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Journal articles on the topic 'Scalar dependent diffusion'
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1
Winters, Kraig B., and Eric A. D'Asaro. "Diascalar flux and the rate of fluid mixing." Journal of Fluid Mechanics 317 (June 25, 1996): 179–93. http://dx.doi.org/10.1017/s0022112096000717.
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We define the rate at which a scalar θ mixes in a fluid flow in terms of the flux of θ across isoscalar surfaces. This flux θd is purely diffusive and is, in principle, exactly known at all times given the scalar field and the coefficient of molecular diffusivity. In general, the complex geometry of isoscalar surfaces would appear to make the calculation of this flux very difficult. In this paper, we derive an exact expression relating the instantaneous diascalar flux to the average squared scalar gradient on an isoscalar surface which does not require knowledge of the spatial structure of the surface itself. To obtain this result, a time-dependent reference state θ(t,z*.) is defined. When the scalar is taken to be density, this reference state is that of minimum potential energy. The rate of change of the reference state due to diffusion is shown to equal the divergence of the diffusive flux, i.e. (∂/∂z*)θd.This result provides a mathematical framework that exactly separates diffusive and advective scalar transport in incompressible fluid flows. The relationship between diffusive and advective transport is discussed in relation to the scalar variance equation and the Osborn–Cox model. Estimation of water mass transformation from oceanic microstructure profiles and determination of the time-dependent mixing rate in numerically simulated flows are discussed.
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GHOSH, S., A. LEONARD, and S. WIGGINS. "Diffusion of a passive scalar from a no-slip boundary into a two-dimensional chaotic advection field." Journal of Fluid Mechanics 372 (October 10, 1998): 119–63. http://dx.doi.org/10.1017/s0022112098002249.
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Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local ‘hot spot’ on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Péclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity field further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Péclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Péclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field.
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WEICHMAN, PETER B., and ROMAN E. GLAZMAN. "Passive scalar transport by travelling wave fields." Journal of Fluid Mechanics 420 (October 10, 2000): 147–200. http://dx.doi.org/10.1017/s0022112000001452.
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We study turbulent transport of passive tracers by random wave fields of a rather general nature. A formalism allowing for spatial inhomogeneity and anisotropy of an underlying velocity field (such as that caused by a latitudinally varying Coriolis parameter) is developed, with the aim of treating problems of large-scale ocean transport by long internal waves. For the special case of surface gravity waves on deep water, our results agree with the earlier theory of Herterich & Hasselmann (1982), though even in that case we discover additional, off-diagonal elements of the diffusion tensor emerging in the presence of a mean drift. An advective diffusion equation including all components of the diffusion tensor D plus a mean, Stokes-type drift u is derived and applied to the case of baroclinic inertia–gravity (BIG) waves. This application is of particular interest for ocean circulation and climate modelling, as the mean drift, according to our estimates, is comparable to ocean interior currents. Furthermore, while on the largest (100 km and greater) scales, wave-induced diffusion is found to be generally small compared to classical eddy-induced diffusion, the two become comparable on scales below 10 km. These scales are near the present limit on the spatial resolution of eddy-resolving ocean numerical models. Since we find that uz and Dzz vanish identically, net vertical transport is absent in wave systems of this type. However, for anisotropic wave spectra the diffusion tensor can have non-zero off-diagonal vertical elements, Dxz and Dyz, and it is shown that their presence leads to non-positive definiteness of D, and a negative diffusion constant is found along a particular principal axis. However, the simultaneous presence of a depth-dependent mean horizontal drift u(z) eliminates any potential unphysical behaviour.
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Kay, Alison L., Jonathan A. Sherratt, and J. B. McLeod. "Comparison theorems and variable speed waves for a scalar reaction–diffusion equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 5 (October 2001): 1133–61. http://dx.doi.org/10.1017/s030821050000130x.
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This paper concerns the reaction-diffusion equation ut = uxx + u2(1 − u). Previous numerical solutions of this equation have demonstrated various different types of wave front solutions, generated by different initial conditions. In this paper, the authors use a phase-plane form of comparison theorems for partial differential equations (PDEs) to confirm analytically these numerical results. In particular, they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave, while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave, or to the exponentially decaying wave, or to a perpetually accelerating wave, dependent upon the exact form of the decay of the initial conditions. We then focus on the case of accelerating waves and investigate their form in more detail, by approximating the full equation in this case with a hyperbolic PDE, which we solve using the method of characteristics. We use this approximate solution to derive a leading-order approximation to the wave speed.
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APPLEBY, JOHN A. D. "DECAY AND GROWTH RATES OF SOLUTIONS OF SCALAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY AND STATE DEPENDENT NOISE." Stochastics and Dynamics 05, no. 02 (June 2005): 133–47. http://dx.doi.org/10.1142/s0219493705001353.
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This paper studies the growth and decay rates of solutions of scalar stochastic delay differential equations of Itô type. The equations studied have a linear drift which contains an unbounded delay term, and a nonlinear diffusion term, which depends on the current state only. We show that when the nonlinearity at zero or infinity is sufficiently weak, the same non-exponential decay and growth rates found in the corresponding underlying linear deterministic equation are recovered, in an almost sure sense.
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Michioka, Takenobu, and Fotini Katopodes Chow. "High-Resolution Large-Eddy Simulations of Scalar Transport in Atmospheric Boundary Layer Flow over Complex Terrain." Journal of Applied Meteorology and Climatology 47, no. 12 (December 1, 2008): 3150–69. http://dx.doi.org/10.1175/2008jamc1941.1.
