High-Reynolds Number Viscous Flow Computations Using an Unstructured-Grid Method

A fast, matrix-free implicit method has been developed to solve the three-dimensional compressible Euler and Navier?Stokes equations on unstructured meshes. An approximate system of linear equations arising from the Newton linearization is solved by the GMRES (generalized minimum residual) algorithm with a LU-SGS (lower?upper symmetric Gauss?Seidel) preconditioner. A remarkable feature of the present GMRES+LU-SGS method is that the storage of the Jacobian matrix can be completely eliminated by approximating the Jacobian with numerical fluxes, resulting in a matrix-free implicit method. The method developed has been used to compute the compressible flows around 3D complex aerodynamic configurations for a wide range of flow conditions, from subsonic to supersonic. The numerical results obtained indicate that the use of the GMRES+LU-SGS method leads to a significant increase in performance over the best current implicit methods, GMRES+ILU and LU-SGS, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from eight to more than one order of magnitude for all test cases in comparison with the explicit method is demonstrated.

A hybrid Cartesian grid and gridless method for compressible flows

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

The capability of an unstructured hybrid grid method to compute the compressible Navier-Stokes equations is discussed for vortical flows over slender delta wings at low subsonic, high-alpha conditions as a basic study of offdesign aerodynamics of HSCT (High-Speed Civil Transport) configurations. Computations of these flows require high resolutions of the boundary layers on the wing and of the vortical flow region above the wing. The unstructured hybrid grid, which is composed of prismatic grid in boundary layers and tetrahedral and pyramidal grids for the remaining region, is used to treat such viscous flows. The compressible Navier-Stokes equations are solved on the hybrid grid by a cell-vertex, upwind finite volume method. The LU-SGS implicit time integration scheme is used for the convergence acceleration to the steady state. The numerical accuracy of the present method is discussed by comparing with the experimental data of the same configurations. It is demonstrated that the present hybrid grid method is efficient and accurate enough to predict the vortical flow fields over the delta wings. The capability of the method is also shown for a HSCT configuration at a low subsonic, high alpha condition.

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

A method and results of numerical modeling of radiative heating of the back surface of the MSRO (Mars Sample Return Orbiter) space vehicle of the European Space Agency are presented. To determine radiation heat fluxes, the method of discrete ordinates on unstructured tetrahedral grids is used. The radiative model is based on the radiation-transfer equation in a multigroup approximation. Numerical calculation has been performed for the most thermally stressed point of the assumed trajectory of the entry of an MSRO-type space vehicle into the Mars atmosphere. A comparison with the discrete ordinates method on structured grids is made. Good agreement between the results of calculations on structured and unstructured grids is demonstrated. The level of radiation heat fluxes to the back surface of the MSRO space vehicle is predicted.

In this article, to solve the frequency-domain airborne electromagnetic (EM) modeling problem, we express the electric and magnetic field by the vector magnetic potential and the scalar electric potential and derive the governing equation using the Galerkin weighted residual method under the Coulomb gauge. To avoid the strong singularity of the solution near the transmitting source, we directly solve the relatively slow-changing secondary potential. By introducing the spectral element method (SEM) based on unstructured tetrahedral grids, in which the EM field in each element is characterized by high-order Proriol–Koornwinder–Dubiner (PKD) orthogonal polynomials, we can obtain stable and accurate numerical solutions. By establishing the mapping relationship between the physical domain, the right-angled tetrahedral reference domain, and the orthogonal hexahedral domain, we can easily accomplish the element matrix analysis and calculation. The numerical examples show that, when the SEM is used to simulate the airborne EM response, a high accuracy can be obtained even if a very coarse grid is used. Furthermore, since the tetrahedral grids can easily model the complex boundaries, the SEM based on unstructured grids has significant advantages for simulating complex underground structures.

Simulation of two-phase flow in 3D fractured reservoirs using a projection-based Embedded Discrete Fracture Model on Unstructured tetrahedral grids (pEDFM-U)

The present paper describes the numerical methods to compute flows around the NAL Scaled Supersonic Airplane in ascending flight and its separation process from the rocket booster. The Euler equations are solved on unstructured tetrahedral grid. A seamless procedure from CAD data to grid generation is developed to treat the complex configuration of the NAL airplane piggybacked on a rocket for ascending flight. The separation process of the airplane from the rocket is efficiently simulated by the overset unstructured grid method. Comparisons of the computed results with the experiments show good agreements in the lift and pitching moment histories. It is also shown that the small parts attached between the rocket and the airplane significantly change the aerodynamic coefficients of the airplane at transonic regime.

