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NASA NTRSDate not supplied

High-Order Shock Fitting with Finite Element Methods

A moving-grid, shock-tracking, finite element method has been implemented that can achieve high-order accuracy for flow simulations with shocks. In this approach, element edges in the computational mesh are fitted to the shock front and moved with the shock throughout the simulation. The Euler equations are solved on the moving mesh in an arbitrary Lagrangian-Eulerian framework. Three different methods for specifying the shock motion have been tested,and the order of accuracy of the resulting solutions was verified. It was found that one of the methods is applicable only for problems with a constant upstream flow state, whereas the other two methods converge to the expected solution for general shock problems. When using a finite element method with polynomial degree p, the two general shock motion methods converged with the expected order of accuracy of (p+1). Finally, accurate solutions are obtained for supersonic nozzle and blunt body bow shock problems. Based on the results, it is concluded that the shock tracking method can preserve high-order solution accuracy in the vicinity of a shock wave.

Luke D'Aquila↗
NASA NTRS1984

Modification to shock fitting program

A modified form of the Lepping - Argentiero single spacecraft, shock normal determination procedure is presented. The modified method incorporates a simple predictor-corrector algorithm which allows a faster convergence rate and the use of average values of the parameters for the starting vector.

Acuna, M. H.↗
NASA NTRS1984

Modification to shock fitting program

A modified form of the Lepping-Argentiero single spacecraft, shock normal determination procedure is presented. The modified method incorporates a simple predictor-corrector algorithm which allows a faster convergence rate and the use of average values of the parameters for the starting vector.

Acuna, M. H.↗
NASA NTRS1976

Inviscid transonic flow computations with shock fitting

First- and second-order numerical procedures for calculating two-dimensional transonic flows that treat shock waves as discontinuities are discussed. This short communication illustrates their application to a simple but non-trivial problem for which there are limited theoretical results.

Yu, N. J.↗
NASA NTRS1975

Inviscid transonic flow computations with shock fitting

First-and second-order numerical procedures are presented for calculating two-dimensional transonic flows that treat shock waves as discontinuities. Their application to a simple but nontrivial problem for which there are limited theoretical results is discussed.

Yu, N. J.↗
NASA NTRS1991

Spectral fitting, shock layer modeling, and production of nitrogen oxides and excited nitrogen

An analysis was made of N2 emission from 8.72 MJ/kg shock layer at 2.54, 1.91, and 1.27 cm positions and vibrational state distributions, temperatures, and relative electronic state populations was obtained from data sets. Other recorded arc jet N2 and air spectral data were reviewed and NO emission characteristics were studied. A review of operational procedures of the DSMC code was made. Information on other appropriate codes and modifications, including ionization, were made as well as a determination of the applicability of codes reviewed to task requirement. A review was also made of computational procedures used in CFD codes of Li and other codes on JSC computers. An analysis was made of problems associated with integration of specific chemical kinetics applicable to task into CFD codes.

Blackwell, H. E.↗
NASA NTRS1983

Spectral methods for the Euler equations

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of both Fourier and Chebyshev spectral collocation methods are provided. Filtering strategies for both shock-fitting and shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one-dimensional, periodic astrophysical 'nozzle' problem. Examples of shock-fitting approaches include a shock/acoustic wave interaction, shock/vortex interaction, and the classical blunt body problem. While the shock capturing spectral method does not yet show a clear advantage over second-order finite differences, equivalent accuracy can be obtained using shock fitting with far fewer grid points.

Hussaini, M. Y.↗
NASA NTRS1989

Supercritical flow past a symmetrical bicircular arc airfoil

A numerical scheme is developed for computing steady supercritical flow about symmetrical airfoils, applying it to an ellipse for zero angle of attack. An algorithmic description of this new scheme is presented. Application to a symmetrical bicircular arc airfoil is also proposed. The flow field before the shock is region 1. For transonic flow, singularity can be avoided by integrating the resulting ordinary differential equations away from the body. Region 2 contains the shock which will be located by shock fitting techniques. The shock divides region 2 into supersonic and subsonic regions and there is no singularity problem in this case. The Method of Lines is used in this region and it is advantageous to integrate the resulting ordinary differential equation along the body for shock fitting. Coaxial coordinates have to be used for the bicircular arc airfoil so that boundary values on the airfoil body can be taken with one direction of the coaxial coordinates fixed. To avoid taking boundary values at + or - infinity in the coaxial co-ordinary system, approximate analytical representation of the flow field near the tips of the airfoil is proposed.

Holt, Maurice↗
NASA NTRS1990

Extension of transonic flow computational concepts in the analysis of cavitated bearings

An analogy between the mathematical modeling of transonic potential flow and the flow in a cavitating bearing is described. Based on the similarities, characteristics of the cavitated region and jump conditions across the film reformation and rupture fronts are developed using the method of weak solutions. The mathematical analogy is extended by utilizing a few computational concepts of transonic flow to numerically model the cavitating bearing. Methods of shock fitting and shock capturing are discussed. Various procedures used in transonic flow computations are adapted to bearing cavitation applications, for example, type differencing, grid transformation, an approximate factorization technique, and Newton's iteration method. These concepts have proved to be successful and have vastly improved the efficiency of numerical modeling of cavitated bearings.