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Abstract This paper presents high-resolution numerical simulations of the atmospheric flow and concentration fields accompanying scalar transport and diffusion from a point source in complex terrain. Scalar dispersion is affected not only by mean flow, but also by turbulent fluxes that affect scalar mixing, meaning that predictions of scalar transport require greater attention to the choice of numerical simulation parameters than is typically needed for mean wind field predictions. Large-eddy simulation is used in a mesoscale setting, providing modeling advantages through the use of robust turbulence models combined with the influence of synoptic flow forcing and heterogeneous land surface forcing. An Eulerian model for scalar transport and diffusion is implemented in the Advanced Regional Prediction System mesoscale code to compare scalar concentrations with data collected during field experiments conducted at Mount Tsukuba, Japan, in 1989. The simulations use horizontal grid resolution as fine as 25 m with up to eight grid nesting levels to incorporate time-dependent meteorological forcing. The results show that simulated ground concentration values contain significant errors relative to measured values because the mesoscale wind typically contains a wind direction bias of a few dozen degrees. Comparisons of simulation results with observations of arc maximum concentrations, however, lie within acceptable error bounds. In addition, this paper investigates the effects on scalar dispersion of computational mixing and lateral boundary conditions, which have received little attention in the literature—in particular, for high-resolution applications. The choice of lateral boundary condition update interval is found not to affect time-averaged quantities but to affect the scalar transport strongly. More frequent updates improve the simulated ground concentration values. In addition, results show that the computational mixing coefficient must be set to as small a value as possible to improve scalar dispersion predictions. The predicted concentration fields are compared as the horizontal grid resolution is increased from 190 m to as fine as 25 m. The difference observed in the results at these levels of grid refinement is found to be small relative to the effects of computational mixing and lateral boundary updates.
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Lastdrager, Boris, Barry Koren, and Jan Verwer. "Solution of Time-dependent Advection-Diffusion Problems with the Sparse-grid Combination Technique and a Rosenbrock Solver." Computational Methods in Applied Mathematics 1, no. 1 (2001): 86–98. http://dx.doi.org/10.2478/cmam-2001-0006.
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Abstract In the current paper the efficiency of the sparse-grid combination tech- nique applied to time-dependent advection-diffusion problems is investigated. For the time-integration we employ a third-order Rosenbrock scheme implemented with adap- tive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coe±cient problem and a system of 2D non- linear Burgers' equations. In short, the combination technique proved more efficient than a single grid approach for the simpler linear problem. For the Burgers' equations this gain in efficiency was only observed if one of the two solution components was set to zero, which makes the problem more grid-aligned.
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CHEREMNYKH, O. K., J. W. EDENSTRASSER, and V. V. GORIN. "Relaxation of a non-ideal incompressible plasma with mass flow." Journal of Plasma Physics 62, no. 2 (August 1999): 195–202. http://dx.doi.org/10.1017/s0022377899007849.
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The time evolution of an incompressible non-ideal magnetohydrodynamic (MHD), current-carrying plasma with mass flow is investigated. An approach for the reduction of the nonlinear vector MHD equations to a set of scalar partial differential equations is supposed. Analytical time-dependent solutions of this system are presented. They describe kinetic plasma equilibria both with well-defined nested-in magnetic and velocity surfaces and in the form of vortices. The obtained solutions may be called ‘diffusion-like’, since their temporal structure is very similar to the solutions of the diffusion problem. It is shown that the magnetic field and the velocity have different dumping rates. In the asymptotic limit t→∞, the plasma slowly relaxes towards the hydrostatic equilibrium of gravitating systems.
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Aiyer, Aditya K., Kandaswamy Subramanian, and Pallavi Bhat. "Passive scalar mixing and decay at finite correlation times in the Batchelor regime." Journal of Fluid Mechanics 824 (July 11, 2017): 785–817. http://dx.doi.org/10.1017/jfm.2017.364.
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An elegant model for passive scalar mixing and decay was given by Kraichnan (Phys. Fluids, vol. 11, 1968, pp. 945–953) assuming the velocity to be delta correlated in time. For realistic random flows this assumption becomes invalid. We generalize the Kraichnan model to include the effects of a finite correlation time, $\unicode[STIX]{x1D70F}$, using renewing flows. The generalized evolution equation for the three-dimensional (3-D) passive scalar spectrum $\hat{M}(k,t)$ or its correlation function $M(r,t)$, gives the Kraichnan equation when $\unicode[STIX]{x1D70F}\rightarrow 0$, and extends it to the next order in $\unicode[STIX]{x1D70F}$. It involves third- and fourth-order derivatives of $M$ or $\hat{M}$ (in the high $k$ limit). For small-$\unicode[STIX]{x1D70F}$ (or small Kubo number), it can be recast using the Landau–Lifshitz approach to one with at most second derivatives of $\hat{M}$. We present both a scaling solution to this equation neglecting diffusion and a more exact solution including diffusive effects. To leading order in $\unicode[STIX]{x1D70F}$, we first show that the steady state 1-D passive scalar spectrum, preserves the Batchelor (J. Fluid Mech., vol. 5, 1959, pp. 113–133) form, $E_{\unicode[STIX]{x1D703}}(k)\propto k^{-1}$, in the viscous–convective limit, independent of $\unicode[STIX]{x1D70F}$. This result can also be obtained in a general manner using Lagrangian methods. Interestingly, in the absence of sources, when passive scalar fluctuations decay, we show that the spectrum in the Batchelor regime at late times is of the form $E_{\unicode[STIX]{x1D703}}(k)\propto k^{1/2}$ and also independent of $\unicode[STIX]{x1D70F}$. More generally, finite $\unicode[STIX]{x1D70F}$ does not qualitatively change the shape of the spectrum during decay. The decay rate is however reduced for finite $\unicode[STIX]{x1D70F}$. We also present results from high resolution ($1024^{3}$) direct numerical simulations of passive scalar mixing and decay. We find reasonable agreement with predictions of the Batchelor spectrum during steady state. The scalar spectrum during decay is however dependent on initial conditions. It agrees qualitatively with analytic predictions when power is dominantly in wavenumbers corresponding to the Batchelor regime, but is shallower when box-scale fluctuations dominate during decay.
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Panna, Neelufar. "An Exact Solution of the Reaction-Diffusion Equation for the Speed of the Interface Propagation in Superconductors." Chittagong University Journal of Science 41, no. 1 (February 8, 2021): 85–95. http://dx.doi.org/10.3329/cujs.v41i1.51916.
