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NASA NTRS2014

DNS of Flows over Periodic Hills using a Discontinuous-Galerkin Spectral-Element Method

Direct numerical simulation (DNS) of turbulent compressible flows is performed using a higher-order space-time discontinuous-Galerkin finite-element method. The numerical scheme is validated by performing DNS of the evolution of the Taylor-Green vortex and turbulent flow in a channel. The higher-order method is shown to provide increased accuracy relative to low-order methods at a given number of degrees of freedom. The turbulent flow over a periodic array of hills in a channel is simulated at Reynolds number 10,595 using an 8th-order scheme in space and a 4th-order scheme in time. These results are validated against previous large eddy simulation (LES) results. A preliminary analysis provides insight into how these detailed simulations can be used to improve Reynoldsaveraged Navier-Stokes (RANS) modeling

DNS↗

NASA NTRS2019

Scale–Resolving Simulations of a Fundamental Trailing–Edge Cooling Slot Using a Discontinuous–Galerkin Spectral–Element Method

The accurate prediction of turbulent mixing in high-pressure turbines that incorporate various airfoil surface-cooling strategies is becoming increasing critical to the design of modern gas turbine engines where the quest for improved efficiency is driving compressor overall pressure ratios and turbine inlet temperatures to much higher levels than ever before. In the present paper, a recently developed computational capability for accurate and efficient scale–resolving simulations of turbomachinery is extended to study the turbulent mixing mechanism of a simplified abstraction of an airfoil trailing-edge cooling slot - a plane wall jet with finite lip thickness discharging into an ambient flow. The computational capability is based on an entropy– stable, discontinuous–Galerkin approach that extends to arbitrarily high orders of spatial and temporal accuracy. The numerical results show that the present simulations capture the trends observed in the experiments. Discrepancies between the simulations and experiments are believed to be due to differences in the inflow profiles and tunnel side–wall effects. The thick lip configuration leads to a thicker wake and higher unsteadiness in the wall jet compared to the thin lip. A detailed comparison of the turbulent flowfields is presented to highlight differences arising due to lip thickness variations.

Scale Resolving↗

NASA NTRS1993

Incompressible spectral-element method: Derivation of equations

A fractional-step splitting scheme breaks the full Navier-Stokes equations into explicit and implicit portions amenable to the calculus of variations. Beginning with the functional forms of the Poisson and Helmholtz equations, we substitute finite expansion series for the dependent variables and derive the matrix equations for the unknown expansion coefficients. This method employs a new splitting scheme which differs from conventional three-step (nonlinear, pressure, viscous) schemes. The nonlinear step appears in the conventional, explicit manner, the difference occurs in the pressure step. Instead of solving for the pressure gradient using the nonlinear velocity, we add the viscous portion of the Navier-Stokes equation from the previous time step to the velocity before solving for the pressure gradient. By combining this 'predicted' pressure gradient with the nonlinear velocity in an explicit term, and the Crank-Nicholson method for the viscous terms, we develop a Helmholtz equation for the final velocity.

Deanna, Russell G.↗

NASA NTRS1988

Nonconforming mortar element methods: Application to spectral discretizations

Spectral element methods are p-type weighted residual techniques for partial differential equations that combine the generality of finite element methods with the accuracy of spectral methods. Presented here is a new nonconforming discretization which greatly improves the flexibility of the spectral element approach as regards automatic mesh generation and non-propagating local mesh refinement. The method is based on the introduction of an auxiliary mortar trace space, and constitutes a new approach to discretization-driven domain decomposition characterized by a clean decoupling of the local, structure-preserving residual evaluations and the transmission of boundary and continuity conditions. The flexibility of the mortar method is illustrated by several nonconforming adaptive Navier-Stokes calculations in complex geometry.

Maday, Yvon↗

NASA NTRS1989

Model-free simulations of turbulent reactive flows

The current computational methods for solving transport equations of turbulent reacting single-phase flows are critically reviewed, with primary attention given to those methods that lead to model-free simulations. In particular, consideration is given to direct numerical simulations using spectral (Galerkin) and pseudospectral (collocation) methods, spectral element methods, and Lagrangian methods. The discussion also covers large eddy simulations and turbulence modeling.

