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.     

Approximation of Flat <Emphasis Type="Italic">W</Emphasis><Superscript>2,2</Superscript> Isometric Immersions by Smooth Ones

Let \({S\subset\mathbb{R}^2}\) be a bounded Lipschitz domain and denote by \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\) the set of mappings \({u\in W^{2,2}(S;\mathbb{R}^3)}\) which satisfy \({(\nabla u)^T(\nabla u) = Id}\) almost everywhere. Under an additional regularity condition on the boundary \({\partial S}\) (which is satisfied if \({\partial S}\) is piecewise continuously differentiable), we prove that the strong W ^2,2 closure of \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)\cap C^{\infty}(\overline{S};\mathbb{R}^3)}\) agrees with \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}\), \({u(r) > 0~\forall r \ge 0}\), and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\). This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}\), and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}\). For each \({n \ge 1}\), a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\). We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\), \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\). Several open problems are stated.     

A semi-locally scaled eddy viscosity formulation for LES wall models and flows at high speeds

We show that the mean wall-shear stresses in wall-modeled large-eddy simulations (WMLES) of high-speed flows can be off by up to \(\approx 100\%\) with respect to a DNS benchmark when using the van-Driest-based damping function, i.e., the conventional damping function. Errors in the WMLES-predicted wall-shear stresses are often attributed to the so-called log-layer mismatch, which, albeit also an error in wall-shear stresses \(\tau _\mathrm{w}\), is an error of about \(15\%\). The larger error identified here cannot be removed using the previously developed remedies for the log-layer mismatch. This error may be removed by using the semi-local scaling, i.e., \(l_\nu =\mu /\sqrt{\rho \tau _\mathrm{w}}\), in the damping function, where \(\mu \) and \(\rho \) are the local mean dynamic viscosity and density, respectively.     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

Displacement and Dissolution Characteristics of CO$$_{}$$/Brine System in Unconsolidated Porous Media

This study investigated the dynamic displacement and dissolution of \(\hbox {CO}_{2}\) in porous media at 313 K and 6/8 MPa. Gaseous (\(\hbox {gCO}_{2}\)) at 6 MPa and supercritical \(\hbox {CO}_{2 }(\hbox {scCO}_{2}) \) at 8 MPa were injected downward into a glass bead pack at different flow rates, following upwards brine injection. The processes occurring during \(\hbox {CO}_{2}\) drainage and brine imbibition were visualized using magnetic resonance imaging. The drainage flow fronts were strongly influenced by the flow rates, resulting in different gas distributions. However, brine imbibition proceeded as a vertical compacted front due to the strong effect of gravity. Additionally, the effects of flow rate on distribution and saturation were analyzed. Then, the front movement of \(\hbox {CO}_{2}\) dissolution was visualized along different paths after imbibition. The determined \(\hbox {CO}_{2}\) concentrations implied that little \(\hbox {scCO}_{2}\) dissolved in brine after imbibition. The dissolution rate was from \(10^{-8}\) to \(10^{-9}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) and from \(10^{-6}\) to \(10^{-8}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) for \(\hbox {gCO}_{2}\) at 6 MPa and \(\hbox {scCO}_{2 }\) at 8 MPa, respectively. The total time for the \(\hbox {scCO}_{2}\) dissolution was short, indicating fast mass transfer between the \(\hbox {CO}_{2}\) and brine. Injection of \(\hbox {CO}_{2}\) under supercritical conditions resulted in a quick establishment of a steady state with high storage safety.     

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.     

Damköhler-Shelkin Paradox in the Theory of Turbulent Flame Propagation,and a Concept of the Premixed Flame at the Intermediate Asymptotic Stage

We formulated a paradox in the theory of turbulent premixed flame in the flamelet regime: discrepancy between the Damköhler (1940) and Shelkin (1943) estimate of the turbulence flame speed \(U_{t} \sim {u}^{\prime }\) in the case of strong turbulence (\({u}^{\prime }>>S_{L} \)) and numerous experiments that show a strong dependence of U_t on the speed of the instantaneous flame S_L. We name this discrepancy the Damköhler-Shelkin paradox. The first aim of the research is to validate and clarify this estimate, which is based on intuitive considerations, as the paradox must be a statement that seems contradictory to observations but is actually true. We analysed the turbulent flame in the context of the original hyperbolic combustion equation that directly describes the leading edge of the flame, which is a locus of the Zel’dovich “leading points” controlling the speed of the turbulent flame. Analysis of the corresponding characteristic equations results in the expression for speed on the steady-state turbulent flame \(U_{t} ={u}^{\prime }\sqrt {1+(S_{L} /{u}^{\prime })^{2}} \), which is the case when \({u}^{\prime }>>S_{L} \) becomes \(U_{t} \cong {u}^{\prime }\). This result confirms and improves the Damköhler-Shelkin estimate \(U_{t} \sim {u}^{\prime }\). The second aim is to resolve the Damköhler-Shelkin paradox. We explain the discrepancy with observations by the fact that turbulent flames are transient due to insufficient residence time in the real burners to reach statistical equilibrium of wrinkle structures of the random flame surface. We consider the transient flame in the intermediate asymptotic stage when the small-scales wrinkles are in statistical equilibrium, while at the same time the large-scale wrinkles are far from equilibrium. The expressions for the flame speed and width, which we deduce using the dimensional analysis and general properties of the ransom surface, \(U_{t} \sim ({u}^{\prime }S_{L})^{1/2}\) and \(\delta _{t} \sim ({u}^{\prime }Lt)^{1/2}\), show that this transient flame is in fact a turbulent mixing layer travelling with constant speed U_t depending on S_L, the intermediate steady propagation (ISP) flame. Qualitative estimations of the times required for the small-scale and large-scale wrinkles to reach statistical equilibrium show that the turbulent Bunsen- and V-flames correspond to the intermediated asymptotic stage, and the turbulent flames with a complete equilibrium structure of the wrinkled flamelet surface are not attainable under laboratory conditions. We present the results of numerical simulations of the impingent flames, which count in favour of the belief that these flames are also transient.     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

