Results for “Linear Solvers” — Page 18

NASA NTRS2023

Measurements and Computations of Natural Transition on the NASA Juncture-Flow Model with a Symmetric Wing

Experiments were performed in the 14- by 22-Foot Subsonic Tunnel to assess natural transition on the symmetric-airfoil wings of the NASA Juncture-Flow Model. Infrared thermography was used to visualize the heating on the upper surface of both wings of the full-span model, and on the fuselage, for angles of incidence ranging from -10° to 10° at a fixed Reynolds number of 2.4E6 based on the chord length at the wing planform break. The fuselage boundary layer transitioned well upstream of the wing-root leading edge for all conditions. Transition fronts were identified by a steep rise in the surface temperature, and the transition coordinates were transformed from an image-based to a body-fixed system. Additionally, the state of the boundary layer was estimated at pressure ports distributed on the wings through observation of the pressure coefficient as a function of the angle of incidence. For increasing angles of incidence, the transition front was observed to advance upstream, in a mostly spanwise-uniform fashion, from near midchord at α = 0°; however, for increasingly negative angles of incidence, the transition front first receded and then advanced in a nonuniform jagged manner that is typically observed with stationary crossflow. The transition wedges first appeared inboard of the wing break and then spread outboard to near the tip by α = -6°. The upstream shift in transition at positive angles of incidence and the outboard progression of crossflow-dominated transition at increasingly negative angles of incidence are consistent with trends identified in a computational assessment of the boundary-layer transition based on both linear stability analysis and Reynolds-averaged-Navier-Stokes-based transition models. The stability results obtained from the Langley Stability and Transition Analysis Code were used to recalibrate a dual N-factor criterion, which allowed for the prediction of transition fronts that showed excellent agreement with the experiment. The Reynolds-averaged-Navier-Stokes-based models, from the NASA OVERFLOW 2.3 solver, that accounted for the crossflow instability showed mixed results in comparison with the experiment, with the helicity-based Langtry-Menter model performing the best. The experimental data, particularly the cases involving strong influence from both Tollmien-Schlichting and crossflow instabilities, will be valuable for the continued validation and improvement of transition models.

Boundary layer transition↗

NASA NTRS2023

Measurements and Computations of Natural Transition on the NASA Juncture-Flow Model with a Symmetric Wing

Experiments were performed in the 14- by 22-Foot Subsonic Tunnel to assess natural transition on the symmetric-airfoil wings of the NASA Juncture-Flow Model. Infrared thermography was used to visualize the heating on the upper surface of both wings of the full-span model, and on the fuselage, for angles of incidence ranging from -10° to 10° at a fixed Reynolds number of 2.4E6 based on the chord length at the wing planform break. The fuselage boundary layer transitioned well upstream of the wing-root leading edge for all conditions. Transition fronts were identified by a steep rise in the surface temperature, and the transition coordinates were transformed from an image-based to a body-fixed system. Additionally, the state of the boundary layer was estimated at pressure ports distributed on the wings through observation of the pressure coefficient as a function of the angle of incidence. For increasing angles of incidence, the transition front was observed to advance upstream, in a mostly spanwise-uniform fashion, from near midchord at α = 0°; however, for increasingly negative angles of incidence, the transition front first receded and then advanced in a nonuniform jagged manner that is typically observed with stationary crossflow. The transition wedges first appeared inboard of the wing break and then spread outboard to near the tip by α = -6°. The upstream shift in transition at positive angles of incidence and the outboard progression of crossflow-dominated transition at increasingly negative angles of incidence are consistent with trends identified in a computational assessment of the boundary-layer transition based on both linear stability analysis and Reynolds-averaged-Navier-Stokes-based transition models. The stability results obtained from the Langley Stability and Transition Analysis Code were used to recalibrate a dual N-factor criterion, which allowed for the prediction of transition fronts that showed excellent agreement with the experiment. The Reynolds-averaged-Navier-Stokes-based models, from the NASA OVERFLOW 2.3 solver, that accounted for the crossflow instability showed mixed results in comparison with the experiment, with the helicity-based Langtry-Menter model performing the best. The experimental data, particularly the cases involving strong influence from both Tollmien-Schlichting and crossflow instabilities, will be valuable for the continued validation and improvement of transition models.

