3D Direct Numerical Simulations Research Articles

Thermodiffusively unstable laminar hydrogen flame in a sufficiently large 3D computational domain – Part II: NOx formation mechanism and flamelet modeling

In this work, the NOx formation mechanism in three-dimensional (3D) thermodiffusively unstable premixed hydrogen flames is investigated through direct numerical simulations (DNS) and a reaction path analysis based on the DNS dataset. In part I of this study (Wen et al., 2024), the characteristic patterns of the instabilities were studied using the same dataset. Here, first, the effects of the computational setup (2D vs. 3D) and curvature on the flux ratio of nitrogen atom are quantified. In addition, the composition space model (CSM) proposed in previous works is extended to account for the NOx chemical reaction mechanism to predict NOx formation in thermodiffusively unstable premixed hydrogen flames. The flamelet solutions are obtained from the premixed flamelet equations in composition space so that the wide range of curvatures associated with the strongly corrugated flame front can be considered. The performance of the CSM in predicting the NOx species and the important radicals involved in the NOx formation pathways is evaluated through an a priori analysis. To investigate the effects of curvature on the NOx formation pathways and the performance of the composition space model, the positively- and negatively-curved regions in the DNS are studied separately. The results show that different from the 2D simulation, the NNH reaction pathway becomes dominant in the 3D simulation due to the increased range of curvature, which promotes the accumulation of the highly diffusive H radical in the positively-curved regions. The NNH reaction pathway is dominant in the positively-curved regions, while the N2O reaction pathway is more important in the negatively-curved regions. The flamelet model based on the composition space solutions that consider the effects of curvature yields accurate predictions for the radicals that are sensitive to the effects of curvature.Novelty and significance statementThe novelty of the present work includes the following aspects,(i) Analysis of NOx formation mechanism based on the large-scale 3D DNS dataset of a laminar fuel-lean premixed hydrogen flame on a sufficiently large domain;(ii) Quantification of the effects of computational setup (2D and 3D) and curvature on the NOx formation pathways;(iii) Extension of the previous composition space model (CSM) by incorporating the NOx chemical reaction mechanism to consider the effects of curvature on the NOx formation.

Quantitative Evaluation of the Onset and Evolution for the Non‐Darcy Behavior of the Partially Filled Rough Fracture

AbstractThe non‐Darcy flow behavior in unfilled and fully filled rough fractures has been investigated thoroughly for decades. Natural fractures usually be partially filled with porous media due to long‐term internal and external dynamic disturbances and water flow erosion. However, how to evaluate the nonlinear seepage characteristics of the partially filled rough fracture (PFRF) is never scrutinized. In the present study, 2D direct numerical simulations are conducted to investigate the non‐Darcy behavior for flow through 6 PFRF models under different filling conditions (i.e., filling degree, roughness, and hydraulic conductivity capability of the filling medium), which included coupling between the free and seepage flow in the unfilled and filled region, respectively. The results show that the critical non‐Darcy factor E = 0.02 can be used to judge the onset of the non‐Darcy regime in PFRF, which is also the critical point of eddy region (ER) volume mutation. The higher the filling degree and the weaker the hydraulic conductivity of the filled medium, the more significant the non‐Darcy behavior. Furthermore, the variation of ER volume can be divided into three stages with the increasing inertia effect, that is, stable stage (E < 0.02), slow nonlinear growth stage (0.02 ≤ E ≤ 0.1), and rapid linear growth stage (E > 0.1). In addition, a new model for quantitatively evaluating the non‐Darcy behavior of PFRF is developed via gene expression programming based on 735 simulation data set, which is further employed to establish the relationship between the friction factor.

