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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenSat, 13 Apr 2024 01:19:02 +0000Evaluation of the importance of the relative velocity during evaporation of drops in sprays
https://resolver.caltech.edu/CaltechAUTHORS:20171025-111923147
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1986
DOI: 10.1016/0017-9310(86)90099-2
Evaporation and combustion of liquid sprays in power systems invariably occurs in environments where there is a convective flow past the spray. This convective flow influences evaporation and combustion in at least to ways. First, it changes the heat and mass transfer rates between the spray as an entity, and the ambience. Secondly, it changes the geometry of the spray by entrainment of the spray periphery and recirculation of the gases surrounding the spray. These processes are all very complex and difficult to model. For this reason, guidance was sought initially from the study of individual drop evaporation and combustion. These studies [1-6] concurred with the experimental observation that a…https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ae6a1-j4b81Ignition of Non Dilute Clusters of Drops in Convective Flows
https://resolver.caltech.edu/CaltechAUTHORS:20171025-111436572
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1987
DOI: 10.1080/00102208708947021
A global model has been developed for the qualitative prediction of ignition of clusters of drops evaporating in a convective flow. This model incorporates the description of convective droplet-cluster evaporation through a model which is valid for both dense and dilute clusters. The model takes into account drop interactions and the resulting possible limitations on evaporation in the limit of dense clusters. An Eulerian description is used to predict both drop and gas velocities. To complement the fluid mechanics model which is self-contained, the bulk interaction between the convective flow around the cluster and the cluster is evaluated using a penetration ratio criterion. The penetration distance itself is calculated in a Lagrangian frame. The model of droplet-cluster ignition can predict both the ignition time of the cluster and the location of the flame rs) at that time (under the, assumption of a spherical cluster). The ignition-timing part of the ignition criterion is valid only for diffusion-controlled ignition. The various possible combustion regimes for droplet-clusters are identified using a two dimensional map which compares convective and diffusive effects. Further numerical calculations show that in practical systems dense droplet-cluster ignition is always diffusion-dominated. The dependence of the ignition time upon both the initial drop temperature and gas temperature is studied as well. It is shown that the initial conditions determine whether a cluster ignites in anyone of the regimes previously identified.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/v88rj-gv063Analysis of the convective evaporation of nondilute clusters of drops
https://resolver.caltech.edu/CaltechAUTHORS:20171025-111709573
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1987
DOI: 10.1016/0017-9310(87)90065-2
A model for the convective evaporation of nondilute clusters of drops has been developed. The critical parameter which controls the different evaporation modes has been identified to be the penetration distance of the outer flow into the cluster volume. A dynamic criterion has been developed to differentiate between penetration and no penetration. Convective evaporation was modeled using a Reynolds number correlation between the evaporation rate with and without convection. Other equations, previously developed [Combust. Flame51, 55–67 (1983)] for quiescent, nondilute-spray evaporation, have been used here as well, with the exception of a new kinetic-evaporation law at the droplet surface and a nonuniform interior temperature model which have both been developed here.
The model is shown to perform well for low penetration distances which are obtained for dense clusters in hot environments and low relative velocities between outer gases and cluster. For dense clusters with low penetration distances the results of the model predict that for the same initial velocity the evaporation time is shorter as the cluster becomes more dilute. For dilute clusters and large penetration distances, the opposite was found. Since for large penetration distances the predictive ability of the model deteriorates, these last trends are questionable. Furthermore, the evaporation time was found to be a weak function of the initial relative velocity and a strong function of the initial drop temperature. The initial surrounding gas temperature was found to have a strong influence in the lower temperature regime, 750–1500 K, whereas in the higher temperature regime the influence was very weak. The vitiation of the ambient gas by fuel vapor was found to have a very small influence upon the evaporation time for rich mixtures when the cluster is introduced in a strongly convective, high temperature surroundings. In all cases the results show that the interior drop-temperature was transient throughout the drop lifetime, but nonuniformities in the temperature persisted up to at most the first third of the total evaporation time.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/wsfce-tq129The details of the convective evaporation of dense and dilute clusters of drops
https://resolver.caltech.edu/CaltechAUTHORS:20171025-111155291
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1987
DOI: 10.1016/0017-9310(87)90038-X
A global model describing the convective evaporation of dense and dilute clusters of drops has been formulated starting from first principles. The volume of the cluster and the number of drops in a given cluster are fixed and the drops do not move with respect to each other. The model has been tested for three different drag models and shows less than 10% sensitivity in the prediction of the droplet lifetime. A thorough parametric study has been performed and the results show that the control parameters are very different in order of importance for dense and dilute clusters. The initial relative velocity between drops and gases is a weak control parameter in the 40–1000cm s^(-1) regime.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ffbnt-a6235Turbulence effects during evaporation of drops in clusters
https://resolver.caltech.edu/CaltechAUTHORS:20171025-105336722
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1988
DOI: 10.1016/0017-9310(88)90278-5
A model of droplet evaporation in clusters and the exchange processes between the cluster and the gas phase surrounding it are presented. This model is developed for use as a subscale model in calculations of spray evaporation and combustion and thus described only global features of cluster behavior. The gas pressure in the cluster remains constant during evaporation and as a result the volume of the cluster and the drop number density inside the cluster vary. Two turbulence models are considered. The first one describes cluster evaporation in surroundings initially devoid of turbulence and turbulence is allowed to build up with time. The second model describes cluster evaporation in surroundings where turbulence is present initially. The results obtained with these models show that turbulence enhances evaporation and is a controlling factor in the evaporation of very dense clusters; examples are shown where with the first turbulence model saturation was obtained before complete evaporation whereas the opposite was obtained with the second turbulence model. As the initial air/fuel mass ratio increases, both turbulence history and the initial relative velocity between drops and gases can control evaporation. It is shown that the evaporation time decreases with an initial increase in turbulence levels or relative velocity. When the initial air/fuel mass ratio increases further and the initial drop number density falls within the dilute regime, neither of the above parameters can control evaporation. Moreover, the evaporation time decreases with the decreasing size of the cluster for dense clusters of drops, whereas for dilute clusters of drops the size is not a controlling factor. The practical implications of these results are discussed.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/mm82c-7wg30Electrostatic Dispersion of Drops in Clusters
https://resolver.caltech.edu/CaltechAUTHORS:20171025-104721118
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1989
DOI: 10.1080/00102208908947125
A theory of evaporation and dispersion of electrostatically charged drops has been developed for drops belonging to a spherical cluster exposed to a flow Under the assumption of constant atmospheric pressure, the quasi-steady approximation was made for the gas phase whereas the drop-temperature history is unsteady. The model lakes into account interdrop interactions (in terms of heal and mass transfer) due to drop proximity, turbulence exchange processes between the cluster and its surroundings and electrostatic force effects due to the charge on the drops. Calculations based upon this model were made for charged as well as uncharged clusters of drops. The charge was varied from a null value to the maximum possible charge found empirically for hydrocarbon sprays. Moreover, the turbulence model was varied in such a way as to simulate the cluster embedded into a flow where turbulence develops with time (Model 1) or a flow with pre-existing turbulence (Model 2). The results show that the control parameters for the evaporation of charged drops are different from those for uncharged drops in dense clusters; turbulence levels which were shown lo be crucial for the latter in the dense cluster regime do not affect the former in the same regime. For dilute clusters turbulence is unimportant in both cases. Moreover, drop charging docs not affect dilute clusters of drops whereas dense clusters of drops are substantially affected. Based upon existing experimental data, inferences are made about how electrostatic spray dispersion can affect soot control in powersystems using fuel sprays. Limited results perlainingto the ignition of nearly-dense clusters of electrostatically charged drops are discussed as well.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/f6gz3-y4663Transport-related phenomena for clusters of drops
https://resolver.caltech.edu/CaltechAUTHORS:20171025-104133863
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1989
DOI: 10.1016/0017-9310(89)90253-6
Measurements performed in sprays characteristic of power systems show that sprays are composed of several regions [1]. Near the atomizer the drops might not be entirely formed and liquid sheets and filaments might still exist. There follows a region where the drops are already formed but have not yet been dispersed, so that they cluster together with a typical distance between the drops that is of the same order of magnitude as that of the average radius of the drops themselves. This region of the spray is called the dense spray region. Finally, further from this dense spray region there exists a region where the drops might still cluster, but in these clusters the distance between drops is much larger than the average radius of the drops. This region is called the dilute spray region.
In the dilute spray region drops are far apart from each other and thus when the spray is exposed to a convective flow, these drops practically behave like isolated drops in a convective flow. In contrast, in the dense spray regime, the drops are close to each other and thus their history is controlled by how much of the surrounding gas can enter in contact with them. This is to say that, unlike for drops belonging to dilute clusters of drops, transport phenomena are crucial in determining the behavior of drops belonging to dense clusters of drops because transport imposes limits on heat and mass transfer between the two phases. These phenomena pertain to indirect interactions and they can control the motion of drops, their heat-up time, evaporation, ignition and combustion.
Previous work [2-5] pointed out some important consequences of these indirect interactions. Two models of turbulent transport were used in ref. [5] in order to investigate the importance of turbulent transport from the surroundings to the cluster. Because of the global aspect of the model in which all the drops were assumed to behave identically, the transport from the cluster to surroundings was modeled using a 'trapping factor'. Basically, the 'trapping factor' is a weighing factor which allows the modeling of intermediary situations between those of dilute clusters where evaporated mass was assumed to be trapped in the cluster and that of dense clusters where evaporated mass was assumed to escape to ambient. It was found [5] that whereas in the dilute regime turbulence is not a controlling parameter, in the dense regime it becomes the crucial control parameter. This is a fact well known by experimentalists and design engineers who locate turbulent enhancement devices near the injector where the spray is dense, rather than further down the combustor where the spray is dilute.
Since the transport processes between the cluster and its surroundings were found to be so important in the case of dense clusters, it was thought very important to improve the description of the transport of heat, mass and species from the cluster and its surroundings. This new model is described in detail in ref. [6] for electrostatically charged drops, and is used to calculate the results presented below for the special case of null charge. Due to the brief nature of the Technical Note, the nomenclature used here is the same as in refs. [5, 6].
