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Numerical and experimental analysis of shallow turbulent flow over complex roughness beds

TL;DR: In this article, a set of shallow water equations based on a k−e^ Reynold stress model is established to simulate the turbulent flows over a complex roughness bed, and the fundamental equations are discretized.
Abstract: A set of shallow-water equations (SWEs) based on a k^−e^ Reynold stress model is established to simulate the turbulent flows over a complex roughness bed. The fundamental equations are discretized ...

Summary (3 min read)

1 Introduction

  • Large and small turbulent swirling flows are often clearly observable when dealing with hydraulic structures in rivers and coastal areas and they are a key factor that influence the frequency and magnitude of natural processes such as the sediment transport, mixing of pollutants and/or riverbed deformation.
  • 2D shallow-turbulence flow models have been extensively developed over the last decades.
  • The standard  k model has been demonstrated to provide satisfactory results after numerous comparisons with the measured data [11, 14] and it is also relatively simple to use and very fast.

2.2 Vertical turbulent production kV

  • In the context of homogeneous, open-channel turbulence, they represent all the turbulent energy produced.
  • The theory can also be applied to the transfer of energy from the bottom of the river to the free surface [20] .
  • To rigorously and precisely quantify the energy transfer across various scales, this study presents an additional model that estimates the generation of vertical turbulent features in a different manner, considering the interaction between points at different levels and the total energy spectrum.
  • Bottom roughness elements markedly affect the amount of turbulence.
  • Cea et al. [11] found a similar degree of accuracy between the  k model, which is based on the concise isotropic hypothesis, and the DASM model, which includes complex model structures considering anisotropy.

2.2.1 Mean velocity profiles

  • For a uniform and fully-developed turbulent flow in a wide open channel, the Reynolds equation in the z-direction (as derived from the Navier-Stokes equations) is reduced to: l over the whole depth can be obtained, Eq. (9) can be easily integrated to yield the distribution of u .
  • According to their contribution to the turbulent structure, the vertical turbulence fields of u are divided into two regions: the inner region and the outer region [20] .
  • The wake law provided by Coles [31] , is typically used to extend the log-law to the outer region.
  • П has been investigated in various studies where its value has been suggested to be around 0.08-0.20.
  • П also shows no distinct value for flows with different bed roughness conditions [32] .

2.2.2 Depth-averaged vertical production

  • In the inertial subrange, energy transfer is the only significant process; there is no energy production or dissipation.
  • In other words, the energy of open-channel flows is predominantly dissipated in the free surface region.
  • Observed  values in this region are less accurate due to the constraints of free-surface fluctuations.
  • By integrating Eq. (20) from the bed to free surface, the authors obtained the depth-averaged turbulent viscosity ˆt  as follows: EQUATION.

2.4 Boundary conditions

  • To perform the staggered-grid difference method, ghost cells are typically imposed around the outmost computational domain.
  • The boundary conditions selected for this study mainly include open boundaries and no-slip boundaries.
  • Open boundary conditions are applied mainly to inflow and outflow.
  • For the tests conducted, subcritical flow is the most frequent, hence the boundary conditions assumed include a specific flow rate assigned at the upstream boundary location; in addition, uniform water depth is applied as the downstream boundary condition.
  • The depth-averaged statistic characteristics in the small region near the wall are assumed to be analogous to the turbulent features in the core region near the bed.

3 Model Validations

  • To validate the model previously described to quantify complex turbulence, its performance has been verified against experimental turbulent flows obtained under various circumstances: 1) a uniform gravel bed, 2) a  90 bend, and 3) a suddenly expanding section.
  • Numerical results were then compared against measured datasets as well as the calculated values from the standard  k model and other numerical schemes.
  • To distinguish among the different models' results, "PF" and "RRF" are used to represent the model constructed by the proposed formula and by Rastogi and Rodi's formula, respectively.

3.1 Turbulent flow in a straight channel with gravel bed

  • To verify the accuracy of the proposed model in replicating bed roughness and Reynolds number effects on the formation of turbulent features, a series of experiments on open-channel flows over rough beds conducted by Wang et al [38] , were used for comparison.
  • Throughout all measurements, the simultaneous high-frequency velocities in the middle of the flume were obtained with an acoustic Doppler velocimeter (ADV).
  • The depth-averaged data calculated from vertical measured regions was taken to represent the entire depth at the corresponding horizontal coordinate due to the operation constraints of the ADV.
  • Show that the proposed model can effectively simulate turbulent flows over most of the gravel bed.

