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At least 289 records · Page 16

Oxgen-burning hydrodynamics. 1: Steady shell burning

With new hydrodynamic techniques, the relatively fast evolutionary stages of a star prior to core collapse may be explicitly computed in two spatial dimensions, with a treatment of the microphysics (e.g., nuclear reactions, equation of state, neutrino cooling) which is comparable to typical one-dimensional simulations. The nature of shell oxygen burning in a massive star, prior to core collapse, is used as a first example; it is of particular interest because it is (1) the region in which Ni-56 will be produced by the supernova shock, (2) the region of the 'mass cut', which will separate the collapsed core from the ejected mantle, (3) the site of much of the explosive nucleosynthesis, and (4) a suggested source of symmetry breaking to drive mixing instabilities which were observed in SN 1987A. The nature of the shell burning affects the size of the core which will collapse. The method is illustrated on this test case, and the character of the convection is examined.

Arnett, David↗

Postcollapse hydrodynamics of SN 1987A - Two-dimensional simulations of the early evolution

The first few seconds of the explosion of SN 1987A are modeled here using a 2D cylindrical geometry smooth particle hydrodynamics code. The success of the explosion is determined to be sensitive to the duration of the infall, the timing of the bounce, and the subsequent neutrino heating. A semianalytical model for the initial structure of the collapsed object is used to present two simulations that differ by the mass that has been allowed to collapse into a neutron star prior to the bounce. In the case of a short initial infall, the explosion fails due to excessive cooling. For a longer initial infall, the cooling is less and the explosion is successful although relatively weak. It is shown that in this case, a successful explosion is brought about by the presence of an entropy gradient which, combined with the gravitational pull of the neutron star, leads to extremely strong instabilities. The critical importance of the global circulation for the success of the explosion is demonstrated.

Herant, Marc↗

The clumpy circumstellar medium around young supernova remnants

Each of the youngest supernova remnants known in the Milky Way, Cas A, Kepler's SNR, and Tycho's SNR, shows a different morphological structure caused by different conditions in the progenitor stars and their surroundings. In all three cases, however, the observed shells have a thickness of about 1/4 the radius, a sharp outer edge, and significant brightness irregularities. These features require that the circumstellar medium be highly clumped. To investigate the phenomenon, models of the expansion have been constructed using a one-dimensional spherical hydrodynamic code. As a supernova shock moves down the external density gradient of the star, material behind the shock begins to go into free expansion. Then as surrounding material is encountered a reverse shock moving back into the ejectum will be formed. Until the expansion has swept up about eight times the ejected mass when the situation can be considered as a point explosion in its surroundings, the dynamics are controlled by conditions between the shocks. The region is also where the synchrotron radio emission from relativistic electrons trapped in magnetic fields arises. Initial particles and fields are accelerated and amplified by eddy motion at the interface between the ejected and swept-up material and at the boundaries of clumps. Polarimetry shows that these SNR have a net radial orientation of their magnetic fields apparently from stretching by Rayleigh-Taylor instabilities at the contact surfaces. Without clumps the observed shell is much too narrow and steep on the inside.

Dickel, J. R.↗

Hydrodynamic Tests of a 1/10-size Model of the Hull of the Latecoere 521 Flying Boat : NACA Model 83

A 1/10-size model of the hull of the French flying boat Latecoere 521 was tested in the NACA tank. This model is one of a series of models of the hulls of actual flying boats of both foreign and domestic type that are being tested in the NACA tank to provide information regarding the water characteristics of a variety of forms of hull and to illustrate the development of present-day types of flying boat. The lines and the offsets of the hull were obtained from the manufacturer through the Paris office of the NACA. The form of the stub-wing stabilizers was not furnished and, therefore, the model was tested without them. The model was tested free to trim at the design initial load (initial load coefficient of 0.428) and by the general method at load coefficients from 0.025 to 0.6. The spray characteristics of the model are good. The form of the bow would be particularly desirable for rough water use. The interference of the afterbody and the tail extension is excessive, causing very high resistance at high speeds. A violent vertical instability is present at trims of 4 degrees and 6 degrees with light loads and high speeds.