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The speed of interface propagation in superconductors for the scalar reaction-diffusion equation ut = ∇2u+ F(u) is studied in detail. Here the non linear reaction term F (u) is the time-dependent Ginzburg-Landau or TDGL equation F(u)=u-u3 which describes the dynamics of the order-disorder transition. In contrast to what has been done in previous work [1] here an improved exact solution has derived by using TDGL equation to determine the speed of the front propagation. The analytical treatment of this study has been found in good agreement with the numerical simulation of V. Mendez et al. [2] and Di Bartolo and Dorsey [3].
The Chittagong Univ. J. Sci. 40(1) : 85-95, 2019
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Margerin, Ludovic, Andres Bajaras, and Michel Campillo. "A scalar radiative transfer model including the coupling between surface and body waves." Geophysical Journal International 219, no. 2 (August 12, 2019): 1092–108. http://dx.doi.org/10.1093/gji/ggz348.
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SUMMARY To describe the energy transport in the seismic coda, we introduce a system of radiative transfer equations for coupled surface and body waves in a scalar approximation. Our model is based on the Helmholtz equation in a half-space geometry with mixed boundary conditions. In this model, Green’s function can be represented as a sum of body waves and surface waves, which mimics the situation on Earth. In a first step, we study the single-scattering problem for point-like objects in the Born approximation. Using the assumption that the phase of body waves is randomized by surface reflection or by interaction with the scatterers, we show that it becomes possible to define, in the usual manner, the cross-sections for surface-to-body and body-to-surface scattering. Adopting the independent scattering approximation, we then define the scattering mean free paths of body and surface waves including the coupling between the two types of waves. Using a phenomenological approach, we then derive a set of coupled transport equations satisfied by the specific energy density of surface and body waves in a medium containing a homogeneous distribution of point scatterers. In our model, the scattering mean free path of body waves is depth dependent as a consequence of the body-to-surface coupling. We demonstrate that an equipartition between surface and body waves is established at long lapse-time, with a ratio which is predicted by usual mode counting arguments. We derive a diffusion approximation from the set of transport equations and show that the diffusivity is both anisotropic and depth dependent. The physical origin of the two properties is discussed. Finally, we present Monte Carlo solutions of the transport equations which illustrate the convergence towards equipartition at long lapse-time as well as the importance of the coupling between surface and body waves in the generation of coda waves.
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Appleby, John A. D. "On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations." International Journal of Differential Equations 2010 (2010): 1–25. http://dx.doi.org/10.1155/2010/508217.
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We consider the zero crossings and positive solutions of scalar nonlinear stochastic Volterra integrodifferential equations of Itô type. In the equations considered, the diffusion coefficient is linear and depends on the current state, and the drift term is a convolution integral which is in some sense mean reverting towards the zero equilibrium. The state dependent restoring force in the integral can be nonlinear. In broad terms, we show that when the restoring force is of linear or lower order in the neighbourhood of the equilibrium, or if the kernel decays more slowly than a critical noise-dependent rate, then there is a zero crossing almost surely. On the other hand, if the kernel decays more rapidly than this critical rate, and the restoring force is globally superlinear, then there is a positive probability that the solution remains of one sign for all time, given a sufficiently small initial condition. Moreover, the probability that the solution remains of one sign tends to unity as the initial condition tends to zero.
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ABDEL KAREEM, WALEED, TAMER NABIL, SEIICHERIO IZAWA, and YU FUKUNISHI. "MULTIRESOLUTION AND NONLINEAR DIFFUSION FILTERING OF HOMOGENEOUS ISOTROPIC TURBULENCE." International Journal of Computational Methods 11, no. 01 (September 2, 2013): 1350054. http://dx.doi.org/10.1142/s0219876213500540.
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The multiresolution (Gabor and wavelet transforms) and nonlinear diffusion filtering (NDF) methods are investigated to extract the coherent and incoherent parts of a forced homogeneous isotropic turbulent field. The aim of this paper is to apply two different analyses to decompose the turbulent field into organized coherent and random incoherent parts. The first analysis filtering process (Gabor and wavelet transforms) is based on the frequency domain; however the second NDF filtering analysis is implemented in the spatial domain. The turbulent field is generated using the Lattice Boltzmann method (LBM) with a resolution of 1283, and the Q-identification method is used to extract the elongated vortical structures. The three filtering methods are applied against the scalar Q-field rather than a vector field (velocity or vorticity fields). The Gabor transform and the orthogonal wavelet with approximately symmetric basis are applied to filter out incoherent noise. Filtering in the Gabor domain is done in the highest quarter frequency values, whereas filtering in the wavelet domain is done using sub-band dependent thresholding. The NDF method is based on explicit finite-difference discretization in the spatial domain. Results indicate that the three filtering methods smoothly identify the coherent and incoherent parts. Although the NDF method isolates the incoherent part more smoothly, the cross sections of the vortices in the coherent part are changed. Also, the Gabor filtering method can remove few incoherent points from the flow field, compared with the other two methods. The wavelet method tends to identify the coherent vortices and remove the incoherent noise without any change in the physical structure of the turbulent field.
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BONN, JAMES, and RICHARD M. McLAUGHLIN. "Sensitive enhanced diffusivities for flows with fluctuating mean winds: a two-parameter study." Journal of Fluid Mechanics 445 (October 16, 2001): 345–75. http://dx.doi.org/10.1017/s002211200100564x.
