17 results
Effects of thermophoresis on high-pressure binary-species boundary layers with uniform and non-uniform compositions
- Takahiko Toki, Josette Bellan
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- Journal:
- Journal of Fluid Mechanics / Volume 952 / 10 December 2022
- Published online by Cambridge University Press:
- 01 December 2022, A37
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Direct numerical simulations of high-pressure binary-species temporal boundary layers are performed to investigate the flow physics for three situations: (1) uniform and equal composition, (2) uniform but unequal compositions and (3) non-uniform composition. Both colder- and hotter-wall situations compared with the free stream are simulated. The working fluid is a nitrogen/methane mixture. The analysis is performed at a case-specific self-similar state. Even when the initial composition is uniform, the methane mean mass fraction decreases near the colder wall, whereas it increases near the hotter wall and the mass fraction fluctuates in the entire boundary layer. Analysis of the species-mass diffusion balance and flow structures reveal that both mass-fraction variation and fluctuations are induced by the Soret effect. When the initial composition is non-uniform and the wall is colder, the methane mean mass fraction monotonically increases from the wall akin to its initial profile. However, when the wall is hotter the mean mass fraction decreases near the wall in contrast to its initial profile, a fact traced through the species-mass diffusion balance to the Soret effect being large and enriching methane near the wall. In contrast, the direction of the Soret flux is opposite for the colder wall, thus keeping the methane concentration small. Although the initial magnitude of the difference between the wall and free-stream temperature is the same in all cases, the situation is not symmetric between colder- and hotter-wall cases; the flow structure exhibits much smaller scales when the wall is hotter than when the wall is colder.
Investigation of species-mass diffusion in binary-species boundary layers at high pressure using direct numerical simulations
- Takahiko Toki, Josette Bellan
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- Journal:
- Journal of Fluid Mechanics / Volume 928 / 10 December 2021
- Published online by Cambridge University Press:
- 06 October 2021, A18
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Direct numerical simulations of single-species and binary-species temporal boundary layers at high pressure are performed with special attention to species-mass diffusion. The working fluids are nitrogen or a mixture of nitrogen and methane. Mean profiles and turbulent fluctuations of mass fraction show that their qualitative characteristics are different from those of streamwise velocity and temperature, due to the different boundary conditions. In a wall-parallel plane near the wall, the streamwise velocity and temperature have streaky patterns and the fields are similar. However, the mass fraction field at the same location is different from the streamwise velocity and temperature fields indicating that species-mass diffusion is not similar to the momentum and thermal diffusion. In contrast, at the centre and near the edge of the boundary layer, the mass fraction and temperature fields have almost the same pattern, indicating that the similarity between thermal and species-mass diffusion holds away from the wall. The lack of similarity near the wall is traced to the Soret effect that induces a temperature-gradient-dependent species-mass flux. As a result, a new phenomenon has been identified for a non-isothermal binary-species system – uphill diffusion, which in its classical isothermal definition can only occur for three or more species. A quadrant analysis for the turbulent mass flux reveals that near the wall the Soret effect enhances the negative contributions of the quadrants. Due to the enhancement of the negative contributions, small species-concentration fluid tends to be trapped near the wall.