boundary layer transition↗

NASA NTRS2023

Measurements and Computations of Natural Transition on the NASA Juncture-Flow Model with a Symmetric Wing

Experiments were performed in the 14- by 22-Foot Subsonic Tunnel to assess natural transition on the symmetric-airfoil wings of the NASA Juncture-Flow Model. Infrared thermography was used to visualize the heating on the upper surface of both wings of the full-span model, and on the fuselage, for angles of incidence ranging from -10° to 10° at a fixed Reynolds number of 2.4E6 based on the chord length at the wing planform break. The fuselage boundary layer transitioned well upstream of the wing-root leading edge for all conditions. Transition fronts were identified by a steep rise in the surface temperature, and the transition coordinates were transformed from an image-based to a body-fixed system. Additionally, the state of the boundary layer was estimated at pressure ports distributed on the wings through observation of the pressure coefficient as a function of the angle of incidence. For increasing angles of incidence, the transition front was observed to advance upstream, in a mostly spanwise-uniform fashion, from near midchord at α = 0°; however, for increasingly negative angles of incidence, the transition front first receded and then advanced in a nonuniform jagged manner that is typically observed with stationary crossflow. The transition wedges first appeared inboard of the wing break and then spread outboard to near the tip by α = -6°. The upstream shift in transition at positive angles of incidence and the outboard progression of crossflow-dominated transition at increasingly negative angles of incidence are consistent with trends identified in a computational assessment of the boundary-layer transition based on both linear stability analysis and Reynolds-averaged-Navier-Stokes-based transition models. The stability results obtained from the Langley Stability and Transition Analysis Code were used to recalibrate a dual N-factor criterion, which allowed for the prediction of transition fronts that showed excellent agreement with the experiment. The Reynolds-averaged-Navier-Stokes-based models, from the NASA OVERFLOW 2.3 solver, that accounted for the crossflow instability showed mixed results in comparison with the experiment, with the helicity-based Langtry-Menter model performing the best. The experimental data, particularly the cases involving strong influence from both Tollmien-Schlichting and crossflow instabilities, will be valuable for the continued validation and improvement of transition models.

boundary layer transition↗

NASA NTRS2003

Toward Immersed Boundary Simulation of High Reynolds Number Flows

In the immersed boundary (IB) method, the surface of an object is reconstructed with forcing terms in the underlying flow field equations. The surface may split a computational cell removing the constraint of the near wall gridlines to be aligned with the surface. This feature greatly simplifies the grid generation process which is cumbersome and expensive in particular for structured grids and complex geometries. The IB method is ideally suited for Cartesian flow solvers. The flow equations written in Cartesian coordinates appear in a very simple form and several numerical algorithms can be used for an efficient solution of the equations. In addition, the accuracy of numerical algorithms is dependent on the underlying grid and it usually deteriorates when the grid deviates from a Cartesian mesh. The challenge for the IB method lies in the representation of the wall boundaries and in providing an adequate near wall flow field resolution. The issue of enforcing no-slip boundary conditions at the immersed surface has been addressed by several authors by imposing a local reconstruction of the solution. Initial work by Verzicco et al. was based on a simple linear, one-dimensional operator and this approach proved to be accurate for boundaries largely aligned with the grid lines. Majumdar et al. used various multidimensional and high order polynomial interpolations schemes. These high order schemes, however, are keen to introduce wiggles and spurious extrema. Iaccarino & Verzicco and Kalitzin & Iaccarino proposed a tri-linear reconstruction for the velocity components and the turbulent scalars. A modified implementation that has proven to be more robust is reported in this paper. The issue of adequate near wall resolution in a Cartesian framework can initially be addressed by using a non-uniform mesh which is stretched near the surface. In this paper, we investigate an unstructured approach for local grid refinement that utilizes Cartesian mesh features. The computation of high Reynolds number wall bounded flows is particularly challenging as it requires the consideration of thin turbulent boundary layers, i.e. near wall regions with large gradients of the flow field variables. For such flows, the representation of the wall boundary has a large impact on the accuracy of the computation. It is also critical for the robustness and convergence of the flow solver.

