Scalar Flux Models Research Articles

Evaluations of SGS Combustion, Scalar Flux and Stress Models in a Turbulent Jet Premixed Flame

A newly developed fractal dynamic SGS (FDSGS) combustion model and a scale self-recognition mixed (SSRM) SGS stress model are evaluated along with other SGS combustion, scalar flux and stress models in a priori and a posteriori manners using DNS data of a hydrogen-air turbulent plane jet premixed flame. A posteriori tests reveal that the LES using the FDSGS combustion model can predict the combustion field well in terms of mean temperature distributions and peak positions in the transverse distributions of filtered reaction progress variable fluctuations. A priori and a posteriori tests of the scalar flux models show that a model proposed by Clark et al. accurately predicts the counter-gradient transport as well as the gradient diffusion, and introduction of the model of Clark et al. into the LES yields slightly better predictions of the filtered progress variable fluctuations than that of a gradient diffusion model. Evaluations of the stress models reveal that the LES with the SSRM model predicts the velocity fluctuations well compared to that with the Smagorinsky model.

Application of algebraic anisotropic turbulence models to film cooling flows

The complex structures in the flow field of gas turbine film cooling increase the anisotropy of turbulence making it difficult to accurately compute turbulent eddy viscosity and scalar diffusivity. An algebraic anisotropic turbulence model is developed while aiming at a more accurate modeling of the Reynolds stress and turbulent scalar flux. The developed algebraic anisotropic turbulence model is validated by a classic film cooling experiment and further verified by a series of in-house experiments with different hole geometries, density ratios and blowing ratios. On one hand, the anisotropic eddy viscosity method improves the modeling of Reynolds stress and the predictive flow field. On the other hand, the anisotropic turbulent scalar-flux model includes the role of anisotropic eddy viscosity in modeling of scalar flux and improves the turbulent scalar flux prediction. Compared with the experimental results, the algebraic anisotropic turbulence model is available and accurate quantitatively. The algebraic anisotropic turbulence model is prospective to improve the RANS predictive capability of film cooling.

Large Eddy Simulation of the Mixing of a Passive Scalar in a High-Schmidt Turbulent Jet

Accurate subgrid-scale (SGS) scalar flux models are essential when large eddy simulation (LES) is used to represent flow, mixing and transport of passive and active scalars in engineering, and environmental applications in turbulent regime. Many SGS scalar flux models have been developed for flows with low Schmidt numbers (Sc), but their application to high Sc flows has important limitations. In high Sc flows, the behavior of the scalar field becomes anisotropic because of intermittency effects, phenomenon that must be accounted for by SGS scalar flux models. The objective of this paper is to evaluate the ability of three SGS scalar flux models to predict the scalar behavior of a high Sc-number flow configuration, namely the anisotropy-resolved SGS scalar flux model: (1) appropriate for high Sc-number flow configurations, and two additional SGS models (linear eddy diffusivity based SGS models) with (2) constant, and (3) dynamically calculated turbulent Schmidt number. The LES simulation results accomplished by these models are compared to each other and to experimental data of a turbulent round jet discharging a diluted scalar into a low-velocity coflowing water stream. The comparison of simulation results and experimental observations shows that, in general, all SGS models reproduce the mean filtered concentration distribution in radial direction. The dynamic eddy diffusivity and anisotropy models reproduce the rms of the concentration and SGS scalar fluxes distribution. In particular, the anisotropy model improves the prediction reliability of LES. However, the three models evaluated in this study cannot accurately predict the scalar behavior at the superviscous layer. Finally, this work demonstrates that complex models can achieve reliable predictions on reasonable grids using less computational effort, while simple models require fine grids with increased computational costs.

Assessment of the performances of sub-grid scalar flux models for premixed flames with different global Lewis numbers: A Direct Numerical Simulation analysis