Investigation of high-pressure turbulent jets using direct numerical simulation
- Nek Sharan, Josette Bellan
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- Journal:
- Journal of Fluid Mechanics / Volume 922 / 10 September 2021
- Published online by Cambridge University Press:
- 13 July 2021, A24
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Direct numerical simulations of free round jets at a Reynolds number ($Re_{D}$) of $5000$, based on jet diameter ($D$) and jet-exit bulk velocity ($U_{e}$), are performed to study jet turbulence characteristics at supercritical pressures. The jet consists of nitrogen ($\mathrm {N}_{2}$) that is injected into $\mathrm {N}_{2}$ at the same temperature. To understand turbulent mixing, a passive scalar is transported with the flow at unity Schmidt number. Two sets of inflow conditions that model jets issuing from either a smooth contraction nozzle (laminar inflow) or a long pipe nozzle (turbulent inflow) are considered. By changing one parameter at a time, the simulations examine the jet-flow sensitivity to the thermodynamic condition (characterized in terms of the compressibility factor ($Z$) and the normalized isothermal compressibility), inflow condition and ambient pressure ($p_{\infty }$) spanning perfect- to real-gas conditions. The inflow affects flow statistics in the near field (containing the potential core closure and the transition region) as well as further downstream (containing fully developed flow with self-similar statistics) at both atmospheric and supercritical $p_{\infty }$. The sensitivity to inflow is larger in the transition region, where the laminar-inflow jets exhibit dominant coherent structures that produce higher mean strain rates and higher turbulent kinetic energy than in turbulent-inflow jets. Decreasing $Z$ at a fixed supercritical $p_{\infty }$ enhances pressure and density fluctuations (non-dimensionalized by local mean pressure and density, respectively), but the effect on velocity fluctuations depends also on the local flow dynamics. When $Z$ is reduced, large mean strain rates in the transition region of laminar-inflow jets significantly enhance velocity fluctuations (non-dimensionalized by local mean velocity) and scalar mixing, whereas the effects are minimal in jets from turbulent inflow.
Fluid density effects in supersonic jet-induced cratering in a granular bed on a planetary body having an atmosphere in the continuum regime
- Kaushik Balakrishnan, Josette Bellan
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- Journal:
- Journal of Fluid Mechanics / Volume 915 / 25 May 2021
- Published online by Cambridge University Press:
- 11 March 2021, A29
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To investigate the effect of atmospheric density and jet-fluid density during supersonic jet-induced cratering on a granular particle bed, large eddy simulation (LES) for the fluid phase coupled with a granular flow model based on kinetic theory is used. Coupling accounts for momentum and energy interaction between particles and fluid. Several simulations are conducted that enable discriminating between the crater characteristics which depend on the atmospheric (or jet) density, and those which depend on the ratio between jet-fluid density and atmospheric density. The crater cross-sectional shapes are controlled by the density ratio between jet fluid and atmosphere. For large such ratios, the craters have a parabolic cross-section shape, are relatively shallow, have large diameters and have subdued ejecta. In contrast, for ratios close to unity, the craters have a conical cross-section, are relatively deep, have relatively small diameters and have substantial ejecta; these craters also display ripples on their walls. The physics leading to these differences is explained. Considerations of particle momentum flux indicate that at same fraction of the crater depth and same radial distance from the jet axis, the radial and vertical components have larger magnitudes for conical craters than for parabolic craters. The spatial distribution of the ratio between the drag force and the force due to the particles resistance to compaction are compared and it is found to be dependent on the jet-to-atmospheric fluid density ratio. A few features of the crater, in particular its depth, only depend on the atmospheric (or jet) fluid density.
The influence of the chemical composition representation according to the number of species during mixing in high-pressure turbulent flows
- Luca Sciacovelli, Josette Bellan
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- Journal:
- Journal of Fluid Mechanics / Volume 863 / 25 March 2019
- Published online by Cambridge University Press:
- 24 January 2019, pp. 293-340
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Mixing of several species in high-pressure (high-$p$) turbulent flows is investigated to understand the influence of the number of species on the flow characteristics. Direct numerical simulations are conducted in the temporal mixing layer configuration at approximately the same value of the momentum ratio for all realizations. The simulations are performed with mixtures of two, three, five and seven species to address various compositions at fixed number of species, at three values of initial vorticity-thickness-based Reynolds number, $Re_{0}$, and two values of the free-stream pressure, $p_{0}$, which is supercritical for each species except water. The major species are C7H16, O2 and N2, and the minor species are CO, CO2, H2 and H2O. The extensive database thus obtained allows the study of the influence not only of $Re_{0}$ and $p_{0}$, but also of the initial density ratio and of the initial density difference between streams, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$. The results show that the layer growth is practically insensitive to all of the above parameters; however, global vortical aspects increase with $Re_{0},p_{0}$ and the number of species; nevertheless, at the same $Re_{0},p_{0}$ and density ratio, vorticity aspects are not influenced by the number of species. Species mixing produces strong density gradients which increase with $p_{0}$ and otherwise scale with $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ but, when scaled by $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, are not affected by the number of species. Generalized Korteweg-type equations are developed for a multi-species mixture, and a priori estimates based on the largest density gradient show that the Korteweg stresses, which account for the influence of the density gradient, have negligible contribution in the momentum equation. The species-specific effective Schmidt number, $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$, is computed and it is found that negative values occur for all minor species – particularly for H2 – thus indicating uphill diffusion, while the major species experience only regular diffusion. The probability density function (p.d.f.) of $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ shows strong variation with $p_{0}$ but weak dependence on the number of species; however, the p.d.f. substantially varies with the identity of the species. In contrast, the p.d.f. of the effective Prandtl number indicates dependence on both $p_{0}$ and the number of species. Similar to $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$, the species-specific effective Lewis-number p.d.f. depends on the species, and for all species the mean is smaller than unity, thus invalidating one of the most popular assumptions in combustion modelling. Simplifying the mixture composition by reducing the number of minor species does not affect the crucial species–temperature relationship of the major species that, for accuracy, must be retained in combustion simulations, but this relationship is affected for the minor species and in regions of uphill diffusion, indicating that the reduction is nonlinear in nature.
