# The Reason for the Rise in Critical Shear Stress on Sloping Beds

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## Abstract

**:**

## 1. Introduction

## 2. Turbulence Damping with Low Water Coverage

**Figure 4.**Measured data from torrents (Rickenmann [23]) and solutions of Equation (8).

## 3. Beginning of Sediment Movement

#### 3.1. Measurement Data on the Beginning of Movement at a Large Gradient and Low Water Cover

#### 3.2. Shields Curve

#### 3.3. Analytical Solution of the Shields Curve

#### 3.3.1. Turbulence-Free Flow

_{d}is effective.

#### 3.3.2. Turbulent Flow

#### 3.4. Angle of Internal Friction Determining the Onset of Motion

#### 3.5. Turbulence Damping and Critical Shear Stress

#### 3.5.1. Finite Critical Shear Stress with Complete Damping of Turbulence

#### 3.5.2. Turbulence Damping for Exposed Particles

#### 3.5.3. Degree of Movement at Start of Movement

#### 3.6. Effect of Dynamic Friction Angle

## 4. Critical Shear Stress at Large Slope and Low Water Cover

## 5. Influence of Sediment Density

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

B | Integration constant of log. velocity profile | - |

d | Grain diameter | m |

${D}_{\tau}$ | $\tau $-Damping coefficient with low water coverage | - |

${D}_{\tau ,c}$ | $\tau $-Damping coefficient with respect to initiation of sediment motion | - |

g | Coefficient of gravity | m/s^{2} |

h | Water depth | m |

${k}_{s}$ | Equivalent sand roughness height, where ${k}_{s}=d$ | m |

${k}_{s}^{+}$ | = $R{e}_{ks}^{\u2605}=R{e}^{*}\phantom{\rule{3.33333pt}{0ex}}{k}_{s}/d=\phantom{\rule{3.33333pt}{0ex}}{k}_{s}\frac{{v}^{\u2605}}{\nu}$ | - |

$R{e}^{\u2605}$ | = ${v}^{\u2605}\phantom{\rule{0.277778em}{0ex}}d/\nu $, Reynolds number of grain | - |

n | Grain motion triggering multiple of ${v}_{rms}^{\prime}$ | - |

$v,{v}_{m}$ | Mean current velocity | m/s |

${v}^{\prime}$ | Fluctuation values of the flow velocity | m/s |

${v}_{m,c}$ | Mean critical velocity | m/s |

${v}_{rms}^{\prime}$ | Standard deviation of ${v}^{\prime}$ | m/s |

${v}_{rms,b}^{\prime}$ | Standard deviation of ${v}^{\prime}$ at the grain level | m/s |

${v}^{\u2605}$ | Shear velocity = $\sqrt{\tau /\rho}=\sqrt{{\tau}^{\u2605}{\rho}^{\prime}gd}$ | m/s |

y | Distance from the bed | m |

${y}^{+}$ | = $y\frac{{v}^{\u2605}}{\nu}$, dimensionless distance from the bed | - |

$\alpha $ | Inclination angle of the bed, positive downhill | ${}^{\circ}$ |

$\delta $ | Thickness of the viscous sublayer of the boundary layer | m |

$\eta $ | Factor for ${\tau}_{c}^{\u2605}$ caused by a bottom slope | - |

${\phi}_{o}$ | Static angle of internal friction | |

Angle of repose of a heap of cohesionless grains | ||

Friction angle above which avalanches occur | ${}^{\circ}$ | |

${\phi}_{d}$ | Dynamic angle of internal friction | |

= Angle of internal friction on the sediment surface, | ||

which is decisive for the beginning of sediment movement | ${}^{\circ}$ | |

$\nu $ | Kinematic viscosity of the fluid | m^{2}/s |

$\kappa $ | von Karman constant = 0.4 | - |

$\rho $ | Density of fluid | kg/m^{3} |

${\rho}_{s}$ | Density of sediment | kg/m^{3} |

${\rho}^{\prime}$ | $=({\rho}_{s}-\rho )/\rho $, relative density | - |

$\tau $ | $=\rho ghI=\rho {v}^{\u26052}$, shear stress at the bed | N/m^{2} |

${\tau}^{\prime}$ | Fluctuation values of shear stress due to turbulence | N/m^{2} |