Olson, Roland E↗

Evolution of the solar nebula. I - Nonaxisymmetric structure during nebula formation

Numerical solutions of the equations of hydrodynamics, gravitation, and radiative transfer in three spatial dimensions are used to model the formation and time evolution of the early solar nebula in order to learn whether or not gravitational torques between nonaxisymmetric structures in the solar nebula can transport angular momentum rapidly enough to produce nebula clearing on astronomically indicated (10 to the 5 to 10 to the 7 yr) time scales. The models involve solutions for the collapse of spherical clouds with assumed initial density and rotation profiles onto protosuns of variable mass. Most of the models assume uniform initial density and rotation, and have variations in the initial parameters of cloud mass, cloud rotation rate, and protosun mass which are chosen to simulate a range of possible phases of early solar nebula evolution. The models show little tendency for directly forming small numbers of giant gaseous protoplanets through gaseous gravitational instability.

Boss, Alan P.↗

Thresholds for the onset of fluid and magnetofluid turbulence

Linear stability calculations conducted in plasma physics are based on the theory of hydrodynamic stability of neutral fluids. The present investigation is concerned with the validity of procedures based on linear stability analysis in six much-studied situations. All have the property that the fluid goes from laminar to turbulent at critical values of some dimensionless number, such as the Reynolds number or the Rayleigh number. The question is, at what threshold values of the dimensionless number do the unstable motions set in, and can these thresholds be predicted by a linear analysis of the stability of the laminar state as the threshold is approached from the stable side. In three cases linear stability analysis clearly seems to fail. These cases include Plane Poiseuille Flow, Plane Couette Flow, and Cylindrical Pipe Flow (Hagen-Poiseuille Flow). Situations in which predictions provided by linear stability analysis are correct are related to Rotating Couette Flow, Thermally-Driven Convection, and Instability of Laminar Boundary Layers.

Montgomery, D.↗

Stability Analysis of Streaks Induced By Optimized Vortex Generators

Numerical computations are performed to investigate the potential for transition control in an axisymmetric boundary layer via fully realizable, streamwise stationary streaks induced by an azimuthally periodic array of surface mounted vortex generators (VGs). Previous work has shown that suitable streaks of this type can significantly reduce the growth of Mack’s second mode instabilities, but large streak amplitudes can make the flow susceptible to previously absent streak instabilities that can become the leading cause of transition. Here, we use the adjoint capabilities of the SU2 flow solver to optimize the VG shape to maximize the reduction in the growth of second-mode disturbances while also preventing the streak amplitudes from reaching large enough values to precipitate an earlier onset of transition via streak instabilities. The geometry and the freestream flow conditions are selected to match a relevant trajectory lo-cation from the HIFiRE-1 flight experiment. Results show that the optimized VGs can increase the mean streak amplitude by 117% with respect to a manually developed baseline design. The stability of this optimized basic state is analyzed via the plane-marching parabolized stability equations, predicting a fully laminar flow over the entire cone, or equivalently, yielding transition delay of 130% versus the 17% for the baseline VGs.

Boundary layer transition↗

Stability Analysis of Streaks Induced by Optimized Vortex Generators

Numerical computations are performed to investigate the potential for transition control in an axisymmetric boundary layer via fully realizable, streamwise stationary streaks induced by an azimuthally periodic array of surface mounted vortex generators (VGs). Previous work has shown that suitable streaks of this type can significantly reduce the growth of Mack’s second mode instabilities, but large streak amplitudes can make the flow susceptible to previously absent streak instabilities that can become the leading cause of transition. Here, we use the adjoint capabilities of the SU2 flow solver to optimize the VG shape to maximize the reduction in the growth of second-mode disturbances while also preventing the streak amplitudes from reaching large enough values to precipitate an earlier onset of transition via streak instabilities. The geometry and the freestream flow conditions are selected to match a relevant trajectory lo-cation from the HIFiRE-1 flight experiment. Results show that the optimized VGs can increase the mean streak amplitude by 117% with respect to a manually developed baseline design. The stability of this optimized basic state is analyzed via the plane-marching parabolized stability equations, predicting a fully laminar flow over the entire cone, or equivalently, yielding transition delay of 130% versus the 17% for the baseline VGs.