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Enhanced diffusion coefficients arising from the theory of periodic homogenized averaging for a passive scalar diffusing in the presence of a large-scale, fluctuating mean wind superimposed upon a small-scale, steady flow with non-trivial topology are studied. The purpose of the study is to assess how the extreme sensitivity of enhanced diffusion coefficients to small variations in large-scale flow parameters previously exhibited for steady flows in two spatial dimensions is modified by either the presence of temporal fluctuation, or the consideration of fully three-dimensional steady flow. We observe the various mixing parameters (Péclet, Strouhal and periodic Péclet numbers) and related non-dimensionalizations. We document non-monotonic Péclet number dependence in the enhanced diffusivities, and address how this behaviour is camouflaged with certain non-dimensional groups. For asymptotically large Strouhal number at fixed, bounded Péclet number, we establish that rapid wind fluctuations do not modify the steady theory, whereas for asymptotically small Strouhal number the enhanced diffusion coefficients are shown to be represented as an average over the steady geometry. The more difficult case of large Péclet number is considered numerically through the use of a conjugate gradient algorithm. We consider Péclet-number-dependent Strouhal numbers, S = QPe−(1+γ), and present numerical evidence documenting critical values of γ which distinguish the enhanced diffusivities as arising simply from steady theory (γ < −1) for which fluctuation provides no averaging, fully unsteady theory (γ ∈ (−1, 0)) with closure coefficients plagued by non-monotonic Péclet number dependence, and averaged steady theory (γ > 0). The transitional case with γ = 0 is examined in detail. Steady averaging is observed to agree well with the full simulations in this case for Q [les ] 1, but fails for larger Q. For non-sheared flow, with Q [les ] 1, weak temporal fluctuation in a large-scale wind is shown to reduce the sensitivity arising from the steady flow geometry; however, the degree of this reduction is itself strongly dependent upon the details of the imposed fluctuation. For more intense temporal fluctuation, strongly aligned orthogonal to the steady wind, time variation averages the sensitive scaling existing in the steady geometry, and the present study observes a Pe1 scaling behaviour in the enhanced diffusion coefficients at moderately large Péclet number. Finally, we conclude with the numerical documentation of sensitive scaling behaviour (similar to the two-dimensional steady case) in fully three dimensional ABC flow.
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Akanji, Lateef, and Gabriel Falade. "Closed-Form Solution of Radial Transport of Tracers in Porous Media Influenced by Linear Drift." Energies 12, no. 1 (December 22, 2018): 29. http://dx.doi.org/10.3390/en12010029.
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A new closed-form analytical solution to the radial transport of tracers in porous media under the influence of linear drift is presented. Specifically, the transport of tracers under convection–diffusion-dominated flow is considered. First, the radial transport equation was cast in the form of the Whittaker equation by defining a set of transformation relations. Then, linear drift was incorporated by considering a coordinate-independent scalar velocity field within the porous medium. A special case of low-intensity tracer injection where molecular diffusion controls tracer propagation but convection with linear velocity drift plays a significant role was presented and solved in Laplace space. Furthermore, a weak-form numerical solution of the nonlinear problem was obtained and used to analyse tracer concentration behaviour in a porous medium, where drift effects predominate and influence the flow pattern. Application in enhanced oil recovery (EOR) processes where linear drift may interfere with the flow path was also evaluated within the solution to obtain concentration profiles for different injection models. The results of the analyses indicated that the effect of linear drift on the tracer concentration profile is dependent on system heterogeneity and progressively becomes more pronounced at later times. This new solution demonstrates the necessity to consider the impact of drift on the transport of tracers, as arrival times may be significantly influenced by drift intensity.
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Fischer, Manfred M. "Spatial Externalities and Growth in a Mankiw-Romer-Weil World." International Regional Science Review 41, no. 1 (February 25, 2016): 45–61. http://dx.doi.org/10.1177/0160017616628602.
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This article presents a theoretical growth model that accounts for technological interdependence among regions in a Mankiw-Romer-Weil world. The reasoning behind the theoretical work is that technological ideas cannot be fully appropriated by investors and these ideas may diffuse and increase the productivity of other firms. We link the diffusion of ideas to spatial proximity and allow ideas to flow to nearby regional economies. Through the magic of solving for the reduced form of the theoretical model and the magic of spatial autoregressive processes, the simple dependence on a small number of neighboring regions leads to a reduced form theoretical model and an associated empirical model where changes in a single region can potentially impact all other regions. This implies that conventional regression interpretations of the parameter estimates would be wrong. The proper way to interpret the model has to rely on matrices of partial derivatives of the dependent variable with respect to changes in the Mankiw-Romer-Weil variables, using scalar summary measures for reporting the estimates of the marginal impacts from the model. The summary impact measure estimates indicate that technological interdependence among the European regions works through physical rather than human capital externalities.
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Du, Yongle, and John A. Ekaterinaris. "On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations." Communications in Computational Physics 20, no. 2 (July 21, 2016): 486–511. http://dx.doi.org/10.4208/cicp.oa-2016-0032.
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AbstractImplicit time integration schemes are popular because their relaxed stability constraints can result in better computational efficiency. For time-accurate unsteady simulations, it has been well recognized that the inherent dispersion and dissipation errors of implicit Runge-Kutta schemes will reduce the computational accuracy for large time steps. Yet for steady state simulations using the time-dependent governing equations, these errors are often overlooked because the intermediate solutions are of less interest. Based on the model equationdy/dt=(μ+iλ)yof scalar convection diffusion systems, this study examines the stability limits, dispersion and dissipation errors of four diagonally implicit Runge-Kutta-type schemes on the complex (μ+iλ)Δtplane. Through numerical experiments, it is shown that, as the time steps increase, the A-stable implicit schemes may not always have reduced CPU time and the computations may not always remain stable, due to the inherent dispersion and dissipation errors of the implicit Runge-Kutta schemes. The dissipation errors may decelerate the convergence rate, and the dispersion errors may cause large oscillations of the numerical solutions. These errors, especially those of high wavenumber components, grow at large time steps. They lead to difficulty in the convergence of the numerical computations, and result in increasing CPU time or even unstable computations as the time step increases. It is concluded that an optimal implicit time integration scheme for steady state simulations should have high dissipation and low dispersion.
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Christmas, Kevin M., and James B. Bassingthwaighte. "Equations for O2 and CO2 solubilities in saline and plasma: combining temperature and density dependences." Journal of Applied Physiology 122, no. 5 (May 1, 2017): 1313–20. http://dx.doi.org/10.1152/japplphysiol.01124.2016.