${\tau}^{\u2605}$ | $=\tau /\left(({\rho}_{s}-\rho )\right)gd={v}^{\u26052}/\left({\rho}^{\prime}gd\right)$, dimensionless shear stress | - |

${\tau}_{c}^{\u2605}$ | $={\tau}_{c,h/d=\infty ,\alpha =0}^{\u2605}$ = reference value for the dimensionless shear stress | |

at the beginning of the sediment movement according to Shields | - | |

Indices | ||

$h/ks,h/d$: | Case with low water cover | |

$h/ks>100$ or $h/d>100$ | Case with high water cover | |

$\alpha $: | with inclination of the bed by the $\alpha $ angle |

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**Figure 1.**${v}_{rms,b}^{\prime}/{v}^{\u2605}$ at bed level for different values of relative coverage ($h/{k}_{s}$); graphical representation of Equation (2). $\delta $ = thickness of viscous boundary layer $=11.63\phantom{\rule{3.33333pt}{0ex}}\nu /{v}^{\u2605}$. Modified from [19].

**Figure 3.**Solutions of Equation (7) for different values of the parameters ${k}_{s}/d$ and ${k}_{s}/{d}_{90}$.

**Figure 5.**Damping factors of shear stress under low water coverage. Blue curves represent Equation (6) for different values of $R{e}_{ks}^{\u2605}={v}^{\u2605}\phantom{\rule{3.33333pt}{0ex}}{k}_{s}/\nu $. The bold blue curve ($R{e}_{ks}^{\u2605}>70$) results from Equation (6) and Equation (7). The dashed red curve shows the course of Equations (10) and (11) according to Bezzola [20].

**Figure 6.**Schematic representation of the definition of the roughness sublayer according to Bezzola (2002) [20].

**Figure 8.**Effect of turbulence damping at small relative coverage ($h/d$) as a function of $R{e}^{\u2605}$ calculated with Equation (15). Blue: $n=1.8$; red: $n=3$.

**Figure 9.**${v}_{y}/{v}^{\u2605}$ at different relative wall distances ($y/{k}_{s}$) as a function of $R{e}_{y}^{\u2605}$ [37,38]. The yellow region marks the transition from viscous to turbulent flow as a function of ${y}^{+}$ and between hydraulically smooth and rough flow as a function of ${k}_{s}^{+}$. Assignment of the curves to the descriptive equations: A = viscous sublayer, $\frac{{v}_{y}}{{v}^{\u2605}}=\frac{{v}^{\u2605}}{\nu}y$; B = solution hydraulically rough, Equation (16) with integration constant $B=8.5$ (green curves); C = classical smooth–rough transition approach, Equation (16) with B according to Equation (17) (blue curves); D = continuous solution Equation (20) (red curves).

**Figure 10.**Reduced turbulence damping on protruding particles. The blue arrow shows the damping at the protruding grain. This is lower than at the bed (modified after Bezzola [20]).

**Figure 11.**Influence of ${k}_{S}/d$ on the effect of turbulence damping on critical shear stress under hydraulically rough conditions. Blue: ${k}_{S}/d=1$; red: ${k}_{S}/d=0.5$. Data reported by Ashida and Bayazit (1973) [2] were adjusted for the influence of bed slope. (Note that Equation (23) for $h/d\to 0$ goes against a finite value, unlike Equation (7)).

**Figure 12.**Effect of ${\phi}_{d}$ on ${D}_{\tau ,c}$ according to Equation (23) presented for ${\phi}_{d}={10}^{\circ},{20}^{\circ},{30}^{\circ}$ for $n=1.8$ (corresponding to a motion state like the Shields curve) and for $n=3$ (corresponding to start of motion of separated grains).

**Figure 13.**Example of the influence of slope and shear stress damping on the critical shear stress under hydraulically rough conditions. In the range of $\eta /{D}_{\tau ,c}<1$, the slope effect is dominant. At $\eta /{D}_{\tau ,c}=1$, (in this example) the effects of slope and turbulence damping cancel each other out, and at $\eta /{D}_{\tau ,c}>1$, the effect of turbulence damping predominates (Equation (7), with ${k}_{S}/d=0.5$).