Towards the Development of an Evolution Equation for Flame Turbulence Interaction in Premixed Turbulent Combustion

Flame turbulence interaction is one of the leading order terms in the scalar dissipation \(\left (\widetilde {\varepsilon }_{c}\right )\) transport equation [35] and is thus an important phenomenon in premixed turbulent combustion. Swaminathan and Grout [36] and Chakraborty and Swaminathan [15, 16] have shown that the effect of strain rate on the transport of \(\widetilde {\varepsilon }_{c}\) is dominated by the interaction between the fluctuating scalar gradients and the fluctuating strain rate, denoted here by \(\overline {\rho }\widetilde {\Delta }_{c}= \overline {\rho {\alpha }\nabla c^{\prime \prime }S_{ij}^{\prime \prime }\nabla c^{\prime \prime }}\) ; this represents the flame turbulence interaction. In order to obtain an accurate representation of this phenomenon, a new evolution equation for \(\widetilde {\Delta }_{c}\) has been proposed. This equation gives a detailed insight into flame turbulence interaction and provides an alternative approach to model the important physics represented by \(\widetilde {\Delta }_{c}\) . The \(\widetilde {\Delta }_{c}\) evolution equation is derived in detail and an order of magnitude analysis is carried out to determine the leading order terms in the \(\widetilde {\Delta }_{c}\) evolution equation. The leading order terms are then studied using a Direct Numerical Simulation (DNS) of premixed turbulent flames in the corrugated flamelet regime. It is found that the behaviour of \(\widetilde {\Delta }_{c}\) is determined by the competition between the source terms (pressure gradient and the reaction rate), diffusion/dissipation processes, turbulent strain rate and the dilatation rate. Closures for the leading order terms in \(\widetilde {\Delta }_{c}\) evolution equation have been proposed and compared with the DNS data.     

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force

$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$

where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.

     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

Topological Pressure of Generic Points for Amenable Group Actions

Let (X, G) be a G-action topological dynamical system (t.d.s. for short), where G is a countably infinite discrete amenable group. In this paper, we study the topological pressure of the sets of generic points. We show that when the system satisfies the almost specification property, for any G-invariant measure \(\mu \) and any continuous map \(\varphi \),

$$\begin{aligned} P\left( X_{\mu },\varphi ,\{F_n\}\right) = h_{\mu }(X)+\int \varphi d\mu , \end{aligned}$$

where \(\{F_n\}\) is a Følner sequence, \(X_{\mu }\) is the set of generic points of \(\mu \) with respect to (w.r.t. for short) \(\{F_n\}\), \(P(X_{\mu },\varphi ,\{F_n\})\) is the topological pressure of \(X_{\mu }\) for \(\varphi \) w.r.t. \(\{F_n\}\) and \(h_{\mu }(X)\) is the measure-theoretic entropy.

     

DNS Study of the Optimal Chemical Markers for Heat Release in Syngas Flames

The purpose of this study is to identify a quantitative marker of the heat release rate (HRR) distribution using experimentally measurable species. Turbulent syngas (CO/H₂/air) flames with different equivalence ratios, H₂/CO ratios, and turbulence intensities are computed by Direct Numerical Simulations (DNS) in order to obtain an indirect but accurate estimation of heat release profiles. To check the robustness of the estimation, two different kinetic mechanisms have been considered. Based on a direct image analysis of the DNS results, normalized species concentrations combined with exponents are systematically tested in an attempt to reconstruct as accurately as possible the field of heat release rate. A systematic comparison is used to identify the best possible exponents associated with each species combination. Differing from previous studies, the present analysis takes into account the local thickness of the turbulent heat release zone. As a consequence, the obtained optimal species combinations represent not only the position of peak heat release but also local changes in the topology of the reaction zone (thickness, curvature). In the end, the heat release rate of atmospheric syngas flames can, in general, be best approximated using the concentrations of HCO and OH, using \(\overline {c}_{HCO}^{1.5}\times \overline {c}_{OH}^{0.75}\), when considering only species that are measurable by Laser-Induced Fluorescence. Another excellent reconstruction would be \(\overline {c}_{CH_{2}O}^{0.32}\times \overline {c}_{OH}^{0.8}\), for cases where CH₂O is preferred to HCO.     

