Inhomogeneous Mixture Research Articles

Morphological patterns and interface instability during withdrawal of liquid-particle mixtures

HypothesisThe stability of fluid–fluid interface is key to control the displacement efficiency in multiphase flow. The existence of particles can alter the interfacial dynamics and induce various morphological patterns. Moreover, the particle aggregations are expected to have a significant impact on the interface stability and patterns. ExperimentsMonodisperse polyethylene particles of different sizes are uniformly mixed in silicone oil to form the granular mixtures, which are injected into a transparent radial Hele-Shaw cell through different strategies to obtain the homogeneous and inhomogeneous (with particle aggregations) initial states. Subsequently, a systematic study of morphology and interface stability during the withdrawal of granular mixtures is performed. FindingsFor homogeneous mixtures, we observe earlier onset of fingering, more fingers and lower gas saturation at breakthrough than for pure fluid with equivalent viscosity. This effect can be attributed to the particle-induced perturbations. For inhomogeneous mixtures, particle clusters and bands significantly enhance the interface instability. Furthermore, we find that particle deposition due to liquid film entrainment occurs above a critical local flow velocity, and we elucidate the responsible mechanism through force balance analysis and the thin film theory. This work could be of practical significance in geoenergy and industrial applications.

Detailed investigation of the mixing field and stability of natural gas and propane in highly turbulent planar flames

In most practical combustion devices, the actual combustion process occurs within different mixture inhomogeneity levels. Investigating the mixture fraction field upstream of the reaction zones of these flames is an essential step toward understanding their structure, stability, and emission formation. In this study, the mixture fraction fields were measured for turbulent non-reacting inhomogeneous mixtures immediately downstream from the slot burner exit, using Rayleigh scattering imaging. The slot burner had two concentric slots. The inner air slot can be recessed at distances upstream from the exit of the outer fuel slot, allowing various degrees of mixture inhomogeneity. Mixture fraction field statistics and the two-dimensional gradient were utilized to characterize the impact of the air-to-fuel velocity ratio, global equivalence ratio, fuel composition, Reynolds number, and the premixing length on the mixture mixing field, and thus flame stability. These impacts were evaluated by tracking the normalized mean mixture fraction and mixture fraction fluctuation transition across the regime diagram for partially premixed flames. The results showed that the air-to-fuel velocity ratio was the critical parameter affecting the mixture fraction field for the short premixing length. Stability results showed that the level of mixture inhomogeneity mainly influenced the flame stability. High flame stability is achieved if a large portion of the inhomogeneous mixture fraction is within the fuel flammability limits.

High-energy ball milling and spark plasma sintering of molybdenum - lanthanum oxide (Mo-La2O3) and molybdenum – lanthanum zirconate (Mo-La2Zr2O7) composite powders

The current study is focused on the preparation of Mo-10 vol%La2O3 and Mo-10 vol% La2Zr2O7 composite powders via low- and high-energy ball milling approaches as potential candidates for near-future high-temperature structural applications. The mechanical milling parameters play a critical role on the final powder's microstructure. When using the high-energy milling mode (using 800 rpm, ball-to-powder ratio (BPR) 100: 6), the homogeneous powder agglomerates are formed with refined laminated microstructure and more uniform ceramic phase distribution in both Mo-La2O3 and Mo-La2Zr2O7 systems compared to the powders produced by means of the low-energy milling mode (using 350 rpm, BPR 100: 6), where inhomogeneous powder mixture with less embedding of ceramic phases into Mo agglomerates was obtained. This study also focuses on the evaluation of high-temperature phase and microstructural stability of the produced composite powders treated at the temperature of 1300 °C under the different gaseous environments, including ambient, inert and reducing atmospheres. The Mo-10 vol% La2Zr2O7 composite powder exhibited better thermal stability during the high-temperature exposure in all tested atmospheres in comparison with the Mo-La2O3 composite powder, since it revealed less intensive formation of the intermediate phases, such as lanthanum oxymolybdates. Therefore, the Mo-10 vol% La2Zr2O7 composite powder was used further for consolidation by means of spark plasma sintering at 1600 °C. The successful production of Mo-La2Zr2O7 composite with homogeneous distribution of ceramic phase, the grain size about of 5 μm, and hardness of 3.4 GPa was not reported so far.

