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| | '''Abstract.''' The last decades there is a strong interest in predicting cavitation dynamics as it is a prerequisite in order to predict cavitation erosion. Industrial applications require accurate results in an acceptable time span and as a result there is a focus on large scale dynamics. In this paper the RANS equations are used to investigate the shedding frequency of sheet cavities in two-dimensional simulations. First a verification study is made for the NACA 0015 in 6 degrees angle of incidence. A grid sensitivity study is conducted in wetted flow and in steady (non-shedding) cavitating condition (σ=1.6). Then an investigation is conducted in order to capture the shedding frequency. The results show that only when a correction for turbulent viscosity at the cavity-water interface is used it was possible to capture the shedding frequency as found in other numerical studies. Furthermore, a validation study is conducted on a NACA66-312 α=0.8 for two different angles of attack. The obtained results are compared and validated with the experimental data from Leroux ''et al''. They indicate that the 2D shedding frequency predicted by the numerical simulations is in good agreement with the frequency obtained in the experiment. | | '''Abstract.''' The last decades there is a strong interest in predicting cavitation dynamics as it is a prerequisite in order to predict cavitation erosion. Industrial applications require accurate results in an acceptable time span and as a result there is a focus on large scale dynamics. In this paper the RANS equations are used to investigate the shedding frequency of sheet cavities in two-dimensional simulations. First a verification study is made for the NACA 0015 in 6 degrees angle of incidence. A grid sensitivity study is conducted in wetted flow and in steady (non-shedding) cavitating condition (σ=1.6). Then an investigation is conducted in order to capture the shedding frequency. The results show that only when a correction for turbulent viscosity at the cavity-water interface is used it was possible to capture the shedding frequency as found in other numerical studies. Furthermore, a validation study is conducted on a NACA66-312 α=0.8 for two different angles of attack. The obtained results are compared and validated with the experimental data from Leroux ''et al''. They indicate that the 2D shedding frequency predicted by the numerical simulations is in good agreement with the frequency obtained in the experiment. |
| − |
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| − | ==4 RESULTS==
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| − |
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| − | ===4.1 NACA 0015 ===
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| − |
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| − | The flow around the frequently used NACA 0015 hydrofoil is investigated in three conditions: wetted flow, steady cavitating flow (σ=1.6) and unsteady cavitating flow (σ=1.0). The results in wetted flow are compared with experimental data [8] and the ones in cavitating flow only with other numerical works [13-18].
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| − |
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| − | ====4.1.1 Wetted flow ====
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| − |
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| − | A grid sensitivity study is conducted in wetted flow for the drag (<span id='cite-_Ref476045028'></span>[[#_Ref476045028|Figure 3]]) and lift (<span id='cite-_Ref476045038'></span>[[#_Ref476045038|Figure 4]]) coefficients and the results are compared with experimental data available from the VIRTUE Workshop [8]. The results are shown in detail in <span id='cite-_Ref476045902'></span>[[#_Ref476045902|Table 3]]. An error estimation has been made by using an approach with power series expansion proposed by Eca and Hoekstra [9]. The expansions are fitted to the data in the least-squares sense.
