# Deriving Six Components of Reynolds Stress Tensor from Single-ADCP Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method Description

#### General Framework

_{0}between the beam and vertical is 25°. As for the angle 2α between any pair of beams, its value may be determined by the following expression, derived from pure geometrical analysis:

_{0}. Later on the correspondent matrix is denoted by

**N**.

_{i}}, measured directly at points A, B, C, is connected to the orthogonal projections u

_{1}, u

_{2}, u

_{3}of the velocity $\overrightarrow{u}$ at the same points by the linear relations, e.g., ${b}_{1}\left(A\right)=\left({\overrightarrow{n}}_{1}\stackrel{\rightharpoonup}{u}\left(A\right)\right)$, or, equivalently (note the summation over the repeated indexes):

**M**is derived directly from Equations (2) and (6) and similar ones:

- 1.
- After proper choice of time averaging interval, the mean beam velocities $\u2329{b}_{i}\u232a$, pulsation intensities $\u2329{b}_{i}^{\prime 2}\u232a$ (Equation (2)) and correlations $\u2329{b}_{i}^{\prime}{b}_{j}^{\prime}\u232a$ (Equation (5), i ≠ j) are calculated directly from experimental data.
- 2.
- For each beam the function ${D}_{LL}$ is calculated. After revealing the inertial interval, its extent is estimated, with the special attention to its upper scale limit l.
- 3.
- The range of depths is chosen in such a way that the distance between beams does not exceed the scale l. The maximum depth h is derived from inequality AB < l (see Figure 1): $h<l/\left(\sqrt{3}\mathit{tan}{\mathsf{\alpha}}_{0}\right)$.
- 4.
- For chosen depths, the turbulent stresses are calculated directly by solving the system (7):$${\mathrm{R}}_{i}={M}_{ij}^{-1}{}_{ij}{\mathrm{B}}_{j},i,j=1\dots 6$$

**M**

^{−1}looks like (here tan(α

_{0}) is shortly denoted as t):

## 3. Experimental Setup and Results

_{i}were used. The root-mean-square error of b

_{i}values varied in the range (0.1–0.5) mm/s.

^{2}/s

^{2}, whereas interbeam correlations, remaining statistically significant, varied roughly from one third to one half of their limits.

^{2}/s

^{2}correspondently. The value of the anisotropy coefficient $\u2329{u}_{3}^{\prime 2}\u232a/\u2329{u}^{\prime 2}\u232a$ was subjected to irregular oscillations within the range (0.05–0.30).

^{2}achieved the value 0.98 (Table 1).

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Relationships between Structure Functions

_{NN}=4 D

_{LL}/3, so one obtains the following presentation of ${\tilde{D}}_{12}$ through the longitudinal SF: ${\tilde{D}}_{12}=\left(\frac{4}{3}{\mathit{cos}}^{2}\mathsf{\alpha}-{\mathit{sin}}^{2}\mathsf{\alpha}\right){D}_{LL}$.

## References

- Lhermitte, R. Turbulent air motion as observed by Doppler radar. In Proceedings of the 13th Conference on Radar Meteorology, Montreal, QC, Canada, 20–23 August 1968; American Meteorological Society: Boston, MA, USA, 1968; pp. 498–503. [Google Scholar]
- Lhermitte, R. Doppler sonar observation of tidal flow. J. Geophys. Res.
**1983**, 88, 725–742. [Google Scholar] [CrossRef] - Lohrmann, A.; Hackett, B.; Roed, L. High-resolution measurements of turbulence, velocity, and stress using a pulse-to-pulse coherent sonar. J. Atmos. Ocean. Technol.
**1990**, 7, 19–37. [Google Scholar] [CrossRef][Green Version] - Guerra, M.; Thomson, J. Turbulence Measurements from Five-Beam Acoustic Doppler Current Profilers. J. Atmos. Ocean. Technol.
**2017**, 34, 1267–1284. [Google Scholar] [CrossRef] - Bouffard, D.; Zdorovennova, G.; Bogdanov, S.; Efremova, T.; Lavanchy, S.; Palshin, N.; Terzhevik, A.; Vinnå, L.R.; Volkov, S.; Wüest, A.; et al. Under-ice convection dynamics in a boreal lake. Inland Waters
**2019**, 9, 142–161. [Google Scholar] [CrossRef] - Bogdanov, S.; Zdorovennova, G.; Volkov, S.; Zdorovennov, R.; Palshin, N.; Efremova, T.; Terzhevik, A.; Bouffard, D. Structure and dynamics of convective mixing in Lake Onego under ice-covered conditions. Inland Waters
**2019**, 9, 177–192. [Google Scholar] [CrossRef][Green Version] - Wiles, P.J.; Rippeth, T.P.; Simpson, J.H.; Hendricks, P.J. A novel technique for measuring the rate of turbulent dissipation in the marine environment. Geophys. Res. Lett.
**2006**, 33, L21608. [Google Scholar] [CrossRef] - Lucas, N.S.; Simpson, J.H.; Rippeth, T.P.; Old, C.P. Measuring Turbulent Dissipation Using a Tethered ADCP. J. Atmos. Ocean. Technol.
**2014**, 31, 1826–1837. [Google Scholar] [CrossRef][Green Version] - Volkov, S.; Bogdanov, S.; Zdorovennov, R.; Zdorovennova, G.; Terzhevik, A.; Palshin, N.; Bouffard, D.; Kirillin, G. Fine scale structure of convective mixed layer in ice-covered lake. Environ. Fluid Mech.
**2019**, 19, 751–764. [Google Scholar] [CrossRef][Green Version] - Lu, Y.; Lueck, R.G. Using a broadband ADCP in a tidal channel. Part II: Turbulence. J. Atmos. Ocean. Technol.
**1999**, 16, 1568–1579. [Google Scholar] [CrossRef] - Stacey, M.T.; Monismith, S.G.; Burau, J.R. Measurements of Reynolds stress profiles in unstratified tidal flow. J. Geophys. Res.
**1999**, 104, 10933–10949. [Google Scholar] [CrossRef] - Rippeth, T.P.; Simpson, J.H.; Williams, E.; Inall, M.E. Measurement of the rates of production and dissipation of turbulent kinetic energy in an energetic tidal flow: Red Wharf Bay revisited. J. Phys. Oceanogr.
**2003**, 33, 1889–1901. [Google Scholar] [CrossRef][Green Version] - Vermeulen, B.; Hoitink, A.J.F.; Sassi, M.G. Coupled ADCPs can yield complete Reynolds stress tensor profiles in geophysical surface flows. Geophys. Res. Lett.
**2011**, 38, L06406. [Google Scholar] [CrossRef] - Bogdanov, S.R.; Zdorovennov, R.E.; Palshin, N.I.; Zdorovennova, G.E.; Terzhevik, A.Y.; Gavrilenko, G.G.; Volkov, S.Y.; Efremova, T.V.; Kuldin, N.A.; Kirillin, G.B. Deriving of turbulent stresses in a convectively mixed layer in a shallow lake under ice by coupling two ADCPs. Fundame. I Prikl. Gidrofiz.
**2021**, 14, 17–28. [Google Scholar] [CrossRef] - Hinze, J.O. Turbulence; McGrawHill: New York, NY, USA, 1975. [Google Scholar]
- Monin, A.S.; Yaglom, A.M. Statistical Fluid Mechanics. Volume II: Mechanics of Turbulence, 1st ed.; Courier Corporation: Cambridge, UK; The MIT Press: Cambridge, MA, USA, 1971. [Google Scholar]

