# A Moment-Based Depth-Averaged K-ε Model for Predicting the True Turbulence Intensity over Bedforms

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

- -
- Investigate the typical spatial variability in the depth-averaged turbulent kinetic energy field over a train of bedforms based on experimental studies reported in the literature;
- -
- Assess the accuracy and limitations of the standard k-ε depth-averaged turbulence SDAKE model in reproducing the measured values of turbulent kinetic energy in both the benchmark uniform flow over a flatbed case as well as the case for flow over a train of bedforms;
- -
- Introduce a new k-ε turbulent (MDAKE) model that is based on the moment concept and suitable for depth-averaged VAM models and can be used to reasonably predict the true spatial variation in turbulence intensity over varied bed topography.

## 2. Method Statement

#### 2.1. The Concept of the Moment of Momentum

_{b}is the local bed level and h is the local water depth at a given location x.

#### 2.2. Turbulent Kinetic Energy (TKE) and Turbulence Intensity

#### 2.2.1. TKE in Case of Uniform Flow

_{b}is the bed shear stress, U

_{o}is the depth-averaged water velocity, ρ is the water density and ${C}_{*}$ is the dimensionless Chezy coefficient, which could be estimated using the Colebrook equation (Equation (6)) or the Manning equation (Equation (7)) as follows:

_{s}is the equivalent sand grain roughness height, n is Manning’s roughness coefficient and g is the acceleration of gravity.

_{r}is a correction factor that will be described later on. It was noticed that the exponential laws for the turbulent kinetic energy match better with the measurements, especially within the region 0.1 < z/h < 0.6. However, it was concluded that the exponential function could be used to predict the longitudinal turbulence intensity over the entire flow depth in the case of uniform and smooth open-channel flow [30].

_{r}, is added to Equation (8) to make it applicable for both smooth and rough boundaries.

_{r}will be very close to, but less than, unity.

_{r}ranges from 0.97 to 1. Therefore, a value of unity will be used in this work. By integrating Equation (8) using Equations (4), the following equation (Equation (9)) could be obtained for the depth-averaged turbulent kinetic energy, in the case of uniform flow over a flatbed:

#### 2.2.2. TKE in Case of a Train of Bedforms

#### 2.3. Assumptions and Simplifications

- The flow is shallow, and the channel stream width is generally wide.
- The fluid is Newtonian.
- The flow is fully developed and turbulent over a train of uniformly spaced bedforms. The bedforms belong to the low regime, such as dunes and ripples. Bars, antidunes, pools and chutes are not included.
- Transversal variations in the bedform’s topography are not considered.
- In the case of having a uniform flow over a flat bed, the log-law could be used to predict the velocity profile in the inner region.
- All experiments used for calibration are for fixed bed boundaries; therefore, the effect of bedload and suspended load on turbulence is not considered in this study.
- The k values deduced from the turbulence measurements are based on the assumption that the value of the lateral turbulent intensity component generally lies in the midway between the corresponding values of the longitudinal and the vertical components (Equation (24)).

## 3. TKE Model Development

#### 3.1. General

#### 3.2. Rastogi and Rodi’s Turbulence (SDAKE) Model

_{μ}, C

_{1ε}, C

_{2ε}, σ

_{k}and σ

_{ε}are constants and equal 0.09, 1.44, 1.92, 1.0 and 1.3, respectively, and C

_{ε}can be determined by using Equation (12).

#### 3.3. A New Moment-Based Depth-Averaged k-ε (MDAKE) Model

_{s}represents the timescale of energy transported through the spectrum.

_{1}

^{3}/ℓ

#### 3.4. Model Descritization

#### 3.4.1. Preface

#### 3.4.2. GCI

## 4. Calibration of MDAKE Model

#### 4.1. Introduction

_{1}, that is important for the evaluation of the production and generation of turbulent kinetic energy and dissipation rate, respectively. The model equations (Equations (22) and (23)) contain a number of dimensionless coefficients r, Φ and ξ

_{k}. Some of these coefficients can be determined analytically, whereas others require the use of measurements for calibration.

- -
- The depth-averaged value of k automatically reduces to the true value of k for the benchmark case of uniform flow over a flatbed;
- -
- The solution of k becomes independent on the value of the coefficient ζ
_{k}.

_{o}, in the case of uniform flow over a flat bed [3]. It is known that the log-law could be used to predict the velocity profile in the inner region in the case of uniform flow. Therefore, α can be obtained using the following equation.

