Lower Bounds on Scalar Dissipation in Bounded Rectilinear Flows

Lower bounds on the scalar dissipation rate in rectilinear flows are found for three sets of constraints. The bounding dissipation rates provide upper bound estimates of the scalar concentration variance. One of the lower bounds on the dissipation rate is close to the dissipation rate assessed from detailed numerical simulation. This proximity enables a simplified model of concentration variance. For a pulse input of solute, the predictions of concentration variance by the simplified model are in agreement with the results of detailed simulation. The large-time bimodality of the concentration variance and its rate of decay observed in the detailed numerical simulations is predicted by the simplified model.     

Computation of Conditional Average Scalar Dissipation in Turbulent Jet Diffusion Flames

The modelling of conditional scalar dissipation in locally self-similar turbulent reacting jets is considered. The streamwise dependence in the transport equation of the conserved scalar pdf is represented by a function solely dependent on centreline mixture fraction. This procedure provides a simple model suitable for non-homogeneous flows and ensures positive values for conditional scalar dissipation. It has been tested in pure hydrogen-air jet diffusion flames using a Conditional Moment Closure method with detailed 12species, 23 reactions chemistry. The calculations show good agreement of the averaged scalar dissipation with reference values and the model proves to be superior to previous models based on homogeneous flows if the distribution of the conditional scalar dissipation in mixture fraction space is compared with experimental results. A dependence of NO predictions on the model of conditional scalar dissipation can be observed. This revised version was published online in July 2006 with corrections to the Cover Date.     

A-Priori Validation of Scalar Dissipation Rate Models for Turbulent Non-Premixed Flames

The modelling of scalar dissipation rate in conditional methods for large-eddy simulations is investigated based on a priori direct numerical simulation analysis using a dataset representing an igniting non-premixed planar jet flame. The main objective is to provide a comprehensive assessment of models typically used for large-eddy simulations of non-premixed turbulent flames with the Conditional Moment Closure combustion model. The linear relaxation model gives a good estimate of the Favre-filtered scalar dissipation rate throughout the ignition with a value of the related constant close to the one deduced from theoretical arguments. Such value of the constant is one order of magnitude higher than typical values used in Reynolds-averaged approaches. The amplitude mapping closure model provides a satisfactory estimate of the conditionally filtered scalar dissipation rate even in flows characterised by shear driven turbulence and strong density variation.

     

Measurements of Scalar Variance, Scalar Dissipation, and Length Scales in Turbulent Piloted Methane/Air Jet Flames

One-dimensional (line) measurements of mixture fraction, temperature, and scalar dissipation in piloted turbulent partially premixed methane/air jet flames (Sandia flames C, D, and E) are presented. The experimental facility combines line imaging of Raman scattering, Rayleigh scattering, and laser-induced CO fluorescence. Simultaneous single-shot measurements of temperature and the mass fractions of all the major species (N₂, O₂, CH₄, CO₂, H₂O, CO, and H₂) are obtained along 7 mm segments with a nominal spatial resolution of 0.2 mm. Mixture fraction, ξ, is then calculated from the measured mass fractions. Ensembles of instantaneous mixture fraction profiles at several streamwise locations are analyzed to quantify the effect of spatial averaging on the Favre average scalar variance, which is an important term in the modeling of turbulent nonpremixed flames. Results suggest that the fully resolved scalar variance may be estimated by simple extrapolation of spatially filtered measurements. Differentiation of the instantaneous mixture fraction profiles yields the radial contribution to the scalar dissipation, χ _r = 2D _ξ(?ξ/?r)², and radial profiles of the Favre mean and rms scalar dissipation are reported. Scalar length scales, based on autocorrelation of the spatial profiles of ξ and χ _r , are also reported. These new data on this already well-documented series of flames should be useful in the context of validating models for sub-grid scalar variance and for scalar dissipation in turbulent flames.     

DNS and Modelling of Passive Scalar Transport in Turbulent Channel Flow with a Focus on Scalar Dissipation Rate Modelling

A direct numerical simulation of turbulent channel flow with an imposed mean scalar gradient is analyzed with a focus on passive scalar flux modelling and in particular the treatment of the passive scalar dissipation equation. The Prandtl number is 0.71 and the Reynolds number based on the wall friction velocity and the channel half width is 265. Budgets are presented for the passive scalar variance and its dissipation rate, as well as for the individual scalar flux components. These form a basis for a discussion of modelling issues related to explicit algebraic scalar flux modelling. This revised version was published online in July 2006 with corrections to the Cover Date.     