DNS of a non-premixed CH4/O2 flame in a supercritical CO2 environment

3D Direct Numerical Simulation (DNS) is performed to investigate turbulent non-premixed methane oxy-combustion in a CO2-rich environment at supercritical conditions (300 bar). The study focuses on a shear-layer configuration typical of slot burners, where the central slot carries the fuel and the oxidant mixture O2/CO2 flows on both sides of the central jet. The simulation solves the compressible Navier–Stokes equations coupled with the translated volume formulation of the Peng–Robinson cubic equation of state (VTPR-EoS), utilizing a reduced 23-species kinetic scheme derived from the ARAMCO 2.0 complete mechanism. The extremely high pressure results in turbulent scales smaller than those observed at lower pressures. Both premixed and diffusion flame zones are identified, with heat primarily released in non-premixed rich regions, leading to a CO mass fraction of up to 0.55. The flow exhibits uphill (counter-gradient) diffusion for both CO2 and H2O, primarily influenced by the interaction of binary diffusion coefficients and individual species chemical potential gradients. Values in the range of 2.52−2.62 are obtained for the fractal dimension of temperature iso-surfaces. Mixture fraction probability density functions (PDFs) are compared with the standard presumed β−PDF, revealing an underestimation of mixing. Differently from flames at atmospheric pressures, the Dufour effect in the energy diffusion flux, is not negligible. Furthermore, fugacity coefficients must be considered in the calculation of chemical species kinetic rates.

Low-Prandtl-number effects on global and local statistics in two-dimensional Rayleigh–Bénard convection

We present global and local statistical properties of turbulent Rayleigh–Bénard (RB) convection at low Prandtl numbers in this work. A series of high resolution two-dimensional (2D) direct numerical simulations are carried out in a square box for the Prandtl number ranges 0.005≤Pr≤0.07 and 0.01≤Pr≤0.15 at Rayleigh numbers Ra=107 and Ra=108, respectively. The global heat and momentum transport expressed as Nusselt number Nu and Reynolds number Re are found to scale as Nu∼Pr0.14 and Re∼Pr−0.82 for Ra=107, and Nu∼Pr0.11Re∼Pr−0.93 for Ra=108. The local velocity fluctuation at the cell center shows larger amplitudes at lowered Pr, indicating a stronger turbulence in the bulk. The magnitudes of kinetic and thermal energy dissipation rates in the bulk also increase with the decreasing of Pr, due to the intensified velocity gradient and larger thermal diffusivity, respectively. In the cell central region, probability density functions (PDFs) of velocity show a bimodal distribution, and it approaches the Gaussian distribution at higher Pr, while the PDFs of temperature display a stretched exponential shape with intermittent behavior. The kinetic energy spectra further reveal that the velocity cascade follows the Bolgiano–Obukhov scaling in the bulk of the convective flow.

Thermodiffusively unstable laminar hydrogen flame in a sufficiently large 3D computational domain – Part I: Characteristic patterns

Thermodiffusive instabilities can have a leading order effect on flame propagation for lean premixed hydrogen flames. Many simulation studies have been performed to study this effect, but almost exclusively in two-dimensional (2D) or domain sizes too small to support the characteristic large-scale features of the instability. The main purpose of this study is to quantify the differences of 3D and 2D flames using simulations on sufficiently large domains. To this end, direct numerical simulations (DNS) of thermodiffusively unstable laminar premixed hydrogen flames stabilized in 3D domains were performed. The effects of confinement on the flame dynamics are rigorously investigated by varying the domain size. The flame burning velocity shows a strong dependence on the domain size when the lateral domain width is less than 50 laminar flame thicknesses. The characteristic patterns of the thermodiffusively unstable flame are analyzed in detail, including the instantaneous flame structure, global burning velocity, flame surface area, stretch factor, curvature distribution, and the cell size. The effects of the computational setup (2D vs. 3D) on the local flame front curvature and the distributions of the thermo-chemical quantities are quantified through a conditional analysis. The formation and destruction mechanisms of the distinct cellular structures observed in the 3D domain are analyzed focusing on the interactions between the flame dynamics and the flow field, and the contributions of the production of flame surface density, kinematic restoration, and curvature dissipation are quantified. Compared with the corresponding 2D simulation, the flame surface area in the cut plane of the 3D configuration is similar, yet the burning velocity and the stretch factor are increased. The reason for this is the difference in curvature statistics in 2D and 3D. In 3D, larger extreme curvature values are obtained, leading to quantitatively different distributions of the thermo-chemical variables. For example, the peak H radical concentration in the 3D simulation is about twice higher than in the 2D simulation.Novelty and significance statementThe novelty of the present work includes the following aspects:(i) First large-scale 3D DNS of laminar lean premixed hydrogen flame on domain large enough to obtain domain-independent results.(ii) First quantitative assessment of characteristic patterns of 3D thermodiffusively unstable premixed hydrogen flames.(iii) First quantitative assessment of differences between 2D and 3D thermodiffusively unstable premixed hydrogen flames for domain-independent results.