Boundary layer transition↗

The stability of thick, self-gravitating disks in protostellar systems

Using numerical hydrodynamics techniques, we perform a nonlinear stability analysis of accretion disk systems that contain thick, self-gravitating disks. The systems are initially represented by a point mass M(sub c) at the center and a geometrically thick, axisymmetric disk of mass M(sub d) that supports uniform specific angular momentum and obeys an n = 3/2, polytropic equation of state. The equilibrium disk structure is uniquely defined upon the specification of two key dimensionless system parameters: M(sub d)/M(sub c) and T/absolute value of W (the ratio of rotational kinetic energy if the disk to the gravitational potential energy of the system). The focus of this work is on the identification of systems within this two-dimensional parameter space that are marginally unstable toward the development of nonaxisymmetric distortions. The geometric form and relative pattern speed of the disk's distortion as well as the likelihood of disk fragmentation as a result of such instabilities is examined, particularly in the context of protostellar systems. The value of T/absolute value of W at which thick disks first become dynamically unstable to nonaxisymmetric distortions is found to vary significantly with the mass ratio of the accertion disk system. Nonaxisymmetric eigenmodes with four distinctly different characters are identified in systems with mass ratios in the range 0.2 less than or equal to (M(sub d)/M(sub c)) less than or equal to 5.

Woodward, John W.↗

PSE-Based Aerodynamic Flow Transition Prediction Using Automated Unstructured CFD Integration

Accurate, robust, and efficient prediction of transition in viscous flows is a significant challenge in computational fluid dynamics. We present a coupled, high-fidelity, iterative approach that leverages the FUN3D flow solver and the LASTRAC stability code to predict transition in low-disturbance environments, initiated by the linear growth of boundary-layer instability modes. Our method integrates the ability of FUN3D to compute mixed laminar-transitional-turbulent mean flows via transition-sensitized Reynolds-Averaged Navier-Stokes equations with the ability of LASTRAC to perform linear stability analysis, all within an automated framework that requires no intermediate user involvement. Unlike conventional frameworks that rely on classical stability theory or reduced-order metamodels, our approach employs the parabolized stability equations to provide more accurate and reliable estimates of disturbance growth for multiple instability mechanisms, including Tollmien-Schlichting, Kelvin-Helmholtz, and crossflow modes. By accounting for the effects of mean-flow nonparallelism as well as the surface curvature, this approach lays the foundation for improved N-factor correlations for transition onset prediction in a broad class of flows. We apply this method to three distinct flow configurations: 1) flow over a zero-pressure-gradient flat plate, 2) the NLF-0416 airfoil with both natural and separation-induced transitions, and 3) a 6:1 prolate spheroid, where transition is primarily driven by crossflow instability. For the two-dimensional cases, a formulated intermittency distribution is used to model the transition zone in between the laminar and fully turbulent flows. The results include comparisons with experimental measurements, similar numerical approaches, and transport-equations-based models, demonstrating good agreement in surface pressure coefficients, transition onset locations, and skin-friction coefficients for all three configurations. Besides contributing a couple of new insights into boundary-layer transition in these canonical cases, this study provides a powerful tool for transition modeling in both research and design applications in aerodynamics.

Transition Prediction↗

Mixing and Transport in the Solar Nebula

Boss & Vanhala (2000, 2001) prepared reviews of triggered collapse and injection models, using Prudence Foster's finite differences code at very high spatial resolution (440 x 1440 cells) to demonstrate the convergence of the R-T fingers in triggered injection models. A two dimensional hydrodynamical calculation with unprecedentedly high spatial resolution (960 x 2880 zones, or almost 3 million grid points) demonstrated that it suitable shock front can both trigger the collapse of an otherwise stable presolar cloud, and inject shock front particles into the collapsing cloud through the formation of what become Rayleigh-Taylor fingers of compressed fluid layers falling into the gravitational potential well of the growing protostar. These calculations suggest that heterogeneity derived from these R-T fingers will persist down to the scale of their injection onto the surface of the solar nebula. Haghighipour developed a numerical code capable of calculating the orbital evolution of dust grains of varied sizes in a gaseous nebula, subject to Epstein and Stokes drag as well as the self-gravity of the disk. In collaboration with the PI and George W. Wetherill, Haghighipour has been involved in development of a new idea on the possibility of rapid formation of ice giant planets via the disk instability mechanism. Haghighipour studied the stability of a five-body system consisting of the Sun and four protoplanets by numerically integrating their equations of motions. Using Levison and Duncan s SWIFT integrator, Haghighipour showed that, depending on the orbital parameters of the bodies, such a system can be stable for 0.1-10 Myr. Time periods of 1 Myr or more are long enough to be consistent with the time scale proposed for the formation of giant planets by the disk instability mechanism and the photoevaporation of the gaseous envelopes of the outermost protoplanets by a nearby OB star, resulting in the formation of ice giant planets. The PI has used his three dimensional models of marginally gravitationally unstable disks to study the preservation of isotopic heterogeneity in evolving protoplanetary disks. Such heterogeneity might arise from the infall onto the disk s surface of solids processed in the X-wind region of the disk, or derived from stellar nucleosynthesis and injected by R-T fingers. The technique used consists of solving a color equation, identical to the gas continuity equation, which follows the time evolution in three space dimensions of an arbitrarily placed initial color field, i.e., a dye inserted the disk. The models show that significant concentrations of color could persist for time periods of about a thousand years or more, even in the most dynamically active region of such a disk. Such a time period might be long enough for solids to coagulate and grow to significant sizes while retaining the isotopic signature of their birth region in the nebula.

