Numerical analysis and modeling of plume meandering in passive scalar dispersion downstream of a wall-mounted cube

A DNS database is employed to examine the onset of plume meandering downstream of a wall-mounted cube and to address the impact of large-scale unsteadiness in modeling dispersion using the RANS equations. The cube is immersed in a uniform stream where the thin boundary-layer developing over the flat plate is responsible for the onset of vortex-shedding in the wake of the bluff-body. Spectra of velocity and concentration fluctuations exhibit a prominent peak in the energy content at the same frequency, showing that the plume meandering is established by the action of the vortex-shedding. The vortex-shedding and plume meandering display a low-frequency modulation where coherent fluctuations are suppressed at times with a quasi-regular period. The onset of the low-frequency modulation is indicated by a secondary peak in the energy spectrum and confirmed by the autocorrelation of velocity and scalar fluctuations. Unsteady RANS simulations performed with the v² − f model are able to detect the onset of the plume meandering and show remarkable improvement of the predicted decay rate and rate of spread of the scalar plume when compared to steady RANS solutions. By computing explicitly the periodic component of velocity and scalar fluctuations, the unsteady v² − f model is able to provide a representation of scalar flux components consistent with DNS statistics, where the counter-gradient transport mechanism that takes place in the streamwise component is also captured by URANS results. Nonetheless, the agreement with DNS statistics for the mean concentration and the plume width is limited by the onset of the low-frequency modulation in the vortex-shedding and plume meandering, giving a challenging modeling issue in the simulation of dispersion using the RANS equations.     

Direct numerical simulation of turbulent mixing of a passive scalar in pipe flow

Turbulent mixing of a passive scalar in fully developed turbulent pipe flow has been investigated by means of a Direct Numerical Simulation (DNS). The scalar is released from a point source located on the centreline of the pipe. The domain size of the concentration field has been chosen large enough to capture the different stages of turbulent mixing. Results are presented for mean concentration profiles, turbulent fluxes, concentration fluctuations, probability density functions and higher-order moments. To validate the numerical simulations the results are compared with experimental data on mixing in grid-turbulence that have been reported in the literature. The agreement between the experimental measurements and the computations is satisfactory. We have also considered the Probability Density Function (PDF). For small diffusion times and positions not on the plume centreline, our results lead to a PDF of an exponential form with a large peak at zero concentration. When the diffusion time increases, the PDF shifts from a exponential to a more Gaussian form.     

Scalar transport from a point source in flows over wavy walls

Simultaneous measurements of the velocity and concentration field in fully developed turbulent flows over a wavy wall are described. The concentration field originates from a low-momentum plume of a passive tracer. PLIF and digital particle image velocimetry are used to make spatially resolved measurements of the structure of the scalar distribution and the velocity. The measurements are performed at three different Reynolds numbers of Re _b = 5,600, Re _b = 11,200 and Re _b = 22,400, respectively, based on the bulk velocity u _b and the total channel height 2h. The velocity field and the scalar field are investigated in a water channel with an aspect ratio of 12:1, where the bottom wall of the test section consists of a train of sinusoidal waves. The wavy wall is characterized by the amplitude to wavelength ratio α = 0.05 and the ratio β between the wave amplitude and the half channel height where β = 0.1. The scalar is released from a point source at the wave crest. For the concentration measurements, Rhodamine B is used as tracer dye. At low to moderate Reynolds number, the flow field is characterized through a recirculation zone which develops after the wave crest. The recirculation zone induces high intensities of the fluctuations of the streamwise velocity and wall-normal velocity. Furthermore, large-scale structures are apparent in the flow field. In previous investigations it has been shown that these large-scale structures meander laterally in flows over wavy bottom walls. The investigations show a strong effect of the wavy bottom wall on the scalar mixing. In the vicinity of the source, the scalar is transported by packets of fluid with a high scalar concentration. As they move downstream, these packets disintegrate into filament-like structures which are subject to strong gradients between the filaments and the surrounding fluid. The lateral scale of the turbulent plume is smaller than the lateral scale of the large-scale structures in the flow field and the plume dispersion is dominated by the structures in the flow field. Due to the lateral meandering of the large-scale structures of the flow field, also the scalar plume meanders laterally. Compared to turbulent plumes in plane channel flows, the wavy bottom wall enhances the mixing effect of the turbulent flow and the spreading rate of the scalar plume is increased.     