Inferring the Magnetic Field from the Rayleigh–Taylor Instability

By extending the potential approach of Layzer and Goncharov to the magnetohydrodynamics equations, we find the nonlinear solutions to the single-mode Rayleigh–Taylor instability subjected to uniform magnetic fields at various inclinations. This allows us to derive the analytical prediction of the terminal bubble and spike velocities at arbitrary Atwood numbers, which are assessed against various 2D and 3D direct numerical simulations. Contrary to the linear phase, where the magnetic field inhibits or delays the instability, the growth rate may be enhanced in the nonlinear regime, exhibiting velocities faster than the Alfvén speed. This sheds light on the importance of the nondimensional number expressing the competition between the magnetic and buoyancy effects. Conversely, we show how the orientation and the intensity of the magnetic field can be simply inferred from these solutions.

Natural convective loops heat transfer scaling analysis

Heat transfer arising in natural convective loops is analyzed numerically and theoretically. Combining 3D direct numerical simulations, unidirectional heat-transfer semi-analytical computations, scaling arguments and asymptotic analysis new universal scaling laws are proposed to account for dimensionless heat transfer behavior of natural convective loops. It is found that the Reynolds number Re resulting from the buoyancy driven rotating convection (as well as the Péclet number) scales as Grashof number Gr as Re∼Gr3/4 for imposed temperature boundary conditions and Re∼Gr3/5 for mixed imposed temperature/flux heating conditions. These scaling laws are successfully confronted with experimental measurements in both heating configurations. It is shown that these scaling result from boundary layers dominated transfers without noticeable influence of possibly complex 3D flow patterns found in the fluid flow. These simple scaling laws constitute a progress over previously existing correlations which have disregarded the effect of boundary layers, heating configurations as well as fluid to solid variable diffusivity/conductivity.

Compressible Rayleigh–Taylor instability subject to isochoric initial background state

The effect of compressibility on the single-mode Rayleigh–Taylor instability is examined using two (2D) and three-dimensional (3D) direct numerical simulations. To isolate compressibility from background stratification effects, this work employs a constant density profile on each side of the interface. The numerical simulations are performed at various Reynolds numbers using the gas kinetic method for static Mach numbers up to M = 0.4. The most important finding is that compressibility acting in isolation enhances the instability and perturbations grows faster with increasing Mach number, unlike previous results with background isothermal state, which show suppression of the instability at higher static Mach numbers. In addition, compressibility is also shown to increase the bubble-spike asymmetry. While the instability grows faster for the 3D case, the findings are qualitatively similar in 2D and 3D. The dynamical reasons underlying the effect of compressibility are elucidated by examining the evolution of vorticity and turbulent kinetic energy transport equations.

Analysis of knock onset based on two-dimensional direct numerical simulation and theory of explosive transition of deflagration

In this study, we analyzed data from a two-dimensional (2D) direct numerical simulation (DNS) that reproduced the knocking experiment in order to elucidate the knocking phenomenon. First, it was confirmed that the reaction front behavior in 2D DNS could be reproduced as a one-dimensional (1D) laminar premixed flame simulation at extreme conditions. Furthermore, a detailed study using a 1D laminar premixed flame revealed a strong relation between the timing of knock onset and the flame propagation limit of the 1D laminar premixed flame at elevated temperature and pressure conditions. To clarify this relation, we introduced the theory of “explosive transition of deflagration.” This theory shows that when the Lewis number is unity, the time evolution of the normalized fuel mass fraction and temperature in a 0D homogeneous ignition is equal to the temporal evolution observed in a 1D laminar premixed flame, if the spatiotemporal transformation is properly applied. Furthermore, the rate at which the normalized fuel mass fraction decreases in the preheat zone was found to depend on the Lewis number, and when the Lewis number is greater than unity, no flame structure exists above a certain threshold temperature. Finally, the mechanism of knock onset was explained by considering the theory of explosive transition of deflagration and explosive transition boundary plotted on a pressure-temporal diagram.

