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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

A linearized Euler analysis of unsteady flows in turbomachinery

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Hall, Kenneth C.↗
NASA NTRS1989

Calculation of unsteady flows in turbomachinery using the linearized Euler equations

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Hall, Kenneth C.↗
NASA NTRS1988

Calculation of unsteady Euler flows in turbomachinery using the linearized Euler equations

The model used by the present unsteady cascade flow calculation method is based on the linearized unsteady Euler equations and accounts for blade loading and geometry, shock motion, and wake motion. The steady flow that must be determined prior to the unsteady flow is ascertained by means of a Newton iteration procedure. A noteworthy feature of the procedure is the use of shock-fitting to determine steady and unsteady shock positions; the use of the Euler equations in conjunction with the Rankine-Hugoniot shock-jump conditions is found to directly model the generation of entropy and vorticity at shocks. Results are presented for both channel and cascade flows.

Hall, Kenneth C.↗
NASA NTRS1972

Dynamics of the inverse pinch.

Experiments are performed on the plasma produced in a large inverse pinch using argon with no applied magnetic fields or preheating. A direct observation is made of the precursor ionization which initially raises the electron temperature and electrical conductivity. Calculations show that most of the mass is entrained in the current sheet, but that less than complete sweeping occurs at the lower velocities. The shock which is embedded in the current sheet closely resembles an ordinary gasdynamic shock. Consideration of the shock jump condition derived from the induction equation is shown to determine the frequent inability of the shock to separate from the current sheet.

Weisbach, M. F.↗
NASA NTRS1990

Properties of large scale plasma flow during the early stage of the plasmaspheric refilling

The objective is to better characterize the macroscopic properties of the interhemisphere plasma flow by solving a more complete set of hydrodynamic equations than that solved previously. Specifically, the ion continuity, momentum and energy equations were solved for the plasma flow along the closed magnetic field lines. During the initial stage of the supersonic outflow in the equatorial region, the ions cool substantially. Using the hydrodynamic model for the large-scale plasma flow, the dynamics of shocks was examined which form in the geomagnetic flux tubes during the early stages of refilling. These shocks are more like those forming in neutral gases than the electrostatic shocks driven by microinstabilities involving ion-ion interaction. Therefore, the shocks seen in the hydrodynamic model are termed as hydrodynamic shocks. Such shocks are generally unsteady and therefore the usual shock jump conditions given by Rankine-Hugoniot relations are not strictly applicable to them. The density, flow velocity and temperature structures associated with the shocks are examined for both asymmetrical and symmetrical flows. In the asymmetrical flow the outflow from one of two conjugate ionospheres is dominant. On the other hand, in the symmetrical case outflows from the two ionospheric sources are identical. Both cases are treated by a two-stream model. In the late type of flow, the early-time refilling shows a relaxation type of oscillation, which is driven by the large-scale interactions between the two identical streams. After this early stage, the resulting temperature structure shows some interesting features. In the equatorial region the streams are isothermal, but in the off-equatorial regions the streams have quite different temperatures, and also densities and flow velocities. The dense and slow stream is found to be warmer than the low-density fast stream. In the late stage of refilling, the temperature is found to steadily increase from the conjugate ionospheres towards the equator; the equatorial temperature is found to be as high as about 8000 K compared to the ionospheric temperature of 3600 K.

Singh, Nagendra↗
NASA NTRS1992

Relativistic, perpendicular shocks in electron-positron plasmas

One-dimensional particle-in-cell plasma simulations are used to examine the mechanical structure and thermalization properties of collisionless relativistic shock waves in electron-positron plasmas. Shocks propagating perpendicularly to the magnetic field direction are considered. It is shown that these shock waves exist, and that they are completely parameterized by the ratio of the upstream Poynting flux to the upstream kinetic energy flux. The way in which the Rankine-Hugoniot shock jump conditions are modified by the presence of wave fluctuations is shown, and they are used to provide a macroscopic description of these collisionless shock flows. The results of a 2D simulation that demonstrates the generality of these results beyond the assumption of the 1D case are discussed. It is suggested that the thermalization mechanism is the formation of a synchrotron maser by the coherently reflected particles in the shock front. Because the downstream medium is thermalized, it is argued that perpendicular shocks in pure electron-positron plasmas are not candidates as nonthermal particle accelerators.