CFD-PBE coupled model for size-driven segregation in polydisperse granular flows

Mixtures of granular materials made of different-sized particles may segregate when subjected to vibration or shear rate and in the presence of a gravitational field. This leads to highly inhomogeneous mixtures, which are undesirable in many industrial processes. This work focuses on size-driven segregation in polydisperse mixtures. We described the evolution of the particle size distribution through a Population Balance Equation (PBE), which we solved numerically with the Direct Quadrature Method of Moments. To allow segregation and micromixing to occur, we closed the size-conditioned velocity of the particles with a segregation-remixing model. The PBE was then included in an Eulerian-Eulerian model and solved in a commercial Computational Fluid Dynamics (CFD) code. We used the model to study granular flows down inclined planes. The numerical results were then compared with those obtained from Discrete Element Method simulations. The CFD-PBE model predicts well segregation and micromixing in packed beds of polydisperse powders.

About motion of inhomogeneous grain mixture on vertical vibrosieve

About motion of inhomogeneous grain mixture on vertical vibrosieve

Hydrogen-air explosion with concentration gradients in a cubic enclosure

An inhomogeneous hydrogen-air mixture is induced once hydrogen leakage occurs, and an explosion may be triggered due to an accidental ignition with sufficient strength. The effects of different hydrogen concentration gradients on flame behavior and overpressure were experimentally investigated in a closed cubic vessel with a volume of 0.125 m3. The transverse concentration gradients of hydrogen injected from the top face of the vessel were recorded using five oxygen sensors. A high-speed photography was employed to capture the flame shape evolution and the flame tip velocity was derived. Three piezoelectric pressure transducers were mounted on the top and side walls to measure the pressure-time profiles. The results show that the maximum overpressure in homogeneous mixtures is larger than that in inhomogeneous mixtures for fuel-lean hydrogen-air mixtures. However, the maximum overpressure in inhomogeneous mixtures is equal to or larger than that in homogeneous mixtures for fuel-rich hydrogen-air mixtures. Due to a larger difference between the upward and downward flame speeds, a mushroom-like shape flame was recorded in inhomogeneous hydrogen-air mixtures in fuel-lean to slightly fuel-rich hydrogen-air mixtures. The flame speed in two directions has the almost same value in inhomogeneous fuel-richer mixtures. The average flame speed in inhomogeneous mixtures is significantly larger than that in homogeneous mixtures for fuel-rich hydrogen air mixtures.

AN APPROXIMATE METHOD FOR CALCULATING GRAIN FLOW IN A VERTICAL CYLINDRICAL VIBROSIEVE

A modified hydrodynamic model of a stable grain flow of an inhomogeneous mixture over the surface of a vertical cylindrical vibrating sieve is proposed under the assumption that the porosity of the mixture in the moving annular layer depends on the velocity of movement. A linear dependence of the porosity of the mixture on the velocity of movement are accepted, where higher speed corresponds to higher porosity.. The calculation of the velocity is reduced to solving an inhomogeneous differential equation of the Bessel type. Further, by "freezing" the variable coefficient of this equation, the problem has been simplified. This simplification is permissible due to the fact that the thickness of the moving layer of the mixture is much less than the radius of the vibrating sieve. As a result, the dependence of the velocity on the radial coordinate is expressed through the elementary functions. A compact formula for determining the maximum grain flow velocity is obtained. By integrating, in elementary functions, the formula for the average velocity in the layer is obtained. An approximate formula for the performance of the vibrating sieve by the mass of the exit fraction is derived. For this, it is proposed to calculate the corresponding integral approximately by the Simpson formula, so as not to calculate the values of special functions of large argument using the asymptotic formulas. It is shown that the named productivity significantly depends on the porosity of the grain mixture. In order to obtain information about the actual errors of the approximate formulas, we additionally carried out the numerical integration of the original non-simplified Bessel-type equation on a computer. A comparative analysis of the calculation results confirmed the small errors of the simplifications introduced into the equation of motion, as well as the adequacy of the theoretical results obtained. By passing to a simplified differential equation, approximate formulas were derived and tested for calculating the main characteristics of the grain flow along a vertical cylindrical vibrating sieve, taking into account the change in porosity in the grain mixture layer from the velocity of movement. The work summarizes the known theoretical results obtained using hydrodynamic models of the motion of grain mixtures fluidized by vibrations. The generalization carried out slightly complicated the theory, because the final calculation formulas are quite compact and convenient in practical implementation.

