High-Order Finite-Element Method and Dynamic Adaptation for Two-Dimensional Laminar and Turbulent Navier-Stokes

Ahrabi, Behzad Reza; Anderson, W. Kyle; Newman, James C.

doi:10.2514/6.2014-2983

Cited by 11 publications

(18 citation statements)

References 52 publications

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“…In recent years, this approach has been extensively used in DG finite element schemes (e.g., see [34][35][36][37][38]). In our previous study [39], this adjoint-based approach was employed in a Petrov-Galerkin discretization of compressible turbulent flows. It is followed here, with the difference that, in the present work, hierarchical basis functions are used instead of Lagrangian basis functions.…”

Section: Results Obtained On Adaptively Refined Gridsmentioning

confidence: 99%

Finite Element Solutions for Turbulent Flow over the NACA 0012 Airfoil

2016

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A high-order Petrov-Galerkin finite element scheme is used to compute turbulent flow over a NACA 0012 airfoil at a freestream Mach number of 0.15, an angle of attack of 10 deg, and a Reynolds number based on the airfoil chord of 6 million. Results are obtained on a series of grids available on the NASA Turbulence Modeling Resource Web site and are compared with reference solutions that have been obtained using the FUN3D and CFL3D finite volume solvers on meshes with as many as 14.6 million degrees of freedom. Forces, moments, pressure distributions, skin friction, and profiles of velocity and turbulence working variable for the Spalart-Allmaras turbulence model are compared between the finite element and the finite volume solutions. It is demonstrated that the finite element scheme shows similar results as the finite volume schemes for most of the comparisons, but demonstrates significantly less dissipation of the wake profiles downstream of the airfoil. It is shown that, when the same number of degrees of freedom is used in the simulations, solutions obtained with quadratic elements are more accurate than those obtained using linear elements with only minor increases in computational cost. Finally, initial results with an adjoint-based hpadaptive methodology are obtained that further demonstrate that the high-order finite element framework can efficiently yield accurate results. NomenclatureA, B, k = flux Jacobian matrices C D = total drag coefficientviscous flux vector N = Lagrangian or hierarchical basis function p = polynomial order Q = solution variables (ρ, u, v, T) S = source term t = time x, y = Cartesian coordinates Γ = surface of control volume τ = stabilization matrix ϕ = weighting function Ω = control volume

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Section: Results Obtained On Adaptively Refined Gridsmentioning

confidence: 99%

Finite Element Solutions for Turbulent Flow over the NACA 0012 Airfoil

2016

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“…These capabilities have been incorporated into the software reported in [21,40]. The two-dimensional version of this software 17 was also enhanced with the multi-level h-adaptive refinement capability reported in [18]. Additionally, a modification to the elliptic hole cutting 17 method, which is amenable to parallel implementation and significantly reduces the amount of searching and interpolation required, was introduced.…”

Section: Discussionmentioning

confidence: 99%

Three-Dimensional Dynamic Overset Method for Stabilized Finite Elements

Liu

Newman

Anderson

et al. 2015

22nd AIAA Computational Fluid Dynamics Conference

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In this paper enhancements to a stabilized finite-element overset scheme are presented. First, the recently proposed grid overlapping method referred to as elliptic hole cutting has been modified to be more suitable for parallel implementation. These modifications dramatically reduce the amount of searching and interpolation required to establish the overset grid network. The second enhancement entails combining the overset approach with an adaptive grid refinement capability. For stabilized finite-elements, the adaptive methodology must ensure that the state-variables remain continuous across element boundaries. Finally, the overset methodology has been extended to three-dimensions with moving domains. Order of accuracy is examined via the method of manufactured solutions for linear and quadratic elements. Overset grid results of a sinusoidally oscillating with and without dynamic hole cutting are compared with single grid. A wing-store separation simulation is carried out to demonstrate the overset capability of current methodology.

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“…[12,13,18,31,34,36]). In our previous study [21], this adjoint-based approach was employed in a Petrov-Galerkin discretization of compressible turbulent flows. It is followed here, with the difference that in the present work, instead of Lagrange, hierarchical basis functions are used.…”

Section: -Discrete Adjoint-based Error Estimationmentioning

confidence: 99%

“…Another important consideration regarding the complexity of the implementation is the choice of the shape functions. In a previous study [21], Lagrange polynomials were used. Although the implementation was successful up to cubic elements for two dimensional problems, the extension of the method to three dimensions was not feasible.…”

Section: Constrained Approximationmentioning

confidence: 99%

“…Clearly, hanging nodes can violate this requirement across the interface between refined and unrefined elements. In a previous study [21], a technique known as constrained approximation [49][50][51][52][53][54][55][56] was employed in which a function value at a hanging node is constrained by the function values at adjacent nodes such that a continuous solution is obtained across all element interfaces. In that study, Lagrange polynomials were used as shape functions.…”

Section: Introductionmentioning

confidence: 99%

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An Adjoint-Based hp-Adaptive Petrov-Galerkin Method for Turbulent Flows

Ahrabi

Anderson

Newman

2015

22nd AIAA Computational Fluid Dynamics Conference

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In this study, an adjoint-based hp-adaptation algorithm has been developed within a Petrov-Galerkin finite-element method. The developed mesh adaptation algorithm is able to perform non-conformal mesh adaptations. To consistently account for hanging nodes, the constrained approximation method has been utilized. Hierarchical basis functions have been employed to facilitate the implementation of the constrained approximation. The methodology has been demonstrated on numerous cases using the Euler and Reynolds Average Navier-Stokes (RANS) equations, equipped with a modified SpalartAllmaras (SA) turbulence model. Also, a PDE-based artificial viscosity has been added to the governing equations, to stabilize the solution in the vicinity of shock waves. For accurate representation of the geometric surfaces, high-order curved boundary meshes have been generated and the interior meshes have been deformed through the solution of a modified linear elasticity equation. Fully implicit linearization has been used to advance the solution toward a steady-state. Dirichlet boundary conditions have been imposed weakly and the functional outputs have been modified according to the weak boundary conditions in order to provide a smooth adjoint solution near the boundaries.To accelerate the error reduction in presence of singularity points, an enhanced hrefinement, based on solution's smoothness, has been used. Numerical results illustrate consistent accuracy improvement of the functional outputs for both h-and hpadaptation, and also capability enhancement in capturing complex viscous effects such as shock-wave/turbulent boundary layer interaction.

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High-Order Finite-Element Method and Dynamic Adaptation for Two-Dimensional Laminar and Turbulent Navier-Stokes

Cited by 11 publications

References 52 publications

Finite Element Solutions for Turbulent Flow over the NACA 0012 Airfoil

Finite Element Solutions for Turbulent Flow over the NACA 0012 Airfoil

Three-Dimensional Dynamic Overset Method for Stabilized Finite Elements

An Adjoint-Based hp-Adaptive Petrov-Galerkin Method for Turbulent Flows

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