Boss, Alan P.↗

How Very Massive Metal-Free Stars Start Cosmological Reionization

The initial conditions and relevant physics for the formation of the earliest galaxies are well specified in the concordance cosmology. Using ab initio cosmological Eulerian adaptive mesh refinement radiation hydrodynamical calculations, we discuss how very massive stars start the process of cosmological reionization. The models include nonequilibrium primordial gas chemistry and cooling processes and accurate radiation transport in the case B approximation using adaptively ray-traced photon packages, retaining the time derivative in the transport equation. Supernova feedback is modeled by thermal explosions triggered at parsec scales. All calculations resolve the local Jeans length by at least 16 grid cells at all times and as such cover a spatial dynamic range of approx.10(exp 6). These first sources of reionization are highly intermittent and anisotropic and first photoionize the small-scale voids surrounding the halos they form in, rather than the dense filaments they are embedded in. As the merging objects form larger, dwarf-sized galaxies, the escape fraction of UV radiation decreases and the H II regions only break out on some sides of the galaxies, making them even more anisotropic. In three cases, SN blast waves induce star formation in overdense regions that were formed earlier from ionization front instabilities. These stars form tens of parsecs away from the center of their parent DM halo. Approximately five ionizing photons are needed per sustained ionization when star formation in 10(exp 6) stellar Mass halos is dominant in the calculation. As the halos become larger than approx.10(exp 7) Stellar Mass, the ionizing photon escape fraction decreases, which in turn increases the number of photons per ionization to 15-50, in calculations with stellar feedback only. Radiative feedback decreases clumping factors by 25% when compared to simulations without star formation and increases the average temperature of ionized gas to values between 3000 and 10,000 K.

Wise, John H.↗

Naturally occurring and forced azimuthal modes in a turbulent jet

Naturally occurring instability modes in an axisymmetric jet were studied using the modal frequency technique. The evolution of the modal spectrum was obtained for a jet with a Reynolds number based on a diameter of 400,000 for both laminar and turbulent nozzle boundary layers. In the early evolution of the jet the axisymmetric mode was predominant, with the azimuthal modes growing rapidly but dominating only the end of the potential core. The growth of the azimuthal was observed closer to the nozzle exit for the jet in the laminar boundary layer case than for the turbulent. Target modes for efficient excitation of the jet were determined and two cases of excitation were studied. First, a jet was excited simultaneously by two helical modes, m equals plus 1 and m equals minus 1 at a Strouhal number based on jet diameter of 0.15 and the axisymmetric mode, m equals 0 at a jet diameter of 0.6. Second, m equals plus one and m equals minus 1 at jet diameter equals 0.3 and m equals 0 at jet diameter equals 0.6 were excited simultaneously. The downstream evolution of the hydrodynamic modes and the spreading rate of the jet were documented for each case. Higher jet spreading rates, accompanied by distorted jet cross sections were observed for the cases where combinations of axisymmetric and helical forcings were applied.