Kalitzin, Georgi↗

NASA NTRS1990

Computation of convective flow with gravity modulation in rectangular cavities

In this work, a computational study is presented for the investigation of gravity modulation (g-jitter) effects in thermally driven cavity flows at terrestrial and microgravity environments. The two-dimensional, time-dependent Navier-Stokes equations are numerically integrated by a time-split method using direct matrix solvers. Computations at terrestrial gravity are utilized to assess the effects of adiabatic side-wall boundary conditions as well as the full nonlinearity of the governing equations on the sinusoidally forced Benard problem studied by Gresho and Sani. The low-g calculations focus on the establishment of critical frequency ranges and consider the effects of modulation direction and randomness. The applicability of linear analysis in the excitable frequency range at low g is also discussed.

Biringen, S.↗

NASA NTRS2005

Parallel Finite Element Domain Decomposition for Structural/Acoustic Analysis

A domain decomposition (DD) formulation for solving sparse linear systems of equations resulting from finite element analysis is presented. The formulation incorporates mixed direct and iterative equation solving strategics and other novel algorithmic ideas that are optimized to take advantage of sparsity and exploit modern computer architecture, such as memory and parallel computing. The most time consuming part of the formulation is identified and the critical roles of direct sparse and iterative solvers within the framework of the formulation are discussed. Experiments on several computer platforms using several complex test matrices are conducted using software based on the formulation. Small-scale structural examples are used to validate thc steps in the formulation and large-scale (l,000,000+ unknowns) duct acoustic examples are used to evaluate the ORIGIN 2000 processors, and a duster of 6 PCs (running under the Windows environment). Statistics show that the formulation is efficient in both sequential and parallel computing environmental and that the formulation is significantly faster and consumes less memory than that based on one of the best available commercialized parallel sparse solvers.

Nguyen, Duc T.↗

NASA NTRS2010

Atmospheric Delay Reduction Using KARAT for GPS Analysis and Implications for VLBI

We have been developing a state-of-the-art tool to estimate the atmospheric path delays by raytracing through mesoscale analysis (MANAL) data, which is operationally used for numerical weather prediction by the Japan Meteorological Agency (JMA). The tools, which we have named KAshima RAytracing Tools (KARAT)', are capable of calculating total slant delays and ray-bending angles considering real atmospheric phenomena. The KARAT can estimate atmospheric slant delays by an analytical 2-D ray-propagation model by Thayer and a 3-D Eikonal solver. We compared PPP solutions using KARAT with that using the Global Mapping Function (GMF) and Vienna Mapping Function 1 (VMF1) for GPS sites of the GEONET (GPS Earth Observation Network System) operated by Geographical Survey Institute (GSI). In our comparison 57 stations of GEONET during the year of 2008 were processed. The KARAT solutions are slightly better than the solutions using VMF1 and GMF with linear gradient model for horizontal and height positions. Our results imply that KARAT is a useful tool for an efficient reduction of atmospheric path delays in radio-based space geodetic techniques such as GNSS and VLBI.

Ichikawa, Ryuichi↗

NASA NTRS2003

Aeroelastic System Development Using Proper Orthogonal Decomposition and Volterra Theory

This research combines Volterra theory and proper orthogonal decomposition (POD) into a hybrid methodology for reduced-order modeling of aeroelastic systems. The out-come of the method is a set of linear ordinary differential equations (ODEs) describing the modal amplitudes associated with both the structural modes and the POD basis functions for the uid. For this research, the structural modes are sine waves of varying frequency, and the Volterra-POD approach is applied to the fluid dynamics equations. The structural modes are treated as forcing terms which are impulsed as part of the uid model realization. Using this approach, structural and uid operators are coupled into a single aeroelastic operator. This coupling converts a free boundary uid problem into an initial value problem, while preserving the parameter (or parameters) of interest for sensitivity analysis. The approach is applied to an elastic panel in supersonic cross ow. The hybrid Volterra-POD approach provides a low-order uid model in state-space form. The linear uid model is tightly coupled with a nonlinear panel model using an implicit integration scheme. The resulting aeroelastic model provides correct limit-cycle oscillation prediction over a wide range of panel dynamic pressure values. Time integration of the reduced-order aeroelastic model is four orders of magnitude faster than the high-order solution procedure developed for this research using traditional uid and structural solvers.

