Modeling, Discontinuous Galerkin Approximation and Simulation of the 1-D Compressible Navier Stokes Equations

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Virginia Tech


In this thesis we derive time dependent equations that govern the physics of a thermal fluid flowing through a one dimensional pipe. We begin with the conservation laws described by the 3D compressible Navier Stokes equations. This model includes all residual terms resulting from the 1D flow approximations. The final model assumes that all the residual terms are negligible which is a standard assumption in industry. Steady state equations are obtained by assuming the temporal derivatives are zero. We develop a semi-discrete model by applying a linear discontinuous Galerkin method in the spatial dimension. The resulting finite dimensional model is a differential algebraic equation (DAE) which is solved using standard integrators. We investigate two methods for solving the corresponding steady state equations. The first method requires making an initial guess and employs a Newton based solver. The second method is based on a pseudo-transient continuation method. In this method one initializes the dynamic model and integrates forward for a fixed time period to obtain a profile that initializes a Newton solver. We observe that non-uniform meshing can significantly reduce model size while retaining accuracy. For comparison, we employ the same initialization for the pseudo-transient algorithm and the Newton solver. We demonstrate that for the systems considered here, the pseudo-transient initialization algorithm produces initializations that reduce iteration counts and function evaluations when compared to the Newton solver. Several numerical experiments were conducted to illustrate the ideas. Finally, we close with suggestions for future research.



Navier Stokes, Derivation, DG Method, Simulation, Pseudo-Transient