Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Generalized relaxation equation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2

Three-dimensional elliptic grid generation technique with application to turbomachinery cascades

Described is a numerical method for generating 3-D grids for turbomachinery computational fluid dynamic codes. The basic method is general and involves the solution of a quasi-linear elliptic partial differential equation via pointwise relaxation with a local relaxation factor. It allows specification of the grid point distribution on the boundary surfaces, the grid spacing off the boundary surfaces, and the grid orthogonality at the boundary surfaces. A geometry preprocessor constructs the grid point distributions on the boundary surfaces for general turbomachinery cascades. Representative results are shown for a C-grid and an H-grid for a turbine rotor. Two appendices serve as user's manuals for the basic solver and the geometry preprocessor.

Chen, S. C.↗

The development of methods for the prediction of primary creep behavior in metals

The applicability of a thermodynamic constitutive theory of deformation to the prediction of primary creep and creep strain relaxation behavior in metals is examined. Constitutive equations derived from the theory are subjected to a parametric analysis in order to determine the influence of several parameters on the curve forms generated by the equations. A computer program is developed which enables the solution of a generalized constitutive equation using experimental data as input. Several metals were tested to form a data base of primary creep and relaxation behavior. The extent to which these materials conformed to the constitutive equation showed wide variability, with the alloy Ti-6Al-4V exhibiting the most consistent results. Accordingly, most of the analysis is concentrated upon data from that alloy, although creep and relaxation data from all the materials tested are presented. Experimental methods are outlined as well as some variations in methods of analysis. Various theoretical and practical implications of the work are discussed.

Zerwekh, R. P.↗

Navier-Stokes computations for aerodynamic configurations at high angles of attack

Navier-Stokes computations at high angles of attack over aerodynamic configurations are presented using an implicit finite-volume algorithm. The spatial differencing is upwind-biased for the convective and pressure terms and central for the shear stress and heat transfer terms. The equations are relaxed to steady state with a spatially factored implicit algorithm. In order to treat general geometries, a multiblock patched-grid framework is implemented. Applications and detailed comparisons with experimental data are made for two simple but representative geometric shapes: (1) a highly swept delta wing and (2) a prolate spheroid of 6:1 length-to-diameter. Recent extensions of the algorithm to compute the flow over an F-18 forebody-strake configuration are shown, including comparisons with wind tunnel and flight test results. Comparisons across the range of Reynolds number for all cases indicate that either low Reynolds number or high Reynolds number flows are simulated well, but that flows at intermediate Reynolds number cannot be simulated accurately without a detailed knowledge of the transitional zone between laminar and turbulent flow.

Thomas, J. L.↗

A kinematically driven anisotropic viscoelastic constitutive model applied to tires

Aircraft tires are composite structures manufactured with viscoelastic materials such as carbon black filled rubber and nylon cords. When loaded they experience large deflections and moderately large strains. Detailed structural models of tires require the use of either nonlinear shell or nonlinear three dimensional solid finite elements. Computational predictions of the dynamic response of tires must consider the composite viscoelastic material behavior in a realistic fashion. We describe a modification to a nonlinear anisotropic shell finite element so it can be used to model viscoelastic stresses during general deformations. The model is developed by introducing internal variables of the type used to model elastic strain energy. The internal variables are strains, curvatures, and transverse shear angles which are in a one-to-one correspondence with the generalized coordinates used to model the elastic strain energy for nonlinear response. A difference-relaxation equation is used to relate changes in the observable strain field to changes in the internal strain field. The internal stress state is introduced into the equilibrium equations by converting it to nodal loads associated with the element's displacement degrees of freedom. In this form the tangent matrix in the Newton-Raphson solution algorithm is not modified from its form for the nonlinear statics problem. Only the gradient vector is modified and the modification is not computationally costly. The existing finite element model for the Space Shuttle nose gear tire is used to provide examples of the algorithm. In the first example, the tire's rim is displaced at a constant rate up to a fixed value. In the second example, the tire's rim is enforced to follow a saw tooth load and unload curve to generate hysteresis loops.