On models for predicting thermodynamic regimes in high-pressure turbulent mixing and combustion of multispecies mixtures
- Giacomo Castiglioni, Josette Bellan
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- Journal:
- Journal of Fluid Mechanics / Volume 843 / 25 May 2018
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- 23 March 2018, pp. 536-574
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The thermodynamic regime of a complex mixture depends on the composition, the pressure and the temperature; the spinodal locus separates the regime of thermodynamic instability from the remainder of the phase space. Since diffusion is one of the phenomena affecting the local chemical composition, the first focus is here on evaluating diffusion models in the context of high-pressure (high-$p$) multispecies mixing and combustion. It is shown that the diffusion model equations previously used to create two high-$p$ direct numerical simulation (DNS) databases can reproduce classical experimental observations of uphill diffusion in an accurate spatiotemporal manner, whereas the popular model which has a diagonal diffusion matrix and uses a velocity correction lacks spatiotemporal accuracy. Further, a mathematical formalism is used to compute the spinodal locus for mixtures for which either experimental data or previous computations from the literature are available, and it is shown that the agreement of the present calculations with that previously existing information is excellent. Using the spinodal-calculation mathematical formalism, the aforementioned DNS databases are then examined to determine the thermodynamic regime of the mixture at important stages of the simulations. In the first subset of the DNS databases that portrays mixing of five species under high-$p$ conditions, this stage is that of the transitional state representing the individual time station at which each simulation, having been initiated in a laminar state, transitions to a state having turbulent characteristics. In the second subset of the DNS databases that portrays high-$p$ turbulent combustion, this stage represents the individual time station at the peak $p$ achieved during the calculations. In both databases, the influence of the initial Reynolds number, the free-stream composition and the free-stream $p$ is studied. The results show that in all cases the mixture is in the single-phase regime. The present DNS databases have only five species, but it is shown that the methodology for computing the spinodal locus can be applied to very complex mixtures, with examples given for a twelve-species mixture and surrogate diesel fuels, thereby boding well for determining the thermodynamic regime of practical mixtures in high-$p$ turbulent flow simulations for engineering applications. According to these calculations, diesel-fuel surrogates are always in the single-phase regime at injection-conditions $p$ and temperatures existing in diesel-engine combustion chambers.
Multi-species turbulent mixing under supercritical-pressure conditions: modelling, direct numerical simulation and analysis revealing species spinodal decomposition
- Enrica Masi, Josette Bellan, Kenneth G. Harstad, Nora A. Okong’o
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- Journal:
- Journal of Fluid Mechanics / Volume 721 / 25 April 2013
- Published online by Cambridge University Press:
- 19 March 2013, pp. 578-626
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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 mass-diffusion 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 $(\mathit{Sc})$ and of an effective Prandtl number $(\mathit{Pr})$ based on effective species-specific 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 species-specific $\mathit{Sc}$ 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 $\mathit{Pr}$ 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.
