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Review Article

Ventilation of Lifting Bodies: Review of the Physics and Discussion of Scaling Effects

[+] Author and Article Information
Y. L. Young

Professor
Department of Naval Architecture and
Marine Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: ylyoung@umich.edu

C. M. Harwood

Department of Naval Architecture and
Marine Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: cmharwoo@umich.edu

F. Miguel Montero

Maritime Research Institute Netherlands (MARIN),
Wageningen, The Netherlands
e-mail: f.miguelmontero@marin.nl

J. C. Ward

Department of Naval Architecture and
Marine Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: jcward@umich.edu

S. L. Ceccio

ABS Professor of Marine and Offshore Design
and Performance Chair
Professor
Department of Naval Architecture and
Marine Engineering,
University of Michigan,
Ann Arbor, MI 48109;
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: ceccio@umich.edu

1Corresponding author.

Manuscript received February 3, 2016; final manuscript received November 29, 2016; published online January 12, 2017. Assoc. Editor: Ardeshir Hanifi.

Appl. Mech. Rev 69(1), 010801 (Jan 12, 2017) (38 pages) Paper No: AMR-16-1015; doi: 10.1115/1.4035360 History: Received February 03, 2016; Revised November 29, 2016

Ventilation is relevant to the performance, safety, and controllability of marine vessels, propulsors, and control surfaces that operate at or near the free surface. The objectives of this work are to (1) review the fundamental physics driving ventilation and its impact upon the hydrodynamic and structural response, and (2) discuss the scaling relations and its implications on the design and interpretation of reduced-scale studies. Natural ventilation occurs when the flow around a body forms a cavity that is open to the free surface. The steady flow regimes, hydrodynamic loads, and unsteady transition mechanisms of naturally ventilated flows are reviewed. Forced ventilation permits control of the cavity pressure and cavity shape, but can result in unsteady cavity pulsations. When a lifting surface is flexible, flow-induced deformations can increase the loading and the size of cavities, as well as lead to earlier ventilation formation. Ventilation tends to reduce the susceptibility of a lifting surface to static divergence. However, fluctuations of fluid added mass, damping, and disturbing forces caused by unsteady ventilation will change the structural resonance frequencies and damping, and may accelerate hydroelastic instabilities. Scaling relations are developed for both the hydrodynamic and hydroelastic response. Similarity in the three-dimensional (3D) ventilation pattern and hydrodynamic response requires simultaneous satisfaction of Froude number, cavitation number, and geometric similarity. However, Froude scaling complicates the selection of suitable model-scale material to achieve similarity in the dynamic hydroelastic response and material failure mechanisms between the model and full scale.

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References

Figures

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Fig. 1

Cavitation regimes on a 2D hydrofoil section: (a) fully wetted, (b) base-cavitation or base-ventilation, (c) partial cavitation or partial ventilation, and (d) supercavitation or superventilation. (Reprinted with permission from Harwood et al. with modifications [32]. Copyright 2016 by Cambridge University Press.)

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Fig. 5

Schematic depictions of lifting surfaces that may be subject to ventilation. (a) Vertical surface-piercing hydrofoil/strut, (b) surface-piercing hydrofoil with dihedral angle Γ, (c) horizontal fully submerged hydrofoil, (d) shallowly submerged propeller, and (e) superventilated or supercavitating body with control surfaces penetrating the cavity boundary. Note that (e) depicts two different appendage configurations—one axisymmetric and one with a dihedral angle Γ relative to the local cavity normal. In all the cases, s ≡ geometric span, h ≡ normal distance from phase boundary to lifting surface tip or centerline, n ≡ unit normal vector on phase interface, directed toward dense fluid, and DP ≡ propeller diameter.

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Fig. 4

Data from experiments by Kawakami et al. [33] (3.5 × 105 ≤ Rec ≤ 1.02 × 106), as well as experimental and numerical simulations by Ducoin coworkers [37,38] (Rec = 7.5 × 105) are shown as symbols. Lines representing Eqs. (11) and (12) are shown, with σc estimated for α2D = 4 deg and κ = 0.33.

