Higher-Order Unstructured Finite Volume Method for Computational Fluid Dynamics

ABSTRACT This dissertation presents a higher-order finite volume method (FVM) for computational fluid dynamics (CFD) for unstructured mesh topologies using Moving Least-Squares (MLS) as the backbone of the method. The MLS method is improved in several ways. First, the local stencil is weighed using a minimum volume enclosing ellipsoid (MVEE), which better encapsulates the local nodal topology than traditional spherical descriptions. Furthermore, a novel approach, known herein as Affine MLS, uses a spherical transformation of the ellipsoidal weights to map to the unit ball, where direct application of orthogonal polynomial bases can be used. This approach dramatically reduces the condition number of the MLS Moment/Gram matrix, especially on stretched grids which are commonly used for viscous flows and where traditional methods fail. All the MLS methods are also extended to use the Pivoting QR method for matrix inversion. The MLS method and improvements are extensively tested with several analytical functions for the full MLS reconstruction and fully diffuse derivatives. Optimal scaling parameters are also determined for the MLS method. Additionally, from work with MLS, the boundary conditions of the higher-order method are enforced with ghost nodes, an approach more commonly used in Immersed Boundary Methods. These boundary conditions do a better job of enforcing the boundary states, since they are included directly into the fluxes and gradients. Non-reflecting ghost nodes are implemented using the Navier–Stokes Characteristic Boundary Condition (NSCBC) for the inlet, outlet, and freestream boundary conditions for the first time in a finite volume ghost boundary node context. A higher-order viscous state reconstruction is presented as well wherein the MLS method is used to deter-mine the state and derivatives at the quadrature location. Some simple test cases are presented that highlight the benefits of the ghost node boundary conditions and viscous flux reconstruction. Finally, the higher-order CFD method is applied to the Taylor-Green Vortex (TGV) problem, a benchmark Large-Eddy Simulation (LES) case