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NASA NTRS2003

Mode I, Mode II, and Mixed-Mode Fracture of Plasma-sprayed Thermal Barrier Coatings at Ambient and Elevated Temperatures

The mixed-mode fracture behavior of plasma-sprayed ZrO2-8 wt% Y2O3 thermal barrier coatings was determined in air at 25 and 1316 C in asymmetric four-point flexure with single edge v-notched beam (SEVNB) test specimens. The mode I fracture toughness was found to be K(sub Ic) = 1.15 plus or minus 0.07 and 0.98 plus or minus 0.13 MPa the square root of m, respectively, at 25 and 1316 C. The respective mode II fracture toughness values were K(sub IIc) = 0.73 plus or minus 0.10 and 0.65 plus or minus 0.04 MPa the square root of m. Hence, there was an insignificant difference in either K(sub Ic or K(sub IIc) between 25 and 1316 C for the coating material, whereas there was a noticeable distinction between K(sub Ic) and K(sub IIc), resulting in K(sub IIc) per K(sub Ic) = 0.65 at both temperatures. The empirical mixed-mode fracture criterion best described the coatings' mixed-mode fracture behavior among the four mixed-mode fracture theories considered. The angle of crack propagation was in reasonable agreement with the minimum strain energy density criterion. The effect of the directionality of the coating material in on K(sub Ic) was observed to be insignificant, while its sintering effect at 1316 C on K(sub Ic) was significant.

Choi, Sung R.↗

NASA NTRS2003

Control of Jet Noise Through Mixing Enhancement

The idea of using mixing enhancement to reduce jet noise is not new. Lobed mixers have been around since shortly after jet noise became a problem. However, these designs were often a post-design fix that rarely was worth its weight and thrust loss from a system perspective. Recent advances in CFD and some inspired concepts involving chevrons have shown how mixing enhancement can be successfully employed in noise reduction by subtle manipulation of the nozzle geometry. At NASA Glenn Research Center, this recent success has provided an opportunity to explore our paradigms of jet noise understanding, prediction, and reduction. Recent advances in turbulence measurement technology for hot jets have also greatly aided our ability to explore the cause and effect relationships of nozzle geometry, plume turbulence, and acoustic far field. By studying the flow and sound fields of jets with various degrees of mixing enhancement and subsequent noise manipulation, we are able to explore our intuition regarding how jets make noise, test our prediction codes, and pursue advanced noise reduction concepts. The paper will cover some of the existing paradigms of jet noise as they relate to mixing enhancement for jet noise reduction, and present experimental and analytical observations that support these paradigms.

Bridges, James↗

NASA NTRS2002

Element-Element Correlations Among Martian Meteorite Bulk Compositions: Peculiarities Explained(?) by Mixing, with Implications for the Composition of Mars

Most estimates of the chemical corn position of Mars' mantle and core derive from elementelement correlations bulk analyses of Martian meteorites [I-31. To use the correlations in this manner assumes implicitly that they reflect mineraVmelt fractionations (e.g., partial melting, fractional crystallization, accumulation) among a cogenetic suite of igneous rocks. However, similar correlations can zppear from mixing of chemical components, and need not imply that the components were cogenetic nor that they had an igneous origin. Several aspects of the elemental and isotopic chem istries of the Martian meteorites can be explained by mixing of geochemical components (mantle or crustal), e.g. [5,6]. Thus, it is worth asking whether mixing rektions might be seen in bulk chemical data, and (if so) whether the mixing components can be used to infer average planet or mantle compositions.

Treiman, A. H.↗

NASA NTRS2004

Assessing Jet-Induced Spatial Mixing in a Rich, Reacting Crossflow

In many advanced low NOx gas turbine combustion techniques, such as rich-burn/quick-mix/lean-burn (RQL), jet mixing in a reacting, hot, fuel-rich crossflow plays an important role in minimizing all pollutant emissions and maximizing combustion efficiency. Assessing the degree of mixing and predicting jet penetration is critical to the optimization of the jet injection design strategy. Different passive scalar quantities, including carbon, oxygen, and helium are compared to quantify mixing in an atmospheric RQL combustion rig under reacting conditions. The results show that the O2-based jet mixture fraction underpredicts the C-based mixture fraction due to jet dilution and combustion, whereas the He tracer overpredicts it possibly due to differences in density and diffusivity. The He-method also exhibits significant scatter in the mixture fraction data that can most likely be attributed to differences in gas density and turbulent diffusivity. The jet mixture fraction data were used to evaluate planar spatial unmixedness, which showed good agreement for all three scalars. This investigation suggests that, with further technique refinement, O2 or a He tracer could be used instead of C to determine the extent of reaction and mixing in an RQL combustor.