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Solubilities of respiratory gasses in water, saline, and plasma decrease with rising temperatures and solute concentrations. Henry’s Law, C = α·P, states that the equilibrium concentration of a dissolved gas is solubility times partial pressure. Solubilities in the water of a solution depend on temperature and the content of other solutes. Blood temperatures may differ more than 20°C between skin and heart, and an erythrocyte will undergo that range as blood circulates. The concentrations of O2 and CO2 are the driving forces for diffusion, exchanges, and for reactions. We provide an equation for O2 and CO2 solubilities, α, that allows for continuous changes in temperature, T, and solution density, ρ, in dynamically changing states:[Formula: see text] This two-exponential expression with a density scalar γ, and a density exponent β, accounts for solubility changes due to density changes of an aqueous solution. It fits experimental data on solubilities in water, saline, and plasma over temperatures from 20 to 40°C, and for plasma densities, ρsol up to 1.020 g/ml with ~0.3% error. The amounts of additional bound O2 (to Hb) and CO2 (bicarbonate and carbamino) depend on the concentrations in the local water space and the reaction parameters. During exercise, solubility changes are large; both ρsol and T change rapidly with spatial position and with time. In exercise hemoconcentration plasma, ρsol exceeds 1.02, whereas T may range over 20°C. The six parameters for O2 and the six for CO2 are constants, so solubilities are calculable continuously as T and ρsol change. NEW & NOTEWORTHY Solubilities for oxygen and carbon dioxide are dependent on the density of the solution, on temperature, and on the partial pressure. We provide a brief equation suitable for hand calculators or mathematical modeling, accounting for these factors over a wide range of temperatures and solution densities for use in rapidly changing conditions, such as extreme exercise or osmotic transients, with better than 0.5% accuracy.
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Sharma, Arjun, Irina I. Rypina, Ruth Musgrave, and George Haller. "Analytic reconstruction of a two-dimensional velocity field from an observed diffusive scalar." Journal of Fluid Mechanics 871 (May 24, 2019): 755–74. http://dx.doi.org/10.1017/jfm.2019.301.
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Inverting an evolving diffusive scalar field to reconstruct the underlying velocity field is an underdetermined problem. Here we show, however, that for two-dimensional incompressible flows, this inverse problem can still be uniquely solved if high-resolution tracer measurements, as well as velocity measurements along a curve transverse to the instantaneous scalar contours, are available. Such measurements enable solving a system of partial differential equations for the velocity components by the method of characteristics. If the value of the scalar diffusivity is known, then knowledge of just one velocity component along a transverse initial curve is sufficient. These conclusions extend to the shallow-water equations and to flows with spatially dependent diffusivity. We illustrate our results on velocity reconstruction from tracer fields for planar Navier–Stokes flows and for a barotropic ocean circulation model. We also discuss the use of the proposed velocity reconstruction in oceanographic applications to extend localized velocity measurements to larger spatial domains with the help of remotely sensed scalar fields.
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Sukoriansky, S., B. Galperin, and V. Perov. "A quasi-normal scale elimination model of turbulence and its application to stably stratified flows." Nonlinear Processes in Geophysics 13, no. 1 (February 3, 2006): 9–22. http://dx.doi.org/10.5194/npg-13-9-2006.
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Abstract. Models of planetary, atmospheric and oceanic circulation involve eddy viscosity and eddy diffusivity, KM and KH, that account for unresolved turbulent mixing and diffusion. The most sophisticated turbulent closure models used today for geophysical applications belong in the family of the Reynolds stress models. These models are formulated for the physical space variables; they consider a hierarchy of turbulent correlations and employ a rational way of its truncation. In the process, unknown correlations are related to the known ones via "closure assumptions'' that are based upon physical plausibility, preservation of tensorial properties, and the principle of the invariant modeling according to which the constants in the closure relationships are universal. Although a great deal of progress has been achieved with Reynolds stress closure models over the years, there are still situations in which these models fail. The most difficult flows for the Reynolds stress modeling are those with anisotropy and waves because these processes are scale-dependent and cannot be included in the closure assumptions that pertain to ensemble-averaged quantities. Here, we develop an alternative approach of deriving expressions for KM and KH using the spectral space representation and employing a self-consistent, quasi-normal scale elimination (QNSE) algorithm. More specifically, the QNSE procedure is based upon the quasi-Gaussian mapping of the velocity and temperature fields using the Langevin equations. Turbulence and waves are treated as one entity and the effect of the internal waves is easily identifiable. This model implies partial averaging and, thus, is scale-dependent; it allows one to easily introduce into consideration such parameters as the grid resolution, the degree of the anisotropy, and spectral characteristics, among others. Applied to turbulent flows affected by anisotropy and waves, the method traces turbulence anisotropization and shows how the dispersion relationships for linear waves are modified by turbulence. In addition, one can derive the internal wave frequency shift and the threshold criterion of internal wave generation in the presence of turbulence. The spectral method enables one to derive analytically various one-dimensional and three-dimensional spectra that reflect the effects of waves and anisotropy. When averaging is extended to all scales, the method yields a Reynolds-averaged, Navier-Stokes equations based model (RANS). This RANS model shows that there exists a range of Ri, approximately between 0.1 and 1, in which turbulence undergoes remarkable anisotropization; the vertical mixing becomes suppressed while the horizontal mixing is enhanced. Although KH decreases at large Ri and tends to its molecular value, KM remains finite and larger than its molecular value. This behavior is attributable to the effect of internal waves that mix the momentum but do not mix a scalar. In the Reynolds stress models, this feature is not replicated; instead, all Reynolds stress models predict KM→0 at some value of Ri≤1 which varies from one model to another. The presented spectral model indicates that there is no a single-valued critical Richardson number Ri at which turbulence is fully suppressed by stable stratification. This result is in agreement with large volume of atmospheric, oceanic and laboratory data. The new spectral model has been implemented in the K-ε format and tested in simulations of the stably stratified atmospheric boundary layers. The results of these simulations are in good agreement with the data collected in BASE, SHEBA and CASES99 campaigns. Implementation of the QNSE-derived KM and KH in the high-resolution weather prediction system HIRLAM results in significant improvement of its predictive skills.
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Yang, Yantao, Roberto Verzicco, and Detlef Lohse. "Scaling laws and flow structures of double diffusive convection in the finger regime." Journal of Fluid Mechanics 802 (August 8, 2016): 667–89. http://dx.doi.org/10.1017/jfm.2016.484.