Givi, Peyman↗

NASA NTRS1996

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization

Fast methods are proposed for solving the system K(sub N)x = b resulting from the discretization of self-adjoint elliptic equations in three dimensional domains by the spectral element method. The domain is decomposed into hexahedral elements, and in each of these elements the discretization space is formed by polynomials of degree N in each variable. Gauss-Lobatto-Legendre (GLL) quadrature rules replace the integrals in the Galerkin formulation. This system is solved by the preconditioned conjugate gradients method. The conforming finite element space on the GLL mesh consisting of piecewise Q(sub 1) elements produces a stiffness matrix K(sub h) that is spectrally equivalent to the spectral element stiffness matrix K(sub N). The action of the inverse of K(sub h) is expensive for large problems, and is therefore replaced by a Schwarz preconditioner B(sub h) of this finite element stiffness matrix. The preconditioned operator then becomes B(sub h)(exp -l)K(sub N). The technical difficulties stem from the nonregularity of the mesh. Tools to estimate the convergence of a large class of new iterative substructuring and overlapping Schwarz preconditioners are developed. This technique also provides a new analysis for an iterative substructuring method proposed by Pavarino and Widlund for the spectral element discretization.

Casarin, Mario↗

NASA NTRS1987

Coupling finite element and spectral methods: First results

A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares, a finite element approximation is used on the first square and a spectral discretization is used on the second one. Two kinds of matching conditions on the interface are presented and compared. In both cases, error estimates are proved.

Bernardi, Christine↗

NASA NTRS1990

Coupling finite element and spectral methods - First results

Bernardi, Christine↗

NASA NTRS1983

Spectral methods for flows in complex geometries

Three methods for the application of spectral methods to flows in complex geometries are discussed. They are the spectral iteration method, the spectral embedding method, and the spectral element method. Applications are given to the flow over wavy walls, past a large-eddy breakup device (LEBU), and to flow over a step.

Orszag, S. A.↗

NASA NTRS1989

Spectral element simulations of laminar and turbulent flows in complex geometries

Spectral element methods are high-order weighted residual techniques based on spectral expansions of variables and geometry for the Navier-Stokes (NS) and transport equations. Here, practical aspects of these methods and their efficient implementation are examined, and several examples of flows in truly complex geometries are presented. The spectral element discretization for NS equations is introduced, and the convergence of the method is addressed. An efficient data management scheme is discussed in the context of parallel processing computations. The method is validated by comparing the spectral element solutions with the exact eigensolutions for the Orr-Sommerfeld equations in two and three dimensions. Computer-aided flow visualizations are presented for an impulsive flow past a sharp edge wedge. Three-dimensional states of channel flow disrupted by an array of cylindrical eddy promoters are studied, and the results of a direct simulation of the turbulent flow in a plane channel are presented.

Karniadakis, George EM↗

NASA NTRS2003

Aircraft Engine Noise Scattering By Fuselage and Wings: A Computational Approach

The paper presents a time-domain method for computation of sound radiation from aircraft engine sources to the far-field. The effects of nonuniform flow around the aircraft and scattering of sound by fuselage and wings are accounted for in the formulation. The approach is based on the discretization of the inviscid flow equations through a collocation form of the Discontinuous Galerkin spectral element method. An isoparametric representation of the underlying geometry is used in order to take full advantage of the spectral accuracy of the method. Large-scale computations are made possible by a parallel implementation based on message passing. Results obtained for radiation from an axisymmetric nacelle alone are compared with those obtained when the same nacelle is installed in a generic configuration, with and without a wing.

FUSELAGE-NACELLE CONFIGURATION↗

NASA NTRS2003

Computation of Engine Noise Propagation and Scattering Off an Aircraft

The paper presents a comparison of experimental noise data measured in flight on a two-engine business jet aircraft with Kulite microphones placed on the suction surface of the wing with computational results. Both a time-domain discontinuous Galerkin spectral method and a frequency-domain spectral element method are used to simulate the radiation of the dominant spinning mode from the engine and its reflection and scattering by the fuselage and the wing. Both methods are implemented in computer codes that use the distributed memory model to make use of large parallel architectures. The results show that trends of the noise field are well predicted by both methods.