The model developed in ref. [6] is similar to that of ref. [5] in that the drops and gas have two velocity components: a uniform axial component along the trajectory direction and a radial component. The difference between the two models is in the description of the radial velocity component. Whereas in ref. [5] a 'trapping factor' was used as discussed above, the new formulation uses the assumption of self-similarity in the radial direction as explained in detail in ref. [6].https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/f60xd-f1t97Evaporation, ignition, and combustion of nondilute clusters of drops
https://resolver.caltech.edu/CaltechAUTHORS:20171019-123707161
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1990
DOI: 10.1016/0010-2180(90)90139-I
A theory of evaporation, ignition, and burning of moderately dense spherical drop clusters has been developed. The theory takes into account burning of premixed air and fuel internal to the cluster at ignition and subsequent burning of fuel emitted from the cluster by a flame sheet surrounding it. The model considers interdrop interaction, momentum exchange between drops and gas, and turbulent exchange processes between the cluster and its surroundings. Calculations are performed for varying initial air-to-fuel-mass ratios, initial cluster radii, ambient gas temperatures and initial drop temperatures. Results are presented for ratios of fuel mass burned to fuel mass lost from the cluster between drop ignition and drop disappearance, fuel burned fractions at ignition and at the moment of drop disappearance, and jump conditions at ignition.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4jkcq-sz128The dynamics of dense and dilute clusters of drops evaporating in large, coherent vortices
https://resolver.caltech.edu/CaltechAUTHORS:20171027-101011089
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1991
DOI: 10.1016/S0082-0784(06)80403-0
The behavior of evaporating clusters of drops embedded into large, coherent vortices is described using a formulation which is valid for both dense and dilute clusters. Drops and gas interact both dynamically and thermodynamically. Dynamic coupling occurs through a force on the drops due to drag resulting from a slip velocity between the two phases The net interaction force on the gas with drops is due to a source thrust from evaporation plus drag on each drop. The drag coefficient accounts for blowing from the drop surface. Thermodynamic coupling is a result of drop heating and evaporation. Limitations due to drop proximity on heating and evaporation are taken into account.
The vortical motion of the drops in the cluster results in the formation of a core region devoid of drops at the center of the vortex, and a shell region containing the drops and surrounding the inner core. Results are presented showing the dependence of the evaporation time, the final to initial volume ratio and the final to initial shell thickness ratio upon the initial air/fuel mass ratio and as a function of the initial tangential velocities, upon the initial Stokes number, initial drop radius and initial outer cluster radius. Differences in behavior between and control parameters of dense and dilute clusters are pointed out by these new results. It is found that for dense clusters the final to initial volume ratio and final to initial shell thickness scale with the initial Stokes number, a new result which must be validated experimentally.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/27z9t-sdv79A Model of the Evaporation of Binary-Fuel Clusters of Drops
https://resolver.caltech.edu/CaltechAUTHORS:20171027-160538311
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1991
DOI: 10.1615/AtomizSpr.v1.i4.20
A formulation has been developed to describe the evaporation of dense or dilute clusters of binary-fuel drops. The binary fuel is assumed to be made of a solute and a solvent whose volatility is much lower than that of the solute. Convective flow effects, inducing a circulatory motion inside the drops, are taken into account, as well as turbulence external to the cluster volume. Results obtained with this model show that, similar to the conclusions for single, isolated drops, the evaporation of the volatile is controlled by liquid mass diffusion when the cluster is dilute. In contrast, when the cluster is dense, the evaporation of the volatile is controlled by surface layer stripping, that is, by the regression rate of the drop, which is, in fact, controlled by the evaporation rate of the solvent. These conclusions are in agreement with existing experimental observations. Parametric studies show that these conclusions remain valid with changes in ambient temperature, initial slip velocity between drops and gas, initial drop size, initial cluster size, initial liquid mass fraction of the solute, and various combinations of solvent and solute. The implications of these results for computationally intensive combustor calculations are discussed.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/dqkrx-xe925Entrainment and Evaporation of Drops in the Laminar Part of a Two-Dimensional Developing Mixing Layer
https://resolver.caltech.edu/CaltechAUTHORS:20171027-101441171
Authors: {'items': [{'id': 'Fichot-F', 'name': {'family': 'Fichot', 'given': 'F.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1994
DOI: 10.1016/S0082-0784(06)80667-3
A formulation has been developed that combines the simplicity of an experimentally derived well-establishedcorrelation for describing the development of a mixing layer, and a rigorous approach for the description of the dynamics and evaporation of dense or dilute clusters of drops in large coherent vortices.
An extensive parametric study has been performed by varying the radius of the drops in the drop-ladenstream both for high and low air/fuel mass ratio, as well as for constant initial drop number density, but at varying air/fuel mass ratio. The air/fuel mass ratio has also been varied at fixed drop radius in the drop-laden stream. Additional parameters independently varied were the temperature of the hot air stream, its velocity, and the velocity ratio between the two streams.
The results show that it is possible to optimize the relative number of drops (with respect to the initial value) entrained into the coherent vortices of the mixing layer by using the velocity ratio as control parameter. The eventual liquid mass entrained in the cluster is an increasing function of the drop radius in the drop-carrying stream for typical drop number densities in sprays. The mass fraction of the evaporated fuel in the clusters can be optimized by using the velocity of the hot air stream as a control parameter.
It is also shown that, in agreement with existing observations, the average drop radius may increase withaxial distance from the mixing layer inception point, and the reasons for this are explained.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1821h-pv885Unsteady evaporation and combustion of a drop cluster inside a vortex
https://resolver.caltech.edu/CaltechAUTHORS:20171019-131442063
Authors: {'items': [{'id': 'Fichot-F', 'name': {'family': 'Fichot', 'given': 'F.'}}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1994
DOI: 10.1016/0010-2180(94)90194-5
A model has been developed that describes the evaporation, ignition, and combustion of a drop cluster embedded in a large vortex. The purpose of this model is to simulate the behavior of drops in large coherent vortices produced in the shear layer of a jet. The model treats the dynamic interactions between the drops and the vortex, and also takes into account the drop proximity to calculate the heat and mass transfer between drops and ambient gas. The gas phase outside the cluster is treated as an unsteady, reacting phase, whereas quasi-steadiness is assumed between the drops and surrounding gas inside the cluster. It is assumed that drops will not burn individually, but as a group. The results show a very complex interaction between the dynamics of the drop-loaded vortex, the flame, and the evaporation process. A quasi-steady state is not always reached, depending upon the drop number density or the vortex intensity. In most cases, the flame is located very close to the cluster. The mass ratio of burned fuel (at complete evaporation) to initial fuel is generally less than 10%.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/69wmq-ez255Unsteady injection of sequences of drop clusters in vortices depicting portions of a spray
https://resolver.caltech.edu/CaltechAUTHORS:20171025-091855457
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1995
DOI: 10.1615/AtomizSpr.v5.i1.20
A model of unsteady injection of sequences of drop clusters embedded in jet vortices was applied to describe both vortices in the shear layer of a spray and small-scale vortical structures in the core of a spray. In the first case, the vortices are large compared to the size of the spray, they rotate fast with respect to the injection rate, and their number per area of spray is small. In the second case, the vortices are small compared to the size of the spray, they rotate slowly with respect to the injection rate, and their number per area of spray is large.
Results were obtained for injection sequences where either the drop size or the air/fuel mass ratio varied from cluster to cluster in the injection sequence. The variation was a mono-tonic increase, a monotonic decrease, or a sinusoidal variation. The results thus obtained were compared to baseline results from steady-state calculations. Additionally, both the entrainment from the ambient into the jet and the initial number of clusters per jet area were varied so as to ascertain their influence on cluster penetration and jet properties.
The results show that penetration of a cluster into the ambient is a function of the characteristics of the cluster sequence following the cluster, that the jet temperature is controlled by entrainment from the ambient into the jet in the shear-layer application whereas conduction also becomes important in the spray-core application, and that the fuel mass fraction in the jet is a function of the initial characteristics of the clusters as well as entrainment.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/jqnvh-4tf52Steady injection of identical clusters of evaporating drops embedded in jet vortices
https://resolver.caltech.edu/CaltechAUTHORS:20171025-091601090
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1995
DOI: 10.1615/AtomizSpr.v5.i1.10
A model has been developed that describes the evaporation of clusters of drops in a flowing gaseous jet. Each one of these clusters is embedded into a coherent vortex and the drops evaporate as the clusters convect downstream together with the vortex. Because there is a continuous injection of clusters, each cluster represents in fact a statistical average of clusters at that particular location. Thus, the formulation contains a conservation equation for the cluster number density, conservation equations for the gas in the jet, and conservation equations for the drops in the cluster and the vortex containing the cluster. The cluster and vortex models are coupled to the gaseous jet model through boundary conditions. The heat necessary to evaporate the drops comes from the surroundings of the gaseous jet, and this is described through a global, diffusive entrainment model. It is assumed that the turbulent diffusion coefficient is proportional either to the local vortex strength or to the cluster velocity and the multiplier is named the entrainment coefficient.
Results are presented here for the stationary case representing the situation when identical clusters are continuously injected and the injection rate is constant. Thus, if a "snapshot" of the calculation is taken at any time, the cluster is observed at that time and the clusters in its wake represent the history of the cluster at previous times. Parametric studies cover the influence of the initial air/fuel mass ratio, the entrainment coefficient, and the initial drop and gas velocities inside the vortices. The results show that quantitative predictions of the evaporation time, the penetration of the clusters into the ambient, and the temperature of the jet depend on details of the entrainment of hot gas into the jet.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/yamdh-h3j37Ignition of a Binary-fuel (Solvent-Solute) Cluster of Drops
https://resolver.caltech.edu/CaltechAUTHORS:20171025-103852612
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1995
DOI: 10.1080/00102209508951939
Evaporation and ignition of a binary-fuel cluster of drops is described by models under the assumptions that the volatile compound has infinite volatility with respect to the solvent and that the chemistries of the two compounds are independent. A Damköhler number criterion developed for use in sprays is utilized to determine the ignition time. Another criterion is used to determine the ignition location which can be either around individual drops, or around groups of drops inside the cluster, or around the entire cluster.
Results show that except for very dilute situations where the initial liquid mass fraction of the volatile is very small, ignition always occurs around the entire cluster. Otherwise, ignition occurs around groups of drops inside the cluster but never around individual drops even though the ratio of the distance between the centers of two adjacent drops by the drop diameter is greater than thirty five.