Direct numerical simulations of methane, ammonia-hydrogen and hydrogen turbulent premixed flames

Three turbulent premixed flames with the same unstretched laminar flame speeds and thicknesses are analyzed and compared using 3D Direct Numerical Simulation (DNS) in a slot burner configuration at atmospheric conditions: a CH4/air flame at ϕ=1, an NH3−H2/air (46% vol. H2) at ϕ=1 and an H2/air flame at ϕ=0.45.While both stoichiometric methane and ammonia-hydrogen flames behave similarly, the lean hydrogen flame brush is twice as short with less flame surface, and exhibits significant alteration of its local flame structure: for the H2 flame, the thickness of flamelets decreases significantly while burning rates increase drastically. This is observed for flame elements convexly curved with respect to fresh reactants (positive curvature) because of preferential diffusion, where thermo-diffusive instabilities generate long-tail structures that continuously head toward the fresh gases, and also for near-flat flame elements because of local strain effects. Opposite behaviors are observed for flame elements concavely curved (negative curvature) where fuel depletion caused by unfocusing of hydrogen and strain effects are unable to increase the consumption, even leading to near-extinction of the flame. The turbulent methane and ammonia-hydrogen flames studied in this work do not exhibit the instabilities seen in the pure hydrogen flame. However, local flame-flame interactions are observed in the cusp regions of all three flames, which significantly increase the displacement speed as well as flame surface destruction.A 1D-3D comparison indicates that strain effects in regions of low curvature and preferential diffusion effects in regions of positive curvature can be modeled using counterflow flamelets to account for stretch effects on the turbulent hydrogen flame propagation in addition to turbulent wrinkling.

Suppression of vortex shedding in the wake of a circular cylinder through high-frequency in-line oscillation

This paper presents a new flow control approach to suppress the vortex shedding in the wake of a circular cylinder through high-frequency oscillation. The circular cylinder is forced to oscillate in the streamwise direction at high-frequency and low amplitude, corresponding to a high Stokes number (β = 100–1000) and low Keulegan–Carpenter number (KC = 0.001–4). Two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations of an oscillating circular cylinder in steady current have been carried out in the parameter space of KC, Rec, and β. Our numerical results show that when the flow remains in the two-dimensional vortex shedding regime, the cylinder wake sequentially experiences transitions from the vortex shedding regime to the suppression of the vortex shedding regime and finally to the symmetry breaking regime, with increasing KC. Corresponding wake characteristics and variations of hydrodynamic forces over the three wake regimes are explored. Three quantities that represent shear-layer characteristics, including the length of separating shear layers, the circulation of shear layers and wake recirculation length, reach maxima at the onset of suppression. The physical mechanisms for the suppression of vortex shedding and occurrence of symmetry breaking are also explained. Once the flow becomes 3-D, vortex shedding from the cylinder cannot be suppressed, primarily because the outer shear layers induced by the steady approaching flow are enhanced in 3-D flows. The cylinder oscillation over the frequency range investigated in the present study delays wake transition to 3-D. The cylinder oscillation alters the 3-D vortical structure and its spanwise wavelength significantly.

FEM Analysis of Buckled Dielectric Thin-Film Packaging Based on 3D Direct Numerical Simulation.

This paper presents a 3D direct numerical simulation of buckled thin-film packaging based on transferred elastic thin-film wrinkling bonded on a compliant polymer ring. The mode change of the fabricated thin-film cap is found by measuring the thin-film cap shape at different times after Si substrate debonding. The conventional linear and nonlinear buckling simulations are not adequate to understand the behavior of the thin-film buckling mechanism creating such packaging cap mode change. Direct buckling simulation is recently reported as an easy and useful numerical wrinkling simulation method. A novel 3D FEM model of a thin-film package suitable for direct 3D buckling simulation is built to reduce the mode mixture between different buckling modes. Buckling modes of the packaging cap are investigated in terms of elastic moduli of package materials and applied strain due to thermal expansion coefficient difference. Based on the simulation results, it is found that there are two main modes in the fabricated thin-film buckling package determining the shape of the transferred thin-film packaging cover depending on the elasticity ratio between the cap and sealing ring materials. The mode shift from wrinkling cap mode to out-of-plane cap mode due to applied strain along a polymeric sealing ring is found.

