Mark H Carpenter
NASA Langley Research CEnter, Computational AeroSciences, Department Member
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In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as... more
In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates. The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version. With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes. We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.
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Transition analysis is performed for a swept wing at a Mach number of 0.75 and chord Reynolds number of approximately 1.7×107, with a focus on roughness-based crossflow-transition control at high Reynolds numbers relevant to subsonic... more
Transition analysis is performed for a swept wing at a Mach number of 0.75 and chord Reynolds number of approximately 1.7×107, with a focus on roughness-based crossflow-transition control at high Reynolds numbers relevant to subsonic flight. The roughness-based transition control involves controlled seeding of suitable, subdominant crossflow modes to weaken the growth of naturally occurring, linearly more unstable instability modes via a nonlinear modification of the mean boundary-layer profiles. Therefore, a synthesis of receptivity, linear and nonlinear growth of crossflow disturbances, and high-frequency secondary instabilities becomes desirable to model this form of control. Because experimental data are currently unavailable for passive crossflow-transition control on high-Reynolds-number configurations, a holistic computational approach is used to assess the feasibility of roughness-based-control methodology. The potential challenges inherent to this control application, as well as the associated di...
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Research Interests: Mechanical Engineering, Chemical Engineering, Materials Science, Mechanics, Chemistry, and 15 moreConvection, Statistical Analysis, Combustion, Numerical Method, Automotive Engineering, Air flow, Computation Fluid Dynamics, Gas Dynamics, Navier Stokes, High Speed, Growth rate, Incompressible Flow, Mixed layer, Mach Number, and Higher order
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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... 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:
ρ = p RT0 The flow is governed by the Navier-Stokes equations with a Prandtl number of 0.71, specific heat ratio γ = 1.4 and the bulk viscosity is assumed to be zero. Furthermore, the Mach number V0/c0 = 0.1 and the Reynolds number Re =... more
ρ = p RT0 The flow is governed by the Navier-Stokes equations with a Prandtl number of 0.71, specific heat ratio γ = 1.4 and the bulk viscosity is assumed to be zero. Furthermore, the Mach number V0/c0 = 0.1 and the Reynolds number Re = ρ0V0L μ = 1600. The initial temperature is uniform, T0 = p0 ρ0R . The solution is computed on the periodic domain Ω = {−πL ≤ x, y, z ≤ πL} which is discretized using four uniform structured grids containing 65, 129, 257 and 513 vertices respectively. For the 65, 129 and 257 grids it was possible to use our local Linux cluster, while the 513 grid was run on up to 512 processors of the LISA machine of SARA, the Dutch Supercomputer Center. With a convective time scale tc = L V0 , the final time in the simulation is tfinal = 20tc. The classical 4 th
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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... 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).
The decoupled solver was used to get an initial solution on the coarse grid and as main solver on the medium and fine grids (see below for a description of the grid and the refinement strategy). The coupled solver was mainly used on the... more
The decoupled solver was used to get an initial solution on the coarse grid and as main solver on the medium and fine grids (see below for a description of the grid and the refinement strategy). The coupled solver was mainly used on the coarse grid. Since all development, tests, and runs have been performed in a relative short time, no tuning for perfomance has been carried out; hence, we expect the solvers not to perform at peak efficiency. For this test case, a fine O-grid of 513 × 169 vertices was generated with the commercial software Pointwise R © [2]. The farfield is located at 240 chords, and all results are obtained by applying a vortex correction [3]. This correction turned out to be crucial in order to limit the effect of the farfield position on lift and drag coefficients to less than 0.01 counts. For the given location, this is indeed the case. However, when the vortex correction is not applied, the farfield should be located several thousands of chord lengths away in or...
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... 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) ...
Abstract A systematic approach based on a diagonal-norm summation-by-parts (SBP) framework is presented for implementing entropy stable (SS) formulations of any order for the compressible Navier–Stokes equations (NSE). These SS... more
Abstract A systematic approach based on a diagonal-norm summation-by-parts (SBP) framework is presented for implementing entropy stable (SS) formulations of any order for the compressible Navier–Stokes equations (NSE). These SS formulations discretely conserve mass, momentum, energy and satisfy a mathematical entropy equality for smooth problems. They are also valid for discontinuous flows provided sufficient dissipation is added at shocks and discontinuities to satisfy an entropy inequality. Admissible SBP operators include all centred diagonal-norm finite-difference (FD) operators and Legendre spectral collocation-finite element methods (LSC-FEM). Entropy stable multiblock FD and FEM operators follows immediately via nonlinear coupling operators that ensure conservation, accuracy and preserve the interior entropy estimates. Nonlinearly stable solid wall boundary conditions are also available. Existing SBP operators that lack a stability proof (e.g. weighted essentially nonoscillatory) may be combined with an entropy stable operator using a comparison technique to guarantee nonlinear stability of the pair. All capabilities extend naturally to a curvilinear form of the NSE provided that the coordinate mappings satisfy a geometric conservation law constraint. Examples are presented that demonstrate the robustness of current state-of-the-art entropy stable SBP formulations.
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Research Interests: Engineering, Mathematics, Applied Mathematics, Computer Science, Computational Physics, and 13 moreConvergence, Computational Mathematics, Mathematical Sciences, Physical sciences, Finite differences, Accuracy, Adaptivity, Grid, Finite Difference, Finite Difference Method, Least Squares, Stencil, and Truncation Error
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Research experience in constructing and applying higher order temporal schemes with error controllers for solving unsteady Reynolds-averaged Navier-Stokes equations is reported. The baseline flow solver under consideration uses a... more
Research experience in constructing and applying higher order temporal schemes with error controllers for solving unsteady Reynolds-averaged Navier-Stokes equations is reported. The baseline flow solver under consideration uses a second-order backward difference scheme with a dual time stepping algorithm for advancing the flow solutions in time. The accuracy and efficiency of the new scheme is assessed by comparing the computational results with either analytical or highly accurate numerical solutions for aerodynamic problems of interest.