Raman, Ganesh↗

Collapse of primordial gas clouds and the formation of quasar black holes

The formation of quasar black holes during the hydrodynamic collapse of protogalactic gas clouds is discussed. The dissipational collapse and long-term dynamical evolution of these systems is analyzed using three-dimensional numerical simulations. The calculations focus on the final collapse stages of the inner baryonic component and therefore ignore the presence of dark matter. Two types of initial conditions are considered: uniformly rotating spherical clouds, and iirotational ellipsoidal clouds. In both cases the clouds are initially cold, homogeneous, and not far from rotational support (T/(absolute value of W) approximately equals 0.1). Although the details of the dynamical evolution depend sensitively on the initial conditions, the qualitative features of the final configurations do not. Most of the gas is found to fragment into small dense clumps, that eventually make up a spheroidal component resembling a galactic bulge. About 5% of the initial mass remains in the form of a smooth disk of gas supported by rotation in the gravitational potential potential well of the outer spheroid. If a central seed black hole of mass approximately greater than 10(exp 6) solar mass forms, it can grow by steady accretion from the disk and reach a typical quasar black hole mass approximately 10(exp 8) solar mass in less than 5 x 10(exp 8) yr. In the absence of a sufficiently massive seed, dynamical instabilities in a strongly self-gravitating inner region of the disk will inhibit steady accretion of gas and may prevent the immediate formation of quasar.

Loeb, Abraham↗

A Recent Impact Origin of Saturn’s Rings and Mid-Sized Moons

We simulate the collision of precursor icy moons analogous to Dione and Rhea as a possible origin forSaturn’s remarkably young rings. Such an event could have been triggered a few hundred million years ago by resonant instabilities in a previous satellite system. Using high-resolution SPH simulations, we find that this kind of impact can produce a wide distribution of massive objects and scatter material throughout the system. This includes the direct placement of pure-ice ejecta onto orbits that enterSaturn’s Roche limit, which could form or rejuvenate rings. In addition, fragments and debris of rock and ice totalling more than the mass of Enceladus can be placed onto highly eccentric orbits that would intersect with any precursor moons orbiting in the vicinity of Mimas, Enceladus, or Tethys. This could prompt further disruption and facilitate a collisional cascade to distribute more debris for potential ring formation, the re-formation of the present-day moons, and to evolve into an eventual cratering population of planeto-centric impactors.

Saturnian satellites↗

Contact and pressure balance structures in two-fluid cosmic-ray hydrodynamics

The role of cosmic-ray-modified contact discontinuities and pressure balance structures in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension are investigated by means of analytic and numerical solution examples, as well as by weakly nonlinear asymptotics. The fundamental wave modes of the two-fluid cosmic-ray hydrodynamic equations in the long-wavelength limit consist of the backward and forward propagating cosmic-ray-modified sound waves, with sound speed dependent on both the cosmic-ray and thermal gas pressures; the contact discontinuity; and a pressure balance mode in which the sum ofthe cosmic ray and thermal gas pressure perturbations is zero. The pressure balance mode, like the contact discontinuity is advected with the background flow. The interaction of the pressure balance mode with the contact discontinuity is investigated by means of the method of multiple scales. The thermal gas and cosmic-ray pressure perturbations satisfy a linear diffusion equation, and entropy perturbations arising from nonisentropic initial conditions for the thermal gas are frozen into the fluid. The contact discontinuity and pressure balance eigenmodes both admit nonzero perturbations in the thermal gas, whereas the cosmic-ray-modified sound waves are isentropic. The total entropy perturbation is shared between the contact discontinuity and pressure balance eigenmodes, and examples are given in which there is a transfer of entropy between the two modes. In particular, N-wave type density disturbances are obtained which arise as a result of the entropy transfer between the two modes. A weakly nonlinear geometric optics perturbation expansion is used to study the long timescale evolution of the short-wavelength entropy wave and the thermal gas sound waves in a slowly varying, large-scale background flow. The weakly nonlinear geometric optics expansion is also used to generalize previous studies of squeezing instability for short-wavelength sound waves in the two fluid model, by including a weakly nonlinear wave steepening term that leads to shock formation, as well as the effect of long time and space dependence of the background flow. Implications of cosmic-ray-modified pressure balance structures and contact discontinuities in models of the interaction of traveling interplanetary shocks and compression and rarefraction waves with the solar wind termination shock are briefly discussed.

