Ph.D. Defense
Chase Leibenguth
(Advisor: Prof. Dimitri Mavris)
"VIDEC-CFD: A Methodology for Variational Integration of a Discrete Exterior Calculus-Based Computational Fluid Dynamics Formulation"
Thursday, October 31
9:30 a.m.
Weber SST, CoVE
Abstract
Computational Fluid Dynamics has revolutionized the design process of aerospace systems in the decades since its introduction. The improvements provided by CFD in conjunction with physical experiments have enabled database creation for aerospace design, design cycle cost and time reductions, and experimental extrapolation corrections for full-scale flight vehicles at in-flight Reynolds Numbers. Despite such valuable applications, there are critical shortcomings in current CFD methodologies.
Turbulent vortex formation, shedding, and interactions with the surrounding flow field are difficult to resolve without significant financial and computational cost and complexity. Furthermore, fundamental invariants, such as circulation and angular momentum, are not conserved in discretized space-time. These quantities may be conserved at the limit of a steady-state solution or an infinitely refined mesh, however, not all problems have such properties. Failing to preserve these invariants will affect the final obtained solution.
An area of research that shows promise involves Consistent Spatial Discretization and Discrete Exterior Calculus-based CFD solvers. The governing equations of fluid mechanics are re-derived using operators from Discrete Differential Geometry and Topology. These operators enable the inherent, exact conservation of theorems, such as Stokes' or Noether's, in the flow domain. Discrete analogs have been constructed that take these operators from a continuous domain to a discretized space.
So far, DEC-based solvers have been derived and applied to viscous and inviscid incompressible flows using only the continuity and momentum equations. No turbulence models were required to resolve turbulent phenomena that arose as flows moved from inviscid to highly viscous. The inherent preservation of underlying geometric quantities related to the governing equations and fundamental physical invariants on a discretized domain mitigated the need for additional models. Application of variational integrators enabled discrete time integration that preserves fundamental invariants during numerical integration in time. However, all of these methodologies relied specifically on the divergence-free velocity field present in an incompressible flow.
The goal of the following research is to extend the cited methodologies to include the energy and entropy equations for modeling viscous, compressible, subsonic flows. The energy equation is essential to this extension to account for viscous dissipation effects on a compressible fluid. The entropy equation is included alongside specific variable definitions to ensure the combined set of continuous and discrete governing equations are Thermodynamically Consistent. That requirement ensures the governing equations inherently satisfy the 1st and 2nd Laws of Thermodynamics topologically.
The first major research contribution is the derivation of the full set of governing equations of fluid mechanics from a Least Action Principle obtained from a symmetric Lie-Bracket and anti-symmetric Poisson Bracket. The second contribution is the derivation of a coordinate-free representation of the viscous dissipation function in terms of DEC operators for a viscous, compressible, homogenous, single-phase Newtonian fluid.
Committee
- Prof. Dimitri Mavris – School of Aerospace Engineering (advisor)
- Prof. Marilyn Smith – School of Aerospace Engineering
- Prof. Graeme Kennedy – School of Aerospace Engineering
- Prof. John Etnyre – School of Mathematics
- Dr. Neil Weston – Aerospace Systems Design Laboratory