_{α}is found to be very close to 1. A more accurate way to get the coefficient C

_{α}is to assume a log-wake relation, which gives a value of C

_{α}= (1 + 8 Π/π

^{2}), where Π is the wake parameter and has a range from 0 to 0.2. Nezu and Rodi found that Π remains nearly constant and equals 0.2 when the Reynolds number is larger than 10

^{5}[49]. However, it should be mentioned that there is no specific and accurate relation to calculate the wake parameter, Π. For the upper value of Π = 0.2, the corresponding upper value of C

_{α}is ≈1.16.

_{k}.

#### 4.2. Lab Experiments for Model Calibration

_{n}) varying from 0.12 to 0.71 and a channel width (b) that varies from 0.08 to 1.5 m.

#### 4.3. Calculation of $\overline{k}$ from Experimental Data

_{eτ}is the Reynolds number based on the bed shear velocity. The table shows that adopting the assumption (given by Equation (27)) tends to overpredict the actual depth-averaged value by 7 to 11%. Despite the crudeness of the assumption but because of the depth-averaging process, it tends to reduce its effect on the calculated averaged value.

_{k}.

#### 4.4. Model Calibration

_{o}and ${u}_{1}$, should be determined. Despite the fact these variables could be calculated using VAM models (similar to [58,59,60,61,62]), in this work, they were derived from the measurements and were best fitted using spline curves. Via the spline curves, the spatial variations in h, U

_{o}and ${u}_{1}$ were determined over the bedform wavelength.

_{k,}is adjusted in order to get the best match with the experimental data. The value of ξ

_{k}for each experiment is given in Table 1.

## 5. Results and Discussion

#### 5.1. Uniform Flow over Flatbed BenchMark Case

#### 5.2. Nonuniform Flow over a Train of Bedforms Case

_{t}, (Equation (29)) over one wavelength of a bedform. It is interesting to notice that Fν

_{t}is somewhat larger than the values reported in the case of uniform flow over a flat bed (Fν

_{t}= 0.06–0.07). The large values of Fν

_{t}(shown on Figure 7) generally refer to the large values for the flow eddy viscosity for the case of flow over bedforms (that is characterized by the existence of flow separation and high shear flow zones downstream of the crest) compared to the case of uniform flow over a flat bed.

_{t}could be determined using the following equations:

_{t}is the depth-averaged eddy viscosity, C

_{μ}is a constant and equals 0.09.

#### 5.3. Limitations of the MDAKE Model

_{α}. As was discussed before in Section 4.1, this coefficient depends on the wake parameter, Π; unfortunately, there is no specific universal and accurate relation to describe this parameter. However, it is noticed that C

_{α}might range from a minimum value of 1 and a maximum value not larger than 1.2. The laboratory experiments used in this study suggest a value for C

_{α}closer to 1.15. It was noticed that most of the experimental runs produce a good match with the data for C

_{α}= 1.15 and one run (run2) provides a good match at C

_{α}= 1.0. Nevertheless, more laboratory experiments are required to investigate the factors that affect C

_{α}.

_{k}, varies from about 0.004 to 0.025 and it can be related with the depth to wavelength ratio, h/λ, via a parabolic relation, as shown on Figure 8. It should be mentioned that the proposed relation also satisfies the uniform flow case over a flatbed (the intersection point with the vertical axis).

_{k}are presented for each experiment: the medium value, which gives a good match with the data, the upper limit value of each bar, which gives, at most, 25% over-prediction of k

_{max}and finally, the lower limit value, which gives, at most, 20% under-prediction of k

_{max}.

_{k}for all the experiments. Figure 9 suggests a global value of about ξ

_{k}= 0.013, as shown by the dashed line. Figure 10a–f presents the predictions of the MDAKE model using this global constant value for the calibration coefficient for all the experiments (ζ

_{k}= 0.013). In general, good agreement is still obtained, except for experiment T5, where the model appears to over-predict kmax by about 50%. Despite this, it is interesting to notice that the model generally provides good predictions for $\overline{k}$ over the crest.

_{k}= 0.075) might be more suitable.