Measurements of the Statistical Distribution of the Scalar Dissipation Rate in Turbulent Axisymmetric Plumes

In this paper we reconsider the high resolution Planar Laser-Induced Fluorescence (PLIF) mixing measurements that were presented in Markides and Mastorakos (Chem. Eng. Sci. 61:2835–2842, 2006). The PLIF experiments were performed in flows created by the continuous injection, with various velocity ratios, of a passive scalar (acetone) from a finite sized round nozzle, into uniform turbulent co-flows confined within a cylindrical tube with a range of turbulent Reynolds numbers. Here, we extend our study of these flows to include an investigation of the Probability Density Functions (PDFs) of the scalar dissipation rate (χ), as well as of the scalar dissipation rate conditional on mixture fraction (χ|ξ). To this effect, we have further processed the resulting data from the earlier work, including an additional correction for density variations in the flow. The results were then reprocessed for the accurate recovery of the spatial gradients of the normalized scalar concentration (ξ), which resulted in direct measurements of χ. All results that are presented in this paper involve the two-dimensional scalar dissipation rate (χ _2D ) evaluated in the plane of imaging, and furthermore, results are only shown and discussed along the centreline of our axisymmetric configuration. The PDFs of χ were calculated and found to deviate slightly from a lognormal distribution, consistent with other published work. Moreover, the PDFs of χ|ξ followed a similar distribution, and showed a similar deviation. This deviation was more pronounced closer to the source and decayed downstream. The ratio of the standard deviation over the mean of χ and χ|ξ increased with streamwise distance and reached values of 2.8 and 1.2 respectively.     

A Posteriori Assessment of Algebraic Scalar Dissipation Models for RANS Simulation of Premixed Turbulent Combustion

This paper examines the effects of scalar dissipation rate modelling on mean reaction rate predictions in turbulent premixed flames. The sensitivity of the mean reaction rate is explored by using different closures for scalar dissipation and the sensitivity of the scalar dissipation models themselves is also examined with respect to their defining constants. The influence of different scalar dissipation models on the flame location and mean velocities is reported and compared with experimental results. The predicted reaction rate is found to be sensitive to the choice of closure used for scalar dissipation and also to the respective constants used in the scalar dissipation models. It is also found that the scalar dissipation models involving chemical and turbulent time scales yield a more physically plausible reaction rate when compared with the scalar dissipation models relying only on the turbulent time scale.     

Linear-Eddy Model Formulated Probability Density Function and Scalar Dissipation Rate Models for Premixed Combustion

A tabulated, pseudo-turbulent Probability Density Function (PDF) model for premixed combustion is proposed. The Linear-Eddy Model (LEM) is used to construct the PDFs for a temperature-based progress variable in a premixed, turbulent methane/air V-flame produced by the Cambridge slot burner. As a second case study, the LEM PDFs are similarly compared to PDFs extracted from Direct Numerical Simulations (DNS) of a turbulent premixed flame. LEM demonstrates the ability to reproduce the salient features from experimental and DNS PDFs; moreover, it is able to better capture turbulent effects than previously suggested laminar flamelet PDF models. The Scalar Dissipation Rate (SDR) for premixed combustion is likewise investigated. The stochastic nature of LEM enables it to mimic the overall behaviors of turbulent reactions inexpensively and qualitatively. Crucially, LEM appears to be well suited for the preprocessing tabulation of PDF and SDR models for a number of premixed combustion simulation strategies.     

Some Fundamental Aspects of Surface Modelling

Attention is drawn to four aspects of surface modelling: (1) delineation of the (scale-dependent) geometrical boundary of a body via molecular considerations, (2) identification of the highly inhomogeneous interfacial region between a body and its exterior, and its modelling as a bidimensional continuum involving interfacial excess quantities, (3) the utility of co-ordinate-free notation for surfaces, and (4) the importance of surface effects for small-scale bodies exemplified within a thermoelastic context.     

Noise Correction and Length Scale Estimation for Scalar Dissipation Rate Measurements in Turbulent Partially Premixed Flames

A recently developed conditional sampling-based method for correcting noise effects in scalar dissipation rate measurements and for estimating the extent of resolution of the dissipation rate is employed to analyze the data obtained in turbulent partially premixed (Sandia) flames. The method uses conditional sampling to select instantaneous fully resolved local scalar fields, which are analyzed to determine the measurement noise and to correct the Favre mean, conditional, and conditionally filtered dissipation rates. The potentially under-resolved local scalar fields, also selected using conditional sampling, are corrected for noise and are analyzed to examine the extent of resolution. The error function is used as a model for the potentially under-resolved local scalar to evaluate the scalar dissipation length scales and the percentage of the dissipation resolved. The results show that the Favre mean dissipation rate, the mean dissipation rate conditional on the mixture fraction, and dissipation rate filtered conditionally on the mixture fraction generally are well resolved in the flames. Analyses of the dissipation rates filtered conditionally on the mixture fraction and temperature show that the length scale increases with temperature, due to lower dissipation rate and higher diffusivity. The dissipation rate is well resolved for temperatures above 1,300 K but is less resolved at lower temperatures, although the probability of very low temperature events is low. To fully resolve these rare events the sample spacing needs to be reduced by approximately one half. The present study further demonstrates the effectiveness of the new noise correction and length scale estimation method.     