Gallant, Yves A.↗
NASA NTRS1987

Shock structure measured in a transonic fan using laser anemometry

Shock structure measurements acquired in a low aspect ratio transonic fan rotor are presented and analyzed. The rotor aspect ratio is 1.56 and the design tip relative Mach number is 1.38. The rotor flowfield was surveyed at near maximum efficiency and near stall operating conditions. Intra-blade velocity measurements acquired with a laser fringe anemometer on blade-to-blade planes in the supersonic region from 10 to 60 percent span are presented. The three-dimensional shock surface determined from the velocity measurments is used to determine the shock surface normal Mach number in order to properly calculate the ideal shock jump conditions. The ideal jump conditions are calculated based upon the Mach numbers measured on a surface of revolution and based upon the normal Mach number to indicate the importance of accounting for shock three dimensionality in turbomachinery design. Comparison of the shock locations with those predicted by a 3-D Euler code showed very good agreement and indicated the usefulness of integrating computational and experimental work to enhance the understanding of the flow physics occurring in transonic turbomachinery passages.

Wood, Jerry R.↗
NASA NTRS1985

Planetary bow shocks

Planetary bow shocks provide insight into both the behavior of collisionless shocks and the nature of the planetary obstacle responsible for creating those bow shocks. This review paper first presents a survey of the microstructure of planetary bow shocks using data obtained at Mercury, Venus, the earth, Mars, Jupiter, and Saturn. When it examines the correspondence of the behavior of the shock jumps with that predicted by theory, and the differences between gas dynamic and magnetohydrodynamic solutions. Finally, it examines the information available from observations of the location of the bow shock about the nature of the solar wind-planetary interaction.

Russell, C. T.↗
NASA NTRS1982

Unsteady Transonic Small Disturbance theory with strong shock waves

Transonic Small Disturbance (TSD) theory is modified to yield steady and unsteady solutions in the case of strong shock waves (where the local Mach number ahead of the shock exceeds 1.3). The modification consists of an additional adjustable nonlinear term which allows the exact Rankine-Hugoniot shock jump relation to be satisfied at all times. The modified TSD theory is applied to steady and unsteady oscillatory transonic flows, and yields results which are in good agreement with solutions of the unsteady Euler equations, and which compare favorably with solutions of the full potential equation when shock waves are weak.

Kerlick, G. D.↗
DOE OSTI2021

The antesonic condition for the explosion of core-collapse supernovae – II. Rotation and turbulence

In the problem of steady free fall on to a standing shockwave around a central mass, the ‘antesonic’ condition limits the regime of stable accretion to $c_T^2/v_\mathrm{esc}^2\le 3/16$, where cT is the isothermal sound speed in the subsonic post-shock flow, and vesc is the escape velocity at the shock radius. Above this limit, it is impossible to satisfy both the Euler equation and the shock jump conditions, and the system transitions to a wind. This physics explains the existence of a critical neutrino luminosity in steady-state models of accretion in the context of core-collapse supernovae. Here, we extend the antesonic condition to flows with rotation and turbulence using a simple one-dimensional formalism. Both effects decrease the critical post-shock sound speed required for explosion. While quite rapid rotation is required for a significant change to the critical condition, we show that the level of turbulence typically achieved in supernova simulations can greatly impact the critical value of $c_T^2/v_\mathrm{esc}^2$. A core angular velocity corresponding to a millisecond rotation period after contraction of the proto-neutron star results in only a ~5 percent reduction of the critical curve. In contrast, near-sonic turbulence with specific turbulent kinetic energy $K/c_T^2=0.5-1$, leads to a decrease in the critical value of $c_T^2/v_{\rm esc}^2$ by ~20 to 40 percent. This analysis provides a framework for understanding the role of post-shock turbulence in instigating explosions in models that would otherwise fail and helps explain why multidimensional simulations explode more easily than their one-dimensional counterparts.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1998