The statistical behaviours of sub-grid flux of reaction progress variable has been assessed for premixed turbulent flames with global Lewis number Le (=thermal diffusivity/mass diffusivity) ranging from 0.34 to 1.2 using a Direct Numerical Simulation (DNS) database of freely propagating statistically planar flames. It is known that the sub-grid scalar flux shows counter-gradient transport when the velocity jump across the flame due to heat release overcomes the effects of turbulent velocity fluctuation. The results show that the sub-grid scalar flux components exhibit counter-gradient transport for all cases considered here. The extent of counter-gradient transport increases with increasing filter width Δ and decreasing value of Le. This is due to the fact that flames with Le≪1 (e.g. Le=0.34) exhibit thermo-diffusive instabilities, which in turn increases the extent of counter-gradient transport. The effects of heat release and flame normal acceleration weaken with increasing Le. Several established algebraic models have been assessed in comparison to the sub-grid scalar flux components extracted from explicitly filtered DNS data in terms of their correlation coefficients at the vector level and their mean variation conditional on the Favre-filtered progress variable. The gradient transport closure does neither capture the quantitative nor the qualitative behaviour of the different sub-grid scalar flux components for all filter widths in all cases considered here. Models which account for local flame normal acceleration perform better, especially when the flame remains completely unresolved. In particular those models that account for the alignment of local resolved velocity and scalar gradients by using a tensor diffusivity, perform relatively better than the other alternative models irrespective of Le.

A stochastic extension of the explicit algebraic subgrid-scale models

The explicit algebraic subgrid-scale (SGS) stress model (EASM) of Marstorp et al. [“Explicit algebraic subgrid stress models with application to rotating channel flow,” J. Fluid Mech. 639, 403–432 (2009)] and explicit algebraic SGS scalar flux model (EASFM) of Rasam et al. [“An explicit algebraic model for the subgrid-scale passive scalar flux,” J. Fluid Mech. 721, 541–577 (2013)] are extended with stochastic terms based on the Langevin equation formalism for the subgrid-scales by Marstorp et al. [“A stochastic subgrid model with application to turbulent flow and scalar mixing,” Phys. Fluids 19, 035107 (2007)]. The EASM and EASFM are nonlinear mixed and tensor eddy-diffusivity models, which improve large eddy simulation (LES) predictions of the mean flow, Reynolds stresses, and scalar fluxes of wall-bounded flows compared to isotropic eddy-viscosity and eddy-diffusivity SGS models, especially at coarse resolutions. The purpose of the stochastic extension of the explicit algebraic SGS models is to further improve the characteristics of the kinetic energy and scalar variance SGS dissipation, which are key quantities that govern the small-scale mixing and dispersion dynamics. LES of turbulent channel flow with passive scalar transport shows that the stochastic terms enhance SGS dissipation statistics such as length scale, variance, and probability density functions and introduce a significant amount of backscatter of energy from the subgrid to the resolved scales without causing numerical stability problems. The improvements in the SGS dissipation predictions in turn enhances the predicted resolved statistics such as the mean scalar, scalar fluxes, Reynolds stresses, and correlation lengths. Moreover, the nonalignment between the SGS stress and resolved strain-rate tensors predicted by the EASM with stochastic extension is in much closer agreement with direct numerical simulation data.

A priori evaluation of large eddy simulation subgrid-scale scalar flux models in isotropic passive-scalar and anisotropic buoyancy-driven homogeneous turbulence

Direct numerical simulation (DNS) of passive (non-buoyant) and active (buoyant) scalar homogeneous turbulence is carried out using a standard pseudo-spectral numerical method. The flow settings simulated include stationary forced and decaying passive-scalar turbulence, as well as decaying anisotropic active-scalar turbulence. The Schmidt number is unity in all cases. The results are compared with, and are found to be in very good agreement with, previous similar DNS studies. The well-validated DNS data are divided into 19 sets, and are employed to study different large eddy simulation (LES) subgrid-scale (SGS) models for the SGS scalar flux. The models examined include three eddy-viscosity-type models (Smagorinsky, Vreman and Sigma with a constant SGS Schmidt number), a Dynamic Structure model and two versions of the Gradient (Gradient and Modulated Gradient) model. The models are investigated with respect to their ability to predict the orientation, and the magnitude, of the SGS scalar flux. Eddy-viscosity models are found to predict the magnitude of the SGS scalar flux accurately, but are poor at predicting the orientation of the SGS scalar flux. The Dynamic Structure and Gradient models are better than eddy-viscosity models at predicting both the magnitude and direction. However, neither of them can be realised in an actual LES, without carrying additional transport equations. Based on these observations, four new models are proposed – combining directions from Dynamic Structure and Gradient models, and magnitudes from Smagorinsky and Vreman eddy-viscosity models. These models are expected to be better than eddy-viscosity and Modulated Gradient models, and this is confirmed by preliminary a posteriori tests.