3.2 Turbulence of open-channel flow in a  90 bend

  • Furthermore, the performance of numerical models PF and RRF and other models presented by Cea et al. [11] was compared against turbulent flows in open channel with a  90 bend based on the experimental conditions described by Bonillo [40] .
  • There are varying degrees of deviation in the numerical curves of the other three models.
  • This may be because the basic assumption of the shallow water equations is difficult to satisfy in the strong shear and bend zone, which includes intense 3D turbulence.
  • The proposed model yielded accurate results overall despite some data scattering in the shear region.

3.3 Turbulent characteristics in an expanding section

  • An additional experiment was conducted to determine whether the proposed model can simulate the turbulent flows on an expanding channel [24] .
  • The experiment was carried out in a flat expanding flume at the Hydraulic Engineering Laboratory at National University of Singapore.
  • The sudden expansion in the profile of the sidewall induces strong non-uniformity to the velocity profiles, which was clearly observable per the flow separation and wake region in the detached flow.
  • By contrast, the maximum forward and backward velocities for both methods closely matched experimental data after the change in channel width, especially in the circulation region.
  • PF yielded slightly more accurate results than RRF overall.

4.1 Site description and numerical setup

  • The Yangliu moraine is located in a relatively straight gorge on the upper reach of the Yangtze River, approximately 1017.8 km upstream of Yichang City, a prefecture-level city in Hubei Province, China.
  • Another upstream moraine section of the main reach, the Huangjia moraine, is affected by a shorter lateral flow area with 300 m.
  • The transitional region, in which the flow is relatively slow, has a flat and straight geometric bed.
  • The local waterway bureau measured the bed topography of the reach, the water surface elevation on the shipping route, and the flow velocity in three streamlines on January 15 th , 2010, to investigate the effects of the two groins on the waterway (Fig. 10 ) and the consequent riverbed erosion.
  • The mixture of boulders and sand-cobbles typically forms a covering layer 2.3 m thick above the base layer composed by sandstones.

4.2 Results and performance

  • Figure 11 shows the comparisons between observed and simulated water levels along the ship route.
  • Especially in the transition section, both have discrepancies from the measured data.
  • Therefore, there is huge potential to greatly improve the results if 50 d was obtained at each specific site for the numerical calculations.
  • Overall, the calculated curves agree well with the experimental data except in the transition region of Streamline 2, where the maximum absolute deviation was about 0.3 m/s. Similar to the measured water surface elevations, the surveyed points of Streamline 2 were mainly distributed on the route.
  • As shown in Fig. 13 , the numerical values of RRF are about 2 times the PF results in the mainstream area and approximately 1.5 times PF's in the circulation zone behind the groin.

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Numerical and experimental analysis of shallow
turbulent flow over complex roughness beds
Item Type Article
Authors Zhang, Y.; Rubinato, M.; Kazemi, E.; Pu, Jaan H.; Huang, Y.; Lin, P.
Citation Zhang Y, Rubinato M, Kazemi E (2019) Numerical and
experimental analysis of shallow turbulent flow over complex
roughness beds. International Journal of Computational Fluid
Dynamics. Accepted for Publication.
Rights © 2019 Taylor & Francis. This is an Author's Original Manuscript
of an article published by Taylor & Francis in International
Journal of Computational Fluid Dynamics on 24 July 2019
available online at https://doi.org/10.1080/10618562.2019.1643845
Download date 10/08/2022 04:29:47
Link to Item http://hdl.handle.net/10454/17144