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Direct numerical simulations are conducted for double diffusive convection (DDC) bounded by two parallel plates. The Prandtl numbers, i.e. the ratios between the viscosity and the molecular diffusivities of scalars, are similar to the values of seawater. The DDC flow is driven by an unstable salinity difference (here across the two plates) and stabilized at the same time by a temperature difference. For these conditions the flow can be in the finger regime. We develop scaling laws for three key response parameters of the system: the non-dimensional salinity flux $\mathit{Nu}_{S}$ mainly depends on the salinity Rayleigh number $\mathit{Ra}_{S}$, which measures the strength of the salinity difference and exhibits a very weak dependence on the density ratio $\unicode[STIX]{x1D6EC}$, which is the ratio of the buoyancy forces induced by two scalar differences. The non-dimensional flow velocity $Re$ and the non-dimensional heat flux $\mathit{Nu}_{T}$ are dependent on both $\mathit{Ra}_{S}$ and $\unicode[STIX]{x1D6EC}$. However, the rescaled Reynolds number $Re\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{u}^{eff}}$ and the rescaled convective heat flux $(\mathit{Nu}_{T}-1)\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{T}^{eff}}$ depend only on $\mathit{Ra}_{S}$. The two exponents are dependent on the fluid properties and are determined from the numerical results as $\unicode[STIX]{x1D6FC}_{u}^{eff}=0.25\pm 0.02$ and $\unicode[STIX]{x1D6FC}_{T}^{eff}=0.75\pm 0.03$. Moreover, the behaviours of $\mathit{Nu}_{S}$ and $Re\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{u}^{eff}}$ agree with the predictions of the Grossmann–Lohse theory which was originally developed for the Rayleigh–Bénard flow. The non-dimensional salt-finger width and the thickness of the velocity boundary layers, after being rescaled by $\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{u}^{eff}/2}$, collapse and obey a similar power-law scaling relation with $\mathit{Ra}_{S}$. When $\mathit{Ra}_{S}$ is large enough, salt fingers do not extend from one plate to the other and horizontal zonal flows emerge in the bulk region. We then show that the current scaling strategy can be successfully applied to the experimental results of a heat–copper–ion system (Hage & Tilgner, Phys. Fluids, vol. 22, 2010, 076603). The fluid has different properties and the exponent $\unicode[STIX]{x1D6FC}_{u}^{eff}$ takes a different value $0.54\pm 0.10$.
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Pozrikidis, C. "Reciprocal identities and integral formulations for diffusive scalar transport and Stokes flow with position-dependent diffusivity or viscosity." Journal of Engineering Mathematics 96, no. 1 (March 13, 2015): 95–114. http://dx.doi.org/10.1007/s10665-015-9783-0.
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Shpund, J., M. Pinsky, and A. Khain. "Microphysical Structure of the Marine Boundary Layer under Strong Wind and Spray Formation as Seen from Simulations Using a 2D Explicit Microphysical Model. Part I: The Impact of Large Eddies." Journal of the Atmospheric Sciences 68, no. 10 (October 1, 2011): 2366–84. http://dx.doi.org/10.1175/2011jas3652.1.
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Abstract The effects of large eddies (LE) on the marine boundary layer (MBL) microphysics and thermodynamics is investigated using a 2D Lagrangian model with spectral bin microphysics including effects of sea spray. The 600 m × 400 m MBL computational area is covered by 3750 adjacent interacting Lagrangian parcels moving in a turbulent-like flow. A turbulent-like velocity field is designed as a sum of a high number of harmonics with random time-dependent amplitudes and different wavelengths including large eddies with scales of several hundred meters. The model explicitly calculates diffusion growth/evaporation, collisions, and sedimentation of droplets forming both as sea spray droplets and background aerosols, as well as aerosol masses within droplets. The turbulent mixing between parcels is explicitly taken into account. Sea spray generation is determined by a source function depending on the background wind speed assumed in the simulations to be equal to 20 m s−1. The results of simulations obtained by taking into account the effects of LE are compared to those obtained under the assumption that the vertical transport of droplets and passive scalars is caused by small-scale turbulent diffusion. Small-scale turbulence diffusion taken alone leads to an unrealistic MBL structure. Nonlocal mixing of the MBL caused by LE leads to the formation of a well-mixed MBL with a vertical structure close to the observed one. LE lead to an increase in the sensible and latent heat surface fluxes by 50%–100% and transport a significant amount of large spray droplets upward. Microphysical processes lead to formation of spray-induced drizzling clouds with cloud base near the 200-m level.
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Veron, Fabrice, W. Kendall Melville, and Luc Lenain. "The Effects of Small-Scale Turbulence on Air–Sea Heat Flux." Journal of Physical Oceanography 41, no. 1 (January 1, 2011): 205–20. http://dx.doi.org/10.1175/2010jpo4491.1.
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Abstract The air–sea exchange of heat is mainly controlled by the molecular diffusive layer adjacent to the surface. With an order of magnitude difference between the kinematic viscosity and thermal diffusivity of water, the thermal sublayer is embedded within its momentum analog: the viscous sublayer. Therefore, the surface heat exchange rates are greatly influenced by the surface kinematics and dynamics; in particular, small-scale phenomena, such as near-surface turbulence, have the greatest potential to affect the surface fluxes. Surface renewal theory was developed to parameterize the details of the turbulent transfer through the molecular sublayers. The theory assumes that turbulent eddies continuously replace surface water parcels with bulk fluid, which is not in equilibrium with the atmosphere and therefore is able to transfer heat. The so-called controlled-flux technique gives direct measurements of the mean surface lifetime of such surface renewal events. In this paper, the authors present results from field experiments, along with a review of surface renewal theory, and show that previous estimates of air–sea scalar fluxes using the controlled-flux technique may be erroneous if the probability density function (PDF) of surface renewal time scales is different from the routinely assumed exponential distribution. The authors show good agreement between measured and estimated heat fluxes using a surface renewal PDF that follows a χ distribution. Finally, over the range of forcing conditions in these field experiments, a clear relationship between direct surface turbulence measurements and the mean surface renewal time scale is established. The relationship is not dependent on the turbulence generation mechanism. The authors suggest that direct surface turbulence measurements may lead to improved estimates of scalar air–sea fluxes.
