Evaluations of SGS Combustion,Scalar Flux and Stress Models in a Turbulent Jet Premixed Flame

A newly developed fractal dynamic SGS (FDSGS) combustion model and a scale self-recognition mixed (SSRM) SGS stress model are evaluated along with other SGS combustion, scalar flux and stress models in a priori and a posteriori manners using DNS data of a hydrogen-air turbulent plane jet premixed flame. A posteriori tests reveal that the LES using the FDSGS combustion model can predict the combustion field well in terms of mean temperature distributions and peak positions in the transverse distributions of filtered reaction progress variable fluctuations. A priori and a posteriori tests of the scalar flux models show that a model proposed by Clark et al. accurately predicts the counter-gradient transport as well as the gradient diffusion, and introduction of the model of Clark et al. into the LES yields slightly better predictions of the filtered progress variable fluctuations than that of a gradient diffusion model. Evaluations of the stress models reveal that the LES with the SSRM model predicts the velocity fluctuations well compared to that with the Smagorinsky model.     

Effects of Lewis Number on Head on Quenching of Turbulent Premixed Flames: A Direct Numerical Simulation Analysis

The head on quenching of statistically planar turbulent premixed flames by an isothermal inert wall has been analysed using three-dimensional Direct Numerical Simulation (DNS) data for different values of global Lewis number Le(0.8, 1.0 and 1.2) and turbulent Reynolds number Re_t. The statistics of head on quenching have been analysed in terms of the wall Peclet number Pe (i.e. distance of the flame from the wall normalised by the Zel’dovich flame thickness) and the normalised wall heat flux Φ. It has been found that the maximum (minimum) value of Φ(Pe) for the turbulent Le=0.8 cases are greater (smaller) than the corresponding laminar value, whereas both Pe and Φ in turbulent cases remain comparable to the corresponding laminar values for Le=1.0 and 1.2. Detailed physical explanations are provided for the observed Le dependences of Pe and Φ. The existing closure of mean reaction rate \(\overline {\dot {\omega }}\) using the scalar dissipation rate (SDR) in the near wall region has been assessed based on a-priori analysis of DNS data and modifications to the existing closures of mean reaction rate and SDR have been suggested to account for the wall effects in such a manner that the modified closures perform well both near to and away from the wall.     

Strain,Rotation and Curvature of Non-material Propagating Iso-scalar Surfaces in Homogeneous Turbulence

This research aims at gaining some physical insight into the problem of scalar mixing, following the time evolution of propagating iso-surfaces, Y (x, t) = constant, where Y (x, t) stands for any scalar field (e.g., species mass fraction or temperature). First, a rigorous kinematic analysis of non-material line, surface and volume elements, related to propagating iso-scalar surfaces, is presented; this formalism is valid for both constant and variable density flows. Time rates of change of the normal distance and volume between two adjacent iso-surfaces and of area elements, rotation rates of lines and surface elements and an evolution equation for the local mean curvature are obtained. Line and area stretch rates, which encompass additive contributions from the flow and the displacement speed (due to diffusion and reaction), are identified as total strain rates, normal and tangential to the iso-surfaces. Volumetric dilatation rates, addition of line plus area stretch rates, include the mass entrainment rate per unit mass into the non-material volume. Flow and added vorticities, the latter due to gradients of the displacement speed, yield the total vorticity, which provides the real angular velocity of lines and surface elements. A 512³ DNS database for the mixing of inert and reactive scalars in a box of forced statistically stationary and homogeneous turbulence of a constant-density fluid is then examined. A strongly segregated scalar field is prescribed as initial condition. A one-step reaction rate with a characteristic chemical time one order of magnitude greater than the Kolmogorov time micro-scale is used. Data are analyzed at 1.051 large-eddy turnover times after initialization of velocity and scalar fields. Mean negative normal (contractive) and positive tangential (stretching) flow strain rates occur over all mass fractions and scalar-gradient magnitudes. However, means of the total normal strain rate, conditional upon mass fraction, scalar-gradient and mean curvature, are positive everywhere and tend to destroy scalar-gradients for small times. Negative conditioned mean total tangential strain rates (area stretch factor) contract local areas, except for large values of scalar-gradients. Conditional averages of total and added enstrophies are almost identical, which implies a negligible contribution of the flow vorticity to the observed rotation of non-material line and surface elements. The added vorticity is exactly tangential to the iso-surfaces, whereas the flow and total ones are predominantly tangential. Flow sources/sinks of the mean curvature transport equation are much smaller than the added contributions; for this particular DNS database, the local mean curvature development is self-induced by spatial changes of the displacement speed.     