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Fig. 3

Schematic illustration of 2D cavity shedding caused by a re-entrant jet. Horizontal rows indicate each of the three steps of the cycle, labeled on the left. Dark gray indicates the solid surface of a flat-plate hydrofoil, light gray represents the vaporous cavity, and (blue) arrows undercutting the cavity represent liquid streamlines.

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Fig. 2

Cavity length (Lc/c) and section lift-slope (a0), approximated with free-streamline theory and plotted as functions of σc/α2D for α2D = 4 deg. Experimental results from Ref. [31] are shown, with open symbols denoting steady cavity flow and filled symbols denoting unsteady cavity flow. The gray region indicates the neighborhood of Lc/c ≈ 1, in which the linearized flat plate solutions are regarded as nonphysical [9]. The theoretical three-quarter chord instability develops where the partially cavitating solution bifurcates at the lower boundary of the gray-shaded region. Rational polynomial approximations to the linear theory are shown as solid black lines, which fair across the three-quarter chord instability.

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Fig. 6

Fully wetted (FW) flow on 3D hydrofoils in vertical, surface-piercing configurations. Flow over the suction surfaces (pictured) is all liquid, with any air entrainment constrained to the separated wake. All the foils are pictured moving from right to left. (a) Fully wetted flow on a vertical surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm) in a towing tank at α = 10 deg, Fnh = 2.0, ARh = 1.0, and Rec = 4.5 × 105. (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.) (b) The effect of forward speed on the free-surface profile along the suction surface of a vertical surface-piercing strut with a symmetric bi-ogival section shape (c = 30.5 cm) in fully wetted flow at a yaw angle of α = 10 deg. The lines denote the free surface profiles at speeds of 3.05, 6.1, and 9.14 m/s (10, 20, and 30 ft/s). (Reprinted from Waid [13].)

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Fig. 7

A depiction of the flow in the neighborhood of cavity-closure lines (a) normal to the inflow and (b) at an angle to the inflow. Conservation of momentum in the tangential direction causes incoming flow to be reflected about the local cavity-closure line. (Reprinted with permission from De Lange and De Bruin [68]. Copyright 1998 by Springer.)

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Fig. 9

Examples of fully ventilated (FV) flow on surface-piercing hydrofoils. (a) Fully ventilated (FV) flow on a vertical, surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm) at α = 15 deg, Fnh = 3.0, and ARh = 1.0 in a towing tank. Most of the suction surface is contained within the cavity, while the pressure surface remains wetted. The cavity is open to the atmosphere, which admits air at atmospheric pressure everywhere in the cavity (Pc = Patm). (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.) (b) A sketch of a typical naturally ventilated cavity on a vertical surface-piercing hydrofoil with a symmetric circular-arc section shape in a rotating-arm tank. The cavity extends along the entire immersed span. Note that the cavity length, as depicted, is not less than the chord length at any point along the span as a result of the large Froude number. (Reprinted with modifications from Breslin and Skalak [3].)

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Fig. 10

A photo of a fully ventilated propeller of diameter DP = 0.25 m operating just beneath the free surface in a towing tank at conditions of h/R = 1.4 and J = 0.133. The low-pressure field around the propeller blade back has drawn down a small “hollow” in the free-surface, through which the tip of the blade protrudes, so the configuration could be fairly argued to be surface-piercing. The entrainment of air around each blade persists through the entire blade revolution. (Reprinted with permission from Califano and Steen [63]. Copyright 2011 by Elsevier.)

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Fig. 11

Measured coefficient of lift/side-force on a vertical surface piercing hydrofoil with a circular-arc section shape (c = 6.4 cm) in a rotating-arm tank at an immersed aspect ratio of ARh = 2.0. Results are shown for values of chord-based Froude number (Fnc) of 2, 4, 6, and 8. Note that at the lower Froude numbers, the lift in the FV regime exhibits a Froude-number dependency, which manifests as a change in the effective camber of the hydrofoil. At the smallest Froude number, laminar separation causes a small increase in the FW lift at substall angles of attack. (Reprinted with modifications from Breslin and Skalak [3].)