Explicit filtering to obtain grid-spacing-independent and discretization-order-independent large-eddy simulation of two-phase volumetrically dilute flow with evaporation
- Senthilkumaran Radhakrishnan, Josette Bellan
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- Journal:
- Journal of Fluid Mechanics / Volume 719 / 25 March 2013
- Published online by Cambridge University Press:
- 19 February 2013, pp. 230-267
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Predictions from conventional large-eddy simulation (LES) are known to be grid-spacing and spatial-discretization-order dependent. In a previous article (Radhakrishnan & Bellan, J. Fluid Mech., vol. 697, 2012a, pp. 399–435), we reformulated LES for compressible single-phase flow by explicitly filtering the nonlinear terms in the governing equations so as to render the solution grid-spacing and discretization-order independent. Having shown in Radhakrishnan & Bellan (2012a) that the reformulated LES, which we call EFLES, yields grid-spacing-independent and discretization-order-independent solutions for compressible single-phase flow, we explore here the potential of EFLES for evaporating two-phase flow where the small scales have an additional origin compared to single-phase flow. Thus, we created a database through direct numerical simulation (DNS) that when filtered serves as a template for comparisons with both conventional LES and EFLES. Both conventional LES and EFLES are conducted with two gas-phase SGS models; the drop-field SGS model is the same in all these simulations. For EFLES, we also compared simulations performed with the same SGS model for the gas phase but two different drop-field SGS models. Moreover, to elucidate the influence of explicit filtering versus gas-phase SGS modelling, EFLES with two drop-field SGS models but no gas-phase SGS models were conducted. The results from all these simulations were compared to those from DNS and from the filtered DNS (FDNS). Similar to the single-phase flow findings, the conventional LES method yields solutions which are both grid-spacing and spatial-discretization-order dependent. The EFLES solutions are found to be grid-spacing independent for sufficiently large filter-width to grid-spacing ratio, although for the highest discretization order this ratio is larger in the two-phase flow compared to the single-phase flow. For a sufficiently fine grid, the results are also discretization-order independent. The absence of a gas-phase SGS model leads to build-up of energy near the filter cut-off indicating that while explicit filtering removes energy above the filter width, it does not provide the correct dissipation at the scales smaller than this width. A wider viewpoint leads to the conclusion that although the minimum filter-width to grid-spacing ratio necessary to obtain the unique grid-independent solution might be different for various discretization-order schemes, the grid-independent solution thus obtained is also discretization-order independent.
Explicit filtering to obtain grid-spacing-independent and discretization-order-independent large-eddy simulation of compressible single-phase flow
- Senthilkumaran Radhakrishnan, Josette Bellan
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- Journal of Fluid Mechanics / Volume 697 / 25 April 2012
- Published online by Cambridge University Press:
- 06 March 2012, pp. 399-435
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In large-eddy simulation (LES), it is often assumed that the filter width is equal to the grid spacing. Predictions from such LES are grid-spacing dependent since any subgrid-scale (SGS) model used in the LES equations is dependent on the resolved flow field which itself varies with grid spacing. Moreover, numerical errors affect the flow field, especially the smallest resolved scales. Thus, predictions using this approach are affected by both modelling and numerical choices. However, grid-spacing-independent LES predictions unaffected by numerical choices are necessary to validate LES models through comparison with a trusted template. First, such a template is created here through direct numerical simulation (DNS). Then, simulations are conducted using the conventional LES equations and also LES equations which are here reformulated so that the small-scale-producing nonlinear terms in these equations are explicitly filtered (EF) to remove scales smaller than a fixed filter width; this formulation is called EFLES. First, LES is conducted with four SGS models, then EFLES is performed with two of the SGS models used in LES; the results from all these simulations are compared to those from DNS and from the filtered DNS (FDNS). The conventional LES solution is both grid-spacing and spatial discretization-order dependent, thus showing that both of these numerical aspects affect the flow prediction. The solution from the EFLES equations is grid independent for a high-order spatial discretization on all meshes tested. However, low-order discretizations require a finer mesh to reach grid independence. With an eighth-order discretization, a filter-width to grid-spacing ratio of two is sufficient to reach grid independence, while a filter-width to grid-spacing ratio of four is needed to reach grid independence when a fourth- or a sixth-order discretization is employed. On a grid fine enough to be utilized in a DNS, the EFLES solution exhibits grid independence and does not converge to the DNS solution. The velocity-fluctuation spectra of EFLES follow those of FDNS independent of the grid spacing used, in concert with the original concept of LES. The reasons for the different predictions of conventional LES or EFLES according to the SGS model used, and the different characteristics of the EFLES predictions compared to those from conventional LES are analysed.