Demayo, T. N.↗

NASA NTRS2003

Enhancement of Mixing and Heat Transfer by Corrugated Permeable Walls

A re-circulation method that is based on the static pressure recovery in the divergent part of a converging-diverging flow channel is discussed. This method can be used to design a static device that can be used to increase the residence time of the fluid and the mixing in the transverse and/or flow directions in compressible and incompressible fluids flowing at subsonic velocities. The device with external reverse flow channels will produce mixing in both transverse and flow directions. If mixing in the flow direction is not desired, the concept can be used to design an insert to provide only transverse mixing. Theoretical relationships for a re-circulating one-dimensional incompressible flow around a converging-diverging channel were obtained. Attempts at numerical simulation of re-circulation in compressible fluids were not successful. Qualitative experimental verification of re-circulating flows for incompressible and compressible fluids were obtained.

Gilinsky, M.↗

NASA NTRS2004

Analysis of Fuel Vaporization, Fuel-Air Mixing, and Combustion in Integrated Mixer-Flame Holders

Requirements to limit pollutant emissions from the gas turbine engines for the future High-Speed Civil Transport (HSCT) have led to consideration of various low-emission combustor concepts. One such concept is the Integrated Mixer-Flame Holder (IMFH). This report describes a series of IMFH analyses performed with KIVA-II, a multi-dimensional CFD code for problems involving sprays, turbulence, and combustion. To meet the needs of this study, KIVA-II's boundary condition and chemistry treatments are modified. The study itself examines the relationships between fuel vaporization, fuel-air mixing, and combustion. Parameters being considered include: mixer tube diameter, mixer tube length, mixer tube geometry (converging-diverging versus straight walls), air inlet velocity, air inlet swirl angle, secondary air injection (dilution holes), fuel injection velocity, fuel injection angle, number of fuel injection ports, fuel spray cone angle, and fuel droplet size. Cases are run with and without combustion to examine the variations in fuel-air mixing and potential for flashback due to the above parameters. The degree of fuel-air mixing is judged by comparing average, minimum, and maximum fuel/air ratios at the exit of the mixer tube, while flame stability is monitored by following the location of the flame front as the solution progresses from ignition to steady state. Results indicate that fuel-air mixing can be enhanced by a variety of means, the best being a combination of air inlet swirl and a converging-diverging mixer tube geometry. With the IMFH configuration utilized in the present study, flashback becomes more common as the mixer tube diameter is increased and is instigated by disturbances associated with the dilution hole flow.

Deur, J. M.↗

NASA NTRS1996

Mixing and Transition Control Studied

Considerable progress in understanding nonlinear phenomena in both unbounded and wallbounded shear flow transition has been made through the use of a combination of high- Reynolds-number asymptotic and numerical methods. The objective of this continuing work is to fully understand the nonlinear dynamics so that ultimately (1) an effective means of mixing and transition control can be developed and (2) the source terms in the aeroacoustic noise problem can be modeled more accurately. Two important aspects of the work are that (1) the disturbances evolve from strictly linear instability waves on weakly nonparallel mean flows so that the proper upstream conditions are applied in the nonlinear or wave-interaction streamwise region and (2) the asymptotic formulations lead to parabolic problems so that the question of proper out-flow boundary conditions--still a research issue for direct numerical simulations of convectively unstable shear flows--does not arise. Composite expansion techniques are used to obtain solutions that account for both mean-flow-evolution and nonlinear effects. A previously derived theory for the amplitude evolution of a two-dimensional instability wave in an incompressible mixing layer (which is in quantitative agreement with available experimental data for the first nonlinear saturation stage for a plane-jet shear layer, a circular-jet shear layer, and a mixing layer behind a splitter plate) have been extended to include a wave-interaction stage with a three-dimensional subharmonic. The ultimate wave interaction effects can either give rise to explosive growth or an equilibrium solution, both of which are intimately associated with the nonlinear self-interaction of the three dimensional component. The extended theory is being evaluated numerically. In contrast to the mixing-layer situation, earlier comparisons of theoretical predictions based on asymptotic methods and experiments in wall-bounded shear-flow transition have been somewhat lacking in one aspect or another. The current work strongly suggests that the main weakness is the underlying asymptotic representation of the linear "part" of the problem and not the explicit modeling of the nonlinear/wave-interaction effects. Consequently, the long-wave-length/high-Reynolds-number asymptotic limit for the Blasius boundary-layer stability problem was reexamined, and a new dispersion relationship for the instability waves that is uniformly valid for both the upper- and lower branch regions to the required order of approximation was obtained. A comparison with numerical results, obtained by solving the Orr-Sommerfeld stability problem, shows that the asymptotic formula provides surprisingly good results, even for values of the frequency parameter usually encountered in experimental investigations. This is particularly evident in the dynamically important upper-branch region, where much of the nonlinear interactions in transition experiments are believed to take place. The result is important in that it can be used to greatly improve the accuracy of weakly nonlinear critical-layer-based theories, and a consistent nonlinear theory is currently under evaluation.

