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NASA NTRS1996

Test Outline for Flutter Analysis of Rectangular Panels in Rarefied Flow Conditions

Jet plume impingement forces acting on large flexible space structures may precipitate dynamically unstable behavior during space flights. Typical operating conditions in space involve rarefied gas flow regimes which are intrinsically distinct from continuum gas flow and are normally modeled using the kinetic theory of gas flow. Docking and undocking operations of the Space Shuttle with the Russian Mir space laboratory represent a scenario in which the stability boundaries of solar panels may be of interest. Extensive literature review of research work on the dynamic stability of rectangular panels in rarefied gas flow conditions indicated the lack of published reports dealing with this phenomenon. A recently completed preliminary study for NASA JSC dealing with the mathematical analysis of the stability of two-degree-of-freedom elastically supported rigid panels under the effect of rarefied gas flow was reviewed. A test plan outline is prepared for the purpose of conducting a series of experiments on four rectangular rigid test articles in a vacuum chamber under the effect of continuous and pulsating Nitrogen jet plumes. The purpose of the test plan is to gather enough data related to a number of key parameters to allow the validation of the two-degree-of-freedom mathematical model. The hardware required careful design to select a very lightweight material while satisfying rigidity and frequency requirements within the constraints of the test environment. The data to be obtained from the vacuum chamber tests can be compared with the predicted behavior of the theoretical two-degree-of-freedom model. Using the data obtained in this study, further research can identify the limitations of the mathematical model. In addition modifications to the mathematical model can be made, if warranted, to accurately predict the behavior of rigid panels under rarefied gas flow regimes.

Akl, Fred A.↗
NASA NTRS1996

Active Control of Fan Noise: Feasibility Study: Numerical Computation of Acoustic Mode Reflection Coefficients for an Unflanged Cylindrical Duct - Volume 5

A computational method to predict modal reflection coefficients in cylindrical ducts has been developed based on the work of Homicz, Lordi, and Rehm, which uses the Wiener-Hopf method to account for the boundary conditions at the termination of a thin cylindrical pipe. The purpose of this study is to develop a computational routine to predict the reflection coefficients of higher order acoustic modes impinging on the unflanged termination of a cylindrical duct. This effort was conducted wider Task Order 5 of the NASA Lewis LET Program, Active Noise Control of aircraft Engines: Feasibility Study, and will be used as part of the development of an integrated source noise, acoustic propagation, ANC actuator coupling, and control system algorithm simulation. The reflection coefficient prediction will be incorporated into an existing cylindrical duct modal analysis to account for the reflection of modes from the duct termination. This will provide a more accurate, rapid computation design tool for evaluating the effect of reflected waves on active noise control systems mounted in the duct, as well as providing a tool for the design of acoustic treatment in inlet ducts. As an active noise control system design tool, the method can be used preliminary to more accurate but more numerically intensive acoustic propagation models such as finite element methods. The resulting computer program has been shown to give reasonable results, some examples of which are presented. Reliable data to use for comparison is scarce, so complete checkout is difficult, and further checkout is needed over a wider range of system parameters. In future efforts the method will be adapted as a subroutine to the GEAE segmented cylindrical duct modal analysis program.

Kraft, R. E.↗
NASA NTRS1995

SRM Internal Flow Tests and Computational Fluid Dynamic Analysis: RSRM Full Scale Motor Analyses - Volume 2

The purpose of the RSRM Nozzle Slag Ejection Precursor Test is to investigate the effect that slag ejection from the RSRM nozzle has on the chamber pressure and trust of the SRB's. In past firings of the Reusable Solid Rocket Motor (RSRM) both static test and flight motors have shown small pressure perturbations occurring primarily between 65 and 80 seconds. A joint NASA/Thiokol team investigation concluded that the cause of the pressure perturbations was the periodic ingestion and ejection of molten aluminum oxide slag from the cavity around the submerged nozzle nose which tends to trap and collect individual aluminum oxide droplets from the approach flow. The conclusions of the team were supported by numerous data and observations from special tests including high speed photographic films, real time radiography, plume calorimeters, accelerometers, strain gauges, nozzle TVC system force gauges, and motor pressure and thrust data. A simplistic slag ballistics model was formulated to relate a given pressure perturbation to a required slag quantity. Also, a cold flow model using air and water was developed to provide data on the relationship between the slag flow rate and the chamber pressure increase. Both the motor and the cold flow model exhibited low frequency oscillations in conjunction with periods of slag ejection. Motor and model frequencies were related to scaling parameters. The data indicate that there is a periodicity to the slag entrainment and ejection phenomena which is possibly related to organized oscillations from instabilities in the dividing streamline shear layer which impinges on the underneath surface of the nozzle.