Visual Investigation of Improvement in Extra-Heavy Oil Recovery by Borate-Assisted $$\hbox {CO}_{2}$$-Foam Injection

The significant reduction in heavy oil viscosity when mixed with \(\hbox {CO}_{2}\) is well documented. However, for \(\hbox {CO}_{2}\) injection to be an efficient method for improving heavy oil recovery, other mechanisms are required to improve the mobility ratio between the \(\hbox {CO}_{2}\) front and the resident heavy oil. In situ generation of \(\hbox {CO}_{2}\)-foam can improve \(\hbox {CO}_{2}\) injection performance by (a) increasing the effective viscosity of \(\hbox {CO}_{2}\) in the reservoir and (b) increasing the contact area between the heavy oil and injected \(\hbox {CO}_{2}\) and hence improving \(\hbox {CO}_{2}\) dissolution rate. However, in situ generation of stable \(\hbox {CO}_{2}\)-foam capable of travelling from the injection well to the production well is hard to achieve. We have previously published the results of a series of foam stability experiments using alkali and in the presence of heavy crude oil (Farzaneh and Sohrabi 2015). The results showed that stability of \(\hbox {CO}_{2}\)-foam decreased by addition of NaOH, while it increased by addition of \(\hbox {Na}_{2}\hbox {CO}_{3}\). However, the highest increase in \(\hbox {CO}_{2}\)-foam stability was achieved by adding borate to the surfactant solution. Borate is a mild alkaline with an excellent pH buffering ability. The previous study was performed in a foam column in the absence of a porous medium. In this paper, we present the results of a new series of experiments carried out in a high-pressure glass micromodel to visually investigate the performance of borate–surfactant \(\hbox {CO}_{2}\)-foam injection in an extra-heavy crude oil in a transparent porous medium. In the first part of the paper, the pore-scale interactions of \(\hbox {CO}_{2}\)-foam and extra-heavy oil and the mechanisms of oil displacement and hence oil recovery are presented through image analysis of micromodel images. The results show that very high oil recovery was achieved by co-injection of the borate–surfactant solution with \(\hbox {CO}_{2}\), due to in-situ formation of stable foam. Dissolution of \(\hbox {CO}_{2}\) in heavy oil resulted in significant reduction in its viscosity. \(\hbox {CO}_{2}\)-foam significantly increased the contact area between the oil and \(\hbox {CO}_{2}\) significantly and thus the efficiency of the process. The synergy effect between the borate and surfactant resulted in (1) alteration of the wettability of the porous medium towards water wet and (2) significant reduction of the oil–water IFT. As a result, a bank of oil-in-water (O/W) emulsion was formed in the porous medium and moved ahead of the \(\hbox {CO}_{2}\)-foam front. The in-situ generated O/W emulsion has a much lower viscosity than the original oil and plays a major role in the observed additional oil recovery in the range of performed experiments. Borate also made \(\hbox {CO}_{2}\)-foam more stable by changing the system to non-spreading oil and reducing coalescence of the foam bubbles. The results of these visual experiments suggest that borate can be a useful additive for improving heavy oil recovery in the range of the performed tests, by increasing \(\hbox {CO}_{2}\)-foam stability and producing O/W emulsions.     

On the Damped Harmonic Oscillator with Time Dependent Damping Coefficient

Conditions guaranteeing asymptotic stability for the differential equation

$$\begin{aligned} x''+h(t)x'+\omega ^2x=0 \qquad (x\in \mathbb {R}) \end{aligned}$$

are studied, where the damping coefficient \(h:[0,\infty )\rightarrow [0,\infty )\) is a locally integrable function, and the frequency \(\omega >0\) is constant. Our conditions need neither the requirement \(h(t)\le \overline{h}<\infty \) (\(t\in [0,\infty )\); \(\overline{h}\) is constant) (“small damping”), nor \(0< \underline{h}\le h(t)\) (\(t\in [0,\infty )\); \(\underline{h}\) is constant) (“large damping”); in other words, they can be applied to the general case \(0\le h(t)<\infty \) (\(t\in [0,\infty \))). We establish a condition which combines weak integral positivity with Smith’s growth condition

$$\begin{aligned} \int ^\infty _0 \exp [-H(t)]\int _0^t \exp [H(s)]\,\mathrm{{d}}s\,\mathrm{{d}}t=\infty \qquad \left( H(t):=\int _0^t h(\tau )\,\mathrm{{d}}\tau \right) , \end{aligned}$$

so it is able to control both the small and the large values of the damping coefficient simultaneously.