Aero-thermal characteristics of transitional flow through a highly loaded transonic linear turbine are studied at the design incidence using computational methods. The three-dimensional compressible turbulent flow through a turbine inlet guide vane is simulated using finite-volume method-based fluid flow solutions using both Reynolds-averaged Navier Stokes (RANS) equations-based turbulence models, and a dynamic large eddy simulation (LES) approach which is based on a Smagorinsky-Lilly subgrid scale (SGS) model. The calculations are made for exit isentropic Mach numbers ranging from 0.7 to 1.1, and for Reynolds numbers ranging from 5 × 105 to 2 × 106. The numerical results obtained from different turbulence model simulations are compared with each other, and with the experimental data available in the open literature in terms of integrated flow parameters, such as pressure and heat transfer coefficients for varying exit Mach and Reynolds numbers. The present numerical approach indicates a guide line that the present LES approach can be a reliable choice for turbomachinery flows.

Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems

Based on recent work by Guo et al. [“An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice,” J. Comput. Phys. 418, 109570 (2020)], an improved thermal hybrid recursive regularized lattice Boltzmann model (iHRR-ρ) on a regular lattice is developed for two- and three-dimensional compressible laminar and turbulent flows. To enhance the numerical stability in a broad range of Courant–Friedrichs–Lewy numbers and in under-resolved simulations, a new equilibrium density distribution function is proposed to enlarge its positivity region in the Mach–temperature plane. An embedded hybridizing procedure is introduced in the quasi-symmetry correction terms, which allow for a decoupled treatment of unphysical modes and physical under-resolved turbulent scales on coarse grids. To handle compressible turbulent flows, the under-resolved scales are modeled using the original hybrid recursive regularized collision model given by Jacob et al. [“A new hybrid recursive regularised Bhatnagar–Gross–Krook collision model for Lattice Boltzmann method-based large eddy simulation,” J. Turbul. 19, 1051–1076 (2018)] equipped with Vreman’s subgrid model for the large-eddy simulation. The validity and accuracy of the present method for laminar and turbulent compressible flows are assessed by considering six test cases: (I) viscous shock wave internal structure, (II) isentropic vortex convection in a supersonic regime, (III) non-isothermal acoustic pulse, (IV) vortex–shock wave interaction, (V) supersonic flow over NACA airfoil at Re = 10 000 and Ma = 1.5, and (VI) compressible Taylor–Green vortex at Ma = 0.29.

: This final report documents the major developments and findings during the grant period from March 2009 to November 2012. The main objective of this project was to develop a new discontinuous formulation named correction procedure via reconstruction (CPR) for hyperbolic conservation laws, and demonstrate its capability for the Euler and Navier-stokes equations on hybrid 3D unstructured prismatic and tetrahedral grids. We achieved the following accomplishments: * Extended the CPR formulation to 3D hybrid meshes, including tetrahedral, hexahedral, prismatic elements; * Extended the CPR formulation to the Navier-Stokes equations on hybrid elements, and demonstrate the method for benchmark 3D problems; * Implemented the CPR method on clusters of CPUs and GPUs, and achieved up to two orders of magnitude speedup on the GPU than the CPU; * Extended the method for dynamic moving grids satisfying the so-called geometric conservation laws, and demonstrated the capability for bio-inspired flow problems; * Implemented solution-based hp-adaptations using a variety of adaptation criteria including residual, adjoint and entropy based adaptation criteria.

A new code aimed at the solution of three-dimensional internal laminar and turbulent compressible flows is presented. The solution procedure is based on the scalar form of the approximate factorization method. The Navier-Stokes equations in mass-averaged form are iterated in time until a steady solution is reached. Evidence is given to the implicit treatment of diffusive terms that proved able to speed up convergence and enhance the algorithm robustness. Turbulence effects are accounted for by the Baldwin-Lomax turbulence model and the q - ω two-equation model. For the first, an investigation on the model behavior in case of multiple boundaries is performed. The numerical code is validated by computing the flow in the Stanitz 90-deg turning elbow. Finally the code is applied to the computation of the flow in the VKI LS-82 three-dimensional linear cascade.

This paper presents the development of finite-volume multiscale methods for quadrilateral and triangular unstructured grids. Families of Darcy-flux approximations have been developed for consistent approximation of the general tensor pressure equation arising from Darcy's law together with mass conservation. The schemes are control-volume distributed (CVD) with flow variables and rock properties sharing the same control-volume location and are comprised of a multipoint flux family formulation (CVD-MPFA). The schemes are used to develop a CVD-MPFA based multiscale finite-volume (MSFV) formulation applicable to both structured and unstructured grids in two dimensions. The basis functions are a key component of the MSFV method, and are a set of local solutions, usually defined subject to Dirichlet boundary conditions. A generalization of the Cartesian grid Dirichlet basis functions described in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] is presented here for unstructured grids. Whilst the transition from a Cartesian grid to an unstructured grid is largely successful, use of Dirichlet basis functions can still lead to pressure fields that exhibit spurious oscillations in areas of strong heterogeneity. New basis functions are proposed in an attempt to improve the pressure field solutions where Neumann boundary conditions are imposed almost everywhere, except corners which remain specified by Dirichlet values.