1
Numerical and Experimental Analysis of Shallow Turbulent Flow over 1
Complex Roughness Beds 2
Yong Zhang, State Key Laboratory of Hydroscience and Engineering, Tsinghua University, 3
Beijing 100084, China 4
Email: zhang.yong.sichuan@163.com
5
6
Matteo Rubinato, Faculty of Engineering, Environment & Computing, School of Energy, 7
Construction and Environment, Coventry University, Coventry, CV1 5FB, UK 8
Email: matteo.rubinato@coventry.ac.uk ORCiD orcid.org/0000-0002-8446-4448
9
10
Ehsan Kazemi, Department of Civil and Structural Engineering, The University of Sheffield, 11
Sheffield S1 3JD, UK 12
Email: e.kazemi@sheffield.ac.uk, ORCiD: orcid.org/0000-0002-1780-1846
13
14
Jaan H. Pu, Faculty of Engineering and Informatics, University of Bradford, Bradford, BD7 15
1DP, UK 16
Email: J.H.Pu1@bradford.ac.uk ORCiD: orcid.org/0000-0002-3944-8801
17
18
Yuefei Huang, State Key Laboratory of Hydroscience and Engineering, Tsinghua University, 19
Beijing 100084, China 20
Email: yuefeihuang@tsinghua.edu.cn
21
22
Pengzhi Lin, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan 23
University,
Chengdu 610065, China 24
Email: cvelinpz@scu.edu.cn (corresponding author)
25
26
ABSTRACT 27
A set of shallow-water equations (SWEs) based on a
ˆ
ˆ
k
Reynold stress model is established 28
to simulate the turbulent flows over a complex roughness bed. The fundamental equations are 29
discretized by the second-order finite-difference method (FDM), in which spatial and temporal 30
discretization are conducted by staggered-grid and leap-frog schemes, respectively. The 31
turbulent model in this study stems from the standard
ˆ
ˆ
k
model, but is enhanced by replacing 32
the conventional vertical production with a more rigorous and precise generation derived from 33
the energy spectrum and turbulence scales. To verify its effectiveness, the model is applied to 34

2
compute the turbulence in complex flow surroundings (including a rough bed) in an abrupt 35
bend and in a natural waterway. The comparison of the model results against experimental data 36
and other numerical results shows the robustness and accuracy of the present model in 37
describing hydrodynamic characteristics, especially turbulence features on the complex 38
roughness bottom. 39
Keywords: Energy spectrum, Roughness bed, SWE model, Shallow flows, Turbulent flows 40
41
1 Introduction 42
Large and small turbulent swirling flows are often clearly observable when dealing with 43
hydraulic structures in rivers and coastal areas and they are a key factor that influence the 44
frequency and magnitude of natural processes such as the sediment transport, mixing of 45
pollutants and/or riverbed deformation. Therefore, to better manage the rivers and design 46
reliable hydraulic structures, it is fundamental to understand these features and facilitate their 47
predictions. However, certain external factors (e.g. various and inconsistent boundary 48
conditions) make the characterizations of turbulent flows very challenging. To the author’s 49
knowledge, there is not yet a fully accurate, time-convenient, or general numerical model to 50
completely replicate the turbulent flows and their impacts over a natural roughness bed. Despite 51
that, effective simulations of the turbulent flow for some specific scenarios have been made due 52
to the rapid progress on the numerical modeling techniques and the computing powers. 53
54
The nature of turbulence is fundamentally three-dimensional (3D). Historically, 3D approaches 55
on the turbulence modeling mainly included the Direct Numerical Simulation (DNS), Large-56
Eddy Simulation (LES), and Reynolds-Averaged Navier-Stokes (RANS) modeling [1-4]. 57
Along the rivers and coastal regions, the flow domain is quite complex and spacious, and hence 58
to characterize the flow structures it would be excessively time-consuming to apply any of these 59
three approaches, which would require a large number of grid nodes in order to provide the 60
accurate results [5]. The two-dimensional (2D) Shallow Water Equations (SWE) coupled with 61
the benchmark turbulence closure model is much faster and enables the interpretation of 62
turbulent characteristics using a smaller vertical length scale (z) as compared with the two 63
horizontal ones (x and y) in those regions [6-9]. Furthermore, to obtain more accurate and 64
repeatable results, it is also critical to select the appropriate coefficients in these turbulence 65
closure equations. 66
67