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NASA NTRS2002

Fundamental Mixing and Combustion Experiments for Propelled Hypersonic Flight

The first experiment is a study of a coaxial jet discharging into stagnant laboratory air, with center jet of a mixture of 5% oxygen and 95% helium by volume and coflow jet of air. The exit flow pressure of both center-jet and coflow nozzles is 1 atmosphere. The presence of oxygen in the center jet is to allow the use of an oxygen flow-tagging technique (RELIEF4) to obtain non-intrusive velocity measurements. Both jets are nominally Mach 1.8, but, because of the greater speed of sound, the center jet velocity is more than twice that of the coflow. The mixing layer which forms between the center jet and the coflow near the nozzle exit is compressible, with a calculated convective Mach number of approximately 0.7. This geometry has several advantages: The streamwise development of the flow is generally dominated by turbulent stresses (rather than pressure forces), and thus calculations are sensitive to turbulence modeling. It includes features present in supersonic combustors, including a compressible mixing layer near the nozzle exit and a light-gas/air plume downstream. Since it is a free jet, it provides easy access for both optical instrumentation and probes. Since it is axisymmetric, it requires fewer experimental measurements to fully characterize, and calculations can be performed with more modest computer resources. However, weak shock waves formed at the nozzle exit strengthen and turn normal as they approach the axis, complicating the flow. Care is thus taken in the design of the facility to provide as near as possible to 1-D flow at the exit of both center and coflow nozzles, and to minimize the strength of waves generated at the nozzle exit. Results from this experiment are compared to CFD solutions obtained by VULCAN, a previously developed code used in engine analysis. The second experiment is a study of a supersonic combustor consisting of a diverging duct with single downstream-angled wall injector. Thus, the geometry is relatively simple and large regions of subsonic recirculating flow are avoided. The nominal entrance Mach number is 2 and the enthalpy of the test gas (hot air "simulant") is nominally that of Mach 7 flight. It was believed, on the basis of calculations performed that this would produce mixing-limited flow, that is to say, one for which chemical reaction to equilibrium proceeds at a much greater rate than mixing. It later proved that this was not the case. The primary experimental technique employed is coherent anti-Stokes Raman spectroscopy, known by its acronym CARS. The species probed is molecular nitrogen and the quantity measured is temperature. Intrusive probes, such as Pitot, total temperature, hot-wire, etc., are not used due to access difficulty and high heat flux in the combustor, and because they may alter the flow. CARS has several advantages over other optical methods. It is a relatively mature and well-understood technique. Signal levels are relatively high and the signal is in the form of a coherent (laser) beam that can be collected through small windows. Incoherent (non-CARS) interferences are rejected by spatial filtering.

Diskin, G. S.↗

NASA NTRS2008

Assessment of Crack Path Prediction in Non-Proportional Mixed-Mode Fatigue

Non-proportional mixed-mode loading is present in many systems and a growing crack can experience any manner of mixed-mode loading. Prediction of the resulting crack path is important when assessing potential failure modes or when performing a failure investigation. Current crack path selection criteria are presented along with data for Inconel 718 under non-proportional mixed-mode loading. Mixed-mode crack growth can transition between path deflection mechanisms with very different orientations. Non-proportional fatigue loadings lack a single parameter for input to current crack path criteria. Crack growth transitions were observed in proportional and non-proportional FCG tests. Different paths displayed distinct fracture surface morphologies. New crack path drivers & transition criteria must be developed.