Webb, G. M.↗

Length Scale and Gravity Effects on Microgravity Boiling Heat Transfer

Boiling is a complex phenomenon where hydrodynamics, heat transfer, mass transfer, and interfacial phenomena are tightly interwoven. An understanding of boiling and critical heat flux in microgravity environments is of importance to space based hardware and processes such as heat exchange, cryogenic fuel storage and transportation, electronic cooling, and material processing due to the large amounts of heat that can be removed with relatively little increase in temperature. Although research in this area has been performed in the past four decades, the mechanisms by which heat is removed from surfaces in microgravity are still unclear. In earth gravity, buoyancy is an important parameter that affects boiling heat transfer through the rate at which bubbles are removed from the surface. A simple model describing the bubble departure size based on a quasistatic force balance between buoyancy and surface tension is given by the Fritz [I] relation: Bo(exp 1/2) = 0.0208 theta where Bo is the ratio between buoyancy and surface tension forces. For small, rapidly growing bubbles, inertia associated with the induced liquid motion can also cause bubble departure. In microgravity, the magnitude of effects related to natural convection and buoyancy are small and physical mechanisms normally masked by natural convection in earth gravity such as Marangoni convection can substantially influence the boiling and vapor bubble dynamics. CHF (critical heat transfer) is also substantially affected by microgravity. In 1 g environments, Bo has been used as a correlating parameter for CHF. Zuber's CHF model for an infinite horizontal surface assumes that vapor columns formed by the merger of bubbles become unstable due to a Helmholtz instability blocking the supply of liquid to the surface. The jets are spaced lambda(sub D) apart, where lambda(sub D) = 2pi square root of 3[(sigma)/(g(rho(sub l) - rho(sub v)](exp 1/2) = 2pi square root of 3 L Bo(exp -1/2) = square root of 3 lambda(sub c) and is the wavelength that amplifies most rapidly. The critical wavelength, lambda(sub c), is the wavelength below which a vapor layer underneath a liquid layer is stable. For heaters with Bo smaller than about 3 (heaters smaller than lambda(sub D)), the above model is not applicable, and surface tension effects dominate. Bubble coalescence is thought to be the mechanism for CHF under these conditions. Small Bo can result by decreasing the size of a heater in earth gravity, or by operating a large heater in a lower gravity environment. In the microgravity of space, even large heaters can have low Bo, and models based on Helmholtz instability should not be applicable. The macrolayer model of Haramura and Katto is dimensionally equivalent to Zuber's model and has the same dependence on gravity, so it should not be applicable as well. The goal of this work is to determine how boiling heat transfer mechanisms in a low-g environment are altered from those at higher gravity levels. Boiling data using a microheater array was obtained under gravity environments ranging from 1.8 g to 0.02 g with heater sizes ranging from 2.7 mm to 1 mm. The boiling behavior for 2.7 mm at 0.02 g looked quite similar to boiling on the 1 mm heater at 1 g-the formation of a large primary bubble surrounded by smaller satellite bubbles was observed under both conditions. The similarity suggests that for heaters smaller than some fraction of I(sub c), coalescence and surface tension dominate boiling heat transfer. It also suggests that microgravity boiling can be studied by studying boiling on very small heaters.

Kim, Jungho↗

High-Order Simulation of Non-Linear Oscillations and Shocks in the Solar Atmosphere

The solar atmosphere presents a rich source of highly non-linear magneto-hydrodynamic phenomena: strong gradients and forcing terms result in both large shocks and oscillations. The additional requirements of energy balance and initialization in hydrostatic equilibrium compound the challenge of this problem. A wealth of observational data allows us to check the results of our simulations. The problem of simulating the solar atmosphere provides, in addition to an interesting system in its own right, a challenging testbed for high-order shock-capturing methods. We discuss the challenge of simulating solar atmospheric phenomena, concentrating on various high-order central methods ranging from second to fourth order. Our method is based on the central-upwind scheme of Kurganov, Noelle and Petrova, which we extend to high order via various interpolants. We investigate various initial data for our simulations, corresponding to observed conditions in different regions of the solar surface: the normal quiet sun and sunspots. When non-oscillatory using second- and third-order methods, we are able to reproduce non-trivial observational results. In particular we find a correlation between initial data and both the shock speeds and particle oscillation spectra that match observations in the corresponding regions. When using fourth-order WENO interpolants, we find that while the individual shock profiles at any given time appear non-oscillatory, spurious oscillations appear in the fields after long time integrations. The issue of initialization in hydrostatic equilibrium raises difficult issues. Careful treatment of the gravitational source term can reduce violations of hydrostatic equilibrium, but difficulties remain, primarily due to discontinuities in the piecewise-polynomial reconstructions. In the case of the Euler equations in gravity, the use of high-order methods reduces the violation of hydrostatic equilibrium to a sufficiently low level for the above described results to be obtained. Including various energy flux terms in the equations, however, leads to severe instability when hydrostatic balance is violated. We discuss various strategies to address this problem, including adaptive grid methods.

Bryson, S.↗