Provably stable flux reconstruction (FR) schemes are derived for partial differential equations c... more Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms are examined as they enable the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and nonconservative forms are inherently different-even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders
The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral... more The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss-Legendre-Lobatto collocation points [1, 2, 3] and their equivalence with summation-by-parts (SBP) finite difference operators [4]. In this work, we show how to efficiently generalize the construction of semi-discretely entropy stable schemes on tensor product elements to Gauss points and generalized SBP operators. Numerical experiments suggest that the use of Gauss points significantly improves accuracy on curved meshes.
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
The problem of crossflow receptivity is considered in the context of a canonical 3D boundary laye... more The problem of crossflow receptivity is considered in the context of a canonical 3D boundary layer (viz., the swept Hiemenz boundary layer) and a swept airfoil used recently in the SWIFT flight experiment performed at Texas A&M University. First, Hiemenz flow is used to analyze localized receptivity due to a spanwise periodic array of small amplitude roughness elements, with the goal of quantifying the effects of array size and location. Excitation of crossflow modes via nonlocalized but deterministic distribution of surface nonuniformity is also considered and contrasted with roughness induced acoustic excitation of Tollmien-Schlichting waves. Finally, roughness measurements on the SWIFT model are used to model the effects of random, spatially distributed roughness of sufficiently small amplitude with the eventual goal of enabling predictions of initial crossflow disturbance amplitudes as functions of surface roughness parameters.
A combination of parabolized stability equations and secondary instability theory has been applie... more A combination of parabolized stability equations and secondary instability theory has been applied to a low-speed swept airfoil model with a chord Reynolds number of 7.15 million, with the goals of (i) evaluating this methodology in the context of transition prediction for a known configuration for which roughness based crossflow transition control has been demonstrated under flight conditions and (ii) of analyzing the mechanism of transition delay via the introduction of discrete roughness elements (DRE). Roughness based transition control involves controlled seeding of suitable, subdominant crossflow modes, so as to weaken the growth of naturally occurring, linearly more unstable crossflow modes. Therefore, a synthesis of receptivity, linear and nonlinear growth of stationary crossflow disturbances, and the ensuing development of high frequency secondary instabilities is desirable to understand the experimentally observed transition behavior. With further validation, such higher fidelity prediction methodology could be utilized to assess the potential for crossflow transition control at even higher Reynolds numbers, where experimental data is currently unavailable.
TECHNICAL NOTES are short manuscripts describing new developments or important results of a preli... more TECHNICAL NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes cannot exceed 6 manuscript pages and 3 figures; a page of text may be substituted for a figure and vice versa. After informal review by the editors, they may be published within a few months of the date of receipt. Style requirements are the same as .for regular contributions (see inside back cover).
This work examines the development of an entropy conservative (for smooth solutions) or entropy s... more This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semidiscrete level ignoring the temporal dependence. In this work, we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semidiscrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space-time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space-time DG method derived herein are validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations.
This classic test case aims at characterizing the solver’s ability to preserve vorticity in an in... more This classic test case aims at characterizing the solver’s ability to preserve vorticity in an inviscid flow. The unsteady 2D Euler equations govern the simulation, which consists in a 2D vortex transported by a uniform flow across a rectangular computational domain of dimensions (x, y) = (0, Lx)× (0, Ly). The initial configuration of the vortex, centered in (xc, yc) and superimposed onto the uniform (infinity) flow, is given by the following equations:
We submit 5 sets of results: 1 for the inviscid subsonic case, 2 for the viscous case (with sharp... more We submit 5 sets of results: 1 for the inviscid subsonic case, 2 for the viscous case (with sharp and with rounded trailing edge), 2 for the transonic case (with and without shock capturing, see below for the detailed description of the shock capturing scheme). For this test case, we generated a fine O-grid of 577 × 513 vertices using the hyperbolic grid generation capabilities of the commercial software Pointwise [1]. The farfield is located at 1000 chords, as requested. The trailing edge is sharp, unless stated otherwise. For the subsonic configurations, the vertex distribution on the airfoil is the same on pressure and suction side, as shown in fig. 1. Instead, for the transonic configuration, vertices are clustered on the suction side, in particular close to the shock region, fig. 2. Initial guesses are obtained via grid sequencing, where appropriate. The coarser grids are obtained by deleting every other grid line from the finer grid (regular coarsening).