**Figure 14.**Increase in critical shear stress according to experiments conducted by Ashida and Bayazit (1973) [2] and by Prancevic et al. (2014) [8,32] in comparison with calculated values. In all cases, the influence of turbulence damping overcomes the influence of slope (${D}_{{\tau}_{c}}<\eta $). Also shown are the values of $\eta $ associated with the individual measured values.

**Figure 15.**Some examples for critical bed slopes where the effects of slope (increasing sediment mobility) and the effects of low water coverage (decreasing sediment mobility) are balanced. In the case of $R{e}^{\u2605}>70$ based on Equation (6) or (7) and in the case of $R{e}^{\u2605}=10$ based on Equation (23), lower $R{e}^{\u2605}\to $ lower near bed turbulence → lower turbulence damping → smaller effect of $h/d$.

**Figure 16.**Critical shear stress (${\tau}_{c,h/d,\alpha}^{\u2605}\sim 1/{D}_{{\tau}_{c}}$) as a function of slope. The values for the acrylic sediment reported by Prancevic and Lamb (2015) obtained relative to gravel using Equation (7) increased by about three times for the same slope, as also observed by in the present study. Data by Ashida & Bayazit [2], Prancevic, Lamb & Fuller [8], Prancevic & Lamb [9,10].

Lowland Rivers | Mountain Streams | |
---|---|---|

${d}_{90}/{d}_{50}$ | $\approx 1.8$ | $\approx 3$ |

**Table 2.**Measured influence of slope and low water coverage on the initiation of movement and calculated values (${\tau}_{c,h/d,\alpha}^{\u2605}/{\tau}_{c,\infty ,0}^{\u2605}=\eta /{D}_{{\tau}_{c}}$). Green background: direct data information or extracted from the data. yellow: estimated; red: calculated on the basis of the data and estimated values.

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Nr. | Author | $\mathit{\alpha}$ | $\mathit{\phi}$ | $\mathit{\eta}$ | h | d | ${\mathit{\rho}}^{\prime}$ | h/d | $\frac{\mathbf{1}}{{\mathit{D}}_{{\mathit{\tau}}_{\mathit{c}}}}$ | $\frac{\mathit{\eta}}{{\mathit{D}}_{{\mathit{\tau}}_{\mathit{c}}}}$ | ${\mathit{\tau}}_{\mathit{c},\mathit{h}/\mathit{d},\mathit{\alpha}}^{\u2605}$ | ${\mathit{\tau}}_{\mathit{c},\infty ,\mathbf{0}}^{\u2605}$ | ${\mathit{\phi}}_{\mathit{d}}$ | $\mathit{\alpha}/{\mathit{\phi}}_{\mathit{d}}$ | $\frac{\mathbf{1}}{{\mathit{D}}_{{\mathit{\tau}}_{\mathit{c}}}}$ | $\frac{\mathit{\eta}}{{\mathit{D}}_{{\mathit{\tau}}_{\mathit{c}}}}$ | P/J |

- | ^{o} | ^{o} | - | m | m | - | - | - | - | - | - | ^{o} | - | - | - | - | |

1 | Ashida | 1.15 | 45.0 | 0.959 | 0.091 | 0.0225 | 1.65 | 4.07 | 1.182 | 1.134 | 0.0431 | 0.038 | 26.5 | 0.043 | 1.2416 | 1.191 | 1.051 |

2 | & | 2.86 | 45.0 | 0.899 | 0.036 | 0.0225 | 1.65 | 1.58 | 1.542 | 1.385 | 0.0527 | 0.038 | 26.5 | 0.108 | 1.7467 | 1.570 | 1.133 |

3 | Bayazit | 4.30 | 45.0 | 0.847 | 0.025 | 0.0225 | 1.65 | 1.11 | 1.891 | 1.601 | 0.0607 | 0.038 | 26.5 | 0.162 | 2.2078 | 1.869 | 1.168 |

4 | [2] | 5.70 | 45.0 | 0.796 | 0.022 | 0.0225 | 1.65 | 0.96 | 2.451 | 1.950 | 0.0743 | 0.038 | 26.5 | 0.215 | 2.5116 | 1.998 | 1.025 |