Source record↗
NASA NTRS1995

Effects of Slag Ejection on Solid Rocket Motor Performance

In past firings of the Reusable Solid Rocket Motor (RSRM) both static test and flight motors have shown small pressure perturbations occurring primarily between 65 and 80 seconds. A joint NASA/Thiokol team investigation concluded that the cause of the pressure perturbations was the periodic ingestion and ejection of molten aluminum oxide slag from the cavity around the submerged nozzle nose which tends to trap and collect individual aluminum oxide droplets from the approach flow. The conclusions of the team were supported by numerous data and observations from special tests including high speed photographic films, real time radiography, plume calorimeters, accelerometers, strain gauges, nozzle TVC system force gauges, and motor pressure and thrust data. A simplistic slag ballistics model was formulated to relate a given pressure perturbation to a required slag quantity. Also, a cold flow model using air and water was developed to provide data on the relationship between the slag flow rate and the chamber pressure increase. Both the motor and the cold flow model exhibited low frequency oscillations in conjunction with periods of slag ejection. Motor and model frequencies were related to scaling parameters. The data indicate that there is a periodicity to the slag entrainment and ejection phenomena which is possibly related to organized oscillations from instabilities in the dividing streamline shear layer which impinges on the underneath surface of the nozzle.

Whitesides, R. Harold↗
NASA NTRS1994

A Comparison of Metallographic Cooling Rate Methods Used in Meteorites

The primary objective of this study was to test the postulate that cooling rates acquired from metal grains in chondrites are consistent with those from iron meteorites. Both types of metal occur in some Group IAB meteorites, which are mixtures of massive metal with well-developed Widmanstatten structures and chondritic inclusions with dispersed metal grains. The grains have textures and compositions similar to chondritic metal, including negligible P. The meteorites studied show little or no sign of shock reheating and textural evidence indicates that silicates and metal were mixed before Widmanstatten patterns formed during cooling. Cooling rates were obtained by comparing measured to modeled taenite grain or lamellae dimensions and central Ni contents. Modeling entails solving diffusion equations using experimental diffusion coefficients, phase relations, and bulk or local Ni and P contents, taking into account geometry, undercooling, and impingement. There is one set of parameters for grains and another, quite different set for Widmanstatten lamellae, including a factor of 30 difference in diffusion coefficients. Yet cooling rates obtained from Widmanstatten structures and metal grains in chondritic inclusions of the same meteorite are consistent; uncertainties in the best data are +/- 10 K/Ma, equivalent to a factor of 1 +/- 0.25. This agreement implies that the data and models are correct or contain fortuitously offsetting errors, which is quite unlikely. Cooling rates range from 40 K/Ma to 70 K/Ma in IAB meteorites that contain both grains and Widmanstatten structures. Rates based on grains in Ni-poor and Ni-rich meteorites lacking Widmansatten patterns expand the range from 30 K/Ma to perhaps 200 K/Ma. Cooling rates correlate with Ni content; Ni-poor meteorites have slower rates than Ni-rich ones. Evidently, IAB meteorites were radially distributed over greater than 30km in a body with a radius less than 50km. A comparison of the available Ar ages with cooling times inferred from the cooling rates suggests that the parent body cooled more slowly after the metallographic cooling rates were established.