Structural response analysis of explosions in hydrogen-air mixtures in tunnel-like geometries

This paper presents a structural response analysis based on the results of experiments involving gas explosions in inhomogeneous hydrogen-air mixtures in a channel. The channel was 3 m long, with a cross-section of 100 mm by 100 mm, and open at one end. Hydrogen was released into the channel using 0.25 mm and 0.5 mm ID nozzles.Strong and relatively prolonged pressure oscillations were recorded during the gas explosions. Based on the pressure recordings from the explosions, the structural damage potential of the pressure was analyzed using shock response spectra. These revealed frequency-dependent dynamic amplifications of relatively high magnitudes. This information must be taken into consideration when designing components and systems that are intended to remain structurally intact and operational in the aftermath of gas explosion events in confined spaces with high aspect ratios, such as tunnels, channels and enclosed passageways.

Quantification of Segregation in Portland Cement Concrete Based on Spatial Distribution of Aggregate Size Fractions

Segregation is one of the quality standards that must be monitored during the fabrication and placement of Portland cement concrete. Segregation refers to separation of coarse aggregate from the cement paste, resulting in inhomogeneous mixture. This study introduces a digital imaging based technique to quantify the segregation of Portland cement concrete from 2D digital images of cut sections. In the previous studies, segregation was evaluated based on the existence of coarse aggregate fraction at different geometrical regions of a sample cross section without considering its distribution characteristics. However, it is shown that almost all particle fractions can form clusters and increase the degree of segregation, thus deteriorating the structural performance of concrete. In the proposed methodology, a segregation index is developed by based on the spatial distribution of different size fractions of coarse aggregate within a sample cross section. It is shown that degradation in mixture’s homogeneity is controlled by the combined effect of particle distribution and their relative proportions in the mixture. Hence, a segregation index characterizing the mixture inhomogeneity is developed by considering not only spatial distribution of aggregate particles, but also their size fractions in the mixture. The proposed methodology can be successfully used as a quality control tool for monitoring the segregation level in hardened concrete samples.

Modelling heterogeneous dust particles: an application to cometary polarization

ABSTRACT In this work, we introduce a comet dust model that incorporates multiple dust morphologies along with inhomogeneous mixture of silicate minerals and carbonaceous materials under power-law size distribution, to replicate the standard polarization-phase curve observed in several comets in the narrow-band continuum. Following the results from Rosetta/midas and COSIMA, we create high porosity hierarchical aggregates (HA) and low porosity (<10 per cent) Solids in the form of agglomerated debris. We also introduce a moderate porosity structure with solids in the core, surrounded by fluffy aggregates called fluffy solids (FS). We study the mixing combinations, (HA and Solids), (HA and FS), and (HA, FS, and Solids) for a range of power-law index n= 2.0 to 3.0 for different sets of mixing percentage of silicate minerals and carbonaceous materials. Polarimetry of the short period comets 1P/Halley and 67P/Churyumov-Gerasimenko match best with the polarization resulting from the combination of HA and Solids while the combinations (HA and FS) and (HA, FS, and Solids) provide the best-fitting results for the long period comets C/1995 O1 (Hale-Bopp) and C/1996 B2 (Hyakutake). The best-fitting model results also recreate the observed wavelength dependence of polarization. Our dust model agree with the idea that the long period comets may have high percentage of loose particles (HA and FS) compared to those in the case of short period comets as the short period comets experience more frequent and/or higher magnitude of weathering.

Deformation patterns and potential active movements of the Fansipan mountain range, northern Vietnam