Studies performed by varying the air/fuel mass ratio for a variety of parametric combinations show that: (1) At typical gas temperatures for combustion devices, the ignition of very dense and very dilute clusters of drops is evaporation-controlled for identical chemistries; it is strongly-controlled by solvent ignition in the very dense cluster regime, it is strongly-controlled by ignition of the volatile in the very dilute regime. In the intermediary regime, ignition is controlled by the relative ignition chemistries of the compounds. These conclusions are independent of the amount of volatile initially present in the liquid. (2) The concept of volatile is more strongly associated with the latent heat of evaporation in the dense regime, and more strongly associated with the saturation pressure curve in the very dilute regime. (3) By increasing the surrounding gas temperature one gradually gains control of ignition in the dense and dilute regimes through the evaporation of solvent and volatile respectively. (4) The initial slip velocity between phases affects ignition only in the very dilute regime. (5) Changes in the cluster size affect the ignition time only in the very dense regime.
Conclusions (3) and (4) are valid under the assumption of identical kinetics for the two compounds; when different kinetics are considered, it turns out that kinetic effects overwhelmingly dominate ignition.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/znx67-xzw70Electrostatic dispersions and evaporation of clusters of drops of high-energy fuel for soot control
https://resolver.caltech.edu/CaltechAUTHORS:20171019-132350571
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1996
DOI: 10.1016/S0082-0784(96)80396-1
A model is presented for the electrostatic dispersion of a poly disperse cluster of evaporating drops embedded into an inviscid vortex. Results from this model obtained for dense clusters of drops show that electrostatic dispersion decreases the mass fraction of the evaporating compound as well as the gas density in side the cluster. Since the sooting tendency of a fuel (through coagulation) is an increasing function of the partial density of the fuel vapor, it is inferred that electrostatic charging decreases the sooting tendency. Results indicate that the sooting tendency is a monotonically decreasing function of the charge. By using this model for different fuels, it is shown that the sooting tendency of a fuel is associated with two competing characteristic times: that of drop dispersion and that of drop evaporation. It is also shown that the drop evaporation time is directly related to the latent heat of the fuel, thereby providing a simple way to relate sooting propensity to fuel-specific properties.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/z29jt-w4435Efficient high-pressure state equations
https://resolver.caltech.edu/CaltechAUTHORS:20171024-144448068
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Miller-R-S', 'name': {'family': 'Miller', 'given': 'Richard S.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1997
DOI: 10.1002/aic.690430624
A method is presented for a relatively accurate, noniterative, computationally efficient calculation of high-pressure fluid-mixture equations of state, especially targeted to gas turbines and rocket engines. Pressures above 1 bar and temperatures above 100 K are addressed. The method is based on curve fitting an effective reference state relative to departure functions formed using the Peng-Robinson cubic state equation. Fit parameters for H_2, O_2, N_2, propane, methane, n-heptane, and methanol are given.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/wk7ky-v8266Behavior of a polydisperse cluster of interacting drops evaporating in an inviscid vortex
https://resolver.caltech.edu/CaltechAUTHORS:20171019-125537950
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1997
DOI: 10.1016/S0301-9322(97)00011-6
The dynamics and evaporation of polydisperse collections of liquid drops in an axisymmetric, infinite, cylindrical vortex are described using a statistical model. This model describes both the dense regime where inter-particle effects are important and the dilute regime. The initial size distribution is partitioned into size classes and each initial size-class is followed dynamically and thermodynamically using a class-defined, drop-frame coordinate system. Each initial-size-class develops a continuum of sizes as drops centrifuge towards hotter surroundings and evaporate. A separate coordinate system tracks the gas phase. Because larger drops experience larger centrifugal force, they approach the hotter gas faster. However, for appropriate liquid heating times, the large drops might evaporate at a faster rate, and so the size-differentiated centrifugation previously observed and calculated for cold flow situations does not occur. Instead, a radially peaked drop size distribution is developed in the gas vortex. The centrifugal motion forms a drop-free inner vortex core bound by a cylindrical shell containing all the drops. This shell of gas and drops is called the drop cluster. Numerical calculations show that more parameters control dense clusters than dilute clusters; examples of these parametric relations include: (i the gas vortex, whereas drop size distribution controls the outer region; and (ii increases the maximum mass fraction of the evaporated compound and enhances penetration of the evaporated compound into the surroundings. Except for dilute clusters, the assumption of uniform drop number distribution in the cluster is found to be inappropriate. Instead, the drop size distribution always becomes non-uniform even if the initial size distribution is monodisperse and the initial drop number distribution is uniform. This development of non-uniformity is caused by drops at the cluster peripheries preventing heat conduction/convection to drops in the central cluster.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/adc1j-z2288Dispersion (Electrostatic/Mechanical) and Fuel Properties Effects on Soot Propensity in Clusters of Drops
https://resolver.caltech.edu/CaltechAUTHORS:20171024-145141363
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1998
DOI: 10.1615/AtomizSpr.v8.i6.10
Soot propensity is studied numerically for an initially binary size, axisymmetric cluster of evaporating drops by defining it as the propensity for nucleation reactions to occur; the study does not address physical or chemical processes ensuing after soot nucleation, such as soot oxidation effects resulting from the fuel molecular structure. The relative magnitude of the fuel vapor partial density is taken as an indication of the soot nucleation magnitude; thus, the effect of drop dispersion on soot (precursor) formation is isolated from that of soot production resulting from formation/destruction by oxidation. The cluster is embedded in an inviscid vortex and exchanges mass, momentum, species, and energy with its surroundings. The vortical motion disperses the drops and the initial cluster evolves into a cylindrical shell with an inner and an outer boundary. In addition to the forces resulting from the vortical motion, an electrostatic force acts on the cluster when the drops are charged; in this situation, the drops might become small enough to reach the Rayleigh limit. Results are obtained for typical vortical motion times having the same order of magnitude as the drop lifetime. Analysis of the results shows that the motion of uncharged drops is determined primarily by centrifugation, whereas for charged drops the electrostatic dispersion becomes the dominant influence in the outer part of the cluster. In the range of parameters investigated, mechanical dispersion cannot rival electrostatically induced dispersion for decreasing the fuel vapor partial density. An additional feature of drop charging is the maintenance of a finite slip velocity in the outer part of the cluster, thereby compounding the advantage of increased dispersion to enhanced evaporation. The results also show that mechanical dispersion combined with electrostatic dispersion does not have a substantial advantage over electrostatic dispersion alone. For uncharged drops it has been found that the latent heat governs soot propensity at small drop dispersion, whereas the liquid density becomes increasingly important with increasing drop dispersion. Drop charging does not affect the influence of fuel physical properties on soot propensity.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/sdbts-x2t78Evaluation of equilibrium and non-equilibrium evaporation models for many-droplet gas-liquid flow simulations
https://resolver.caltech.edu/CaltechAUTHORS:20171019-110525024
Authors: {'items': [{'id': 'Miller-R-S', 'name': {'family': 'Miller', 'given': 'R. S.'}}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1998
DOI: 10.1016/S0301-9322(98)00028-7
A variety of liquid droplet evaporation models, including both classical equilibrium and non-equilibrium Langmuir–Knudsen formulations, are evaluated through comparisons with experiments with particular emphasis on computationally efficient procedures for gas–liquid flow simulations. The models considered are those used in droplet laden flow calculations such as direct numerical simulations for which large numbers of individual (isolated) droplet solutions are obtained. Diameter and temperature evolution predictions are made for single-component droplets of benzene, decane, heptane, hexane and water with relatively large initial sizes ∼1 mm vaporizing in convective air flows. All of the models perform nearly identically for low evaporation rates at gas temperatures significantly lower than the boiling temperature. For gas temperatures at and above the boiling point, large deviations are found between the various model predictions. The simulated results reveal that non-equilibrium effects become significant when the initial droplet diameter is <50 μm and that these effects are enhanced with increasing slip velocity. It is additionally observed that constant properties can be used throughout each simulation if both the gas and vapor values are calculated at either the wet-bulb or boiling temperature. The models based on the Langmuir–Knudsen law and a corrected (for evaporation effects) analytical heat transfer expression derived from the quasi-steady gas phase assumption are shown to agree most favorably with a wide variety of experimental results. Since the experimental droplet sizes are all much larger than the limit for non-equilibrium effects to be important, for these conditions the most crucial aspect of the current Langmuir–Knudsen models is the corrected analytical form for the heat transfer expression as compared to empirical relations used in the remaining models.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/cdqqb-z4q64Isolated fluid oxygen drop behavior in fluid hydrogen at rocket chamber pressures
https://resolver.caltech.edu/CaltechAUTHORS:20171025-073649264
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1998
DOI: 10.1016/S0017-9310(98)00049-0
A model has been developed for the behavior of an isolated fluid drop of a single compound immersed into another compound in finite, quiescent surroundings at supercritical conditions. The model is based upon fluctuation theory which accounts for both Soret and Dufour effects in the calculation of the transport matrix relating molar and heat fluxes to the transport properties and the thermodynamic variables. The transport properties have been modeled over a wide range of pressure and temperature variation applicable to LO_x–H_2 conditions in rocket chambers, and the form of the chemical potentials is valid for a general fluid. The equations of state have been calculated using a previously-derived, computationally-efficient and accurate protocol. Results obtained for the LO_x–H_2 system show that the supercritical behavior is essentially one of diffusion. The temperature profile relaxes fastest followed by the density and lastly by the mass fraction profile. An effective Lewis number calculated using theory derived elsewhere shows that it is larger by approximately a factor of 40 than the traditional Lewis number. The parametric variations show that gradients increasingly persist with increasing fluid drop size or pressure, and with decreasing temperature. The implication of these results upon accurate measurements of fluid drop size under supercritical conditions is discussed.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/xek6k-x5373Interactions of fluid oxygen drops in fluid hydrogen at rocket chamber pressures
https://resolver.caltech.edu/CaltechAUTHORS:20171025-074109330
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1998
DOI: 10.1016/S0017-9310(98)00048-9
