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In this paper of Lighthill, the author studies the motion of a fish in a constant flow field $U$, modelling the movement of the fish as a deforming, rotationally symmetric, rigid surface in an inviscid fluid. The cross section of the solid volume delimited by the surface, is called $S_x$ and depends on $x$, along the body of the fish.

The fish is supposed to be Slender, that is the displacements and the transversal dimensions of the surface are small compared to the length of the fish in the $x$-direction.

At a certain point, Lighthill says:

Locally, the body shape differs little from that of an infinite cylinder $C$, whose cross-section is $S_x$, all the way along. Accordingly, to the slender-body approximation, the flow component due to the displacements of the cross section near $S_x$, is identical with the two-dimensional potential flow that would result from the motion of the cylinder $C$, through fluid at rest with velocity $V(x, > t)$.

We suppose now that this flow has momentum $$\rho\ A(x)\ V(x,t)$$

per unit length of cylinder, where $\rho$ is the density of the water. In the usual terminology, $\rho A(x)$ is the ‘virtual mass’ of the cylinder $C$, per unit length for motions in the $z$-direction. Thus, the coefficient $A(x)$ has the dimensions of area; for example, it is equal to the area of the cross-section $S_x$, when the latter is circular, while for an ellipse with minor axis in the z-direction $A(x)$ is the area of its circumscribing circle.

My questions are basically two:

  • What does Lighthill refer to, when he writes usual terminology? Is there a classical paper or textbook you could suggest me to read?

  • How is $A(x)$ related, in general, to $S_x$? With which procedure could I get this dependence?

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The quantity Lighthill referred to as the 'virtual mass' is usually called 'added mass' in ocean engineering and naval architecture. It is basically the force on the body due to its acceleration in the fluid. The term 'added' is a bit misleading, as in certain cases, it can be negative. Note that the added mass depends also on the mode of motion. So, for example, for an ellipse whose major axis is horizontal, the added mass due to its motion in the vertical direction is larger than that due to its motion in the horizontal direction.

A classical textbook you could read is Marine Hydrodynamics by J. N. Newman. See, in particular, page 145, where he provides a table of formulas to calculate the added mass of some simple geometries. If you do not have access to this book, you could see these lecture notes.

Analytical formulas for added mass, such as those given by Lighthill, only exist for simple body geometries, where analytical solution for the velocity potential in the fluid domain can be obtained. However, for other general body geometries, a numerical procedure must be employed to obtain the added mass of the body. A popular method is the boundary element method or the panel method, which involves discretizing the body surface into small elements.

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