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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.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

A mass conservative, well balanced, tangency preserving and energy decaying method for the shallow water equations on a sphere

Here a fully discrete surface finite element method is proposed for solving the viscous shallow water equations in a bounded Lipschitz domain on the sphere based on a general triangular mesh. The method consists of a modified Crank–Nicolson method in time and a Galerkin surface finite element method in space for the fluid thickness H and the fluid velocity u. A finite element space tangential to the sphere at all finite element nodes is proposed to approximate the fluid velocity u. The proposed method has second-order accuracy in time and first-order accuracy in space, and preserves mass conservation, well balancedness, tangency of velocity to the sphere, and energy decay. Numerical experiments are presented to illustrate the accuracy of the proposed method and the preservation of the physical properties, including mass conservation, well balancedness, and energy decay. A numerical simulation of ocean mesoscale activity on a circular basin with a continental shelf is provided.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2014

Conservation of Mass and Preservation of Positivity with Ensemble-Type Kalman Filter Algorithms

This paper considers the incorporation of constraints to enforce physically based conservation laws in the ensemble Kalman filter. In particular, constraints are used to ensure that the ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. In certain situations filtering algorithms such as the ensemble Kalman filter (EnKF) and ensemble transform Kalman filter (ETKF) yield updated ensembles that conserve mass but are negative, even though the actual states must be nonnegative. In such situations if negative values are set to zero, or a log transform is introduced, the total mass will not be conserved. In this study, mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate non-negativity constraints. Simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. In two examples, an update that includes a non-negativity constraint is able to properly describe the transport of a sharp feature (e.g., a triangle or cone). A number of implementation questions still need to be addressed, particularly the need to develop a computationally efficient quadratic programming update for large ensemble.

EnKF↗
NASA NTRS1992

Restoring and Maintaining Bone in Osteopenic Female Rat Skeleton: Changes in Bone Mass and Structure - Part 1

This experiment contains the crucial data for the lose, restore, and maintain (LRM) concept, a practical approach for reversing existing osteoporosis. The LRM concept uses anabolic agents to restore bone mass and architecture (+ phase) and then switches to an agent with the established ability to maintain bone mass, to keep the new bone (+/- phase). The purpose of this study was to learn whether switching to an agent known chiefly for its ability to maintain existing bone mass preserves new bone induced by PGE2, in osteopenic,estrogen-depleted rats. The current study had three phases, the bone loss (-), restore (+), and maintain (+/-) phases. We ovariectomized (OX) or sham ovariectomized (sham-OX) 5.5 month-old female rats (- phase). The OX rats were treated 5 months postovariectomy with 1-6 mg PGE2, per kg/day for 75 days to restore lost cancellous bone mass (+ phase), and then PGE2, treatment was stopped and treatment began with 1 or 5 micro-g/kg of risedronate, a bisphosphonate, twice a week for 60 days (+/- phase). During the loss (-) phase, the cancellous bone volume of the proximal tibial metaphysis in the OX rat fell to 19% of initial and 30% of age-matched control levels. During the restore (+) phase, the cancellous bone volume in OX rats doubled. When PGE2 treatment was stopped, however, and no special maintenance efforts were made during the maintain (+/-) phase, the PGE2-induced cancellous bone disappeared. In contrast, the PGE2-induced cancellous bone persisted when the PGE2 treatment was followed by either a 1 or 5 micro-g treatment of risedronate per kg given twice a week for 60 days during the maintain (+/-) phase. The tibial shaft demonstrated very little cortical bone loss during the loss (-) phase in OX rats. The tibial shaft cortical bone fell some 8%. During the restore (+) phase, new cortical bone in OX rats increased by 22%. When PGE2 treatment was stopped and nothing was given during the maintain (+/-) phase, however, all but the PGE2-induced subperiosteal bone disappeared. In contrast, when PGE2 treatment was stopped and 1 micro-g risedronate per kg twice a week for 60 days was administered during the maintenance (+/-) phase, the PGE2-induced subperiosteal bone and some of the subendocortical bone and marrow trabeculae persisted. When 5 micro-g risedronate per kg was given twice a week, all the PGE2-induced bone persisted. The study shows that most of the new cancellous and cortical bone induced by PGE2, can be maintained for at least 60 days after discontinuing PGE2 by administering enough of the resorption inhibitor, risedronate. The lower dose of risedronate was not adequate to save most of the PGE2-induced endocortical bone.