Finite element solution to passive scalar transport behind line sources under neutral and unstable stratification

This study employed a direct numerical simulation (DNS) technique to contrast the plume behaviours and mixing of passive scalar emitted from line sources (aligned with the spanwise direction) in neutrally and unstably stratified open‐channel flows. The DNS model was developed using the Galerkin finite element method (FEM) employing trilinear brick elements with equal‐order interpolating polynomials that solved the momentum and continuity equations, together with conservation of energy and mass equations in incompressible flow. The second‐order accurate fractional‐step method was used to handle the implicit velocity–pressure coupling in incompressible flow. It also segregated the solution to the advection and diffusion terms, which were then integrated in time, respectively, by the explicit third‐order accurate Runge–Kutta method and the implicit second‐order accurate Crank–Nicolson method. The buoyancy term under unstable stratification was integrated in time explicitly by the first‐order accurate Euler method. The DNS FEM model calculated the scalar‐plume development and the mean plume path. In particular, it calculated the plume meandering in the wall‐normal direction under unstable stratification that agreed well with the laboratory and field measurements, as well as previous modelling results available in literature. Copyright © 2005 John Wiley & Sons, Ltd.     

DNS study of passive plume interference emitting from two parallel line sources in a turbulent channel flow

Turbulent mixing of dual plumes emitting simultaneously from line sources in a turbulent channel flow has been studied using direct numerical simulation (DNS). Three test cases have been compared to investigate the effects of the source separation on turbulent mixing of the two instantaneous plumes. The dispersion and interference of dual plumes are investigated in both physical and spectral spaces, which include an analysis of statistical moments of the concentration field, cross-correlation between the two instantaneous plumes, pre-multiplied spectra of the velocity and concentration fields, and co-spectrum and coherency spectrum of the dual plumes. As the downstream distance from the line source increases, the plume development associated with a single source emission transitions from a turbulent convective stage to a turbulent diffusive stage. It is observed that a plume released from a ground-level source reaches the turbulent diffusive stage faster than that released from an elevated source. It is also observed that a smaller separation between the two line sources tends to facilitate a more rapid growth in the cross-correlation coefficient of two instantaneous plumes. In the near-source region, the maximum coherency spectrum is produced at lower frequencies indicating that dual-plume mixing is dominated by the external flapping effects of large-scale eddy motions. However, in the far downstream region of the sources, the coherency spectrum in the higher frequency range increases significantly, indicating that the spread of the total plume is larger than all scales of turbulent eddies, such that they all contribute to the in-plume mixing of the dual plumes.     

A turbulent time scale based k–ε model for probability density function modeling of turbulence/chemistry interactions: Application to HCCI combustion

Homogenous Charge Compression Ignition (HCCI) engine technology is known as an alternative to reduce NO_x and particulate matter (PM) emissions. As shown by several experimental studies published in the literature, the ideally homogeneous mixture charge becomes stratified in composition and temperature, and turbulent mixing is found to play an important role in controlling the combustion progress. In a previous study, an IEM model (Interaction by Exchange with the Mean) has been used to describe the micromixing in a stochastic reactor model that simulates the HCCI process. The IEM model is a deterministic model, based on the principle that the scalar value approaches the mean value over the entire volume with a characteristic mixing time. In this previous model, the turbulent time scale was treated as a fixed parameter. The present study focuses on the development of a micro-mixing time model, in order to take into account the physical phenomena it stands for. For that purpose, a (k–ε) model is used to express this micro-mixing time model. The turbulence model used here is based on zero dimensional energy cascade applied during the compression and the expansion cycle; mean kinetic energy is converted to turbulent kinetic energy. Turbulent kinetic energy is converted to heat through viscous dissipation. Besides, in this study a relation to calculate the initial heterogeneities amplitude is proposed. The comparison of simulation results against experimental data shows overall satisfactory agreement at variable turbulent time scale.     

