Radically New, Hyperbolic Navier-Stokes Method for Complex Simulations, Year 3

In our approach, we make gradients as independent variables, and reformulate the original governing equations as a system of equations that is hyperbolic in pseudo-time. We then construct innovative efficient and compact high-order schemes for the reformulated governing equations, and solve the system, including the solution gradients, directly with our proposed high-order schemes. In this approach not only we evaluate the solution gradients with higher order of accuracy but also we predict them with a superior quality on irregular elements. In FY15, we proposed a new Design Principle, which dictates how the fluxes should be discretized to ensure computed gradients are accurate and smooth, and constructed up to third-order Residual-Distribution (RD) scheme for irregular triangular elements [J. Comput. Phys. 300(2015), pp. 455-491; Computers and Fluids 131(2016), pp. 29-44]. The FY15 proposed high-order schemes increase the degrees-of-freedom (DoF) compared to the conventional schemes, that are designed based on the target governing equations. In FY16, we extended our approach to Discontinuous Galerkin (DG) framework for the two following primary reasons: 1) to be able to construct arbitrary compact high-order schemes, and 2) to eliminate the extra DoF that was initiated with our earlier proposed high-order RD schemes. We introduced, for the first-time, the construction of arbitrary high-order DG schemes with a suitable reformulated governing equation, and showed that the proposed schemes do not impose additional DoF, cost the same as the traditional schemes, do not require second-derivative diffusion operators, and produce excellent quality solution gradients on irregular triangular elements [J. Comput. Phys. 321(2016), pp. 729-754]. We also extended our approach, for the first time, to dispersive partial differential equations (such as those occur in Shallow Water analysis) and obtained high-order accurate Hessian (second derivatives) [J. Comput. Phys. 321(2016), pp. 593-605]. In FY17, we extend the proposed arbitrary high-order DG schemes to three-dimensional (3D) compressible Navier-Stokes (NS) equations. We will also construct a suitable NS reformulation that is required for the proposed DG schemes. We will demonstrate the capabilities of the newly constructed high-order schemes against conventional high-order schemes for various flow conditions ranging from subsonic to hypersonic flows. We will also look into time accurate simulations, parallelization, and efficiency (time permitting).

Conventional computational fluid dynamics (CFD) schemes suffer from ability in predicting accurate and smooth solution gradients for fully tetrahedral unstructured elements (on the boundaries and/or within the computational domain). Drag prediction, adjoint error estimation, turbulence modeling and analysis, transition prediction, vorticity prediction, surface heat flux estimation, etc. rely on accurate and smooth predictions of gradients (e.g., velocity gradients, shear stresses, heat flux). A successful outcome of this project will make a far-reaching impact on numerical methods for the Navier-Stokes equations through the entire flight regime, and could dramatically change the way flow simulations are performed today for many applications; e.g., CFD, Magneto-Hydro-Dynamics (MHD), Computational Structural Mechanics, and Shallow Water Analysis.

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