Computations of the Numerical Solutions of Higher Class of Navier-Stokes Equations: 2D Newtonian Fluid Flow

Abstract

Abstract This paper presents formulations, computations, investigations and consequences of the various aspects of the numerical solutions of classes C00 and C11 of the two dimensional Navier-Stokes equations in primitive variables u, v, p, τxx, τxy and τyy for incompressible, isothermal and laminar Newtonian fluid flows using p-version Least Squares Finite Element Formulations (LSFEF). The stick-slip problem is used as a model problem in all investigations since this model problem is typical of many other flow situations like contraction, expansion etc. The major thrust of the work presented is to attempt to resolve the local behavior of the solutions in the immediate vicinity of the stick-slip point. The investigations reveal the following: a) The manner in which the stresses are non-dimensionalized in the governing differential equations (GDEs) influences the performance of the iterative procedure of solving non-linear algebraic equations and thus, computational efficiency. b) Solutions of the class C00 are always the wrong class of solutions and thus are always spurious. c) In the flow domains, containing sharp gradients of dependent variables, conservation of mass is difficult to achieve specially at lower p-levels. d) C11 solutions of the Navier-Stokes equations are in conformity with the continuity considerations in the GDEs. e) An augmented form of the Navier-Stokes equations is proposed that always ensures conservation of mass regardless of mesh, p-levels and the nature of the solution gradients. This approach yields the most desired class of C11 solutions. f) It is mathematically established and numerically demonstrated using stick-slip problem that τij are in fact zero at the stick-slip point and the peak values of τxx and τyy must occur, and in fact do, past the stick-slip point in the free field and that peak values of τxy must occur before the stick-slip point on the no-slip boundary. Thus, there is no singularity of τij in the stick-slip problem at the stick-slip point. A significant finding is that imposition of symmetry boundary condition (necessary based on physics) at the stick-slip point even in C11 interpolations is not possible without deteriorating τij behavior in the vicinity of the stick-slip point. However, with the boundary condition, the peak of τxy does occur before the stick-slip point, while the locations of τxx and τyy remain past the stick-slip point in the free field. h) A significant feature of our research work is that we utilize straightforward p-version LSFEF with C00 and C11 type interpolation without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used. All numerical studies are conducted and presented using three different meshes (progressively refined and graded) for two different velocities (0.01 and 100 m/s).