最小均方差条件和非线性IEM模型的分析

对最小均方差估计条件在构造混合模型中的应用作了进一步的讨论，证明了最小均方差条件可以代替工方程作为推导混合模型的约束条件，同时得到了混合项的一个严格的展开式。作为一般情况的一个特例，重新得到了NLIME模型，最后利用NLIEM模型计算了两个基本流场，并和IEM模型的计算结果作了比较。进一步表明了NLIEM模型的优越性。     

Structural inclination angle of near-field scalar fluctuations in a turbulent boundary layer

Two-point concentration measurements are obtained in a meandering passive-scalar plume released at five different heights within the fully-turbulent region of a high-Reynolds number turbulent boundary layer (TBL). Mean statistics of two-point concentration measurements are found to agree very well with the single-point measurements previously reported in Talluru et al. (2017a). The two-point correlation results of concentration indicate strong coherence in the scalar field similar to the large-scale coherence observed in the streamwise velocity fluctuations in a TBL (Marusic and Heuer, 2007). Particularly, the isocontours in the two-dimensional correlation map of concentration fluctuations illustrate that the scalar structures are inclined at 30^∘ to the direction of the flow; such a trend is consistently observed for all the elevated plumes below z/δ ≤ 0.33. This observation of steeper inclination angle of scalar structures relative to the inclination angle of large-scale velocity fluctuations in a TBL is explained using the physical model put forth by Talluru et al. (2018). Most importantly, these results provide insights on the differences in the structural organisation of a passive scalar plume in the near- and the far-field regions.     

Large eddy simulation (2D) of spatially developing mixing layer using vortex‐in‐cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion‐velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte‐Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations profiles and negative cross‐stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root‐mean‐square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

A Comprehensive Study of Effects of Mixing and Chemical Kinetic Models on Predictions of n-heptane Jet Ignitions with the PDF Method

The composition Probability Density Function (PDF) model is coupled with a Reynolds-averaged k???ε turbulence model and three computationally efficient, yet widely used chemical mechanisms to simulate transient n-heptane spray injection and ignition in a high temperature and high density ambient fluid. Molecular diffusion is modelled by three mixing models, namely the interaction by exchange with the mean (IEM), modified Curl (MC) and Euclidean minimum spanning trees (EMST) models. The liquid phase is modelled by a discrete phase model (DPM). This represents among the first applications of the PDF method in practical diesel engine conditions. A non-reacting case is first considered, with the focus on the ability of the model to capture the spray structure, e.g., vapour penetration and liquid length, fuel mixture fraction and its variance. Reacting cases are then investigated to compare and evaluate the three different chemical mechanisms and the three mixing models. It is concluded that the EMST mixing model in conjunction with a reduced chemical kinetic model (Lu et al., Combust Flame 156(8):1542–1551, 2009) performs the best among the options considered. The sensitivity of the results to the choice of the mixing constant is also studied to understand its effect on the flame ignition and stabilisation. Finally, the PDF model is compared to a well-mixed model that assumes turbulent fluctuations are negligible, which has been widely used in the diesel spray combustion community. Significant structural differences in the modelled flame are revealed comparing the PDF method with the well-mixed model. Quantitatively, the PDF model exhibits excellent agreement with the measurements and shows much better results than the well-mixed model in all ambient O₂ and temperature conditions.     

Sensitivity of plume dynamics to the parameterization of vertical mixing

A three‐dimensional primitive equation, baroclinic numerical model incorporating a range of turbulence closure schemes is used to investigate the effects of vertical diffusion of momentum and density upon the spread of a freshwater plume, with particular reference to the Ebro plume. Initial calculations show that there are some differences in the horizontal spread and vertical mixing of the plume when diffusion coefficients are computed from a two‐equation turbulence energy model compared with a one‐equation model. To understand results from the turbulence energy models, the sensitivity of the plume dynamics to variations in the coefficient of vertical eddy viscosity and diffusivity is also considered, with increases in these parameters having a significant effect upon the cross‐shore and along‐shore spread of the plume. Also, increasing these parameters changes the plume characteristics from supercritical to subcritical and reduces the occurrence of meandering and baroclinic instability along the plume's off‐shore edge. However, differences in the southerly spread (the direction of Kelvin wave propagation in the model) of the plume (although not its northerly spread) produced by changes in diffusion coefficients are small compared with the influence of changes in the bottom slope, upon the along‐shore southerly spread of the plume, which moves in the direction of Kelvin wave propagation in the near coastal region. Results from the series of calculations are used as a guide in experimental design, with reference to a planned experiment in the Ebro region involving a coastal HF Radar deployment, as well as off‐shore measurements. Calculations suggest that surface current measurements from a coastal HF Radar, together with a detailed survey of the density field associated with the plume, may be an appropriate, although indirect, means of determining suitable mixing coefficients to use in plume discharge problems. Detailed measurements of water depth variation will also be required. Copyright © 1999 John Wiley & Sons, Ltd.     