The Local ISM and its Interaction with the Winds of Nearby Late-type Stars

We present new Goddard High-Resolution Spectrograph (GHRS) observations of the Ly-alpha and Mg II absorption lines seen toward the nearby stars 61 Cyg A and 40 Eri A. We use these data to measure interstellar properties along these lines of sight and to search for evidence of circumstellar hydrogen walls, which are produced by collisions between the stellar winds and the Local InterStellar Medium (LISM). We were able to model the Ly-alpha lines of both stars without hydrogen-wall absorption components, but for 61 Cyg A the fit required a stellar Ly-alpha, line profile with an improbably deep self-reversal, and for 40 Eri A the fit required a very low deuterium-to-hydrogen ratio that is inconsistent with previous GHRS measurements. Since these problems could be rectified simply by including stellar hydrogen-wall components with reasonable attributes, our preferred fits to the data include these components. We have explored several ways in which the hydrogen-wall properties measured here and in previous work can be used to study stellar winds and the LISM. We argue that the existence of a hydrogen wall around 40 Eri A and a low H I column density along that line of sight imply that either the interstellar density must decrease toward 40 Eri A or the hydrogen ionization fraction (chi) must increase. We find that hydrogen-wall temperatures are larger for stars with faster velocities through the LISM. The observed temperature-velocity relation is consistent with the predictions of hydromagnetic shock jump conditions. More precise comparison of the data and the jump conditions suggests crude upper limits for both chi and the ratio of magnetic to thermal pressure in the LISM (alpha): chi less than 0.6 and alpha less than 2. The latter upper limit corresponds to a limit on the LISM magnetic field of B less than 5 micro G. These results imply that the plasma Mach number of the interstellar wind flowing into the heliosphere is M(sub A) greater than 1.3, which indicates that the collision is supersonic and that there should therefore be a bow shock outside the heliopause in the upwind direction. Finally, we estimate stellar wind pressures (P sub wind) from the measured hydrogen-wall column densities. These estimates represent the first empirical measurements of wind properties for late-type main-sequence stars. The wind pressures appear to be correlated with stellar X-ray surface fluxes, F(x), in a manner consistent with the relation P(wind) varies as F(x)(exp -1/2), a relation that is also consistent with the variations of P(sub wind) and F(sub x) observed during the solar activity cycle. If this relation can in fact be generalized to solar-like stars, as is suggested by our data, then it is possible to estimate stellar wind properties simply by measuring stellar X-rays. One implication of this is that stellar wind pressures and mass-loss rates are then predicted to increase with time, since F(sub x) is known to decrease with stellar age.

Wood, Brian E.↗
DOE OSTI2022

Kinetic simulations verifying reconnection rates measured in the laboratory, spanning the ion-coupled to near electron-only regimes

The rate of reconnection characterizes how quickly flux and mass can move into and out of the reconnection region. In the Terrestrial Reconnection EXperiment (TREX), the rate at which the antiparallel asymmetric reconnection occurs is modulated by the presence of a shock and a region of flux pileup in the high-density inflow. Simulations utilizing a generalized Harris-sheet geometry have tentatively shown agreement with TREX's measured reconnection rate scaling relative to system size, which is indicative of the transition from ion-coupled toward electron-only reconnection. Here, in this study, we present simulations tailored to reproduce the specific TREX geometry, which confirm both the reconnection rate scale and the shock jump conditions previously characterized experimentally in TREX. The simulations also establish an interplay between the reconnection layer and the Alfvénic expansions of the background plasma associated with the energization of the TREX drive coils; this interplay has not yet been experimentally observed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1974

