  In this paper, we analyze the numerical approximation of the Navier-Stokes
problem over a bounded polygonal domain in $\mathbb{R}^2$, where the initial
condition is modeled by a log-normal random field. This problem usually arises
in the area of uncertainty quantification. We aim to compute the expectation
value of linear functionals of the solution to the Navier-Stokes equations and
perform a rigorous error analysis for the problem. In particular, our method
includes the finite element, fully-discrete discretizations, truncated
Karhunen-Lo\'eve expansion for the realizations of the initial condition, and
lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values
over the parameter space. Our QMC analysis is based on randomly-shifted lattice
rules for the integration over the domain in high-dimensional space, which
guarantees the error decays with $\mathcal{O}(N^{-1+\delta})$, where $N$ is the
number of sampling points, $\delta>0$ is an arbitrary small number, and the
constant in the decay estimate is independent of the dimension of integration.