Herpfer, Marc A.↗
NASA NTRS1996

Annual Research Briefs - 1996, Center for Turbulence Research

Partial contents include: (1) Simulation of jet impingement heat transfer with the k-e-V2 model; (2) A new methodology for turbulence modelers using Direct Numerical Simulation (DNS) database analysis; (3) A particle representation model for the deformation of homogeneous turbulence; (4) Modeling of inhomogeneous compressible turbulence using a two-scale statistical theory; (5) Lewis number effects on partially premixed flames; (6) Conditional moment closure of mixing and reaction in turbulent non-premixed combustion; (7) Dynamic models for LES of turbulent front propagation with a spectral method; (8) Jet noise using large eddy simulation; (9) Towards numerical simulations of trailing-edge aeroacoustics; (10) A model for the onset of vortex breakdown; (11) Compressing an elliptic vortex: transition to turbulence by tumble breakdown; (12) A new method for the adaptive control of vortex-wall interactions; (13) Symmetries in turbulent boundary layer flows; (14) Near-wall models in large eddy simulations of flow behind a backward-facing step; (15) Assessment of the subgrid-scale models at low and high Reynolds numbers; (16) Large-eddy simulations of flow around a NACA 4412 airfoil using unstructured grids; (17) Progress on LES of flow past a circular cylinder; (18) Experimental investigation of flow through an asymmetric plane diffuser; (19) Progress in the large-eddy simulation of an asymmetric plane diffuser; (20) A numerical study of self-similarity in a turbulent plane wake using large-eddy simulation; (21) On the generation of turbulent inflow conditions for boundary layer simulations, and (22) A new class of finite difference schemes.

COMPUTATIONAL FLUID DYNAMICS↗
NASA NTRS1997

Transition and Turbulence Modeling for Blunt-Body Wake Flows

Aerobraking has been proposed as an efficient means of decelerating spacecraft for planetary missions. Most current aerobrake designs feature a blunt forebody shielding the payload from the intense heat generated during atmospheric entry. Although this forebody will absorb the largest portion of the heat pulse, accurate prediction of heating in the near wake is of great importance, since large local heating values can occur at points of shear-layer impingement. In order to address the various issues associated with these blunt-body wake flowfields, the Advisory Group for Aerospace Research and Development (AGARD) formed Working Group 18 in 1992. One of the objectives of this activity was to examine real-gas effects in high-speed flow fields around a 70 deg. blunted cone. To date, many researchers have conducted experiments using this geometry in various facilities, such as the Large Energy National Shock (LENS) tunnel at Cubric/Calspan and the HEG shock tunnel at DLR-Goettingen. Several computational studies have also been conducted in concert with these tests. Many of the experimental results have indicated the possible presence of a transitional shear layer through a large increase in heat transfer downstream of the reattachment point. The presence of transition could in fact lead to much higher peak heating than if the separated flow is entirely laminar or turbulent. In the shock-tunnel tests, however, it is difficult to separate such viscous-flow phenomena from real-gas effects. In order to help make this distinction, Horvath et al. recently conducted a set of experiments in the NASA Langley 20-Inch Mach 6 Tunnel, and compared the results to laminar Navier-Stokes calculations. They found heat-transfer distributions similar to those obtained in the high-enthalpy facilities, with the measured peak heating along the sting support markedly greater than that predicted by the laminar computations. These trends point to the need to find transitional and turbulent computational solutions for these flowfields.

Nance, Robert P.↗
NASA NTRS1997

DRA/NASA/ONERA Collaboration on Icing Research: Prediction of Airfoil Ice Accretion - Part 2

This report presents results from a joint study by DRA, NASA, and ONERA for the purpose of comparing, improving, and validating the aircraft icing computer codes developed by each agency. These codes are of three kinds: (1) water droplet trajectory prediction, (2) ice accretion modeling, and (3) transient electrothermal deicer analysis. In this joint study, the agencies compared their code predictions with each other and with experimental results. These comparison exercises were published in three technical reports, each with joint authorship. DRA published and had first authorship of Part 1 - Droplet Trajectory Calculations, NASA of Part 2 - Ice Accretion Prediction, and ONERA of Part 3 - Electrothermal Deicer Analysis. The results cover work done during the period from August 1986 to late 1991. As a result, all of the information in this report is dated. Where necessary, current information is provided to show the direction of current research. In this present report on ice accretion, each agency predicted ice shapes on two dimensional airfoils under icing conditions for which experimental ice shapes were available. In general, all three codes did a reasonable job of predicting the measured ice shapes. For any given experimental condition, one of the three codes predicted the general ice features (i.e., shape, impingement limits, mass of ice) somewhat better than did the other two. However, no single code consistently did better than the other two over the full range of conditions examined, which included rime, mixed, and glaze ice conditions. In several of the cases, DRA showed that the user's knowledge of icing can significantly improve the accuracy of the code prediction. Rime ice predictions were reasonably accurate and consistent among the codes, because droplets freeze on impact and the freezing model is simple. Glaze ice predictions were less accurate and less consistent among the codes, because the freezing model is more complex and is critically dependent upon unsubstantiated heat transfer and surface roughness models. Thus, heat transfer prediction methods used in the codes became the subject for a separate study in this report to compare predicted heat transfer coefficients with a limited experimental database of heat transfer coefficients for cylinders with simulated glaze and rime ice shapes. The codes did a good job of predicting heat transfer coefficients near the stagnation region of the ice shapes. But in the region of the ice horns, all three codes predicted heat transfer coefficients considerably higher than the measured values. An important conclusion of this study is that further research is needed to understand the finer detail of of the glaze ice accretion process and to develop improved glaze ice accretion models.