Direct Numerical Simulation of Inert Droplet Effects on Scalar Dissipation Rate in Turbulent Reacting and Non-Reacting Shear Layers

Three-dimensional direct numerical simulation has been performed to investigate the effects of inert evaporating droplets on scalar dissipation rate χ in temporally-developing turbulent reacting and non-reacting mixing layers with the Reynolds number based on the vorticity thickness up to 8000 and the number of traced Lagrangian droplets up to 10⁷. The detailed instantaneous field analysis and ensemble-averaged statistics reveal complex interactions among combustion, droplet dynamics and evaporation, all of which have a considerable influence on χ. The presence of inert evaporating droplets promotes χ in both non-reacting and reacting mixing layers. In the latter, combustion reduces χ, so when combustion is suppressed by evaporating droplets, χ is enhanced. The transport equation of χ has been analyzed to investigate the various effects on χ in detail. The terms in the equation contain explicitly the evaporation rate and its spatial derivative, acting as a sink and a source for χ, respectively. On the whole, the net effect of the evaporation-rate terms is to promote χ. However, the production and dissipation terms are the dominant source and sink terms, respectively.     

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 Algebraic Closures of the Mean Scalar Dissipation Rate of the Progress Variable Applied to Stagnating Turbulent Flames

In the turbulent premixed reactive flows considered in this study, i.e. large Damköhler and Reynolds numbers, the flamelet regime of turbulent combustion applies and the scalar dissipation rate and mean reaction rate are inter related. In this situation various algebraic models for the mean chemical rate that are obtained from an equilibrium of the dominant terms of the transport equation for the scalar dissipation rate, are evaluated through their application to flames stabilized in a turbulent stagnating flow. An asymptotic analysis is first performed and results obtained through the resulting one-dimensional calculation are compared with the experimental data of Li et al. (Proc Combust Inst 25:1207–1214, 1994). Eventually, three-dimensional CFD calculations including suited algebraic closures to represent the turbulent transport terms are carried out. Results are satisfactorily compared to the experimental data of Cho et al. (Proc Combust Inst 22:739–745, 1988). As a first outcome, the analysis confirms the interest and the relevance of the corresponding algebraic closures to deal with turbulent premixed combustion in such conditions. In the search of a satisfactory representation of such premixed impinging flames, the computational results also clearly emphasize the strong intertwinment that exits between the mean reaction rate, i.e. scalar dissipation rate or micro-mixing taking place at the smallest scale of the reactive flowfield, and the Reynolds fluxes modelling, i.e. turbulent macro-mixing.     

Scalar Dissipation Rate Transport in the Context of Large Eddy Simulations for Turbulent Premixed Flames with Non-Unity Lewis Number

The effects of global Lewis number Le on the statistical behaviour of the unclosed terms in the transport equation of the Favre-filtered scalar dissipation rate (SDR) Ñ _c have been analysed using a Direct Numerical Simulation (DNS) database of freely propagating statistically planer turbulent premixed flames with Le ranging from 0.34 to 1.2. The DNS data has been explicitly filtered to analyse the statistical behaviour of the unclosed terms in the SDR transport equation arising from turbulent transport T ₁, density variation due to heat release T ₂, scalar-turbulence interaction T ₃, reaction rate gradient T ₄, molecular dissipation (?D ₂) and diffusivity gradients f(D) in the context of Large Eddy Simulations (LES). It Le has significant effects on the magnitudes of T ₁, T ₂, T ₃, T ₄, (?D ₂) and f(D). Moreover, both qualitative and quantitative behaviours of the unclosed terms T ₁, T ₂, T ₃, T ₄, (?D ₂) and f(D) are found to be significantly affected by the LES filter width Δ, which have been explained based on a detailed scaling analysis. Both scaling analysis and DNS data suggest that T ₂, T ₃, T ₄, (?D ₂) and f(D) remain leading order contributors to the SDR \(\tilde {{N}}_{c} \) transport for LES. The scaling estimates of leading order contributors to the SDR \(\tilde {{N}}_{c} \) transport has been utilised to discuss the possibility of extending an existing SDR model for Reynolds Averaged Navier Stokes (RANS) simulation for SDR \(\tilde {{N}}_{c} \) closure in the context of LES of turbulent premixed combustion.     

