Verication and validation of the sheet cavitation model
5.1 Steady sheet cavitation model
5.1.1 NACA66 2 -415 hydrofoil
This section investigates the flow around the NACA662-415 hydrofoil. The flow around the NACA662-415 foil was studied in detail by the Potsdam Model Basin (SVA) and their results were summed up in the report by Richter (2011). The geometry data of this hydrofoil is listed in Table 5.1 and its shape is illustrated in Figure 5.1. The foil has a chord length of 0.2025m and a span width of 0.6m.
Characteristics Notation Value Unit
Span length b 0.6 m
Chord length c 0.2025 m
Foil area Aref 0.1215 m2
Maximum thickness tmax 0.0304 m Aspect ratio Λ=b2/A 2.963
-Scale factor λ 2.5
-Table 5.1: NACA662-415 geometry data (in model scale)
−0.01 0 0.01 0.02
0 0.2 0.4 0.6 0.8 1
x/c[-]
y[m]
Figure 5.1: Scaled geometry of the NACA662-415 hydrofoil prole
In the first part of this section the non-cavitating flow around the NACA662-415 foil with different grid resolutions of the hydrofoil’s surface is simulated. The aim of this study is to detect grid resolutions that provide convergent and grid-independent results for the lift coefficients of the hydrofoil flow. For this purpose the lift coefficients calculated by panMARE with different grid discretisation levels are evaluated and validated against measured data from the report by Richter (2011).
The second part of this section focuses on the simulation of the cavitating flow around the NACA662-415 foil. Analogous to the non-cavitating case, different grid resolutions are used in this study to analyse their influence on the calculated sheet cavity shapes. For the verification and validation of the results obtained by the numerical sheet cavitation model, the scaled pressure distributions and sheet cavity extents are examined and the latter ones are compared with the measurements by Richter (2011). Additionally, the lift
Chapter 5. Verication and validation of the sheet cavitation model and drag coefficients are computed for several cavitation numbers and the influence of the cavitation amount on the hydrofoil’s flow characteristics is emphasised.
Non-cavitating NACA662-415 hydrofoil ow
In the present study the non-cavitating flow around the NACA662-415 foil is simulated for the angles of attackα+α0 =−5◦-10◦, whereαis the geometrical angle of attack and α0 is the zero-lift angle of attack. The zero-lift angle of attack calculated by panMARE is α0 = −3.34◦. All relevant input data of the simulations is summarised in Table 5.2.
As mentioned above, different grid resolutions are used in the calculations (s. Table 5.3).
The surface is discretised in the circumferential and in the span-wise direction. Figure 5.2 shows two example grid resolutions: a coarse grid (s. Figure 5.2a) and a fine grid (s.
Figure 5.2b).
Characteristics Notation Value Unit Angle of attack α −5- 10 ◦ Inow velocity Vinf low 5.5 m/s Water density ρ 997.46 kg/m3
Table 5.2: Input data for the non-cavitating case study (NACA662-415 hydrofoil)
Characteristics Notation Value
Number of panels in span-wise direction Nk 4/7/10/13/16 Number of panels in circumferential
direc-tion Nl 10/18/26/34
Table 5.3: Grid resolutions for the non-cavitating case study (NACA662-415 hydro-foil)
(a)Nk×Nl= 4×10 (b)Nk×Nl= 16×34
Figure 5.2: Grid resolution examples
Chapter 5. Verication and validation of the sheet cavitation model
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 α+α0[◦]
Cl[−]
panMARE,Nk= 7,Nl= 10 panMARE,Nk= 7,Nl= 18 panMARE,Nk= 7,Nl= 26 panMARE,Nk= 7,Nl= 34 measurements (taken from Richter (2011))
Figure 5.3: Inuence of the grid resolution in the circumferential direction on the calculated lift coecients of the NACA662-415 hydrofoil
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 α+α0[◦]
Cl[−]
panMARE,Nk= 4,Nl= 18 panMARE,Nk= 7,Nl= 18 panMARE,Nk= 10,Nl= 18 panMARE,Nk= 13,Nl= 18 measurements (taken from Richter (2011))
Figure 5.4: Inuence of the grid resolution in the span-wise direction on the calcu-lated lift coecients of the NACA662-415 hydrofoil
Figures 5.3-5.4 and D.1-D.6 show the measured and simulated results for the dimension-less lift coefficient Cl as functions of the angle of attack. The lift coefficients calcu-lated bypanMARE coincide very well with the measured results for the angles of attack α+α0 ∈ [−5◦ : 6◦]. For the angles within the rangeα+α0 ∈ [7◦ : 10◦]the calculated
Chapter 5. Verication and validation of the sheet cavitation model values are overestimated compared to the measured ones. This deviation can be explained by the friction effects that play an important role for high or low angles of attack. Friction effects cannot be captured by a potential flow solver in an accurate manner and must be included in the numerical method via semi-empirical formulas.