Vijayaraghavan, D.↗
NASA NTRS1990

Extension of transonic flow computational concepts in the analysis of cavitated bearings

An analogy between the mathematical modeling of transonic potential flow and the flow in a cavitating bearing is described. Based on the similarities, characteristics of the cavitated region and jump conditions across the film reformation and rupture fronts are developed using the method of weak solutions. The mathematical analogy is extended by utilizing a few computational concepts of transonic flow to numerically model the cavitating bearing. Methods of shock fitting and shock capturing are discussed. Various procedures used in transonic flow computations are adapted to bearing cavitation applications, for example, type differencing, grid transformation, an approximate factorization technique, and Newton's iteration method. These concepts have proved to be successful and have vastly improved the efficiency of numerical modeling of cavitated bearings.

Vijayaraghavan, D.↗
NASA NTRS1978

A computer program for the calculation of the flow field in supersonic mixed-compression inlets at angle of attack using the three-dimensional method of characteristics with discrete shock wave fitting

The calculation procedure is based on the method of characteristics for steady three-dimensional flow. The bow shock wave and the internal shock wave system were computed using a discrete shock wave fitting procedure. The general structure of the computer program is discussed, and a brief description of each subroutine is given. All program input parameters are defined, and a brief discussion on interpretation of the output is provided. A number of sample cases, complete with data deck listings, are presented.

Vadyak, J.↗
NASA NTRS1992

Numerical simulation of entry flow over blunt swept-wing planes

Viscous, heat-conducting flow with chemical and vibrational relaxation processes of the constituent gases surrounding winged spacecraft is considered in the continuum regime. The Navier-Stokes equations are appended by additional vibrational energy and species rate equations and supplemented by the equations of state and the phenomenological laws based on mixture rules or collisional cross sections. Numerical convective flux can be obtained from several forms of one-dimensional Riemann solver, with or without entropy correction. High-order accuracy is obtained from two types of reconstructive interpolation. A number of explicit and implicit numerical schemes have been implemented as a means to yield converged solutions. Both shock-fitting, finite-difference and shock-capturing, finite-volume techniques have been tested for configurations such as a sphere, double ellipsoid, blunt-edge delta wing, a European Hermes vehicle, and the U.S. Shuttle Orbiter. The shock-fitting code provides excellent results only for simple configurations, whereas the shock-capturing code leads to overall satisfying solutions for complex geometries.

Li, C. P.↗
NASA NTRS1979

Calculation of the flow field in supersonic inlets using a bicharacteristics method with shock wave fitting

The results of weakly viscous flow analysis are presented. The flow field, including molecular transport, is computed with the aid of a bicharacteristic method. The bow shock wave and the internal shock wave are computed with the aid of a three-dimensional shock wave fitting procedure. Characteristic equations are presented, and numerical integration procedure is discussed. Here, an inverse marching scheme is employed in which the solution is obtained on space-like planes of constant x and on space curves defined by the intersections of the internal shock wave with the solid boundaries. The distance between solution planes is arrived at by the Courant-Friedrichs-Lewy stability criterion.

Vadyak, J.↗
NASA NTRS2015

Wind Magnetic Clouds for 2010-2012: Model Parameter Fittings, Associated Shock Waves, and Comparisons to Earlier Periods

We fitted the parameters of magnetic clouds (MCs) as identified in the Wind spacecraft data from early 2010 to the end of 2012 using the model of Lepping, Jones, and Burlaga (J. Geophys. Res. 95, 1195, 1990). The interval contains 48 MCs and 39 magnetic cloud-like (MCL) events. This work is a continuation of MC model fittings of the earlier Wind sets, including those in a recent publication, which covers 2007 to 2009. This period (2010 - 2012) mainly covers the maximum portion of Solar Cycle 24. Between the previous and current interval, we document 5.7 years of MCs observations. For this interval, the occurrence frequency of MCs markedly increased in the last third of the time. In addition, over approximately the last six years, the MC type (i.e. the profile of the magnetic-field direction within an MC, such as North-to-South, South-to-North, all South) dramatically evolved to mainly North-to-South types when compared to earlier years. Furthermore, this evolution of MC type is consistent with global solar magnetic-field changes predicted by Bothmer and Rust (Coronal Mass Ejections, 139, 1997). Model fit parameters for the MCs are listed for 2010 - 2012. For the 5.7 year interval, the observed MCs are found to be slower, weaker in estimated axial magnetic-field intensity, and shorter in duration than those of the earlier 12.3 years, yielding much lower axial magnetic-field fluxes. For about the first half of this 5.7 year period, i.e. up to the end of 2009, there were very few associated MC-driven shock waves (distinctly fewer than the long-term average of about 50 % of MCs). But since 2010, such driven shocks have increased markedly, reflecting similar statistics as the long-term averages. We estimate that 56 % of the total observed MCs have upstream shocks when the full interval of 1995 - 2012 is considered. However, only 28 % of the total number of MCLs have driven shocks over the same period. Some interplanetary shocks during the 2010 - 2012 interval are seen to apparently occur without an obvious MC-driver, probably indicating an encounter with a distant flank of a MC-driven shock. Some of these may be driven by a different kind of structure, however.

Magnetic clouds↗