# Advanced 3D Mapping of Hydrodynamic Parameters for the Analysis of Complex Flow Motions in a Submerged Bedrock Canyon of the Tocantins River, Brazil

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}/s used for rivers. It is concluded that turbulent values can be calculated to better represent physical processes with the intention to improve hydrodynamic numerical models calibrations.

## 1. Introduction

_{0}) and eddy viscosity (υ

_{t}) were assessed in this river reach, based on extensive ADCP measurements collected from a moving boat to better understand the river dynamics in this submerged bedrock canyon. Using this real field-scale case study, three different approaches were used to calculate bed shear stress: the total kinetic energy (TKE) method, law of the wall method and the depth–slope product method. With the results of the TKE method, the eddy viscosity was estimated with the Boussinesq approach.

## 2. Materials and Methods

#### 2.1. Bedrock Canyon of Tocantins River

^{2}, smaller only than the Amazon River Basin.

^{3}/s, with high seasonal variations ranging from 1898 to 45,717 m

^{3}/s. During the flood season, the river width is around 1.5 km, with the presence of a few rocky outcrops, high velocities and strong turbulent flow (large shear layers and longitudinal vortices). During the drought season, the flow is confined along the canyon area (150 m width), exposing the river bedrock at the reach. Although having smaller diameters, the presence of vortices is constant, even under low flow conditions.

#### 2.2. Equipment and Field Data

_{x}, σ

_{y}, σ

_{z}). Some ADCP brands supply all the velocity samples from the time series, allowing the user to calculate the statistics; however, the M9 only provides the results of mean velocity and standard deviation, which will be used here.

_{cel}) varied between 0.1–0.5 m for the 3 MHz and 0.5–2.0 m for the 1 MHz. The horizontal sampling dimension is determined as a function of the boat’s movement in time, and the data acquisition frequency (f = 1 s). Also, there is a vertical beam of 0.5 MHz in the middle of the instrument head to improve bed level measurements. The Sontek M9 maximum depth for velocity measurements is 40 m, and some parts of the canyon were more than 50 m deep, thus resulting in larger blanking areas near the bottom. Nonetheless, the bed level was measured throughout the canyon using the vertical beam.

^{3}/s. Several transversal sections and a longitudinal zig-zag type of measurement were undertaken along Lourenço’s Rock Canyon during a short period (short enough to have discharge changes of less than 3%). The measurement resulted in 10,531 velocity profiles (ADCP ensembles) at different points of the canyon, each having on average about 14 cells, thus resulting in approximately 140,000 3D velocity vectors. Figure 2 shows the location of the spatial measurements. Three Elevation Reference (ER) stations were installed and geo-referenced along the study area using the Global Navigation Satellite System (GNSS). The ADCP’s compass was calibrated three times during the measurement. The water surface elevation measured at each point is presented in Figure 2, and the resulting mean water surface slope (S

_{WS}) was 2.36 × 10

^{−4}m/km. A detailed bathymetry measurement of the area was available from previous measurements in 2012, with both single and multi-beam echosounder equipment. A DTM (Digital Terrain Model) was created (Figure 1), combining the bathymetry data with SRTM (Shuttle Radar Topographic Mission) data to complete the margin information.

#### 2.3. Initial ADCP Data Processing

^{2}= σ

_{t}

^{2}+ σ

_{n}

^{2}

_{t}is the velocity fluctuation from the local turbulence (m/s); σ

_{n}is the acoustic noise effect from the equipment on the standard deviation (m/s). The acoustic noise is a result of the physical process by which sound waves are dispersed by the solid particles in the water. To estimate the turbulent part of the standard deviation, previous approaches calculated the acoustic noise using different assumptions. Vachtman and Laronne [27] considered the acoustic noise equal to the minimum variance of calculated water velocities for all depth cells (σ

_{n}

^{2}= σ

^{2}

_{min}). Another approach was presented by Rennie and Church [6], who applied Equation (1), but used the average ADCP error velocity (σ

_{error}), instead of the acoustic noise. The average ADCP error velocity considered other uncertainties and can be calculated by the velocity difference (V

_{d}) from 2 vertical velocity measurements and the horizontal velocity error (V

_{e}). According to Sontek [24], although being a random effect, it can be assumed that acoustic noise follows a Gaussian distribution. The magnitude of the noise can vary according to the equipment frequency (F), the vertical dimension of the cell (ΔZ

_{cel}) and the number of pings (N) emitted in the period measured per cell. For a standard 3D ADCP measurement, the acoustic noise can be estimated with Equation (2).

_{n}= 235 (F ΔZ

_{cel}N

^{0.5})

^{−1}

_{tx}, σ

_{ty}, σ

_{tz}) and computed flow parameters (turbulent kinetic energy k and shear stress τ), as shown in Figures 5–7, and explained in the following section. The velocity distributions show the part of the canyon with highest velocities and highest longitudinal velocities occurring in the deeper part of the channel. These effects follow the observations by Venditti et al. [16], where similar flow features were observed for a bedrock canyon of the Fraser River.