A model of fluid drop behavior in clusters has been developed including the interactions induced by the drop proximity. The model is based upon the global conservation equations for the interstitial cluster region coupled to isolated fluid drop equations previously developed. Heat and mass transfer to the cluster are modeled using the Nusselt number concept. Results from calculations for the LO_x–H_2 system show the predictions to be insensitive to the value of the Nusselt number over three orders of magnitude. The results also show that at fixed pressure, increased drop proximity induces increased accumulation of LO_x in the interstitial space inside the cluster. At fixed initial drop proximity, the gradients of the dependent variables become increasingly smeared as the pressure increases; an opposite result from that obtained for isolated drops (Harstad K, Bellan J. Isolated fluid oxygen drop behavior in fluid hydrogen at rocket chamber pressure. Int J Heat Mass Transfer 1998; 41:3537–50). It is thus inferred that clusters of drops might be a desirable aspect in supercritical combustion because they aid interdiffusion of the reactive components.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/kmj4g-ypp76High-Energy-Density Fuel Blending Strategies and Drop Dispersion for Fuel Cost Reduction and Soot Propensity Control
https://resolver.caltech.edu/CaltechAUTHORS:20171024-080951487
Authors: {'items': [{'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}]}
Year: 1999
DOI: 10.1615/AtomizSpr.v9.i4.40
High-energy-density (HED) liquid fuels have high soot propensity and are expensive. The idea of mitigating these characteristics by adding a less expensive, low soot propensity liquid fuel to the HED is tested through numerical simulations. The model represents an axisymmetric, polydisperse, dense cluster of binary-fuel (solvent/solute) spherical drops embedded into a vortex. Since soot propensity depends on the partial density of the evaporated fuel, this partial density is compared for uncharged and electrostatically charged drops; charging is used here as an effective way to increase dispersion and reduce sooting propensity. Results from the simulations show that while the solvent soot propensity indeed decreases with drop charging, contrary to simplistic expectations, addition of HED as a solute increases sooting propensity of the solute with increased drop dispersion. This is due to the additional dispersion maintaining the slip velocity at the drop surface and preferentially evaporating the solute. These counterintuitive but correct physical effects are independent of the initial solvent/solute mass ratio, and the soot propensity decreases with decreasing solute volatility. Based on these results, blending strategies are suggested for minimizing sooting propensity and decreasing fuel costs.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0kjtm-0vj74The Lewis number under supercritical conditions
https://resolver.caltech.edu/CaltechAUTHORS:20171024-144744226
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 1999
DOI: 10.1016/S0017-9310(98)00230-0
An effective Lewis number is calculated for situations where temperature and mass fraction gradients are very large by defining effective thermal and mass diffusivities; such situations may occur in systems where there is more than one chemical component, and in particular under supercritical conditions. The definitions evolve from a model assuming that derivatives of certain functions are small with respect to those of the dependent variables. In the model, Soret and Dufour effects are included and Shvab–Zeldovich-like variables are defined to remove the coupling between the operators of the differential equations for temperature and mass fractions. Results from calculations using binary systems of chemical components, using both isolated fluid drops and interacting fluid drops, show that under supercritical conditions, depending upon the compounds, the effective Lewis number can be 2–40 times larger than the traditionally calculated Lewis number and that the spatial variation of the two numbers is different. For the values of the thermal diffusion factor used in the calculations, the Soret and Dufour effects are negligible; the discrepancy between the traditional and effective Lewis numbers is due to the combined effect of the small mass diffusion factor and the difference between the specific enthalpies of the two compounds. Parametric variations show that the effective Lewis number increases with increasing pressure and decreasing surrounding gas temperature. Closer drop proximity in clusters results in sharper peaks in the effective Lewis number due to the increased gradients of the dependent variables.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3cfbd-fan30An all-pressure fluid drop model applied to a binary mixture: heptane in nitrogen
https://resolver.caltech.edu/CaltechAUTHORS:20171019-111417758
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2000
DOI: 10.1016/S0301-9322(99)00108-1
The differences between subcritical liquid drop and supercritical fluid drop behavior are shown to be a direct consequence of the length scales near the fluid drop boundary. Under subcritical, evaporative high emission rate conditions, a film layer is present in the inner part of the drop surface which contributes to the unique determination of the boundary conditions; it is this film layer in conjunction with evaporation which gives to the solution its convective–diffusive character. In contrast, under supercritical conditions the boundary conditions contain a degree of arbitrariness due to the absence of a physical surface, and the solution has then a purely diffusive character. Results from simulations of a free fluid drop under no-gravity conditions are compared to microgravity experimental data from suspended, large drop experiments at high, low and intermediary temperatures and in a range of pressures encompassing the sub- and supercritical regime. Despite the difference between the conditions of the simulations and the experiments, the time rate of variation of the drop diameter square is remarkably well predicted in the linear curve regime. Consistent with the optical measurements, in the simulations the drop diameter is determined from the location of the maximum density gradient. Detailed time-wise comparisons between simulations and data show that this location is very well predicted at 0.1 MPa. As the pressure increases, the data and simulations agreement becomes good to fair, and the possible reasons for this discrepancy are discussed. Simulations are further conducted for a small drop, such as that encountered in practical applications, over a wide range of specified, constant far field pressures. Additionally, a transient pressure simulation crossing the critical point is also conducted. Results from these simulations are analyzed and major differences between the sub- and supercritical behavior are explained. In particular, it is shown that the classical calculation of the Lewis number gives erroneous results at supercritical conditions, and that an effective Lewis number previously defined gives correct estimates of the length scales for heat and mass transfer at all pressures.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/qkc2w-16m17The D^2 variation for isolated LOX drops and polydisperse clusters in hydrogen at high temperature and pressures
https://resolver.caltech.edu/CaltechAUTHORS:20171019-111811362
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2001
DOI: 10.1016/S0010-2180(00)00217-0
A study of the d^2 variation for isolated fluid drops and for fluid drops belonging to polydisperse clusters has been conducted at a high temperature and elevated pressures. The mathematical formulation is based on a previously validated model of subcritical/supercritical isolated fluid drop behavior. Coupled with the isolated drop equations, a set of conservation equations has been developed to describe the global cluster behavior. All these equations are based on the general transport matrix including Soret and Dufour terms and they are consistent with nonequilibrium thermodynamics and at low pressure with kinetic theory. Moreover, the model also accounts for real gas effects through accurate equations of state and for correct values of the transport properties in the high pressure, high temperature regime. The model has been first exercised for isolated LOX drops in H_2 at pressures ranging from 1.5 MPa (subcritical pressure for O_2) to 20 MPa (supercritical pressure for O_2). The results show that while at subcritical pressures the d^2 variation is nearly linear, with increasing pressure it departs considerably from the linear behavior; the largest departure occurs in the vicinity of the oxygen critical point. The slope of d^2(t) was fitted using both a constant and a linear fit, and it was shown that the linear fit provides a better alternative for correlation purposes. Simulations were also conducted for clusters of LOX drops in H_2 in the range 6 to 40 MPa (reduced pressures of 1.2–8 with respect to pure O_2). Parametric studies of the effect of the thermal diffusion factor value reveal that it is minor at 10 MPa and moderate at 40 MPa, and that although the Soret term is dominated by the Fick, Dufour, and Fourier terms, it is not negligible. The influence of a cluster Nusselt number is also shown to be relatively small in the range 10^3 to 10^4, consistent with the supercritical behavior being essentially a diffusive one. All of the results show a nonlinear d^2 variation with curves having a positive curvature independent of the values of the thermal diffusion factor, the Nusselt number or the LOX/H_2 mass ratio. The approximation of a binary size cluster containing relatively a much larger number of small drops by a monodisperse cluster with a drop size based upon the surface average of the drops in the polydisperse cluster yields a good evaluation of the thermodynamic quantities in the interstitial drop region but an underestimate of the lifetime of the drops in the cluster.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/6f3gw-3te36Direct numerical simulations of supercritical fluid mixing layers applied to heptane–nitrogen
https://resolver.caltech.edu/CaltechAUTHORS:20171024-154826158
Authors: {'items': [{'id': 'Miller-R-S', 'name': {'family': 'Miller', 'given': 'Richard S.'}}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2001
DOI: 10.1017/S0022112001003895
Direct numerical simulations (DNS) are conducted of a model hydrocarbon–nitrogen mixing layer under supercritical conditions. The temporally developing mixing layer configuration is studied using heptane and nitrogen supercritical fluid streams at a pressure of 60 atm as a model system related to practical hydrocarbon-fuel/air systems. An entirely self-consistent cubic Peng–Robinson equation of state is used to describe all thermodynamic mixture variables, including the pressure, internal energy, enthalpy, heat capacity, and speed of sound along with additional terms associated with the generalized heat and mass transport vectors. The Peng–Robinson formulation is based on pure-species reference states accurate to better than 1% relative error through comparisons with highly accurate state equations over the range of variables used in this study (600 ⩽ T ⩽ 1100 K, 40 ⩽ p ⩽ 80 atm) and is augmented by an accurate curve fit to the internal energy so as not to require iterative solutions. The DNS results of two-dimensional and three-dimensional layers elucidate the unique thermodynamic and mixing features associated with supercritical conditions. Departures from the perfect gas and ideal mixture conditions are quantified by the compression factor and by the mass diffusion factor, both of which show reductions from the unity value. It is found that the qualitative aspects of the mixing layer may be different according to the specification of the thermal diffusion factors whose value is generally unknown, and the reason for this difference is identified by examining the second-order statistics: the constant Bearman–Kirkwood (BK) thermal diffusion factor excites fluctuations that the constant Irwing–Kirkwood (IK) one does not, and thus enhances overall mixing. Combined with the effect of the mass diffusion factor, constant positive large BK thermal diffusion factors retard diffusional mixing, whereas constant moderate IK factors tend to promote diffusional mixing. Constant positive BK thermal diffusion factors also tend to maintain density gradients, with resulting greater shear and vorticity. These conclusions about IK and BK thermal diffusion factors are species-pair dependent, and therefore are not necessarily universal. Increasing the temperature of the lower stream to approach that of the higher stream results in increased layer growth as measured by the momentum thickness. The three-dimensional mixing layer exhibits slow formation of turbulent small scales, and transition to turbulence does not occur even for a relatively long non-dimensional time when compared to a previous, atmospheric conditions study. The primary reason for this delay is the initial density stratification of the flow, while the formation of strong density gradient regions both in the braid and between-the-braid planes may constitute a secondary reason for the hindering of transition through damping of emerging turbulent eddies.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nmxjw-wa012Evaluation of commonly used assumptions for isolated and cluster heptane drops in nitrogen at all pressures