Mixing and Entrainment of Transitional Non-Circular Buoyant Reactive Plumes

Three-dimensional spatial direct numerical simulation is used to investigate the evolution of reactive plumes established on non-circular sources. Simulations are performed for three cases: a rectangular plume with an aspect ratio of 2:1, a square plume, and the square plume in a corner configuration. Buoyancy-induced large scale vortical structures evolve spatially in the flow field. A stronger tendency of transition to turbulence is observed for the free rectangular plume than the free square case due to the aspect ratio effect. Dynamics of the corner square plume differs significantly from the corresponding free case due to the enhanced mixing by the side-wall effects. A turbulent inertial subrange has been observed for the free rectangular and corner square plumes. Mean flow properties are also calculated. The study shows significant effects of source geometry and side-wall boundary on the flow transition and entrainment of reactive plumes. This revised version was published online in July 2006 with corrections to the Cover Date.     

Large eddy simulation of hot and cold fluids mixing in a T-junction for predicting thermal fluctuations

Temperature fluctuations in a mixing T-junction have been simulated on the FLUENT platform using the large eddy simulation （LES） turbulent flow model and a sub-grid scale Smagorinsky-Lilly model. The normalized mean and root mean square temperatures for describing time-averaged temperature and temperature fluctuation intensity, and the velocity are obtained. The power spectrum densities of temperature fluctuations, which are key parameters for thermal fatigue analysis and lifetime evaluation, are analyzed. Simulation results are consistent with experimental data published in the literature, showing that the LES is reliable. Several mixing processes under different conditions are simulated in order to analyze the effects of varying Reynolds number and Richardson number on the mixing course and thermal fluctuations.     

A comparison between a plume path model and a virtual point source model for a stack plume

Some years ago we developed a theoretical model for the calculation of the rise of a stack plume in the atmosphere. Recently we have extended this model in such a way that the ground level concentration of gases emitted by a stack can also be calculated. The cross-section of the plume, which was originally assumed to be a circle, is now assumed to be an ellipse. The height-to-width ratio of the ellipse is chosen in accordance with full-scale data. The entrainment of air into the plume due to atmospheric turbulence can now be calculated not only for neutral atmospheric conditions, but also for stable and unstable conditions.Predictions of the ground level concentration made with this model have been compared with predictions made with the often used virtual point source model. For neutral and unstable atmospheric conditions the agreement is rather good. For stable conditions the differences can be rather large. The sensitivity of the ground level concentration with respect to the atmospheric stability condition, the stack height, the plume exit momentum flux and the plume exit buoyancy flux has also been investigated with the aid of the model. The influence of the plume exit momentum flux is negligible; however, the influence of the other parameters can be large.     

Vortex-In-Cell and Probability Density Function Approach for a Passive Scalar Field in a Mixing Layer

A Reynolds-averaged simulation based on the vortex-in-cell (VIC) and the transport equation for the probability density function (PDF) of a scalar has been developed to predict the passive scalar field in a two-dimensional spatially growing mixing layer. The VIC computes the instantaneous velocity and vorticity fields. Then the mean-flow properties, i.e. the mean velocity, the root-mean-square (rms) longitudinal and lateral velocity fluctuations, the Reynolds shear stress, and the rms vorticity fluctuations are computed and used as input to the PDF equation. The PDF transport equation is solved using the Monte Carlo technique. The convection term uses the mean velocities from the VIC. The turbulent diffusion term is modeled using the gradient transport model, in which the eddy diffusivity, computed via the Boussinesq's postulate, uses the Reynolds shear stress and gradients of mean velocities from the VIC. The molecular mixing term is closed by the modified Curl model.

