Passive Scalar Equation Research Articles

A unified lattice Boltzmann - phase field scheme for simulations of solutal dendrite growth in the presence of melt convection

A unified lattice Boltzmann (LB) - phase field (PF) scheme is proposed to simulate the dendrite growth of binary alloys in the presence of melt convection. The solute transfer with phase transition is modeled by using the passive scalar equation fitting into the LB framework, and the melt convection is governed by the multiple-relaxation-time (MRT) LB equation. The evolutions of all the fields during solutal dendrite growth are described by the LB equations, and the macroscopic equations can be recovered by the Chapman-Enskog analysis. Model validation and verification are accomplished through comparisons with benchmark solutions and mesh-independence examination. The growths of the single/multiple equiaxial and columnar dendrites are numerically investigated by using the present model. The effects of supersaturation, anisotropy, thermal gradient and forced convection are concluded in terms of morphologies, growing kinetics and solute segregation.

Indices for ventilation efficiency and their numerical computation

The paper addresses methods to evaluate the ventilation efficiency both in regard to air exchange and to contaminant removal in a room. Extra variables are introduced via transport differential equations which are solved by Computational Fluid Dynamics (CFD). Ventilation efficiency indices are defined and computed by alternative methods with an excellent agreement. For the numerical calculations a test chamber with a fume cupboard has been modelled and simulated. The paper discusses peculiarities in the calculation of the passive scalar equations to ensure mass conservation. Guidelines for optimal locations of the exhaust grilles of ventilation systems have been proposed to maximize ventilation efficiency.

Enhanced dissipation, hypoellipticity for passive scalar equations with fractional dissipation

We consider the passive scalar equations subject to shear flow advection and fractional dissipation. The enhanced dissipation estimates are derived. For the classical passive scalar equation (γ=1), our result agrees with the sharp one obtained in [46].

New symmetry-induced scaling laws of passive scalar transport in turbulent plane jets

Abstract

Heat diffusion in a channel under white noise modeling of turbulence

<abstract><p>A passive scalar equation for the heat diffusion and transport in an infinite channel is studied. The velocity field is white noise in time, modelling phenomenologically a turbulent fluid. Under the driving effect of a heat source, the phenomenon of eddy dissipation is investigated: the solution is close, in a weak sense, to the stationary deterministic solution of the heat equation with augmented diffusion coefficients.</p></abstract>

Investigation of the Horizontal Motion of Particle-Laden Jets

Particle-laden jet flows can be observed in many industrial applications. In this investigation, the horizontal motion of particle laden jets is simulated using the Eulerian–Lagrangian framework. The two-way coupling is applied to the model to simulate the interaction between discrete and continuum phase. In order to track the continuum phase, a passive scalar equation is added to the solver. Eddy Life Time (ELT) is employed as a dispersion model. The influences of different non-dimensional parameters, such as Stokes number, Jet Reynolds number and mass loading ratio on the flow characteristics, are studied. The results of the simulations are verified with the available experimental data. It is revealed that more gravitational force is exerted on the jet as a result of the increase in mass loading, which deflects it more. Moreover, with an increase in the Reynolds number, the speed of the jet rises, and consequently, the gravitational force becomes less capable of deviating the jet. In addition, it is observed that by increasing the Stokes number, the particles leave the jet at higher speed, which causes a lower deviation of the jet.

Mixing dynamics in an uncovered unbaffled stirred tank using Large-Eddy Simulations and a passive scalar transport equation

In this study, several large eddy simulations were performed to solve the free-surface flow in an uncovered unbaffled stirred tank driven by a Rushton turbine using different Froude numbers and impeller clearance settings. The numerical code developed is based on the pseudo-compressibility method. To solve the flow equations and track the free surface, compact finite difference schemes and the level set method were used. For the treatment of the boundary conditions at the interface and for the tank internals representation, the ghost fluid method was used. The liquid surface was studied and showed excellent agreement when compared with previous experimental studies. The trailing vortices behind the impeller blades were visualized. Additionally, other important turbulent structures were captured moving over the impeller and around the critical radius. Finally, a passive scalar equation was added to show part of the mixing dynamics, in which an important segregated zone appeared under the impeller.

On geometric and analytic mixing scales: comparability and convergence rates for transport problems

In this article we are interested in the geometric and analytic mixing scales of solutions to passive scalar problems. Here, we show that both notions are comparable after possibly removing large scale projections. In order to discuss our techniques in a transparent way, we further introduce a dyadic model problem. In a second part of our article we consider the question of sharp decay rates for both scales for Sobolev regular initial data when evolving under the transport equation and related active and passive scalar equations. Here, we show that slightly faster rates than the expected algebraic decay rates are optimal.