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Caputi, Dani J., Ian Faloona, Justin Trousdell, Jeanelle Smoot, Nicholas Falk, and Stephen Conley. "Residual layer ozone, mixing, and the nocturnal jet in California's San Joaquin Valley." Atmospheric Chemistry and Physics 19, no. 7 (April 9, 2019): 4721–40. http://dx.doi.org/10.5194/acp-19-4721-2019.
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Abstract. The San Joaquin Valley of California is known for excessive ozone air pollution owing to local production combined with terrain-induced flow patterns that channel air in from the highly populated San Francisco Bay area and stagnate it against the surrounding mountains. During the summer, ozone violations of the National Ambient Air Quality Standards (NAAQS) are notoriously common, with the San Joaquin Valley having an average of 115 violations of the current 70 ppb standard each year between 2012 and 2016. Because regional photochemical production peaks with actinic radiation, most studies focus on the daytime, and consequently the nocturnal chemistry and dynamics that contribute to these summertime high-ozone events are not as well elucidated. Here we investigate the hypothesis that on nights with a strong low-level jet (LLJ), ozone in the residual layer (RL) is more effectively mixed down into the nocturnal boundary layer (NBL) where it is subject to dry deposition to the surface, the rate of which is itself enhanced by the strength of the LLJ, resulting in lower ozone levels the following day. Conversely, nights with a weaker LLJ will sustain RLs that are more decoupled from the surface, retaining more ozone overnight, and thus lead to more fumigation of ozone the following mornings, giving rise to higher ozone concentrations the following afternoon. The relative importance of this effect, however, is strongly dependent on the net chemical overnight loss of Ox (here [Ox] ≡ [O3] + [NO2]), which we show is highly uncertain, without knowing the ultimate chemical fate of the nitrate radical (NO3). We analyze aircraft data from a study sponsored by the California Air Resources Board (CARB) aimed at quantifying the role of RL ozone in the high-ozone events in this area. By formulating nocturnal scalar budgets based on pairs of consecutive flights (the first around midnight and the second just after sunrise the following day), we estimate the rate of vertical mixing between the RL and the NBL and thereby infer eddy diffusion coefficients in the top half of the NBL. The average depth of the NBL observed on the 12 pairs of flights for this study was 210(±50) m. Of the average −1.3 ppb h−1 loss of Ox in the NBL during the overnight hours from midnight to 06:00 PST, −0.2 ppb h−1 was found to be due to horizontal advection, −1.2 ppb h−1 due to dry deposition, −2.7 ppb h−1 to chemical loss via nitrate production, and +2.8 ppb h−1 from mixing into the NBL from the RL. Based on the observed gradients of Ox in the top half of the NBL, these mixing rates yield eddy diffusivity estimates ranging from 1.1 to 3.5 m2 s−1, which are found to inversely correlate with the following afternoon's ozone levels, providing support for our hypothesis. The diffusivity values are approximately an order of magnitude larger than the few others reported in the extant literature for the NBL, which further suggests that the vigorous nature of nocturnal mixing in this region, due to the LLJ, may have an important control on daytime ozone levels. Additionally, we propose that the LLJ is a branch of what is colloquially referred to as the Fresno eddy, which has been previously proposed to recirculate pollutants. However, vertical mixing from the LLJ may counteract this effect, which highlights the importance of studying the LLJ and Fresno eddy as a single interactive system. The synoptic conditions that are associated with strong LLJs are found to contain deeper troughs along the California coastline. The LLJs observed during this study had an average centerline height of 340 m, average speed of 9.9 m s−1 (σ=3.1 m s−1), and a typical peak timing around 23:00 PST. A total of 7 years of 915 MHz radioacoustic sounding system and surface air quality network data show an inverse correlation between the jet strength and ozone the following day, further suggesting that air quality models need to forecast the strength of the LLJ in order to more accurately predict ozone violations.
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Yadav, Rakesh, Pravin Nakod, and Pravin Rajeshirke. "NO Prediction in Turbulent Diffusion Flame Using Multiple Unsteady Laminar Flamelet Modeling." Journal of Engineering for Gas Turbines and Power 136, no. 10 (May 9, 2014). http://dx.doi.org/10.1115/1.4026801.
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The steady laminar flamelet model (SLFM) (Peters, 1984, “Laminar Diffusion Flamelet Models in Non-Premixed Turbulent Combustion,” Prog. Energy Combust. Sci., 10(3), pp. 319–339; Peters, 1986, “Laminar Flamelet Concepts in Turbulent Combustion,” Symp. (Int.) Combust., 21(1), pp. 1231–1250) has been shown to be reasonably good for the predictions of mean temperature and the major species in turbulent flames (Borghi, 1988, “Turbulent Combustion Modeling,” Prog. Energy Combust. Sci., 14(4), pp. 245–292; Veynante and Vervisch, 2002, “Turbulent Combustion Modeling,” Prog. Energy Combust. Sci., 28(3), pp. 193–266). However, the SLFM approach has limitations in the prediction of slow chemistry phenomena like NO formation (Benim and Syed, 1998, “Laminar Flamelet Modeling of Turbulent Premixed Combustion,” Appl. Math. Model., 22(1–2), pp. 113–136; Heyl and Bockhorn, 2001, “Flamelet Modeling of NO Formation in Laminar and Turbulent Diffusion Flames,” Chemosphere, 42(5–7), pp. 449–462). In the case of SLFM, the turbulence and chemistry are coupled through a single variable called scalar dissipation, which is representative of the strain inside the flow. The SLFM is not able to respond to the steep changes in the scalar dissipation values and generally tends to approach to the equilibrium solution as the strain relaxes (Haworth et al., 1989, “The Importance of Time-Dependent Flame Structures in Stretched Laminar Flamelet Models for Turbulent Jet Diffusion Flames,” Symp. (Int.) Combust., 22(1), pp. 589–597). A pollutant like NO is formed in the post flame zones and with a high residence time, where the scalar dissipation diminishes and hence the NO is overpredicted using the SLFM approach. In order to improve the prediction of slow forming species, a transient history of the scalar dissipation evolution is required. In this work, a multiple unsteady laminar flamelet approach is implemented and used to model the NO formation in two turbulent diffusion flames using detailed chemistry. In this approach, multiple unsteady flamelet equations are solved, where each flamelet is associated with its own scalar dissipation history. The time averaged mean variables are calculated from weighted average contributions from different flamelets. The unsteady laminar flamelet solution starts with a converged solution obtained from the steady laminar flamelet modeling approach. The unsteady flamelet equations are, therefore, solved as a post processing step with the frozen flow field. The domain averaged scalar dissipation for a flamelet at each time step is obtained by solving a scalar transport equation, which represents the probability of occurrence of the considered flamelet. The present work involves the study of the effect of the number of flamelets and also the different methods of probability initialization on the accuracy of NO prediction. The current model predictions are compared with the experimental data. It is seen that the NO predictions improves significantly even with a single unsteady flamelet and further improves marginally with an increase in number of unsteady flamelets.