The computational results were compared with two-dimensional experimental results for passive scalar. The predicted turbulent flow characteristics, i.e. mean velocity and rms longitudinal fluctuations in the self-preserving region, show good agreement with the experimental measurements. The profiles of the mean scalar and the rms scalar fluctuations are also in reasonable agreement with the experimental measurements. Comparison between the mean scalar and the mean velocity profiles shows that the scalar mixing region extends further into the free stream than does the momentum mixing region, indicating enhanced transport of scalar over momentum. The rms scalar profiles exhibit an asymmetry relative to the concentration centerline, and indicate that the fluid on the high-speed side mixes at a faster rate than the fluid on the low-speed side. The asymmetry is due to the asymmetry in the mixing frequency cross-stream profiles. Also, the PDFs have peaks biased toward the high-speed side.     

Effects of Mass Reduction,Flow Reduction and Enhanced Biodecay of DNAPL Source Zones

A mathematical model and an analytical solution are presented to describe field-scale dense non-aqueous phase liquid (DNAPL) source dissolution and source zone biodecay rates coupled with advective–dispersive dissolved plume transport. The model is employed to investigate various source remediation options on source zone mass depletion, net source mass flux, and dissolved plume attenuation for different source zone “architectures” (i.e., pools versus residual DNAPL) and compliance criteria. Remediation options considered include partial source mass removal, source flow reduction, and source zone enhanced biodecay. Partial mass reduction reduces the source zone mass flux and downgradient concentrations for residual DNAPL sources and pools, which can significantly reduce dissolved plume size and time to reach compliance criteria. Source zone flow reduction decreases the rate of source mass depletion, but can facilitate compliance, if concentrations at compliance locations are not too high initially. Increase in source biodecay rate, especially with concomitant increases in dissolution kinetics, can decrease the time to achieve compliance criteria over biodecay alone.     

Source Characteristics and Contaminant Plume Evolution

This study investigated how features of mass distribution within a source area control the pattern of contaminant plume evolution. A series of numerical trials was conducted which simulated contaminant migration in a three-dimensional saturated porous media with multiple source patches representing zones of contamination. Results showed that the extent of mixing near the source was greatly affected by both advection and dispersive mixing processes. The factors affecting the concentration distribution in the dissolved plume were size and distribution of source patches, dispersivity values, and the ratio of total patch area to the source area. The influence of dispersive mixing increased as the size of source patches decreased. However, dispersion was less important when patches were clustered or the ratio of total patch area to the source area increased. The variations in concentration values near the source, which were caused by differences in sizes and distribution of patches, diminished as the contaminant traveled further from the source. This result implies that detailed information on source characteristics generally would not be required beyond a certain travel distance. The exception in this regard is the ratio of total patch area to the source area, i.e., a zero-difference point. However, the location of this zero-difference point cannot be predicted easily because the location was affected by several source characteristics.     

Taylorian diffusion in mildly inhomogeneous turbulence

Taylor’s theory of dispersion was extended to produce estimates of the far–field growth rate of the plume of a passive scalar in grid turbulence (GT) and in uniformly sheared flow (USF), both of which evolve in the streamwise direction. Expressions for the evolution of the plume width relative to the integral length scale of turbulence were also derived. The predictions of plume growth rate were tested against available data in both of these types of flows and were found to be accurate in an extensive region of USF and compatible with an extrapolated trend in GT, in which the available data did not extend very far from the scalar source. Although in both cases the measured half-width of the plume was comparable in magnitude to the streamwise integral length scale of the turbulence, the far–field approximation seemed to hold.     

Simple analytical model for depth-averaged velocity in meandering compound channels

A simple but applicable analytical model is presented to predict the lateral distribution of the depth-averaged velocity in meandering compound channels. The governing equation with curvilinear coordinates is derived from the momentum equation and the flow continuity equation under the condition of quasi-uniform flow. A series of experiments are conducted in a large-scale meandering compound channel. Based on the experimental data, a magnitude analysis is carried out for the governing equation, and two lower-order shear stress terms are ignored. Four groups of experimental data from different sources are used to verify the predictive capability of this model, and good predictions are obtained. Finally, the determination of the velocity parameter and the limitation of this model are discussed.     

Large Scale Mixing for Immiscible Displacement in Heterogeneous Porous Media

We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.