Matched asymptotics for two-dimensional planing hydrofoils with spoilers

1998 ◽  
Vol 358 ◽  
pp. 259-281 ◽  
Author(s):  
G. M. FRIDMAN
Keyword(s):  
Asymptotic Solution ◽  
Flat Plate ◽  
Linear Theory ◽  
Asymptotic Expansions ◽  
Flow Problem ◽  
Matched Asymptotic Expansions ◽  
Asymptotic Solutions ◽  
Two Dimensional ◽  
Trailing Edge ◽  
Matched Asymptotics

The purpose of the paper is to demonstrate the effectiveness of the matched asymptotic expansions (MAE) method for the planing flow problem. The matched asymptotics, taking into account the flow nonlinearities in those regions where they are most pronounced (i.e. in the vicinity of the edges), are shown to significantly extend the range where the linear theory gives good results. Two model problems are used: the planing flat plate with a spoiler on the trailing edge and the curved planing foil. Asymptotic solutions obtained by the MAE method are compared with those obtained using linear and exact nonlinear theories. Based on the results, the asymptotic solution to the planing problem under the gravity is proposed.

Asymptotic Methods for an Infinite Slider Bearing With a Discontinuity in Film Slope

1976 ◽  
Vol 98 (3) ◽  
pp. 446-452 ◽  
Author(s):  
J. A. Schmitt ◽  
R. C. DiPrima
Keyword(s):  
Boundary Layer ◽  
Film Thickness ◽  
Asymptotic Expansions ◽  
Asymptotic Expression ◽  
Asymptotic Methods ◽  
Matched Asymptotic Expansions ◽  
Slider Bearing ◽  
Trailing Edge ◽  
Slider Bearings ◽  
Special Cases

The method of matched asymptotic expansions is used to develop an asymptotic expression for the pressure for large bearing numbers for the case of an infinite slider bearing with a general film thickness that has a discontinuous slope at a point. It is shown that, in addition to the boundary layer of the pressure at the trailing edge, there is also a boundary layer in the derivative of the pressure at the point of discontinuity. The corresponding load formula is also derived. The special cases of the taper-flat and taper-taper slider bearings are discussed.

Asymptotic solutions of the Orr—Sommerfeld equation

1975 ◽  
Vol 280 (1295) ◽  
pp. 271-316
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansions ◽  
Asymptotic Properties ◽  
Matched Asymptotic Expansions ◽  
Asymptotic Solutions ◽  
Asymptotic Approximations ◽  
General Kind ◽  
Work Done ◽  
Sommerfeld Equation ◽  
Special Case

A rigorous justification is given of work done by Eagles (1969), in which he applied the method of matched asymptotic expansions to the Orr-Sommerfeld equation to obtain formal uniform asymptotic approximations to a certain pair of solutions. (Somewhat more polished formal expansions of the same general kind were subsequently obtained by Reid (1972).) First, a study is made of the asymptotic properties of solutions of a certain differential equation which admits the Orr—Sommerfeld equation as a special case. Previous work on this differential equation by Lin & Rabenstein ( i960, 1969) is extended to develop a theory suited to our main purpose: to prove the validity of Eagles’s approximations. It is then shown how this theory can be used to prove the existence of actual solutions of the Orr—Sommerfeld equation approximated by these formal expansions. In addition, it is verified that these solutions have the properties assumed by Eagles (1969).

scholarly journals Two-dimensional resonators with small openings

1982 ◽  
Vol 24 (1) ◽  
pp. 2-27 ◽  
Author(s):  
G. R. Bigg ◽  
E. O. Tuck
Keyword(s):  
Natural Frequency ◽  
Asymptotic Expansions ◽  
Velocity Potential ◽  
Low Frequency ◽  
Matched Asymptotic Expansions ◽  
Two Dimensional ◽  
Acoustic Response ◽  
Closed Cavity ◽  
Small Opening