Statistical Analysis of Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

Statistically planar turbulent premixed and partially premixed flames for different initial turbulence intensities are simulated for global equivalence ratios ??>?=?0.7 and ??>?=?1.0 using three-dimensional Direct Numerical Simulations (DNS) with simplified chemistry. For the simulations of partially premixed flames, a random distribution of equivalence ratio following a bimodal distribution of equivalence ratio is introduced in the unburned reactants ahead of the flame. The simulation parameters in all of the cases were chosen such that the combustion situation belongs to the thin reaction zones regime. The DNS data has been used to analyse the behaviour of the dissipation rate transports of both active and passive scalars (i.e. the fuel mass fraction Y _F and the mixture fraction ξ) in the context of Reynolds Averaged Navier–Stokes (RANS) simulations. The behaviours of the unclosed terms of the Favre averaged scalar dissipation rates of fuel mass fraction and mixture fraction (i.e. \(\widetilde {\varepsilon }_Y =\overline {\rho D\nabla Y_F^{\prime \prime } \cdot \nabla Y_F^{\prime \prime } } /\overline{\rho }\) and \(\widetilde {\varepsilon }_\xi =\overline {\rho D\nabla \xi ^{\prime \prime }\cdot \nabla \xi ^{\prime \prime }} /\overline {\rho })\) transport equations have been analysed in detail. In the case of the \(\widetilde {\varepsilon }_Y \) transport, it has been observed that the turbulent transport term of scalar dissipation rate remains small throughout the flame brush whereas the terms due to density variation, scalar–turbulence interaction, reaction rate and molecular dissipation remain the leading order contributors. The term arising due to density variation remains positive throughout the flame brush and the combined contribution of the reaction and molecular dissipation to the \(\widetilde {\varepsilon }_Y \) transport remains negative throughout the flame brush in all cases. However, the behaviour of scalar–turbulence interaction term of the \(\widetilde {\varepsilon }_Y \) transport equation is significantly affected by the relative strengths of turbulent straining and the straining due to chemical heat release. In the case of the \(\widetilde {\varepsilon }_\xi \) transport, the turbulent transport term remains small throughout the flame brush and the density variation term is found to be negligible in all cases, whilst the reaction rate term is exactly zero. The scalar–turbulence interaction term and molecular dissipation term remain the leading order contributors to the \(\widetilde {\varepsilon }_\xi \) transport throughout the flame brush in all cases that have been analysed in the present study. Performances of existing models for the unclosed terms of the transport equations of \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) are assessed with respect to the corresponding quantities obtained from DNS data. Based on this exercise either suitable models have been identified or new models have been proposed for the accurate closure of the unclosed terms of both \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) transport equations in the context of Reynolds Averaged Navier–Stokes (RANS) simulations.     

Analysis of the Combined Modelling of Sub-grid Transport and Filtered Flame Propagation for Premixed Turbulent Combustion

Flame surface density (FSD) based reaction rate closure is an important methodology of turbulent premixed flame modelling in the context of Large Eddy Simulations (LES). The transport equation for the Favre-filtered reaction progress variable needs closure of the filtered reaction diffusion imbalance (FRDI) term (i.e. filtered value of combined reaction rate and molecular diffusion rate) and the sub-grid scalar flux (SGSF). A-priori analysis of the FRDI and SGSF terms has in the past revealed advantages and disadvantages of the specific modelling attempts. However, it is important to understand the interaction of the FRDI and SGSF closures for a successful implementation of the FSD based closure. Furthermore, it is not known a-priori if the combination of the best SGSF model with the best FRDI model results in the most suitable overall modelling strategy. In order to address this question, a variety of SGSF models is analysed in this work together with one well established and one recent FRDI closure based on a-priori analysis. It is found that the success of the combined FRDI and SGSF closures depends on subtle details like the co-variances of the FRDI and SGSF terms. It is demonstrated that the gradient hypothesis model is not very successful in representing the SGSF term. However the gradient hypothesis provides satisfactory performance in combination of a recently proposed FRDI closure, whereas unsatisfactory results are obtained when used in combination with another existing closure, which was shown to predict the FRDI term satisfactorily in several previous analyses.     

Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations

Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component u _j of the velocity field u is determined by the scalar θ through \({u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}\) , where \({\mathcal{R}}\) is a Riesz transform and Λ = (?Δ)^1/2. The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound \({\|\nabla u||_{L^\infty}}\) for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log(I + log(I ? Δ))) ^γ with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ ^β with 0 ≦ β ≦ 1 is also obtained.     

Pre-Chamber Ignition Mechanism: Simulations of Transient Autoignition in a Mixing Layer Between Reactants and Partially-Burnt Products

The structure of autoignition in a mixing layer between fully-burnt or partially-burnt combustion products from a methane-air flame at ? = 0.85 and a methane-air mixture of a leaner equivalence ratio has been studied with transient diffusion flamelet calculations. This configuration is relevant to scavenged pre-chamber natural-gas engines, where the turbulent jet ejected from the pre-chamber may be quenched or may be composed of fully-burnt products. The degree of reaction in the jet fluid is described by a progress variable c (c = taking values 0.5, 0.8, and 1.0) and the mixing by a mixture fraction ξ (ξ = 1 in the jet fluid and 0 in the CH₄-air mixture to be ignited). At high scalar dissipation rates, N₀, ignition does not occur and a chemically-frozen steady-state condition emerges at long times. At scalar dissipation rates below a critical value, ignition occurs at a time that increases with N₀. The flame reaches the ξ = 0 boundary at a finite time that decreases with N₀. The results help identify overall timescales of the jet-ignition problem and suggest a methodology by which estimates of ignition times in real engines may be made.     

Direct Numerical Simulation of Head-On Quenching of Statistically Planar Turbulent Premixed Methane-Air Flames Using a Detailed Chemical Mechanism

A three-dimensional compressible Direct Numerical Simulation (DNS) analysis has been carried out for head-on quenching of a statistically planar stoichiometric methane-air flame by an isothermal inert wall. A multi-step chemical mechanism for methane-air combustion is used for the purpose of detailed chemistry DNS. For head-on quenching of stoichiometric methane-air flames, the mass fractions of major reactant species such as methane and oxygen tend to vanish at the wall during flame quenching. The absence of \(\text {OH}\) at the wall gives rise to accumulation of carbon monoxide during flame quenching because \(\text {CO}\) cannot be oxidised anymore. Furthermore, it has been found that low-temperature reactions give rise to accumulation of \(\text {HO}_{2}\) and \(\mathrm {H}_{2}\mathrm {O}_{2}\) at the wall during flame quenching. Moreover, these low temperature reactions are responsible for non-zero heat release rate at the wall during flame-wall interaction. In order to perform an in-depth comparison between simple and detailed chemistry DNS results, a corresponding simulation has been carried out for the same turbulence parameters for a representative single-step Arrhenius type irreversible chemical mechanism. In the corresponding simple chemistry simulation, heat release rate vanishes once the flame reaches a threshold distance from the wall. The distributions of reaction progress variable c and non-dimensional temperature T are found to be identical to each other away from the wall for the simple chemistry simulation but this equality does not hold during head-on quenching. The inequality between c (defined based on \(\text {CH}_{4}\) mass fraction) and T holds both away from and close to the wall for the detailed chemistry simulation but it becomes particularly prominent in the near-wall region. The temporal evolutions of wall heat flux and wall Peclet number (i.e. normalised wall-normal distance of \(T = 0.9\) isosurface) for both simple and detailed chemistry laminar and turbulent cases have been found to be qualitatively similar. However, small differences have been observed in the numerical values of the maximum normalised wall heat flux magnitude \(\left ({\Phi }_{\max } \right )_{\mathrm {L}}\) and the minimum Peclet number \((Pe_{\min })_{\mathrm {L}}\) obtained from simple and detailed chemistry based laminar head-on quenching calculations. Detailed explanations have been provided for the observed differences in behaviours of \(\left ({\Phi }_{\max }\right )_{\mathrm {L}}\) and \((Pe_{\min })_{\mathrm {L}}\). The usual Flame Surface Density (FSD) and scalar dissipation rate (SDR) based reaction rate closures do not adequately predict the mean reaction rate of reaction progress variable in the near-wall region for both simple and detailed chemistry simulations. It has been found that recently proposed FSD and SDR based reaction rate closures based on a-priori DNS analysis of simple chemistry data perform satisfactorily also for the detailed chemistry case both away from and close to the wall without any adjustment to the model parameters.     