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"Mixing regimes in a spatially confined two-dimensional compressible mixing layer." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 449, no. 1936 (May 9, 1995): 351–80. http://dx.doi.org/10.1098/rspa.1995.0049.
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The evolution of a high-speed compressible confined temporally evolving supersonic mixing layer between hydrogen and oxygen gas streams is examined using time-dependent two-dimensional numerical simulations that include the effects of viscosity, molecular diffusion and thermal conduction. The flow shows three distinct mixing regimes: an apparently ordered, laminar stage in which the structures grow due to the initial perturbation; a convective-mixing regime in which vortices begin to interact and structures grow; and a diffusive-mixing regime in which vortical structures break down and diffusive mixing dominates. Varying the strength of the diffusion terms shows that diffusion is important in the laminar and diffusive-mixing stages, but not in the convective-mixing stage. Varying the convective Mach shows that compressiblity does not change the general structural features of the mixing process, although higher compressibility results in a slower transition between the various flow regimes. Increasing the size of the computational domain increases the absolute time of transition from convective to diffusive mixing, but does not affect the dimensionless time normalized to the system size. Comparisons between full Navier–Stokes computations at different levels of numerical resolution show that the measurements of scalar mixing converge for resolutions at an order of magnitude greater than the Kolmogorov scale, although measurements of turbulence intensity are more sensitive to grid size.
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Chen, Ziyu, Yifei Li, Xinrong Su, and Xin Yuan. "Scalar Diffusion Equation-Based Model to Predict 2-Dimensional Film Cooling Effectiveness of a Shaped Hole." Journal of Turbomachinery 143, no. 4 (March 22, 2021). http://dx.doi.org/10.1115/1.4049782.
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Abstract One-dimensional laterally averaged adiabatic film cooling effectiveness η¯lat-based correlations have been widely employed in the cooling design of the modern gas turbine and aero-engine; however, the flow field of the discrete film cooling is fully three dimensional, and thus, the cooling effectiveness distribution on the solid surface is two dimensional. Accurate prediction of the cooling effectiveness distribution in the lateral direction would help to optimize the film cooling design, but few paid attention to this issue in the literature. In this study, a simple yet accurate scalar diffusion equation based model is proposed to extend the one-dimensional correlation into two dimensional. The model is proved to be accurate and efficient. According to the accuracy analysis, the R2 value is larger than 0.95 for the two-dimensional prediction and over 0.93 along the centerline. With given input parameters, the calculation cost for solving a certain case is in the magnitude of 1 × 10−3s in time using the space-marching method. There is only the effective diffusion coefficient left to be modeled in the control equation. It represents the balance between the diffusion and the passive transportation by the main flow. Analyses conducted within the typical experimental range show that κ~eff is only dependent on the velocity ratio and the main-flow turbulence.
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Hupkes, H. J., and E. S. Van Vleck. "Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness." Journal of Dynamics and Differential Equations, July 14, 2021. http://dx.doi.org/10.1007/s10884-021-10013-5.
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AbstractIn this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations.
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von Wahl, Henry, Thomas Richter, and Christoph Lehrenfeld. "An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains." IMA Journal of Numerical Analysis, July 5, 2021. http://dx.doi.org/10.1093/imanum/drab044.
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Abstract We analyse a Eulerian finite element method, combining a Eulerian time-stepping scheme applied to the time-dependent Stokes equations with the CutFEM approach using inf-sup stable Taylor–Hood elements for the spatial discretization. This is based on the method introduced by Lehrenfeld & Olshanskii (2019, A Eulerian finite element method for PDEs in time-dependent domains. ESAIM: M2AN, 53, 585–614) in the context of a scalar convection–diffusion problems on moving domains, and extended to the nonstationary Stokes problem on moving domains by Burman et al. (2019, arXiv:1910.03054 [math.NA]) using stabilized equal-order elements. The analysis includes the geometrical error made by integrating over approximated level set domains in the discrete CutFEM setting. The method is implemented and the theoretical results are illustrated using numerical examples.
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Tumelero, Fernanda, Celso M. F. Lapa, Bardo E. J. Bodmann, and Marco T. Vilhena. "Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain." Brazilian Journal of Radiation Sciences 7, no. 2B (June 25, 2019). http://dx.doi.org/10.15392/bjrs.v7i2b.389.
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In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the first recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution.
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Uecker, Hannes. "Optimal spatial patterns in feeding, fishing, and pollution." Discrete & Continuous Dynamical Systems - S, 2021, 0. http://dx.doi.org/10.3934/dcdss.2021099.
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<p style='text-indent:20px;'>Infinite time horizon spatially distributed optimal control problems may show so–called optimal diffusion induced instabilities, which may lead to patterned optimal steady states, although the problem itself is completely homogeneous. Here we show that this can be considered as a generic phenomenon, in problems with scalar distributed states, by computing optimal spatial patterns and their canonical paths in three examples: optimal feeding, optimal fishing, and optimal pollution. The (numerical) analysis uses the continuation and bifurcation package <inline-formula><tex-math id="M1">\begin{document}$\mathtt{pde2path} $\end{document}</tex-math></inline-formula> to first compute bifurcation diagrams of canonical steady states, and then time–dependent optimal controls to control the systems from some initial states to a target steady state as <inline-formula><tex-math id="M2">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>. We consider two setups: The case of discrete patches in space, which allows to gain intuition and to compute domains of attraction of canonical steady states, and the spatially continuous (PDE) case.</p>