Highsmith, Shelby, Jr.↗

NASA NTRS2008

Kinetic Temperature and Electron Density Measurement in an Inductively Coupled Plasma Torch using Degenerate Four-Wave Mixing

Laser wave mixing is presented as an effective technique for spatially resolved kinetic temperature measurements in an atmospheric-pressure radio-frequency inductively-coupled plasma. Measurements are performed in a 1 kW, 27 MHz RF plasma using a continuous-wave, tunable 811.5-nm diode laser to excite the 4s(sup 3)P2 approaches 4p(sup 3)D3 argon transition. Kinetic temperature measurements are made at five radial steps from the center of the torch and at four different torch heights. The kinetic temperature is determined by measuring simultaneously the line shape of the sub-Doppler backward phase-conjugate degenerate four-wave mixing and the Doppler-broadened forward-scattering degenerate four-wave mixing. The temperature measurements result in a range of 3,500 to 14,000 K+/-150 K. Electron densities measured range from 6.1 (+/-0.3) x 10(exp 15)/cu cm to 10.1 (+/-0.3) x 10(exp 15)/cu cm. The experimental spectra are analyzed using a perturbative treatment of the backward phase-conjugate and forward-geometry wave-mixing theory. Stark width is determined from the collisional broadening measured in the phase-conjugate geometry. Electron density measurements are made based on the Stark width. The kinetic temperature of the plasma was found to be more than halved by adding deionized water through the nebulizer.

Schafer, Julia↗

NASA NTRS2007

The Effects of Air Preheat and Number of Orifices on Flow and Emissions in an RQL Mixing Section

This study was motivated by a goal to understand the mixing and emissions in the rich-burn/quick-mix/lean-burn (RQL) combustor scheme that has been proposed to minimize the formation of oxides of nitrogen (NOx) in gas turbine combustors. The study reported in this paper was a reacting jet-in-crossflow experiment at atmospheric pressure in a cylindrical duct. The jets were injected from the perimeter of the duct through round-hole orifices into a fuel-rich mainstream flow. The number of orifices investigated in this study gave over- to optimum to underpenetrating jets at a jet-to-mainstream momentum-flux ratio of 57. The size of individual orifices was decreased as their number increased to maintain a constant total area. The jet-to-mainstream mass-flow ratio was held constant at 2.5. The experiments focused on the effects of the number of orifices and inlet air preheat and were conducted in a facility that provided the capability for independent variation of jet and main inlet air preheat temperature. The number of orifices was found to have a significant effect on mixing and the distributions of species, but very little effect on overall NOx emissions, suggesting that an aerodynamically optimum mixer may not minimize NOx emissions. Air preheat was found to have very little effect on mixing and the distributions of major species, but preheat did increase NOx emissions significantly. Although the air jets injected in the quick-mix section of a RQL combustor may comprise over 70% of the total air flow, the overall NOx emission levels were found to be more sensitive to mainstream air preheat than to jet stream air preheat.

Holdeman, James D.↗

NASA NTRS2010

Increasing Springtime Ozone Mixing Ratios in the Free Troposphere Over Western North America

In the lowermost layer of the atmosphere - the troposphere - ozone is an important source of the hydroxyl radical, an oxidant that breaks down most pollutants and some greenhouse gases. High concentrations of tropospheric ozone are toxic, however, and have a detrimental effect on human health and ecosystem productivity1. Moreover, tropospheric ozone itself acts as an effective greenhouse gas. Much of the present tropospheric ozone burden is a consequence of anthropogenic emissions of ozone precursors resulting in widespread increases in ozone concentrations since the late 1800s. At present, east Asia has the fastest-growing ozone precursor emissions. Much of the springtime east Asian pollution is exported eastwards towards western North America. Despite evidence that the exported Asian pollution produces ozone, no previous study has found a significant increase in free tropospheric ozone concentrations above the western USA since measurements began in the late 1970s. Here we compile springtime ozone measurements from many different platforms across western North America. We show a strong increase in springtime ozone mixing ratios during 1995-2008 and we have some additional evidence that a similar rate of increase in ozone mixing ratio has occurred since 1984. We find that the rate of increase in ozone mixing ratio is greatest when measurements are more heavily influenced by direct transport from Asia. Our result agrees with previous modelling studies, which indicate that global ozone concentrations should be increasing during the early part of the twenty-first century as a result of increasing precursor emissions, especially at northern mid-latitudes, with western North America being particularly sensitive to rising Asian emissions. We suggest that the observed increase in springtime background ozone mixing ratio may hinder the USA s compliance with its ozone air quality standard.

