- Research
- Open Access
Experimental and numerical investigations of primary flow patterns and mixing in laboratory meandering channel
- Sung Won Park^{1} and
- Jungkyu Ahn^{1}Email author
- Received: 3 September 2018
- Accepted: 17 April 2019
- Published: 10 May 2019
Abstract
Considering fundamental hydraulics, fluid dynamics, and experimental analysis should be analyzed simultaneously with mathematical methods due to the effects of hydraulic properties such as meandering form, sediment, and so on. In this research, to reveal the effects of hydraulic characteristics and longitudinal and transverse coefficients based on the two-dimensional advection-dispersion equation, a laboratory experimental channel has been conducted and analyzed. Additionally, results of experiments have been compared with horizontal two dimensional distributions of flow velocity and concentration fields with respect to the water depth and inlet discharge using two-dimensional depth-averaged numerical models based on FEM (Finite Element Method). SMS and RAMS have been applied with the same experimental conditions and compared. From the analysis of velocity profiles, primary and secondary flows have been visualized. Also the result of pollutant clouds illustrated from the results of tracer tests with an instantaneous and centered injection of solute transport, separation, superposition and stagnation could be deduced outwards of two meandering sections. And same types of characteristics of velocity and pollutant transport could be defined with two-dimensional numerical models.
Keywords
- Meandering
- Two-dimensional advection-dispersion equation
- Longitudinal and transverse coefficients
- Two-dimensional depth averaged numerical models
- Velocity profiles
- Pollutant transport
Introduction
Experimental Researches about Mixing in Channels
Researcher | Channel | Results |
---|---|---|
Elder (1959) | Straight | e_{y}/HU_{∗} = 0.23 |
Fischer (1969) | Meander | Theoretical form Derived |
Chang (1971) | Meander | Developed |
Miller & Richardson (1974) | Straight | D_{L}/e_{y} > 100 |
Krishnappan & Lau (1977) | Meander | 0.222 < D_{T}/WU_{∗}, 0.416 < D_{T}/WU_{∗} |
Lau & Krishnappan (1977) | Straight | Instead of D_{T}/HU_{∗}, using D_{T}/WU_{∗} |
Webel & Schatzmann (1984) | Straight | Criticized results of Lau & Krishnappan (1977) |
Almquist & Holley (1985) | Meander | Transverse dispersion coefficient increasing in natural cross section |
Nokes & Wood (1988) | Straight | Transverse dispersion coefficient dependent of a friction factor |
Boxall & Guymer (2003) | Meander | Conducted with setups natural cross-section |
Boxall et al. (2003) | Meander | Transverse dispersion coefficient varied in direction of channel curvature |
Numerical approaches have been developed for describing circulation phenomena and the solute transport in the meandering open channel flows. McGuirk and Rodi (1978) developed a depth-averaged model for the near field calculations of the flow and concentration distribution by side discharges of the pollutant into open-channel flow. Duan (2004) has derived the dispersion term for the depth-averaged 2D model in Cartesian coordinates and used the Schmidt number as a calibration parameter to simulate mass dispersion in meandering channels. The model by Duan (2004) was applied to the experiments of Chang (1971) to test the capability of the model to simulate mass transport in meandering channels.
Experimental analysis
Governing equation
The three-dimensional Fickian diffusion equation under a non-buoyant tracer being transported in an unsaturated incompressible laminar flow in the Cartesian coordinates can be time-averaged and rewritten as three-dimensional advection-diffusion equation.
where s is concentration; t is time; x, y, and z is the longitudinal, transverse, and vertical distances measured from the injection point respectively. v_{x}, v_{y}, and v_{z} are directional velocities, and e_{m} is the molecular diffusion coefficient. Time averaging three-dimensional advection-diffusion equation can be derived by the Reynolds averaging. And in a mathematical model, the dispersion term is occurred as transforming the three-dimensional advection-diffusion equation into two-dimensional advection-dispersion equation by integration with respect to depth. In rivers, diffusion coefficients are negligible compared to dispersion coefficients with using the continuity equation (Baek 2004).
where D_{L} and D_{T} is the longitudinal and transverse dispersion coefficient which accounts for the effects on the depth-averaged tracer concentration of depth variations in the longitudinal and transverse velocity.
Instantaneous injection condition of tracer can be presented with a procedure by Beltaos (1975). Following this suggestion, integrating Eq. 2 with respect to time from t = 0 to t = ∞, gives
Observing C|_{∞} = C|_{0} = 0 for instantaneous injection and introducing the concept of a dosage,
Eq. 3 becomes
The integral equations of conservation of tracer mass are as follows.
where A is a cross-sectional area and Eq. 6 can be also re-written as follows with constant injection, W_{s}
With these procedures, the amount of tracer needed to perform an instantaneous test is much less than what is required for a continuous test. And injection of a slug is rather simple whereas constant injection requires special equipment and it may be difficult in rivers with poor accessibility. Additionally an instantaneous test provides information on not only the transverse mixing but also the longitudinal and temporal characteristics. This kind of information is not able to be obtained from a continuous injection condition due to decrease of time variation.
