The Linearized Theory of a Supercavitating Hydrofoil With a Jet Flap

1964 ◽  
Vol 86 (4) ◽  
pp. 851-858 ◽  
Author(s):  
Hung-Ta Ho
Keyword(s):  
Theoretical Analysis ◽  
Momentum Flux ◽  
Vortex Sheet ◽  
Cavity Flow ◽  
Cavitation Number ◽  
Pitching Moment ◽  
Cavity Shape ◽  
Small Deflection ◽  
Small Incidence ◽  
Linearized Theory

A theoretical analysis is carried out for the steady cavity flow about thin hydrofoil sections at small incidence α, at the trailing edge of which a thin jet emerges at a small deflection τ. The flow is assumed to be inviscid and incompressible and the cavitation number is taken to be zero. The jet is assumed so thin that it can be considered as a vortex sheet across which the velocity is discontinuous. For the case of a flat plate, expressions have been obtained for lift, drag, pressure distribution, pitching moment, the jet shape, and the cavity shape. Numerical calculations are made for a number of jet momentum-flux coefficients CJ lying between 0.01 and 5.

The Role of Eigensolutions in Nonlinear Inverse Cavity-Flow Theory

1988 ◽  
Vol 110 (3) ◽  
pp. 315-324
Author(s):  
B. R. Parkin
Keyword(s):  
Inverse Problem ◽  
Free Surface ◽  
Direct Problem ◽  
Cavity Flow ◽  
Cavitation Number ◽  
The Novel ◽  
Cavity Shape ◽  
Linearized Theory ◽  
Exact Point

The method of Levi Civita is applied to an isolated fully cavitating body at zero cavitation number and adapted to the solution of the inverse problem in which one prescribes the pressure distribution on the wetted surface and then calculates the shape. The novel feature of this work is the finding that the exact theory admits the existence of a “point drag” function or eigensolution. While this fact is of no particular importance in the classical direct problem, we already know from the linearized theory that the eigensolution plays an important role. In the present discussion, the basic properties of the exact “point-drag” solution are explored under the simplest of conditions. In this way, complications which arise from non-zero cavitation numbers, free surface effects, or cascade interactions are avoided. The effects of this simple eigensolution on hydrodynamic forces and cavity shape are discussed. Finally, we give a tentative example of how this eigensolution might be used in the design process.

Hydrodynamic Characteristics of a Cambered Hydrofoil With a Jet Flap

1972 ◽  
Vol 39 (2) ◽  
pp. 337-344 ◽  
Author(s):  
Hung-Ta Ho
Keyword(s):  
Pressure Distribution ◽  
Steady Flow ◽  
Polynomial Function ◽  
Cavitation Number ◽  
Numerical Results ◽  
Geometric Parameters ◽  
Hydrodynamic Characteristics ◽  
Pitching Moment ◽  
Cavity Shape ◽  
Linearized Problem

The linearized problem of an infinite steady flow about a thin cambered hydrofoil with a jet flap is solved. The flow is assumed to be inviscid and incompressible and the cavitation number is taken to be zero. It is also assumed that the profile of the foil can be expressed in terms of a polynomial function. Expressions have been obtained for lift, drag, pressure distribution, pitching moment, the jet shape, and the cavity shape. Numerical results are presented for a general “polynomial” foil, and, in particular, Tulin’s low drag foil to show the effects of the geometric parameters of the foil on these hydrodynamic characteristics.

The lift coefficient of a thin, jet-flapped wing

1956 ◽  
Vol 238 (1212) ◽  
pp. 46-68 ◽  
Keyword(s):  
Fourier Series ◽  
Momentum Flux ◽  
Lift Coefficient ◽  
Elliptic Cylinder ◽  
Vortex Sheet ◽  
Main Stream ◽  
Trailing Edge ◽  
Vortex Sheets ◽  
Small Incidence ◽  
Limiting Case