Tang, Li Ya↗
NASA NTRS1992

Restoring and maintaining bone in osteopenic female rat skeleton: I. Changes in bone mass and structure

This experiment contains the crucial data for the lose, restore, and maintain (LRM) concept, a practical approach for reversing existing osteoporosis. The LRM concept uses anabolic agents to restore bone mass and architecture (+ phase) and then switches to an agent with the established ability to maintain bone mass, to keep the new bone (+/- phase). The purpose of this study was to learn whether switching to an agent known chiefly for its ability to maintain existing bone mass preserves new bone induced by PGE2 in osteopenic, estrogen-depleted rats. The current study had three phases, the bone loss (-), restore (+), and maintain (+/-) phases. We ovariectomized (OX) or sham ovariectomized (sham-OX) 5.5-month-old female rats (- phase). The OX rats were treated 5 months postovariectomy with 1-6 mg PGE2 per kg/day for 75 days to restore lost cancellous bone mass (+ phase), and then PGE2 treatment was stopped and treatment began with 1 or 5 micrograms/kg of risedronate, a bisphosphonate, twice a week for 60 days (+/- phase). During the loss (-) phase, the cancellous bone volume of the proximal tibial metaphysis in the OX rat fell to 19% of initial and 30% of age-matched control levels. During the restore (+) phase, the cancellous bone volume in OX rats doubled. When PGE2 treatment was stopped, however, and no special maintenance efforts were made during the maintain (+/-) phase, the PGE2-induced cancellous bone disappeared. In contrast, the PGE2-induced cancellous bone persisted when the PGE2 treatment was followed by either a 1 or 5 micrograms treatment of risedronate per kg given twice a week for 60 days during the maintain (+/-) phase. The tibial shaft demonstrated very little cortical bone loss during the loss (-) phase in OX rats. The tibial shaft cortical bone fell some 8%. During the restore (+) phase, new cortical bone in OX rats increased by 22%. When PGE2 treatment was stopped and nothing was given during the maintain (+/-) phase, however, all but the PGE2-induced subperiosteal bone disappeared. In contrast, when PGE2 treatment was stopped and 1 micron risedronate per kg twice a week for 60 days was administered during the maintenance (+/-) phase, the PGE2-induced subperiosteal bone and some of the subendocortical bone and marrow trabeculae persisted.(ABSTRACT TRUNCATED AT 400 WORDS).

NASA Discipline Musculoskeletal↗
DOE OSTI2021

Numerical Investigation of Enhanced Dehumidification Processes By Using Dielectrophoresis Principles in Moist Airflows

