^{1}

^{2}

^{1}

^{2}

The dispersion coefficient tensor including off-diagonal components was introduced in the flow with secondary currents, which is called skewed shear flow dispersion (SSFD) coefficient tensor, in this paper. To observe the detailed effect of cross-dispersion terms in SSFD model on solute dispersion, mathematical analysis of eigenvalue problem with respect to the equation with SSFD coefficient tensor was performed. The analysis results show the several differences of SSFD model compared to CSFD (conventional shear flow dispersion) model: the oblique direction of principal dispersion with respect to the streamline, the increase of peak concentration, and the change in the eccentricity of elliptical tracer cloud. SSFD coefficient tensor in a streamwise curvilinear coordinate system of curved channel was transformed to those components of fixed Cartesian coordinate system, and 2D numerical model with finite element method was established in the Eulerian-Cartesian coordinate. Through this process, the transformation equation using the depth-averaged velocity field was derived. Several numerical tests were performed to assure the results obtained in the mathematical analysis and to show the applicability of the derived transformation equation on the flow with continuously changing flow direction.

The advection and dispersion of passive solutes in open channels—which includes pollutant transport in artificial canals, natural streams, and rivers—is an important topic in environmental hydraulics. In open channels, once vertical mixing is completed in the initial period of solute transport, the vertical shear velocity profile increases the longitudinal spreading in the streamline direction [

Schematic diagram for CSFD model:

However, the secondary current around pronounced curvatures in many open channels introduces a large magnitude of transverse circulation combined with the principal longitudinal flow. Hence, the solute dispersion by the secondary current cannot be described by only the dispersion in the longitudinal direction; there is a dispersion effect in the transverse direction that is much more effective than the transverse turbulent diffusion. A flow with secondary currents, as that in Figure

Schematic diagram for SSFD model:

Fischer [

In this study, we introduced the full dispersion coefficient tensor to the solute dispersion model with respect to the stream wise curvilinear frame of reference as described in (

Two-dimensional advection-dispersion equation is simply expressed below in (

Because

Another remarkable effect of the off-diagonal terms in the SSFD model is the difference in the resultant peak concentration. When we present the analytical solution of the instantaneously dumped point mass with respect to the

The final characteristic of the SSFD model compared to the CSFD model is the change in the eccentricity of ellipses. The dispersive scale in the direction of the symmetric axes of concentration depends on the pair of principal dispersion coefficients, which are

Including (

To deal with diverse flow directions in natural streams and rivers with irregular boundaries, conventional river hydrodynamics and mass transport models are usually established in a fixed Eulerian coordinate system, where implementing a horizontal unstructured grid is convenient. In computational models established for such curved channels with continuously changing flow directions, the principal direction of anisotropic dispersion is usually not parallel to the axes of the Cartesian coordinates. Therefore, in commonly used CSFD models, components of

The full dispersion coefficient tensor introduced by Fischer [

Using the transformed component of the nodal CSFD and SSFD coefficient tensor with respect to the global coordinate system, we present expanded the Cartesian forms of (

In order to observe the oblique direction of the principal dispersion with respect to the longitudinal streamline, a solute mixing in uniform oscillatory flow was simulated by the established CSFD and SSFD models. The direction of the oscillatory flow was varied at ^{2}/s:^{2}/s. With these longitudinal and transverse coefficients, the off-diagonal components of the SSFD coefficient were determined as ^{2}/s, following the dispersion tensor for the application example in Fischer [^{2}/s by (^{2}/s.

Representation of the shear flow on the continental shelf of the middle Atlantic bight [

In Figure ^{2}/s were computed (

Concentration distribution in uniform oscillatory flow (unit: ppm).

CSFD model,

SSFD model,

CSFD model,

SSFD model,

CSFD model,

SSFD model,

The results in Figure

In order to investigate the performance of the SSFD model in a flow field with secondary currents, an example case similar to the classic teacup experiment was considered. First, we assumed a solid-body rotation of water in a coaxial cylindrical container, as in Figure

Rotating water in coaxial cylindrical container.