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| − |
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| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | [[Image:draft_Melissaris_493103718-image3.png|600px]] </div>
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| − |
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| − | <span id='_Ref476045028'>'''Figure 3''': Drag coefficient for different grid density and comparison with the experimental value.</span>
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| − |
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| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | [[Image:draft_Melissaris_493103718-image4.png|600px]] </div>
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| − |
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| − | <span id='_Ref476045038'>'''Figure 4''': Lift coefficient for different grid density and comparison with the experimental value.</span>
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| − |
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| − | <span id='_Ref476045902'>'''Table 3''': Drag and Lift coefficient values for all the grids and comparison with experimental values (Δexp). The uncertainty (U<sub>φ</sub>) of the values is also shown.</span>
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| − |
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| − | {| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Grid</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">C<sub>D</sub></span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">U<sub>φ</sub></span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">% Δexp</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">C<sub>L</sub></span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">U<sub>φ</sub></span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">% Δexp</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Experiment</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''0.014'''</span>
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| − | | style="border: 1pt solid black;text-align: center;"|
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''0.658'''</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G1 (Coarse)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01459</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">8.47%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">4.20%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.67018</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.40%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.85%</span>
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| − | |-
| |
| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G2 (Medium)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01435</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3.48%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.47%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.66838</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.62%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.58%</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G3 (Fine)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01430</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2.36%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.16%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.66838</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.69%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.58%</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G4 (Very Fine)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01432</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3.05%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.28%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.66703</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.05%</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.37%</span>
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| − | |}
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| − |
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| − | The results show a good agreement with the experimental data and the assessment of the uncertainty shows that there is a small sensitivity of the drag coefficient to the grid density, however already with the medium mesh (G2) the solution is quite consistent.
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| − |
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| − | ====4.1.2 Steady cavitating flow ====
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| − |
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| − | In the steady cavitating condition a steady sheet cavity is expected that covers approximately the 20% of the foil. The time step and the number of inner iterations per time step (I<sub>N</sub>) are investigated in the coarse mesh by comparing the pressure distribution, the vapor volume fraction and the time history of the total vapor volume. A grid sensitivity study for the lift and drag coefficient is conducted for this condition as well (<span id='cite-_Ref476045834'></span>[[#_Ref476045834|Table 4]]).
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| − |
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| − | <span id='_Ref476045834'>'''Table 4''': Drag and Lift coefficient values in cavitating flow for all the grids and the computed uncertainty (U<sub>φ</sub>). </span>
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| − |
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| − | {| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Grid</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">C<sub>D</sub></span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">U<sub>φ</sub></span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">C<sub>L</sub></span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">U<sub>φ</sub></span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G1 (Coarse)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01916</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">15.67%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.6352</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.24%</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G2 (Medium)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01849</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">6.02%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.6361</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.68%</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G3 (Fine)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01852</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">6.00%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.6358</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.51%</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G4 (Very Fine)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.01838</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3.61%</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.6348</span>
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| − | | style="border: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.35%</span>
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| − | |}
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| − |
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| − | <span id='_Ref476045953'>'''Table 5''': Numerically obtained frequencies from various sources (6 deg angle of incidence and σ=1.0) [12].</span>
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| − |
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| − | {| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Author</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">V<sub>inlet</sub></span><span style="text-align: center; font-size: 75%;">(m/s)</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Shedding Frequency</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Koop [13]</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">≈24</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Sauer [14]</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">≈11</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Schnerr ''et al''. [15]</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
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| − |
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| − | <span style="text-align: center; font-size: 75%;">12</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">≈11.18 (incompressible)</span>
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| − |
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| − | <span style="text-align: center; font-size: 75%;">≈9 (compressible)</span>
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| − | |-
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Oprea [16]</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">≈14</span>
| |
| − | |-
| |
| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Hoekstra & Vaz [17]</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">≈15.4</span>
| |
| − | |-
| |
| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Ziru Li [18]</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
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| − | | style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">≈11.4</span>
| |
| − | |}
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| − |
| |
| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | [[Image:draft_Melissaris_493103718-image5.png|600px]] </div>
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| − |
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| − | <span id='_Ref476045132'>'''Figure 5''': Pressure distribution along the foil for different I<sub>N</sub>per time step (σ=1.6).</span>
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| − |
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| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | [[Image:draft_Melissaris_493103718-image6.png|600px]] </div>
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| − |
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| − | <span id='_Ref476045101'>'''Figure 6''': Vapor volume fraction along the foil for different time steps (σ=1.6).</span>
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| − |
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| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | [[Image:draft_Melissaris_493103718-image7.png|600px]] </div>
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| − |
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| − | '''Figure 7''': Total vapor volume for different time steps with 5 and 100 inner iterations per time step.