Dispersed particle-laden flows are encountered in many building and industrial applications, such as flow in a fluidized bed, hydrocarbon transportation in pipelines, and the fouling of air-cooled heat exchangers (Kuruneru et al., 2016; Ray et al., 2019; Wang et al., 2019). Computational fluid dynamic (CFD) models have been developed in recent years to depict particle-fluid and particle-particle interactions in laminar or turbulent flows with increasing accuracy and stability. One particular particle-laden system of interest for moisture control is electrically-enhanced condensation in air and water droplet flows. Electrically-enhanced condensation consists of the use of highly charged water droplets injected in the moist air. The droplets become electric seeds that attract polar water vapor molecules to their surfaces and promote condensation. The nucleation and growth of the charged droplets deplete the vapor phase near a droplet, which is compensated for by the dielectrophoresis flow and diffusion. Dielectrophoresis flow involves surrounding vapor at a distance of about 10 to 100 nm for droplets charged by an electrospray compared to ~2 nm for a single electron charge in a droplet. As the vapor molecules collapse on the surface of the droplets, their initial electrical charge decreases with time due to the neutralization of the ions. While the physics of this phenomena is well known, engineering models for predicting the condensation rates are not available. This work computationally investigates dehumidification of moist airflow in a converging rectangular duct. The objective is to develop an engineering model that predicts water vapor condensation by employing dielectrophoresis principles. We construct a Computational Fluid Dynamics (CFD) model of the duct with electrically-enhanced condensation. The model is implemented in the open-source software OpenFOAM. We utilize the Multi-Phase Particle-In-Cell (MP-PIC) method coupled with a Population Balance Equation (PBE) approach to simulate the particle-laden system. This methodology is an Eulerian-Lagrangian approach used to simulate the droplets' behavior in the humid air. The MP-PIC approach (Andrews and O'Rourke, 1996) mitigates the computational cost by parceling several fundamental particles with similar properties (such as types, sizes, and temperature) into one computational particle. Thus, the billions of particles can be substituted by millions of computational particles without significant loss of information. The PBE was considered with the Lagrangian frame to combine the particle distribution function used in MP-PIC (Kim et al., 2020). This approach preserves mass and energy conservation between the phases in the Eulerian and Lagrangian structures. The PBE in this procedure was directly linked to the discrete parcels, making the simulation of the particle distribution computationally efficient and robust. The MP-PIC-PBE approach used in the present work was applied to the dehumidification of air. Water droplets were injected in the air stream and forced to grow according to experimentally derived correlation. The experiments were conducted on a converging duct with the same geometry and boundary conditions used to build the CFD model. This approach enabled us to approximate the effect of dielectrophoresis phenomena on the droplet and air interface. This presentation will discuss the details of the new CFD model built for the duct, the implementation of the model in OpenFOAM CFD programming language, and the experimental validation of the newly developed model. The results revealed a moderate yet measurable increase in droplet diameter due to water vapor condensation at the vapor-liquid interface of the electrically charged droplets' surface. The seed water droplet particles grew in size by capturing the water vapor in the surrounding air. The OpenFOAM model predicted reductions of humidity in the air from 5 to 10 percent.

Yel Mahi, Maliha↗
NASA NTRS1981

On various treatments of potential equations at shocks

The potential equation that governs transonic inviscid flowfields is elliptic in subsonic regions and hyperbolic in supersonic regions. The transition from supersonic to subsonic flow may occur either continuously or across a surface of discontinuity known as a shock. Because of the changing type of governing equation and the presence of the surfaces of discontinuity, special treatments of the governing equations are needed at shocks to preserve mass conservation and to ensure satisfaction of the governing equation. In the so-called shock-capturing techniques, shocks are captured by smearing the discontinuity over several mesh cells in the computational domain. Most existing conservative schemes conserve mass flux throughout the flowfield, including shock boundaries; however, they also introduce zero-order errors in the approximation to the governing equation. It is shown that the zero-order errors, which do not diminish as the mesh spacing approaches zero, can cause discrepancies in the prediction of shock strength and location. These parameters are extremely sensitive features of transonic inviscid flowfield calculations. It is also shown that the zero-order errors can be avoided by developing higher-order schemes that properly model the governing equation at shocks.

Chen, L. T.↗
NASA NTRS1997

Life Sciences Issues for a Mission to Mars

Session MP5 includes short reports on: (1) Cardiovascular Concerns for a Mars Mission: Autonomic and Biomechanical Effects; (2) Reducing the Risk of Space Radiation Induced Bioeffects: Vehicle Design and Protectant Molecules; (3) Musculoskeletal Issues for Long Duration Mission: Muscle Mass Preservation, Renal Stone Risk Factors, Countermeasures, and Contingency Treatment Planning; (4) Psychological Issues and Crew Selection for a Mars Mission: Maximizing the Mix for the Long Haul; and (5) Issues in Crew Health, Medical Selection and Medical Officer (CMO) Training for a Mission to Mars.