The following section concerns the discussion of the influence of the grid discretisation level on the calculated lift coefficients. Figures 5.3 and D.1-D.3 emphasise the influence of the grid refinement in the circumferential direction and Figures 5.4 and D.4-D.6 illus-trate the impact of the mesh refinement in the span-wise direction. The grid resolution has a relatively small influence on the calculated lift coefficients. Only the results calcu-lated with very coarse grids, i.e. Nl = 10andNk = 4, deviate from the results obtained with the finer grids. By increasing the number of panels, the calculated lift coefficients converge to the measurements forα+α0 ∈[−5◦ : 6◦].
Cavitating NACA662-415 hydrofoil ow
In the present study the cavitating flow around the NACA662-415 foil is simulated for dif-ferent inflow and cavitation conditions. The input data of the simulations is summarised in Table 5.4. Altogether two simulation cases with different angles of attack are investi-gated. In the simulations the angles of attack are adjusted in such way that they are equal to the lift coefficients measured in experiments under non-cavitating conditions. The anal-ysed grids are presented in Table 5.5. Equivalent to the non-cavitating case study, the grid resolution is varied in the circumferential and span-wise direction. For sheet cavitation calculations a finer grid resolution is used than in the non-cavitating case since an accurate pressure distribution is important for the detection of sheet cavitation.
Characteristics Notation Case 1 Case 2 Unit Lift coecient Clnoncav 0.84 1.04
-Angle of attack α+α0 14.4 18 ◦
Inow velocity Vinf low 5.5 5.5 m/s
Water density ρ 997.46 997.46 kg/m3 Cavitation number σv 1.5/ 2.0 3.0
-Table 5.4: Input data for the cavitating case study (NACA662-415 hydrofoil)
Characteristics Notation Value
Number of panels in span-wise
direc-tion Nk 10/13/16
Number of panels in circumferential
direction Nl 24/30/36/42/48/54/60/66
Table 5.5: Grid resolutions for the cavitating case study (NACA662-415 hydrofoil)
Chapter 5. Verication and validation of the sheet cavitation model
(a)σv= 1.5 (b)σv= 2.0
Figure 5.5: Sheet cavitation extents measured in experiment for the simulation case 1 with air content of 80% (taken from Richter (2011, p. 4.9))
-1
Figure 5.6: Scaled pressure coecients and sheet cavitation extents calculated by panMARE with dierent grid resolutions for the simulation case 1 with σv = 1.5
Firstly, the results of the simulation case 1 are discussed. The flow around the hydrofoil is simulated at the angle of attack α+α0 = 14.4◦ (corresponds toα+α0 = 10◦ in ex-periments) for two different cavitation numbers, namelyσv = 1.5andσv = 2.0. Figures 5.5a and 5.5b illustrate the sheet cavity extents measured by Richter (2011, p. 4.9) in experiments with the air content in water of 80%. According to the photographs, sheet cavitation extends until approximately70%of the chord length for the cavitation number
Chapter 5. Verication and validation of the sheet cavitation model
Figure 5.7: Comparison of the scaled pressure coecients calculated by panMARE with dierent grid resolutions for the simulation case 1 withσv = 1.5
σv = 1.5. For the cavitation number σv = 2.0 sheet cavitation extends until approxi-mately30%of the chord length.
Figures 5.6-5.7 and D.7 visualise the results calculated by panMARE for the cavitation
Chapter 5. Verication and validation of the sheet cavitation model
Figure 5.8: Scaled pressure coecients and sheet cavitation extents calculated by panMARE with dierent grid resolutions for the simulation case 1 with σv = 2.0
number σv = 1.5. The evaluations present the scaled pressure distributions −Cp,v and sheet cavity thickness10ηcav obtained with different grid resolutions. Independent of the grid refinement, the scaled pressure within the cavity sheet is equal to the cavitation num-ber, i.e. the physics of the implemented sheet cavitation model are reproduced correctly.
Furthermore, the constant part of the scaled pressure distribution identifies the length of the cavity sheet. The sheet cavitation length and thickness calculated by panMARE strongly depend on the discretisation of the foil’s surface. The impact of the grid refine-ment in the circumferential direction is significant (s. Figures 5.6). For example, the cavity thickness calculated with the grid 13×24is twice as big as the cavity thickness obtained with the grid13×66. Compared to the measured cavity length, the computed re-sults are overestimated when the number of panels isNl ≤48. The results calculated with the grid resolutionsNl = 54/60/66converge to the measured length, namely0.7cwhere cis the foil chord length. The refinement of the grid in the span-wise direction shows al-most no influence on the calculated pressure distributions and corresponding sheet cavity extents.