Algebraic Anisotropic Turbulence Modeling of Compound Angled Film Cooling Validated by Particle Image Velocimetry and Pressure Sensitive Paint Measurements

The complex structures in the film cooling flow field of gas turbines lead to the anisotropic property of the turbulent eddy viscosity and scalar diffusivity. An algebraic anisotropic turbulence model is developed aiming at a more accurate modeling of the Reynolds stress and turbulent scalar flux. In this study, the algebraic anisotropic model is validated by a series of in-house experiments for cylindrical film cooling with compound angle injection of 0, 45, and 90 deg. Adiabatic film cooling effectiveness and flow field are measured using pressure sensitive paint and particle image velocimetry techniques on film cooling test rig in Tsinghua University. Detailed analyses of computational simulations are performed. The algebraic anisotropic model gives a good prediction of the secondary vortices associated with the jet and the trajectory of the jet, therefore improves the prediction of the scalar field. On one hand, the anisotropic eddy viscosity improves the modeling of Reynolds stress and the predictive flow field. On the other hand, the anisotropic turbulent scalar-flux model includes the role of anisotropic eddy viscosity in modeling of scalar flux and directly improves the turbulent scalar flux prediction.

Analysis of a dynamic model for subfilter scalar dissipation rate in large eddy simulation based on the subfilter scalar variance transport equation

Accurate prediction of non-premixed turbulent combustion using large eddy simulation (LES) requires detailed modelling of the mixing between fuel and oxidizer that occurs at scales smaller than the LES filterwidth. The small-scale mixing process can be quantitatively characterized by two related variables, the subfilter scalar variance and the subfilter scalar dissipation rate. A recently proposed alternative dynamic modelling procedure for the subfilter scale dissipation rate, designed for use with transport equation based models for subfilter scalar variance, is analysed in this work. This new dynamic non-equilibrium modelling approach produces a nonlinear interaction between variance and dissipation rate predictions that makes it difficult to isolate the performance of any single modelling component in a conventional LES simulation. To gain a better understanding of the new model, a three-part study is undertaken here. The first part of the study uses a priori analysis to examine some novel aspects of the model’s computation and guide its practical implementation. In the second part of the study, detailed a posteriori analysis of the model is performed. This analysis suggests that the dynamic estimate of the dissipation rate model coefficient helps to compensate for over-prediction of variance production rates and improves the accuracy of variance prediction. However, improved modelling of the variance production term, which in turn depends on the accuracy of models for the subfilter scalar flux, is necessary to allow both the scalar variance and dissipation rate to be predicted accurately. Therefore, the third part of the study examines the effect of the scalar flux model on the predictions of the dynamic non-equilibrium model. Use of a mixed model for the fluxes, rather than a gradient-diffusion-only model, is found to improve variance predictions in some cases.

Experimentally informed optimization of turbulent diffusivity for a discrete hole film cooling geometry

A process has been developed by which mean velocity and concentration measurements can be used to determine optimal turbulent diffusivity values for an angled jet in cross-flow configuration. This configuration has applications in film cooling for gas turbine blades. The measurements, obtained by magnetic resonance imaging techniques, provide 3D time-averaged velocity and concentration fields. The mean velocity field is fed into a Reynolds-Averaged Advection Diffusion solver, which uses a turbulent diffusivity model to solve for the mean coolant concentration distribution. This distribution can be compared to the experimentally-obtained concentration field by means of an error metric that quantifies the difference between the computational and experimental concentration fields. By minimizing this error, an optimal value of the turbulent diffusivity can be determined. This optimized distribution is then compared to a RANS simulation to evaluate the relative contribution to error of the turbulent momentum flux model versus the turbulent scalar flux model.

On filtering in the viscous-convective subrange for turbulent mixing of high Schmidt number passive scalars

In the present work, we investigate the possibility of performing velocity-resolved, scalar-filtered (VR-SF) numerical simulations of turbulent mixing of high Schmidt number scalars, by using a Large Eddy Simulation (LES)-type filter in the viscous-convective subrange. The only requirement for this technique is the large scale separation between the Kolmogorov and Batchelor length scales, which is a direct outcome of the high Schmidt number of the scalar. The present a priori analysis using high fidelity direct numerical simulation data leads to two main observations. First, the missing triadic interactions between (resolved) velocity and (filtered-out) scalar modes in the viscous-convective subrange do not affect directly the large scales. Second, the magnitude of the subgrid term is shown to be extremely small, which makes it particularly susceptible to numerical errors associated with the scalar transport scheme. A posteriori tests indicate that upwinded schemes, generally used for LES in complicated geometries, are sufficiently dissipative to overwhelm any contribution from the subgrid term. This renders the subgrid term superfluous, and as a result, VR-SF simulations run without subgrid scalar flux models are able to preserve large scale transport characteristics with remarkable accuracy.