Source record↗
NASA NTRS1999

GH/IGF-I Transgene Expression on Muscle Homeostasis

We propose to test the hypothesis that the growth hormone/ insulin like growth factor-I axis through autocrine/paracrine mechanisms may provide long term muscle homeostasis under conditions of prolonged weightlessness. As a key alternative to hormone replacement therapy, ectopic production of hGH, growth hormone releasing hormone (GHRH), and IGF-I will be studied for its potential on muscle mass impact in transgenic mice under simulated microgravity. Expression of either hGH or IGF-I would provide a chronic source of a growth-promoting protein whose biosynthesis or secretion is shut down in space. Muscle expression of the IGF-I transgene has demonstrated about a 20% increase in hind limb muscle mass over control nontransgenic litter mates. These recent experiments, also establish the utility of hind-limb suspension in mice as a workable model to study atrophy in weight bearing muscles. Thus, transgenic mice will be used in hind-limb suspension models to determine the role of GH/IGF-I on maintenance of muscle mass and whether concentric exercises might act in synergy with hormone treatment. As a means to engineer and ensure long-term protein production that would be workable in humans, gene therapy technology will be used by to monitor muscle mass preservation during hind-limb suspension, after direct intramuscular injection of a genetically engineered muscle-specific vector expressing GHRH. Effects of this gene-based therapy will be assessed in both fast twitch (medial gastrocnemius) and slow twitch muscle (soleus). End-points include muscle size, ultrastructure, fiber type, and contractile function, in normal animals, hind limb suspension, and reambutation.

Schwartz, Robert J.↗
NASA NTRS2002

Competing Classical and Quantum Effects in Shape Relaxation of a Metallic Island

Pb islands grown on a silicon substrate transform at room temperature from the initially flattop facet geometry into an unusual ring, shape with a volume-preserving mass transport process catalysed by the tip electrical field of a scanning tunnelling microscope. The formation of such ring shape morphology results from the competing classical and quantum effects in the shape relaxation. The latter also leads to a sequential regrowth on alternating, strips of the same facet defined by the underlying substrate steps, showing for the first time the dynamical impact of the quantum size effect on the stability of a nanostructure.

Okamoto, Rowland H.↗
NASA NTRS2005

Unitized Regenerative Fuel Cell System Gas Storage-Radiator Development

High-energy-density regenerative fuel cell systems that are used for energy storage require novel approaches to integrating components in order to preserve mass and volume. A lightweight unitized regenerative fuel cell (URFC) energy storage system concept is being developed at the NASA Glenn Research Center. This URFC system minimizes mass by using the surface area of the hydrogen and oxygen storage tanks as radiating heat surfaces for overall thermal control of the system. The waste heat generated by the URFC stack during charging and discharging is transferred from the cell stack to the surface of each tank by loop heat pipes, which are coiled around each tank and covered with a thin layer of thermally conductive carbon composite. The thin layer of carbon composite acts as a fin structure that spreads the heat away from the heat pipe and across the entire tank surface. Two different-sized commercial-grade composite tanks were constructed with integral heat pipes and tested in a thermal vacuum chamber to examine the feasibility of using the storage tanks as system radiators. The storage tank-radiators were subjected to different steady-state heat loads and varying heat load profiles. The surface emissivity and specific heat capacity of each tank were calculated. In the future, the results will be incorporated into a model that simulates the performance of similar radiators using lightweight, spacerated carbon composite tanks.

Burke, Kenneth A.↗
NASA NTRS2013

Mass Conservation and Positivity Preservation with Ensemble-type Kalman Filter Algorithms

Maintaining conservative physical laws numerically has long been recognized as being important in the development of numerical weather prediction (NWP) models. In the broader context of data assimilation, concerted efforts to maintain conservation laws numerically and to understand the significance of doing so have begun only recently. In order to enforce physically based conservation laws of total mass and positivity in the ensemble Kalman filter, we incorporate constraints to ensure that the filter ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. We show that the analysis steps of ensemble transform Kalman filter (ETKF) algorithm and ensemble Kalman filter algorithm (EnKF) can conserve the mass integral, but do not preserve positivity. Further, if localization is applied or if negative values are simply set to zero, then the total mass is not conserved either. In order to ensure mass conservation, a projection matrix that corrects for localization effects is constructed. In order to maintain both mass conservation and positivity preservation through the analysis step, we construct a data assimilation algorithms based on quadratic programming and ensemble Kalman filtering. Mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate constraints. Some simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. The results show clear improvements in both analyses and forecasts, particularly in the presence of localized features. Behavior of the algorithm is also tested in presence of model error.