Located in northern Vietnam, the Fansipan mountain range is the highest topography in the Indochina area. Until recently, there has been limited research regarding the tectonic deformation of the Fansipan mountain range and whether there are notable surface ruptures or active faults in the area. This study provides the first insight into the deformation patterns and potential active movements of the Fansipan mountain range, and may improve our understanding of the overall tectonic activities in northern Vietnam. Our observations from ASTER 30-m DEM show a symmetrical mountain form in the north and an asymmetrical form in the south. The analysis of river profiles indicates similar patterns on both sides in the north and different patterns in the south with short and steep rivers on the SW side and long and gentle rivers on the NE side. The results of the normalized steepness index (ksn) also show higher values on the SW side of the south part than the values from the other sides. These results suggest a spatial variation in rock uplift patterns of the Fansipan mountain range. Notably, the mountain front of the SW side of the southern Fansipan mountain range is associated with the Phong Tho-Nam Pia normal fault, suggesting that the normal faulting activity likely played an important role in tectonic uplift of this high mountain range. Our investigation of the stress field using fault kinematics indicates that the Fansipan mountain range and its surrounding areas are undergoing an inhomogeneous mixture of strike-slip and normal faulting. It is proposed that the strike-slip and normal motions alternated because of a permutation of σ1/σ2 under the same extensional stress regime of approximately NE–SW σ3. This kinematic pattern may effectively control the recent tectonic movements in northern Vietnam.

Structure and stability characteristics of turbulent planar flames with inhomogeneous jet in a concentric flow slot burner

Turbulent flames with compositionally inhomogeneous mixtures are common in many combustion systems. Turbulent jet flames with a circular nozzle burner were used earlier to study the impact of inhomogeneous mixtures, and these studies showed that the nozzle radius affects the flame stability. Accordingly, planar turbulent flames with inhomogeneous turbulent jet are created in a concentric flow slot burner (CFSB) to avoid this effect in the present study. The stability characteristics, the mixing field structure, and the flame front structure were measured, and the correlations between stability and the mixing field structure were investigated. The mixture fraction field was measured in non-reacting jets at the nozzle exit using highly resolved Rayleigh scattering technique, and the flame front was measured in some selected turbulent flames using high-speed Planar Laser-Induced Fluorescence (PLIF) of OH technique. The data show strong correlations between flame stability and the range of mixture fraction fluctuations. The flames are highly stabilized within a mixing field environment with the range of fluctuation in mixture fraction close to the range of the flammability limits. The mixing field structure is also illustrated and discussed using a mixing regime diagram and showed that the scatter of the data of the different cases is consistent with the classified mixing regimes. Lean flames are stabilized in the current slot burner. The flame front structure topology varies consistently from thin, small curvature at the low level of turbulence and higher equivalence ratio to more wrinkled, larger curvature, but a thicker structure at a higher level of turbulence and lower equivalence ratio.

Experimental Study on Pulse Detonation Engine with Two-Phase Inhomogeneous Mixture

In order to investigate the effects of fuel distribution on the operation of two-phase pulse detonation engine (PDE), a series of cold flow and multicycle PDE experiments was carried out with 9 mixing schemes. Homogeneity degree with fuel distribution considered in terms of space and time was proposed to quantitatively evaluate the mixing of liquid fuel and air by particle image velocimetry (PIV) in cold flow experiments. Operation stability of multicycle PDE was presented by statistical analysis of peak pressure at the outlet of a detonation tube. The relationship between operation stability and homogeneity degree was quantitatively elaborated. These experimental results indicated that not only using mixing reinforcement devices (such as pore plate and reed valve) was fuel distribution improved but also the effect of inlet ways on the homogeneity degree was weakened. The homogeneity degree of fuel distribution ζ=0.72 was a critical value for stable working of multicycle PDE. When homogeneity degree was lower than 0.72, stable state was not maintained and detonation wave in some cycles was not established due to poor fuel distribution. Therefore, it is necessary to hold homogeneity degree larger than 0.72 to achieve stable operation of PDE. These results contribute to enhancing the operation stability and offering guidelines for the design of PDE’s mixing scheme.

Propagation of gaseous detonation across inert layers

In rotating detonation engines and explosion accidents, detonation may propagate in an inhomogeneous mixture with inert layers. This study focuses on detonation propagation in a stoichiometric H2/O2/N2 mixture with multiple inert layers normal to the detonation propagation direction. One- and two-dimensional simulations considering detailed chemistry are conducted. The emphasis is placed on assessing the effects of inert layer on detonation reinitiation/failure, detonation propagation speed, detonation cell structure and cell size. Specifically, the inert layer thickness and the spacing between two consecutive inert layers are varied. Either detonation reinitiation or failure across the inert layers is observed. It is found that successful detonation reinitiation occurs only at relatively small values of the inert layer thickness and spacing. For each given value of the inert layer spacing, there is a critical inert layer thickness above which detonation fails after crossing the inert layers. This critical inert layer thickness is found to decrease as the inert layer spacing increases. The detailed process of detonation reinitiation across the inert layers is analyzed. The interaction between the transverse shock waves is shown to induce local autoignition/explosion and eventually over-driven detonation development in the reactive layer. The averaged detonation propagation speed in the inhomogeneous mixture is compared to the CJ speed and very good agreement is achieved. This indicates that the inert layer does not affect the detonation propagation speed once successful detonation reinitiation happens. Unlike the detonation speed, the detonation cell structure and cell size are greatly affected by the inert layer results. For the first time, large cellular structure with size linearly proportional to the inert layer spacing is observed for detonation propagation across inert layers. Besides, a double cellular structure is observed for relatively large spacing between inert layers. The formation of double cellular structure is interpreted.