3
2D shallow-turbulence flow models have been extensively developed over the last decades. 68
Most of the available ones are based on the Boussinesq approximations. For example, the depth-69
averaged eddy viscosity model suggests that eddy viscosity is the simple product of the bed 70
shear velocity and water depth [10] and the depth-averaged mixing length model accounts for 71
the influence of vertical turbulence [11]. Rastogi and Rodi [12] established the 2D standard 72
depth-averaged
ˆ
ˆ
k
turbulence model based on the 3D version described by Launder and 73
Spalding [13]. To widen the range of practical applications, some coefficients in the model 74
developed by Rastogi and Rodi were modified and new
ˆ
ˆ
k
models were introduced, as 75
presented in the other studies [14-18]. The standard
ˆ
ˆ
k
model has been demonstrated to 76
provide satisfactory results after numerous comparisons with the measured data [11,14] and it 77
is also relatively simple to use and very fast. Despite this, more progresses have been made in 78
this field and recently, Cea et al. [11] established a depth-averaged algebraic stress turbulence 79
model (DASM) to solve a single transport equation for each Reynolds stress without requiring 80
an isotropic assumption. 81
82
For the shallow turbulent flows on a roughness bed, the vertical velocity gradient distribution 83
is the main source of turbulence. The accuracy of its depth-averaged process directly determines 84
the numerical performance of the aforementioned
ˆ
ˆ
k
models. To date there is still area for 85
improvement due to the fact that preliminary results of the standard
ˆ
ˆ
k
model only partially 86
agree with the experimental data [12]. According to Rastogi and Rodi [12], this may be related 87
to the bottom shear stress, via the friction velocity, under the assumption of similarity in the 88
vertical velocity profiles. The depth-averaging process is a consideration only of the 89
macroscopic effects of the bottom roughness on the turbulent generations. In the classic 90
“cascade” theory of energy introduced by Richardson [19], the turbulent motion is a process of 91
energy transfer among various scales including not only the macroscale, but also various 92
microscales [2, 20, 21]. The contribution of these features should be rigorously and precisely 93
replicated within turbulent closure models. 94
95
The turbulent structure of various scales based on the framework of energy cascade can be 96
divided into the energy-containing, inertial, and dissipative regions, respectively. The kinetic 97
energy, produced in the energy-containing region, is considered to be transferred by inertial 98
forces to smaller scales until the energy is typically dissipated by the molecular viscosity [2]. 99
The energy spectrum in the inertial region has a universal statistical form, i.e. the Kolmogorov 100
-5/3 spectrum [22], and can be applied to larger or smaller wave numbers as the Reynolds 101
number increases [8]. Nezu and Nakagawa [20] analogized the energy transfer processes to 102

4
open-channel flows and divided the whole water depth into the three regions: wall, 103
intermediate, and free-surface zones. A universal log-law in the intermediate region was 104
verified by the extensive experimental and numerical results and they confirmed this can be 105
applied to a wider range of bed roughness and Reynolds numbers. 106
107
Taking into considerations all of these previous works and the new insights that have been 108
provided, this study aims to improve the performance of the standard
ˆ
ˆ
k
model to 109
characterise the generation of turbulent conditions at various scales over complex roughness 110
beds. The numerical SWE model utilised in this work includes a second-order leap-frog finite-111
difference method (FDM) and it is built into a staggered-grid system. This model was initially 112
developed by Cho [23] to calculate the evolution of the long waves and was further extended 113
by Lin [24] to simulate the turbulent structures within an experimental zone. Lately this 114
turbulence model has been widely used to compute the complex flows induced by irregular 115
geometries and the results obtained have proven that it is a robust numerical technique [25-27]. 116
In the present study, we emphasized the improvement of the model for application to the 117
complex roughness beds. 118
119
The paper is organized as follows: Section 2 presents the description of the mathematical and 120
numerical shallow turbulence model considered for this study, clarifying various assumptions 121
and hypothesis: the governing equations are presented in Section 2.1, then the vertical turbulent 122
production is formulated in Section 2.2 by incorporating the energy transfer information 123
between various scales into P
kV
and P
εV
based on the two universal semi-theoretical formulas 124
of Kolmogorov -5/3 scaling law and log-law. Section 3 includes the validation of the numerical 125
model against the experimental data collected on a flume, where varying roughness on the bed 126
was tested with a complex sharp bend. Finally, to further demonstrate the potentials of the 127
model, its performance was more vigorously examined in the transport of moraine along the 128
Yangtze River under dry seasons, and the results are explained in Section 4. Section 5 provides 129
a brief summary and concluding remarks of the whole study. 130
2 Numerical Model 131
2.1 Governing equations 132
This section presents the governing equations and the boundary conditions utilised in this study. 133
2.1.1 Shallow-water equations 134
The time-dependent SWEs are the fundamental hydrodynamic equations described in this 135

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TL;DR: In this paper, the authors present a review of the applicability and applicability of numerical predictions of turbulent flow, and advocate that computational economy, range of applicability, and physical realism are best served by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy k and its dissipation rate ϵ, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour.

