A least‐squares finite element method for time‐dependent incompressible flows with thermal convection

The time‐dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least‐squares finite element method based on a velocity–pressure–vorticity–temperature–heat‐flux (u–P–ω–T–q) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l²‐norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 10⁶, lid‐driven cavity flow at Reynolds numbers up to 10⁴ and flow over a square obstacle at Reynolds number 200, are presented to validate the method.