A solution is given for the inviscid, incompressible flow past a thin, two-dimensional wing at a small incidence α , at the trailing edge of which a thin jet emerges at a small deflexion ז . The flow inside the jet is assumed to be irrotational, and bounded by vortex sheets across which it is prevented from mixing with the main stream. The effect of the jet on the outside flow is the same as that of a vortex sheet of strength proportional to its curvature and to the jet momentum flux, together with a doublet distribution proportional to the jet thickness (and vanishing in the present limiting case). An integral equation is obtained for the slope of the jet, by aligning the trailing vortex sheet in the undisturbed-stream direction. The solution is expressed as the sum of a Fourier series, together with a function possessing the correct (i. e. logarithmic) form of singular behaviour at the trailing edge. The coefficients of a 9-term interpolation to the series have been calculated on an electronic computer, for a number of momentum-flux coefficients C J lying between 0.01 and 10. The convergence is sufficient to justify the truncation of the Fourier series a posteriori . Simple expressions have been obtained for lift, pressure distribution and pitching moment. In particular, the lift coefficient is given by C L = 4 πA 0 τ + 2 π (1+ 2 B 0 ) α , where A 0 and B 0 are the leading Fourier coefficients associated with deflexion and incidence respectively. Close interpolations to their values are provided by the formulae 4 πA 0 = 3.54 C ½ J + 0.325 C J + 0.156 C 3/2 J , 4 πB 0 = 1.152 C ½ J + 1.106 C J + 0.051 C 3/2 J . For small C J these exhibit the dependence on C ½ J which has been predicted by the empirical theories of Stratford (1956) and Woods (1955). Close agreement with the theory is found in tests made at the National Gas Turbine Establishment, Pyestock, by Dimmock (1955) on an 8:1 elliptic cylinder with a narrow deflected jet exit close to its trailing edge, both for τ ≑ 30° and for τ ≑ 60°. The measured lift coefficients lie within 5% of the corresponding calculated curves (an allowance of 12½% having been made for thickness), and the observed centres of lift lie within 2% of the chord of those calculated for a thin aerofoil, over the whole range of C J . The ‘shallow jet’ approxi­mations used in the theory had not been expected to apply for τ as large as 60°, but the experimental agreement still found in this case resembles that obtained at large deflexion angles by the Glauert (1927) small angle theory for hinged flaps, in which similar approxima­tions occur.

A First and Second-Order Theory on a Supercavitating Hydrofoil With a Jet Flap

1974 ◽  
Vol 41 (3) ◽  
pp. 575-580 ◽  
Author(s):  
T. Kida ◽  
Y. Miyai
Keyword(s):  
Asymptotic Expansions ◽  
Deflection Angle ◽  
Order Theory ◽  
Second Order ◽  
Pitching Moment ◽  
Order Problem ◽  
Momentum Coefficient ◽  
Small Deflection ◽  
Small Incidence ◽  
Poincaré Problem

The basic relations for an infinite steady flow around a thin hydrofoil with a jet flap are obtained as the solution of the Riemann-Hilbert-Poincare problem, and the first and second-order problem at small incidence and small deflection angle of the jet is analytically solved by means of the matched asymptotic expansions in the case of the jet-momentum coefficient small. Expressions for lift, drag, pitching-moment, and the cavity shape have been obtained as the asymptotic expansions in powers of the jet-momentum coefficient together with its logarithm. From the comparison of the lift with numerical results of Ho, the analytical method in this paper is seen to be very useful and reasonable.

Two-Dimensional, Steady, Cavity Flow About Slender Bodies in Channels of Finite Breadth

1957 ◽  
Vol 24 (2) ◽  
pp. 170-176
Author(s):  
Hirsh Cohen ◽  
Robert Gilbert
Keyword(s):  
Cavity Length ◽  
Solid Wall ◽  
Cavity Flow ◽  
Cavitation Number ◽  
Integral Equation Formulation ◽  
Slender Bodies ◽  
Linearized Theory ◽  
Finite Breadth ◽  
Arbitrary Body ◽  
Cavity Width

Abstract The steady, cavitating flow past slender symmetrical bodies placed in a solid-wall channel is studied by means of the linearized theory of Tulin. The free-boundary condition is linearized and boundary conditions are applied on the line of symmetry of the flow in analogy with thin-air-foil theory. A singular integral equation formulation of the boundary-value problem is obtained and can be solved to yield expressions for cavity length, maximum cavity width, and drag coefficient as functions of the cavitation number and the channel breadth. These expressions are given for an arbitrary body and evaluated for the case of a wedge.

Linearized Theory of Supercavitating Hydrofoils in Subsonic Liquid Flow

1977 ◽  
Vol 99 (2) ◽  
pp. 311-318
Author(s):  
Tetsuo Nishiyama
Keyword(s):  
Incompressible Flow ◽  
Liquid Flow ◽  
Sound Speed ◽  
Gas Flow ◽  
Three Dimensional ◽  
Cavitation Number ◽  
Compressibility Effect ◽  
Linearized Theory ◽  
Velocity Pressure ◽  
Subsonic Region

In order to clarify the compressibility effect, the perturbed flow field of the supercavitating hydrofoil in subsonic region is examined by a linearized technique and, as a result, the general corresponding rule of the compressible flow to the incompressible one is proposed to obtain the characteristics of the supercavitating hydrofoil. The main contents are summarized as follows: (i) Basic relations between velocity, pressure, and sound speed are shown in subsonic liquid flow within the framework of linearization. (ii) The correspondence of the steady, characteristics of the two and three dimensional supercavitating hydrofoils in subsonic liquid flow to ones in incompressible flow is clarified. Hence we can readily calculate the characteristics by simple correction to ones in incompressible flow. (iii) Numerical calculations are made to show the essential differences of the compressibility effect between liquid and gas flow, and also the interrelated effect between cavitation number and Mach number on the characteristics of the supercavitating hydrofoils.

