Experimental and numerical investigation for energy dissipation of supercritical flow in sudden contractions
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Abstract:
Dealing with kinetic energy is one of the most important problems in hydraulic structures, and this energy can damage downstream structures. This study aims to study energy dissipation of supercritical water flow passing through a sudden contraction. The experiments were conducted on a sudden contraction with 15 cm width. A 30 cm wide flume was installed. The relative contraction ranged from 8.9 to 9.7, where relative contraction refers to the ratio of contraction width to initial flow depth. The Froude value in the investigation varied from 2 to 7. The contraction width of numerical simulation was 5~15 cm, the relative contraction was 8.9~12.42, and the Froude value ranged from 8.9~12.42. In order to simulate turbulence, the k-ε RNG model was harnessed. The experimental and numerical results demonstrate that the energy dissipation increases with the increase of Froude value. Also, with the sudden contraction, the rate of relative depreciation of energy is increased due to the increase in backwater profile and downstream flow depth. The experimentation verifies the numerical results with a correlation coefficient of 0.99 and the root mean square error is 0.02.
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Key words:
- Relative energy dissipation /
- Hydraulic jump /
- Sudden contraction
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Table 1. Discharge and Froude number used in the experiments
Num 10 9 8 7 6 5 4 3 2 1 Q (L/m) 750 700 650 600 550 500 450 400 350 300 Fr value 6.36 5.97 5.67 5.28 4.90 4.52 4.13 3.74 3.36 2.97 
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Table 2. Comparing experimental with numerical Froude value in turbulence models
Methods RNG k-ε k-ω R2 0.995 0.956 0.937 NRMSE 0.02 0.09 0.11
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Table 3. Range of measured variables
Contraction (cm) Discharge (L/s) Upstream depth (cm) Froude value Downstream depth (cm) Reynolds value 5 5.5~12.4 2.01~2.05 2.48~5.18 4.4~8.9 75 000~141 260 10 5.5~12.4 2.01~2.05 2.44~4.91 4.55~9.2 76 500~159 000 15 5.5~12.4 1.54~1.62 2.72~6.53 4.72~10.2 59 000~238 000
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Table 4. Details of the computational mesh
Size of cells X direction Mesh plane Y direction Mesh plane Z direction Mesh plane 0.005 (m) 1 2 1 2 1 2 1.00 3.80 0.00 0.30 0.00 0.40
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Table 5. Boundary conditions
X boundaries Y boundaries Z boundaries Min Max Min Max Min Max Volume Flow Rate (VFR) Outflow Wall Wall Wall Symmetry
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Table 6. Classical equations of energy and hydraulic jump
$ {E_0} = {y_0} + \frac{{V_\mathit{0}^2}}{{{2_g}}} = {y_0} + \frac{{{q^2}}}{{2gy_0^2}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(3 \right) $
$ {E_1} = {y_1} + \frac{{V_1^2}}{{{2_g}}} = {y_1} + \frac{{{q^2}}}{{2gy_1^2}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(4 \right) $
$ {y_1} = \frac{{{y_0}}}{2} = \left({ - 1 + \sqrt {1 + 8Fr_0^2} } \right)\;\;\;\;\;\;\;\;\;\left(5 \right) $
$ {y_{cr}} = \sqrt[3]{{\frac{{{q^2}}}{g}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(6 \right) $
$ {F_{cr}} = \frac{{{V_0}}}{{\sqrt {g{y_0}} }}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(7 \right) $
$ {E_r} = \frac{{{x_{Exp}} - {x_{Num}}}}{{{E_{Exp}}}} \times 100\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(8 \right) $
$ RMSE = \sqrt {\frac{1}{n}\sum {{{\left[ {{x_{Exp}} - {x_{Num}}} \right]}^2}} }\;\;\;\;\;\;\left(9 \right) $
$ NRMSE = \frac{{RMSE}}{{{x_{Max}} - {x_{Min}}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left({10} \right) $
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