Acceleration of turbulent combustion DNS via principal component transport

We investigate the implementation of principal component (PC) transport to accelerate the direct numerical simulation (DNS) of turbulent combustion flows. The acceleration is achieved using the transport of PCs and the tabulation of the closure terms in the PC-transport equations using machine learning. Further acceleration is achieved by a treatment for bottlenecks associated with the acoustic time steps for low Mach number flows. The approach is implemented in 2D and 3D on a laboratory scale lean premixed methane-air flame stabilized on a slot burner. DNS based on the transport of thermochemical scalars (species and energy) is also carried out, first to develop a 2D DNS database for PC-transport equations’ closure terms and, second, to validate the approach against species DNS in 2D and 3D, a principal goal of the present effort. The results show that surrogate PC DNS can reproduce instantaneous profiles as well as statistics associated with turbulence, flame topology properties and measures of flame-turbulence interactions. The study also demonstrates that parametric simulations with surrogate PC DNS can be implemented at a fraction of the cost of a full 3D DNS with species and energy transport.

DNS study on reactivity stratification with prechamber H[formula omitted]/air turbulent jet flame to enhance NH[formula omitted]/air combustion in gas engines

The present study investigates turbulent flame dynamics and pollutant emission characteristics in a prechamber/main chamber system. A H2/air turbulent jet flame issued from the prechamber penetrates into the NH3/air mixture found in the main chamber. Recent experimental studies have demonstrated that this configuration is very promising for future NH3 combustion in gas engines. In the present work, 3D direct numerical simulations (DNS) with detailed chemistry have been carried out in the above configuration. The dynamics of the turbulent premixed flame in the prechamber (flame thickness, flame propagation speed) and the structure of the turbulent stratified flame in the main chamber are investigated. It is found that the stratified flame thickness is becoming thicker as the flame propagates within the NH3/air mixture. The flame displacement speed is slowing down in a first phase, before becoming faster again after jet-induced turbulence starts to decay. A chemical explosive mode analysis (CEMA) has been done to distinguish between ignition, flame propagation, and local extinction events both for the premixed flame in the prechamber and the stratified flame in the main chamber. The temporal evolution of the identified modes are consistent with the flame dynamics and help to explain the later increase in flame displacement speed. NO emission characteristics in the main chamber have also been investigated. The NO production speed first increases, reaches a plateau, before being reduced during later flame propagation; stratification is important to explain the NO emission characteristics. Compared to the laminar NH3/air flame, the NO production speed is obviously larger in the present configuration, mainly due to the reduced contribution of the NO destruction reactions in the fuel pathway. Compared to premixed NH3/H2/air flames with H2 volume ratios found in practical systems (typically ≥ 0.4), the NO production speed is lower in the present configuration, demonstrating its superiority for practical applications. Finally, conventional heat release rate (HRR) markers are evaluated for the stratified flame. They approximate quite well the general structure of the flame front, but lead to noticeable quantitative errors.

Effective performance analysis of stiffened honeycomb sandwich panels using VAM-based equivalent model

The stiffened honeycomb sandwich panel (stiffened HSP) is an improved honeycomb structure by adding the stiffeners into the traditional honeycomb core to enhance the structural performance. The novelty of this work is that the constitutive parameters of the panel are obtained by asymptotical analysis of the strain energy stored in the periodic unit cell, and a 2D dimensional reduced model (2D-DRM) is constructed on this basis. The efficiency and accuracy of 2D-DRM are verified by comparing with the results of 3D direct numerical simulations (3D-DNS) under different working conditions (including static displacement, local field distribution, free vibration and global buckling). In addition, the effects of the structural parameters of the stiffened HSP and the layup configuration of the facesheet on the equivalent stiffness, specific stiffness as well as static and dynamic performance are systematically studied. The results reveal that the facesheet thickness-to-core height ratio had the greatest influence on the buckling load, while the side length-to-height ratio of core cell had the greatest influence on the fundamental frequency. Compare with the traditional and vertical stiffened HSPs, the horizontal stiffened HSP has larger buckling load and lower natural frequency.