A framework for epistemic uncertainty quantification of turbulent scalar flux models for Reynolds-averaged Navier-Stokes simulations

Reynolds-averaged Navier-Stokes (RANS) simulations are a practical approach for solving complex multi-physics turbulent flows, but the underlying assumptions of the turbulence models introduce errors and uncertainties in the simulation outcome. The flow in scramjet combustors is an example of such a complex flow and the accurate characterization of safety and operability limits of these engines using RANS simulations requires an assessment of the model uncertainty. The objective of this paper is to present a framework for the epistemic uncertainty quantification of turbulence and mixing models in RANS simulations. The capabilities of the methodology will be demonstrated by performing simulations of the mixing of an underexpanded jet in a supersonic cross flow, which involves many flow features observed in scramjet engines. The fundamental sources of uncertainty in the RANS simulations are the models used for the Reynolds stresses in the momentum equations and the turbulent scalar fluxes in the scalar transport equations. The methodology consists in directly perturbing the modeled quantities in the equations, thereby establishing a method that is completely independent of the initial model form to overcome the limitations of traditional sensitivity studies. The perturbations are defined in terms of the decomposed Reynolds stress tensor, i.e., the tensor magnitude and the eigenvalues and eigenvectors of the normalized anisotropy tensor. The turbulent scalar fluxes are perturbed by using the perturbed Reynolds stresses in a generalized gradient diffusion model formulation and by changing the model constant. The perturbations were parameterized based on a comparison between the Reynolds stresses obtained from a baseline RANS simulation and those obtained from a large-eddy simulation database. Subsequently an optimization problem was solved, varying the parameters in the perturbation functions to maximize a quantity of interest that quantifies the downstream mixing. The result encompasses the value for the quantity of interest obtained from the LES database. It is shown that a traditional sensitivity study, in which the turbulent Schmidt number is varied, cannot capture this uncertainty, which further demonstrates the effectiveness of the proposed approach.

An explicit algebraic Reynolds-stress and scalar-flux model for stably stratified flows

AbstractThis work describes the derivation of an algebraic model for the Reynolds stresses and turbulent heat flux in stably stratified turbulent flows, which are mutually coupled for this type of flow. For general two-dimensional mean flows, we present a correct way of expressing the Reynolds-stress anisotropy and the (normalized) turbulent heat flux as tensorial combinations of the mean strain rate, the mean rotation rate, the mean temperature gradient and gravity. A system of linear equations is derived for the coefficients in these expansions, which can easily be solved with computer algebra software for a specific choice of the model constants. The general model is simplified in the case of parallel mean shear flows where the temperature gradient is aligned with gravity. For this case, fully explicit and coupled expressions for the Reynolds-stress tensor and heat-flux vector are given. A self-consistent derivation of this model would, however, require finding a root of a polynomial equation of sixth-order, for which no simple analytical expression exists. Therefore, the nonlinear part of the algebraic equations is modelled through an approximation that is close to the consistent formulation. By using the framework of a$K\text{{\ndash}} \omega $model (where$K$is turbulent kinetic energy and$\omega $an inverse time scale) and, where needed, near-wall corrections, the model is applied to homogeneous shear flow and turbulent channel flow, both with stable stratification. For the case of homogeneous shear flow, the model predicts a critical Richardson number of 0.25 above which the turbulent kinetic energy decays to zero. The channel-flow results agree well with DNS data. Furthermore, the model is shown to be robust and approximately self-consistent. It also fulfils the requirements of realizability.