Pioneer 10 observations of the solar wind interation with Jupiter

Pioneer 10 Plasma Analyzer experiment flight data during the Jupiter flyby are presented. The observations show that the interaction of Jupiter's magnetic field with the solar wind is similar in many ways to that at earth, but the scale size is over 100 times larger. Jupiter is found to have a detached standing bow shock wave of high Alfven Mach number. Jupiter has a prominent magnetopause which deflects the magnetosheath plasma and excludes its direct entry into the Jovian magnetosphere. The sunward hemisphere of Jupiter's outer magnetosphere is found to be highly inflated with thermal plasma and a high beta region which is highly responsive to changes in solar wind dynamic pressure. Observational arguments are presented which tend to discount a thin disklike magnetosphere but, rather, favor a Jovian magnetosphere, albeit probabily considerably flattened as compared to the earth's magnetosphere, yet still with reasonable thickness. Results concerning the shock jump conditions, the magnetosheath flow field and inferred internal magnetospheric plasma are presented.

Wolfe, J. H.↗
NASA NTRS1975

On the computation of two- and three-dimensional steady transonic flows by relaxation methods

The paper is concerned with the application of the Murman and Cole (1971) relaxation scheme to steady, inviscid transonic flow problems in two and three dimensions. This scheme, which automatically accounts for weak shock waves, uses separate difference operators in elliptic and hyperbolic regions. The details of the scheme are described in terms of the original small disturbance formulation of Murman and Cole. In particular, Murman's recent (1973) introduction of fully conservative difference operators to obtain the correct shock jumps is examined. The extension to treating the exact isentropic equation is then covered with special attention given to Jameson's (to appear) rotated difference scheme for supersonic flow regions. The bulk of the discussion is related to two-dimensional procedures, and some comparisons with experiment are made, with emphasis on the effects of viscosity and wind-tunnel walls. Application of the Murman-Cole scheme is then discussed for small disturbances in three dimensions.

Bailey, F. R.↗
NASA NTRS1976

Numerical computation of transonic flows with shock waves

Some recent developments in numerical methods for calculating solutions to the transonic potential flow equation are reviewed, including (1) the construction of the stable coordinate independent difference schemes; (2) the use of conservation form to insure proper shock jump conditions; (3) analysis of the relaxation method by the time dependent analogy; (4) accelerated iterative schemes.

Jameson, A.↗
NASA NTRS1983

Corequake and shock heating model of the 5 March 1979 gamma ray burst

Ramatry, et al. proposed a model to account for the 5 March 1979 gamma ray burst in terms of a neutron star corequake and subsequent shock heating of the neutron star atmosphere. This model is extended by examining the overall energetics and characteristics of these shocks, taking into account the e(+)-e(-) pair production behind the shock. The effects of a dipole magnetic field in the shock jump conditions are also examined and it is concluded that the uneven heating produced by such a field can account for the temperature difference between pole and equator implied by the pulsating phase of the burst. The overall energetics and distribution of energy between e(+)-(-) pairs and photons appears to be in agreement with observations if this event is at a distance of 55 kpc as implied by its association with the Large Magellanic Cloud.

Ellison, D. C.↗
NASA NTRS1983

Corequake and shock heating model of the 5 March 1979 gamma ray burst

Ramatry, et al. proposed a model to account for the 5 March 1979 gamma ray burst in terms of a neutron star corequake and subsequent shock heating of the neutron star atmosphere. This model is extended by examining the overall energetics and characteristics of these shocks, taking into account the e(+)-e(-) pair production behind the shock. The effects of a dipole magnetic field in the shock jump conditions are also examined and it is concluded that the uneven heating produced by such a field can account for the temperature difference between pole and equator implied by the pulsating phase of the burst. The overall energetics and distribution of energy between e(+)-e(-) pairs and photons appear to be in agreement with observations if this event is at a distance of 55 kpc as implied by its association with the Large Magellanic Cloud. Previously announced in STAR as N83-31568

Ellison, D. C.↗