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Fig. 12

Measured hydrodynamic loads on a vertical surface-piercing strut with a symmetric semi-ogival section shape (c = 27.9 cm) in a towing tank at ARh = 1.0 and Fnh = 2.5. Load coefficients were also modeled using a modified lifting line method, plotted as dashed lines. In all the three plots, αb = 2.5 deg denotes the bifurcation angle and αs = 15 deg denotes the stall angle. Flow below α = 2.5 deg is exclusively fully wetted and flow above α = 15 deg is exclusively fully ventilated, leaving a range of bistable yaw angles of 2.5 deg ≤ α ≤ 15 deg. (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.) (a) Three-dimensional lift coefficient. Lift in the fully ventilated regime is significantly lower than that in the fully wetted regime at large yaw angles. At small yaw angles, lift coefficients in the two regimes are similar as a result of short, thin cavities, which neither reduce a0 significantly nor induce significant apparent camber. (b) Three-dimensional drag coefficient. The onset of ventilation reduces frictional and lift-induced drag, but incurs larger form and parasitic drag components. (c) Three-dimensional yawing moment coefficient, measured about midchord. Ventilation reduces both the lift and the moment arm of the lift about midchord. As a result, the yawing moment behaves as a sublinear (convex) function of α.

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Fig. 8

Three examples of partially ventilated (PV) flow on vertical, surface-piercing hydrofoils. The foils are all moving from right to left. (a) PV flow on a vertical surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm) at α = 20 deg, Fnh = 1.0, and ARh = 1.0 in a towing tank. The pictured cavity does not extend the entire immersed span of the hydrofoil. (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.) (b) PV flow on a vertical surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm) at α = 7.5 deg, Fnh = 1.54, and ARh = 1.0 in a towing tank. The angle of the cavity-closure line (shown as a solid white line) exceeds the 45 deg criterion (shown as a dashed white line) proposed by Harwood et al. [32] for cavity stability. (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.) (c) Sketch of typical PV flow on a vertical surface-piercing strut with a symmetric circular-arc section shape in a rotating-arm tank. (Reprinted with modifications from Breslin and Skalak [3].)

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Fig. 15

Overlay of observed flow regimes of a surface-piercing strut with a symmetric semi-ogival section shape (c = 27.9 cm) in a towing tank with ARh = 1.0. Overlapping zones indicate bistable flow regimes. Open symbols and lines indicate experimentally observed ventilation transition processes (covered in Sec. 4). (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.)

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Fig. 16

Taxonomy of ventilation formation and elimination mechanisms on a surface-piercing hydrofoil. A detailed depiction of the initial paths of air-ingress is shown in Fig. 17. The hydrofoil surface is shaded light gray. The solid curvy (blue) lines near the free surface and in the cross section at the left-hand side of the figure lines indicate streamlines and free surface profiles. The (green) hatching and shading indicate atmospheric air entrained below the initial free surface or disturbed free surface. The (blue) lines and hatching above the free surface on the right-most drawing indicate spray sheets, which detach from the suction side of the hydrofoil and extend vertically downward to join with the immersed cavity boundary. Black stippling indicates wetted flow separation. The thin (red) arrows indicate paths of air ingress or egress. In the case of cavitation-induced formation, (light blue) shading and hatching indicate water vapor.

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Fig. 18

Examples of suction-surface streamline patterns preceding spontaneous ventilation formation on surface-piercing hydrofoils with sharp leading edges. Large-scale flow separation from the leading edge is conducive to ventilation in both cases. (a) Surface flow visualization on a vertical surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm) at α = 14 deg, Fnh = 2.5, and ARh = 1.0. Dots of a (yellow) shear-thinning paint mixture were applied to the suction surface prior to towing-tank tests. A large separation bubble is indicated by the forward-swept paint streaks. The (red) arrow indicates the entry point of air from the free surface. (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.) (b) Sketch of flow separation on a vertical surface-piercing hydrofoil with a symmetric circular-arc section shape (c = 6.4 cm) at α = 10 deg, Fnh = 1.41, and ARh = 2.0. Separation was visualized by ultraviolet illumination of oil-smear patterns on the body. (Reprinted from Breslin and Skalak [3].)

