Numerical and experimental analysis of shallow turbulent flow over complex roughness beds
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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References
11,866 citations
"Numerical and experimental analysis..." refers background or methods in this paper
...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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6,495 citations
"Numerical and experimental analysis..." refers background or methods in this paper
...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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6,063 citations
"Numerical and experimental analysis..." refers methods in this paper
...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)....
[...]
...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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...The energy spectrum in the inertial region has a universal statistical form, i.e. 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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2,347 citations
"Numerical and experimental analysis..." refers methods in this paper
...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)
Q2. How many groins were surveyed in the transitional region of a straight water?
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.
Q3. What is the effect of the horizontal bursts of turbulent activity on the k?
530 The horizontal bursts of turbulent activity propagated downstream and expanded on both sides, 531forcing the k̂ values towards uniformity.
Q4. How many regular points were set between the longitudinal coordinates x = -0.10,?
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.
Q5. what is the turbulence intensity in x -, y - and?
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.
Q6. What is the effect of the anisotropic 438 tendency?
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.
Q7. What was the cost of the proposed 2D 655 Shallow Water Equations?
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.
Q8. How many times the PF results in the circulation zone?
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.
Q9. How many points were surveyed in Streamline 2?
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.