5 | “ | 8.50 | 45.0 | 0.692 | 0.015 | 0.0225 | 1.65 | 0.68 | 3.395 | 2.350 | 0.0894 | 0.038 | 26.5 | 0.321 | 3.6478 | 2.525 | 1.074 |

6 | “ | 11.30 | 45.0 | 0.587 | 0.013 | 0.0225 | 1.65 | 0.58 | 5.279 | 3.099 | 0.1178 | 0.038 | 26.5 | 0.427 | 4.5863 | 2.693 | 0.869 |

7 | “ | 0.57 | 45.0 | 0.980 | 0.097 | 0.0125 | 1.65 | 8.15 | 1.079 | 1.057 | 0.0402 | 0.038 | 26.5 | 0.022 | 1.1144 | 1.092 | 1.033 |

8 | “ | 1.43 | 45.0 | 0.950 | 0.036 | 0.0125 | 1.65 | 3.04 | 1.183 | 1.124 | 0.0427 | 0.038 | 26.5 | 0.054 | 1.3355 | 1.268 | 1.129 |

9 | “ | 2.86 | 45.0 | 0.899 | 0.020 | 0.0125 | 1.65 | 1.71 | 1.565 | 1.406 | 0.0535 | 0.038 | 26.5 | 0.108 | 1.6738 | 1.504 | 1.070 |

10 | “ | 4.30 | 45.0 | 0.847 | 0.014 | 0.0125 | 1.65 | 1.21 | 1.891 | 1.601 | 0.0608 | 0.038 | 26.5 | 0.162 | 2.0715 | 1.754 | 1.095 |

11 | “ | 5.70 | 45.0 | 0.796 | 0.012 | 0.0125 | 1.65 | 1.00 | 2.283 | 1.816 | 0.0691 | 0.038 | 26.5 | 0.215 | 2.4109 | 1.918 | 1.056 |

12 | “ | 7.10 | 45.0 | 0.744 | 0.011 | 0.0125 | 1.65 | 0.96 | 2.993 | 2.227 | 0.0846 | 0.038 | 26.5 | 0.268 | 2.5049 | 1.864 | 0.837 |

13 | “ | 0.57 | 52.0 | 0.983 | 0.054 | 0.0064 | 1.65 | 8.00 | 1.033 | 1.016 | 0.0386 | 0.038 | 30.6 | 0.019 | 1.1089 | 1.090 | 1.073 |

14 | “ | 1.43 | 52.0 | 0.957 | 0.024 | 0.0064 | 1.65 | 3.75 | 1.265 | 1.211 | 0.0461 | 0.038 | 30.6 | 0.047 | 1.2645 | 1.211 | 1.000 |

15 | “ | 2.86 | 52.0 | 0.914 | 0.013 | 0.0064 | 1.65 | 2.03 | 1.571 | 1.437 | 0.0536 | 0.038 | 30.6 | 0.094 | 1.5422 | 1.410 | 0.981 |

16 | Prancevic | 1.8 | 45.6 | 0.946 | 0.021 | 0.015 | 1.65 | 1.420 | 1.198 | 1.133 | 0.034 | 0.030 | 30.4 | 0.059 | 1.8584 | 1.758 | 1.551 |

17 | & | 3.2 | 45.6 | 0.903 | 0.016 | 0.015 | 1.65 | 1.073 | 1.734 | 1.567 | 0.047 | 0.030 | 30.4 | 0.105 | 2.2702 | 2.051 | 1.309 |

18 | Lamb | 5.6 | 45.6 | 0.829 | 0.010 | 0.015 | 1.65 | 0.693 | 2.453 | 2.033 | 0.061 | 0.030 | 30.4 | 0.184 | 3.5581 | 2.949 | 1.450 |

19 | & | 5.9 | 45.6 | 0.819 | 0.010 | 0.015 | 1.65 | 0.680 | 2.571 | 2.107 | 0.063 | 0.030 | 30.4 | 0.194 | 3.6478 | 2.989 | 1.419 |

20 | Fuller | 6.8 | 45.6 | 0.791 | 0.011 | 0.015 | 1.65 | 0.713 | 3.160 | 2.500 | 0.075 | 0.030 | 30.4 | 0.224 | 3.4337 | 2.717 | 1.087 |