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Fig. 13

Top: the instantaneous measured speed. Middle: measured lift, drag, and yawing moment. Lower left: detailed view of the estimated lift, drag, and moment. Bottom right: estimated normalized location of the center of pressure. Results are from a surface-piercing strut with a symmetric semi-ogival section shape (c = 27.9 cm) at ARh = 1.0, Fnh = 3.0, and α = 15 deg. The formation of a ventilated cavity is caused by natural aeration of a tip-vortex. The flow transitions from fully wetted to fully ventilated in approximately 0.2 s, which leads to significant changes in both the magnitude and distribution of pressures on the hydrofoil's suction surface. From authors' experimental data, collected in a towing-tank.

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Fig. 14

The characteristic lift curves for the wetted and ventilated flows and the notional stability zones of a surface-piercing hydrofoil. The shaded overlapping range of angles of attack is referred to by Fridsma [54] as the “unstable region,” stemming from the fact that stability of either flow regime can be perturbed to initiate a jump to the alternate stable regime. In the present context, it can be understood as the region of bistability, rather than one of true instability. Figure styled after Ref.[54].

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Fig. 17

Depiction of a surface-piercing hydrofoil prior to the inception of a ventilated cavity. The suction side may be covered in a vaporous cavity or may experience wetted separation at the leading or trailing edges. Initial air ingress (inception) occurs through any of the paths indicated by (red) arrows pointing downward from the free surface and pointing toward the foil leading edge from the ventilated tip vortex, including upstream disturbances, artificially introduced perturbations, wave-breaking, Taylor-type instabilities, the tip vortex, or secondary interactions between the free-surface and vaporous cavity shedding. Figure composited by the authors from sketches found in Refs. [7,13,32,47,55].

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Fig. 20

Process sketch showing ventilation formation on a shallowly submerged flat plate with a geometric aspect ratio of ARs = s/c = 0.25. Air ingress occurs through aerated tip vortices. (Reprinted from Ramsen [50].)

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Fig. 21

Ventilation formation on a shallowly submerged propeller with a diameter of DP = 0.25 m in a towing tank at h/R = 2.04 and J = 0.075. Air-ingress occurs through the interactions of the propeller tip vortices with the free surface. (Reprinted with permission from Califano and Steen [63]. Copyright 2011 by Elsevier.)

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Fig. 22

The effects of an attached vaporous cavity on the formation mechanisms of natural ventilation for the case of a surface-piercing strut. Cavitation contributes vorticity to the wake via vortex shedding, promotes growth of Taylor instabilities in the thin film of water separating the cavity from the free surface, and causes the tip vortex core to cavitate, making it more buoyant. (Reprinted with Waid [13].)

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Fig. 23

A vertical surface-piercing hydrofoil with a symmetric bi-ogival section shape at α = 9 deg, Fnh = 8.4, ARh = 2, Rec = 5.5 × 106, and σv = 0.47. The vapor clouds in the wake of the foil indicate the shedding of the attached partial cavity. (Reprinted from Rothblum et al. [7].)

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Fig. 24

The angle of the cavity-closure line at the washout condition is approximately Φ¯wo=45deg just prior to the washout of the cavity by an upstream re-entrant jet across tested values of ARh and α. (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.) (a) Impending cavity washout on a vertical surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm) at α=20deg, Fnh=1.5, and ARh=1.0 in a towing tank. The approximate cavity angle at the washout condition is dubbed Φ¯wo. An approximation of the cavity-closure line is used to determine the approximate cavity closure angle (Φ¯) at the time of cavity washout. (b) Values of Φ¯wo as a function of α and ARh for vertical surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm).

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Fig. 19

Taylor-type instabilities are visible on the free surface along the suction side of a vertical surface-piercing hydrofoil with a symmetric bi-ogival section shape in a towing tank at Fnh = 7.14 and ARh = 2.0. The yaw angle was unreported. (Reprinted from Rothblum et al. [7].)

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Fig. 25

Scaled washout boundary in α − Fnh space. Data from experiments are plotted as symbols; B&S: [3]; S&W: [14] experiments in variable-pressure water tunnel. 1Tests conducted at atmospheric pressure. 2Tests conducted at a reduced pressure of 4.67 kPa in a cavitation tunnel. (Reprinted with permission from Harwood et al. [32]. Copyright 2016 by Cambridge University Press.)