An explicit algebraic model for the subgrid-scale passive scalar flux

AbstractIn Marstorpet al. (J. Fluid Mech., vol. 639, 2009, pp. 403–432), an explicit algebraic subgrid stress model (EASSM) for large-eddy simulation (LES) was proposed, which was shown to considerably improve LES predictions of rotating and non-rotating turbulent channel flow. In this paper, we extend that work and present a new explicit algebraic subgrid scalar flux model (EASSFM) for LES, based on the modelled transport equation of the subgrid-scale (SGS) scalar flux. The new model is derived using the same kind of methodology that leads to the explicit algebraic scalar flux model of Wikströmet al. (Phys. Fluids, vol. 12, 2000, pp. 688–702). The algebraic form is based on a weak equilibrium assumption and leads to a model that depends on the resolved strain-rate and rotation-rate tensors, the resolved scalar-gradient vector and, importantly, the SGS stress tensor. An accurate prediction of the SGS scalar flux is consequently strongly dependent on an accurate description of the SGS stresses. The new EASSFM is therefore primarily used in connection with the EASSM, since this model can accurately predict SGS stresses. The resulting SGS scalar flux is not necessarily aligned with the resolved scalar gradient, and the inherent dependence on the resolved rotation-rate tensor makes the model suitable for LES of rotating flow applications. The new EASSFM (together with the EASSM) is validated for the case of passive scalar transport in a fully developed turbulent channel flow with and without system rotation. LES results with the new model show good agreement with direct numerical simulation data for both cases. The new model predictions are also compared to those of the dynamic eddy diffusivity model (DEDM) and improvements are observed in the prediction of the resolved and SGS scalar quantities. In the non-rotating case, the model performance is studied at all relevant resolutions, showing that its predictions of the Nusselt number are much less dependent on the grid resolution and are more accurate. In channel flow with wall-normal rotation, where all the SGS stresses and fluxes are non-zero, the new model shows significant improvements over the DEDM predictions of the resolved and SGS quantities.

CFD Modeling of Pollution Dispersion in Building Array: Evaluation of turbulent scalar flux modeling in RANS model using LES results

Modeling of turbulent scalar flux in CFD (Computational Fluid Dynamics) for near-field dispersion around buildings is examined by investigating both velocity and concentration fields obtained by two modeling approaches, i.e., RANS RNG k-ε and LES. A building array model with a point source located in between the central buildings is adopted as a target configuration. First, the prediction accuracy of LES is confirmed by comparing with the RNGcomputation and the results from an experiment conducted by the authors. LES gives better results than RNG, in terms of time-averaged velocity and concentration distribution in comparison with the wind tunnel experimental results. Next, the eddy viscosity and the eddy diffusivity are determined by LES data using a least square approach as suggested in the dynamic sub-grid scale model. Large differences can be observed between the distributions of the estimated eddy viscosity by using LES data and the eddy diffusivity obtained by RNG, since the eddy diffusivity is not always proportional to the eddy viscosity.

Numerical Simulation of 3D Density Flow by an Improved EASM Model

Reynolds stress turbulence models are still an effective method to solve thermal density flow in reservoir. Presently, these turbulence models have extensive research and play an important role in the project. However, the pursuit of a more general, economic and accurate Reynolds stress model is still necessary for buoyant turbulent flow. In this paper, an improved explicit algebraic Reynolds stress model was established accounting for buoyancy. The tensor representation of the Reynolds stresses are divided into two parts, the former is composed of EASM derived by Wallin & Johansson (2000) and latter is the formulations of buoyant stress. In the derivation of explicit algebraic active scalar flux model, WWJ (Wikström, Wallin and Johansson, 2000) model only for explicit tensor representation of passive scalar flux is improved by affiliating external buoyancy effects terms. The current EASM is discretized on three-dimensional unstructured grids and compared with experimental data of 3D thermal density flow. The calculations show that the model yields better results than LAHM and LARM. In addition, EASM model also has good stability and relative economic calculation CPU time-consuming. The current model would be promising in hydraulic engineering and environmental engineering.