The outer and inner cylinders had radii of 10 and 3 m, respectively, and the container rotated at an angular speed of ^{2}/s of which the magnitude of the off-diagonal component was smaller than those of (^{2}/s. An initial concentration of 100 was dropped at the nodal point

Grid system for coaxial cylindrical container.

The simulation results at

Concentration distribution in coaxial cylindrical container.

CSFD

SSFD

For another example case of flow with secondary currents, the dispersion problem in a strongly curved channel with secondary flow was solved by the SSFD and CSFD modes. The famous experiments provided by Rozovski [

The velocity field in the whole domain was reproduced by the commonly used depth-averaged flow analysis model RMA-2 in TABS-MD [^{2}/s was arbitrarily selected for the SSFD model, and ^{2}/s was used for the corresponding CSFD model.

Comparison between computed and measured flow fields in Rozovski’s channel.

Concentration distributions at

Concentration distribution in Rozovski’s channel at

CSFD

SSFD

Concentration distribution in Rozovski’s channel at

CSFD

SSFD

The rotation of the major dispersion axis in the SSFD model and the nonuniform advection along the curved streamline had a combined effect on the increase in the peak concentration. The peak concentrations at

In this study, it was proposed that the SSFD coefficient tensor should be applied for 2D passive solute transport modeling in the flow with secondary current because of its vertically skewed shear flow structure. Mathematical analysis of eigenvalue problem pointed out several significant effects of the off-diagonal terms of dispersion tensor: the rotation of principal direction of dispersion with respect to the streamline, the increase of peak concentration, and the change in eccentricity of elliptical concentration. To apply full dispersion coefficient tensor defined in a stream-wise curvilinear coordinate system to the numerical model on the Eulerian-Cartesian coordinates, transformation relationship was derived with given depth-averaged velocity field. With the derived transformation equation, 2D numerical model was established with finite element method on the Eulerian coordinate system. Numerical tests show that the coordinate transformation relationship derived in this study successfully introduced the SSFD coefficient tensor in the numerical grid of Eulerian coordinates. It was also shown that there is a possibility of overestimation in dilution of pollutant if CSFD model was applied instead of SSFD model in the dispersion process affected by secondary currents. The conventional 2D solute mixing modules equipped in the various hydrodynamic modeling packages are expected to predict more reliable mixing patterns of pollutants by including off-diagonal terms as in SSFD model, when it is applied to flow field with secondary currents.

Depth-averaged concentration

Peak concentration

Components of the full dispersion coefficient tensor defined in stream-wise curvilinear coordinates

Mean value of

Components of the nodal dispersion coefficient tensor in the Eulerian-Cartesian coordinates

Matrix notation of SSFD coefficient tensor in curvilinear coordinates

Matrix notation of CSFD coefficient tensor in curvilinear coordinates

Symmetric version of SSFD coefficient tensor, which takes

Principal dispersion coefficient tensor, which takes eigenvalues

Flow depth

Jacobian matrix for coordinate transformation

Isotropic diffusion coefficient

Total mass of tracer

Axis normal to the streamline in the stream-wise curvilinear coordinate system

Axis along the streamline in the stream-wise curvilinear coordinate system

Time

Discretization size in time

Magnitude of longitudinal and transverse velocity deviations in Figure

Horizontal velocities on the

Longitudinal depth-averaged velocity

Vertical deviations of the point velocities with respect to depth-averaged

Ddepth-averaged velocity components in the

Axes of the Eulerian-Cartesian coordinate system

Axes with identical directions as the principal axes of

Discretization size in the

Discretization size in the

Matrix that takes the eigenvectors of

Axis of the vertical direction

Transverse turbulent diffusion coefficient

Vertical turbulent diffusion coefficient

Angular frequency of oscillatory flow

Angle of oscillatory direction with respect to the

Eigenvalues of

Angular speed of rotation of coaxial cylindrical container

Angle of counterclockwise rotation from the