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  • ...According to the boundary-layer theory, if the distance s between l and l + 1/2 is sufficiently small, the shear stress and the turbulent production at l can be balanced approximately with the wall shear stress and the dissipation at l + 1/2, respectively (Launder and Spalding 1974)....

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  • ...Rastogi and Rodi (1978) established the 2D standard depthaveraged k̂ − ε̂ turbulence model based on the 3D version described by Launder and Spalding (1974)....

    [...]

  • ...…ν + ν̂t σε ) ∂(Hε̂) ∂x ] + ∂ ∂y [( ν + ν̂t σε ) ∂(Hε̂) ∂y ] − Cε2H ε̂ 2 k̂ (5b) where Cμ, σk, σε,C1εandC2ε are the empirical constants and their values, as recommended by Launder and Spalding (1974), are Cμ = 0.09, σk = 1.0, σε = 1.3, C1ε = 1.44,C2ε = 1.92 (6) and, as suggested in (Launder…...

    [...]

  • ...…+ ∂(Pε̂) ∂x + ∂(Qε̂) ∂y = Cε1 ε̂ k̂ Hν̂t [ 2 ( ∂U ∂x )2 + 2 ( ∂V ∂y )2 + ( ∂U ∂y + ∂V ∂x )2] + PεV + ∂ ∂x [( ν + ν̂t σε ) ∂(Hε̂) ∂x ] + ∂ ∂y [( ν + ν̂t σε ) ∂(Hε̂) ∂y ] − Cε2H ε̂ 2 k̂ (5b) where Cμ, σk, σε,C1εandC2ε are the empirical constants and their values, as recommended by Launder and…...

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  • ...Considering the comparisons made by Pope (2000) and Nezu and Nakagawa (1993), the profile of 〈u〉 can be effectively approximated by Equation (11) over the whole depth except in a small region near the bed....

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  • ...Turbulence tends to be isotropy as the bed roughness andReynolds numbers increase (Nezu and Nakagawa 1993; Pope 2000; Abbaspour and Kia 2014)....

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  • ...The kinetic energy, produced in the energy-containing region, is considered to be transferred by inertial forces to smaller scales until the energy is typically dissipated by the molecular viscosity (Pope 2000)....

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  • ...The component τ/τw tends towards unity and l+m can be specified as l+m = κz+ in this region, so the solution to Equation (9) (the ‘log-law’), can be obtained as follows: 〈u〉+ = 1 κ ln z+ + A (10) where κ = 0.40 − 0.43 and B = 5.2 are the empirical constants (Pope 2000; Pu, Wei, and Huang 2017)....

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  • ...In the classic ‘cascade’ theory of energy introduced by Richardson (1922), the turbulent motion is a process of energy transfer among various scales including not only the macroscale, but also various microscales (Nezu and Nakagawa 1993; Pope 2000; Hunt et al. 2010)....

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Abstract: §1. We shall denote by uα ( P ) = uα ( x 1, x 2, x 3, t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1, x 2, x 3. In considering the turbulence it is natural to assume the components of the velocity uα ( P ) at every point P = ( x 1, x 2, x 3, t ) of the considered domain G of the four-dimensional space ( x 1, x 2, x 3, t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ 2 α and (d uα /d xβ )2― are finite and bounded in every bounded subdomain of the domain G .

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  • ...Based on Kolmogorov’s scaling theory outlined at (Kolmogorov 1941), the dissipation rate ε could be related to ||u′|| by using the macroscale of turbulence described in (Nezu and Nakagawa 1993; Hunt et al. 2010) as follows: ε = K ||u ′|...

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  • ...the Kolmogorov −5/3 spectrum (Kolmogorov 1941), and can be applied to larger or smaller wave numbers as the Reynolds number increases (Pu, Shao, and Huang 2014)....

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  • ...Based on Kolmogorov’s scaling theory outlined at (Kolmogorov 1941), the dissipation rate ε could be related to ||u′|| by using the macroscale of turbulence described in (Nezu and Nakagawa 1993; Hunt et al....