Assessment of Five SGS Models for Low-Rem MHD Turbulence

Assessment of three regularization-based and two eddy-viscosity-based subgrid-scale (SGS) turbulence models for large eddy simulations (LES) are carried out in the context of magnetohydrodynamic (MHD) decaying homogeneous turbulence (DHT) with a Taylor scale Reynolds number (Re_λ) of 120 and a MHD transition-to-turbulence Taylor-Green vortex (TGV) problems with a Reynolds number of 3000, through direct comparisons to direct numerical simulations (DNS). Simulations are conducted using the low-magnetic Reynolds number approximation (Re_m<<1). LES predictions using the regularization-based Leray- α,LANS- α, and Clark- α SGS models, along with the eddy viscosity-based non-dynamic Smagorinsky and the dynamic Smagorinsky models are compared to in-house DNS for DHT and previous results for TGV. With regard to the regularization models, this work represents their first application to MHD turbulence. Analyses of turbulent kinetic energy decay rates, energy spectra, and vorticity fields made between the varying magnetic field cases demonstrated that the regularization models performed poorly compared to the eddy-viscosity models for all MHD cases, but the comparisons improved with increase in magnitude of magnetic field, due to a decrease in the population of SGS eddies within the flow field.     

Experimental and Theoretical Sensitivity Analysis of Turbulent Flow Past a Square Cylinder

The Siemens SGT-800 3^rd generation DLE burner fitted to an atmospheric combustion rig has been numerically investigated. Pure methane and methane enriched by 80 vol% hydrogen flames have been considered. A URANS (Unsteady Reynolds Averaged Navier-Stokes) approach was used in this study along with the k ? ω SST and the k ? ω SST-SAS models for the turbulence transport. The chemistry is coupled to the turbulent flow simulations by the use of a laminar flamelet library combined with a presumed PDF. The effect of the mesh density in the mixing and the flame region and the effect of the turbulence model and reaction rate model constant are first investigated for the methane/air flame case. The results from the k ? ω SST-SAS along with flamelet libraries are shown to be in excellent agreement with experimental data, whereas the k ? ω SST model is too dissipative and cannot capture the unsteady motion of the flame. The k ? ω SST-SAS model is used for simulation of the 80 vol% hydrogen enriched flame case without further adjusting the model constants. The global features of the hydrogen enrichment are very well captured in the simulations using the SST-SAS model. With the hydrogen enrichment the time averaged flame front location moves upstream towards the burner exit nozzle. The results are consistent with the experimental observations. The model captures the three dominant low frequency unsteady motion observed in the experiments, indicating that the URANS/LES hybrid model indeed is capable of capturing complex, time dependent, features such as an interaction between a PVC and the flame front.     

Effects of Fuel Lewis Number on Localised Forced Ignition of Turbulent Mixing Layers

