09-04-2019 | Lingquan Li: 123rd NIA CFD Seminar: Hyperbolic Navier-Stokes Method Based on Reconstructed-Discontinuous-Galerkin or Reconstructed-Finite-Volume Formulation with Variational Reconstruction

123rd NIA CFD Seminar: Hyperbolic Navier-Stokes Method Based on Reconstructed-Discontinuous-Galerkin or Reconstructed-Finite-Volume Formulation with Variational  Reconstruction

Date: Wednesday, Sept. 4, 2019
Time: 11 a.m.-noon (EDT)
Room: NIA, Rm. 137
Speaker: Lingquan Li

Speaker’s Bio:
Lingquan Li received a Bachelor’s Degree of Aerospace Engineering at Xi’an Jiaotong University, Xi’an, Shaanxi Province, China, in July 2011. She then attended the graduate school at Fudan University, Shanghai, China, in September 2013 and studied under the supervision of Dr. Aiming Yang. She received a Master’s degree of Aerospace Engineering in July 2016.

The author was then recruited by the Department of Mechanical and Aerospace Engineering at North Carolina State University (NCSU), Raleigh, North Carolina, USA, and studied in the doctoral program of Aerospace Engineering since August 2016. Her academic advisor is Dr. Hong Luo.

The objective of the presented work is to develop an efficient, accurate and compact method for solving compressible Navier-Stokes (NS) equations by combining the hyperbolic Navier-Stokes (HNS) formulation and the reconstructed discontinuous Galerkin method (rDG), which includes the finite-volume (FV) and discontinuous Galerkin methods.

A new HNS formulation is derived so that an efficient high-order construction for compressible NS equations can be derived. The gradients of the primitive variables such as density, velocity, and temperature are introduced as additional unknowns. The newly introduced gradients can be recycled to get a higher-order polynomial solution for these primitive variables. An even more accurate method is obtained when reconstruction is performed on these gradient variables. These reconstructed variables are also reused as higher-order derivatives of the primitive variables. In the presented work, a variational formulation is used for reconstruction. This variational reconstruction (VR) can be seen as an extension of the compact finite difference schemes to unstructured grids.

The reconstructed variables are obtained by solving an extreme value problem, which minimizes the jumps at cell interfaces and therefore maximizes the smoothness of the reconstructed polynomials. The spatial discretization is performed by multiplying the HNS system by a test function matrix. If the matrix is taken as a diagonal matrix, then the primitive variables and auxiliary variables are regarded as decoupled. This will generate an FV type formulation, which is denoted as HNS+rFV method. If the matrix is taken as the primitive variables and auxiliary variables are coupled, a Galerkin type formulation is obtained, which is denoted as HNS+rDG method.

All the primitive variables, auxiliary variables and reconstructed variables are stored in a consistent way with the Taylor-basis DG counterpart. The fully implicit method is implemented for steady problems, while a third-order implicit Runge-Kutta (IRK), i.e., ESDIRK3 time marching method is implemented for unsteady flows. In these implicit methods, an automatic differentiation tool TAPENADE is used to obtain the resulting flux Jacobian matrices.

The approximate system of linear equations arising from the Newton iteration is solved with two methods: symmetric gauss-seidel (SGS) and general minimum residual (GMRES) algorithm with lower-upper symmetric gauss-seidel (LU-SGS) preconditioning.