Wright, William B.↗
NASA NTRS1997

High-Flux, High-Temperature Thermal Vacuum Qualification Testing of a Solar Receiver Aperture Shield

As part of the International Space Station (ISS) Phase 1 program, NASA Lewis Research Center (LERC) and the Russian Space Agency (RSA) teamed together to design, build and flight test the world's first orbital Solar Dynamic Power System (SDPS) on the Russian space station Mir. The Solar Dynamic Flight Demonstration (SDFD) program was to operate a nominal 2 kWe SDPS on Mir for a period up to 1-year starting in late 1997. Unfortunately, the SDFD mission was demanifested from the ISS phase 1 shuttle program in early 1996. However, substantial flight hardware and prototypical flight hardware was built including a heat receiver and aperture shield. The aperture shield comprises the front face of the cylindrical cavity heat receiver and is located at the focal plane of the solar concentrator. It is constructed of a stainless steel plate with a 1-m outside diameter, a 0.24-m inside diameter and covered with high-temperature, refractory metal Multi-Foil Insulation (MFI). The aperture shield must minimize heat loss from the receiver cavity, provide a stiff, high strength structure to accommodate shuttle launch loads and protect receiver structures from highly concentrated solar fluxes during concentrator off-pointing events. To satisfy Mir operational safety protocols, the aperture shield was required to accommodate direct impingement of the intensely concentrated solar image for a 1-hour period. To verify thermal-structural durability under the anticipated high-flux, high-temperature loading, an aperture shield test article was constructed and underwent a series of two tests in a large thermal vacuum chamber configured with a reflective, point-focus solar concentrator and a solar simulator. The test article was positioned near the focal plane and exposed to concentrated solar flux for a period of 1-hour. In the first test, a near equilibrium temperature of 1862 K was attained in the center of the shield hot spot. In the second test, with increased incident flux, a near equilibrium temperature of 2072 K was achieved. The aperture shield sustained no visible damage as a result of the exposures. This paper describes the aperture shield thermal-vacuum qualification test program including the test article, test facility, procedures, data collection, test success criteria, results and conclusions.

Kerslake, Thomas W.↗
NASA NTRS1996

Shock Tunnel Studies of Scramjet Phenomena

Work focussed on a large number of preliminary studies of supersonic combustion in a simple combustion duct - thrust nozzle combination, investigating effects of Mach number, equivalence ratio, combustor divergence, fuel injecting angle and other parameters with an influence on the combustion process. This phase lasted for some three or four years, during which strongest emphasis was placed on responding to the request for preliminary experimental information on high enthalpy effects, to support the technology maturation activities of the NASP program. As the need for preliminary data became less urgent, it was possible to conduct more systematic studies of high enthalpy combustion phenomena, and to initiate other projects aimed at improving the facilities and instrumentation used for studying scramjet phenomena at high enthalpies. The combustion studies were particularly directed towards hypersonic combustion, and to the effects of injecting fuel along the combustion chamber wall. A substantial effort was directed towards a study of the effect of scale on the supersonic combustion process. The influence of wave phenomena (both compression waves and expansion waves) on the realization of thrust from a supersonic combustion process was also investigated. The effect of chemical kinetics was looked into, particularly as it affected the composition of the test flow provided by a ground facility. The effect of injection of the fuel through wall orifices was compared with injection from a strut spanning the stream, and the effect of heating the fuel prior to injection was investigated. Studies of fuel-air mixing by shock impingement were also done, as well as mass spectrometer surveys of a combustion wake. The use of hypersonic nozzles with an expansion tube was investigated. A new method was developed for measuring the forces acting of a model in less than one millisecond. Also included in this report are listings of published journal papers and conference presentations.

