An analytical solution for two slender bodies of revolution translating in very close proximity

Abstract

The irrotational flow past two slender bodies of revolution at angles of yaw, translating in parallel paths in very close proximity, is analysed by extending the classical slender body theory. The flow far away from the two bodies is shown to be a direct problem, which is represented in terms of two line sources along their longitudinal axes, at the strengths of the variation rates of their cross-section areas. The inner flow near the two bodies is reduced to the plane flow problem of the expanding (contracting) and lateral translations of two parallel circular cylinders with different radii, which is then solved analytically using conformal mapping. Consequently, an analytical flow solution has been obtained for two arbitrary slender bodies of revolution at angles of yaw translating in close proximity. The lateral forces and yaw moments acting on the two bodies are obtained in terms of integrals along the body lengths. A comparison is made among the present model for two slender bodies in close proximity, Tuck & Newman's (1974) model for two slender bodies far apart, and VSAERO (AMI)–commercial software based on potential flow theory and the boundary element method (BEM). The attraction force of the present model agrees well with the BEM result, when the clearance, h0, is within 20% of the body length, whereas the attraction force of Tuck & Newman is much smaller than the BEM result when h0 is within 30% of the body length, but approaches the latter when h0 is about half the body length. Numerical simulations are performed for the three typical manoeuvres of two bodies: (i) a body passing a stationary body, (ii) two bodies in a meeting manoeuvre (translating in opposite directions), and (iii) two bodies in a passing manoeuvre (translating in the same direction). The analysis reveals the orders of the lateral forces and yaw moments, as well as their variation trends in terms of the manoeuvre type, velocities, sizes, angles of yaw of the two bodies, and their proximity, etc. These irrotational dynamic features are expected to provide a basic understanding of this problem and will be beneficial to further numerical and experimental studies involving additional physical effects.

Higher order hydrodynamic interaction between two slender bodies in potential flow

“…A range of separation distances g 0 =L ¼ ð0:1; 1Þ is considered. Figure 4 shows that while the first-order solution is in good agreement with the numerical calculations only at relatively large separation distances g 0 =L [ 0:7, and the solution by Wang [34] is in good agreement with the numerical calculations only at relatively small lateral separation distances g 0 =L\0:3, our proposed second order approximation gives a satisfactory agreement at the entire range of separation distances \g 0 =L\1.…”

“…The second-order effects are considered and compared with the first-order approximation, and other solutions found in the literature. The solid curve represents the current second-order approximation; the dashed curve represents the first-order approximation solution (the same solution is also given by Tuck [22]); the open circles are numerical results using a boundary element method [35], which was originally presented in [34]; and the dotted curve is the analytical solution of two slender bodies moving in very close proximity by Wang [34]. A range of separation distances g 0 =L ¼ ð0:1; 1Þ is considered.…”

“…Some special applications also exist, such as in the case of fish swimming [31,32], or in the case of dolphin mother-calf interaction which explains the extra gain of thrust by the calf [33]. More recently, Wang [34] studied the hydrodynamic interaction (lateral force and yawing moment) of two slender bodies translating in very close proximity, using matched asymptotic expansions and conformal mapping for the inner solution. The analytical solution by Wang [34] agrees well with the boundary element model (BEM) by Nathman [35], for very small lateral separation distances.…”

“…More recently, Wang [34] studied the hydrodynamic interaction (lateral force and yawing moment) of two slender bodies translating in very close proximity, using matched asymptotic expansions and conformal mapping for the inner solution. The analytical solution by Wang [34] agrees well with the boundary element model (BEM) by Nathman [35], for very small lateral separation distances. However, an increasing deviation from the BEM is observed with increasing the separation distance.…”

“…Empirical evidence and theoretical analyses show that the effect on moving bodies is less than on stationary ones (e.g. [22,34] ). We extend the general solution by Tuck [22] who assumed large lateral separation distance between the two bodies, compared with their lateral dimensions.…”