Xu, J.↗

NASA NTRS1991

Spectral simulations of reacting turbulent flows

Spectral methods for simulating flows are reviewed, emphasizing their recent applications to reacting flow problems. Various classifications of spectral methods and their convergence properties are described and the 'spectral element' method is presented, highlighting its flexibility in dealing with complex flow geometries. The applications considered include chemical reactions in homogeneous turbulence, temporally evolving mixing layers, variable-density simulations, nonequilibrium chemistry, and spatially evolving mixing layers.

Mcmurtry, Patrick A.↗

NASA NTRS1993

Triangular spectral elements for incompressible fluid flow

We discuss the use of triangular elements in the spectral element method for direct simulation of incompressible flow. Triangles provide much greater geometric flexibility than quadrilateral elements and are better conditioned and more accurate when small angles arise. We employ a family of tensor product algorithms for triangles, allowing triangular elements to be handled with comparable arithmetic complexity to quadrilateral elements. The triangular discretizations are applied and validated on the Poisson equation. These discretizations are then applied to the incompressible Navier-Stokes equations and a laminar channel flow solution is given. These new triangular spectral elements can be combined with standard quadrilateral elements, yielding a general and flexible high order method for complex geometries in two dimensions.

Mavriplis, C.↗

NASA NTRS1993

Triangular spectral elements for incompressible fluid flow

We discuss the use of triangular elements in the spectral element method for direct simulation of incompressible flow. Triangles provide much greater geometric flexibility and are better conditioned and more accurate than quadrilateral elements when small angles arise. We employ a family of tensor product algorithms for triangles, allowing triangular elements to be handled with comparable arithmetic complexity to quadrilateral elements. The triangular discretizations are applied to the Poisson equation and are validated. The triangular discretizations are then applied to the incompressible Navier-Stokes equations, and a laminar channel flow solution is given. The new triangular spectral elements can be combined with standard quadrilateral elements, yielding a general and flexible high order method for complex geometries in two dimensions.

Mavriplis, Catherine↗

NASA NTRS2003

Aircraft Engine Noise Scattering by Fuselage and Wings: A Computational Approach

Stanescu, D.↗

NASA NTRS2003

Aircraft Engine Noise Scattering by Fuselage and Wings: A Computational Approach

The paper presents a time-domain method for computation of sound radiation from aircraft engine sources to the far field. The effects of non-uniform flow around the aircraft and scattering of sound by fuselage and wings are accounted for in the formulation. The approach is based on the discretization of the inviscid flow equations through a collocation form of the discontinuous Galerkin spectral element method. An isoparametric representation of the underlying geometry is used in order to take full advantage of the spectral accuracy of the method. Large-scale computations are made possible by a parallel implementation based on message passing. Results obtained for radiation from an axisymmetric nacelle alone are compared with those obtained when the same nacelle is installed in a generic configuration, with and without a wing. 0 2002 Elsevier Science Ltd. All rights reserved.

Farassat, F.↗

NASA NTRS2001

Design of Unstructured Adaptive (UA) NAS Parallel Benchmark Featuring Irregular, Dynamic Memory Accesses

We describe the design of a new method for the measurement of the performance of modern computer systems when solving scientific problems featuring irregular, dynamic memory accesses. The method involves the solution of a stylized heat transfer problem on an unstructured, adaptive grid. A Spectral Element Method (SEM) with an adaptive, nonconforming mesh is selected to discretize the transport equation. The relatively high order of the SEM lowers the fraction of wall clock time spent on inter-processor communication, which eases the load balancing task and allows us to concentrate on the memory accesses. The benchmark is designed to be three-dimensional. Parallelization and load balance issues of a reference implementation will be described in detail in future reports.

Feng, Hui-Yu↗