Figures 5.8-5.9 and D.8 illustrate the calculated results obtained for the cavitation
num-Chapter 5. Verication and validation of the sheet cavitation model
Figure 5.9: Comparison of the scaled pressure coecients calculated by panMARE with dierent grid resolutions for the simulation case 1 withσv = 2.0
ber σv = 2.0. One can explicitly see that the results computed with the grid resolu-tionsNl = 54/60/66converge to the measured cavity length, namely 0.3c. Grids with Nl = 36/42/48panels lead to overestimated cavity shapes. By using a grid resolution
Chapter 5. Verication and validation of the sheet cavitation model
withNl = 24/30panels no cavitation is detected. This can be explained by the fact that a coarse grid involves an imprecise pressure distribution at the leading edge. Indeed, the value of the pressure at the leading edge is very important for the sheet cavity detection and the grid there should be fine enough to calculate the pressure values accurately. The grid refinement in the span-wise direction has a relatively small influence on the calcu-lated scaled pressure distributions and corresponding cavity shapes (s. Figures 5.9). Only the simulation results withNk = 10panels differ slightly from the results obtained with finer grid resolutions.
Figure 5.10: Sheet cavitation extent measured in experiment for the simulation case 2 and σ = 3.0 with air content of 80% (taken from Richter (2011, p. 4.10))
The following section concerns the discussion of the second simulation study. The flow around the hydrofoil is simulated at the angle of attack α+α0 = 18◦ (corresponds to α +α0 = 15◦ in experiments) for the cavitation number σv = 3.0. Figure 5.10 illus-trates the sheet cavity extent measured by Richter (2011, p. 4.10) in experiments with the air content in water of 80%. The photograph shows that sheet cavitation extends un-til approximately 30% of the foil chord length. Figures 5.11-5.12 and D.9 present the calculated results in form of the scaled pressure distributions and sheet cavity extents for different grid resolutions. Independent of the surface grid resolution, the scaled pressure distribution −Cp,v within the cavity sheet is equal to the cavitation number σv = 3.0.
This observation verifies the implemented sheet cavitation model. The calculated sheet cavitation length and thickness strongly depend on the refinement of the surface grid in the circumferential direction. By using the grid resolutionsNl = 24/30/36, no cavitation is detected by the underlying numerical method (s. Figure 5.11). ForNl = 48/54/60/66 the calculated cavity lengths converge to the same value, namely 30% of the chord length, which coincides very well with the measured cavity length. According to Figure 5.12, the grid refinement in the span-wise direction does not influence the calculated results.
All simulated cases show that the calculated sheet cavitation lengths are in agreement with measurements and that the scaled pressure distribution within the cavitating area is
Chapter 5. Verication and validation of the sheet cavitation model
-0.50.51.52.53.54.5-1012345
−Cp,v[-]
x/c[-]
-0.50.51.52.53.54.5-1012345
-0.050.050.150.250.350.450.10.20.30.40 10ηcav [m]
x/c[-]
-0.050.050.150.250.350.450.10.20.30.40 σv = 3.0
σv = 3.0
Nk= 13, Nl= 48 Nk= 13, Nl= 54 Nk= 13, Nl= 60 Nk= 13, Nl= 66 Nk= 13, Nl= 24 Nk= 13, Nl= 30 Nk= 13, Nl= 36 Nk= 13, Nl= 42
Figure 5.11: Scaled pressure coecients and sheet cavitation extents calculated by panMARE with dierent grid resolutions for the simulation case 2 with σv = 3.0
reproduced correctly. The present study points out that the choice of a suitable grid is a very important preparatory step to be taken before the computations. The employed sheet cavitation model delivers reliable results only for particular grid resolutions. The refinement of the hydrofoil grid in the span-wise direction should have at least 13 panels.
The number of panels in the circumferential direction should be higher or equal to 54.
The results with such grids were reliable and matched well with the measurements. The user of the program must find a compromise between the required level of accuracy and tolerable computation time. On the one hand, the hydrofoil grid should be fine enough to achieve grid-independent results. But on the other hand, the grid should be not too fine since the computational effort increases rapidly and the simulation results will not improve significantly.
In a final step, the influence of sheet cavitation on the hydrofoil flow characteristics is in-vestigated. Figures 5.13 and 5.14 illustrate the relative lift and drag coefficientsCl/Clnoncav
as well asCd/Cdnoncavas functions of the cavitation numberσv. The results are presented for the surface grid resolution13×66for both simulation cases. It can be observed that a de-crease of the cavitation number (which corresponds to a higher amount of sheet cavitation
Chapter 5. Verication and validation of the sheet cavitation model
Figure 5.12: Comparison of the scaled pressure coecients calculated by panMARE with dierent grid resolutions for the simulation case 2 with σv = 3.0
on the foil) leads to an increase of the relative lift and drag coefficients. Phoemsapthawee et al. (2009, p. 2) report that partial sheet cavitation leads to a slight increase of the lift and drag coefficients. When the cavitation number is decreased so far that supercavitation
Chapter 5. Verication and validation of the sheet cavitation model
Figure 5.13: Relative lift and drag coecients as functions of the cavitation number calculated by panMARE for the simulation case 1
0.95
Figure 5.14: Relative lift and drag coecients as functions of the cavitation number calculated by panMARE for the simulation case 2
occurs on the hydrofoil, an abrupt decrease of the lift and drag coefficients is expected. In the underlying study no supercavitation was simulated and the calculated increase of the lift and drag coefficients in presence of partial sheet cavitation seems plausible.