Drift, partial drift and Darwin's proposition

Abstract

A body moves at uniform speed in an unbounded inviscid fluid. Initially, the body is infinitely far upstream of an infinite plane of marked fluid; later, the body moves through and distorts the plane and, finally, the body is infinitely far downstream of the marked plane. Darwin (1953) suggested that the volume between the initial and final positions of the surface of marked fluid (the drift volume) is equal to the volume of fluid associated with the 'added-mass' of the body.We re-examine Darwin's (1953) concept of drift and, as an illustration, we study flow around a sphere. Two lengthscales are introduced: pmax, the radius of a circular plane of marked particles; and XO, the initial separation of the sphere and plane. Numerical solutions and asymptotic expansions are derived for the horizontal Lagrangian displacement of fluid elements. These calculations show that depending on its initial position, the Lagrangian displacement of a fluid element can be either positive -a Lagrangian drift -or negative -a Lagrangian reflux. By contrast, previous investigators have found only a positive horizontal Lagrangian displacement, because they only considered the case of infinite XO. For finite XO, the volume between the initial and final positions of the plane of marked fluid is defined to be the 'partial drift volume', which is calculated using a combination of the numerical solutions and the asymptotic expansions. Our analysis shows that in the limit corresponding to Darwin's study, namely that both xo and pmax become infinite, the partial drift volume is not well-defined: the ordering of the limit processes is important. This explains the difficulties Darwin and others noted in trying to prove his proposition as a mathematical theorem and indicates practical, as well as theoretical, criteria that must be satisfied for Darwin's result to hold.We generalize our results for a sphere by re-considering the general expressions for Lagrangian displacement and partial drift volume. It is shown that there are two contributions to the partial drift volume. The first contribution arises from a reflux of fluid and is related to the momentum of the flow; this part is spread over a large area. It is well-known that evaluating the momentum of an unbounded fluid is problematic since the integrals do not converge; it is this first term which prevented Darwin from proving his proposition as a theorem. The second contribution to the partial drift volume is related to the kinetic energy of the flow caused by the body: this part is Darwin's concept of drift and is localized near the centreline. Expressions for partial drift volume are generalized for flow around arbitrary-shaped two-and three-dimensional bodies. The partial drift volume is shown to depend on the solid angles the body subtends with the initial and final positions of the plane of marked fluid. This result explains why the proof of Darwin's proposition depends on the ratio t Present address: Department of Meteorology, University of Reading, Reading, RG6 2AU, UK Pmax 1x0. 202u...

On the estimation of swimming and flying forces from wake measurements

“…To address the issue of a finite surrounding flow volume, Eames et al (1994) introduced the concept of partial drift to describe drift volume measurements in which the body travels only a finite distance through the plane of Lagrangian particles being tracked. The concept of partial drift also accounts for the fact that the size of the plane of Lagrangian particles is limited by the measurement window size, and therefore cannot be infinite in practice as Darwin's method assumes.…”

“…where r L is the (finite) radius of the plane of Lagrangian fluid particles being tracked and d 0 is the (finite) distance upstream from the Lagrangian plane at which the motion of the body toward the plane is initiated (Eames et al, 1994; see also Fig.·4A). Although this equation is strictly valid only for spherical wake vortices, we will see that it also provides a useful approximation for a wider class of vortex wake geometries as well.…”

“…for infinite domain) was computed using measured partial drift in the measurement window and an analytical asymptotic correction factor (i.e. Eq.·12; Eames et al, 1994). The oscillation in drift volume corresponds to changes in direction of Lagrangian particle translation during looping elastica trajectories.…”

“…Darwin, 1953;Batchelor, 1967;Daniel, 1984;Eames et al, 1994Eames et al, , 1997Bush and Eames, 1998;Eames and Flor, 1998;Eames, 2003). Qualitatively, the concept of added-mass accounts for the resistance force that a body faces when it is accelerated through a surrounding medium of non-zero density.…”

“…for infinite domain) was computed using measured partial drift in the computational domain and an analytical asymptotic correction factor (i.e. Eq.·12; Eames et al, 1994). (A) Sphere approaches initially planar Lagrangian surface from left; t=0·s.…”