Johnson, Arthur R.↗

An implicit finite element method for high speed flows

A fast algorithm is presented for constructing continuous lines, consisting of element sides, on general unstructured two-dimensional triangular meshes. The lines must pass through each node of the mesh once and only once. The discussion focuses on the use of these lines in a relaxation method for the solution of the equation system arising from an implicit algorithm for the solution of two-dimensional Euler and Navier-Stokes equations on general unstructured grids. It is also shown that the method can be used for solving three-dimensional Navier-Stokes equations on a grid composed of both structured and unstructured regions.

Hassan, O.↗

Relaxation solutions for inviscid axisymmetric transonic flow over blunt or pointed bodies.

A finite-difference relaxation method is presented for numerical solution of the full potential equation and exact boundary conditions for general axisymmetric bodies is inviscid, steady transonic flow. Body-normal coordinates are used in the nose region and sheared cylindrical coordinates are used on the afterbody to accommodate corners such as boattails and flares. An improved difference scheme is used which does not require that the flow be nearly alined with a coordinate direction in supersonic regions, and which treats either subsonic or supersonic free streams. Numerical results are illustrated for some simple classical shapes such as spheres and ellipsoids, and for more practical shapes like tangent-ogives with boattails. Special attention is given to bodies which have been studied for area-rule applications. Agreement with available experimental results is good in cases where viscous effects and wind-tunnel wall interference are not important.

South, J. C., Jr.↗

Impact of Satellite Sea Surface Salinity Observations on ENSO Predictions from the GMAO S2S Forecast System

Sea surface salinity (SSS) observations from space allow us to investigate if improved estimates of near-surface density stratification and associated mixing can positively impact seasonal to interannual variability of tropical Pacific Ocean dynamics as well as dynamical ENSO forecasts. For the first part of the presentation, we utilize our intermediate-complexity coupled model. Baseline experiments assimilate satellite sea level (multi-satellite gridded AVISO, 2013), SST (Reynolds et al., 2004), and in situ subsurface temperature and salinity observations (GTSPP NODC, 2006). These baseline experiments are then compared with experiments that additionally assimilate Aquarius (V5.0 Lilly and Lagerloef, 2008) and SMAP (V4.0 Fore et al., 2016) SSS. Twelve-month forecasts are initialized for each month from September 2011 to September 2017. For initialization of the coupled forecast, the positive impact of SSS assimilation is brought about by surface freshening near the eastern edge of the western Pacific warm pool and density changes that lead to shallower mixed layer between 10S-5N. This pattern enhances air/sea interaction and amplifies the equatorial Kelvin wave signal. We find that including satellite SSS significantly improves NINO3.4 sea surface temperature anomaly validation over most forecast lead times. We next assess how different satellite SSS products impact the validation of ENSO forecasts. SMAP V4 reduces the salty bias in the western Pacific and so is an improvement upon SMAP V2 and SMOS V2 (Boutin et al., 2017) has similar validation characteristics as a combination of Aquarius and SMAP V4. Next we shift to present results from the NASA GMAO Sub-seasonal to seasonal (S2S_v2.1) production coupled model (i.e. the same model that contributes ENSO forecasts to the North American Multi-Model Ensemble Experiment). From March to June 2015, the availability of two overlapping satellite SSS instruments, Aquarius and SMAP, allows a unique opportunity to compare and contrast forecasts initialized with the benefit of these two satellite SSS observation types. We assess the impact of satellite sea surface salinity (SSS) observations on dynamical ENSO forecasts for the big 2015 El Nino event. We will present distinct experiments for the overlap period that include 1) freely evolving SSS (i.e. no satellite SSS as the production system), 2) Aquarius, and 3) SMAP initialization. Our results show that using Aquarius slightly improves validation of the reanalysis (including sea level and temperature statistics). Our production system without SSS assimilation generated too warm forecasts for the 2015 El Nino from March initial conditions. Incorporating Aquarius into initialization of the coupled system leads to a deeper, more realistic MLD that acts to damp the downwelling Kelvin signal and slightly cool NINO3.4 SST. With Aquarius the forecasts better match the observed amplitude of the 2015 event. On the other hand, SMAP V2 relaxation generally degrades validation statistics. At forecast initialization, SMAP is much too salty within 10o of the equator, leading to deeper MLD east of 165W. This deeper MLD leads to over-damping of the downwelling signal (i.e. relative upwelling), in turn leading to relatively too cool ENSO forecasts.