## 6. Conclusions and Challenges

_{k}) in such a case.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

b | Channel width |

C_{r} | Correction factor in the turbulent kinetic energy profile in case of uniform flow over a flat bed (Equation (5)) |

C_{*} | Dimensionless Chezy coefficient |

C_{α} | The wake coefficient ≈ 1.15, however it varies from 1 to 1.2 |

C_{μ}, C_{1ε}, C_{2ε} | Are universal constants in the Rastogi and Rodi’s k-ε Model |

DNS | Direct numerical simulation CFD model |

F_{νt} | Eddy viscosity coefficient ≅ 0.07 for uniform flow over flat bed |

g | Acceleration due to gravity |

h | Depth of flow measured vertically |

k | The time averaged turbulent kinetic energy per unit mass |

$\tilde{k}$ | The virtual value of turbulent kinetic energy as given by Rastogi and Rodi’s k-ε Model |

$\overline{k}$ | The depth-averaged turbulent kinetic energy per unit mass |

${\overline{k}}_{crst}$ | The depth-averaged turbulent kinetic energy at the crest |

${\overline{k}}_{u}$ | The depth-averaged turbulent kinetic energy in case of uniform flow over a flat bed |

ks | Effective sand roughness height |

ks+ | The dimensionless sand grain roughness (ks+ = u_{∗}ks/ν) |

$\ell $ | Turbulence length scale (or the length scale of the most energetic eddy) |

MDAKE | Moment-based depth-averaged k-ε model (Equations (22) and (23)) |

P.R. | The point of reattachment |

q | Longitudinal discharge per unit width of the channel (q = u_{o}·h) |

q_{1} | =u_{1}·h |

SDAKE | Standard depth-averaged k-ε model (Rastogi and Rodi model, Equations (10) and (11)) |

TKE | Turbulence kinetic energy |

u(z) | Longitudinal velocity at elevation z |

u_{1} | Velocity at the surface in excess of the mean u_{o} |

u_{1log} | The equivalent u_{1} velocity in case of logarithmic velocity profile |

u_{1o} | u_{1} velocity over the crest of a train of bedforms |

u* | The skin friction shear velocity |

U_{o}, u_{o} | Depth-averaged longitudinal velocity |

$\sqrt{\overline{u{\prime}^{2}}}$ | r.m.s. of turbulence in the downstream direction |

v_{1} | Lateral velocity at the surface in excess of the mean value V_{o} |

$v$ | Turbulence velocity scale |

$\sqrt{\overline{v{\prime}^{2}}}$ | The r.m.s. of turbulence in the lateral direction |

VAM | Vertically averaged and moment set of equations |

V_{o} | Depth-averaged lateral/transverse velocity |

$\sqrt{\overline{w{\prime}^{2}}}$ | The r.m.s. of turbulence in the vertical direction |

W_{o} | Depth-averaged vertical velocity |

x | Horizontal coordinate |

z | Vertical coordinate |

$\overline{z}$ | =z_{b} + h/2 |

z_{b} | Bed elevation from an arbitrary horizontal plane |

z_{o} | Roughness parameter (z_{o} = k_{s}/30 + 0.11ν_{t}/u_{*}) |

z+ | The vertical distance normalized by the viscous scale ν/u_{*} |

α | The ratio between u_{1} and the mean velocity, u_{o} in case of uniform flow over a flat bed |

δk | The net increase in the depth-averaged turbulent kinetic energy over bedforms |

δΦ | Changes in the nodal values of Φ |

∆ | The height of bedform |

∆t | Time discretization |

∆x | Distance discretization |

ε | Dissipation of turbulent kinetic energy by viscous effects |

$\tilde{\epsilon}$ | The approximate (virtual) value of turbulent dissipation as given by SDAKE Model |

κ | von Karman constant ≅ 0.41 |

λ | Bedform wavelength |

ν | Kinematic viscosity of fluid |

ν_{t} | The eddy viscosity |

Π | The wake parameter and it ranges from 0 to 0.2 for uniform flow |

ρ | Mass density of water (ρ = 1000 kg/m^{3}) |

σ_{k}, σ_{ε} | Constants related to k-ε models |

ζ_{k} | Calibration coefficient for the modified k-ε model |

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**Figure 2.**Variation in the Integral Moment Velocity Scale u

_{1}for Accelerating and Decelerating Flow. (

**a**) Decelerating Flow, (

**b**) Accelerating Flow.

**Figure 3.**(

**a**) A typical spatial variation for the integral velocity u

_{1}over one wavelength of a train of bedforms; (

**b**) a typical variation for the depth-averaged turbulent kinetic energy, for fully developed turbulent flow; (

**c**) bed geometry of one wavelength for a train of dunes.

**Figure 4.**Spatial distribution of depth-averaged turbulent kinetic energy. The graph also shows a comparison between different models with data from experiment (T5).

