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A three dimensional panel method for free surface flows around ships of small angles of yaw

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Z. J. Zou —

inavERaiBr

Institut & Schiffijau LaboraSorium voor

Universität Hamburg Scheepshydromsohanlca

Mekelweg 2. 2628

CD Delft

Teï.: 015 • 786873 • Faic 01B - 781833

Introduction

During the last decade, three-dimensional flow computations using panel methods witk either Rankine singularities or Kelvin singulari^ties have been developed rapidly in tiae field of marme hydrodynamics. Great success has been adiieved mainly for the wave resistztnee problem. There are, however, only a few successful ^orts reported with respect to the seakeeping problem and ship manoeuvrability investigation.

This papCT discribes a numerical approach for computing the hydrodynamica! forces on yawed conventional ships in deep and/or shallow water. Based on potential theory, a three-dmiensional panel method using Rankine sources and a semi-infinite dipole sheet was developed. The non-linear free surface conditions and the non-non-linear Kutta condition at the trailing edge were introduced and. linearized. The flow around the siiip with small angles cf yaw was splitted into a symmetric one due to the longitudinal motion of the ship and an antisymmetric one due to the lateral motion. The symmetric fiow and the antisymmetric flow were then determined respectively. Initial numerical results obtained for a Wigley model axe presented.

Forinui^ion ' .

The flow is assumed to be irrotational and steady. The water depth is ass.umed to be constant. A disturbed velocity potential^(z,y,z) which satisfies Laplace's equation is introduced. It must satisfy the following boimdary conditions:

(V^ — y,) • V f = ^, oa the surface, (1)

Ç = -{-Vs • V<f> -h ^ V ^ . V^) on the free surface, (2)

ff. 2

• n = V, • n on the ship's huU, (3) ^, = 0 on the water bottom, (4)

V ^ = (0,0,0) at infinity. (5) where V, = {u, u, 0) is the speed of the ship and f the elevation of the free surface.

Moreover, the vdocity potential should satisfy a

- Radiation condition, which stateS: that np waves appear ahead of the ship at a great distance;

- Kutta condition, which demand? a finite vdocity at the trailing edge of the ship.

Eliminating the unknown f &om (1) and. (2), the combined condition on the free surface follows

(V^ - Vs) • V ( - K - V4> + . V^) = g4>.. (6)

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Followiiig a procedure proposed by Jensen |1], the non-linear conditions. (2) ' and (6) are lin-earized. It is assumed that the yaw angle is smiall, so that the velocity potential ^ can be divided into a symmetric part \$ due to the longitudinal motion and an antisymmetric part tp due to the lateral motion^ and that Vy> <C V ^ . " ,

The longitudinal flow is first determined by aa iterative scheme starting from \$ = 0 and ^ = 0 like in Jensen^s method. Symmetric Rankine sources are distributed on the shipis huU and on a plane above the undisturbed free surface. The strengths of the singulariti^ axe deterxnined so that the corresponding conditions arefuMUed at the collocation points on the ship's hull and on the non-linear free surface. Meanwhile the radiation condition is satisfied by Jensen's technique of staggered grids. The shallow water effect is taken into'accoimt by a mirror method.

Unlike Jensen's single source points method which bases on the assumption of a dosed body, a constant source stirface distribution (see Bai'& Yeuag [2], appendix B) was used.

In regard to the antisymmetric flow, it deals with the three-dimensional lifting problem. In order to determine this fiow, an antisymmetric Rankine source distribution on the the ship's hull and on a plane above the imdisturbed free surface, and a dipole sheet on tke shifts lateral plane and on the symmetric plane downstream of the trwling edge are used. The dipole sheet win provide the required vortidty to produce the lifting forces.

Taldng the solution for the longitudinal flow as an approximation for the o r i ^ a l flow, the linearized condition on the free surface'is obtained forthe antisymmtric velocity potential ip. This condition is used to determine the strengths of the singularities.

As Hess [3] remarked, the mathematical description of the lifting probliem is merely a model to describe by means of an invisdd flow a phöiomjsnon that is idtimately due to viscosity. It is essentiell to impose explidtiy the Kutta condition at tbe trailing edge of three-dimensional lifting bodies to make the lifting prohiba unique.

In the present paper, the Kutta condition is applied by indirect means, namely, the equal-pressure condition is satisfied on both sides of the trailing edge. This concUtion is primitively non-Hnear and requires an iterative procedure (see Cig. Nakatake, et al.[4]). However, by msLking use of the symmetry and antisymmetry of the flows^ the Kutta condition can be linearized as foiiows:

• Vip -utp^^v^y. (7) (7) was fulfilled approximatdy at the collocation points of the pands adjacent to the trailing

edge.

In order to ensure that there are enough conditions to determine the unknown strengths of the sources and the dipole sheet, the semi-infinte dipole sheet is divided horizontally into some strips. It is assumed that the dipple strength for each strip changes proportionally from zero at the leading edge to the maximal value at the trailing edge, and remains constant downstream the trailing edge, so that there is (mly one unknown for each strip. Consequently this dipole sheet can be exactly replaced by a syatem of horseshoe vortices, while the bound vortices at the same strip have the same strength.

Numerical results ' . .

A numerical investigation was implemented for a.Wigl^ xnodel. The wave-making resistance, the steady sinkage and trim, the lateral hydrodynamical force and the yawing moment were calculated. Fig.l shows the free surface contours of the linear and the non-Knear calculation for the steady forward motion with Fn = 0.27. In Figv2 the wave system around the Wigley model at 5** yaw is shown*

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The results presented here are for deep water. At the time of writing, this work is still in progress. It is expected that by the time of the Workshop results for shallow water can also be presented.

Acknowledgnient

I am very grateful to Prof. H. Söding for his continual guidance during this work. The finandal support by DFG (Deutsche îbrschungsgemdnschaft) is also acknowledged.

References.

1] Jensen,G., Bertram,V. and Söding,H.(1989) . Ship Wave-Resistance Cowputatitms 5th International Con£. Numerical Ship Hydrodynamics, Hiroshima

2] BaijKJ. & Yeung,R.W.(1974) Numerical solutions to fret-surface flow probleiJis Proc l&th Symposium on Naval Hydrodynamics, Cambridge, Massachusetts

3} Hess,J.L.(1974) The problem of the three-dimensional lifting potential flow and its

solution hy meaTts of surface singularity distribution

Comput. Methods Appl. Mech. Eng. 4, pp. 283-319

4j Nakatake,K., Ando,J., Komura,A. and Kataofca,K.(1990) On the Flow Field and the

Hydrodynamic Forces of an Obliquing Ship

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WIGLEY £ ; = 2.64(Tn/5) L„ = 10(m) i^n = 0.27 N F = 1 0 1 2 I T E R = 5 n II t I i i f 1

I

Fig.1 Contour lines of surface devation for the Wigley modd at 0" yaw lower half the linear result, upper half the non-linear result

i

WIGLEY £ / o = 2 . 6 6 ( m / s ) F n = 0.27

i

Fig.2 Contour lines of siirface elevation for the Wigley model at 5* yaw

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