This article introduces a block preconditioner to solve large-scale block structured saddle point systems using a Krylov-based method. Such saddle point systems arise, e.g., in the Riccati-based feedback stabilization approach for multi-field flow problems as discussed in [2]. Combining well known approximation methods like a least-squares commutator approach for the Navier-Stokes Schur complement, an algebraic multigrid method, and a Chebyshev-Semi-Iteration, an efficient preconditioner is derived and tested for different parameter sets by using a simplified reactor model that describes the spread concentration of a reactive species forced by an incompressible velocity field.