A thermodynamic description of the glass state and the glass transition

Many properties of solids, such as the glass state, hysteresis, and memory effects, are commonly treated as nonequilibrium phenomena, which involve numerous conceptual difficulties. However, few studies have addressed the problem of understanding equilibrium itself. Equilibrium is often assessed based on the assumption that its thermodynamic state should be solely determined using temperature and pressure. However, this assumption must be fundamentally reappraised from the beginning because no rigorous proof for this assumption exists. Previous work showed that for solids, the time-averaged positions of all constituent atoms of the solid are thermodynamic coordinates (TCs). In this study, this conclusion is further elaborated starting from the principles of solid-state physics. This theory is applied to glass materials, for which many challenges remain. Results show that first, the glass state, which is the state after freezing, is an equilibrium state under given constraints. Accordingly, the properties of a glass can be solely described using the present time-averaged atom positions, irrespective of their history, which is consistent with the definition of a state in the thermodynamic context. Any quantity that is determined using TCs can be used as an order parameter. Second, although the traditional view that the transition state during the glass transition is a nonequilibrium state is correct, whether it reaches the supercooled-liquid state given sufficient time is highly questionable. The final state must be an inhomogeneous mixture of solid and liquid phases. In the liquid phase, the atom positions are missing from the set of TCs. The degeneration of the state space occurs, which corresponds to an increase in configuration entropy. Conversely, when the phonon and structural parts of the atom positions are well separated, the time-averaged atom positions in the solid part remain as TCs. In this manner, the transition state can be described using a reduced set of TCs as a function of time. An important outcome of this theory is that the hysteresis of the glass transition can be described using the present values of TCs through the structural-dependent energy barrier. Complicated response functions describing the history, which are widely used in glass literature, are unnecessary. The expense of this simplification is that, in a rigorous sense, all the atom positions are needed for TCs. However, in most cases, only a few TCs are sufficient to describe the transition state, if these TCs are suitably chosen.

Effects of radiation models on steady and flickering laminar non-premixed flames

Four laminar non-premixed methane/air flames with increasing Reynolds numbers are investigated using a newly developed conjugate combustion-radiation laminar flame solver. The baseline model includes a forward Monte Carlo ray tracing solver coupled with a line-by-line spectral model, a 16-species skeletal chemical mechanism and no soot model. A backward Monte Carlo solver is developed for the nongray inhomogeneous combustion mixtures and verified using the forward Monte Carlo solver. A 25-species skeletal chemical mechanism is adopted for parametric studies. Effects of soot radiation on unsteady flames are investigated using a two-equation soot model. Good agreement with experiments in radiative flux and radiant fraction is observed for all flames using the baseline model. The radiative heat flux predicted by the backward Monte Carlo solver agrees well with that obtained from the forward Monte Carlo solver, with well-controlled standard deviations. Contours of radiation-related scalars indicate that the laminar flames are within the optically-thin limit. Results obtained from conjugate combustion-radiation simulations with various radiation solvers and spectral models show minor differences for the two smaller flames. For the two unsteady flames, radiative re-absorption has a more significant impact on the overall flame structure, especially when soot is considered. Parametric studies using the longest flickering flame suggest that the choice of chemical mechanism, the consideration of radiation, and the inclusion of soot radiation, affect the prediction of the instantaneous flame height. The prediction of flame puffing frequency is insensitive to the variations in chemistry, soot and radiation models. The radiative heat flux prediction is sensitive to the thermal boundary condition of the fuel nozzle, the inclusion of soot and its radiation with re-absorption. It is recommended that soot should be considered for future studies when assessing radiation models using this set of target flames.