Stalker, R. J.↗
NASA NTRS1996

Supersonic Gas-Liquid Cleaning System

The Supersonic Gas-Liquid Cleaning System Research Project consisted mainly of a feasibility study, including theoretical and engineering analysis, of a proof-of-concept prototype of this particular cleaning system developed by NASA-KSC. The cleaning system utilizes gas-liquid supersonic nozzles to generate high impingement velocities at the surface of the device to be cleaned. The cleaning fluid being accelerated to these high velocities may consist of any solvent or liquid, including water. Compressed air or any inert gas is used to provide the conveying medium for the liquid, as well as substantially reduce the total amount of liquid needed to perform adequate surface cleaning and cleanliness verification. This type of aqueous cleaning system is considered to be an excellent way of conducting cleaning and cleanliness verification operations as replacements for the use of CFC 113 which must be discontinued by 1995. To utilize this particular cleaning system in various cleaning applications for both the Space Program and the commercial market, it is essential that the cleaning system, especially the supersonic nozzle, be characterized for such applications. This characterization consisted of performing theoretical and engineering analysis, identifying desirable modifications/extensions to the basic concept, evaluating effects of variations in operating parameters, and optimizing hardware design for specific applications.

Kinney, Frank↗
NASA NTRS1997

F-16XL Wing Pressure Distributions and Shock Fence Results from Mach 1.4 to Mach 2.0

Chordwise pressure distributions were obtained in-flight on the upper and lower surfaces of the F-16XL ship 2 aircraft wing between Mach 1.4 and Mach 2.0. This experiment was conducted to determine the location of shock waves which could compromise or invalidate a follow-on test of a large chord laminar flow control suction panel. On the upper surface, the canopy closure shock crossed an area which would be covered by a proposed laminar flow suction panel. At the laminar flow experiment design Mach number of 1.9, 91 percent of the suction panel area would be forward of the shock. At Mach 1.4, that value reduces to 65 percent. On the lower surface, a shock from the inlet diverter would impinge on the proposed suction panel leading edge. A chordwise plate mounted vertically to deflect shock waves, called a shock fence, was installed between the inlet diverter and the leading edge. This plate was effective in reducing the pressure gradients caused by the inlet shock system.

Landers, Stephen F.↗
NASA NTRS1997

Creating a Simple Single Computational Approach to Modeling Rarefied and Continuum Flow About Aerospace Vehicles

We proposed to create a single computational code incorporating methods that can model both rarefied and continuum flow to enable the efficient simulation of flow about space craft and high altitude hypersonic aerospace vehicles. The code was to use a single grid structure that permits a smooth transition between the continuum and rarefied portions of the flow. Developing an appropriate computational boundary between the two regions represented a major challenge. The primary approach chosen involves coupling a four-speed Lattice Boltzmann model for the continuum flow with the DSMC method in the rarefied regime. We also explored the possibility of using a standard finite difference Navier Stokes solver for the continuum flow. With the resulting code we will ultimately investigate three-dimensional plume impingement effects, a subject of critical importance to NASA and related to the work of Drs. Forrest Lumpkin, Steve Fitzgerald and Jay Le Beau at Johnson Space Center. Below is a brief background on the project and a summary of the results as of the end of the grant.

Goldstein, David B.↗
NASA NTRS1997

An Experimental Study of Characteristic Combustion-Driven Flow for CFD Validation

