A comparison of transient and steady bem computations for free surface flows

A COMPARISON OF TRANSIENT AND STEADY BEM COMPUTATIONS FOR FREE-SURFACE FLOWS

William Schultz Volker Bertram

University of Michigan University of Hamburg, IfS Ann Arbor, USA Hamburg, Germany

ABSTRACT

The steady potential flow around an elongated spheroid under a free water surface is computed by various linear and nonlinear methods. Results are in good agreement and can serve as a validation test for future methods in computing wave resistance.

PROBLEM AND METHOD OF SOLUTION

A spheroid moves steadily under a free surface. The length-diameter ratio is L/D = 5. A Cartesian coordinate system with z pointing upwards and z (and L) aligned with the motion is used. The flow is assumed to be irrotational and incompressible in an ideal fluid. Then Laplace's equation holds in whole fluid domain:

(1) The steady boundary conditions on the free surface (z _{= ) are:}

JV2

_-

and

= (3)

The no-penetration boundary condition on the body is

7.

j4)

where is the velocity potential, the unit normal vector on the body, U the spheroid's speed, and g is the acceleration of gravity. H denotes the depth of submergence of the spheroid's center under the undisturbed free surface. Far from the body the flow tends to a uniform flow. The radi-ation condition prohibits propogating waves ahead of the body. The open-boundary condition lets waves pass undistrubed through the computational boundary. All computations use port/starboard symmetry. The steady-state hydrodynamic coefficients are given by

F

CL.= _{tpU2S and CM} M

-pU2SL' where S is the wetted area of the spheroid.

Doctors and Beck [1] linearize the free-surface condition about uniform flow and solved the resulting classical Neumann-Kelvin problem for depths of submergence H/L 0.16 and H/L =

0.245. They used a panel method with extrapolation from 36, 64 and 100 panels of constant strength Kelvin sources over each panel.

Bertram and Laudan [2] also use a Rankine source method. Discrete point sources are distributed above a finite section of the free surface ship to fulfill the boundary condition at the free surface.

-_{sto - csaa-cio -i} ea co a ¡o1Ip:v uqoewoipAqsdooq, JOOA wflJOBJOfl 111ISH3A!Mfl 3FI3SINFI31

On the hull, polygonal panels of constant strength are distributed. The involved integrals are evaluated by simple numerical integration. The nonlinear free-surface boundary condition is met in an iterative scheme that linearizes around arbitrary approximations of and Ç The solution for potentiad and surface elevation is improved in each step. The radiation and open-boundary conditions are enforced by shifting sources versus collocation points on the free surface. Bertram

[3] gives results for F = U//I = 0.4 for H/L = 0.16. The results presented here used 356

collocation points on the half-body and about 700 on the, free surface. The residual r = max j

V(Vç5)2

-

gI/gU

iteration (from the initial Neumann-Kelvin solution) the three leading digits in the coefficients Cw, CL, and CM do not change. Various free-surface grid modifications had little effect.

Cao, Schultz and Beck [4] use a mixed Eulerian-Lagrangian time stepping procedure combined with a desingularized boundary integral equation method to solve the unsteady problem in which the spheroid starts from rest and gradually reaches a constant speed. The fully nonlinear free-surface boundary conditions and the exáct body boundary condition are used. Rankine sources,

which are distributed inside the body and above the free surface, desingularize the boundary integral equation. Collocation is used to determine the strength of the sources. A fourth-order Runge-Kutta-Fehlberg method is used in the time stepping. An initial time increment is set, but is modified by the Runge-Kutta-Fehlberg subroutine where appropriate. A computational domain on the free surface moves with the body. At each time step, a row of material nodal points is discarded downstream and a row of new material points with specified values of wave elevation and potential is added upstream. For the cases presented here, homogeneous values for the new points are sufficient to prevent spurious results at the upstream computational boundary. Fig.1 shows results for varying Froude number based on L obtained with 137 points on the half-body and 61><16 points on the free surface for F = 0.6, 0.7 and 0.8 and 71 x 16 points for F,. = 0.4 and 0.5. The steady state coefficients are estimated by extrapolating to infinite time to avoid the r = 1/4 oscillation. The size and persistence of these oscillations generally increase with decreasing F and the faster start up of U from rest. The convergence was confirmed (for some Froude numbers) by the variationsof

a) the size of the computational domain on the free surface, b) the mesh spacing of both the free surface and the body surface and c) the time increment.

RESULTS

Fig.1, Table I compare these nonlinear results for H/L = 0.245 to the original linear results of Doctors and Beck [1]. The bow-up moments are taken about the center of the spheroid projected onto the undisturbed free surface. The time-marching values are closer to Doct&rs & Beck's linear results for lower F7, (0.4 and 0.5) and further for larger F7, when compared to the method of Jensen et al. (which are very similar to the steady linear results produced by the same method).

- J I

0.4 0.6 0.8 0.4 0.6 0.8

Fig.1: Results for spheroid at H/L = 0.245 ± Doctors and Beck, linear

E Bertram, linear

REFERENCES

Doctors, L. and Beck, R., Convergence properties of the Neumann-Kelvin problem for a submerged body, J. Ship Research 31, (1987)

Bertram, V. and Laudan, J., Computation of viscous and inviscid flows around ship hulls, Ship Technology Research 40/3, (1993)

Bertram, V., A Rankine source method for the forward-speed diffraction problem, IfS-Rep. 508, Univ. Hamburg, (1990)

Cao, Y., Schultz, W. and Beck, R., Three-dimensional, unsteady computations of nonlinear waves caused by underwater disturbances, 18th Symp. on Naval Hydrodyn., Ann Arbor,

(1990)

Table I: Results for spheroid as in Fig.1

0.4 0.6 0.8

Cao et aL, nonlinear o Bertram, nonlinear

F, lO3Cw lO3Ct -1O3CM

+ E O + _E + _E ° 0.4 2.59 2.76 2.67 2.50 7.94 7.54 7.52 7.71 2.02 2.31 2.53 2.22 0.5 7.34 7.30 7,21 6.84 4.71 4.22 3.85 3.95 5.22 5.27 5.49 4.85 0.6 6.47 6.52 6.67 6.21 0.0 -0.38 -0.58 -1.01 -4.98 -4.90 -4.91 -4.32 0.7 4.60 4.45 4.48 4.24 -2.06 -2.45 -2.44 -3.08 -3.80 -3.61 -3.49 -3.11 0.8 3.02 2.92 2.85 2.54 -2.65 -3.14 -3.03 -3.77 -2.73 -2.64 -2.45 -2.06 CLA 0.006 0.006 0.006 0.005 0.005 0.005 0.004 0.004 Ee 0.004 J-0.003 0.003 0.003 0.002 0.002 0.002 I 0.001 0.00 1 0.00 1