Direct numerical simulation (DNS) of forced plumes arising from input of both momentum and buoyancy into an ambient fluid is presented. The large vortical structures in the near field of thermal and reactive plumes are investigated. Boundary conditions associated with the spatial DNS of open-boundary buoyant flows that are compatible with the modern non-dissipative, high-order, finite-difference schemes have been developed. The governing equations for flow and combustion at the plume centerline are put into a special form to circumvent the singularity at the axis associated with the cylindrical coordinates. Mixing is found to be stronger in the planar thermal plume than in the axisymmetric case. An explanation is provided based on the vorticity budget. Axisymmetric reactive plumes with a one-step reaction governed by the Arrhenius kinetics have also been studied. The unsteady effects of chemical heat release and combustion-induced buoyancy on the flow structures are investigated. Budgets of the vorticity transport are examined to reveal the mechanisms leading to the formation and evolution of large vortical structures in forced plumes. It is found that volumetric expansion due to chemical heat release tends to destroy vorticity, while combustion-induced buoyancy under the gravitational effect generates vorticity. The gravitational term in the vorticity transport equation is found to be the main mechanism for the buoyant flow instability and the development of counter-rotating vortices in reactive plumes.