We submit two sets of results (in Tecplot format): one set obtained imposing an inviscid (slip) w... more We submit two sets of results (in Tecplot format): one set obtained imposing an inviscid (slip) wall boundary condition on walls, and one set obtained imposing the exact solution on walls. This test case has been computed on four structured grids containing 129 × 65, 257 × 129, 513 × 257 and 1025 × 513 vertices respectively. The finest grid is obtained by applying elliptic smoothing to an algebraically created grid. Both a second and a fourth order discretization of the Laplace equation is used for this smoothing. It turned out that the resulting grids produced virtually indistinguishable results. The point distribution on the boundary is uniform and the coarse grids are obtained by deleting recursively every other grid line from the fine grid. The coarser grids are obtained by deleting every other grid line from the finer grid. The coarsest grid used is shown in figure 1. When using the slip wall boundary condition, it was found that at least a 3(on the coarsest grid at least a 4) ...
A concerted effort is underway at NASA c Langley Research Center to create a benchmark for cf Com... more A concerted effort is underway at NASA c Langley Research Center to create a benchmark for cf Computational Fluid Dynamic (CFD) codes, both Cp unstructured and structured, against a data set for the c_, hump model with actuation. The hump model was <c,>
In this paper we discuss the issue of conservation and convergence to weak solutions of several g... more In this paper we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used compact schemes and spectral collocation schemes, for solving hyperbolic conservation laws. It is shown that such s c hemes, if convergent boundedly almost everywhere, will converge to weak solutions. The results are extensions of the classical Lax-Wendro theorem concerning conservative s c hemes.
An extension of the reacting Hz-air computer code SPARK is presented, which enables the code to b... more An extension of the reacting Hz-air computer code SPARK is presented, which enables the code to be used on any reacting flow problem. Routines are developed that calculate in a general fashion, the reaction rates, and chemical Jacobians of any reacting system. In addition, an equilibrium routine is added so that the code will have frozen, finite rate, and equilibrium capabilities. The reaction rates for the species are determined from the law of mass action using Arrhenius expressions for the rate constants. The Jacobian routines are determined by numerically or analytically differentiating the law of mass action for each species. The equilibrium routine is based on a Gibbs free energy minimization routine. The routines are written in Fortran 77, with special consideration given to vectorization. Run times for the generalized routines are generally 20% slower than reaction specific routines. The numerical efficiency of the generalized analytical Jacobian, however, is nearly 300% better than the reaction specific numerical Jacobian used in SPARK. I P NOMENCLATURE Aj : reaction rate constant for jth reaction b, : body force of species i Ci : concentrak-ion of species i C, : time rate of change of Cj Cp : specific heat at constant pressure E : total internal energy; activation energy *Presently, Research Scientist working under contract for NASA Langley Research Center
It has been previously shown that the temporal integration of hyperbolic partial differ-17. SECUR... more It has been previously shown that the temporal integration of hyperbolic partial differ-17. SECURITY CLASSIFICATION OF REPORT Unclassified
The numerical study of aeroacoustic problems places stringent demands on the choice of a computat... more The numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm, because it requires the ability to propagate disturbances of small amplitude and short wavelength. The demands are particularly high when shock waves are involved, because the chosen algorithm must also resolve discontinuities in the solution.
Numerical Heat Transfer, Part B: Fundamentals, 2001
A higher order a.(:curat(' mnnerical t)roce(ture has been deveh)l)ed for solving incompressibh' •... more A higher order a.(:curat(' mnnerical t)roce(ture has been deveh)l)ed for solving incompressibh' • *The authors were partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the first two authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASI£,), NASA Langley Resear(:h Center, Hampton, VA 23681-2199. Additional sllpport was provided l)y the NASA Graduate Studenl ll.esearch Program. • _l)epartment of M(,ehanical Engineering, Ohl Dominion l!niversfly, N_)rfolk, Vir_zinia 2352.(t. • {Acouslic and Flow Met hods Branch,
Provably stable flux reconstruction (FR) schemes are derived for partial differential equations c... more Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms are examined as they enable the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and nonconservative forms are inherently different-even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders
The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral... more The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss-Legendre-Lobatto collocation points [1, 2, 3] and their equivalence with summation-by-parts (SBP) finite difference operators [4]. In this work, we show how to efficiently generalize the construction of semi-discretely entropy stable schemes on tensor product elements to Gauss points and generalized SBP operators. Numerical experiments suggest that the use of Gauss points significantly improves accuracy on curved meshes.