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  • ...To widen the range of practical applications, some coefficients in the model developed by Rastogi and Rodi weremodified and new k̂ − ε̂models were introduced, as presented in the other studies (Chen and Kim 1987; Booij 1989; Babarutsi and Chu 1991; Yakhot et al. 1992;Wu,Wang, and Chiba 2004)....

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Frequently Asked Questions (9)
Q1. What contributions have the authors mentioned in the paper "Numerical and experimental analysis of shallow turbulent flow over complex roughness beds" ?

The 31 turbulent model in this study stems from the standard ̂ ˆ k model, but is enhanced by replacing 32 the conventional vertical production with a more rigorous and precise generation derived from 33 the energy spectrum and turbulence scales. 

In the transitional region of a straight waterway, the 627water depth was at least 3.5 m and the width of the water surface was greater than 80 m. 

530 The horizontal bursts of turbulent activity propagated downstream and expanded on both sides, 531forcing the k̂ values towards uniformity. 

137 regular points of the intersection 482 were set between the longitudinal coordinates x = -0.10, 0.25, 0.50, 0.75,1.00, 1.25, 1.50, 1.75, 483 2.00, 2.25, 2.50, 2.75, and 3.00 m and the lateral coordinates y = 0.05, 0.10, 0.15, 0.20, 0.25, 484 0.30, 0.35, 0.40, 0.45, 0.50, and 0.55 m to observe velocity profiles and further inspect 485 turbulence changes. 

675Notation 676f = bed friction factor (-) 677 g = gravitational acceleration (ms-2) 678 H = water depth (m) 679 k = turbulent energy (m2s-2) 680 k̂ = depth-averaged turbulent energy (m2s-2) 681 sk = equivalent sand roughness (m) 682 ml = length scale of turbulent flow (m) 683 m = Manning’s roughness coefficient (sm-1/3) 684 QP, = unit volume flux in x - and y -directions, respectively (m2s-1) 685 R = hydraulic radius (m) 686 Re = Reynolds number (-) 687 yyxx TT , = depth-averaged normal stress in x - and y -directions, respectively (Pa) 688 xyyx TT , = depth-averaged shear stress in x - and y -directions, respectively (Pa) 689 *u = velocity scale of turbulent flow (ms-1) 690 *u = friction velocity (ms-1) 691 WVU ,, = depth-averaged velocity in x -, y - and z -directions, respectively (ms-1) 692wvu ,, = instantaneous velocity in x -, y - and z -directions, respectively (ms-1) 693 wvu ,, = ensemble-averaged velocity in x -, y - and z -directions, respectively (ms-1) 694 wvu ,, = fluctuating velocity in x -, y - and z -directions, respectively (ms-1) 695 wvu ,, = mean velocity in x -, y - and z -directions, respectively (ms-1) 696wvu ,, = turbulence intensity in x -, y - and z -directions, respectively (ms-1) 697 zyx ,, = streamwise, spanwise, and vertical coordinates, respectively (-) 698 bz = bed elevation (m) 699 = turbulent dissipation (m2s-3) 700 ̂ = depth-averaged turbulent dissipation (m2s-3) 701 = free surface elevation (m) 702 = turbulent viscosity (m2s-1) 703 k = kinematic viscosity (m2s-1) 704 t̂ = depth-averaged turbulent viscosity (m2s-1) 705 = fluid density (kgm-3) 706 = total shear stress (Pa) 707 b = bed shear stress (Pa) 708 709References 7101. 

This behaviour can be attributed to the anisotropic 438 tendency under which turbulent energy is redistributed over a smooth bed more slowly than 439 over a rough one, which may be enhanced as roughness size decreases. 

By using the proposed 2D 655 Shallow Water Equations with improved turbulence modelling techniques, it was possible to 656 achieve reasonably engineering accuracy but with a much lower CPU cost. 

As shown in 621 Fig. 13, the numerical values of RRF are about 2 times the PF results in the mainstream area 622 and approximately 1.5 times PF’s in the circulation zone behind the groin. 

the calculated curves agree well with the 610 experimental data except in the transition region of Streamline 2, where the maximum absolute 611 deviation was about 0.3 m/s. Similar to the measured water surface elevations, the surveyed 612 points of Streamline 2 were mainly distributed on the route.