https://resolver.caltech.edu/CaltechAUTHORS:20171019-111028665
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2001
DOI: 10.1016/S0010-2180(01)00292-9
A study is performed to assess commonly used assumptions in the modeling of drop behavior in moderate to high temperature surroundings and at all pressures. The model employed for this evaluation has been previously validated for isolated drops by using microgravity data, and is very general: it contains Soret and Dufour effects, does not assume mass transfer quasi-steadiness at the drop boundary, or necessarily the existence of a drop surface (i.e., phase discontinuity). Moreover, the numerical simulations are performed with accurate equations of state and transport properties over a wide range of thermodynamic variables. Consistent with low pressure conditions, the drop boundary is identified a posteriori of the calculations with the location of the largest density change. Simulations are here performed for isolated drops, and for monodisperse as well as binary size drop clusters. The results show that at locations arbitrarily near the boundary, the drop does not reach the mixture critical point within the wide range of conditions investigated (far-field temperatures of 470–1000 K and pressures ranging from 0.1 to 5 MPa). However, the state arbitrarily near the boundary is closer to the critical condition for smaller drops in a cluster than for the larger drops. Evaluations of the effect of the relaxation time at the drop boundary show that quasi-steadiness of the mass transfer prevails for drops of radius as small as 2 × 10^(−3) cm. Finally, the diameter squared exhibits a linear time variation only at atmospheric pressure. At all other pressures investigated (1–5 MPa), the diameter squared displays a negative curvature with time which never becomes linear. In agreement with existing experimental data, the drop lifetime increases monotonically with pressure at low far field temperatures (470 K), but exhibits a maximum as a function of pressure at high temperatures (1000 K). On an appropriate scale, the slope of the diameter squared versus time is shown to be independent of the drop size at all pressures.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/gac0g-xj548Direct Numerical Simulations of O_2/H_2 Temporal Mixing Layers Under Supercritical Conditions
https://resolver.caltech.edu/CaltechAUTHORS:20171023-144446319
Authors: {'items': [{'id': "Okong'o-N-A", 'name': {'family': "Okong'o", 'given': 'Nora'}}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2002
DOI: 10.2514/2.1728
Direct numerical simulations of a supercritical oxygen/hydrogen temporal three-dimensional mixing layer are conducted to explore the features of high-pressure transitional mixing behavior. The conservation equations are
formulated according to fluctuation–dissipation theory and are coupled to a modified Peng–Robinson equation
of state. The boundary conditions are periodic in the streamwise and spanwise directions and of nonreflecting
outflow type in the cross-stream direction. Simulations are conducted with initial Reynolds numbers of 6 x 10^2
and 7.5 x 10^2, initial pressure of 100 atm, and temperatures of 400 K in the O_2 and 600 K in the H_2 stream. Each
simulation encompasses the rollup and pairing of four initial spanwise vortices into a single vortex. The layer eventually
exhibits distorted regions of high density-gradient-magnitude similar to the experimentally observed wisps
of fluid at the boundary of supercritical jets. Analysis of the data reveals that the higher-Reynolds-number layer
reaches transition, whereas the other one does not. The transitional layer is analyzed to elucidate its characteristics.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/7bwpj-twg20Statistical Model of Multicomponent-Fuel Drop Evaporation for Many-Drop Flow Simulations
https://resolver.caltech.edu/CaltechAUTHORS:20171024-145459492
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K. G.'}}, {'id': 'Le-Clercq-P-C', 'name': {'family': 'Le Clercq', 'given': 'P. C.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2003
DOI: 10.2514/2.1894
A statistical formulation is developed describing the composition in an evaporating multicomponent-fuel liquid
drop and in the gas phase surrounding it. When a complementary discrete-component model is used, it is shown
that, when drops are immersed in a carrier gas containing fuel vapor, condensation of species onto the drop results in
the development of a minor peak in the liquid composition probability distribution function (PDF). This peak leads
to a PDF shape that can be viewed as a combination of two gamma PDFs, which is determined by five parameters.
A model is developed for calculating the parameters of the two combined gamma PDFs. Extensive tests of the
model for both diesel and gasoline show that the PDF results replicate accurately the discrete model predictions.
Most important, the mean and variance of the composition at the drop surface are in excellent agreement with
the discrete model. Results from the model show that although the second peak is minor for the liquid PDF, its
corresponding peak for the vapor distribution at the drop surface has a comparable magnitude to and sometimes
exceeds that corresponding to the first peak. Four-parameter models are also exercised, and it is shown that they are unable to capture the physics of the problem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/j16f6-6n993High-Pressure Binary Mass Diffusion Coefficients for Combustion Applications
https://resolver.caltech.edu/CaltechAUTHORS:20171019-155454998
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2004
DOI: 10.1021/ie0304558
A scaling for binary mass diffusion coefficients is developed using a corresponding states expression based on kinetic theory. The scaling is used to form nondimensional diffusion coefficients. Available data for high-pressure binary mass diffusion coefficients related to combustion applications are processed in conjunction with the scaling, leading to recommended scaled coefficient fits as a function of the reduced density. Data uncertainties and possible interpretation difficulties are examined. A means for comprehensive diffusion coefficient modeling over a broad (3 orders of magnitude) pressure range is suggested.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4k6k9-czh57Mixing rules for multicomponent mixture mass diffusion coefficients and thermal diffusion factors
https://resolver.caltech.edu/CaltechAUTHORS:20171023-131503487
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K. G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2004
DOI: 10.1063/1.1650296
Mixing rules are derived for mass diffusion coefficient and thermal diffusion factor matrices by developing compatibility conditions between the fluid mixture equations obtained from nonequilibrium thermodynamics and Grad's 13-moment kinetic theory. The mixing rules are shown to be in terms of the species mole fractions and binary processes. In particular, the thermal diffusion factors for binary mixtures obtained by the Chapman–Enskog expansion procedure are suitably generalized for many-component mixtures. Some practical aspects of the results are discussed including the utilization of these mixing rules for high pressure situations.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/jnfhn-hje51Modeling evaporation of Jet A, JP-7, and RP-1 drops at 1 to 15 bars
https://resolver.caltech.edu/CaltechAUTHORS:20171019-101642932
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2004
DOI: 10.1016/j.combustflame.2004.01.012
A model describing the evaporation of an isolated drop of a multicomponent fuel containing hundreds of species has been developed. The model is based on continuous thermodynamics concepts wherein the composition of a fuel is statistically described using a probability distribution function (PDF). Following previous studies, this PDF is parametrized on the species molar weight. However, unlike in previous studies, a unified formulation is developed wherein the same PDF holds for three major homologous hydrocarbon classes. The new PDF is a double-Gamma-PDF that is parametrized on the square root of the molar weight. The additional advantage of the formulation is that it is valid in the subcritical region from 1 to 15 bars. Discrete species distributions for Jet A, JP-7, and RP-1 are fitted using this novel PDF and extensive calculations for isolated drops of these kerosenes are performed. The results show that under the quasi-steady gas phase assumption, the D^2 law is recovered after an initial transient. The evaporation constant is an increasing function of the far field temperature and pressure and a complex function of far field composition according to the values of the far field temperature and pressure. The difference between the surface and the far field vapor molar fraction is nearly independent of the far field pressure. The composition of the vapor at the drop surface is kerosene-fuel specific. A comparison between results obtained with a model assuming the drop interior to be well mixed and a model wherein the drop may evaporate either in a well-mixed mode or at unchanging composition shows that the percentage difference between the evaporation constant predicted by the two models is within the range of uncertainty in the transport properties.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/y5x64-rfj52Modeling of multicomponent homogeneous nucleation using continuous thermodynamics
https://resolver.caltech.edu/CaltechAUTHORS:20171019-103222230
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2004
DOI: 10.1016/j.combustflame.2004.08.012
A theory of homogeneous nucleation in a multicomponent vapor is developed by combining classic nucleation and continuous thermodynamics concepts. The perfect gas equation of state is used in conjunction with this theory to obtain a model valid at low vapor pressures. The theory is applied to kerosenes used for fuels in aeronautics (Jet A, JP-7, and RP-1) at temperatures from 220 to 360 K and at vapor pressures up to 1 bar. The results show that although the overall nucleation trends regarding the dependency on vapor pressure and temperature are similar for all kerosenes, Jet A has a distinct behavior compared with JP-7 and RP-1. For all kerosenes, as pressure increases, the nucleus size rapidly decreases and is smallest at low temperatures.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/xfhmk-y2136Global analysis and parametric dependencies for potential unintended hydrogen-fuel releases
https://resolver.caltech.edu/CaltechAUTHORS:20171019-095150347
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2006
DOI: 10.1016/j.combustflame.2005.07.005
Global, simplified analyses of gaseous-hydrogen releases from a high-pressure vessel and liquid-hydrogen pools are conducted for two purposes: (1) establishing order-of-magnitude values of characteristic times and (2) determining parametric dependencies of these characteristic times on the physical properties of the configuration and on the thermophysical properties of hydrogen. According to the ratio of the characteristic release time to the characteristic mixing time, two limiting configurations are identified: (1) a rich cloud exists when this ratio is much smaller than unity, and (2) a jet exists when this ratio is much larger than unity. In all cases, it is found that the characteristic release time is proportional to the total released mass and inversely proportional to a characteristic area. The approximate size, convection velocity, and circulation time of unconfined burning-cloud releases scale with the cloud mass at powers 1/3, 1/6, and 1/6, respectively, multiplied by an appropriately dimensional constant; the influence of cross flow can only be important if its velocity exceeds that of internal convection. It is found that the fireball lifetime is approximately the maximum of the release time and thrice the convection-associated characteristic time. Transition from deflagration to detonation can occur only if the size of unconfined clouds exceeds by a factor of O(10) that of a characteristic detonation cell, which ranges from 0.015 m under stoichiometric conditions to approximately 1 m under extreme rich/lean conditions. For confined vapor pockets, transition occurs only for pocket sizes larger than the cell size. In jets, the release time is inversely proportional to the initial vessel pressure and has a square root dependence on the vessel temperature. Jet velocities are a factor of 10 larger than convective velocities in fireballs and combustion is possible only in the subsonic, downstream region where entrainment may occur.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/fve1b-2y226On possible release of microbe-containing particulates from a Mars lander spacecraft