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
The problem of crossflow receptivity is considered in the context of a canonical 3D boundary laye... more The problem of crossflow receptivity is considered in the context of a canonical 3D boundary layer (viz., the swept Hiemenz boundary layer) and a swept airfoil used recently in the SWIFT flight experiment performed at Texas A&M University. First, Hiemenz flow is used to analyze localized receptivity due to a spanwise periodic array of small amplitude roughness elements, with the goal of quantifying the effects of array size and location. Excitation of crossflow modes via nonlocalized but deterministic distribution of surface nonuniformity is also considered and contrasted with roughness induced acoustic excitation of Tollmien-Schlichting waves. Finally, roughness measurements on the SWIFT model are used to model the effects of random, spatially distributed roughness of sufficiently small amplitude with the eventual goal of enabling predictions of initial crossflow disturbance amplitudes as functions of surface roughness parameters.
A combination of parabolized stability equations and secondary instability theory has been applie... more A combination of parabolized stability equations and secondary instability theory has been applied to a low-speed swept airfoil model with a chord Reynolds number of 7.15 million, with the goals of (i) evaluating this methodology in the context of transition prediction for a known configuration for which roughness based crossflow transition control has been demonstrated under flight conditions and (ii) of analyzing the mechanism of transition delay via the introduction of discrete roughness elements (DRE). Roughness based transition control involves controlled seeding of suitable, subdominant crossflow modes, so as to weaken the growth of naturally occurring, linearly more unstable crossflow modes. Therefore, a synthesis of receptivity, linear and nonlinear growth of stationary crossflow disturbances, and the ensuing development of high frequency secondary instabilities is desirable to understand the experimentally observed transition behavior. With further validation, such higher fidelity prediction methodology could be utilized to assess the potential for crossflow transition control at even higher Reynolds numbers, where experimental data is currently unavailable.
TECHNICAL NOTES are short manuscripts describing new developments or important results of a preli... more TECHNICAL NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes cannot exceed 6 manuscript pages and 3 figures; a page of text may be substituted for a figure and vice versa. After informal review by the editors, they may be published within a few months of the date of receipt. Style requirements are the same as .for regular contributions (see inside back cover).
This work examines the development of an entropy conservative (for smooth solutions) or entropy s... more This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semidiscrete level ignoring the temporal dependence. In this work, we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semidiscrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space-time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space-time DG method derived herein are validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations.
This classic test case aims at characterizing the solver’s ability to preserve vorticity in an in... more This classic test case aims at characterizing the solver’s ability to preserve vorticity in an inviscid flow. The unsteady 2D Euler equations govern the simulation, which consists in a 2D vortex transported by a uniform flow across a rectangular computational domain of dimensions (x, y) = (0, Lx)× (0, Ly). The initial configuration of the vortex, centered in (xc, yc) and superimposed onto the uniform (infinity) flow, is given by the following equations:
We submit 5 sets of results: 1 for the inviscid subsonic case, 2 for the viscous case (with sharp... more We submit 5 sets of results: 1 for the inviscid subsonic case, 2 for the viscous case (with sharp and with rounded trailing edge), 2 for the transonic case (with and without shock capturing, see below for the detailed description of the shock capturing scheme). For this test case, we generated a fine O-grid of 577 × 513 vertices using the hyperbolic grid generation capabilities of the commercial software Pointwise [1]. The farfield is located at 1000 chords, as requested. The trailing edge is sharp, unless stated otherwise. For the subsonic configurations, the vertex distribution on the airfoil is the same on pressure and suction side, as shown in fig. 1. Instead, for the transonic configuration, vertices are clustered on the suction side, in particular close to the shock region, fig. 2. Initial guesses are obtained via grid sequencing, where appropriate. The coarser grids are obtained by deleting every other grid line from the finer grid (regular coarsening).