**Figure 6.**Spatial variation in the depth-averaged turbulent kinetic energy over a train of 5 bedforms for fully developed turbulent flow for: (

**a**) experiment (T5), (

**b**) experiment (Run2), (

**c**) experiment (Run5).

**Figure 7.**Spatial distribution of eddy viscosity coefficient over one wavelength of bedforms (data from experiment T5).

**Figure 8.**(

**a**) Effect of bedform height on the equilibrium value of the depth-averaged turbulent kinetic energy over the crest. (

**b**) Effect of bedform height on the net increase in the depth-averaged turbulent kinetic energy, dk, over bedforms (refer to Figure 3 for definitions).

**Figure 9.**Calibration coefficient for the MDAKE model. The upper limit of each bar gives at most 25% over-prediction of k

_{max}whereas the lower limit gives at most 20% under-prediction of k

_{max}.

**Figure 10.**Depth-averaged predictions of MDAKE model using ζ

_{k}= 0.013 (solid line). Solid circles represent data of: (

**a**) experiment T5, (

**b**) experiment T6, (

**c**) Run2, (

**d**) Run4, (

**e**) Run5, (

**f**) experiment by Bennett and Best.

**Table 1.**Summary of the geometrical and flow parameters of selected fixed-bedform experiments reported in the literature.

T5 ^{(}^{1)} | T6 ^{(1)} | Run2 ^{(2)} | Run3 ^{(2)} | Run4 ^{(2)} | Run5 ^{(2)} | Run6 ^{(2)} | Lyn2 ^{(3)} | Lyn3 ^{(3)} | Bennett ^{(4)} | |
---|---|---|---|---|---|---|---|---|---|---|

λ (m) | 1.60 | 1.60 | 0.80 | 0.80 | 0.40 | 0.40 | 0.40 | 0.15 | 0.15 | 0.63 |

∆ (m) | 0.080 | 0.080 | 0.040 | 0.040 | 0.040 | 0.040 | 0.040 | 0.012 | 0.012 | 0.040 |

h_{av} (m) | 0.252 | 0.334 | 0.158 | 0.546 | 0.159 | 0.159 | 0.300 | 0.061 | 0.061 | 0.120 |

∆/h | 0.317 | 0.240 | 0.253 | 0.073 | 0.252 | 0.252 | 0.133 | 0.197 | 0.197 | 0.333 |

∆/λ | 0.050 | 0.050 | 0.050 | 0.050 | 0.100 | 0.100 | 0.100 | 0.080 | 0.080 | 0.063 |

F_{n} | 0.25 | 0.28 | 0.30 | 0.12 | 0.30 | 0.16 | 0.31 | 0.35 | 0.71 | 0.44 |

k_{u} (m^{2}/s^{2}) | 0.00281 | 0.00371 | 0.00226 | 0.00071 | 0.00305 | 0.00100 | 0.00430 | 0.00159 | 0.00571 | 0.00320 |

ζ_{k} | 0.004 | 0.009 | 0.014 | 0.020 | 0.008 | 0.009 | 0.025 | - | - | 0.019 |

Simulation Run | $\overline{k}$ | $\overline{k}$* | % of Variance |

2D ripples (R_{et} = 180) | 5.04 | 5.58 | 11% |

2D ripples (R_{et} = 400) | 6.58 | 7.05 | 7% |

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**MDPI and ACS Style**

Elgamal, M.
A Moment-Based Depth-Averaged K-ε Model for Predicting the True Turbulence Intensity over Bedforms. *Water* **2022**, *14*, 2196.
https://doi.org/10.3390/w14142196

**AMA Style**

Elgamal M.
A Moment-Based Depth-Averaged K-ε Model for Predicting the True Turbulence Intensity over Bedforms. *Water*. 2022; 14(14):2196.
https://doi.org/10.3390/w14142196

**Chicago/Turabian Style**

Elgamal, Mohamed.
2022. "A Moment-Based Depth-Averaged K-ε Model for Predicting the True Turbulence Intensity over Bedforms" *Water* 14, no. 14: 2196.
https://doi.org/10.3390/w14142196