Hackert, E.↗

An efficient numerical method for solving the incompressible Navier-Stokes equations

This paper describes an efficient numerical method for solving the steady incompressible Navier-Stokes equations. The method is a fully implicit method based on the generalized Galerkin method, and the resulting system of equations is solved in a sweeping mode by iterative line relaxation. Results of the present method are compared with published results for separating and reattaching flows, and parametric studies showing the effects of step size, boundary locations, and Reynolds number are presented. The present method is substantially faster than previously published methods; typical run times range from 10 sec to 1 min of 7600 CPU time; and, based on results obtained to date, it is stable at any Reynolds number.

Murphy, J. D.↗

Vectorizable implicit algorithms for the flux-difference split, three-dimensional Navier-Stokes equations

The computational efficiency of four vectorizable implicit algorithms is assessed when applied to calculate steady-state solutions to the three-dimensional, incompressible Navier-Stokes equations in general coordinates. Two of these algorithms are characterized as hybrid schemes; that is, they combine some approximate factorization in two coordinate directions with relaxation in the remaining spatial direction. The other two algorithms utilize an approximate factorization approach which yields two-factor algorithms for three-dimensional systems. All four algorithms are implemented in identical high-resolution upwind schemes for the flux-difference split Navier-Stokes equations. These highly nonlinear schemes are obtained by extending an implicit Total Variation Diminishing (TVD) scheme recently developed for linear one-dimensional systems of hyperbolic conservation laws to the three-dimensional Navier-Stokes equations. The computation of vortical flow over a sharp-edged, thin delta wing has been chosen as a common numerical test case. The convergence of the algorithms is discussed and the accuracy of the computed flow-field results is assessed. The validity of the present results are demonstrated by a comparison with experimental data.

Hartwich, P. M.↗

Using High Frequency Passive Microwave, A-train, and TRMM Data to Evaluate Hydrometer Structure in the NASA GEOS-5 Data Assimilation System

Validating water vapor and prognostic condensate in global models remains a challenging research task. Model parameterizations are still subject to a large number of tunable parameters; furthermore, accurate and representative in situ observations are very sparse, and satellite observations historically have significant quantitative uncertainties. Progress on improving cloud / hydrometeor fields in models stands to benefit greatly from the growing inventory ofA-Train data sets. ill the present study we are using a variety of complementary satellite retrievals of hydrometeors to examine condensate produced by the emerging NASA Modem Era Retrospective Analysis for Research and Applications, MERRA, and its associated atmospheric general circulation model GEOS5. Cloud and precipitation are generated by both grid-scale prognostic equations and by the Relaxed Arakawa-Schubert (RAS) diagnostic convective parameterization. The high frequency channels (89 to 183.3 GHz) from AMSU-B and MRS on NOAA polar orbiting satellites are being used to evaluate the climatology and variability of precipitating ice from tropical convective anvils. Vertical hydrometeor structure from the Tropical Rainfall Measuring Mission (TRMM) and CloudSat radars are used to develop statistics on vertical hydrometeor structure in order to better interpret the extensive high frequency passive microwave climatology. Cloud liquid and ice water path data retrieved from the Moderate Resolution Imaging Spectroradiometer, MODIS, are used to investigate relationships between upper level cloudiness and tropical deep convective anvils. Together these data are used to evaluate cloud / ice water path, gross aspects of vertical hydrometeor structure, and the relationship between cloud extent and surface precipitation that the MERRA reanalysis must capture.

Robertson, Franklin↗

A semi-direct procedure using a local relaxation factor and its application to an internal flow problem

Generally, fast direct solvers are not directly applicable to a nonseparable elliptic partial differential equation. This limitation, however, is circumvented by a semi-direct procedure, i.e., an iterative procedure using fast direct solvers. An efficient semi-direct procedure which is easy to implement and applicable to a variety of boundary conditions is presented. The current procedure also possesses other highly desirable properties, i.e.: (1) the convergence rate does not decrease with an increase of grid cell aspect ratio, and (2) the convergence rate is estimated using the coefficients of the partial differential equation being solved.