Lucia, David J.↗

NASA NTRS1999

Reduced-Order Models Based on Linear and Nonlinear Aerodynamic Impulse Responses

This paper discusses a method for the identification and application of reduced-order models based on linear and nonlinear aerodynamic impulse responses. The Volterra theory of nonlinear systems and an appropriate kernel identification technique are described. Insight into the nature of kernels is provided by applying the method to the nonlinear Riccati equation in a non-aerodynamic application. The method is then applied to a nonlinear aerodynamic model of RAE 2822 supercritical airfoil undergoing plunge motions using the CFL3D Navier-Stokes flow solver with the Spalart-Allmaras turbulence model. Results demonstrate the computational efficiency of the technique.

Silva, Walter A.↗

NASA NTRS2017

Parallel Computation of the Jacobian Matrix for Nonlinear Equation Solvers Using MATLAB

Demonstrating speedup for parallel code on a multicore shared memory PC can be challenging in MATLAB due to underlying parallel operations that are often opaque to the user. This can limit potential for improvement of serial code even for the so-called embarrassingly parallel applications. One such application is the computation of the Jacobian matrix inherent to most nonlinear equation solvers. Computation of this matrix represents the primary bottleneck in nonlinear solver speed such that commercial finite element (FE) and multi-body-dynamic (MBD) codes attempt to minimize computations. A timing study using MATLAB's Parallel Computing Toolbox was performed for numerical computation of the Jacobian. Several approaches for implementing parallel code were investigated while only the single program multiple data (spmd) method using composite objects provided positive results. Parallel code speedup is demonstrated but the goal of linear speedup through the addition of processors was not achieved due to PC architecture.

Rose, Geoffrey K.↗

NASA NTRS1999

Reduced Order Models Based on Linear and Nonlinear Aerodynamic Impulse Responses

This paper discusses a method for the identification and application of reduced-order models based on linear and nonlinear aerodynamic impulse responses. The Volterra theory of nonlinear systems and an appropriate kernel identification technique are described. Insight into the nature of kernels is provided by applying the method to the nonlinear Riccati equation in a non-aerodynamic application. The method is then applied to a nonlinear aerodynamic model of an RAE 2822 supercritical airfoil undergoing plunge motions using the CFL3D Navier-Stokes flow solver with the Spalart-Allmaras turbulence model. Results demonstrate the computational efficiency of the technique.

Silva, Walter A.↗

NASA NTRS1993

Grid adaption using Chimera composite overlapping meshes

The objective of this paper is to perform grid adaptation using composite over-lapping meshes in regions of large gradient to capture the salient features accurately during computation. The Chimera grid scheme, a multiple overset mesh technique, is used in combination with a Navier-Stokes solver. The numerical solution is first converged to a steady state based on an initial coarse mesh. Solution-adaptive enhancement is then performed by using a secondary fine grid system which oversets on top of the base grid in the high-gradient region, but without requiring the mesh boundaries to join in any special way. Communications through boundary interfaces between those separated grids are carried out using tri-linear interpolation. Applications to the Euler equations for shock reflections and to a shock wave/boundary layer interaction problem are tested. With the present method, the salient features are well resolved.

Kao, Kai-Hsiung↗

NASA NTRS1993

Grid adaptation using Chimera composite overlapping meshes

The objective of this paper is to perform grid adaptation using composite over-lapping meshes in regions of large gradient to capture the salient features accurately during computation. The Chimera grid scheme, a multiple overset mesh technique, is used in combination with a Navier-Stokes solver. The numerical solution is first converged to a steady state based on an initial coarse mesh. Solution-adaptive enhancement is then performed by using a secondary fine grid system which oversets on top of the base grid in the high-gradient region, but without requiring the mesh boundaries to join in any special way. Communications through boundary interfaces between those separated grids are carried out using tri-linear interpolation. Applications to the Euler equations for shock reflections and to a shock wave/boundary layer interaction problem are tested. With the present method, the salient features are well resolved.