The influences of fuel Lewis number Le _F on localised forced ignition of inhomogeneous mixtures are analysed using three-dimensional compressible Direct Numerical Simulations (DNS) of turbulent mixing layers for Le _F  = 0.8, 1.0 and 1.2 and a range of different root-mean-square turbulent velocity fluctuation u′ values. For all Le _F cases a tribrachial flame has been observed in case of successful ignition. However, the lean premixed branch tends to merge with the diffusion flame on the stoichiometric mixture fraction isosurface at later stages of the flame evolution. It has been observed that the maximum values of temperature and reaction rate increase with decreasing Le _F during the period of external energy addition. Moreover, Le _F is found to have a significant effect on the behaviours of mean temperature and fuel reaction rate magnitude conditional on mixture fraction values. It is also found that reaction rate and mixture fraction gradient magnitude \(\vert \nabla \xi \vert \) are negatively correlated at the most reactive region for all values of Le _F explored. The probability of finding high values of \(\vert \nabla \xi \vert \) increases with increasing Le _F . For a given value of u′, the extent of burning decreases with increasing Le _F . A moderate increase in u′ gives rise to an increase in the extent of burning for Le _F  = 0.8 and 1.0, which starts to decrease with further increases in u′. For Le _F  = 1.2, the extent of burning decreases monotonically with increasing u′. The extent of edge flame propagation on the stoichiometric mixture fraction ξ = ξ _st isosurface is characterised by the probability of finding burned gas on this isosurface, which decreases with increasing u′ and Le _F . It has been found that it is easier to obtain self-sustained combustion following localised forced ignition in case of inhomogeneous mixtures than that in the case of homogeneous mixtures with the same energy input, energy deposition duration when the ignition centre is placed at the stoichiometric mixture. The difficultly to sustain combustion unaided by external energy addition in homogeneous mixture is particularly prevalent in the case of Le _F  = 1.2.     

Spray-Flame Dynamics in a Rich Droplet Array

We numerically study spray-flame dynamics. The initial state of the spray is schematized by alkane droplets located at the nodes of a face-centered 2D-lattice. The droplets are surrounded by a gaseous mixture of alkane and air. The lattice spacing s reduced by the combustion length scale is large enough to consider that the chemical reaction occurs in a heterogeneous medium. The overall spray equivalence ratio is denoted by ?_T, with ?_T = ?_L + ?_G, where ?_G corresponds to the equivalence ratio of the gaseous surrounding mixture at the initial saturated partial pressure, while ?_L is the so-called liquid loading. To model such a heterogenous combustion, the retained chemical scheme is a global irreversible one-step reaction governed by an Arrhenius law, with a modified heat of reaction depending on the local equivalence ratio. ?_T is chosen in the range 0.9 ≤ ?_T ≤ 2. Three geometries (s = 3, s = 6, s = 12) and four liquid loadings, ?_L = 0.3, ?_L = 0.5, ?_L = 0.7, ?_L = 0.85 are studied. In the rich sprays, our model qualitatively retrieves the recent experimental measurements: the rich spray-flames can propagate faster than the single-phase flames with the same overall equivalence ratio. To analyse the conditions for this enhancement, we introduce the concept of “spray Peclet number”, which compares the droplet vaporization time with the combustion propagation time of the single-phase flame spreading in the fresh surrounding mixture.     

Kovasznay Mode Decomposition of Velocity-Temperature Correlation in Canonical Shock-Turbulence Interaction

The correlation coefficient R_uT between the streamwise velocity and temperature is investigated for the case of canonical shock-turbulence interaction, motivated by the fact that this correlation is an important component in compressible turbulence models. The variation of R_uT with the Mach number, the turbulent Mach number, and the Reynolds number is predicted using linear inviscid theory and compared to data from DNS. The contributions from the individual Kovasznay modes are quantified. At low Mach numbers, the peak post-shock R_uT is determined by the acoustic mode, which is correctly predicted by the linear theory. At high Mach numbers, it is determined primarily by the vorticity and entropy modes, which are strongly affected by nonlinear and viscous effects, and thus less well predicted by the linear theory.     

Effects of Fuel Lewis Number on Localised Forced Ignition of Globally Stoichiometric Stratified Mixtures: a Numerical Investigation