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| − |
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| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
| |
| − | [[Image:draft_Melissaris_493103718-image8.png|600px]] </div>
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| − |
| |
| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | [[Image:draft_Melissaris_493103718-image9.png|600px]] </div>
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| − |
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| − | <span id='_Ref476055924'>'''Figure '''8: Sheet cavity visualization obtained by the coarse (top) and the very fine mesh (bottom).</span>
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| − |
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| − | It is concluded that a time step according to a courant number even higher than one can be sufficient (see <span id='cite-_Ref476045101'></span>[[#_Ref476045101|Figure 6]]) and the number of inner iterations does not affect the development of the cavity although the vapor volume might not be fully converged in every time step (for instance despite the fact that using a time step corresponding to courant number 0.75 the total vapor volume is converged after 20 iterations per time step, the results between 5, 20 and 100 inner iterations are identical, <span id='cite-_Ref476045132'></span>[[#_Ref476045132|Figure 5]]). The uncertainty assessment is in line with the results in wetted flow showing dependence to the mesh (stronger this time) for the drag coefficient. In addition to that, a slight grid sensitivity regarding the shape at the trailing edge of the sheet cavity is observed (see <span id='cite-_Ref476055924'></span>[[#_Ref476055924|Figure 8]]).
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| − |
| |
| − | ====4.1.3 Unsteady cavitating flow ====
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| − |
| |
| − | An investigation on the impact of the number of inner iterations per time step is conducted first. Using 100 inner iterations and a time step corresponding to Courant number 0.75 the solution shows that a number of 40 inner iterations per time step is sufficient (<span id='cite-_Ref476045192'></span>[[#_Ref476045192|Figure 9]]). An unsteady periodic cycle is predicted giving a shedding frequency of about 3.6 Hz (<span id='cite-_Ref476045215'></span>[[#_Ref476045215|Figure 11]]). However, according to other numerical studies such a frequency seems to be very low (<span id='cite-_Ref476045953'></span>[[#_Ref476045953|Table 5]]). To this end the modification for the turbulent viscosity is applied. The results show that a higher frequency can be achieved with a second order temporal discretization scheme or with lower time step. Eventually a shedding frequency of about 13.6 Hz is computed, using 40 inner iterations, courant number 0.75 and a second order temporal discretization scheme (<span id='cite-_Ref476045215'></span>[[#_Ref476045215|Figure 11]]) together with Reboud’s correction for eddy viscosity.
| |
| − |
| |
| − | The instantaneous images of the volume fraction are shown in <span id='cite-_Ref476045998'></span>[[#_Ref476045998|Figure 10]]. First, as the cloud cavities from the previous cycle are moving downstream, a sheet cavity starts to grow at the leading edge in combination with some cavities growing at the trailing edge (steps 1-3). The re-entrant jet is formed and moves towards the leading edge as the bubbly cloud from the previous cycle collapses (steps 4-5). Then the sheet cavity starts to shed (step 6) and as it becomes smaller and smaller it continues shedding (steps 6-8) till it completely disappears (step 9). Finally all the shedding parts are combined into a bubbly cloud and move downstream to the trailing edge as the sheet cavity starts to grow again at the leading edge and the new cycle starts (steps 10-12).
| |
| − |
| |
| − | A grid sensitivity study for the shedding frequency has also been conducted. The results are shown in <span id='cite-_Ref476046030'></span>[[#_Ref476046030|Table 6]]. With every mesh a shedding frequency between 13 and 14 Hz has been computed. The high uncertainty of the solution can be explained by the unsteadiness and randomness of the shedding.