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

Statistically planar turbulent partially premixed flames for different initial intensities of decaying turbulence have been simulated for global equivalence ratios <????> = 0.7 and <????> = 1.0 using three-dimensional simplified chemistry based Direct Numerical Simulations (DNS). The simulation parameters are chosen such that the combustion situation belongs to the thin reaction zones regime and a random bi-modal distribution of equivalence ratio ?? is introduced in the unburned gas ahead of the flame to account for mixture inhomogeneity. The DNS data has been used to analyse the statistical behaviour of the transport of the cross-scalar dissipation rate based on the fuel mass fraction Y _F and the mixture fraction ?? fluctuations $\,\tilde{\varepsilon}_{Y\xi}={\overline{\rho D\nabla Y_{F}^{\prime\prime}.\nabla \xi^{\prime\prime}} } \big/ {\bar {\rho }}$ (where $\bar{q}$ , $\tilde{q}={\overline{\rho q} } \big/ {\bar {\rho }}$ and $q^{\prime\prime} =q-\tilde {q}$ are Reynolds average, Favre mean and Favre fluctuation of a general quantity q) in the context of Reynolds Averaged Navier?CStokes simulations where ?? is the gas density and D is the gas diffusivity. The statistical behaviours of the unclosed terms in the $\tilde{\varepsilon }_{Y\xi } $ transport equation originating from turbulent transport T ₁, density variation T ₂, scalar?Cturbulence interaction T ₃, chemical reaction rate T ₄ and the molecular dissipation rate D ₂ have been analysed in detail. It has been observed that the contributions of T ₂, T ₃, T ₄ and D ₂ play important roles in the $\tilde{\varepsilon }_{Y\xi } $ transport for the globally stoichiometric cases, but in the globally fuel-lean cases the contributions of T ₂ and T ₄ become relatively weaker in comparison to the contributions of T ₃ and D ₂. The term T ₁ remains small in comparison to the leading order contributions of T ₃ and D ₂ for all cases, but the contribution of T ₁ plays a more important role in the low Damköhler combustion cases. The term T ₂ behaves as a sink term towards the unburned gas side but becomes a source term towards the burned gas side. The scalar?Cturbulence interaction term T ₃ has been found to be generally positive throughout the flame brush, but in globally stoichiometric cases the contribution of T ₃ becomes negative in regions of intense heat release. The combined contribution of (T ₄ ?C D ₂) remains mostly as a sink in all cases studied here. Models are proposed for the unclosed terms of the $\tilde{\varepsilon }_{Y\xi } $ transport equation in the context of Reynolds Averaged Navier?CStokes simulations, which are shown to satisfactorily predict the corresponding quantities extracted from the DNS data for all cases.     

Some Thermodynamical Aspects of the Solidification of Two-phase Flows

The solidification of a two-phase flow is analyzed. It is shown how fluid-to-fluid phase transformations and kinetic stresses due to relative diffusion between fluid components influence dissipation of energy, friction and heat flux at the interface solid–fluid.     

Some Aspects of the Gyroscopic Stabilization in Acoustic Fields

Consideration is given to the results of an analysis of the additional drift in a three-axial gyroplatform under the concurrent action of kinematic and acoustic stimuli.     

Some Aspects of the Lift Force on a Spherical Bubble

Several situations in which a spherical bubble experiences a lift force are examined, especially through the use of computational results obtained by solving the full Navier–Stokes equations. The lift force is computed over a wide range of Reynolds number for the case of pure shear flow, pure strain and solid body rotation. Using these results, the validity of asymptotic solutions derived in the limit of low Reynolds number or inviscid flow is discussed. A general expression of the lift force valid for low to moderate shears is proposed. It is shown that for such shears, the lift force in a complex flow can be predicted by superposing the results obtained in pure strain flow and solid body rotation flow. Finally, the interaction force experienced by two bubbles rising side-by-side is studied. The computational results reveal that, at variance with the predictions of potential theory, the sign of this force changes when the Reynolds number or the separation distance between the bubbles decreases below a critical value. All these results are discussed in terms of vorticity. The respective role played by the vorticity generated at the bubble surface and by the one that is eventually present in the unperturbed flow is emphasized.     

Some Aspects of Stability for Semigroup Actions and Control Systems

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\) , and let \(\mathcal {F}\) be a family of subsets of \(S\) . For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\) . The point \(x\) is forward \(\mathcal {F}\) -Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \) . The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\) -nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \) -prolongational limit set \(J\left( x,\mathcal {F}\right) \) . The paper contains applications to control systems.