J. Fluid Mech. (2008), vol. 600, pp. 95–104. c  2008 Cambridge University Press doi:10.1017/S0022112008000396 Printed in the United Kingdom 95 A many-body force decomposition with applications to flow about bluff bodies CHIEN-C. CHANG 1,2 , SHIH-HAO YANG 2,1 AND CHIN-CHOU CHU 2 1 Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan 2 Institute of Applied Mechanics & Taida Institute of Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan mechang@gate.sinica.edu.tw; chucc@iam.ntu.edu.tw (Received 13 July 2007 and in revised form 3 January 2008) The study presents a force theory for incompressible flow about several solid bodies, which enables us to examine the force contribution to each body from individual fluid elements. By employing auxiliary potential functions, we decompose hydrodynamic forces in terms of the unsteadiness of the incoming stream, vorticity within the flow, and surface vorticity on the solid bodies. The usefulness of this force decomposition is illustrated by examining separated flow about several circular cylinders. Guidelines were obtained for finding an optimal arrangement to achieve significantly small drag exerted on the cylinders. 1. Introduction It has always been of interest to see how forces exerted on a body for flow in nature (such as bird flight) or in engineering (such as flow about an aircraft) are related to the structures of the flow (see e.g. Lighthill 1986a). In view of advances in numerical and measurement techniques, a theory that helps to clarify the relationship between the force and the various flow structures is useful and necessary as a large part of the flow or the entire flow field can be measured or computed. For example, if we consider a flow about a circular cylinder or sphere, the fluid meets a buffer layer in front of the body, develops a boundary layer when flowing along its shoulder, and may separate further downstream, evolving into a pair of wake vortices. Can we tell in quantity how the individual flow structures, or more precisely fluid elements, contribute to the drag exerted on the cylinder or sphere? In the literature, there are several useful force theories which shed light on different aspects of hydrodynamic or aerodynamic forces. Circulation theory is the early attempt at predicting the lift (see e.g. Howarth 1935; Sears 1956, 1976). These authors provided insightful relationships between forces and inviscid models in terms of boundary-layer separation, vortex shedding and conservation of circulation. Subsequent studies are meant to provide exact means or theories for hydrodynamic forces through a rigorous analysis of the equation for viscous flow; see e.g. Phillips (1956), Payne (1958), Wu (1981), Quartapelle & Napolitano (1983), and in particular, Howe and colleagues (1989a,b, 1991, 1995, 2001), Kambe (1986) as well as Wells (1996) for inviscid and viscous flow. On the other hand, Lighthill (1986b) developed the ideas which validate a separation of hydrodynamic loadings into potential flow forces and vortex-flow forces. The applied force was deliberately explained as the rate of change of a momentum, defined by an absolutely convergent integral.