Modification of vortex dynamics and transport properties of transitional axisymmetric jets using zero-net-mass-flux actuation

We study the near field of a zero-net-mass-flux (ZNMF) actuated round jet using direct numerical simulations. The Reynolds number of the jet ReD = 2000 and three ZNMF actuators are used, evenly distributed over a circle, and directed towards the main jet. The actuators are triggered in phase, and have a relatively low momentum coefficient of Cμ = 0.0049 each. We study four different control frequencies with Strouhal numbers ranging from StD = 0.165 to StD = 1.32; next to that, also two uncontrolled baseline cases are included in the study. We find that this type of ZNMF actuation leads to strong deformations of the near-field jet region that are very similar to those observed for non-circular jets. At the end of the jet's potential core (x/D = 5), the jet-column cross section is deformed into a hexagram-like geometry that results from strong modifications of the vortex structures. Two mechanisms lead to these modifications, i.e., (i) self-deformation of the jet's primary vortex rings started by distortions in their azimuthal curvature by the actuation, and (ii) production of side jets by the development and subsequent detachment of secondary streamwise vortex pairs. Further downstream (x/D = 10), the jet transforms into a triangular pattern, as the sharp corner regions of the hexagram entrain fluid and spread. We further investigate the global characteristics of the actuated jets. In particular when using the jet preferred frequency, i.e., StD = 0.33, parameters such as entrainment, centerline decay rate, and mean turbulent kinetic energy are significantly increased. Furthermore, high frequency actuation, i.e., StD = 1.32, is found to suppress the mechanisms leading to large scale structure growth and turbulent kinetic energy production. The simulations further include a passive scalar equation, and passive scalar mixing is also quantified and visualized.

Persistence in advection of a passive scalar

We consider the persistence phenomenon in advected passive scalar equation in one dimension. The velocity field is random with the v(k,omega)v(-k,-omega) approximately mid R:kmid R:;{-(2+alpha)} . In the presence of the nonlinearity the complete Green's function becomes G;{-1}=-iomega+Dk2+Sigma . We determine Sigma self-consistently from the correlation function which gives Sigma approximately k;{beta} , with beta=(1-alpha)2 . The effect of the nonlinear term in the equation in the O(;{2}) is to replace the diffusion term due to molecular viscosity by an effective term of the form Sigma_{0}k;{beta} . The stationary correlator for this system is [sech(T2)];{1beta} . Using the self-consistent theory we have determined the relation between beta and alpha . Finally, the independent interval approximation (IIA) method is used to determine the persistent exponent.

A hybrid grid/particle filter for Lagrangian data assimilation. I: Formulating the passive scalar approximation

AbstractIn this two‐part paper, we present a new nonlinear method for the assimilation of Lagrangian data. In part I, we formulate the method as a generalization of other particle filters. A particularly novel feature of the formulation is the use of a hybrid discretisation of the probability density function (PDF) in physical/phase space. Moreover, we show that, under the assumption that the drifters are uncorrelated, the projection of the Fokker–Planck equation onto the observation space associated with the drifter positions reduces to a set of passive scalar equations. This property allows us to efficiently compute the transitional PDF. To compute the analysis states, we present a grid/particle filter specifically formulated for use with the hybrid representation of our PDF. In common with other particle filters, our filter can suffer from sample impoverishment. To remedy this problem, we extend the Gaussian resampling procedure of Xiong et al. to produce a very efficient filter. This produces a fully functional scheme for Lagrangian data assimilation when combined with our forecasts of the prior. Copyright © 2008 Royal Meteorological Society

Wiener chaos solutions of linear stochastic evolution equations

A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.

Resonant interactions in rotating homogeneous three-dimensional turbulence

Direct numerical simulations of three-dimensional (3D) homogeneous turbulence under rapid rigid rotation are conducted to examine the predictions of resonant wave theory for both small Rossby number and large Reynolds number. The simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by 2D Navier-Stokes equations for resonant interactions of slow modes. As the rotation rate increases, the vertically-averaged horizontal velocity field from 3D Navier-Stokes converges to the velocity field from 2D Navier-Stokes, as measured by the energy in their difference field. Likewise, the vertically-averaged vertical velocity from 3D Navier-Stokes converges to a solution of the 2D passive scalar equation. The energy flux directly into small wave numbers in the $k_z=0$ plane from non-resonant interactions decreases, while fast-mode energy concentrates closer to that plane. The simulations are consistent with an increasingly dominant role of resonant triads for more rapid rotation.