Steady Performance of a Flexible Hydrofoil Near a Free Surface

1990 ◽  
Vol 34 (04) ◽  
pp. 302-310
Author(s):  
Salwa M. Rashad ◽  
Theodore Green
Keyword(s):  
Mathematical Model ◽  
Free Surface ◽  
Cavity Length ◽  
Leading Edge ◽  
Cavity Flow ◽  
Cavitation Number ◽  
Deformation Characteristics ◽  
Flow Depth ◽  
Lift And Drag ◽  
The Galerkin Method

A linearized cavity-flow theory is used to develop a mathematical model to study the steady characteristics of a flexible hydrofoil near a free surface. The Galerkin method is employed to account for the mutual interaction between the fluid and structure forces. Cheng and Rott's method [1]2 is used to derive general expressions for the deformation characteristics in steady flow of an arbitrarily shaped hydrofoil, with a clamped trailing edge and free leading edge. From the analysis it is possible to determine the lift and drag coefficients, cavity length, and the foil steady deformation for any given specific foil shape, cavitation number, angle of attack, flow depth/chord ratio and rigidity. Sample numerical results are given, and the effects of flexibility and the proximity of the free surface are discussed. Chordwise flexibility tends to increase drag and decrease lift coefficients. This effect is more serious near the free surface. A slight increase of the thickness near the leading edge diminishes the flexibility effects.

Numerical Simulation and Analysis for Influence of Wall Effect on Cavity Shape

2013 ◽  
Vol 385-386 ◽  
pp. 400-403
Author(s):  
Fu Yuan Li ◽  
Yu Wen Zhang ◽  
Xi Zhao Du
Keyword(s):  
Numerical Simulation ◽  
Simulation Model ◽  
Empirical Formula ◽  
Diameter Ratio ◽  
Cavitation Number ◽  
Wall Effect ◽  
Water Tunnel ◽  
Cavity Shape ◽  
Model Size ◽  
The Impact

In the experiment of cavitation, the same water tunnel with different model size will get cavity shape that is different from the result of the empirical formula under the same cavitation number. In this article, we studied the impact of wall effect on natural cavity shape and the resistance of cavitator. We get the cavity shape and resistance of cavitator under different diameter ratio. We also get the law how cavity shape and resistance of cavitator change with the diameter ratio. The results provide a reference for experiment in water tunnel and the simulation model.

Analysis of Rotating Cavitation in a Finite Pitch Cascade Using a Closed Cavity Model and a Singularity Method

1999 ◽  
Vol 121 (4) ◽  
pp. 834-840 ◽  
Author(s):  
Satoshi Watanabe ◽  
Kotaro Sato ◽  
Yoshinobu Tsujimoto ◽  
Kenjiro Kamijo
Keyword(s):  
Cavity Length ◽  
Gain Factor ◽  
Cavitation Number ◽  
Cavity Model ◽  
Closed Cavity ◽  
Cavity Shape ◽  
Actuator Disk ◽  
Singularity Method ◽  
Eigen Value ◽  
The Stability

A new method is proposed for the stability analysis of cavitating flow. In combination with the singularity method, a closed cavity model is employed allowing the cavity length freely to oscillate. An eigen-value problem is constituted from the boundary and supplementary conditions. This method is applied for the analysis of rotating cavitation in a cascade with a finite pitch and a finite chordlength. Unlike previous semi-actuator disk analyses (Tsujimoto et al., 1993 and Watanabe et al., 1997a), it is not required to input any information about the unsteady cavitation characteristics such as mass flow gain factor and cavitation compliance. Various kinds of instability are predicted. One of them corresponds to the forward rotating cavitation, which is often observed in experiments. The propagation velocity ration of this mode agrees with that of experiments, while the onset range in terms of cavitation number is larger than that of experiments. The second solution corresponds to the backward mode, which is also found in semi-actuator disk analyses and identified in an experiments. Other solutions are found to be associated with higher order cavity shape fluctuations, which have not yet been identified in experiments.

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