https://resolver.caltech.edu/CaltechAUTHORS:20171019-101258515
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2006
DOI: 10.1016/j.pss.2005.12.007
Due to possible planet contamination, before Earth-departure, Mars landers and/or rovers are subject to strict requirements on the maximum number of attached spores or particles that carry viable microbes. Estimates of the release rates of these particles on Mars are made considering the three mechanisms of wind shear, collision with suspended dust, and collision with saltating sand particles. The first mechanism is found to apply only to particles of size greater than 10μm, the second mechanism has a characteristic particle adhesion half life that is so long as to be of no concern, and the third mechanism is deemed of possible importance, vitally depending on attached particle size and detailed surface characteristics of sand and spacecraft. While not investigated in detail, dust devils are shown to be possible contributors to release of microbe-containing particles.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/h168x-cy007Novel Subgrid Modeling of the LES Equations Under Supercritical Pressure
https://resolver.caltech.edu/CaltechAUTHORS:20200506-071823327
Authors: {'items': [{'id': 'Selle-L-C', 'name': {'family': 'Selle', 'given': 'Laurent C.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}]}
Year: 2007
DOI: 10.2514/6.2007-568
Transitional states obtained from Direct Numerical Simulation (DNS) of a supercritical mixing layer are analyzed for studying small-scale behavior and assessing the ability of Subgrid Scale (SGS) models to duplicate that behavior. Initially, the mixing layer contains a single chemical species in each of the two streams, and a perturbation promotes rollup and a double pairing of the four spanwise vortices initially present. The database encompasses three combinations of chemical species, several perturbation wavelengths and amplitudes, and several initial Reynolds numbers specifically chosen for the sole purpose of achieving transition. The Large Eddy Simulation (LES) equations are derived from the DNS ones through filtering. This filtering leads to two types of additional terms in the LES compared to the DNS equations : SGS fluxes and other terms for which either assumptions or models are necessary. The magnitude of all terms in the LES conservation equations is analyzed on the DNS database, with special attention to terms that could possibly be neglected. It is shown that in contrast to atmospheric-pressure gaseous flows, there are two new terms that must be modeled: one in each of the momentum and the energy equations. Discussed is a model for the momentum-equation additional term. This model performs well at small filter size but deteriorates as the filter size increases, highlighting the necessity of ensuring appropriate grid resolution in LES.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/dc26g-0ct41A New Method in Modeling and Simulations of Complex Oxidation Chemistry
https://resolver.caltech.edu/CaltechAUTHORS:20200505-073936057
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2007
DOI: 10.2514/6.2007-1433
A simplified model is proposed for the kinetics of alkane oxidation in air, based on a decomposition of heavy (carbon number ≥3) hydrocarbons into a 13 constituent radical base. The behavior of this base is examined in test computations for heptane utilizing Chemkin II with LLNL data inputs. Emphasis is placed on prediction of the heat release and temperature evolution. At stoichiometric conditions, the total constituent molar density was found to follow a quasi-steady rate which is a simplification in the modeling of its reaction rate.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/sf8zg-8g646Modelling of subgrid-scale phenomena in supercritical transitional mixing layers: an a priori study
https://resolver.caltech.edu/CaltechAUTHORS:SELjfm07
Authors: {'items': [{'id': 'Selle-L-C', 'name': {'family': 'Selle', 'given': 'Laurant C.'}}, {'id': "Okong'o-N-A", 'name': {'family': "Okong'o", 'given': 'Nora A.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}]}
Year: 2007
DOI: 10.1017/S0022112007008075
A database of transitional direct numerical simulation (DNS) realizations of a supercritical mixing layer is analysed for understanding small-scale behaviour and examining subgrid-scale (SGS) models duplicating that behaviour. Initially, the mixing layer contains a single chemical species in each of the two streams, and a perturbation promotes roll-up and a double pairing of the four spanwise vortices initially present. The database encompasses three combinations of chemical species, several perturbation wavelengths and amplitudes, and several initial Reynolds numbers specifically chosen for the sole purpose of achieving transition. The DNS equations are the Navier-Stokes, total energy and species equations coupled to a real-gas equation of state; the fluxes of species and heat include the Soret and Dufour effects. The large-eddy simulation (LES) equations are derived from the DNS ones through filtering. Compared to the DNS equations, two types of additional terms are identified in the LES equations: SGS fluxes and other terms for which either assumptions or models are necessary. The magnitude of all terms in the LES conservation equations is analysed on the DNS database, with special attention to terms that could possibly be neglected. It is shown that in contrast to atmospheric-pressure gaseous flows, there are two new terms that must be modelled: one in each of the momentum and the energy equations. These new terms can be thought to result from the filtering of the nonlinear equation of state, and are associated with regions of high density-gradient magnitude both found in DNS and observed experimentally in fully turbulent high-pressure flows. A model is derived for the momentum-equation additional term that performs well at small filter size but deteriorates as the filter size increases, highlighting the necessity of ensuring appropriate grid resolution in LES. Modelling approaches for the energy-equation additional term are proposed, all of which may be too computationally intensive in LES. Several SGS flux models are tested on an a priori basis. The Smagorinsky (SM) model has a poor correlation with the data, while the gradient (GR) and scale-similarity (SS) models have high correlations. Calibrated model coefficients for the GR and SS models yield good agreement with the SGS fluxes, although statistically, the coefficients are not valid over all realizations. The GR model is also tested for the variances entering the calculation of the new terms in the momentum and energy equations; high correlations are obtained, although the calibrated coefficients are not statistically significant over the entire database at fixed filter size. As a manifestation of the small-scale supercritical mixing peculiarities, both scalar-dissipation visualizations and the scalar-dissipation probability density functions (PDF) are examined. The PDF is shown to exhibit minor peaks, with particular significance for those at larger scalar dissipation values than the mean, thus significantly departing from the Gaussian behaviour.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/kjrmf-9rg92Modeling of the Energy Equation for LES of Flows at Supercritical Pressure
https://resolver.caltech.edu/CaltechAUTHORS:20200330-131934118
Authors: {'items': [{'id': 'Selle-L-C', 'name': {'family': 'Selle', 'given': 'L. C.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'J.'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'K. G.'}}]}
Year: 2008
DOI: 10.2514/6.2008-948
A database of transitional Direct Numerical Simulation (DNS) realizations of a supercritical mixing layer is analyzed for understanding small-scale behavior and examining Subgrid Scale (SGS) models duplicating that behavior. Initially, the mixing layer contains a single chemical species in each of the two streams, and a perturbation promotes roll-up and a double pairing of the four spanwise vortices initially present. The database encompasses three combinations of chemical species, several perturbation wavelengths and amplitudes, and several initial Reynolds numbers specifically chosen for the sole purpose of achieving transition. The DNS equations are the Navier Stokes, total energy and species equations coupled to a real gas equation of state; the fluxes of species and heat include the Soret and Dufour effects. The Large Eddy Simulation (LES) equations are derived from the DNS ones through Altering. Compared to the DNS equations, two types of additional terms are identified in the LES equations: SGS fluxes and other terms for which either assumptions or models are necessary. The focus is here on the energy equation. The magnitude of all terms in this filtered DNS equation is analyzed on the DNS database, with special attention to terms that could possibly be neglected. It is shown that in contrast to atmospheric-pressure gaseous flows, there is a new term that must be modeled in this equation. This new term can be thought to result from the filtering of the strongly nonlinear equation of state, and is associated with high density-gradient magnitude regions both found in DNS and observed experimentally in fully-turbulent high-pressure flows. A priori modeling approaches for the energy-equation additional term are proposed, all of which must ultimately be tested in LES to show viability.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/kt3q6-frf90A Simplified Model of Alkane Oxidation
https://resolver.caltech.edu/CaltechAUTHORS:20200421-083907781
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2008
DOI: 10.2514/6.2008-975
A simplified model is proposed for the kinetics of alkane oxidation in air, based on a decomposition of heavy (carbon number greater or equal to 3) hydrocarbons into a 13 constituent radical base. The behavior of this base is examined in test computations for n-heptane utilizing Chemkin II with LLNL data inputs, placing emphasis on modeling to predict the heat release and temperature evolution. A normalized temperature was constructed which when used to plot the total constructed molar density divided by the product of the equivalence ratio and a nondimensional pressure, reveals a self-similar behavior of the plotted variable over a wide range of initial pressures and equivalence ratios. Examination of the LLNL kinetics shows that the total constituent molar density rate follows a quasi-steady behavior. This reaction rate was curve fitted along with the corresponding enthalpy production. The fits are shown against the normalized temperature for various equivalence ratios and initial nondimensional pressures and comparisons with the LLNL kinetics are very favorable. The model reduces the LLNL n-heptane mechanism from 160 species (progress variables) and 1540 reactions to 12 progress variables, 16 quasi-steady rates (associated with heavy species), 162 conventional reaction rates (light species) and 11 other functional forms. (i.e. fits for the mean heavy-species heat capacity at constant pressure, the enthalpy release rate of the heavy species, and the molar fraction of quasi-steady light species). The proposed kinetic mechanism is valid over a pressure range from atmospheric to 60 bar, temperatures from 600 K to 2500 K and equivalence ratios from 0.125 to 8. This range encompasses diesel, HCCI and gas turbine engines, including cold ignition; and NO_x, CO and soot pollutant formation in the lean and rich regimes, respectively.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/g9qg0-nf078Modeling of Alkane Oxidation using Constituents and Species
https://resolver.caltech.edu/CaltechAUTHORS:20200117-081103466
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2009
DOI: 10.2514/6.2009-1368
A chemical kinetics reduction model is proposed for alkane oxidation in air that is baaed on a parallel methodology to that used in turbulence modeling in the context of Large Eddy Simulation. The objective of kinetic modeling is to predict the heat release and temperature evolution. In an a priori step, a categorization of time scales is first conducted to identify scales that must be modeled and scales that must be computed using progress variables based on the model for the other scales. First, a decomposition of heavy (carbon number greater or equal to 3) hydrocarbons into constituents is proposed. Examination of results obtained using the LLNL heptane-oxidation database in conjunction with Chemkin II shows that (i) with appropriate scaling, the total constituent mole fraction behaves in a self-similar manner and the total constituent molar density rate follows a quasi-steady behavior, and (ii) the light species can be partitioned into two subsets according to whether