Kao, Kai-Hsiung↗

NASA NTRS2011

The Prediction of Broadband Shock-Associated Noise Including Propagation Effects

An acoustic analogy is developed based on the Euler equations for broadband shock- associated noise (BBSAN) that directly incorporates the vector Green's function of the linearized Euler equations and a steady Reynolds-Averaged Navier-Stokes solution (SRANS) as the mean flow. The vector Green's function allows the BBSAN propagation through the jet shear layer to be determined. The large-scale coherent turbulence is modeled by two-point second order velocity cross-correlations. Turbulent length and time scales are related to the turbulent kinetic energy and dissipation. An adjoint vector Green's function solver is implemented to determine the vector Green's function based on a locally parallel mean flow at streamwise locations of the SRANS solution. However, the developed acoustic analogy could easily be based on any adjoint vector Green's function solver, such as one that makes no assumptions about the mean flow. The newly developed acoustic analogy can be simplified to one that uses the Green's function associated with the Helmholtz equation, which is consistent with the formulation of Morris and Miller (AIAAJ 2010). A large number of predictions are generated using three different nozzles over a wide range of fully expanded Mach numbers and jet stagnation temperatures. These predictions are compared with experimental data from multiple jet noise labs. In addition, two models for the so-called 'fine-scale' mixing noise are included in the comparisons. Improved BBSAN predictions are obtained relative to other models that do not include the propagation effects, especially in the upstream direction of the jet.

Miller, Steven↗

NASA NTRS1988

The solution of linear systems of equations with a structural analysis code on the NAS CRAY-2

Two methods for solving linear systems of equations on the NAS Cray-2 are described. One is a direct method; the other is an iterative method. Both methods exploit the architecture of the Cray-2, particularly the vectorization, and are aimed at structural analysis applications. To demonstrate and evaluate the methods, they were installed in a finite element structural analysis code denoted the Computational Structural Mechanics (CSM) Testbed. A description of the techniques used to integrate the two solvers into the Testbed is given. Storage schemes, memory requirements, operation counts, and reformatting procedures are discussed. Finally, results from the new methods are compared with results from the initial Testbed sparse Choleski equation solver for three structural analysis problems. The new direct solvers described achieve the highest computational rates of the methods compared. The new iterative methods are not able to achieve as high computation rates as the vectorized direct solvers but are best for well conditioned problems which require fewer iterations to converge to the solution.

Poole, Eugene L.↗

NASA NTRS1998

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized demoralization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin demoralization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS, and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobean linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Discrete maximum principle theory will be presented for general finite volume approximations on unstructured meshes. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc, will. be addressed as needed.

Barth, Timothy J.↗

NASA NTRS2012

Prediction of Broadband Shock-Associated Noise Including Propagation Effects Originating NASA

An acoustic analogy is developed based on the Euler equations for broadband shock-associated noise (BBSAN) that directly incorporates the vector Green s function of the linearized Euler equations and a steady Reynolds-Averaged Navier-Stokes solution (SRANS) to describe the mean flow. The vector Green s function allows the BBSAN propagation through the jet shear layer to be determined. The large-scale coherent turbulence is modeled by two-point second order velocity cross-correlations. Turbulent length and time scales are related to the turbulent kinetic energy and dissipation rate. An adjoint vector Green s function solver is implemented to determine the vector Green s function based on a locally parallel mean flow at different streamwise locations. The newly developed acoustic analogy can be simplified to one that uses the Green s function associated with the Helmholtz equation, which is consistent with a previous formulation by the authors. A large number of predictions are generated using three different nozzles over a wide range of fully-expanded jet Mach numbers and jet stagnation temperatures. These predictions are compared with experimental data from multiple jet noise experimental facilities. In addition, two models for the so-called fine-scale mixing noise are included in the comparisons. Improved BBSAN predictions are obtained relative to other models that do not include propagation effects.

Miller, Steven↗

NASA NTRS1989

Application of a full-potential solver to bending-torsion flutter in cascades

A two-dimensional, unsteady, full-potential cascade aerodynamic code has been used for flutter calculations. The code, which uses a time-marching implicit scheme with internal Newton-iterations at each time step, allows oscillatory blade motions with phase lag between adjacent blades to be modeled using a single inter-blade passage. This aerodynamic code has been used with a two-degrees-of-freedom typical section structural model to determine the aeroelastic stability of a bladed disk. The unsteady pressure results calculated from the aerodynamic code have been verified by comparison with results from the linear and linearized potential theories. Flutter results have been presented for two cases. A comparison with linear theory shows identical trends; however, the present calculations show a lower flutter frequency.

Bakhle, Milind A.↗