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

SUMMARY 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-w- T-q) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the 12-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 lo6, lid-driven cavity flow at Reynolds numbers up to lo4 and flow over a square obstacle at Reynolds number 200, are presented to validate the method. The past decade has witnessed a great deal of progress in the area of computational fluid dynamics. Numerous flow problems have been successfully solved by finite difference, finite volume and finite element methods. Most finite element methods are based on the Galerkin method, 9 ' the Taylor-Galerkin method and the Petrov-Galerkin method. 3-5 Mixed-order interpolation and penalty approach are commonly used in these methods. It is well known that these methods often lead to large, sparse, unsymmetric linear systems which are difficult to solve numerically. This explains why finite element analysis for three-dimensional fluid flow problems is not a common practice. To overcome this difficulty, we propose and develop a Least-Squares Finite Element Method (LSFEM) for time-dependent incompressible flow problems. The linear systems resulting from the discretization of LSFEM are always symmetrical and positive-definite. Therefore, they can be solved more easily and efficiently. This is the main reason for investigating the least-squares finite element approach. Least-squares finite element methods have already been applied with some success to com- pressible Euler and hyperbolic equations. Jiang and Carey6- ' and Jiang and Povinelli' used an implicit method for compressible flows. To further the capabilities of the method, Lefebvre et al.' applied unstructured triangular meshes to compressible flow problems. For transient advection problems, Donea and Quartapelle lo classified four different LSFEM approaches: characteristic LSFEM proposed by Li," LSFEM by Carey and Jiang," Taylor LSFEM by Park and Liggett l3 and space-time LSFEM by Nguyen and Reynen.I4