We submit two sets of results (in Tecplot format): one set obtained imposing an inviscid (slip) w... more We submit two sets of results (in Tecplot format): one set obtained imposing an inviscid (slip) wall boundary condition on walls, and one set obtained imposing the exact solution on walls. This test case has been computed on four structured grids containing 129 × 65, 257 × 129, 513 × 257 and 1025 × 513 vertices respectively. The finest grid is obtained by applying elliptic smoothing to an algebraically created grid. Both a second and a fourth order discretization of the Laplace equation is used for this smoothing. It turned out that the resulting grids produced virtually indistinguishable results. The point distribution on the boundary is uniform and the coarse grids are obtained by deleting recursively every other grid line from the fine grid. The coarser grids are obtained by deleting every other grid line from the finer grid. The coarsest grid used is shown in figure 1. When using the slip wall boundary condition, it was found that at least a 3(on the coarsest grid at least a 4) ...
A concerted effort is underway at NASA c Langley Research Center to create a benchmark for cf Com... more A concerted effort is underway at NASA c Langley Research Center to create a benchmark for cf Computational Fluid Dynamic (CFD) codes, both Cp unstructured and structured, against a data set for the c_, hump model with actuation. The hump model was <c,>
In this paper we discuss the issue of conservation and convergence to weak solutions of several g... more In this paper we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used compact schemes and spectral collocation schemes, for solving hyperbolic conservation laws. It is shown that such s c hemes, if convergent boundedly almost everywhere, will converge to weak solutions. The results are extensions of the classical Lax-Wendro theorem concerning conservative s c hemes.
An extension of the reacting Hz-air computer code SPARK is presented, which enables the code to b... more An extension of the reacting Hz-air computer code SPARK is presented, which enables the code to be used on any reacting flow problem. Routines are developed that calculate in a general fashion, the reaction rates, and chemical Jacobians of any reacting system. In addition, an equilibrium routine is added so that the code will have frozen, finite rate, and equilibrium capabilities. The reaction rates for the species are determined from the law of mass action using Arrhenius expressions for the rate constants. The Jacobian routines are determined by numerically or analytically differentiating the law of mass action for each species. The equilibrium routine is based on a Gibbs free energy minimization routine. The routines are written in Fortran 77, with special consideration given to vectorization. Run times for the generalized routines are generally 20% slower than reaction specific routines. The numerical efficiency of the generalized analytical Jacobian, however, is nearly 300% better than the reaction specific numerical Jacobian used in SPARK. I P NOMENCLATURE Aj : reaction rate constant for jth reaction b, : body force of species i Ci : concentrak-ion of species i C, : time rate of change of Cj Cp : specific heat at constant pressure E : total internal energy; activation energy *Presently, Research Scientist working under contract for NASA Langley Research Center
It has been previously shown that the temporal integration of hyperbolic partial differ-17. SECUR... more It has been previously shown that the temporal integration of hyperbolic partial differ-17. SECURITY CLASSIFICATION OF REPORT Unclassified
The numerical study of aeroacoustic problems places stringent demands on the choice of a computat... more The numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm, because it requires the ability to propagate disturbances of small amplitude and short wavelength. The demands are particularly high when shock waves are involved, because the chosen algorithm must also resolve discontinuities in the solution.
Numerical Heat Transfer, Part B: Fundamentals, 2001
A higher order a.(:curat(' mnnerical t)roce(ture has been deveh)l)ed for solving incompressibh' •... more A higher order a.(:curat(' mnnerical t)roce(ture has been deveh)l)ed for solving incompressibh' • *The authors were partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the first two authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASI£,), NASA Langley Resear(:h Center, Hampton, VA 23681-2199. Additional sllpport was provided l)y the NASA Graduate Studenl ll.esearch Program. • _l)epartment of M(,ehanical Engineering, Ohl Dominion l!niversfly, N_)rfolk, Vir_zinia 2352.(t. • {Acouslic and Flow Met hods Branch,
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