21 | [8] | 8.0 | 45.6 | 0.753 | 0.009 | 0.015 | 1.65 | 0.613 | 3.541 | 2.667 | 0.080 | 0.030 | 30.4 | 0.263 | 4.1987 | 3.162 | 1.186 |

22 | [32] | 9.8 | 45.6 | 0.695 | 0.009 | 0.015 | 1.65 | 0.620 | 4.459 | 3.100 | 0.093 | 0.030 | 30.4 | 0.322 | 4.1345 | 2.875 | 0.927 |

23 | “ | 11.5 | 45.6 | 0.640 | 0.009 | 0.015 | 1.65 | 0.587 | 5.832 | 3.733 | 0.113 | 0.030 | 30.4 | 0.378 | 4.4817 | 2.869 | 0.768 |

24 | “ | 12.4 | 45.6 | 0.611 | 0.007 | 0.015 | 1.65 | 0.487 | 5.950 | 3.633 | 0.109 | 0.030 | 30.4 | 0.408 | 6.0996 | 3.725 | 1.025 |

25 | “ | 13.5 | 45.6 | 0.574 | 0.007 | 0.015 | 1.65 | 0.467 | 6.731 | 3.867 | 0.116 | 0.030 | 30.4 | 0.444 | 6.5911 | 3.786 | 0.979 |

26 | “ | 14.2 | 45.6 | 0.551 | 0.007 | 0.015 | 1.65 | 0.480 | 7.557 | 4.167 | 0.125 | 0.030 | 30.4 | 0.467 | 6.2547 | 3.448 | 0.828 |

27 | “ | 15.6 | 45.6 | 0.505 | 0.007 | 0.015 | 1.65 | 0.487 | 9.179 | 4.633 | 0.139 | 0.030 | 30.4 | 0.513 | 6.0996 | 3.079 | 0.665 |

28 | “ | 16.9 | 45.6 | 0.461 | 0.006 | 0.015 | 1.65 | 0.433 | 10.260 | 4.733 | 0.141 | 0.030 | 30.4 | 0.556 | 7.6199 | 3.515 | 0.743 |

29 | “ | 19.6 | 45.6 | 0.370 | 0.007 | 0.015 | 1.65 | 0.493 | 16.023 | 5.933 | 0.178 | 0.030 | 30.4 | 0.645 | 5.9523 | 2.204 | 0.371 |

30 | Prancevic | 0.7 | - | - | 0.0164 | 0.023 | 0.15 | 0.714 | - | - | 0.058 | - | - | - | 3.4354 | - | - |

31 | & | 0.9 | - | - | 0.0162 | 0.023 | 0.15 | 0.704 | - | - | 0.074 | - | - | - | 3.4882 | - | - |

32 | Lamb | 1.4 | - | - | 0.0155 | 0.023 | 0.15 | 0.676 | - | - | 0.110 | - | - | - | 3.6907 | - | - |

33 | [10] | 0.7 | - | - | 0.0135 | 0.023 | 0.15 | 0.588 | - | - | 0.048 | - | - | - | 4.4784 | - | - |

34 | 0.9 | - | - | 0.0159 | 0.023 | 0.15 | 0.690 | - | - | 0.072 | - | - | - | 3.5714 | - | - | |

35 | 1.4 | - | - | 0.0155 | 0.023 | 0.15 | 0.676 | - | - | 0.110 | - | - | - | 3.6907 | - | - |

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Zanke, U.; Roland, A.; Wurpts, A.
The Reason for the Rise in Critical Shear Stress on Sloping Beds. *Water* **2023**, *15*, 2976.
https://doi.org/10.3390/w15162976

**AMA Style**

Zanke U, Roland A, Wurpts A.
The Reason for the Rise in Critical Shear Stress on Sloping Beds. *Water*. 2023; 15(16):2976.
https://doi.org/10.3390/w15162976

**Chicago/Turabian Style**

Zanke, Ulrich, Aron Roland, and Andreas Wurpts.
2023. "The Reason for the Rise in Critical Shear Stress on Sloping Beds" *Water* 15, no. 16: 2976.
https://doi.org/10.3390/w15162976