## 3. Theory and Analytical Methods

#### 3.1. TKE Method

_{0}= C

_{0}TKE = C

_{0}ρ k

^{2}> + <v’

^{2}> + <w’

^{2}>)/2

_{0}is a dimensionless constant (i.e., C

_{0}= 0.18–0.21) [13]; ρ is the water density (i.e., ρ = 1 g/cm

^{3}at 15 °C). Using the ADCP in Narrowband mode, it is possible to combine the Standard Deviation (STD, Equation (1)) from the velocity data (for each component), and the Reynolds Decomposition to determine the temporal root mean square of the velocity fluctuations (<u’

^{2}>).

_{tx}= (N

^{−1}∑(u − <u>)

^{2})

^{0.5}= (<(u − <u>)

^{2}>)

^{0.5}= (<u’

^{2}>)

^{0.5}

_{0-TKEcel}) can be calculated using the velocity STD components of the ADCP data.

_{0-TKEcel}= C

_{0}ρ (σ

_{tx}

^{2}+ σ

_{ty}

^{2}+ σ

_{tz}

^{2})/2

_{0-TKE}), calculated by the TKE method, can be determined by the integration of the shear stress calculated in each cell (τ

_{0-TKEcel}) of the vertical measurement (ensemble). For the vertical average the variance sum law was applied, where the covariance of the velocity time series between each cell should be calculated; however without the samples, this study considered that the sum of the covariance among the ensemble is equal to zero. Togneri et al. [35] used the same consideration to compare the ADCP observations with a 3D model simulation. Besides, Lu and Lueck [21] calculated the distribution (histogram) of the covariance in velocity profiles from an ADCP fixed to the bottom of a tidal channel at a depth of 30 m, during 20-min intervals in 2 different flow events. The results showed a balance between positive and negative covariance, with a mean near to zero.

#### 3.2. Law of the Wall Method

_{0})

_{0}is the roughness length. For natural rivers, considered as a hydraulically rough flow, Equation (7) can be described using the characteristic roughness length-scale (k

_{s}).

_{s})

_{s}) = m ln(30/k

_{s}).

_{0-log}), by the log-law, is given by:

_{0-log}= ρ (u*)

^{2}

#### 3.3. Depth–Slope Product Method

_{0}) is parallel to the energy slope (S

_{e}) and the water level slope (S

_{ws}), and the hydraulic radius (R

_{h}) nearly equals the transect-averaged water depth (<h>). The application of these assumptions within the mean boundary shear stress equation is often referred to as the “depth-slope product”, as presented in:

_{0-ds}= ρ g R

_{h}S

_{e}= ρ g <h> S

_{0}

_{0-ds}is bed shear stress, by the depth-slope product method; g is the gravitational acceleration (i.e., g = 9.81 m

^{2}/s); <h> is the transect-averaged water depth (m); S

_{e}is the local energy slope. The transect-averaged water depth is a standard result of the ADCP measurement, and the bed slope (S

_{0}) was considered equal to the water surface slope (S

_{ws}) measured in the field. Because of the depth difference, the ADPC cross-section data were subdivided, selecting a mean hydraulic radius of the canyon area and another value for the rest of the cross-section. Unfortunately, the simplifications presented in the depth-slope product method (steady and uniform flow) are an unrealistic representation of the spatial data condition of the measurements in Lourenço’s Rock Canyon. Nonetheless, Sime et al. [11] considered it worthwhile to compare the method with other results because of its high usability. However, it has not been followed further in this manuscript.

#### 3.4. Boussinesq’s Hypothesis

_{t}) and is not a physical property but is dependent on the current flow condition and turbulence levels. The correlation between the turbulent shear stress of the flow and the eddy viscosity (for the x and z coordinate axes) is described by:

_{xz}/ρ = − <u’w’> = υ

_{t}(∂<u>/∂z + ∂<w>/∂x) − 2/3 δ

_{xz}k

_{xz}is the turbulent shear stress (x and z-axes), in kg/ms

^{2}; <u’w’> is the Reynolds Stress component (x and z-axes); υ

_{t}is the eddy viscosity, in m

^{2}/s; k is the turbulent kinetic energy, in m

^{2}/s

^{2}; δ is the dimensionless Kronecker delta.

_{1}TKE

_{1}is a dimensionless constant (i.e., C

_{1}≈ 0.19–0.21). The specific value of C

_{1}may not apply at all levels within the water column because it requires confirmation before using the method universally [13]. Nonetheless, this paper considered the turbulent shear stress for the water column (τ’

_{ensemble}) equal to bed shear stress, by the TKE method (C

_{1}= C

_{0}= 0.20).

_{t ensemble}= τ’

_{ensemble}/(ρ ω

_{ensemble})

_{t ensemble}is the turbulence eddy viscosity, in m

^{2}s

^{−1}, τ’

_{ensemble}is shear stress, by the TKE method (Nm

^{−2}); ω

_{ensemble}is the vertical average of the 3D vorticity along the cross-section (s

^{−1}).