Cooper, O. R.↗

NASA NTRS2011

A Human-in-the-Loop Evaluation of Multi-Sector Planning in Mixed Equipage Airspace (MSP III)

A human-in-the-loop (HITL) simulation was conducted in May 2010 to determine the feasibility and value 01 conducting multi-sector planning (MSP) operations in a mixed equipage environment. Aircraft were categorized as equipped or unequipped based on the presence or absence of an air-ground data communications (Data Comm) capability for receiving auto-loadable clearances and transfer of communication messages from the air navigation service provider (ANSP). The purpose of the study was to determine the feasibility and possible benefits of introducing multi-sector planning in a mixed equipage context, or whether Data Comm equipage was required for MSP operations. Each test scenario presented one of three different equipage levels to the controllers (10%, 50% or 90% equipped aircraft), so that the operational impact of different equipage levels could be observed. Operational feasibility assessment addressed two related questions: (1) are MSP operations feasible for unequipped aircraft, and (2) are they feasible in a mixed equipage context. Similarly, two categories of potential benefits were explored: (1) system performance improvements (e.g., throughput, workload) associated with MSP at different equipage levels, and (2) the possibility of providing differential service for equipage through MSP operations. Tool requirements (for both planning and controller stations), as well as planning and coordination procedures - within facility (traffic management unit/operational area) and within sector (R-Side/D-Side) - were two other topics addressed in the study. Overall, results suggested that MSP operations were feasible in a mixed equipage environment and that the tools were effective with both equipped and unequipped aircraft. Using the MSP tools, traffic management coordinators were able to manage controller task load, effectively balancing throughput with complexity and controller task load at each of the three equipage levels tested.

Smith, Nancy↗

NASA NTRS2011

New Observational Evidence of Flash Mixing on the White Dwarf Cooling Curve

Blue hook stars are a class of subluminous extreme horizontal branch stars that were discovered in UV images of the massive globular clusters w Cen and NGC 2808. These stars occupy a region of the HR diagram that is unexplained by canonical stellar evolution theory. Using new theoretical evolutionary and atmospheric models, we have shown that the blue hook stars are very likely the progeny of stars that undergo extensive internal mixing during a late helium-core flash on the white dwarf cooling curve. This "flash mixing" produces hotter-than-normal EHB stars with atmospheres significantly enhanced in helium and carbon. The larger bolometric correction, combined with the decrease in hydrogen opacity, makes these stars appear sub luminous in the optical and UV. Flash mixing is more likely to occur in stars born with a high helium abundance, due to their lower mass at the main sequence turnoff. For this reason, the phenomenon is more common in those massive globular clusters that show evidence for secondary populations enhanced in helium. However, a high helium abundance does not, by itself, explain the presence of blue hook stars in massive globular clusters. Here, we present new observational evidence for flash mixing, using recent HST observations. These include UV color-magnitude diagrams of six massive globular clusters and far-UV spectroscopy of hot subdwarfs in one of these clusters (NGC 2808).

Brown, T. M.↗

NASA NTRS2003

Turbulence in Supercritical O2/H2 and C7H16/N2 Mixing Layers

This report presents a study of numerical simulations of mixing layers developing between opposing flows of paired fluids under supercritical conditions, the purpose of the study being to elucidate chemical-species-specific aspects of turbulence. The simulations were performed for two different fluid pairs O2/H2 and C7H16/N2 at similar reduced initial pressures (reduced pressure is defined as pressure divided by critical pressure). Thermodynamically, O2/H2 behaves more nearly like an ideal mixture and has greater solubility, relative to C7H16/N2, which departs strongly from ideality. Because of a specified smaller initial density stratification, the C7H16/N2 layers exhibited greater levels of growth, global molecular mixing, and turbulence. However, smaller density gradients at the transitional state for the O2/H2 system were interpreted as indicating that locally, this system exhibits enhanced mixing as a consequence of its greater solubility and closer approach to ideality. These thermodynamic features were shown to affect entropy dissipation, which was found to be larger for O2/H2 and concentrated in high-density-gradient-magnitude regions that are distortions of the initial density-stratification boundary. In C7H16/N2, the regions of largest dissipation were found to lie in high-density-gradient-magnitude regions that result from mixing of the two fluids.