EnKR↗
NASA NTRS1972

Thermal scale modeling of radiation-conduction-convection systems.

Investigation of thermal scale modeling applied to radiation-conduction-convection systems with particular emphasis on the spacecraft cabin atmosphere/cabin wall thermal interface. The 'modified material preservation,' 'temperature preservation,' 'scaling compromises,' and 'Nusselt number preservation' scale modeling techniques and their inherent limitations and problem areas are described. The compromised scaling techniques of mass flux preservation and heat transfer coefficient preservation show promise of giving adequate thermal similitude while preserving both gas and temperature in the scale model. The use of these compromised scaling techniques was experimentally demonstrated in tests of full scale and 1/4 scale models. Correlation of test results for free and forced convection under various test conditions shows the effectiveness of these scaling techniques. It is concluded that either mass flux or heat transfer coefficient preservation may result in adequate thermal similitude depending on the system to be modeled. Heat transfer coefficient preservation should give good thermal similitude for manned spacecraft scale modeling applications.

Shannon, R. L.↗
NASA NTRS1997

Cryogenics and the Human Exploration of Mars

Current plans within NASA involve extending the human exploration of space from low earth orbit into the solar system, with the first human exploration of Mars presently planned in 2011. Integral to all hum Mars mission phases is cryogenic fluid management. Cryogenic fluids will be required both as propellant and for In-Situ Resource Utilization (ISRU). Without safe and efficient cryogen storage human Mars missions will not be possible. Effective control and handling of cryogenic fluids is the key to affordable Mars missions, and advancing active thermal control technology is synergistic with all of NASA's exploration initiatives and with existing and future instrument cooling programs, including MTPE and Origins. Present mission scenarios for human exploration require cryogenic propellant storage for up to 1700 days and for up to 60 metric tons. These requirements represent increases of an order of magnitude over previous storage masses and lifetimes. The key cryogenic terminology areas to be addressed in human Mars missions are long-term propellant storage, cryogenic refrigeration, cryogenic liquefaction, and zero gravity fluid management. Long-term storage for the thermal control of cryogenic propellants is best accomplished with a mix of passive and active technologies. Passive technologies such as advanced multilayer insulation (MLI) concepts will be combined with the development of active coolers (cryogenic refrigerators). Candidates for long-life active cooling applications include Reverse Turbo-Brayton, Stirling, and Pulse-Tube coolers. The integration of passive and active technologies will form a hybrid system optimized to minimize the launch mass while preserving the cryogenic propellants. Since cryogenic propellants are the largest mass that Mars missions must launch from earth, even a modest reduction in the percentage of propellant carried results in a significant weight saving. This paper will present a brief overview of cryogenic fluid management technology as it applies to the current human Mars mission scenarios.

Salerno, Louis J.↗
NASA NTRS1992

The density profile of the elliptical planetary nebula NGC 3242

We present the three-dimensional density structure of the elliptical planetary nebula NGC 3242, deconvolved from its H-alpha image. Using the simplistic assumptions that each mass element preserves its original velocity, which is radial and depends only on latitude, we deduce from this density profile the variation of mass-loss rate from the progenitor of NGC 3242 with latitude and time. The resulting somewhat qualitative mass-loss geometry and history are used to constrain models for the formation of the elliptical structure of NGC 3242. We argue that a triple system, with a very close brown dwarf companion and a more massive distant tertiary star, is compatible with the morphology of NGC 3242. In this model the brown dwarf, of about 0.01 solar mass, shared a common envelope with the progenitor star, and spun up the envelope through deposition of angular momentum. The oblate rotating envelope blew an axisymmetrical wind. We suggest that the presence of a third star, with a mass of about 1 solar mass and an orbital period of about 4000 years, could have caused the large scale deviation from axial symmetry seen in the density structure.