## 4. Results

#### 4.1. Single Cross-Section

#### 4.2. Spatial Distribution

## 5. Discussion and Conclusions

^{2}/s used for rivers. In most of the areas of the stretch, the values were between 0.8 and 1.8 m

^{2}/s. In the canyon, analyzing each section, some points reached values higher than 20 m

^{2}/s. For spatial analysis, the higher value considered was 10 times higher than the other parts of the stretch.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

_{b}+ |PB| cos (α), ψ (|PB| sin (α)]

_{b}is the longitudinal distance of B from the arbitrary beginning; B is the planimetric coordinate [E

_{B}, N

_{B}] of the upstream point; P is the planimetric coordinate [E

_{P}, N

_{P}] of the surveyed point; α is the angle between the vectors |CP| and |CB|; C is the planimetric coordinate [E

_{C}, N

_{C}] of the downstream point; ψ is a parameter that equals −1 for points on the right side and 1 for points to the left.

**Figure A1.**(

**a**) Bathymetry data (below 63 m) in a regular coordinate system grid (E-N); (

**b**) Bathymetry data transformed in the s-d grid (by s-line direction); (

**c**) Kriging interpolation using the s-d grid; (

**d**) Interpolated bathymetry grid along the canyon central path in a regular grid.

**Figure A2.**(

**a**) All the Bathymetry data and the results from the first step at the coordinate system grid (E-N) were used as an input for the second step; (

**b**) Bathymetry results from kriging interpolation after the second step.

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**Figure 1.**(

**A**) Geographic location of the Tocantins River and the canyon location at Lourenço’s Rock; (

**B**) Bathymetrical map (above sea level); (

**C**) Image looking downstream during low flow condition; (

**D**) Image looking upstream during low flow conditions and indicating the navigation channel; (

**E**) 3D bathymetrical visualization (vertical exaggeration = 15) overlaid on a satellite image and (

**C**,

**D**) viewing directions.

**Figure 2.**Measurements in Lourenço’s Rock Canyon; (

**A**) ADCP trajectories with colored depth-averaged velocity vectors (April 2015); (

**B**) River banks during high flow are shown in blue, and the location of the three gaging stations with measured values of the water surface elevation (April 2015) is presented.

**Figure 3.**Large vortices in the canyon shear-layer where steep bathymetrical gradients exist; (

**A**) Flood event—April 2015; (

**B**) Drought event—October 2015.

**Figure 4.**Measured vertical velocity profile of longitudinal velocity (+), and fitted logarithmic curves using a different number of measured points.

**Figure 6.**Flow velocity profiles over the canyon part (April 2015); (

**A**–

**C**) Magnitude of mean velocity components <u> (longitudinal), <v> (transversal), <w> (vertical) over cross-section; (

**D**–

**F**) Magnitude of turbulent velocity fluctuations.

**Figure 7.**Turbulent kinetic energy (k) before (

**A**) and after (

**B**) vertical interpolation; (

**C**) Shear-stress distribution using the TKE method.

**Figure 8.**Bed shear stress distribution along the cross-section, using the TKE method, the log-law method, and the depth–slope product method.

**Figure 9.**(

**A**) Distribution plot of the <vw> direction (angle from up) resulting velocity in the vertical plane (<vw> = (<v>

^{2}+ <w>

^{2})

^{0.5}) and schematized identification of vortex structure; (

**B**) Vorticity (ω) of the canyon section part.

**Figure 11.**(

**A**) Velocity magnitude and vector direction within the river section; (

**B**) Shear stress distribution using the log method; (

**C**) Shear stress distribution using the TKE method; (

**D**) Eddy viscosity distribution based on the shear stress computed by TKE method.

Measurement | Stress Component | Plane | Direction | Velocity Gradient |
---|---|---|---|---|

−ρ <u’w’> | τ’_{xz} | yz | z | ∂<w>/∂x |

τ’_{zx} | yx | x | ∂<u>/∂z | |

−ρ <u’v’> | τ’_{xy} | yz | y | ∂<v>/∂x |

τ’_{yx} | zx | x | ∂<u>/∂y | |

−ρ <v’w’> | τ’_{yz} | zx | z | ∂<w>/∂y |

τ’_{zy} | yx | y | ∂<v>/∂z |

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**MDPI and ACS Style**

Tomas, G.; Bleninger, T.; Rennie, C.D.; Guarneri, H.
Advanced 3D Mapping of Hydrodynamic Parameters for the Analysis of Complex Flow Motions in a Submerged Bedrock Canyon of the Tocantins River, Brazil. *Water* **2018**, *10*, 367.
https://doi.org/10.3390/w10040367

**AMA Style**

Tomas G, Bleninger T, Rennie CD, Guarneri H.
Advanced 3D Mapping of Hydrodynamic Parameters for the Analysis of Complex Flow Motions in a Submerged Bedrock Canyon of the Tocantins River, Brazil. *Water*. 2018; 10(4):367.
https://doi.org/10.3390/w10040367

**Chicago/Turabian Style**

Tomas, Gustavo, Tobias Bleninger, Colin D. Rennie, and Henrique Guarneri.
2018. "Advanced 3D Mapping of Hydrodynamic Parameters for the Analysis of Complex Flow Motions in a Submerged Bedrock Canyon of the Tocantins River, Brazil" *Water* 10, no. 4: 367.
https://doi.org/10.3390/w10040367