The influences of fuel Lewis number Le_F on localised forced ignition of globally stoichiometric stratified mixtures have been analysed using three-dimensional compressible Direct Numerical Simulations (DNS) for cases with Le_F ranging from 0.8 to 1.2. The globally stoichiometric stratified mixtures with different values of root-mean-square (rms) equivalence ratio fluctuation (i.e. ?^′= 0.2, 0.4 and 0.6) and the Taylor micro-scale l_? of equivalence ratio ? variation (i.e. l_?/l_f= 2.1, 5.5 and 8.3 with l_f being the Zel’dovich flame thickness of the stoichiometric laminar premixed flame) have been considered for different initial rms values of turbulent velocity u^′. A pseudo-spectral method is used to initialise the equivalence ratio variation following a presumed bi-modal distribution for prescribed values of ?^′ and l_?/l_f for global mean equivalence ratio 〈?〉=1.0. The localised ignition is accounted for by a source term in the energy transport equation that deposits energy for a stipulated time interval. It has been observed that the maximum values of temperature and the fuel reaction rate magnitude increase with decreasing Le_F during the period of external energy deposition. The initial values of Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f have been found to have significant effects on the extent of burning of the stratified mixtures following localised ignition. For a given value of u^′/S_b(?=1), the extent of burning decreases with increasing Le_F. An increase in u^′ leads to a monotonic reduction in the burned gas mass for all values of Le_F in all stratified mixture cases but an opposite trend is observed for the Le_F=0.8 homogeneous mixture. It has been found that an increase in ?^′ has adverse effects on the burned gas mass, whereas the effects of l_?/l_f on the extent of burning are non-monotonic and dependent on ?^′ and Le_F. Detailed physical explanations have been provided for the observed Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f dependences.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Turbulent Drag Reduction by a Near Wall Surface Tension Active Interface

In this work we study the turbulence modulation in a viscosity-stratified two-phase flow using Direct Numerical Simulation (DNS) of turbulence and the Phase Field Method (PFM) to simulate the interfacial phenomena. Specifically we consider the case of two immiscible fluid layers driven in a closed rectangular channel by an imposed mean pressure gradient. The present problem, which may mimic the behaviour of an oil flowing under a thin layer of different oil, thickness ratio h₂/h₁ =?9, is described by three main flow parameters: the shear Reynolds number Re _τ (which quantifies the importance of inertia compared to viscous effects), the Weber number We (which quantifies surface tension effects) and the viscosity ratio λ = ν₁/ν₂ between the two fluids. For this first study, the density ratio of the two fluid layers is the same (ρ₂ = ρ₁), we keep Re _τ and We constant, but we consider three different values for the viscosity ratio: λ =?1, λ =?0.875 and λ =?0.75. Compared to a single phase flow at the same shear Reynolds number (Re _τ =?100), in the two phase flow case we observe a decrease of the wall-shear stress and a strong turbulence modulation in particular in the proximity of the interface. Interestingly, we observe that the modulation of turbulence by the liquid-liquid interface extends up to the top wall (i.e. the closest to the interface) and produces local shear stress inversions and flow recirculation regions. The observed results depend primarily on the interface deformability and on the viscosity ratio between the two fluids (λ).     

Application of the γ-Reθ Transition Model to Simulations of the Flow Past a Circular Cylinder

Based on the finite volume method, the flow past a two-dimensional circular cylinder at a critical Reynolds number (Re = 8.5 × 10⁵) was simulated using the Navier-Stokes equations and the γ-Re_θ transition model coupled with the SST k ? ω turbulence model (hereinafter abbreviated as γ-Re_θ model). Considering the effect of free-stream turbulence intensity decay, the SST k ? ω turbulence model was modified according to the ambient source term method proposed by Spalart and Rumsey, and then the modified SST k ? ω turbulence model is coupled with the γ-Re_θ transition model (hereinafter abbreviated as γ-Re_θ-SR model). The flow past a circular cylinder at different inlet turbulence intensities were simulated by the γ-Re_θ-SR model. At last, the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers were each simulated by the γ-Re_θ-SR model, and the three flow states were analyzed. It was found that compared with the SST k ? ω turbulence model, the γ-Re_θ model could simulate the transition of laminar to turbulent, resulting in better consistency with experimental result. Compared with the γ-Re_θ model, for relatively high inlet turbulence intensities, the γ-Re_θ-SR model could better simulate the flow past a circular cylinder; however the improvement almost diminished for relatively low inlet turbulence intensities The γ-Re_θ-SR model could well simulate the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers.     

Families of rational solutions of the <Emphasis Type="Italic">y</Emphasis>-nonlocal Davey–Stewartson II equation

The y-nonlocal Davey–Stewartson II equation is an extension of the usual DS II equation involving a partially parity-time-symmetric potential only with respect to the spatial variable y. By using the Hirota bilinear method, families of n-order rational solutions are obtained, which include lumps in the (x, y)-plane and the (y, t)-plane, growing-and-decaying line waves in the (x, t)-plane, and hybrid solutions of interacting line rogue waves and lumps in the (x, y)-plane.