Spectral properties of decaying turbulence in electron magnetohydrodynamics

The spectral properties of decaying turbulence in 212-dimensional electron magnetohydrodynamics are studied numerically. In the range kde<1 the energy exhibits a direct cascade while mean square momentum exhibits an inverse cascade. Their spectra are characterized by k−7/3 and k−13/3, respectively. The self-similar decay state of the turbulence is reached after an initial phase of fast exchange between the axial and poloidal magnetic energies. The time behavior t−2/3 of the total energy is found to be consistent with that obtained from selective decay. The maximum of the energy spectrum shifts towards low mode numbers and decays in time as t−1, in agreement with the infrared scaling of the turbulence. In the large de limit, both energy and mean square generalized momentum exhibit direct cascades. No stationary turbulent state could be found as long as the axial kinetic energy is large as compared to the poloidal kinetic energy initially. The global physical quantities decay well before turbulent macroscopic quantities have established similar space–time behavior, and the turbulence is infected by the lack of stationarity. The system decouples into a Navier–Stokes equation and a passive scalar equation only if the poloidal kinetic energy is larger than or equal to the axial kinetic energy. In this limit the k−5/3 and k−3 spectra of the poloidal kinetic energy are recovered.

Nonlocal transport of passive scalars in turbulent penetrative convection

We present a Green's function approach for quantifying the transport of a passive scalar (tracer) field in three-dimensional simulations of turbulent convection. Nonlocal, nondiffusive behavior is described by a transilient matrix (the discretized Green's function), whose elements contain the fractional tracer concentrations moving from one subvolume to another as a function of time. The approach was originally developed for and applied to geophysical flows, but here we extend the formalism and apply it in an astrophysical context to three-dimensional simulations of turbulent compressible convection with overshoot into convectively stable bounding regions. We introduce a novel technique to compute this matrix in a single simulation by advecting labeled particles rather than solving the passive scalar equation for a large number of different initial conditions. The transilient matrices thus computed are used as a diagnostic tool to quantitatively describe nonlocal transport via matrix moments and transport coefficients in a generalized, multiorder diffusion equation. Results indicate that transport in both the vertical and horizontal directions is strongly influenced by the presence of coherent velocity structures, generally resembling ballistic advection more than diffusion. The transport of a small fraction of tracer particles deep into the underlying stable region is reasonably efficient, a result which has possible implications for the problem of light-element depletion in late-type stars.

The equations of nearly incompressible fluids. I. Hydrodynamics, turbulence, and waves

A unified analysis delineating the conditions under which the equations of classical incompressible and compressible hydrodynamics are related in the absence of large-scale thermal, gravitational, and field gradients is presented. By means of singular expansion techniques, a method is developed to derive modified systems of fluid equations in which the effects of compressibility are admitted only weakly in terms of the incompressible hydrodynamic solutions (hence ‘‘nearly incompressible hydrodynamics’’). Besides including molecular viscosity self-consistently, the role of thermal conduction in an ideal fluid is also considered. With the inclusion of heat conduction, it is found that two distinct routes to incompressibility are possible, distinguished according to the relative magnitudes of the temperature, density, and pressure fluctuations. This leads to two distinct models for thermally conducting, nearly incompressible hydrodynamics—heat-fluctuation-dominated hydrodynamics (HFDH’s) and heat-fluctuation-modified hydrodynamics (HFMD’s). For the HFD case, the well-known classical passive scalar equation for temperature is derived as one of the nearly incompressible fluid equations and temperature and density fluctuations are predicted to be anticorrelated. For HFM fluids, a new thermal transport equation, in which compressible acoustic effects are present, is obtained together with a more-complicated ‘‘correlation’’ between temperature, density, and pressure fluctuations. Although the equations of nearly incompressible hydrodynamics are envisaged principally as being applicable to homogeneous turbulence and wave propagation in low Mach number flow, it is anticipated that their applicability is likely to be far greater.

Nearly incompressible hydrodynamics and heat conduction.

By means of an asymptotic analysis, two distinct approaches to incompressibility are found for a low-Mach-number ideal fluid, distinguished according to the relative magnitudes of temperature, density, and pressure fluctuations. For heat-conduction-dominated-fluids, temperature and density fluctuations are predicted to be anticorrelated, and the classical passive scalar equation for temperature is recovered, whereas a generalized ``pseudosound'' relationship for the fluctuations is found for heat-conduction-modified fluids, together with a modified thermal equation. The full set of nearly incompressible dynamical equations is described.