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Fig. 26

The figure compares a vaporous cavity (top) to a ventilated cavity created by injection of noncondensable gas (bottom) for similar σc. The vaporous cavity is steady, while the ventilated cavity is pulsating. (Reproduced with permission from Verron and Michel [95]. Copyright 1984 by Society of Naval Architects and Marine Engineers.)

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Fig. 27

Plot of the dimensionless volumetric airflow (CQ,V) and the resulting cavitation number (σc) for a given Froude number. The rate of airflow was increased, and then decreased, revealing hysteresis in the artificial cavity pulsation mechanism. (Reprinted with permission from Michel [94]. Copyright 1984 by American Society of Mechanical Engineers.)

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Fig. 28

A schematic illustration of 2D cavity pulsation in the absence of gravitational effects. The case shown is the time-evolution of a two-wavelength mode of pulsation behind a ventilated wedge. Here, λ is the wavelength on the cavity interface; lm and lM, respectively, denote the minimum and maximum cavity lengths; and T is the pulsation period. (Reprinted with permission from Michel [94]. Copyright 1984 by American Society of Mechanical Engineers.)

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Fig. 31

A two degree-of-freedom (2DOF) model of a cantilevered, rectangular hydrofoil with spanwise bending and twisting flexibilities only. The gray lines indicate the deformed position of the foil and its chord line under hydrodynamic load. Sketch styled after Refs. [75,89].

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Fig. 29

Sketch of 3D cavity topology on a trapezoidal hydrofoil at large α. A base-cavity, cavitating tip vortex, and leading-edge cavity exist simultaneously. Note the non-normal cavity closure angle (this time a result by the tapered planform) results in the nonimpinging re-entrant flow described in Sec. 3. (Reprinted with permission from Verron and Michel [95]. Copyright 1984 by Society of Naval Architects and Marine Engineers.)

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Fig. 30

The nondimensional pulsation frequency (ϕ) is shown as a function of σc/σA for rectangular hydrofoils of varying aspect ratios (λ). As σc/σA increases, the influence of the noncondensable gas within the cavity increases, as does the resulting pulsation frequency. (Reprinted with permission from Verron and Michel [95]. Copyright 1984 by Society of Naval Architects and Marine Engineers.)

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Fig. 36

The dependency of the flutter speed (UF) on mass ratio (μ) for fully wetted bare struts (bending) and struts with pods (torsion) in FW conditions. Note that the most stable operating point (highest flutter speed) coincides at the transition between bending and torsional flutter at μ = 0.66. (Reprinted from Besch and Liu [133].)

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Fig. 32

Effect of varying material properties on the steady-state FSI response of a rectangular, cantilevered surface piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm). A nonlinear lifting-line model was coupled to an FEA model to simulate the two-way FSI coupling. All the cases shown are at conditions of α = 2.5 deg, Fnh = 3, and ARh = 1.0. The elastic axis of the hydrofoil is specified at a location of 0.91 c aft of the leading edge in order to simulate a case where a reinforcing-rod has been installed at the trailing edge. All the materials are assumed to be isotropic and linearly elastic, and PVC stands for polyvinyl chloride. (a) Dimensionless spanwise bending deflections, (b) dimensionless spanwise twisting deflections, (c) ventilated cavity patterns, and (d) spanwise distribution of sectional Cl2D and locations of the center of lift.

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Fig. 33

Behavior of flexible, cantilevered surface-piercing hydrofoil with a symmetric semi-ogival section shape (c = 27.9 cm), near the static-divergence boundaries in FW and FV flows. Results were obtained using a coupled LL-FEM solver developed by the authors [32,42,72]: (a) Extrapolation to divergence condition of a polyvinyl chloride (PVC) surface-piercing hydrofoil at ARh = 2.5 and e = 1.32. The results were calculated with the coupled LL-FEM model used to generate Fig. 32. In the FW regime, the linear relationship of Eq. (41) holds, indicating a divergence speed of UD ≈ 8.1 m/s. In the FV regime, the result approaches linearity as all the sections become supercavitating, leading to an approximate divergence speed of UD = 14.1 m/s. (b) Effective angle of attack αe normalized by the initial geometric angle of attack (α) of a PVC surface-piercing hydrofoil at ARh = 2.5 as speeds approach the divergence velocity. The lower and upper X-axes are nondimensionalized by the qD in FW and FV flows, respectively. The Y-axis shows the amplification of the initial geometric angle of attack. αe = α + θ, as defined in Fig. 31.

