A least-squares finite element method for 3D incompressible Navier-Stokes equations

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

[1]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[2]  G. Carey,et al.  Element‐by‐element linear and nonlinear solution schemes , 1986 .

[3]  K. Morgan,et al.  LEAST SQUARES FINITE ELEMENT SOLUTION OF COMPRESSIBLE AND INCOMPRESSIBLE FLOWS , 1992 .

[4]  R. Glowinski,et al.  FINITE ELEMENT METHODS FOR NAVIER-STOKES EQUATIONS , 1992 .

[5]  B. Jiang.  A least‐squares finite element method for incompressible Navier‐Stokes problems , 1992 .

[6]  Robert L. Street,et al.  The Lid-Driven Cavity Flow: A Synthesis of Qualitative and Quantitative Observations , 1984 .

[7]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[8]  J. E. Akin,et al.  Semi‐implicit and explicit finite element schemes for coupled fluid/thermal problems , 1992 .

[9]  Robert L. Street,et al.  Non-linear transient phenomena in a complex recirculating flow: A numerical investigation , 1988 .

[10]  T. Taylor,et al.  A Pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations , 1987 .

[11]  Jae Min Hyun,et al.  Numerical simulation of three-dimensional flow structure in a driven cavity , 1989 .

[12]  Robert L. Street,et al.  On End Wall Effects in a Lid-Driven Cavity Flow , 1984 .

[13]  Philip M. Gresho,et al.  On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory , 1990 .