A series of uni-element rocket injector studies were completed to provide benchmark quality data needed to validate computational fluid dynamic models. A shear coaxial injector geometry was selected as the primary injector for study using gaseous hydrogen/oxygen and gaseous hydrogen/liquid oxygen propellants. Emphasis was placed on the use of nonintrusive diagnostic techniques to characterize the flowfields inside an optically-accessible rocket chamber. Measurements of the velocity and species fields were obtained using laser velocimetry and Raman spectroscopy, respectively. Qualitative flame shape information was also obtained using laser-induced fluorescence excited from OH radicals and laser light scattering studies of aluminum oxide particle seeded combusting flows. The gaseous hydrogen/liquid oxygen propellant studies for the shear coaxial injector focused on breakup mechanisms associated with the liquid oxygen jet under subcritical pressure conditions. Laser sheet illumination techniques were used to visualize the core region of the jet and a Phase Doppler Particle Analyzer was utilized for drop velocity, size and size distribution characterization. The results of these studies indicated that the shear coaxial geometry configuration was a relatively poor injector in terms of mixing. The oxygen core was observed to extend well downstream of the injector and a significant fraction of the mixing occurred in the near nozzle region where measurements were not possible to obtain. Detailed velocity and species measurements were obtained to allow CFD model validation and this set of benchmark data represents the most comprehensive data set available to date. As an extension of the investigation, a series of gas/gas injector studies were conducted in support of the X-33 Reusable Launch Vehicle program. A Gas/Gas Injector Technology team was formed consisting of the Marshall Space Flight Center, the NASA Lewis Research Center, Rocketdyne and Penn State. Injector geometries studied under this task included shear and swirl coaxial configurations as well as an impinging jet injector.

Santoro, Robert J.↗
NASA NTRS1997

Inflight performance of the Ulysses reaction control system

The Ulysses spacecraft has been exploring the heliosphere since October 1990 in a six-year polar orbit. Despite varying operational demands, the pressure-fed monopropellant hydrazine reaction control system (RCS) has experienced few problems. The observed anomalies, having minimal operational impact, include plume impingement effects, electrical power overload effects and hydrazine gas generation effects. These anomalies are presented and discussed, with emphasis on the first observation of gas in the hydrazine propellant. The relatively low gas generation rate is attributed to: the use of high purity hydrazine; the configuration of the spin-stabilized spacecraft; the extensive use of titanium alloys; and the efficiency of the thermal control of the propellant tank which maintains a temperature of 21 C.

McGarry, Andrew↗
NASA NTRS1998

Development of Tripropellant CFD Design Code

A tripropellant, such as GO2/H2/RP-1, CFD design code has been developed to predict the local mixing of multiple propellant streams as they are injected into a rocket motor. The code utilizes real fluid properties to account for the mixing and finite-rate combustion processes which occur near an injector faceplate, thus the analysis serves as a multi-phase homogeneous spray combustion model. Proper accounting of the combustion allows accurate gas-side temperature predictions which are essential for accurate wall heating analyses. The complex secondary flows which are predicted to occur near a faceplate cannot be quantitatively predicted by less accurate methodology. Test cases have been simulated to describe an axisymmetric tripropellant coaxial injector and a 3-dimensional RP-1/LO2 impinger injector system. The analysis has been shown to realistically describe such injector combustion flowfields. The code is also valuable to design meaningful future experiments by determining the critical location and type of measurements needed.

Farmer, Richard C.↗
NASA NTRS1996

Computation of Large Turbulence Structures and Noise of Supersonic Jets

Our research effort concentrated on obtaining an understanding of the generation mechanisms and the prediction of the three components of supersonic jet noise. In addition, we also developed a computational method for calculating the mean flow of turbulent high-speed jets. Below is a short description of the highlights of our contributions in each of these areas: (a) Broadband shock associated noise, (b) Turbulent mixing noise, (c) Screech tones and impingement tones, (d) Computation of the mean flow of turbulent jets.

Tam, Christopher↗
NASA NTRS1998

Performance of Low Dissipative High Order Shock-Capturing Schemes for Shock-Turbulence Interactions

Accurate and efficient direct numerical simulation of turbulence in the presence of shock waves represents a significant challenge for numerical methods. The objective of this paper is to evaluate the performance of high order compact and non-compact central spatial differencing employing total variation diminishing (TVD) shock-capturing dissipations as characteristic based filters for two model problems combining shock wave and shear layer phenomena. A vortex pairing model evaluates the ability of the schemes to cope with shear layer instability and eddy shock waves, while a shock wave impingement on a spatially-evolving mixing layer model studies the accuracy of computation of vortices passing through a sequence of shock and expansion waves. A drastic increase in accuracy is observed if a suitable artificial compression formulation is applied to the TVD dissipations. With this modification to the filter step the fourth-order non-compact scheme shows improved results in comparison to second-order methods, while retaining the good shock resolution of the basic TVD scheme. For this characteristic based filter approach, however, the benefits of compact schemes or schemes with higher than fourth order are not sufficient to justify the higher complexity near the boundary and/or the additional computational cost.

Sandham, N. D.↗