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Fig. 34

Mode shapes and modal frequencies of a PVC hydrofoil partially submerged in water (ρs/ρf = 1.33 and μ = 0.146). The hydrofoil is cantilevered at its root and has a rectangular planform with a symmetric semi-ogival section shape (c = 27.9 cm) and a geometric aspect ratio of ARs = 3.27. FEM analysis is used to predict the mode shapes and modal frequencies (solid lines for FW conditions and dashed lines for FV conditions) and are compared with experimental measurements (for FW conditions only). (Reprinted from Harwood et al. [125].) (a) Dry mode shapes, computed by Harwood et al. [125], (b) frequencies of PVC hydrofoil, and (c) wet–dry frequency ratio of PVC hydrofoil.

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Fig. 35

Comparison of theoretical and experimental flutter speeds (UF) as a function of the square root of mass ratio, (μ), for cantilevered, rectangular foils in FW conditions. The results show that the theoretical predictions are unconservative in the region of low mass ratios (μ), where most hydrodynamic lifting bodies operate. (Reprinted with permission from Abramson and Chu [131]. Copyright 1959 by Society of Naval Architects and Marine Engineers.)

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Fig. 37

Total damping ratio (ζ) and vibration frequencies for a bending-type strut in water in FW conditions. The top plot shows the variation in the damping ζ versus flow speed U, while vibration frequency is shown on the bottom. The hydroelastic instability occurs in the “new mode,” which appears highly damped at intermediate speeds. The damping of the new mode decreases rapidly toward zero as the flutter speed is approached, and flutter occurs in the first bending mode shape. (Reprinted from Besch and Liu [133].)

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Fig. 38

Total damping ratio (ζ) and vibration frequencies for a torsion-type strut with an attached pod in water in FW conditions. The top plot shows the variation in the damping ζ versus flow speed U, while the vibration frequency is shown on the bottom. The hydroelastic instability occurs in the second mode of the system, corresponding to the first torsional mode. The damping of this mode starts out small and rises slightly until dropping below zero at the inception of flutter. (Reprinted from Besch and Liu [133].)

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Fig. 39

Measured flutter boundary of a cavitating hydrofoil. The test fixture was designed in such a way so that the bending ωh and pitch ωα natural frequencies in vacuo could be varied. As evidenced here, the critical solid-to-fluid mass ratio μ increases as ωh/ωα decreases, or as the foil becomes stiffer in torsion. Flutter was detected by observation of the foil displacement. (Reprinted from Song and Almo [143].)

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Fig. 40

Cavitation inception index (σi) for a von Kármán vortex street versus the square root of the Reynolds number (Reh, where h is the thickness of the foil trailing edge) for both lock-in and lock-off regimes. (Reprinted with permission from Ausoni et al. [146]. Copyright 2007 by American Society of Mechanical Engineers.)

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Fig. 41

The influence of the effective cavitation number σv/(2αeff) on the measured frequency spectrums of a flexible NACA66 foil (left) and a NACA0015 foil (right). Both foils are made of POM polyacetate and are cantilevered at the root. αeff=α−αCL=0 is the linear effective angle of attack for a cambered foil, fcavrigid is the modeled cavity shedding frequency of a rigid hydrofoil, and fhw is the natural spanwise bending frequency of the flexible hydrofoil in water. (Reprinted with permission from Akcabay and Young [75]. Copyright 2015 by Elsevier.)

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Fig. 42

Chart showing Young's modulus (E) versus density ρs of common engineering materials. Image courtesy of Granta Design [155] with permission from Professor Ashby and Granta Design.

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Fig. 43

Chart showing the material failure strength (σf) versus solid density ρs of common engineering materials. Image courtesy of Granta Design2 [155] with permission from Professor Ashby and Granta Design.

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