further, the power-law fluid will changes from steady flow to
periodic flow and then to turbulent flow. For the steady flow, the lager the
{\theta}, the more complicated the vortices. And the critical Re number from
steady to periodic state decreases with the decrease of power-law index n.