Subgrid-scale models and large-eddy simulation of oxygen stream disintegration and mixing with a hydrogen or helium stream at supercritical pressure
- EZGI S. TAŞKINOĞLU, JOSETTE BELLAN
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- Journal of Fluid Mechanics / Volume 679 / 25 July 2011
- Published online by Cambridge University Press:
- 11 May 2011, pp. 156-193
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For flows at supercritical pressure, p, the large-eddy simulation (LES) equations consist of the differential conservation equations coupled with a real-gas equation of state, and the equations utilize transport properties depending on the thermodynamic variables. Compared to previous LES models, the differential equations contain not only the subgrid-scale (SGS) fluxes but also new SGS terms, each denoted as a ‘correction’. These additional terms, typically assumed null for atmospheric pressure flows, stem from filtering the differential governing equations and represent differences, other than contributed by the convection terms, between a filtered term and the same term computed as a function of the filtered flow field. In particular, the energy equation contains a heat-flux correction (q-correction) which is the difference between the filtered divergence of the molecular heat flux and the divergence of the molecular heat flux computed as a function of the filtered flow field. We revisit here a previous a priori study where we only had partial success in modelling the q-correction term and show that success can be achieved using a different modelling approach. This a priori analysis, based on a temporal mixing-layer direct numerical simulation database, shows that the focus in modelling the q-correction should be on reconstructing the primitive variable gradients rather than their coefficients, and proposes the approximate deconvolution model (ADM) as an effective means of flow field reconstruction for LES molecular heat-flux calculation. Furthermore, an a posteriori study is conducted for temporal mixing layers initially containing oxygen (O) in the lower stream and hydrogen (H) or helium (He) in the upper stream to examine the benefit of the new model. Results show that for any LES including SGS-flux models (constant-coefficient gradient or scale-similarity models; dynamic-coefficient Smagorinsky/Yoshizawa or mixed Smagorinsky/Yoshizawa/gradient models), the inclusion of the q-correction in LES leads to the theoretical maximum reduction of the SGS molecular heat-flux difference; the remaining error in modelling this new subgrid term is thus irreducible. The impact of the q-correction model first on the molecular heat flux and then on the mean, fluctuations, second-order correlations and spatial distribution of dependent variables is also demonstrated. Discussions on the utilization of the models in general LES are presented.
A posteriori study using a DNS database describing fluid disintegration and binary-species mixing under supercritical pressure: heptane and nitrogen
- EZGI S. TASKINOGLU, JOSETTE BELLAN
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- Journal:
- Journal of Fluid Mechanics / Volume 645 / 25 February 2010
- Published online by Cambridge University Press:
- 09 February 2010, pp. 211-254
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A large eddy simulation (LES) a posteriori study is conducted for a temporal mixing layer which initially contains different species in the lower and upper streams and in which the initial pressure is larger than the critical pressure of either species. A vorticity perturbation, initially imposed, promotes roll-up and a double pairing of four initial spanwise vortices to reach a transitional state. The LES equations consist of the differential conservation equations coupled with a real-gas equation of state, and the equations utilize transport properties depending on the thermodynamic variables. Unlike all LES models to date, the differential equations contain, additional to the subgrid-scale (SGS) fluxes, a new SGS term denoted a ‘pressure correction’ (p correction) in the momentum equation. This additional term results from filtering the Navier–Stokes equations and represents the gradient of the difference between the filtered p and p computed from the filtered flow field. A previous a priori analysis, using a direct numerical simulation (DNS) database for the same configuration, found this term to be of leading order in the momentum equation, a fact traced to the existence of regions of high density-gradient magnitude that populated the entire flow; in that study, the appropriateness of several SGS-flux models was assessed, and a model for the p-correction term was proposed.