AbstractThe acoustic response of a two-dimensional nearly-closed cavity to an excitation through a small opening is examined, using the method of matched asymptotic expansions. The Helmholtz mode of vibration is discussed using a low-frequency expansion of the velocity potential in the cavity interior. The variation in frequency and magnitude of the resonator response is explored, both for the Helmholtz and the natural-frequency modes.

scholarly journals Eigenvalues in trailing edge flows

1975 ◽  
Vol 19 (2) ◽  
pp. 210-221
Author(s):  
K. Capell
Keyword(s):  
Asymptotic Formula ◽  
Flat Plate ◽  
Similarity Solution ◽  
Shear Flows ◽  
Two Dimensional ◽  
Trailing Edge ◽  
Uniform Shear

Abstract A wake similarity solution for symmetric uniform shear flows merging at the trailing edge of a flat plate has associated with it an eigenfunction problem which was overlooked by Hakkinen and O'Neil (1967). An asymptotic formula for large eigenvalues is obtained and compared with another such formula related to both the Goldstein (1930) inner wake solution and Tillett's (1968) similarity solution for a jet emerging from a two-dimensional channel.

Flow along a horizontal plate near a free surface

1993 ◽  
Vol 252 ◽  
pp. 399-418
Author(s):  
Milan Hofman
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Flat Plate ◽  
Viscous Liquid ◽  
Gravity Waves ◽  
Asymptotic Expansions ◽  
Small Amplitude ◽  
Surface Effect ◽  
Matched Asymptotic Expansions ◽  
Horizontal Plate

The problem of flow along a horizontal semi-infinite flat plate moving in its own plane through a viscous liquid just below the free surface is considered. The method of matched asymptotic expansions is used to analyse the interaction between the free surface and the boundary layer formed on the plate. It is found that, due to viscosity, small-amplitude gravity waves on the free surface can be formed. The formulae for the resistance of the plate containing the free-surface effect and for the lift, appearing as a new phenomenon, are derived.

A uniformly asymptotic solution for incompressible flow past thin sharp-edged aerofoils at zero incidence

1971 ◽  
Vol 70 (1) ◽  
pp. 135-155
Author(s):  
A. F. Sheer
Keyword(s):  
Asymptotic Solution ◽  
Incompressible Flow ◽  
Asymptotic Expansions ◽  
Matched Asymptotic Expansions ◽  
Proper Choice ◽  
Circular Arc ◽  
Flow Past ◽  
Sharp Edges

Expansions obtained from classical subsonic thin-aerofoil theory break down in the neighbourhood of the aerofoil edges. At sharp edges the method of matched asymptotic expansions fails to remedy this. Here this failure is explained, and in the case of incompressible flow past a symmetric aerofoil at zero incidence it is shown that by proper choice of the dependent variable an expansion may be obtained which is uniformly asymptotic. Finally, the case of a circular-arc aerofoil is considered in more detail.

scholarly journals Flow Past a Flat Plate With a Vortex/Sink Combination

1996 ◽  
Vol 63 (2) ◽  
pp. 543-550 ◽  
Author(s):  
N. J. Mourtos ◽  
M. Brooks
Keyword(s):  
Flat Plate ◽  
Current Model ◽  
Angle Of Attack ◽  
Separated Flow ◽  
Leading Edge ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Trailing Edge ◽  
Kutta Condition ◽  
Attached Flow

This paper presents a potential flow model for the leading edge vortex over a two-dimensional flat plate at an angle of attack. The paper is an extension of a model by Saffman and Sheffield (1977). A sink has been added in this model in an effort to satisfy the Kutta condition at both the leading edge and the trailing edge of the plate. The introduction of the sink was inspired by the fact that most steady vortices in nature appear in combination with a flow feature which can be interpreted as a sink at their cores when the flow is analyzed in a two-dimensional observation plane. As in the Saffman and Sheffield model, the presence of a vortex results in increased lift; however, in the current model a unique vortex/sink position is found at each angle of attack. A comparison has also been made between the lift and the drag of this model and the corresponding results for two classical solutions of flow over a flat plate: (a) the fully attached flow with the Kutta condition satisfied at the trailing edge only and (b) the Helmholtz solution of fully separated flow.

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