  We present simulation friendly detectability conditions for 2D Navier-Stokes
Equation (NSE) with periodic boundary conditions, and describe a generic class
of ``detectable'' observation operators: it includes pointwise evaluation of
NSE's solution at interpolation nodes, and spatial average measurements. For
``detectable'' observation operators we design a global infinite-dimensional
observer for NSE with uncertain possibly destabilizing inputs: in our numerical
experiments we illustrate $H^1$-sensitivity of NSE to small perturbations of
initial conditions, yet the observer converges for known and uncertain inputs.