| |
| − |
| |
| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
| |
| − | {|
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| − | |-
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| − | | [[Image:draft_Melissaris_493103718-image10.png|324px]]
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| − | | [[Image:draft_Melissaris_493103718-image11.png|center|240px]]
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| − | |}
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| − | </div>
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| − |
| |
| − | <div id="_Ref476045192" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | <span style="text-align: center; font-size: 75%;">'''Figure 9''': Convergence of the residuals and the total vapor volume in every time step. </span></div>
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| − |
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| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | [[Image:draft_Melissaris_493103718-image12.png|600px]] </div>
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| − |
| |
| − | <div id="_Ref476045998" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
| |
| − | <span style="text-align: center; font-size: 75%;">'''Figure 10''': Total Vapor volume, drag and lift coefficient during a typical shedding cycle.</span></div>
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| − |
| |
| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | <span style="text-align: center; font-size: 75%;">'' [[Image:draft_Melissaris_493103718-image13.png|600px]] ''</span></div>
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| − |
| |
| − | <span id='_Ref476045215'>'''Figure 11''': Total vapor volume in time and frequency domain for NACA 0015 with (right) and without (left) Reboud’s correction for eddy viscosity.</span>
| |
| − |
| |
| − | ===4.2 NACA66 (mod.)-312===
| |
| − |
| |
| − | The NACA66 hydrofoil as described before is used to validate the computational set-up. The results are compared with experimental data obtained by Leroux ''et al''. for two different conditions. The experimental and the computational obtained data are shown in <span id='cite-_Ref476045499'></span>[[#_Ref476045499|Table 7]].
| |
| − |
| |
| − | <span id='_Ref476046030'>'''Table 6''': Grid sensitivity study on the shedding frequency and assessment of the uncertainty for each mesh for the NACA 0015.</span>
| |
| − |
| |
| − | {| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"
| |
| − | |-
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Grid density'''</span>
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| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Shedding (Hz)'''</span>
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| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''U<sub>φ</sub>'''</span>
| |
| − | |-
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G1 (Coarse)</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">13.59</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">15.40%</span>
| |
| − | |-
| |
| − | | <span style="text-align: center; font-size: 75%;">G2 (Medium)</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">13.92</span>
| |
| − | | style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">22.44%</span>
| |
| − | |-
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">G3 (Fine)</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">13.50</span>
| |
| − | | style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">12.32%</span>
| |
| − | |-
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">G4 (Very Fine)</span>
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">13.23</span>
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| − | | style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">7.39%</span>
| |
| − | |}
| |
| − |
| |
| − | <span id='_Ref476045499'>'''Table 7''': Experimental frequency and Strouhal number based on chord length as measured by Leroux ''et al''. V<sub>ref</sub>=5.33m/s, Re = 0.8 x 10<sup>6</sup>[10].</span>
| |
| − |
| |
| − | {| style="width: 100%;border-collapse: collapse;"
| |
| − | |-
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
| |
| − | | colspan='3' style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Experiment'''</span>
| |
| − | | colspan='3' style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''CFD'''</span>
| |
| − | |-
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''α'''</span>
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''f (Hz)'''</span>
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''σ'''</span>
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''st<sub>c</sub>'''</span>
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''f (Hz)'''</span>
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''σ'''</span>
| |
| − | | style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''st<sub>c</sub>'''</span>
| |
| − | |-
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''5.5'''</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">2.88</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.88</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.081</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
| |
| − | | style="border-top: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
| |
| − | |-
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''6.0'''</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3.50</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.99</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.098</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3.60</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.00</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.101</span>