Aspect ratio and interference effects on flow over two rectangular cylinders in tandem arrangement

Flow past twin tandem rectangular cylinders for various gap ratios (L* = 1.0–8.0) and aspect ratios (B* = 0.3–4.0) at a Reynolds number of 150 was investigated via 2D direct numerical simulation. Results showed that based on the variation trend of fluid force with B*, three types of gaps spacing are categorized, i.e., narrow gap (L* = 1.0, 2.0), medium gap (L* = 3.0, 4.0), and wide gap (L* = 6.0, 8.0). Higher B* reduces the fluctuating force of downstream cylinder (DC) for narrow gaps, but amplifies that of the DC for wide gaps. For medium gaps, variation of fluid forces with B* < 1 is in line with that of wide gaps, while it resembles that of narrow gap type at B* > 1. Moreover, gap flow is classified into three flow regimes, i.e., extended body, reattachment, and co-shedding. Wake flow structures of DC show 2S vortex type at narrow gaps for all B* cases, whereas they transfer from synchronized C(2S) to 2S type when B* > 0.3 for medium and wide gaps. Proper orthogonal decomposition analysis captures the spatial distribution and temporal evolution of coherent structure hidden in flow field; thus, it well explains the fundamental mechanism that fluid force, gap flow and wake vortex change with B* and L*.

Investigation of the derivation and consistency of the quasi-two-dimensional flamelet models for non-premixed flames

Three non-premixed quasi-two-dimensional flamelet (Q2DF) models can be derived via integrating one-dimensional flamelet libraries, which are generated by premixing the third stream with the fuel stream or (and) the oxidizer stream as the premix fraction (η) varied from zero to unity. These models are attractive and considered effective substitutes for two-dimensional flamelet (2DF) formulations; however, the connections between these Q2DF models and 2DF equations have not been clearly illustrated formerly. In this work, the 2DF formulations are converted from the space defined by two mixture fractions (Z1×Z2) to that used for each Q2DF model (ξ×η) in which ξ designates a newly defined mixture fraction. For moderate Reynolds number conditions, it is considered that the Q2DF models and the 2DF formulations are equivalents as the one-dimensional (1D) flamelet elemental libraries are integrated upon sufficient grids of η. To confirm this, the two-dimensional (2D) direct numerical simulations (DNS) and three-dimensional (3D) large-eddy simulation (LES) are performed to provide reference data since it is readily to associate the 2DF formulations with variable transports in physical space. The 2D DNS cases present various three-feed systems characterized by different third streams and ensure a well-mixed mixture in moderate η, while the 3D LES case is performed with a large flux of pure CO2 as the third stream. Three Q2DF models are compared by using the same tracking parameters obtained from, respectively, the DNSs and LES. The comparisons based on the DNS cases demonstrate that the Q2DF models can reproduce the DNS data with quite satisfying accuracy, and it is confirmed that the accuracy has nothing to do with the type of the third stream. Through the LES results and the related comparisons, it can be concluded that the Q2DF models are reliable to predict the experimental measurements, and on the condition of a sufficient η resolution, the three Q2DF models can give the same prediction in the whole range of η.

The flow dynamics in a short annulus with rotating end-walls

The paper reports on the 3D direct numerical simulation (DNS) results of the flow in a short annulus of aspect ratios 3.8–4.8 and radius ratios 0.45, 0.5 with end-walls attached to a rotating inner cylinder. The research is focused on the bifurcation processes occurring in the local unsteady area, existing at very low Reynolds numbers (Re = 90–164) - this unsteady area is associated with the presence of the codimension-2 point occurring in this range of parameters. The study has revealed many interesting phenomena, e.g. the period-doubling, the homoclinic collision and the second modulated wave. New bifurcation lines have been determined. The DNS results are analyzed in the light of experimental data published so far. At higher Reynolds numbers the next unsteady area occurs, dynamical features of which have been studied in detail for Re up to 1000. The computations have also been performed for the configurations with cylinders rotating in the co- and counter-rotating systems with rotational rate Ωout/Ωin = ± 0.1 and ± 0.2, to determine the impact of these new boundary conditions on the bifurcation processes.

A simplified two-mixture-fraction-based flamelet modelling and its validation on a non-premixed staged combustion system