**Figure 1.**The coordinate system and main notations for three-beam ADCP configuration. Point O corresponds to the device head. Axis X lies in the plane AOO’.

**Figure 2.**(

**a**) Bathymetry of the Lake Vendyurskoe with indication of the measuring complex (yellow triangle). (

**b**) Two ADCPs anchored on the ice of Lake Vendyurskoe in Spring 2020. (

**c**) Schematic vertical distribution of temperature during springtime underice convection and scheme of the measuring complex. Indexes 1, 2, 3—the beams of the first ADCP, and 4, 5, 6—the second. Red dashed lines serve as the markers of beams’ intersection points.

**Figure 3.**The calculated dynamics of beam velocity intensities and interbeam velocity correlations for both devices. Time readings from 4 April, 00:00.

**Figure 4.**The set of longitudinal SF (averaged over six beams) calculated with a two-hour step for the time interval (09:00–18:00) 4 April 2020. Time averaging over 100 min. Solid lines represent the series of Kolmogorov curves D

_{LL}= C ε

^{2/3}r

^{2/3}with ε increasing from 0.2·10

^{−9}to 0.5·10

^{−9}. The labels correspond to the measurement time (a.m.).

**Figure 5.**The dynamics of turbulent stresses calculated independently for each device. Top panel—pulsation intensities along axes X, Y, Z. Bottom panel—off-diagonal stresses. The time intervals, when realizability conditions were violated, are marked by the vertical red lines on the middle image of top panel.

**Figure 6.**Correlations between the pulsation intensities, derived by two independent methods. Projection of each point on the X and Y axes represent the values, obtained by the coupled-ADCPs and single-ADCP methods correspondently. The linear regression curves are presented by black dashed lines.

**Figure 7.**Correlations between the TKE values, derived by two independent methods. Projection of each point on the X and Y axes represent the values, obtained by the coupled-ADCPs and single-ADCP methods respectively.

Stress Component | Correlation Coefficient, r | Coefficient of Determination, R ^{2} | Linear Regression Coefficient |
---|---|---|---|

$\u2329{u}_{1}^{\prime 2}\u232a$ | 0.96 | 0.92 | 1.31 |

$\u2329{u}_{2}^{\prime 2}\u232a$ | 0.98 | 0.96 | 1.03 |

$\u2329{u}_{3}^{\prime 2}\u232a$ | 0.92 | 0.85 | 0.61 |

$\u2329{u}_{i}^{\prime 2}\u232a$ | 0.99 | 0.98 | 1.14 |

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**MDPI and ACS Style**

Bogdanov, S.; Zdorovennov, R.; Palshin, N.; Zdorovennova, G. Deriving Six Components of Reynolds Stress Tensor from Single-ADCP Data. *Water* **2021**, *13*, 2389.
https://doi.org/10.3390/w13172389

**AMA Style**

Bogdanov S, Zdorovennov R, Palshin N, Zdorovennova G. Deriving Six Components of Reynolds Stress Tensor from Single-ADCP Data. *Water*. 2021; 13(17):2389.
https://doi.org/10.3390/w13172389

**Chicago/Turabian Style**

Bogdanov, Sergey, Roman Zdorovennov, Nikolay Palshin, and Galina Zdorovennova. 2021. "Deriving Six Components of Reynolds Stress Tensor from Single-ADCP Data" *Water* 13, no. 17: 2389.
https://doi.org/10.3390/w13172389