3D scanner technology implementation to numerical modeling of GPR

This study presents an integration of 3D scanned objects into numerical modeling for ground-penetrating radar (GPR). The characteristics of antenna, features of surrounding medium and the realistic design of a buried object are decisive for the success of the numerical modeling. Realistic antenna design has become one of the remarkable developments of the numerical modeling by Finite-Difference Time Domain (FDTD) method. The antenna models with the specific features of commercial GPR antennas, allow researchers to obtain much closer synthetic response to measurements. In addition, lossy and inhomogeneous soil mixtures with different water content can be included in numerical models. Despite the realistic description of GPR antennas and soil types, buried objects are still roughly defined using basic geometric shapes. This article presents an alternative approach using outputs of the 3D scanner as an input into gprMax software to overcome more complex and realistic shapes of the buried objects. For this purpose, a bone fragment belonging to a bovine was firstly simulated with a basic geometric object (cylinder) and then modeled with the realistic shape of the bone, which was obtained from 3D scanner. In this way, the numerical modeling can be performed with the real shape of the object in the lossy, inhomogeneous surrounding medium by using a realistic modeled commercial antenna.

Effects of fluctuations on correlation functions in inhomogeneous mixtures

Approximate expressions for correlation functions in binary inhomogeneous mixtures are derived in a framework of the mesoscopic theory [Ciach A., Mol. Phys., 2011, {\textbf{109}}, 1101]. Fluctuation contribution is taken into account in a Brazovskii-type approximation. Explicit results are obtained for two model systems. In the two models, the diameters of the hard cores of particles are equal, and the interactions favour a periodic arrangement of alternating species A and B. However, the optimal distance between the species A and B is much different in the two models. Theoretical results for different temperature and volume fractions of the two components are compared with the results of Monte Carlo simulations, and the structure is illustrated by simulation snapshots. Despite different interaction potentials and different length scale of the local ordering, properties of the correlation functions in the two models are very similar.

Evaporation of free falling droplets of binary alkane-ethanol blends

The admixture of ethanol to gasoline has become a key factor to reduce the reliance on fossil fuels. In this context, the distinct occurrence of component-by-component evaporation compromises the efficient and the low-emission operation of internal combustion engines due to inhomogeneous mixture formation. In order to meet these challenges, a more precise understanding of the mechanisms causing the component-by-component evaporation is inevitable. Therefore, in the present study the evaporation behaviour of binary alkane-ethanol blends has been experimentally investigated. A high-resolution Raman-setup is combined with a fast trigger system to quantify the substance composition of the gaseous phase directly behind free falling droplets. Considerable preparatory work has been carried out to achieve controlled and consistent measurement conditions and to achieve a quantitative and a substance related evaluation under cold start conditions. Binary blends of ethanol and n-hexane, iso-octane or n-decane are studied. A systematic variation of the droplet temperature, the falling distance, the droplet generating frequency and the relative ratios of the applied substances are accomplished. The evaluation demonstrates partially strong component-by-component evaporation dependent on the measurement conditions. Blends of ethanol and hexane in a mixture of 85/15 vol-% and of ethanol and decane with 10/90 vol-% show a distinct influenceability of the varied conditions. Furthermore, the results indicate a depletion of the more volatile component at the droplet surface. With increasing temperature, the depletion and the component-by-component evaporation are promoted. Finally, a decisive influence of the droplet spacing, the droplet temperature and the falling distance is described.

Моделирование физической неустойчивости на контактных границах в течениях многокомпонентных сжимаемых газов гибридным методом крупных частиц

Статья посвящена развитию гибридного метода крупных частиц применительно к двумерным течениям с развитием физической неустойчивости на поверхностях раздела неоднородных газовых смесей. Высокая разрешающая способность метода продемонстрирована при решении задач взаимодействия ударной волны с цилиндрическим пузырем легкого или тяжелого газов в сравнении с экспериментом и расчетами по другим схемам повышенного порядка аппроксимации. This paper is devoted to the development of a large-particle hybrid method for two-dimensional flows with physical instability on the interface of inhomogeneous gas mixtures. The high resolving capacity of the method is shown for problems of shock wave interaction with a cylindrical bubble of a light or heavy gas in comparison with experiments and simulations using other schemes of higher-order approximation.