Computations of viscous compressible flows in h, p, k finite element framework with variationally consistent integral forms

This thesis presents mathematical models for time dependent and stationary viscous compressible flows based on conservation laws, constitutive equations and equations of state using Eulerian description. In the presence of physical viscosity, conductivity and other transport properties, the mathematical models are well recognized Navier-Stokes equations. Variable transport properties as well as ideal and real gas models are considered for equations of state. The mathematical models are a highly non-linear coupled partial differential equations in space and time. The mathematical and computational infrastructure using finite element method is presented for obtaining numerical solutions of the Boundary Value Problems and Initial Value Problems associated with the mathematical models. This infrastructure is based on h, p, k (h-characteristic length, p-degree of local approximation, k-order of approximation space) as independent computational parameters with an additional requirement that the integral form be variationally consistent in case of Boundary Value Problems and space-time variationally consistent in case of Initial Value Problems. All methods of approximation except Least Squares and Space-Time Least Squares Processes are Variationally Inconsistent. Variational Consistency and Space-Time Variational Consistency of integral forms ensure unconditionally stable computational processes.; A variety of numerical studies are presented for Initial Value Problems as well as Boundary Value Problems. 1-D transient viscous form of Burgers equation, 1-D Riemann shock tube with ideal and real gas models and Boundary Value Problems in 2-D compressible flow: Carter's plate with Mach 1, 2, 3 and 5 flows and Mach 1 flow past a circular cylinder are used as model problems. Shock evolution, propagation, interactions and reflection are quantified based on the rate of entropy production using Air as a medium for 1-D Riemann shock tube. It is clearly established that rarefaction shocks are not possible for FC70 for any choice of initial conditions. In all studies evolution of a shock is presented (unlike the published work). Its existence and sustained propagation is established based on Sr, the rate of entropy production per unit volume. In case of transient Burgers equation it is demonstrated that time accurate evolutions can be computed for any finite Reynolds number. Contrary to the common belief, the work presented here shows that solutions of Boundary Value Problems in compressible flows present no special problems.; In Summary: (i) the mathematical models for the compressible flow are based on Navier-Stokes equations. (ii) computational infrastructure is based on hpk and unconditionally stable integral forms with higher order global differentiability in space and time. (iii) All numerical studies utilize actual transport properties of the medium. (iv) Up-winding methods such as SUPG, SUPG/DC, SUPG/DC/LS are neither needed nor used. (v) existence of shocks is established through evolution and not using Rankine-Hugoniot relations. (vi) Governing Differential Equations in the mathematical models are neither linearized nor altered in any form during the entire process of formulation and computations.; The work presented here clearly demonstrates that the numerical simulations of Boundary Value Problems and Initial Value Problems based on Navier-Stokes equations describing viscous compressible flows can be done in a straight forward manner in h, p, k framework with Variational Consistent and Space-Time Variationally Consistent integral forms. The computational processes always remain unconditionally stable. The mathematical models based on Euler's equations lack physics, computational methods for Euler's equations use problem dependent up-winding methods which lack mathematical basis and rigor and thus in our view are of little merit if at all for numerical simulations of Boundary Value Problems and Initial Value Problems in compressible fl...