In the present study, the constant-coefficient SGS-flux models of the a priori investigation are tested a posteriori in LES devoid of, or including, the SGS p-correction term. A new p-correction model, different from that of the a priori study, is used, and the results of the two p-correction models are compared. The results reveal that the former is less computationally intensive and more accurate than the latter in reproducing global and structural features of the flow. The constant-coefficient SGS-flux models encompass the Smagorinsky (SMC) model, in conjunction with the Yoshizawa (YO) model for the trace, the gradient (GRC) model and the scale similarity (SSC) models, all exercised with the a priori study constant-coefficient values calibrated at the transitional state. Further, dynamic SGS-flux model LESs are performed with the p correction included in all cases. The dynamic models are the Smagorinsky (SMD) model, in conjunction with the YO model, the gradient (GRD) model and ‘mixed’ models using SMD in combination with GRC or SSC utilized with their theoretical coefficient values. The LES comparison is performed with the filtered-and-coarsened DNS (FC-DNS) which represents an ideal LES solution. The constant-coefficient models including the p correction (SMCP, GRCP and SSCP) are substantially superior to those devoid of it; the SSCP model produces the best agreement with the FC-DNS template. For duplicating the local flow structure, the predictive superiority of the dynamic mixed models is demonstrated over the SMD model; however, even better predictions in capturing vortical features are obtained with the GRD model. The GRD predictions improve when LES is initiated at a time past the initial range in which the p-correction term rivals in magnitude the leading-order term in the momentum equation. Finally, the ability of the LES to predict the FC-DNS irreversible entropy production is assessed. It is shown that the SSCP model is the best at recovering the domain-averaged irreversible entropy production. The sensitivity of the predictions to the initial conditions and grid size is also investigated.
Modelling of subgrid-scale phenomena in supercritical transitional mixing layers: an a priori study
- LAURENT C. SELLE, NORA A. OKONG'O, JOSETTE BELLAN, KENNETH G. HARSTAD
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- Journal:
- Journal of Fluid Mechanics / Volume 593 / 25 December 2007
- Published online by Cambridge University Press:
- 23 November 2007, pp. 57-91
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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.
Direct numerical simulation of gaseous mixing layers laden with multicomponent-liquid drops: liquid-specific effects
- PATRICK C. LE CLERCQ, JOSETTE BELLAN
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- Journal:
- Journal of Fluid Mechanics / Volume 533 / 25 June 2005
- Published online by Cambridge University Press:
- 15 June 2005, pp. 57-94
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A representation of multicomponent-liquid (MC-liquid) composition as a linear combination of two single-Gamma probability distribution functions (PDFs) is used to describe a large number of MC-liquid drops evaporating in a gas flow. The PDF, called the double-Gamma PDF, depends on the molar mass. The gas-phase conservation equations are written in an Eulerian frame and the drops are described in a Lagrangian frame. Gas conservation equations for mass, momentum, species and energy are combined with differential conservation equations for the first four moments of the gas-composition PDF and coupled to the perfect gas equation of state. Source terms in all conservation equations account for the gas/drop interaction. The drop governing equations encompass differential conservation statements for position, mass, momentum, energy and four moments of the liquid-composition PDF. Simulations are performed for a three-dimensional mixing layer whose lower stream is initially laden with drops colder than the surrounding gas. Initial perturbations excite the layer to promote the double pairing of its four initial spanwise vortices to an ultimate vortex. During the layer evolution, the drops heat and evaporate. The results address the layer evolution, and the state of the gas and drops when layers reach a momentum-thickness maximum past the double vortex pairing. Of interest is the influence of the liquid composition on the development of the vortical features of the flow, on the vortical state reached after the second pairing, and on the gas temperature and composition. The MC-liquid simulations are initiated with a single-Gamma PDF composition so as to explore the development of the double-Gamma PDF. Examination of equivalent simulations with n-decane, diesel and three kerosenes as the liquid, permits assessment of the single-species versus the MC-liquid aspect, and of mixture composition specific effects. Global layer growth and global rotational characteristics are unaffected by liquid specificity; however, the global mixing is highly liquid-specific. Also liquid-specific is the evolution of the ensemble-averaged drop characteristics and of the volumetric averages representing the gas composition. Visualized rotational characteristics show that the small-scale vortical activity increases with increased fuel volatility, which is confirmed by analysis of the vorticity budgets. Homogeneous-plane-average budgets of the vorticity and vorticity-magnitude equations indicate that the stretching and tilting, and momentum-source terms are responsible for the difference among simulations. For all MC liquids, the gas displays a high level of composition heterogeneity, which can directly be traced to the original PDF representing the MC-liquid composition. Under most conditions, the single-Gamma PDF develops into a double-Gamma PDF; however, the extent of this transformation, indicative of vapour condensation onto drops, is not readily parametrized by the liquid volatility, initial carrier-gas temperature or trace vapour in the initial gas.
Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Part 2. A posteriori modelling
- ANTHONY LEBOISSETIER, NORA OKONG'O, JOSETTE BELLAN
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- Journal:
- Journal of Fluid Mechanics / Volume 523 / 25 January 2005
- Published online by Cambridge University Press:
- 21 January 2005, pp. 37-78
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Large-eddy simulation (LES) is conducted of a three-dimensional temporal mixing layer whose lower stream is initially laden with liquid drops which may evaporate during the simulation. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, species, momentum and energy source terms. The drop evolution is modelled using physical drops, or using computational drops to represent the physical drops. Simulations are performed using various LES models previously assessed on a database obtained from direct numerical simulations (DNS). These LES models are for: (i) the subgrid-scale (SGS) fluxes and (ii) the filtered source terms (FSTs) based on computational drops. The LES, which are compared to filtered-and-coarsened (FC) DNS results at the coarser LES grid, are conducted with 64 times fewer grid points than the DNS, and up to 64 times fewer computational than physical drops. It is found that both constant-coefficient and dynamic Smagorinsky SGS-flux models, though numerically stable, are overly dissipative and damp generated small-resolved-scale (SRS) turbulent structures. Although the global growth and mixing predictions of LES using Smagorinsky models are in good agreement with the FC-DNS, the spatial distributions of the drops differ significantly. In contrast, the constant-coefficient scale-similarity model and the dynamic gradient model perform well in predicting most flow features, with the latter model having the advantage of not requiring a priori calibration of the model coefficient. The ability of the dynamic models to determine the model coefficient during LES is found to be essential since the constant-coefficient gradient model, although more accurate than the Smagorinsky model, is not consistently numerically stable despite using DNS-calibrated coefficients. With accurate SGS-flux models, namely scale-similarity and dynamic gradient, the FST model allows up to a 32-fold reduction in computational drops compared to the number of physical drops, without degradation of accuracy; a 64-fold reduction leads to a slight decrease in accuracy.
Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Part 1. Direct numerical simulation, formulation and a priori analysis
- NORA A. OKONG'O, JOSETTE BELLAN
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- Journal:
- Journal of Fluid Mechanics / Volume 499 / 25 January 2004
- Published online by Cambridge University Press:
- 30 January 2004, pp. 1-47
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Large-eddy simulation (LES) models are presented and evaluated on a database obtained from direct numerical simulation (DNS) of a three-dimensional temporal mixing layer with evaporating drops. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, momentum and energy source terms. The DNS database consists of transitional states attained by layers with different initial Reynolds numbers and initial liquid-phase mass loadings. Budgets of the LES equations at the transitional states show that, for the mass loadings considered, the filtered source terms (FSTs) are smaller than the resolved inviscid terms and some subgrid scale (SGS) terms, but larger than the resolved viscous stress, heat flux and mass flux terms. The irreversible entropy production (i.e. the dissipation) expression for a two-phase flow with phase change is derived, showing that the dissipation contains contributions due to viscous stresses, heat and species-mass fluxes, and source terms. For both the DNS and filtered flow fields at transition, the two leading contributions are found to be the dissipation due to the energy source term and that due to the chemical potential of the mass source. Therefore, the modelling effort is focused on both the SGS fluxes and the FSTs in the LES equations. The FST models considered are applicable to LES in which the grid is coarser than the DNS grid and, consistently, ‘computational’ drops represent the DNS physical drops. Because the unfiltered flow field is required for the computation of the source terms, but would not be available in LES, it was approximated using the filtered flow field or the filtered flow field augmented by corrections based on the SGS variances. All of the FST models were found to overestimate DNS-field FSTs, with the relative error of modelling the unfiltered flow field compared to the error of using computational drops showing a complex dependence on filter width and number of computational drops. For modelling the SGS fluxes and (where possible) SGS variances, constant-coefficient Smagorinsky, gradient and scale-similarity models were assessed on the DNS database, and calibrated coefficients were statistically equivalent when computed on single-phase or two-phase flows. The gradient and scale-similarity models showed excellent correlation with the SGS quantities. An a posteriori study is proposed to evaluate the impact of the studied models on the flow-field development, so as to definitively assess their suitability for LES with evaporating drops.
