MEAN FLOW AND TURBULENT STATISTICS OF THE FLOW OVER A SUBMERGED SHARP-CRESTED WEIR.
Chaewoong Ban, Ph.D Candidate, Department of Civil & Environmental engineering, Yonsei University, Seoul, Korea
Sung-Uk Choi Professor, Department of Civil & Environmental engineering, Yonsei University, Seoul, Korea

1. Introduction

2. GOVERNING EQUATIONS§ Flow over a sharp-crested weir shows the free flow, where the critical flow occurs
at the tip of the weir. However, as the discharge increases, the downstream waterlevel affects the overflow and the flow over the weir becomes the submerged flow
§ Wu and Rajaratnam (1996) divided the submerged flow into impinging jet andsurface flow regimes. Surface flow regimes can further be classified into the surface
jump (or breaking wave), surface wave, and surface jet.

§ Filtered Navier-Stokes equations
¶ui=0
¶xir

§ Figures 5(a) and 5(b) show the toe of jump and trough of surface wave and Figure5(c) and 5(d) show the dimensions of the first wave. In the figure, vertical line
means the boundary of threshold between the flow regimes.
§ Positions of the toe of jump and trough of surface wave are well predicted by theLES.

¶ui
¶u¶p¶
+ ru j i = +t ij + t ijsgs + r gi¶t¶x j¶xi ¶x j()§ Subgird (SGS) stress tensor
13§ The SGS stress tensor is modeled by the wall-adapting local eddy viscosity (WALE)
model, proposed by Nicoud & Ducros (1999).

()

t ijsgs = r ui u j - ui u j = 2 rn t Sii + d ijt kksgs

Figure 5. Position of the toe of jump/trough of surface wave (xw/h, yw/h) and dimensions of
the first wave (a/h, b/h).

3. Numerical framework

Figure 1. Classification of submerged flow regimes.

§ Wu and Rajaratnam (1996) proposed a criterion to distinguish the impinging jet andthe surface flow. The following relationships present the transition region from
impinging jet to surface flow:t / h = 1 - 0.215l + 0.0142l 2 - 0.00031l 3 (for lower curve)t / h = 1 - 0.126l + 0.0076l 2 - 0.00017l 3 (for upper curve)l = g ( h - t ) / ( q / y t ) , g = gravity, h = head above the crest of the weir,§ The computations, three computational
grids were used, namely coarse,
intermediate, and fine grids for thesurface jet case, and fine grids for theother cases. For the fine grid resolution,
the maximum grid spacing is Δ = 5 x 10-3
m. In the near-wall region, the grid size is
Δ = 2.5 x 10-3 m with a spatial average ofΔ+≅ 13. The number of grid cells is
about 10.8 x 106 for the fine grid.

Figure 3. Computational domains:(a) 3D view; (b) side view.

§ The inflow condition comes from a preliminary LES computation in a periodic
channel of an identical cross-section. The velocity fields did not include
fluctuations, but time-averaged velocity fields were fed into the inflow.
§ For the free surface tracking, volume of fluid (VOF) method is used. Interface
corresponding α = 0.5 is considered as free surface.

t = depth of tailwater above the weir crest, q = unit discharge, yt = tailwater depth§ which are the lower and upper bounds of the transition, respectively. The region
between the upper curve and lower curve indicates the transition stage, where theflow regime is affected by the history of the flow, so-called the hysteresis effect.
§ Azimi et al. (2016) also presented classification of submerged flows for 0.146 < h/P< 0.389 based on t/h, where P = weir height. That is, impinging jet, surface jump,surface wave, and surface jet occur in the respective ranges of t/h < 0.2, 0.2 < t/h< 0.48, 0.48 < t/h < 0.78, and t/h < 0.78. The boundary between the surface wave
and surface jet is defined when the amplitude of the wave becomes less than 10% of
the approach flow head.