Generalised Langevin Model in Correspondence with a Chosen Standard Scalar-Flux Second-Moment Closure

In the context of transported joint velocity-scalar probability density function methods, the correspondence between Generalised Langevin Models (GLM) for Lagrangian particle velocity evolution and Eulerian Reynolds-stress turbulence models has been established in the 1990’s by S.B. Pope. It was shown that the GLM representation of a given Reynolds stress model is not unique. It was also shown that a given GLM together with a given mixing model for particle composition evolution implies a differential scalar-flux model. In this paper, we study how extra constraints can be applied on the choice of the GLM coefficients in order to imply a chosen scalar-flux model. This correspondence between GLM-implied and standard scalar-flux models is based on the linear relaxation term and on the mean velocity gradient contributions in the rapid term. In general, GLM-implied models possibly involve more terms (including anisotropy effects in the scalar-flux decay rate and some high-order terms in the rapid-pressure-scrambling term). The proposed form of the GLM supposes a non-constant value for the diffusion coefficient C 0, originally identified as a Kolmogorov constant. Here, the value of C 0 is determined in order to yield the Monin model for linear relaxation of the scalar-flux, and the constant in the rapid-pressure contribution is related to the choice of the parameter β ∗ in the GLM. We finally show how GLM-implied scalar-flux models are in general dependent on the choice of the mixing model and how the proposed GLM can reduce this dependency. These developments are illustrated by results obtained from calculations of the Sydney bluff-body stabilised flame HM1.

Modeling the Effects of System Rotation on the Turbulent Scalar Fluxes

A proposal for modeling the effects of system rotation on the turbulent scalar fluxes is presented. It is based on extension to rotating frames of an explicit algebraic model derived using tensor-representation theory. The model is formulated to allow for the turbulent scalar fluxes to depend on the details of the turbulence field and on the gradients of both the mean-velocity and the scalar. Such dependence, which is absent from conventional models, is required by the exact equations governing the transport of the scalar fluxes. The model’s performance is assessed, both a priori and by actual computations, by comparisons with results from recent direct numerical simulations (DNS) of flows in heated channels rotated about their streamwise, spanwise, and wall-normal axes. To place the new model’s performance in context, additional comparisons are made with predictions obtained from three alternative models, namely, the conventional gradient-transport model, a model that is implicit in the scalar fluxes derived by simplification of the modeled transport equations for the scalar fluxes, and a differential scalar-flux transport model. The results show that the present model yields predictions that are substantially in better agreement with the DNS results than the algebraic models, and which are indistinguishable from those obtained with the more complex differential model. However, important differences remain and reasons for these are discussed.

Prediction of second-order concentration statistics for dispersing plumes in obstacle arrays

In this paper, the turbulent dispersion of a passive scalar released from a continuous ground-level point-source in two obstacle arrays is simulated using the Reynolds-averaged Navier-Stokes (RANS) method. An explicit algebraic nonlinear turbulent stress model and a tensor-diffusivity scalar-flux model are used for the closure of the ensemble-averaged momentum and scalar transport equations, respectively. The principal physical mechanisms governing the transport of the second-order concentration statistics of a passive scalar are modelled following Yee et al. (2009). The results of the simulation for the flow and concentration statistics are compared against two sets of high-quality water-channel measurement data obtained using laser induced fluorescence (for concentration) and laser Doppler anemometry (for velocity).

Application of a Subfilter-Scale Flux Model over the Ocean Using OHATS Field Data

Abstract Simple rate equation models for subfilter-scale scalar and momentum fluxes have previously been developed for application in the so-called “terra incognita” of atmospheric simulations, where the model resolution is comparable to the scale of turbulence. The models performed well over land, but only the scalar flux model appeared to perform adequately over the ocean. Analysis of data from the Ocean Horizontal Array Turbulence Study (OHATS) reveals a need to account for the moving ocean–air interface in the subfilter stress model. The authors develop simple parameterizations for the effect of surface-induced pressure fluctuations on the subfilter stress, leading to good predictions of subfilter momentum flux both over land and in OHATS.

Scalar flux modeling of solute transport in open channel flows: Numerical tests and effects of secondary currents

Numerical experiments involving various algebraic scalar flux models for solute transport in open channel flows are presented. Five algebraic scalar flux models including these of Daly and Harlow, Abe and Suga, Suga and Abe, Sommer and So, and Wikstrom et al. are tested. For the flow computation, a Reynolds stress model is used. The models are applied to laboratory experiments of solute transport in rectangular and compound open channel flows. The performance of each model is evaluated both qualitatively and quantitatively. It is found that Daly and Harlow's model, although simple, predicts the solute transport most accurately. Further, with reference to the simulation results, the roles of the Reynolds fluxes and secondary currents in the solute transport equation are investigated. It is found that the Reynolds fluxes and secondary currents reduce and move the peak concentration, respectively.