Hyperbolic Navier-Stokes Solver for Three-Dimensional Flows

Abstract

In this paper, we present a hyperbolic Navier-Stokes solver for three-dimensional compressible viscous flows. Hyperbolic Navier-Stokes systems, which have been constructed and demonstrated in two dimensions, are extended here to three dimensions and the eigenstructure of the hyperbolized viscous terms is derived. The system is discretized by the second-order node-centered edge-based method on unstructured grids. The resulting residual equations are solved by an implicit defect-correction solver based on a compact Jacobian derived exactly from a first-order edge-based discretization. Numerical results are presented to examine the practical applicability and potential advantages of the hyperbolic Navier-Stokes method, accurate gradients in particular, through three-dimensional viscous flow problems with unstructured grids.

Efficient high-order discontinuous Galerkin schemes with first-order hyperbolic advection–diffusion system approach

“…(1). For the Navier-Stokes equations in three-dimension, for example, we will have 14/5 = 2.8, 17/5 = 3.4, or 20/5 = 4 times more DoF compared to the conventional DG schemes that are not constructed from the hyperbolic Navier-Stokes equations [10,48].…”

Third-Order Inviscid and Second-Order Hyperbolic Navier-Stokes Schemes for Three-Dimensional Inviscid and Viscous Flows

“…The default FUN3D viscous scheme (i.e., the P 1 continuous Galerkin discretization) is added to the inviscid scheme. The second approach is the hyperbolic Navier-Stokes (HNS) method [23,24,25,26], which reformulates the viscous terms as a hyperbolic system with the solution gradients introduced as additional unknowns. In the HNS method, third-order accuracy in the inviscid terms can be achieved without quadratic LSQ methods [24] because second-order accurate gradients are directly obtained as unknowns.…”

“…The HNS schemes have been demonstrated for 3D flows in Ref. [26]. This paper introduces some key improvements to the methodologies presented in Ref.…”

“…This paper introduces some key improvements to the methodologies presented in Ref. [26]. Both approaches are implemented in FUN3D, verified, and applied for inviscid and viscous flow problems.…”

“…Another important aspect is that the HNS schemes are known to produce high-order and highquality gradients (e.g., viscous stresses, heat fluxes, and vorticity) on irregular unstructured grids [23,24,26] while the first approach can produce only first-order accurate gradients (unless the inviscid terms dominate) polluted by numerical noise on irregular grids. Moreover, the HNS solvers are known to eliminate numerical stiffness arising from second derivatives in the governing equations and converge faster on fine grids than conventional solvers [23,24,25,26]. Efficiency comparison is, thus, not a simple matter, and will be even more complicated if other aspects, such as solver constructions, code optimizations, other discretization alternatives, etc., are brought into discussion.…”