they are quasi-steady (nine species) or unsteady (11 species). The twelve progress variables represented by the total constituent molar density and the molar densities of the unsteady light species are defined to be a base from which the system's behavior can be reproduced. This is a dramatic reduction from the 160 species (progress variables) and 1540 reactions in the LLNL set to 12 progress variables, 16 quasi-steady rates (associated with heavy species), 162 conventional reaction rates (light species) and 11 other functional forms (i.e. fits for the mean heavy-species heat capacity at constant pressure, the enthalpy release rate of the heavy species, and the molar fraction of quasi-steady light species). A summary of the model is presented explaining the curve fits that constitute the model, namely (1) for the constituent molar density rate a long with the corresponding enthalpy production rate, (2) for the quasi-steady species mole fraction, and (3) for the contribution from the heavy species to the unsteady light species reaction rates. The proposed kinetic mechanism is valid over a pressure range from atmospheric to 60 bar, temperatures from 600 K to 2500 K and equivalence ratio1 from 0.125 to 8. This range encompasses diesel, HCCI and gas turbine engines, including cold ignition; and NO_x, CO and soot pollutant formation in the lean and rich regimes, respectively. Highlights of the a priori model results are illustrated for a variety of initial conditions. Results from a posteriori tests are shown in which the model predictions for the unsteady light species and the temperature are compared to the equivalent quantities baaed on the LLNL dataset.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4jve6-r0p63Alkane Kinetics Reduction Consistent with Turbulence Modeling using Large Eddy Simulation
https://resolver.caltech.edu/CaltechAUTHORS:20191016-111202608
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2010
DOI: 10.2514/6.2010-1514
A methodology for deriving a reduced kinetic mechanism for alkane oxidation is described, inspired by n-heptane oxidation. The model is based on partitioning the species of the skeletal kinetic mechanism into lights, defined as those having a carbon number smaller than 3, and heavies, which are the complement of the species ensemble. For modeling purposes, the heavy species are mathematically decomposed into constituents, which are similar but not identical to groups in the group additivity theory. From analysis of the n-heptane LLNL skeletal mechanism in conjunction with CHEMKIN II, it is shown that a similarity variable can be formed such that the appropriately non-dimensionalized global constituent molar density exhibits a self-similar behavior over a very wide range of equivalence ratios, initial pressures and initial temperatures that is of interest for predicting n-heptane oxidation. Furthermore, the oxygen and water molar densities are shown to display a quasi-linear behavior with respect to the similarity variable. The light species ensemble is partitioned into quasi-steady and unsteady species. The reduced model is based on concepts consistent with those of Large Eddy Simulation in which functional forms are used to replace the small scales eliminated through Altering of the governing equations; these small scales are unimportant as far as dynamic energy is concerned. Here, we remove the scales deemed unimportant for recovering the thermodynamic energy. The concept is tested by using tabular information from the n-heptane LLNL skeletal mechanism in conjunction with CHEMKIN II utilized as surrogate ideal functions replacing the necessary functional forms. The test reveals that the similarity concept is indeed justified and that the combustion temperature is well predicted, but that the ignition time is overpredicted, which is traced to neglecting a detailed description of the processes lending to the heavies chemical decomposition. To palliate this deficiency, functional modeling is incorporated into our conceptual reduction. This functional modeling includes the global constituent molar density, the enthalpy evolution of the heavies, the contribution to the reaction rate of the unsteady lights from other light species and from the heavies, the molar density evolution of oxygen and water, and the mole &actions of the quasi-steady light species. The model is compact in that there are only nine species-related progress variables. Results are presented showing the performance of the model for predicting the temperature and species evolution for n-heptane. The model reproduces the ignition time over a wide range of equivalence ratios, initial pressure and initial temperature. Preliminary results for iso-octane using the full mechanism are also presented, showing encouragingly that the concept may be generalized to other alkanes. The utility of the model and possible improvements are discussed.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/31y6z-srt68A model of reduced kinetics for alkane oxidation using constituents and species: Proof of concept for n-heptane
https://resolver.caltech.edu/CaltechAUTHORS:20171019-095909602
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2010
DOI: 10.1016/j.combustflame.2010.02.013
A methodology for deriving a reduced kinetic mechanism for alkane oxidation is described and applied to n-heptane. The model is based on partitioning the species of the skeletal kinetic mechanism into lights, defined as those having a carbon number smaller than 3, and heavies, which are the complement in the species ensemble. For modeling purposes, the heavy species are mathematically decomposed into constituents, which are similar but not identical to groups in the group additivity theory. From analysis of the LLNL skeletal mechanism in conjunction with CHEMKIN II, it is shown that a similarity variable can be formed such that the appropriately scaled global constituent molar density exhibits a self-similar behavior over a very wide range of equivalence ratios, initial pressures and initial temperatures that is of interest for predicting n-heptane oxidation. Furthermore, the oxygen and water molar densities are shown to display a quasi-linear behavior with respect to the similarity variable. The light species ensemble is partitioned into quasi-steady and unsteady species. The concept is tested by using tabular information from the LLNL skeletal mechanism in conjunction with CHEMKIN II. The test reveals that the similarity concept is indeed justified and that the combustion temperature is well predicted, but that the ignition time is overpredicted. To palliate this deficiency, functional modeling is incorporated into our conceptual reduction. Due to the reduction process, models are also included for the global constituent molar density, the kinetics-induced enthalpy evolution of the heavy species, the contribution to the reaction rate of the unsteady lights from the heavies, the molar density evolution of oxygen and water, the mole fractions of the quasi-steady light species and the mean molar heat capacity of the heavy species. The model is compact in that there are only nine species-related progress variables. Results are presented comparing the performance of the model for predicting the temperature and species evolution with that of the skeletal mechanism. The model reproduces the ignition time over a wide range of equivalence ratios, initial pressure and initial temperature.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/76cjw-5zz26A model of reduced oxidation kinetics using constituents and species: Iso-octane and its mixtures with n-pentane, iso-hexane and n-heptane
https://resolver.caltech.edu/CaltechAUTHORS:20171019-095549725
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2010
DOI: 10.1016/j.combustflame.2010.06.010
A previously described methodology for deriving a reduced kinetic mechanism for alkane oxidation and tested for n-heptane is here shown to be valid, in a slightly modified version, for iso-octane and its mixtures with n-pentane, iso-hexane and n-heptane. The model is still based on partitioning the species into lights, defined as those having a carbon number smaller than 3, and heavies, which are the complement in the species ensemble, and mathematically decomposing the heavy species into constituents which are radicals. For the same similarity variable found from examining the n-heptane LLNL mechanism in conjunction with CHEMKIN II, the appropriately scaled total constituent molar density still exhibits a self-similar behavior over a very wide range of equivalence ratios, initial pressures and initial temperatures in the cold ignition regime. When extended to larger initial temperatures than for cold ignition, the self-similar behavior becomes initial temperature dependent, which indicates that rather than using functional fits for the enthalpy generation due to the heavy species' oxidation, an ideal model based on tabular information extracted from the complete LLNL kinetics should be used instead. Similarly to n-heptane, the oxygen and water molar densities are shown to display a quasi-linear behavior with respect to the similarity variable, but here their slope variation is no longer fitted and instead, their rate equations are used with the ideal model to calculate them. As in the original model, the light species ensemble is partitioned into quasi-steady and unsteady species; the quasi-steady light species mole fractions are computed using the ideal model and the unsteady species are calculated as progress variables using rates extracted from the ideal model. Results are presented comparing the performance of the model with that of the LLNL mechanism using CHEMKIN II. The model reproduces excellently the temperature and species evolution versus time or versus the similarity variable, with the exception of very rich mixtures, where the predictions are still very good but the multivalued aspect of these functions at the end of oxidation is not captured in the reduction. The ignition time is predicted within percentages of the LLNL values over a wide range of equivalence ratios, initial pressures and initial temperatures.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/m34nn-p8y20Computation of Laminar Premixed Flames Using Reduced Kinetics Based on Constituents and Species
https://resolver.caltech.edu/CaltechAUTHORS:20190930-110459291
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2011
DOI: 10.2514/6.2011-415
A model is proposed for quasi-one-dimensional steady flame development in the configuration of an inviscid, premixed fuel jet injected into air. The governing equations are written within the framework of a reduced kinetic model based on constituents and species. The reduced kinetic model, previously exercised in a constant-volume perfectly-stirred reactor mode, has been successful at predicting ignition and combustion product and temperature evolution for n-heptane, iso-octane, PRF fuel combinations, and mixtures of iso-octane with either n-pentane or iso-hexane. The differential governing equations have the option of an axially variable area and they are coupled with a real gas equation of state. The flame development model accounts for a full diffusion matrix, and thermal conductivity computed for the species mixture. Preliminary results are presented.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/k5cjx-eej12Modeling of Steady Laminar Flames for One-dimensional Premixed Jets of Heptane/Air and Octane/Air Mixtures
https://resolver.caltech.edu/CaltechAUTHORS:20190828-102318692
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2012
DOI: 10.2514/6.2012-340
A model is proposed for quasi-one-dimensional steady flame development in the configuration of an inviscid, premixed fuel jet injected into air. The governing equations are written within the framework of a reduced kinetic model based on constituents and species. The reduced kinetic model, previously exercised in a constant-volume perfectlystirred reactor mode, has been successful at predicting ignition and combustion product and temperature evolution for n-heptane, iso-octane, PRF fuel combinations, and mixtures of iso-octane with either n-pentane or iso-hexane. The differential governing equations have the option of an axially variable area and they are coupled with a real gas equation of state. The flame development model accounts for a full diffusion matrix, and thermal conductivity computed for the species mixture. Results from four simulations at various conditions are presented.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/7b2pw-t5720Direct Numerical Simulation of High-Pressure Multispecies Turbulent Mixing in the Cold Ignition Regime
https://resolver.caltech.edu/CaltechAUTHORS:20190828-102318779
Authors: {'items': [{'id': 'Masi-E', 'name': {'family': 'Masi', 'given': 'Enrica'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}]}
Year: 2012
DOI: 10.2514/6.2012-351
