NUMERICAL SIMULATION OF FREE OVERFALL IN A ROUGH
CHANNEL
Yakun Guo1*, Lixiang Zhang2 and Jisheng Zhang3
1Department of Engineering, University of Aberdeen, Aberdeen, AB24 3UE, UK. Tel: ++44 1224
272936, Fax: ++44 1224 272497, Email: y.guo@abdn.ac.uk.
2Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming
650051, Yunnan, China.
3 Department of Engineering, University of Aberdeen, Aberdeen, AB24 3UE, UK.
Key words: Turbulent Model, Volume of Fluid Technique, Free Overfall, Rough Channel Abstract. This paper reports the results from a turbulent numerical modelling study on the free overfall in a rectangular channel. A wide range of model parameters (i.e. discrete square bar bed roughness, channel slope, and incoming upstream Froude number) is investigated. The water surface profiles, velocity fields and end-depth are simulated and compared with experimental results for various input conditions. The influence of the bed slope, separation of the bar roughness and upstream flow Froude number on the flow structure and free overfall is discussed. The computational results agree well with the experimental measurements.
1 INTRODUCTION
Since the pioneering work of Rouse1, free overfall has been studied extensively for the last several decades, primarily through the laboratory experiments and analytical and theoretical study. However, the problem still provides great challenges to engineers and researchers, as evidenced by the continuous heavy publications. On the other hand, the practical reason for such study is that free overfall can be used as a simple flow measuring device since the flow discharge per unit width q in a rectangular channel can be estimated by q=[ghe3(hc/he)3]1/2 if
the end water depth ratio (EDR=hc/he) can be determined, where g=gravity acceleration, hc=
critical water depth, and he= water depth at the brink of channel which can be easily
conditions, many studies have been carried out on smooth channel2-7 and on rough channel8-14. More studies on free overfall for different channel cross section can be found in a review article by Dey15.
Though these studies demonstrated some features of the effects of bed roughness on EDR, references to turbulent numerical modelling studies for free overfall over a rough channel and studies of the effects of separation distance of discrete bar roughness on the free overfall are still lacking. For example, the work by Guo14 applied Darcy-Weisbach equation to estimate the energy loss due to the roughness that was converted into Manning’s coefficient using the formula given by Rajaratnam et al. 10. Obviously this method is not valid for treating the discrete bar roughness with different separation spacing (λ, see Figure 1). Furthermore the approach proposed there is potential flow theory and cannot be applied to treat the turbulent flow. In this study, instead of considering the bed as rough, we treat every single discrete roughness element as a part of the solid boundary of the flow domain in the numerical model. Therefore such approach can automatically account for the effects of discrete bar roughness on the free overfall as either pure frictional resistance16, 17 or geometrical aspect10 or both. Using this approach, we seek to improve the understanding of the effects of the bar roughness on free overfall. Given that the natural rivers and man-made channels are generally rough, the study has immediate practical importance and engineering application.
λ
Figure 1 Sketch of the arrangement of discrete bar roughness 2 NUMERICAL MODEL
2.1 Governing equations
The flow in question can be described by 2D incompressible Reynolds-averaged Navier-Stokes (RANS) and continuity equations, which can be written as following:
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ∂ ∂ − = ∂ ∂ + ∂ ∂ + ∂ ∂ z u z x u x x p z wu x uu t u e e µ µ ρ ρ ρ ) ( ) ( ) ( (2) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ∂ ∂ − − = ∂ ∂ + ∂ ∂ + ∂ ∂ z w z x w x z p g z ww x uw t w e e µ µ ρ ρ ρ ρ ) ( ) ( ) ( (3) To accurately predict the free surface volume of fluid (VOF) technique is used in the simulation18. The time-averaged volume fraction equation is:
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ Φ ∂ ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ Φ ∂ ∂ ∂ = ∂ Φ ∂ + ∂ Φ ∂ + ∂ Φ ∂ Φ Φ x z z x z w x u t e e σ µ σ µ ρ ρ ρ ) ( ) ( ) ( (4) where ρ =Φρ1 +(1−Φ)ρ2 , µ =Φµ1+(1−Φ)µ2 , and = the horizontal and vertical Cartesian co-ordinates, u and = the velocity components in the
x z
w x and − z−direction, Φ = the volume fraction of first phase, g= acceleration of gravity, p= pressure, ρ1, ρ2 and ρ = the first phase (air), second phase (water) and mixture fluid density, µ1, µ2 and µ = the first phase, second phase and mixture fluid dynamic viscosity, µe = the effective dynamic viscosity defined by the k−ε turbulence model as following
ε ρ µ
µe = + cµ k2 (5)
In which k = turbulence energy, ε= turbulence energy dissipation rate and can be described by the standard transport equations:
ρε σ µ σ µ ρ ρ ρ − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ P z k z x k x z wk x uk t k k e k e ) ( ) ( ) ( (6) k c P k c z z x x z w x u t e e 2 2 1 ) ( ) ( ) ( ε ε ρ ε σ µ ε σ µ ε ρ ε ρ ρε ε ε ε ε − + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ (7) where = the turbulence production term and can be written in tensor notation, P
j i i j j i e x u x u x u P ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ =µ (8)
where , 2 corresponding to the , co-ordinates. The values of the turbulence constants used here are
1 = i x z 19: =0.09, µ c σk =1.0, 3σε =1. , σΦ =1.0, 44c1ε =1. and c2ε =1.92.
