Finite element analysis of incompressible laminar boundary layer flows

A numerical procedure was developed to solve the two‐dimensional and axisymmetric incompressible laminar boundary layer equations using the semi‐discrete Galerkin finite element method. Linear Lagrangian, quadratic Lagrangian, and cubic Hermite interpolating polynomials were used for the finite element discretization; the first‐order, the second‐order backward difference approximation, and the Crank‐Nicolson method were used for the system of non‐linear ordinary differential equations; the Picard iteration and the Newton‐Raphson technique were used to solve the resulting non‐linear algebraic system of equations. Conservation of mass is treated as a constraint condition in the procedure; hence, it is integrated numerically along the solution line while marching along the time‐like co‐ordinate. Among the numerical schemes tested, the Picard iteration technique used with the quadratic Lagrangian polynomials and the second‐order backward difference approximation case turned out to be the most efficient to achieve the same accuracy. The advantages of the method developed lie in its coarse grid accuracy, global computational efficiency, and wide applicability to most situations that may arise in incompressible laminar boundary layer flows.