Bellan, Josette↗

NASA NTRS2011

Comparison of Four Mixed Layer Mesoscale Parameterizations and the Equation for an Arbitrary Tracer

In this paper we discuss two issues, the inter-comparison of four mixed layer mesoscale parameterizations and the search for the eddy induced velocity for an arbitrary tracer. It must be stressed that our analysis is limited to mixed layer mesoscales since we do not treat sub-mesoscales and small turbulent mixing. As for the first item, since three of the four parameterizations are expressed in terms of a stream function and a residual flux of the RMT formalism (residual mean theory), while the fourth is expressed in terms of vertical and horizontal fluxes, we needed a formalism to connect the two formulations. The standard RMT representation developed for the deep ocean cannot be extended to the mixed layer since its stream function does not vanish at the ocean's surface. We develop a new RMT representation that satisfies the surface boundary condition. As for the general form of the eddy induced velocity for an arbitrary tracer, thus far, it has been assumed that there is only the one that originates from the curl of the stream function. This is because it was assumed that the tracer residual flux is purely diffusive. On the other hand, we show that in the case of an arbitrary tracer, the residual flux has also a skew component that gives rise to an additional bolus velocity. Therefore, instead of only one bolus velocity, there are now two, one coming from the curl of the stream function and other from the skew part of the residual flux. In the buoyancy case, only one bolus velocity contributes to the mean buoyancy equation since the residual flux is indeed only diffusive.

Canuto, V. M.↗

NASA NTRS2012

Sensitivity of Cirrus and Mixed-phase Clouds to the Ice Nuclei Spectra in McRAS-AC: Single Column Model Simulations

The salient features of mixed-phase and ice clouds in a GCM cloud scheme are examined using the ice formation parameterizations of Liu and Penner (LP) and Barahona and Nenes (BN). The performance of LP and BN ice nucleation parameterizations were assessed in the GEOS-5 AGCM using the McRAS-AC cloud microphysics framework in single column mode. Four dimensional assimilated data from the intensive observation period of ARM TWP-ICE campaign was used to drive the fluxes and lateral forcing. Simulation experiments where established to test the impact of each parameterization in the resulting cloud fields. Three commonly used IN spectra were utilized in the BN parameterization to described the availability of IN for heterogeneous ice nucleation. The results show large similarities in the cirrus cloud regime between all the schemes tested, in which ice crystal concentrations were within a factor of 10 regardless of the parameterization used. In mixed-phase clouds there are some persistent differences in cloud particle number concentration and size, as well as in cloud fraction, ice water mixing ratio, and ice water path. Contact freezing in the simulated mixed-phase clouds contributed to transfer liquid to ice efficiently, so that on average, the clouds were fully glaciated at T approximately 260K, irrespective of the ice nucleation parameterization used. Comparison of simulated ice water path to available satellite derived observations were also performed, finding that all the schemes tested with the BN parameterization predicted 20 average values of IWP within plus or minus 15% of the observations.

Betancourt, R. Morales↗

NASA NTRS2014

Optics of Water Cloud Droplets Mixed with Black-Carbon Aerosols

We use the recently extended superposition T-matrix method to calculate scattering and absorption properties of micrometer-sized water droplets contaminated by black carbon. Our numerically exact results reveal that, depending on the mode of soot-water mixing, the soot specific absorption can vary by a factor exceeding 6.5. The specific absorption is maximized when the soot material is quasi-uniformly distributed throughout the droplet interior in the form of numerous small monomers. The range of mixing scenarios captured by our computations implies a wide range of remote sensing and radiation budget implications of the presence of black carbon in liquid-water clouds. We show that the popular Maxwell-Garnett effective-medium approximation can be used to calculate the optical cross sections, single-scattering albedo, and asymmetry parameter for the quasi-uniform mixing scenario, but is likely to fail in application to other mixing scenarios and in computations of the elements of the scattering matrix.

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