2.2 Boundary conditions
At the inflow boundary, the velocity, turbulent energy, energy dissipation and free surface displacement are specified according to laboratory measurements20, while the outflow boundary is set as free. On the free water surface the zero pressure condition is applied. When the wind effect is absent the continuity of stress condition gives rise to the zero shear and normal stresses on the free surface. The wall shear stress is calculated by using the standard wall function. On all solid boundaries (comprising the bed and all elements of bar roughness) the non-slip boundary condition is applied.
3 RESULTS AND DISCUSSION
3.1 Effects of roughness on the free water profile
Effects of the bar roughness on the free water surface upstream of the channel brink are studied for a broad range of model input parameters. The results show, in general, that the free water profile for rough bed is higher (having greater water depth) than that of smooth bed for otherwise identical input conditions. This water depth difference becomes insignificant as the spacing of the bar roughness increases. Figure 2 is an example of simulated water free surface (solid line) comparing with the experimental data (close circles, Guo, et al.20) for Fr=0.57, bed slope s= 1/400, and λ/d=λ/w=6. It is seen that the numerical simulation results are in
good agreement with the laboratory experimental results, though some small discrepancy can be seen immediately upstream of the channel brink, which may be ascribed to the turning of the flow pattern from subcritical to supercritical near the brink.
3.2 Effects of roughness on the flow structure
Figures 3 and 4 are the comparison of simulated flow field with the PIV measurements21 for λ/d=λ/w=3 and 10, respectively. It is seen that eddies are generated in the pitch between
two bar elements. In general the simulated flow fields agree well with the experimental measurements. It is seen in Fig 3a and b (λ/d=λ/w=3, the so-called d-type roughness22 that a pair eddies are formed in the cavity between two bars. The large clockwise eddy generated has the size of the cavity, which makes the downstream net flow rate from the cavities negligible. On the other hand, the vertical velocity profile in cavities is altered by the presence of the eddy in which there is a slip velocity (see Fig 5) at the level of the height of the bar elements, which may favor the flow discharge. In general, the results of this study show that the flow discharge for rough bed is smaller than that of the smooth bed for the same upstream flow depth and bed slope. Fig 4a and b show that a single large eddy is formed immediately behind the upstream bar. Downstream of this eddy the flow reattaches the bed and the flow pattern is similar to that in the smooth bed. For this configuration (large λ/d) the downstream
flow discharge from the cavities is not trivial, thus reducing the effects of the bar roughness on the flow.
(a) Simulation
(b) Measurement
Figure 3. Comparison of simulated (a) and measured (b) flow velocity in the cavity between two bars for
Fr=0.39, s= 1/400, λ/d=λ/w=3. Both simulated and measured results show a pair of eddies formed in the
(a) Simulation
(b) Measurement
Figure 4. Comparison of simulated (a) and measured (b) flow streams in the cavity for Fr=0.50, s= 1/400, λ/d=λ/w=10. The flow reattachment is seen to take place downstream of the eddy. Unit is mm.
Figs 5a and b are the vertical streamwise velocity profiles deduced from Figs 3 and 4, respectively. The velocity profiles are taken through the center of the large eddies. It is seen from figs 5a, b that there exists an adverse velocity near the bed, corresponding to the lower part of the large clockwise eddy. Comparing Fig 5a and b indicates that the eddy generated for small spacing (λ/d=3) is stronger than that formed in large spacing (λ/d=10). The results
(a)
(b)
3.3 Effects of bed slope and roughness on EDR
Figure 6 is the plot of EDR versus bed slope for various spacing and Froude number. In general, the EDR decreases with the increase of bed slope for otherwise identical input model parameters, as found in previous studies8, 9,12. It is seen that for a given dimension of bed roughness the relative spacing of roughness (λ/d) has a significant effect on EDR with the
decrease of λ/d the EDR increases. The influence of the bar roughness on EDR is negligible
when λ/d ≥18 in this study. The results also show that the numerical simulated results
favorably compare with the measurements with the relative maximum error being around
10%. 0.6 0.7 0.8 he/hc -0.002 0 0.002 0.004 0.006 0.008 0.01 S
Figure 6. Effects of bed roughness and slope on EDR. Symbols are experiments: λ/d=λ/w=0 ○ (smooth bed), 3 ●, 6 □, 9 ▲, 18 ∆; lines are simulated results: λ/d=λ/w=0 dash and dot line, 3 dash line and 6 solid line.
4 CONCLUSION
A combined numerical and laboratory experimental model study is proposed to evaluate the effects of bar roughness on the free overfall. The water free surface profile, flow velocity structure and EDR are measured and simulated for both smooth and rough beds with a broad range of bed slope and upstream incoming flow Froude number. The results indicate that the sparing of bar elements (λ/d) has a significant influence on EDR, whose value increases with
the decrease of λ/d in this study. Eddies are generated in the cavities between two bars. For
scale of the cavity, thus reduce the flow discharge conveyed in the channel through occupying the space and conveying small downstream net flow. Such ‘geometric’ effect becomes negligible as the spacing between two bar elements (λ/d) increases to a certain value for the
range of parameters investigated in this study.
Acknowledgements: The writers would like to thank Dr L Campbell for providing the PIV measurements for the comparison.
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