| |
| − | |-
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''7.0'''</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4.50</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.13</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.127</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
| |
| − | | style="text-align: center;"|<span style="text-align: center; font-size: 75%;">-</span>
| |
| − | |-
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''8.0'''</span>
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">18.00</span>
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1.28</span>
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.507</span>
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">18.29</span>
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1.28</span>
| |
| − | | style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">0.515</span>
| |
| − | |}
| |
| − |
| |
| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
| |
| − | [[Image:draft_Melissaris_493103718-image14.png|600px]] </div>
| |
| − |
| |
| − | '''Figure 12''': Total vapor volume in time and frequency domain for 6 deg (top) and 8 deg (bottom) angle of incidence.'' ''
| |
| − |
| |
| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
| |
| − | [[Image:draft_Melissaris_493103718-image15.png|600px]] </div>
| |
| − |
| |
| − | <span id='_Ref476045296'>'''Figure 13''': Experimental-numerical comparison on the NACA66 for 6 deg angle of incidence. Experimental images are computed from an average of three instantaneous periods (Δt=1/50 s) and compared with computed instantaneous void fraction images with the same period.</span>
| |
| − |
| |
| − | <span id='_GoBack'></span>The results show that in both cases a frequency similar to this in the experiment is obtained. There is a difference less than 3% for the low frequency case and 2% for the high frequency case. A comparison between the computations and the experimental data of the foil in 6 deg angle of incidence is shown in Fig. 13. As illustrated by Leroux ''et al''. two steps can be identified during a typical shedding cycle: The first step consists of the growth of the sheet cavity (<span id='cite-_Ref476045296'></span>[[#_Ref476045296|Figure 13]] a-e) till it is slowed down and counterbalanced by the shedding of vapor structures (secondary clouds) in the wake (<span id='cite-_Ref476045296'></span>[[#_Ref476045296|Figure 13]] f-i).After the shedding of secondary clouds, the detachment of a large vapor cloud (main cloud) occurs (<span id='cite-_Ref476045296'></span>[[#_Ref476045296|Figure 13]] j). It is followed by the roll-up and convection of the main cloud (Fig. 13 k) together with the growth of the residual cavity. The second step occurs just after the cavity break-off. Indeed, the growth of the residual cavity is abruptly stopped at nearly the sametime the main cloud of vapor collapses (<span id='cite-_Ref476045296'></span>[[#_Ref476045296|Figure 13]] l), and the residual cavity almost entirely disappears (<span id='cite-_Ref476045296'></span>[[#_Ref476045296|Figure 13]] m). Then the cavity starts to grow again.
| |
| − |
| |
| − | Similar cavitation dynamics are calculated by the simulations. The growth of the cavity and the secondary clouds are captured as well as the detachment of the large vapor cloud and the sudden vanishing of the cavity after the collapse of the cloud. Discrepancies can only be observed on the growth of the residual cavity, where a larger expansion of the residual cavity is predicted in the computations (the same behaviour was also predicted in the computations by Leroux ''et al'').
| |
| | | | |
| | ==5 CONCLUSIONS AND RECOMMENDATIONS== | | ==5 CONCLUSIONS AND RECOMMENDATIONS== |
In this study an attempt was made to verify the incompressible RANS solver in StarCCM+ in cavitating flow. Despite the three-dimensional character of cavitation dynamics a first investigation was conducted on the grid and numerical (time step, inner iterations etc.) sensitivity with the intension to predict the shedding frequency using two-dimensional domain. For the current computational set-up and the tested conditions the following conclusions are drawn:
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J.-B. Leroux, J. A. Astolfi and J. Y. Billard, "An Experimental Study of Unsteady Partial Cavitation," ASME J. Fluids Eng., vol. 126, pp. 94-101, (2004).
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J. Sauer, "Instationär kavitierende strömungen - Ein neues modell, basierend auf fron capturing (VoF) und blasendynamik," Ph.D. Thesis, Karlsruhe University, Karlsruhe, Germany, (2000).
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G. H. Schnerr, S. J. Schmidt, I. H. Sezal and M. Thalhamer, "Shock and Wave Dynamics of Compressible Liquid Flows With Special Emphasis on Unsteady Load on Hydrofoils and Cavitation in Injection Nozzles," in Proceedings of 6th International Symposium on Cavitation, Wageningen, The Netherlands, (2006).
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M. Hoekstra and G. Vaz, "FreSCo Exercises for NACA0015 Foil," VIRTUE WP4 Workshop, Contribution from MARIN, (2008).
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Z. Li, Assessment of Cavitation Erosion with Multiphase Reynolds-Averaged Navier-Stokes Method. PhD Thesis, Technische Universiteit Delft, (2012).
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