A model is proposed for describing mixing of several species under high pressure conditions that relies on a previously proposed model based on governing equations for multispecies mixing that has so far only been exercised for two-species mixing. For the two-species mixing simulations, transport properties were computed from correlated Schmidt (Sc) and Prandtl (Pr) numbers, accurately calculated as functions of the thermodynamic variables, and from a specified Reynolds number value from which an adjusted viscosity value was calculated so as to enable Direct Numerical Simulation (DNS). One of the novelties of the present study is the modeling and computation of multispecies mixing based on a full mass-diffusion matrix, a full thermal-diffusion-factor matrix necessary to include Soret and Dufour effects, and thermal conductivity computed for the species mixture. The scaling of the viscosity necessary for conducting DNS induces a scaling of the other transport properties that respects the accurate values of the Sc numbers and of the Pr number. Computations are performed with five species in the configuration of a temporal mixing layer and the effect of transport properties on species mixing and layer development are analyzed and discussed.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/y01wa-vt468Pressure Effects from Direct Numerical Simulation of High-Pressure Multispecies Mixing
https://resolver.caltech.edu/CaltechAUTHORS:20190826-092411849
Authors: {'items': [{'id': 'Masi-E', 'name': {'family': 'Masi', 'given': 'Enrica'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}]}
Year: 2013
DOI: 10.2514/6.2013-711
The focus of this study is the understanding of effects of pressure increase or Reynolds number increase in supercritical-pressure flows. To this effect, Direct Numerical Simulations are conducted for supercritical-pressure flows in which five species undergo mixing. The computation of multispecies mixing is based on a full mass-diffusion matrix, a full thermal-diffusion-factor matrix necessary to include Soret and Dufour effects, and both viscosity and thermal conductivity computed for the species mixture. The scaling of the physical viscosity, necessary for conducting DNS, induces a scaling of the other transport properties that respects the accurate values of the Schmidt (Sc) numbers and of the Prandtl (Pr) number. Computations are performed in the configuration of a temporal mixing layer and the results are analyzed to reveal the separate effect of pressure or Reynolds number increase on the flow. The analysis consists of examining vortical aspects of the flow, the fluxes and relevant thermodynamic properties. It is found that a larger pressure has an opposite effect to a larger Reynolds number, mainly by increasing the fluid density and making it more difficult to entrain and mix.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/esqwj-88k62Modeling of Steady High-Pressure Laminar Premixed Flames of n-Heptane and Iso-Octane
https://resolver.caltech.edu/CaltechAUTHORS:20190826-092411436
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2013
DOI: 10.2514/6.2013-1168
A model is proposed for quasi-one-dimensional steady flame development in the configuration of an inviscid, premixed fuel jet injected into air. The governing equations are written within the framework of a reduced kinetic model based on constituents and species. The reduced kinetic model, previously exercised in a constant-pressure perfectlystirred reactor mode, has been successful at predicting ignition and combustion product and temperature evolution for n-heptane, iso-octane, PRF fuel combinations, and mixtures of iso-octane with either n-pentane or iso-hexane. The differential governing equations are coupled with a real gas equation of state. The flame development model accounts for a full diffusion matrix, and thermal conductivity computed for the species mixture. Results are presented for both n-heptane and iso-octane at stoichiometric conditions over the pressure range of 1 - 40 bar. Additionally, an equivalence ratio study is conducted for iso-octane at a pressure of 40 bar.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ze2ws-2gd52Multi-species turbulent mixing under supercritical-pressure conditions: modelling, direct numerical simulation and analysis revealing species spinodal decomposition
https://resolver.caltech.edu/CaltechAUTHORS:20130627-094307167
Authors: {'items': [{'id': 'Masi-E', 'name': {'family': 'Masi', 'given': 'Enrica'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth G.'}}, {'id': "Okong'o-N-A", 'name': {'family': "Okong'o", 'given': 'Nora A.'}}]}
Year: 2013
DOI: 10.1017/jfm.2013.70
A model is developed for describing mixing of several species under high-pressure
conditions. The model includes the Peng–Robinson equation of state, a full massdiffusion
matrix, a full thermal-diffusion-factor matrix necessary to incorporate the
Soret and Dufour effects and both thermal conductivity and viscosity computed for
the species mixture using mixing rules. Direct numerical simulations (DNSs) are
conducted in a temporal mixing layer configuration. The initial mean flow is perturbed
using an analytical perturbation which is consistent with the definition of vorticity
and is divergence free. Simulations are performed for a set of five species relevant
to hydrocarbon combustion and an ensemble of realizations is created to explore the
effect of the initial Reynolds number and of the initial pressure. Each simulation
reaches a transitional state having turbulent characteristics and most of the data
analysis is performed on that state. A mathematical reformulation of the flux terms
in the conservation equations allows the definition of effective species-specific Schmidt
numbers (Sc) and of an effective Prandtl number (Pr) based on effective speciesspecific
diffusivities and an effective thermal conductivity, respectively. Because these
effective species-specific diffusivities and the effective thermal conductivity are not
directly computable from the DNS solution, we develop models for both of these
quantities that prove very accurate when compared with the DNS database. For two
of the five species, values of the effective species-specific diffusivities are negative
at some locations indicating that these species experience spinodal decomposition; we
determine the necessary and sufficient condition for spinodal decomposition to occur.
We also show that flows displaying spinodal decomposition have enhanced vortical
characteristics and trace this aspect to the specific features of high-density-gradient
magnitude regions formed in the flows. The largest values of the effective speciesspecific
Sc numbers can be well in excess of those known for gases but almost
two orders of magnitude smaller than those of liquids at atmospheric pressure. The
effective thermal conductivity also exhibits negative values at some locations and the
effective Pr displays values that can be as high as those of a liquid refrigerant.
Examination of the equivalence ratio indicates that the stoichiometric region is thin
and coincides with regions where the mixture effective species-specific Lewis number
values are well in excess of unity. Very lean and very rich regions coexist in the
vicinity of the stoichiometric region. Analysis of the dissipation indicates that it is dominated by mass diffusion, with viscous dissipation being the smallest among the
three dissipation modes. The sum of the heat and species (i.e. scalar) dissipation is
functionally modelled using the effective species-specific diffusivities and the effective
thermal conductivity. Computations of the modelled sum employing the modelled
effective species-specific diffusivities and the modelled effective thermal conductivity
shows that it accurately replicates the exact equivalent dissipation.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/greny-kgs28Prediction of premixed, n-heptane and iso-octane unopposed jet flames using a reduced kinetic model based on constituents and light species
https://resolver.caltech.edu/CaltechAUTHORS:20131003-081047378
Authors: {'items': [{'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}]}
Year: 2013
DOI: 10.1016/j.combustflame.2013.06.005
A model of steady, quasi one-dimensional premixed laminar jet flame developing unopposed into a uniform flow has been formulated using a previously successful reduced chemical-kinetics model [10] and [11]. A detailed derivation of the steady quasi one-dimensional conservation equations revealed that it is only under very restrictive conditions – probably very difficult to achieve experimentally and the validity of which is not reported in detail in experimental studies – that the quasi one-dimensional concept is meaningful. The governing equations have been mathematically manipulated to be consistent with the framework of the reduced chemical-kinetics model which relied on constituents representing the heavy species, and on quasi-steady light species and unsteady light species. The flame model includes accurate transport property calculation for high-pressure conditions and a real-gas equation of state. Based on a found self-similarity [10] and [11] which deteriorates at increasingly rich conditions, the chemistry model consists of tables of kinetic rates, quasi-steady species molar fractions and the heavy species mean molar mass extracted from the LLNL model in the framework of the reduced kinetics. The progress variables are only the mass fractions of the unsteady light species and the temperature. The values of the dependent variables are specified at the inflow location and null gradients are specified at the outflow. Simulations were performed for both n-heptane and iso-octane air oxidation over a wide range of pressures and equivalence ratios. The limited documentation of experimental conditions not specifying the inflow velocity (or flux) made it impossible to use this data for detailed comparison. In the one case where the inflow velocity was available for a burner experiment, those conditions were adopted for the simulation and the configuration was changed to a constant-area jet to approach the burner configuration. Results from this simulation compared favorably with the data, considering the different configurations. Results from parametric studies not associated with experimental data showed that at stoichiometric conditions the flame temperature, flame velocity and strain rate are not sensitive to the pressure, although flames become increasingly thinner with increasing pressure and the yield of the unsteady light species is different. Computations conducted at 40 bar for various equivalence ratios and for velocities differing with the equivalence ratio showed that the maximum flame velocity, flame strain and flame temperature were obtained at stoichometric conditions. Finally, we discuss the limitations of utilizing a priori obtained reduced chemical-kinetic models in flames calculations.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/m1aka-tvq58Dimensionality Reduction Using a Dominant Dynamic Variable, Self Similarity and Data Tabulation: Application to Hydrocarbon Oxidation
https://resolver.caltech.edu/CaltechAUTHORS:20190819-150530040
Authors: {'items': [{'id': 'Kourdis-P-D', 'name': {'family': 'Kourdis', 'given': 'Panagiotis D.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}]}
Year: 2014
DOI: 10.2514/6.2014-0819
A dimensionality reduction method is developed for autonomous dynamical systems exploiting the local (near) self similarity due to the presence of a dominant dynamic variable. The method is coupled with a simple tabulation scheme to take advantage of the computationally more efficient local partial self similarity. The proposed methodology is used to construct reduced kinetics models of hydrocarbon oxidation and is tested for a n-dodecane/air mixture.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0gwhe-4d922Towards direct simulations of counterflow flames with consistent differential-algebraic boundary conditions
https://resolver.caltech.edu/CaltechAUTHORS:20190816-144341538
Authors: {'items': [{'id': 'Kourdis-P-D', 'name': {'family': 'Kourdis', 'given': 'Panagiotis D.'}}, {'id': 'Bellan-J', 'name': {'family': 'Bellan', 'given': 'Josette'}, 'orcid': '0000-0001-9218-7017'}, {'id': 'Harstad-K-G', 'name': {'family': 'Harstad', 'given': 'Kenneth'}}]}
Year: 2015
DOI: 10.2514/6.2015-1383
A new approach for the formulation of boundary conditions for the counterflow configuration is presented. Upon discretization of the steady-state Navier-Stokes equations at the inflow boundaries, numerically algebraic equations are imposed as boundary conditions, while upon discretization of the unsteady Navier-Stokes equations at the outflow, differential boundaries result. It is demonstrated that the resulting numerical differential-algebraic boundary conditions are suitable to account for the multi-directional character of the flow at the boundaries of the counterflow configuration.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/dag14-qec60