Chang, S. C.↗

Three-dimensional adaptive grid generation for body-fitted coordinate system

This report describes a numerical method for generating 3-D grids for general configurations. The basic method involves the solution of a set of quasi-linear elliptic partial differential equations via pointwise relaxation with a local relaxation factor. It allows specification of the grid spacing off the boundary surfaces and the grid orthogonality at the boundary surfaces. It includes adaptive mechanisms to improve smoothness, orthogonality, and flow resolution in the grid interior.

Chen, S. C.↗

Three-dimensional adaptive grid generation for body-fitted coordinate system

This report describes a numerical method for generating 3-D grids for general configurations. The basic method involves the solution of a set of quasi-linear elliptic partial differential equations via pointwise relaxation with a local relaxation factor. It allows specification of the grid spacing off the boundary surfaces and the grid orthogonality at the boundary surfaces. It includes adaptive mechanisms to improve smoothness, orthogonality, and flow resolution in the grid interior.

Chen, S. C.↗

Kinetic relaxation of a non-Maxwellian monatomic gas in a state of gross rest

The kinetic relaxation of a monatomic gas in a state of gross rest is studied by use of Maxwell's second-order moment equation. The molecular models considered range from the hard sphere to the Maxwell molecule. The development is exact for an ellipsoidal distribution and becomes a good approximation for more general distributions. The results show that the relaxation of second-order moments is nearly exponential for power-law molecules and that the characteristic time mu/p appears as the principal controlling parameter while the power-law constant plays a secondary role. Comparisons between results of numerical simulations using Bird's direct simulation Monte Carlo method and theory show excellent agreement for both hard-sphere and Maxwell molecules.

Baganoff, D.↗

The practical application of a finite difference method for analyzing transonic flow over oscillating airfoils and wings

Separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady equations for small disturbances was performed. The steady velocity potential was obtained first from the well known nonlinear equation for steady transonic flow. The unsteady velocity potential was then obtained from a linear differential equation in complex form with spatially varying coefficients. Since sinusoidal motion is assumed, the unsteady equation is independent of time. The results of an investigation into the relaxation-solution-instability problem was discussed. Concepts examined include variations in outer boundary conditions, a coordinate transformation so that the boundary condition at infinity may be applied to the outer boundaries of the finite difference region, and overlapping subregions. The general conclusion was that only a full direct solution in which all unknowns are obtained at the same time will avoid the solution instabilities of relaxation. An analysis of the one-dimensional form of the unsteady transonic equation was studied to evaluate errors between exact and finite difference solutions. Pressure distributions were presented for a low-aspect-ratio clipped delta wing at Mach number of 0.9 and for a moderate-aspect-ratio rectangular wing at a Mach number of 0.875.

Weatherill, W. H.↗

Flux-split algorithms for flows with non-equilibrium chemistry and vibrational relaxation

The present consideration of numerical computation methods for gas flows with nonequilibrium chemistry thermodynamics gives attention to an equilibrium model, a general nonequilibrium model, and a simplified model based on vibrational relaxation. Flux-splitting procedures are developed for the fully-coupled inviscid equations encompassing fluid dynamics and both chemical and internal energy-relaxation processes. A fully coupled and implicit large-block structure is presented which embodies novel forms of flux-vector split and flux-difference split algorithms valid for nonequilibrium flow; illustrative high-temperature shock tube and nozzle flow examples are given.

Grossman, B.↗

Comparison of solutions of the Navier-Stokes and interacting boundary layer equations for separated turbulent flow

Comparisons are presented between numerical solutions of the Navier-Stokes and interacting boundary-layer equations for the subsonic separated turbulent flow over an axisymmetric boattail configuration for which experimental data was available. Comparisons were made using a standard algebraic turbulence model and its modification using a streamwise relaxation model. Grid size studies were performed to provide an approximate assessment of the sensitivity of each method to mesh size. In general, good agreement was observed between the solutions of the Navier-Stokes and interacting boundary-layer equations; however, the interaction approach responded more strongly to the modified turbulence model thereby giving better agreement with experimental data.

Vatsa, V. N.↗