The reliable and effective assimilation of measurements and numerical simulations in
engineering applications involving computational fluid dynamics is an emerging problem as
soon as new devices provide more data. In this paper we are mainly driven by hemodynamics
applications, a field where the progressive increment of measures and numerical tools
makes this problem particularly up-to-date. We adopt a Bayesian approach to the inclusion
of noisy data in the incompressible steady Navier-Stokes equations (NSE). The purpose is
the quantification of uncertainty affecting velocity and flow related variables of
interest, all treated as random variables. The method consists in the solution of an
optimization problem where the misfit between data and velocity - in a convenient norm -
is minimized under the constraint of the NSE. We derive classical point estimators, namely
the maximum a posteriori – MAP – and the maximum likelihood – ML – ones.
In addition, we obtain confidence regions for velocity and wall shear stress, a flow
related variable of medical relevance. Numerical simulations in 2-dimensional and
axisymmetric 3-dimensional domains show the gain yielded by the introduction of a complete
statistical knowledge in the assimilation process.