### A 3D Panel Code for Wave Drag Calculations - Part. I: General Formulation and Discretization

#### Abstract

We present a BEM/panel code to compute 3D potential flows about ship-forms with lineaarized free-surface conditions in order to compute the wave drag as a function of the Froude number.

The basic governing equations of potential flow with free surface are the Laplace equation for the velocity potential with appropriated boundary conditions and the free surface condition. The last is based on the Bernoulli equation which relates the surface elevation with the local absolute value of velocity. However this problem is ill-posed in the sense that allows multiple solutions, associated with the existence of a system of trailing gravity waves propagating in both (upstream and

downstream) directions. Solutions with upstream propagating trailing waves should be considered non-physical and should be discarded. This is done by means of the addition of an upwind or artificial viscosity term. Details of the upwind technique will be given in another paper [1].

The standard BEM/panel discretization is based in the Green's third theorem and an integral representation of the potential velocity is obtained by means the Morino's formulation. On the surface of the domain, this representation reduces to an integral equation for the source (or monopolar) and the doublet (or dipolar) density layers. In this problem the first is found by application of a linearized boundary condition and the second is the unknown over the surface of the domain.

A low order panel method is used for the analytic integrations of the monopolar and dipolar influence

coefficients. Then a non-symmetric dense linear system is obtained which is solved by preconditioned Krylov iterative methods, where the coefficient matrix is the sum of the dipolar influence matrix, plus the product of the monopolar influence and the difussive matrices. Both the

dipolar and the monopolar influence matrices are evaluated with an exact field integral and they are full populated in general, whereas the difussive matrix is sparse. These properties are taken into account for an iterative solution where the principal CPU time cost is the evaluation of the

coefficient matrix.

The basic governing equations of potential flow with free surface are the Laplace equation for the velocity potential with appropriated boundary conditions and the free surface condition. The last is based on the Bernoulli equation which relates the surface elevation with the local absolute value of velocity. However this problem is ill-posed in the sense that allows multiple solutions, associated with the existence of a system of trailing gravity waves propagating in both (upstream and

downstream) directions. Solutions with upstream propagating trailing waves should be considered non-physical and should be discarded. This is done by means of the addition of an upwind or artificial viscosity term. Details of the upwind technique will be given in another paper [1].

The standard BEM/panel discretization is based in the Green's third theorem and an integral representation of the potential velocity is obtained by means the Morino's formulation. On the surface of the domain, this representation reduces to an integral equation for the source (or monopolar) and the doublet (or dipolar) density layers. In this problem the first is found by application of a linearized boundary condition and the second is the unknown over the surface of the domain.

A low order panel method is used for the analytic integrations of the monopolar and dipolar influence

coefficients. Then a non-symmetric dense linear system is obtained which is solved by preconditioned Krylov iterative methods, where the coefficient matrix is the sum of the dipolar influence matrix, plus the product of the monopolar influence and the difussive matrices. Both the

dipolar and the monopolar influence matrices are evaluated with an exact field integral and they are full populated in general, whereas the difussive matrix is sparse. These properties are taken into account for an iterative solution where the principal CPU time cost is the evaluation of the

coefficient matrix.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**