Soker, Noam↗
NASA NTRS1998

Cytoskeletal role in the transition from compensated to decompensated hypertrophy during adult canine left ventricular pressure overloading

Increased microtubule density causes cardiocyte contractile dysfunction in right ventricular (RV) pressure-overload hypertrophy, and these linked phenotypic and contractile abnormalities persist and progress during the transition to failure. Although more severe in cells from failing than hypertrophied RVs, the mechanical defects are normalized in each case by microtubule depolymerization. To define the role of increased microtubule density in left ventricular (LV) pressure-overload hypertrophy and failure, in a given LV we examined ventricular mechanics, sarcomere mechanics, and free tubulin and microtubule levels in control dogs and in dogs with aortic stenosis both with LV hypertrophy alone and with initially compensated hypertrophy that had progressed to LV muscle failure. In comparing initial values with those at study 8 weeks later, dogs with hypertrophy alone had a very substantial increase in LV mass but preservation of a normal ejection fraction and mean systolic wall stress. Dogs with hypertrophy and associated failure had a substantial but lesser increase in LV mass and a reduction in ejection fraction, as well as a marked increase in mean systolic wall stress. Cardiocyte contractile function was equivalent, and unaffected by microtubule depolymerization, in cells from control LVs and those with compensated hypertrophy. In contrast, cardiocyte contractile function in cells from failing LVs was quite depressed but was normalized by microtubule depolymerization. Microtubules were increased only in failing LVs. These contractile and cytoskeletal changes, when assayed longitudinally in a given dog by biopsy, appeared in failing ventricles only when wall stress began to increase and function began to decrease. Thus, the microtubule-based cardiocyte contractile dysfunction characteristic of pressure-hypertrophied myocardium, originally described in the RV, obtains equally in the LV but is shown here to have a specific association with increased wall stress.

Non-NASA Center↗
DOE OSTI2024

A phase-field diffraction model for thermo-hydro-mechanical propagating fractures

This paper introduces a novel diffraction based thermo-hydraulic–mechanical (THM) model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables—displacements, phase-field, pressure, and temperature—each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to a new formulation of a coupled variational inequality system. A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation—crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor–corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. Furthermore, these new model and algorithmic developments represent significant advances in the field and have been substantiated through rigorous numerical tests.

Diffraction systems↗
DOE OSTI2021

Conservative High-Order Time Integration for Lagrangian Hydrodynamics

In tis study, we develop novel time integration methods for the compressible Euler equations in the Lagrangian frame that are of arbitrary high order and exactly preserve the mass, momentum, and total energy of the system. The equations are considered in nonconservative form, that is, common for staggered grid hydrodynamics (SGH) methods; namely, the evolved quantities are mass, momentum, and internal energy. A general family of time integration schemes is formulated, and practical pairs for orders three and four are derived. Numerical results on standard hydrodynamics benchmarks confirm the high-order convergence on smooth problems and the exact numerical preservation of all physically conserved quantities.

97 MATHEMATICS AND COMPUTING↗
NASA NTRSDate not supplied

A New Approach for a Wider Class of Entropy Split Methods for Compressible Gas Dynamics and MHD

The high order entropy split methods of Sjögreen & Yee [1, 2] by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function [3, 4, 5]. Their derivation takes advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, DRP (dispersion relation-preserving) [6, 7, 8] or Padé (compact) spatial discretizations [9] with summation-by-parts (SBP) operators [10]. Our entropy split methods have been proven entropy conserving and stable [1, 11, 12]. Our proofs do not rely on a two-point numerical flux, but rather only a linear difference operator is required to derive these methods. To extend the entropy split method for the MHD, we used the Godunov symmetrizable non-conservative MHD form [12, 13, 14]. These high order entropy split methods not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation. In our previous published work, extensive error norm comparison with grid refinement was performed to show the high accuracy performance of these methods. These studies also showed how well the entropy split methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows [1, 2, 12, 13, 14]. The objective of the present work is to use a new approach to obtain a wider class of entropy split methods consisting of a two-point numerical flux portion and a non-conservative portion in such a way that the homogeneity property of the compressible Euler flux is not required. For high order classical spatial central, DRP (dispersion relation-preserving) or Padé (compact) spatial discretizations, this new approach can be proven to be entropy conservative with conservative spatial dsicretizations while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. We also use this generalization to derive an entropy split scheme that is entropy conserving for the equations of MHD without the homogeneity property using the Godunov symmetrizable ideal MHD formulation [15].

High Order Physical Preserving Methods↗