Flows of Incompressible Newtonian and Generalized Newtonian Fluids over a Circular Cylinder

Abstract

This thesis presents numerical solutions of the boundary value problems describing the isothermal and non-isothermal steady flows of incompressible Newtonian, power-law and Carreau fluids over a circular cylinder using the hpk-finite element process based on the residual functional (least squares process). This computational framework yields unconditionally stable algebraic systems for non-linear partial differential equations that result from the mathematical models, regardless of the choices of h, p and k and the dimensionless parameters in the mathematical models. It is shown that for such fluids, the energy equation and heat flux equations are decoupled (or weakly coupled) from the rest of the mathematical model resulting from the conservation of mass, balance of momenta and the constitutive theory for the deviatoric Cauchy stress tensor. Thus, one could solve for velocities, pressure and deviatoric Cauchy stress independent of the energy equation and heat flux equations. However, the weak coupling between the energy equation and the heat flux equations and the remaining mathematical model permits numerical solutions of the combined mathematical model in the present computational framework. Numerical studies are presented for progressively increasing flow rates corresponding to Re = 20, 40, 60, 100 and 200 for Newtonian and Carreau fluids and Re_n = 15.6, 37.2, 64.2, 118.8 and 285.0 for power-law fluids. The inlet length and the height of the domain are established so that boundaries of the domain do not influence the flow feature around and in the neighborhood of the cylinder for all Reynolds numbers considered. The choice of discretization and p-levels are determined such that the integrated sum of the squares of the residuals for the whole domain are always of the order of O(10^-6) or lower for converged solutions. The choice of

g_i

max <= O(10^-6) always ensures that Newton's linear method with line search yields an accurate solution of the system of non-linear algebraic equations resulting from the least squares process. The residual functional values of the order of O (10^-6) or lower ensure that GDEs are satisfied accurately over the entire domain and, thus the numerical solutions presented in this thesis can be viewed as benchmark quality solutions. In cases of generalized Newtonian fluids (power-law and Carreau models) only shear thinning fluids are considered. Numerical studies demonstrate decoupled behavior of the temperature field from the rest of the deformation field. Shear thinning behavior and viscous dissipation for progressively increasing Reynolds numbers are simulated accurately without any difficulty.