A simplified two-mixture-fraction-based flamelet model is proposed in this work for a non-premixed staged combustion mode in which three feeds are introduced into the flow field. As the third stream is injected downstream of the main port, the system can be considered a special case of the two-mixture-fraction-based systems. It is considered that if a well-mixing is achieved in the upper stream prior to the third-stream injection, the simplification of the two-mixture-fraction-based flamelet model is feasible. In this work, a simplified model is proposed which can greatly reduce the library size compared to the complete two-mixture-fraction-based flamelet library providing identical resolutions for the fuel stream mixture fraction and the progress variable, respectively. Analysis associated with the interpolation strategy has been implemented. Extension of the current model for the cases in which the well-mixing state is not attained is also analysed, as well as the consideration of heat loss. To validate the model, two adiabatic cases of two-dimensional (2D) direct numerical simulations (DNS) have been performed in this work. To describe the chemical events, one is using finite-rate chemistry (FRC), while the other one is realised by means of the current flamelet model (FLM). An a priori test case that directly looks up the libraries by using the tracking parameters obtained from the FRC case is also considered. It is observed that the interpolation along the primary oxidiser mixture fraction direction outperforms that along directions for both fuel and primary oxidiser mixture fractions. It is also found that three one-mixture-fraction-based flamelet libraries which form the current model are sufficient for simulation of the non-premixed staged combustion, while the extended one which composes five libraries is expected to gain higher accurateness. In the FLM case, although the distributions of tracking parameters deviate from the FRC case slightly, good agreements can be obtained in terms of temperature and species mass fractions. The a priori test shows that the current model can reproduce the reacting flow accurately when the tracking parameters are identical to the FRC case. It is confirmed that the current model can be used to predict the characteristics of the reacting flow in the non-premixed staged combustion.

Spiral- and scroll-wave dynamics in mathematical models for canine and human ventricular tissue with varying Potassium and Calcium currents

We conduct a systematic, direct-numerical-simulation (DNS) study, in mathematical models for ventricular tissue, of the dependence of spiral- and scroll-wave dynamics on GKr, the maximal conductance of the delayed rectifier Potassium current (IKr) channel, and the parameter γCao, which determines the magnitude and shape of the current ICaL for the L-type calcium-current channel, in both square and anatomically realistic, whole-ventricle simulation domains. We study canine and human models. In the former, we use a canine-ventricular geometry, with fiber-orientation details, obtained from diffusion-tensor-magnetic-resonance-imaging (DTMRI) data; and we employ the physiologically realistic Hund-Rudy-Dynamic (HRD) model for a canine ventricular myocyte. To focus on the dependence of spiral- and scroll-wave dynamics on GKr and γCao, we restrict ourselves to an HRD-model parameter regime, which does not produce spiral- and scroll-wave instabilities because of other, well-studied causes like a very sharp action-potential-duration-restitution (APDR) curve or early after depolarizations (EADs) at the single-cell level. We find that spiral- or scroll-wave dynamics are affected predominantly by a simultaneous change in ICaL and IKr from their original values in the model, rather than by a change in any one of these currents; other currents do not have such a large effect on these wave dynamics in this parameter regime of the HRD model. In particular, we examine spiral-wave dynamics for ten different values of GKr and ten different values of γCao in our 2D DNSs. For our 3D DNSs in an anatomically realistic domain, we chose 16 parameter sets. In the parameter regime we begin with, the system displays broken spiral or scroll states with S1–S2 initial conditions (see below). We show that, by simultaneously increasing GKr and reducing γCao, we can get to a parameter regime in which the system displays single, stable rotating spirals or scroll waves. We obtain stability diagrams (or phase diagrams) in the GKr−γCao plane; and we find that these diagrams are significantly different in our 2D and 3D studies. In the 3D case, the geometry of the domain itself supports the confinement of the scroll waves and makes them more stable compared to their spiral-wave counterparts in our flat, 2D simulation domain. Thus, a combination of functional and geometrical mechanisms produce different dynamics for 3D scroll waves and their 2D spiral-wave counterparts. In particular, the former do not break easily because, in an anatomically realistic ventricular geometry, they are not easily absorbed at boundaries, nor do they break near boundaries. We have also carried out a comparison of our HRD results with their counterparts for the human-ventricular TP06 model; and we have found important differences between wave dynamics in these two models. The region in parameter space, where we obtain broken spiral or scroll waves in the HRD model is the region of stable rotating waves in the TP06 model; the default parameter values produce broken waves in the HRD model, but stable scrolls in the TP06 model. In both these models, to make a transition, (most simply, from broken-wave to stable-scroll states) we must simultaneously increase IKr and decrease ICaL; a modification of only one of these currents is not enough to effect this transition. Furthermore, the converse, i.e., an increase in ICaL along with a decrease in IKr does not yield any interesting dynamical transitions in the HRD model, for, in this range of currents, this model does not sustain spiral or scroll waves or broken waves.