### A 3D Panel Code for Wave Drag Calculations - Part. II: The Upwind Technique

#### Abstract

We present a 3D BEM/panel code to compute potential flows about ship-forms with linearized free-surface conditions in order to compute the wave drag. Details of the 3D panel discretization and numerical results, will be given in a companion paper [1], so that this one deals with those aspects related to the implementation of the upwind technique in order to capture the "physical" admissible solutions, i.e. those ones satisfying an

appropriated radiation boundary condition at infinity downstream.

The basic governing equations of potential flow with free surface are the Laplace equation for the velocity potential with appropriated botmda.ty conditions, and the free surface condition, which is based on the Bernoulli equation and 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. In real life, this wave system originates in the ship and propagates to infinity downstream. In 2D situations the trailing waves propagates to infinity downstream without damping, whereas in 3D situations the amplitude of the wave pattern decreases due to the spreading in the transversal direction, but keeping constant some quadratic norm of the transverse profile, associated to the wave-drag. The expenditure of energy in creating this wave pattern is at the cause of the wave drag. The set of governing

equations, as described so far, allows solutions with trailing waves propagating in both (upstream and downstream) directions. Solutions with upstream propagating trailing waves should be considered non-physical and, consequently, discarded. This is done by

means' of the addition of an "upwind" or "artificial viscosity" term. Once this term is added, the unicity of the solution for the problem is recovered. In this paper we discuss the theoretical and practical aspects of the implementation of such term.

appropriated radiation boundary condition at infinity downstream.

The basic governing equations of potential flow with free surface are the Laplace equation for the velocity potential with appropriated botmda.ty conditions, and the free surface condition, which is based on the Bernoulli equation and 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. In real life, this wave system originates in the ship and propagates to infinity downstream. In 2D situations the trailing waves propagates to infinity downstream without damping, whereas in 3D situations the amplitude of the wave pattern decreases due to the spreading in the transversal direction, but keeping constant some quadratic norm of the transverse profile, associated to the wave-drag. The expenditure of energy in creating this wave pattern is at the cause of the wave drag. The set of governing

equations, as described so far, allows solutions with trailing waves propagating in both (upstream and downstream) directions. Solutions with upstream propagating trailing waves should be considered non-physical and, consequently, discarded. This is done by

means' of the addition of an "upwind" or "artificial viscosity" term. Once this term is added, the unicity of the solution for the problem is recovered. In this paper we discuss the theoretical and practical aspects of the implementation of such term.

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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**