Direct numerical simulation of a transitional supercritical binary mixing layer: heptane and nitrogen
- NORA A. OKONG'O, JOSETTE BELLAN
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- Journal:
- Journal of Fluid Mechanics / Volume 464 / 10 August 2002
- Published online by Cambridge University Press:
- 21 August 2002, pp. 1-34
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Direct numerical simulations (DNS) of a supercritical temporal mixing layer are conducted for the purpose of exploring the characteristics of high-pressure transitional mixing behaviour. The conservation equations are formulated according to fluctuation-dissipation (FD) theory, which is consistent with non-equilibrium thermodynamics and converges to kinetic theory in the low-pressure limit. According to FD theory, complementing the low-pressure typical transport properties (viscosity, diffusivity and thermal conductivity), the thermal diffusion factor is an additional transport property which may play an increasingly important role with increasing pressure. The Peng–Robinson equation of state with appropriate mixing rules is coupled to the dynamic conservation equations to obtain a closed system. The boundary conditions are periodic in the streamwise and spanwise directions, and of non-reflecting outflow type in the cross-stream direction. Due to the strong density stratification, the layer is considerably more difficult to entrain than equivalent gaseous or droplet-laden layers, and exhibits regions of high density gradient magnitude that become very convoluted at the transitional state. Conditional averages demonstrate that these regions contain predominantly the higher-density, entrained fluid, with small amounts of the lighter, entraining fluid, and that in these regions the mixing is hindered by the thermodynamic properties of the fluids. During the entire evolution of the layer, the dissipation is overwhelmingly due to species mass flux followed by heat flux effects with minimal viscous contribution, and there is a considerable amount of backscatter in the flow. Most of the species mass flux dissipation is due to the molecular diffusion term with significant contributions from the cross-term proportional to molecular and thermal diffusion. These results indicate that turbulence models for supercritical fluids should primarily focus on duplicating the species mass flux rather than the typical momentum flux, which constitutes the governing dissipation in atmospheric mixing layers. Examination of the passive-scalar probability density functions (PDFs) indicates that neither the Gaussian, nor the beta PDFs are able to approximate the evolution of the DNS-extracted PDF from its inception through transition. Furthermore, the temperature–species PDFs are well correlated, meaning that their joint PDF is not properly approximated by the product of their marginal PDFs; this indicates that the traditional reactive flow modelling based on replacing the joint PDF representing the reaction rate by the product of the marginal PDFs is not appropriate. Finally, the subgrid-scale temperature–species PDFs are also well correlated, and the species PDF exhibits important departures from the Gaussian. These results suggest that classic PDFs used in atmospheric pressure flows would not capture the physics of this supercritical mixing layer, either in an assumed PDF model at the larger scale, or at the subgrid scale.
Direct numerical simulations of supercritical fluid mixing layers applied to heptane–nitrogen
- RICHARD S. MILLER, KENNETH G. HARSTAD, JOSETTE BELLAN
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- Journal:
- Journal of Fluid Mechanics / Volume 436 / 10 June 2001
- Published online by Cambridge University Press:
- 22 June 2001, pp. 1-39
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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 [les ] T [les ] 1100 K, 40 [les ] p [les ] 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.