Experimental setup
Properties of Meander Pattern of Previous Studies and This Research
Velocity data and solute concentration was measured by micro-ADV (Acoustic Doppler Velocimetry) and the electrode conductivity meter (KENEK: Model NST-30) respectively. The density of the salt solution was adjusted to that of the flume water by adding methanol. The tracer material can be instantaneously injected into water flow as a full-depth vertical line source by using the injecting acrylic cylinder. The initial concentration of the tracer was 100,000 mg/ℓ. The interval of concentration measuring points at each section was about 7 cm in the transverse direction. Four cases of tracer tests have been proposed and performed in this test. The tracer was instantaneously injected into the flow as a full-depth vertical line source either at the centerline of the channel. Vertically, at a point located 60% of the water depth above the bottom, concentration was measured under the shallow water cases (H = 15, 20 cm). And at two points located 20, 80% of the water depth above the channel bottom, concentration was measured under the deep water cases (H = 30, 40 cm).
Numerical analysis
Flow models
where ρ is density of fluid, E_{xx} is eddy viscosity coefficient for normal direction on x axis surface, E_{yy} is for normal direction on y axis surface, E_{xy} and E_{yx} is for shear direction on each surface, g is acceleration due to gravity, b is elevation of bottom, n is roughness coefficient in Manning’s formula, ζ is empirical wind shear coefficient, V_{a} is wind speed, ω is rate of earth’s angular rotation, and φ_{l} is local latitude.
RAMS (River Analysis and Modeling System) is consist of river flow analysis model (RAM2), pollutant transport model (RAM4), bed elevation change model (RAM6), and graphic user interface (RAMS-GUI) is combined with above numerical programs to develop commercial package. This software can simulate physical phenomena in natural rivers with complex topography by 2D finite element numerical calculations with underlying consistency and generality, which would provide accurate and stable solutions to open channel flow equation, and mass transport equation for various types of problems.
Flow model, RAM2 is a finite element model based on Streamline Upwind / Petrov-Galerkin (SU/PG) scheme for analyzing and simulating two-dimensional flow characteristics of irregular natural rivers with complex geometries. The type of elements in a mesh can be a triangular, quadrilateral or mixed one. A triangular element could have either 3-nodes or 6-nodes and a quadrilateral element either 4–nodes or 8-nodes. This mesh can constructed from DEM and TIN format in the tools of GIS.
Pollutant transport models
where x and t are space and time variable respectively, c is unknown variable which denotes solute concentration in this model, q the fluid velocity vector, and D the diffusion/dispersion tensor, Ω a bounded problem domain in 2D or 3D space, and T is objective time defined in the problem. Three types of boundary conditions can be defined for the problem domains. The first type can be defined on a boundary segment.
The governing equation modeled in RAM4 is a 2D advection-dispersion equation which is given as
where is the first order decay coefficient and is some sink or source function. From researches about equations of tensor-forming dispersion coefficients (Fischer 1979; Alavian 1986) by applying Taylor’s method to a 2D advection-dispersion model, the following dispersion coefficient tensor forms can represent the longitudinal and transverse dispersion coefficients of the curved channel in fixed Cartesian coordinate system.
The SUPG (Streamline-Upwind/ Petrov - Galerkin) method with finite difference time discretization was used for the finite element formulation of RAM4. When the advective transport is dominant compared with dispersive transport, numerical solutions by conventional numerical methods exhibit excessive nonphysical oscillation or artificial smearing. SUPG method can overcome the generally well-known numerical errors in the modeling of advection dominated conditions.
Results and analysis
Primary flow
Observed data
Simulated data
Properties of geometries for simulation (SMS and RAMS)
RMA-2 | RAM2 | |
---|---|---|
Number of nodes | 7729 | 2625 |
Number of element | 2480 | 2480 |
Element type | Rectangular (quadratic basis element) | Rectangular (linear basis element) |
Tracer transport
Observed data
Simulated data
Conclusions
In this research, velocity profiles of experiment method and numerical simulation have been com-pared and analyzed with same hydraulic conditions. The depth-averaged primary velocity vector field of the representative case was plotted in Fig. 10. As shown in this figure, the velocity distribution of the primary flow for all cases skewed toward the inner bank at the first bend, and was almost symmetric at the crossovers, and then shifted toward the inner bank again at the next alternating bend. Thus, one can clearly notice that the maximum velocity occurs taking the shortest course along the channel, irrespective of the flow conditions. This behavior of the primary flow in meandering channels with a rectangular cross-section was reported by several researchers (Chang 1971; Almquist and Holley 1985; Shiono and Muto 1998). The result of the tracer tests shows that pollutant clouds are spreading following the maximum velocity lines in each case.
Secondly, several various characteristics were expressed by velocity structures such as the secondary flow, shear flow, and path of primary flow. Separation and stagnation have been occurred from asymmetric velocity fields. As velocity increases, superposition and secondary separation of concentration cloud repeatedly occur with progressive movements.
Furthermore numerical simulation data can make expectation of the similar developments such as separation, stagnation, and superposition of pollutant clouds. In the future, characteristics of solute transport in the channel can be compared with 2-D numerical analysis using quantitative analysis methods of plotting. For the longitudinal progress, transverse spatial distribution, and dimensional hydraulic index can be estimated with the further researches.
Declarations
Acknowledgments
This work was supported by National Research Foundation of Korea (NRF) grant funded by Korean government Ministry of Science, ICT & Future Planning 2016R1C1B1014280.
Funding
This work was supported by National Research Foundation of Korea (NRF) grant funded by Korean government Ministry of Science, ICT & Future Planning 2016R1C1B1014280.
Availability of data and materials
Not applicable, because this paper is not Biological nor Medical.
Authors’ contributions
The first author performed modeling works and the second author finalized the results. The figures were prepared by the first author and the manuscript was written by both authors. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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