§ Hydraulic conditions for the surface jet are from the experiments by Rajaratnamand Muralidhar (1969). We made up the other three flow regimes satisfying thethreshold proposed by both Wu and Rajaratnam (1996) and Azimi et al. (2016).
Table 1. Hydraulic conditions.
Flow

∆
/λ



regime(m) (m) (m) (m) (m)(m2/s) (m/s)
SurfaceA
0.457 0.302 0.141 0.136 0.005 0.965 2.955 0.033 0.0741 0.061 11,170
jetCaseBCD

Surface
0.457 0.302 0.075 0.052 0.023 0.693 5.142 0.033 0.0867 0.073 12,340
wave
Surface
0.457 0.302 0.071 0.028 0.043 0.394 6.554 0.033 0.0877 0.074 12,422
jumpImpinging
0.457 0.302 0.067 0.010 0.057 0.149 7.135 0.033 0.0886 0.075 12,505
jet§ This research aims to study the flow characteristics of submerged flows over sharpcrested weirs in various flow regimes. The flows downstream of the weir are
featured by highly 3D unsteady nature shed from the weir, requiring highlyresolved large eddy simulation (LES) to demonstrate the detailed flow structures.
This study presents results of the LES of four different flow regimes to investigate
the mean flow and turbulent structures.

§ The velocity profiles showthat the flow accelerates bypassing over the weir andthe backward flows occur
near the bed.
§ Theoverall
agreement
betweenthesimulated
velocity
profilesandmeasured data is good.
§ Threedifferent
grid
resolutions are tested andFigure 4 confirms that thefine-mesh grid is accurateenough to converge thesolution.

Figure 6. Contours of time-averaged streamwise velocity with streamlines:(a) Case A; (b) Case B; (c) Case C; (d) Case D.

5. Conclusions

4. Results

Figure 2. Boundaries of impinging jet andsurface flow regimes.

§ Figure 6 present contours of time-averaged streamwise velocity with streamlines
for the four different flow regimes. The LES results show that the four flow
regimes show distinctive characteristics of the recirculation zones downstream of
the weir and the free surface.
§ Figure 6(a), a jet is made at the tip of the weir, forming a large primary vortex
with a small weak vortex at the lower corner of the re-circulation zone. The
contour plot indicates that reattachment takes place at x/P = 7.6.
§ As t/h decreases, the surface wave is generated as can be seen in Figure 6(b). The
jet flow becomes more intensive with the flow depth reduced over the tip of theweir. This resulted in stronger backward flows, increasing the size of the weak
vortex at the lower corner.
§ Figure 6(c) shows the surface jump, which is made as t/h decreases from thesurface wave. The generation of the surface jump is confirmed by the presence ofthe surface vortex. Strong backward flows are observed in the vicinity of the bed.
§ If t/h further decreases, then the impinging jet is generated. The overflow isheading the bed, with counter-rotating vortices formed about the overflow. The
size of the weak vortex at the lower corner is observed to be larger than that of theweak vortices made in the other three flow regimes.

§ Large eddy simulation was carried out for turbulent flows over a submerged weir.
Four different overflows, namely surface jet, surface wave, surface jump, andimpinging jet, were simulated.
§ The LES results were compared with the measured data by Rajaratnam &Muralidhar (1969), showing good agreement.
§ Distinctive characteristics of the recirculation zones and the free surface werefound and discussed.

References§§

Figure 4. Time-averaged streamwise velocity profiles
along the streamwise direction.

§§

Azimi, A. H., Rajaratnam, N., and Zhu, D. Z. (2016). Water surface characteristics of submerged rectangular sharpcrested weirs. Journal of Hydraulic Engineering, 142(5), 06016001.
Nicoud, F., and Ducros, F. (1999). Subgrid-scale stress modelling based on the square of the velocity gradienttensor. Flow, Turbulence and Combustion, 62(3), 183-200.
Rajaratnam, N., and Muralidhar, D. (1969). Flow below deeply submerged rectangular weirs. Journal of HydraulicResearch, 7(3), 355-374.
Wu, S., and Rajaratnam, N. (1996). Submerged flow regimes of rectangular sharp-crested weirs. Journal ofHydraulic Engineering, 122(7), 412-414.